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HANDBOOK OF MATERIALS MODELING

HANDBOOK OF MATERIALS MODELING Part B. Models Editor Sidney Yip, Massachusetts Institute of Technology

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-3287-0 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-10 1-4020-3286-2 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3287-5 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3286-8 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved

© 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in The Netherlands

CONTENTS PART A – METHODS Preface

xii

List of Subject Editors

ix

List of Contributors

xi

Detailed Table of Contents

xxix

Introduction

1

Chapter 1.

Electronic Scale

7

Chapter 2.

Atomistic Scale

449

Chapter 3.

Mesoscale/Continuum Methods

1069

Chapter 4.

Mathematical Methods

1215

PART B – MODELS Preface

xii

List of Subject Editors

ix

List of Contributors

xi

Detailed Table of Contents

xxix

Chapter 5.

Rate Processes

1565

Chapter 6.

Crystal Defects

1849

Chapter 7.

Microstructure

2081

Chapter 8.

Fluids

2409

Chapter 9.

Polymers and Soft Matter

2553

Plenary Perspectives

2657

Index of Contributors

2943

Index of Keywords

2947 v

PREFACE This Handbook contains a set of articles introducing the modeling and simulation of materials from the standpoint of basic methods and studies. The intent is to provide a compendium that is foundational to an emerging field of computational research, a new discipline that may now be called Computational Materials. This area has become sufficiently diverse that any attempt to cover all the pertinent topics would be futile. Even with a limited scope, the present undertaking has required the dedicated efforts of 13 Subject Editors to set the scope of nine chapters, solicit authors, and collect the manuscripts. The contributors were asked to target students and non-specialists as the primary audience, to provide an accessible entry into the field, and to offer references for further reading. With no precedents to follow, the editors and authors were only guided by a common goal – to produce a volume that would set a standard toward defining the broad community and stimulating its growth. The idea of a reference work on materials modeling surfaced in conversations with Peter Binfield, then the Reference Works Editor at Kluwer Academic Publishers, in the spring of 1999. The rationale at the time already seemed quite clear – the field of computational materials research was taking off, powerful computer capabilities were becoming increasingly available, and many sectors of the scientific community were getting involved in the enterprise. It was felt that a volume that could articulate the broad foundations of computational materials and connect with the established fields of computational physics and computational chemistry through common fundamental scientific challenges would be timely. After five years, none of the conditions have changed; the need remains for a defining reference volume, interest in materials modeling and simulation is further intensifying, the community continues to grow. In this work materials modeling is treated in 9 chapters, loosely grouped into two parts. Part A, emphasizing foundations and methodology, consists of three chapters describing theory and simulation at the electronic, atomistic, and mesoscale levels, and a chapter on analysis-based methods. Part B is more concerned with models and basic applications. There are five chapters describing basic problems in materials modeling and simulation, rate-dependent phenomena, crystal defects, microstructure, fluids, polymers and soft matter. In vii

viii

Preface

addition this part contains a collection of commentaries on a range of issues in materials modeling, written in a free-style format by experienced individuals with definite views that could enlighten the future members of the community. See the opening Introduction for further comments on modeling and simulation and an overview of the Handbook contents. Any organizational undertaking of this magnitude cans only be a collective effort. Yet the fate of this volume would not be so certain without the critical contributions from a few individuals. My gratitude goes to Liesbeth Mol, Peter Binfield’s successor at Springer Science + Business Media, for continued faith and support, Ju Li and Xiaofeng Qian for managing the websites and manuscript files, and Tim Kaxrias for stepping in at a critical stage of the project. To all the authors who found time in your hectic schedules to write the contributions, I am deeply appreciative and trust you are not disappointed. To the Subject Editors I say the Handbook is a reality only because of your perseverance and sacrifices. It has been my good fortune to have colleagues who were generous with advice and assistance. I hope this work motivates them even more to continue sharing their knowledge and insights in the work ahead. Sidney Yip Department of Nuclear Science and Engineering, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

LIST OF SUBJECT EDITORS Martin Bazant, Massachusetts Institute of Technology (Chapter 4) Bruce Boghosian, Tufts University (Chapter 8) Richard Catlow, Royal Institution, UK (Chapter 6) Long-Qing Chen, Pennsylvania State University (Chapter 7) William Curtin, Brown University (Chapter 1, Chapter 2, Chapter 4) Tomas Diaz de la Rubia, Lawrence Livermore National Laboratory (Chapter 6) Nicolas Hadjiconstantinou, Massachusetts Institute of Technology (Chapter 8) Mark F. Horstemeyer, Mississippi State University (Chapter 3) Efthimios Kaxiras, Harvard University (Chapter 1, Chapter 2) L. Mahadevan, Harvard University (Chapter 9) Dimitrios Maroudas, University of Massachusetts (Chapter 4) Nicola Marzari, Massachusetts Institute of Technology (Chapter 1) Horia Metiu, University of California Santa Barbara (Chapter 5) Gregory C. Rutledge, Massachusetts Institute of Technology (Chapter 9) David J. Srolovitz, Princeton University (Chapter 7) Bernhardt L. Trout, Massachusetts Institute of Technology (Chapter 1) Dieter Wolf, Argonne National Laboratory (Chapter 6) Sidney Yip, Massachusetts Institute of Technology (Chapter 1, Chapter 2, Chapter 6, Plenary Perspectives)

ix

LIST OF CONTRIBUTORS Farid F. Abraham IBM Almaden Research Center, San Jose, California [email protected] P20

Robert Averback Accelerator Laboratory, P.O. Box 43 (Pietari Kalmin k. 2), 00014, University of Helsinki, Finland; Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois, USA [email protected] 6.2

Francis J. Alexander Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 8.7

D.J. Bammann Sandia National Laboratories, Livermore, CA, USA [email protected] 3.2

N.R. Aluru Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3

K. Barmak Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA [email protected] 7.19

Filippo de Angelis Istituto CNR di Scienze e Tecnologie Molecolari ISTM, Dipartimento di Chimica, Universit´a di Perugia, Via Elce di Sotto $, I-06123, Perugia, Italy [email protected] 1.4

Stefano Baroni DEMOCRITOS-INFM, SISSA-ISAS, Trieste, Italy [email protected] 1.10

Emilio Artacho University of Cambridge, Cambridge, UK [email protected] 1.5

Rodney J. Bartlett Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA [email protected] 1.3

Mark Asta Northwestern University, Evanston, IL, USA [email protected] 1.16

Corbett Battaile Sandia National Laboratories, Albuquerque, NM, USA [email protected] 7.17

xi

xii Martin Z. Bazant Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.1, 4.10 Noam Bernstein Naval Research Laboratory, Washington, DC, USA [email protected] 2.24 Kurt Binder Institut fuer Physik, Johannes Gutenberg Universitaet Mainz, Staudinger Weg 7, 55099 Mainz, Germany [email protected] P19 Peter E. Bl¨ohl Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6 Bruce M. Boghosian Department of Mathematics, Tufts University, Bromfield-Pearson Hall, Medford, MA 02155, USA [email protected] 8.1 Jean Pierre Boon Center for Nonlinear Phenomena and Complex Systems, Universit´e Libre de Bruxelles, 1050-Bruxelles, Belgium [email protected] P21

List of contributors Russel Caflisch University of California at Los Angeles, Los Angeles, CA, USA [email protected] 7.15 Wei Cai Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-4040, USA [email protected] 2.21 Roberto Car Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, NJ, USA [email protected] 1.4 Paolo Carloni International School for Advanced Studies (SISSA/ISAS) and INFM Democritos Center, Trieste, Italy [email protected] 1.13 Emily A. Carter Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 1.8

Iain D. Boyd University of Michigan, Ann Arbor, MI, USA [email protected] P22

C.R.A. Catlow Davy Faraday Laboratory, The Royal Institution, 21 Albemarle Street, London W1S 4BS, UK; Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK [email protected] 2.7, 6.1

Vasily V. Bulatov Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA [email protected] P7

Gerbrand Ceder Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 1.17, 1.18

List of contributors

xiii

Alan V. Chadwick Functional Materials Group, School of Physical Sciences, University of Kent, Canterbury, Kent CT2 7NR, UK [email protected] 6.5

Marvin L. Cohen University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA [email protected] 1.2

Hue Sun Chan University of Toronto, Toronto, Ont., Canada [email protected] 5.16

John Corish Department of Chemistry, Trinity College, University of Dublin, Dublin 2, Ireland [email protected] 6.4

James R. Chelikowsky University of Minnesota, Minneapolis, MN, USA [email protected] 1.7 Long-Qing Chen Department of Materials Science and Engineering, Penn State University, University Park, PA 16802, USA [email protected] 7.1 I-Wei Chen Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6282, USA [email protected] P27 Sow-Hsin Chen Department of Nuclear Engineering, MIT, Cambridge, MA 02139, USA [email protected] P28 Christophe Chipot Equipe de dynamique des assemblages membranaires, Unit´e mixte de recherche CNRS/UHP 7565, Institut nanc´een de chimie mol´eculaire, Universit´e Henri Poincar´e, BP 239, 54506 Vanduvre-l`es-Nancy cedex, France 2.26 Giovanni Ciccotti INFM and Dipartimento di Fisica, Universit`a “La Sapienza,” Piazzale Aldo Moro, 2, 00185 Roma, Italy [email protected] 2.17, 5.4

Peter V. Coveney Centre for Computational Science, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK [email protected] 8.5 Jean-Paul Crocombette CEA Saclay, DEN-SRMP, 91191 Gif/Yvette cedex, France [email protected] 2.28 Darren Crowdy Department of Mathematics, Imperial College, London, UK [email protected] 4.10 G´abor Cs´anyi Cavendish Laboratory, University of Cambridge, UK [email protected] P16 Nguyen Ngoc Cuong Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15 Christoph Dellago Institute of Experimental Physics, University of Vienna, Vienna, Austria [email protected] 5.3

xiv J.D. Doll Department of Chemistry, Brown University, Providence, RI, USA Jimmie [email protected] 5.2 Patrick S. Doyle Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 9.7

List of contributors Diana Farkas Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, USA [email protected] 2.23 Clemens J. F¨orst Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6

Weinan E Department of Mathematics, Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA [email protected] 4.13

Glenn H. Fredrickson Department of Chemical Engineering & Materials, The University of California at Santa, Barbara Santa Barbara, CA, USA [email protected] 9.9

Jens Eggers School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK [email protected] 4.9

Daan Frenkel FOM Institute for Atomic and Molecular Physics, Amsterdam, The Netherlands [email protected] 2.14

Pep Espanol ˜ Dept. Física Fundamental, Universidad Nacional de Educaci´on a Distancia, Aptdo. 60141, E-28080 Madrid, Spain [email protected] 8.6 J.W. Evans Ames Laboratory - USDOE, and Department of Mathematics, Iowa State University, Ames, Iowa, 50011, USA [email protected] 5.12 Denis J. Evans Research School of Chemistry, Australian National University, Canberra, ACT, Australia [email protected] P17 Michael L. Falk University of Michigan, Ann Arbor, MI, USA [email protected] 4.3

Julian D. Gale Nanochemistry Research Institute, Department of Applied Chemistry, Curtin University of Technology, Perth, 6845, Western Australia [email protected] 1.5, 2.3 Giulia Galli Lawrence Livermore National Laboratory, CA, USA [email protected] P8 Venkat Ganesan Department of Chemical Engineering, The University of Texas at Austin, Austin, TX, USA [email protected] 9.9 Alberto García Universidad del País Vasco, Bilbao, Spain [email protected] 1.5

List of contributors C. William Gear Princeton University, Princeton, NJ, USA [email protected] 4.11 Timothy C. Germann Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11 Eitan Geva Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, USA [email protected] 5.9 Nasr M. Ghoniem Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597, USA [email protected] 7.11, P11, P30 Paolo Giannozzi Scuola Normale Superiore and National Simulation Center, INFM-DEMOCRITOS, Pisa, Italy [email protected] 1.4, 1.10 E. Van der Giessen University of Groningen, Groningen, The Netherlands [email protected] 3.4 Daniel T. Gillespie Dan T Gillespie Consulting, 30504 Cordoba Place, Castaic, CA 91384, USA [email protected] 5.11 George Gilmer Lawrence Livermore National Laboratory, P.O. box 808, Livermore, CA 94550, USA [email protected] 2.10

xv William A. Goddard III Materials and Process Simulation Center, California Institute of Technology, Pasadena, CA 91125, USA [email protected] P9 Axel Groß Physik-Department T30, TU M¨unchen, 85747 Garching, Germany [email protected] 5.10 Peter Gumbsch Institut f¨ur Zuverl¨assigkeit von Bauteilen und Systemen izbs, Universit¨at Karlsruhe (TH), Kaiserstr. 12, 76131Karlsruhe, Germany and Fraunhofer Institut f¨ur Werkstoffmechanik IWM, W¨ohlerstr. 11, D-79194 Freiburg, Germany [email protected] P10 Fran¸cois Gygi Lawrence Livermore National Laboratory, CA, USA P8 Nicolas G. Hadjiconstantinou Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA [email protected] 8.1, 8.8 J.P. Hirth Ohio State and Washington State Universities, 114 E. Ramsey Canyon Rd., Hereford, AZ 85615, USA [email protected] P31 K.M. Ho Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA 1.15

xvi

List of contributors

Wesley P. Hoffman Air Force Research Laboratory, Edwards, CA, USA [email protected] P37

C.S. Jayanthi Department of Physics, University of Louisville, Louisville, KY 40292 [email protected] P39

Wm.G. Hoover Department of Applied Science, University of California at Davis/Livermore and Lawrence Livermore National Laboratory, Livermore, California, 94551-7808 [email protected] P34

Raymond Jeanloz University of California, Berkeley, CA, USA [email protected] P25

M.F. Horstemeyer Mississippi State University, Mississippi State, MS, USA [email protected] 3.1, 3.5 Thomas Y. Hou California Institute of Technology, Pasadena, CA, USA [email protected] 4.14 Hanchen Huang Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590, USA [email protected] 2.30 Gerhard Hummer National Institutes of Health, Bethesda, MD, USA [email protected] 4.11 M. Saiful Islam Chemistry Division, SBMS, University of Surrey, Guildford GU2 7XH, UK [email protected] 6.6 Seogjoo Jang Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA [email protected] 5.9

Pablo Jensen Laboratoire de Physique de la Mati´ere Condens´ee et des Nanostructures, CNRS and Universit´e Claude Bernard Lyon-1, 69622 Villeurbanne C´edex, France [email protected] 5.13 Yongmei M. Jin Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA [email protected] 7.12 Xiaozhong Jin Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3 J.D. Joannopoulos Francis Wright Davis Professor of Physics, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] P4 Javier Junquera Rutgers University, New Jersey, USA [email protected] 1.5 Jo˜ao F. Justo Escola Polit´ecnica, Universidade de S˜ao Paulo, S˜ao Paulo, Brazil [email protected] 2.4

List of contributors Hideo Kaburaki Japan Atomic Energy Research Institute, Tokai, Ibaraki, Japan [email protected] 2.18 Rajiv K. Kalia Collaboratory for Advanced Computing and Simulations, Department of Physics & Astronomy, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ont. M5S 3H6, Canada [email protected] 2.17, 5.4 Alain Karma Northeastern University, Boston, MA, USA [email protected] 7.2 Johannes K¨astner Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6 Markos A. Katsoulakis Department of Mathematics and Statistics, University of Massachusetts - Amherst, Amherst, MA 01002, USA [email protected] 4.12 Efthimios Kaxiras Department of Nuclear Science and Engineering and Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 2.1, 8.4

xvii Ronald J. Kerans Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, Ohio, USA [email protected] P38 Ioannis G. Kevrekidis Princeton University, Princeton, NJ, USA [email protected] 4.11 Armen G. Khachaturyan Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA [email protected] 7.12 T.A. Khraishi University of New Mexico, Albuquerque, NM, USA [email protected] 3.3 Seong Gyoon Kim Kunsan National University, Kunsan 573-701, Korea [email protected] 7.3 Won Tae Kim Chongju University, Chongju 360-764, Korea [email protected] 7.3 Michael L. Klein Center for Molecular Modeling, Chemistry Department, University of Pennsylvania, 231 South 34th Street, Philadelphia, PA 19104-6323, USA [email protected] 2.26 Walter Kob Laboratoire des Verres, Universit´e Montpellier 2, 34095 Montpellier, France [email protected] P24

xviii David A. Kofke University at Buffalo, The State University of New York, Buffalo, New York, USA [email protected] 2.14 Maurice de Koning University of S˜ao Paulo, S˜ao Paulo, Brazil [email protected] 2.15 Anatoli Korkin Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA 1.3 Kurt Kremer MPI for Polymer Research, D-55021 Mainz, Germany [email protected] P5

List of contributors C. Leahy Department of Physics, University of Louisville, Louisville, KY 40292, USA P39 R. LeSar Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 7.14 Ju Li Department of Materials Science and Engineering, Ohio State University, Columbus, OH, USA [email protected] 2.8, 2.19, 2.31 Xiantao Li Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 4.13

Carl E. Krill III Materials Division, University of Ulm, Albert-Einstein-Allee 47, D-89081 Ulm, Germany [email protected] 7.6

Gang Li Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3

Ladislas P. Kubin LEM, CNRS-ONERA, 29 Av. de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France [email protected] P33

Vincent L. Lign`eres Department of Chemistry, Princeton University, Princeton, NJ 08544, USA 1.8

D.P. Landau Center for Simulational Physics, The University of Georgia, Athens, GA 30602, USA [email protected] P2 James S. Langer Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA [email protected] 4.3, P14

Turab Lookman Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 7.5 Steven G. Louie Department of Physics, University of California at Berkeley and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA [email protected] 1.11

List of contributors

xix

John Lowengrub University of California, Irvine, California, USA [email protected] 7.8

Richard M. Martin University of Illinois at Urbana, Urbana, IL, USA [email protected] 1.5

Gang Lu Division of Engineering and Applied Science, Harvard University, Cambridge, Massachusetts, USA [email protected] 2.20

Georges Martin ´ Commissariat a` l’Energie Atomique, Cab. H.C., 33 rue de la F´ed´eration, 75752 Paris Cedex 15, France [email protected] 7.9

Alexander D. MacKerell, Jr. Department of Pharmaceutical Sciences, School of Pharmacy, University of Maryland, 20 Penn Street, Baltimore, MD, 21201, USA [email protected] 2.5

Nicola Marzari Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 1.1, 1.4

Alessandra Magistrato International School for Advanced Studies (SISSA/ISAS) and INFM Democritos Center, Trieste, Italy 1.13

Wayne L. Mattice Department of Polymer Science, The University of Akron, Akron, OH 44325-3909 [email protected] 9.3

L. Mahadevan Division of Engineering and Applied Sciences, Department of Organismic and Evolutionary Biology, Department of Systems Biology, Harvard University Cambridge, MA 02138, USA [email protected] Dionisios Margetis Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 4.8

V.G. Mavrantzas Department of Chemical Engineering, University of Patras, Patras, GR 26500, Greece [email protected] 9.4 D.L. McDowell Georgia Institute of Technology, Atlanta, GA, USA [email protected] 3.6, 3.9

E.B. Marin Sandia National Laboratories, Livermore, CA, USA [email protected] 3.5

Michael J. Mehl Center for Computational Materials Science, Naval Research Laboratory, Washington, DC, USA [email protected] 1.14

Dimitrios Maroudas University of Massachusetts, Amherst, MA, USA [email protected] 4.1

Horia Metiu University of California, Santa Barbara, CA, USA [email protected] 5.1

xx R.E. Miller Carleton University, Ottawa, ON, Canada [email protected] 2.13 Frederick Milstein Mechanical Engineering and Materials Depts., University of California, Santa Barbara, CA, USA [email protected] 4.2 Y. Mishin George Mason University, Fairfax, VA, USA [email protected] 2.2 Francesco Montalenti INFM, L-NESS, and Dipartimento di Scienza dei Materiali, Universit`a degli Studi di Milano-Bicocca, Via Cozzi 53, I-20125 Milan, Italy [email protected] 2.11 Dane Morgan Massachusetts Institute of Technology, Cambridge MA, USA [email protected] 1.18 John A. Moriarty Lawrence Livermore National Laboratory, University of California, Livermore, CA 94551-0808 [email protected] P13 J.W. Morris, Jr. Department of Materials Science and Engineering, University of California, Berkeley, CA, USA [email protected] P18 Raymond D. Mountain Physical and Chemical Properties Division, Chemical Science and Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899-8380, USA [email protected] P23

List of contributors Marcus Muller ¨ Department of Physics, University of Wisconsin, Madison, WI 53706-1390, USA [email protected] 9.5 Aiichiro Nakano Collaboratory for Advanced Computing and Simulations, Department of Computer Science, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 A. Needleman Brown University, Providence, RI, USA [email protected] 3.4 Abraham Nitzan Tel Aviv University, Tel Aviv, 69978, Israel [email protected] 5.7 Kai Nordlund Accelerator Laboratory, P.O. Box 43 (Pietari Kalmin k. 2), 00014, University of Helsinki, Finland; Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois, USA 6.2 G. Robert Odette Department of Mechanical Engineering and Department of Materials, University of California, Santa Barbara, CA, USA [email protected] 2.29 Shigenobu Ogata Osaka University, Osaka, Japan [email protected] 1.20

List of contributors Gregory B. Olson Department of Materials Science and Engineering, Northwestern University, Evanston, IL, USA [email protected] P3 Pablo Ordej´on Instituto de Materiales, CSIC, Barcelona, Spain [email protected] 1.5 Tadeusz Pakula Max Planck Institute for Polymer Research, Mainz, Germany and Department of Molecular Physics, Technical University, Lodz, Poland [email protected] P35 Vijay Pande Department of Chemistry and of Structural Biology, Stanford University, Stanford, CA 94305-5080, USA [email protected] 5.17 I.R. Pankratov Russian Research Centre, “Kurchatov Institute”, Moscow 123182, Russia [email protected] 7.10 D.A. Papaconstantopoulos Center for Computational Materials Science, Naval Research Laboratory, Washington, DC, USA [email protected] 1.14 J.E. Pask Lawrence Livermore National Laboratory, Livermore, CA, USA [email protected] 1.19 Anthony T. Patera Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15

xxi Mike Payne Cavendish Laboratory, University of Cambridge, UK [email protected] P16 Leonid Pechenik University of California, Santa Barbara, CA, USA [email protected] 4.3 Joaquim Peir´o Department of Aeronautics, Imperial College, London, UK [email protected] 8.2 Simon R. Phillpot Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611, USA [email protected] 2.6, 6.11 G.P. Potirniche Mississippi State University, Mississippi State, MS, USA [email protected] 3.5 Thomas R. Powers Division of Engineering, Brown University, Providence, RI, USA thomas [email protected] 9.8 Dierk Raabe Max-Planck-Institut f¨ur Eisenforschung, Max-Planck-Str. 1, D-40237 D¨usseldorf, Germany [email protected] 7.7, P6 Ravi Radhakrishnan Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, USA [email protected] 5.5

xxii Christian Ratsch University of California at Los Angeles, Los Angeles, CA, USA [email protected] 7.15 John R. Ray 1190 Old Seneca Road, Central, SC 29630, USA [email protected] 2.16 William P. Reinhardt University of Washington Seattle, Washington, USA [email protected] 2.15 Karsten Reuter Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany [email protected] 1.9 J.M. Rickman Department of Materials Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA [email protected] 7.14, 7.19

List of contributors Tomonori Sakai Centre for Computational Science, Queen Mary, University of London, Mile End Road, London E1 4NS, UK 8.5 Deniel S´anchez-Portal Donostia International Physics Center, Donostia, Spain [email protected] 1.5 Joachim Sauer Institut f¨ur Chemie, Humboldt-Universit¨at zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany 1.12 Avadh Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 7.5 Matthias Scheffler Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany [email protected] 1.9

Angel Rubio Departamento Física de Materiales and Unidad de Física de Materiales Centro Mixto CSIC-UPV, Universidad del País Vasco and Donosita Internacional Physics Center (DIPC), Spain [email protected] 1.11

Klaus Schulten Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

Robert E. Rudd Lawrence Livermore National Laboratory, University of California, L-045 Livermore, CA 94551, USA [email protected] 2.12

Steven D. Schwartz Departments of Biophysics and Biochemistry, Albert Einstein College of Medicine, New York, USA [email protected] 5.8

Gregory C. Rutledge Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 9.1

Robin L.B. Selinger Physics Department, Catholic University, Washington, DC 20064, USA [email protected] 2.23

List of contributors Marcelo Sepliarsky Instituto de Física Rosario, Facultad de Ciencias Exactas, Ingenieria y Agrimensura, Universidad Nacional de Rosario, 27 de Febreo 210 Bis, (2000) Rosario, Argentina [email protected] 2.6 Alessandro Sergi Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ont. M5S 3H6, Canada [email protected] 2.17, 5.4 J.A. Sethian Department of Mathematics, University of California, Berkeley, CA, USA [email protected] 4.6 Michael J. Shelley Courant Institute of Mathematical Sciences, New York University, New York, NY, USA [email protected] 4.7 C. Shen The Ohio State University, Columbus, Ohio, USA [email protected] 7.4 Spencer Sherwin Department of Aeronautics, Imperial College, London, UK [email protected] 8.2 Marek Sierka Institut f¨ur Physikalische Chemie, Lehrstuhl f¨ur Theoretische Chemie, Universit¨at Karlsruhe, Kaiserstraße 12, D-76128 Karlsruhe, Germany [email protected] 1.12 Asimina Sierou University of Cambridge, Cambridge, UK [email protected] 9.6

xxiii Grant D. Smith Department of Materials Science and Engineering, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, USA [email protected] 9.2 Fr´ed´eric Soisson CEA Saclay, DMN-SRMP, 91191 Gif-sur-Yuette, France [email protected] 7.9 Jos´e M. Soler Universidad Aut´onoma de Madrid, Madrid, Spain [email protected] 1.5 Didier Sornette Institute of Geophysics and Planetary Physics and Department of Earth and Space Science, University of California, Los Angeles, California, USA and CNRS and Universit´e des Sciences, Nice, France [email protected] 4.4 David J. Srolovitz Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA [email protected] 7.1, 7.13 Marcelo G. Stachiotti Instituto de Física Rosario, Facultad de Ciencias Exactas, Ingenieria y Agrimensura, Universidad Nacional de Rosario, 27 de Febreo 210 Bis, (2000) Rosario, Argentina [email protected] 2.6 Catherine Stampfl Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany; School of Physics, The University of Sydney, Sydney 2006, Australia [email protected] 1.9

xxiv

List of contributors

H. Eugene Stanley Center for Polymer Studies and Department of Physics Boston, University, Boston, MA 02215, USA [email protected] P36

Meijie Tang Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 [email protected] 2.22

P.A. Sterne Lawrence Livermore National Laboratory, Livermore, CA, USA [email protected] 1.19

Mounir Tarek Equipe de dynamique des assemblages membranaires, Unit´e mixte de recherche CNRS/UHP 7565, Institut nanc´eien de chimie mol´eculaire, Universit´e Henri Poincar´e, BP 239, 54506 Vanduvre-l`es-Nancy cedex, France 2.26

Howard A. Stone Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 01238, USA [email protected] 4.8 Marshall Stoneham Centre for Materials Research, and London Centre for Nanotechnology, Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK [email protected] P12 Sauro Succi Istituto Applicazioni Calcolo, National Research Council, viale del Policlinico, 137, 00161, Rome, Italy [email protected] 8.4 E.B. Tadmor Technion-Israel Institute of Technology, Haifa, Israel [email protected] 2.13 Emad Tajkhorshid Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

DeCarlos E. Taylor Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA [email protected] 1.3 Doros N. Theodorou School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou Campus, 157 80 Athens, Greece [email protected] P15 Carl V. Thompson Department of Materials Science and Engineering, M.I.T., Cambridge, MA 02139, USA [email protected] P26 Anna-Karin Tornberg Courant Institute of Mathematical Sciences, New York University, New York, NY, USA [email protected] 4.7 S. Torquato Department of Chemistry, PRISM, and Program in Applied & Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 4.5, 7.18

List of contributors Bernhardt L. Trout Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 5.5 Mark E. Tuckerman Department of Chemistry, Courant Institute of Mathematical Science, New York University, New York, NY 10003, USA [email protected] 2.9 Blas P. Uberuaga Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11, 5.6 Patrick T. Underhill Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 9.7 V.G. Vaks Russian Research Centre, “Kurchatov Institute”, Moscow 123182, Russia [email protected] 7.10 Priya Vashishta Collaboratory for Advanced Computing and Simulations, Department of Chemical Engineering and Materials Science, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 A. Van der Ven Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 1.17 Karen Veroy Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15

xxv Alessandro De Vita King’s College London, UK, Center for Nanostructured, Materials (CENMAT) and DEMOCRITOS National Simulation Center, Trieste, Italy alessandro.de [email protected] P16 V. Vitek Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA [email protected] P32 Dionisios G. Vlachos Department of Chemical Engineering, Center for Catalytic Science and Technology, University of Delaware, Newark, DE 19716, USA [email protected] 4.12 Arthur F. Voter Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11, 5.6 Gregory A. Voth Department of Chemistry and Henry Eyring Center for Theoretical Chemistry, University of Utah, Salt Lake City, Utah 84112-0850, USA [email protected] 5.9 G.Z. Voyiadjis Louisiana State University, Baton Rouge, LA, USA [email protected] 3.8 Dimitri D. Vvedensky Imperial College, London, United Kingdom [email protected] 7.16 G¨oran Wahnstr¨om Chalmers University of Technology and G¨oteborg University Materials and Surface Theory, SE-412 96 G¨oteborg, Sweden [email protected] 5.14

xxvi

List of contributors

Duane C. Wallace Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] P1

Brian D. Wirth Department of Nuclear Engineering, University of California, Barkeley, CA, USA [email protected] 2.29

Axel van de Walle Northwestern University, Evanston, IL, USA [email protected] 1.16

Dieter Wolf Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA [email protected] 6.7, 6.9, 6.10, 6.11, 6.12, 6.13

Chris G. Van de Walle Materials Department, University of California, Santa Barbara, California, USA [email protected] 6.3

C.Z. Wang Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA [email protected] 1.15

Y. Wang The Ohio State University, Columbus, Ohio, USA [email protected] 7.4

Yu U. Wang Department of Materials Science and Engineering, Virginia Tech., Blacksburg, VA 24061, USA [email protected] 7.12

Hettithanthrige S. Wijesinghe Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA [email protected] 8.8

Chung H. Woo The Hong Kong Polytechnic University, Hong Kong SAR, China [email protected] 2.27 Christopher Woodward Northwestern University, Evanston, Illinois, USA [email protected] P29 S.Y. Wu Department of Physics, University of Louisville, Louisville, KY 40292, USA [email protected] P39 Yang Xiang Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong [email protected] 7.13 Sidney Yip Department of Physics, Harvard University, Cambridge, MA 02138, USA [email protected] 2.1, 2.10, 6.7, 6.8, 6.11 M. Yu Department of Physics, University of Louisville, Louisville, KY 40292, USA P39

List of contributors H.M. Zbib Washington State University, Pullman, WA, USA [email protected] 3.3 Fangqiang Zhu Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

xxvii M. Zikry North Carolina State University, Raleigh, NC, USA [email protected] 3.7

DETAILED TABLE OF CONTENTS PART A – METHODS Chapter 1. Electronic Scale 1.1

Understand, Predict, and Design Nicola Marzari 1.2 Concepts for Modeling Electrons in Solids: A Perspective Marvin L. Cohen 1.3 Achieving Predictive Simulations with Quantum Mechanical Forces Via the Transfer Hamiltonian: Problems and Prospects Rodney J. Bartlett, DeCarlos E. Taylor, and Anatoli Korkin 1.4 First-Principles Molecular Dynamics Roberto Car, Filippo de Angelis, Paolo Giannozzi, and Nicola Marzari 1.5 Electronic Structure Calculations with Localized Orbitals: The Siesta Method Emilio Artacho, Julian D. Gale, Alberto García, Javier Junquera, Richard M. Martin, Pablo Ordej´on, Deniel S´anchez-Portal, and Jos´e M. Soler 1.6 Electronic Structure Methods: Augmented Waves, Pseudopotentials and the Projector Augmented Wave Method Peter E. Bl¨ochl, Johannes K¨astner, and Clemens J. F¨orst 1.7 Electronic Scale James R. Chelikowsky 1.8 An Introduction to Orbital-Free Density Functional Theory Vincent L. Lign`eres and Emily A. Carter 1.9 Ab Initio Atomistic Thermodynamics and Statistical Mechanics of Surface Properties and Functions Karsten Reuter, Catherine Stampfl, and Matthias Scheffler 1.10 Density-Functional Perturbation Theory Paolo Giannozzi and Stefano Baroni

xxix

9 13

27

59

77

93 121 137

149 195

xxx

Detailed table of contents

1.11 Quasiparticle and Optical Properties of Solids and Nanostructures: The GW-BSE Approach Steven G. Louie and Angel Rubio 1.12 Hybrid Quantum Mechanics/Molecular Mechanics Methods and their Application Marek Sierka and Joachim Sauer 1.13 Ab Initio Molecular Dynamics Simulations of Biologically Relevant Systems Alessandra Magistrato and Paolo Carloni 1.14 Tight-Binding Total Energy Methods for Magnetic Materials and Multi-Element Systems Michael J. Mehl and D.A. Papaconstantopoulos 1.15 Environment-Dependent Tight-Binding Potential Models C.Z. Wang and K.M. Ho 1.16 First-Principles Modeling of Phase Equilibria Axel van de Walle and Mark Asta 1.17 Diffusion and Configurational Disorder in Multicomponent Solids A. Van der Ven and G. Ceder 1.18 Data Mining in Materials Development Dane Morgan and Gerbrand Ceder 1.19 Finite Elements in Ab Initio Electronic-Structure Calculations J.E. Pask and P.A. Sterne 1.20 Ab Initio Study of Mechanical Deformation Shigenobu Ogata

215

241

259

275 307 349

367 395 423 439

Chapter 2. Atomistic Scale 2.1 2.2 2.3 2.4 2.5 2.6

2.7 2.8

Introduction: Atomistic Nature of Materials Efthimios Kaxiras and Sidney Yip Interatomic Potentials for Metals Y. Mishin Interatomic Potential Models for Ionic Materials Julian D. Gale Modeling Covalent Bond with Interatomic Potentials Jo˜ao F. Justo Interatomic Potentials: Molecules Alexander D. MacKerell, Jr. Interatomic Potentials: Ferroelectrics Marcelo Sepliarsky, Marcelo G. Stachiotti, and Simon R. Phillpot Energy Minimization Techniques in Materials Modeling C.R.A. Catlow Basic Molecular Dynamics Ju Li

451 459 479 499 509

527 547 565

Detailed table of contents 2.9 2.10 2.11

2.12

2.13 2.14 2.15 2.16

2.17 2.18

2.19 2.20

2.21 2.22

2.23 2.24 2.25

2.26

Generating Equilibrium Ensembles Via Molecular Dynamics Mark E. Tuckerman Basic Monte Carlo Models: Equilibrium and Kinetics George Gilmer and Sidney Yip Accelerated Molecular Dynamics Methods Blas P. Uberuaga, Francesco Montalenti, Timothy C. Germann, and Arthur F. Voter Concurrent Multiscale Simulation at Finite Temperature: Coarse-Grained Molecular Dynamics Robert E. Rudd The Theory and Implementation of the Quasicontinuum Method E.B. Tadmor and R.E. Miller Perspective: Free Energies and Phase Equilibria David A. Kofke and Daan Frenkel Free-Energy Calculation Using Nonequilibrium Simulations Maurice de Koning and William P. Reinhardt Ensembles and Computer Simulation Calculation of Response Functions John R. Ray Non-Equilibrium Molecular Dynamics Giovanni Ciccotti, Raymond Kapral, and Alessandro Sergi Thermal Transport Process by the Molecular Dynamics Method Hideo Kaburaki Atomistic Calculation of Mechanical Behavior Ju Li The Peierls–Nabarro Model of Dislocations: A Venerable Theory and its Current Development Gang Lu Modeling Dislocations Using a Periodic Cell Wei Cai A Lattice Based Screw-Edge Dislocation Dynamics Simulation of Body Center Cubic Single Crystals Meijie Tang Atomistics of Fracture Diana Farkas and Robin L.B. Selinger Atomistic Simulations of Fracture in Semiconductors Noam Bernstein Multimillion Atom Molecular-Dynamics Simulations of Nanostructured Materials and Processes on Parallel Computers Priya Vashishta, Rajiv K. Kalia, and Aiichiro Nakano Modeling Lipid Membranes Christophe Chipot, Michael L. Klein, and Mounir Tarek

xxxi 589 613

629

649 663 683 707

729 745

763 773

793 813

827 839 855

875 929

xxxii

Detailed table of contents

2.27 Modeling Irradiation Damage Accumulation in Crystals Chung H. Woo 2.28 Cascade Modeling Jean-Paul Crocombette 2.29 Radiation Effects in Fission and Fusion Reactors G. Robert Odette and Brian D. Wirth 2.30 Texture Evolution During Thin Film Deposition Hanchen Huang 2.31 Atomistic Visualization Ju Li

959 987 999 1039 1051

Chapter 3. Mesoscale/Continuum Methods 3.1 3.2

3.3 3.4 3.5 3.6 3.7 3.8 3.9

Mesoscale/Macroscale Computational Methods M.F. Horstemeyer Perspective on Continuum Modeling of Mesoscale/Macroscale Phenomena D.J. Bammann Dislocation Dynamics H.M. Zbib and T.A. Khraishi Discrete Dislocation Plasticity E. Van der Giessen and A. Needleman Crystal Plasticity M.F. Horstemeyer, G.P. Potirniche, and E.B. Marin Internal State Variable Theory D.L. McDowell Ductile Fracture M. Zikry Continuum Damage Mechanics G.Z. Voyiadjis Microstructure-Sensitive Computational Fatigue Analysis D.L. McDowell

1071

1077 1097 1115 1133 1151 1171 1183 1193

Chapter 4. Mathematical Methods 4.1 4.2

4.3

4.4

Overview of Chapter 4: Mathematical Methods Martin Z. Bazant and Dimitrios Maroudas Elastic Stability Criteria and Structural Bifurcations in Crystals Under Load Frederick Milstein Toward a Shear-Transformation-Zone Theory of Amorphous Plasticity Michael L. Falk, James S. Langer, and Leonid Pechenik Statistical Physics of Rupture in Heterogeneous Media Didier Sornette

1217

1223

1281 1313

Detailed table of contents 4.5 4.6

4.7 4.8 4.9 4.10 4.11 4.12

4.13 4.14

4.15

Theory of Random Heterogeneous Materials S. Torquato Modern Interface Methods for Semiconductor Process Simulation J.A. Sethian Computing Microstructural Dynamics for Complex Fluids Michael J. Shelley and Anna-Karin Tornberg Continuum Descriptions of Crystal Surface Evolution Howard A. Stone and Dionisios Margetis Breakup and Coalescence of Free Surface Flows Jens Eggers Conformal Mapping Methods for Interfacial Dynamics Martin Z. Bazant and Darren Crowdy Equation-Free Modeling for Complex Systems Ioannis G. Kevrekidis, C. William Gear, and Gerhard Hummer Mathematical Strategies for the Coarse-Graining of Microscopic Models Markos A. Katsoulakis and Dionisios G. Vlachos Multiscale Modeling of Crystalline Solids Weinan E and Xiantao Li Multiscale Computation of Fluid Flow in Heterogeneous Media Thomas Y. Hou Certified Real-Time Solution of Parametrized Partial Differential Equations Nguyen Ngoc Cuong, Karen Veroy, and Anthony T. Patera

xxxiii 1333

1359 1371 1389 1403 1417 1453

1477 1491

1507

1529

PART B – MODELS Chapter 5. Rate Processes 5.1 5.2 5.3 5.4 5.5

5.6

Introduction: Rate Processes Horia Metiu A Modern Perspective on Transition State Theory J.D. Doll Transition Path Sampling Christoph Dellago Simulating Reactions that Occur Once in a Blue Moon Giovanni Ciccotti, Raymond Kapral, and Alessandro Sergi Order Parameter Approach to Understanding and Quantifying the Physico-Chemical Behavior of Complex Systems Ravi Radhakrishnan and Bernhardt L. Trout Determining Reaction Mechanisms Blas P. Uberuaga and Arthur F. Voter

1567 1573 1585 1597

1613 1627

xxxiv 5.7 5.8

5.9 5.10

5.11 5.12

5.13 5.14 5.15

5.16 5.17

Detailed table of contents Stochastic Theory of Rate Processes Abraham Nitzan Approximate Quantum Mechanical Methods for Rate Computation in Complex Systems Steven D. Schwartz Quantum Rate Theory: A Path Integral Centroid Perspective Eitan Geva, Seogjoo Jang, and Gregory A. Voth Quantum Theory of Reactive Scattering and Adsorption at Surfaces Axel Groß Stochastic Chemical Kinetics Daniel T. Gillespie Kinetic Monte Carlo Simulation of Non-Equilibrium Lattice-Gas Models: Basic and Refined Algorithms Applied to Surface Adsorption Processes J.W. Evans Simple Models for Nanocrystal Growth Pablo Jensen Diffusion in Solids G¨oran Wahnstr¨om Kinetic Theory and Simulation of Single-Channel Water Transport Emad Tajkhorshid, Fangqiang Zhu, and Klaus Schulten Simplified Models of Protein Folding Hue Sun Chan Protein Folding: Detailed Models Vijay Pande

1635

1673 1691

1713 1735

1753 1769 1787

1797 1823 1837

Chapter 6. Crystal Defects 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Point Defects C.R.A. Catlow Point Defects in Metals Kai Nordlund and Robert Averback Defects and Impurities in Semiconductors Chris G. Van de Walle Point Defects in Simple Ionic Solids John Corish Fast Ion Conductors Alan V. Chadwick Defects and Ion Migration in Complex Oxides M. Saiful Islam Introduction: Modeling Crystal Interfaces Sidney Yip and Dieter Wolf

1851 1855 1877 1889 1901 1915 1925

Detailed table of contents 6.8 6.9 6.10

6.11 6.12 6.13

Atomistic Methods for Structure–Property Correlations Sidney Yip Structure and Energy of Grain Boundaries Dieter Wolf High-Temperature Structure and Properties of Grain Boundaries Dieter Wolf Crystal Disordering in Melting and Amorphization Sidney Yip, Simon R. Phillpot, and Dieter Wolf Elastic Behavior of Interfaces Dieter Wolf Grain Boundaries in Nanocrystalline Materials Dieter Wolf

xxxv 1931 1953

1985 2009 2025 2055

Chapter 7. Microstructure 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

7.9

7.10 7.11 7.12 7.13

Introduction: Microstructure David J. Srolovitz and Long-Qing Chen Phase-Field Modeling Alain Karma Phase-Field Modeling of Solidification Seong Gyoon Kim and Won Tae Kim Coherent Precipitation – Phase Field Method C. Shen and Y. Wang Ferroic Domain Structures using Ginzburg–Landau Methods Avadh Saxena and Turab Lookman Phase-Field Modeling of Grain Growth Carl E. Krill III Recrystallization Simulation by Use of Cellular Automata Dierk Raabe Modeling Coarsening Dynamics using Interface Tracking Methods John Lowengrub Kinetic Monte Carlo Method to Model Diffusion Controlled Phase Transformations in the Solid State Georges Martin and Fr´ed´eric Soisson Diffusional Transformations: Microscopic Kinetic Approach I.R. Pankratov and V.G. Vaks Modeling the Dynamics of Dislocation Ensembles Nasr M. Ghoniem Dislocation Dynamics – Phase Field Yu U. Wang, Yongmei M. Jin, and Armen G. Khachaturyan Level Set Dislocation Dynamics Method Yang Xiang and David J. Srolovitz

2083 2087 2105 2117 2143 2157 2173

2205

2223 2249 2269 2287 2307

xxxvi

Detailed table of contents

7.14 Coarse-Graining Methodologies for Dislocation Energetics and Dynamics J.M. Rickman and R. LeSar 7.15 Level Set Methods for Simulation of Thin Film Growth Russel Caflisch and Christian Ratsch 7.16 Stochastic Equations for Thin Film Morphology Dimitri D. Vvedensky 7.17 Monte Carlo Methods for Simulating Thin Film Deposition Corbett Battaile 7.18 Microstructure Optimization S. Torquato 7.19 Microstructural Characterization Associated with Solid–Solid Transformations J.M. Rickman and K. Barmak

2325 2337 2351 2363 2379

2397

Chapter 8. Fluids 8.1 8.2

8.3

8.4 8.5

8.6 8.7

8.8

Mesoscale Models of Fluid Dynamics Bruce M. Boghosian and Nicolas G. Hadjiconstantinou Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations Joaquim Peir´o and Spencer Sherwin Meshless Methods for Numerical Solution of Partial Differential Equations Gang Li, Xiaozhong Jin, and N.R. Aluru Lattice Boltzmann Methods for Multiscale Fluid Problems Sauro Succi, Weinan E, and Efthimios Kaxiras Discrete Simulation Automata: Mesoscopic Fluid Models Endowed with Thermal Fluctuations Tomonori Sakai and Peter V. Coveney Dissipative Particle Dynamics Pep Espa˜nol The Direct Simulation Monte Carlo Method: Going Beyond Continuum Hydrodynamics Francis J. Alexander Hybrid Atomistic–Continuum Formulations for Multiscale Hydrodynamics Hettithanthrige S. Wijesinghe and Nicolas G. Hadjiconstantinou

2411

2415

2447 2475

2487 2503

2513

2523

Chapter 9. Polymers and Soft Matter 9.1 9.2

Polymers and Soft Matter L. Mahadevan and Gregory C. Rutledge Atomistic Potentials for Polymers and Organic Materials Grant D. Smith

2555 2561

Detailed table of contents 9.3 9.4 9.5 9.6 9.7

9.8 9.9

Rotational Isomeric State Methods Wayne L. Mattice Monte Carlo Simulation of Chain Molecules V.G. Mavrantzas The Bond Fluctuation Model and Other Lattice Models Marcus M¨uller Stokesian Dynamics Simulations for Particle Laden Flows Asimina Sierou Brownian Dynamics Simulations of Polymers and Soft Matter Patrick S. Doyle and Patrick T. Underhill Mechanics of Lipid Bilayer Membranes Thomas R. Powers Field-Theoretic Simulations Venkat Ganesan and Glenn H. Fredrickson

xxxvii 2575 2583 2599 2607

2619 2631 2645

Plenary Perspectives P1 P2 P3 P4 P5 P6

P7

Progress in Unifying Condensed Matter Theory Duane C. Wallace The Future of Simulations in Materials Science D.P. Landau Materials by Design Gregory B. Olson Modeling at the Speed of Light J.D. Joannopoulos Modeling Soft Matter Kurt Kremer Drowning in Data – A Viewpoint on Strategies for Doing Science with Simulations Dierk Raabe Dangers of “Common Knowledge” in Materials Simulations Vasily V. Bulatov

Quantum Simulations as a Tool for Predictive Nanoscience Giulia Galli and François Gygi P9 A Perspective of Materials Modeling William A. Goddard III P10 An Application Oriented View on Materials Modeling Peter Gumbsch P11 The Role of Theory and Modeling in the Development of Materials for Fusion Energy Nasr M. Ghoniem

2659 2663 2667 2671 2675

2687

2695

P8

2701 2707 2713

2719

xxxviii

Detailed table of contents

P12 Where are the Gaps? Marshall Stoneham P13 Bridging the Gap between Quantum Mechanics and Large-Scale Atomistic Simulation John A. Moriarty P14 Bridging the Gap between Atomistics and Structural Engineering J.S. Langer P15 Multiscale Modeling of Polymers Doros N. Theodorou P16 Hybrid Atomistic Modelling of Materials Processes Mike Payne, G´abor Cs´anyi, and Alessandro De Vita P17 The Fluctuation Theorem and its Implications for Materials Processing and Modeling Denis J. Evans P18 The Limits of Strength J.W. Morris, Jr. P19 Simulations of Interfaces between Coexisting Phases: What Do They Tell us? Kurt Binder P20 How Fast Can Cracks Move? Farid F. Abraham P21 Lattice Gas Automaton Methods Jean Pierre Boon P22 Multi-Scale Modeling of Hypersonic Gas Flow Iain D. Boyd P23 Commentary on Liquid Simulations and Industrial Applications Raymond D. Mountain P24 Computer Simulations of Supercooled Liquids and Glasses Walter Kob P25 Interplay between Materials Theory and High-Pressure Experiments Raymond Jeanloz P26 Perspectives on Experiments, Modeling and Simulations of Grain Growth Carl V. Thompson P27 Atomistic Simulation of Ferroelectric Domain Walls I-Wei Chen

2731

2737

2749 2757 2763

2773 2777

2787 2793

2805 2811

2819 2823

2829

2837 2843

Detailed table of contents

xxxix

P28 Measurements of Interfacial Curvatures and Characterization of Bicontinuous Morphologies Sow-Hsin Chen

2849

P29 Plasticity at the Atomic Scale: Parametric, Atomistic, and Electronic Structure Methods Christopher Woodward P30 A Perspective on Dislocation Dynamics Nasr M. Ghoniem P31 Dislocation-Pressure Interactions J.P. Hirth P32 Dislocation Cores and Unconventional Properties of Plastic Behavior V. Vitek P33 3-D Mesoscale Plasticity and its Connections to Other Scales Ladislas P. Kubin P34 Simulating Fluid and Solid Particles and Continua with SPH and SPAM Wm.G. Hoover P35 Modeling of Complex Polymers and Processes Tadeusz Pakula P36 Liquid and Glassy Water: Two Materials of Interdisciplinary Interest H. Eugene Stanley P37 Material Science of Carbon Wesley P. Hoffman P38 Concurrent Lifetime-Design of Emerging High Temperature Materials and Components Ronald J. Kerans P39 Towards a Coherent Treatment of the Self-Consistency and the Environment-Dependency in a Semi-Empirical Hamiltonian for Materials Simulation S.Y. Wu, C.S. Jayanthi, C. Leahy, and M. Yu

2865 2871 2879

2883 2897

2903 2907

2917 2923

2929

2935

5.1 INTRODUCTION: RATE PROCESSES Horia Metiu University of California, Santa Barbara, CA, USA

We can divide the time evolution of a system into two classes. In one, a part of the system changes its state from time to time; chemical reactions, polaron mobility, diffusion of adsorbates on a surface, and protein folding belong to this class. In the other, the change of state takes place continuously; electrical conductivity, the diffusion of molecules in gases, and the thermoelectric effect in doped semiconductors belong to this class. Chemical kinetics deals with phenomena of the first kind; the second kind is studied by transport theory. It is in the nature of a many-body system that its parts share energy with each other, creating a state of approximate equality. This leads to stagnation: each part tends to hover near the bottom of a bowl in the potential energy surface. Occasionally, the inherent thermal fluctuations put enough energy in a part of the many-body system to cause it to escape from its bowl and travel away from home. But the tendency to lose energy rapidly, once a part acquires more than its average share, will trap the traveler in another bowl. When this happens, the system has undergone a chemical reaction, or the polaron took a jump to another lattice site, or an impurity in a solid changed location. The rate of these events is described by well known, generic, phenomenological rate equations. The parameter characterizing the rate of a specific system is the rate constant k. In the past 30 years great progress has been made in our ability to calculate the rate constant by atomic simulations. The machinery for performing such calculations is described in the first articles in this chapter. Doll presents the modern view on the old and famous transition state theory, which is still one of the most useful and widely used procedures for calculating rate constants. The atomic motion in a many-body system takes place on a scale of femtoseconds, while the lifetime of a system in a potential energy bowl is much longer. This discrepancy led to the misconception that the dynamics of a chemical reaction is slow. 1567 S. Yip (ed.), Handbook of Materials Modeling, 1567–1571. c 2005 Springer. Printed in the Netherlands.

1568

H. Metiu

The main insight of TST is that a system acquires enough energy to undergo a reaction only “once in a blue moon”. If enough energy is acquired, in the right coordinates, the dynamics of the reaction is very rapid. The rate of reaction is low not because its dynamics is slow, but because the system has enough energy very rarely. In modern parlance the reaction is a rare event. This causes problems for a brute-force simulation of a reaction. One can follow a group of atoms, in the many-body system, for a nanosecond, because of limitations in computing power, and not observe a reactive event. The second insight of TST is that the only parameter out of equilibrium, in a chemical kinetics experiment, is the concentration. Each molecule participating in the reaction is in equilibrium with its environment at all times. Therefore, one can calculate, from equilibrium statistical mechanics, the probability that the system reaches the transition state and the rate with which the system crosses the ridge separating the bowl in which the system is initially located from the one that is the final destination. This is all it takes to build a theory of the rate constant. The only approximation is the assumption that once the system crosses the ridge, it will turn back only in a long time, on the order of k−1 . This late event is part of the backward reaction and it does not affect the forward rate constant. Given the propensity of many-body systems to share energy among degrees of freedom, this is not a bad assumption: once it crosses the ridge the system has a high energy in the reaction coordinate and it is likely to lose it. There are, however, cases in which the shape of the potential energy around the ridge is peculiar or the reaction coordinate is weakly coupled to the other degrees of freedom. When this happens, recrossing is not negligible and TST makes errors. In my experience these errors are small and rarely affect the prefactor A, in the expression k =A exp[−E/RT], by more than 30%. Given the fact that we are unable to calculate the activation energy E accurately and that the latter appears at the exponent it seems unwise to try to obtain an accurate value for A when one makes substantial errors in E (a 0.2-eV error is not rare). This is why TST is still popular in spite of the fact that one could calculate the rate constant exactly, sometimes without a great deal of additional computational effort. The TST reduces the calculation of k to the calculation of partition functions, which can be performed by Monte Carlo simulations. There is no longer any need to perform extremely long Molecular Dynamics calculations in the hope of observing a transition of the system from one bowl to another. Because recrossing is neglected, the rate constant calculated by TST is always larger than the exact rate constant. This does not mean that the TST rate constant is always larger than the measured one. It is only larger than the rate constant calculated exactly on the potential energy surface used in the TST calculations. This inequality led to the development of variational transition state theory, developed and used extensively in Truhlar’s work. In this procedure one

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varies the position of the surface dividing the initial and the final bowls, until the transition theory rate constant has a minimum. The rate constant obtained in this way is more accurate (assuming that the potential energy is accurate) than the one calculated by placing the dividing surface on the ridge separating the two bowls. These issues are discussed and explained in Doll’s article. The next two articles, by Dellago and by Ciccotti, Kapral and Sergi, describe the methods used for exact calculations of the rate constant k. Here “exact” means the exact rate constant for a given potential energy surface. If the potential energy surface is erroneous, the exact rate constant has nothing to do with reality. However, it is important to have an exact theory, since our ability to generate reasonable (and sometimes accurate) potential energy surfaces is improving each year. The exact theory of the rate constant is based on the so-called correlation function theory, which first appeared in a paper by Yamamoto. Since this theory does not assume that recrossing does not take place, it must use molecular dynamics to determine which trajectories recross and which do not. It does this very cleverly, to avoid the “rare event” trap. It uses equilibrium statistical mechanics to place the system on the dividing surface, with the correct probability. Then it lets the system evolve to cross the dividing surface and follows its evolution to determine whether it will recross the dividing surface. If it does, that particular crossing event is discarded. If it does not, it is kept as a reactive event. Averages over many such reactive events, used in a specific equation provided by the theory, give the exact rate constant. The advantage of this method, over ordinary molecular dynamics, is that it must follow the trajectory only for the time when the reaction coordinate loses energy and the system becomes unable to recross the dividing surface. As many experiments and simulations show, this time is shorter than a picosecond, which is quite manageable in computations. Moreover, the procedure generates a large of number of reactive trajectories with the appropriate probability. Since reactive trajectories are very improbable, a brute-force molecular dynamics simulation, starting with the reactants, will generate roughly one reactive trajectory in 100 000 calculations, each requiring a very long trajectory. This is why brute-force calculations of the rate constant are not possible. The two articles mentioned above discuss two different ways of implementing the theory. The theory presented by Dellago is new and has not been extensively tested. The one presented by Ciccotti, Kapral, and Sergi is the workhorse used for all systems that can be described by classical mechanics. While in principle the method is simple, the implementation is full of pitfalls and “small” technical difficulties, and these are clarified in the articles. Application of the correlation function theory to the diffusion of impurities in solids is discussed by Wahnstrom.

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The statements made above, about the time scales reached by molecular dynamics, were true until a few years ago, when Voter proposed several methods that allow us to accelerate molecular dynamics to the point that we can follow the evolution of a complex system for microseconds. This has brought unexpected benefits. To use the transition state theory, or the correlation function theory of rare events, one must know what the events are; we need to know the initial and final state of the system. There are systems for which this is not easy to do. For example, Johnsson discovered, while studying the evolution of the shape of an “island” made by adsorbed atoms, that six atoms move in concert with relative ease. It is very unlikely that anyone would have proposed the existence of this “reaction” on the basis of chemical intuition. In general, in the complex systems encountered in biology and materials science, a group of molecules may move coherently and rapidly together in ways that are not intuitively expected. The accelerated dynamics method often finds such events, since it does not make assumptions about the final state of the system. The article of Blas, Uberuaga, and Voter discusses this aspect of kinetics. Since Kramers’ classic work, it has been realized that in many systems chemical reactions can be described by a stochastic method that involves the Brownian motion of the representative point of the system on the potential energy surface. Since then, the theory has been expanded and used to explain chemical kinetics in condensed phases. Its advantage is that it expresses chemical kinetics in complex media in terms of a few parameters, the strength of thermal fluctuations in the system and the “friction” causing the system to lose energy from the reaction coordinate. This reductionist approach appeals to many experimentalists who have used it to analyze chemical kinetics of molecules in liquids. Much work has also been done to connect the friction and the fluctuations to the detailed dynamics of the system. Nitzan’s article reviews the status of this field. All theories mentioned above assume that the motion of the system can be described by classical mechanics. This is not the case in reactions involving proton or electron transfer. The generalization of the correlation function theory of the rate constant to a fully quantum theory has been made by Miller, Schwartz, and Tromp, who extended considerably the early work of Yamamoto. Some of the first computational methods using this theory were proposed by Wahnstrom and Metiu. Since then, approximate methods, that allow calculations for systems with many degrees of freedom, have been invented. These are reviewed by Schwartz and Voth, who have both contributed substantially to this field. The review of quantum theory of rates is rounded off by an article by Gross, on reactive scattering and adsorption at surfaces. This discusses the dynamics of such reactions in more detail than usual in kinetics, since it examines the rate of reaction (dissociation or adsorption) when the molecule approaching the surface has a well-defined quantum

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state. One can obtain the rate constant from this information, by averaging the state-specific rates over a thermal distribution of initial states. Many people familiar with statistical mechanics have realized that chemical kinetics is, like any other phenomenon in a many-body system, subject to fluctuations that might be observable if one could detect the kinetic behavior of a small number of molecules. It was believed that light scattering may be able to study such fluctuations, since it can detect the evolution of concentration in the very small volume illuminated by light. It turned out that the volume was not small enough and, as far as I know, the fluctuations have not been detected by this method. Undaunted by this lack of experimental observations, Gillespie went ahead and developed the methodology needed for studying the stochastic evolution of the concentration in a system undergoing chemical reactions. This methodology assumed that the rate constants are known and examined the evolution of the concentrations in space and time. Later on, scanning tunneling microscopy studies of the evolution of atoms deposited on a surface and a variety of single molecule kinetic experiments provided examples of systems in which fluctuations in rate processes play a very important role. Gillespie’s article reviews the methods dealing with fluctuating chemical kinetics. Evans reviews the stochastic algorithms needed for studying the kinetics of adsorbates, with applications to crystal growth and catalysis. Jensen’s article studies specific kinetic models used in crystal growth. The chapter ends with three articles on kinetic phenomena of interest in biology. The rate of protein folding, studied with minimalist models that try to capture the essential features causing proteins to fold, is reviewed by Chan. Pande examines the use of detailed models in which the interatomic interactions are treated in detail. The two approaches are complementary and much can be learned by comparing their conclusions. Tajkhorshid, Zhu, and Schulten review the transport of water through the pores of cell membranes. A dominant feature of this transport is that water forms a quasi one-dimensional “wire”. For this reason, transport in biological channels is closely related to water transport through a carbon nanotube and the article reviews both. Kinetics, one of the oldest and most useful branches of chemical physics, is undergoing a quiet revolution and is penetrating in all areas of materials science and biochemistry. There is a very good reason for this: most systems we are interested in are metastable. To understand what they are, we need to use kinetics to simulate how they are made. Moreover, we need to use kinetics to understand how they function and how they are degraded by outside influences or by inner instabilities. Finally, a well-formulated kinetic model contains thermodynamics as the long-time limit.

5.2 A MODERN PERSPECTIVE ON TRANSITION STATE THEORY J.D. Doll Department of Chemistry, Brown University, Providence, RI, USA

Chemical rates, the temporal evolution of the populations of species of interest, are of fundamental importance in science. Understanding how such rates are determined by the microscopic forces involved is, in turn, a basic focus of the present discussion. Before delving into the details, it is valuable to consider the general nature of the problem we face when considering the calculation of chemical rates. In what follows we shall assume that we know: • • • •

the relevant physical laws (classical or quantum) governing the system, the molecular forces at work, the identity of the chemical species of interest, and the formal statistical-mechanical expressions for the desired rates.

Given all that, what is the “problem?” In principle, of course, there is none. “All” that we need do is to work out the “details” of our formal expressions and we have our desired rates. The kinetics of any conceivable physical, chemical, or biologic process are thus within our reach. We can predict fracture kinetics in complex materials, investigate the effects of arbitrary mutations on protein folding rates, and optimize the choice of catalyst for the decomposition/storage of hydrogen in metals, right? Sadly, “no.” Even assuming that all of the above information is at our disposal, at present it is not possible in practice to carry out the “details” at the level necessary to produce the desired rates for arbitrary systems of interest. Why not? The essential problem we face when discussing chemical rates is one of greatly differing time scales. If, for example, a species is of sufficient interest that it makes sense to monitor its population, it is, by default, generally relatively “stable.” That is, it is a species that tends to live a “long” time on the scale 1573 S. Yip (ed.), Handbook of Materials Modeling, 1573–1583. c 2005 Springer. Printed in the Netherlands.

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of something like a molecular vibration. On the other hand, if we are to understand the details of chemical events of interest, then we must be able to describe the dynamics of those events on a time scale that is “short” on the molecular level. If we do otherwise , we risk losing the ability to understand how those detailed molecular motions influence and/or determine the rates at issue. What happens then when we confront the problem of describing a rate process whose natural time scale is on the order of seconds? If we are not careful we end up drowning in the detail imposed by being forced to describe events on macroscopic time scales using microscopic dynamical methods. In short, we spend a great deal of time (and effort) watching things “not happen.” Is there a better way to proceed? Fortunately, “yes.” Using methods developed by a number of investigators [1–9], it is possible to formulate practical and reliable methods for estimating chemical rates for systems of realistic complexity. While there are often assumptions involved in the practical implementation of these approaches, it is increasingly feasible to quantify and often remove the effects of these assumptions albeit at the expense of additional work. It is our purpose to review and illustrate these methods. Our discussion will focus principally on classical level implementations. Quantum formulations of these methods are possible and are considered elsewhere in this monograph. While much effort has been devoted to the quantum problem, it remains a particularly active area of current research. In the present discussion, we purposely utilize a sometimes nonstandard language in order to unify the discussion of a number of historically separate topics and approaches. The starting point for any discussion of chemical rates is the identification of various species of interest whose population will be monitored as a function of time. While there are many possible ways in which to do this, it is convenient to consider an approach based on the Stillinger/Weber inherent structure ideas [10, 11]. In this formulation, configuration space is partitioned by assigned each position to a unique potential energy basin (“inherent structure”) based on a steepest descent quench procedure. The relevant mental image is that of watching a “ball” roll slowly “downhill” on the potential energy surface under the action of an over-damped dynamics. In many applications the Stillinger/Weber inherent structures are themselves of primary interest. Although the number of such structures grows rapidly (exponentially) with system size [12], this type of analysis and the associated graphical tools it has spawned [13], provide a valuable language for characterizing potential energy surfaces. Wales, in particular, has utilized variations of the technique to great advantage in their study of the minimization problem [14]. In our discussion, it is the evolution of the populations of the inherent structures rather than the structures themselves that are of primary concern. Inherent structures, by construction, are associated with local minima in the

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potential energy surface. They thus have an intrinsic equilibrium population that can, if desired, be estimated using established statistical–mechanical techniques. Since the dynamics in the vicinity of the inherent structures is locally stable, the inherent structure populations tend to be (relatively) slowly varying and thus provide us with a natural set of populations for kinetic study. If followed as a function of time under the action of the dynamics generated by potential energy surface to which the inherent structures belong, the populations of the inherent structures will, aside from fluctuations, tend to remain constant at their various equilibrium values. Fluctuations in these populations, on the other hand, will result in a net flow of material between the various inherent structures. Such flows are the mechanism by which such fluctuations, either induced or spontaneous, “relax.” Consequently, they contain sufficient information to establish the desired kinetic parameters. To make the discussion more explicit, we consider the simple situation of a particle moving on the bistable potential energy depicted in Fig. 1. Performing a Stillinger/Weber quench on this potential energy will obviously produce two inherent structures. Denoted A and B in the figure, these correspond to the regions to the left and right of the potential energy maximum, respectively. We now imagine that we follow the dynamics of a statistical ensemble of N particles moving on this potential energy surface. For the purposes of discussion, we assume that the physical dynamics involved includes a solvent or “bath” (here unspecified) that provides fluctuating forces that act on the system

V(x)

A

B x

Figure 1. A prototypical, bistable potential energy. The two inherent structures, A and B, are separated by an energy barrier.

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of interest. The bath dynamics acts both to energize the system (permitting it to acquire sufficient energy to sometimes cross the potential barrier) as well as to dissipate that energy once it has been acquired. It is important to note that these fluctuations and dissipations must, in some sense, be balanced if an equilibrium state is to be produced and sustained [7]. Were the dynamics in our example purely conservative and one-dimensional in nature, for example, the notion of rates would be ill-posed. We now assume in what follows that we can monitor the populations of the inherent structures as a function of time. Denoting these populations NA (t) and NB (t), we further assume, following Chandler [7], that the overall kinetics of the system can described by the phenomenological rate equations dNA (t) = −kA→B NA (t) + kB→A NB (t) dt (1) dNB (t) = +kA→B NA (t) − kB→A NB (t). dt If the total number of particles is conserved, then the two inherent structure populations are trivially related: the fluctuation in the population of one inherent structure is the negative of that for the other. Assuming a fixed number of particles, it is thus a relatively simple matter to show that dδ NA (t) = −(kA→B + kB→A )δ NA (t), (2) dt where δ NA (t) indicates the deviation of NA (t) from its equilibrium value. The decay of a fluctuation in the population of inherent structure A, relative to an initial value at time zero, is thus given by δ NA (t) = δ NA (0) e−keff t ,

(3)

where keff is given by the sum of the “forward” and “backward” rate constants keff = (kA→B + kB→A ).

(4)

As noted by Onsager [15], it is physically reasonable to assume that if they are small, fluctuations, whether induced or spontaneous, are damped in a similar manner. Accepting this hypothesis, we conclude from the above analysis that the decay of the equilibrium population autocorrelation function, denoted here by , is given in terms of keff by δ NA (0)δ NA (t) = e−keff t . δ NA (0)δ NA (0)

(5)

Equivalently, taking the time derivative of both sides of this expression, we see that keff is given explicitly as keff = −

δ NA (0)δ N˙ A (t) . δ NA (0)δ NA (t)

(6)

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Equations (5) and (6) are formally exact expressions that relate the sum of the basic rate constants of interest to various dynamical objects that can be computed. Since we also know the ratio of these two rate constants (it is given by the corresponding ratio of the equilibrium populations), the desired rate parameters can be obtained from either expression provided that we can obtain the relevant time correlation functions involved. Although formally equivalent, Eqs. (5) and (6) differ with respect to their implicit computational demands. Computing the rate parameters via Eq. (5), for example, entails monitoring the decay of the population autocorrelation function. To obtain reliable estimates of the rate parameters from Eq. (5), we have to follow the system dynamics over a time-scale that is an appreciable fraction of the reciprocal of keff . If the barriers separating the inherent structures involved are “large”, this time scale can become macroscopic. Simply stated, the disparate time-scale problem makes it difficult to study directly the dynamics of infrequent events using the approach suggested by Eq. (5). Equation (6), on the other hand, offers a more convenient route to the desired kinetic parameters. In particular, it indicates that we might be able to obtain these parameters from short as opposed to long-time dynamical information. If the phenomenological rate expressions are formally correct for all times, then the ratio of the two time correlation functions in Eq. (6) is time-independent. However, since it is generally likely that the phenomenological rate expressions accurately describe only the longer-time motion between inherent structures, we expect in practice that the ratio on the right hand side of Eq. (6) will approach a constant “plateau” value only at times long on the scale of detailed molecular motions. The critical point, however, is that this transient period will be of molecular not macroscopic duration. With Eq. (6), we thus have a route to the desired kinetic parameters that requires only molecular or short time-scale dynamical input. A valuable practical point concerning kinetic formulations based on Eq. (6) is that for many applications the final plateau value of the correlation function ratio involved is often relatively well approximated by its zero time value. Because the correlation functions required depend only on time differences, such zero-time quantities are purely equilibrium objects. Consequently, an existing and extensive set of equilibrium tools can be invoked to produce approximations to kinetic parameters. The approach to the calculation of chemical rates based on Eq. (6) has several desirable characteristics. Most importantly, it has a refinable nature and can be implemented in stages. At the simplest level, we can estimate chemical rate parameters using purely zero-time, or equilibrium methods. Such approximate methods alone may be adequate for many applications. We are, however, not restricted to accepting such approximations blindly. With additional effort we can “correct” such preliminary estimates by performing additional dynamical studies. Because such calculations involve “corrections” to

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equilibrium estimates of rate parameters, as opposed to the entire rate parameters themselves, the dynamical input required is only that necessary to remove the errors induced by the initial equilibrium assumptions. Because such errors tend to involve simplified assumptions concerning the nature of transition state dynamics, the input required to estimate the corrections is of a molecular, not macroscopic time scale. We now focus our discussion on some of practical issues involved in generating equilibrium estimates of the rates. We shall illustrate these using the simple two-state example described above. We begin by imagining that we have at our disposal the time history of a reaction coordinate of interest, x(t). As a function of time, x(t) moves back-and-forth between inherent structures A and B, which we assume to be separated by the position x = q. Using one of the basic properties of the delta function, δ(ax) =

1 δ(x), |a|

(7)

it is easy to show that N (τ , [x(t)]), defined by N (τ, [x(t)]) =

τ

dx(t) δ(x(t) − q), dt

dt

0

(8)

is a functional of the path whose value is equal to the (total) number of crossings of the x(t) = q surface in the interval (0,τ ). Every time x(t) crosses q, the delta function argument takes on a zero value. Because the delta function in Eq. (8) is in coordinate space while the integral is with respect to time, the Jacobian factor into Eq. (8) creates a functional whose value jumps by unity each time x(t) − q sweeps through a value of zero. If we form a statistical ensemble corresponding to various possible histories of the motion of our system and bath, we can compute the average number of crossings of the x(t) = q surface in the (0,τ ) interval, N(τ , [x(t)]), using the expression N (τ, [x(t)]) =

τ

dt x(t) ˙ δ(x(t) − q) .

(9)

0

Here represents the time derivative of x(t). Because are dealing with a “stationary” or equilibrium process, the time correlation function that appears on the right hand side of Eq. (9) can be function only of time differences. Consequently, the integrand on the right hand side of Eq. (9) is time-independent and can be brought outside the integral. The result thus becomes τ dt, N (τ, [x(t)]) = x˙ δ(x − q) 0

(10)

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where the (now unnecessary) time labels have been dropped. We thus see that the number of crossings of the x(t) = q surface in this system per unit time is given by N (τ, [x(t)]) = x˙ δ(x − q) . (11) τ Recalling that N measures the total number of crossings, the number of crossings per unit time in the direction from A to B (the number of “up zeroes” of x(t) − q in the language of Slater) is half the value in Eq. (11). Thus, the equilibrium estimate of the rate constant for the A to B transition, (i.e., the number of crossings per unit time from A to B per atom in inherent structure A) is given by

1 TST = kA→B 2

x˙ δ(x − q)

NA

.

(12)

Equation (12) gives an approximate expression to the rate constant that involves an equilibrium flux between the relevant inherent structures. Because the relevant flux is associated with the “transition” of one inherent structure into another, the approach to chemical rates suggested by Eq. (12) is typically termed “transition state” theory (TST). Along with its multi-dimensional generalizations, it represents a convenient and useful approximation to the desired chemical rate constants. Being an equilibrium approximation to the dynamical objects of interest, it permits the powerful machinery of Monte Carlo methods [16, 17] to be brought to bear on the computational problem. The significance of this is that the required averages can be computed to any desired accuracy for arbitrary potential energy models. One can proceed analytically by making secondary, simplifying assumptions concerning the potential. Such approximations are, however, controllable in that their quality can be tested. Furthermore, Eq. (12) provides a unified treatment of the problem that is independent of the nature of the statistical ensemble that is involved. Applications involving canonical, microcanonical and other ensembles are treated within a common framework. It is historically interesting in this regard to note that if the reaction coordinate of interest is expressed as a superposition of normal modes, Eq. (12) leads naturally to the unimolecular reaction expressions of Ref. [4]. There is a technical aspect concerning the calculation of the averages appearing in Eq. (12) that merits discussion. In particular, it is apparent from the nature of the average involved that, if they are to be computed accurately, the numerical methods involved must be capable of accurately describing the reactant’s concentration profile in the vicinity of the transition state. If we are dealing with with activated processes where the difference between transition state in inherent structure energies are “large”, then such concentrations can become quite small and difficult to treat by standard methods. This is

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simply the equilibrium, “sparse-sampling” analog of the disparate time-scale dynamical problem. Fortunately, there are a number well-defined techniques for coping with this technical issue. These include, to name a few, umbrella methods [18], Bennett/Voter techniques [19, 20], J-walking [21, 22], and parallel tempering approaches [23]. These and related methods make it possible to compute the the required, transition-state-constrained averages. The basic approach outlined above can be extended in a number of ways. One immediate extension involves problems in which there are multiple, rather than two states involved. Adams has considered such problems in the context of his studies on the effects of precursor states on thermal desorption [24]. A second extension involves using the fundamental kinetic parameters produced to study more complex events. Voter, in a series of developments, has formulated a computationally viable method for studying diffusion in solids based on such an approach [25]. In its most complete form (including dynamical corrections), this approach produces a computationally exact procedure for surface or bulk diffusion coefficients of a point defect at arbitrary temperatures in a periodic system [26]. In related developments, Voter [25] and Henkelmen and J´onsson [27] have discussed using “on-the-fly” determinations of TST kinetic parameters in kinetic Monte Carlo studies. Such methods make it possible to explore a variety of lattice dynamical problems without resorting to ad hoc assumptions concerning mechanisms of various elementary events. In a particularly promising development, they also appear to offer a valuable tool for the study of long-time dynamical events [28, 29]. An important practical issue in the calculation of TST approximations to rates is the identification of the transition state itself. In many problems, such as the simple two-state problem discussed previously, locating the transition state is trivial. In others, it is not. Techniques designed to locate explicit transition states in complex systems have been discussed in the literature. One popular technique, developed by Cerjan and Miller [30] and extended by others [31–33], is based on an “eigenvector following” method. In this approach, one basically moves “up-hill” from a selected inherent structure using local mode information to determine the transition state. Other approaches, including methods that do not require explicit second-order derivatives of the potential, have been discussed [34]. It is also important to mention a different class of methods suggested by Pratt [35]. Borrowing a page from path integral applications, this technique attempts to locate transition states by working with paths that build in proper initial and final inherent structure character from the outset. Expanding upon the spirit of the original Pratt suggestion, recent efforts have considered sampling barrier crossing paths directly [36]. We wish to close by pointing out what we feel may prove to be a potentially useful link between inherent structure decomposition methods and the problem of “probabilistic clustering” [37, 38]. An important problem in applied mathematics is the reconstruction of an unknown probability distribution given a

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known statistical sampling of that distribution. So stated, the probabilistic clustering problem is effectively the inverse of the Monte Carlo sampling problem. Rather than producing a statistical sampling of a given distribution, we seek instead to reconstruct the unknown distribution from a known statistical sampling. This clustering problem is of broad significance in information technology and has received considerable attention. Our point in emphasizing the link between probabilistic clustering and inherent structure methods is that our increased ability to sample arbitrary, sparse distributions would appear to offer an alternative to the Stillinger/Weber quench approach to the inherent structure decomposition problem. In particular, one could use clustering methods both to “identify” and to “measure” the concentrations of inherent structures present in a system.

Acknowledgments The author would like to thank the National Science Foundation for support through awards CHE-0095053 and CHE-0131114 and the Department of Energy through award DE-FG02-03ER46704. He also wishes to thank the Center for Advanced Scientific Computing and Visualization (TCASCV) at Brown University for valuable assistance with respect to some of the numerical simulations described in the present paper.

References [1] M. Polanyi and E. Wigner, “The interference of characteristic vibrations as the cause of energy fluctuations and chemical change,” Z. Phys. Chem., 139(Abt. A), 439, 1928. [2] H. Eyring, “Activated complex in chemical reaction,” J. Chem. Phys., 3, 107, 1935. [3] H.A. Kramers, “Brownian motion in a field of force and the diffusion model of chemical reactions,” Physica (The Hague), 7, 284, 1940. [4] N.B. Slater, Theory of Unimolecular Reactions, Cornell University Press, Ithaca, 1959. [5] P.J. Robinson and K.A. Holbrook, Unimolecular Reactions, Wiley-Interscience, 1972. [6] D.G. Truhlar and B.C. Garrett, “Variational transition state theory,” Ann. Rev. Phys. Chem., 35, 159, 1984. [7] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford, New York, 1987. [8] P. H¨anggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Rev. Mod. Phys., 62, 251, 1990. [9] M. Garcia-Viloca, J. Gao, M. Karplus, and D.G. Truhlar, “How enzymes work: analysis by modern rate theory and computer simulations,” Science, 303, 186, 2004.

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[10] F.H. Stillinger and T.A. Weber, “Dynamics of structural transitions in liquids,” Phys. Rev. A, 28, 2408, 1983. [11] F.H. Stillinger and T.A. Weber, “Packing structures and transitions in liquids and solids,” Science, 225, 983, 1984. [12] F.H. Stillinger, “Exponential multiplicity of inherent structures,” Phys. Rev. E, 59, 48, 1999. [13] O.M. Becker and M. Karplus, “The topology of multidimensional potential energy surfaces: theory and application to peptide structure and kinetics,” J. Chem. Phys., 106, 1495, 1997. [14] D.J. Wales and J.P.K. Doye, “Global optimization by basin-hopping and the lowest energy structures of Lennard–Jones clusters containing up to 110 atoms,” J. Phys. Chem. A, 101, 5111, 1997. [15] L. Onsager, “Reciprocal relations in irreversible processes, II,” Phys. Rev., 38, 2265, 1931. [16] M.H. Kalos and P.A. Whitlock, Monte Carlo Methods, Wiley-Interscience, New York, 1986. [17] M.P. Nightingale and C.J. Umrigar, Quantum Monte Carlo Methods in Physics and Chemistry, Kluwer, Dordrecht, 1998. [18] J.P. Valleau and G.M. Torrie, “A guide to Monte Carlo for statistical mechanics: 2. byways,” In: B.J. Berne (ed.), Statistical Mechanics: Equilibrium Techniques, Plenum, New York, 1969, 1977. [19] C.H. Bennett, “Exact defect calculations in model substances,” In: A.S. Nowick and J.J. Burton (eds.), Diffusion in Solids: Recent Developments, Academic Press, New York, pp. 73, 1975. [20] A.F. Voter, “A Monte Carlo method for determining free-energy differences and transition state theory rate constants,” J. Chem. Phys., 82,1890, 1985. [21] D.D. Frantz, D.L. Freeman, and J.D. Doll, “Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking: applications to atomic clusters,” J. Chem. Phys., 93, 2769, 1990. [22] J.P. Neirotti, F. Calvo, D.L. Freeman, and J.D. Doll, “Phase changes in 38 atom Lennard-Jones clusters: I: a parallel tempering study in the canonical ensemble,” J. Chem. Phys., 112, 10340, 2000. [23] C.J. Geyer and E.A. Thompson, “Anealing Markov chain Monte Carlo with applications to ancestral inference,” J. Am. Stat. Assoc., 90, 909, 1995. [24] J.E. Adams and J.D. Doll, “Dynamical aspects of precursor state kinetics,” Surf. Sci., 111, 492, 1981. [25] J.D. Doll and A.F. Voter, “Recent developments in the theory of surface diffusion,” Ann. Revi. Phys. Chem., 38, 413, 1987. [26] A.F. Voter, J.D. Doll, and J.M. Cohen, “Using multistate dynamical corrections to compute classically exact diffusion constants at arbitrary temperature,” J. Chem. Phys., 90, 2045, 1989. [27] G. Henkelman and H. J´onsson, “Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table,” J. Chem. Phys., 115, 9657, 2001. [28] A.F. Voter, F. Montalenti, and T.C. Germann, “Extending the time scale in atomistic simulation of materials,” Ann. Rev. Mater. Res., 32, 321, 2002. [29] V.S. Pande, I. Baker, J. Chapman, S.P. Elmer, S. Khaliq, S.M. Larson, Y.M. Rhee, M.R. Shirts, C.D. Snow, E.J. Sorin, and B. Zagrovic, “Atomistic protein folding simulations on the submillisecond time scale using worldwide distributed computing,” Biopolymers, 68, 91, 2003.

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[30] C.J. Cerjan and W.H. Miller, “On finding transition states,” J. Chem. Phys., 75, 2800, 1981. [31] C.J. Tsai and K.D. Jordan, “Use of an eigenmode method to locate the stationary points on the potential energy surfaces of selected argon and water clusters,” J. Phys. Chem., 97, 11227, 1993. [32] J. Nichols, H. Taylor, P. Schmidt, and J. Simons, “Walking on potential energy surfaces,” J. Chem. Phys., 92, 340, 1990. [33] D.J. Wales, “Rearrangements of 55-atom Lennard–Jones and (C60) 55 clusters,” J. Chem. Phys., 101, 3750, 1994. [34] G. Henkelman and H. J´onsson, “A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives,” J. Chem. Phys., 111, 7010, 1999. [35] L.R. Pratt, “A statistical method for identifying transition states in high dimensional problems,” J. Chem. Phys., 85, 5045–5048, 1986. [36] P.G. Bolhuis, D. Chandler, C. Dellago, and P.L. Geissler, “Transition path sampling: throwing ropes over rough mountain passes, in the dark,” Ann. Rev. Phys. Chem., 53, 291, 2002. [37] B.G. Mirkin, Mathematical Classification and Clustering, Kluwer, Dordrecht, 1996. [38] D. Sabo, D.L Freeman, and J.D. Doll, “Stationary tempering and the complex quadrature problem,” J. Chem. Phys., 116, 3509, 2002.

5.3 TRANSITION PATH SAMPLING Christoph Dellago Institute of Experimental Physics, University of Vienna, Vienna, Austria

Often, the dynamics of complex condensed materials is characterized by the presence of a wide range of different time scales, complicating the study of such processes with computer simulations. Consider, for instance, dynamical processes occurring in liquid water. Here, the fastest molecular processes are intramolecular vibrations with periods in the 10–20 fs range. The translational and rotational motions of water molecules occur on a significantly longer time scale. Typically, the direction of translational motion of a molecule persist for about 500 fs, corresponding to 50 vibrational periods. Hydrogen bonds, responsible for many of the unique properties of liquid water, have an average lifetime of about 1 ps and the rotational motion of water molecules stays correlated for about 10 ps. Much longer time scales are typically involved if covalent bonds are broken and formed. For instance, the average lifetime of a water molecule in liquid water before it dissociates and forms hydroxide and hydronium ions is on the order of 10 h. This enormous range of time scales, spanning nearly 20 orders of magnitude, is a challenge for the computer simulator who wants to study such processes. In general, the dynamics of molecular systems can be explored on a computer with molecular dynamics simulation (MD), a method in which the underlying equations of motion are solved in small time steps. In such simulations the size of the time step must be shorter than the shortest characteristic time scale in the system. Thus, many molecular dynamics steps must be carried out to explore the dynamics of a molecular system for times that are long compared with the basic time scale of molecular vibrations. Depending on specific system properties and the available computer equipment, one can carry out from 10 000 to millions of such steps. In ab initio simulations where interatomic forces are determined by solving the electronic structure problem on the fly, total simulation times typically do not exceed dozens of picoseconds. Longer simulations of nanosecond, or, in some rare cases, microsecond length can be achieved if forces are determined from computationally less expensive empirical force fields often 1585 S. Yip (ed.), Handbook of Materials Modeling, 1585–1596. c 2005 Springer. Printed in the Netherlands.

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used to simulate biological systems. But many interesting and important processes still lay beyond the time scale accessible with MD simulations even on today’s fastest computers. Indeed, an ab initio molecular dynamics simulation of liquid water long enough to observe a few dissociations of water molecules would require a multiple of the age of the universe of computing time even on state-of-the-art parallel high performance computers. The computational effort needed to study many other interesting processes, ranging from protein folding to the nucleation of phase transitions and transport in and on solids, in straightforward molecular dynamics simulations with atomistic resolution may be less extreme, but still surpasses the capabilities of current computer technology. Fortunately, many processes occurring on long time scale are rare rather than slow. Consider, for instance, a chemical reaction during which the system has to overcome a large energy barrier on its way from reactants to products. Before the reaction occurs, the system typically spends a long time in the reactant state and only a rare fluctuation can drive the system over the barrier. If this fluctuation happens, however, the barrier is crossed rapidly. For example, it is now known from transition path sampling simulations that the dissociation of a water molecule in liquid water takes place in a few hundred femtoseconds once a rare solvent fluctuation drives the transition between the stable states, the intact water molecule and the separated ion pair. As mentioned earlier, the waiting time for this event, however, is of the order of 10 h. Other examples of rare but fast transitions between stable states include the nucleation of first order phase transitions, conformational transitions of biopolymers, and transport in and on solids. In such cases it is computationally advantageous to focus on those segments of the time evolution during which the rare event takes place rather than wasting large amounts of computing time following the dynamics of the system waiting for the rare event to happen. Several computational techniques to accomplish that have been put forward [1–4]. One approach consists in locating (or postulating) the bottleneck separating the stable states between which the rare transition occurs. Molecular dynamics trajectories initiated at this bottleneck, or transition state, can then be used to study the reaction mechanism in detail and to calculate reaction rate constants [5]. In small or highly ordered systems transition states can often be associated with saddle points on the potential energy surface. Such saddle points can be located with appropriate algorithms. Particularly in complex, disordered systems such as liquids, however, such an approach is frequently unfeasible. The number of saddle points on the potential energy surface may be very large and most saddle points may be irrelevant for the transition one wants to study. Entropic effects can further complicate the problem. In this case, a technique called transition path sampling provides an alternative approach [6]. Transition path sampling is a computational methodology based on a statistical mechanics of trajectories. It is designed to study rare transitions between

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known and well defined stable states. In contrast to other methods, transition path sampling does not require any a priori knowledge of the mechanism. Instead, it is sufficient to unambiguously define the stable states between which the transition occurs. The basic idea of transition path sampling consists in assigning a probability, or weight, to every pathway. This probability is a statistical description of all possible reactive trajectories, the transition path ensemble. Then, trajectories, are generated according to their probability in the transition path ensemble. Analysis of the harvested pathways yields detailed mechanistic information on the transition mechanism. Reaction rate constants can be determined within the framework of transition path sampling by calculating “free energies” between different ensembles of trajectories. In the following, we will give a brief overview of the basic concepts and algorithms of the transition path sampling technique. For a detailed description of the methodology and for practical issues regarding the implementation of transition path sampling simulations the reader is referred to two recent review articles [7, 8].

1.

The Transition Path Ensemble

Imagine a system with two long-lived stable states, call them A and B, between which rare transitions occur (see Fig. 1). The system spends much of its time fluctuating in the stable states A and B but rarely transitions between A and B occur. In the transition path sampling method one focuses on short

B

A

Figure 1. Several transition pathways connecting stable states A and B which are separated by a rough free energy barrier.

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trajectories x(T ) of length T (in time) represented by a time-ordered discrete sequence of states: x(T ) ≡ {x0 , xt , x2t , . . . , xT }.

(1)

Here, xt is the state of the system at time t. Each trajectory may be thought of as a chain of states obtained by taking snapshots at regular time intervals of length t as the system evolves according to the rules of the underlying dynamics. If the time evolution of the system follows Newton’s equations of motion, x ≡ {r, p} is a point in phase space and consists of the coordinates, r, and momenta, p, of all particles. For systems evolving according to a high friction Langevin equation or a Monte Carlo procedure the state x may include only coordinates and no momenta. The probability of a certain trajectory to be observed depends on the probability ρ(x0 ) of its initial point x0 and on the probability to observe the subsequent sequence of states starting from that initial point. For a Markovian process, that is for a process in which the probability of state xt to evolve into state xt +t over a time t depends only on xt and not on the history of the system prior to t, the probability P[x(T )] of a trajectory x(T ) can simply be written as a product of single step transition probabilities p(xt → xt +t ): P[x(T )] = ρ(x0 )

T /t −1

p(xit → x(i+1)t ).

(2)

i=0

For an equilibrium system in contact with a heat bath at temperature T the distribution of starting points is canonical, i.e., ρ(x0 ) ∝ exp{−H (x)/kB T }, where H (x) is the Hamiltonian of the system and kB is Boltzmann’s constant. Depending on the process under study other distributions of initial conditions may be appropriate. The path distribution of Eq. (2) describes the probability to observe a particular trajectory regardless of whether it connects the two stable states A and B. Since in the transition path approach the focus is on reactive trajectories, the path distribution P[x(T )] is restricted to the subset of pathway starting in A and ending in B: PAB [x(T )] ≡ h A (x0 )P[x(T )]h B (xT )/Z AB (T ).

(3)

The functions h A (x) and h B (x) are unity if their argument x lies in region A or B, respectively, and they vanish otherwise. Accordingly, only reactive trajectories starting in A and ending in B can have a weight different from zero in the path distribution PAB [x(T )]. The factor Z AB (T ) ≡

Dx(T ) h A (x0 )P[x(T )]h B (xT ),

(4)

which has the form of a partition function, normalizes the path distribution of Eq. (3). The notation Dx(T ), familiar from path integral theory, denotes a

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summation over all pathways. The function PAB [x(T )], which is a probability distribution function in the high dimensional space of all trajectories, describes the set of all reactive trajectories with their correct weight. This set of pathways is the transition path ensemble. In transition path sampling simulations care must be exercised in defining the stable states A and B. Both A and B need to be large enough to accommodate most equilibrium fluctuations, i.e., the system should spend the overwhelming fraction of time in either A or B. At the same time, A and B should not overlap with each other’s basin of attraction. Here, the basin of attraction of region A consist of all configurations that relax predominantly into that region. The basin of attraction of region B is defined analogously. If state A is incorrectly defined in such a way that it contains also points belonging to the basin of attraction of B, the transition path ensemble includes pathways only apparently connecting the two stable states. This situation is illustrated in Fig. 2. In many cases the stable states A and B can be defined through specific limits of a one-dimensional order parameter q(x). Although there is no general rule guiding the construction of such order parameters, this step in setting up a

q'

TS

A

B qA

qB

q

Figure 2. Regions A and B must be defined in a way to avoid overlap of A and B with each other’s basin of attraction. On this two dimensional free energy surface region A defined through q < q A includes points belonging to the basin of attraction of B (defined through q > q B ). Thus, the transition path ensemble PAB [x(T )] contains paths which start in A and end in B, but which never cross the transition state region marked by TS (dashed line). This problem can be avoided by using also the variable q in the definition of the stable states.

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transition path sampling simulation can be usually completed quite easily with a trial and error procedure. Note, however, that an appropriate order parameter is not necessarily a good reaction coordinate capable of describing the whole transition. In general, finding such a reaction coordinate is a difficult problem.

2.

Sampling the Transition Path Ensemble

In the transition path sampling method a biased random walk through path space is performed in such a way that pathways are visited according to their weight in the transition path ensemble PAB [x(T )]. This can be accomplished in an efficient way with Monte Carlo methods proceeding in analogy to a conventional Monte Carlo simulation of, say, a simple liquid at a given temperature T [9]. In that case a random walk through configuration space is constructed by carrying out a sequence of discrete steps. In each step, a new configuration is generated from an old one, for instance by displacing a single particle in a random direction by a random amount. Then, the new configuration, also called trial configuration, is accepted or rejected depending on how the probability of the new configuration compares to that of the old one. This is most easily done by applying the celebrated Metropolis rule [10], designed to enforce detailed balance between the move and its reverse. As a result, the trial move is always accepted if the energy of the new configuration is lower than that of the old one and accepted with a probability exp(−E/kB T ) if the trial move is energetically uphill (here, E is the energy difference between the new and the old configuration). Execution of a long sequence of such random moves followed by the acceptance or rejection step yields a random walk of the system through configuration space during which configurations are sampled with a frequency proportional to their weight in the canonical ensemble. Ensemble averages of structural and thermodynamics quantities can then straightforwardly computed by averaging over this sequence of configurations. In a transition path sampling simulation one proceeds analogously. But in contrast to a conventional Monte Carlo simulation, the random walk is carried out in the space of all trajectories and the result is a sequence of pathways instead of a sequence of configurations. In each step of this random walk a new pathway x (n) (T ), the trial path, is generated from an an old one, x (o) (T ). Then, the trial pathway is accepted or rejected according to how its weight PAB [x (n) (T )] in the transition path ensemble compares to the weight of the old one, PAB [x (o) (T )]. Correct sampling of the transition path ensemble is guaranteed by enforcing the detailed balance condition which requires the probability of a path move from x (o) (T ) to x (n) (T ) to be balanced exactly by the probability of the reverse path move. This detailed balance condition can be satisfied using the Metropolis criterion. Iterating this procedure of path generation followed by acceptance or rejection, one obtains a sequence of pathways in which

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each pathway is visited according to its weight in the transition path ensemble. It is important to note that while pathways are sampled with a Monte Carlo procedure, each single pathway is a genuinely dynamical pathway generated according to the rules of the underlying dynamics. To implement the procedure outlined above, one needs to specify how to generate a new pathway from an old one. This can be done efficiently with algorithms called shooting and shifting. For simplicity we will explain these algorithms for Newtonian dynamics (as used in most MD simulations) although they can be easily applied to other types of dynamics as well. So, imagine a Newtonian trajectory of length T as obtained from a molecular dynamics simulation of L = T /t steps starting in region A and ending in region B (see Fig. 3). From this existing transition pathway a new trajectory is generated by first randomly selecting a state of the existing trajectory. Then, the momenta belonging to the selected state are changed by a small random amount. Starting from this state with modified momenta the equations of motion are integrated forward to time T and backward to time 0. As a result, one obtains a complete new trajectory of length T which crosses (in configuration space) the old trajectory at one point. By keeping the momentum displacement small the new trajectory can be made to resemble the old one closely. As a consequence, the new pathway is likely to be reactive as well and to have a nonzero weight in the transition path ensemble. Any new trajectory with starting point in A and ending point in B can be accepted with high likelihood (in fact, for constant energy trajectories with a microcanonical distribution of initial conditions all new trajectories connecting A and B can be accepted). If the new trajectory does not begin in A or does not end in B it is rejected. For optimum efficiency, the magnitude of the momentum displacement should be selected such that the average acceptance probability is in the range from 40 to 60%. Shooting moves can be complemented with shifting moves, which consist in shifting the starting point of the path in time. This kind of move is computationally inexpensive since typically only a small part of the pathway needs to

A

B

Figure 3. In a shooting move one generates a new trajectory (dashed line) from an old one (solid line) by integrating the equations of motion forward and backward starting from a point with random momenta randomly selected along the old trajectory. The acceptance probability of the newly generated path can be controlled by varying the magnitude of the momentum displacement (thin arrow).

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be regrown. If the starting point of the path is shifted forward in time, a path segment of appropriate length has to be appended at the end of the path by integration of the equation of motion. If, on the other hand, the starting point is shifted backward in time, the trajectory must be completed by integrating the equations of motion backward in time starting from the initial point of the original pathway. Depending on the time by which the path is shifted, the new path can have large parts in common with the old path. Since ergodic sampling is not possible with shifting moves alone, path shifting always needs to be combined with path shooting. Although shifting moves cannot generate a truly new path, they can increase sampling efficiency especially for the calculation of reaction rate constants. To start the Monte Carlo path sampling procedure one needs a pathway that already connects A with B. This initial pathway is not required to be a high-weight dynamical trajectory, but can be an artificially constructed chain of states. Shooting and shifting will then rapidly relax this initial pathway towards regions of higher probability in path space. The generation of an initial trajectory is strongly system dependent and usually does not pose a serious problem.

3.

Analyzing Transition Pathways

Pathways harvested with the transition path sampling method are full dynamical trajectories in the space spanned by positions and momenta of all particles. In such high-dimensional many-particle systems it is usually difficult to identify the relevant degrees of freedom and to distinguish them from those which might be regarded as random noise. In the case of a chemical reaction occurring in a solvent, for instance, the specific role of solvent molecules during the reaction is often unclear. Although direct inspection of transition pathways with molecular visualization tools may yield some insight, detailed knowledge of the transition mechanism can only be gained through systematic analysis of the collected pathways. In the following, we will briefly review two approaches to carry out such an analysis: the transition state ensemble and the distribution of committors. In simple systems of a few degrees of freedom, for instance a small molecule undergoing an isomerization in the gas phase, one can study transition mechanisms by locating minima and saddle points on the potential energy surface of the system. While the potential energy minima are the stable states in which the system spends most of its time, the saddle point are configurations the system must cross on its way from one potential energy well to another. These so called transition states are the lowest points on ridges separating the stable states from each other. From the transition states the system can relax

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into either one of the two stable states depending on the initial direction of motion. In a high dimensional complex system local potential energy minima and saddle points do not carry the same significance as in simple systems. In a large, disordered system many local potential energy minima and saddle points may belong to one single stable state, and free energy barriers may not be related to a single saddle point. Nevertheless, the concept of a transition state is still meaningful if defined in a statistical way. In this definition, configurations are considered to be transition states if trajectories started from them with random initial momenta have equal probability to relax to either one of the stable states between which transitions occur. Naturally, along each transition pathway there is at least one (but sometimes several) configuration with this property. Performing such an analysis for many transition pathways yields the transition state ensemble, the set of all configurations on transition pathways which relax into A and B with equal probability. Inspection of this set of configurations is simpler than scrutiny of the set of harvested complete pathways. As a result of the analysis described above one may be led to guess which degrees of freedom are most important during the transition, or, in other words, which degrees of freedom contribute to the reaction coordinate. Such a guessed reaction coordinate, q(x), can be tested with the following procedure. The first step consists in calculating the free energy F(q), for instance by using umbrella sampling [9] or constrained molecular dynamics [11]. The free energy profile F(q) will possess minima at values of q typical for the stables states A and B and a barrier located at q = q ∗ separating these two minima. If q is a good reaction coordinate, trajectories started from configurations with q = q ∗ relax into A and B with equal probability. To verify the quality of the postulated reaction coordinate, a set of configurations with q = q ∗ is generated. Then, for each of these configurations one calculates p B , the probability to relax into state B, also called the committor. This can be done by initiating many short trajectories at the configuration and observing which state they relax to. As a result, one obtains a distribution P( p B ) of committors. For a good reaction coordinate, this distribution should peak at a value of p B ≈ 1/2. If this is not case, other degrees of freedom need to be taken into account for a correct description of the transition [7].

4.

Reaction Rate Constants

Since trajectories collected in the transition path sampling method are genuine dynamical trajectories, they can be used to study the kinetics of reactions. The phenomenologic description of the kinetics in terms of reaction rate constants is related to the underlying microscopic dynamics by time correlation functions of appropriate population functions that describe how the system

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relaxes after a perturbation [12]. In particular, for transitions from A to B the relevant correlation function is h A (x0 )h B (xt ) , (5) C(t) = h A where the angular brackets · · · denote equilibrium averages. The correlation function C(t) is the conditional probability to observe the system in region B at time t provided it was in region A at time 0. To understand the general features of this function, let us imagine that we prepare a large number of identical and independent systems in a way that at time t = 0 all of them are located in A. Then, we let all systems evolve freely and observe the fraction of systems in region B as a function of time. This fraction is the correlation function C(t). Initially, all systems are in A and, therefore, C(0) = 0. As time goes on, some systems cross the barrier due to random fluctuations and contribute to the population in region B. So C(t) grows and it keeps growing until equilibrium sets in, i.e., until the flow of systems from A to B is compensated by the flow of system moving from B back to A. For very long times, correlations are lost and the probability to find a system in B is just given by the equilibrium population h B . For first order kinetics C(t) approaches its asymptotic value exponentially, C(t) = h B [1 − exp(−t/τrxn )], where the reaction time τrxn can be written in terms of the forward and backward reaction rate constants, τrxn = (k AB + k B A )−1 . For times short compared to the reaction time τrxn (but longer than the time necessary to cross the barrier) the correlation function C(t) grows linearly, C(t) ≈ k AB t, and the slope of this curve is the forward rate constant k AB . Thus, to determine reaction rate constants one has to calculate the correlation function C(t). To determine the correlation form C(t) in the transition path sampling method we rewrite it in the suggestive form

C(t) =

Dx(t) h A (x0 )P[x(t)]h B (xt ) . Dx(t) h A (x0 )P[x(t)]

(6)

Here, both numerator and denominator are integrals over path distributions and can be viewed as partition functions belonging to two different path ensembles. The integral in the denominator has the form of a partition function of the ensemble of pathways starting in region A and ending somewhere. The integral in the numerator, on the other hand, is more restrictive and places a condition also on the final point of the path. This integral can be viewed as the partition function of the ensemble of pathways starting in region A and ending in region B. Thus, the path ensemble in the numerator is a subset of the path ensemble in the denominator. The ratio of partition functions can be related to the free energy difference F between the two ensembles of pathways, C(t) ≡ exp(−F).

(7)

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This free energy difference is the generalized reversible work necessary to confine the endpoints of pathways starting in A to region B. Exploiting this viewpoint, one can calculate the time correlation function C(t) and hence determine reaction rate constants by adapting conventional free energy estimation methods to work in trajectory space. So far, reaction rate constants have been calculated in the framework of transition path sampling with umbrella sampling, thermodynamic integration, and fast switching methods. In principle, the forward reaction rate constant k AB can be determined by carrying out a free energy calculation for different times t and taking a numerical derivative. In the time range where C(t) grows linearly, this derivative has a plateau which coincides with k AB . Proceeding in such a way one has to perform several computationally expensive free energy calculations. Fortunately, C(t) can be factorized in a way so that only one such calculation needs to be carried out for a particular time t . The value of C(t) at all other times in the range [0, T ] can then be determined from a single transition path sampling simulation of trajectories with length T . Thus, calculating reaction rate constants in the transition path sampling method is a two-step procedure. First, C(t ) is determined for a particular time t using a free energy estimation method in path space. In a second step, one additional transition path sampling simulation is carried out to determine C(t) at all other times. The reaction rate constant can finally be calculated by determining the time derivative of C(t).

5.

Outlook

Transition path sampling is a practical and very general methodology to collect and analyze rare pathways. In equilibrium, such rare but important trajectories may arise due to free energetic barriers impeding the motion of the system through configuration space. Transition path sampling, however, can be used equally well to study rare trajectories occurring in non-equilibrium processes such as solvent relaxation following excitation or rare pathways arising in new methodologies for the computation of free energy differences. Different types of dynamics ranging from Monte Carlo and Brownian dynamics to Newtonian and non-equilibrium dynamics can be treated on the same footing. To date, the transition path sampling method has been applied to many processes in physics, chemistry and materials science. Examples include chemical reactions in solution, conformational changes of biomolecules, isomerizations of small cluster, the dynamics of hydrogen bonds, ionic dissociation, transport in solids, proton transfer in aqueous systems, the dynamics of non-equilibrium systems, base pair binding in DNA, hydrophobic collapse, and cavitation between solvophobic surfaces. Furthermore, the transition path sampling has been combined with other approaches such as parallel tempering, master equation methods, and the Jarzynski method for the computation of free energy

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differences. Due to the generality of the transition path sampling method it is likely that in the future this approach will be used fruitfully to study new problems in a variety of complex systems.

References [1] D. Wales, Energy Landscapes, Applications to Clusters, Biomolecules and Glasses, Cambridge University Press, Cambridge, 2003. [2] R. Elber, A. Ghosh, and A. C´ardenas, Long time dynamics of complex systems, Acc. Chem. Res., 35, 396, 2002. [3] H. J´onsson, G. Mills, and K.W. Jacobsen, “Nudged elastic band method for finding minimum energy paths of transitions,” In: B.J. Berne, G. Ciccotti, and D.F. Coker, (eds.), Computer Simulation of Rare Events and Dynamics of Classical and Quantum Condensed-Phase Systems – Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore, p. 385, 1998. [4] W.E.W. Ren and E. Vanden-Eijnden, String method for the study of rare events, Phys. Rev. B, 66, 052301, 2002. [5] D. Chandler, “Barrier crossings: classical theory of rare but important events,” In: B.J. Berne, G. Ciccotti, and D.F. Coker (eds.), Computer Simulation of Rare Events and Dynamics of Classical and Quantum Condensed-Phase Systems – Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore, p. 3, 1998. [6] C. Dellago, P.G. Bolhuis, F.S. Csajka, and D. Chandler, “Transition path sampling and the calculation of rate constants,” J. Chem. Phys., 108, 1964, 1998. [7] C. Dellago, P.G. Bolhuis, and P.L. Geissler, “Transition path sampling,” Adv. Chem. Phys., 123, 1, 2002. [8] P.G. Bolhuis, D. Chandler, C. Dellago, and P.L. Geissler, “Transition path sampling: throwing ropes over mountain passes in the dark,” Ann. Rev. Phys. Chem., 53, 291, 2002. [9] D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, 2nd edn. Academic, San Diego, 2002. [10] N. Metropolis, A.W. Metropolis, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations for fast computing machines,” J. Chem. Phys., 21, 1087, 1953. [11] G. Ciccotti, “Molecular dynamics simulations of nonequilibrium phenomena and rare dynamical events,” In: M. Meyer and V. Pontikis (eds.), Proceedings of the NATO ASI on Computer Simulation in Materials Science, Kluwer, Dordrecht, p. 119, 1991. [12] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, 1987.

5.4 SIMULATING REACTIONS THAT OCCUR ONCE IN A BLUE MOON Giovanni Ciccotti1 , Raymond Kapral2 , and Alessandro Sergi2 1

INFM and Dipartimento di Fisica, Universit`a “La Sapienza”, Piazzale Aldo Moro, 2, 00185 Roma, Italy 2 Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON, Canada, M5S 3H6

The computation of the rates of condensed phase chemical reactions poses many challenges for theory. Not only do condensed phase systems possess a large number of degrees of freedom so that computations are lengthy, but typically chemical reactions are activated processes so that transitions between metastable states are rare events that occur on time scales long compared to those of most molecular motions. This time scale separation makes it almost impossible to determine reaction rates by straightforward simulation of the equations of motion. Furthermore, condensed phase reactions often involve collective degrees of freedom where the solvent participates in an important way in the reactive process. Consequently, the choice of a reaction coordinate to describe the reaction is often far from a trivial task. Various methods for determining reaction paths have been devised (see Refs. [1, 2] and references therein). These methods have the goal of determining how the system passes from one metastable state to another and thus finding the reaction path or reaction coordinate. In many situations one has some knowledge of how to construct a reaction coordinate (or set of reaction coordinates) for a particular physical problem. One example is the use of the many-body solvent polarization reaction coordinate to describe electron or proton transfer in solution. In almost all situations investigated to date the dynamics of condensed phase activated reaction rates can be described in terms of a small number of reaction coordinates (often involving collective degrees of freedom). In this chapter, we describe how to simulate the rates of activated chemical reactions that occur on slow time scales, assuming that some set of suitable reaction coordinates is known. In order to compute the rates of rare reactive 1597 S. Yip (ed.), Handbook of Materials Modeling, 1597–1611. c 2005 Springer. Printed in the Netherlands.

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events we need to be able to sample regions of configuration space that are rarely visited since the interconversion between reactants and products entails passage through such regions of low probability. We show that by applying holonomic constraints to the reaction coordinate in a molecular dynamics simulation we can force the system to visit unfavorable configuration space regions. Through such constraints we can generate an ensemble of configurations (the Blue Moon ensemble) that allows one to efficiently estimate the rate constant for activated chemical processes [3].

1.

Reactive Flux Correlation Function Formalism

We begin with a sketch of the reactive flux correlation function formalism in order to specify the quantities that must be computed to obtain the reaction rate constant. In order to simplify the notation, we consider a molecular system containing N atoms with Hamiltonian H = K (p) + V (r), where K (p) is the kinetic energy, V (r) is the potential energy and (p, r) denotes the 6N momenta and coordinates defining the phase space of the system. A chemical reaction A B is assumed to take place in the system. The reaction dynamics is described phenomenologically by the mass action rate law dn A (t) = −kf n A (t) + kr n B (t), dt

(1)

where n A (t) is the mean number density of species A. The task is to compute −1 the forward kf and reverse kr = kf K eq (K eq is the equilibrium constant) rate constants by molecular dynamics simulation. (The formalism is easily generalized to other reaction schemes.) To this end, we assume that the progress of the reaction can be characterized on a microscopic level by a scalar reaction coordinate ξ(r) which is a function of the positions of the particles in the system. A dividing surface at ξ ‡ serves to partition the configuration space of the system into two A and B domains that contain the metastable A and B species. The microscopic variable corresponding to the fraction of systems in the A domain is n A (r) = θ(ξ ‡ − ξ(r)), where θ is the Heaviside function. Similarly, the fraction of systems in the B domain is n B (r) = θ(ξ(r) − ξ ‡ ). The time rate of change of n A (r) is n˙ A (r) = −ξ˙ (r)δ(ξ(r) − ξ ‡ ).

(2)

The rate at which the A and B species interconvert can be determined from the well-known reactive flux formula for the rate constant [4–6]. Using this formalism the time-dependent forward rate coefficient can be expressed in terms of the equilibrium correlation function of the initial flux of A with the

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A species density at time t as 1 1 ˙ A (r)n A (r, t) = eq ξ˙ δ(ξ(r) − ξ ‡ ) θ(ξ(r(t)) − ξ ‡ ) . (3) eq n nA nA

kf (t) =

Here, the angular brackets denote an equilibrium canonical average, · · · = eq Q −1 dr dr exp{−β H } · · · , where Q is the partition function and n A is the equilibrium density of species A. The forward rate constant can be determined from the plateau value of this time-dependent forward rate coefficient [6]. We can separate the static and dynamic contributions to the rate coefficient by multiplying and dividing each term on the right-hand side of Eq. (3) by δ(ξα (r) − ξ ‡ ) to obtain kf (t) =

ξ˙ δ(ξ(r) − ξ ‡ )θ(ξ ‡ − ξ(r(t))) δ(ξ(r) − ξ ‡ )

δ(ξ(r) − ξ ‡ )

eq

nA

.

(4)

The equilibrium average δ(ξ(r) − ξ ‡ ) = P(ξ ‡ ) is the probability density of finding the value of the reaction coordinate ξ(r) = ξ ‡ . We may introduce the free energy W(ξ ) associated with the reaction coordinate by the definition W(ξ ) = − β −1 ln(P(ξ )/Pu ), where Pu is a uniform probability density of ξ . For an activated process the free energy will have the form shown schematically in Fig. 1. A high free energy barrier at ξ = ξ ‡ separates the metastable reactant and product states. The equilibrium density of species A is

n A = θ(ξ ‡ − ξ(r)) = eq

=

dξ θ(ξ ‡ − ξ ) δ(ξ(r) − ξ )

dξ P(ξ ).

(5)

ξ 0 and u(x) = − for x < 0, and is discontinuous at the origin, u(0± ) = ±. A phase-field model for cracks can be formulated by introducing a scalar field φ(x) which describes the state of the material [27]. The model retains the same parametrization of linear elasticity where u(x) measures the displacement of mass points from their original positions. Hence, φ measures the

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⫹1

Solid

Crack

Solid ψ

x

0

or x ⫹ u (x ) u /∆

⫺1 Figure 2. Schematic phase-field profiles vs. the material coordinate x (thick solid line) and vs. the spatial coordinate x + u(x) (dashed line), where u(x) (thin solid line) is the displacement of mass points with respect to their original positions in the unstretched material. The thick vertical solid lines denote the spatial locations of the two crack surfaces.

state of the material at a spatial location x + u(x). The unbroken solid, which behaves purely elastically, corresponds to φ = 1, whereas the fully broken material that cannot support stress corresponds to φ = 0. The total energy per unit area of crack surfaces is taken to be

E=

κ dx 2

dφ dx

2

µ + h f (φ) + g(φ) 2 − c2 , 2

(10)

where = du/dx is the strain, f (φ) = φ 2 (1 − φ)2 is the same double-well potential as before with minima at φ =1 and φ =0, µ is the elastic modulus, and c is the critical strain magnitude such that the unbroken (broken) state is energetically favored for | | < c (| | > c ). The function g(φ) is a monotonously increasing function of φ with limits g(0) = 0 and g(1) = 1, which controls the softening of the elastic energy at large strain. In equilibrium, the energy must be a minimum, which implies that δ E/δφ = 0 and δ E/δu = 0. The second condition is equivalent to uniform stress in the material. It implies that d(g(φ) )/dx = 0, and hence that = 0 /g(φ) where 0 is the value of the remanent strain in the bulk of the material far from the crack. The first condition δ E/δφ = 0, after the substitution = 0 /g(φ), can

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be written in the form of a one-dimensional mechanical problem of a rolling ball with coordinate φ and mass κ dVeff (φ) d2 φ , =− 2 dx dφ in an effective potential κ

(11)

µ 2 2 c g(φ) + 0 Veff (φ) = −h f (φ) + 2 g(φ)

(12)

This potential (Fig. 3) has a repulsive part because g(φ) vanishes for small φ. In this mechanical analog, the stationary phase-field profile φ(x) shown in Fig. 2 corresponds to the ball rolling down this potential, starting from φ = 1 at x = − W , to the turning point located close to φ = 0, and then back to φ = 1 at x = +W . This mechanical problem must be solved under the constraint that the

+Wintegral of the strain equals the total displacement of the fracture surfaces, −W dx 0 /g(φ)=2. An analysis of the solutions in the large system size limit [27] shows that the remanent strain is determined by the behavior of the function g(φ) for small φ. If this function has the form of a power law g ∼ φ 2+α

0.25 ε0 ⫽ 0.01 ε0 ⫽ 0.001 ε0 ⫽ 0.0001

Veff

0.15

0.05

⫺0.05

0

0.2

0.4

0.6

0.8

1

ψ Figure 3. Plots of the effective potential for different values of the remanent strain in the bulk material 0 for one-dimensional static fracture (µ = h = 1 and c = 1/2).

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near φ = 0, the result is 0 ∼ −(2+α)/α . Hence, as long as α is positive, 0 will vanish in the large system size limit such that the local contribution of the crack to the overall displacement is dominant compared to the bulk contribution, which scales √ as 0 W . In this limit, the width of φ-profile remains finite and scales ∼ κ/µ. The u-profile is also continuous in the diffuse interface region, but its width vanishes has an inverse power of the system size, such that the strain = du/dx becomes a Dirac delta function in the large system size limit. In addition, this analysis yields the expression for the surface energy [27] γ=

µ c2 κ

1

dφ 0

1 − g(φ) + 2

h f (φ) µ c2

(13)

In contrast to the interface energy for a phase boundary (Eq. (6)), γ for a crack remains finite when the height h of the double well potential vanishes. Therefore, the inclusion of this potential is not a prerequisite to model cracks within this model.

3.

Interface Dynamics

The preceding sections focused on flat static interfaces and their energies. This section examines the application of the phase-field method to simulate the motion of curved interfaces outside of equilibrium, when spatially inhomogeneous distributions of temperature, alloy concentration, or stress are present in the material. The effect of these inhomogeneities are straightforward to incorporate into the model by adding bulk internal energy and entropic contributions to the free-energy functional. Furthermore, the Ginzburg–Landau form [15, 18] of the equations is prescribed by conservation laws and by requiring that the total free-energy relaxes to a minimum. Three illustrative examples are considered: the solidification of a pure substance, the solidification of a binary alloy, and crack propagation. For the solidification of a pure melt [32], the total free-energy that includes the contribution due to the variation of the temperature field is a generalization of Eq. (1)

F[φ, T ] =

κ 2 α dV |∇φ| + h f (φ) + (T − Tm )2 , 2 2

(14)

which is minimum at the melting point T = Tm . Dynamical equations which guarantee that F decreases monotonically in time (dF/dt ≤ 0), and which

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conserve the total energy dV e in a closed system with no energy flux through the boundaries, are [32] δF ∂φ = −K φ , ∂t δφ δF ∂e = Ke ∇ 2 , ∂t δe

(15) (16)

where the energy density e = C(T − Tm ) − p(φ)L and φ are chosen as the independent field variables in Eq. (14), C is the specific heat per unit volume, L is the latent heat of melting per unit volume, and p(φ) is a function that increases monotonously with φ with limits p(0) = 0 and p(1) = 1. The energy equation (Eq. (16)) yields L ∂ p(φ) ∂T = D∇ 2 T + ∂t C ∂t

(17)

where we have defined the thermal diffusivity D = α K e /C 2 . This is the standard heat diffusion equation with an extra source term ∼ ∂ p/∂t corresponding to latent heat production. The equation of motion for the phase-field (Eq. (15)), in turn, gives K φ−1

∂φ = κ∇ 2 φ − h f (φ) − α(L/C) p (φ)(T − Tm ), ∂t

(18)

where the prime denotes differentiation with respect to φ. In the region near the interface, where T is locally constant, Eq. (18) implies that the phase change is driven isothermally by the modified double-well potential h f (φ) + α(L/c) p(φ)(T − Tm ). This potential has a “bias” introduced by the undercooling of the interface, which lowers the free-energy of the solid well relative to that of the liquid well. A one-dimensional analysis of this equation [9, 32] shows that the velocity of the interface is simply proportional to the undercooling, V = µsl (Tm − T ), where the interface kinetic coefficient µsl ∼ α K φ (κ/ h)1/2(L/c). Further refinement of this phase-field model [24] and algorithmic developments have made it possible to simulate dendritic evolution quantitatively both in a low undercooling regime where the scale of the diffusion field is much larger than the scale of the dendrite tip [45, 47, 48], and in the opposite limit of rapid solidification [6]. Parameter free results obtained for the latter case using anisotropic forms of γsl and µsl computed from atomistic simulations [20, 21] are compared with experimental data in Fig. 4. Next, let us consider the isothermal solidification of a binary alloy [5, 26, 30, 59, 61]. The total free-energy of the system can be written in the form

F[φ, c, T ] =

dV

κ 2 |∇φ| + f pure (φ, T ) + f solute(φ, c, T ) , 2

(19)

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80

V (m/s)

60

40

20

0 0

100

200 ∆T (K)

300

400

Figure 4. Example of application of the phase-field method to simulate the dendritic crystallization of deeply undercooled nickel [6]. A snapshot of the solid–liquid interface is shown for an undercooling of 87 K. The dendrite growth rate versus undercooling obtained from the simulations (filled triangles and solid line) is compared to two sets of experimental data from Ref. [37] (open squares) and Ref. [64] (open circles).

where c denotes the solute concentration defined as the mole fraction of B in a binary mixture of A and B molecules, f pure = h f (φ) + α(L/c) p(φ)(T − Tm ) is the double-well potential of the pure material, and f solute(φ, c, T ) is the contribution due to solute addition. Dynamical equations that relax the system to a free-energy minimum are δF ∂φ = −K φ , ∂t δφ ∂c δF , = ∇ · Kc∇ ∂t δc

(20) (21)

where Eq. (21) is equivalent to the mass continuity relation with µc ≡ δ F/δc c as the solute current density. identified as the chemical potential and −K c ∇µ The smooth spatial variation of φ in the diffuse interface can be exploited to interpolate between known forms of the free-energy densities in solid and liquid ( f s and f l , respectively), by writing f solute(φ, c, T ) = g(φ) f s (c, T ) + (1 − g(φ)) f l (c, T ),

(22)

where g(φ) has limits g(0) = 0 and g(1) = 1. For example, for a dilute alloy, f s,l = s,l c + (RTm /v 0 )(c ln c − c) where s,l c is the change of internal energy density due to solute addition in solid or liquid, and the second term is the standard entropy of mixing, where R is the gas constant and v 0 is the molar volume. This interpolation describes the thermodynamic properties of the diffuse interface region as an admixture of the thermodynamic properties of the bulk

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solid and liquid phases. The static phase-field and solute profiles through the interface are then obtained from the equilibrium conditions ∂φ/∂t = ∂c/∂t = 0. The limits of c in bulk solid (cs ) and liquid (cl ) are the same as the equilibrium values obtained by the standard common tangent construction of the alloy phase diagram. The method has been extended to non-isothermal conditions, multicomponent alloys, and polyphase transformations, as illustrated in Fig. 5 for the solidification of a ternary eutectic alloy. The first models of polyphase solidification used either the concentration field [23] or a second non-conserved order parameter [35, 62] to distinguish between the two solid phases in addition to the usual phase field that distinguishes between solid and liquid. The more recent multi-phase-field approach interprets the phase fields as local phase fractions and therefore assigns one field to each phase present [14, 42, 53, 54]. This approach provides a more general formulation of multi-phase solidification. The simplest nontrivial example of dynamic brittle fracture is antiplane shear (mode III) crack propagation where the displacement field u(x, y) perpendicular to the x–y plane is purely scalar. The total energy (defined here

Figure 5.

Phase-field simulation of two-phase cell formation in a ternary eutectic alloy [46].

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per unit length of the crack front) must now include both kinetic and elastic contributions to this energy, yielding the form

E=

dx dy

µ ρ 2 κ 2 u˙ + |∇φ| + h f (φ) + g(φ) | |2 − c2 , 2 2 2

(23)

is where dot denotes derivative with respect to time, ρ is the density, ≡ ∇u the strain and all the other functions and parameters are as previously defined. The dynamical equations of motion are then obtained variationally from this energy in the forms δE ∂φ = −χ ∂t δφ 2 δE ∂ u ρ 2 =− ∂t δu

(24) (25)

These equations describe both the microscopic physics of material failure and macroscopic linear elasticity. Figure 6 shows examples of cracks obtained in phase-field simulations of this model in a strip of width 2W with a fixed displacement u(x, ±W ) = ± at the strip edges. The stored elastic energy per unit area ahead of the crack tip is G = µ2 /W . The Griffith’s threshold for the onset fracture is well reproduced in this model [27]. This approach has been recently used to study instabilities of mode III [28] and mode I cracks [19]. (a)

(b)

(c)

Figure 6. Example of dynamic crack patterns for mode III brittle fracture [28] with increasing load from (a) to (c). Plots correspond to φ = 1/2 contours at different times.

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Discussion

The preceding examples illustrate the power of the phase-field method to simulate a host of complex interfacial pattern formation phenomena in materials. Making quantitative predictions on experimentally relevant length and time scales, however, remains a major challenge. This challenge stems from the fact that, in most applications, the interface thickness and the time scale of the phase field kinetics need to be chosen orders of magnitude larger than in a real material for the simulations to be feasible. Because of this constraint, phase-field results often depend on interface thickness and are only qualitative. Over the last decade, progress has been made in achieving quantitative simulations despite this constraint [12, 24, 26, 51, 66]. One important property of the phase-field model is that the interfacial energy (Eq. (6)) scales as W h. Hence, the correct magnitude of capillary effects can be modeled even with a thick interface by lowering the height h of the double-well potential. For alloys, the coupling of the phase field and solute profiles through the diffuse interface makes the interface energy dependent on interface thickness. This dependence, however, can be eliminated by a suitable choice of freeenergy density [12, 26]. More difficult to eliminate are nonequilibrium effects that become artificially magnified because of diffusive transport across a thick interface. These effects can compete with, or even supersede, capillary effects, and dramatically alter microstructural evolution. To illustrate these nonequilibrium effects, consider the solidification of a binary alloy. The effect best characterized experimentally and theoretically is solute trapping [1, 4], which is associated with a chemical potential jump across the interface. The magnitude of this effect scales with the interface thickness. Since W is orders of magnitude larger in simulations than in reality, solute trapping will prevail at growth speeds where it is completely negligible in a material. Additional effects modify the mass conservation condition at the interface cl (1 − k)Vn = −D

∂c + ··· ∂n

(26)

where cl is the concentration on the liquid side of the interface, k is the partition coefficient, Vn is the normal interface velocity, and “· · · ” is the sum of a correction ∼ cl (1 − k)W Vn κ, where κ is the local interface curvature, a correction ∼ W D∂ 2 cl /∂s 2 , corresponding to surface diffusion, and a correction ∼ kcl (1 − k)W Vn2 /D proportional to the chemical potential jump at the interface. All three corrections can be shown to originate physically from the surface excess of various quantities [12], such as the excess of solute illustrated in Fig. 7. These corrections are negligible in a real material. For this reason, they have not been traditionally considered in the standard free-boundary problem of alloy solidification. For a mesoscopic interface thickness, however, the

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c1

Solid

Liquid

cs 0 r Figure 7. Illustration of surface excess associated with a diffuse interface. The excess of solute is the integral, along the coordinate r normal to the interface, of the actual solute profile (thick solid line) minus its step profile idealization (thick dashed line) with the Gibbs dividing surface at r = 0. This excess is negative in the depicted example. The thin solid line depicts the phasefield profile. The use of a thick interface in simulations artificially magnifies the surface excess of several quantities and alters the results [12].

magnitude of these corrections becomes large. Thus, the phase-field model must be formulated to make these corrections vanish. Achieving this goal requires a detailed asymptotic analysis of the thin-interface limit of diffuse interface models [2, 12, 24, 26, 39]. This analysis provides the formal guide to formulate models free of these corrections. So far, however, progess has only been possible for dilute [12, 26] and eutectic alloys [14]. Thus, it is not yet clear whether or not it will always be possible to make phase-field models quantitative in more complicated applications.

5.

Outlook

The phase-field method has emerged as a powerful computational tool to model a wide range of interfacial pattern formation phenomena. The success of the approach can be judged by the rapidly growing list of fields in which it has been used from materials science to biology. It can also be judged by the wide range of scales that have been modeled from crystalline defects to nanostructures to microstructures. Like with any simulation method, however, obtaining quantitative results remains a major challenge. The core of this challenge is the disparity of length and time scales between phenomena on the

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scale of the diffuse interface and on the scale of energy or mass transport in the bulk material. For well-established problems like solidification, and a few others, quantitative simulations have been achieved in a few cases following two decades of research since the introduction of the first models. In more recent applications like fracture, with no clear separation between microscopic and macroscopic scales, results remain so far qualitative. In the future, one can expect phase-field simulations to be useful both to gain new qualitative insights into pattern formation mechanisms and to make quantitative predictions in mature applications.

Acknowledgments The author thanks the US Department of Energy and NASA for financial support.

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A. Karma A. Karma and W.-J. Rappel, Phys. Rev. E, 57, 4323, 1998. A. Karma and M. Plapp, Phys. Rev. Lett., 81, 4444, 1998. A. Karma, Phys. Rev. Lett., 87, 115701, 2001. A. Karma, D. Kessler, and H. Levine, Phys. Rev. Lett., 87, 045501, 2001. A. Karma and A. Lobkovsky, Phys. Rev. Lett., 92, 245510, 2004. K. Kassner and C. Misbah, Europhys. Lett., 46, 217, 1999. S.-G. Kim, W.T. Kim, and T. Suzuki, Phys. Rev. E, 60, 7186, 1999. R. Kobayashi, J.A. Warren, and W.C. Carter, Physica D, 140, 141, 2000. J. S. Langer, In: G. Grinstein and G. Mazenko (eds.), Directions in Condensed Matter, World Scientific, Singapore, p. 164, 1986. F. Liu and H. Metiu, Phys. Rev. E, 49, 2601, 1994. Z.R. Liu, H.J. Gao, L.Q. Chen, and K.J. Cho, Phys. Rev. B, 035429, 2003. T.-S. Lo, A. Karma, and M. Plapp, Phys. Rev. E, 63, 031504, 2001. A.E. Lobkovsky and J.A. Warren, Phys. Rev. E, 63, 051605, 2001. J.W. Lum, D.M. Matson, and M.C. Flemings, Metall. Mater. Trans. B, 27, 865, 1996. G.B. McFadden, A.A. Wheeler, R.J. Braun, S.R. Coriell, and R.F. Sekerka, Phys. Rev. E, 48, 2016, 1993. G.B. McFadden, A.A. Wheeler, and D.M. Anderson, Physica D, 154, 144, 2000. L.V. Mikheev and A.A. Chernov, J. Cryst. Growth, 112, 591, 1991. J. Muller and M. Grant, Phys. Rev. Lett., 82, 1736, 1999. B. Nestler and A.A. Wheeler, of growth structures: 114–133, Physica D, 138, 114, 2000. D.W. Oxtoby and P.R. Harrowell, J. Chem. Phys., 96, 3834, 1992. O. Pierre-Louis, Phys. Rev. E, 68, 021604, 2003. M. Plapp and A. Karma, Phys. Rev. Lett., 84, 1740, 2000; M. Plapp and A. Karma, J. Comp. Phys., 165, 592, 2000. M. Plapp and A. Karma, Phys. Rev. E, 66, 061608, 2002. N. Provatas, N. Goldenfeld, and J.A. Dantzig, Phys. Rev. Lett, 80, 3308, 1998. N. Provatas, N. Goldenfeld, and J.A. Dantzig, J. Comp. Phys., 148, 265, 1999. D. Rodney, Y. Le Bouar, and A. Finel, Acta Mater., 51, 17, 2003. Y. Shen and D.W. Oxtoby, J. Chem. Phys., 104, 4233, 1996. C. Shen, Q. Chen Q, and Y.H. Wen et al., Scripta Mater., 50, 1029, 2004. C. Shen C and Y. Wang, Acta Mater., 52, 683, 2004. I. Steinbach, F. Pezzolla, B. Nestler, M. BeeBelber, R. Prieler, G.J. Schmitz, and J.L.L. Rezende, Physica D, 94, 135, 1996. J. Tiaden, B. Nestler, H.J. Diepers, and I. Steinbach, Physica D, 115, 73, 1998. Y.U. Wang, Y.M. Jin, A.M. Cuitino, and A.G. Khachaturyan, Acta Mater., 49, 1847, 2001. Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, J. Appl. Phys., 91, 6435, 2002. J. Wang, S.Q. Shi, L.Q. Chen et al., Acta Mater., 52, 749, 2004. Y.U. Wang, Y.M.M. Jin, and A.G. Khachaturyan, Acta Mater., 52, 81, 2004. J.A. Warren and W.J. Boettinger, Acta Metall. Mater. A, 43, 689–703, 1995. J.A. Warren, R. Kobayashi, A.E. Lobkovsky, and W.C. Carter, Acta Mater., 51, 6035, 2003. A.A. Wheeler, W.J. Boettinger, and G.B. McFadden, Phys. Rev. A, 45, 7424, 1992. A.A. Wheeler, G.B. McFadden, and W.J. Boettinger, Proc. Royal Soc. Lond. A, 452, 495–525, 1996.

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[63] A.A. Wheeler and G.B. McFadden, Euro J. Appl. Math., 7, 367, 1996. [64] R. Willnecker, D.M. Herlach, and B. Feuerbacher, Phys. Rev. Lett., 62, 2707, 1989. [65] R.K.P. Zia and D.J. Wallace, Phys. Rev. B, 31, 1624, 1985. [66] J.Z. Zhu, T. Wang, S.H. Zhou, Z.K. Liu, and L.Q. Chen, Acta Mater., 52, 833, 2004.

7.3 PHASE-FIELD MODELING OF SOLIDIFICATION Seong Gyoon Kim1 and Won Tae Kim2 1 Kunsan National University, Kunsan 573-701, Korea 2

Chongju University, Chongju 360-764, Korea

1.

Pattern Formation in Solidification and Classical Model

Pattern formation in solidification is one of the most well known freeboundary problems [1, 2]. During solidification, solute partitioning and release of the latent heat take place at the moving solid/liquid interface, resulting in a build-up of solute atoms and heat ahead of the interface. The diffusion field ahead of the interface tends to destabilize the plane-front interface. Conversely, the role of the solid/liquid interface energy, which tends to decrease by reducing the total interface area, is to stabilize the plane solid/liquid interface. Therefore the solidification pattern is determined by a balance between the destabilizing diffusion field effect and the stabilizing capillary effect. Anisotropy of interfacial energy or interface kinetics in a crystalline phase contributes to form an ordered pattern with a unique characteristic length scale rather than a fractal pattern. The key ingredients in pattern formation during solidification thus are contributions of diffusion field, interfacial energy and crystalline anisotropy [2]. The classical moving boundary problem for solidification of alloys assumes that the interface is mathematically sharp. The governing equations for isothermal alloy solidification [1] are given by ∂c L = DL ∇ 2cL ∂t ∂cS ∂c L V (ciL − ciS ) = D S − DL ∂n ∂n Hm 1 S i L i e βV + σ κ f c (cS ) = f c (c L ) = f c − e c L − ceS Tm ∂cS = DS ∇ 2 cS ; ∂t

2105 S. Yip (ed.), Handbook of Materials Modeling, 2105–2116. c 2005 Springer. Printed in the Netherlands.

(1) (2) (3)

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where c is composition and D is diffusivity. The subscripts S and L under c and D denote the values of solid and liquid phase, respectively. The superscripts i and e on composition denote the interfacial and equilibrium compositions, respectively. f cS and f cL are the chemical potentials of bulk solid and liquid, respectively, and f ce is the equilibrium chemical potential. Here the chemical potential denotes the difference between the chemical potentials of solute and solvent. β is the interface kinetics coefficient, V the interface velocity, Hm the latent heat of melting, Tm the melting point of the pure solvent, σ the interface energy, κ the interface curvature, ∂/∂t and ∂/∂n are the time and the interface normal derivatives, respectively. The solidification of pure substances involving the latent heat release at interface, instead of solute partitioning, can be described by the same set of Eqs. (1)–(3), which can be expressed by replacing variables: c → H/Hm , f c → T Hm /Tm , D → DT , where T , H and DT are temperature, enthalpy density and thermal diffusivity, respectively, with the same meanings for the superscripts and subscripts L, S and i.

2.

Phase-field Model

Many numerical approaches have been proposed to solve the Eqs. (1)– (3). These include direct front tracking methods and boundary integral formulations, where the interface is treated to be mathematically sharp. However these sharp interface approaches lead to significant difficulties due to the requirement of tracking interface position every time step, especially in handling topological changes in interface pattern or extending to 3D computation. An alternative technique for modeling the pattern formation in solidification is the phase-field model (PFM) [3, 4]. This approach adopts a diffuse interface model, where a solid phase changes gradually into a liquid phase across an interfacial region of a finite width. The phase state is defined by an order parameter φ as a function of position and time. The phase field φ takes on a constant value in each bulk phase, e.g., φ = 0 in liquid phase and φ = 1 in solid phase, and it changes smoothly from φ = 0 to φ = 1 across the interfacial region. Any point within the interfacial region is assumed to be a mixture of the solid and liquid phases, whose fractions are varying gradually from 0 to 1 across the transient region. All the thermodynamic and kinetic variables then are assumed to follow a mixture rule. A set of equations for PFM can be derived in a thermodynamically consistent way. Let us consider an isothermal solidification of an alloy. It is

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assumed that the total free energy of the system of volume is given by a functional

F=

[ 2 |∇φ|2 + ωg(φ) + f (φ, c, T )]d

(4)

During solidification, which is a non-equilibrium process, the system evolves toward a more stable state by reducing the total free energy. To decrease the total free energy, the first term (phase-field gradient energy) in the functional (4) makes the phase-field profile to spread out, i.e., to widen the transient region. The second term (double-well potential ωg(φ)) makes the bulk phases stable, i.e., to sharpen the transient region. The diffuse interface maintains a stable width by a balance between these two opposite effects. Once the stable diffuse interface is formed, the two terms start to cooperate to decrease the total volume (in 3D) or area (in 2D) of the diffuse interfacial region where|∇φ| and g(φ) are not vanishing. This is corresponding to the curvature effect in the classical sharp interface model. Thus the gradient energy and the doublewell potential play two-fold roles; formation of stable diffuse interface and incorporation of the curvature effect. √ As the result, the interface width scales as the ratio of the coefficients (/ √ ω), whereas the interface energy scales as the multiplication of them ( ω). The last term in the functional (4) is a thermodynamic potential assumed to follow a mixture rule f (φ, c, T ) = h(φ) f S (cS , T ) + [1 − h(φ)] f L (c L , T )

(5)

where c is the average composition of the mixture, cS and c L are the compositions of coexisting solid and liquid phases in the mixture, respectively, and f S and f L are the free energy densities of the solid and liquid phases, respectively. It is natural to take c(x) at a given point x to be c(x) = h(φ)cS (x) + [1 − h(φ)]c L (x)

(6)

The monotonic function h(φ) satisfying h(0) = 0 and h(1) = 1 has the meaning of solid fraction. One more restriction is required for h(φ) to ensure that the solid and liquid phases are stable or metastable(i.e., exhibit energy minima), the function ωg(φ) + f in the functional (4) must have local minima at φ = 0 and φ = 1. It then leads to the condition h (0) = h (1) = 0 which confines the phase change occurring within the interfacial region. Finally the anisotropy effect in interface energy can be incorporated into the functional (4) by allowing to depend on the local orientation of the phase-field gradient [5]. Note that all thermodynamic components controlling pattern formation during solidification are incorporated into a single functional (4). The kinetic components controlling pattern formation are incorporated into the dynamic equations of the phase and diffusion fields. In a solidifying

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system where its total free energy decreases monotonically with time, the total amount of solute is conserved, whereas the total volume of each phase is not conserved. Therefore the phase field and concentration are assumed to follow relaxation dynamics of δF ∂φ = −Mφ ∂t δφ

(7)

δF ∂c = −∇ Mc · ∇ ∂t δc

(8)

where Mφ and Mc are mobilities of the phase and concentration fields, respectively. From the variational derivatives of the functional (4), it follows 1 ∂φ = 2 ∇ 2 φ − ωg (φ) − f φ Mφ ∂t ∂c = ∇ Mc · ∇ f c ∂t

(9) (10)

where the subscripts in f denote the partial derivatives by the specific variables. Mc is related to the chemical diffusivity D(φ) by Mc = D/ f cc , where f cc ≡ ∂ 2 f /∂c2 , D(1) = D S and D(0) = D L . The PFM for isothermal solidification of alloys thus consists of Eqs. (9) and (10). To solve these equations, we need f φ , f c and f cc . For the given thermodynamic data f S (cS ) and f L (c L ) at a given temperature, the above functions are obtained by differentiating Eq. (5). For this differentiation, relationships between c(x), cS (x) and c L (x) are required. Two alternative ways have been proposed for these relationships [6]: equal composition condition; and equal chemical potential condition. In the former case, which has been widely adopted [3], it is assumed that cS (x) = c L (x) and so f c S (cS (x)) =/ f c L (c L (x)), resulting in c = cS = c L from Eq. (6). Under this condition, it is straightforward to find fφ , f c and fcc from Eq. (5). In the latter case, it is assumed that fc S (cS (x)) = f c L (c L (x)) and so cS (x) =/ c L (x), resulting in f c = f c S = f c L from Eqs. (5) and (6). Under this condition, f φ in the phase-field Eq. (9) is given by f φ = h (φ)[ f S (cS , T ) − f L (c L , T ) − (cS − c L ) f c ]

(11)

and the diffusion Eq. (10) can be modified into the form ∂c = ∇ · D(φ)[h(φ)∇cS + (1 − h(φ))∇c L ] (12) ∂t Note that this diffusion equation is derived in a thermodynamically consistent way, even though the same equation has been introduced in an ad hoc manner previously [3]. In case of nonisothermal solidification of alloys, the evolution equations for thermal, solutal and phase fields can also be derived in a thermodynamically

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consistent way, where positive entropy production is guaranteed [7]. The resulting evolution equations are dependent on the detailed form of the adopted entropy functional. With a simple form of the entropy functional, the thermal diffusion equation is given by ∂H = ∇k(φ) · ∇T (13) ∂t where H is the enthalpy density, k(φ) is the thermal conductivity, and the phase field and chemical diffusion equations remain identical with Eqs. (9) and (10). In the simplest case where the thermal conductivities and the specific heats of the liquid and solid are same and independent of temperature, the thermal diffusion equation can be written into ∂φ ∂T Hm h (φ) (14) − = ∇ DT · ∇T ∂t cp ∂t where c p is the specific heat.

3.

Sharp Interface Limit

Equations (9) and (10) in the PFM of alloy solidification can be mapped onto the classical free boundary problem, described in Eqs. (1)–(3). The relationships between the parameters in the phase-field equations and material’s parameters are obtained from the mapping procedure. It can be done at two different limit conditions: a sharp interface limit where the interface width 2ξ p is vanishingly small; and a thin interface limit where the interface width is finite, but much smaller than the characteristic length scales of diffusion field and the interface curvature. At first we deal with the sharp interface analysis. To find the interface width, consider an alloy system at equilibrium, with a 1D diffuse interface between solid (φ = 1 at x < − ξ p ) and liquid (φ = 0 at x > ξ p ) phases. Then the phase-field equation can be integrated and the equilibrium phase-field profile φ0 (x) [8] is the solution of

2 dφ0 2 = ωg(φ0 ) + Z (φ0 ) − Z (0) (15) 2 dx where Z (φ0 ) = f −c fc . The function Z (φ0 )−Z (0) in the right side of this equation has a double-well form under the equal composition condition, whereas it disappears under the equal chemical potential condition for alloys or the equal temperature condition for pure substances [6]. Integrating Eq. (15) again gives the interface width 2ξ p , corresponding to a length over which the phase field changes from φa to φb ; 2ξ p = √ 2

φb φa

√

dφ0 ωg(φ0 ) + Z (φ0 ) − Z (0)

(16)

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The interface energy is obtained by considering an equilibrium system with a cylindrical solid in liquid matrix, maintaining a diffuse interface between them. Integrating the phase-field equation in the cylindrical coordinate gives the chemical potential shift from the equilibrium value, which recovers the curvature effect in Eq. (3), if the interface energy σ is given by σ =

∞ 2 −∞

dφ0 dr

2

dr =

√

2

1

ωg(φ0 ) + Z (φ0 ) − Z (0) dφ0

(17)

0

The same expression for σ can be directly obtained from the excess free energy arising from the nonuniform phase-field distribution in the functional (4). In sharp interface limit, the interface width is vanishingly small, while the interface energy should remain finite. From Eqs. (16) and (17), it appears that the limit can be attained when → 0 and ω → ∞. This leads to ωg(φ0 ) Z (φ0 ) − Z (0) and then the interface width and the energy in the sharp interface limit are given by √ (18) 2ξ p = √ 2 2 ln 3 ω √ ω σ= √ (19) 3 2 when we used φa = 0.1, φb = 0.9 and g(φ) = φ 2 (1 − φ)2 . In sharp interface limit, Eq. (10) for chemical diffusion recovers not only the usual diffusion equations in bulk phases, but also the mass conservation condition at the interface. Similarly, the thermal diffusion Eq. (13) also reproduces the usual thermal diffusion equation in bulk phases and the energy balance condition at the interface. The remaining procedure is to find a relationship between the mobility Mφ and the kinetic coefficient β. Consider a moving plane-front interface with a steady velocity V . The 1D phase-field equation in a moving coordinate system can be integrated over the interfacial region, in which the chemical potential at the interface is regarded as a constant because its variation within the interfacial region can be ignored in the sharp interface limit. The integration yields a linear relationship between the interface velocity and the thermodynamic driving force, which recovers the kinetic effect in Eq. (3) if we put √ ωTm 1 (20) β= √ 3 2 Hm Mφ For given 2ξ p , σ and β, all the parameters , ω and Mφ in the phase-field Eq. (9) thus are determined from the three relationships (18)–(20). For the model of pure substances consisting of Eqs. (9) and (13), exactly same relationships between phase-field parameters and material’s parameters are maintained. When the phase-field parameters are determined with these equations,

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special care should be taken to avoid the interface width effect on the computational results. It is often computationally too stringent to choose 2ξ p small enough to resolve the desired sharp interface limit.

4.

Thin Interface Limit

Remarkable progress has recently been made to overcome the stringent restriction on interface width by using a thin-interface analysis of the PFM [5, 9]. This analysis maps the PFM with a finite interface width, 2ξ p , onto the classi˜ V and ξ p R, where D˜ cal moving boundary problem at the limit of ξ p D/ and R are the average diffusivity in the interfacial region and the local radius of interface curvature, respectively. Furthermore, this makes it possible to eliminate the interface kinetic effect by a specific choice of the phase-field mobility. The mapping of the thin interface PFM onto the classical moving boundary problem is based on the following two ideas. First, due to the finite interface width, there can exist anomalous effects in (1) interface energy, (2) diffusion in the interfacial region, (3) release of the latent heat and (4) solute partitioning. Crossing the interface with a finite width, 2ξ p , the anomalous effects vary sigmoidally and change their signs around the middle position of the interface. By specific choices of the functions in the PFM such as h(φ) and D(φ), these anomalous effects can be eliminated by summing them over the interface width. Second, the thermodynamic variables such as temperature T and chemical potential f c at the interface are not uniquely defined, but rather varying smoothly ˜ V is satisfied, within the finite interface width. When the condition ξ p D/ ˜ V are linhowever, their profiles at the solid and liquid sides of ξ p |x| D/ ear. Extrapolating two straight profiles into the interfacial region, we get a value of the thermodynamic variable at the intersection point. The value corresponds to that in sharp interface limit. In this way, we can find the unique thermodynamic driving force for the thin interface. First we deal with a symmetric model [5] for pure substances, where the specific heat, c p , and the thermal diffusivity, DT , are constant throughout the whole system. In this case, all the anomalous effects arising from the finite interface width are vanishing when φ0 (x) − 1/2 and h(φ0 (x)) are odd functions of x. Because the extra potential disappears in Eq. (15) for pure substances, usual choices of g(φ) and h(φ) satisfy these conditions, for example, g(φ) = φ 2 (1 − φ)2 and h(φ) = φ 3 (6φ 2 − 15φ + 10 ); furthermore the relationships (18) and (19) remain unchanged. The next step of the thin interface analysis is to find the linear relationship between the interface velocity and the thermodynamic driving force, which leads to √ ωTm Hm 1 √ J − (21) β= √ DT c p 2ω 3 2 Hm Mφ

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and J is a constant given by 1

J= 0

h p (φ)[1 − h d (φ)] √ dφ g(φ)

(22)

where the subscripts p and d under h(φ) are added to discriminate solid fractions from the phase-field and diffusion equations, respectively. The discrimination was made because a model with h p (φ) =/ h d (φ) can also be mapped onto the classical moving boundary problem, although both functions become identical when the model is derived from the functional (4). The second term in the right side of Eq. (21) is the correction from the finite √ interface width effect, which disappears in sharp interface limit 2ξ p ∼ / ω → 0. For given 2ξ p , σ and β, all the parameters , ω and Mφ in the phase-field Eq. (9) thus are determined from the three relationships (18)–(20) in thin interface limit. Note that Mφ can be determined at the vanishing interface kinetic condition by putting β = 0 in Eq. (21).

5.

One-sided Model

When the specific heat c p and thermal diffusivity DT in solid and liquid phases are different from each other, the thin interface analysis is more deliberate because one must take care of the anomalous effects associated with asymmetric functions of c p (φ) and DT (φ). There exists similar difficulties in the analysis for the PFM of alloys. The analysis requires additional care of the solute trapping arising from a finite relaxation time for solute partitioning in the interfacial region. The thin interface analysis, however, is still tractable for a one-sided system where the diffusion in solid phase is ignored, which is described below. When the interface width is finite, the interface width and energy are given by Eqs. (16) and (17), respectively. They are influenced by the extra potential Z (φ)− Z (0). The potential imposes a restriction on the interface width [8, 10], for a given interface energy. The restriction is often so tight that it prevents us from taking the merit of the thin interface analysis – enhancing the computational efficiency by increasing the interface width. For high computational efficiency, therefore, it is desirable to take the equal chemical potential condition instead of the equal composition condition, under which the extra potential Z (φ0 ) − Z (0) disappears [6, 10]. In a dilute solution, the equal chemical potential condition is reduced to a simple relationship cS (x)/c L (x) = ceS /ceL ≡ k,

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and the diffusion equation and the phase-field equation are as follows [9, 10]; c = [1 − (1 − k)h d (φ)]c L ≡ A(φ)c L

(23)

∂c = ∇ · D(φ)A( φ)∇c L ∂t

(24)

RT (1 − k) e 1 ∂φ = 2 ∇ 2 φ − ωg (φ) − (c L − c L )h p (φ) Mφ ∂t vm

(25)

where the last term in Eq. (25) is the dilute solution approximation of Eq. (11) and v m is the molar volume. The coefficient RT (1−k)/v m may be replaced by Hm /(m e Tm ), following the van’t Hoff relation, where m e is the equilibrium liquidus slope in the phase diagram. The mapping of Eqs. (24) and (25) in thin interface limit onto the classical moving boundary problem can be performed under the assumption of D S D L [9]. The following are the results obtained to remove anomalous interfacial effects in thin interface limit: Anomalous interface energy is vanishing if dφ0 (x)/dx is an even function of x, where the origin x = 0 is taken as the position with φ = 1/2. This is fulfilled by taking a symmetric potential, such as g(φ) = φ 2 (1−φ)2 . Anomalous solute partitioning is vanishing if h d (φ0 ) dφ0 /dx is an even function of x. This requirement is fulfilled when h d (φ0 (x)) is an even function of x because dφ0 (x)/dx also is an even function following the first condition. Usual choice for h d (φ) satisfies this condition, for example, h d (φ) = φ or h d (φ) = φ 3 (6φ 2 − 15φ + 10). Anomalous surface diffusion in the interfacial region is vanishing if D(φ(x))A(φ(x)) − D L /2 is an odd function of x, which can be fulfilled by putting D(φ) A(φ) = (1 − φ)D L . Also a condition for vanishing chemical potential jump is required at the imaginary sharp interface at x = 0. The chemical potential jump is directly related with the solute trapping effects arising from the finite interface width. Even though the solute trapping is one of the important physical phenomena in rapid solidification of alloys, it is negligible in normal slow solidification conditions. This often leads to a strong artificial solute trapping effect in such normal conditions, however, when a thick interface width is adopted for high efficiency in the phase-field computation. These artificial effects can be remedied by introducing an anti-trapping mass flux into the diffusion Eq. (24) [4], which is proportional to the interface velocity (∼ ∂φ/∂t) and directed toward the normal direction (∇φ/|∇φ|) to the interface. The modified diffusion equation then has the form

∂φ ∇φ ∂c = ∇ · D(φ) A(φ)∇c L + α(c L ) ∂t ∂t |∇φ|

(26)

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The coupling coefficient α(c L ) can be found from the condition for vanishing chemical potential jump; α(c L ) = √ (1 − k)c L (27) 2ω with the previous choices g(φ)=φ 2 (1−φ)2 , h d (φ)=φ and D(φ)A(φ)=(1−φ) D L . The linear relationship between the thermodynamic driving force and the interface velocity leads to a similar relationship between β and Mφ as for symmetric model, but with a replacement of Hm /(DT c p ) by m e ceL (1−k)/D L in the second term of the right side of Eq. (21).

6.

Multiphase and/or Multicomponent Models

The PFM explained above is for solidification of binary alloys into a single solid phase. Solidification of industrial alloys often involves more solid phases and/or more components. In multiphase systems, eutectic and peritectic solidification involving one liquid and two solid phases are of particular importance not only from engineering aspects, but also from scientific aspects because of their richness in interface patterns. Extending the number of phases for eutectic/peritectic solidification can be done by several ways; introducing three phase fields to denote each phase, introducing two phase fields where one is to distinguish between the solid and liquid phases and the other between two different solid phases, or coupling the PFM with the spinodal decomposition model where two solids phases are discriminated by two different compositions. Each approach has its own merits, yielding fruitful information for understanding pattern formation. For quantitative computation in real alloys with enhanced numerical efficiency, however, it is desirable for the models to have the following properties. First, thermodynamic and kinetic properties for three different interfaces in the system should be controlled independently. Second, the force balance at the triple interface junction should be maintained because it plays an essential role in pattern formation. Third, imposing the equal chemical potential condition is preferable because it significantly improves the numerical efficiency, as compared to the equal composition condition. Fourth, all the parameters should be determined to map the model onto the classical moving boundary problem of the eutectic/peritectic solidification. Such multiphase-field models are at the stage of development [10, 11]. The PFMs for binary alloys can be straightforwardly extended to the multicomponent system under the equal composition or equal chemical potential conditions. However, utilizing the advantage of the latter condition requires extra computation to find the compositions of coexisting solid and liquid phases having an equal chemical potential. If the thermodynamic database that are usually given by functions of the compositions are transformed into

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functions of the chemical potential as a preliminary step of computation, the extra cost may be significantly reduced. When the dilute solution approximation is adopted, in particular, the cost is negligible because the condition is reduced to the constant partition coefficients for a reference phase, e.g., liquid phase. Although multicomponent PFMs have been developed with the constant partition coefficients, the complete mapping of the models onto the classical sharp interface model has not yet been done. Presently, the multicomponent PFMs remain as tools for qualitative simulation.

7.

Simulations

The PFMs can be easily implemented into a numerical code by finite difference or finite element schemes, and various simulations for dendritic, eutectic, peritectic and monotectic solidifications have been performed. Examples of them can be found in [3]. The large disparity between the interface width, the microstructural scale and the diffusion boundary layer width hinders the simulation in physically relevant growth conditions. Therefore, early simulations have focused on the qualitative computations of the basic patterns. However, recent advances in hardware resources and the thin interface analysis greatly improved the computational power and efficiency in phasefield simulation. For modeling the free dendritic growth at low undercooling, where the diffusion field reaches far beyond the dendrite itself, computationally efficient methods such as adaptive mesh refining methods [12, 13] and the multi-scale hybrid method [14] of the finite difference scheme and the Monte Carlo scheme have been developed. Through a combination of such advances, not only qualitative but also quantitative phase-field simulation are possible in experimentally relevant growth conditions. The earliest quantitative phase-field simulation [5] was on the free dendritic growth of the symmetric model in 2D and 3D. This was the first numerical test of the microscopic solvability theory for free dendritic growth, which left little doubt about its validity. Quantitative 3D simulations of free dendritic growth in pure substance are further being refined to answer long-standing questions, for examples, the role of fluid flow for dendritic growth [3] and the origin of the abrupt changes of growth velocity and morphology in highly undercooled pure melt [4]. In spite of the variety of simulations for alloy solidification, the quantitative simulation for alloys have been limited. Recent advances in thin interface analysis for a one-sided model opened the window for quantitative calculations. One example is the 2D multiphase-field simulations of directional eutectic solidification in CBr4 −C2 Cl6 alloys [10]. The 2D experimental results of solidification in this alloy may be used for benchmarking the quantitative simulations because all the materials’ parameters were not only measured with reasonable accuracy, but also the various oscillatory/tilting

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instabilities occur with varying lamella spacing, growth velocity and composition. The 2D phase-field simulations of the eutectic solidification under real experimental conditions quantitatively reproduced all the lamella patterns and the morphological changes observed in experiments. In views of the recent success in the thin interface analysis and importance of the alloy solidification in both the engineering and scientific aspects, application of the one-sided PFM will soon be one of the most active fields in modeling alloy solidification. The quantitative application of PFMs to solidification of real alloys is hindered by the lack of information on thermo-physical properties such as interface energy, interface kinetic coefficient and their anisotropies. Combining the PFMs with atomistic modeling to determine these properties will provide powerful tools for studying the solidification behavior in real alloys.

References [1] J.S. Langer, “Instabilities and pattern formation in crystal growth,” Rev. Mod. Phys., 52, 1–28, 1980. [2] P. Meakin, “Fractals, scaling and growth far from equilibrium,” 1st edn., Cambridge Press, UK, 1998. [3] Boettinger, W.J. Warren, J.A., C. Beckermann, and A. Karma, “Phase-field simulation of solidification,” Annu. Rev. Mater. Res., 32, 163–194, 2002. [4] W.J. Hoyt, M. Asta, and A. Karma, “Atomistic and continuum modeling of dendritic solidification,” Mater. Sci. Eng. R, 41, 121–163, 2003. [5] A. Karma and W.-J. Rappel, “Quantitative phase-field modeling of dendritic growth in two and three dimension,” Phys. Rev. E, 57, 4323–4349, 1998. [6] S.G. Kim, W.T. Kim, and T. Suzuki, “Phase-field model for binary alloys,” Phys. Rev. E, 60, 7186–7197, 1999. [7] A.A. Wheeler, G.B. McFadden, and W.J. Boettinger, “Phase-field model for solidification of a eutectic alloy,” Proc. R. Soc. London. A, 452, 495–525, 1996. [8] S.G. Kim, W.T. Kim, and T. Suzuki, “Interfacial compositions of solid and liquid in a phase-field model with finite interface thickness for isothermal solidification in binary alloys,” Phys. Rev. E, 58, 3316–3323, 1998. [9] A. Karma, “Phase-field formulation for quantitative modeling of alloy solidification,” Phys. Rev. Lett., 87, 115701, 2001. [10] S.G. Kim, W.T. Kim, T. Suzuki, and M. Ode, “Phase-field modeling of eutectic solidification,” J. Cryst. Growth, 261, 135–158, 2004. [11] R.Folch and M. Plapp, “Toward a quantitative phase-field modeling of two-solid solidification,” Phys. Rev. E, 68, 010602, 2003. [12] N. Provatas, N. Goldenfeld, and J. Dantzig, “Adaptive mesh refinement computation of solidification microstructures using dynamic data structures,” J. Comp. Phys., 148, 265–290, 1999. [13] C.W. Lan, C.C. Liu, and C.M. Hsu, “An adaptive finite volume method for incompressible heat flow problem in solidification,” J. Comp. Phys., 178, 464–497, 2002. [14] M. Plapp and A. Karma, “Multiscale random-walk algorithm for simulating interfacial pattern formation,” Phys. Rev. Lett., 84, 1740–1743, 2000.

7.4 COHERENT PRECIPITATION – PHASE FIELD METHOD C. Shen and Y. Wang The Ohio State University, Columbus, Ohio, USA

Phase transformation is still the most efficient and effective way to produce various microstructures at mesoscales, and to control their evolution over time. In crystalline solids, phase transformations are usually accompanied by coherency strain generated by lattice misfit between coexisting phases. The coherency strain accommodation alters both thermodynamics and kinetics of the phase transformations and, in particular, produces various self-organized, quasi-periodical array of precipitates such as the tweed [1], twin [2], chessboard structures [3], and fascinating morphological patterns such as the stars, fans and windmill patters [4], to name a few (Fig. 1). These microstructures have puzzled materials researchers for decades. Incorporation of the strain energy in models of phase transformations not only allows for developing a fundamental understanding of the formation of these microstructures, but also provides the opportunity to engineer new microstructures of salient features for novel applications. Therefore, it is desirable to have a model that is able to predict the formation and time-evolution of coherent microstructural patterns. Yet coherent transformation in solid is the toughest nut to crack [5]. In a non-uniform (either compositionally or structurally) coherent solid where lattice planes are continuous on passing from one phase to another (Fig. 2), the lattice misfit between the adjacent non-uniform regions has to be accommodated by displacement of atoms from their regular positions along the boundaries. This sets up elastic strain fields within the solid. Being long-range and strongly anisotropic, the mechanical interactions among these strain fields are very different from the short-range chemical interactions. For example, the bulk chemical free energy and interfacial energy, both of which are associated with the short-range chemical interactions, depend solely on the volume fraction and the total area and inclination of interfaces of the precipitates, respectively. The elastic strain energy, on the other hand, depends on the size, shape, spatial orientation and mutual arrangement of the precipitates. When 2117 S. Yip (ed.), Handbook of Materials Modeling, 2117–2142. c 2005 Springer. Printed in the Netherlands.

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(a)

(b)

10µm (c)

(d)

30mm

50mm

70mm

50mm

70mm

Figure 1. Various strain-accommodating morphological patterns produced by coherent precipitation: (a) tweed, (b) twin, (c) chessboard structures, and (d) stars, fans and windmill patterns.

the elastic strain energy is included in the total free energy, every single precipitate (its size, shape and spatial position) contributes to the morphological changes of all other precipitates in the system through its influence on the stress field and the corresponding diffusion process. Therefore, many of the thermodynamic principles and rate equations obtained for incoherent precipitation may not be applicable anymore to coherent precipitation. A rigorous treatment of coherent precipitation requires a self-consistent description of microstructural evolution without any a priori assumptions about possible particle shapes and their spatial arrangements along a phase transformation path. The phase field method seems to satisfy this requirement. Over the past two decades, it has been demonstrated to have the ability to deal with arbitrary coherent microstructures produced by diffusional and

Coherent precipitation – phase field method (a)

2119

(b)

Figure 2. Schematic drawing of coherent interfaces (dashed lines). In (a) the precipitate (in grey) and the matrix have the same crystal structure but different lattice parameters while in (b) the precipitate has different crystal structure from the matrix.

displacive transformations with arbitrary transformation strains. Many complicated strain-induced morphological patterns such as those shown in Fig. 1 have been predicted (for recent reviews see [6–8]). A variety of new and intriguing kinetic phenomena underlying the development of these morphological patterns have been discovered, which include the correlated and collective nucleation [6, 7, 9, 10], local inverse coarsening, precipitate drifting and particle splitting [11–14]. These predictions have contributed significantly to our fundamental understanding of many experimental observations [15]. The purpose of this article is to provide an overview of the phase field method in the context of its applications to coherent transformations. We shall start with a discussion of the fundamentals of coherent precipitation, including how the coherency strain affects phase equilibrium (e.g., equilibrium compositions of coexisting phases and their equilibrium volume fractions), driving forces for nucleation, growth and coarsening, thermodynamic factors in diffusivity, and precipitate shape and spatial distribution. This will be followed by an introduction to microelasticity of an arbitrary coherent heterogeneous microstructure and its incorporation in the phase field method. Finally, implementation of the method in modeling coherent precipitations will be illustrated through three examples with progressively increasing complexity. For the purpose of simplicity and clarity, we limit our discussions to bulk materials of homogeneous modulus (i.e., all coexisting phases have the same elastic constants). Applications to more complicated problems such as small confining systems (such as thin films, multi-layers, and nano-particles) and elastically inhomogeneous systems will not be presented. For interested readers, these applications can be found in the references listed under Further Reading.

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Fundamentals of Coherent Precipitation

In depth coverage of this subject can be found in the monograph by Khachaturyan [16] and the book chapter by Johnson [17]. Below we discuss some of the basic concepts related to coherent precipitation. In a series of classical papers [18–21], Cahn laid the theoretical foundation for coherent transformations in crystalline solids. He distinguished the atomic misfit energy (part of the mixing energy of a solid solution) from the coherency elastic strain energy, and incorporated the latter into the total free energy to study coherent processes. He analyzed the effect of coherency strain energy on phase equilibrium, nucleation, and spinodal decomposition. Since the free energy is formulated within the framework of gradient thermodynamics [22], these studies are actually the earliest applications of the phase field method to coherent transformations.

1.1.

Atomic-misfit Energy and Coherency Strain Energy

A macroscopically stress-free solid solution with uniform composition can be in a “strained” state if the constituent atoms differ in size. The elastic energy associated with this microscopic origin is often referred to as the atomic-misfit energy in solid-solution theory [23]. It is the difference between the free energy of a real, homogeneous solution and the free energy of a hypothetical solution of the same system in which all the atoms have the same size. This atomicmisfit energy, even though mechanical in origin and long-range in character, is part of the physically measurable chemical free energy (e.g., free energy of mixing) and is included in thermodynamic databases in literature. The elastic energy associated with composition or structure non-uniformity (such as fluctuations and precipitates) in a coherent system is referred to as the coherency strain energy. The reference state for the measure of the coherency strain energy is a system of identical fluctuations or precipitate–matrix mixture, but with the fluctuations or precipitates/matrix separated into stressfree portions [21] (i.e., the incoherent counterpart). Since the coherency strain energy is in general a function of size, shape, spatial orientation and mutual arrangement of precipitates [16], it cannot be incorporated into the chemical free energy except for very special cases [18]. Thus the coherency strain energy is usually not included in the free energy from thermodynamic databases.

1.2.

Coherent and Incoherent Phase Diagrams

Different from the atomic misfit energy, the coherency strain energy is zero for homogeneous solid solutions and positive for any non-uniform coherent systems. It always promotes a homogeneous solid solution and suppresses

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phase separation. For a given system, the phase diagram determined by minimizing solely the bulk chemical free energy (including the contribution from the atomic misfit energy), or measured from a stage when precipitates already loose their coherency with the matrix, is referred to as incoherent phase diagram. Correspondingly the phase diagram determined by minimizing the sum of the bulk chemical free energy and the coherency strain energy, or measured from coherent stages of the system is referred to as coherent phase diagram. A coherent phase diagram, which is relevant to the study of coherent precipitation, could differ significantly from an incoherent one. This has been demonstrated clearly by Cahn [18] using an elastically isotropic system with a linear dependence of lattice parameter on composition. In this particular case the equilibrium compositions and volume fractions of coherent precipitates can be determined by the common-tangent rule with respect to the total bulk free energy (Fig. 3). Cahn showed that a coherent miscibility gap lies within an incoherent miscibility gap, with the differences in critical point and width of the miscibility gap determined by the amount of lattice misfit. In an elastically anisotropic system, the coherency strain energy becomes a function of precipitate size, shape and spatial location. In this case precipitates of different configurations will have different coherency strain energies, leading to a series of miscibility gaps lying within the incoherent one.

Incoherent free energy coherent free energy

c 1 c '1

c0

c

c'2 c 2

Figure 3. Incoherent (solid line) and coherent (dotted line) free energy as a function of composition for a regular solution that is elastically isotropic and its lattice parameter depends linearly on concentration. The equilibrium compositions in both cases (c1 ,c2 , c1 and c2 ) are determined by the common tangent construction. c0 is the average composition of the solid solution (after [21]).

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Coherent Precipitation

Precipitation involves typically phenomena of nucleation and growth of new phase particles out of a parent phase matrix, and subsequent coarsening of the resulting two-phase mixture. In the absence of coherency strain, nucleation is controlled by the interplay between the bulk chemical free energy and the interfacial energy, while growth and coarsening are dominated, respectively, by the bulk chemical free energy and the interfacial energy. For coherent precipitation, the coherency strain energy enters the driving forces for all three processes because it depends on both volume and morphology of the precipitates. In this case, nucleation is determined by the interplay among the bulk chemical free energy, the coherency strain energy, and the interfacial energy, while growth is dominated by the interplay between the chemical free energy and the coherency strain energy, and coarsening dominated by the interplay between the coherency strain energy and the interfacial energy. Therefore, many of the thermodynamic principles and rate equations derived for incoherent precipitation have to be modified for coherent processes. First of all, one has to pay attention to how the phase diagram and thermodynamic database for a given system were developed. For an incoherent phase diagram the thermodynamic data do not include the contribution from the coherency strain energy. In this case one needs to add the coherency strain energy to the chemical free energy from the database to obtain the total free energy for coherent transformations. However, if the phase diagram is determined for coherent precipitates and the thermodynamic database is developed by fitting the “chemical” free energy model to the coherent phase diagram, the “chemical” free energy already includes the coherency strain energy corresponding to the precipitate configuration encountered in the experiment. Adding again the coherency strain energy to such a “chemical” free energy will overestimate its contribution. Extra effort has to be made to formulate correctly the total free energy function in this case (see next section). Phase diagrams reported in literature are usually incoherent phase diagrams, but exceptions are not uncommon. For example, most existing Ni–Ni3 Al (γ /γ ) phase diagrams are actually coherent ones because the incoherent equilibrium between γ and γ are rarely observed in usual experiment [24]. To have an accurate chemical free energy model is essential for the construction of an accurate total free energy in the phase field method, which determines the coherent phase diagram and the driving forces for coherent precipitation. Even though the coherency strain energy always suppresses phase separation, reducing the driving force for nucleation and growth, coherent precipitation is still the preferred path at early stages of transformations in many material systems. This is because the nucleation barrier for a coherent precipitate is usually significantly lower than that for an incoherent precipitate because of the order-of-magnitude difference in interfacial energy between a

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coherent and an incoherent interface. Precipitates may loose their coherency at later stages when they grow to certain sizes; by then the strain-induced interactions among the coherent fluctuations and precipitates may have already fixed the spatial distribution of the precipitates. Therefore, developing any model for coherent precipitation has to start with coherent nucleation. Classical treatments of strain energy effect on nucleation (for reviews see, [5, 25, 26]) considered an isolated precipitate and calculate the strain energy per unit volume of the precipitate as a function of its shape. The strain energy was then added to the chemical free energy. In these approaches, the interaction of a nucleating particle with the existing microstructure was ignored. However, the strain fields associated with coherent particles interact strongly with each other in elastically anisotropic crystals. In this case the strain energy of a coherent precipitate depends not only on the strain field of its own but also on the strains due to all other particles in the system (for review, see [16]). This may have a profound influence on the nucleation process, e.g., making certain locations preferred nucleation sites [21]. In fact, many of the strain-induced morphological patterns observed (e.g., Fig. 1) may have been inherited from the nucleation stages and further developed during growth and coarsening. For example, the correlated (the position of a nucleus is determined by its interaction with the existing microstructure) [3, 6, 9] and collective (particles appear in groups) nucleation phenomena [10, 27] have been predicted for the formation of various self-organized, quasi-periodical morphological patterns as those shown in Fig. 1. Cahn [18, 21] analyzed coherent nucleation using the phase field method. He showed that one could derive analytical expressions for coherent interfacial energy, activation energy and critical size of a coherent nucleus for an elastically isotropic system. These expressions have exactly the same forms as those derived for incoherent precipitation, but with the chemical free energy replaced by a sum of the chemical free energy and the coherency strain energy. Although no solution is given for coherent nucleation in elastically anisotropic systems, Cahn illustrated qualitatively the effect of elastic interactions among coherent precipitates on the nucleation process in an elastically anisotropic cubic crystal. The driving force for nucleation reaches maximum at a nearby location in an elastically soft direction to an existing precipitate. In computer simulations using the phase field method, nucleation has been implemented in two ways: (a) solving numerically the stochastic phase field equations with the Langevin noise terms [6] and (b) stochastically seeding nuclei in an evolving microstructure according to the nucleation rates calculated as a function of local concentration and temperature [28] following the classical or non-classical nucleation theory. Recently the latter has been extended to coherent nucleation where the effect of elastic interaction of a nucleating particle with an existing microstructure is considered [29]. The Langevin approach is self-consistent with the phase field method but computationally

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intensive, because observation of nucleation requires sampling at very high frequency in the simulation. It has been applied successfully to the study of collective and correlated nucleation under site-saturation conditions [6, 9, 10, 27]. The explicit algorithm is computationally more efficient and has been applied successfully in concurrent nucleation and growth processes under either isothermal or continuous cooling conditions [28, 30]. Because the interfacial energy scales with surface area while the coherency strain energy scales with volume, the shape of a precipitate tends to be dominated by the interfacial energy when it is small and by the coherency strain energy when it grows to larger sizes. Therefore, shape transitions during growth and coarsening of coherent precipitates are expected. The long-range and highly anisotropic elastic interactions give rise to directionality in precipitate growth and coarsening, promoting spatial correlation among precipitates. Extensive discussions on these subjects can be found in the references listed in the Further Reading section. Indeed, significant shape transition (including splitting) and strong spatial alignment of precipitate have been observed (See reviews [6, 15, 31]). The shape transition of a growing particle may further induce growth instability, leading to faceted dendrite [32]. One of the major advantages of the phase field mode is that it describes growth and coarsening seamlessly in a single, self-consistent methodology. Incorporation of the coherency strain energy in the phase field model allows for capturing all possible microstructural features developing during growth and coarsening of coherent precipitates. For example, precipitate drifting, local inverse coarsening, and particle splitting have been predicted during growth and coarsening of coherent precipitates [11–14]. Incorporation of the coherency strain energy will also alter the thermodynamic factor in diffusivity, which is the second derivative of the total free energy with respect to concentration. Since atomic mobility rather than diffusivity is employed in the phase field model, the effect of coherency strain on the thermodynamic factor is included automatically. Note that the thermodynamic factor used in the calculation of atomic mobility from diffusivity should include the elastic energy contribution if the diffusivity was measured from a coherent system.

2. 2.1.

Theoretical Formulation Phase Field Microelasticity of Coherent Precipitation

In the phase field approach [7, 8], microstructural evolution during phase transformation is characterized self-consistently by the tempero-spatial evolution of a set of continuum order parameters or phase fields. One of the major

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advantages of the method is its ability to describe effectively and efficiently an arbitrary microstructure at mesoscale without exp-licitly tracking moving interfaces. In order to apply such a method to describe coherent transformations, one need to formulate the coherency strain energy as a functional of the phase fields without any a priori assumptions about possible particle shapes and their spatial arrangements along the transformation path. The theoretical treatment of such an elasticity problem was due to Khachaturyan and Shatalov [16, 33, 34] who derived a close form of the coherency strain energy for an arbitrary coherent multi-phase mixture in an elastically anisotropic crystal under the homogenous modulus assumption. The theory essentially solves the equation of mechanical equilibrium in the reciprocal space for the well-known virtual process by Eshelby [35, 36] (Fig. 4). The process consists of five steps: (1) isolate portions of a parent phase matrix; (2) transform the isolated portions into precipitate phases in a stress-free state (e.g., outside the parent phase matrix). The deformation involved in this step by assuming certain lattice correspondence between the precipitate and parent phases is defined as the stress-free transformation strain (SFTS) εi0j ; (3) apply an opposite stress −Ci j kl εkl0 to the precipitates to restore their original shapes and sizes; (4) placed them back into the spaces they occupied originally in the matrix; (5) allow both the precipitates and matrix to relax to minimize the elastic strain energy subject to the requirement of interface coherency. Step (1) is traditionally taken prior to the phase transformation. If the precipitates

(5)

(1)

(2)

ε0ij

(4) (3)

σij ⫽⫺Cijkl ε0kl

Figure 4. The Eshelby’s virtual procedures for calculating the coherency strain energy of coherent precipitates.

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differ in composition from the matrix the transformation in Step (2) will change the matrix composition as well because of mass conservation. To be consistent with the definition of the coherency strain energy given in Section 2, we may modify the Eshelby cycle as follows: (1 ) consider a coherent microstructure consisting of arbitrary concentration or structural non-uniformity produced along a phase transformation path; (2 ) decompose the microstructure into its incoherent counterpart (i.e., with all the microstructural constituents being in their stress-free states); (3 ) apply counter stress to force the lattices of all the constituents to be identical to nullify SFTS; (4 ) put them back together by re-stitching their corresponding lattice planes at interfaces; (5 ) let the system relax to minimize the elastic strain energy. The SFTS field associated with arbitrary compositional or structural inhomogeneities can be expressed either in terms of shape functions for sharpinterface approximation [16] of an arbitrary multi-phase mixture, or in terms of phase fields for diffuse-interface approximation of arbitrary concentration or structural non-uniformities: εi0j (x) =

N

εi00j ( p)φ p (x),

(1)

p=1

which is a linear superposition of all N types of non-uniformities with φ p (x) being the phase fields characterizing the p-th type non-uniformity and εi00j ( p) the corresponding SFTS measured from a given reference state. Note that εi00j ( p)(i, j = 1, 2, 3) depends on the lattice correspondence between the precipitate and parent phases. The calculation of εi00j ( p) is an important step towards formulating the coherency strain energy and it will be described in details later in several examples. The equilibrium elastic strain and hence the strain energy can be found from the condition of mechanical equilibrium [37] ∂σi j (x) + f i (x) = 0, ∂x j

(2)

subject to boundary conditions. Here σi j (x) is the ij component of the coherency stress at position x. f i (x) is a body force per unit volume exerted by, e.g., an external field. In Eq. (2) we have used the convention by Einstein where the repeated index j implies a summation over all its possible values. The boundary conditions include constraints on external surfaces and internal interfaces. At external surfaces the boundary conditions are determined by physical constraints on the macroscopic body of a sample, such as shape, surface traction, or a combination of the two. At internal interfaces,

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continuities of both displacement and coherency stress are required to ensure the coherency of the interfaces. The Green’s function solution of Eq. (2) under the homogeneous modulus assumption, gives the equilibrium elastic strain [16, 38]:

ei j (x) = ε¯ i j +

N 1 dg − [n j ki (n) + n i kj (n)]n l σkl00 ( p)φ˜ p (g) 2 (2π )3 p=1

−

N

εi00j ( p)φ p (x)

(3)

p=1

where ε¯ i j is a homogeneous strain that represents the macroscopic shape change of the material body, g is a vector in the reciprocal space and n ≡ g/g. [−1 (n)]ik ≡ Ci j kl n j n l is the inverse of the Green’s function in the reciprocal 00 ˜ space. σi00 j ( p) ≡ C i j kl εkl ( p), φ p (g) is the Fourier transform of φ p (x). – represents a principle value of the integral that excludes a small volume in the reciprocal space (2π )3 / V at g = 0, where V is the total volume of the system. The total coherency strain energy of the system at equilibrium is then readily obtained as

1 Ci j kl ei j (x)ekl (x)dx E = 2 N N V 1 Ci j kl εi00j ( p)εkl00 (q)φ p (x)φq (x)dx + Ci j kl ε¯ i j ε¯ kl = 2 p=1 q=1 2 el

− ε¯ i j

N

Ci j kl εkl00 ( p)φ p (x)dx

p=1

−

1 2

N N

00 ˜∗ ˜ − n i σi00 j ( p) j k (n)σkl (q)n l φ p (g)φq (g)

p=1 q=1

dg (2π )3

(4)

The asterisk in the last term stands for the complex conjugate. Equations (3) and (4) contain the homogeneous strain, ε¯ i j , which is suitable if the external boundary condition is given for a constrained macroscopic shape. Corresponding to the Eshelby circle aforementioned, the first term in the right-hand side of Eq. (4) is the energy required to “squeeze” the microstructural constituents to nullify the stress-free transformation strain in Step (3 ), and the remaining terms represent the energy reductions associated with relaxations of the “squeezed” state in Step (5 ). In particular, the second and third terms describe the homogeneous (macroscopic shape) relaxation and the fourth term

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describes the local heterogeneous relaxation. For a constrained stress condition at the external surface, ε¯ i j is determined by the minimization of the total elastic energy with respect to itself, which yields [38]. ε¯ i j =

appl Si j kl σkl

N 1 + εi00j ( p)φ p (x)dx V p=1

(5) appl

where Si j kl is the elastic compliance tensor and σi j is the applied stress that appl is related to the surface traction T and the surface normal s by Ti = σi j s j . Combining Eqs. (3)–(5) gives 1 E = 2 el

− −

N N

Ci j kl εi00j ( p)εkl00 (q)φ p (x)φq (x)dx

p=1 q=1

1 Ci j kl 2V 1 2

N

εi0j ( p)φ p (x)dx

p=1

N

εkl00 (q)φq (x )dx

q=1

N N p=1

appl − σi j

dg 00 ˜∗ ˜ − n i σi00 j ( p) j k (n)σkl (q)n l φ p (g)φq (g) (2π )3 q=1

N p=1

εi00j ( p)φ p (x)dx −

V appl appl Si j kl σi j σkl 2

(6)

The expression for the mixed constrained shape and surface traction boundary conditions can be derived in a similar way [38]. The above equations were derived under an assumption of constant Ci j kl , i.e., the homogeneous modulus assumption. In cases with spatially dependent Ci j kl the solution is found to be contained in an implicit equation and thus requires a suitable solver, such as an iteration method. Readers are referred to the recent development by Wang et al. [39]. Equations (2)–(6) provide the close forms of the coherency strain energy for a general elastically anisotropic system with arbitrary coherent precipitates described by the phase fields. In such formulations, the coherency strain energy can be added directly to the chemical free energy in the phase field method, because both of them are functionals of the same phase field variables. As mentioned earlier, the “chemical” free energy contains part of the coherency strain energy if it is obtained by fitting the free energy model to a coherent phase diagram. In order to avoid possible double counting, it is necessary to subtract this part of the coherency strain energy from Eq. (4) or (6). Therefore, it is useful to separate the coherency strain energy into self-energy

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and interaction-energy. Following the same treatment as that presented in the microscopic elasticity theory of solid solutions [16], we can rewrite Eq. (4) as el el E el = E sel f + E int , el E sel f =

N N 1 2 p=1 q=1

−¯εi j

N

Ci j kl εi00j ( p)εkl00 (q)φ p (x)φq (x)dx +

V Ci j kl ε¯ i j ε¯ kl 2

Ci j kl εkl00 ( p)φ p (x)dx

p=1

− el =− E int

1 2

N N p=1 q=1

dg Qδ pq − φ˜ p (g)φ˜ q∗ (g) , (2π )3

N N 1 00 − n i σi00 j ( p) j k (n)σkl (q)n l −Qδ pq 2 p=1 q=1

dg × φ˜ p (g)φ˜ q∗ (g) , (2π )3 00 00 00 where Q = n i σi00 j ( p) j k (n)σkl ( p)n l g is the average of n i σi j ( p) j k (n)σkl ( p)n l over the entire reciprocal space and δ pq is the Kronecker delta that equals el unity when p = q or zero otherwise. E sel f is configuration-independent and equals the elastic energy of placing a coherent precipitate of unit-volume multiplying the total volume of the precipitate (small as compared to the volume el of the system) into a uniform matrix. E int is configuration-dependent and contains the pair-wise interactions between precipitates and between volume elements within a finite precipitate. Since the self-energy depends only on the total volume of the precipitates and is independent of their morphology and spatial arrangement, it could be incorporated into and renormalizes the chemical free energy. Clearly, the self-energy should not be included in the calculation of the coherently strain energy if the “chemical” free energy of a system is obtained by fitting to a coherent phase diagram.

2.2.

Incorporation of Coherency Strain Energy into Phase Field Equations

The chemical free energy of a non-uniform system in the phase field approach is formulated as a functional of the field variable based on gradient thermodynamics [22]

F ch =

[ f (φ(x)) + κ|∇φ(x)|2 ]dx,

(7)

where the first term in the integrand is the local chemical free energy density that depends only on local values of the field, φ(x), while the second term is

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the gradient energy that accounts for contributions from spatial variation of φ(x). More complex system may require multiple phase fields, as will be seen in the examples given in the next section. For a coherent system, the total free energy is a sum of the chemical free energy, F ch , and the coherency strain energy, E el , F = F ch + E el ,

(8)

where the chemical free energy is usually measured from a stress-free reference state mentioned earlier, and the coherency strain energy contains both the self- and interaction-energy discussed above. The time evolution of the phase fields, and thus the coherent microstructure, is described by the Onsager-type kinetic equation that assumes a linear dependence of the rate of evolution, ∂φ/∂t, on the driving force, δ F/δφ, ∂φ(x, t) ˆ δ F + ξ(x, t), = −M ∂t δφ(x, t)

(9)

ˆ is a kinetic coefficient matrix and ξ is the Langevin where the operator M random force term that describes thermal fluctuation. The kinetic coefficient ˆ = M if the phase field is non-conserved and matrix is often simplified to M 2 ˆ M = − M∇ if the phase field is conserved, where M is a scalar. Note that the total free energy, F, is a functional of the spatial distribution of the phase field and the energy minimization is a variational process.

3.

Examples of Applications Cubic → Cubic Transformation

3.1.

For a simple cubic → cubic transformation the SFTS is dilatational. If we assume that the coherency strain is caused by concentration inhomogeneity, which is the case for most cubic alloys, the SFTS tensor becomes a function of concentration, e.g., εi0j = ε 0 (c)δi j . The compositional dependence of ε 0 (c) can be written in a Taylor series around the average composition of the parent phase matrix, c¯

dε 0 ε (c) = ε (c) ¯ + ¯ + ··· . (c − c) dc c=c¯ 0

0

(10)

¯ c), ¯ and By choosing a reference state at c(stress-free), ¯ ε 0 (c) = [a(c) − a(c)]/a( the leading term at the right hand side of Eq. (10) vanishes. The SFTS may be approximated by taking the first non-vanishing term

εi0j (x) =

1 da [c(x) − c]δ ¯ ij , a(c) ¯ dc c=c¯

(11)

Coherent precipitation – phase field method

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where we have added the explicit dependence of the SFTS on the spatial posi¯ tion x. Accordingly, εi00j = a −1 (c)(da/dc) c=c¯ δi j . With the stress-free condition for the external boundary applied in this and the subsequent examples, the coherency strain energy is reduced from Eq. (4) with substituting φ p by c(x) − c¯ to

1 V ¯ 2 dx + Ci j kl ε¯ i j ε¯ kl E el = Ci j kl εi00j εkl00 [c(x) − c] 2 2 −¯εi j Ci j kl εkl00

[c(x) − c]dx ¯

dg 1 00 ˜ c˜∗ (g) , − − n i σi00 j j k (n)σkl n l c(g) 2 (2π )3 ˜ is the Fourier where ε¯ i j is determined by the boundary condition and c(g) transform of c(x). The kinetics of coherent precipitates is then described by Eqs. (7)–(9). A typical example of such a cubic → cubic coherent transformation is the precipitation of an ordered intermetallic phase (γ -L12 (Ni3 Al)) from a disordered matrix (γ-fcc solid solution) in Ni–Al (Fig. 5). The coherency strain is caused by the difference in composition between γ and γ that modifies the lattice parameters of the two phases. Since the two-phase equilibrium is coherent equilibrium in the system, the coherency strain energy should include only the configuration-dependent part, as discussed earlier:

1 dg el 00 00 00 00 ˜ c˜∗ (g) . E = − − n i σi j j k (n)σkl n l − n i σi j j k (n)σkl n l c(g) g 2 (2π )3 Figure 6 shows the simulated microstructural evolution during coherent precipitation by the phase field method [40]. The chemical free energy is

(a)

(b)

Figure 5. Crystal structures of γ (fcc solid solution) (a) and γ (ordered L12 ) (b) phases in nickel-aluminum alloy. In (b) the solid circles indicate nickel atoms and the open circles indicate aluminum atoms.

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approximated by a Landau-type expansion polynomial, which provides appropriate descriptions of the equilibrium thermodynamic properties (such as equilibrium compositions and driving force) and reflects the symmetry relationship between the parent and product phases (for general discussion see [41, 42]. The elastic constants of the cubic crystal c11 (=C1111), c12 (=C1122), c44 (=C2323 ) are 231, 149, 117 GPa, respectively [43]. εi00j is chosen as 0.049δi j which corresponds to a SFTS of 0.56%. The simulation is performed on a 512 × 512 mesh with grid size of 1.7 nm. The starting microstructure is a homogeneous supersaturated solid solution of an average composition of 0.17at%Al. The nucleation processes in this and the subsequent examples was simulated by the Langevin noise terms described by ξ in Eq. (9). The noise terms were applied only for a short period of time at the beginning, corresponding to the site-saturation approximation. According to the group and subgroup relationship of crystal lattice symmetry of the parent and precipitate phases, three long-range order parameter fields were used in addition to the concentration field, which introduces automatically four anti-phase domains of the ordered γ phase. Periodical boundary conditions were employed. Because of the strong elastic anisotropy, the precipitates evolved into cuboidal shapes and align themselves into a quasi-periodical array, with both the interface inclination and spatial alignment along the elastically soft 100 directions. The simulated γ /γ microstructure agrees well with experimental observations (Fig. 6(b)). Through this example it becomes clear that the phase field method is able to handle high volume fractions of diffusionally and elastically interacting precipitates of complicated shapes and spatial distributions.

3.2.

Hexagonal → Orthorhombic Transformation

The hexagonal → orthorhombic transformation is a typical example of structural transformations with crystal lattice symmetry reduction. Different from a cubic → cubic transformation, there are several symmetry related orientation variants of the precipitate phase. Experimental observations [44–46] have shown remarkably similar morphological patterns formed by the low symmetry orthorhombic phase in different materials systems, indicating that accommodation of coherency strain among different orientation variants dominate the microstructural evolution during the precipitation reaction. In this example we present a generic transformation of a disordered hexagonal phase to an ordered orthorhombic phase with three lattice correspondence variants [27]. The atomic rearrangement during ordering occurs primarily on the (0001) plane of the parent hexagonal phase and, therefore, the essential features of the microstructural evolution can be well represented by ordering of the (0001) planes (Fig. 7) and effectively modeled in two-dimension.

Coherent precipitation – phase field method (a)

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(b)

0.2µm

Figure 6. (a) Simulated γ /γ microstructure by the phase field method. The lattice misfit is taken as (aγ − aγ ) / aγ ≈ 0.0056. (b) Experimental observation in Ni–Al–Mo alloy (Courtesy of M. F¨ahrmann). (a)

[010]O

[12 10]H

(b)

bO [12 10]H [100]O

aH

[100]O

bO

[12 10]H

Figure 7. Correspondence of the lattices of (a) the disordered hexagonal phase and (b) the ordered orthorhombic phase (with three orientation variants).

The lattice correspondence between the parent and product phases is shown in Fig. 7. For the first variant in Fig. 7(b) we have, 1 ¯¯ [2110]H 3 1 ¯ H [1120] 3

→ [100]O ,

→ 12 [110]O , [0001]H → [001]O , and the corresponding STFS tensor is

√ α 0 0 a b cO − cH − a − 3aH O H O 0 ,β = √ ,γ= , εi j = 0 β 0 , where α = aO cH 3aH 0 0 γ

where aH and cH are the lattice parameters of the hexagonal phase and aO , bO and cO are the lattice parameters of the orthorhombic phase. If we assume

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no volume change for the transformation and the lattice parameter difference between the hexagonal and orthorhombic phases along the c-axis is negligible, the SFTS is simplified to

1 0 0 εi0j = ε 0 0 −1 0 , 0 0 0

(12)

where ε 0 = (aO − aH )/aO is the magnitude of the shear deformation. The three lattice correspondence variants of the orthorhombic phase are related by 120◦ rotation with respect to each other around the c-axis (Fig. 7b). The SFTS of the remaining two variants thus can be obtained by rotational operation (±120◦ around [100]0 ) on the strain tensor given in (Eq. (12). Furthermore, since the deformation along the c-axis is assumed zero, the SFTS of the three variants can be written as 2 × 2 tensors: √ 0 −1/2 3/2 00 0 1 00 0 , εi j (2) = ε √ , εi j (1) = ε 0 −1 3/2 1/2 √ −1/2 − 3/2 00 0 √ . (13) εi j (3) = ε 1/2 − 3/2 In the phase field method, the three variants are described by three longrange order (lro) parameters (η1 , η2 , η3 ), with each representing one variant. Since there is no composition change during the ordering reaction, the structural inhomogeneity is solely characterized by the lro parameters. Correspondingly, the chemical free energy is formulated as a Landau polynomial expansion with respect to the lro parameters. Substituting φ p by η2p ( p = 1, 2, 3) in Eq. (4) the elastic energy becomes, 3 3 1 Ci j kl εi00j ( p)εkl00 (q) E = 2 p=1 q=1 el

− ε¯ i j −

1 2

3

Ci j kl εkl00 ( p)

p=1 3 3

η2p (x)ηq2 (x)dx +

V Ci j kl ε¯ i j ε¯ kl 2

η2p (x)dx

00 2 2∗ − n i σi00 j ( p) j k (n)σkl (q)n l η p (g)ηq (g)

p=1 q=1

. dg (2π )3

Figure 8 shows the simulated microstructures by the phase field method [27]. The system was discretized into a 1024 × 1024 mesh with grid size 0.5 nm. The initial microstructure is a homogeneous hexagonal phase. Strong spatial correlation among the orthorhombic phase particles was developed during the nucleation (Fig. 8(a)). The subsequent growth and coarsening of the orthorhombic phase particles produced various special domain patterns

Coherent precipitation – phase field method (a)

(b)

t* = 20

2135 (c)

t* = 1000

t* = 3000

Figure 8. Microstructures obtained during hexagonal → orthorhombic ordering by 2D phase field simulation. Specific patterns (highlighted by circles, ellipses, and squares) are also found in experimental observations (Fig. 1d).

as a result of elastic strain accommodation among different orientation variants. These patterns show excellent agreements with experimental observations (Fig. 1(d)). Typical sizes of these configurations were also found in good agreement with the experimental observations. If the coherency strain energy was not considered, completely different domain pattern were observed. This indicates that the elastic strain accommodation among different orientation variants dominates the morphological pattern formation during the hexagonal → orthorhombic transformations. The coarsening kinetics of the domain structure deviates significantly from the one observed for an incoherent system [47].

3.3.

Cubic → Trigonal (ζ2 ) Martensitic Transformation in Polycrystalline Au–Cd Alloy

In the two examples presented above, single crystals with relative simple lattice rearrangements during precipitation are considered. In this example we present one of the most complicated cases that have been studied by the phase field method [48]. The trigonal lattice of the ζ2 martensite in Au–Cd can be visualized as a stretched cubic lattice in one of the body diagonal (i.e., [111]) directions. Four lattice correspondence variants are associated with the transformation, which correspond to the four 111 directions of the cube. In the phase field method, the spatial distribution of the four variants is characterized by four lro parameter fields and the chemical free energy is approximated by a Landau expansion polynomial with respect to the lro parameters. If we represent the trigonal phase in hexagonal indices, the lattice correspondence

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between the parent and product phases are [49]: ¯ ς , [121] ¯ β2 → [12 ¯ 10] ¯ ς , [111]β2 → [0001]ς , ¯ β2 → [21¯ 10] Variant 1: [211] 2 2 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ Variant 2: [121]β2 → [2110]ς2 , [211]β2 → [1210]ς2 , [111]β2 → [0001]ς2 , ¯ ς , [121] ¯ β2 → [12 ¯ 10] ¯ ς , [1¯ 11] ¯ β2 → [0001]ς , ¯ β2 → [21¯ 10] Variant 3: [211] 2 2 2 ¯ ¯ ¯ ¯ ¯ ¯ ¯ Variant 4: [121]β2 → [2110]ς , [211]β2 → [1210]ς , [111]β2 → [0001]ς , 2

2

2

Correspondingly, the SFTS for the four lattice correspondence variants are:

α β β εi00j (1) = β α β , β β α

α −β −β β , εi00j (2) = −β α −β β α

α −β β α β −β α −β , (14) εi00j (3) = −β α −β , εi00j (4) = β β −β α −β −β α √ √ √ √ where α = ( 6ah + 3ch − 9ac )/9ac , β = (− 6ah + 2 3ch )/18ac , ac is the lattice parameter of the cubic parent phase, ah and ch are the lattice parameters of the trigonal phase represented in the hexagonal indices. The SFTS field that characterizes the structural inhomogeneity is a linear superposition of the SFTS of each variant, as given by Eq. (2). Thus the elastic energy (Eq. (4)) reduces to

4 4 1 00 − Ci j kl εi00j ( p)εkl00 (q) − n i σi00 E = j ( p) j k (n)σkl (q)n l 2 p=1 q=1 el

×η˜ p (g)η˜ q∗ (g)

dg . (2π )3

Figure 9(a) shows the 3D microstructure simulated in a 128 × 128 × 128 mesh for a single crystal. The grid size is 0.5 µm. The simulation started with a homogeneous cubic solid solution characterized by η1 (x) = η2 (x) = η3 (x) = η4 (x) = 0. The four orientation variants are represented by four shades of gray in the figure. The typical “herring-bone” feature of the microstructure formed by self-assembly of the four variants is readily seen, which agrees well with experimental observations (Fig. 9(c)). The treatment for a polycrystalline material may take the strain tensors in Eq. (14) as the ones in the local coordinate of each constituent single crystal grain. The SFTS expressed in the global coordinate thus requires applying a rotational operation 0,g

εi j (x) = Rik (x)R j l (x)εi0j (x),

(15)

where Ri j (x) is a 3×3 matrix that defines the orientation of the grain in the global coordinate, which has a constant value within a grain but differs from

Coherent precipitation – phase field method (a)

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(b)

(c)

Figure 9. Microstructures developed in a cubic → trigonal (ζ2 ) martensitic transformation in (a) single crystal and (b) polycrystal from 3D phase field simulations. The “herring-bone” structure observed in the simulation (a) agrees well with experiment observations (c).

one grain to another. The microstructure in Fig. 9(b) is obtained for a polycrystal with eight randomly oriented grains. The produced multi-domain structure is found to be quite different from the one obtained from the single crystal. Because of the constraint from neighboring randomly oriented grains, the martensitic transformation does not go to completion and the multi-domain structure is stable against further coarsening, which is in contrary to the case with single crystal where the martensitic transformation goes to completion and the multi-domain microstructure undergoes coarsening till a single domain state for the entire system is reached. This example demonstrates well the capability of the phase field method in predicting very complicated strain accommodating microstructural patterns produced by a coherent transformation in polycrystals.

4.

Summary

In this article we reviewed some of the fundamentals related to coherent transformations, the microelasticity theory of coherent precipitation and

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its implementation in the phase field method. Through three examples, the formulations of the stress-free transformation strain field associated with compositional or structural non-uniformity produced by diffusional and diffusionless transformations are discussed. For any given coherent transformations, if the lattice correspondence between the parent and product phases, their lattice parameters and elastic constants are known, the coherency strain energy can be formulated in a rather straightforward fashion as a functional of the same field variables chosen to characterize the microstructure in the phase field method. The flexibility of the method in treating various coherent precipitations involving simple and complex atomic rearrangements has been well demonstrated through these examples. The description of microstructures in terms of phase fields allows for complexities at a level close to that encountered in real materials. The evolution of the microstructures is treated in a self-consistent framework where the variational principle is applied to the total free energy of the system. It would not be surprising to see in the near future a significant increase in the attempts of exploring various kinds of complex coherent phenomena with phase field method owing to these benefits. The formulation of the chemical free energy for solid state phase transformations is not emphasized in this review, but can be found in other reviews (see e.g., [6–8]). The numerical techniques employed in current phase field modeling of coherent transformations involve uniform finite difference schemes, which pose serious limitations on length scales. As a physical model, the affordable system size that can be considered in a phase field simulation is limited by the thickness of the actual interfaces when real material parameters are used as inputs. In order to overcome this length scale limit, one has to either employ more efficient algorithms such as the adaptive [50] and wavelet method [51] that are currently under active development, or produce artificially diffuse interfaces at length scales of interest without altering the velocity of interface motion by modifying properly certain model parameters [52–55]. Since the close form of the coherency strain energy is given in the reciprocal space, Fourier transform is required in solving the partial differential equations, which may impose serious challenges to the adaptive or wavelet method. A common approach to scale up the length scale of phase field modeling of a coherent transformation is to increase the contribution of the coherency strain energy relative to the chemical free energy [40, 56]. While it seems to be a reasonable approach for qualitative studies, it may result in serious artifacts in quantitative studies. For example, it may produce artificially high strain-induced concentration non-uniformity which may affect the kinetics of nucleation, growth and coarsening. This issue has received increasing attentions as the phase field method is being applied to quantitative simulation studies.

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5.

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Further Reading

Monographs and Reviews on Coherent Phase Transformations 1. A.G. Khachaturyan, Theory of structural transformations in solids, John Wiley & Sons, New York, 1983. 2. Y. Wang, L.Q. Chen, and A.G. Khachaturyan, “Computer simulation of microstructure evolution in coherent solids,” Solid phase transformations, Warrendale, PA, TMS, 1994. 3. W.C. Johnson, “Influence of elastic stress on phase transformations,” In: H.I. Aaronson (ed.), Lectures on the theory of phase transformations, The Minerals, Metals & Materials Society, 35–134, 1999. 4. L.Q. Chen, “Phase field models for microstructure evolution,” Annu. Rev. Mater. Res., 32, 113–140, 2002. Articles on Elastically Inhomogeneous Solids and Thin Films 5. A.G. Khachaturyan, S. Semenovskaya, and T. Tsakalokos, “Elastic strain energy of inhomogeneous solids,” Phys. Rev. B, 52, 15909–15919, 1995. 6. S.Y. Hu and L.Q. Chen, “A phase-field model for evolving microstructures with strong elastic inhomogeneity,” Acta Mater., 49, 1879, 2001. 7. Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, “Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid,” J. Appl. Phys., 92, 1351–1360, 2002. 8. Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, “Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin films,” Acta Mater., 51, 4209–4223, 2003.

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[50] N. Provatas, N. Goldenfield et al., “Efficient computation of dendritic microstructures using adaptive mesh refinement,” Phys. Rev. Lett., 80, 3308–3311, 1998. [51] D. Wang and J. Pan, “A wavelet-galerkin scheme for the phase field model of microstructural evolution of materials,” Computat. Mat. Sci., 29, 221–242, 2004. [52] A. Karma and W.-J. Rappel, “Quantitative phase-field modeling of dendritic growth in two and three dimensions,” Phys. Rev. E, 57(4), 4323–4349, 1998. [53] K.R. Elder and M. Grant, “Sharp interface limits of phase-field models,” Phys. Rev. E, 64, 021604, 2001. [54] C. Shen, Q. Chen et al., “Increasing length scale of quantitative phase field modeling of growth-dominant or coarsening-dominant process,” Scripta Mater., 50, 1023– 1028, 2004. [55] C. Shen, Q. Chen et al., “Increasing length scale of quantitative phase field modeling of concurrent growth and coarsening processes,” Scripta Mater., 50, 1029–1034, 2004. [56] J.Z. Zhu, Z.K. Liu et al., “Linking phase-field model to calphad: Application to precipitate shape evolution in Ni-base alloys,” Scripta Mater., 46, 401–406, 2002.

7.5 FERROIC DOMAIN STRUCTURES USING GINZBURG–LANDAU METHODS Avadh Saxena and Turab Lookman Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

We present a strain-based formalism of domain wall formation and microstructure in ferroic materials within a Ginzburg–Landau framework. Certain components of the strain tensor serve as the order parameter for the transition. Elastic compatibility is explicitly included as an anisotropic, long-range interaction between the order parameter strain components. Our method is compared with the phase-field method and that used by the applied mathematics community. We consider representative free energies for a twodimensional triangle to rectangle transition and a three-dimensional cubic to tetragonal transition. We also provide illustrative simulation results for the two-dimensional case and compare the constitutive response of a polycrystal with that of a single crystal. Many minerals and materials of technological interest, in particular martensites [1] and shape memory alloys [2], undergo a structural phase transformation from one crystal symmetry to another crystal symmetry as the temperature or pressure is varied. If the two structures have a simple group–subgroup relationship then such a transformation is called displacive, e.g., cubic to tetragonal transformation in FePd. However, if the two structures do not have such a relationship then the tranformation is referred to as replacive or reconstructive [3, 4]. An example is the body-centered cubic (BCC) to hexagonal closepacked (HCP) transformation in titanium. Structural phase transitions in solids [5, 6] have aroused a great deal of interest over a century due to the crucial role they play in the fundamental understanding of physical concepts as well as due to their central importance in developing technologically useful properties. Both the diffusion-controlled replacive (or reconstructive) and the diffusionless displacive martensitic transformations have been studied although the former have received far more

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attention simply because their reaction kinetics is much more conducive to control and manipulation than the latter. We consider here a particular class of materials known as ferroelastic martensites. Ferroelastics are a subclass of materials known as ferroics [4], i.e., a non-zero tensor property appears below a phase transition. Some examples include ferromagnetic and ferroelectric materials. In some cases more than one ferroic property may coexist, e.g., magnetoelectrics. Such materials are called multi-ferroics. The term martensitic refers to a diffusionless first order phase transition which can be described in terms of one (or several successive) shear deformation(s) from a parent to a product phase [1]. The transition results in a characteristic lamellar microstructure due to transformation twinning. The morphology and kinetics of the transition are dominated by the strain energy. Ferroelasticity is defined by the existence of two or more stable orientation states of a crystal that correspond to different arrangements of the atoms, but are structurally identical or enantiomorphous [4, 5]. In addition, these orientation states are degenerate in energy in the absence of mechanical stress. Salient features of ferroelastic crystals include mechanical hysteresis and mechanically (reversibly) switchable domain patterns. Usually ferroelasticity occurs as a result of a phase transition from a non-ferroelastic high-symmetry “prototype” phase and is associated with the softening of an elastic modulus with decreasing temperature or increasing pressure in the prototype phase. Since the ferroelastic transition is normally weakly first order, or second order, it can be described to a good approximation by the Landau theory [7] with spontaneous strain as the order parameter. Depending on whether the spontaneous strain, which describes the deviation of a given ferroelastic orientation state from the prototype phase is the primary or a secondary order parameter, the low symmetry phase is called a proper or an improper ferroelastic, respectively. While martensites are proper ferroelastics, examples of improper ferroelastics include ferroelectrics and magnetoelastics. There is a small class of materials (either metals or alloy systems) which are both martensitic and ferroelastic and exhibit shape memory effect [2]. They are characterized by highly mobile twin boundaries and (often) show precursor structures (such as tweed and modulated phases) above the transition. Furthermore, these materials have small Bain strain, elastic shear modulus softening, and a weakly to moderately first order transition. Some examples include In1−x Tlx , FePd, CuZn, CuAlZn, CuAlNi, AgCd, AuCd, CuAuZn2 , NiTi and NiAl. In many of these transitions intra-unit cell distortion modes (or shuffles) can couple to the strain either as a primary or secondary order parameter. NiTi and titanium represent two such examples of technological importance. Additional examples include actinide alloys: UNb6 shape memory alloy and Ga-stabilized δ-Pu.

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Landau Theory

To understand the thermodynamics of the phase transformation and the phase diagram a free energy of the transformation is needed. This Landau free energy (LFE) is a symmetry allowed polynomial expansion in the order parameter that characterizes the transformation [7], e.g., strain tensor components and/or (intra-unit cell) shuffle modes. A minimization of this LFE with respect to the order parameter components leads to conditions that give the phase diagram. Derivatives of the LFE with respect to temperature, pressure and other relevant thermodynamic variables provide information about the specific heat, entropy, susceptibility, etc. To study domain walls between different orientational variants (i.e., twin boundaries) or diferent shuffle states (i.e., antiphase boundaries) symmetry allowed strain gradient terms or shuffle gradient terms must be added to the Landau free energy. These gradient terms are called Ginzburg terms and the augmented free energy is referred to as the Ginzburg–Landau (GLFE) free energy. Variation of the GLFE with respect to the order parameter components leads to (Euler–Lagrange) equations [8] whose solution leads to the microstruture. In two dimensions we define the symmetry-adapted dilatation (area change), deviatoric and shear strains [8, 9], respectively, as a function of the Lagrangian strain tensor components i j : 1 e1 = √ (x x + yy ), 2

1 e2 = √ (x x − yy ), 2

e3 = x y .

(1)

As an example, the Landau free energy for a triangular to (centered) rectangular transition is given by [10, 11] F(e2 , e3 ) =

A 2 B C A1 2 (e2 + e32 ) + (e23 − 3e2 e32 ) + (e22 + e32 )2 + e , 2 3 4 2 1

(2)

where A is the shear modulus, A1 is the bulk modulus, B and C are third and fourth order elastic constants, respectively. This free energy without the non-order parameter strain (e1 ) term below the transition temperature (Tc ) has three minima in (e2 , e3 ) corresponding to the three rectangular variants. Above Tc it has only one global minimum at e2 = e3 = 0 associated with the stable triangular lattice. Since the shear modulus softens (partially) above Tc , we have A = A0 (T − Tc ). In three dimensions we define symmetry-adapted strains as [8] 1 1 e1 = √ (x x + yy + zz ), e2 = √ (x x − yy ), 3 2 1 e3 = √ (x x + yy − 2zz ), e4 = x y , e5 = yz , 6

e6 = x z .

(3)

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As an example, the Landau part of the elastic free energy for a cubic to tetragonal transition in terms of the symmetry-adapted strain components is given by [8, 12, 13] F(e2 , e3 ) =

A 2 B C A1 2 (e + e32 ) + (e23 − 3e2 e32 ) + (e22 + e32 )2 + e 2 2 3 4 2 1 A4 2 + (e + e52 + e62 ), 2 4

(4)

where A1 , A and A4 are bulk, deviatoric and shear modulus, respectively, B and C denote third and fourth order elastic constants and (e2 , e3 ) are the order parameter deviatoric strain components. The non-order parameter dilatation (e1 ) and shear (e4 , e5 , e6 ) strains are included to harmonic order. For studying domain walls (i.e., twinning) and microstructure this free energy must be augmented [12] by symmetry allowed gradients of (e2 , e3 ). The plot of the free energy in Eq. (4) without the non-order parameter strain contributions (i.e., compression and shear terms) is identical to the two-dimensional case, Eq. (2), except that the three minima in this case correspond to the three tetragonal variants. The coefficients in the GLFE are determined from a combination of experimental structural (lattice parameter variation as a function of temperature or pressure), vibrational (e.g., phonon dispersion curves along different high symmetry directions) and thermodynamic data (entropy, specific heat, elastic constants, etc.). Where sufficient experimental data is not available, electronic structure calculations and molecular dynamics simulations (using appropriate atomistic potentials) can provide the relevant information to determine some or all of the coefficients in the GLFE. For simple phase transitions (e.g., two-dimensional square to rectangle [8, 9] or those involving only one component order parameter [14]) the GLFE can be written down by inspection (from the symmetry of the parent phase). However, in general the GLFE must be determined by group theoretic means which are now readily available for all 230 crystallographic space groups in three dimensions and (by projection) for all 17 space groups in two dimensions [14] (see the computer program ISOTROPY by Stokes and Hatch [15]).

2.

Microstructure

There are several different but related ways of modeling the microstructure in structural phase transformations: (i) GLFE based as described above [8], (ii) phase-field model in which strain variables are coupled in a symmetry allowed manner to the morphological variables [6], (iii) sharp interface models used by applied mathematicians [16, 17].

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The natural order parameters in the GLFE are strain tensor components. However, until recent years researchers have simulated the microstructure in displacement variables by rewriting the free energy in displacement variables [10, 13]. This procedure leads to the microstructure without providing direct physical insight into the evolution. A natural way to bring out the insight is to work in strain variables only. However, if the lattice integrity is maintained during the phase transformation, that is no dislocation (or topological defect) generation is allowed, then one must obey the St. Venant elastic compatibility constraints because various strain tensor components are derived from the displacement field and are not all independent. This can be achieved by minimizing the free energy with compatibility constraints treated with Lagrangian multipliers [9, 11]. This procedure leads to an anisotropic long-range interaction between the order parameter strain components. The interaction (or compatibility potential) provides direct insight into the domain wall orientations and various aspects of the microstructure in general. Mathematically, the elastic compatibility condition on the “geometrically linear” strain tensor is given by [18]: × (∇ × ) = 0. ∇ (5) which is one equation in two dimensions connecting the three components of the symmetric strain tensor: x x,yy + yy,x x = 2x y,x y . In three dimensions it is two sets of three equations each connecting the six components of the symmetric strain tensor ( yy,zz + zz,yy = 2 yz,yz and two permutations of x, y, z; x x,yz + yz,x x = x y,x z + x z,x y and two permutations of x, y, z). For periodic boundary conditions in Fourier space it becomes an algebraic equation which is then easy to incorporate as a constraint. For the free energy in Eq. (2), the Euler–Lagrange variation of [F−G] with respect to the non-O P strain, e1 is then [11, 14] δ(F c −G)/δe1 = 0, where G denotes the constraint equation, Eq. (5), is a Lagrange multiplier and F c = ( A1 /2)e12 is identically equal to k F c (k). The variation gives (in k space assuming periodic boundary conditions) (k x2 + k 2y )(k) . (6) A1 We then put e1 (k) back into the compatibility constraint condition, Eq. (5), and solve for the Lagrange multiplier (k). Thus e1 (k) is expressed in terms of e2 (k), e3 (k) and e1 (k) =

A1 F (k) = 2 c

2 (k 2 − k 2 )e2 2k x2 k 2y e3 x y + , k2 k2

(7)

(k)e (k) with l = 2, 3, which is used in a identically equal to (1/2) A1 U (k)e (static) free energy variation of the order parameter strains. The (static) “comˆ is independent of |k| and therefore only orientationally patibility kernel” U (k)

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→ U (k). ˆ In coordinate space this is an anisotropic long-range dependent: U (k) 2 (∼ 1/r ) potential mediating the elastic interactions of the primary order parameter strain. From these compatibility kernels one can obtain domain wall orientations, parent product interface (i.e., “habit plane”) orientations and local rotations [14] consistent with those obtained previously using macroscopic matching conditions and symmetry considerations [19, 20]. The concept of elastic compatibility in a single crystal can be readily generalized to polycrystals by defining the strain tensor components in a global frame of reference [21]. By adding a stress term (bilinear in strain) to the free energy one can compute the stress–strain constitutive response in the presence of microstructure for both single and polycrystals and compare the recoverable strain upon cycling. The grain rotation and grain boundaries play an important role when polycrystals are subject to external stress in the presence of a structural transition. Similarly, the calculation of the constitutive response can be generalized to improper ferroelastic materials such as those driven by shuffle modes, ferroelectrics and magnetoelastics.

3.

Dynamics and Simulations

The overdamped (or relaxational) dynamics can be used in simulations to obtain equilibrium microstructure e˙ = −1/ A δ(F + F c )/δe, where A is a friction coefficient and F c is the long-range contribution to the free energy due to elastic compatibility. However, if the evolution of an initial non-equilibrium structure to the equilibrium state is important, one can use inertial strain dynamics with appropriate dissipation terms included in the free energy. The strain dynamics for the order parameter strain tensor components εl is given by [11]

c2 2 δ(F + F c ) δ(R + R c ) + , ρ0 ¨l = l ∇ 4 δl δ ˙l

(8)

where ρ0 is a scaled mass density, cl is a symmetry-specific constant, R = ( A /2)˙εl2 is Rayleigh dissipation and R c is contribution to the dissipation due to the long-range elastic interaction. We replace the compressional free energy in Eq. (2) with the corresponding long-range elastic energy in the order parameter strains and include a gradient term FG = (K /2)[(∇e2 )2 + (∇e3 )2 ], where the gradient coefficient K determines the elastic domain wall energy and can be estimated from the phonon dispersion curves. Simulations performed with the full underdamped dynamics for the triangle to centered rectangular transition are depicted in Fig. 1. The equilibrium microstructure is essentially the same as that found from the overdamped dynamics. The three shades of gray represent the three rectangular variants (or orientations) in the martensite phase. A similar microstrucure has

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Figure 1. A simulated microstructure below the transition temperature for the triangle to rectangle transition. The three shades of gray represent the three rectangular variants.

been observed in lead orthovanadate Pb3 (VO4 )2 crystals [22]. This has also been simulated in the overdamped limit by phase–field [23] and displacement based simulations of Ginzburg–Landau models [10]. The 3D cubic to tetragonal transition (free energy in Eq. (4)) can be simulated either using the strain based formalism outlined here [12] or directly using the displacements [13]. In Fig. 2 we depict microstructure evolution for the cubic to tetragonal transition in FePd mimicked by a square to rectangle transition. To simulate mechanical loading of a polycrystal [21], an external tensile stress σ is applied quasi-statically, i.e., starting from the unstressed configuration of left panel (a), the applied stress σ is increased in steps of 5.13 MPa, after allowing the configurations relax for t ∗ = 25 time steps after each increment. The loading is continued till a maximum stress of σ = 200 MPa is reached in panel (e). Thereafter, the system is unloaded by

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Figure 2. Comparison of the constitutive response for a single crystal and a polycrystal for FePd parameters. The four right panels show the single crystal microstructure and the four left panels depict the polycrystal microstructure.

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decreasing σ to zero at the same rate at which it was loaded; see panel (g). Panel (c) relates to a stress level of σ = 46.15 MPa during the loading process. The favored (rectangular) variants have started to grow at the expense of the unfavored (differently oriented rectangular) variants. The orientation distribution does not change much. As the stress level is increased further, the favored variants grow. Even at the maximum stress of 200 MPa, some unfavored variants persist, as is clear from panel (e). We note that the grains with large misorientation with the loading direction rotate. Grains with lower misorientation do not undergo significant rotation. The mechanism of this rotation is the tendency of the system to maximize the transformation strain in the direction of loading so that the total free energy is minimized [21]. Within the grains that rotate, sub-grain bands are present which correspond to the unfavored strain variants that still survive. Panel (g) depicts the situation after unloading to σ = 0. Upon removing the load, a domain structure is nucleated again due to the local strains at the grain boundaries and the surviving unfavored variants in the loaded polycrystal configuration in panel (e). This domain structure is not the same as that prior to loading, see panel (a), and thus there is an underlying hysteresis. The unloaded configuration has non-zero average strain. This average strain is recovered by heating to the austenite phase, as per the shape memory effect [2]. Note also that the orientation distribution reverts to its preloading state as the grains rotate back when the load is removed. We compare the above mechanical behavior of the polycrystal to the corresponding single crystal. The recoverable strain for the polycrystal is smaller than that for the single crystal due to nucleation of domains at grain boundaries upon unloading. In addition, the transformation in the stress–strain curve for the polycrystal is not abrupt because the response of the polycrystal is averaged over all grain orientations.

4.

Comparison with Other Methods

We compare our approach that is based on the work of Barsch and Krumhansl [8] with two other methods that make use of Landau theory to model structural transformations. Here we provide a brief outline of the differences, the methods are compared and reviewed in detail in Ref. [24]. Khachaturyan and coworkers [6, 23, 25] have used a free energy in which a “structural” or “morphological” order parameter, η, is coupled to strains. This order parameter is akin to a “shuffle” order parameter [26] and the inhomogeneous strain contribution is evaluated using the method of Eshelby [6]. The strains are then effectively removed in favor of the η’s and the minimization is carried out for these variables. This approach (sometimes referred to as

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“phase-field”) applied to improper ferroelastics is essentially the same as our approach with minor differences in the way the inhomogeneous strain contribution is evaluated. However, for the proper ferroelastics that are driven by strain, rather than shuffle, essentially the same procedure is used with phasefield, that is, the minimization (through relaxation methods) is ultimately for the η’s, rather than the strains. In our approach, the non-linear free energy is written up front in terms of the relevant strain order parameters with the discrete symmetry of the transformation taken into account. Here terms that are gradients in strains, which provide the costs of creating domain walls, are also added according to the symmetries. The free energy is then minimized with respect to the strains. That the microstructure for proper ferroelastics obtained from either method would appear qualitatively similar is not surprising. Although the free energy minima or equilibrium states are the same from either procedure, differences in the details of the free energy landscape would be expected to exist. These could affect, for example, the microstructure associated with metastable states. Our method and that developed by the Applied Mechanics community [16, 17] share the common feature of minimizing a free energy written in terms of strains. The method is ideally suited for laminate microstructures with domain walls that are atomistically sharp. This sharp interface limit means that incoherent strains are incorporated through the use of the Hadamard jump condition [16, 17]. The method takes into account finite deformation and has served as an optimization procedure for obtaining static, equilibrium structures, given certain volume fractions of variants. Our approach differs in that we use a continuum formulation with interfaces that have finite width and therefore the incoherent strains are taken into account through the compatibility relation [9, 11]. In addition, we solve the full evolution equations so that we can study kinetics and the effects of inertia.

5.

Ferroic Transitions

Above we considered proper ferroelastic transitions. This method can be readily extended (including the Ginzburg–Landau free energy and elastic compatibility) to the study of improper ferroelastics (e.g., shuffle driven transitions such as in NiTi [26]), proper ferroelectrics such as BaTiO3 [27–29], improper ferroelectrics such as SrTiO3 [30] and magnetoelastics and magnetic shape memory alloys, e.g., Ni2 GaMn [31], by including symmetry allowed coupling between the shuffle modes (or polarization or magnetization) with the appropriate strain tensor components. However, now the elastic energy is considered only up to the harmonic order whereas the primary order parameter has anharmonic contributions. For example for a two-dimensional

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ferroelectric transition on a square lattice the Ginzburg–Landau free energy is given by [25, 32]: = α1 (Px2 + Py2 ) + α11 (Px4 + Py4 ) + α12 Px2 Py2 + α111 (Px6 + Py6 ) F( P) g1 2 g2 2 2 2 + Py,y ) + (Px,y + Py,x ) + α112 (Px2 Py4 + Px4 Py2 ) + (Px,x 2 2 1 1 1 + g3 Px,x Py,y + A1 e12 + A2 e22 + e32 + β1 e1 (Px2 + Py2 ) 2 2 2 2 2 + β2 e2 (Px − Py ) + β3 e3 Px Py , where Px and Py are the polarization components. The free energy for a twodimensional magnetoelastic transition is very similar with magnetization (m x , m y ) replacing the polarization (Px , Py ). For specific physical geometries the long-range electric (or magnetic) dipole interaction must be included. Certainly ferroelectric (and magnetoelastic) transitions can be modeled by phasefield [33] and other methods [34]. We have presented a strain-based formalism for the study of domain walls and microstructure in ferroic materials within a Ginzburg–Landau free energy framework with elastic compatibility constraint explicitly taken into account. The latter induces an anisotropic long-range interaction in the primary order parameter (strain in proper ferroelastics such as martensites and shape memory alloys [9, 11] or shuffle, polarization or magnetization in improper ferroelastics [28, 32]). We compared this method with the widely used phase-field method [6, 23, 25] and the formalism used by applied mathematics and mechanics community [16, 17, 34]. We also discussed the underdamped strain dynamics for the evolution of microstructure and compared the constitutive response of a single crystal with that of a polycrystal. Finally, we briefly mention four other related topics that can be modeled within the Ginzburg–Landau formalism. (i) Some martensites show strain modulation (or tweed precursors) above the martensitic phase transition. These are believed to be caused by disorder such as compositional fluctuations. They can be modeled and simulated by including symmetry allowed coupling of strain to compositional fluctuations in the free energy [9, 35, 36]. Similarly, symmetry allowed couplings of polarization (magnetization) with polar (magnetic) disorder can lead to polar [37] (magnetic [38]) tweed precursors. (ii) Some martensites exhibit supermodulated phases [39] (e.g., 5R, 7R, 9R) which can be modeled within the Landau theory in terms of a particular phonon softening [40] (and its harmonics) and coupling to the transformation shear. (iii) Elasticity at nanoscale can be different from macroscopic continuum elasticity. In this case one must go beyond the usual elastic tensor components and include intra-unit cell modes [41]. (iv) The results presented here are relevant for displacive transformations, i.e., when the parent and product crystal

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structures have a group-subgroup symmetry relationship. However, reconstructive transformations [3], e.g., BCC to HCP transitions, do not have a group– subgroup relationship. Nevertheless, the Ginzburg–Landau formalism can be generalized to these transformations [42]. Notions of a transcendental order parameter [3] and irreversibility [43] have also been invoked to model the reconstructive transformations.

Acknowledgments We acknowledge collaboration with R. Ahluwalia, K.H. Ahn, R.C. Albers, A.R. Bishop, T. Cast´an, D.M. Hatch, A. Planes, K.Ø. Rasmussen and S.R. Shenoy. This work was supported by the US Department of Energy.

References [1] Z. Nishiyama, Martensitic Transformations, Academic, New York, 1978. [2] K. Otsuka and C.M. Wayman (eds.), Shape Memory Materials, Cambridge University Press, Cambridge, 1998; MRS Bull., 27, 2002. [3] P. Tol´edano and V. Dimitriev, Reconstructive Phase Transitions, World Scientific, Singapore, 1996. [4] V.K. Wadhawan, Introduction to Ferroic Materials, Gordon and Breach, Amsterdam, 2000. [5] E.K.H. Salje, Phase Transformations in Ferroelastic and Co-elastic Solids, Cambridge University Press, Cambridge, UK, 1990. [6] A.G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, 1983. [7] J.C. Tol´edano and P. Tol´edano, The Landau Theory of Phase Transitions, World Scientific, Singapore, 1987. [8] G.R. Barsch and J.A. Krumhansl, Phys. Rev. Lett., 53, 1069, 1984; G.R. Barsch and J.A. Krumhansl, Metallurg. Trans., A18, 761, 1988. [9] S.R. Shenoy, T. Lookman, A. Saxena, and A.R. Bishop, Phys. Rev. B, 60, R12537, 1999. [10] S.H. Curnoe and A.E. Jacobs, Phys. Rev. B, 63, 094110, 2001. [11] T. Lookman, S.R. Shenoy, K. Ø. Rasmussen, A. Saxena, and A.R. Bishop, Phys. Rev. B, 67, 024114, 2003. [12] K. Ø. Rasmussen, T. Lookman, A. Saxena, A.R. Bishop, R.C. Albers, and S.R. Shenoy, Phys. Rev. Lett., 87, 055704, 2001. [13] A.E. Jacobs, S.H. Curnoe, and R.C. Desai, Phys. Rev. B, 68, 224104, 2003. [14] D.M. Hatch, T. Lookman, A. Saxena, and S.R. Shenoy, Phys. Rev. B, 68, 104105, 2003. [15] H.T. Stokes and D.M. Hatch, Isotropy Subgroups of the 230 Crystallographic Space Groups, World Scientific, Singapore, 1988. (The software package ISOTROPY is available at http://www.physics.byu.edu/∼ stokesh/isotropy.html, ISOTROPY (1991)). [16] J.M. Ball and R.D. James, Arch. Rational Mech. Anal., 100, 13, 1987. [17] R.D. James and K.F. Hane, Acta Mater., 48, 197, 2000.

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[18] S.F. Borg, Fundamentals of Engineering Elasticity, World Scientific, Singapore, 1990; M. Baus and R. Lovett, Phys. Rev. Lett., 65, 1781, 1990; M. Baus and R. Lovett, Phys. Rev. A, 44, 1211, 1991. [19] J. Sapriel, Phys. Rev. B, 12, 5128, 1975. [20] C. Boulesteix, B. Yangui, M. Ben Salem, C. Manolikas, and S. Amelinckx, J. Phys., 47, 461, 1986. [21] R. Ahluwalia, T. Lookman, and A. Saxena, Phys. Rev. Lett., 91, 055501, 2003; R. Ahluwalia, T. Lookman, A. Saxena, and R.C. Albers, Acta Mater., 52, 209, 2004. [22] C. Manolikas and S. Amelinckx, Phys. Stat. Sol., (a) 60, 607, 1980; C. Manolikas and S. Amelinckx, Phys. Stat. Sol., 61, 179, 1980. [23] Y.H. Wen, Y.Z. Wang, and L.Q. Chen, Philos. Mag. A, 80, 1967, 2000. [24] T. Lookman, S.R. Shenoy, and A. Saxena, to be published. [25] H.L. Hu and L.Q. Chen, Mater. Sci. Eng., A238, 182, 1997. [26] G.R. Barsch, Mater. Sci. Forum, 327–328, 367, 2000. [27] W. Cao and L.E. Cross, Phys. Rev. B, 44, 5, 1991. [28] S. Nambu and D.A. Sagala, Phys. Rev. B, 50, 5838, 1994. [29] A.J . Bell, J. Appl. Phys., 89, 3907, 2001. [30] W. Cao and G.R. Barsch, Phys. Rev. B, 41, 4334, 1990. [31] A.N. Vasil’ev, A.D. Dozhko, V.V. Khovailo, I.E. Dikshtein, V.G. Shavrov, V.D. Buchelnikov, M. Matsumoto, S. Suzuki, T. Takagi, and J. Tani, Phys. Rev. B, 59, 1113, 1999. [32] R. Ahluwalia and W. Cao, Phys. Rev. B, 63, 012103, 2001. [33] Y.L. Li, S.Y. Hu, Z.K. Liu, and L.Q. Chen, Appl. Phys. Lett., 78, 3878, 2001. [34] Y.C. Shu and K. Bhattacharya, Phil. Mag. B, 81, 2021, 2001. [35] S. Kartha, J.A. Krumhansl, J.P. Sethna, and L.K. Wickham, Phys. Rev. B, 52, 803, 1995. [36] T. Cast´an, A. Planes, and A. Saxena, Phys. Rev. B, 67, 134113, 2003. [37] O. Tikhomirov, H. Jiang, and J. Levy, Phys. Rev. Lett., 89, 147601, 2002. [38] Y. Murakami, D. Shindo, K. Oikawa, R. Kainuma, and K. Ishida, Acta Mater., 50, 2173, 2002. [39] K. Otsuka, T. Ohba, M. Tokonami, and C.M. Wayman, Scr. Matallurg. Mater., 19, 1359, 1993. [40] R.J. Gooding and J.A. Krumhansl, Phys. Rev. B, 38, 1695, 1988; R.J. Gooding and J.A. Krumhansl, Phys. Rev. B, 39, 1535, 1989. [41] K.H. Ahn, T. Lookman, A. Saxena, and A.R. Bishop, Phys. Rev. B, 68, 092101, 2003. [42] D.M. Hatch, T. Lookman, A. Saxena, and H.T. Stokes, Phys. Rev. B, 64, 060104, 2001. [43] K. Bhattacharya, S. Conti, G. Zanzotto, and J. Zimmer, Nature, 428, 55, 2004.

7.6 PHASE-FIELD MODELING OF GRAIN GROWTH Carl E. Krill III Materials Division, University of Ulm, Albert-Einstein-Allee 47, D–89081 Ulm, Germany

When a polycrystalline material is held at elevated temperature, the boundaries between individual crystallites, or grains, can migrate, thus permitting some grains to grow at the expense of others. Planar sections taken through such a specimen reveal that the net result of this phenomenon of grain growth is a steady increase in the average grain size and, in many cases, the evolution toward a grain size distribution manifesting a characteristic shape independent of the state prior to annealing. Recognizing the tremendous importance of microstructure to the properties of polycrystalline samples, materials scientists have long struggled to develop a fundamental understanding of the microstructural evolution that occurs during materials processing. In general, this is an extraordinarily difficult task, given the structural variety of the various elements of microstructure, the topological complexities associated with their spatial arrangement and the range of length scales that they span. Even for single-phase samples containing no other defects besides grain boundaries, experimental and theoretical efforts have met with surprisingly limited success, with observations deviating significantly from the predictions of the best analytic models. Consequently, researchers are turning increasingly to computational methods for modeling microstructural evolution. Perhaps the most impressive evidence for the power of the computational approach is found in its application to single-phase grain growth, for which several successful simulation algorithms have been developed, including Monte Carlo Potts and cellular automata models (both discussed elsewhere in this chapter), and phase-field, front-tracking and vertex approaches. In particular, the phase-field models have proven to be especially versatile, lending themselves to the simulation of growth occurring not only in single-phase systems, but also in the presence of multiple phases or gradients of concentration, strain or temperature. It is no exaggeration to claim that these simulation techniques 2157 S. Yip (ed.), Handbook of Materials Modeling, 2157–2171. c 2005 Springer. Printed in the Netherlands.

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have revolutionized the study of grain growth, offering heretofore unavailable insight into the statistical properties of polycrystalline grain ensembles and the detailed nature of the microstructural evolution induced by grain boundary migration.

1.

Fundamentals of Grain Growth

From a thermodynamic standpoint, grain growth occurs in a polycrystalline sample because the network of grain boundaries is a source of excess energy with respect to the single-crystalline state. The interfacial excess free energy G int can be written as the product of the total grain boundary area AGB and the average excess energy per unit boundary area, γ: G tot = G bulk + G int = G X (T, P) + AGB γ (T, P),

(1)

where G X (T, P, . . .) denotes the free energy of the single-crystalline grain interiors at temperature T and pressure P. Because the specific grain boundary energy γ is a positive quantity, there is a thermodynamic driving force to reduce AGB or, owing to the inverse relationship between AGB and the average grain size R, to increase R. Consequently, grain boundaries tend to migrate such that smaller grains are eliminated in favor of larger ones, resulting in steady growth of the average grain size. The kinetics of this process of grain growth follow one of two qualitatively different pathways [1]: during so-called normal grain growth, the grain size distribution f (R, t) maintains a unimodal shape, shifting to larger R with increasing time t. In abnormal grain growth, on the other hand, only a subpopulation of grains in the sample coarsens, leading to the development of a bimodal size distribution. Although abnormal grain growth is far from rare, the factors responsible for its occurrence are poorly understood at best, depending strongly on properties specific to the sample in question [2]. In contrast, normal grain growth obeys two laws of apparently universal character: power-law evolution of the average grain size and the establishment of a quasistationary scaled grain size distribution [1, 3]. The first entails a relationship of the form Rm (t) − Rm (t0 ) = k (t − t0 ),

(2)

where k is a rate constant (with a strong dependence on temperature), and m denotes the growth exponent [Fig. 1(a)]. Experimentally, m is found to take on a value between 2 and 4, tending toward the lower end of this scale in materials of the highest purity annealed at temperatures near the melting point [2]. The second feature of normal grain growth encompasses the fact that, with increasing annealing time, f (R, t) evolves asymptotically toward a

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Figure 1. Normal grain growth in polycrystalline Fe. [Data obtained from Ref. [30].] (a) Plot of the average grain size as a function of time in samples annealed at the indicated temperatures. Dashed lines are fits of Eq. (2) with m = 2 (fit function modified slightly to take ‘size effect’ into account). (b) Self-similar evolution of the grain size distribution in the sample annealed at 800 ◦ C for the indicated times. Solid line is a least-squares fit of a lognormal function to the scaled distributions. Dashed line is the prediction of Hillert’s analytic model for grain growth in 3D.

time-invariant shape when plotted as a function of the normalized grain size R/R [Fig. 1(b)]; that is, f (R, t) −→ f˜(R/R),

(3)

with the quasistationary distribution f˜(R/R) generally taking on a lognormal shape [4]. Analytical efforts to explain the origin of Eqs. (2) and (3) generally begin with the assumption that the migration rate v GB of a given grain boundary is proportional to its local curvature, with the proportionality factor defining the grain boundary mobility M [5]. Hillert [6] derived a simple expression for the resulting growth kinetics of a single grain embedded in a polycrystalline matrix. Solving the Hillert model self-consistently for the entire ensemble of grains leads directly to a power-law growth equation with m = 2 and to selfscaling behavior of f (R, t), but the shape predicted for f˜(R/R)–plotted in Fig. 1(b)–has never been confirmed experimentally. This failure is typical of all analytic growth models, which, owing to their statistical mean-field nature, do not properly account for the influence of the grain boundary network’s local topology on the migration of individual boundaries. Computer simulations are able to circumvent this limitation, either by calculating values for v GB from instantaneous local boundary curvatures (cellular automata, vertex, front-tracking methods) or by determining the excess free energy stored in the grain boundary network and then allowing this energy to relax in a physically plausible manner (Monte Carlo, phase-field approaches) [7, 8].

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Phase-field Representation of Polycrystalline Microstructure

The phase-field model for simulating grain growth takes its cue from Eq. (1), expressing the total free energy Ftot as the sum of contributions arising from the grain interiors, Fbulk , and the grain boundary (interface) regions, Fint [9]: Ftot = Fbulk + Fint =

f bulk({φi }) + f int ({φi }, {∇φi }) dr.

(4)

Both Fbulk and Fint are specified as functionals of a set of phase fields {φi (r, t)} (also called order parameters), which are continuous functions defined for all times t at all points r in the simulation cell. The energy density f bulk describes the free energy per unit volume of the grain interior regions, whereas f int accounts for the free energy contributed by the grain boundaries. As discussed below, grain boundaries in the phase-field model have a finite (i.e., non-zero) thickness; therefore, the interfacial energy density f int –like f bulk–is an energy per unit volume and must be integrated over the entire volume of the simulation cell to recover the total interfacial energy. The function f bulk({φi }) can be constructed such that each of the phase fields φi takes on one of two constant values–such as zero or unity–in the interior region of each crystallite [9]. Only when a boundary between two crystallites is crossed do one or, generally, more order parameters change continuously from value to the other; consequently, grain boundaries are locations of large gradients in one or more φi , suggesting that the grain boundary energy term f int should be defined as a function of {∇φi }. The specific functional forms chosen for f bulk and f int, however, depend on considerations of computational efficiency, the physics underlying the growth model and, to a certain extent, personal taste. Over the past several years, two general approaches have emerged in the literature for simulating grain growth by means of Eq. (4).

2.1.

Discrete-orientation Models

In the discrete-orientation approach [10, 11], each order parameter φi is viewed as a continuous-valued component of a vector φ (r, t) = φ (r, t), φ2 1 (r, t), . . . , φ Q (r, t) specifying the local crystalline orientation throughout the simulation cell. Stipulating that the phase fields φi take on constant values of 0 or 1 within the interior of a grain, this model clearly allows at most 2 Q distinct grain orientations, with Q denoting the total number of phase fields. In the most common implementation of the discrete-orientation method, f bulk({φi }) is defined to have local minima when one and only one component of φ equals unity in a grain interior, thus reducing the total number of allowed

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orientations to Q. For example, in a simulation with Q = 4, a given grain might be represented by the contiguous set of points at which φ = (0, 0, 1, 0), and a neighboring grain by φ = (0, 1, 0, 0) [Fig. 2(a)]. As illustrated in Fig. 2(b), upon crossing from one grain to the other, φ2 changes continuously from 0 from 1 to 0; minimization of f int , which is defined to be proporto 1 and φ 3 Q tional to i=1 (∇φi )2 , leads to a smooth–rather than instantaneous–variation in the order-parameter values. The width of the resulting interfacial region is prevented from expanding without bound by the increase in f bulk that occurs when φ deviates from the orientations belonging to the set of local minima of f bulk. Thus, the mathematical representation of each grain boundary is determined by a competition between the bulk and interfacial components of Ftot – a common feature of phase-field representations of polycrystalline microstructures.

(a)

(b) (0, 0, 0, 1) 1

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Figure 2. Phase-field representations of polycrystalline microstructure. (a) Discreteorientation model: grain orientations are specified by a vector-valued phase field φ having four components in this example. (b) Smooth variation of φ2 and φ3 along the dashed arrow in (a). (c) Continuous-orientation model: grain orientations are specified by the angular order parameter θ, and local crystalline order by the value of φ. (d) Smooth variation of θ and φ along the dashed arrow in (c).

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Restricting the grains to a set of discrete orientations may simplify the task of constructing expressions for f bulk and f int in Eq. (4), but it also introduces some conceptual as well as practical limitations to the model. Clearly, it is unphysical for the free energy density of the grain interiors, f bulk, to favor specific grain orientations defined relative to a fixed reference frame, for the free energy of the bulk phase must be invariant with respect to rotation in laboratory coordinates [12]. Even more seriously, the energy barrier in f bulk that separates allowed orientations prohibits the rotation of individual grains during a simulation of grain growth. Since the rotation rate rises dramatically with decreasing grain size [13], grain rotations may be important to the growth process even when R is large, given that there is always a subpopulation of smaller grains losing volume to their growing nearest neighbors.

2.2.

Continuous-orientation Models

In an effort to avoid the undesirable consequences of a finite number of allowed grain orientations, a number of researchers have attempted to express Eq. (4) in terms of continuous, rather than discrete, grain orientations [14–16]. In two dimensions, the orientation of a given grain can be specified completely by a single continuous parameter θ representing, say, the angle between the normal to a particular set of atomic planes and a fixed direction in the laboratory reference frame [Fig. 2(c)]. In 3D, the same specification can be accomplished with three such angular fields. By choosing f bulk to be independent of the orientational order parameters, one ensures that grains are free to take on arbitrary orientations rather than only those corresponding to local minima of the bulk energy density. Because of this independence, however, there is no orientational energy penalty preventing grain boundaries from widening without bound during a growth simulation; thus, it is necessary to introduce an additional phase field that couples the width of the interfacial region to the value of f bulk. Generally, one defines an order parameter φ specifying the degree of crystallinity at each point in the simulation cell, with a value of unity signifying perfect crystalline order (such as obtains in the grain interior) and lower values (0 ≤ φ

m max pmax

(11) Except for the probabilistic evaluation of the analytically calculated transformation probabilities, the approach is entirely deterministic. Thermal fluctuations other than already included via Turnbull’s rate equation are not permitted. The use of realistic or even experimental input data for the grain boundaries enables one to make predictions on a real time and space scale. The switching rule is scalable to any mesh size and to any spectrum of boundary mobility and driving force data. The state update of all cells is made in synchrony.

3.3.

Simulation of Primary Static Recrystallization and Comparison to Avrami-type Kinetics

Figure 2 shows the kinetics and 3D microstructures of a recrystallizing aluminum single crystal. The initial deformed crystal had a uniform Goss orientation (011)[100] and a dislocation density of 1015 m−2 . The driving force was due to the stored elastic energy provided by the dislocations. In order to compare the predictions with analytical Avrami kinetics recovery

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100 recrystallized volume fraction [%]

90 80 70 60 50 40 30 20 10 0 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 time [s] Figure 2. Kinetics and microstructure of recrystallization in a plastically strained aluminum single crystal. The deformed crystal had a (011)[100] orientation and a uniform dislocation density of 1015 m−2 . Simulation parameter: site saturated nucleation, lattice size: 10 × 10 × 10 × µ m3 , cell size: 0.1 µm, activation energy of large angle grain boundary mobility: 1.3 eV, pre–exponential factor of large angle boundary mobility: m 0 = 6.2 ×10−6 m3 /(N s), temperature: 800 K, time constant: 0.35 s.

and driving forces arising from local boundary curvature were not considered. The simulation used site saturated nucleation conditions, i.e., the nuclei were at t =0 s statistically distributed in physical space and orientation space. The grid size was 10 × 10 × 10 µm3 . The cell size was 0.1 µm. All grain boundaries had the same mobility using an activation energy of the grain boundary mobility of 1.3 eV and a pre–exponential factor of the boundary mobility of m 0 = 6.2 · 10−6 m3 /(N s) [37]. Small angle grain boundaries had a mobility of zero. The temperature was 800 K. The time constant of the simulation was 0.35 s. Figure 3 shows the kinetics for a number of 3D recrystallization simulations with site saturated nucleation conditions and identical mobility for all grain boundaries. The different curves correspond to different initial numbers of nuclei. The initial number of nuclei varied between 9624 (pseudo–nucleation energy of 3.2 eV) and 165 (pseudo–nucleation energy of 6.0 eV). The curves (Fig. 3a) all show a typical Avrami shape and the logarithmic plots (Fig. 3b)

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Figure 3. Kinetics for various 3D recrystallization simulations with site saturated nucleation conditions and identical mobility for all grain boundaries. The different curves correspond to different initial numbers of nuclei. The initial number of nuclei varied between 9624 (pseudo– nucleation energy of 3.2 eV) and 165 (pseudo–nucleation energy of 6.0 eV). (a) Avrami diagrams. (b) Logarithmic diagrams showing Avrami exponents between 2.86 and 3.13.

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reveal Avrami exponents between 2.86 and 3.13 which is in very good accord with the analytical value of 3.0 for site saturated conditions. The simulations with a very high initial density of nuclei reveal a more pronounced deviation of the Avrami exponent with values around 2.7 during the beginning of recrystallization. This deviation from the analytical behavior is due to lattice effects: while the analytical derivation assumes a vanishing volume for newly formed nuclei the cellular automaton has to assign one lattice point to each new nucleus. Figure 4 shows the effect of grain boundary mobility on growth selection. While in Fig. 4a all boundaries had the same mobility, in Fig. 4b one grain boundary had a larger mobility than the others (activation energy of the mobility of 1.35 eV instead of 1.40 eV) and consequently grew much faster than the neighboring grains which finally ceased to grow. The grains in this simulation all grew into a heavily deformed single crystal. (a)

temporal evolution

deformed single crystal

growing nucleation front

(b)

temporal evolution

deformed single crystal

growing nucleation front

Figure 4. Effect of grain boundary mobility on growth selection. All grains grow into a deformed single crystal. (a) All grain boundaries have the same mobility. (b) One grain boundary has a larger mobility than the others (activation energy of the mobility of 1.35 eV instead of 1.40 eV) and grows faster than the neighboring grains.

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4.1.

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Examples of Coupling Cellular Automata with Crystal Plasticity Finite Element Models for Predicting Recrystallization Motivation for Coupling Different Spatially Discrete Microstructure and Texture Simulation Methods

Simulation approaches such as the crystal plasticity finite element method or cellular automata are increasingly gaining momentum as tools for spatial and temporal discrete prediction methods for microstructures and textures. The major advantage of such approaches is that they consider material heterogeneity as opposed to classical statistical approaches which are based on the assumption of material homogeneity. Although the average behavior of materials during deformation and heat treatment can sometimes be sufficiently well described without considering local effects, prominent examples exist where substantial progress in understanding and tailoring material response can only be attained by taking material heterogeneity into account. For instance in the field of plasticity the quantitative investigation of ridging and roping or related surface defects observed in sheet metals requires knowledge about local effects such as the grain topology or the form and location of second phases. In the field of heat treatment, the origin of the Goss texture in transformer steels, the incipient stages of cube texture formation during primary recrystallization of aluminum, the reduction of the grain size in microalloyed low carbon steel sheets, and the development of strong {111}uvw textures in steels can hardly be predicted without incorporating local effects such as the orientation and location of recrystallization nuclei and the character and properties of the grain boundaries surrounding them. Although spatially discrete microstructure simulations have already profoundly enhanced our understanding of microstructure and texture evolution over the last decade, their potential is sometimes simply limited by an insufficient knowledge about the external boundary conditions which characterize the process and an insufficient knowledge about the internal starting conditions which are, to a large extent, inherited from the preceding process steps. It is thus an important goal to improve the incorporation of both types of information into such simulations. External boundary conditions prescribed by real industrial processes are often spatially non-homogeneous. They can be investigated using experiments or process simulations which consider spatial resolution. Spatial heterogeneities in the internal starting conditions, i.e., in the microstructure and texture, can be obtained from experiments or microstructure simulations which include spatial resolution.

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Coupling, Scaling and Boundary Conditions

In the present example the results obtained from a crystal plasticity finite element simulation were used to map a starting microstructure for a subsequent discrete recrystallization simulation carried out with a probabilistic cellular automaton. The finite element model was used to simulate a plane strain compression test conducted on aluminum with columnar grain structure to a total logarithmic strain of ε = –0.434. Details about the finite element model are given elsewhere [34, 35, 38, 39]. The values of the state variables (dislocation density, crystal orientation) given at the integration points of the finite element mesh were mapped on the regular lattice of a 2D cellular automaton. While the original finite element mesh consisted of 36 977 quadrilateral elements, the cellular automaton lattice consisted of 217 600 discrete points. The values of the state variables (dislocation density, crystal orientation) at each of the integration points were assigned to the new cellular automaton lattice points which fell within the Wigner–Seitz cell corresponding to that integration point. The Wigner–Seitz cells of the finite element mesh were constructed from cell walls which were the perpendicular bisecting planes of all lines connecting neighboring integration points, i.e., the integration points were in the centers of the Wigner–Seitz cells. In the present example the original size of the specimen which provided the input microstructure to the crystal plasticity finite element simulations gave a lattice point spacing of λm = 61.9 µm. The maximum driving force in the region arising from the stored dislocation density amounted to about 1 MPa. The temperature dependence of the shear modulus and of the Burgers vector was considered in the calculation of the driving force. The grain boundary mobility in the region was characterized by an activation energy of the grain boundary mobility of 1.46 eV and a pre-exponential factor of the grain boundary mobility of m0 = 8.3 × 10−3 m3 /(N s). Together with the scaling length λm = 61.9 µm these data were used for the calculation of the time step t = 1/ν0min and of the local switching probabilities wˆ local. The annealing temperature was 800 K. Large angle grain boundaries were characterized by an activation energy for the mobility of 1.3 eV. Small angle grain boundaries were assumed to be immobile.

4.3.

Nucleation Criterion

The nucleation process during primary static recrystallization has been explained for pure aluminum in terms of discontinuous subgrain growth [40]. According to this model nucleation takes place in areas which reveal high misorientations among neighboring subgrains and a high local driving force

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for curvature driven discontinuous subgrain coarsening. The present simulation approach works above the subgrain scale, i.e., it does not explicitly describe cell walls and subgrain coarsening phenomena. Instead, it incorporates nucleation on a more phenomenological basis using the kinetic and thermodynamic instability criteria known from classical recrystallization theory (see e.g., [40]). The kinetic instability criterion means that a successful nucleation process leads to the formation of a mobile large angle grain boundary which can sweep the surrounding deformed matrix. The thermodynamic instability criterion means that the stored energy changes across the newly formed large angle grain boundary providing a net driving force pushing it forward into the deformed matter. Nucleation in this simulation is performed in accord with these two aspects, i.e., potential nucleation sites must fulfill both, the kinetic and the thermodynamic instability criterion. The used nucleation model does not create any new orientations: at the beginning of the simulation the thermodynamic criterion, i.e., the local value of the dislocation density was first checked for all lattice points. If the dislocation density was larger than some critical value of its maximum value in the sample, the cell was spontaneously recrystallized without any orientation change, i.e., a dislocation density of zero was assigned to it and the original crystal orientation was preserved. In the next step the ordinary growth algorithm was started according to Eqs. (1)–(11), i.e., the kinetic conditions for nucleation were checked by calculating the misorientations among all spontaneously recrystallized cells (preserving their original crystal orientation) and their immediate neighborhood considering the first, second, and third neighbor shell. If any such pair of cells revealed a misorientation above 15◦ , the cell flip of the unrecrystallized cell was calculated according to its actual transformation probability, Eq. (8). In case of a successful cell flip the orientation of the first recrystallized neighbor cell was assigned to the flipped cell.

4.4.

Predictions and Interpretation

Figures 5–7 show simulated microstructures for site saturated spontaneous nucleation in all cells with a dislocation density larger than 50% of the maximum value (Fig. 5), larger than 60% of the maximum value (Fig. 6), and larger than 70% of the maximum value (Fig. 7). Each figure shows a set of four subsequent microstructures during recrystallization. The upper graphs in Figs. 5–7 show the evolution of the stored dislocation densities. The gray areas are recrystallized, i.e., the stored dislocation content of the affected cells was dropped to zero. The lower graphs represent the microtexture images where each color represents a specific crystal orientation.

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(a)

(b)

(c)

(d)

Figure 5. Consecutive stages of a 2D simulation of primary staticrecrystallization in a deformed aluminum polycrystal on the basis of crystal plasticity finite element starting data. The figure shows the change in dislocation density (top) and in microtexture (bottom) as a function of the annealing time during isothermal recrystallization. The texture is given in terms of the magnitude of the Rodriguez orientation vector using the cube component as reference. The gray areas in the upper figures indicate a stored dislocation density of zero, i.e., these areas are recrystallized. The fat white lines in both types of figures indicate grain boundaries with misorientations above 15◦ irrespective of the rotation axis. The thin green lines indicate misorientations between 5◦ and 15◦ irrespective of the rotation axis. The simulation parameters are: 800 K; thermodynamic instability criterion: site-saturated spontaneous nucleation in cells with at least 50% of the maximum occurring dislocation density (threshold value); kinetic instability criterion for further growth of such spontaneous nuclei: misorientation above 15◦ ; activation energy of the grain boundary mobility: 1.46 eV; pre-exponential factor of the grain boundary mobility: m0 = 8.3 × 10−3 m3 /(N s); mesh size of the cellular automaton grid (scaling length): λm = 61.9 µm.

The color level is determined as the magnitude of the Rodriguez orientation vector using the cube component as reference. The fat white lines in both types of figures indicate grain boundaries with misorientations above 15◦ irrespective of the rotation axis. The thin green lines indicate misorientations between 5◦ and 15◦ irrespective of the rotation axis.

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Figure 6. Parameters like in Fig. 5, but site-saturated spontaneousnucleation occurred in all cells with at least 60% of the maximum occurring dislocation density.

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Figure 7. Parameters like in Fig. 5, but site-saturated spontaneousnucleation occurred in all cells with at least 70% of the maximum occurring dislocation density.

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The incipient stages of recrystallization in Fig. 5 (cells with 50% of the maximum occurring dislocation density undergo spontaneous nucleation without orientation change) reveal that nucleation is concentrated in areas with large accumulated local dislocation densities. As a consequence the nuclei form clusters of similarly oriented new grains (e.g., Fig. 5a). Less deformed areas between the bands reveal a very small density of nuclei. Logically, the subsequent stages of recrystallization (Fig. 5 b–d) reveal that the nuclei do not sweep the surrounding deformation structure freely as described by Avrami– Johnson–Mehl theory but impinge upon each other and thus compete at an early stage of recrystallization. Figure 6 (using 60% of the maximum occurring dislocation density as threshold for spontaneous nucleation) also reveals strong nucleation clusters in areas with high dislocation densities. Owing to the higher threshold value for a spontaneous cell flip nucleation outside of the deformation bands occurs vary rarely. Similar observations hold for Fig. 7 (70% threshold value). It also shows an increasing grain size as a consequence of the reduced nucleation density. The deviation from Avrami–Johnson–Mehl type growth, i.e., the early impingement of neighboring crystals is also reflected by the overall kinetics which differ from the classical sigmoidal curve which is found for homogeneous nucleation conditions. Figure 8 shows the kinetics of recrystallization

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Figure 8. Kinetics of the recrystallization simulations shown in Figs. 5–7, annealing temperature: 800 K; scaling length λm = 61.9 µm.

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(for the simulations with different threshold dislocation densities for spontaneous nucleation, Figs. 5–7). Al curves reveal a very flat shape compared to the analytical model. The high offset value for the curve with 50% critical dislocation density is due to the small threshold value for a spontaneous initial cell flip. This means that 10% of all cells undergo initial site saturated nucleation. Figure 9 shows the corresponding Cahn–Hagel diagrams. It is found that the curves increasingly flatten and drop with an increasing threshold dislocation density for spontaneous recrystallization. It is an interesting observation in all three simulation series that in most cases where spontaneous nucleation took place in areas with large local dislocation densities, the kinetic instability criterion was usually also well enough fulfilled to enable further growth of these freshly recrystallized cells. In this context one should take notice of the fact that both instability criteria were treated entirely independent in this simulation. In other words only those spontaneously recrystallized cells which subsequently found a misorientation above 15◦ to at least one non-recrystallized neighbor cell were able to expand further. This makes the essential difference between a potential nucleus and a successful nucleus. Translating this observation into the initial deformation microstructure means that in the present example high dislocation densities

interface area between recrystallized and non-recrystallized matter devided by sample volume [cellsize1]

0.020 0.018 0.016 0.014 0.012 0.010 0.008 0.006 50% max. disloc. density

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Figure 9. Simulated interface fractions between recrystallized and non-recrystallized material for the recrystallization simulations shown in Figs. 5–7, annealing temperature: 800 K; scaling length λm = 61.9 µm.

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and large local lattice curvatures typically occurred in close neighborhood or even at the same sites. Another essential observation is that the nucleation clusters are particularly concentrated in macroscopical deformation bands which were formed as diagonal instabilities through the sample thickness. Generic intrinsic nucleation inside heavily deformed grains, however, occurs rarely. Only the simulation with a very small threshold value of only 50% of the maximum dislocation density as a precondition for a spontaneous energy drop shows some successful nucleation events outside the large bands. But even then nucleation is only successful at former grain boundaries where orientation changes occur naturally. Summarizing this argument means that there might be a transition from extrinsic nucleation such as inside bands or related large scale instabilities to intrinsic nucleation inside grains or close to existing grain boundaries. It is likely that both types of nucleation deserve separate attention. As far as the strong nucleation in macroscopic bands is concerned, future consideration should be placed on issues such as the influence of external friction conditions and sample geometry on nucleation. Both aspects strongly influence through thickness shear localization effects. Another result of relevance is the partial recovery of deformed material. Figures 5d, 6d, and 7d reveal small areas where moving large angle grain boundaries did not entirely sweep the deformed material. An analysis of the state variable values at these coordinates and of the grain boundaries involved substantiates that not insufficient driving forces but insufficient misorientations between the deformed and the recrystallized areas–entailing a drop in grain boundary mobility– were responsible for this effect. This mechanisms is referred to as orientation pinning.

4.5.

Simulation of Nucleation Topology within a Single Grain

Recent efforts in simulating recrystallization phenomena on the basis of crystal plasticity finite element or electron microscopy input data are increasingly devoted to tackling the question of nucleation. In this context it must be stated clearly that mesoscale cellular automata can neither directly map the physics of a nucleation event nor develop any novel theory for nucleation at the sub-grain level. However, cellular automata can predict the topological evolution and competition among growing nuclei during the incipient stages of recrystallization. The initial nucleation criterion itself must be incorporated in a phenomenological form. This section deals with such as an approach for investigating nucleation topology. The simulation was again started using a crystal plasticity finite

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element approach. The crystal plasticity model set-up consisted in a single aluminum grain with face centered cubic crystal structure and 12 {111}110 slip systems which was embedded in a plastic continuum which had the elasticplastic properties of an aluminum polycrystal with random texture. The crystallographic orientation of the aluminum grain in the center was ϕ1 = 32◦ , φ = 85◦ , ϕ2 = 85◦ . The entire aggregate was plane strain deformed to 50% thickness reduction (given as d/d0 , where d is the actual sample thickness and d0 its initial thickness). The resulting data (dislocation density, orientation distribution) were then used as input data for the ensuing cellular automaton recrystallization simulation. The distribution of the dislocation density taken from all integration points of the finite element simulation is given in Fig. 10. Nucleation was initiated as outlined in detail in Section 4.3, i.e., each lattice point which had a dislocation density above some critical value (500 × 1013 m−2 in the present case, see Fig. 10) of the maximum value in the sample was

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Figure 10. Distribution of the simulated dislocation density in a deformed aluminum grain embedded in a plastic aluminum continuum. The simulation was performed by using a crystal plasticity finite element approach. The set-up consisted of a single aluminum grain (orientation: ϕ1 = 32◦ , φ = 85◦ , ϕ2 =85◦ in Euler angles) with face centered cubic crystal structure and 12 {111}110 slip systems which was embedded in a plastic continuum which had the elasticplastic properties of an aluminum polycrystal with random texture. The sample was plane strain deformed to 50% thickness reduction. The resulting data (dislocation density, orientation distribution) were used as input data for a cellular automaton recrystallization simulation.

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spontaneously recrystallized without orientation change. In the ensuing step the growth algorithm was started according to Eqs. (1)–(11), i.e., a nucleus could only expand further if it was surrounded by lattice points of sufficient misorientation (above 15◦ ). In order to concentrate on recrystallization in the center grain the nuclei could not expand into the surrounding continuum material. Figures 11a–c show the change in dislocation density during recrystallization (Fig. 11a: 9% of the entire sample recrystallized, 32.1 s; Fig. 11b: 19% of the entire sample recrystallized, 45.0 s; Fig. 11c: 29.4% of the entire sample recrystallized, 56.3 s). The color scale marks the dislocation density of each lattice point in units of 1013 m−2 . The white areas are recrystallized. The surrounding blue area indicates the continuum material in which the grain is embedded (and into which recrystallization was not allowed to proceed). Figures 12a–c show the topology of the evolving nuclei without coloring the as-deformed volume. All recrystallized grains are colored indicating their crystal orientation. The non-recrystallized material and the continuum surrounding the grain are colored white. Figure 13 shows the volume fractions of the growing nuclei during recrystallization as a function of annealing time (800 K). The data reveal that two groups of nuclei occur. The first class of nuclei shows some growth in the beginning but no further expansion during the later stages of the anneal. The second class of nuclei shows strong and steady growth during the entire recrystallization time. One could refer to the first group as non-relevant nuclei while the second group could be termed relevant nuclei. The reasons of such a spread in the evolution of nucleation topology after their initial formation are nucleation clustering, orientation pinning, growth selection, or driving force selection phenomena. Nucleation clustering means that areas which reveal localization of strain and misorientation produce high local nucleation rates. This entails clusters of newly formed nuclei where competing crystals impinge on each other at an early stage of recrystallization so that only some of the newly formed grains of each cluster can expand further. Orientation pinning is an effect where not insufficient driving forces but insufficient misorientations between the deformed and the recrystallized areas – entailing a drop in grain boundary mobility – are responsible for the limitation of further growth. In other words some nuclei expand during growth into areas where the local misorientation drops below 15◦ . Growth selection is a phenomenon where some grains grow significantly faster than others due to a local advantage originating from higher grain boundary mobility such as shown in Fig. 4b. Typical examples are the 40◦ 111 rotation relationship in aluminum or the 27◦ 110 rotation relationship in iron–silicon which are known to have a growth advantage (e.g., Ref. [40]). Driving force selection is a phenomenon where some grains grow significantly faster than others due to a local advantage in driving force (shear bands, microbands, heavily deformed grain).

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(b)

(c)

Figure 11. Change in dislocation density during recrystallization (800 K).The color scale indicates the dislocation density of each lattice point in units of 1013 m−2 . The white areas are recrystallized. The surrounding blue area indicates the continuum material in which the grain is embedded. (a) 9% of the entire sample recrystallized, 32.1 s; (b) 19% of the entire sample recrystallized, 45.0 s; (c) 29.4% of the entire sample recrystallized, 56.3 s.

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(a)

(b)

(c)

Figure 12. Topology of the evolving nuclei of the microstructure given inFig. 11 without coloring the as-deformed volume. All newly recrystallized grains are colored indicating their crystal orientation. The non-recrystallized material and the continuum surrounding the grain are colored white. (a) 9% of the entire sample recrystallized, 32.1 s; (b) 19% of the entire sample recrystallized, 45.0 s; (c) 29.4% of the entire sample recrystallized, 56.3 s.

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Figure 13. Volume fractions of the growing nuclei in Fig. 11 during recrystallization as a function of annealing time (800 K).

5.

Conclusions and Outlook

A review was given about the fundamentals and some applications of cellular automata in the field of microstructure research. Special attention was placed on reviewing the fundmentals of mapping rate formulations for interfaces and driving forces on cellular grids. Some applications were discussed from the field of recrystallization theory. The future of the cellular automaton method in the field of mesoscale materials science lies most likely in the discrete simulation of equilibrium and non-equilibrium phase transformation phenomena. The particular advantage of automata in this context is their versatility with respect to the constitutive ingredients, to the consideration of local effects, and to the modification of the grid structure and the interaction rules. In the field of phase transformation simulations the constitutive ingredients are the thermodynamic input data and the kinetic coefficients. Both sets of input data are increasingly available from theory and experiment rendering cellular automaton simulations more and more realistic. The second advantage, i.e., the incorporation of local effects will improve our insight into cluster effects, such as arising from the spatial competition of expanding neighboring spheres already in the incipient stages of transformations. The third advantage, i.e., the flexibility of automata with respect to the grid structure and the interaction rules is probably the most

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important aspect for novel future applications. By introducing more global interaction rules (in addition to the local rules) and long-range or even statistical elements in addition to the local rules for the state update might establish cellular automata as a class of methods to solve some of the intricate scale problems that are often encountered in the materials sciences. It is conceivable that for certain mesoscale problems such as the simulation of transformation phenomena in heterogeneneous materials in dimensions far beyond the grain scale cellular automata can occupy a role between the discrete atomistic approaches and statistical Avrami-type approaches. The mayor drawback of the cellular automaton method in the field of transformation simulations is the absence of solid approaches for the treatment of nucleation phenomena. Although basic assumptions about nucelation sites, nucleation rates, and nucelation textures can often be included on an empirical basis as a function of the local values of the state variables, intrinsic physically based phenomenological concepts such as available to a certain extent in the Ginzburg–Landau framework (in case of the spinodal mechanism) are not yet available for automata. It might hence be beneficial in future work to combine Ginzburg–Landau-type phase field approaches with the cellular automaton method. For instance the (spinodal) nucleation phase could then be treated with a phase field method and the resulting microstructure could be further treated with a cellular automaton simulation.

References [1] J. von Neumann, “The general and logical theory of automata,” In: W. Aspray and A. Burks (eds.), Papers of John von Neumann on Computing and Computer Theory, vol. 12 in the Charles Babbage Institute Reprint Series for the History of Computing, MIT Press, Cambridge, 1987, 1963. [2] S. Wolfram, Theory and Applications of Cellular Automata, Advanced Series on Complex Systems, selected papers 1983–1986, vol. 1, World Scientific Publishing Co. Pte. Ltd, Singapore, 1986. [3] S. Wolfram, “Statistical mechanics of cellular automata,” Rev. Mod. Phys., 55, 601– 622, 1983. [4] M. Minsky, Computation: Finite and Infinite Machines, Prentice-Hall, Englewood Cliffs, NJ, 1967. [5] J.H. Conway, Regular Algebra and Finite Machines, Chapman & Hall, London, 1971. [6] D. Raabe, Computational Materials Science, Wiley-VCH, Weinheim, 1998. [7] H.W. Hesselbarth and I.R. G¨obel, “Simulation of recrystallization by cellular automata,” Acta Metall., 39, 2135–2144, 1991. [8] C.E. Pezzee and D.C. Dunand, “The impingement effect of an inert, immobile second phase on the recrystallization of a matrix,” Acta Metall., 42, 1509–1522, 1994. [9] R.K. Sheldon and D.C. Dunand, “Computer modeling of particle pushing and clustering during matrix crystallization,” Acta Mater., 44, 4571–4582, 1996. [10] C.H.J. Davies, “The effect of neighbourhood on the kinetics of a cellular automaton recrystallisation model,” Scripta Metall. et Mater., 33, 1139–1154, 1995.

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[11] V. Marx, D. Raabe, and G. Gottstein, “Simulation of the influence of recovery on the texture development in cold rolled BCC-alloys during annealing,” In: N. Hansen, D. Juul Jensen, Y.L. Liu, and B. Ralph (eds.), Proceedings 16th RISøInt. Sympos. on Mat. Science: Materials: Microstructural and Crystallographic Aspects of Recrystallization, RISø Nat. Lab, Roskilde, Denmark, pp. 461–466, 1995. [12] D. Raabe, “Cellular automata in materials science with particular reference to recrystallization simulation,” Ann. Rev. Mater. Res., 32, 53–76, 2002. [13] V. Marx, D. Raabe, O. Engler, and G. Gottstein, “Simulation of the texture evolution during annealing of cold rolled bcc and fcc metals using a cellular automaton approach,” Textures Microstruct., 28, 211–218, 1997. [14] V. Marx, F.R. Reher, and G. Gottstein, “Stimulation of primary recrystallization using a modified three-dimensional cellular automaton,” Acta Mater., 47, 1219–1230, 1998. [15] C.H.J. Davies, “Growth of nuclei in a cellular automaton simulation of recrystallisation,” Scripta Mater., 36, 35–46, 1997. [16] C.H.J. Davies and L. Hong, “Cellular automaton simulation of static recrystallization in cold-rolled AA1050,” Scripta Mater., 40, 1145–1152, 1999. [17] D. Raabe, “Introduction of a scaleable 3D cellular automaton with a probabilistic switching rule for the discrete mesoscale simulation of recrystallization phenomena,” Philos. Mag. A, 79, 2339–2358, 1999. [18] D. Raabe and R. Becker, “Coupling of a crystal plasticity finite element model with a probabilistic cellular automaton for simulating primary static recrystallization in aluminum,” Modell. Simul. Mater. Sci. Eng., 8, 445–462, 2000. [19] D.Raabe, “Yield surface simulation for partially recrystallized aluminum polycrystals on the basis of spatially discrete data,” Comput. Mater. Sci., 19, 13–26, 2000. [20] D. Raabe, F. Roters, and V. Marx, “Experimental investigation and numerical simulation of the correlation of recovery and texture in bcc metals and alloys,” Textures Microstruct., 26–27, 611–635, 1996. [21] M.B. Cortie, “Simulation of metal solidification using a cellular automaton,” Metall. Trans. B, 24, 1045–1052, 1993. [22] S.G.R. Brown, T. Williams, and JA. Spittle, “A cellular automaton model of the steady-state free growth of a non-isothermal dendrite,” Acta Metall., 42, 2893–2906, 1994. [23] C.A. Gandin and M. Rappaz, “A 3D cellular automaton algorithm for the prediction of dendritic grain growth,” Acta Metall., 45, 2187–2198, 1997. [24] C.A. Gandin, “Stochastic modeling of dendritic grain structures,” Adv. Eng. Mater., 3, 303–306, 2001. [25] C.A. Gandin, J.L. Desbiolles, and P.A. Thevoz, “Three-dimensional cellular automaton-finite element model for the prediction of solidification grain structures,” Metall. Mater. Trans. A, 30, 3153–3172, 1999. [26] J.A. Spittle and S.G.R. Brown, “A cellular automaton model of steady-state columnardendritic growth in binary alloys,” J. Mater. Sci., 30, 3989–3402, 1995. [27] S.G.R. Brown, G.P. Clarke, and A.J. Brooks, “Morphological variations produced by cellular automaton model of non-isothermal free dendritic growth,” Mater. Sci. Technol., 11, 370–382, 1995. [28] J.A. Spittle and S.G.R. Brown, “A 3D cellular automation model of coupled growth in two component systems,” Acta Metallurgica, 42, 1811–1820, 1994. [29] M. Kumar, R. Sasikumar, P. Nair, and R. Kesavan, “Competition between nucleation and early growth of ferrite from austenite-studies using cellular automaton simulations,” Acta Mater., 46, 6291–6304, 1998.

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[30] S.G.R. Brown, “Simulation of diffusional composite growth using the cellular automaton finite difference (CAFD) method,” J. Mater. Sci., 33, 4769–4782, 1998. [31] T. Yanagita, “Three-dimensional cellular automaton model of segregation of granular materials in a rotating cylinder,” Phys. Rev. Lett., 3488–3492, 1999. [32] E.M. Koltsova, I.S. Nenaglyadkin, A.Y. Kolosov, and V.A. Dovi, “Cellular automaton for description of crystal growth from the supersaturated unperturbed and agitated solutions,” Rus. J. Phys. Chem., 74, 85–91, 2000. [33] J. Geiger, A. Roosz, and P. Barkoczy, “Simulation of grain coarsening in two dimensions by cellular-automaton,” Acta Mater., 49, 623–629, 2001. [34] Y. Liu, T. Baudin, and R. Penelle, “Simulation of grain growth by cellular automata,” Scripta Mater., 34, 1679–1686, 1996. [35] T. Karapiperis, “Cellular automaton model of precipitation/sissolution coupled with solute transport,” J. Stat. Phys., 81, 165–174, 1995. [36] M.J. Young and C.H.J. Davies, “Cellular automaton modelling of precipitate coarsening,” Scripta Mater., 41, 697–708, 1999. [37] O. Kortluke, “A general cellular automaton model for surface reactions,” J. Phys. A, 31, 9185–9198, 1998. [38] G. Gottstein and L.S. Shvindlerman, Grain Boundary Migration in Metals– Thermodynamics, Kinetics, Applications, CRC Press, Boca Raton, 1999. [39] R.C. Becker, “Analysis of texture evoltuion in channel die compression-I. Effects of grain interaction,” Acta Metall. Mater., 39, 1211–1230, 1991. [40] R.C. Becker and S. Panchanadeeswaran, “Effects of grain interactions on deformation and local texture in polycrystals,” Acta Metall. Mater., 43,2701–2719, 1995. [41] F.J. Humphreys and M. Hatherly, Recrystallization and Related Annealing Phenomena, Pergamon Press, New York, 1995.

7.8 MODELING COARSENING DYNAMICS USING INTERFACE TRACKING METHODS John Lowengrub University of California, Irvine, California, USA

In this paper, we will discuss the current state-of-the-art in numerical models of coarsening dynamics using a front-tracking approach. We will focus on coarsening during diffusional phase transformations. Many important structural materials such as steels, aluminum and nickel-based alloys are products of such transformations. Diffusional transformations occur when the temperature of a uniform mixture of materials is lowered into a regime where the uniform mixture is unstable. The system responds by nucleating second phase precipitates (e.g., crystals) that then evolve diffusionally until the process either reaches equilibrium or is quenched by further reducing the temperature. The diffusional evolution consists of two phases – growth and coarsening. Growth occurs in response to a local supersaturation in the primary (matrix) phase and a local mass balance relation is satisfied at each precipitate interface. Coarsening occurs when a global mass balance is achieved and involves a dynamic rearrangement of the fixed total mass in the system so as to minimize a global energy. Typically, the global energy consists of the surface energy. If the transformation occurs between components in the solid state, there is also an elastic energy that arises due to the presence of a misfit stress between the precipitates and the matrix as their crystal structures are often slightly different. Diffusional phase transformations are responsible for producing the material microstructure, i.e., the detailed arrangement of distinct constituents at the microscopic level. The details of the microstructure greatly influence the material properties of the alloy (i.e., stiffness, strength, and toughness). In many alloys, an in situ coarsening process can occur at high temperatures in which a dispersion of very small precipitates evolves to a system consisting of a few very large precipitates in order to decrease the surface energy of the system. This coarsening severely degrades the properties of the alloy and can lead to in service failures. The details of this coarsening process depend strongly 2205 S. Yip (ed.), Handbook of Materials Modeling, 2205–2222. c 2005 Springer. Printed in the Netherlands.

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on the elastic properties and crystal structure of the alloy components. Thus, one of the goals of this line of research is to use elastic stress to control the evolution process so as to achieve desirable microstructures. Numerical simulations of coarsening two-phase microstructures have followed two directions – interface capturing and interface tracking. In capturing methods, the precipitate/matrix interfaces are implicitly determined through an auxiliary function that is introduced to delineate between the precipitate and matrix phases. Examples include phase-field and level-set methods. Typically, sharp interfaces are smoothed out and the elasticity and diffusion systems are replaced by mesoscopic approximations that mimic the true field equations together with interface jump conditions. These methods have the advantage that topological changes such as precipitate coalescence and splitting are easily described. A disadvantage of this approach is that the results can be sensitive to the parameters that determine the thickness of the interfacial regions and care needs to be taken reconcile the results using sharp interfaces and tracking methods. In interface tracking methods, which are the subject of this article, a specific mesh is introduced to approximate the interface. The evolution of the interface is tracked by explicitly evolving the interface mesh in time. Examples include boundary integral, immersed interface [1], ghost-fluid [2], front-tracking [3, 4]. In boundary integral, immersed interface and ghost-fluid methods, for example, the interfaces remain sharp and the true field equations and jump conditions are solved. These methods have the advantage that high order accurate solutions can be obtained. Thus, in addition to their intrinsic value, results from these algorithms can also be used as benchmarks to validate interface-capturing methods. Boundary integral methods have the additional advantage that the field equations and jump conditions are mapped to the precipitate/matrix interfaces thereby reducing the dimensionality of the problem. However, boundary integral methods typically apply only in the limited situation where the physical domains and parameters are piecewise homogeneous. The other tracking methods listed above do not suffer from this difficulty although they are generally not as accurate as boundary integral methods. A general disadvantage of the tracking approach is that ad-hoc cut-and-connect procedures are required to handle changes in interface topologies. In this article, we will focus primarily on a description of the state-of-theart in boundary integral methods.

1.

Coarsening

One of the central assumptions of mathematical models of coarsening is that the system evolves so as to decrease the total energy. This energy consists of an interfacial part, associated with the precipitate/matrix interfaces and a

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bulk part due to the elasticity of the constituent materials. In the absence of the elastic stress, precipitates tend to be roughly spherical and interfacial area is reduced by the diffusion of matter from regions of high interfacial curvature to regions of low curvature. During coarsening, this leads to a survival of the fattest since large precipitates grow at the expense of small ones. This coarsening process may severely degrade the properties of the alloy. In the early 1960s, an asymptotic theory, now referred to as the LSW theory, was developed by Lifshitz and Slyosov [5], and Wagner [6] to predict the temporal power law of precipitate growth and in particular the scaling at long times of the precipitate radius distribution. In this LSW theory, only surface energy is considered and it is found that the average precipitate radius R ∼ t 1/3 at long times. The LSW theory has two major restrictions, however. First, precipitates are assumed to be circular (spherical in 3-D) and second, the theory is valid only in the zero (precipitate) volume fraction limit. Extending the results of LSW to account for non-spherical precipitates, finite volume fractions and elastic interactions has been a subject of intense research interest and is one of the primary reasons for the development of accurate and efficient numerical methods to study microstructure evolution. See the recent reviews by Johnson and Voorhees [7], Voorhees [7] and Thornton et al. [8].

2.

Governing Equations

For the purposes of illustration, let us focus a two-phase microstructure in a binary alloy. We further assume that the matrix phase M extends to infinity (or in 2D may be contained in a large domain ∞ ), while the precipitate phase P consists of isolated particles occupying a finite volume. The interface between the two phases is a collection of closed surfaces . The evolution of the precipitate matrix interface is controlled by diffusion of matter across the interface. Assuming quasi-static diffusion, the normalized composition c is governed by Laplace’s equation c = 0

(1)

in both phases. The composition on a precipitate-matrix interface is given by the Gibbs–Thomson boundary condition [9] c = −(τ I + ∇n ∇n τ ) : K − Zg el − λVn ,

(2)

where τ = τ (n) is the non-dimensional anisotropic surface tension, n is the normal vector directed towards M , I is the identity tensor, (∇n ∇n τ )i j = ∂ 2 τ/ ∂n i ∂n j ,

K=−

L M N s1 s1 + √ (s1 s2 + s2 s1 ) + s2 s2 E F EG

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is the curvature tensor where s1 and s2 are tangent vectors to the interface and the definitions of L, E, M, G, F and N depend on the interface parametrization and can be found in standard differential geometry texts [10]. Note that ˆ = 2H where H is the mean curvature. In addition, Z characterizes the tr(E) relative strength of the surface and elastic energies, g el is the elastic energy density (defined below), Vn is the normal velocity of the precipitate/matrix interface and λis a non-dimensional linear kinetic coefficient. Roughly speaking, this boundary condition reflects the idea that changing the shape of a precipitate changes the energy of the system both through the additional surface energy, (τ I + ∇n ∇n τ ) : K, and also through the change in elastic energy of the system, Zg el . We note that the composition is normalized differently in the precipitate and matrix, so that the normalized composition is continuous across the interface; the actual dimensional composition undergoes a jump. The normal velocity is given by the flux balance Vn = k

∂c ∂c − , ∂n P ∂n M

(3)

and (∂c/∂n) P and (∂c/∂n) M denote the values of normal derivative of c evaluated on the precipitate side and the matrix side of the interface, respectively, and k is the ratio of thermal diffusivities. Two different far-field conditions for the diffusion problem can be posed. In the first, the mass flux J into the system is specified: 1 J= 4π

1 Vn d = 4π

∂∞

∂c d∂∞ , ∂n

(4)

where ∞ is a large domain containing all the precipitates. As a second, alternative boundary condition, the far-field composition c∞ is specified lim c(x) = c∞ .

|x|→∞

(5)

In 2D, the limit in Eq. (5) is taken only to ∂∞ since c diverges logarithmically at infinity (see the 2D Green’s function below). Since the elastic energy density g el appears in the Gibbs–Thomson boundary condition (1), one must solve for the elastic fields in the system before finding the diffusion fields. The elastic fields arise if there is a misfit strain, denoted by ε T between the precipitate and matrix. This misfit is taken into account through the constitutive relations between the stress σi j and strain εi j . These are σiPj = CiPj kl εlkP − εlkT in the precipitate and σiMj = CiMj kl εlkM in the matrix, where we have taken the matrix lattice as the reference. The superscripts P and M refer to the values in the precipitate and matrix respectively. The elastic stiffness tensor Ci j kl may be different in the matrix and precipitate (elastically inhomogeneous) and may also reflect different material symmetries of the two phases.

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The equations of elastic equilibrium require that σi j, j = 0,

(6)

in both phases (in the absence of body forces). We also assume the interfaces are coherent, so the displacement u (i.e., εi j =(u i, j +u j,i )/2) and the traction t (i.e., ti = σi j n j ) are continuous across them. For simplicity, we suppose that the far-field tractions and displacements vanish. Finally, the elastic energy density g el is given by 1 P P σi j εi j − εiTj − σiMj εiMj + σiMj εiMj − εiPj . (7) 2 Finally, the total energy of the system is the sum of the surface and elastic energies

g el =

Wtot = Ws + Wel . where

Ws =

(8)

Z τ (n) d, and Wel = 2

σiPj εiPj − εiTj d+

P

σiMj εiMj d.

M

(9) For details on the isotropic formulation, derivation and nondimensionalization, see Li et al. [11], the review articles [8, 12] and the references therein.

3.

The Boundary Integral Formulation

We first consider the diffusion problem. If the interface kinetics λ > 0, then a single-layer potential can be used. That is, the composition is given by

c(x) =

σ (x ) G(x − x ) d(x ) + c¯∞ ,

(10)

where σ (x) is the single-layer potential, G(x)is the Green’s function (i.e., 2D: G(x) = (1/2π ) log |x|, 3D: G(x) = (1/4π |x|) and c¯∞ is a constant. Then, taking the limit as x → , and using Eq. (2), we get the Fredholm boundary integral equation

σ (x ) G(x − x ) d(x ) + λV n + c¯∞

(11)

where the normal velocity Vn is related to σ (x). In fact, if the ratio of diffusivities k = 1, then Vn = σ (x) and the equation is a 2nd kind Fredholm integral

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equation. See [13]. For simplicity, let us suppose this is the case. Then, if the flux is specified, Eq. (11) is solved together with Eq. (4) to determine Vn and c¯∞ . In 3D, if far-field condition (5) is imposed, then c¯∞ = c∞ . In 2D, if (5) is imposed, then another single layer potential must be introduced at the far-field boundary ∂∞ [14]. If the interface kinetics λ = 0, then a double-layer potential should be used: c(xi ) =

µi (x )

∂G np (xi − x )d(x ) + k=1 Ak G(xi − Sk ), ∂n

(12)

in each domain i where i = p, m, and n p is the number of precipitates and Sk is a point inside the kth precipitate. In the limit x → leads to the system of 2nd kind Fredholm equations

µi (x )

∂G µi np (xi − x ) d(x ) + k=1 Ak G(xi − Sk ) ± ∂n 2

= − (τ I + ∇n ∇n τ ) : K − Zg el ,

(13)

where the plus sign is taken when i = m [13]. The Ak are determined from the equations

µi (x) d(x ) = 0,

for i = 1, n p − 1,

and

n

p k=1 Ak = J.

The normal velocity Vn is obtained by taking the normal derivative of Eq. (12), taking care to treat the singularity of the Green’s function [13], and thus depends on µi (x ). Equation (13) is then solved together with the far-field conditions in either Eq. (4) or (5) to obtain µi (x ) and c¯∞ and Vn . We note that in 3D, we have recently found that a vector potential formulation [15] rather than a dipole formulation gives better numerical accuracy for computing Vn in this case (Pham, Lowengrub, Nie, Cristini, in preparation). Finally, once Vn is known, the interface is updated by n•

dx = Vn . dt

(14)

To actually solve the boundary integral equations, the elastic energy density g el must be determined first. This requires the solution of the elasticity equations. The boundary integral equations for the continuous displacement field u(x), and traction field t(x) on the interface involve Cauchy-principal-value

Modeling coarsening dynamics using interface tracking methods

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integrals over the interface. The equations can, using a direct formulation, be written as

(u i (y) − u i (x))Ti Pj k (y − x)n k (y) d(y) −

=

ti (y)G iPj k (y − x) d(y)

tiT (y)G iPj k (y

− x) d(y),

(15)

and u i (x) −

(u i (y) − u i (x))Ti M j k (y − x)n k (y) d(y)

−

ti (y)G iMj k (y − x) d(y) = 0,

(16)

where Ti j k and G ikj are the Green’s functions associated with the traction T and displacement respectively and tiT = CiPj kp εkp n j is the misfit traction. For isotropic elasticity, the Green’s functions are given by the Kelvin solution. For general 3D anisotropic materials, the Green’s functions cannot be written explicitly and are formulated in terms of line integrals. In 2D, explicit formulas exist for the Green’s functions. See for example [16, 17]. From the components of the displacements and tractions, the elastic energy density g el can be calculated [12].

4.

Numerical Implementation

The numerical procedure to simulate the evolution is as follows. Given the precipitate shapes, the elasticity Eqs. (15) and (16) are solved and the elastic energy g el is determined. Then diffusion equation is solved, the normal velocity is calculated and the interfaces are advanced one step in time. Precipitates whose volume falls below a certain tolerance are removed from the simulation. In 2D, very efficient and spectrally numerical methods have been developed to solve this problem [12]. The integrals with smooth integrands are discretized with spectral accuracy using the trapezoid rule. The Cauchy principal value integrals are discretized with spectral accuracy using the alternating point trapezoid rule. The fast multipole method [18] is used to evaluate the discrete sums in O(N ) work where N is the total number of collocation points on all the interfaces. Further efficiency is gained by neglecting particle–particle interactions if the particles are well-separated. The iterative method GMRES is then used to solve the discrete nonsymmetric, non-definite elasticity and diffusion matrix systems. The surface tension introduces a severe third order time step constraint for stability: t ≤ Cs 3 where C is a constant and s is the minimum spacing in

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arclength along all the interfaces. To overcome this difficulty, Hou, Lowengrub and Shelley [12] performed a mathematical analysis of the equations of motion at small length-scales (the “small-scale decomposition”). This analysis shows that when the equations of motion are properly formulated, surface tension acts through a linear operator at small length-scales. This contribution, when combined with a special reference frame in which the collocation points remain equally spaced in arclength, can then be treated implicitly and efficiently in a time-integration scheme, and the high-order constraints removed. In 3D, efficient algorithms have been recently developed by Li et al. [11] and Cristini and Lowengrub [19]. In these approaches, the surfaces are discretized using an adaptive surface triangulated mesh [20]. As in 2D, the integral equations are solved using the collocation method and GMRES. In Li et al. [11], local quadratic Lagrange interpolation is used to represent field quantities (i.e., u, t, Vn , and the position of the interface x ) in triangle interiors. The normal vector is derived from the local coordinates using the Lagrange interpolants of the interface position. The curvature is determined by performing a local quadratic fit to the triangulated surface. This combination was found to yield the best accuracy for a given resolution. On mesh triangles where the integrand is singular, a nonlinear change of variables (Duffy’s transformation) is used to map the singular triangle to a unit square and to remove the 1/r divergence of the integrand. For triangles in a region close to the singular triangle, the integrand is nearly singular, and, so, each of these triangles is divided into four smaller triangles, and a high-order quadrature is used on each subtriangle individually. On all other mesh triangles, the highorder quadrature is used to approximate the integrals. In Cristini and Lowengrub [19], there are no effects of elasticity (Z = 0) the collocation method is used to solve the diffusion integral equation together with GMRES and the nonlinear Duffy transformation to remove the singularity of the integrand in the singular triangle. Away from the singular triangle, the trapezoid rule is used and no interpolations are used to represent the field quantities in triangle interiors. As in Li et al., the curvature is still determined by performing a local quadratic fit to the triangulated surface. In both Li et al., and Cristini and Lowengrub, a second-order Runge–Kutta method is used to advance the triangle nodes. The time-step size is proportional to the smallest diameter of the triangular elements raised to the 3/2 power:t = Ch 3/2 . This scaling is due to the fact that the adaptive mesh uniformly resolves the solid angle. Since the shape of the precipitate can change substantially during its evolution, one of the keys to the success of these algorithms is the use of the adaptive-mesh refinement algorithm developed originally by Cristini, Blawzdzieweicz, and Loewenberg [20]. In this algorithm, the solid angle is uniformly resolved throughout the simulation using the following local-mesh restructuring operations to achieve an optimal mesh density: grid equilibration,

Modeling coarsening dynamics using interface tracking methods

2213

edge-swapping, and node addition and subtraction. This results in a density of node points that is proportional to the maximum of the curvature (in absolute value), so that grid points cluster in highly curved regions of the interface. Further, each of the mesh triangles is nearly equilateral. Finally, to further increase efficiency, a parallelization algorithm is implemented for the diffusion and elasticity solvers. The computational strategy for the parallelization is similar to the one designed for the microstructural evolution in 2D elastic media [12]. A new feature of the algorithm implemented by Li et al. is that the diffusion and elasticity matrices are also divided among the different processors in order to reduce the amount of memory required on each individual processor.

5.

Two-dimensional Results

The state-of-the-art in 2D simulations of purely diffusional evolution in the absence of elastic stress (Z = 0) is the work of [21]. In metallic alloy systems, this corresponds to simulating systems of very small precipitates where the surface energy dominates the elastic energy. Using the methods described above, Akaiwa and Meiron performed simulations containing over 5000 precipitates. Akaiwa and Meiron divided the computational domain into subdomains each containing 50–150 precipitates. Inside each sub-domain, the full diffusion field is computed. The influence of particles outside each subdomain is restricted to only involve those lying within a distance of 6–7 times the average precipitate radius from the sub-domain. This was found to give at most a 1% error in the diffusion field and significantly reduces the computational cost. In Fig. 1, two snapshots of a typical simulation are shown at the very late stages of coarsening. In this simulation, the precipitate area fraction is 0.5 and periodic boundary conditions are applied. In Fig. 1(left), there are approximately 130 precipitates remaining, while in Fig. 1(right) there are only approximately 70 precipitates left. Note that there is no discernible alignment of precipitates. Further, as the system coarsens, the typical shape of a precipitate shows significant deviation from a circle. The simulation results of Akaiwa and Meiron agree with the classical Lifshitz–Slyozov–Wagner (LSW) theory in which the average precipitate radius R is predicted to scale as R ∝ t 1/3 at large times t. It was found that certain statistics, such as the particle size distribution functions, are insensitive to the non-circular particle shapes at even at moderate volume fractions. Simulations were restricted to volume fractions less than 0.5 due to the large computational costs associated with refining the space and time scales to resolve particle-particle near contact interactions at larger volume fractions. The current state of the art in simulating diffusional evolution in homogeneous, anisotropic elastic media is the recent work of [22] who studied alloys

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Figure 1. The late stages of coarsening in the absence of elastic forces (Z = 0). Left: Moderate time; Right: Late time. After [21]. Reproduced with permission.

with cubic symmetry. In metallic alloys, such a system can be considered as a model for nickel–aluminum alloys. In the homogeneous case, one need not solve Eqs. (15)–(16). Instead, the derivatives of the displacement field and hence the elastic energy density g el due to a misfitting precipitate may be evaluated directly from the Green’s function tensor via the boundary integral [22]

u j,k (x) = Ci j + Ci j 22

gi j,k (x, x )n l (x )d(x ),

(17)

where the misfit is a unit dilatation and x is either in the matrix or precipitate and Ci j kl is the stiffness tensor. Using the methods described above together with a fast summation method to calculate the integral in Eq. (17), Akaiwa, Thornton and Voorhees, 2001 have performed simulations involving over 4000 precipitates. See Fig. 2 for results with isotropic surface tension and dilatational misfits. The value of Z is allowed to vary dynamically through an average precipitate radius. Thus, as precipitates coarsen and grow larger, Z increases correspondingly. The initial volume fraction of precipitates is 0.1. Thornton, Akaiwa and Voorhees find that the morphological evolution is significantly different in the presence of elastic stress. In particular, large-scale alignment of particles is seen in the 100 and 010 directions during the evolution process. In addition, there is significant shape dependence as nearly circular precipitates are seen at small Z and as Z increases, precipitates become squarish and then rectangular. It is found that in the elastically homogeneous system, elastic stress does not modify the 1/3 temporal exponent of the LSW coarsening law even though the precipitate morphologies are far from circular. Surprisingly, as long as the

Modeling coarsening dynamics using interface tracking methods

2215

Figure 2. Coarsening in homogeneous, cubic elasticity. The volume fraction is 10%. The left column shows the computational domain, while the right column is scaled with the average particle size. After Thornton, Akaiwa and Voorhees, 2001. Reproduced with permission.

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J. Lowengrub

shapes remain fourfold symmetric, the kinetics (coefficient of temporal factor) remains unchanged also. It is only when a majority of the particles have a two-fold rectangular shape that the coarsening kinetics changes [23]. The inhomogeneous elasticity problem is much more difficult to solve than the homogeneous problem because in the inhomogeneous case, the integral Eqs. (15)–(16) must be solved in order to obtain the inhomogeneous elastic fields and the elastic energy density g el . For this reason, the state of theory and simulations are less well-developed for the inhomogeneous case compared to the homogeneous problem. The current state-of-the-art in simulating microstructure evolution in inhomogeneous, anisotropic elastic media is the work of Leo, Lowengrub and Nie 2000. Although the system (15), (16) is a Fredholm equation of mixed type with smooth, logarithmic, and Cauchy-type kernels, it was shown by Leo, Lowengrub and Nie 2000, in the anisotropic case, that the system may be transformed directly to a second kind Fredholm system with smooth kernels. The transformation relies on an analysis of the equations at small spatial scales. Leo, Lowengrub and Nie, 2000 found that even small elastic inhomogeneities may have a strong effect on precipitate evolution in systems with small numbers of precipitates. For instance, in systems where the elastic constants of the precipitates are smaller than those of the matrix (soft precipitates), the precipitates move toward each other. In the opposite case (hard precipitates), the precipitates tend to repel one another. The rate of approach or repulsion depends on the amount of inhomogeneity. Anisotropic surface energy may either enhance or reduce this effect. The evolutions of two sample inhomogeneous systems in 2D are shown in Fig. 3. The solid curves correspond to Ni3 Al precipitates (soft, elastic constants less than the Ni matrix) and the dashed curves correspond to Ni3 Si precipitates (hard, elastic constants larger than the Ni matrix). In both cases, the matrix is Ni. Note that only the Ni3 Si precipitates are shown at time t = 20.09 for reasons explained below. From a macroscopic point of view, there seems to be little difference in the results of the two simulations over the times considered. The precipitates become squarish at very early times and there is only a small amount of particle translation. One can observe that the upper and lower two relatively large pairs of precipitates tend to align along the horizontal direction locally. The global alignment of all precipitates on the horizontal and vertical directions appears to occur on a longer time scale. On the time scale presented, the kinetics appears to be primarily driven by the surface energy which favors coarsening–the growth of large precipitates at the expense of the small precipitates to reduce the surface energy. Upon closer examination, differences between the simulations are observed. For example, consider the result at time t = 15.77 which is shown in Fig. 3. In the Ni3 Al case, the two upper precipitates attract one another and likely merge. In the Ni3 Si case, on the other hand, it does not appear that these two

Modeling coarsening dynamics using interface tracking methods t 0

t 2.5

t 5.0

t 15.0

t 15.77

t 20.09

2217

Figure 3. Evolution of 10 precipitates in a Ni matrix. Solid, Ni3 Al; dashed, Ni3 Si, Z=1. After Leo, Lowengrub and Nie 2000. Reproduced with permission.

precipitates will merge. This is consistent with the results of smaller precipitate simulations [24]. In addition, the interacting pairs of Ni3 Al precipitates tend to be “flatter” than their Ni3 Si counterparts. Also observe that the lower two precipitates in the Ni3 Al case attract one another. In the process, the lower right precipitate develops very high curvature (note its flat bottom) that ultimately prevents the simulation to be continued much beyond this time. This is why no Ni3 Al precipitates are shown in Fig. 3 at time t = 20.09. Finally, more work needs to be done in order to simulate larger inhomogeneous systems in order to reliably determine coarsening rate constants.

6.

Three-dimensional Results

Because of the difficulties in simulating the evolution of 2D surfaces in 3D, the simulation of microstructure evolution in 3D is much less developed than the 2D counterpart. Nevertheless, there has been promising recent work that is beginning to bridge the gap. The state-of-the-art in 3D boundary integral simulations is the work of Cristini and Lowengrub, 2004 and Li et al., 2003. Using the adaptive simulation algorithms described above, Cristini and Lowengrub, 2004 simulated the diffusional evolution of systems with a

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single precipitate growing under the influence of a driving force consisting of either an imposed far-field heat flux or a constant undercooling in the far-field. Under conditions of constant heat flux, Cristini and Lowengrub demonstrated that the Mullins–Sekerka instability can be suppressed and precipitates can be grown with compact shapes. An example simulation from Cristini and Lowengrub, 2004 is shown in Fig. 4. In this figure, the precipitate morphologies together with the shape factor δ/R are shown for precipitates grown under constant undercooling and constant flux conditions. R is the effective precipitate radius (i.e., radius of a (equivalent) sphere with the same volume enclosed) and δ/Rmeasures the shape deviation from the equivalent sphere. In Fig. 5, the coarsening of a system of 8 precipitates in 3D is shown in the absence of elastic effects (Z = 0), from Li, Lowengrub and Cristini, 2004. This adaptive simulation uses the algorithms described above and is performed an infinitely large domain. Because the precipitates are spaced relatively far from one another, there is little apparent deviation of the morphologies from spherical. However, this is not assumed or required by the algorithm. In Fig. 5, we see the classical survival of the fattest as mass is transferred from small precipitates to large ones. Work is ongoing to develop simulations at finite

Figure 4. Precipitate morphologies grown under constant undercooling and constant flux conditions. After Cristini and Lowengrub, 2004. Reproduced with permission.

Modeling coarsening dynamics using interface tracking methods t0

t 0.75

t 1.5

t 4.0

t 6.75

t 7.5

2219

Figure 5. The coarsening of a system of 8 precipitates in 3D in the absence of elastic effects (Z = 0). Figure courtesy of Li, Lowengrub and Cristini, 2004.

Figure 6. The evolution of a Ni3 Al precipitate in a Ni matrix (Z = 4). Left: early time. Right: late time (equilibrium). After Li et al., 2003. Reproduced with permission.

volume fractions of precipitate coarsening in periodic geometries [25] in order to determine statistically meaningful coarsening rate statistics. The current state-of-the-art in simulations of coarsening in 3D with elastic effects is the work of Li et al., 2003. To date, simulations have been performed with single precipitates. A sample simulation from Li et al., 2003 is shown in Fig. 6 for the evolution of a Ni3 Al precipitate in a Ni matrix with Z = 4. For this value of Z , and those above it (for Ni3 Al precipitates), there is a transition from cuboidal shapes to elongated shapes as seen in the figure. Such elongated

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Figure 7. Left and Middle: Growth shapes of a Ni3 Al precipitate in a Ni matrix. After Li et al., 2003. Reproduced with permission. A. Right: An experimental precipitate from a Ni-based superalloy after Yoo, Yoon and Henry, 1995. Reproduced with permission.

shapes are often seen in experiments. Finally, in Fig. 7, we present growth shapes (left and middle) of a Ni3 Al precipitate in a Ni matrix with Z = 4 under a driving force consisting of a constant flux of Al atoms [11]. In contrast to the precipitate in Fig. 6, under growth, the Ni3 Al precipitate retains its cuboidal shape although it develops concave faces. On the right, an image is shown from an experiment [26] showing Ni-based precipitates with concave faces similar to those observed in the simulation.

7.

Outlook

In this paper, we have presented a brief description of the state-of-theart in simulating microstructure evolution, and in particular coarsening, using boundary integral interface tracking methods. In general, the methods are quite well-developed in 2D. In particular, large-scale coarsening studies have been performed in the absence of elastic effects and when the elastic media is homogeneous and anisotropic. Although methods have been developed to study coarsening in fully inhomogeneous, anisotropic elastic media, so far the computational expense of the current methods have prevented large-scale studies to be performed. There have been exciting developments in 3D and although the state-ofthe-art in 3D simulations is still well behind those in 2D, this direction looks very promising for the future. This is also an important future direction as coarsening in metallic alloys, for example, is a fully 3D phenomenon. Efforts in this direction will have a significant potential payoff in that they will allow, for the first time, not only a rigorous check of the LSW coarsening kinetics in 3D but also will allow the effects of finite volume fraction and elastic forces on the coarsening kinetics to be assessed.

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References [1] Z. Li and R. Leveque, “Immersed interface methods for Stokes flow with elastic boundaries or surface tension,” SIAM J. Sci. Comput., 18, 709, 1997. [2] S. Osher and R. Fedkiw, “Level set methods: An overview and some recent results,” J. Comp. Phys., 169, 463, 2001. [3] J. Glimm, M.J. Graham, J. Grove et al., “Front tracking in two and three dimensions,” Comput. Math. Appl., 35, 1, 1998. [4] G. Tryggvason, B. Bunner, A. Esmaeeli et al., “A front tracking method for the computations of multiphase flow,” J. Comp. Phys., 169, 708, 2001. [5] I.M. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids, 19, 35, 1961. [6] C. Wagner, Z. Elektrochem., 65, 581, 1961. [7] W.C. Johnson and P.W. Voorhees, “Elastically-induced precipitate shape transitions in coherent solids,” Solid State Phenom, 23, 87, 1992. [8] K. Thornton, J. Agren, and P.W. Voorhees, “Modelling the evolution of phase boundaries in solids at the meso- and nano-scales,” Acta Mater., 51(3), 5675–5710, 2003. [9] C. Herring “Surface tension as a motivation for sintering,” In: W. E. Kingston, (ed.), The Physics of Powder Metallurgy, Mcgraw-Hill, p. 143, 1951. [10] M. Spivak, “A Comprehensive Introduction to Differential Geometry,” Vol. 4, Publish or Perish, 3rd edn., 1999. [11] Li Xiaofan, J.S. Lowengrub, Q. Nie et al., “Microstructure evolution in threedimensional inhomogeneous elastic media,” Metall. Mater. Trans. A, 34A, 1421, 2003. [12] T.Y. Hou, J.S. Lowengrub, and M.J. Shelley, “Boundary integral methods for multicomponent fluids and multiphase materials,” J. Comp. Phys., 169, 302–362, 2001. [13] S.G. Mikhlin, “Integral equations and their applications to certain problems in mechanics, mathematical physics, and technology,” Pergamon, 1957. [14] P.W. Voorhees, “Ostwald ripening of two phase mixtures,” Annu. Rev. Mater. Sci., 22, 197, 1992. [15] W.T. Scott, “The physics of electricity and magnetism,” Wiley, 1959. [16] A.E.H. Love, “A treatise on the mathematical theory of elasticity,” Dover, 1944. [17] T. Mura, “Micromechanics of defects in solids,” Martinus Nijhoff, 1982. [18] J. Carrier, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm,” SIAM J. Sci. Stat. Comput., 9, 669, 1988. [19] V. Cristini and J.S. Lowengrub, “Three-dimensional crystal growth II. Nonlinear simulation and control of the Mullins-Sekerka instability,” J. Crystal Growth, in press, 2004. [20] V. Cristini, J. Blawzdzieweicz, and M. Loewenberg, “An adaptive mesh algorithm for evolving surfaces: Simulations of drop breakup and coalescence,” J. Comp. Phys., 168, 445, 2001. [21] N. Akaiwa and D.I. Meiron, “Two-dimensional late-stage coarsening for nucleation and growth at high-area fractions,” Phys. Rev. E, 54, R13, 1996. [22] N. Akaiwa, K. Thornton, and P.W. Voorhees, “Large scale simulations of microstructure evolution in elastically stressed solids,” J. Comp. Phys., 173, 61–86, 2001. [23] K. Thornton, N. Akaiwa, and P.W. Voorhees, “Dynamics of late stage phase separation in crystalline solids,” Phys. Review Lett., 86(7), 1259–1262, 2001. [24] P.H. Leo, J.S. Lowengrub, and Q. Nie, “Microstructure evolution in inhomogeneous elastic media,” J. Comp. Phys., 157, 44, 2000.

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[25] Li Xiangrong, J.S. Lowengrub, and V. Cristini, “Direct numerical simulations of coarsening kinetics in three-dimensions,” In preparation, 2004. [26] Y.S. Yoo, D.Y. Yoon, and. M.F. Henry, “The effect of elastic misfit strain on the morphological evolution of γ -precipitates in a model Ni-base superalloy,” Metals Mater., 1, 47, 1995.

7.9 KINETIC MONTE CARLO METHOD TO MODEL DIFFUSION CONTROLLED PHASE TRANSFORMATIONS IN THE SOLID STATE Georges Martin1 and Fr´ed´eric Soisson2 1

´ Commissariat a` l’Energie Atomique, Cab. H.C., 33 rue de la F´ed´eration, 75752 Paris Cedex 15, France 2 CEA Saclay, DMN-SRMP, 91191 Gif-sur-Yuette, France

The classical theories of diffusion-controlled transformations in the solid state (precipitate-nucleation, -growth, -coarsening, order-disorder transformation, domain growth) imply several kinetic coefficients: diffusion coefficients (for the solute to cluster into nuclei, or to move from smaller to larger precipitates. . . ), transfer coefficients (for the solute to cross the interface in the case of interface-reaction controlled kinetics) and ordering kinetic coefficients. If we restrict to coherent phase transformations, i.e., transformations, which occur keeping the underlying lattice the same, all such events (diffusion, transfer, ordering) are nothing but jumps of atoms from site to site on the lattice. Recent progresses have made it possible to model, by various techniques, diffusion controlled phase transformations, in the solid state, starting from the jumps of atoms on the lattice. The purpose of the present chapter is to introduce one of the techniques, the Kinetic Monte Carlo method (KMC). While the atomistic theory of diffusion has blossomed in the second half of the 20th century [1], establishing the link between the diffusion coefficient and the jump frequencies of atoms, nothing as general and powerful occurred for phase transformations, because of the complexity of the latter at the atomic scale. A major exception is ordering kinetics (at least in the homogeneous case, i.e., avoiding the question of the formation of microstructures), which has been described by the atomistic based Path Probability Method [2]. In contrast, supercomputers made it possible to simulate the formation of microstructures by just letting the lattice sites occupancy change in course of time following a variety of rules: the Kinetic Ising model (KIM) in particular has been (and 2223 S. Yip (ed.), Handbook of Materials Modeling, 2223–2248. c 2005 Springer. Printed in the Netherlands.

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still is) extensively studied and is summarized in the appendix [3]; other models include “Diffusion Limited Aggregation”, etc. . . Such models stimulate a whole field of the statistical physics of non-equilibrium processes. However, we choose here a distinct point of view, closer to materials science. Indeed, a unique skill of metallurgists is to master the formation of a desired microstructure simply by well controlled heat treatments, i.e., by imposing a strictly defined thermal history to the alloy. Can we model diffusion controlled phase transformations at a level of sophistication capable of reproducing the expertise of metallurgists? Since Monte Carlo techniques were of common use in elucidating delicate points of the theory of diffusion in the solid state [4, 5], it has been quite natural to use the very same technique to simulate diffusion controlled coherent phase transformations. Doing so, one is certain to retain the full wealth that the intricacies of diffusion mechanisms might introduce in the kinetic pathways of phase transformations. In particular, the question of the time scale is a crucial one, since the success of a heat treatment in stabilizing a given microstructure, or in insuring the long-term integrity of that microstructure, is of key importance in Materials Science. In the following, we first recall the physical foundation of the expression for the atomic jump frequency, we then recall the connection between jump frequencies and kinetic coefficients describing phase transformation kinetics; the KMC technique is then introduced and typical results pertaining to metallurgy relevant issues are given in the last section.

1.

Jumps of Atoms in the Solid State

With a few exceptions, out of the scope of this introduction, atomic jump in solids is a thermally activated process. Whenever an atom jumps, say from site α to α , the configuration of the alloy changes from i to j . The probability per unit time, for the transition to occur, writes:

Wi, j = νi, j

Hi, j exp − kB T

(1)

In Eq. (1), ν i, j is the attempt frequency, kB is the Boltzmann’s constant, T is the temperature and Hi, j is the activation barrier for the transition between configurations i and j . According to the rate theory [6], the attempt frequency writes, in the (quasi-) harmonic approximation: 3N−3

νi, j = k=1 3N−4 k=1

νk νk

(2)

In Eq. (2), νk and νk are the vibration eigen-frequencies of the solid, respectively in the initial configuration, i, and at the saddle point between configurations i and j . Notice that for a solid with N atoms, the number of eigen modes

Diffusion controlled phase transformations in the solid state

2225

is 3N . However, the vibrations of the centre of mass (3 modes) are irrelevant in the diffusion process, hence the upper bound 3N −3 in the product at the numerator. At the saddle point position between configurations i and j , one of the modes is a translation rather than a vibration mode, hence the upper bound 3N −4 in the denominator. Therefore, provided we know the value of Hi, j and νi, j for each pair of configurations, i and j , we need to implement some algorithm which would propagate the system in its configuration space, as the jumps of atoms actually do in the real solid. Notice that the algorithm must be probabilistic since Wi, j in Eq. (1) is a jump probability per unit time. Before we discuss this algorithm, we give some more details on diffusion mechanisms in solids, since the latter deeply affect the values of Wi, j in Eq. (1). The most common diffusion mechanisms in crystalline solids are vacancy-, interstitial- and interstitialcy-diffusion [7]. Vacancies (a vacant lattice site) allow for the jumps of atoms from site to site on the lattice; in alloys, vacancy diffusion is responsible for the migration of solvent- and of substitutional solute- atoms. Therefore, the transition from configuration i to j implies that one atom and one (nearest neighbor) vacancy exchange their position. As a consequence, the higher the vacancy concentration, the more numerous are the configurations, which can be reached from configuration i: indeed, starting from configuration i, any jump of any vacancy destroys that configuration. Therefore the transformation rate depends both on the jump frequencies of vacancies, as given by Eq. (1), and on the concentration of vacancies in the solid. This fact is commonly taken advantage of, in practical metallurgy. At equilibrium, the vacancy concentration depends on the temperature, the pressure and, in alloys, of the chemical potential differences between the species:

Cve

gf = exp − v kB T

(3)

In Eq. (3), Cve = Nv /(N + Nv ), with N the number of atoms, and gvf is the free enthalpy of formation of the vacancy. At equilibrium, the probability for an atom to jump equals the product of the probability for a vacancy to be nearest neighbor of that atom (deduced from Eq. 3), times the jump frequency given by Eq. (1). In real materials, vacancies form and annihilate at lattice discontinuities (free surfaces, dislocation lines and other lattice defects). If, in course of the phase transformation the equilibrium vacancy concentration changes, e.g., because of vacancy trapping in one of the phases, it takes some time for the vacancy concentration to adjust to its equilibrium value. This point, of common use in practical metallurgy, is poorly known from the basic point of view [8] and will be discussed later. Interstitial diffusion occurs when an interstitial atom (like carbon or nitrogen in steels) jumps to a nearest neighbor unoccupied interstitial site.

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Interstitialcy diffusion mechanism implies that a substitutional atom is “pushed” into an interstitial position by a nearest neighbor interstitial atom, which itself, becomes a substitutional one. This mechanism prevails, in particular, in metals under irradiation, where the collisions of lattice atoms with the incident particles produce Frenkel pairs; a Frenkel pair is made of one vacancy and one dumb-bell interstitial (two atoms competing for one lattice site). The migration of the dumb-bell occurs by the interstitialcy mechanism. The concentration of dumb-bell interstitials results from the competition between the production of Frenkel pairs by nuclear collisions and of their annihilation either by recombination with vacancies or by elimination on some lattice discontinuity. The interstitialcy mechanism may also prevail in some ionic crystals, and in the diffusion of some gas atoms in metals.

2.

From Atomic Jumps to Diffusion and to the Kinetics of Phase Transformations

The link between the jump frequencies and the diffusion coefficients has been established in details in limiting cases [1]. The expressions are useful for adjusting the values of the jump frequencies to be used, to experimental data. As a matter of illustration, we give below some expressions for the vacancy diffusion mechanism in crystals with cubic symmetry (with a for the lattice parameter): – In a pure solvent, the tracer diffusion coefficient writes: D ∗ = a 2 f 0 W0 Cve ,

(4a)

with f 0 for the correlation factor (a purely geometrical factor) and W0 , the jump frequency of the vacancy in the pure metal. – In a dilute solution with Face Centered Cubic (FCC) lattice, with non interacting solutes, and assuming that the spectrum of the vacancy jump frequencies is limited to 5 distinct values (Wi , i = 0 to 4, for the vacancy jumps respectively in the solvent, around one solute atom, toward the solute, toward a solvent atom nearest neighbor of the solute, and away from the solute atom, see Fig. 1), the solute diffusion coefficient writes: W4 f 2 W2 , (4b) W3 where the correlation factor f 2 can be expressed as a function of the Wi ’s. In dilute solutions, the solvent- as well as the solute-diffusion coefficient depends linearly on the solute concentration, C, as: Dsolute = a 2 Cve

D ∗ (C) = D ∗ (0)(1 + bC). The expression of b is given in [1, 9].

(4c)

Diffusion controlled phase transformations in the solid state W3

2227

W1

W3 W2

W3

W1

W4 W0

Figure 1. The Five-frequency model in dilute FCC alloys: the five types of vacancy jumps are represented in a (111) plane (light gray: solvent atoms, dark gray: solute atom, open square: vacancies).

– In concentrated alloys, approximate expressions have been recently derived [10]. The atomistic foundation of the classical models of diffusion controlled coherent phase transformation is far less clear. For precipitation problems, two main techniques are of common use: the nucleation theory (and its atomistic variant sometimes named “cluster dynamics”) and Cahn–Hilliard diffusion equation [11]. In the nucleation theory, one defines the formation free energy (or enthalpy, if the transformation occurs under fixed pressure), F(R) of a nucleus with size R (volume vR 3 and interfacial area sR2 , v and s being geometric factors computed for the equilibrium shape): F(R) = δµvR 3 +σ sR 2 .

(5)

In Eq. (5), δµ and σ are respectively the gain of chemical potential on forming one unit volume of second phase, and the interfacial free energy (or free enthalpy) per unit area. If the solid solution is supersaturated, δµ is negative and F(R) first increases as a function of R, then goes through a maximum for the critical size R ∗ (R ∗ = (2s/3v) (σ/|δµ|)) and then decreases (Fig. 2). F(R) can be given a more precise form, in particular for small values of R. More details may be found in Perini et al. [12]. For the critical nucleus, F ∗ = F(R ∗ ) ≈ σ 3 /(δµ)2 .

(6)

F(R) can be seen as an energy hill which opposes the growth of sub-critical nuclei (R< R ∗ ) and which drives the growth of super-critical nuclei (R >R ∗ ). The higher the barrier, i.e., the larger F ∗ , the more difficult the nucleation is. F ∗ is very sensitive to the gain in chemical potential: the higher the supersaturation, the larger the gain, the shallower the barrier, and the easier the

2228

G. Martin and F. Soisson

F (R )

F R

R Figure 2. Free energy change on forming a nucleus with radius R.

nucleation. F ∗ also strongly depends on the interfacial energy, a poorly known quantity, which, in principle depends on the temperature. With the above formalism, the nucleation rate (i.e., the number of supercritical nuclei which form per unit time in a unit volume) writes, under stationary conditions: F∗ ∗ (7a) Jsteady = β Z N0 exp − kB T with N0 for the number of lattice sites and Z for the Zeldovich’s constant:

1 Z= − 2π kT

∂2F ∂n 2

1/2

,

(7b)

n=n ∗

n for the number of solute atoms in a cluster and θ ∗ for the sticking rate of solute atoms on the critical nucleus. If the probability of encounter of one solute atom with one nucleus is diffusion controlled: β(R) = 4πDRC

(7c)

For a detailed discussion, see Waite [13]. In Eq. (7c), D is the solute diffusion coefficient in the (supersaturated) matrix with the solute concentration C. An interesting quantity is the incubation time for precipitation, τinc , i.e., the time below which the nucleation current is much smaller than Jsteady . The former writes: 1 (7d) τinc ∝ ∗ 2 β Z When the supersaturation is small and/or the interfacial energy is high, the incubation time gets very large. Also the incubation time is scaled to the diffusion coefficient of the solute.

Diffusion controlled phase transformations in the solid state

2229

The nucleation process can be described also by the technique named “cluster dynamics”. The microstructure is described, at any time, by the number density, ρn, of clusters made of n solute atoms. The latter varies in time as: dρn = − ρn (αn + βn ) + ρn+1 αn+1 + ρn−1 βn−1 dt

(8)

where α n and β n are respectively the rate of solute evaporation and sticking at a cluster of n solute atoms. Again, α n and β n can be expressed in terms of solute diffusion or transfer coefficients. At later stages, when the second phase precipitation has exhausted the solute supersaturation, Ostwald ripening takes place: because the chemical potential of the solute close to a precipitate increases with the curvature of the precipitate-matrix interface (δµ(R) = 2σ/R), the smaller precipitates dissolve to the benefit of the larger ones. According to Lifschitz and Slyosov and to Wagner [14], the mean precipitate volume increases linearly with time, or the mean radius (as well as the mean precipitate spacing) goes as: R(t) − R(0) = k t 1/3

(9a)

with k3 =

(8/9)Dσ Cs kB T

(9b)

In Eq. (9b), D is again the solute diffusion coefficient, Cs the solubility limit, and the atomic volume. The problem of multicomponent alloys has been addressed by several authors [15]. The above models do not actually generate a full microstructure: they give the size distribution of precipitates as a function of time, as well as the mean precipitate spacing, since the total amount of solute is conserved, provided that the precipitates do not change composition in the course of the phase separation process. The formation of a full microstructure (i.e., including the variability of precipitate shapes, the correlation in the positions of precipitates etc.) is best described by Cahn’s diffusion equation [16]. In the latter, the chemical potential, the gradient of which is the driving force for diffusion, includes an inhomogeneity term, i.e., is a function, at each point, both of the concentration and of the curvature of the concentration field. The diffusion coefficient was originally given the form due to Darken. Based on a simple model of Wi, j and a mean field approximation, an atomistic based expression of the mobility has been proposed, both for binary [17] and multicomponent alloys [18]. When precipitation occurs together with ordering, Cahn’s equation is complemented with an equation for the relaxation of the degree of order; the latter relaxation occurs at a rate proportional to the gain in free energy due to the onsite relaxation of the degree order. The rate constant is chosen arbitrarily [19]. Since in a crystalline

2230

G. Martin and F. Soisson

sample the ordering reaction proceeds by the very same diffusion mechanism as the precipitation, both rate constants (for the concentration- and for the degree of order fields) should be expressed from the same set of Wi, j . This introduces some couplings, which have been ignored by classical theories [20]. As a summary, despite their efficiency, the theories of coherent phase separation introduce rate constants (diffusion coefficients, interfacial transfer coefficients, rate constants for ordering) the microscopic definition of which is not fully settled. The KMC technique offers a means to by-pass the above difficulties and to directly simulate the formation of a microstructure in an alloy where atoms jump with the frequencies defined by Eq. (1).

3.

Kinetic Monte Carlo Technique to Simulate Coherent Phase Transformations

The KMC technique can be implemented in various manners. The one we present here has a transparent physical meaning.

3.1.

Algorithm

Consider a computational cell with Ns sites, Na atoms and Nv = Ns − Na vacancies; each lattice site is linked to Z neighbor sites with which atoms may be exchanged (usually, but not necessarily, nearest neighbor sites). A configuration is defined by the labels of the sites occupied respectively by A, B, C, . . . atoms and by vacancies. Each configuration “i” can be escaped by Nch channels (Nch = Nv Z minus the number of vacancy–vacancy bounds if any), leading to Nch new configurations “ j1 ” to “ j Nch ”. The probability that the transition “i; jq ” occurs per unit time is given by Eq. (1) which can be computed a priori provided a model is chosen for Hi, j and νi, j . Since the configuration “i” may disappear by Nch independent channels, the probability for the configuration to disappear per unit time, Wiout , is the sum of the probabilities it decays by each channel (Wi, j q , q = 1 to Nch ), and the life time τ i of the configuration is the inverse of Wiout :

τi =

Nch

−1

Wi, jq

(10a)

q=1

The probability that the configuration “ jq ” is reached among the Nch target configurations is simply given by: Wi, jq Pi jq = N = Wi, jq × τi (10b) ch

Wi, jq q=1

Diffusion controlled phase transformations in the solid state

2231

Assuming all possible values of Wi, jq are known (see below), the code proceeds as follows: Start at time t = 0 from the configuration “i 0 ”, set i = i 0 ;

1. Compute τi (Eq. (10a)) and the Nch values of Si,k = kq=1 Pi jq , k = 1 to Nch . 2. Generate a random number R on ]0; 1]. 3. Find the value of f to be given to k such that Si,k−1 < R ≤ Si,k . Choose f as the final configuration. 4. Increment the time by τi (t MC => t MC + τi ) and repeat the process from step 1, giving to i the value f .

3.2.

Models for the Transition Probabilities Wi , j (Eq. (1))

For a practical use of the above algorithm, we need a model for the transitions probabilities per unit time, Wi, j . In principle, at least, given an interatomic potential, all quantities appearing in Eqs. (1)–(3) can be computed for any pair of configurations, hence Wi, j . The computational cost for this is so high that most studies use simplified models for the parameters entering Eqs. (1)–(3); the values of the parameters are obtained by fitting appropriate quantities to available experimental data, such as phase boundaries and tie lines in the equilibrium phase diagram, vacancy formation energy and diffusion coefficients. We describe below the most commonly used models, starting from the simplest one. Model (a) The energy of any configuration is a sum of pair interactions ε with a finite range (nearest- or farther neighbors). The configurational energy is the sum of the contributions of two types of bounds: those which are modified by the jump, and those which are not. We name esp the contribution of the bounds created in the saddle point configuration. This model is illustrated in Fig. 3. The simplest version of this model is to assume that esp depends neither on the atomic species undergoing the jump, nor on the composition in the surrounding of the saddle point [17]. Model (b) Same as above, but with esp depending on the atomic species at the saddle point. This approximation turned out to be necessary to account for the contrast in diffusivities in the ternary Ni–Cr–Al [21]. Model (c) Same as above, but with esp written as a sum of pair interactions [22]. This turned out to provide an excellent fit to the activation barriers computed in Fe(Cu) form fully relaxed atomistic simulations based on an EAM potential. As shown on Fig. (4), the

2232

G. Martin and F. Soisson

non broken bonds

0

esp broken bonds

Saddle-Point position

∆Hi;j

Hj Hi

(

)

(

)

i

j

Figure 3. Computing the migration barrier between configurations i and j (Eq. (1)), from the contribution of broken- and restored bounds. 7.5 8 2

eFe(SP)

8.5

6

9

eCu(SP)

5 3 1

V

4

9.5 10

0

1

2

3

4

5

6

NCu(SP)

Figure 4. The six nearest-neighbors (labeled 1 to 6) of the saddle-point in the BCC lattice (left). Contribution to the configurational energy, of one Fe atom, eFe (SP), or one Cu atom, eCu (SP), at the saddle point, as a function of the number of Cu atoms nearest neighbor of the saddle point (right).

contribution to the energy of one Cu atom at the saddle point, eCu (SP), does not depend on the number of Cu atoms around the saddle point, while that of one Fe atom, eFe (SP), increases linearly with the latter. Model (d) The energy of each configuration is a sum of pair and multiple interactions [18]. Taking into account higher order interactions permits to reproduce phase diagrams beyond the regular solution

Diffusion controlled phase transformations in the solid state

2233

model. The attempt frequency (Eq. 2) was adjusted, based on an empirical correlation between the pre-exponential factor and the activation enthalpy. Complex experimental interdiffusion profiles in four components alloys (AgInCdSn) could be reproduced successfully. Multiplet interactions have been used in KMC to model phase separation and ordering in AlZr alloys [23]. Model (e) The energies of each configuration and at the saddle point, as well as the vibration frequency spectrum (entering Eq. (2)) are computed from a many body interaction potential [24]. The vibration frequency spectrum can be estimated either with Einstein’s model [25] or Debye approximation [26, 27]. The above list of approximations pertains to the vacancy diffusion mechanism. Fewer studies imply also interstitial diffusion, as carbon in iron, or dumbbell diffusion, in metals under irradiation, as will be seen in the next section. The models for the activation barrier are of model (b) described above.

3.3.

Physical Time and Vacancy Concentration

Consider the vacancy diffusion mechanism. If the simulation cell only contains one vacancy, the vacancy concentration is 1/Ns , often much larger than a typical equilibrium vacancy concentration Cve . From Eq. (10), we conclude that the time evolution in the cell is faster than the real one, by a factor equal to the vacancy supersaturation in the cell: (1/Ns )/Cve . The physical time, t is therefore longer than the Monte Carlo time, tMC , computed above: t = tMC /(Ns Cve )

(11)

Equation (11) works as long as the equilibrium vacancy concentration does not vary much in the course of the phase separation process, a point which we discuss now. Consider an alloy made of N A atoms A, N B atoms B on Ns lattice sites. For any atomic configuration of the alloy, there is an optimum number of lattice sites, Nse , that minimizes the configurational free energy; the vacancy concentration in equilibrium with that configuration is: Cve = (Nse − N A − N B )/Nse . For example assume that the configurations can be described by K types of sites onto which the vacancy is bounded by an energy E bk (k = 1 to K ), with k = 1 corresponding to sites surrounded by pure solvent (E b1 = 0). We name N1 , . . . , N K the respective numbers of such sites. The equilibrium concentrations of vacancies on the sites of type 1 to K are respectively:

e Cvk =

Nvk E f + E bk = exp − Nk + Nvk kB T

(12a)

2234

G. Martin and F. Soisson

In Eq. (11), E f is the formation energy of a vacancy in pure A. The total vacancy concentration, in equilibrium with the configuration as defined by N1 , . . . , N K is thus (in the limit of small vacancy concentrations):

e Nk Cvk e ≈ = Cv0 k Nk X k = Nk /N1

Cve

k

1+

X k exp(−E bk /kB T )

; 1 + k=2,K X k

k=2,K

(12b)

e is the equilibrium vacancy concentration in the pure solvent, In Eq. (12), Cv0 and X k depends on the advancement of the phase separation process: e.g., in the early stages of solute clustering, we expect the proportion of sites surrounded by a small number of solute atoms to decrease. The overall vacancy equilibrium concentration thus changes in time (Eq. (12b)), while it remains unaffected for each type of site (Eq. (12a)). Imposing a fixed number of vacancies in the simulation cell, creates the opposite situation: in the simulation, the overall vacancy concentration is kept constant, thus the vacancy concentration on each type of site must change in course of time: the kinetic pathway will be altered. This problem can be faced in various ways. We quote below two of them:

– Rescaling the time from an estimate of the free vacancy concentration, i.e., the concentration of those vacancies with no solute as neighbor [22]. The vacancy concentration in the solvent is estimated in the course of the simulation, at a certain time scale, t, from the fraction of the time, where the vacancy is surrounded by solvent atoms only. Each time interval t is rescaled by the vacancy super saturation, which prevails during that time interval. – Modeling a vacancy source (sink) in the simulation cell [28]: in real materials, vacancies are formed and destroyed at lattice discontinuities (extended defects), such as dislocation lines (more precisely jogs on the dislocation line), grain boundaries, incoherent interfaces and free surfaces. The simplest scheme is as follows: creating one vacancy implies that one atom on the lattice close to the extended defect jumps into the latter in such a way as to extend the lattice by one site; eliminating one vacancy implies that one atom at the extended defect jumps into the nearby vacancy. Vacancy formation and elimination are a few more channels by which a configuration may change. The transition frequencies are still given by Eq. (1) with appropriate activation barriers: Fig. 5 gives a generic energy diagram for the latter transitions. As shown by the above scheme, while the vacancy equilibrium concentration is dictated by the formation energy, E f , the time to adjust to a change in the equilibrium vacancy concentration implies the three parameters E f ,

Diffusion controlled phase transformations in the solid state

2235

Em

Ef

Figure 5. Configurational energy as a function of the position of the vacancy. When one vacancy is added to the crystal, the energy is increased by E f .

E m and δ. In other words, a given equilibrium concentration can be achieved either by frequent or by rare vacancy births and deaths. The consequences of this fact on the formation of metastable phases during alloy decomposition are not yet fully understood.

3.4.

Tools to Characterize the Results

The output of a KMC simulation is a string of atomistic configurations as a function of time. The latter can be observed by the eye (e.g., to recognize specific features in the shape of solute clusters); one can also measure various characteristics (short range order, cluster size distribution, cluster composition and type of ordering. . . ); one can simulate signals one would get from classical techniques such as small- or large-angle scattering, or use the very same tools as used in Atom Probe Field Ion Microscopy to process the very same data, namely the location of each type of atom. Some examples are given below.

3.5.

Comparison with the Kinetic Ising Model

The KIM, of common use in the Statistical Physics community, is summarized in the appendix. It is easily checked that the models presented above for the transition probabilities introduce new features, which are not addressed by the KIM. In particular, the only energetic parameter to appear in KIM is what is named, in the community of alloys thermodynamics, the ordering energy: ω = ε AB − (ε A A + ε B B )/2 (for the sake of simplicity, we restrict, here, to two

2236

G. Martin and F. Soisson

component alloys). While ω is indeed the only parameter to enter equilibrium thermodynamics, the models we introduced show that the kinetic pathways are affected by a second independent energetic parameter, the asymmetry between the cohesive energies of the pure elements: ε A A − ε B B . This point is discussed into details, by Ath`enes and coworkers [29–31]. Also, the description of the activated state between two configurations is more flexible in the present model as compared to KIM. For these reasons, the present model offers unique possibilities to study complex kinetic pathways, a common feature in real materials.

4.

Typical Results: What has been Learned

In the 70s the early KMC simulations have been devoted to the study of simple ordering and phase separation kinetics in binary systems with conserved or non-conserved order parameters. Based on the Kinetic Ising model and so called “Kawazaki dynamics” (direct exchange between nearest neighbor atoms, with a probability proportional to exp [−(Hfinal − Hinitial)/2kB T ]), with no point defects and no migration barriers, they could mainly reproduce some generic features of intermediate time behaviors, taking the number of Monte Carlo step as an estimate of physical time: the coarsening regime of precipitation with R − R0 ∝ t 1/3 ; the growth rate of ordered domains R − R0 ∝ t 1/2 , dynamical scaling laws, etc. [3, 32]. However, such models cannot reproduce important metallurgical features such as the role of distinct solute and solvent mobilities, of point defect trapping, or of correlations among successive atomic jumps etc. In the frame of the models (a)–(e) previously described, these features are mainly controlled by the asymmetry parameters for the stable configurationsp sp and saddle-point energies (respectively ε A A − ε B B , and e A − e B ). We give below typical results, which illustrate the sensitivity, to the above features, of the kinetic pathways of phase transformations.

4.1.

Diffusion in Ordered Phases

Since precipitates are often ordered phases, the ability of the transition probability models to well describe diffusion in ordered phases must be assessed. As an example, diffusion in B2 ordered phases presents specific features which have been related to the details of the diffusion mechanism: at a given composition, the Arrhenius plot displays a break at the order/disorder temperature and an upward curvature in the ordered phase; at a given temperature, the tracer diffusion coefficients are minimum close to the stoichiometric composition. The reason for that is as follows: starting from a perfectly

Diffusion controlled phase transformations in the solid state

2237

ordered B2 phase, any vacancy jump creates an antisite defect, so that the most probable next jump is the reverse one which annihilates the defect. As a consequence, it has been proposed that diffusion in B2 phases occurs via highly correlated vacancy jump sequences, such as the so-called 6-jump cycle (6JC) which corresponds to 6 effective vacancy jumps (resulting from many more jumps, most of them being canceled by opposite jumps). Based on the above “model (a)” for the jump frequency, Ath`enes’ KMC simulations [29] show that other mechanisms (e.g., the antisite-assisted 6JC) contribute to long-range diffusion, in addition to the classical 6JC (see Figure 6). Their relative importance increases with the asymmetry parameter u = ε A A − ε B B , which controls the respective vacancy concentrations on the two B2 sublattices and the relative mobilities of A and B atoms. Moreover while diffusion by 6JC only would implies a D ∗A /D ∗B ratio between 1/2 and 2, the newly discovered antisite-assisted cycles yield to a wider range, as observed experimentally in some B2 alloys, such as Co–Ga. Moreover, high asymmetry parameters produce an upward curvature of the Arrhenius plot in the B2 domain. Similar KMC model has been applied to the L12 ordered structures and successfully explains some particular diffusion properties in these phases [30].

4.2.

Simple Unmixing: Iron–Copper Alloys

Copper precipitation in α-Fe has been extensively studied with KMC: although pure copper has an FCC structure, experimental observations show that the first step of precipitation is indeed fully coherent, up to precipitate radii of the order of 2 nm, with a Cu BCC lattice parameter very close to that of iron. The composition of the small BCC copper clusters has long been debated: early atom probe or field ion microscopy studies or small angle neutron scattering experiments suggested that they might contain more than 50% (a)

(b) 0

4

3

1

4

3

1 5

5 2

0

6

2

6

Figure 6. Classical Six Jump Cycle (a) and Antisite assisted Six Jump Cycle (b) in B2 compounds [29].

2238

G. Martin and F. Soisson

iron, while others experimental techniques suggested pure copper clusters. Using the above simple “model (a)”, KMC suggest almost pure copper precipitates, but with very irregular shapes [33]: the significant iron content measured in some experiments could then be due to the contribution of atoms at the precipitate matrix interface if a simple smooth shape is attributed to the precipitate while the small Cu clusters have very irregular shapes. This explanation is in agreement with the most direct observations using a 3D atom probe [34]. The simulations have also shown that, with the parameter values we used, fast migration of small Cu clusters occurs: the latter induces direct coagulation between nuclei, yielding ramified precipitate morphologies. On the same Fe–Cu system, Le Bouar and Soisson [22] have used an EAM potential to parameterize the activation barriers in Eq. (1). In dilute alloys, the EAM computed energies of stable and saddle-point relaxed configurations, can be reproduced with pair interactions on a rigid lattice (including some vacancy-atom interactions). The saddle-point binding energies of Fe and Cu are shown in Fig. 4 and have already been discussed. Such a dependence of the SP binding energies does not modify the thermodynamic properties of the system (the solubility limit, the precipitation driving force, the interfacial energies, the vacancy concentrations in various phases do not depend on the SP properties) and it slightly affects the diffusion coefficients of Fe and Cu in pure iron. Nevertheless such details strongly affect the precipitation kinetic pathway, by changing the diffusion coefficients of small Cu clusters and thus the balance between the two possible growth mechanisms: classical emissionadsorption of single solute atoms and direct coagulation between precipitates. This is illustrated by Fig. 7, where two simulations of copper precipitation Fe on the are displayed: one which takes into account the dependence of esp Fe local atomic composition and one with a constant esp . In the second case small copper clusters (with typically less than 10 Cu atoms) are more mobile than in the first case, which results in an acceleration of the precipitation. Moreover, the nucleation regime in Fig. 7(b) almost vanishes, because two small clusters can merge as- or even more rapidly than a Cu monomer and a precipitate. The dashed line of Fig. 7 represents the results obtained with the empirical parameter values described in the previous paragraph [33]: as can be seen these results do not differ qualitatively from those obtained by Le Bouar et al. [22], so that the qualitative interpretation of the experimental observations is conserved. The competition between the classical solute emission–adsorption and direct precipitate coagulation mechanisms observed in dilute Fe–Cu alloys appears indeed to be quite general and to have important consequences on the whole kinetic pathway. First studies [35] focused on the role of the atomic jump mechanism (Kawasaki dynamics versus vacancy jump), but recent KMC simulations based on the transition probability models (a)–(c) above have shown that both single solute atom- and cluster-diffusion are observed when

Diffusion controlled phase transformations in the solid state

2239

t (year) 3

10

(a) 0,8

101

100

101

101

DSPE ISPE

0,6 Cu

10

2

0,4 0,2 0,0

(b)

1600 1200 Np(i 1) 800 400 0

(c) 102

101

100 104

105

106

107

108

109

1010

t (s)

Figure 7. Precipitation kinetics in a Fe-3at.%Cu alloy at T = 573 K [22]. Evolution of (a) the degree of the copper short-range order parameter, (b) the number of supercritical precipFe itates and (c) the averaged size of supercritical precipitates. Monte Carlo simulations with esp depending on the local atomic configuration (•) or not (♦). The dashed lines corresponds to the results of Soisson et al. [33].

vacancy diffusion is carefully modeled. Indeed the balance between both mechanisms is controlled by: – the asymmetry parameter which controls the relative vacancy concentrations in the various phases [31]. A vacancy trapping in the precipitates (e.g., in Fe–Cu alloys) or at the precipitate-matrix interface tends to favor direct coagulation, while if the vacancy concentration is higher in the matrix, as is the case for Co precipitation in Cu, [36], the migration of monomers and emission-adsorption of single solute atoms are dominant.

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G. Martin and F. Soisson

– the saddle-point energies which, together with the asymmetry parameter, control the correlation between successive vacancy jumps and the migration of solute clusters [22].

4.3.

Nucleation/Growth/Coarsening: Comparison with Classical Theories

The classical theories of nucleation, growth or coarsening, as well as the theory of spinodal decomposition in highly supersaturated solid solutions, can be assessed using KMC simulations [37]. For the nucleation regime, the thermodynamic and kinetic data involved in Eqs. (5)–(7) (the driving force for precipitation, δµ, the interfacial energy, σ , the adsorption rate β, etc.) can be computed from the atomistic parameters used in KMC (pair interaction, saddle-point binding energies, attempt frequencies): a direct assessment of the classical theories is thus possible. For low supersaturations and in cases where only the solute monomers are mobile, the incubation time and the steady–state nucleation rate measured in the KMC simulations are very close to those predicted by the classical theory of nucleation. On the contrary, when small solute clusters are mobile (keeping the overall solute diffusion coefficient the same), the classical theory strongly overestimates the incubation time and weakly underestimates the nucleation rate, as exemplified on Fig. 8.

100 10 5 4 1014

ψi Cv(s)

109

3

2.5

2.1

1012

1010

F

1011

W

1010 J 108

1012

106

13

10

1014 3 10

T = 0.5 Ω/2k b

st

T = 0.4 Ω/2k

104 102 1/(S0 (ln S0)3)

101

102

b

T = 0.3 Ω/2kb

0

0.5

1

1.5

2

1/ (ln S0)2

Figure 8. Incubation time and steady-state nucleation rate, in a binary model alloy A–B, as eq a function of supersaturation S0 = C 0B /C B (initial/equilibrium B concentration in the solid solution). Comparison of KMC (symbols) and Classical Theory of Nucleation (lines). On the left: the dotted lines refer to two classical expressions of the incubation time (Eq. (7d)), the plain line is obtained by numerical integration of Eq. (8); KMC with mobile monomers only, KMC with small mobile clusters. On the right: the dotted and plain lines refer to Eq. (7a) with respectively Z = 1 or Z from Eq. (7b); ♦, ◦ and refer to KMC with mobile monomers. For more details, see Ref. [37].

Diffusion controlled phase transformations in the solid state

2241

The above general argument has been assessed in the case of Al3 Zr and Al3 Sc precipitation in dilute aluminum alloys: the best estimates of the parameters suggest that diffusion of Zr and Sc in Al occurs by monomer migration [38]. When the precipitation driving force and interfacial energy are computed in the frame of the Cluster Variation Method, the classical theory of nucleation predicts nucleation rates in excellent agreement with the results of the KMC simulations, for various temperatures and supersaturations. Similarly, the distribution of cluster sizes in the solid solution ρn ∼ exp(−Fn /kB T ), with Fn given by the capillarity approximation (Eq. (5)) is well reproduced, even for very small precipitate sizes.

4.4.

Precipitation in Ordered Phases

The kinetic pathways become more complex when ordering occurs in addition to simple unmixing. Such kinetics have been explored by Ath`enes [39] in model BCC binary alloys, in which the phase diagram displays a tricritical point and a two-phase field (between a solute rich B2 ordered phase and a solute depleted A2 disordered phase). The simulation was able to reproduce qualitatively the main experimental features reported from transmission electron microscopy observations during the decomposition of Fe–Al solid solutions: (i) for small supersaturations, a nucleation-growth-coarsening sequence of small B2 ordered precipitates in the disordered matrix occurs; (ii) for higher supersaturations, a short range ordering starts before any modification of the composition field, followed by a congruent ordering with a very high density of antiphase boundaries (APB). In the center of the two phase field, this homogeneous state then decomposes by a thickening of the APBs which turns into the A2 phase. Close to the B2 phase boundary, the decomposition process also involves a nucleation of iron rich A2 precipitates inside the B2 phase. Varying the asymmetry parameter u mainly affects the time scale. However qualitative differences are observed, at very early stages, in the formation of ordered microstructures: if the value of u enhances preferentially the vacancy exchanges with the majority atoms (u > 0), ordering proceeds everywhere, in a diffuse manner; while if u favors vacancy exchanges with the solute atoms (u < 0), ordering proceeds locally by patches. This could explain the experimental observation of small B2 ordered domains in as-quenched Fe-Al alloys, in cases where phenomenological theories predict a congruent ordering [39]. Precipitation and ordering in Ni(Cr,Al) FCC alloys have been studied by Pareige et al. [21], with MC parameters fitted to thermodynamic and diffusion properties of Ni-rich solid solutions (Fig. 9a). For relatively small Cr and Al

2242

G. Martin and F. Soisson

30 nm

(a)

(b)

Figure 9. (a) Microstructure of a Ni-14.9at.%Cr-5.2at%Al alloy after a thermal ageing of 1 h at 600◦ C. Monte Carlo simulation (left) and 3D atom probe image (right). Each dot represents an Al atom (for the sake of clarity, Ni and Cr atoms are not represented). One observes the Al-rich 100 planes of γ precipitates, with an average diameter of 2 nm [21]. (b) Monte Carlo simulation of NbC precipitation in ferrite with transient precipitation of a metastable iron carbide, shown in faint in the snapshots at 1.5, 11 and 25 seconds [28].

Diffusion controlled phase transformations in the solid state

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contents, at 873 K, the phase transformation occurs in three stages: (i) a short range ordering of the FCC solid solution, with two kinds of ordering symmetry (a “Ni3 Cr” symmetry corresponding to the one observed at high temperature in binary Ni–Cr alloys, and an L12 symmetry) followed by a nucleation-growthcoarsening sequence, (ii) the formation of the Al-rich γ precipitates (with L12 structure), (iii) the growth and coarsening of the precipitates. In the γ phase Cr atoms substitute for both Al and Ni atoms, with a preference for the Al sublattice. The simulated kinetics of precipitation are in good agreement with 3D-atom probe observations during a thermal ageing of the same alloy, at the same temperature [21]. For higher Cr and Al contents, MC simulations predict an congruent L12 ordering (with many small antiphase domains) followed by the γ − γ decomposition, as in the A2/B2 case discussed above.

4.5.

Interstitial and Vacancy Diffusion in Parallel

Advanced high purity steels offer a field of application of KMC with practical relevance. In so called High-Strength Low-Alloy (HSLA) steels, Nb is used as a means to retain carbon in niobium carbide precipitates, out of solution in the BCC ferrite. The precipitation of NbC implies the migration, in the BCC Fe lattice, of both Nb, by vacancy mechanism, and C, by direct interstitial mechanism. At very early stages, the formation of coherent NbC clusters on the BCC iron lattice is documented from 3D atom probe observations. The very same Monte Carlo technique can be used [28]; the new feature is the large value of the number of channels by which a configuration can decay, because of the many a priori possible jumps of the numerous carbon atoms. This makes step 3 of the algorithm above, very time consuming. A proper grouping of the channels, as a function of their respective decay time, helps speeding up this step. Among several interesting features, KMC simulations revealed the possibility for NbC nucleation to be preceded by the formation of a transient iron carbide, due to the rapid diffusion of C atoms by comparison with Nb and Fe diffusion (Fig. 9b). This latter kinetic pathway is found to be sensitive to the ability of the microstructure to provide the proper equilibrium vacancy concentration during the precipitation process.

4.6.

Driven Alloys

KMC offers a unique tool to explore the stability and the evolution of the microstructure in “Driven Alloys”, i.e., alloys exposed to a steady flow of energy, such as alloys under irradiation, or ball milling, or cyclic loading. . . [40]. Atoms in such alloys, change position as a function of time because of two mechanisms acting in parallel: one of the thermal diffusion mechanisms as discussed above, on the one hand, and forced, or “ballistic jumps”

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on the other hand. The latter occur with a frequency imposed by the coupling with the surrounding of the system: their frequency is proportional to some “forcing intensity” (e.g., the irradiation flux). This situation is reminiscent of the “Kinetic Ising Model with two competing dynamics”, much studied in the late 80s. However, one observes a strong sensitivity of the results to the details of the diffusion mechanism and of the ballistic jumps. The main results are : – a solubility limit which is a function both of the temperature and of the ratio of the frequencies of ballistic to thermally activated jumps (i.e., on the forcing intensity); – at given temperature and forcing intensity, the solubility limit may also depend on the number of ballistic jumps to occur at once (“cascade size effect”); – the “replacement distance”, i.e., the distance of ballistic jumps has a crucial effect on the phase diagrams as shown in Fig. 10. For appropriate replacement distances, self-patterning can occur, with a characteristic length, which depends on the forcing intensity and on the replacement distance [41]. What has been said of the solubility limit also applies to the kinetic pathways followed by the microstructure when the forcing conditions are changed. Such KMC studies and the associated theoretical work helped to understand, for alloys under irradiation, the respective effects of the time and space structure of the elementary excitation, of the dose rate and of the integrated dose (or “fluence”). (a)

(A) G 5 104 s1

(B) 103 s1

(b) 2

1 Patterning

Solid Solution

(C) 102 s1

(D) 1 s1

R (ann)

10

1

Macroscopic Phase Separation 102 101 100

101

102

103

104

105

(s1)

Figure 10. (a) Steady–state microstructures in KMC simulations of the phase separation in a binary alloy, for different ballistic jump frequencies . (b) Dynamical phase diagram showing the steady–state microstructure as a function of the forcing intensity and the replacement distance R [41].

Diffusion controlled phase transformations in the solid state

5.

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Conclusion and Future Trends

The above presentation is by no means exhaustive. It aimed mainly at showing the necessity to model carefully the diffusion mechanism, and the techniques to do so, in order to have a realistic kinetic pathway for solid state transformations. All the examples we gave are based on a rigid lattice description. The latter is correct as long as strain effects are not too large, as shown by the discussion of the Fe(Cu) alloy. Combining KMC for the configuration together with some technique to handle the relaxation of atomic positions is quite feasible, but for the time being requires a heavy computation cost if the details of the diffusion mechanism are to be retained. Interesting results have been obtained e.g., for the formation of strained hetero-epitaxial films [42]. A field of growing interest is the first principle determination of the parameters entering the transition probabilities. In view of the lack of experimental data for relevant systems, and of the fast improvement of such techniques, no doubt such calculations will be of extreme importance. Finally, at the atomic scale, all the transitions modeled so far are either thermally activated or forced at some imposed frequency. A field of practical interest is where “stick and slip” type processes are operating: such is the case in shear transformations, in coherency loss etc. Incorporating such processes in KMC treatment of phase transformations has not yet been attempted to our knowledge, and certainly deserves attention.

Acknowledgments We gratefully acknowledge many useful discussions with our colleagues at Saclay and at the Atom Probe Laboratory in the University of Rouen, as well as with Prs. Pascal Bellon (UICU) and David Seidman (NWU).

Appendix: The Kinetic Ising Model In the KIM, the kinetic version of the model proposed by Ising for magnetic

materials, the configurational Hamiltonian writes H = i=/ j Ji j σi σ j + i h i σi , with σ ι = ± 1, the spin at site i, Ji j , the interaction parameter between spins at sites i and j , and h i the external field on site i. The probability of a transition per unit time, between two configurations {σι } and {σι } is chosen as: W{σ },{σ } = w exp[−(H − H )/2kB T ], with w for the inverse time unit. Two models are studied:

KIM with conserved total spin, for which i σi = so that the configuration after the transition is obtained by permuting the spins on two (nearest neighbor) sites;

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KIM with non-conserved total spin, for which the new configuration is obtained by flipping one spin on one given site. When treated by Monte Carlo technique, two types of algorithms are currently applied to KIM: Metropolis’ algorithm, where the final configuration is accepted with probability one if (H − H ) ≤ 0, and with probability exp[−(H − H )/kB T ] if (H − H ) > 0. Glauber’s the final configuration is accepted with proba algorithm, where bility 1/2 1 + tanh(−(H − H )/2kB T ) .

References [1] A.R. Allnatt and A.B. Lidiard, “Atomic transport in solids,” Cambridge University Press, Cambridge, 1994. [2] T. Morita, M. Suzuki, K. Wada, and M. Kaburagi, “Foundations and Applications of Cluster Variation Method and Path Probability Method,” Prog. Theor. Phys. Supp., 115, 1994. [3] K. Binder, “Applications of Monte Carlo methods to statistical physics,” Rep. Prog. Phys., 60, 1997. [4] Y. Limoge and J.-L. Bocquet, “Monte Carlo simulation in diffusion studies: time scale problems,” Acta Met., 36, 1717, 1988. [5] G.E. Murch and L. Zhang, “Monte Carlo simulations of diffusion in solids: some recent developments,” In: A.L. Laskar et al. (eds.), Diffusion in Materials, Kluwer Academic Publishers, Dordrecht, 1990. [6] C.P. Flynn, “Point defects and diffusion,” Clarendon Press, Oxford, 1972. [7] J. Philibert, “Atom movements, diffusion and mass transport in solids,” Les Editions de Physique, Les Ulis, 1991. [8] D.N. Seidman and R.W. Balluffi, “Dislocations as sources and sinks for point defects in metals,” In: R.R. Hasiguti (ed.), Lattice Defects and their Interactions, GordonBreach, New York, 1968. [9] J.-L. Bocquet, G. Brebec, and Y. Limoge, “Diffusion in metals and alloys,” In: R.W. Cahn and P. Haasen (eds.), Physical Metallurgy, North-Holland, Amsterdam, 1996. [10] M. Nastar, V.Y. Dobretsov, and G. Martin, “Self consistent formulation of configurational kinetics close to the equilibrium: the phenomenological coefficients for diffusion in crystalline solids,” Philos. Mag. A, 80, 155, 2000. [11] G. Martin, “The theories of unmixing kinetics of solids solutions,” In: Solid State Phase Transformation in Metals and Alloys, pp. 337–406. Les Editions de Physique, Orsay, 1978. [12] A. Perini, G. Jacucci, and G. Martin, “Interfacial contribution to cluster free energy,” Surf. Sci., 144, 53, 1984. [13] T.R. Waite, “Theoretical treatment of the kinetics of diffusion-limited reactions,” Phys. Rev., 107, 463–470, 1957. [14] I.M. Lifshitz and V.V. Slyosov, “The kinetics of precipitation from supersaturated solid solutions,” Phys. Chem. Solids, 19, 35, 1961. [15] C.J. Kuehmann and P.W. Voorhees, “Ostwald ripening in ternary alloys,” Metall. Mater Trans., 27A, 937–943, 1996.

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[16] J.W. Cahn, W. Craig Carter, and W.C. Johnson (eds.), The selected works of J.W. Cahn., TMS, Warrendale, 1998. [17] G. Martin, “Atomic mobility in Cahn’s diffusion model,” Phys. Rev. B, 41, 2279– 2283, 1990. [18] C. Desgranges, F. Defoort, S. Poissonnet, and G. Martin, “Interdiffusion in concentrated quartenary Ag–In–Cd–Sn alloys: modelling and measurements,” Defect Diffus. For., 143, 603–608, 1997. [19] S.M. Allen and J.W. Cahn, “A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsening,” Acta Metal., 27, 1085–1095, 1979. [20] P. Bellon and G. Martin, “Coupled relaxation of concentration and order fields in the linear regime,” Phys. Rev. B, 66, 184208, 2002. [21] C. Pareige, F. Soisson, G. Martin, and D. Blavette, “Ordering and phase separation in Ni–Cr–Al: Monte Carlo simulations vs Three-Dimensional atom probe,” Acta Mater., 47, 1889–1899, 1999. [22] Y. Le Bouar and F. Soisson, “Kinetic pathways from EAM potentials: influence of the activation barriers,” Phys. Rev. B, 65, 094103, 2002. [23] E. Clouet and N. Nastar, “Monte Carlo study of the precipitation of Al3 Zr in Al–Zr,” Proceedings of the Third International Alloy Conference, Lisbon, in press, 2002. [24] J.-L. Bocquet, “On the fly evaluation of diffusional parameters during a Monte Carlo simulation of diffusion in alloys: a challenge,” Defect Diffus. For., 203–205, 81–112, 2002. [25] R. LeSar, R. Najafabadi, and D.J. Srolovitz, “Finite-temperature defect properties from free-energy minimization,” Phys. Rev. Lett., 63, 624–627, 1989. [26] A.P. Sutton, “Temperature-dependent interatomic forces,” Philos. Mag., 60, 147– 159, 1989. [27] Y. Mishin, M.R. Sorensen, F. Arthur, and A.F. Voter, “Calculation of point-defect entropy in metals,” Philos. Mag. A, 81, 2591–2612, 2001. [28] D. Gendt, Cin´etiques de Pr´ecipitation du Carbure de Niobium dans la ferrite, CEA Report, 0429–3460, 2001. [29] M. Ath`enes, P. Bellon, and G. Martin, “Identification of novel diffusion cycles in B2 ordered phases by Monte Carlo simulations,” Philos. Mag. A, 76, 565–585, 1997. [30] M. Ath`enes and P. Bellon, “Antisite diffusion in the L12 ordered structure studied by Monte Carlo simulations,” Philos. Mag. A, 79, 2243–2257, 1999. [31] A. Ath`enes, P. Bellon, and G. Martin, “Effects of atomic mobilities on phase separation kinetics: a Monte Carlo study,” Acta Mater., 48, 2675, 2000. [32] R. Wagner and R. Kampmann, “Homogeneous second phase precipitation,” In: P. Haasen (ed.), Phase Transformations in Materials, VCH, Weinhem, 1991. [33] F. Soisson, A. Barbu, and G. Martin, “Monte Carlo simulations of copper precipitation in dilute iron-copper alloys during thermal ageing and under electron irradiation,” Acta Mater., 44, 3789, 1996. [34] P. Auger, P. Pareige, M. Akamatsu, and D. Blavette, “APFIM investigation of clustering in neutron irradiated Fe–Cu alloys and pressure vessel steels,” J. Nucl. Mater., 225, 225–230, 1995. [35] P. Fratzl and O. Penrose, “Kinetics of spinodal decomposition in the Ising model with vacancy diffusion,” Phys. Rev. B, 50, 3477–3480, 1994. [36] J.-M. Roussel and P. Bellon, “Vacancy-assisted phase separation with asymmetric atomic mobility: coarsening rates, precipitate composition and morphology,” Phys. Rev. B, 63, 184114, 2001. [37] F. Soisson and G. Martin, Phys. Rev. B, 62, 203, 2000.

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[38] E. Clouet, M. Nastar, and C. Sigli, “Nucleation of Al3 Zr and Al3 Sc in aluminiun alloys: from kinetic Monte Carlo simulations to classical theory,” Phys. Rev. B, 69, 064109, 2004. [39] M. Ath`enes, P. Bellon, G. Martin, and F. Haider, “A Monte Carlo study of B2 ordering and precipitation via vacancy mechanism in BCC lattices,” Acta Mater., 44, 4739–4748, 1996. [40] G. Martin and P. Bellon, “Driven alloys,” Solid State Phys., 50, 189, 1997. [41] R.A. Enrique and P. Bellon, “Compositional patterning in immiscible alloys driven by irradiation,” Phys. Rev. B, 63, 134111, 2001. [42] C.H. Lam, C.K. Lee, and L.M. Sander, “Competing roughening mechanisms in strained heteroepitaxy: a fast kinetic Monte Carlo study,” Phys. Rev. Lett., 89, 216102, 2002.

7.10 DIFFUSIONAL TRANSFORMATIONS: MICROSCOPIC KINETIC APPROACH I.R. Pankratov and V.G. Vaks Russian Research Centre, “Kurchatov Institute”, Moscow 123182, Russia

The term “diffusional transformations” is used for the phase transformations (PTs) of phase separation or ordering of alloys as these PTs are realized via atomic diffusion, i.e., by interchange of positions of different species atoms in the crystal lattice. Studies of kinetics of diffusional PTs attract interest from both fundamental and applied points of view. From the fundamental side, the creation and evolution of ordered domains or precipitates of a new phase provide classical examples of the self-organization phenomena being studied in many areas of physics and chemistry. From the applied side, the macroscopic properties of such alloys, such as their strength, plasticity, coercivity of ferromagnets, etc., depend crucially on their microstructure, in particular, on the distribution of antiphase or interphase boundaries separating the differently ordered domains or different phases, while this microstructure, in its turn, sharply depends on the thermal and mechanical history of an alloy, in particular, on the kinetic path taken during the PT. Therefore, the kinetics of diffusional PTs is also an important area of Materials Science. Theoretical treatments of these problems employ usually either Monte Carlo simulation or various phenomenological kinetic equations for the local concentrations and local order parameters. However, Monte Carlo studies in this field are difficult, and until now they provided limited information about the microstructural evolution. The phenomenological equations are more feasible, and they are widely used to describe the diffusional PTs, see, e.g., Turchi and Gonis [1], part I. However, a number of arbitrary assumptions are usually employed in such equations, and their validity region is often unclear [2]. Recently, the microscopic statistical approach has been suggested to treat the diffusional PTs [3–5]. It aims to develop the theoretical methods which can describe the non-equilibrium alloys as consistently and generally as the canonical Gibbs method describes the equilibrium systems. This approach was used for simulations of many different PTs. The simulations revealed a number 2249 S. Yip (ed.), Handbook of Materials Modeling, 2249–2268. c 2005 Springer. Printed in the Netherlands.

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of new and interesting microstructural effects, many of them agreeing well with experimental observations. Below we describe this approach.

1. 1.1.

Statistical Theory of Non-equilibrium Alloys Master Equation Approach: Basic Equations

A consistent microscopical description of non-equilibrium alloys can be based on the fundamental master equation for the probabilities of various atomic distributions over lattice sites [3, 4]. For definiteness, we consider a binary alloy Ac B1−c with c ≤ 0.5. Various distributions of atoms over lattice sites i are described by the sets of occupation numbers {n i } where the operator n i = n Ai is unity when the site i is occupied by atom A and zero otherwise. The interaction Hamiltonian H has the form H=

vi j ni n j +

vi j k ni n j nk + · · ·

(1)

i> j >k

i> j

where v i... j are effective interactions. The fundamental master equation for the probability P of finding the occupation number set {n i } = α is dP(α) = [W (α, β)P(β) − W (β, α)P(α)] ≡ Sˆ P dt β

(2)

where W (α, β) is the β → α transition probability per unit time. Adopting for this probability the conventional “thermally activated atomic exchange model”, we can express the transfer matrix Sˆ in Eq. (2) in terms of the probability WiAB j of an elementary inter-site exchange Ai B j : s ˆ in ˆ in WiAB j = n i n j ωi j exp[−β(E i j − E i j )] ≡ n i n j γi j exp(β E i j ).

(3)

Here n j = n B j = (1 − n j ); ωi j is the attempt frequency; β = 1/T is the reciprocal temperature; E isj is the saddle point energy; γi j is ωi j exp(−β E isj ); and Eˆ iinj is the initial (before the jump) configurational energy of jumping atoms. The most general expression for the probability P{n i } in (2) can be conveniently written in the “generalized Gibbs” form:

P{n i } = exp

β

+

i

λi n i − Q

.

(4)

Diffusional transformations: microscopic kinetic approach

2251

Here the parameters λi can be called the “site chemical potentials”; the “quasiHamiltonian” Q is an analogue of the hamiltonian H in (1); and the generalized grand-canonical potential = {λi , ai... j } is determined by the normalizing condition: Q=

ai j n i n j +

ai j k n i n j n k + · · ·

i> j >k

i> j

= −T ln Tr exp

β

λi n i − Q

(5)

i

where Tr (. . .) means the summation over all configurations {n i }. Multiplying Eq. (2) by operators n i , n i n j , etc., and summing over all configurations, we obtain the set of exact kinetic equations for averages gi j ...k = n i n j . . . n k , in particular, for the mean site occupation ci ≡ gi = n i where . . . means Tr (. . . )P: dgi... j ˆ = n i . . . n j S. (6) dt These equations enable us to derive an explicit expression for the free energy of a non-equilibrium state, F = F{ci , gi... j }, which obeys both the “generalized first” and the second law of thermodynamics: F = H + T ln P = + dF =

iα

λi =

λi dci +

∂F ∂ci

λi ci + H − Q

iα

(v i... j − ai... j ) dgi... j

i>... j

(v i... j − ai... j ) =

dF ≤ 0. dt

∂F ∂gi... j (7)

The stationary state (being not necessarily uniform) corresponds to the minimum of F with respect to its variables ci and gi... j provided the total number of atoms N A = i ci is fixed. Then the relations (7) yield the usual, Gibbs equilibrium equations: λi = µ = constant; ai... j = v i... j ,

or :

(8) Q = H.

(9)

Non-stationary atomic distributions arising under the usual conditions of diffusional PTs appear to obey the “quasi-equilibrium” relations which correspond to an approximate validity in the course of the evolution of the second

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equilibrium Eq. (9), while the site chemical potentials, generally, differ with each other [2]. Then the free energy F in (7) takes the form: F =+

λi ci

(10)

i

while the system of Eq. (6) is reduced to the “quasi-equilibrium” kinetic equation (QKE) for the mean occupations ci = ci (t) [3]:

β(λ j − λi ) dci Mi j 2 sinh . = dt 2 j

(11)

Here the quantities λ j are related to ci by the self-consistency equation:

ci = n i = Tr n i P{λ j }

(12)

while the “generalized mobility” Mi j for the pair interaction case, when the Hamiltonian (1) includes only the first term, can be written as [6]:

Mi j = γi j n i n j exp

β λ + λ − (v + v + u + u )n i j jk ik jk k k ik

2

. (13)

AB BB Here γi j , n i and v i j = ViAA j − 2Vi j + Vi j are the same as in Eqs. (3) and (1), AA BB while u i j = Vi j − Vi j is the so-called asymmetric potential. The description of the diffusional PTs in terms of the mean occupations ci given by Eqs. (11)–(13) seems to be sufficient for the most situations of practical interest, in particular, for the “mesoscopic” stages of such PTs when the local fluctuations of occupations are insignificant. At the same time, to treat the fluctuative phenomena, such as the formation and evolution of critical nuclei in metastable alloys, one should modify the QKE (11), for example, by an addition of some “Langevin-noise”-type terms [4].

1.2.

Kinetic Mean-field and Kinetic Cluster Approximations

To find explicit expressions for the functions F{ci }, λi {c j }, and Mi j {ck } in Eqs. (10)–(12), one should employ some approximate method of statistical physics. Several such methods have been developed [4]. For simplicity we consider the pair interaction model and write the interaction v i j in (1) as δi j,n v n where the symbol δi j,n is unity when sites i and j are nth neighbors in the lattice and zero otherwise, while v n is the interaction constant. Then the simplest,

Diffusional transformations: microscopic kinetic approach

2253

“kinetic mean-field” approximation (KMFA, or simply MFA) corresponds to the following expressions for , λi and Mi j : MFA =

T ln ci −

i

λMFA i

=T

ln (ci /ci )

1 δi j,n v n ci c j 2 i, j,n

+

(14)

δi j,n v n c j

j,n

MiMFA j

= γi j

ci ci c j cj

exp β

1/2

(u ik + u j k )ck )

.

(15)

k

Here ci is 1 − ci , while the free energy F is related to and λi by Eq. (10). For a more refined and usually more accurate, kinetic pair-cluster approximation (KPCA, or simply PCA), the expressions for and λi are more complex but still can be written analytically: PCA =

T ln ci +

i

λPCA i

= T ln (ci /ci ) +

1 δi j,n inj 2 i, j,n

(16)

ij

δi j,n λni .

j,n ij

Here inj = − T ln(1 − ci c j gni j ); λni = −T ln(1 − c j gni j ); and the function gni j is expressed via the Mayer function f n = exp (−βv n ) − 1 and the mean occupations ci and c j : gni j = Rni j

2 fn ij Rn

+ 1 + f n (ci + c j )

= [1 + (ci + c j ) f n ] − 4ci c j f n ( f n + 1) 2

1/2

(17) .

For the weak interaction, βv n 1, the function gni j becomes (−βv n ), inj − ij v i j ci c j , λni v n c j , and the PCA expressions (16) become the MFA ones (14). The MFA or the PCA is usually sufficient to describe the PTs between the disordered phases and/or the BCC-based ordered phases, such as the B2 and D03 phases. However, these simple methods are insufficient to describe the FCC-based L12 and L10 ordered alloys as strong many-particle correlations are characteristic of such systems. These alloys can be adequately described by the cluster variation method (CVM) which takes into account the correlations mentioned within at least 4-site tetrahedron cluster of nearest neighbors. However, the CVM is cumbersome, and it is difficult to use it for the non-homogeneous systems. At the same time, a simplified version of CVM, the tetrahedron cluster-field approximation (TCA), usually combines the high accuracy of CVM with great simplification of calculations [6].

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The TCA expressions for and λi can be written explicitly and are similar to those in Eq. (16), but to find the functions (ci ) and λi (c j ) explicitly one should solve the system of four algebraic equations for each tetrahedron cluster. In practice, these equations can easily be solved numerically using the conjugate gradients method [4, 7]. We can also use the PCA or the TCA methods to more accurately calculate the mobility Mi j in the expression (13) [4]. However, in this expression the above-mentioned correlations of atomic positions result only in some quantitative factors that weakly depend on the local composition and ordering and seem to be of little importance for the microstructural evolution. Therefore, the simple MFA expression (15) for Mi j was employed in the previous KTCAbased simulations of the L12 and L10 -type orderings [4, 7].

1.3.

Deformational Interactions in Dilute and Concentrated Alloys

The effective interaction v i... j in the Hamiltonian (1) includes the “chemic cal” contribution v i... j which describes the energy change under the substitution of some atoms A by atoms B in the rigid lattice, and the “deformational” d term v i... j due to the difference in the lattice deformation under such a substitution. The interaction v d includes the long-range elastic forces which can significantly affect the microstructural evolution, see, e.g., Turchi and Gonis [1]. A microscopical model to calculate the interaction v d in dilute alloys was suggested by Khachaturyan [8]. In the concentrated alloys, the deformational interaction can lead to some new effects, in particular, to the lattice symmetry change under PT, such as the tetragonal distortion under L10 ordering. Below we describe the generalization of the Khachaturyan’s model of deformational interactions to the case of a concentrated alloy [9]. Supposing a displacement uk of site k relative to its position Rk in the “average” crystal Ac B1−c to be small, we can write the alloy energy H as H = Hc {n i } −

u αk Fαk +

k

1 u αk u βl Aαk,βl 2 αk,βl

(18)

where α and β are Cartesian indices and both the Kanzaki force Fk and the force constant matrix Aαk,βl are some functions of occupation numbers n i . For the force constant matrix, the conventional average crystal approximation seems usually to be sufficient: Aαk,βl {n i } → Aαk,βl {c} ≡ A¯ αk,βl . The Kanzaki force Fαk can be written as a series in the occupation numbers n i : Fαk =

i

(1) f αk,i ni +

i> j

(2) f αk,i j ni n j + · · ·

(19)

Diffusional transformations: microscopic kinetic approach

2255

where the coefficients f (n) do not depend on n i . Minimizing the energy (18) with respect to displacements uk we obtain for the deformational Hamiltonian Hd: 1 Fαk ( A¯ −1 )αk,βl , Fβl (20) Hd = − 2 αk,βl where ( A¯ −1 )αk,βl means the matrix inverse to A¯ αk,βl which can be written ¯ explicitly using the Fourier transformation of the force constant matrix A(k). For the dilute alloys, one can retain in (19) only the first sum which corresponds to a pairwise H d by Khachaturyan [8]. The next terms in (19) lead to non-pairwise interactions which describe, in particular, the above-mentioned effects of a lattice symmetry change. To describe these effects, for example, for the case of the L10 ordering in the FCC lattice, we can retain in (19) only terms with f (1) and f (2) and estimate them from the experimental data about the concentration dilatation in the disordered phase and about the lattice parameter changes under the L12 and L10 orderings [6, 9].

1.4.

Vacancy-mediated Kinetics and Equivalence Theorem

In the most theoretical treatments of kinetics of diffusional PT, as well as in the previous part of this paper, the simplified direct exchange model was used which assumes direct exchange of positions between unlike neighboring atoms in an alloy. Actually, the exchange occurs between the main alloy component atom, e.g., A or B atom in an ABv alloy, and the neighboring vacancy “v”. As the vacancy concentration cv in alloys is actually quite small, cv 10−4 , employing the direct exchange model greatly simplifies the theoretical studies by reducing the computation times by several orders of magnitude. However, it is not clear a priori whether using the unrealistic direct exchange model results in some errors or missing some effects. In particular, a notable segregation of vacancies at interphase or antiphase boundaries was observed in a number of simulations, and the problem of possible influence of this segregation on the microstructural evolution was discussed by a number of authors. To clarify these problems, the statistical approach described above has been generalized to the vacancy-mediated kinetics case [5]. In particular, the QKE for an ABv alloy, instead of Eq. (11), takes the form of a set of equations for the A-atom and the vacancy mean occupations, cAi = ci and cvi :

dci Av = γi j Bi j eβ(λA j + λvi ) − eβ(λAi + λv j ) dt j

dcvi vA βλAi Bi j eβλv j γivB = j + γi j e dt j

(21)

− {i → j } .

(22)

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Here Bi j is an analogue of the second factor in Eq. (13), while λAi and λvi are the site chemical potentials for the A atom and the vacancy, respectively, in (14): which in the MFA have the form similar to λMFA i

λMFA Ai

= T ln

ci ci

+

v iAA j cj;

λMFA vi

j

= T ln

cvi ci

+

v ivA j cj

j

(23) where v ivA j is an effective interaction between a vacancy and an A atom. The main alloy components kinetics determined by the QKE (21) can usually be described in terms of a certain equivalent direct exchange model; this statement can be called “the equivalence theorem”. To prove it, we first note that the factor exp(βλvi ) in Eqs. (21) and (22) is proportional to the vacancy concentration cvi , which is illustrated by (22) and is actually a general relation of thermodynamics of dilute solutions. Thus the time derivatives of the mean occupations are proportional to the local vacancy concentration cvi or cv j , which is natural for the vacancy-mediated kinetics. As cvi is quite small, this implies that the main component relaxation times are by a factor 1/cvi larger than the time of the relaxation of vacancies to their “quasi-equilibrium” distribution cvi {ci } minimizing the free energy F{cvi , ci } at the given main component distribution {ci }. Therefore, neglecting the small correction of the relative order of cvi 1, we can find this “adiabatic” vacancy distribution cvi by equating the left-hand side of (22) to zero. Employing for simplicity the vB conventional nearest-neighbor vacancy exchange model: γivB j = δi j,1 γnn and vA vA γi j = δi j,1 γnn , we can solve this equation explicitly. The solution corresponds to the first term in square brackets in (22) to be constant not depending on the site number i, though it can, generally, depend on time: νi =

vB γnn exp(βλvi ) = ν(t) vB Av [γnn + γnn exp(βλρi )]c¯v

(24)

vB and the average concentration of vacancies c¯v where the common factor γnn are introduced for convenience. Relations (24) determine the adiabatic vacancy distribution cvi {ci } mentioned above. Substituting these relations into (21) we obtain the QKE for the main alloy component which has the “direct exchange” form (11) with an effective rate vA γieff j = γi j c¯v ν(t).

(25)

Physically, the opportunity to reduce the vacancy-mediated kinetics to the equivalent direct exchange kinetics is connected with the above-mentioned fact that in the course of the alloy evolution the vacancy distribution adiabatically fast follows that of the main components. Thus it is natural to believe that

Diffusional transformations: microscopic kinetic approach

2257

for the quasi-equilibrium stages of evolution under consideration such equivalence holds not only for the nearest-neighbor vacancy exchange model but is actually a general feature of any vacancy-mediated kinetics. In more detail, features of the vacancy-mediated kinetics for both the phase separation and the ordering case have been discussed by Belashchenko and Vaks [5] who used computer simulations based on Eqs. (21) and (22). The simulations confirmed the equivalence theorem for the “quasi-equilibrium” stages of evolution, t τAB , where τAB is the mean time needed for an exchange of neighboring A and B atoms. The function ν(t) in (24) was found to monotonously increase with the PT time t, and in the course of the PT this function slowly approaches its equilibrium value ν∞ . At the same time, at very early stages of PT, for times t less than the vacancy distribution equilibration time τve , the equivalence theorem does not hold as the spatial fluctuations in the initial vacancy distribution are here important. These fluctuations can lead, in particular, to a peculiar phenomenon of “localized ordering” observed by Allen and Cahn [10] in Fe–Al alloys. However, at later times t τve ∼ τAB · cv1/3 , the vacancy distribution equilibrates and the equivalence theorem holds.

2.

Applications of Statistical Approach for Simulation of Diffusional Transformations

Numerous applications of the above-described statistical methods for simulation of diffusional PTs are discussed and compared to experimental observations in reviews [4, 7]. Below we illustrate these applications with some examples.

2.1.

Methods of Simulation

Most of these simulations were based on the QKE (11). For the mobility Mi j in this equation, the MFA expression (15) with the “nearest-neighbor symmetric atomic exchange”, γi j = δi j,1 γnn and u i j = 0, was usually used. Vaks, Beiden and Dobretsov [11] also considered the effect of an asymmetric potential u i j =/ 0 on spinodal decomposition. For the site chemical potential λi in the disordered phase and in the BCC-based ordered phases, the MFA expression (14) was employed which is usually sufficient to describe PTs between these phases. The simulations of the L12 - and L10 -type orderings in FCC alloys were based on the KTCA expressions. Equations (11) were usually solved by the 4th-order Runge–Kutta method [12] with the dimensionless time variable t = tγnn and the variable time-step t . This time-step was chosen so that the maximum variation | ci | = |ci (t + t ) − ci (t )| for one time-step does not exceed 0.01. The typical t values were 0.01 − 0.1, depending on

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the evolution stage. For the PTs after a quench of a disordered alloy, the initial as-quenched distribution ci =c(Ri ) at t =0 was characterized by its mean value c and small random fluctuations δci ±0.01. The most of simulations were performed on 2D lattices with periodic boundary conditions as it enables us to study more sizable structures. However, some main conclusions were also verified by 3D simulations with periodic boundary conditions.

2.2.

Spinodal Decomposition of Disordered Alloys

Vaks, Beiden and Dobretsov [11] simulated spinodal decomposition (SD) of a disordered alloy after its quench into the spinodal instability area in the c, T plane. The interaction v i j = v(ri j ) = v(Ri − R j ) was assumed to be Gaussian and long-ranged: v(r)=−A exp (−r 2 /rv2 ) with rv2 a 2 and the constant A proportional to the critical temperature Tc . Some results of this simulation are presented in Figs. 1 and 2. The figures illustrate the transition from the initial stage of SD corresponding to the development of non-interacting Cahn’s concentration waves with growing amplitudes (see, e.g., [8]) to the next stages, first to the stage of non-linear interaction of concentration waves (Fig. 1), and then to the stage of interaction and fusion of new-formed precipitates via a peculiar “bridge” mechanism

(a)

(b)

Figure 1. Profiles of the concentration c(r) at spinodal decomposition for the 2D model described in the text at c = 0.35; T = T/Tc = 0.4, u i j = 0 , and the following values of the reduced time t = tγnn : (a) 5; and (b) 10. Distances at the horizontal axes are given in the interaction radius rv units.

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Figure 2. Distribution of c(r) for the same model as in Fig. 1 at the following t : (a) 20, (b) 120, (c) 130, (d) 140, (e) 160, (f) 180, (g) 200, and (h) 5000. The grey level linearly varies with c(r) for c between 0 and 1 from completely dark to completely bright.

illustrated by Fig. 2. This mechanism was discussed in detail by Vaks, Beiden and Dobretsov [11], while the microstructures shown in Fig. 2 reveal a striking similarity with those observed in the recent experimental studies of SD in some liquid mixtures [4].

2.3.

Kinetics of B2 and D03 -type Orderings

The B2 order corresponds to the splitting of the BCC lattice into two cubic sublattices, a and b, with the displacement vector rab = [1, 1, 1]a/2 and the mean occupations ca = c + η and cb = c − η where η is the order parameter. There are two types of antiphase ordered domain (APD) differing with the sign of η, and one type of antiphase boundary (APB) separating these APDs. The inhomogeneously ordered alloy states including APBs can be conveniently described in terms of the local order parameter ηi = η(Ri ) and the local concentration c¯i = c(Ri ) obtained by the averaging of mean occupations ci over site i and its nearest neighbors: c¯i =

1 1 1 1 ci + cj ηi = ci − c j exp(ik1 Ri ). (26) 2 z nn j =nn(i) 2 z nn j =nn(i)

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Here index nn(i) means the summation over nearest-neighbors of site i; z nn is the number of such neighbors, i.e., 4 for the 2D square lattice and 8 for the 3D BCC lattice; and the superstructure vector k1 is (1, 1)2π/a or (1, 1, 1)2π/a for the 2D or 3D case, respectively. Dobretsov, Martin and Vaks [13] investigated kinetics of phase separation with B2 ordering using the KMFA-based 2D simulations on a square lattice of 128 × 128 sites and the Fe–Al-type interaction model. The simulations enabled one to specify the earlier phenomenological considerations [10] and to find a number of new effects. As an illustration, in Fig. 3 we show the evolution after a quench of an alloy from the disordered A2 phase to the two-phase state in which SD into the B2 and the A2 phases takes place. The volume ratio of these two phases in the final mixture is the same as that for the disordered “dark” and “bright” phases in Fig. 2, and so one might expect a similarity of microstructural evolution for these two transformations. However, the formation of numerous APBs at the initial, “congruent ordering” stage of PT A2 → A2 + B2 (which occurs at an approximately unchanged initial concentration c) ¯ and the subsequent “wetting” of these APBs by the A2 phase lead to significant structural differences with the SD into disordered phases. In particular, the concentration c(r) ¯ and the order parameter η(r) at the first stages of SD shown in Figs. 3(a)–3(c) form a “ridge-valley”-like pattern, rather

Figure 3. Temporal evolution of mean occupationals ci =c(ri ) for the Fe–Al-type alloy model under PT A2→A2+B2 at c = 0.175, T = 0.424, and the following t : (a) 50, (b) 100, (c) 200, (d) 1000, (e) 4000, and (f) 9000.

Diffusional transformations: microscopic kinetic approach

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than the “hill-like” pattern seen in Fig. 1. For the PT B2 → A2 + B2, the simulations reveal some peculiar microstructural effects in vicinity of initial APBs, the formation of wave-like distributions, “broken layers” of ordered and disordered domains parallel to the initial APB, and these results agree well with experimental observations for Fe–Al alloys [4, 10]. For the homogeneous D03 phase, the mean occupation ci can be written as ci = c + η exp(ik1 Ri ) + ζ [exp(ik2 Ri )sgn(η) + exp(−ik2 Ri )].

(27)

Here Ri is the BCC lattice vector of site i; k2 = [111]π/a is the D03 superstructure vectors, and η or ζ is the B2- or the D03 -type order parameter. Both η and ζ in (27) can be positive and negative, thus there are four types of ordered domain and two types of APB, which separate either the APDs differing in the sign of η (“η-APB”), or the APDs differing in the sign of ζ (“ζ -APB”). Using the relations analogous to (26), one can also define the local parameters ηi , ζi and c¯i , in particular, the local order parameter ηi2 used in Figs. 4 and 5:

1 2 1 ci − cj + ηi2 = 16 z nn j =nn(i) z nnn

2

cj .

(28)

j =nnn(i)

Here nn(i) or nnn(i) means the summation over nearest or next-nearest neighbors of site i, and z nn or z nnn is the total number of such neighbors. The

a

b

c

d

e

f

Figure 4. Temporal evolution of model I for PT A2 → A2 + D03 at c = 0.187, T = T/Tc = 0.424, and the following t : (a) 10, (b) 30, (c) 100, (d) 500, (e) 1000, and (f) 2000. The grey level linearly varies with ηi2 defined by (28) between its minimum and maximum values from completely dark to completely bright.

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b

c

d

e

f

Figure 5. As Fig. 4, but for model II and PT A2 → A2 + B2 at c = 0.325, T = 0.424.

distribution of ηi2 is similar to that observed in the transmission electron microscopy (TEM) images with the reflection vector k1 [14]. To study kinetics of D03 ordering, Belashchenko, Samolyuk and Vaks [15] simulated PTs A2 → D03 , A2 → A2 + D03 , A2 → B2 + D03 and D03 → B2 + D03 using the Fe-Al-type interaction models. They also considered two more models, I and II, in which the deformational interaction v d was taken into account for the PT A2 → A2 + D03 and A2 → A2 + B2, respectively. The simulations reveal a number of microstructural features related to the “multivariance” of the D03 orderings. Some of them are illustrated in Figs. 4 and 5 where the PT A2 → A2 + D03 for model I is compared to the PT A2 → A2 + B2 for model II. The first stage of both PTs corresponds to congruent ordering at approximately unchanged initial concentration. Frame 4a illustrates the transient state in which only the B2 ordered APDs (“η-APDs”) are present. Frame 4b shows the formation of the D03 -ordered APDs (“ζ -APDs”) within initial η-APDs, and these ζ -APDs are much more regular-shaped than the η-APDs in frame 5b. Frames 4b–4d also illustrate wetting of both the η-APBs and ζ -APBs by the disordered A2 phase. Later on the deformational interaction tends to align the ordered precipitates along elastically soft (100) directions, and frame 4f shows an array of approximately rectangular D03 -ordered precipitates, unlike rod-like structures seen in frame 5f. The microstructure in frame 4f is similar to those observed for the PT A2 → A2 + D03 in alloys Fe– Ga, while the microstructure in frame 5f is similar to those observed for the PT B2 → B2 + D03 in alloys Fe–Si. The latter similarity reflects the topological equivalence of the A2 → A2 + B2 and B2 → B2 + D03 PTs [4].

Diffusional transformations: microscopic kinetic approach

2.4.

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Kinetics of L12 and L10 -type Orderings

For the FCC-based L12 - or L10 -ordered structures, the occupation ci of the FCC lattice site Ri is described by three order parameters ηα corresponding to three superstructure vectors kα : ci = c + η1 exp(ik1 Ri ) + η2 exp(ik2 Ri ) + η3 exp(ik3 Ri ) k1 = (1, 0, 0)2π/a k2 = (0, 1, 0)2π/a k3 = (0, 0, 1)2π/a

(29)

where a is the FCC lattice constant. For the cubic L12 structure |η1 | = |η2 | = |η3 |, η1 η2 η3 > 0, and four types of ordered domain are possible. In the L10 phase with the tetragonal axis α, a single nonzero parameter ηα is present which is either positive or negative. Thus six types of ordered domain are possible with two types of APB. The APB separating two APDs with the same tetragonal axis can be for brevity called the “shift-APB”, and that separating the APDs with perpendicular tetragonal axes can be called the “flip-APB”. The inhomogeneously ordered alloy states can be described by the local 2 similar to those in Eqs. (26) and (30), and by quantities ηi2 parameters ηαi characterizing the total degree of the local order:

2

1 1 2 = ci + c j exp(ikα Ri j ) ; ηαi 16 4 j =nn(i)

2 2 2 ηi2 = η1i + η2i + η3i

(30)

where R j i is R j − Ri . Belashchenko et al. [6, 9] simulated PTs A1 → L12 , A1 → A1 + L12 , and A1 → L10 after a quench of an alloy from the disordered FCC phase A1. The simulations were performed in FCC simulation boxes of sizes Vb = L 2 × H , and the value H = 1 (in the lattice constant a units) corresponds to quasi-2D simulation when the simulation box contains two atomic planes. A number of different models have been considered: the short-range-interaction models 1, 2, and 3; the intermediate-range-interaction model 4 with v n estimated from the experimental data for Ni–Al alloys; and the extended-interaction model 5. In studies of PTs A1 → L10 , the models 1 –5 were also considered in which the deformational interaction v d was added to the “chemical” interactions v n of models 1–5. This v d was found with the use of Eq. (20) and the experimental data for Co–Pt alloys. The simulations revealed many interesting microstructural features for both the L12 and L10 -type orderings. It was found, in particular, that the character of the microstructural evolution strongly depends on the type of the interaction v i j , particularly on its interaction range rint , as well as on temperature T and the degree of non-stoichiometry δc which is (c − 0.25) for the L12 phase, and (c − 0.5) for the L10 phase. With increasing rint , T , or δc, the microstructures become more isotropic and the APBs become more diffuse and mobile. At the same time, for the short-range-interaction systems at not-high T and small δc,

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the microstructures are highly anisotropic while the most of APBs are thin and low-mobile. Figures 6 and 7 illustrate these features for the L12 -type orderings. Figure 6 shows the evolution under the A1 → L12 PT for the intermediate-interactionrange model 4 at non-stoichiometric c = 0.22. We see that the distribution of APBs is virtually isotropic. The main evolution mechanism is the growth of larger domains at the expense of smaller ones which is also typical for the simple B2 ordering. At the same time, one more mechanism, the fusion of in-phase domains, is also important for the multivariant orderings under consideration. For the later stages of evolution, Fig. 6 also reveals many approximately equiangular triple junctions of APDs with angles 120◦ ; it agrees with TEM observations for Cu–Pd alloys [14]. Kinetics of the A1 → L12 PT for the short-range-interaction system is illustrated in Fig. 7. The distribution of APBs here reveals a high anisotropy, a tendency to the formation of thin “conservative” APBs with (100)-type orientation. One also observes many “step-like” APBs with the conservative segments; the triple junctions of APBs with one non-conservative APBs and two conservative APBs; and the “quadruple” junctions of APDs. All these features were

Figure 6. Temporal evolution of model 4 under PT A1 → L12 for the simulation box size Vb = 1282 × 1 at c = 0.22, T = 0.685 and the following t : (a) 5; (b) 50; (c) 120; (d) 125; 2 + η2 + η2 between its mini(e) 140; and (f) 250. The grey level linearly varies with ηi2 = η1i 2i 3i mum and maximum values from completely dark to completely bright. The symbol A, B, C or D indicates the type of the ordered domain, and the thick arrow indicate the fusion-of-domain process.

Diffusional transformations: microscopic kinetic approach a

b

c

d

e

f

2265

Figure 7. As Fig. 6, but for model 1 and Vb = 642 × 1 at c = 0.25, T = 0.57 and the following t : (a) 2, (b) 3, (c) 20, (d) 100, (e) 177 and (f) 350.

observed in the electron microscopy studies of Cu3 Au alloys [14]. Figure 7 also illustrates the peculiar kinetic processes related to conservative APBs and discussed by Vaks [4, 7]. The L10 structure, unlike the cubic L12 structure, is tetragonal and has a tetragonal distortion . Depending on the importance of this distortion, the evolution in the course of the A1 → L10 PT can be divided into three stages. I. The initial stage when the L10 -ordered APDs are quite small, their tetragonal distortion is insignificant, and all six types of APD are present in the same proportion. II. The intermediate stage when the tetragonal distortion of APDs leads to some predominance of the (110)-type orientations of flip-APBs and to decreasing of the portion of APDs with the unfavorable orientation (001). III. The final, “twin” stage when the well-defined twin bands delimited by the flip-APBs with (110)-type orientation are formed. Each band includes only two types of APD with the same tetragonal axis, and these axes in the adjacent bands are “twin” related, i.e., have alternate (100) and (010) orientations. The thermodynamic driving force for the (110)-type orientation of flipAPBs is the gain in the elastic energy: at other orientations this energy increases proportionally to the volume of the adjacent APDs [8].

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The simulations of PTs A1 → L10 [9] revealed a number of peculiar microstructural features for each of the stages mentioned above. Figures 8 and 9 illustrate some of these features. Frame 8a corresponds to stage I ; frames 8b–8c, to stage II; and frames 8d–8f and 9a–9d, to stage III. The following processes and configurations are seen to be characteristic of both the stage I and stage II: (1) The abundant processes of fusion of in-phase domains which are among the main mechanisms of domain growth at these stages. (2) Peculiar long-living configurations, the quadruple junctions of APDs (4-junctions) of the type A1 A2 A1 A3 where A2 and A3 can correspond to any two of four types of APD different from A1 and A1 . (3) Many processes of “splitting” of a shiftAPB into two flip-APBs which leads either to the fusion of in-phase domains or to the formation of a 4-junction. For the final, “nearly equilibrium” twin stage, Figs. 8f and 9a–9d demonstrate a peculiar alignment of shift-APBs: within a (100)-oriented twin band in a (110)-type polytwin the APBs tend to align normally to some direction n = (cos α, sin α, 0) characterized by a “tilting” angle α which is mainly

Figure 8. Temporal evolution of model 4 under PT A1 → L10 for Vb = 1282 × 1 at c = 0.5, T = 0.67, and the following t : (a) 10; (b) 20; (c) 50; (d) 400; (e) 750; and (f) 1100. ¯ B or B ¯ and C or C¯ indicates an APD with the tetragonality axis along The symbol A or A, (100), (010) and (001), respectively. The thick, the thin and the single arrow indicates the fusion-of-domain process, the quadruple junction of APDs, and the splitting APB process, respectively.

Diffusional transformations: microscopic kinetic approach

2267

Figure 9. As Fig. 8, but for model 2 at the following values of c, T , and t : (a) c = 0.5, T = 0.77, and t = 350; (b) c = 0.5, T = 0.95, and t = 300; (c) c = 0.46, T = 0.77, and t = 350; and (d) c = 0.44, T = 0.77, and t = 300.

determined by the type of chemical interaction. For the short-range interaction systems this angle is close to zero, in agreement with observations for CuAu. For the intermediate-interaction-range systems, the scale of α is illus-trated by Fig. 8f, and the alignment of APBs shown in this figure is very similar to that observed for a Co0.4 Pt0.6 alloy [4]. Figure 9 also illustrates sharp changes of the alignment type under variation of temperature T and non-stoichiometry δc, including the “faceting-tilting”-type morphological transitions.

3.

Outlook

For the last decade the statistical theory of diffusional PTs has been formulated in terms of both approximate and exact kinetic equations and was applied to studies of many concrete problems. These applications yielded numerous new results, many of them agreeing well with experimental observations. Many predictions of this theory are still awaiting experimental verification. At the same time, there remain a number of further problems in this approach to be solved, such as the elaboration of a microscopical “phase-fieldtype” approach suitable for treatments of sizeable and complex structures [2]; the consistent treatment of fluctuative effects, including the problem of nucleation of embryos of a new phase within the metastable one, and others. Some of these problems are now underway, and for the nearest future one can expect a further progress in that field.

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References [1] P.E.A. Turchi and A. Gonis (eds.), “Phase transformations and evolution in materials,” TMS, Warrendale, 2000. [2] I.R. Pankratov and V.G. Vaks, “Generalized Ginzburg–Landau functionals for alloys: general equations and comparison to the phase-field method,” Phys. Rev. B, 68, 134208 (in press), 2003. [3] V.G. Vaks, “Master equation approach to the configurational kinetics of nonequilibrium alloys: exact relations, H-theorem and cluster approximations,” JETP Lett., 78, 168–178, 1996. [4] V.G. Vaks, “Kinetics of phase separation and orderings in alloys,” Physics Reports, 391, 157–242, 2004. [5] K.D. Belashchenko and V.G. Vaks, “Master equation approach to configurational kinetics of alloys via vacancy exchange mechanism: general relations and features of microstructural evolution,” J. Phys. Condensed Matter, 10, 1965–1983, 1998. [6] K.D. Belashchenko, V. Yu. Dobretsov, I.R. Pankratov et al., “The kinetic clusterfield method and its application to studes of L12 -type orderings in alloys,” J. Phys. Condens. Matter, 11, 10593–10620, 1999. [7] V.G. Vaks, “Kinetics of L12 -type and L10 -type orderings in alloys,” JETP Lett., 78, 168–178, 2003. [8] A.G. Khachaturyan, “Theory of structural phase transformations in solids,” Wiley, New York, 1983. [9] K.D. Belashchenko, I.R. Pankratov, G.D. Samolyuk et al., “Kinetics of formation of twinned structures under L10 -type orderings in alloys,” J. Phys. Condens. Matter, 14, 565–589, 2002. [10] S.M. Allen and J.W. Cahn, “Mechanisms of phase transformations within the miscibility gap of Fe-rich Fe-Al alloys,” Acta Metall., 24, 425–437, 1976. [11] V.G. Vaks, S.V. Beiden, V. Dobretsov, and Yu., “Mean-field equations for configurational kinetics of alloys at arbitrary degree of nonequilibrium,” JETP Lett., 61, 68–73, 1995. [12] G. Korn and T. Korn, “Mathematical handbook for scientists and engineers,” McGraw-Hill, New York, 1961. [13] V. Yu. Dobretsov, V.G. Vaks, and G. Martin, “Kinetic features of phase separation under alloy ordering,” Phys. Rev. B, 54, 3227–3239, 1996. [14] A. Loiseau, C. Ricolleau, L. Potez, and F. Ducastelle, “Order and disorder at interfaces in alloys,” In: W.C. Johnson, J.M. Howe, D.E. Mc Laughlin, and W.A. Soffa (eds.), Solid–Solid Phase Transformations, pp. 385–400, TMS, Warrendale, 1994. [15] K.D. Belashchenko, G.D. Samolyuk, and V.G. Vaks, “Kinetic features of alloy ordering with many types of ordered domain: D03 -type ordering,” J. Phys. Condens. Matter, 10, 10567–10592, 1999.

7.11 MODELING THE DYNAMICS OF DISLOCATION ENSEMBLES Nasr M. Ghoniem Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597, USA

1.

Introduction

A fundamental description of plastic deformation is under development by several research groups as a result of dissatisfaction with the limitations of continuum plasticity theory. The reliability of continuum plasticity descriptions is dependent on the accuracy and range of available experimental data. Under complex loading situations, however, the database is often hard to establish. Moreover, the lack of a characteristic length scale in continuum plasticity makes it difficult to predict the occurrence of critical localized deformation zones. It is widely appreciated that plastic strain is fundamentally heterogenous, displaying high strains concentrated in small material volumes, with virtually undeformed regions in-between. Experimental observations consistently show that plastic deformation is internally heterogeneous at a number of length scales [1–3]. Depending on the deformation mode, heterogeneous dislocation structures appear with definitive wavelengths. It is common to observe persistent slip bands (PSBs), shear bands, dislocation pile ups, dislocation cells and sub grains. However, a satisfactory description of realistic dislocation patterning and strain localization has been rather elusive. Since dislocations are the basic carriers of plasticity, the fundamental physics of plastic deformation must be described in terms of the behavior of dislocation ensembles. Moreover, the deformation of thin films and nanolayered materials is controlled by the motion and interactions of dislocations. For all these reasons, there has been significant recent interest in the development of robust computational methods to describe the collective motion of dislocation ensembles. Studies of the mechanical behavior of materials at a length scale larger than what can be handled by direct atomistic simulations, and smaller than what allows macroscopic continuum averaging represent particular difficulties. Two 2269 S. Yip (ed.), Handbook of Materials Modeling, 2269–2286. c 2005 Springer. Printed in the Netherlands.

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complimentary approaches have been advanced to model the mechanical behavior in this meso length scale. The first approach, commonly known as dislocation dynamics (DD), was initially motivated by the need to understand the origins of heterogeneous plasticity and pattern formation. In its early versions, the collective behavior of dislocation ensembles was determined by direct numerical simulations of the interactions between infinitely long, straight dislocations [3–9]. Recently, several research groups extended the DD methodology to the more physical, yet considerably more complex 3D simulations. Generally, coarse resolution is obtained by the Lattice Method, developed by Kubin et al. [10] and Moulin et al. [11], where straight dislocation segments (either pure screw or edge in the earliest versions, or of a mixed character in more recent versions) are allowed to jump on specific lattice sites and orientations. Straight dislocation segments of mixed character in the The Force Method, developed by Hirth et al. [12] and Zbib et al. [13] are moved in a rigid body fashion along the normal to their mid-points, but they are not tied to an underlying spatial lattice or grid. The advantage of this method is that the explicit information on the elastic field is not necessary, since closed-form solutions for the interaction forces are directly used. The Differential Stress Method developed by Schwarz and Tersoff [14] and Schwarz [15] is based on calculations of the stress field of a differential straight line element on the dislocation. Using numerical integration, Peach–Koehler forces on all other segments are determined. The Brown procedure [16] is then utilized to remove the singularities associated with the self-force calculation. The method of The Phase Field Microelasticity [17–19] is of a different nature. It is based on Khachaturyan–Shatalov (KS) reciprocal space theory of the strain in an arbitrary elastically homogeneous system of misfitting coherent inclusions embedded into the parent phase. Thus, consideration of individual segments of all dislocation lines is not required. Instead, the temporal and spatial evolution of several density function profiles (fields) are obtained by solving continuum equations in Fourier space. The second approach to mechanical models at the mesoscale has been based on statistical mechanics methods [20–24]. In these developments, evolution equations for statistical averages (and possibly for higher moments) are to be solved for a complete description of the deformation problem. We focus here on the most recent formulations of 3D DD, following the work of Ghoniem et al. We review here the most recent developments in computational DD for the direct numerical simulation of the interaction and evolution of complex, 3D dislocation ensembles. The treatment is based on the parametric dislocation dynamics (PDD), developed by Ghoniem et al. In Section 2, we describe the geometry of dislocation loops with curved, smooth, continuous parametric segments. The stress field of ensembles of such curved dislocation loops is then developed in Section 3. Equations of motion for dislocation loops

Modeling the dynamics of dislocation ensembles

2271

are derived on the basis of irreversible thermodynamics, where the time rate of change of generalized coordinates will be given in Section 4. Extensions of these methods to anisotropic materials and multi-layered thin films are discussed in Section 5. Applications of the parametric dislocation dynamics methods are given in Section 6, and a discussion of future directions is finally outlined in Section 7.

2.

Computational Geometry of Dislocation Loops

Assume that the dislocation line is segmented into (n s ) arbitrary curved segments, labeled (1 ≤ i ≤ n s ). For each segment, we define rˆ (ω)=P(ω) as the position vector for any point on the segment, T(ω) = T t as the tangent vector to the dislocation line, and N(ω) = N n as the normal vector at any point (see Fig. 1). The space curve is then completely described by the parameter ω, if one defines certain relationships which determine rˆ (ω). Note that the position of any other point in the medium (Q) is denoted by its vector r, and that the vector connecting the source point P to the field point is R, thus R = r − rˆ . In the following developments, we restrict the parameter 0 ≤ ω ≤ 1, although we map it later on the interval −1 ≤ ωˆ ≤ 1, and ωˆ = 2ω − 1 in the numerical quadrature implementation of the method. To specify a parametric form for rˆ (ω), we will now choose a set of gen( j) eralized coordinates qi for each segment ( j ), which can be quite general. If one defines a set of basis functions C i (ω), where ω is a parameter, and allows

g3 ⫽ b冫 冩 b 冩 P ω⫽ 0

R

g2 ⫽ t

g2 ⫽ e r

Q

ω⫽ 1

1z

1x

Figure 1. segment.

1y

Differential geometry representation of a generalparametric curved dislocation

2272

N.M. Ghoniem

for index sums to extend also over the basis set (i = 1, 2, . . . , I ), the equation of the segment can be written as ( j) rˆ ( j ) (ω) = qi Ci (ω)

2.1.

(1)

Linear Parametric Segments

The shape functions of linear segments Ci (ω), and their derivatives Ci,ω take the form: C1 = 1 − ω, C2 = ω and C1,ω = −1, C2,ω = 1. Thus, the available degrees of freedom for a free, or unconnected linear segment ( j ) are just the position vectors of the beginning ( j ) and end ( j + 1) nodes. ( j)

( j)

q1k = Pk

2.2.

and

( j)

( j +1)

q2k = Pk

(2)

Cubic Spline Parametric Segments

For cubic spline segments, we use the following set of shape functions, their parametric derivatives, and their associated degrees of freedom, respectively: C1 = 2ω3 − 3ω2 + 1, C2 = −2ω3 + 3ω2 , C3 = ω3 − 2ω2 + ω, and C4 = ω3 − ω2 C1,ω = 6ω2 − 6ω, C2,ω = −6ω2 + 6ω2 , C3,ω = 3ω2 − 4ω + 1, and C4,ω = 3ω2 − 2ω ( j) q1k

=

( j) Pk ,

( j) q2k

=

( j +1) Pk ,

( j) q3k

=

( j) Tk ,

and

( j) q4k

=

( j +1) Tk

(3) (4) (5)

Extensions of these methods to other parametric shape functions, such as circular, elliptic, helical, and composite quintic space curves are discussed by Ghoniem et al. [25]. Forces and energies of dislocation segments are given per unit length of the curved dislocation line. Also, line integrals of the elastic field variables are carried over differential line elements. Thus, if we express the Cartesian ( j) ( j) ( j) differential in the parametric form: dk = rˆk, ω dω = qsk Cs, ω dω. The arc length differential for segment j is then given by

( j)

( j ) 1/2

| d( j ) | = dk dk

( j)

( j)

( j ) ( j ) 1/2

= rˆk, ω rˆk, ω

= q pk C p, ω qsk Cs, ω

1/2

dω

dω

(6) (7)

Modeling the dynamics of dislocation ensembles

3.

2273

Elastic Field Variables as Fast Sums

3.1.

Formulation

In materials that can be approximated as infinite and elastically isotropic, the displacement vector u, strain ε and stress σ tensor fields of a closed dislocation loop are given by deWit [26] ui = −

εi j =

σi j

bi 4π

1 8π

=

Ak dlk +

C

ikl bl R, pp +

C

1 kmn bn R,mi dlk 1−ν

−

1 j kl bi R,l + ikl b j R,l − ikl bl R, j − j kl bl R,i , pp 2

×

kmn bn R,mi j dlk 1−ν

C

µ 4π

1 8π

(8)

C

(9)

1 1 R,mpp j mn dli + imn dl j + kmn 2 1−ν

× R,i j m − δi j R, ppm dlk

(10)

where µ and ν are the shear modulus and Poisson’s ratio, respectively, b is Burgers vector of Cartesian components bi , and the vector potential Ak (R) = i j k X i s j /[R(R+R· s)] satisfies the differential equation: pik Ak, p (R) = X i R −3 , where s is an arbitrary unit vector. The radius vector R connects a source point on the loop to a field point, as shown in Fig. 1, with Cartesian components Ri , successive partial derivatives R,i j k... , and magnitude R. The line integrals are carried along the closed contor C defining the dislocation loop, of differential arc length dl of components dlk . Also, the interaction energy between two closed loops with Burgers vectors b1 and b2 , respectively, can be written as µb1i b2 j EI = − 8π

R,kk C (1) C (2)

2ν dl2 j dl1i + dl2i dl1 j 1−ν

2 + (R,i j − δi j R,ll )dl2k dl1k 1−ν

(11)

The higher order derivatives of the radius vector, R,i j and R,i j k are components of second and third order Cartesian tensors that are explicitly known [27]. The dislocation segment in Fig. 1 is fully determined as an affine mapping on the scalar interval ∈ [0, 1], if we introduce the tangent vector T,

2274

N.M. Ghoniem

the unit tangent vector t, the unit radius vector e, and the vector potential A, as follows T=

dl , dω

t=

T , |T|

e=

R , R

A=

e×s R(1 + e · s)

Let the Cartesian orthonormal basis set be denoted by 1 ≡ {1x , 1 y , 1z }, I = 1 ⊗ 1 as the second order unit tensor, and ⊗ denotes tensor product. Now define the three vectors (g1 = e, g2 = t, g3 = b/|b|) as a covariant basis set for the curvilinear segment, and their contravariant reciprocals as: gi · g j = δ ij , where δ ij is the mixed Kronecker delta and V = (g1 × g2 ) · g3 the volume spanned by the vector basis, as shown in Fig. 1. When the previous relationships are substituted into the differential forms of Eqs. (8), (10), with V1 = (s × g1 ) · g2 , and s an arbitrary unit vector, we obtain the differential relationships (see Ref. [27] for details)

|b||T|V (1 − ν)V1 / V du = g3 + (1 − 2ν)g1 + g1 dω 8π(1 − ν)R 1 + s · g1 V |T| d 1 1 =− −ν g ⊗ g + g ⊗ g 1 1 dω 8π(1 − ν)R 2

+ (1 − ν) g3 ⊗ g3 + g3 ⊗ g3 + (3g1 ⊗ g1 − I) µV |T| dσ 1 1 = g ⊗ g + g ⊗ g 1 1 dω 4π(1 − ν)R 2

+ (1 − ν) g2 ⊗ g2 + g2 ⊗ g2 − (3g1 ⊗ g1 + I)

µ|T1 ||b1 ||T2 ||b2 | d2 E I =− (1 − ν) g2I · g3I g2II · g3II dω1 dω2 4π(1 − ν)R

+ 2ν g2II · g3I

+ g3I · g1

g2I · g3II − g2I · g2II

g3II · g1

g3I · g3II

µ|T1 ||T2 ||b|2 d2 E S =− (1 + ν) g3 · g2I g3 · g2II dω1 dω2 8π R (1 − ν)

− 1 + (g3 · g1 )2

g2I · g2II

(12)

The superscripts I and II in the energy equations are for loops I and II , respectively, and g1 is the unit vector along the line connecting two interacting points on the loops. The self energy is obtained by taking the limit of 1/2 the interaction energy of two identical loops, separated by the core distance. Note that the interaction energy of prismatic loops would be simple, because g3 · g2 = 0. The field equations are affine transformation mappings of the scalar interval neighborhood dω to the vector (du) and second order tensor (d, dσ)

Modeling the dynamics of dislocation ensembles

2275

neighborhoods, respectively. The maps are given by covariant, contravariant and mixed vector, and tensor functions.

3.2.

Analytical Solutions

In some simple geometry of Volterra-type dislocations, special relations between b, e, and t can be obtained, and the entire dislocation line can also be described by one single parameter. In such cases, one can obtain the elastic field by proper choice of the coordinate system, followed by straight-forward integration. Solution variables for the stress fields of infinitely-long pure and edge dislocations are given in Table 1, while those for the stress field along the 1z -direction for circular prismatic and shear loops are shown in Table 2. Note that for the case of a pure screw dislocation, one has to consider the product of V and the contravariant vectors together, since V = 0. When the parametric equations are integrated over z from −∞ to +∞ for the straight dislocations, and over θ from 0 to 2π for circular dislocations, one obtains the entire stress field in dyadic notation as: 1. Infinitely-long screw dislocation µb − sin θ 1x ⊗ 1z + cos θ 1 y ⊗ 1z + cos θ 1z ⊗ 1 y 2πr − sin θ 1z ⊗ 1x }

σ=

(13)

Table 1. Variables for screw and edge dislocations Screw dislocation

Edge dislocation

g2

1 (r cos θ1x + r sin θ1 y + z1z ) R 1z

1 (r cos θ1x + r sin θ1 y + z1z ) R 1z

g3

1z

1x

g1

0

1 1y V

g1

g2 V g3 V T R V

r

r

r 2 + z2 r 2 + z2

dz 1z dω

(− sin θ1x + cos θ1 y ) V (sin θ1x − cos θ1 y ) V

r 2 + z2 r

r 2 + z2

dz 1z dω

0

r 2 + z2

1

r 2 + z2

r sin θ r 2 + z2

(−z1 y + r sin θ1z ) (sin θ1x − cos θ1 y )

2276

N.M. Ghoniem

Table 2. Variables for circular shear and prismatic loops Shear loop 1

Prismatic loop

g2

− sin θ1x + cos θ1 y

− sin θ1x + cos θ1 y

g3

1x

1z

g1

−

r 2 + z2

g2 V g3 V

T R V

(r cos θ1x + r sin θ1 y + z1z )

cos θ 1y V 1

(−z1 y + r sin θ1z )

(−z cos θ1x − z sin θ1 y

r 2 + z2 1 r 2 + z2

−r sin θ

+ r 1z )

dθ dθ 1x + r cos θ 1y dω dω

1

g1

r 2 + z2

1 (cos θ1x + sin θ1 y ) V r (− sin θ1x + cos θ1 y ) V r 2 + z2 1 (−z cos θ1x − z sin θ1 y V r 2 + z2 + r 1z ) −r sin θ

r 2 + z2

(r cos θ1x + r sin θ1 y + z1z )

dθ dθ 1x + r cos θ 1y dω dω

r 2 + z2

z cos θ − r 2 + z2

r

r 2 + z2

2. Infinitely-long edge dislocation µb sin θ(2 + cos 2θ )1x ⊗1x − (sin θ cos 2θ )1 y ⊗1 y 2π(1 − ν)r + (2ν sin θ)1z ⊗ 1z − (cos θ cos 2θ)(1x ⊗ 1 y + 1 y ⊗ 1x ) (14)

σ=−

3. Circular shear loop (evaluated on the 1z -axis)

σ=

µbr 2 2 2 2 (ν − 2)(r + z ) + 3z 4(1 − ν)(r 2 + z 2 )5/2 × 1 x ⊗ 1z + 1z ⊗ 1 x

(15)

4. Circular prismatic loop (evaluated on the 1z -axis)

σ=

µbr 2 (2(1 − ν)(r 2 + z 2 ) − 3r 2 ) 4(1 − ν)(r 2 + z 2 )5/2

× 1x ⊗ 1x + 1 y ⊗ 1 y − 2(4z 2 + r 2 ) 1z ⊗ 1z

(16)

As an application of the method in calculations of self- and interaction energy between dislocations, we consider here two simple cases. First, the

Modeling the dynamics of dislocation ensembles

2277

interaction energy between two parallel screw dislocations of length L and with a minimum distance ρ between them is obtained by making the following substitutions in Eq. (12) g2I = g2II = g3I = g3II = 1z ,

|T| =

dl = 1, dz

z2 − z1 1z · g1 = 2 ρ + (z 2 − z 1 )2

where z 1 and z 2 are distances along 1z on dislocations 1 and 2, respectively, connected along the unit vector g1 . The resulting scalar differential equation for the interaction energy is d2 E I µb2 =− dz 1 dz 2 4π(1 − ν)

ν (z 2 − z 1 )2 − 2 ρ 2 + (z 2 − z 1 )2 [ρ + (z 2 − z 1 )2 ] 3/2

(17) Integration of Eq. (17) over a finite length L yields identical results to those obtained by deWit [26] and by application of the more standard Blin formula [28]. Second, the interaction energy between two coaxial prismatic circular dislocations with equal radius can be easily obtained by the following substitutions g3I = g3II = 1z , g2I = − sin ϕ1 1x + cos ϕ1 1 y , g2II = − sin ϕ2 1x + cos ϕ2 1 y ϕ1 − ϕ2 2 z 1z · g2I = 0, R 2 = z 2 + (2ρ sin ) , 1z · g1 = 2 R Integration over the variables ϕ1 and ϕ2 from (0 − 2π ) yields the interaction energy.

4.

Dislocation Loop Motion

Consider the virtual motion of a dislocation loop. The mechanical power during this motion is composed of two parts: (1) change in the elastic energy stored in the medium upon loop motion under the influence of its own stress (i.e., the change in the loop self-energy), (2) the work done on moving the loop as a result of the action of external and internal stresses, excluding the stress contribution of the loop itself. These two components constitute the Peach– Koehler work [29]. The main idea of DD is to derive approximate equations of motion from the principle of Virtual Power Dissipation of the second law of thermodynamics Ghoniem et al. [27]. Once the parametric curve for the dislocation segment is mapped onto the scalar interval {ω ∈ [0, 1]}, the stress field everywhere is obtained as a fast numerical quadrature sum [30]. The Peach– Koehler force exerted on any other dislocation segment can be obtained from the total stress field (external and internal) at the segment as [30]. F P K = σ · b × t.

2278

N.M. Ghoniem

The total self-energy of the dislocation loop is determined by double line integrals. However, Gavazza and Barnett [31] have shown that the first variation in the self-energy of the loop can be written as a single line integral, and that the majority of the contribution is governed by the local line curvature. Based on these methods for evaluations of the interaction and self-forces, the weak variational form of the governing equation of motion of a single dislocation loop was developed by Ghoniem et al. [25] as

Fkt − Bαk Vα δrk |ds| = 0

(18)

Here, Fkt are the components of the resultant force, consisting of the Peach– Koehler force F P K (generated by the sum of the external and internal stress fields), the self-force Fs , and the Osmotic force F O (in case climb is also considered [25]). The resistivity matrix (inverse mobility) is Bαk , Vα are the velocity vector components, and the line integral is carried along the arc length of the dislocation ds. To simplify the problem, let us define the following dimensionless parameters r r∗ = , a

f∗ =

F , µa

t∗ =

µt B

Here, a is lattice constant, and t is time. Hence Eq. (18) can be rewritten in dimensionless matrix form as dr∗ ∗ ∗ ∗ δr f − ∗ ds = 0 (19) dt ∗

Here, f∗ = [ f 1∗ , f 2∗ , f 3∗ ] and r∗ = [r1∗ , r2∗ , r3∗ ] , which are all dependent on the dimensionless time t ∗ . Following Ghoniem et al. [25], a closed dislocation loop can be divided into Ns segments. In each segment j , we can choose a set of generalized coordinates qm at the two ends, thus allowing parametrization of the form r∗ = CQ

(20)

Here, C = [C1 (ω), C2 (ω), . . . , Cm (ω)], Ci (ω), (i = 1, 2, . . . , m) are shape functions dependent on the parameter (0 ≤ ω ≤ 1) and Q = [q1 , q2 , . . . , qm ] , qi are a set of generalized coordinates. Substituting Eq. (20) into Eq. (19), we obtain Ns j =1

Let,

δQ

dQ C f − C C ∗ |ds| = 0 dt ∗

j

fj = j

C f∗ |ds| ,

kj = j

C C |ds|

(21)

Modeling the dynamics of dislocation ensembles

2279

Following a similar procedure to the FEM, we assemble the EOM for all contiguous segments in global matrices and vectors, as F=

Ns j =1

fj,

K=

Ns

kj

j =1

then, from Eq. (21) we get, dQ =F (22) dt ∗ The solution of the set of ordinary differential Eq. (22) describes the motion of an ensemble of dislocation loops as an evolutionary dynamical system. However, additional protocols or algorithms are used to treat: (1) strong dislocation interactions (e.g., junctions or tight dipoles), (2) dislocation generation and annihilation, (3) adaptive meshing as dictated by large curvature variations [25]. In the The Parametric Method [25, 27, 32, 33] presented above, the dislocation loop can be geometrically represented as a continuous (to second derivative) composite space curve. This has two advantages: (1) there is no abrupt variation or singularities associated with the self-force at the joining nodes in between segments, (2) very drastic variations in dislocation curvature can be easily handled without excessive re-meshing. K

5.

Dislocation Dynamics in Anisotropic Crystals

Extension of the PDD to anisotropic linearly elastic crystals follows the same procedure described above, with the exception of two aspects [34]. First, calculations of the elastic field, and hence forces on dislocations, is computationally more demanding. Second, the dislocation self-force is obtained from non-local line integrals. Thus PDD simulations in anisotropic materials are about an order of magnitude slower than in isotropic materials. Mura [35] derived a line integral expression for the elastic distortion of a dislocation loop, as u i, j (x)= ∈ j nk C pqmn bm

G ip,q (x − x )νk dl(x ),

(23)

L

where νk is the unit tangent vector of the dislocation loop line L, dl is the dislocation line element, ∈ j nh is the permutation tensor, Ci j kl is the fourth order elastic constants tensor, G i j ,l (x − x ) = ∂ G i j (x − x )/∂ xl , and G i j (x − x ) are the Green’s tensor functions, which correspond to displacement component along the xi -direction at point x due to a unit point force in the x j -direction applied at point x in an infinite medium.

2280

N.M. Ghoniem

The elastic distortion formula (23) involves derivatives of the Green’s functions, which need special consideration. For general anisotropic solids, analytical expressions for G i j,k are not available. However, these functions can be expressed in an integral form (see, e.g., Refs. [36–39]), as G i j ,k (x − x ) =

1 2 8π |r|2

Ck

¯ −1 (k) ¯ − r¯k Ni j (k)D

¯ j m (k)D ¯ −2 (k) ¯ dφ + k¯k Clpmq (¯r p k¯q + k¯ p r¯q )Nil (k)N (24) where r = x − x , r¯ = r/|r|, k¯ is the unit vector on the plane normal to r, the integral is taken around the unit circle Ck on the plane normal to r, Ni j (k) and D(k) are the adjoint matrix and the determinant of the second order tensor Cikj l kk kl , respectively. The in-plane self-force at the point P on the loop is also obtained in a manner similar to the external Peach–Koehler force, with an additional contribution from stretching the dislocation line upon a virtual infinitesimal motion [40] F S = κ E(t) − b · σ¯ S · n

(25)

where E(t) is the pre-logarithmic energy factor for an infinite straight dislocation parallel to t: E(t) = 12 b · (t) · n, with (t) being the stress tensor of an infinite straight dislocation along the loop’s tangent at P. σ S is self stress tensor due to the dislocation L, and σ¯ = 12 [σ S (P + m) + σ S (P − m)] is the average self-stress at P, κ is the in-plane curvature at P, and = |b|/2. Barnett [40] and Gavazza and Barnett [31] analyzed the structure of the self-force as a sum 8 S − J (L , P) + Fcore (26) F = κ E(t) − κ E(t) + E (t) ln κ where the second and third terms are line tension contributions, which usually account for the main part of the self-force, while J (L , P) is a non-local contribution from other parts of the loop, and Fcore is due to the contribution to the self-energy from the dislocation core.

6.

Selected Applications

Figure 2 shows the results of computer simulations of plastic deformation in single crystal copper (approximated as elastically isotropic) at a constant strain rate of 100 s−1 . The initial dislocation density of ρ = 2 × 1013 m−2 has been divided into 300 complete loops. Each loop contains a random number

Modeling the dynamics of dislocation ensembles

2281

Figure 2. Results of computer simulations for dislocation microstructure deformation in copper deformed to increasing levels of strain (shown next to each microstructure).

of initially straight glide and superjog segments. When a generated or expanding loop intersects the simulation volume of 2.2 µm side length, the segments that lie outside the simulation boundary are periodically mapped inside the simulation volume to preserve translational strain invariance, without loss of dislocation lines. The number of nodes on each loop starts at five, and is then increased adaptively proportional to the loop length, with a maximum number of 20 nodes per loop. The total number of Degrees of Freedom (DOF) starts at 6000, and is increased to 24 000 by the end of the calculation. However, the number of interacting DOF is determined by a nearest neighbor criterion, within a distance of 400a (where a is the lattice constant), and is based on a binary tree search. The dislocation microstructure is shown in Fig. 2 at different total strain. It is observed that fine slip lines that nucleate at low strains evolve into more pronounced slip bundles at higher strains. The slip bundles are well-separated in space forming a regular pattern with a wavelength of approximately one micron. Conjugate slip is also observed, leading to the formation of dislocation junction bundles and stabilization of a cellular structures. Next, we consider the dynamic process of dislocation dipole formation in anisotropic single crystals. To measure the degree of deviation from elastic isotropy, we use the anisotropy ratio A, defined in the usual manner: A = 2C44 /(C11 − C12 ) [28]. For an isotropic crystal, A = 1. Figure 3(a) shows the configurations (2D projected on the (111)-plane) of two pinned dislocation segments, lying on parallel (111)-planes. The two dislocation segments are

2282

N.M. Ghoniem (a) 300

A⫽1 A⫽2 A ⫽ 0.5

200

[⫺1 ⫺1 2]

b Stable dipole

100

0

⫺500

0

500

[⫺1 1 0]

(b) 0.4 A⫽1 A⫽1

0.35

A⫽2 A⫽1

τ/µ (%)

0.3 A ⫽ 0.5

0.25

0.2

0.15 Backward break up Forward break up Infinite dipole

0.1

0.05

0.04

0.08

0.12

a/h

Figure 3. Evolution of dislocation dipoles without applied loading (a) and dipole break up shear stress (b).

¯ initially straight, parallel, and along [110], but of opposite line directions, ¯ have the same Burgers vector b = 1/2[101], and are pinned √ at both ends. Their 3a, L : d : h = 800 : glide planes are separated by h. In this figure, h = 25 √ 300 : 25 3, with L and d being the length of the initial dislocation segments and the horizontal distance between them, respectively. Without the application of any external loading, the two lines attract one another, and form an equilibrium state of a finite-size dipole. The dynamic shape of the segments during the dipole formation is seen to be dependent on the anisotropy ratio A, while the final configuration appears to be insensitive to A. Under external loading, the dipole may be unzipped, if applied forces overcome binding forces between dipole arms. The forces (resolved shear stresses τ , divided by µ = (C11 − C12 )/2) to break up the dipoles are shown in Fig. 3(b). It can be seen that the break up stress is inversely proportional to the separation distance h, consistent with the results of infinite-size dipoles. It is easier to break up dipoles in crystals with smaller A-ratios (e.g., some BCC crystals). It is also noted that two ways to break up dipoles are possible: in backward direction (where the self-force assists the breakup), or forward direction (where the

Modeling the dynamics of dislocation ensembles

2283

self-force opposes the breakup). For a finite length dipole, the backward break up is obviously easier than the forward one, due to the effects of self forces induced by the two curved dipole arms, as can be seen in Fig. 3(b). As a final application, we consider dislocation motion in multi-layer anisotropic thin films. It has been experimentally shown that the strength of multilayer thin films is increased as the layer thickness is decreased, and that maximum strength is achieved for layer thickness on the order of 10–50 nm. Recently, Ghoniem and Han [41] developed a new computational method for the simulation of dislocation ensemble interactions with interfaces in anisotropic, nanolaminate superlattices. Earlier techniques in this area use cumbersome and inaccurate numerical resolution by superposition of a regular elastic field obtained from a finite element, boundary element, surface dislocation or point force distributions to determine the interaction forces between 3D dislocation loops and interfaces. The method developed by Ghoniem and Han [41] utilizes two-dimensional Fourier Transforms to solve the full elasticity problem in the direction transverse to interfaces, and then by numerical inversion, obtain the solution for 3D dislocation loops of arbitrary complex geometry. Figure 4 shows a comparison between the numerical simulations (stars) for the critical yield strength of a Cu/Ni superlattice, compared to Freund’s analytical solution (red solid line) and the experimental data of the Los Alamos group (solid triangles). The saturation of the nanolayered system strength (and hardness) with a nanolayer thickness less than 10–50 nm is a result of dislocations overcoming the interface Koehler barrier and loss of dislocation confinement within the soft Cu layer.

4.0 Freund critical stress Experiment (Misra, et al.,1998) Simulation, image force

Critical yield stress (GPa)

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1

Figure 4.

10 Cu layer thickness h (nm)

100

Dependence of a Cu/Ni superlattice strength onthe thickness of the Cu layer [41].

2284

7.

N.M. Ghoniem

Future Outlook

As a result of increased computing power, new mathematical formulations, and more advanced computational methodologies, tremendous progress in modeling the evolution of complex 3D dislocation ensembles has been recently realized. The appeal of computational dislocation dynamics lies in the fact that it offers the promise of predicting the dislocation microstructure evolution without ad hoc assumptions, and on sound physical grounds. At this stage of development, many physically-observed features of plasticity and fracture at the nano- and micro-scales have been faithfully reproduced by computer simulations. Moreover, computer simulations of the mechanical properties of thin films are at an advanced stage now that they could be predictive without ambiguous assumptions. Such simulations may become very soon standard and readily available for materials design, even before experiments are performed. On the other hand, modeling the constitutive behavior of polycrystalline metals and alloys with DD computer simulations is still evolving and will require significant additional developments of new methodologies. With continued interest by the scientific community in achieving this goal, future efforts may well lead to new generations of software, capable of materials design for prescribed (within physical constraints) strength and ductility targets.

Acknowledgments Research is supported by the US National Science Foundation (NSF), grant #DMR-0113555, and the Air Force Office of Scientific Research (AFOSR), grant #F49620-03-1-0031 at UCLA.

References [1] H. Mughrabi, “Dislocation wall and cell structures and long-range internal-stresses in deformed metal crystals,” Acta Met., 31, 1367, 1983. [2] H. Mughrabi, “A 2-parameter description of heterogeneous dislocation distributions in deformed metal crystals,” Mat. Sci. & Eng., 85, 15, 1987. [3] R. Amodeo and N.M. Ghoniem, “A review of experimental observations and theoretical models of dislocation cells,” Res. Mech., 23, 137, 1988. [4] J. Lepinoux and L.P. Kubin, “The dynamic organization of dislocation structures: a simulation,” Scripta Met., 21(6), 833, 1987. [5] N.M. Ghoniem and R.J. Amodeo, “Computer simulation of dislocation pattern formation,” Sol. St. Phen., 3&4, 377, 1988. [6] A.N. Guluoglu, D.J. Srolovitz, R. LeSar, and R.S. Lomdahl, “Dislocation distributions in two dimensions,” Scripta Met., 23, 1347, 1989.

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[7] N.M. Ghoniem and R.J. Amodeo, “Numerical simulation of dislocation patterns during plastic deformation,” In: D. Walgreaf and N. Ghoniem (eds.), Patterns, Defects and Material Instabilities, Kluwer Academic Publishers, Dordrecht, p. 303, 1990. [8] R.J. Amodeo and N.M. Ghoniem, “Dislocation dynamics I: a proposed methodology for deformation micromechanics,” Phys. Rev., 41, 6958, 1990a. [9] R.J. Amodeo and N.M. Ghoniem, “Dislocation dynamics II: applications to the formation of persistent slip bands, planar arrays, and dislocation cells,” Phy. Rev., 41, 6968, 1990b. [10] L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, and Y. Brechet, “Dislocation microstructures and plastic flow: a 3D simulation,” Diffusion and Defect Data–Solid State Data, Part B (Solid State Phenomena), 23–24, 455, 1992. [11] A. Moulin, M. Condat, and L.P. Kubin, “Simulation of frank-read sources in silicon,” Acta Mater., 45(6), 2339–2348, 1997. [12] J.P. Hirth, M. Rhee, and H. Zbib, “Modeling of deformation by a 3D simulation of multi pole, curved dislocations,” J. Comp.-Aided Mat. Des., 3, 164, 1996. [13] R.M. Zbib, M. Rhee, and J.P. Hirth, “On plastic deformation and the dynamics of 3D dislocations,” Int. J. Mech. Sci., 40(2–3), 113, 1998. [14] K.V. Schwarz and J. Tersoff, “Interaction of threading and misfit dislocations in a strained epitaxial layer,” Appl. Phys. Lett., 69(9), 1220, 1996. [15] K.W. Schwarz, “Interaction of dislocations on crossed glide planes in a strained epitaxial layer,” Phys. Rev. Lett., 78(25), 4785, 1997. [16] L.M. Brown, “A proof of lothe’s theorem,” Phil. Mag., 15, 363–370, 1967. [17] A.G. Khachaturyan, “The science of alloys for the 21st century: a hume-rothery symposium celebration,” In: E. Turchi and a. G.A. Shull, R.D. (eds.), Proc. Symp. TMS, TMS, 2000. [18] Y.U. Wang, Y.M. Jin, A.M. Cuitino, and A.G. Khachaturyan, “Presented at the international conference, Dislocations 2000, the National Institute of Standards and Technology,” Gaithersburg, p. 107, 2000. [19] Y. Wang, Y. Jin, A.M. Cuitino, and A.G. Khachaturyan, “Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations,” Acta Mat., 49, 1847, 2001. [20] D. Walgraef and C. Aifantis, “On the formation and stability of dislocation patterns. I. one-dimensional considerations,” Int. J. Engg. Sci., 23(12), 1351–1358, 1985. [21] J. Kratochvil and N. Saxlo`va, “Sweeping mechanism of dislocation patternformation,” Scripta Metall. Mater., 26, 113–116, 1992. [22] P. H¨ahner, K. Bay, and M. Zaiser, “Fractal dislocation patterning during plastic deformation,” Phys. Rev. Lett., 81(12), 2470, 1998. [23] M. Zaiser, M. Avlonitis, and E.C. Aifantis, “Stochastic and deterministic aspects of strain localization during cyclic plastic deformation,” Acta Mat., 46(12), 4143, 1998. [24] A. El-Azab, “Statistical mechanics treatment of the evolution of dislocation distributions in single crystals,” Phys. Rev. B, 61, 11956–11966, 2000. [25] N.M. Ghoniem, S.-H. Tong, and L.Z. Sun, “Parametric dislocation dynamics: a thermodynamics-based approach to investigations of mesoscopic plastic deformation,” Phys. Rev., 61(2), 913–927, 2000. [26] R. deWit, “The continuum theory of stationary dislocations,” In: F. Seitz and D. Turnbull (eds.), Sol. State Phys., 10, Academic Press, 1960. [27] N.M. Ghoniem, J. Huang, and Z. Wang, “Affine covariant-contravariant vector forms for the elastic field of parametric dislocations in isotropic crystals,” Phil. Mag. Lett., 82(2), 55–63, 2001.

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[28] J. Hirth and J. Lothe, Theory of Dislocations, 2nd edn, McGraw–Hill, New York, 1982. [29] M.O. Peach and J.S. Koehler, “The forces exerted on dislocations and the stress fields produced by them,” Phys. Rev., 80, 436, 1950. [30] N.M. Ghoniem and L.Z. Sun, “Fast sum method for the elastic field of 3-D dislocation ensembles,” Phys. Rev. B, 60(1), 128–140, 1999. [31] S. Gavazza and D. Barnett, “The self-force on a planar dislocation loop in an anisotropic linear-elastic medium,” J. Mech. Phys. Solids, 24, 171–185, 1976. [32] R.V. Kukta and L.B. Freund, “Three-dimensional numerical simulation of interacting dislocations in a strained epitaxial surface layer,” In: V. Bulatov, T. Diaz de la Rubia, R. Phillips, E. Kaxiras, and N. Ghoniem (eds.), Multiscale Modelling of Materials, Materials Research Society, Boston, Massachusetts, USA, 1998. [33] N.M. Ghoniem, “Curved parametric segments for the stress field of 3-D dislocation loops,” Transactions of ASME. J. Engrg. Mat. & Tech., 121(2), 136, 1999. [34] X. Han, N.M. Ghoniem, and Z. Wang, “Parametric dislocation dynamics of anisotropic crystalline materials,” Phil. Mag. A., 83(31–34), 3705–3721, 2003. [35] T. Mura, “Continuous distribution of moving dislocations,” Phil. Mag., 8, 843–857, 1963. [36] D. Barnett, “The precise evaluation of derivatives of the anisotropic elastic green’s functions,” Phys. Status Solidi (b), 49, 741–748, 1972. [37] J. Willis, “The interaction of gas bubbles in an anisotropic elastic solid,” J. Mech. Phys. Solids, 23, 129–138, 1975. [38] D. Bacon, D. Barnett, and R. Scattergodd, “Anisotropic continuum theory of lattice defects,” In: C.J.M.T. Chalmers, B (ed.), Progress in Materials Science, vol. 23, Pergamon Press, Great Britain, pp. 51–262, 1980. [39] T. Mura, Micromechanics of Defects in Solids, Martinus Nijhoff, Dordrecht, 1987. [40] D. Barnett, “The singular nature of the self-stress field of a plane dislocation loop in an anisotropic elastic medium,” Phys. Status Solidi (a), 38, 637–646, 1976. [41] X. Han and N.M. Ghoniem, “Stress field and interaction forces of dislocations in anisotropic multilayer thin films,” Phil. Mag., in press, 2005.

7.12 DISLOCATION DYNAMICS – PHASE FIELD Yu U. Wang,1 Yongmei M. Jin,2 and Armen G. Khachaturyan2 1 Department of Materials Science and Engineering, Virginia Tech., Blacksburg, VA 24061, USA 2 Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA

Dislocation, as an important category of crystal defects, is defined as a one-dimensional line (curvilinear in general) defect. It not only severely distorts the atomic arrangement in a region (called core) around the mathematical line describing its geometrical configuration, but also in a less severe manner (elastically) distorts the lattice beyond its core region. Dislocation core structure is studied by using the methods and models of atomistic scale (see Chapter 2). The long-range strain and stress fields generated by dislocation are well described by linear elasticity theory. In the elasticity theory of dislocations, dislocation is defined as a line around which a line integral of the elastic displacement yields a non-zero vector (Burgers vector). The elastic fields, displacement, strain and stress, of an arbitrarily curved dislocation are known in the form of line integrals. For complex dislocation configurations, the exact elasticity solution is quite difficult. A conventional alternative is to approximate a curved dislocation by a series of straight line segments or spline fitted curved segments. This involves explicit tracking of each segment of the dislocation ensemble (see “Dislocation Dynamics – Tracking Methods” by Ghoniem). In a finite body, the strains and stresses depend on the external surface. For general surface geometries, the elastic fields of dislocations are difficult to determine. In this article we discuss an alternative to the front-tracking methods in modeling dislocation dynamics. This is the structure density phase field method, which is a more general version of the phase field method used to describe solidification process. Instead of explicitly tracking the dislocation lines, the phase field method describes the slipped (plastically deformed by shear) and unslipped regions in a crystal by using field variables (structure density functions or, less accurately but more conventionally called, phase 2287 S. Yip (ed.), Handbook of Materials Modeling, 2287–2305. c 2005 Springer. Printed in the Netherlands.

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fields). Dislocations are the boundaries between the regions of different degrees of slipping. One of the advantages of the phase field approach is that it treats the system with arbitrarily complex microstructures as a whole and automatically describes the evolution events producing changes of the microstructure topology (e.g., nucleation, multiplication, annihilation and reaction of dislocations) without explicitly tracking the moving segments. Therefore, it is easy for numerical implementation even in three-dimension (a front-tracking scheme often results in difficult and untidy numerical algorithm). No ad hoc assumptions are required on evolution path. The micromechanics theory proposed by Khachaturyan and Shatalov (KS) [1–3] and recently further developed by Wang, Jin and Khachaturyan (WJK) in a series of works [4–9] is formulated in such a form that it is easily incorporated in the phase field theory. It allows one to determine the elastic interactions at each step of the dislocation dynamics. In the case of elastically homogeneous systems, the exact elasticity solution for an arbitrary dislocation configuration can be formulated as a closed-form functional of the Fourier transforms of the phase fields describing the dislocation microstructure irrespective of its geometrical complexity (the number of the phase fields is equal to the number of operative slip systems that is determined by the crystallography instead of by a concrete dislocation microstructure). This fact makes it easy to achieve high computational efficiency by using Fast Fourier Transform technique, which is also suitable for parallel computing. The Fourier space solution is formulated in terms of arbitrary elastic modulus tensor. This means that the solution for dislocations in single crystal of elastic anisotropy practically does not impose more difficulty. By simply introducing a grain rotation matrix function that describes the geometry and orientation of each grain and the entire multi-grain structure, the phase field method is readily extended to model dislocation dynamics in polycrystal composed of elastically isotropic grains. If the grains are elastically anisotropic, their misorientation makes the polycrystal an elastically inhomogeneous body. The limitation of grain elastic isotropy could be lifted without serious complication of the theory and model by an introduction of additional virtual misfit strain field. This field acting in the equivalent system with the homogeneous modulus produces the same mechanical effect as that produced by elastic modulus heterogeneity. The introduction of the virtual misfit strain greatly simplifies a treatment of elastically inhomogeneous system of arbitrary complexity, in particular, a body with voids, cracks, and free surfaces. The structural density phase field model of multi-crack evolution can be developed in the formalism similar to the phase field model of multidislocation dynamics. This development of the theory has been an extension of the corresponding phase field theories of diffusional and displacive phase transformations (e.g., decomposition, ordering, martensitic transformation, etc.). All these structure density field theories are conceptually similar and

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are formulated in the similar theoretical and computational framework. The latter facilitates an integration of multi-physics such as dislocations, cracks and phase transformations into one unified structure density field model, where multiple processes are described by simultaneous evolution of various relaxing density fields. Such a unified model would be highly desirable for simulations of complex materials behaviors. The following sections will discuss the basic ingredients of the phase field model of dislocation dynamics. Single crystalline system is considered first, followed by the extension to polycrystal composed of elastically isotropic grains. Finite body with free surfaces is discussed next. The phase field model of cracks, in many respects, is similar to the dislocation model. It is also discussed. The article concludes with a brief outlook on the structural density field models for integration of multiple solid-state physical phenomena and connections between mesoscale phase field modeling and atomistic as well as continuum models.

1.

Dislocation Loop as Thin Platelet Misfitting Inclusion

Consider a simple two-dimensional lattice of circles representing atoms, as shown in Fig. 1(a). Imagine that we cut and remove from the lattice a thin platelet consisting of two monolayers indicated by shaded circles, deform it by gliding the top layer with respect to the bottom layer by one interatomic distance, as shown in Fig. 1(b), then reinsert the deformed thin platelet back into the original lattice, and allow the whole lattice to relax and reach mechanical equilibrium. In doing so, we create an edge dislocation that is located at (a)

(c)

(d)

(b)

Figure 1. Illustration of dislocations as thin platelet misfitting inclusions. (a) A 2D lattice. (b) A thin platelet misfitting inclusion generated by transformation. (c) Bragg–Nye bubble model of an edge dislocation in mechanical equilibrium (after Ref. [10], reproduced with permission). (d) Continuum presentation of the dislocation line ABC ending on the crystal surface at points A and C and a dislocation loop by the thin platelet misfitting inclusions (after Ref. [4], reproduced with permission). b is the Burgers vector, d is the thickness of the inclusion equal to the interplanar distance of the slip plane, and n is the unit vector normal to the inclusion habit plane coinciding with the slip plane.

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the edge of the thin platelet. The equilibrium state of such a lattice is demonstrated in Fig. 1(c), which shows the Bragg–Nye bubble model of an edge dislocation [10]. In the continuum elasticity theory of dislocations, dislocation loop can be created in the same way by transforming thin platelet in the matrix of untransformed solid. Consider an arbitrary-shaped plate-like misfitting inclusion, whose habit plane (interface between inclusion and matrix) coincides with slip plane, as shown in Fig. 1(d). The misfit strain (also called stress-free transformation strain or eigenstrain describing the homogeneous deformation of the transformed stress-free state) of the platelet is a dyadic prod inclusion under = b n + b n 2d, where b is a Burgers vector, n is the normal and d uct, εidis i j j i j is the platelet thickness equal to the interplanar distance of the slip plane. Such a misfitting thin platelet generates stress that is exactly the same as generated by a dislocation loop of Burgers vector b encircling the platelet [2]. This fact, as will be shown in next two sections, greatly facilitates the description of dislocation microstructure and the solution of the elasticity problem, which is the basis of the WJK phase field microelasticity (PFM) theory of dislocations [4]. This theory was extended by Shen and Wang [11] and WJK [7, 9, 12]. In fact, the dislocation-associated misfit strain εidis j characterizes the plastic strain of the transformed (plastically deformed) platelet inclusion.

2.

Structure Density Field Description of Dislocation Ensemble

As discussed above, by treating dislocation loops as thin platelet misfitting inclusions, instead of describing dislocations by lines, we describe the transformed regions in the untransformed matrix. The transformed regions are the regions that have been plastically deformed by slipping. Dislocations correspond to the boundaries separating the regions of different degrees of slipping. In this description, we track a spatial and temporal evolution of the dislocationassociated misfit strain (plastic strain), which is the structure density field. This field describes the evolution of individual dislocations in an arbitrary ensemble. For an arbitrary dislocation ensemble involving all operative slip systems, the total dislocation-associated misfit strain εidis j (r) is the sum over all slip planes numbered by α: εidis j (r) =

1 α

2

bi (α, r) H j (α) + b j (α, r) Hi (α) ,

(1)

where b(α, r) is the slip displacement vector, H(α) = n(α)/d(α) is the reciprocal lattice vector of the slip plane α, n(α) and d(α) are the normal and interplanar distance, respectively, of the slip plane α. Therefore, a set of

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vector fields, {b(α, r)}, completely characterizes the dislocation configuration. Slipped (plastically deformed) regions are the ones where b(α, r) =/ 0. The vector b(α, r) can be expressed as a sum of the slip displacement vectors numbered by m α corresponding to the operative slip modes within the same slip plane α: b (α, r) =

b (α, m α , r).

(2)

mα

It is convenient to present each field b(α, m α , r) in terms of an order parameter η (α, m α , r) through the following relation b (α, m α , r) = b (α, m α ) η (α, m α , r),

(3)

where η (α, m α , r) is a scalar field, and b (α, m α ) is the corresponding elementary Burgers vector of the slip mode m α in the slip plane α. Thus, an arbitrary dislocation configuration involving all possible slip systems is completely characterized by a set of order parameter fields (phase fields), {η(α, m α , r)}. The number of the fields is equal to the number of the operative slip systems that is determined by the crystallography rather than a concrete dislocation configuration. For example, face-centered cubic (fcc) crystal has four {111} slip planes (α=1, 2, 3, 4) and three 110 slip modes in each slip plane (m α =1, 2, 3), thus has 12 slip systems. A total number of 12 phase fields are used to characterize an arbitrary dislocation ensemble in a fcc crystal if all possible slip systems are involved. An in-depth discussion on the choice of Phase Fields (dislocation density fields) is presented in Ref. [12]. It is noteworthy that the structural density phase field (order parameter) here has the physical meaning of structure (dislocation) density, which is more general than the order parameter used in the phase field model of solidification that assumes 1 in solid and 0 in liquid.

3.

Phase Field Microelasticity Theory

As discussed in the preceding section, the micromechanics of an arbitrary dislocation ensemble involving all operative slip systems is characterized by the dislocation-associated misfit strain εidis j (r) defined in Eq. (1). Substituting Eqs. (2) and (3) into Eq. (1) expresses εidis j (r) as a linear function of a set of phase fields, {η (α, m α , r)}: εidis j (r) =

1 α

mα

2

bi (α, m α ) H j (α) + b j (α, m α ) Hi (α) η (α, m α , r). (4)

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The elastic (strain) energy generated by such a dislocation ensemble is

E

elast

= V

1 Ci j kl εi j (r) − εidis εkl (r) − εkldis (r) d 3r, j (r) 2

(5)

where Ci j kl is elastic modulus tensor, V is body volume, and εi j (r) is the equilibrium strain that minimizes the elastic energy (5) under the compatibility (continuity) condition. The exact elastic energy E elast can be expressed as closed-form functional of εidis j (r). This is obtained by using the KS theory developed for arbitrary multi-phase and multi-domain misfitting inclusions in the homogeneous anisotropic elastic modulus case. The total elastic energy for an arbitrary multidislocation ensemble described by a set of phase fields {η(α, m α , r)} in an appl elastically homogeneous anisotropic body under applied stress σi j is E elast =

1 d 3k α,m α β,m β

−

α,m α

−

2

−

(2π )

3

K α, m α , β, m β , e

∗

×η˜ (α, m α , k) η˜ β, m β , k appl σi j

bi (α, m α ) H j (α)

η (α, m α , r) d 3r

V

V −1 appl appl C σ σkl , 2 i j kl i j

(6)

where η˜ (α, m α , k) = V η (α, m α , r) e−ik·r d 3r is the Fourier transform of η(α, m α , r), the superscript asterisk (*) indicates complex conjugate, e = k/k is

a unit directional vector in the reciprocal (Fourier) space, and the integral as a principal value excluding the point – in the reciprocal space is evaluated k = 0. The scalar function K α, m α , β, m β , e is defined as

K α, m α , β, m β , e = Ci j kl −em Ci j mn np (e) Cklpq eq × bi (α, m α ) H j (α) bk β, m β Hl (β),

(7)

where i j (e) is the Green function tensor inverse to the tensor −1 i j (e) = Cikj l ek el . The elastic energy (6) is a closed-form functional of η(α, m α , r) and their Fourier transform η˜ (α, m α , k) irrespective of dislocation geometrical complexity. This fact makes it easy to achieve high computational efficiency in solving elasticity problem of dislocations. In computer simulations, elasticity solution is obtained numerically. The fields η˜ (α, m α , k) are evaluated by using fast Fourier transform technique, which is also suitable for parallel computing. Since the functional (6) is formulated for arbitrary elastic modulus tensor Ci j kl , a consideration of elastic anisotropy does not impose more difficulty. In fact, in simulations the function K(α, m α , β, m β , e) defined in Eq. (7)

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needs to be evaluated only once and stored in computer memory. Therefore, elastic anisotropy practically does not affect computational efficiency. The elastic energy E elast consists of dislocation self-energy and interaction appl energy as well as the energy generated by the applied stress σi j and the (potential) energy associated with the external loading device. The elastic energy is calculated by using the linear elasticity theory. Equation (6) provides the exact solution for the long-range elastic interactions between individual dislocations in an arbitrary configuration, which is the same as described by the Peach–Koehler equation.

4.

Crystalline Energy and Gradient Energy

In the phase field model, individual dislocations of an arbitrary configuration are completely described by a set of phase fields, {η(α, m α , r)}. For perfect dislocations, each slip displacement vector b(α, m α , r) should relax to a discrete set of values that are multiples of the corresponding elementary Burgers vector b(α, m α ). Thus according to Eq. (3), the order parameter η(α, m α , r) should relax to integer values. The elementary Burgers vectors b(α, m α ) correspond to the shortest crystal lattice translations in the slip planes. For partial dislocations, b(α, m α , r) do not correspond to crystal lattice translations, and η(α, m α , r) may assume non-integer values. The integers η(α, m α , r) are equal to the number of perfect dislocations with Burgers vector b(α, m α ) sweeping through the point r. The sign of the integer determines the slip direction with respect to b(α, m α ). The above-discussed behavior of η(α, m α , r) is automatically achieved by a choice of the Landau-type coarse-grained “chemical” free energy functional of a set of phase fields {η(α, m α , r)}. In the case of dislocations, this free energy is the crystalline energy that reflects the periodic properties of the host crystal lattice:

E

cryst

=

f cryst ({η (α, m α , r)})d 3r,

(8)

V

which should be minimized at {η(α, m α )} equal to integers. The integrand f cryst ({η(α, m α )}) is a periodical function of all parameters {η(α, m α )} with periods equal to any integers. This property follows from the fact that the Burgers vectors b(α, m α , r) in Eq. (3) corresponding to the integers η(α, m α , r) are lattice translation vectors that do not change the crystal lattice. The crystalline energy characterizes an interplanar potential during a homogeneous gliding of one atomic plane above another atomic plane by a slip displacement vector b(α). In the case of one slip mode, say (α1 , m α1 ), the

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local specific crystalline energy function f cryst ({η(α, m α )}) can be reduced to the simplest form by keeping the first non-vanishing term of its Fourier series: f cryst [b(α1 , m α1 )η (α1 , m α1 ) , 0, . . . , 0] = A sin2 π η (α1 , m α1 ),

(9)

where A is a positive constant providing the shear modulus at small strain limit. Its general behavior is schematically illustrated in Fig. 2(a). Any deviation of the slip displacement vector b from the lattice translation vectors is penalized by the crystalline energy. In the case where all slip modes are operative, the general expression of the multi-periodical function f cryst({η(α, m α )}) can also be presented as a Fourier series summed over the reciprocal lattice vectors of the host lattice, which reflects the symmetry of the crystal lattice (see, for detailed discussion, Refs. [4, 9, 11, 12]). The energy E cryst characterizes an interplanar potential of a homogeneous slipping. If the interplanar slipping is inhomogeneous, correction should be made to the crystalline energy (8). This is done by gradient energy E grad that characterizes the energy contribution associated with the inhomogeneity of the slip displacement. For one dislocation loop characterized by the phase field η(α1 , m α1 , r), as shown in Fig. 3(a) where η = 1 inside the disc domain describing the slipped region and 0 outside, E grad is formulated as

E

grad

= V

1 β [n (α1 ) × ∇η (α1 , m α1 , r)]2 d 3r, 2

(10)

where β is a positive coefficient, and ∇ is the gradient operator. As shown in Fig. 3(a), the term n (α) × ∇η (α, m α , r) defines the dislocation sense at point r. The gradient energy (10) is proportional to the dislocation loop perimeter and vanishes over the slip plane. For an arbitrary dislocation configuration characterized by a set of phase fields {η(α, m α )}, the general form of the gradient energy is

E

grad

=

ϕi j (r) d 3r,

(11)

V

(a)

(b) f(b)

f(h)

2γ/d b0

2b0

b

h hc

Figure 2. Schematic illustration of the general behavior of Landau-type coarse-grained “chemical” energy function for (a) dislocation (crystalline energy) and (b) crack (cohesion energy).

Dislocation dynamics – phase field (a)

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Figure 3. (a) A thin platelet domain describing the slipped region. The term n×∇η (r) defines the dislocation sense along the dislocation line (plate edge) and vanishes over the slip plane (plate surface). (b) Schematic of a polycrystal model. Each grain has a different orientation described by its rotation matrix Qi . The rotation matrix function Q (r) completely describes the geometry and orientation of each grain and the entire multi-grain structure.

where the argument of the integrand, ϕi j (r), is defined as ϕi j (r) =

α

[H(α) × ∇η(α, m α , r)]i b j (α, m α ).

(12)

mα

The choice of the tensor ϕi j (r) is dictated by the physical requirements that (i) the gradient energy is proportional to the dislocation length and vanishes over the slip planes and (ii) the gradient energy depends on the total Burgers vector of the dislocation. Following the Landau theory, we can approximate the function ϕi j (r) by the Taylor expansion, which reflects the symmetry of the crystal lattice. As discussed in the preceding section, the elastic energy of dislocations is calculated by using the linear elasticity theory. The nonlinear effects associated with dislocation cores are described in the phase field model by both the crystalline energy E cryst and the gradient energy E grad , which produce significant contributions only near dislocation cores. More detailed discussion on the crystalline and gradient energies is presented in Refs. [4, 9, 11, 12].

5.

Time-dependent Ginzburg–Landau Kinetic Equation

The total energy of a dislocation system is the sum of elastic energy (6), crystalline energy (8) and gradient energy (11): E = E elast + E cryst + E grad ,

(13)

which is a functional of a set of phase fields {η(α, m α , r)}. The temporalspatial dependence of η (α, m α , r, t ) describes the collective motion of the dislocation ensemble. The evolution of η (α, m α , r, t ) is characterized by a

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phenomenological kinetic equation, which is the time-dependent Ginzburg– Landau equation: δE ∂η(α, m α , r, t) = −L + ξ(α, m α , r, t), (14) ∂t δη(α, m α , r, t ) where L is the kinetic coefficient characterizing dislocation mobility, E is the total system energy (13), and ξ(α, m α , r, t) is the Langevin Gaussian noise term reproducing the effect of thermal fluctuations (an in-depth discussion on the invariant form of the time-dependent Ginzburg–Landau kinetic equation is presented in Ref. [12]). A numerical solution η(α, m α , r, t) of the kinetic Eq. (14) automatically takes into account the dislocation multiplication, annihilation, interaction and reaction without ad hoc assumptions. Figure 4 shows one example of the PFM simulation of self-multiplying and self-organizing dislocations during plastic deformation of single crystal ([4]; more simulations are presented therein, and also in Ref. [13] on dislocations in polycrystal, Ref. [11] on network formation, Ref. [14] on solute–dislocation interaction, and Ref. [15] on alloy hardening). The kinetic Eq. (14) is based on the assumption that the relaxation rate of a field is proportional to the thermodynamic driving force. Note that Eq. (14) assumes a linear dependence between dislocation glide velocity v and local resolved shear stress τ along the Burgers vector, i.e., v = mτ b, where m is a constant. In fact, ∂η/∂t = −Lδ E elast/δη −Lδ(E cryst + E grad )/δη, where the first term of the right-hand side gives the linear dependence (L/d) σi j n j bi with σi j being local stress. The second term provides the effect of lattice friction on dislocation motion. It is worth noting that the WJK theory is an interpolational theory providing a bridge between the high and low spatial resolutions. In the high resolution

Figure 4. PFM simulation of stress–strain curve and the corresponding 3D dislocation microstructures during plastic deformation of fcc single crystal under uniaxial loading (after Ref. [4], reproduced with permission).

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limit, it is a 3D generalization of the Peierls–Nabarro (PN) theory [16] to arbitrary dislocation configuration: the WJK theory reproduces the results of the PN theory in a particular case considered in this theory, i.e., a 2D model with a straight single dislocation. The gradient energy (11) is one ingredient that the PN theory lacks. As discussed in the preceding section, the gradient term is necessary as an energy correction associated with slip inhomogeneity and, together with the crystalline energy, describes the core radius and nonlinear core energy. As the PN theory, the WJK theory is applicable in the atomic resolution as well. However, to make the PN and WJK theories fully consistent with atomic resolution modeling, instead of continuum Green function, the atomistic Green function of the crystal lattice statics should be used [2]. To obtain the atomic resolution in the computer simulations, the computational grid sites should be the crystal lattice sites. Another option is to use subatomic scale phase field model where density function models individual atoms [17]. In the low resolution, the WJK theory gives a natural transition to the continuum dislocation theory where local dislocation density εidis j (r), which is related to the dislocation density fields η(α, m α , r) by Eq. (4), is smeared over volume elements corresponding to a computational grid cell, where the grid size l is much larger than the crystal lattice parameter. Then the reciprocal lattice vectors should be defined as H (α) = n (α)/l. In such situations, individual dislocation’s position is uncertain within one grid cell. The dislocation core width, which is the order of crystal lattice parameter, is too small to be resolved by the low resolution computational grids. To effectively eliminate the inaccuracy associated with the Burgers vector relaxation (the core effect) to the dislocation interaction energies at distances exceeding a computational grid length, a non-linear relation between the slip displacement vector b(α, m α , r) and the order parameter η(α, m α , r), rather than the linear relation (3), should be used in the low resolution cases. One simple example of such non-linear relation is [14]:

b(α, m α , r) = b(α, m α ) η(α, m α , r) −

1 2π

sin 2π η (α, m α , r) ,

(15)

which shrinks the effective radius of the dislocation core to improve the accuracy in the mesoscale diffuse-interface modeling. If the resolution of the simulation is microscopic, the use of the non-linear relation becomes unnecessary and the linear dependence (3) of the Burgers vector on the order parameter should be used.

6.

Dislocation Dynamics in Polycrystals

Equation (4) completely characterizes the dislocation configuration in a single crystal, where the elementary Burgers vectors b(α, m α ) and reciprocal

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lattice vectors H(α) are defined in the coordinate system related to the crystallographic axes of crystal. However, it should be modified to characterize a dislocation microstructure in a polycrystal. In the same global coordinate system the components of the vectors b(α, m α ) and H(α) will have different values in different grains because of the mutual rotations of crystallographic axes of grains. In the latter case, we have to describe the orientation of each grain in the polycrystal. To do this, we introduce a static rotation matrix function Q i j (r) that is constant within each grain but assumes different constant values in different grains [13]. In fact, Q i j (r) describes the geometry and orientation of each grain and the entire multi-grain structure, as shown in Fig. 3(b). Then the misfit strain εidis j (r) of a dislocation microstructure in a polycrystal is given by εidis j (r) =

1 α

mα

2

Q ik (r) Q j l (r) [bk (α, m α ) Hl (α)

+ bl (α, m α ) Hk (α)] η (α, m α , r).

(16)

For a single crystal, Q i j (r) = δi j and Eq. (6) is reduced to Eq. (4). Therefore, a dislocation microstructure consisting of all possible slip systems in both single crystal and polycrystal can be completely described by a set of phase fields {η(α, m α , r)}. The elastic energy E elast is still determined by Eq. (6) if the polycrystal is composed of elastically isotropic grains, since the KS theory is applicable to elastically homogeneous body. Otherwise if the grains are elastically anisotropic, their mutual rotations would make the polycrystal an elastically inhomogeneous body. The limitation of grain elastic isotropy could be lifted without serious complication of the theory and computational model by using the PFM theory of elastically inhomogeneous solid [6]. A special case of this theory, viz., a discontinuous body with voids, cracks and free surfaces, will be discussed in the following sections. With the simple modification (16), the above-discussed theory is applicable to dislocation dynamics in polycrystal composed of elastically isotropic grains. Simulation examples are presented in Ref. [13].

7.

Free Surfaces and Heteroepitaxial Thin Films

Free surface is one common type of defects that is shared by all real materials. The stress field is significantly modified near free surfaces (the so-called image force effect). This produces important effects on dislocation dynamics. It is generally a difficult task to calculate the image force corrections to stress field and elastic energy for an arbitrary dislocation configuration in the vicinity of arbitrary-shaped free surfaces. To address this problem, the WJK theory has been extended to deal with finite systems with arbitrary-shaped free

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surfaces based on the theory of a stressed discontinuous body with arbitraryshaped voids and free surfaces [5]. The latter provides an effective method to solve the elasticity problem without sacrificing accuracy. In this section, we discuss the applications of the phase field dislocation dynamics to a system with free surfaces. We first discuss a recently established variational principle that makes this extension possible. A body containing voids is no longer continuous. The elasticity problem for this discontinuous body under applied stress can be solved by using the (r), located following variational principle [5]: if a virtual misfit strain, εivirtual j within the domains of equivalent continuous body minimizes its elastic energy, the generated strain and elastic energy of this equivalent continuous body are the equilibrium strain and elastic energy of original discontinuous body with voids. This variational principle is equally applicable to the cases of voids within a solid and a finite body with arbitrary-shaped free surfaces. The latter can be considered as the body fully “immersed into a void”, where the vacuum around the body can be regarded as the domain defined in the vari(r) ational principle. The position, shape and size of the domains with εivirtual j coincide with those of the voids and surrounding vacuum. Together with the (r), generates externally applied stress, the strain energy minimizer, εivirtual j the stress that vanishes within the domains. The latter allows one to remove the domains without disturbing the strain field and thus return to the initial externally loaded discontinuous body. This variational principle enables one to reduce the elasticity problem of a stressed discontinuous elastically anisotropic body to a much simpler equivalent problem of the continuous body. The above-discussed variational principle leads to the method of determination of the virtual misfit strain εivirtual (r) through a numerical minimization j elast , for the equivalent continuous body with of the strain energy functional, E equiv (r) under external stress. The explicit form of this functional of εivirtual εivirtual (r) j j is given by the KS theory. We may employ a Ginzburg–Landau type equation for energy minimization, which is similar to Eq. (14): elast δ E equiv ∂εivirtual (rd , t) j , (17) = −K i j kl virtual ∂t δεkl (rd , t) where K i j kl is “kinetic” coefficient, t is “time”, and rd represents the points inside the void domains. The “kinetic” Eq. (17) leads to a steady-state solution (r) that is the energy minimizer and generates vanishing stress in the εivirtual j void domains. Equation (17) provides a general approach to determining 3D elastic field, displacement and elastic energy of an arbitrary finite multi-void system in an elastically anisotropic body under applied stress. In particular, it can be used to calculate elasticity solution for a body with mixed-mode cracks of arbitrary configuration, which enables us to develop a phase field model of cracks, as discussed in next section.

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The system with free surface is also structurally inhomogeneous if defects generate a crystal lattice misfit. In the case of dislocations in a heteroepitaxial film, the structural inhomogeneity is characterized by dislocation-associated epitax misfit strain εidis (r) associated with j (r) as well as epitaxial misfit strain εi j crystal lattice misfit between film and substrate. The effective misfit strain εieffect (r) of equivalent system is a sum as j epitax

εieffect (r) = εi j j

virtual (r) + εidis (r). j (r) + εi j

(18)

elast of equivalent system is expressed in terms of The elastic energy E equiv effect εi j (r). For a given dislocation microstructure characterized by εidis j (r), the virtual virtual misfit strain εi j (r) can be determined by using Eq. (17), which has to be solved only at points rd inside the domains corresponding to vacuum (r) generates vanishing stress in around the body. As discussed above, εivirtual j the vacuum domains. Since the whole equivalent system (regions corresponding to vacuum and film/substrate) is in elastic equilibrium, the vanishing stress in the vacuum region automatically satisfies free surface boundary condition. The total energy of a dislocation ensemble near free surfaces is also given elast . Since the role of virby Eq. (13), where the elastic energy is given by E equiv virtual tual misfit strain εi j (r) is just to satisfy the free surface boundary condition, it does not enter crystalline energy (8) or gradient energy (11). As discussed above, the dislocation-associated misfit strain εidis j (r) is a function of a set of phase fields {η (α, m α , r)} given by Eq. (4). Since the epitaxy misfit strain epitax εi j (r) is a static field describing heteroepitaxial structure, the total energy is

a functional of two sets of evolving fields, i.e., E {η (α, m α , r)}, εivirtual (r) . j Following Wang et al. [5], the evolution of dislocations in a heteroepitaxial film is characterized by simultaneous solutions of Eqs. (14) and (17), which is driven by an epitaxial stress relaxation under influence of image forces near free surfaces. Figure 5 shows one example of the PFM simulation of

Figure 5. PFM simulation of motion of a threading dislocation and formation of misfit dislocation at film/substrate interface during stress relaxation in heteroepitaxial film. The numbers indicate the time sequence of dislocation configurations (after Ref. [5], reproduced with permission).

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misfit dislocation formation through threading dislocation motion in epitaxial film [5].

8.

Phase Field Model of Cracks

According to the variational principle discussed in the preceding section, the effect of voids can be fully reproduced by an appropriately chosen virtual misfit strain εivirtual (r) defined inside the domains corresponding to the voids. In j particular, the domains corresponding to cracks are thin platelets of interplanar thickness. To model moving cracks, which can spontaneously nucleate, propagate and coalesce, the virtual misfit strain εivirtual (r) is no longer constrained inside fixed j domains and is allowed to evolve driven by a reduction of total system free energy. In this formalism, εivirtual (r) describes evolving cracks: regions where j virtual εi j (r) =/ 0 are the laminar domains describing cracks. The crack-associated virtual misfit strain is also a dyadic product, εicrack = j (h i n j + h j n i )/2d, where n is the normal and d is the interplanar distance of the cleavage plane, and h(r) is the crack opening vector. As in the phase field model of dislocations, individual cracks of an arbitrary configuration are completely described by a set of fields, {h(α, r)}, where α numbers operative cleavage planes [15]. The total number of the fields is determined by the crystallography rather than a concrete crack configuration. For an arbitrary crack configuration in a polycrystal involving all operative cleavage planes, the total virtual misfit strain is expressed as a function of the fields h(α, r): εicrack (r) = j

1 α

2

Q ik (r) Q j l (r) [h k (α, r) Hl (α) + h l (α, r) Hk (α)], (19)

where H (α) = n (α)/d(α) is the reciprocal lattice vector of the cleavage plane α, and Q i j (r) is the grain rotation matrix field function that describes polycrystalline structure. Under stress, the opposite surfaces of cracks undergo opening displacements h(α, r). For given crack configuration, h(α, r) are a priori unknown and vary under varying stress. The crack-associated virtual misfit strain εicrack (r) j defined in Eq. (18), and thus the fields h(α, r), can be obtained through a numerical minimization procedure similar to that in Eq. (17), where the elaselast tic energy E equiv of such a crack system is also given by the KS elastic energy functional in terms of εicrack (r). j

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The non-linear effect of cohesive forces resisting crack-opening is described by the Landau-type coarse-grained “chemical” energy, which in the case of cracks is the cohesion energy,

f cohes [{h (α, r)}]d 3r,

E cohes =

(20)

V

whose integrand is a function of a set of fields {h(α, r)}. The specific cohesion energy f cohes (h) characterizes an energy that is required to provide a separation of two pieces of crystals by the distance h cut along the cleavage plane. From a microscopic point of view, the energy f cohes (h) is the atomistic energy required for a continuous breaking of atomic bonds across cleavage plane and thus creating two free surfaces during a process of crack formation. A specific approximation of this function similar to the one first proposed by Orowan is formulated by Wang et al. [5]. The general behavior of specific cohesion energy is schematically illustrated in Fig. 2(b), which introduces crack tip cohesive force acting in small crack tip zones. The cohesion energy E cohes defined in Eq. (20) describes a homogeneous separation where both boundaries of crack-opening are kept flat and parallel to cleavage plane. The energy correction associated with the effect of crack surface curvature is taken into account by the gradient energy

E

grad

=

φi j (r) d 3r,

V

(21)

where the argument of the integrand φi j (r) is defined as φi j (r) =

[H (α) × ∇]i h j (α, r),

(22)

α

which is similar to the tensor ϕi j (r) defined in Eq. (12) in the case of dislocations. The choice of the tensor φi j (r) is dictated by similar physical requirement, i.e., the gradient energy is significant only near crack tip where the surface curvature is big and is proportional to the crack front length while vanishes at flat surfaces of homogeneous opening. Following the Landau theory approach, we can also approximate the function φi j (r) by the Taylor expansion, which reflects the symmetry of the crystal lattice (see, for detailed discussion, Refs. [5, 9, 12]). The total free energy of the crack system characterized by the fields h(α, r) (r)), cohesion energy (20) and is the sum of elastic energy (in terms of εicrack j gradient energy (21): E = E elast + E cohes + E grad ,

(23)

which is a functional of a set of fields, {h(α, r)}. The temporal-spatial dependences of h(α, r, t) describe the collective motion of the crack ensemble.

Dislocation dynamics – phase field (a)

2303 (b)

Figure 6. PFM simulation of crack propagation during cleavage fracture in a 2D polycrystal composed of elastically isotropic grains (after Ref. [5], reproduced with permission). Different grain orientations are shown in gray scales.

The evolution of h(α, r, t) is obtained as a solution of the time-dependent Ginzburg–Landau kinetic equation: δE ∂h i (α, r, t) = −L i j + ξi (α, r, t), ∂t δh j (α, r, t)

(24)

where L i j is the kinetic coefficient characterizing crack propagation mobility, E is the system free energy (23), and ξi (α, r, t) is the Gaussian noise term reproducing the effect of thermal fluctuations. As shown by Wang et al. [5], a numerical solution h(α, r, t) of kinetic Eq. (24) automatically takes into account crack evolution without ad hoc assumption on possible path. Figure 6 shows one example of the PFM simulation of self-propagating crack during cleavage fracture in polycrystal [5].

9.

Multi-physics and Multi-scales

This article discusses the recent developments of the phase field theory and models of structurally inhomogeneous systems and their applications to modeling of the multi-dislocation dynamics and multi-crack evolution. The phase field approach can be used to simulate diffusional and displacive phase transformations (see “Phase Field Method–General Description and Computational Issues” by Karma and Chen, “Coherent Precipitation–Phase Field” by Wang, “Ferroic Domain Structures/Martensite” by Saxena and Chen, and the references therein), dislocation dynamics during plastic deformation and cracks development during fracture, as well as dislocation dynamics and morphology evolution [7, 8] of the heteroepitaxial thin films driven by the relaxation of epitaxial stress. These computational models are formulated in the same PFM formalism of the structure density dynamics. The difference between them is only in the analytical form of the Landau-type coarse-grained energy reflecting the physical nature and invariancy properties of the structural heterogeneities. This common analytical framework makes it easy to integrate the models of

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physically different processes into one unified structure density dynamics model. A cost of this would be just an increase in the number of evolving fields. A use of such a unified model allows one to address problems of arbitrary multi-mode microstructure evolution in complex materials systems. In particular, it enables one to investigate structure–property relationships of structurally inhomogeneous materials in situations where the structural heterogeneities of different kinds, which determine the mechanical properties of these materials, simultaneously evolve. The PFM theories and models presented in this article show that while challenges remain, significant advances have been achieved in integrating multiple physical phenomena for simulation of complex materials behavior. The second issue of equal importance is to bridge multiple length and time scales in materials modeling and simulation. Since the PFM approach is based on continuum theory, the PFM simulation is performed at mesoscale from a few nanometers to hundreds of micrometers. The PFM theory can also be applied to the atomic scales, in which case the role of structure density Fields is played by the occupation probabilities of the crystal lattice sites [18]. Recently the Phase Field model has been further extended to the subatomic scale where the field is the subatomic scale continuum density describing individual atoms [17]. The latter model bridges the molecular dynamics approach and the phase field theories discussed in this article. At an intermediate length scale, the mesoscale PFM theory and modeling bridge the gap between the modeling of atomistic level physical processes and macroscopic level material behaviors. The input information to the mesoscale modeling is the macroscopic material constants such as crystallographic data, elastic moduli, bulk chemical energy, interfacial energy, equilibrium composition, domain wall mobility, diffusivity, etc., which could be obtained via either atomistic calculations (first principle, molecular dynamics) or experimental measurements or both. Its output could be directly used to formulate the continuum constitutive relations for macroscopic materials theory and modeling. In particular, the PFM theory and models require a determination of the functional forms of Landau-type energy for different physical processes. This could be obtained through atomistic scale calculations. Incorporation of the results of atomistic simulations into the mesoscale PFM theories is a feasible way for multi-scale modeling.

References [1] A.G. Khachaturyan, Fiz. Tverd. Tela, 8, 2710 (1967. Sov. Phys. Solid State, 8, 2163), 1966. [2] A.G. Khachaturyan, Theory of Structural Transformations in Solids, John Wiley & Sons, New York, 1983. [3] A.G. Khachaturyan and G.A. Shatalov, Sov. Phys. JETP, 29, 557, 1969.

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[4] Y.U. Wang, Y.M. Jin, A.M. Cuiti˜no, and A.G. Khachaturyan, Acta Mater., 49, 1847, 2001. [5] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, J. Appl. Phys., 91, 6435, 2002a. [6] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, J. Appl. Phys., 92, 1351, 2002b. [7] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, Acta Mater., 51, 4209, 2003. [8] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, Acta Mater., 52, 81, 2004. [9] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, “Mesoscale modeling of mobile crystal defects – dislocations, cracks and surface roughening: phase field microelasticity approach,” accepted to Phil. Mag., 2005a. [10] W.L. Bragg and J.F. Nye, Proc. R. Soc. Lond. A, 190, 474, 1947. [11] C. Shen and Y. Wang, Acta Mater., 51, 2595, 2003. [12] Y.U. Wang, Y.M. Jin, and A.G. Khachaturyan, “Structure density field theory and model of dislocation dynamics,” unpublished, 2005b. [13] Y.M. Jin and A.G. Khachaturyan, Phil. Mag. Lett., 81, 607, 2001. [14] S.Y. Hu, Y.L. Li, Y.X. Zheng, and L.Q. Chen, Int. J. of Plast., 20, 403, 2004. [15] D. Rodney, Y. Le Bouar, and A. Finel, Acta Mater., 51, 17, 2003. [16] F.R.N. Nabarro, Proc. Phys. Soc. Lond., 59, 256, 1947. [17] K.R. Elder and M. Grant, “Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals,” unpublished, 2003. [18] L.Q. Chen and A.G. Khachaturyan, Acta Metall. Mater., 39, 2533, 1991.

7.13 LEVEL SET DISLOCATION DYNAMICS METHOD Yang Xiang1 and David J. Srolovitz2 1

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 2 Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA

1.

Introduction

Although dislocation theory had its origins in the early years of the last century and has been an active area of investigation ever since (see [1–3]), our ability to describe the evolution of dislocation microstructures has been limited by the inherent complexity and anisotropy of the problem. This complexity has several contributing features. The interactions between dislocations are extraordinarily long-ranged and depend on the relative positions of all dislocation segments and the orientation of their Burgers vectors and line orientation. Dislocation mobility depends on the orientations of the Burgers vector and line direction with respect to the crystal structure. A description of the dislocation structure within a solid is further complicated by such topological events as annihilation, multiplication and reaction. As a result, analytical descriptions of dislocation structure have been limited to a small number of the simplest geometrical configurations. More recently, several dislocation dynamics simulation methods have been developed that account for complex dislocation geometries and/or the motion of multiple, interacting dislocations. The first class of these dislocation dynamics simulation methods is based upon front tracking methods. Three-dimensional simulations based upon these methods were first performed by Kubin et al. [4, 5] and later augmented by other researchers [6–11]. In these simulation methods, dislocation lines are discretized into individual segments. During the simulations, each segment is tracked and the forces on each segment from all other segments are calculated at each time increment (usually through the Peach–Koehler formula 2307 S. Yip (ed.), Handbook of Materials Modeling, 2307–2323. c 2005 Springer. Printed in the Netherlands.

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[12]). Three-dimensional front tracking methods made it possible to simulate dislocations motion with a degree of reality heretofore not possible. Such methods require, however, large computational investments because they track each segment of each dislocation line and calculate the force on each segment due to all other segments at each time increment. Moreover, special rules are needed to describe the topological changes that occur when segments of the same or different dislocations annihilate or merge [8, 9, 11]. Another class of dislocation dynamics models employs a phase field description of dislocations, as proposed by Khachaturyan, et al. [13, 14]. In their phase field model, density functions are used to model the evolution of a three-dimensional dislocation system. Dislocation loops are described as the perimeters of thin platelets determined by density functions. Since this method is based upon the evolution of a field in the full dimensions of the space, there is no need to track individual dislocation line segments and topological changes occur automatically. However, contributions to the energy that are normally not present in dislocation theory must be included within the phase field model to keep the dislocation core from expanding. In addition, dislocation climb is not easily incorporated into this type of model. Recently, a three-dimensional level set method for dislocation dynamics has been proposed [15, 16]. In this method, dislocation lines in three dimensions are represented as the intersection of zero levels (or zero contors) of two three-dimensional scalar functions (see [17–19] for a description of the level set method). The two three-dimensional level set functions are evolved using a velocity field extended smoothly from the velocity of the dislocation lines. The evolution of the dislocation lines is implicitly determined by the evolution of the two level set functions. Linear elasticity theory is used to compute the stress field generated by solved using a fast Fourier transform (FFT) method, assuming periodic boundary conditions. Since the level set method does not track individual dislocation line segments, it easily handles topological changes associated with dislocation multiplication and annihilation. This level set method for dislocation dynamics is capable of simulating the threedimensional motion of dislocations, naturally accounting for dislocation glide, cross-slip and climb through the choice of the ratio of the glide and climb mobilities. Unlike previous field-based methods [13, 14], no unconventional contributions to the system energy are required to keep the dislocation core localized. Numerical implementation of the level set method is through simple and accurate finite difference schemes on uniform grids. Results of simulation examples using this method agree very well with the theoretic predictions and the results obtained using other methods [15]. This method has also been used to simulate the dislocation-particle bypass mechanisms [16]. Here we shall review this level set dislocation dynamics method and present some of the simulation results in [15, 16].

Level set dislocation dynamics method

2.

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Continuum Dislocation Theory

We first briefly review the aspects of the continuum theory of dislocations that are relevant to the development of the level set description of dislocation dynamics. More complete descriptions of the continuum theory of dislocations can be found in, e.g., [2, 3, 20, 21]. Dislocations are line defects in crystals for which the elastic displacement vector satisfies

du = b,

(1)

L

where L is any contor enclosing the dislocation line with Burgers vector b and u is the elastic displacement vector. We can rewrite Eq. (1) in terms of the distortion tensor w, wi j = ∂u j /∂ xi for i, j = 1, 2, 3, as ∇ × w = ξ δ(γ ) ⊗ b,

(2)

where ξ is the unit vector tangent to the dislocation line, δ(γ ) is the two dimensional delta function in the plane perpendicular to the dislocation and is zero everywhere except on the dislocation, the operator ⊗ implies the tensor product of two vectors. While the Burgers vector is constant along any individual dislocation line, different dislocation lines may have different Burgers vectors. Equation (2) is valid only for dislocations with the same Burgers vector. In crystalline materials, the number of possible Burgers vectors, N , is finite (e.g., typically N = 12 for a FCC metal). Equation (2) may be extended to account for all possible Burgers vectors: ∇×w=

N

ξi δ(γi ) ⊗ bi

(3)

i=1

where γi represents all of the dislocations with Burgers vector bi , and ξi is the tangent to dislocation line i. Next, we consider the tensors describing the strain and stress within the body containing the dislocations. The strain tensor is defined as i j = 12 (wi j + w j i )

(4)

for i, j = 1, 2, 3. The stress tensor σ is determined from the strain tensor by the linear elastic constitutive equations (Hooke’s law) σi j =

3 k,l=1

Ci j kl kl

(5)

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for i, j = 1, 2, 3, where {Ci j kl } is the elastic constant tensor. For an isotropic medium, the constitutive equations can be written as 2ν (6) (11 + 22 + 33 )δi j 1 − 2ν for i, j = 1, 2, 3, where G is the shear modulus, ν is the Poisson ratio, and δi j is equal to 1 if i = j and is equal to 0, otherwise. In the absence of body forces, the equilibrium equation is simply σi j = 2Gi j + G

∇ · σ = 0.

(7)

Finally, the stress and strain tensors associated with a dislocation can be found by combining Eqs. (2), (4), (5) and (7). Dislocations can be driven by stresses within the body. The driving force for dislocation motion, referred to as the Peach–Koehler force, is f = σ tot · b × ξ,

(8)

where the total stress field σ includes the applied stress σ self-stress σ obtained by solving Eqs. (2), (4), (5) and (7): tot

σ tot = σ + σ appl.

appl

and the (9)

Dislocation migration can, at low velocities, be thought of as purely dissipative, such that the local dislocation velocity can be written as v = M · f,

(10)

where M is the mobility tensor. The interpretation of the mobility tensor M is deferred to the next section.

3.

The Level Set Dislocation Dynamics Method

The level set framework was devised by Osher and Sethian [17] in 1987 and and has been successfully applied to a wide range of physical and computer graphics problems [18, 19]. In this section, we present the level set approach to dislocation dynamics. More details and applications of this method can be found in [15, 16]. A level set is defined as a surface on which the level set function has a particular constant value. Therefore, an arbitrary scalar level set function can be used to describe a surface in three dimensional space, a line in two dimensional space, etc. In the level set method for dislocation dynamics, a dislocation in three dimensional space γ (t) is represented by the intersection of the zero levels of two level set functions φ(x, y, z, t) and ψ(x, y, z, t) defined in the three-dimensional space, i.e., where φ(x, y, z, t) = ψ(x, y, z, t) = 0,

(11)

Level set dislocation dynamics method

2311

see Fig. 1. The evolution of the dislocation is described by φt + v · ∇φ = 0 ψt + v · ∇ψ = 0

(12)

where v is the velocity of the dislocation extended smoothly to the threedimensional space, as described below. The reason this system of partial differential equations gives the correct motion of the dislocation can be understood in the following way. Assume that the dislocation γ (s, t), described in parametric form using the variable s, is given by φ(γ (s, t), t) = 0 ψ(γ (s, t), t) = 0,

(13)

where t is time. The derivative of Eq. (13) with respect to t gives ∇φ(γ (s, t), t) · γt (s, t) + φt (γ (s, t), t) = 0 ∇ψ(γ (s, t), t) · γt (s, t) + ψt (γ (s, t), t) = 0.

(14)

Comparing this result with Eq. (12) shows that γt (s, t) = v,

(15)

which means the velocity of the dislocation is equal to v, as required. The velocity field of a dislocation is computed from the stress field using Eqs. (8), (9) and (10). The self-stress field is obtained by solving the elasticity equations: (2), (4), (5) and (7). The unit vector locally tangent to the dislocation line, ξ , in Eqs. (2) and (8), is calculated from the level set functions φ and ψ using ξ=

∇φ × ∇ψ . |∇φ × ∇ψ|

(16)

ψ(x,y,z) ⫽ 0

ψ(x,y,z) ⫽ 0

Figure 1. A dislocation in three-dimensional space γ (t) is the intersection of the zero levels of the two level set functions φ(x, y, z, t) and ψ(x, y, z, t).

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The self-stress obtained by solving the elasticity equations (2), (4), (5) and (7) is singular on the dislocation line. This singularity is artificial because of the discreteness of the atomic lattice and non-linearities in the stress–strain relation not included in the linear elastic formulation. This non-linear region corresponds to the dislocation core. One approach to handling this problem is to use a smeared delta function instead of the exact delta function in Eq. (2) near each point on the dislocation line. The smeared delta function, like the exact one, is defined in the plane perpendicular to the dislocation line, and the vector ξ is defined everywhere in this plane to be the dislocation line tangent vector. This smeared delta function can be considered to be the distribution of the Burgers vector in the plane perpendicular to the dislocation line. The width of the smeared delta function is the diameter of the core region of the dislocation line. We use this approach to treat the dislocation core and its smeared delta function description. More precisely, the smeared delta function in Eq. (2) is given by δ(γ ) = δ(φ)δ(ψ),

(17)

where the delta functions on the right-hand-side are one-dimensional smeared delta functions δ(x) =

1 1 + cos π x

2 0

− ≤ x ≤

,

(18)

otherwise

and scales the distance over which the delta function is smeared. The level set functions φ and ψ are usually chosen to be signed distance functions to their zero levels (i.e., the magnitude of the function is the distance from the closest point on the surface and the sign changes as we cross the zero level) and their zero levels are kept perpendicular to each other. A procedure called reinitialization is used to retain these properties of φ and ψ during their temporal evolution (see the next section for details). Therefore the delta function defined by (17) is a two-dimensional smeared delta function in the plane perpendicular to the dislocation line. Moreover, the size and the shape of the core region do not change during the evolution of the system. We now define the mobility tensor M. A dislocation line can glide conservatively (i.e., without diffusion) only in the plane containing both its tangent vector and the Burgers vector (i.e., the slip plane). A screw segment on a dislocation line can move in any plane containing the dislocation segment, since the tangent vector and Burgers vector are parallel. The switching of a screw segment from one slip plane to another is known as cross-slip. At high temperatures, non-screw segments of a dislocation can also move out of the slip plane by a non-conservative (i.e., diffusive) process; i.e., climb. The

Level set dislocation dynamics method

2313

following form of the mobility tensor satisfies these constraints:

M=

m g (I − n ⊗ n) + m c n ⊗ n

non-screw (ξ not parallel to b)

mgI

screw (ξ parallel to b)

,

(19) where n=

ξ ×b |ξ × b|

(20)

is the unit vector normal to the slip plane (i.e., the plane that contains the tangent vector ξ of the dislocation and its Burgers vector b), I is the identity matrix, I − n ⊗ n is the orthogonal matrix that projects vectors onto the plane with normal vector n, m g is the mobility constant for dislocation glide and m c is the mobility constant for dislocation climb. Typically, mc 1. (21) 0≤ mg The mobility tensor M, defined above, can account for the relatively high glide mobility and slow climb mobility. The present method is equally applicable to all crystal systems and all crystal orientations through appropriate choice of the Burgers vector and the mobility tensor (which can be rotated into any arbitrary orientation). In the present model, the dislocation can slip on all mathematical slip planes (i.e., planes containing the Burgers vector and line direction) and are not constrained to a particular set of crystal plane {hkl}, although it would be relatively simple to impose this constraint. Finally, while we implicitly assume that the glide mobilities of screw and non-screw segments are identical, this restriction is also easily relaxed. For simplicity, we restrict our description of the problem throughout rest of this discussion to the case of isotropic elasticity. While anisotropy will not cause any essential difficulties in the model, the added complexity clouds the description of the method. If we further assume periodic boundary conditions, the stress field can be solved analytically from the elasticity system (2), (4), (6) and (7) in Fourier space. The formulation can be found in [15]. A necessary condition for the elasticity system to have a periodic solution is that the total Burgers vector is equal to zero in the simulation cell. If the total Burgers vector is not equal to zero, the stress is equal to a periodic function plus a linear function in x, y and z [22, 23]. In this case, we also use the above mentioned expression for the stress field, as it only gives the periodic part of that field. This is consistent with the approach suggested by Bulatov et al. for computing periodic image interactions in the front tracking method [22, 23]. The above description of the method can only be applied to the case where all dislocations have the same Burgers vector b. For a more general case,

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where dislocation lines have different Burgers vectors, we would use different level set functions φi and ψi for each of the unique set of Burgers vectors bi , i = 1, 2, . . . , N , where N is the total number of the possible Burgers vectors, and use Eq. (3) instead of Eq. (2) in the elasticity equations.

4. 4.1.

Numerical Implementation Computing the Elastic Fields and the Dislocation Velocity

We solve the elasticity equations associated with the dislocations (2), (4), (6) and (7) using the FFT approach. The first step is to compute the dislocation tangent vector ξ δ(γ ) from the level set functions φ and ψ. The delta function δ(γ ) is computed using Eq. (17) with core radius = 3dx, where dx is the spacing of the numerical grid. The tangent vector ξ is computed using a regularized form of Eq. (16) (to avoid division by zero), i.e., ∇φ × ∇ψ , ξ= |∇φ × ∇ψ|2 + dx 2

(22)

as is standard in level set methods. The gradients of φ and ψ in Eq. (22) are computed using the third order weighted essentially nonoscillatory (WENO) method [24]. Since (WENO) derivatives are one-sided, we switch sides after several time steps to reduce the error caused by asymmetry. After we obtain the stress field, we compute the velocity field using Eqs. (8)–(10). We now use central differencing to compute the gradients of φ and ψ in (22) to get the tangent vector ξ in Eqs. (8) and (20). The mobility tensor in Eq. (10) is computed using Eqs. (19) and (20). We also regularize the denominator in Eq. (20) to avoid division by zero, as we did in Eq. (22). For the mobility tensor (19), we use the mobility for a screw dislocation when |ξ × b| < 0.1 and use the mobility for a non-screw dislocation otherwise.

4.2.

Numerical Implementation of the Level Set Method

4.2.1. Solving the evolution equations The level set evolution equations are commonly solved using high order essentially nonoscillatory (ENO) or WENO methods for the spatial discretization [17, 25, 24] and total variation diminishing (TVD) Runge–Kutta methods for the time discretization [26, 27]. Here we compute the spatial upwind derivatives using the third order WENO method [24] and use the fourth order TVD Runge–Kutta [27] to solve the temporal evolution equations (12).

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4.2.2. Reinitialization In level set methods for three-dimensional curves, the desired level set functions φ and ψ are signed distance functions to their zero levels (i.e., the value at each point in the scalar field is equal to the distance from the closest point on the zero level contor surface with a positive value on one side of the zero level and a minus sign on the other). Ideally, the zero level surfaces of these two functions should be perpendicular to each other. Initially, we choose φ and ψ to be such signed distance functions. However, there is no guarantee that the level set functions will always remain orthogonal signed distance functions during their evolution. This has the potential for causing large numerical errors. Standard level set techniques are used to reconstruct new level set functions from old ones with the dislocations unchanged. The resultant new level set functions are signed distance functions and their zero levels are perpendicular to each other. It has been shown [28, 29, 18, 30] that this procedure does not change the evolution of the lines represented by the intersection of the two level set functions, which are the dislocations here. (1) Signed Distance Functions To obtain a new signed distance function φ˜ from φ, we solve the following evolution equation to steady state [29] φ˜ ˜ − 1) = 0 (|∇ φ| φ˜t +

˜ 2 dx 2 . φ˜ 2 + |∇ φ| ˜ φ(t = 0) = φ

(23)

The new signed distance function ψ˜ from the level set function ψ can be found similarly. We solve for the steady state solutions to these equations using fourth order TVD Runge Kutta [27] in time and Godunov’s scheme [25, 31] combined with third order WENO [24] in space. We iterate these equations several steps of the fourth order TVD Runge Kutta method [27] using a time increment equal to half of the Courant-Friedrichs-Levy (CFL) number (i.e., the numerical stability limit). We solve for the new level set functions φ˜ and ψ˜ at each time step for use in solving the evolution equation (12). (2) Perpendicular Zero Levels Theoretically, the following equation resets the zero level of φ perpendicular to that of ψ [18, 30] ψ

∇ψ · ∇ φ˜ = 0 φ˜t +

2 2 2 ψ + |∇ψ| dx |∇ψ|2 + dx 2 . (24) ˜ φ(t = 0) = φ We solve for the steady state solution to this equation using fourth order TVD Runge Kutta [27] in time and third order WENO [24] for the upwind one˜ The gradient of ψ in the equation is computed using sided derivatives of φ.

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the average of the third order WENO [24] derivatives on both sides. We iterate this equation several steps of the fourth order TVD Runge–Kutta method given in [27] using a time increment of half of the CFL number. We reset the zero level of ψ perpendicular to that of φ similarly. We perform this perpendicular resetting procedure once every few time steps in the integration of the level set evolution equations (Eq. (12)).

4.2.3. Visualization The plotting of the dislocation line configurations is complicated by the fact that the dislocation lines are determined implicitly by the two level set functions. We use the following plotting method, described in more detail in [18]. Each cube in the grid is divided into six tetrahedra. Inside each tetrahedron, the level set functions φ and ψ are approximated by linear functions. The intersection of the zero levels of the two linear functions is a line segment inside the tetrahedron if the intersection is not empty (i.e., we need only compute the two ending points of the line segment on the tetrahedron surface), see Fig. 2. The union of all of these segments is the dislocation configuration.

4.2.4. Velocity interpolation and extension We use a smeared delta function (rather than an exact delta function) to compute the self-stress of the dislocations in order to smooth the singularity in the dislocation self-stress. The region near the dislocations where the smeared delta function is non-zero is the core region of the dislocations. The size of the core region is set by the discretization of space rather than by the physical

A

E G B

D F C

Figure 2. A cube in the grid, a tetrahedron A BC D and a dislocation line segment E F inside the tetrahedron. Point G is on the segment E F and the length of CG is the distance from the grid point C to the segment E F.

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core size. The leading order of the self-stress near the dislocations, when using a smeared delta function, is of the order 1/, where is the dislocation core size. This O(1/) self-stress near the dislocations does not contribute to the motion of the dislocations. We remove this contribution to the self-stress by a procedure which we call velocity interpolation and extension. We first interpolate the velocity on the dislocation line and then extend the interpolated value to the whole space using the fast sweeping method [32–36]. In the velocity interpolation, we use a method similar to that used in the plotting of dislocation lines. For any grid point, the dislocation line segments in nearby cubes can be found by the plotting method. The distance from this grid point to the dislocation line is the minimum distance to any dislocation segment. The remainder of the procedure is most simply described by consideration of the example in Fig. 2. The distance from the grid point of interest, point C for example, to the dislocation line is the distance from C to the segment E F. We locate a point G on the segment E F such that the length of C G is the minimum distance from C to E F. We know the velocity on the grid points of the cube in Fig. 2. We compute the velocity on the points E and F by trilinear interpolation of the velocity on these grid points. Then, we compute the velocity on the point G using a linear interpolation of the velocity on E and F. The velocity of point C is approximated as that on grid point G. To extend the velocities calculated at grid points neighboring the dislocation lines to the whole space, we employ the fast sweeping method [32–36]. The fast sweeping method is an algorithm for obtaining the distance function d(x) to the dislocations at all gridpoints from the distance values at gridpoints neighboring the dislocations (obtained as described above). This involves solving |∇d(x)| = 1

(25)

using the Godunov scheme with Gauss-Seidel iterations [35, 36]. Velocity extension is incorporated into this algorithm by updating the velocity v = (v 1 , v 2 , v 3 ) at each gridpoint after the distance function is determined such that the velocity is constant in the directions normal to the dislocations (the gradient directions of the distance function). This involves solving equations ∇v i (x) · ∇d(x) = 0,

(26)

for i = 1, 2, 3 simultaneously d(x) [32–34].

4.2.5. Initialization Initially, we choose the level set functions φ and ψ such that (1) the intersection of their zero levels gives the initial configuration of the dislocation lines; (2) φ and ψ are signed distance functions to their zero levels, respectively; and (3) the zero levels of φ and ψ are perpendicular to each other.

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Though we solve the elasticity equations assuming periodicity, the level set functions are not necessarily periodic and may be defined in a region smaller than the periodic simulation box.

5.

Applications

Figures 3–10 show several applications of the level set method for dislocation dynamics, described above. Additional simulation details and results can be found in [15, 16]. The simulations were performed within simulation cells that were l × l × l (where l = 2) in arbitrary units. The simulation cell is discretized into 64 × 64 × 64 grid points (For Fig. 6, the simulation cell is 2l ×2l ×l discretized into 128×128×64 grid points). We set the Poisson ratio ν = 1/3 and the climb mobility m c = 0, except in Figs. 3 and 4. The simulations described in Fig. 3, performed with these parameters, required less than five hours on a personal computer with a 450 MHz Pentium II microprocessor. 1 0.8 0.6 0.4 0.2

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Figure 3. A prismatic loop shrinking under its self-stress by climb. The Burgers vector b is pointing out of the paper. The loop is plotted at uniform time intervals starting with the outermost circle. The loop eventually disappears. (a)

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Figure 4. An initially circular glide loop in the x y plane, with a Burgers vector b in the x direction, expanding under a complex applied stress (σx z , σx y =/ 0) with mobility ratios m c /m g of (a) 0, (b) 0.25, (c) 0.5, (d) 0.75, and (e) 1.0. The loop is plotted at regular intervals in time.

Level set dislocation dynamics method

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The computational efficiency is independent of the absolute value of the glide mobility or the absolute value of the grid spacing. Figure 3 shows a prismatic loop (Burgers vector perpendicular to the plane containing the loop) shrinking under its self-stress by climb (the climb mobility m c > 0). The simulation result agrees with the well-known fact that the leading order shrinking force in this case is proportional to the curvature of the loop. Figure 4 shows an initially circular glide loop expanding under a complex applied stress with mobility ratios m c /m g of 0, 0.25, 0.5, 0.75, and 1.0. The applied stress generates a finite force on all the dislocation segments that tends to move them out of the initial slip plane. However, if the climb mobility m c = 0, only the screw segments move out of the slip plane; the non-screw segments cannot because the mobility in such direction is zero (Fig. 4(a)). If the climb mobility m c > 0, both the screw and non-screw segments move out of the slip plane (Fig. 4(b)–(e)). Figure 5 shows the intersection of two initially straight screw dislocations with different Burgers vectors. One dislocation is driven by an applied stress towards the other and then cuts through it. Two pairs of level set functions are used and the elastic fields are described using Eq. (3) instead of Eq. (2). Figure 6 shows the simulation of the Frank-Read source. Initially the dislocation segment is an edge segment. It bends out under an applied stress and generates a new loop outside. The initial configuration in this simulation is a rectangular loop. Of its four segments, two opposite ones are operating as the

b2

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b1 y x

Figure 5. Intersection of two initially straight screw dislocations with Burgers vectors b1 and b2 . Dislocation 1 is driven in the direction of the −x axis by the applied stress σ yz .

Figure 6. Simulation of the Frank-Read source. Initially the dislocation segment is an edge segment in the x y plane (the z axis is pointing out of the paper). The Burgers vector is parallel to the x axis and a stress σx z is applied. The configuration in the slip plane is plotted at different time during the evolution.

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Frank-Read source in the plane perpendicular to the initial loop and the other two are fixed. Figure 7 shows an edge dislocation bypassing a linear array of impenetrable particles, leaving Orowan loops [37] around the particles behind. The dislocation moves towards the particles under an applied stress. The glide plane of the dislocation intersects the centers of the particles (the particles are coplanar). The impenetrable particles are assumed to exert a strong short-range repulsive force on dislocations, see [15] for details. Figure 8 shows a screw dislocation bypassing an impenetrable particle by a combination of Orowan looping [37] and cross-slipping [38]. The dislocation moves towards the particle under an applied stress. It leaves two loops behind on the two sides of the particle. The plane in which the screw dislocation would glide in the absence of the particle is above the particle center. Figure 9 shows an edge dislocation bypassing a misfitting spherical particle by cross-slip [38], where the slip plane of the dislocation is above the particle center. The misfit > 0. The dislocation moves towards the particles under an applied stress. Two loops are left behind: one is behind the particle and the other is around the particle. They have the same Burgers vector but opposite line directions. The stress fields generated by a (dilatational) misfitting spherical particle (isotropic elasticity) were given by Eshelby [39]. Figure 10 shows the critical stress for an edge dislocation to bypass co-planar impenetrable particles by the Orowan mechanism. The stress is 3

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Figure 8. A screw dislocation bypassing an impenetrable particle by a combination of Orowan looping [37] and cross-slipping [38]. The Burgers vector b is in the y direction, the applied stress is σ yz =/ 0, and the plane in which the screw dislocation would glide in the absence of the particle is above the particle center (in the +z direction).

Level set dislocation dynamics method

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Figure 9. An edge dislocation bypassing a misfitting spherical particle by cross-slip [38], where the slip plane of the dislocation is above the particle center. The Burgers vector b is in the x direction, the applied stress is σx z =/ 0. The misfit > 0.

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plotted in the unit (Gb/L) against log(D1 /r0 ), where G is the shear modulus, b is the length of the Burgers vector, L is the inter-particle distance, D is the diameter of the particle, D1 is the harmonic mean of L and D, and r0 is the inner cut-off radius, associated with the dislocation core. The data points represent the simulation results and the straight line is the best fit to our data using the classic equation (Gb/2π L) log(D1 /r0 ) [37, 40, 41]. It shows a good agreement between the simulation results using the level set method and the theoretical estimates.

References [1] V. Volterra, Ann. Ec. Norm., 24, 401, 1905. [2] F.R.N. Nabarro, Theory of Crystal Dislocations, Clarendon Press, Oxford, England, 1967. [3] J.P. Hirth and J. Lothe, Theory of Dislocations, 2nd edition, John Wiley, New York, 1982. [4] L.P. Kubin and G.R. Canova, In: U. Messerschmidt et al. (eds.), Electron Microscopy in Plasticity and Fracture Research of Materials, Akademie Verlag, Berlin, p. 23, 1990. [5] L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, and Y. Brechet, Solid State Phenomena, 23/24, 455, 1992. [6] H.M. Zbib, M. Rhee, and J.P. Hirth, Int. J. Mech. Sci., 40, 113, 1998. [7] M. Rhee, H.M. Zbib, J.P. Hirth, H. Huang, and T. de la Rubia, Modelling Simul. Mater. Sci. Eng., 6, 467, 1998. [8] K.W. Schwarz, J. Appl. Phys., 85, 108, 1999. [9] N.M. Ghoniem, S.H. Tong, and L.Z. Sun, Phys. Rev. B, 61, 913, 2000. [10] B. Devincre, L.P. Kubin, C. Lemarchand, and R. Madec, Mat. Sci. Eng. A-Struct., 309, 211, 2001. [11] D. Weygand, L.H. Friedman, E. Van der Giessen, and A. Needleman, Modelling Simul. Mater. Sci. Eng., 10, 437, 2002. [12] M. Peach and J.S. Koehler, Phys. Rev., 80, 436, 1950. [13] A.G. Khachaturyan, In: E.A. Turchi, R.D. Shull, and A. Gonis (eds.), Science of Alloys for the 21st Century, TMS Proceedings of a Hume-Rothery Symposium, TMS, p. 293, 2000. [14] Y.U. Wang, Y.M. Jin, A.M. Cuitino, and A.G. Khachaturyan, Acta Mater., 49, 1847, 2001. [15] Y. Xiang, L.T. Cheng, D.J. Srolovitz, and W. E, Acta Mater., 51, 5499, 2003. [16] Y. Xiang, D.J. Srolovitz, L.T. Cheng, and W. E, Acta Mater., 52, 1745, 2004. [17] S. Osher and J.A. Sethian, J. Comput. Phys., 79, 12, 1988. [18] P. Burchard, L.T. Cheng, B. Merriman, and S. Osher, J. Comput. Phys., 170, 720, 2001. [19] S. Osher and R.P. Fedkiw, J. Comput. Phys., 169, 463, 2001. [20] R.W. Lardner, Mathematical Theory of Dislocations and Fracture, University of Toronto Press, Toronto and Buffalo, 1974. [21] L.D. Landau and E.M. Lifshitz, Theory of Elasticity, 3rd edn., Pergamon Press, New York, 1986. [22] V.V. Bulatov, M. Rhee, and W. Cai, In: L. Kubin, et al. (eds.), Multiscale Modeling of Materials – 2000, Materials Research Society, Warrendale, PA, 2001.

Level set dislocation dynamics method [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

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W. Cai, V.V. Bulatov, J. Chang, J. Li, and S. Yip, Phil. Mag., 83, 539, 2003. G.S. Jiang and D. Peng, SIAM J. Sci. Comput., 21, 2126, 2000. S. Osher and C.W. Shu, SIAM J. Numer. Anal., 28, 907, 1991. C.W. Shu and S. Osher, J. Comput. Phys., 77, 439, 1988. R.J. Spiteri and S.J. Ruuth, SIAM J. Numer. Anal., 40, 469, 2002. M. Sussman, P. Smereka, and S. Osher, J. Comput. Phys., 114, 146, 1994. D. Peng, B. Merriman, S. Osher, H.K. Zhao, and M. Kang, J. Comput. Phys., 155, 410, 1999. S. Osher, L.T. Cheng, M. Kang, H. Shim, and Y.H.R. Tsai, J. Comput. Phys., 179, 622, 2002. M. Bardi and S. Osher, SIAM J. Math. Anal., 22, 344, 1991. H.K. Zhao, T. Chan, B. Merriman, and S. Osher, J. Comput. Phys., 127, 179, 1996. S. Chen, M. Merriman, S. Osher, and P. Smereka, J. Comput. Phys., 135, 8, 1997. D. Adalsteinsson and J.A. Sethian, J. Comput. Phys., 148, 2, 1999. Y.H.R. Tsai, L.T. Cheng, S. Osher, and H.K. Zhao, SIAM J. Numer. Anal., 41, 673, 2003. H.K. Zhao, Math Comp., to appear. E. Orowan, In: Symposium on Internal Stress in Metals and Alloys, London: The Institute of Metals, p. 451, 1948. P.B. Hirsch, J. Inst. Met., 86, 13, 1957. J.D. Eshelby, In: F. Seitz and D. Turnbull, (ed.), Solid State Physics, vol. 3, Academic Press, New York, 1956. M.F. Ashby, Acta Metall., 14, 679, 1966. D.J. Bacon, U.F. Kocks, and R.O. Scattergood, Phil. Mag., 28, 1241, 1973.

7.14 COARSE-GRAINING METHODOLOGIES FOR DISLOCATION ENERGETICS AND DYNAMICS J.M. Rickman1 and R. LeSar2 1

Department of Materials Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA 2 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

1.

Introduction

Recent computational advances have permitted mesoscale simulations, wherein individual dislocations are the objects of interest, of systems containing on the order of 106 dislocation [1–4]. While such simulations are beginning to to elucidate important energetic and dynamical features, it is worth noting that the large-scale deformation response in, for example, wellworked metals having dislocation densities ranging between 1010 −1014 /m2 can be accurately described by a relatively small number of macrovariables. This reduction in the number of degrees of freedom required to characterize plastic deformation implies that a homogenization, or coarse-graining, of variables is appropriate over some range of length and time scales. Indeed, there is experimental evidence that, at least in some cases, the mechanical response of materials depends most strongly on the macroscopic density of dislocations [5] while, in others, the gross substructural details may also be of importance. A successful, coarse-grained theory of dislocation behavior requires the identification of the fundamental homogenized variables from among the myriad of dislocation coordinates as well as the time scale for overdamped defect motion. Unfortunately, there has been, to date, little effort to devise workable coarse-graining strategies that properly reflect the long-ranged nature of dislocation–dislocation interactions. Thus, in this topical article, we review salient work in this area, highlighting the observation that seemingly unrelated problems are, in fact, part of a unified picture of coarse-grained dislocation behavior that is now emerging. More specifically, a prescription is given for identifying a relevant macrovariable set that describes a collection of mutually interacting dislocations. This set follows from a real-space 2325 S. Yip (ed.), Handbook of Materials Modeling, 2325–2335. c 2005 Springer. Printed in the Netherlands.

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analysis involving the subdivision of a defected system into volume elements and subsequent multipole expansions of the dislocation density. It is found that the associated multipolar energy expansion converges quickly (i.e., usually at dipole or quadrupole order) for well-separated elements. Having formulated an energy functional for the macrovariables, the basic ingredients of temporal coarse-graining schemes are then outlined to describe dislocation–dislocation interactions at finite temperature. Finally, we suggest dynamical models to describe the time evolution of the coarse macrovariables. This article is organized as follows. In Section 2 we outline spatial coarsegraining strategies that permit one to link mesoscale dislocation energetics and dynamics with the continuum. In Section 3 we review some temporal coarsegraining procedures that make it possible to reduce the number of macrovariables needed in a description of thermally induced kinks and jogs on dislocation lines. Section 4 contains a summary of the paper and a discussion of coarse-grained dynamics.

2.

Spatial Coarse-Graining Strategies

A homogenized description of the energetics of a collection of dislocations in, for example, a well-worked metal is complicated by the long-ranged, anisotropic nature of dislocation–dislocation interactions. Such interactions lead to the formation of patterns at multiple length scales as dislocations polygonize to lower the energy of the system [6, 7]. This tendency to form dislocation walls can be quantified via the calculation of an orientationally weighted pair correlation function [8, 9] from a large-scale, two-dimensional mesoscale simulation of edge dislocations, as shown in Fig. 1. As is evident from the figure, both 45◦ and 90◦ walls are dominant (with other orientations also represented), consistent with the propensity to form dislocation dipoles with these relative orientations. Thus, a successful coarse-graining strategy must preserve the essential features of these dislocation structures while reducing systematically the number of degrees of freedom necessary for an accurate description. There are different, although complementary, avenues to pursue in formulating a self-consistent, real-space numerical coarse-graining strategy in which length scales shorter than some prescribed cutoff are eliminated from the problem. One such approach involves subdividing the system into equally sized blocks and then, after integrating out information on scales less than the block size, inferring the corresponding coarse-grained free energy from probability histograms compiled during finite-temperature mesoscale simulations [10–12]. In this context, each block contains many dislocations, and so the free energy extracted from histograms will be a function of a block-averaged dislocation density. This method is motivated by Monte Carlo coarse-graining (MCCG) studies of spin system and can be readily applied, for example, to a

Coarse-graining methodologies for dislocation energetics

2327

Figure 1. An angular pair-correlation function. In the white (black) region there is a relatively high probability of finding a dislocation with positive (negative) Burgers vector, given that a dislocation with positive Burgers vector is located at the origin. From [9].

two-dimensional dislocation system, modeled as a “vector” lattice gas, once the long-ranged nature of the dislocation–dislocation interaction is taken into account. Unfortunately, however, the energy scale associated with dislocation interactions is typically much greater than kB T , where kB is Boltzmann’s constant and T is the temperature, and therefore the finite-temperature sampling inherent in the MCCG technique is not well-suited to the current problem. To develop a more useful technique that reflects the many frustrated, lowenergy states relevant here, consider first the ingredients of a coarse-graining strategy based on continuous dislocation theory. The theory of continuous dislocations follows from the introduction of a coarse-graining volume over which the dislocation density is averaged. The dislocation density is a tensor r ), where k indicates the component of field defined at r with components ρki ( the line direction and i indicates the component of the Burgers vector. In this development it is generally assumed that is large relative to the dislocation spacing, yet small relative to the system size [13]. However, the exact meaning of this averaging prescription is unclear, and it is not obvious at what scales a continuum theory should hold. In particular, if one takes the above assumption that a continuum theory holds for length scales much greater than the typical dislocation spacing, then the applicability of the method is restricted to

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J.M. Rickman and R. LeSar

scales much greater than the dislocation structures known to be important for materials response [14]. Clearly, if the goal is to apply this theory at smaller length scales so as to capture substructures relevant to mechanical response, then one must build the ability to represent such substructures into the formalism. As previous work focused on characterizing these dislocation structures (calculated from two-dimensional simulations) through the use of pair correlation functions [8, 9], we outline here an extension to the continuous dislocation theory that incorporates important spatial correlations. The starting point for this development is the work of Kosevich, who showed that the interaction energy of systems of dislocations (in an isotropic linear elastic medium) can be written in terms of Kr¨oner’s incompatibility tensor [15]. From that form one can derive an energy expression in terms of the dislocation density tensor [16] µ EI = 16π

ipl j mn R,mp ( r , r )

r )ρin ( r ) + δi j ρkl ( r )ρkn (r ) + × ρ j l (

2ν ρil ( r )ρ j n ( r ) d r d r , 1−ν (1)

where the integrals are over the entire system, δi j is the Kronecker delta, and repeated indices are summed. The notation a,i denotes the derivative of a with respect to xi .R,mk indicates the derivative ∂ 2 | r − r | /∂ xm ∂ xk . It should be noted here that the energy expression in Eq. (l) includes very limited information about dislocation structures at scales smaller than the averaging volume. Here we summarize results from one approach to incorporate the effects of lower-scale structures, with a more complete derivation given elsewhere [17]. The basic plan is to divide space into small averaging volumes, calculate the local multipole moments of the dislocation microstructure (as described next), and then to write the energy as an expansion over the multipoles. Consider a small region of space with volume containing n distinct dislocation loops, not necessarily entirely contained within . We can define a set of moment densities of the distribution of loops in as [17] = ρl() j ρl() jα = ···

n 1 (q) b q=1 j

1

n q=1

,

(q)

(q)

dll ,

(2)

C(q),

bj

C(q),

(q)

rα(q) dll ,

(3)

Coarse-graining methodologies for dislocation energetics

2329

where b is the Burgers vector and the notation (C (q) , ) indicates that we integrate over those parts of dislocation line q that lie within the volume . () Here ρl() j is the dislocation density tensor and ρl j α is the dislocation dipole moment tensor for volume . Higher-order moments can also be constructed. Consider next two regions in space denoted by A and B. We can write the interaction energy between the dislocations in the two regions as sums of pair interactions or, equivalently, as line integrals over the dislocation loops [18, 19]. Now, if the volumes are well separated, then the interaction energy can be written as a multipole expansion [17]. Upon truncating this expansion at zeroth order (i.e., the “charge–charge” term) one finds (o) = E AB

µ 8π

A B

×

ipl j mn R,mp

B ) (A ) ρ ( j l ρin

+

(A ) δi j ρkl(B ) ρkn

2ν (B ) (A ) ρ + ρ jn d rA d rB , (4) 1 − ν il

where R connects the centers of the two regions. Summing the interactions between all regions of space and then taking the limit that the averaging volumes A and B go to differential volume elements, the Kosevich form for continuous dislocations in Eq. (l) is recovered and the dislocation density tensor approaches asymptotically the continuous result. Corrections to the Kosevich form associated with a finite averaging volume can now be obtained by including higher-order moments in the expansion. For example, the first-order term (“charge–dipole”) has the form (dipole−charge)

EI

=

µ 16π

ipl j mn R,mpα

ρ j l ( r )ρinα (r )

2ν ρil ( r )ρknα (r ) + r )ρ j nα (r ) + δi j ρkl ( 1−ν

− ρ j lα ( r )ρin (r ) + δi j ρklα ( r )ρkn (r ) 2ν ρil,α ( r )ρ j n (r ) + 1−ν

d r dr

(5)

where R,mpα is the next higher-order derivative of R [17]. We note that inclusion of terms that depend on the local dipole are equivalent to gradient corrections to the Kosevich form. The expression in Eq. (5) (and higher-order terms) can be used as a basis for a continuous dislocation theory with local structure by including the dipole (and higher) dislocation moment tensors as descriptors. For a systematic analysis of the terms in a dislocation multipolar energy expansion and their dependence on coarse-grained cell size, the reader is referred to a review elsewhere [20].

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J.M. Rickman and R. LeSar

Temporal Coarse-graining – Finite-temperature Effects

At finite temperatures dislocation lines may be perturbed by thermally induced kinks and jogs. While such perturbations are inherent in 3D mesoscale dislocation dynamics simulations at elevated temperatures, it is of interest here to explore methods to integrate out these modes to arrive at a simpler description of dislocation interactions. For example, motivated by calculations of the fluctuation-induced coupling of dipolar chains in electrorheological fluids and flux lines in superconductors [21], one can determine the interaction free energy between fluctuating dislocation lines that are in contact with a thermal bath and thereby deduce the effective force between dislocations. Indeed, the impact of temperature-induced fluctuations on the interaction of two (initially) parallel screw dislocations was the focus of a recent paper [16]. In this work it was assumed that perturbations in the dislocation lines that arise from thermal fluctuations in the medium can be viewed as a superposition of modes having screw, edge and mixed character. The impact of these fluctuations on the force between the dislocations at times greater than the those associated with the period of a fluctuation was then examined by integrating out the vibrational modes of the dislocation lines. The procedure employed was similar to that used to construct quaisharmonic models of solids in which vibrational atomic displacements are eliminated in favor of their corresponding frequency spectrum in the canonical partition function [22]. In both cases the resulting free energy then depends on a small set of coarse-grained variables. To see how a finite-temperature force may be constructed, consider a prototypical system in which harmonic perturbations are added to two straight screw dislocation lines without changing the Burgers vector, which remains along the z (i.e., x3 ) axis. We describe those fluctuations by parameterizing the line position in the x1 −x2 plane with a Fourier series with r = xˆ1 F (x3 ) + xˆ2 F⊥ (x3 ) where Fκ (x3 ) =

n max

C+,n,κ einκ π x3 /L + C−,n,κ e−inκ π x3 /L ,

(6)

n κ=1

κ is either ⊥ or , L is a maximal length characterizing the system, and n max is related to a minimum characteristic length. An expression for the dislocation density tensor (ρi j ( r )) in the form of the expansion in Eq. (6) can be written in terms of Dirac delta functions indicating the line position. The next step in the analysis is to calculate the Fourier transform of the dislocation density for the perturbed dislocation lines. While it is possible to write these densities in terms of infinite series expansions, it is more useful here to restrict attention to the lowest-order terms in the fluctuation amplitudes that are excited at low temperatures. Having determined the dislocation density tensor, the aim is then to calculate the interaction energy between two

Coarse-graining methodologies for dislocation energetics

2331

perturbed dislocation lines. This energy will, in turn, determine the corresponding Boltzmann weight for the fluctuating pair of lines and, hence, the equilibrium statistical mechanics of this system. The interaction energy can be obtained from an expression for the total energy, E, based on ideas from continuous dislocation theory [23]. For this purpose it is again convenient to write the Kosevich energy functional, this time as an integral in reciprocal space, [13, 15] as E[ρ] ¯ =

1 2

d3 k ρ˜i j (k) ρ˜kl (−k), K i j kl (k) (2π)3

(7)

where the integration is over reciprocal space (tilde denoting a Fourier transform), the kernel (without core energy contributions)

K i j kl

µ 2ν Ci j Ckl , = 2 Q ik Q j l + Cil Ckj + k 1−ν

(8)

and Q¯ and C¯ are longitudinal and transverse projection operators, respectively. (The energetics of the disordered core regions near each line can be incorporated, at least approximately, by the inclusion of a phenomenological energy penalty term in the kernel above.) The Helmholtz free energy and, therefore, the associated finite-temperature k ) for forces can be obtained by first constructing the partition function Z (k, ˆ ¯ the system of two perturbed screw dislocations with associated k = i k + jˆk¯⊥ and k = iˆ k¯ + jˆk¯ ⊥ . This is accomplished by considering the change in energy,

e(a), associated with fluctuations on the (initially straight) dislocations and noting that it can be written as a sum of contributions, ( e) and ( e)⊥ , corresponding to in-plane and transverse fluctuation modes. One then finds that the factorized partition function k ) = N Z (k,

= Z⊥ Z,

−L( e) dω exp kB T

−L( e)⊥ dω⊥ exp kB T

(9)

where N is a normalization factor and ω is the eight-dimensional configuration space described by the complex fluctuation amplitudes. The Helmholtz free energy associated with the interactions between the fluctuating screws is then given by A = −kB T ln(Z ) = −kB T {ln(Z ) + ln(Z ⊥ )}.

(10)

In our earlier work [16] we gave analytic expressions for both Z ⊥ and Z . Upon integrating A over all possible perturbation wavevectors one finally

2332

J.M. Rickman and R. LeSar 0.002 0.0015

b2/kBT

0.001 0.0005 0 ⫺0.0005 ⫺0.001 ⫺0.0015 ⫺0.002 20

22

24

26

28 a*

30

32

34

36

Figure 2. The contributions to the normalized force versus normalized separation for two perturbed dislocations. The parallel (perpendicular) contribution is denoted by triangles (circles). From [16].

arrives at the total free energy, now a function of coarse-grained variables (i.e., the average line locations.) From the development above it is clear that the average force between the dislocations is obtained by differentiating the total free energy with respect to the line separation a. For the purposes of illustration it is convenient to decompose this force into a sum of components both parallel and perpendicular to a line joining the dislocations. For concreteness, we evaluate the resulting force for dislocations embedded in copper and having the same properties. The maximum size of the system is taken to be L = 200b, where b is the magnitude of the Burgers vector of a dislocation. As can be seen from Fig. 2, a plot of the normalized force contributions versus normalized separation a ∗ (a ∗ = a/b), the parallel (perpendicular) contribution to the force is repulsive (attractive), both components being of similar magnitude. Further analysis indicates that the net thermal force at a temperature of 600 K at a separation of a ∗ = 22 is approximately 1.3 × 10−4 J/m 2 for b = 2.56 Å. This thermal force is approximately 1000 smaller in magnitude than the direct (Peach–Koehler) force for the same separation.

4.

Discussion

Several applications of spatial and temporal coarse graining to systems containing large numbers of dislocations have been outlined here. A common

Coarse-graining methodologies for dislocation energetics

2333

theme linking these strategies is the classification of relevant state variables and the subsequent elimination of a subset of degrees of freedom (via averaging, etc.) in favor of those associated with a coarser description. For example, in the case of the straight screw dislocations interacting with a thermal bath (see Section 3), the vibrational modes of the dislocation lines can be identified as “fast” variables that can be integrated out of the problem, with the resultant free energy based on the long-time, average location of these lines. Furthermore, the spatial coarse graining schemes proposed above involve the identification of a dislocation density, based on localized collections of dislocations, and the separation of interaction length scales (i.e., in terms of a multipolar decomposition and associated gradient expansions) with the aim of developing a model based solely on the dislocation density and other macrovariables. It remains to link coarse-grained dislocation energetics with the corresponding dynamics. While the history of the theory of dislocation dynamics goes back to the early work of Frank [24], Eshelby [25], Mura [26] and others, who deduced the inertial response for isolated edge and screw dislocations in an elastically isotropic medium, we note that the formulation of equations of motion for an ensemble of mutually interacting dislocations at finite temperature is an ongoing enterprise that presents numerous challenges. We therefore merely outline promising approaches here. The construction of a kinetic model is, perhaps, best motivated by earlier work in the field of critical dynamics [27, 28]. More specifically, in this approach, one formulates a set of differential equations that reflect any conservation laws that constrain the evolution of the variables (e.g., conservation of Burgers vector in the absence of sources). Different workers have employed variations of this formalism in dislocation dynamics simulations. For example, in early work in this area, Holt [29] postulated a dissipative equation of motion for the scalar dislocation density, subject to the constraint of conservation of Burgers vector, with a driving force given by gradients of fluctuations in the dislocation interaction energy. Rickman and Vinals [30], following an earlier statistical-mechanical treatment of free dislocation loops [13] and by hydrodynamic descriptions of condensed systems, considered a dynamics akin to a noise-free Model B [28] to track the time evolution of the dislocation density tensor in an elastically isotropic medium. Equations of motion for dislocation densities have also been advanced by Marchetti and Saunders [31] in a description of a viscoelastic medium containing unbound dislocations, by Haataja et al. [32] in a continuum model of misfitting heteroepitaxial films and, recently, by Khachaturyan and coworkers [33–35] in several phase-field simulations. The elegant approach of this group is, however, an alternative formulation of overdamped discrete dislocation models, as opposed to a spatially coarse-grained description. As indicated above, work in this area continues, with some current efforts directed at incorporating dislocation substructural information in the dynamics.

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Acknowledgments J.M. Rickman would like to thank the National Science Foundation for its support under grant number DMR-9975384. The work of R. LeSar was performed under the auspices of the United States Department of Energy (US DOE under Contract No. W-7405-ENG-36) and was supported by the Office of Science/Office of Basic Energy Sciences/Division of Materials Science of the US DOE.

References [1] E. Van der Giessen and A. Needleman, “Micromechanics simulations of fracture,” Ann. Rev. Mater. Res., 32, 141, 2002. [2] R. Madec, B. Devincre, and L. Kubin, “Simulation of dislocation patterns in multislip,” Scripta Mater., 47, 689–695, 2002. [3] M. Rhee, D.H. Lassila, V.V. Bulatov, L. Hsiung, and T.D. de la Rubia, “Dislocation multiplication in BCC molybdenum: a dislocation dynamics simulation,” Phil. Mag. Lett., 81, 595, 2001. [4] M. Koslowski, A.M. Cuitino, and M. Ortiz, “A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals,” J. Mech. Phys. Solids, 50, 2597, 2002. [5] A. Turner and B. Hasegawa, “Mechanical testing for deformation model development,” ASTM, 761, 1982. [6] J.P. Hirth and J. Lothe, Theory of Dislocations, Krieger, Malabar, Florida, 1982. [7] D.A. Hughes, D.C. Chrzan, Q. Liu, and N. Hansen, “Scaling of misorientation angle distributions,” Phys. Rev. Lett., 81, 4664–4667, 1998. [8] A. Gulluoglu, D.J. Srolovitz, R. LeSar, and P.S. Lomdahl, “Dislocation distributions in two dimensions,” Scripta Metall., 23, 1347–1352, 1989. [9] H.Y. Wang, R. LeSar, and J.M. Rickman, “Analysis of dislocation microstructures: impact of force truncation and slip systems,” Phil. Mag. A, 78, 1195–1213, 1998. [10] K. Binder, “Critical properties from Monte Carlo coarse graining and renormalization,” Phys. Rev. Lett., 47, 693–696, 1981. [11] K. Kaski, K. Binder, and J.D. Gunton, “Study of cell distribution functions of the three-dimensional ising model,” Phys. Rev. B, 29, 3996–4009, 1984. [12] M.E. Gracheva, J.M. Rickman, and J.D. Gunton, “Coarse-grained Ginzburg-Landau free energy for Lennard–Jones systems,” J. Chem. Phys., 113, 3525–3529, 2000. [13] D.R. Nelson and J. Toner, “Bond-orientational order, dislocation loops and melting of solids and smectic–a liquid crystals,” Phys. Rev. B, 24, 363–387, 1981. [14] U.F. Kocks, A.S. Argon, and M.F. Ashby, Thermodynamics and Kinetics of Slip, Prog. Mat. Sci., 19, 1975. [15] A.M. Kosevich, In: F.R.N. Nabarro (ed.), Dislocations in Solids, New York, p. 37, 1979. [16] J.M. Rickman and R. LeSar, “Dislocation interactions at finite temperature,” Phys. Rev. B, 64, 094106, 2001. [17] R. LeSar and J.M. Rickman, Phys. Rev. B, 65, 144110, 2002. [18] N.M. Ghoniem and L.Z. Sun, “Fast-sum method for the elastic field of three-dimensional dislocation ensembles,” Phys. Rev. B, 60, 128, 1999.

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[19] R. de Wit, Solid State Phys., 10, 249, 1960. [20] R. LeSar and J.M. Rickman, “Coarse-grained descriptions of dislocation behavior,” to be published in Phil. Mag., 83, 3809–3827, 2003. [21] T.C. Halsey and W. Toor, “Fluctuation-induced couplings between defect lines or particle chains,” J. Stat. Phys., 61, 1257–1281, 1990. [22] J.M. Rickman and D.J. Srolovitz, “A modified local harmonic model for solids,” Phil. Mag. A, 67, 1081–1094, 1993. [23] E. Kr¨oner, Kontinuumstheorie der Versetzungen and Eigenspannungen, Ergeb. Angew. Math. 5 (Springer-Verlag, Berlin 1958). English translation: Continuum Theory of Dislocations and Self-Stresses, translated by I. Raasch and C.S. Hartley, (United States Office of Naval Research), 1970. [24] F.C. Frank, “On the equations of motion of crystal dislocations,” Proc. Phys. Soc., 62A, 131–134, 1949. [25] J.D. Eshelby, “Supersonic dislocations and dislocations in dispersive media,” Proc. Phys. Soc., B69, 1013–1019, 1956. [26] T. Mura, “Continuous distribution of dislocations,” Phil. Mag., 8, 843–857, 1963. [27] J.D. Gunton and M. Droz, “Introduction to the theory of metastable and unstable states,” Springer-Verlag, New York, pp. 34–42, 1983. [28] P.C. Hohenberg and B.I. Halperin, “Theory of dynamic critical phenomena,” in Rev. Mod. Phys., 49, 435–479, 1977. [29] D.L. Holt, “Dislocation cell formation in metals,” J. Appl. Phys., 41, 3197 1970. [30] J.M. Rickman and Jorge Vinals, “Modeling of dislocation structures in materials,” Phil. Mag. A, 75, 1251, 1997. [31] M.C. Marchetti and K. Saunders, “Viscoelasticity from a microscopic model of dislocation dynamics,” Phys. Rev. B 66, 224113, 2002. [32] M. Haataja, J. Miiller, A.D. Rutenberg, and M. Grant, “Dislocations and morphological instabilities: continuum modeling of misfitting heteroepitaxial films,” Phys. Rev. B, 65, 165414, 2002. [33] Y.U. Wang, Y.M. Jin, A.M. Cuitino, and A.G. Khachaturyan, “Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations,” Acta Mater., 49, 1847–1857, 2001. [34] Y.M. Jin and A.G. Khachaturyan, “Phase field microelasticity theory of dislocation dynamics in a polycrystal: model and three-dimensional simulations,” Phil. Mag. Lett., 81, 607–616, 2001. [35] S.Y. Hu and L.-Q. Chen, “Solute segregation and coherent nucleation and growth near a dislocation – a phase-field model for integrating defect and phase microstructures,” Acta Mater., 49, 463–472, 2001.

7.15 LEVEL SET METHODS FOR SIMULATION OF THIN FILM GROWTH Russel Caflisch and Christian Ratsch University of California at Los Angeles, Los Angeles, CA, USA

The level set method is a general approach to numerical computation for the motion of interfaces. Epitaxial growth of a thin film can be described by the evolution of island boundaries and step edges, so that the level set method is applicable to simulation of thin film growth. In layer-by-layer growth, for example, this includes motion of the island boundaries, merger or breakup of islands, and creation of new islands. A system of size 100 × 100 nm may involve hundreds or even thousands of islands. Because it does not require smoothing and or discretization of individual island boundaries, the level set method can accurately and efficiently simulate the dynamics of a system of this size. Moreover, because it does not resolve individual hopping events on the terraces or island boundaries, the level set method can take longer time steps than those of an atomistic method such as kinetic Monte Carlo (KMC). Thus the level set approach can simulate some systems that are computationally intractable for KMC.

1.

The Level Set Method

The level set method is a numerical technique for computing interface motion in continuum models, first introduced by [11]. It provides a simple, accurate way of computing complex interface motion, including merger and pinchoff. This method enables calculations of interface dynamics that are beyond the capabilities of traditional analytical and numerical methods. For general references on level set methods, see the books [12, 21]. The essential idea of the method is to represent the interface as a level set of a smooth function, φ(x) – for example the set of points where φ = 0. For numerical purposes, the interface velocity is smoothly extended to all points x of the domain, as v(x). Then, the interface motion is captured simply by 2337 S. Yip (ed.), Handbook of Materials Modeling, 2337–2350. c 2005 Springer. Printed in the Netherlands.

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R. Caflisch and C. Ratsch

convecting the values of the smooth function φ with the smooth velocity field v. Numerically, this is accomplished by solving the convection equation ∂φ + v · ∇φ = 0 ∂t

(1)

on a fixed, regular spatial grid. The main advantage of this approach is that interface merger or pinch off is captured without special programming logic. The merger of two disjoint level sets into one occurs naturally as this equation is solved, through smooth changes in the function φ(x, t). For example, two disjoint interface loops would be represented by a φ with two smooth humps, and their merging into a single loop is represented by the two humps of φ smoothly coming together to form a single hump. Pinch off is the reverse process. In particular, the method does not involve smoothing out of the interface. The normal component of the velocity v = n · v contains all the physical information of the simulated system, where n is the outward normal of the moving boundary and v · ∇ϕ = v|∇ϕ|. Another advantage of the method is that the local interface geometry – normal direction, n, and curvature, κ – can be easily computed in terms of partial derivatives of φ. Specifically, −∇φ |∇φ| κ =∇·n n=

(2) (3)

provide the normal direction and curvature at points on the interface.

2.

Epitaxial Growth

Epitaxy is the growth of a thin film on a substrate in which the crystal properties of the film are inherited from those of the substrate. Since an epitaxial film can (at least in principle) grow as a single crystal without grain boundaries or other defects, this method produces crystals of the highest quality. In spite of its ideal properties, epitaxial growth is still challenging to mathematically model and numerically simulate because of the wide range of length and time scales that it encompasses, from the atomistic scale of Ångstroms and picoseconds to the continuum scale of microns and seconds. The geometry of an epitaxial surface consists of step edges and island boundaries, across which the height of the surface increases by one crystal layer, and adatoms which are weakly bound to the surface. Epitaxial growth involves deposition, diffusion and attachment of adatoms on the surface. Deposition is from an external source, such as a molecular beam. The principal dimensionless parameter (for growth at low temperature) is the ratio D/(a 4 F),

Level set methods for simulation of thin film growth

2339

in which a is the lattice constant and D and F are the adatom diffusion coefficient and deposition flux. It is conventional to refer to this parameter as D/F, with the understanding that the lattice constant serves as the unit of length. Typical values for D/F are in the range of 104 –108 . The models that are typically used to describe epitaxial growth include the following: Molecular dynamics (MD) consists of Newton’s equations for the motion of atoms on an energy landscape. A typical Kinetic Monte Carlo (KMC) method simulates the dynamics of the epitaxial surface through the hopping of adatoms along the surface. The hopping rate comes from an Arrhenius rate of the form e−E/kB T in which E is the energy barrier for going from the initial to the final position of the hopping atom. Island dynamics and level set methods, the subject of this article, describe the surface through continuum scaling in the lateral directions but atomistic discreteness in the growth direction. Continuum equations approximate the surface using a smooth height function h = h(x, y, t), obtained by coarse graining in all directions. Rate equations describe the surface through a set of bulk variables without spatial dependence. Within the level set approach, the union of all boundaries of islands of height k + 1, can be represented by the level set ϕ = k, for each k. For example, the boundaries of islands in the submonolayer regime then correspond to the set of curves ϕ = 0. A schematic representation of this idea is given in Fig. 1, where two islands on a substrate are shown. Growth of these islands is described by a smooth evolution of the function ϕ (cf. Figs. 1 (a) and (b)). (a) ϕ⫽0

(b) ϕ ⫽0

(c) ϕ⫽0

(d)

ϕ ⫽1 ϕ ⫽0

Figure 1. A schematic representation of the level-set formalism. Shown are island morphologies (left side), and the level-set function ϕ (right side) that represents this morphology.

2340

R. Caflisch and C. Ratsch

The boundary curve (t) generally has several disjoint pieces that may evolve so as to merge (Fig. 1(c)) or split. Validation of the level set method will be detailed in this article by comparison to results from an atomistic KMC model. The KMC model employed is a simple cubic pair-bond solid-on-solid (SOS) model [24]. In this model, atoms are randomly deposited at a deposition rate F. Any surface atom is allowed to move to its nearest neighbor site at a rate that is determined by r = r0 exp{−(E S + n E N )/kB T }, where r0 is a prefactor which is chosen to be 1013 s−1 , kB is the Boltzmann constant, and T is the surface temperature. E S and E N represent the surface and nearest neighbor bond energies, and n is the number of nearest neighbors. In addition, the KMC simulations include fast edge diffusion, where singly bonded step edge atoms diffuse along the step edge of an island with a rate Dedge , to suppress roughness along the island boundaries.

3.

Island Dynamics

Burton, Cabrera and Frank [5] developed the first detailed theoretical description for epitaxial growth. In this “BCF” model, the adatom density solves a diffusion equation with an equilibrium boundary condition (ρ = ρeq ), and step edges (or island boundaries) move at a velocity determined from the diffusive flux to the boundary. Modifications of this theory were made, for example in [9], to include line tension, edge diffusion and nonequilibrium effects. These are “island dynamics” models, since they describe an epitaxial surface by the location and evolution of the island boundaries and step edges. They employ a mixture of coarse graining and atomistic discreteness, since island boundaries are represented as smooth curves that signify an atomistic change in crystal height. Adatom diffusion on the epitaxial surface is described by a diffusion equation of the form 2dNnuc (4) ∂t ρ − D∇ 2 ρ = F − dt in which the last term represents loss of adatoms due to nucleation and desorption from the epitaxial surface has been neglected. Attachment of adatoms to the step edges and the resulting motion of the step edges are described by boundary conditions at an island boundary (or step-edge) for the diffusion equation and a formula for the step-edge velocity v. For the boundary conditions and velocity, several different models are used. The simplest of these is ρ = ρ∗ v=D

∂ρ ∂n

(5)

Level set methods for simulation of thin film growth

2341

in which the brackets indicate the difference between the value on the upper side of the boundary and the lower side. Two choices for ρ∗ are ρ∗ = 0, which corresponds to irreversible aggregation in which all adatoms that hit the boundary stick to it irreversibly, and ρ∗ = ρeq for reversible aggregation. For the latter case, ρeq is the adatom density for which there is local equilibrium between the step and the terrace [5]. Line tension and edge diffusion can be included in the boundary conditions and interface velocity as in ∂ρ = DT (ρ± − ρ∗ ) − µκ, ∂n ±

(6) µ κss , v = DT n · [∇ρ] + βρ∗ ss + DE in which κ is curvature, s is the variable along the boundary, and D E is the coefficient for diffusion along and detachment from the boundary. Snapshots of the results from a typical level-set simulation are shown in Fig. 2. Shown is the level-set function (a) and the corresponding adatom concentration (b) obtained from solving the diffusion Eq. (4). The island boundaries that correspond to the integer levels of panel (a) are shown in (c). Dashed (solid) lines represent the boundaries of islands of height 1. Comparison of panels (a) and (b) illustrates that ρ is indeed zero at the island boundaries (where ϕ takes an integer value). Numerical details on implementation of the level set method for thin film growth are provided in [7]. The figures in this article are taken from [17] and [15].

4.

Nucleation and Submonolayer Growth

For the case of irreversible aggregation, a dimer (consisting of two atoms) is the smallest stable island, and the nucleation rate is dNnuc = Dσ1ρ 2 , (7) dt where · denotes the spatial average of ρ(x, t)2 and σ1 =

4π ln[(1/α)ρD/F]

(8)

is the adatom capture number as derived in [4]. The parameter α reflects the island shape, and α 1 for compact islands. Expression (7) for the nucleation rate implies that the time of a nucleation event is chosen deterministically. Whenever Nnuc L 2 passes the next integer value (L is the system size), a new island is nucleated. Numerically, this is realized by raising the level-set function to the next level at a number of grid points chosen to represent a dimer.

2342

R. Caflisch and C. Ratsch (a)

2.5 2 1.5 1 0.5 0 90

90 60

60 30

30 0

0

(b) 5

z 10 5 4 3 2 1 0 90

90 60

90

60 30

30 0 0

(c)

Figure 2. Snapshots of a typical level-set simulation. Shown are a 3D view of the level-set function (a) and the corresponding adatom concentration (b). The island boundaries as determined from the integer levels in (a) are shown in (c), where dashed (solid) lines correspond to islands of height 1 (2).

Level set methods for simulation of thin film growth

2343

The choice of the location of the new island is determined by probabilistic choice with spatial density proportional to the nucleation rate ρ 2 . This probabilistic choice constitutes an atomistic fluctuation that must be retained in the level set model for faithful simulation of the epitaxial morphology. For growth with compact islands, computational tests have shown additional atomistic fluctuations can be omitted [18]. Additions to the basic level set method, such as finite lattice constant effects and edge diffusion, are easily included [17]. The level set method with these corrections is in excellent agreement with the results of KMC simulations. For example, Fig. 3 shows the scaled island size distribution (ISD)

s ns = 2 g , sav sav

(9)

where n s is the density of islands of size s, sav is the average island size, and g(x) is a scaling function. The top panel of Fig. 3 is for irreversible attachment; the other two panels include reversibility that will be discussed below. All three panels show excellent agreement between the results from level set simulations, KMC and experiment.

5.

Multilayer Growth

In ideal layer-by-layer growth, a layer is completed before nucleation of a new layer starts. In this case, growth on subsequent layers would essentially be identical to growth on previous layers. In reality, however, nucleation on higher layers starts before the previous layer has been completed and the surface starts to roughen. This roughening transition depends on the growth conditions (i.e., temperature and deposition flux) and the material system (i.e., the value of the microscopic parameters). At the same time, the average lateral feature size increases in higher layers, which we will refer to as coarsening of the surface. These features of multilayer growth and the effectiveness of the level set method in reproducing them is illustrated in Fig. 4 that shows the island number density N as a function of time for two different values of D/F from both a level set simulation and from KMC. The results show near perfect agreement. The KMC results were obtained with a value for the edge diffusion that is 1/100 of the terrace diffusion constants. The island density decreases as the film height increases which implies that the film coarsens. The surface roughness w is defined as w 2 = (h i − h)2 ,

(10)

where the index i labels the lattice site. Figure 5 shows the increase of surface roughness for various different values of the edge diffusion, which implies that

2344

R. Caflisch and C. Ratsch 1.4

n s s av 2/ψ

1.2

KMC

1.0

LS

0.8

Exp

0.6 0.4 0.2 0.0 1.4 1.2

n s s av 2/ψ

1.0 0.8 0.6 0.4 0.2 0.0 1.4 1.2

n s s av 2/ψ

1.0 0.8 0.6 0.4 0.2 0.0 0

1

2

3

s /s av

Figure 3. The island size distribution, as given by KMC (squares) and LS (circles) methods, in comparison with STM experiments(triangles) on Fe/Fe(001) [23]. The reversibility increases from top to bottom.

Level set methods for simulation of thin film growth

2345

0.0015 KMC Levelset

N

0.001

0.0005

0

N

0.002

0.001

0

0

2

4 6 Coverage (ML)

8

Figure 4. Island densities N on each layer for D/F =106 (lower panel) and D/F =107 (upper panel) obtained with the level-set method and KMC simulations. For each data set there are 10 curves in the plot, corresponding to the 10 layers.

edge diffusion contributes to roughening, as also observed in KMC studies. It suggests that faster edge diffusion leads to more compact island shapes, and as a result the residence time of an atom on top of compact islands is extended. This promotes nucleation at earlier times on top of higher layers, and thus enhanced roughening. Effects of edge diffusion were included in these simulations through a term of the form κ − κ rather than κss as in (6).

6.

Reversibility

The simulation results presented above have been for the case of irreversible aggregation. If aggregation is reversible the KMC method must simulate a large number of events that do not affect the time-average of the system: Atoms detach from existing islands, diffuse on the terrace for a short period of time and reattach to the same island most of the time. These processes can slow down KMC simulations significantly. On the other hand, in a level set simulation these events can directly be replaced by their time average

2346

R. Caflisch and C. Ratsch D edge 0 D edge 10 D edge 20 D edge 50 D edge 100

0.7

Roughness

0.6

0.5

0.4

0.3

0.2 0

5

10

15

Coverage (ML) Figure 5. Time evolution of the surface roughness w for different values of edge diffusion Dedge .

and therefore the simulation only needs to include detachment events that do not lead to a subsequent reattachment, making the level set method much faster than KMC. Reversibility does not necessarily depend only on purely local conditions (e.g., local bond strength) but often on more global quantities such as strain or chemical environment. To include these kind of effects is a rather hard task in a KMC simulation but can be quite naturally included in a mean field picture. Reversibility can be included in the level set method using the boundary conditions (5) with ρ∗ = ρeq in which ρeq depends on the local environment of the island, in particular the edge atom density [6]. For islands consisting of only of a few atoms, however, the stochastic nature of detachment becomes relevant and is included through random detachment and breakup for small islands, as detailed in [14]. Figure 3 shows that the level set method with reversibility reproduces nicely the trends in the scaled ISD found in the KMC simulations and experiment. In particular, the scaled ISD depends only on the degree of reversibility, and it narrows and sharpens in agreement with the earlier prediction of [19].

Level set methods for simulation of thin film growth 1.4

2347

1.3 1.2

1.2

1.1

log R

1 |

1

1.2

1.4

1

ψ 0.085 ψ 0.16

0.8

0.6 0.5

0

0.5 log t

1

1.5

Figure 6. Time dependence (in seconds) of the average island radius R¯ (in units of the lattice constant) for two different coverages on a log–log plot. The straight lines have slope 1/3, which was the theoretical prediction.

In [15], the level set method with reversibility was used to determine the long time asymptotics of Ostwald ripening. A similar computation was performed in [8]. Figure 6 shows that the average island size R¯ grows as t 1/3 , which was an earlier theoretical prediction. Because reversibility greatly increases the number of hopping events and thus lowers the time step for an atomistic computation, KMC simulations have been unable to reach this asymptotic regime. The longer time steps in the level set simulation give it a significant advantage over KMC for this problem.

7.

Hybrid Methods and Additional Applications

As described above, the level set method does not include island boundary roughness or fractal island shapes, which can be significant in some applications. One way of including boundary roughness is by including additional state variables φ for the density of edge atoms and k for the density of kinks along an island boundary or step edge. A detailed step edge model was derived

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R. Caflisch and C. Ratsch

in [6] and used in determination of ρeq for the level set method with reversibility. While adequate for simulating reversibility, this approach will not extend to fractal island shapes. A promising alternative is a hybrid method that combines island dynamics with KMC; e.g., the adatom density is evolved through diffusion of a continuum density function, but attachment at island boundaries is performed by Monte Carlo [20]. In a different approach [10], where diffusion is described and the adatom density is evolved by explicit solution of the master equation, the atoms are resolved explicitly only once they attach to an island boundary. While this methods do not use a level set method, it is sufficiently similar to the method discussed here to warrant mention in this discussion. Level set methods have been used for a number of thin film growth problems that are related to the applications described above. In [22] a level set method was used to describe spiral growth in epitaxy. A general level set approach to material processing problems, including etching, deposition and lithography, was developed in [1], [2] and [3]. A similar method was used in [13] for deposition in trenches and vias.

8.

Outlook

The simulations described above have established the validity of the level set method for simulation of epitaxial growth. Moreover, the level set method makes possible simulations that would be intractable for atomistic methods such as KMC. This method can now be used with confidence in many applications that include epitaxy along with additional phenomena and physics. Examples that seem promising for future developments include strain, faceting and surface chemistry: Elastic strain is generated in heteroepitaxial growth due to lattice mismatch between the substrate and the film. It modifies the material properties and surface morphology, leading to many interesting growth phenomena such as quantum dot formation. Strained growth could be simulated by combining an elasticity solver with the level set method, and this would have significant advantages over KMC simulations for strained growth. Faceting occurs in many epitaxial systems, e.g., corrugated surfaces and quantum dots, and can be an important factor in the energy balance that determines the kinetic pathways for growth and structure. The coexistence of different facets can be represented in a level set formulation using two level set functions, one for crystal height and the second to mark the boundaries between adjacent facets [16]. Determination of the velocity for a facet boundary, as well for the nucleation of new facets, should be performed using energetic arguments. Similarly, surface chemistry such as the effects of different surface reconstructions could in principle be represented using two level set functions.

Level set methods for simulation of thin film growth

2349

References [1] D. Adalsteinsson and J.A. Sethian, “A level set approach to a unified model for etching, deposition, and lithography 1. Algorithms and two-dimensional simulations,” J. Comp. Phys., 120, 128–144, 1995. [2] D. Adalsteinsson and J.A. Sethian, “A level set approach to a unified model for etching, deposition, and lithography. 2. 3-dimensional simulations,” J. Comp. Phys., 122, 348–366, 1995. [3] D. Adalsteinsson and J.A. Sethian, “A level set approach to a unified model for etching, deposition, and lithography. 3. Redeposition, reemission, surface diffusion, and complex simulations,” J. Comp. Phys., 138, 193–223, 1997. [4] G.S. Bales and D.C. Chrzan, “Dynamics of irreversible island growth during submonolayer epitaxy,” Phys. Rev. B, 50, 6057–6067, 1994. [5] W.K. Burton, N. Cabrera, and F.C. Frank, “The growth of crystals and the equilibrium structure of their surfaces,” Phil. Trans. Roy. Soc. London Ser. A, 243, 299–358, 1951. [6] R.E. Caflisch, W.E, M. Gyure, B. Merriman, and C. Ratsch, “Kinetic model for a step edge in epitaxial growth,” Phys. Rev. E, 59, 6879–87, 1999. [7] S. Chen, M. Kang, B. Merriman, R.E. Caflisch, C. Ratsch, R. Fedkiw, M.F. Gyure, and S. Osher, “Level set method for thin film epitaxial growth,” J. Comp. Phys., 167, 475–500, 2001. [8] D.L. Chopp. “A level-set method for simulating island coarsening,” J. Comp. Phys., 162, 104–122, 2000. [9] B. Li and R.E. Caflisch, “Analysis of island dynamics in epitaxial growth,” Multiscale Model. Sim., 1, 150–171, 2002. [10] L. Mandreoli, J. Neugebauer, R. Kunert, and E. Sch¨oll, “Adatom density kinetic Monte Carlo: A hybrid approach to perform epitaxial growth simulations,” Phys. Rev. B, 68, 155429, 2003. [11] S. Osher and J.A. Sethian, “Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,” J. Comp. Phys., 79, 12–49, 1988. [12] S.J. Osher and R.P. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Springer Verlag, New York, 2002. [13] P.L. O’Sullivan, F.H. Baumann, G.H. Gilmer, J.D. Torre, C.S. Shin, I. Petrov, and T.Y. Lee, “Continuum model of thin film deposition incorporating finite atomic length scales,” J. Appl. Phys., 92, 3487–3494, 2002. [14] M. Petersen, C. Ratsch, R.E. Caflisch, and A. Zangwill, “Level set approach to reversible epitaxial growth,” Phys. Rev. E, 64, #061602, U231–U236, 2001. [15] M. Petersen, A. Zangwill, and C. Ratsch, “Homoepitaxial Ostwald ripening,” Surf. Sci., 536, 55–60, 2003. [16] C. Ratsch, C. Anderson, R.E. Caflisch, L. Feigenbaum, D. Shaevitz, M. Sheffler, and C. Tiee, “Multiple domain dynamics simulated with coupled level sets,” Appl. Math. Lett., 16, 1165–1170, 2003. [17] C. Ratsch, M.F. Gyure, R.E. Caflisch, F. Gibou, M. Petersen, M. Kang, J. Garcia, and D.D. Vvedensky, “Level-set method for island dynamics in epitaxial growth,” Phys. Rev. B, 65, #195403, U697–U709, 2002. [18] C. Ratsch, M.F. Gyure, S. Chen, M. Kang, and D.D. Vvedensky, “Fluctuations and scaling in aggregation phenomena,” Phys. Rev. B, 61, 10598–10601, 2000. [19] C. Ratsch, P. Smilauer, A. Zangwill, and D.D. Vvedensky, “Submonolyaer epitaxy without a critical nucleus,” Surf. Sci., 329, L599–L604, 1995.

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[20] G. Russo, L. Sander, and P. Smereka, “A hybrid Monte Carlo method for surface growth simulations,” preprint, 2003. [21] J.A. Sethian. Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge U. Press, Cambridge, 1999. [22] P. Smereka, “Spiral crystal growth,” Physica D, 138:282–301, 2000. [23] J.A. Stroscio and D.T. Pierce, “Scaling of diffusion-mediated island growth in ironon-iron homoepitaxy,” Phys. Rev. B, 49:8522–8525, 1994. [24] D.D. Vvedensky, “Atomistic modeling of epitaxial growth: comparisons between lattice models and experiment,” Comp. Materials Sci., 6:182–187, 1996.

7.16 STOCHASTIC EQUATIONS FOR THIN FILM MORPHOLOGY Dimitri D. Vvedensky Imperial College, London, United Kingdom

Many physical phenomena can be modeled as particles on a lattice that interact according to a set of prescribed rules. Such systems are called “lattice gases”. Examples include the non-equilibrium statistical mechanics of driven systems [1, 2], cellular automata [3, 4], and interface fluctuations of growing surfaces [5, 6]. The dynamics of lattice gases are generated by transition rates for site occupancies that are determined by the occupancies of neighboring sites at the preceding time step. This provides the basis for a multi-scale approach to non-equilibrium systems in that atomistic processes are expressed as transition rates in a master equation, while a partial differential equation, derived from this master equation, embodies the macroscopic evolution of the coarse-grained system. There are many advantages to a continuum representation of the dynamics of a lattice system: (i) the vast analytic methodology available for identifying asymptotic scaling regimes and performing stability analyses; (ii) extensive libraries of numerical methods for integrating deterministic and stochastic differential equations; (iii) the extraction of macroscopic properties by coarsegraining the microscopic equations of motion, which, in particular, enables (iv) the discrimination between inherently atomistic effects from those that find a natural expression in a coarse-grained framework; (v) the more readily discernible qualitative behavior of a lattice model from a continuum representation than from its transition rules, which (vi) helps to establish connections between different models and thereby facilitate the transferal of concepts and methods across disciplines; and (vii) the ability to examine the effect of apparently minor modifications to the transition rules on the coarse-grained evolution which, in turn, facilitates the systematic reduction of full models to their essential components.

2351 S. Yip (ed.), Handbook of Materials Modeling, 2351–2361. c 2005 Springer. Printed in the Netherlands.

2352

1.

D.D. Vvedensky

Master Equation

The following discussion is confined to one-dimensional systems to demonstrate the essential elements of the methodology without the formal complications introduced by higher dimensional lattices. Every site i of the lattice has a column of h i atoms, so every configuration H is specified completely by the array H = {h 1 , h 2 , . . .}. The system evolves from an initial configuration according to transition rules that describe processes such as particle deposition and relaxation, surface diffusion, and desorption. The probability P(H, t) of configuration H at time t is a solution of the master equation [7], ∂P [W (H − r; r)P(H − r, t) − W (H; r)P(H, t)], = ∂t r

(1)

where W (H; r) is the transition rate from H to H + r, r = {r1 , r2 , . . .} is the array of all jump lengths ri , and the summation over r is the joint summation over all the ri . For particle deposition, H and H + r differ by the addition of one particle to a single column. In the simplest case, random deposition, the deposition site is chosen randomly and the transition rate is W (H; r) =

1 δ(ri , 1) δ(r j , 0), τ0 i j= /i

(2)

where τ0−1 is the deposition rate and δ(i, j ) is the Kronecker delta. A particle may also relax immediately upon arrival on the substrate to a nearby site within a fixed range according to some criterion. The two most common relaxation rules are based on identifying the local height minimum, which leads to the Edwards–Wilkinson equation, and the local coordination maximum, i.e., the site with greatest number of lateral nearest neighbors, which is known as the Wolf-Villain model [5]. If the search range extends only to nearest neighbors, the transition rate becomes W (H; r) =

1 (1) wi δ(ri , 1) δ(r j , 0) + wi(2) δ(ri−1 , 1) δ(r j , 0) τ0 i j= /i j= / i−1

+ wi(3)δ(ri+1 , 1)

δ(r j , 0) ,

(3)

j= / i+1

where the wi(k) embody the rules that determine the final deposition site. The sum rule wi(1) + wi(2) + wi(3) = 1

(4)

Stochastic equations for thin film morphology

2353

expresses the requirement that the deposition rate per site is τ0−1 . The transition rate for the hopping of a particle from a site i to a site j is W (H; r) = k0

wi j δ(ri , −1)δ(r j , 1)

δ(rk , 0),

(5)

k= / i, j

ij

where k0 is the hopping rate and the wi j contain the hopping rules. Typically, hopping is considered between nearest neighbors ( j = i ± 1).

2.

Lattice Langevin Equation

Master equations provide the same statistical information as kinetic Monte Carlo (KMC) simulations [8] and so are not generally amenable to an analytic solution. Accordingly, we will use a Kramers–Moyal–Van Kampen expansion [7] of the master equation to obtain an equation of motion that is a more manageable starting point for detailed analysis. This requires expanding the first term on the right-hand side of Eq. (1) which, in turn, relies on two criteria. The first is that W is a sharply peaked function of r in that there is a quantity δ > 0 such that W (H; r) ≈ 0 for |r| > δ. For the transition rates in Eqs. (2), (3) and (5), this “small jump” condition is fulfilled because the difference between successive configurations is at most a single unit on one site (for deposition) or two sites (for hopping). The second condition is that W is a slowly varying function of H, i.e., W (H + H; r) ≈ W (H; r)

for

|H| < δ.

(6)

In most growth models, the transition rules are based on comparing neighboring column heights to determine, for examine, local height minima or coordination maxima, as discussed above. Thus, an arbitrarily small change in the height of a particular column can lead to an abrupt change in the transition rate at a site, in clear violation of Eq. (6). Nevertheless, this condition can be accommodated by replacing the unit jumps in Eqs. (2), (3) and (5) with rescaled jumps of size −1 , where is a “largeness” parameter that controls the magnitude of the intrinsic fluctuations. The time is then rescaled as t → τ = t/ to preserve the original transition rates. The transformed master equation reads ∂P = ∂τ

(H − r; r)P(H − r, t) − W (H; r)P(H, t) dr, W

(7)

2354

D.D. Vvedensky

corresponding to those in Eqs. (2), (3) and (5) are where the transition rates W given by (H; r) = τ −1 W

δ ri −

i

(H; r) = τ −1 W

i

wi(1)δ

+ wi(2) δ ri−1 −

+ wi(3) δ ri+1 − (H; r) = W

ij

1 δ(r j ), j =/ i

(8)

1 ri − δ(r j ) j =/ i

1 1

δ(r j )

j= / i−1

δ(r j ) ,

j= / i+1

1 1 wi j δ r i + δ rj −

(9) δ(rk ),

(10)

k= / i, j

in which δ(x) is the Dirac δ-function. The central quantities for extracting a Langevin equation from the master : equation in Eq. (7) are the moments of W K i(1) (H) = K i(2) j (H) =

(H; r)dr ∼ O(1), ri W

(11)

(H; r)dr ∼ O(−1 ), ri r j W

(12)

and, in general, K (n) ∼ O(1−n ). With these orderings in , a limit theorem due to Kurtz [9] states that, as → ∞, the solution of the master equation (1) is approximated, with an error of O(ln / ), by that of the Langevin equation dh i (13) = K i(1) (H) + ηi , dτ where the ηi are Gaussian noises that have zero mean, ηi (τ ) = 0, and covariance ηi (τ )η j (τ ) = K i(2) j (H)δ(τ − τ ).

(14)

The solutions of this stochastic equation of motion are statistically equivalent to those of the master equation (1).

3.

The Edwards–Wilkinson Model

There are several applications of the Langevin equation (13). If the occupancy of only a single site is changed with each transition, the correlation

Stochastic equations for thin film morphology

2355

matrix in Eq. (14) is site-diagonal, in which case the numerical integration of Eq. (13) provides a practical alternative to KMC simulations. More important for our purposes, however, is that this equation can be used as a starting point for coarse-graining to extract the macroscopic properties produced by the transition rules. We consider the Edwards–Wilkinson model as an example. The Edwards–Wilkinson model [10], originally proposed as a continuum equation for sedimentation, is one of the standard models used to investigate morphological evolution during surface growth. There are several atomistic realizations of this model, but all are based on identifying the minimum height or heights near a randomly chosen site. In the version we study here, a particle incident on a site remains there only if its height is less than or equal to that of both nearest neighbors. If only one nearest neighbor column is lower than that of the original site, deposition is onto that site. However, if both nearest neighbor columns are lower than that of the original site, the deposition site is chosen randomly between the two. The transition rates in Eq. (3) are obtained by applying these relaxation rules to local height configurations. These configurations can be tabulated by using the step function

θ(x) =

1 if x ≥ 0 0 if x < 0

(15)

to express the pertinent relative heights between nearest neighbors as an identity:

θ(h i−1 − h i ) + (h i−1 − h i ) θ(h i+1 − h i ) + (h i+1 − h i ) = 1, (16)

where (h i − h j ) = 1 − θ(h i − h j ). The expansion of this equation produces four configurations, which are shown in Fig. 1 together with the deposition ( j) rules described above. Each of these is assigned to one of the wi , so the sum rule in Eq. (4) is satisfied by construction, and we obtain the following expressions: wi(1) = θ(h i−1 − h i )θ(h i+1 − h i ),

wi(2) = θ(h i+1 − h i ) 1 − θ(h i−1 − h i ) +

× 1 − θ(h i+1 − h i ) , wi(3) = θ(h i−1 − h i ) 1 − θ(h i+1 − h i ) +

× 1 − θ(h i+1 − h i ) .

1 2

1 2

1 − θ(h i−1 − h i ) 1 − θ(h i−1 − h i )

(17)

The lattice Langevin equation for the Edwards–Wilkinson model is, therefore, from Eq. (13), given by 1 (1) dh i (2) (3) = wi + wi+1 + wi−1 + ηi , dτ τ0

(18)

2356

D.D. Vvedensky (a)

(b)

(c)

(d)

Figure 1. The relaxation rules of the Edwards–Wilkinson model. The rule in (a) corresponds (1) (2) (3) to wi , those in (b) and (d) to wi , and those in (c) and (d) to wi . The broken lines indicates sites where greater heights do not affect the deposition site.

where the ηi have mean zero and covariance ηi (τ )η j (τ ) =

1 (1) (2) (3) wi + wi+1 + wi−1 δi j δ(τ − τ ). τ0

(19)

The statistical equivalence of solutions of this Langevin equation and those of the master equation, as determined by KMC simulations, can be demonstrated by examining correlation functions of the heights. One such quantity is the surface roughness, defined as the root-mean-square of the heights,

W (L , t) = h 2 (t) − h(t)2

1/2

,

(20)

where h k (t) = L −1 i h ki (t) for k = 1, 2, and L is the length of the substrate. For sufficiently long times and large substrate sizes, W is observed to conform to the dynamical scaling hypothesis [5], W (L , t) ∼ L α f (t/L z ), where f (x) ∼ x β for x 1 and f (x) → constant for x 1, α is the roughness exponent, z = α/β is the dynamic exponent, and β is the growth exponent. The comparison of W (L , t) obtained from KMC simulations with that computed from the Langevin equation in (18) is shown in Fig. 2 for systems

Stochastic equations for thin film morphology

2357

(a)

W(lattice units)

L⫽100

Ω⫽1 Ω⫽2

100

Ω⫽50 KMC

100

101

t(ML)

102

(b)

W(lattice units)

L⫽1000

Ω⫽1 Ω⫽20

100

KMC 100

101

102 t(ML)

103

Figure 2. Surface roughness obtained from the lattice Langevin Eq. (18) and KMC simulations for systems of size L = 100 and 1000 for the indicated values of . Data sets for L = 100 were averaged over 200 independent realizations. Those for L = 1000 were obtained from a single realization. The time is measured in units of monolayers (ML) deposited. Figure courtesy of A.L.-S. Chua.

2358

D.D. Vvedensky

of lengths L = 100 and 1000, each for several values of . Most apparent is that the roughness increases with time, a phenomenon known as “kinetic roughening” [5], prior to a system-size-dependent saturation. The roughness obtained from the Langevin equation is greater than that of the KMC simulation at all times, but with the difference decreasing with increasing . The greater roughness is due, in large part, to the noise in Eq. (19): the variance includes information about nearest-neighbors, but the noise is uncorrelated between sites. Thus, as the lattice is scanned, the uncorrelated noise produces a larger variance in the heights than the simulations. But even apart from the rougher growth front the discrepancies for smaller are appreciable. For L = 100 and = 1, 2, the saturation of the roughness is delayed to later times and the slope prior to saturation differs markedly from that of the KMC simulation. There are remnants of these discrepancies for L = 1000, though the slope of the roughness does approach the correct value at sufficiently long times even for = 1.

4.

Coarse-grained Equations of Motion

The non-analyticity of the step functions in Eq. (17), which reflects the threshold character of the relative column heights on neighboring sites, presents a major obstacle to coarse graining the lattice Langevin equation in Eq. (18), as well as those corresponding to other growth models [11, 12]. To address this problem, we begin by observing that θ(x) is required only at the discrete values h k±1 − h k = n, where n is an integer. Thus, we are free to interpolate between these points at our convenience. Accordingly, we use the following representation of θ(x) [13]:

θ(x) = lim+ →0

e(x+1)/ + 1 ln e x/ + 1

.

(21)

For finite , the right-hand side of this expression is a smooth function that represents a regularization of the step function (Fig. 3). This regularization can be expanded as a Taylor series in x and, to second order, we obtain θ(x) = A +

B2x 2 Cx3 Bx − − + ··· , 2 8 62

(22)

where A = ln

1 (1 2

+ e1/ ) ,

B=

e1/ − 1 , e1/ + 1

C=

e1/ (e1/ − 1) . (e1/ )3

As → 0, A → 1 − ln 2 + · · · , B → 1, and C → 0.

(23)

Stochastic equations for thin film morphology

2359

1 ∆⫽1

0.8 ∆⫽0.5

θ (x)

0.6

∆⫽0.25

0.4 0.2 0 ⫺2

⫺1

0

1

x

Figure 3. The regularization in (21) showing how, with decreasing , the step function (shown emboldened) is recovered.

We now introduce the coarse-grained space and time variables x = i and t = z τ/τ0 , where z is to be determined and parametrizes the extent of the coarse-graining, with = 1 corresponding to a smoothed lattice model (with no coarse-graining) and → 0 corresponding to the continuum limit. The coarse-grained height function u is u(x, t) =

α

τ hi − , τ0

(24)

where α is to be determined and τ/τ0 is the average growth rate. Upon applying these transformations and the expansion in Eq. (22) to Eqs. (18) and (19), we obtain the following leading terms in the equation of motion:

z−α

∂u ∂ 2u ∂ 4u ∂ 2 ∂u = ν 2−α 2 + K 4−α 4 + λ1 4−2α 2 ∂t ∂x ∂x ∂x ∂x 3 ∂ ∂u + λ2 4−3α + · · · + (1+z)/2ξ, ∂x ∂x

2

(25)

where ν = B,

K=

1 (4 − 3A), 12

λ1 =

B2 B2 − (1 − A), 8 8

λ2 = −

C , 3 (26)

and ξ is a Gaussian noise with mean zero and covariance ξ(x, t)ξ(x , t ) = δ(x − x )δ(t − t ).

(27)

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The most direct approach to the continuum limit is obtained by requiring (i) that the coefficients of u t , u x x , and ξ have the same scale in and (ii) that these are the dominant terms as → 0. The first of these necessitates setting z = 2 and α = 1/2. To satisfy condition (ii), we first write = δ . A lower bound of the scale of the nth order term in the expansion in Eq. (25) can be estimated from Eq. (22) as

1−n

∂h ∂x

n

1

∼ n(1−α)−(n−1)δ = 2 n−(n−1)δ .

(28)

This yields the condition δ < 1/2, and satisfies condition (ii) for λ1 and λ2 as well. Thus, in the limit → 0, we obtain the Edwards–Wilkinson equation: ∂u ∂ 2 u = + ξ. ∂t ∂ x 2

(29)

The method used to obtain this equation can be applied to other models and in higher spatial dimensions. There have been several simulation studies of the Edwards–Wilkinson [14] and Wolf–Villain [15, 16] models that suggest intriguing and unexpected behavior that is not present for one-dimensional substrates. Taking a broader perspective, if a direct coarse-graining transformation is not suitable, our method can be used to generate an equation of motion as the initial condition for a subsequent renormalization group analysis. This will provide the basis for an understanding of continuum growth models as the natural expression of particular atomistic processes.

5.

Outlook

There are many phenomena in science and engineering that involve a disparity of length and time scales [17]. As a concrete example from materials science, the formation of dislocations within a material (atomic-scale) and their mobility across grain boundaries of the microstructure (“mesoscopic” scale) are important factors for the deformation behavior of the material (macroscopic scale). A complete understanding of mechanical properties thus requires theoretical and computational tools that range from the atomic-scale detail of density functional methods to the more coarse-grained picture provided by continuum elasticity theory. One approach to addressing such problems is a systematic analytic and/or numerical coarse-graining of the equations of motion for one range of length and time scales to obtain equations of motion that are valid over much longer length and time scales. A number of approaches in this direction has already been taken. Since driven lattice models are simple examples of atomic-scale systems, the approach described here may serve as a paradigm for such efforts.

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References [1] C. Godr`eche (ed.), Solids far from Equilibrium, Cambridge University Press, Cambridge, England, 1992. [2] H.J. Jensen, Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems, Cambridge University Press, Cambridge, England, 2000. [3] S. Wolfram (ed.), Theory and Applications of Cellular Automata, World Scientific, Singapore, 1986. [4] G.D. Doolen (ed.), Lattice Gas: Theory Application, and Hardware, MIT Press, Cambridge, MA, 1991. [5] A.-L. Barab´asi and H.E. Stanley, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, England, 1995. [6] J. Krug, “Origins of scale invariance in growth processes,” Adv. Phys., 46, 139–282, 1997. [7] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981. [8] M.E.J. Newman and G.T. Barkema, Monte Carlo Methods in Statistical Physics, Oxford University Press, Oxford, England, 1999. [9] R.F. Fox and J. Keizer, “Amplification of intrinsic fluctuations by chaotic dynamics in physical systems,” Phys. Rev. A, 43, 1709–1720, 1991. [10] S.F. Edwards and D.R. Wilkinson, “The surface statistics of a granular aggregate,” Proc. R. Soc. London Ser. A, 381, 17–31, 1982. [11] D.D. Vvedensky, A. Zangwill, C.N. Luse, and M.R. Wilby, “Stochastic equations of motion for epitaxial growth,” Phys. Rev. E, 48, 852–862, 1993. [12] M. Pˇredota and M. Kotrla, “Stochastic equations for simple discrete models of epitaxial growth,” Phys. Rev. E, 54, 3933–3942, 1996. [13] D.D. Vvedensky, “Edwards–Wilkinson equation from lattice transition rules,” Phys. Rev. E, 67, 025102(R), 2003. [14] S. Pal, D.P. Landau, and K. Binder, “Dynamical scaling of surface growth in simple lattice models,” Phys. Rev. E, 68, 021601, 2003. ˇ [15] M. Kotrla and P. Smilauer, “Nonuniversality in models of epitaxial growth,” Phys. Rev. B, 53, 13777–13792, 1996. [16] S. Das Sarma, P.P. Chatraphorn, and Z. Toroczkai, “Universality class of discrete solid-on-solid limited mobility nonequilibrium growth models for kinetic surface roughening,” Phys. Rev. E, 65, 036144, 2002. [17] D.D. Vvedensky, “Multiscale modelling of nanostructures,” J. Phys.: Condens. Matter, 16, R1537–R1576, 2004.

7.17 MONTE CARLO METHODS FOR SIMULATING THIN FILM DEPOSITION Corbett Battaile Sandia National Laboratories, Albuquerque, NM, USA

1.

Introduction

Thin solid films are used in a wide range of technologies. In many cases, strict control over the microscopic deposition behavior is critical to the performance of the film. For example, today’s commercial microelectronic devices contain structures that are only a few microns in size, and emerging microsystems technologies demand stringent control over dimensional tolerances. In addition, internal and surface microstructures can greatly influence thermal, mechanical, optical, electronic, and many other material properties. Thus it is important to understand and control the fundamental processes that govern thin film deposition at the nano- and micro-scale. This challenge can only be met by applying different tools to explore the various aspects of thin film deposition. Advances in computational capabilities over recent decades have allowed computer simulation in particular to play an invaluable role in uncovering atomic- and microstructure-scale deposition and growth behavior. Ab initio [1] and molecular dynamics (MD) calculations [2, 3] can reveal the energetics and dynamics of processes involving individual atoms and molecules in very fine temporal and spatial resolution. This information provides the fundamentals – the “unit processes” – that work in concert to deposit a solid film. The environmental conditions in the deposition chamber are commonly simulated using either the basic processing parameters directly (e.g., temperature and flux for simple physical vapor deposition systems); or continuum transport/reaction models [4] or direct simulation Monte Carlo methods [5] for more complex chemically active environments. These methods offer a wealth of information about the conditions inside a deposition chamber, but perhaps most important to the modeling of film growth itself are the fluxes and identities of species arriving at the deposition surface. All of 2363 S. Yip (ed.), Handbook of Materials Modeling, 2363–2377. c 2005 Springer. Printed in the Netherlands.

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this information, including atomic-scale information about unit processes and chamber-scale information about surface fluxes and chemistry, must be used to construct a comprehensive model of deposition. Many methods have been used to model film growth. These range from one-dimensional solutions of coupled rate equations, which usually provide only growth rate information; to time-intensive MD simulations of the arrival and incorporation of many atoms at the growth surface, which yield detailed structural and energetic information at the atomic scale. This chapter addresses an intermediate approach, namely kinetic Monte Carlo (KMC) [6], that has been widely and successfully used to model a variety deposition systems. The present discussion is restricted to latticed-based KMC approaches, i.e., those that employ a discrete (lattice) representation of the material, which can provide a wealth of structural information about the deposited material. In addition, the underlying KMC foundation allows the treatment of problems spanning many time and length scales, depending primarily on the nature of the input kinetic data. These kinetic data are often derived using transition state information from experiments or from atomistic simulations. The growth model is often coupled to information about the growth environment such as temperature, pressure, vapor composition, and flux, and these data can be measured experimentally or computed using reactive transport models. The following discussion begins with a brief theoretical background of the Monte Carlo (MC) method in the context of thin film deposition, then continues with a discussion of its implementation, and concludes with an overview of both historical and current applications of KMC (and related variants) to the modeling of thin film growth. The intent is to instill in the reader a basic understanding of the foundations and implementation of the MC method in the context of thin film deposition simulations, and to provide a starting point in their exploration of this broad and rich topic.

2.

The Monte Carlo Method

Many collective phenomena in nature are essentially deterministic. For example, a ball thrown repeatedly with a specific initial velocity (in the absence of wind, altitudinal air density variations, and other complicating factors) will follow virtually the same trajectory each time. Other behaviors appear stochastic, as evidenced by the seemingly random behavior of a pachinko ball. Nanonscopically (i.e., on the time and length scale of atomic motion), most processes behave stochastically rather than deterministically. The vibrations of an atom or molecule as it explores the energetic landscape near the potential energy minimum created by the interactions with its environment are, for all practical purposes, random, i.e., stochastic. When that atom is in the vicinity of others, e.g., in a solid or liquid, the energetic landscape is very

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complex and consists of many potential energy minima separated by energy barriers (i.e., maxima). Given enough time, a vibrating atom will eventually happen to “hop over” one of these barriers and “fall into” an adjacent potential energy “well.” In doing so, the atom has transitioned from one state (i.e., energy) to a new one. The energetics of such a transition are depicted in Fig. 1, where the states are described by their free energies (i.e., both enthalpic and entropic contributions). These concepts apply not only to vibrating atoms but also to the fundamental transitions of any system that has energy minima in configurational space. Transition state theory describes the frequency of any transition that can be described energetically by a curve like the one in Fig. 1. Although a detailed account of transition state theory is beyond the scope of this chapter, suffice it to say that the average rate of transitioning from State A to State B is described by the rate constant

E , kA→B = A exp − kT

(1)

where A is the frequency with which the system attempts the transition, E is the activation barrier, k is Boltzmann’s constant equal to 1.3806503 × 10−23 J K−1 = 8.617269 × 10−5 eV K−1 , and T is the temperature. Likewise, the

Energy

E

A ∆G

B Reaction coordinate Figure 1.

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average rate of the reverse transition from State B to State A is described by the rate constant E − G , (2) kA←B = A exp − kT where G is the change in free energy on transitioning from State A to B (notice from Fig. 1 that G is negative), and the reuse of the symbol, A, implies that the attempt frequencies for the forward (A → B) and reverse (A ← B) transitions are assumed equal. (The rate constants are obviously in the same units as the attempt frequency. If these units are not those of an absolute rate, i.e., sec−1 , then the rate constant can be converted into an absolute rate by multiplying by the appropriate quantity, e.g., concentrations in the case of chemical reactions.) Whereas Eqs. (1) and (2) describe the average rates for the transitions in Fig. 1, the actual rates for each instance of a particular transition will vary because the processes are stochastic. The state of the system will vary (apparently) randomly inside the energy well at State A until, by chance, the system happens to make an excursion that reaches the activated state, at which point (according to transition state theory) the system has a 50% chance of returning to State A and a 50% chance of transitioning into State B. The Monte Carlo (MC) method, named after the casinos in the Principality of Monaco (an independent sovereign state located between the foot of the Southern Alps and the Mediterranean Sea) is ideally suited to modeling not only realistic instantiations of individual state transitions (provided the relevant kinetic parameters are known) but also time- and ensemble-averages of complex and collective phenomena. The MC method is essentially an efficient method for numerically estimating complex and/or multidimensional integrals [7]. It is commonly used to find a system’s equilibrium configuration via energy minimization. Early MC algorithms involved choosing system configurations at random, and weighting each according to its potential energy via the Boltzmann equation, E (3) P = exp − kT where P is the weight (i.e., the probability the configuration would actually be realized). The configuration with the most weight corresponds to equilibrium. Metropolis et al. [7] improved on this scheme with an algorithm that, instead of choosing configurations randomly and Boltzmann-weighting them, chooses configurations with the Boltzmann probability in Eq. (3) and weighting them equally. In this manner, the model system wastes less time in configurations that are highly unlikely to exist. Bortz et al. [8] introduced yet another rephrasing of the MC method, and termed the new algorithm the N-Fold Way (NFW). This algorithm always accepts the chosen changes to the system’s configuration, and shifts the stochastic component of the computation into the time

Monte Carlo methods for simulating thin film deposition

2367

incrementation (which can thereby vary at each MC step). Independent discoveries of essentially the same algorithm were presented shortly thereafter by Gillespie [9], and more recently by Voter [10]. The NFW is only applicable in situations where the Boltzmann probability is nonzero only for a finite and enumerable set of configurational transitions. So, for example, it cannot be used (without adaptation of either the algorithm or the model system) to find the equilibrium positions of atoms in a liquid, since the phase space representing these positional configurations is continuous and thus contains a virtually infinite number of possible transitions. Both the Metropolis algorithm (in its kinetic variation, described below) and the NFW can treat kinetic phenomena, but the NFW is better suited to generating physically realistic temporal sequences of configurational transitions [6] provided the rates of all possible state transitions are known a priori. To illustrate the concepts behind these techniques, it is useful to consider a simple example. Imagine a system that can exist in one of three states: A, B, or C. All the possible transitions for this system are therefore A ↔ B ↔ C. When the system is in State A, it can undergo only one transition, i.e., conversion to State B. When in State C, the system is only eligible for conversion to State B. When in State B, the system can either convert to State A, or convert to State C. Assume that the energetics of the transition paths are described by Fig. 2. The symbol *IJ denotes the activated state for the transition between

AB

CB Energy

EAB

ECB C

A

∆GCB

∆GAB

B Reaction coordinate Figure 2.

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States I and J, E I J is the activation barrier encountered upon the transition from State I to State J, and G IJ is the difference in the free energies between States I and J. (Note that both G AB and G CB are negative because the free energy decreases upon transitioning from State A to B, and from State C to B.) The lowest-energy state is B, the highest is C, and A is intermediate. Simply by examining Fig. 2, it is clear that the thermodynamic equilibrium for this system is State B. However, the kinetic properties of the system depend on the transition rates, which in turn depend not only on the energies but also on the attempt frequencies. If the attempt frequencies of all four transitions are equal, then the state with the maximum residence time (in steady state) would certainly be State B, and that with the minimum time would be State C. Otherwise the residence properties might be quite different. As aforementioned, the Metropolis algorithm proceeds by choosing configurations at random, and accepting or rejecting them based on the change to the system energy that is incurred by changing the system’s configuration. So, in the present example, such an algorithm would randomly choose one of the three states – A, B, or C – and accept or reject the chosen state with a probability based on the energy difference between it and the previous state. Specifically, the probability of accepting a new State J when the system is in State I is PI→J =

G IJ exp − kT

G IJ > 0

.

(4)

G IJ ≤ 0

1

This so-called thermodynamic Metropolis MC approach clearly utilizes only the states’ energy differences, and does not account for the properties of the activated states or the dynamics that lead to transition attempts. As such, it can reveal the equilibrium state of the system, but provides no information about the kinetics of the system’s evolution. However, the same algorithm can be adapted into a kinetic Metropolis MC scheme in order to capture kinetic information. This is accomplished by introducing time into the approach, and by using the transition rate information from Eqs. (1) and (2). Specifically, the rate constants for the “forward” transition, I→J, and the “backward,” I←J, are E IJ and (5) kI→J = AIJ exp − kT

kI←J = AJI exp −

E IJ − G IJ . kT

(6)

Assuming that the rate and the rate constant are the same, the probability of accepting a new State J when the system is in an “adjacent” State I is PI→J = kI→J t,

(7)

Monte Carlo methods for simulating thin film deposition

2369

where t is a constant time increment that is chosen a priori to accommodate the fastest transitions in the problem. Generally, t is chosen to be near 0.5/kmax . Thus, at each step in a kinetic Metropolis MC calculation, a transition is chosen at random from those that are possible given the state of the system, the chosen transition is realized with a probability according to Eq. (7), and the time is incremented by the constant t. Notice that while the thermodynamic Metropolis scheme allows the system to change its configuration to a state that is not directly accessible (e.g., A→C), the kinetic Metropolis approach considers only transitions between accessible states (i.e., the transitions A ↔ C in Fig. 2 would be forbidden). Similarly, the NFW deals only with accessible transitions, but unlike the kinetic Metropolis formulation, the NFW realizes state transitions with unit probability. Specifically, at each step in an NFW computation, a transition is chosen at random from those that are possible given the state of the system. The probability of choosing a particular transition depends on its relative rate. As such, i−1 j =1

kj < ζ ≤

M

kj,

(8)

j =i

where j merely indexes each transition, i denotes the chosen transition, ζ is a random number between zero (inclusive) and one (exclusive) such that ζ ∈ [0, 1), and is the sum of the rates of all the transitions that are possible given the state of the system. (Recall that the transition rates are equal to the rate constants in the present example, as aforementioned.) The chosen transition is always realized, and the time is incremented by ln (ξ ) , (9) t = − where ξ is another random number between zero and one (exclusive of both bounds) such that ξ ∈ (0,1). On closer inspection, it is apparent that the NFW is simply a rearrangement of the kinetic Metropolis MC algorithm [8]. Consider a system in some arbitrary State I. Assume that the system can exist in multiple states, so that the system will eventually (at non-zero temperature) transition out of State I. Because the transitioning process is stochastic, the time that the system spends in State I will vary each time it visits that state. (This “fact” is evident in the kinetic MC algorithms discussed above.) Let P− (dt) denote the probability that the system remains in State I for a time of at least dt, and P+ (dt) be the probability that the system leaves State I before dt has elapsed, where dt = 0 refers to the moment that the system entered State I. Since the system has no other choices but to either stay in State I during dt or leave State I sometime during dt, it is clear that P− (dt) + P+ (dt) = 1.

(10)

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Consider some value of time, t =/ dt, where again t =0 refers to the moment that the system entered State I. Multiplying Eq. (10) by P− (t) yields P− (dt) P− (t) + P+ (dt) P− (t) = P− (t).

(11)

Notice that P− (dt)P− (t) = P− (t)P− (dt), and is simply the probability that the system is still in State I after t and also after the following dt, i.e., it is the probability that the system remains in State I for at least a time of t + dt. Therefore, P− (t + dt) + P+ (dt) P− (t) = P− (t).

(12)

Also notice that P+ (dt) = dt,

(13)

where is the average number of transitions from State I per unit time, i.e., the sum of the rates of all the transitions that the system can make from State I. Substituting Eqs. (12) and (13) into Eq. (11) yields P− (t + dt) + P− (t) dt = P− (t).

(14)

Rearranging Eq. (14) produces P− (t + dt) − P− (t) = − P− (t). dt In the limit that dt → 0, Eq. (15) becomes

(15)

dP−

= − P− (t). dt t

(16)

Integrating Eq. (16), and realizing that P− (0) = 1, yields

ln P− (t) = −t, hence

(17)

ln P− (t) . (18) t = − Let t∗ be the average residence time for State I. On each visit that the system makes to State I, it remains there for a different amount of time, and the associated residence probabilities for each visit follow a uniform random distribution such that P− (t∗ ) ∈ (0,1). Therefore, the individual residence times from visit to visit follow a distribution of the form ln (ξ ) , (19) t∗ = − where ξ is a random number such that ξ ∈ (0,1), and thus Eq. (9) is obtained. Clearly the time that elapses between one transition and the next is stochastic

Monte Carlo methods for simulating thin film deposition

2371

and is a function only of the sum of the rates of all available transitions. When in any given state, the probability that the system will actually make a particular transition, provided it is accessible, is equal to the rate of the transition relative to the sum of the rates of all accessible transitions, as described in Eq. (8).

3.

Implementing the N-Fold Way

One can readily see the utility of the NFW for simulating the fundamental processes involved in thin film deposition. Simply put, one need only apply the algorithms described above, illustrated for the idealized system in Fig. 2, onto each fundamental location on the model deposition surface. For example, consider the simple two-dimensional surface in Fig. 3a. Assume that each square represents a fundamental unit of the solid structure (e.g., an atom), that there is a flux of material toward the substrate, and that gray denotes a static substrate. If the incoming material is appropriate for coherent epitaxy on the substrate, then the evolution of the surface in Fig. 3a will begin by the attachment of material to the surface, i.e., the filling of one of the sites above the surface denoted by dotted squares in Fig. 3b, by a unit of incoming material. Consider only one of these candidate sites, e.g., the one labeled d in Fig. 3c. Site d represents a subsystem of the entire surface, and that subsystem is in a particular state whose configuration is defined by the “empty” site just above the surface. Site d can transition into another state, namely one in which the site contains a deposited unit of material, as depicted in Fig. 3d. This local transition occurs just as described in the simple example for Fig. 2 above. In fact, the evolution of the entire surface can be modeled by collectively considering the local transitions of each fundamental unit (i.e., site) in the system. Consider the behavior of the entire system from the initial configuration shown in Fig. 3c. Each site above the surface can be filled by incoming

(a)

Flux

(b)

(c)

(d)

Figure 3. Deposition of a single practicle onto a simple two-dimensional substrate. Gray squares are substrate sites, white dotted squares are sites into which particles can potentially deposit, and black squares are deposited particles.

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material. The NFW algorithm suggests that the time that passes before a particular site, e.g., Site a transitions is ta = −

X ln (ξ ), F

(20)

where X is the aerial density of surface sites in units of length−2 and F is the deposition flux in length−2 sec−1 (taking into account such factors as the sticking probability), so that F/X is the average rate at which material can deposit into Site a. But how much time passes before any of the sites makes a transition? In other words, how long will the system remain in the configuration of Fig. 3c? By way of analogy, consider rolling six-sided dice. If only one die is rolled, the chance that a particular side faces upwards (after the die comes to rest) is 1/6. So the chance of rolling “a three” is 1/6, as is the chance of rolling a five, etc. If three dice are rolled, the chance that at least one of them shows a three is 3/6 = 1/2. Thus, since there are seven sites in Fig. 3c that can accept incoming material, the probability that at least one of them will transition during some small time increment is seven times the probability that a specific isolated site will transition in the same increment. Because more probable events obviously occur more often, i.e., require less time, then the time that passes before the entire system leaves the configuration in Fig. 3c, i.e., the time it takes for a unit of material to deposit somewhere on the surface, is t =

1X td =− ln (ξ ). 7 7F

(21)

Notice that 7F/X is simply the sum of the rates of all the per-site transitions that can occur in the entire system, i.e., the system’s activity, and thus it is clear that the general form of Eq. (21) is Eq. (9). As described above, the NFW algorithm prescribes that the choice of transition at each time step be randomized, with the probability of choosing a particular transition proportional to its relative rate. Since one of only seven transitions can occur on the surface in Fig. 3c, and each has the same rate, then the selection of a transition from the configuration in Fig. 3c involves simply choosing at random one of the seven sites marked with dotted outlines. Duplicating Fig. 3c as Fig. 4a, and assuming that Site d is randomly selected to transition, then the configuration after the first time step is that in Fig. 4b. If the per-site flux (i.e., F/ X ) is 1 sec−1 , then the time increment that elapses before the first transition is dictated by Eq. (9) to be t1 = −

ln (ξ1 ) sec . 7

(22)

A random number of ξ1 = 0.631935 yields a time increment for the first step of t1 = 0.065567 sec for a total time after the first step of, obviously, t1 = 0.065567 sec (where the starting time is naturally t0 = 0 sec).

Monte Carlo methods for simulating thin film deposition

2373

Assume that the deposited material at Site d can either diffuse to Site c, diffuse to Site e, or desorb. Assume further that the temperature is T = 1160 K such that kT =0.1 eV; the attempt frequency and activation barrier for diffusion are A D = 1x104 sec−1 and E D = 0.70 eV, respectively; and those for desorption are A R = 1x104 sec−1 and E R = 0.85 eV. Then the per-site rate of diffusion is approximately 9 sec−1 , and that of desorption is 2 sec−1 . The set of transitions available to the configuration in Fig. 4b includes seven deposition events at the dotted sites, two diffusion events, and one desorption event. To illustrate the process of choosing one of these ten transitions in the NFW algorithm, it is useful to visualize them on a graph. Figure 5b shows the ten possible transitions on a line plot, with the width of each corresponding to its relative rate. A transition can be selected in accord with Eq. (8) simply by generating a random number ζ2 ∈ [0,1), plotting it on the graph in Fig. 5b, and selecting the appropriate transition. (Figure 5a depicts the same type of plot for the configuration in Fig. 4a, assuming a value of ζ1 = 0.500923.) For example, if a random number of ζ2 = 0.652493 is generated, then the black atom at Site d in Fig. 4b would diffuse into Site e yielding the configuration in Fig. 4c. Since the activity of the system in Fig. 4b is = 27 sec −1 , a random number of ξ2 = 0.548193 yields a time increment for the second step of t2 = 0.022264 sec yielding a time value of t2 = 0.087831 sec. (Notice that when fast transitions are available to the system, as in Fig. 5b, the activity of the system increases and the time resolution in the NFW becomes finer to accommodate the fast processes.) By repeating this recipe, the evolution of the surface from its initial state in Fig. 4a can be simulated, as shown in Figs. 4 and 5. The random numbers (for transition selection) corresponding to the system’s evolution from Fig. 4c are ζ3 = 0.132087, ζ4 = 0.327872, and ζ5 = 0.891473, and the simulation time would be calculated as prescribed above. This NFW approach can be straightforwardly extended into three dimensions, and all manner of complex, collective, and environment- and structure-dependent transitions can be modeled provided their rates are known.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4. Possible configurations for the first few steps of deposition onto a simple twodimensional substrate. Gray squares are substrate sites, white dotted squares are sites into which particles can potentially deposit, and black squares are deposited particles.

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Figure 5. Lists of transitions for use in the NFW algorithm applied to the surface evolution depicted in Fig. 4. The numerals at the upper right of each plot indicate the total rate in sec−1 , those below each plot demark relative rates, and the letters above each plot denote transition classes and locations. The letter F corresponds to particle deposition, D to diffusion, and R to desorption. Lowercase italic letters correspond to the site labels, in Fig. 4, and the notation i ⇒ j indicates diffusion of the particle at site i into site j . The thick gray lines below each plot mark the locations of the random numbers used to select a transition from each configuration.

4.

Historical Perspective

Thousands of papers have been published on Monte Carlo simulations of thin film deposition. They encompass a wide range of thin film applications

Monte Carlo methods for simulating thin film deposition

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and employ a variety of methods. This section contains a brief overview of some selected examples. No attempt is made here to provide a comprehensive review; instead, the goal is to present selected sources for further exploration. Some of the earliest applications of MC to the computer simulation of deposition used simple models of deposition on an idealized surface. One of the first of these is attributed to Young and Schubert [11], who simulated the multilayer adsorption (without desorption or migration) of tungsten onto a tungsten substrate. Chernov and Lewis [12] performed MC calculations of kink migration during binary alloy deposition using a 1000-particle linear chain in one dimension, a 99×49 square grid in two dimensions (where the grid represented a cross-section of the film), and a 99×49×32 cubic grid in three dimensions. Gordon [13] simulated the monolayer adsorption and desorption of particles onto a 10×10 grid of sites with hexagonal close-packing (where the grid represented a plan view of the film). Abraham and White [14] considered the monolayer adsorption, desorption, and migration of atoms onto a 10×10 square grid (again in plan view), with atomic transitions modeled using a scheme that resembles the NFW. (Notice that Abraham’s and White’s publication appeared five years before the first publication of the NFW algorithm.) Leamy and Jackson [15], and Gilmer and Bennema [16], used the solid–on-solid (SOS) model [17–19] to analyze the roughness of the solid– vapor interface on a three-dimensional film represented by a 20×20 square grid. The SOS model represents the film by columns of atoms (or larger solid quanta) so that no subsurface voids or vacancies can exist. One major advantage of this approach is that the three-dimensional film can be represented digitally by a two-dimensional matrix of integers that describe the height of the film at each location on the free surface. Their approach was later extended [20] to alleviate the restrictions of the SOS model so that the structure and properties of the diffuse solid-vapor interface could be examined. Over the years, KMC methods have been applied to a wide range of materials and deposition technologies. These include materials such as simple metals, alloys, semiconductors, oxides, diamond, nanotubes, and quasicrystals; and technologies like molecular beam epitaxy, physical vapor deposition, chemical vapor deposition, electrodeposition, ion beam assisted deposition, and laser assisted deposition. Because of their relative simplicity, lattice KMC models were used in many of the computational deposition studies performed to date. However, MC methods can also be applied to model systems where the basic structural units (e.g., atoms) do not adhere to prescribed lattice positions. For example, continuous-space MC methods [21] allow particles to assume any position inside the computational domain. The motion of the particles is generally simulated by attempting small displacements, computing the associated energy changes via an interparticle potential, and applying the MC algorithms described above to accept or reject the attempted displacements. Alternatively, MC methods can be combined with other techniques within

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the same simulation framework to create a hybrid approach [22]. Common applications of these hybrids involve relaxing atomic positions near the surface, usually by means of energy minimization or molecular dynamics, and performing the MC calculations at off-lattice locations that are identified as potential transition sites on the relaxed structure.

5.

Summary

The preceding discussion should demonstrate clearly that the topic of MC deposition simulations is broad and rich. Unfortunately, a comprehensive review of existing methods and past research is beyond the scope of this article, and the reader is referred to the works mentioned herein and to the numerous reviews on the subject [23–30] for further study. As the techniques for applying MC methods to the study of thin film deposition continue to mature, novel approaches and previously inaccessible technologies will emerge. Hybrid MC methods seem particularly promising, as they allow for a physically based description of the fundamental surface structure, can allow for the real-time calculation of transition rates via physically accurate methods, and are able to access spatial and temporal scales that are well beyond the reach of more fundamental approaches. Whatever the future holds, it is certain that our ability to study thin film processing using computer simulations will continue to evolve and improve, yielding otherwise unobtainable insights into the physics and phenomenology of deposition, and that MC methods will play a crucial role in that process.

References [1] J. Fritsch and U. Schr¨oder, “Density functional calculation of semiconductor surface phonons,” Phys. Lett. C – Phys. Rep., 309, 209–331, 1999. [2] M.P. Allen, “Computer simulation of liquids,” Oxford University Press, Oxford, 1989. [3] J.M. Haile, “Molecular dynamics simulation: elementary methods,” John Wiley and Sons, New York, 1992. [4] C.K. Harris, D. Roekaerts, F.J.J. Fosendal, F.G.J. Buitendijk, P. Daskopoulos, A.J.N. Vreenegoor, and H. Wang, “Computational fluid dynamics for chemical reactor engineering,” Chem. Eng. Sci., 51, 1569–1594, 1996. [5] G.A. Bird, “Molecular gas dynamics and the direct simulation of gas flows,” Oxford University Press, Oxford, 1994. [6] K.A. Fichthorn and W.H. Weinberg, “Theoretical foundations of dynamical Monte Carlo simulations,” J. Chem. Phys., 95, 1090–1096, 1991. [7] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys., 21, 1087–1092, 1953.

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[8] A.B. Bortz, M.H. Kalos, and J.L. Lebowitz, “A new algorithm for Monte Carlo simulation of ising spin systems,” J. Comp. Phys., 17, 10–18, 1975. [9] D.T. Gillespie, “Exact stochastic simulation of coupled chemical reactions,” J. Phys. Chem., 81, 2340–2361, 1977. [10] A.F. Voter, “Clasically exact overlayer dynamics: diffusion of rhodium clusters on Rh(100),” Phys. Rev., B, 34, 6819–6829, 1986. [11] R.D. Young and D.C. Schubert, “Condensation of tungsten on tungsten in atomic detail – Monte Carlo and statistical calculations vs experiment,” J. Chem. Phys., 42, 3943–3950, 1965. [12] A.A. Chernov and J. Lewis, “Computer model of crystallization of binary systems – kinetic phase transitions,” J. Phys. Chem. Solids, 28, 2185–2198, 1967. [13] R. Gordon, “Adsorption isotherms of lattice gases by computer simulation,” J. Chem. Phys., 48, 1408–1409, 1968. [14] F.F. Abraham and G.W. White “Computer simulation of vapor deposition on twodimensional lattices,” J. Appl. Phys., 41, 1841–1849, 1970. [15] H.J. Leamy, and K.A. Jackson, “Roughness of crystal–vapor interface,” J. Appl. Phys., 42, 2121–2127, 1971. [16] G.H. Gilmer and P. Bennema, “Simulation of crystal-growth with surface diffusion,” J. Appl. Phys., 43, 1347–1360, 1972. [17] T.L. Hill, “Statistical mechanics of multimolecular adsorption 3: introductory treatment of horizontal interactions – Capillary condensation and hysteresis,” J. Chem. Phys., 15, 767–777, 1947. [18] W.K. Burton, N. Cabrera, and F.C. Frank, “The growth of crystals and the equilibrium structure of their surfaces,” Phil. Trans. Roy. Soc. A, 243, 299–358, 1951. [19] D.E. Temkin, “Crystallization processes,” Consultant Bureau, New York, 1966. [20] H.J. Leamy, G.H. Gilmer, K.A. Jackson, and P. Bennema, “Lattice–gas interface structure: a Monte Carlo simulation,” Phys. Rev. Lett., 30, 601–603, 1973. [21] B.W. Dodson and P.A. Taylor, “Monte Carlo simulation of continuous-space crystal growth,” Phys. Rev. B, 34, 2112–2115, 1986. [22] M.D. Rouhani, A.M. Gu´e, M. Sahlaoui, and D. Est`eve, “Strained semiconductor structures: simulation of the first stages of the growth,” Surf. Sci., 276, 109–121, 1992. [23] K. Binder, “Monte Carlo methods in statistical physics,” Springer-Verlag, Berlin, 1986. [24] T. Kawamura, “Monte Carlo simulation of thin-film growth on si surfaces,” Prog. Surf. Sci., 44, 67–99, 1993. [25] J. Lapujoulade, “The roughening of metal surfaces,” Surf. Sci. Rep., 20, 195–249, 1994. [26] M. Kotrla, “Numerical simulations in the theory of crystal growth,” Comp. Phys. Comm., 97, 82–100, 1996. [27] G.H. Gilmer, H. Huang, and C. Roland, “Thin film deposition: fundamentals and modeling,” Comp. Mat. Sci., 12, 354–380, 1998. [28] M. Itoh, “Atomic-scale homoepitaxial growth simulations of reconstructed III–V surfaces,” Prog. Surf. Sci., 66, 53–153, 2001. [29] H.N.G. Wadley, A.X. Zhou, R.A. Johnson, and M. Neurock, “Mechanisms, models, and methods of vapor deposition,” Prog. Mat. Sci., 46, 329–377, 2001. [30] C.C. Battaile, and D.J. Srolovitz, “Kinetic Monte Carlo simulation of chemical vapor deposition,” Ann. Rev. Mat. Res., 32, 297–319, 2002.

7.18 MICROSTRUCTURE OPTIMIZATION S. Torquato Department of Chemistry, PRISM, and Program in Applied & Computational Mathematics Princeton University, Princeton, NJ 08544, USA

1.

Introduction

An important goal of materials science is to have exquisite knowledge of structure-property relations in order to design material microstructures with desired properties and performance characteristics. Although this objective has been achieved in certain cases through trial and error, a systematic means of doing so is currently lacking. For certain physical phenomena at specific length scales, the governing equations are known and the only barrier to achieving the aforementioned goal is the development of appropriate methods to attack the problem. Optimization methods provide a systematic means of designing materials with tailored properties for a specific application. This article focuses on two optimization techniques: (1) the topology optimization procedure used to design composite or porous media, and (2) stochastic optimization methods employed to reconstruct or construct material microstructures.

2.

Topology Optimization

A promising method for the systematic design of composite microstructures with desirable macroscopic properties is the topology optimization method. The topology optimization method was developed almost two decades ago by Bendsøe and Kikuchi [1] for the design of mechanical structures. It is now also being used in smart and passive material design, mechanism design, microelectro-mechanical systems (MEMS) design, target optimization, multifunctional optimization, and other design problems [2–7]. Consider a two-phase composite material consisting of a phase with a property K 1 and volume fraction φ1 and another phase with a property K 2 and 2379 S. Yip (ed.), Handbook of Materials Modeling, 2379–2396. c 2005 Springer. Printed in the Netherlands.

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volume fraction φ2 (= 1 − φ1 ). The property K i is perfectly general: it may represent a transport, mechanical or electromagnetic property, or properties associated with coupled phenomena, such as piezoelectricity or thermoelectricity. For steady-state situations, the generalized flux F(r) at some local position r in the composite obeys the following conservation law in the phases: ∇ · F(r) = 0.

(1)

In the case of electrical conduction and elasticity, F represents the current density and stress tensor, respectively. The local constitutive law relates F to a generalized gradient G, which in the special case of a linear relationship is given by F(r) = K (r)G(r),

(2)

where K (r) is the local property. In the case of electrical conduction, relation (2) is just Ohm’s law, and K and G are the conductivity and electric field, respectively. For elastic solids, relation (2) is Hooke’s law, and K and G are the stiffness tensor and strain field, respectively. For piezoelectricity, F is the stress tensor, K embodies the compliance and piezoelectric coefficients, and G embodies both the electric field and strain tensor. The generalized gradient G must also satisfy a governing differential equation. For example, in the case of electrical conduction, G must be curl free. One must also specify the appropriate boundary conditions at the two-phase interface. One can show that the effective properties are found by homogenizing (averaging) the aforementioned local fields [8, 9]. In the case of linear material, the effective property K e is given by F(r) = K e G(r),

(3)

where angular brackets denote a volume average and/or an ensemble average. For additional details, the reader is referred to the other article (“Theory of Random Heterogeneous Materials”) by the author in this encyclopedia.

2.1.

Problem Statement

The basic topology optimization problem can be stated as follows: distribute a given amount of material in a design domain such that an objective function is extremized [1, 2, 4, 7]. The design domain is the periodic base cell and is initialized by discretizing it into a large number of finite elements (see Fig. 2) under periodic boundary conditions. The problem consists in finding the optimal distribution of the phases (solid, fluid, or void), such that the objective function is minimized. The objective function can be any combination of the individual components of the relevant effective property tensor

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subject to certain constraints [2, 7]. For target optimization [5] and multifunctional optimal design [6], the objective function can be appropriately modified, as described below. In the most general situation, it is desired to design a composite material with N different effective properties, which we denote by K e(1) , K e(2) , . . . , K e(N) , given the individual properties of the phases. In principle, one wants to know the region (set) in the multidimensional space of effective properties in which all composites must lie (see Fig. 1). The size and shape of this region depends on how much information about the microstructure is specified and on the prescribed phase properties. One could begin by making an initial guess for the distribution of the two phases among the elements, solve for the local fields using finite elements and then evolve the microstructure to the targeted properties. However, even for a small number of elements, this integer-type optimization problem becomes a huge and intractable combinatorial problem. For example, for a small design problem with N = 100, the number of different distributions of the three material phases would be astronomically large (3100 = 5 · 1047 ). As each function evaluation requires a full finite element analysis, it is hopeless to solve the optimization problem using random search methods such as, genetic algorithms or simulated annealing methods, which use a large number of function evaluations and do not make use of sensitivity information. Following the idea of standard topology optimization procedures, the problem is therefore relaxed by allowing the material at a given point to be a gray-scale mixture of the two phases. This makes it possible to find sensitivities with respect to design changes, which in turn allows one to use linear programming methods to solve the optimization problem. The optimization

Property Ke(2)

All composites

Property Ke(1) Figure 1. Schematic illustrating the allowable region in which all composites with specified phase properties must lie for the case of two different effective properties.

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procedure solves a sequence of finite element problems followed by changes in material type (density) of each of the finite elements, based on sensitivities of the obj-ective function and constraints with respect to design changes. At the end of the optimization procedure, however, we desire to have a design where each element is either phase 1 or phase 2 material (Fig. 2). This is achieved by imposing a penalization for grey phases at the final stages of the simulation. In the relaxed system, let xi ∈ [0, 1] be the local density of the ith element, so that when xi = 0, the element corresponds to phase 1 and when xi = 1, the element corresponds to phase 2. Let x (xi ,i = 1, . . . , n) be the vector of design variables which satisfies the constraint for the fixed volume fraction φ2 = xi . For any x, the local fields are computed using the finite element method and the effective property K e (K ;x), which is a function of the material property K and x, is obtained by the homogenization of the local fields. The optimization problem is specified as follows: Minimize : subject to :

= K e (x) n 1 xi = φ2 n i=1

(4)

0 ≤ xi ≤ 1, i = 1, . . . , n and prescribed symmetries. The objective function K e (x) is generally nonlinear. To solve this problem, the objective function is linearized, enabling one to take advantage of powerful sequential linear programming techniques. Specifically, the objective function is expanded in Taylor series for a given microstructure x0 : K e (X0 ) + ∇ K e · x, Design domain (base cell)

Phase 1:

(5) Periodic material structure

Phase 2:

Figure 2. Design domain and discretization for a two-phase, three-dimensional topology optimization problem. Each cube represents one finite element, which can consist of either phase 1 material or phase 2 material.

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where x = x − x0 is the vector of density changes. In each iteration, the microstructure evolves to the optimal state by determining the small change x. One can use the simplex method [2] or the interior-point method [5] to minimize the linearized objective function in Eq. (5). In each iteration, the homogenization step to obtain the effective property K e (K ; x0 ) is carried out numerically via the finite-element method on the given configuration x0 . Derivatives of the objective function (∇ K e ) are calculated by a sensitivity analysis which requires one finite element calculation for each iteration. One can use the topology optimization to design at will composite microstructures with targeted effective properties under required constraints [5]. The objective function for such a target optimization problem has been chosen to be given by a least-square form involving the effective property K e (x) at any point in the simulation and a target effective property K 0 : = [K e (x) − K 0 ]2 .

(6)

The method can also be employed for multifunctional optimization problems. The objective function in this instance has been chosen to be a weighted average of each of the effective properties [6].

2.2.

Illustrative Examples

The topology optimization procedure has been employed to design composite materials with extremal properties [2, 3, 10], targeted properties [5, 11], and multifunctional properties [6]. To illustrate the power of the method, we briefly describe microstructure designs in which thermal expansion and piezoelectric behaviors are optimized, the effective conductivity achieves a targeted value, and the thermal conduction demands compete with the electrical conduction demands. Materials with extreme or unusual thermal expansion behavior are of interest from both a technological and fundamental standpoint. Zero thermal expansion materials are needed in structures subject to temperature changes such as space structures, bridges and piping systems. Materials with large thermal displacement or force can be employed as “thermal” actuators. A negative thermal expansion material has the counterintuitive property of contracting upon heating. A fastener made of a negative thermal expansion material, upon heating, can be inserted easily into a hole. Upon cooling, it will expand, fitting tightly into the hole. All three types of expansion behavior have been designed [2]. In the negative expansion case, one must consider a three-phase material: a high expansion material, a low expansion material, and a void region. Figure 3 shows the two-dimensional optimal design that was found; the main mechanism behind the negative expansion behavior is the reentrant cell

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Figure 3. Optimal microstructure for minimization of effective thermal expansion coefficient [2]. White regions denote void, black regions consist of low expansion material and cross-hatched regions consist of high expansion material.

structure having bimaterial components which bend (into the void space) and cause large deformation when heated. In the case of piezoelectricity, actuators that maximize the delivered force or displacement can be designed. Moreover, one can design piezocomposites (consisting of an array of parallel piezoceramic rods embedded in a polymer matrix) that maximize the sensitivity to acoustic fields. The topology optimization method has been used to design piezocomposites with optimal performance characteristics for hydrophone applications [3]. When designing for maximum hydrostatic charge coefficient, the optimal transversally isotropic matrix material has negative Poisson’s ratio in certain directions. This matrix material itself turns out be a composite, namely, a special porous solid. Using an autocad file of the three-dimensional matrix material structure and a stereolithography technique, such negative Poisson’s ratio materials have actually been fabricated [3]. For the case of a two-phase, two-dimensional, isotropic composite, the popular effective-medium approximation (EMA) formula for the effective electrical conductivity σ e is given by

φ1

σe − σ1 σe − σ2 + φ2 = 0, σe + σ1 σe + σ2

(7)

where φi and σi are the volume fraction and conductivity of phase i, respectively. Milton [12] showed that the EMA expression is exactly realized by granular aggregates of the two phases such that spherical grains (in any dimension) of comparable size are well separated with self-similarity on all length scales. This is why the EMA formula breaks down when applied to dispersions of identical circular inclusions. An interesting question is the following: Can the EMA formula be realized by simple structures with a single

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length scale? Using the target optimization formulation in which the target effective conductivity σ0 is given by the EMA function (7), Torquato et al. [6] found a class of periodic, single-scale dispersions that achieve it at a given phase conductivity ratio for a two-phase, two-dimensional composite over all volume fractions. Moreover, to an excellent approximation (but not exactly), the same structures realize the EMA for almost the entire range of phase conductivities and volume fractions. The inclusion shapes are given analytically by the generalized hypocycloid, which in general has a non-smooth interface (see Fig. 4). Minimal surfaces necessarily have zero mean curvature, i.e., the sum of the principal curvatures at each point on the surface is zero. Particularly fascinating are minimal surfaces that are triply periodic because they arise in a variety of systems, including block copolymers, nanocomposites, micellar materials, and lipid-water systems [6]. These two-phase composites are bicontinuous in the sense that the surface (two-phase interface) divides space into two disjoint

2 ⫽ 0.001

2 ⫽ 0.05

2 ⫽ 0.089

2 ⫽ 0.3

2 ⫽ 0.5

2 ⫽ 0.7

2 ⫽ 0.911

2 ⫽ 0.95

2 ⫽ 0.999

Figure 4. Unit cells of generalized hypocycloidal inclusions in a matrix that realize the EMA relation (1) for selected values of the volume fraction in the range 0 < φ2 < 1. Phases 1 and 2 are the white and black phase, respectively.

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but intertwining phases that are simultaneously continuous. This topological feature of bicontinuity is rare in two dimensions and therefore virtually unique to three dimensions [8]. Using multifunctional optimization [6], it has been discovered that triply periodic two-phase bicontinuous composites with interfaces that are the Schwartz P and D minimal surfaces (see Fig. 5) are not only geometrically extremal but extremal for simultaneous transport of heat and electricity. More specifically, these are the optimal structures when a weighted sum of the effective thermal and electrical conductivities ( = λe + σ e ) is maximized for the case in which phase 1 is a good thermal conductor but poor electrical conductor and phase 2 is a poor thermal conductor but good electrical conductor with φ1 = φ2 = 1/2. The demand that this sum is maximized sets up a competition between the two effective transport properties, and this demand is met by the Schwartz P and D structures. By mathematical analogy, the optimality of these bicontinuous composites applies to any of the pair of the following scalar effective properties: electrical conductivity, thermal conductivity, dielectric constant, magnetic permeability, and diffusion coefficient. It will be of interest to investigate whether the optimal structures when φ1 =/ φ2 are bicontinuous structures with interfaces of constant mean curvature, which would become minimal surfaces at the point φ1 = φ2 = 1/2. The topological property of bicontinuity of these structures suggests that they would be mechanically stiff even if one of the phases is a compliant solid or a liquid, provided that the other phase is a relatively stiff material. Indeed, it has recently been shown that the Schwartz P and D structures are extremal when a competition is set up between the bulk modulus and electrical (or thermal) conductivity of the composite [13].

Figure 5. Unit cells of two different minimal surfaces with a resolution of 64 × 64 × 64. Left panel: Schwartz simple cubic surface. Right panel: Schwartz diamond surface.

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3.

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Reconstruction Techniques

The reconstruction of realizations of disordered materials, such as liquids, glasses, and random heterogeneous materials, from a knowledge of limited microstructural information (lower-order correlation functions) is an intriguing inverse problem. Clearly, one can never reconstruct the original material perfectly in the infinite-system limit, i.e., such reconstructions are nonunique. Thus, the objective here is not the same as that of data decompression algorithms that efficiently restore complete information, such as the gray scale of every pixel in an image. The generation of realizations of random media with specified lower-order correlation functions can: 1. shed light on the nature of the information contained in the various correlation functions that are employed; 2. ascertain whether the standard two-point correlation function, accessible experimentally via scattering, can accurately reproduce the material and, if not, what additional information is required to do so; 3. identify the class of microstructures that have exactly the same lowerorder correlation functions but widely different effective properties; 4. probe the interesting issue of nonuniqueness of the generated realizations; 5. construct structures that correspond to specified correlation functions and categorize classes of random media; 6. provide guidance in ascertaining the mathematical properties that physically realizable correlation functions must possess [14]; and 7. attempt three-dimensional reconstructions from slices or micrographs of the sample: a poor man’s X-ray microtomography experiment. The first reconstruction procedures applied to heterogeneous materials were based on thresholding Gaussian random fields. This approach to reconstruct random media originated with Joshi [15], and was extended by Adler [16] and Roberts and Teubner [17]. This method is currently limited to the standard two-point correlation function, and is not suitable for extension to non-Gaussian statistics.

3.1.

Optimization Problem

It has recently been suggested that reconstruction problems can posed as optimization problems [18, 19]. A set of target correlation functions are prescribed based upon experiments, theoretical models, or some ansatz. Starting from some initial realization of the random medium, the method proceeds to find a realization by evolving the microstructure such that the calculated correlation functions best match the target functions. This is achieved by minimizing

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an error based upon the distance between the target and calculated correlation functions. The medium can be a dispersion of particles [18] or, more generally, a digitized image of a disordered material [19]. For simplicity, we will introduce the problem for the case of digitized heterogeneous media here and consider only a single two-point correlation function for statistically isotropic two-phase media (the generalization to multiple correlation functions is straightforward [18, 19]). It is desired to generate realizations of two-phase isotropic media that have a target two-point correlation function f 2 (r) associated with phase i, where r is the distance between the two points and i = 1 or 2. Let fˆ2 (r) be the corresponding function of the reconstructed digitized system (with periodic boundary conditions) at some time step. It is this system that we will attempt to evolve towards f 2 (r) from an initial guess of the system realization. Again, for simplicity, we define a fictitious “energy” (or norm-2 error) E at any particular stage as E=

[ fˆ2 (r) − f 2 (r)]2 ,

(8)

r

where the sum is over all discrete values of r. Potential candidates for the correlation functions [8] include: (1) the standard two-point probability function S2(r), lineal path function L(z), pore-size density function P(δ), and twopoint cluster function C2 (r). For statistically isotropic materials, S2 (r) gives the probability of finding the end points of a line segment of length r in one of the phases (say phase 1) when randomly tossed into the system, whereas L(z) provides the probability of finding the entire line segment of length r in phase 1 (or 2) when randomly tossed into the system.

3.2.

Solution of Optimization Problem

The aforementioned optimization problem is very difficult to solve due to the complicated nature of the objective function, which involves complex microstructural information in the form of correlation functions of the material, and due to the combinatorial nature of the feasible set. Standard mathematical programming techniques are therefore most likely inefficient and likely to get trapped in local minima. In fact, the complexity and generality of the reconstruction problem makes it difficult to devise deterministic algorithms of wide applicability. One therefore often resorts to heuristic techniques for global optimization, in particular, the simulated annealing method. Simulated annealing has been applied successfully to many difficult combinatorial problems, including NP-hard ones such as the “traveling salesman” problem. The utility of the simulated annealing method stems from its simplicity in that it only requires “black-box” cost function evaluations, and in its physically designed ability to escape local minima via accepting locally

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unfavorable configurations. In its simplest form, the states of two selected pixels of different phases are interchanged, automatically preserving the volume fraction of both phases. The change in the error (or “energy”) E = E − E between the two successive states is computed. This phase interchange is then accepted with some probability p(E) that depends on E. One reasonable choice is the Metropolis acceptance rule, i.e.,

p(E) =

1, E ≤ 0, exp(−E/T ), E > 0,

(9)

where T is a fictitious “temperature”. The concept of finding the lowest error (lowest energy) state by simulated annealing is based on a well-known physical fact: If a system is heated to a high temperature T and then slowly cooled down to absolute zero, the system equilibrates to its ground state. We note that there are various ways of appreciably reducing computational time. For example, computational cost can be significantly lowered by using other stochastic optimization schemes such as the “Great Deluge” algorithm, which can be adjusted to accept only “downhill” energy changes, and the “threshold acceptance” algorithm [20]. Further savings can be attained by developing strategies that exploit the fact that pixel interchanges are local and thus one can reuse the correlation functions measured in the previous time step instead of recomputing them fully at any step [19]. Additional cost savings have been achieved by interchanging pixels only at the two-phase interface [8].

3.3.

Illustrative Examples

Lower-order correlation functions generally do not contain complete information and thus cannot be expected to yield perfect reconstructions. Of course, the judicious use of combinations of lower-order correlation functions can yield more accurate reconstructions than any single function alone. Yeong and Torquato [19, 21] clearly showed that the two-point function S2 alone is not sufficient to reconstruct accurately random media. By also incorporating the lineal-path function L, they were able to obtain better reconstructions. They studied one-, two- and three-dimensional digitized isotropic media. Each simulation began with an initial configuration of pixels (white for phase 1 and black for phase 2) in the random checkerboard arrangement at a prescribed volume fraction. A two-dimensional example illustrating the insufficiency of S2 in reconstructions is a target system of overlapping disks at a disk volume fraction of φ 2 = 0.5; see Fig. 6(a). Reconstructions that accurately match S2 alone, L alone, and both S2 and L are shown in Fig. 6. The S2-reconstruction is not very accurate; the cluster sizes are too large, and the system actually percolates.

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S. Torquato (a)

(b)

(c)

(d)

Figure 6. (a) Target system: a realization of random overlapping disks. System size = 400 × 400 pixels, disk diameter = 31 pixels, and volume fraction φ2 = 0.5. (b) S2 -reconstruction. (c) Corresponding L-reconstruction. (d) Corresponding hybrid (S2 + L)-reconstruction.

(Note that overlapping disks percolate at a disk area fraction of φ2 ≈ 0.68 [8]). The L-reconstruction does a better job than the S2 -reconstruction in capturing the clustering behavior. However, the hybrid (S2 + L)-reconstruction is the best. The optimization method can be used in the construction mode to find the specific structures that realize a specified set of correlation functions. An interesting question in this regard is the following: Is any correlation function physically realizable or must the function satisfy certain conditions? It turns out that not all correlation functions are physically realizable. For example, what are the existence conditions for a valid (i.e., physically realizable) auto covariance function χ(r) ≡ S2(r)−φ12 for statistically homogeneous twophase media? It is well known that there are certain nonnegativity conditions involving the spectral representation of the auto covariance χ(r) that must be obeyed [14]. However, it is not well known that these nonnegativity conditions are necessary but not sufficient conditions that a valid auto covariance χ(r) of a statistically homogeneous two-phase random medium (i.e., a binary stochastic spatial process) must meet. Some of these “binary” conditions are described by Torquato [8] but the complete characterization is a very difficult problem. Suffice it to say that that the algorithm in the construction mode can be used

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to provide guidance on the development of the mathematical conditions that a valid auto covariance χ(r) must obey. Cule and Torquato [20] considered the construction of realizations having the following autocovariance function: sin(qr) S2(r) − φ12 , = e−r/a φ1 φ2 qr

(10)

where q = 2π/b and the positive parameter b is a characteristic length that controls oscillations in the term sin(qr)/(qr), which also decays with increasing r. This function possesses phase-inversion symmetry [8] and exhibits a considerable degree of short-range order; it generalizes the purely exponentiallydecaying function studied by Debye, et al. [22]. This function satisfies the nonnegativity condition on the spectral function but may not satisfy the “binary” conditions, depending on the values of a, b, and φ1 [14]. Two structures possessing the correlation function (10) are shown in Fig. 7 for φ2 = 0.2 and 0.5, in which a = 32 pixels and b = 8 pixels. For these sets of parameters, all of the aforementioned necessary conditions on the function are met. At φ2 = 0.2, the system resembles a dilute particle suspension with “particle” diameters of order b. At φ2 = 0.5, the resulting pattern is labyrinthine such that the characteristic sizes of the “patches” and “walls” are of order a and b, respectively. Note that S2(r) was sampled in all directions during the annealing process. In all of the previous two-dimensional examples, however, both S2 and L were sampled along two orthogonal directions to save computational time. This time-saving step should be implemented only for isotropic media, provided that there is no appreciable short-range order; otherwise, it leads to unwanted anisotropy [20, 23]. However, this artificial anisotropy can be reduced by optimizing along additional selected directions [24].

2 ⫽ 0.2

2 ⫽ 0.5

Figure 7. Structures corresponding to the target correlation function given by (10) for φ2 = 0.2 and 0.5. Here a = 32 pixels and b = 8 pixels.

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To what extent can information extracted from two-dimensional cuts through a three-dimensional isotropic medium, such as S2 and L , be employed to reproduce intrinsic three-dimensional information, such as connectedness? This question was studied for the aforementioned Fontainebleau sandstone for which we know the full three-dimensional structure via X-ray microtomography [21]. The three-dimensional reconstruction that results by using a single slice of the sample and matching both S2 and L is shown in Fig. 8. The reconstructions reproduce accurately certain three-dimensional properties of the pore space, such as the pore-size functions, the mean survival time of a Brownian particle, and the fluid permeability. The degree of connectedness of the pore space also compares remarkably well with the actual sandstone, although this is not always the case [25]. As noted earlier, the aforementioned algorithm was originally applied to reconstruct realizations of many-particle systems [18]. The hard-sphere system in which pairs of particles only interact with an infinite repulsion when they overlap is one of the simplest interacting particle systems [8]. Importantly, the impenetrability constraint does not uniquely specify the statistical ensemble. The hard-sphere system can be in thermal equilibrium or in one of the infinitely many nonequilibrium states, such as the random sequential addition (or adsorption) (RSA) process that is produced by randomly, irreversibly, and sequentially placing nonoverlapping objects into a volume [8]. While particles in equilibrium have thermal motion such that they sample the configuration space uniformly, particles in an RSA process do not sample the configuration space uniformly, since their positions are forever “frozen” (i.e., do not diffuse) after they have been placed into the system.

Figure 8. Hybrid reconstruction of a sandstone (described in Ref. [8]) using both S 2 and L obtained from a single “slice”. System size is 128 × 128 × 128 pixels. Left panel: Pore space is white and opaque, and the grain phase is black and transparent. Right panel: 3D perspective of the surface cuts.

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The geometrical blocking effects and the irreversible nature of the process results in structures that are distinctly different from corresponding equilibrium configurations, except for low densities. The saturation limit (the final state of this process whereby no particles can be added) occurs at a particle volume fraction φ2 ≈ 0.55 in two dimensions [8]. The reconstruction of the two-dimensional RSA disk system in which the target correlation function is the well-known radial distribution function (RDF) g(r). In two dimensions, the quantity ρ2πrg(r) dr gives the average number of particle centers in an annulus of thickness dr at a radial distance of r from the center of a particle (where ρ is the number density). The RDF is of central importance in the study of equilibrium liquids in which the particles interact with pairwise-additive forces since all of the thermodynamics (a)

(b)

Figure 9. (a) A portion of a sample RSA system at φ2 = 0.543. (b) A portion of the reconstructed RSA system at φ2 = 0.543.

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Figure 10. Configurations of 289 particles for φ2 = 0.2 in two dimensions. The equilibrium hard-disk system (left) shows more clustering and larger pores than the annealed step-function system (right).

can be expressed in terms of the RDF. The RDF can be ascertained from scattering experiments, which makes it a likely candidate for the reconstruction of a real system. The initial configuration was 5000 disks in equilibrium. Figure 9 shows a realization of the the RSA system at φ2 = 0.543 in (a), and the reconstructed system. As a quantitative comparison of how the original and reconstructed systems matched, it was found that the corresponding pore-size distribution functions [8] were similar. This conclusion gives one confidence that a reasonable facsimile of the actual structure can be produced from the RDF for a class of many-particle systems in which there is not significant clustering of the particles. For the elementary unit step-function g2 , previous work [26] indicated that this function was achievable by hard-sphere configurations up to a terminal covering fraction of particle exclusion diameters equal to 2−d in d dimensions. To test whether the unit step g2 is actually achievable by hard spheres for nonzero densities, the aforementioned stochastic optimization procedure was applied in the construction mode. Calculations for d = 1 and 2 confirmed that the step-function g2 is indeed realizable up to the terminal density [27]. Figure 10 compares an equilibrium hard-disk configuration at φ2 = 0.2 to a corresponding annealed step-function system.

4.

Summary

The fundamental understanding of the microstructure/properties connection is the key to designing new materials with the tailored properties for

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specific applications. Optimization methods combined with novel synthesis and fabrication techniques provide a means of accomplishing this goal systematically and could make optimal design of real materials a reality in the future. The topology optimization technique and the stochastic reconstruction (construction) method address only a small subset of optimization issues of importance in materials science, but the results that are beginning to emerge from these relatively new methods bode well for progress in the future.

References [1] M.P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng., 71, 197–224, 1988. [2] O. Sigmund and S. Torquato, “Design of materials with extreme thermal expansion using a three-phase topology optimization method,” J. Mech. Phys. Solids, 45, 1037–1067, 1997. [3] O. Sigmund, S. Torquato, and I.A. Aksay, “On the design of 1-3 piezocomposites using topology optimization,” J. Mater. Res., 13, 1038–1048, 1998. [4] M.P. Bendsøe, Optimization of Structural Topology, Shape and Material, SpringerVerlag, Berlin, 1995. [5] S. Hyun and S. Torquato, “Designing composite microstructures with targeted properties,” J. Mater. Res., 16, 280–285, 2001. [6] S. Torquato, S. Hyun, and A. Donev, “Multifunctional composites: optimizing microstructures for simultaneous transport of heat and electricity,” Phys. Rev. Lett., 89, 266601, 1–4, 2002. [7] M.P. Bendsøe and O. Sigmund, Topology Optimization, Springer-Verlag, Berlin, 2003. [8] S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer-Verlag, New York, 2002. [9] G.W. Milton, The Theory of Composites, Cambridge University Press, Cambridge, England, 2002. [10] U.D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio,” J. Microelectromechanical Systems, 6(2), 99–106, 1997. [11] S. Torquato and S. Hyun, “Effective-medium approximation for composite media: realizable single-scale dispersions,” J. Appl. Phys., 89, 1725–1729, 2001. [12] G.W. Milton, “Multicomponent composites, electrical networks and new types of continued fractions. I and II,” Commun. Math. Phys., 111, 281–372, 1987. [13] S. Torquato and A. Donev, “Minimal surfaces and multifunctionality,” Proc. R. Soc. Lond. A, 460, 1849–1856, 2004. [14] S. Torquato, “Exact conditions on physically realizable correlation functions of random media,” J. Chem. Phys., 111, 8832–8837, 1999. [15] M.Y. Joshi, A Class of Stochastic Models for Porous Media, Ph.D. thesis, University of Kansas, Lawrence, 1974. [16] P.M. Adler, Porous Media – Geometry and Transports, Butterworth-Heinemann, Boston, 1992. [17] A.P. Roberts and M. Teubner, “Transport properties of heterogeneous materials derived from Gaussian random fields: bounds and simulation,” Phys. Rev. E, 51, 4141–4154, 1995.

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[18] M.D. Rintoul and S. Torquato, “Reconstruction of the structure of dispersions,” J. Colloid Interface Sci., 186, 467–476, 1997. [19] C.L.Y. Yeong and S. Torquato, “Reconstructing random media,” Phys. Rev. E, 57, 495–506, 1998a. [20] D. Cule and S. Torquato, “Generating random media from limited microstructural information via stochastic optimization,” J. Appl. Phys., 86, 3428–3437, 1999. [21] C.L.Y. Yeong and S. Torquato, “Reconstructing random media: II. Three-dimensional media from two-dimensional cuts,” Phys. Rev. E, 58, 224–233, 1998b. [22] P. Debye, H.R. Anderson, and H. Brumberger, “Scattering by an inhomogeneous solid. II. The correlation function and its applications,” J. Appl. Phys., 28, 679–683, 1957. [23] C. Manwart and R. Hilfer, “Reconstruction of random media using Monte Carlo methods,” Phys. Rev. E, 59, 5596–5599, 1999. [24] N. Sheehan and S. Torquato, “Generating microstructures with specified correlation function,” J. Appl. Phys., 89, 53–60, 2001. [25] C. Manwart, S. Torquato, and R. Hilfer, “Stochastic reconstruction of sandstones,” Phys. Rev. E, 62, 893–899, 2000. [26] F.H. Stillinger, S. Torquato, J.M. Eroles, and T.M. Truskett, “Iso-g (2) processes in equilibrium statistical mechanics,” J. Phys. Chem. B, 105, 6592–6597, 2001. [27] J.R. Crawford, S. Torquato, and F.H. Stillinger, “Aspects of correlation function realizability,” J. Chem. Phys., 2003.

7.19 MICROSTRUCTURAL CHARACTERIZATION ASSOCIATED WITH SOLID–SOLID TRANSFORMATIONS J.M. Rickman1 and K. Barmak2 1

Department of Materials Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA 2 Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

1.

Introduction

Materials scientists have long been interested in the characterization of complex poly-crystalline systems, as embodied in the distribution of grain size and shape, and have sought to link microstructural features with observed mechanical, electronic and magnetic properties [1]. The importance of detailed microstructural characterization is underscored by systems of limited spatial dimensionality, with length scales of the order of nanometers to microns, as their reliability and performance are greatly influenced by specific microstructural features rather than by average, bulk properties [2]. For example, the functionalities of many electronic devices depend critically on the microstructure of thin metallic films via the film deposition process and the occurrence of reactive phase formation at metallic contacts. Various tools are available for quantitative microstructural characterization. Most notably, microstructural analyses often employ stereological techniques [1] and the related formalism of stochastic geometry [3] to interrogate grain populations and to deduce plausible growth scenarios that led to the observed grain morphologies. In this effort computer simulation is especially valuable, permitting one to implement various growth assumptions and to generate a large number of microstructures for subsequent analysis. The acquisition of comparable grain size and shape information from experimental images is, however, often problematic given difficulties inherent in grain recognition. The case of polycrystalline thin films is illustrative here. In these systems transmission 2397 S. Yip (ed.), Handbook of Materials Modeling, 2397–2408. c 2005 Springer. Printed in the Netherlands.

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electron microscopy (TEM) is necessary to resolve pertinent microstructural features. Unfortunately, complex contrast variations peculiar to TEM images plague grain recognition and therefore image interpretation. As a result, the tedious, state-of-the-art analysis, until quite recently [4, 6], involved human intervention to trace grain boundaries and to collect grain statistics. In this topical article we review methods for quantitative microstructural analysis, focusing first on systems that evolve from nucleation and subsequent growth processes. As indicated above, computer simulation of these processes provides considerable insight into the link between initial conditions and product microstructures, and so we will highlight some recent work in this area of evolving, first-order phase transformations in the absence of grain growth (i.e., coarsening). The analysis here will involve several important descriptors that are sensitive to different microstructural details and that can be used to infer the conditions that led to a given structure. Finally, we conclude with a discussion of new, automated image processing techniques that permit one to acquire information on large grain populations and to make useful comparisons of the associated grain-size distributions with those advanced in theoretical investigations of grain growth [6–8].

2.

Phase Transformations

Computer simulations are particularly useful for investigating the impact of nucleation conditions on product grain morphology resulting from a firstorder phase transformation [9, 3]. Several schemes for modeling such transformations have been discussed in the literature [10, 11], and it is generally possible to use them to describe a variety of nucleation scenarios, including those involving site saturation (e.g., a burst) and a constant nucleation rate. To illustrate a typical microstructural analysis, consider the constant radial growth to impingement of product grains originating from a burst of nuclei that are randomly distributed in two dimensions. The resulting microstructures consists of a set of Voronoi grains that tile the system, as shown in Fig. 1.

2.1.

Grain Area Distribution

Our characterization of this microstructure begins with the compilation of ¯ where the bar a frequency histogram of normalized grain areas, A = A/ A, denotes a microstructural average. The corresponding probability distribution P( A ), as obtained for a relatively large grain population (∼106 grains) is

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Figure 1. A fully coalesced product microstructure produced by a burst of nuclei that subsequently grow at a constant radial rate until impingement.

shown in Fig. 2. While no exact analytical form for this distribution is known, approximate expressions based on the gamma distribution P γ ( A ) =

1 (A )α−1 exp( A ) β α (α)

(1)

follow from stochastic geometry [12, 13], where α and β are parameters such that α = 1/β. For the Voronoi structure β is then the variance, as obtained analytically by Gilbert [14]. As can be seen from Fig. 2, the agreement between the simulated and approximate distributions is excellent. As P( A ) is a quantity of central importance in most microstructural analyses, it is of interest to determine whether it can be used to deduce, a posteriori, nucleation conditions. For this purpose, consider next the product microstructure resulting from nuclei that are randomly distributed on an underlying microstructure. A systematic analysis of such structures follows from a comparison of the relevant length scales here, namely the average underlying cell diameter, lu , and the average internuclear separation along the boundary, lb . For this discussion it is convenient to define a relative measure of these length scales r = lb /lu , and so one would intuitively expect that in the limit r > 1 (r < 1) the product microstructure comprises largely equiaxed (elongated) grains. Several product microstructures corresponding to different values of r, shown in Fig. 3, confirm these expectations.

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Figure 2. The corresponding probability distribution, P(A ), of normalized grain areas, A , and an approximate representation, P γ (A ) (solid line), based on the gamma distribution. Note the excellent agreement between the simulated and approximate distributions.

Figure 3. Product microstructures corresponding, from left to right, to r < 1, r ∼ 1, and r > 1. Note that in the limit r > 1(r < 1) the product microstructure comprises largely equiaxed (elongated) grains.

Upon examining the probability distributions for large collections of grains with these values of r (see Fig. 4), it is evident that, upon decreasing r, the distribution shifts to the left and a greater number of both relatively small and large grains is created. A more detailed analysis of these distributions demonstrates, again, that the gamma distribution is a good approximation in many cases, and a calculation of lower-order moments reveals a scaling regime for intermediate values of r [9]. Despite these features, it is found that, in general, P( A ) lacks the requisite sensitivity to variations in r needed for an unambiguous identification of nucleation conditions. As an alternative to the grain-area distribution, one can obtain descriptors that focus on the spatial distribution of the nucleation sites themselves, regarded here as microstructural generators. The utility of such descriptors depends, of course, on the ability to extract from a product microstructure the spatial distribution of these generators. As a reverse Monte Carlo method was recently devised to accomplish this task in some circumstances [3], we merely

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Figure 4. The probability distribution P(A ) for different values of the ratio of length scales r . Although there is a discernible shift in curve position and attendant change in curve shape upon changing r, the distribution is not very sensitive to these changes.

outline here the use of one such descriptor. Now, from the theory of point processes one can define a neighbor distribution wk (r), the generalization of a waiting-time distribution to space, that measures the probability of finding the k-th neighbor at a distance r (not to be confused with the dimensionless microstructural parameter defined above) away from a given nucleus [15]. Consider then the first moment of this distribution rk for the kth neighbor. For points randomly distributed in d dimensions one finds that

1 d 1+ rk = √ 1/d π(λd ) 2

1/d

(k + 1/d) , (k)

(2)

where λd is the d-dimensional volume density of points. Thus, the dependence of rk on k is a signature of the effective spatial dimensionality of the site-saturated nucleation process. Figure 5 shows the dependence of the normalized first moment on k for several cases of catalytic nucleation on an underlying microstructure, each corresponding to a different value of ζ = 1/r. An interpretation of Fig. 5 follows upon examining Fig. 6, the latter showing the dependence of the moment on k for the small and large ζ along with the predicted results for strictly oneand two-dimensional random distributions of nuclei. For low linear nucleation densities (e.g., ζ = 0.1) the underlying structure is effectively unimportant and so rk follows the theoretical two dimensional random curve for small to intermediate k. By contrast, at high nucleation densities, nuclei have many neighbors along a given edge and so rk initially exhibits pseudo-one-dimensional behavior. As more distant neighbors are considered, rk is consistent with

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Figure 5. The first moment of the neighbor distribution, rk , as a function of neighbor number k for different values of ζ = 1/r.

two-dimensional random behavior as these neighbors are now on other boundary segments distributed throughout the system. With this information it is possible to infer different nucleation scenarios from rk vs. k [3]. Finally, it is worth noting that other, related descriptors are useful in distinguishing different nucleation conditions. For example, as is often done in atomistic simulation, one can calculate the pair correlation function, g(r), for the nuclei. The results of such a calculation are presented in Fig. 7 for nucleation on the corners of an underlying grain structure. A measure of the nonrandomness of this spatial distribution of nuclei at a particular r is given by g(r) − 1. Thus, g(r) is a sensitive measure of deviations from randomness, and has been employed to investigate spatial correlations among nuclei formed at underlying grain junctions [3, 16].

3.

Image Processing and Grain-size Distributions

As indicated above, the acquisition of statistically significant grain size and shape information from experimental micrographs is difficult owing to

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Figure 6. The dependence of rk on k for the small and large ζ along with the predicted results for strictly one- and two-dimensional random distributions of nuclei.

problems associated with grain recognition. Nevertheless, it is essential to obtain such information to enable meaningful comparisons with simulated structures and to investigate various nucleation and growth scenarios. With this in mind, we outline below recent progress toward automated analysis of experimental micrographs. In this short review, our focus will be on assessing models of grain growth (i.e., coarsening) in thin films that describe microstructural kinetics after transformed grains have grown to impingement. The grain size of thin films is known to scale with the thickness of the film. Thus, for films with thicknesses of 1 nm to 1 µm it is necessary to employ transmission electron microscopy (TEM) to image the film grain structure. Although the grain structure of these film is easily discernable by eye from TEM micrographs, the image contrast is often quite complex. Such image contrast arises primarily from variations in the diffraction condition that result from: (1) changes in crystal orientation as a grain boundary is traversed, (2) localized straining of the lattice, and (3) long-wavelength bending of the sample. The latter two sources of contrast cannot be easily deconvoluted from the first, and, as a result, conventional image processing algorithms have been of limited utility in thin film grain structure analysis.

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Figure 7. The pair correlation function g(r ) versus internuclear separation, r , for nucleation on the corners of an underlying grain structure. A measure of the nonrandomness of this spatial distribution of nuclei at a particular r is given by g(r ) − 1.

Recently we have developed and used a practical, automated scheme for the acquisition of microstructural information from a statistically significant population of grains imaged by TEM [4]. Our overall philosophy for automated detection of grain boundaries is to first optimize the microscope operating conditions and the resultant image, and to then eliminate as much as possible false features in the processed images, even sometimes at the expense of real microstructural features. The true information deleted in this manner is recovered by optimally combining processed images of the same field of view taken at different sample tilts. The new algorithms developed to independently process the images taken at different samples tilts are automated thresholding and three filters for removal of (i) short, disconnected pixel segments, (ii) excessively connected or “tangled” lines, and (iii) isolated clusters. The segment and tangle filters employ a length scale specified by the user that is estimated as the length, in

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pixels, of the shortest real grain boundary segment in the TEM image. These newly developed filters are used in combination with other existing image processing routines including, for example, gray scale and binary operations such as the median noise filter, the Sobel and Kirsch edge detection filters, dilation and erosion operations, skeletonization, and opening and closing operations to generate the binary image seen Fig. 8. The images at different sample tilts are then combined to generate a single processed image that can then be analyzed using available software (e.g., NIH image, Rasband, US National Institutes of Health, or Scion Image, http://www.scioncorp.com.) Additional details of our automated image processing methodology can be found elsewhere [4, 5]. The experimentally determined grain size data for 8185 Al grains obtained using our automated methodology is shown in Fig. 9. The figure also shows three continuous probability density functions, corresponding to the lognormal (l), gamma (g), and Rayleigh (r) distributions, respectively, that have been fitted to the experimental data. The functional forms of these distributions are given by pl (d) =

1 2 2 exp −(1n(d) − α) /2β , d(2πβ 2 )1/2

(3)

pg (d) =

d α−1 exp(−d/β), β α (α)

(4)

pr (d) =

αd exp −d 2 /4β , β

(5)

where α and β are fitting parameters that are different for each distribution and, in the case of the Rayleigh distribution, normalization requires that α = 1/2. In these expressions, d represents an equivalent circular grain diameter, i.e., the diameter of a circle with equal area to that of the grain. The figure clearly demonstrates that the Rayleigh density is a poor representation of the experimental data, while both the lognormal and gamma densities fall within the error of the experimental distribution. It should be emphasized that large data sets, acquired here via automated methodologies, are needed to examine quantitatively the predictions of various grain growth models.

4.

Conclusions

Various techniques for the analysis of microstructures generated both experimentally and by computer simulation were described above. In the case

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Figure 8. A bright-field scanning transmission electron micrograph and processed images of a 100 nm thick Al film. (B1–D1) Results from conventional image processing, after (B1) median noise filter and Sobel edge detection operator, (C1) dilation, skeletonization. (B2–D2) Results from using a combination of new and conventional image processing operations, after (B2) hybrid median noise filter and Kirsch edge detection filter, (C2) dilation, skeletonization, segment filter and tangle filter, and (D2) cluster filter and final consolidation. Note that conventional image processing results in a number of false grains.

of computer simulation the focus was on developing descriptors that can be used to infer nucleation and growth conditions associated with a first-order phase transformation from a final, coalesced product microstructure. We also describe a methodology for the automated analysis of experimental TEM

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Figure 9. Fig. 3 (a) Lognormal, (b) Gamma, and (c) Rayleigh distributions fitted to experimental grain size data comprising 8185 Al grains in a thin film. Error bars represent a 95% confidence level.

micrographs. The purpose of such an analysis is to obtain statistically significant size and shape data for a large grain population. Finally, we use the information from the automated analysis to assess the validity of different grain growth models.

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Acknowledgments The authors are grateful for support under DMR-9256332, DMR-9713439 and DMR-9996315.

References [1] E.E. Underwood, Quantitative Stereology, Addison-Wesley, Reading, Massachusetts, 1970. [2] J. Harper and K. Rodbell, J. Vac. Sci. Technol. B, 15, 763, 1997. [3] W.S. Tong, J.M. Rickman, and K. Barmak, Acta Mater., 47, 435, 1999. [4] D.T. Carpenter, J.M. Rickman, and K. Barmak, J. Appl. Phys., 84, 5843, 1998. [5] D.T. Carpenter, J.R. Codner, K. Barmak, and J.M. Rickman, Mater. Lett., 41, 296, 1999. [6] N. Louat, Acta Metall., 22, 721, 1974. [7] P. Feltham, Acta Metall., 5, 97, 1957. [8] W.W. Mullins, Acta Mater., 46, 6219, 1998. [9] W.S. Tong, J.M. Rickman, and K. Barmak, “Impact of boundary nucleation on product grain size distribution,” J. Mater. Res., 12, 1501, 1997. [10] K.W. Mahin, K. Hanson, and J.W. Morris, Jr., Acta Metall., 28, 443, 1980. [11] H.J. Frost and C.V. Thompson, Acta Metall., 35, 529, 1987. [12] T. Kiang, Z. Astrophys, 48, 433, 1966. [13] D. Weaire, J.P. Kermode, and J. Wejchert, Phil. Mag. B, 53, L101–105, 1986. [14] E.N. Gilbert, Ann. Math. Stat., 33, 958, 1962. [15] N.G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, New York, 1992. [16] D. Stoyan and H. Stoyan, Appl. Stoch. Mod. Data Anal., 6, 13, 1990.

8.1 MESOSCALE MODELS OF FLUID DYNAMICS Bruce M. Boghosian1 and Nicolas G. Hadjiconstantinou2 1 Department of Mathematics, Tufts University, Bromfield-Pearson Hall, Medford, MA 02155, USA 2 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

During the last half century, enormous progress has been made in the field of computational materials modeling, to the extent that in many cases computational approaches are used in a predictive fashion. Despite this progress, modeling of general hydrodynamic behavior remains a challenging task. One of the main challenges stems from the fact that hydrodynamics manifests itself over a very wide range of length and time scales. On one end of the spectrum, one finds the fluid’s “internal” scale characteristic of its molecular structure (in the absence of quantum effects, which we omit in this chapter). On the other end, the “outer” scale is set by the characteristic sizes of the problem’s domain. The resulting scale separation or lack thereof as well as the existence of intermediate scales are key to determining the optimal approach. Successful treatments require a judicious choice of the level of description which is a delicate balancing act between the conflicting requirements of fidelity and manageable computational cost: a coarse description typically requires models for underlying processes occuring at smaller length and time scales; on the other hand, a fine-scale model will incur a significantly larger computational cost. When no molecular or intermediate length scales are important, e.g., for simple fluids, modeling the fluid at the outer scale and as a continuum results in the most efficient approach. The most well known example of these “continuum” approaches is the Navier–Stokes description of a viscous fluid. Continuum hydrodynamic descriptions are typically derived from conservation laws which require transport models before they can be solved. The resulting mathematical model is in the form of partial differential equations. A variety of methods have been developed for the solution of these, including finitedifference, finite-element, finite-volume, and spectral-element methods, such 2411 S. Yip (ed.), Handbook of Materials Modeling, 2411–2414. c 2005 Springer. Printed in the Netherlands.

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as are described in Ref. [1]. All of these methods require that the physical domain is discretized using a mesh, the generation of which can be fairly involved, depending on the complexity of the problem. More recent efforts have culminated in the development of meshless methods for solving partial differential equations, an exposition of which can be found in Ref. [2]. In certain circumstances, the separation between the molecular and macroscopic scales of length and time is lost. This happens locally in, inter alia, liquid droplet coalescence, amphiphilic membranes and monolayers, contactline dynamics in immiscible fluids, and shock formation. It may also happen globally, for example if ultra-high frequency waves are excited in a fluid. In these cases, one is forced to use a particulate description, the most well known of which is Molecular Dynamics (MD) in which particle orbits are tracked numerically. An extensive description of MD can be found in Chapter 2 [3], while a discussion of its applications to hydrodynamics can be found in Ref. [4]. The Navier–Stokes equations on one hand and MD on the other, represent two extreme possibilities. Typical problems of interest, and in particular of practical interest involving complex fluids and inhomogeneities, are significantly more complex leading to a wide range of intermediate scales that need to be addressed. For the foreseeable future, MD can be applied only to very small systems and for very short periods of time due to the computational cost associated with this approach. The principal purpose of this Chapter is to describe numerous intermediate or “mesoscale” approaches between these extremes, which attempt to coarse-grain the particulate description to varying degrees to address modeling needs. An example of a mesoscale approach can be found in descriptions of a dilute gas, in which particles travel in straight line orbits for the great majority of the time. In this situation, calculating trajectories between collisions in an exact fashion is unnecessary and therefore inefficient. A particularly ingenious method, known as Direct Simulation Monte Carlo (DSMC) takes advantage of this observation to split particle motion into successive collisionless advection and collision events. The collisionless advection occurs in steps on the order of a collision time, in contrast to MD which may require on the order of 102 time steps per collision time; likewise, collision events are processed in a stochastic manner in DSMC, in contrast to MD which tracks detailed trajectories of colliding particles. The result of this coarse graining is a description which is many orders of magnitude more computationally efficient than MD, but sacrifices atomic-level detail and precise representation of interparticle correlations. The method is described in Ref. [5]. An extension of this method, called Direct Simulation Automata (DSA), includes multiparticle collisions that make it suitable for the description of liquids and complex fluids; this is described in Ref. [6]. For a wider range of materials, including dense liquids and complex fluids, thermal fluctuations and viscous dissipation are among the essential emergent

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properties captured by MD. For example, these are the principle ingredients of a Langevin description of a colloidal suspension. In a physical system, with microscopically reversible particle orbits, these quantities are related by the Fluctuation–Dissipation Theorem of statistical physics. Dissipative Particle Dynamics (DPD) takes advantage of this to include these ingredients in a physically realistic fashion. It modifies the conservative forces of an MD simulation by introducing effective potentials, as well as fluctuating and frictional forces that represent the degrees of freedom that are lost by coarse-graining. The result is a description that is orders of magnitude more computationally efficient than MD, but which sacrifices the precise treatment of correlations and fluctuations, and requires the use of effective potentials. The DPD model is described in Ref. [7]. If one is willing to dispense with all representation of thermal fluctuations and multiparticle correlations, one may retain only the single-particle distribution function, as for example in the Boltzmann equation of kinetic theory. It was discovered in the late 1980’s that the Navier–Stokes limit of the Boltzmann description is surprizingly robust with respect to radical discretizations of the velocity space. In particular, it is possible to adopt a velocity space that consists only of a small discrete set of velocities, coincident with lattice vectors of a particular lattice. For certain choices of lattice and of collision operator, the resulting Boltzmann equation, which describes the transport of particles on a lattice with collisions localized to lattice sites, can be rigorously shown to give rise to Navier–Stokes behavior. These lattice Boltzmann models are described in Ref. [8]. Since their discovery, they have been extended to deal with compressible flow, adaptive mesh refinement on structured and unstructured grids, multiphase flow, and complex fluids. In a number of situations, the hydrodynamics of certain problems evolve in a wide range of length and time scales. If this range of scales is sufficiently wide such that no single description can be used, hybrid methods which combine more than one description can be used. The motivation for hybrid methods stems from the fact that, invariably, the “higher fidelity” description is also more computationally expensive and thus it becomes advantageous to limit its use only in the regions in which it is necessary. Clearly, hybrid methods in this respect make sense only when the “higher fidelity” description is required in small regions of space. Although hybrid methods coupling any of the methods described in this chapter can be envisioned, currently most effort has been focused towards the development of Navier–Stokes/MD and Navier–Stokes/DSMC hybrids. These are described in detail in Ref. [9]. The list of topics chosen for inclusion in this chapter is representative but not exhaustive. In particular, space limitations have precluded us from including much interesting and excellent work in the area of mesh genration, adaptive mesh refinement, and boundary element methods for the Navier–Stokes equations. Also missing are descriptions of certain mesoscale methods, such

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as lattice-gas automata and smoothed-particle hydrodynamics. Nevertheless, we feel that the topics included provide a representative cross section of this fast developing and exciting area of materials modeling research.

References [1] S. Sherwin and J. Peiro, “Finite difference, finite element and finite volume methods for partial differential equations,” Article 8.2, this volume. [2] G. Li, X. Jin, and N.R. Aluru, “Meshless methods for numerical solution of partialdifferential equations,” Article 8.3, this volume. [3] J. Li, “Basic molecular dynamics,” Article 2.8, this volume. [4] J. Koplik and J.R. Banavar, “Continuum deductions from molecular hydrodynamics,” Ann. Rev. Fluid Mech., 27, 257–292, 1995. [5] F.J. Alexander, “The direct simulation Monte Carlo method: going beyond continuum hydrodynamics,” Article 8.7, this volume. [6] T. Sakai and P.V. Coveney, “Discrete simulation automata: mesoscopic fluid models endowed with thermal fluctuations,” Article 8.5, this volume. [7] P. Espa˜nol, “Dissipative particle dynamics,” Article 8.6, this volume. [8] S. Succi, W.E, and E. Kaxiras, “Lattice Boltzmann methods for multiscale fluid problems,” Article 8.4, this volume. [9] H.S. Wijesinghe and N.G. Hadjiconstantinou, “Hybrid atomistic-continuum formulations for multiscale hydrodynamics,” Article 8.8, this volume.

8.2 FINITE DIFFERENCE, FINITE ELEMENT AND FINITE VOLUME METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS Joaquim Peir´o and Spencer Sherwin Department of Aeronautics, Imperial College, London, UK

There are three important steps in the computational modelling of any physical process: (i) problem definition, (ii) mathematical model, and (iii) computer simulation. The first natural step is to define an idealization of our problem of interest in terms of a set of relevant quantities which we would like to measure. In defining this idealization we expect to obtain a well-posed problem, this is one that has a unique solution for a given set of parameters. It might not always be possible to guarantee the fidelity of the idealization since, in some instances, the physical process is not totally understood. An example is the complex environment within a nuclear reactor where obtaining measurements is difficult. The second step of the modeling process is to represent our idealization of the physical reality by a mathematical model: the governing equations of the problem. These are available for many physical phenomena. For example, in fluid dynamics the Navier–Stokes equations are considered to be an accurate representation of the fluid motion. Analogously, the equations of elasticity in structural mechanics govern the deformation of a solid object due to applied external forces. These are complex general equations that are very difficult to solve both analytically and computationally. Therefore, we need to introduce simplifying assumptions to reduce the complexity of the mathematical model and make it amenable to either exact or numerical solution. For example, the irrotational (without vorticity) flow of an incompressible fluid is accurately represented by the Navier–Stokes equations but, if the effects of fluid viscosity are small, then Laplace’s equation of potential flow is a far more efficient description of the problem. 2415 S. Yip (ed.), Handbook of Materials Modeling, 2415–2446. c 2005 Springer. Printed in the Netherlands.

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After the selection of an appropriate mathematical model, together with suitable boundary and initial conditions, we can proceed to its solution. In this chapter we will consider the numerical solution of mathematical problems which are described by partial differential equations (PDEs). The three classical choices for the numerical solution of PDEs are the finite difference method (FDM), the finite element method (FEM) and the finite volume method (FVM). The FDM is the oldest and is based upon the application of a local Taylor expansion to approximate the differential equations. The FDM uses a topologically square network of lines to construct the discretization of the PDE. This is a potential bottleneck of the method when handling complex geometries in multiple dimensions. This issue motivated the use of an integral form of the PDEs and subsequently the development of the finite element and finite volume techniques. To provide a short introduction to these techniques we shall consider each type of discretization as applied to one-dimensional PDEs. This will not allow us to illustrate the geometric flexibility of the FEM and the FVM to their full extent, but we will be able to demonstrate some of the similarities between the methods and thereby highlight some of the relative advantages and disadvantages of each approach. For a more detailed understanding of the approaches we refer the reader to the section on suggested reading at the end of the chapter. The section is structured as follows. We start by introducing the concept of conservation laws and their differential representation as PDEs and the alternative integral forms. We next discusses the classification of partial differential equations: elliptic, parabolic, and hyperbolic. This classification is important since the type of PDE dictates the form of boundary and initial conditions required for the problem to be well-posed. It also permits in some cases, e.g., in hyperbolic equations, to identify suitable schemes to discretise the differential operators. The three types of discretisation: FDM, FEM and FVM are then discussed and applied to different types of PDEs. We then end our overview by discussing the numerical difficulties which can arise in the numerical solution of the different types of PDEs using the FDM and providing an introduction to the assessment of the stability of numerical schemes using a Fourier or Von Neumann analysis. Finally we note that, given the scientific background of the authors, the presentation has a bias towards fluid dynamics. However, we stress that the fundamental concepts presented in this chapter are generally applicable to continuum mechanics, both solids and fluids.

1.

Conservation Laws: Integral and Differential Forms

The governing equations of continuum mechanics representing the kinematic and mechanical behaviour of general bodies are commonly referred

Numerical methods for partial differential equations

2417

to as conservation laws. These are derived by invoking the conservation of mass and energy and the momentum equation (Newton’s law). Whilst they are equally applicable to solids and fluids, their differing behaviour is accounted for through the use of a different constitutive equation. The general principle behind the derivation of conservation laws is that the rate of change of u(x, t) within a volume V plus the flux of u through the boundary A is equal to the rate of production of u denoted by S(u, x, t). This can be written as d dt

u(x, t) dV +

V

f(u) · n dA −

A

S(u, x, t) dV = 0

(1)

V

which is referred to as the integral form of the conservation law. For a fixed (independent of t) volume and, under suitable conditions of smoothness of the intervening quantities, we can apply Gauss’ theorem

∇ · f dV =

V

f · n dA

A

to obtain

V

∂u + ∇ · f (u) − S dV = 0. ∂t

(2)

For the integral expression to be zero for any volume V , the integrand must be zero. This results in the strong or differential form of the equation ∂u + ∇ · f (u) − S = 0. ∂t

(3)

An alternative integral form can be obtained by the method of weighted residuals. Multiplying Eq. (3) by a weight function w(x) and integrating over the volume V we obtain V

∂u + ∇ · f (u) − S w(x) dV = 0. ∂t

(4)

If Eq. (4) is satisfied for any weight function w(x), then Eq. (4) is equivalent to the differential form (3). The smoothness requirements on f can be relaxed by applying the Gauss’ theorem to Eq. (4) to obtain V

∂u − S w(x) − f (u) · ∇w(x) dV + ∂t

f · n w(x) dA = 0.

A

(5) This is known as the weak form of the conservation law.

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J. Peir´o and S. Sherwin

Although the above formulation is more commonly used in fluid mechanics, similar formulations are also applied in structural mechanics. For instance, the well-known principle of virtual work for the static equilibrium of a body [1], is given by δW =

(∇ σ + f ) · δv dV = 0

V

where δW denotes the virtual work done by an arbitrary virtual velocity δv, σ is the stress tensor and f denotes the body force. The similarity with the method of weighted residuals (4) is evident.

2.

Model Equations and their Classification

In the following we will restrict ourselves to the analysis of one-dimensional conservation laws representing the transport of a scalar variable u(x, t) defined in the domain = {x, t : 0 ≤ x ≤ 1, 0 ≤ t ≤ T }. The convection–diffusionreaction equation is given by ∂u ∂u ∂ au − b −r u =s (6) L(u) = + ∂t ∂x ∂x together with appropriate boundary conditions at x = 0 and x = 1 to make the problem well-posed. In the above equation L(u) simply represents a linear differential operator. This equation can be recast in the form (3) with f (u) = au − ∂u/∂ x and S(u) = s + ru. It is linear if the coefficients a, b, r and s are functions of x and t, and non-linear if any of them depends on the solution, u. In what follows, we will use for convenience the convention that the presence of a subscript x or t under a expression indicates a derivative or partial derivative with respect to this variable, for example du ∂u (x); u t (x, t) = (x, t); dx ∂t Using this notation, Eq. (6) is re-written as u x (x) =

u x x (x, t) =

∂ 2u (x, t). ∂x2

u t + (au − bu x )x − ru = s.

2.1.

Elliptic Equations

The steady-state solution of Eq. (6) when advection and source terms are neglected, i.e., a=0 and s =0, is a function of x only and satisfies the Helmholtz equation (bu x )x + ru = 0.

(7)

Numerical methods for partial differential equations

2419

This equation is elliptic and its solution depends on two families of integration constants that are fixed by prescribing boundary conditions at the ends of the domain. One can either prescribe Dirichlet boundary conditions at both ends, e.g., u(0) = α0 and u(1) = α1 , or substitute one of them (or both if r=/ 0) by a Neumann boundary condition, e.g., u x (0) = g. Here α0 , α1 and g are known constant values. We note that if we introduce a perturbation into a Dirichlet boundary condition, e.g., u(0) = α0 + , we will observe an instantaneous modification to the solution throughout the domain. This is indicative of the elliptic nature of the problem.

2.2.

Parabolic Equations

Taking a = 0, r = 0 and s = 0 in our model, Eq. (6) leads to the heat or diffusion equation u t − (b u x )x = 0,

(8)

which is parabolic. In addition to appropriate boundary conditions of the form used for elliptic equations, we also require an initial condition at t = 0 of the form u(x, 0) = u 0 (x) where u 0 is a given function. If b is constant, this equation admits solutions of the form u(x, t) = Aeβt sin kx if β + k 2 b = 0. A notable feature of the solution is that it decays when b is positive as the exponent β < 0. The rate of decay is a function of b. The more diffusive the equation (i.e., larger b) the faster the decay of the solution is. In general the solution can be made up of many sine waves of different frequencies, i.e., a Fourier expansion of the form u(x, t) = Aeβt sin k x u(x, t) =

Am eβm t sin km x,

m

where Am and km represent the amplitude and the frequency of a Fourier mode, respectively. The decay of the solution depends on the Fourier contents of the initial data since βm = −km2 b. High frequencies decay at a faster rate than the low frequencies which physically means that the solution is being smoothed. This is illustrated in Fig. 1 which shows the time evolution of u(x, t) for an initial condition u 0 (x) = 20x for 0 ≤ x ≤ 1/2 and u 0 (x) = 20(1 − x) for 1/2 ≤ x ≤ 1. The solution shows a rapid smoothing of the slope discontinuity of the initial condition at x = 1/2. The presence of a positive diffusion (b > 0) physically results in a smoothing of the solution which stabilizes it. On the other hand, negative diffusion (b < 0) is de-stabilizing but most physical problems have positive diffusion.

2420

J. Peir´o and S. Sherwin u(x)

11 10

t0

9

t T

8

t 2T

7

t 3T t 4T

6 5

t 5T t 6T

4 3 2 1 0 0.0

0.5

1.0

x Figure 1. Rate of decay of the solution to the diffusion equation.

2.3.

Hyperbolic Equations

A classic example of hyperbolic equation is the linear advection equation u t + a u x = 0,

(9)

where a represents a constant velocity. The above equation is also clearly equivalent to Eq. (6) with b = r = s = 0. This hyperbolic equation also requires an initial condition, u(x, 0) = u 0 (x). The question of what boundary conditions are appropriate for this equation can be more easily be answered after considering its solution. It is easy to verify by substitution in (9) that the solution is given by u(x, t) = u 0 (x − at). This describes the propagation of the quantity u(x, t) moving with speed “a” in the x-direction as depicted in Fig. 2. The solution is constant along the characteristic line x − at = c with u(x, t) = u 0 (c). From the knowledge of the solution, we can appreciate that for a > 0 a boundary condition should be prescribed at x = 0, (e.g., u(0) = α0 ) where information is being fed into the solution domain. The value of the solution at x = 1 is determined by the initial conditions or the boundary condition at x = 0 and cannot, therefore, be prescribed. This simple argument shows that, in a hyperbolic problem, the selection of appropriate conditions at a boundary point depends on the solution at that point. If the velocity is negative, the previous treatment of the boundary conditions is reversed.

Numerical methods for partial differential equations

2421 Characteristic x at c

u (x,t ) t

x

u (x,0 )

x Figure 2. Solution of the linear advection equation.

The propagation velocity can also be a function of space, i.e., a = a(x) or even the same as the quantity being propagated, i.e., a = u(x, t). The choice a = u(x, t) leads to the non-linear inviscid Burgers’ equation u t + u u x = 0.

(10)

An analogous analysis to that used for the advection equation shows that u(x, t) is constant if we are moving with a local velocity also given by u(x, t). This means that some regions of the solution advance faster than other regions leading to the formation of sharp gradients. This is illustrated in Fig. 3. The initial velocity is represented by a triangular “zig-zag” wave. Peaks and troughs in the solution will advance, in opposite directions, with maximum speed. This will eventually lead to an overlap as depicted by the dotted line in Fig. 3. This results in a non-uniqueness of the solution which is obviously non-physical and to resolve this problem we must allow for the formation and propagation of discontinuities when two characteristics intersect (see Ref. [2] for further details).

3.

Numerical Schemes

There are many situations where obtaining an exact solution of a PDE is not possible and we have to resort to approximations in which the infinite set of values in the continuous solution is represented by a finite set of values referred to as the discrete solution. For simplicity we consider first the case of a function of one variable u(x). Given a set of points xi ; i = 1, . . . , N in the domain of definition of u(x), as

2422

J. Peir´o and S. Sherwin t

t3

t2

u u>0

t1

t 0

x

u 0 Figure 3. Formation of discontinuities in the Burgers’ equation.

ui ui 1

ui 1

x1

x i 1

xi

xi 1

xn

x

Ωi

xi

1 2

xi 12

Figure 4. Discretization of the domain.

shown in Fig. 4, the numerical solution that we are seeking is represented by a discrete set of function values {u 1 , . . . , u N } that approximate u at these points, i.e., u i ≈ u(xi ); i = 1, . . . , N . In what follows, and unless otherwise stated, we will assume that the points are equally spaced along the domain with a constant distance x = xi+1 − xi ; i = 1, . . . , N − 1. This way we will write u i+1 ≈ u(xi+1 ) = u(xi + x). This partition of the domain into smaller subdomains is referred to as a mesh or grid.

Numerical methods for partial differential equations

3.1.

2423

The Finite Difference Method (FDM)

This method is used to obtain numerical approximations of PDEs written in the strong form (3). The derivative of u(x) with respect to x can be defined as u(xi + x) − u(xi ) x→0 x u(xi ) − u(xi − x) = lim x→0 x u(xi + x) − u(xi − x) . = lim x→0 2x

u x |i = u x (xi ) = lim

(11)

All these expressions are mathematically equivalent, i.e., the approximation converges to the derivative as x → 0. If x is small but finite, the various terms in Eq. (11) can be used to obtain approximations of the derivate u x of the form u i+1 − u i x u i − u i−1 u x |i ≈ x u i+1 − u i−1 . u x |i ≈ 2x

u x |i ≈

(12) (13) (14)

The expressions (12)–(14) are referred to as forward, backward and centred finite difference approximations of u x |i , respectively. Obviously these approximations of the derivative are different.

3.1.1. Errors in the FDM The analysis of these approximations is performed by using Taylor expansions around the point xi . For instance, an approximation to u i+1 using n + 1 terms of a Taylor expansion around xi is given by

u i+1

x 2 dn u x n = u i + u x |i x + u x x |i + · · · + n 2 dx i n! dn+1 u ∗ x n+1 + n+1 (x ) . dx (n + 1)!

(15)

The underlined term is called the remainder with xi ≤ x ∗ ≤ xi+1 , and represents the error in the approximation if only the first n terms in the expansion are kept. Although the expression (15) is exact, the position x ∗ is unknown.

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J. Peir´o and S. Sherwin

To illustrate how this can be used to analyse finite difference approximations, consider the case of the forward difference approximation (12) and use the expansion (15) with n = 1 (two terms) to obtain x u i+1 − u i = u x |i + u x x (x ∗ ). x 2 We can now write the approximation of the derivative as u i+1 − u i + T x where T is given by u x |i =

(16)

(17)

x u x x (x ∗ ). (18) 2 The term T is referred to as the truncation error and is defined as the difference between the exact value and its numerical approximation. This term depends on x but also on u and its derivatives. For instance, if u(x) is a linear function then the finite difference approximation is exact and T = 0 since the second derivative is zero in (18). The order of a finite difference approximation is defined as the power p such that limx→0 (T /x p ) = γ =/ 0, where γ is a finite value. This is often written as T = O(x p ). For instance, for the forward difference approximation (12), we have T = O(x) and it is said to be first-order accurate ( p = 1). If we apply this method to the backward and centred finite difference approximations (13) and (14), respectively, we find that, for constant x, their errors are T = −

x u i − u i−1 + u x x (x ∗ ) ⇒ T = O(x) x 2 x 2 u i+1 − u i−1 − u x x x (x ) ⇒ T = O(x 2 ) u x |i = 2x 12 u x |i =

(19) (20)

with xi−1 ≤ x ∗ ≤ xi and xi−1 ≤ x ≤ xi+1 for Eqs. (19) and (20), respectively. This analysis is confirmed by the numerical results presented in Fig. 5 that displays, in logarithmic axes, the exact and truncation errors against x for the backward and the centred finite differences. Their respective truncation errors T are given by (19) and (20) calculated here, for lack of a better value, with x ∗ = x = xi . The exact error is calculated as the difference between the exact value of the derivative and its finite difference approximation. The slope of the lines are consistent with the order of the truncation error, i.e., 1:1 for the backward difference and 1:2 for the centred difference. The discrepancies between the exact and the numerical results for the smallest values of x are due to the use of finite precision computer arithmetic or round-off error. This issue and its implications are discussed in more detail in numerical analysis textbooks as in Ref. [3].

Numerical methods for partial differential equations

2425

1e 00 Backward FD Total Error Backward FD Truncation Error Centred FD Total Error Centred FD Truncation Error

1e 02 1e 04

1

1e 06

ε

1 2

1e 08

1 1e 10 1e 12 1e 14 1e 00

1e 02

1e 04

1e 06

1e 08

1e 10

1e 12

∆x

Figure 5.

Truncation and rounding errors in the finite difference approximation of derivatives.

3.1.2. Derivation of approximations using Taylor expansions The procedure described in the previous section can be easily transformed into a general method for deriving finite difference schemes. In general, we can obtain approximations to higher order derivatives by selecting an appropriate number of interpolation points that permits us to eliminate the highest term of the truncation error from the Taylor expansions. We will illustrate this with some examples. A more general description of this derivation can be found in Hirsch (1988). A second-order accurate finite difference approximation of the derivative at xi can be derived by considering the values of u at three points: xi−1 , xi and xi+1 . The approximation is constructed as a weighted average of these values {u i−1 , u i , u i+1 } such as u x |i ≈

αu i+1 + βu i + γ u i−1 . x

(21)

Using Taylor expansions around xi we can write x 2 x 3 u x x |i + u x x x |i + · · · 2 6 x 2 x 3 u x x |i − u x x x |i + · · · = u i − x u x |i + 2 6

u i+1 = u i + x u x |i +

(22)

u i−1

(23)

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J. Peir´o and S. Sherwin

Putting (22), (23) into (21) we get 1 αu i+1 + βu i + γ u i−1 = (α + β + γ ) u i + (α − γ ) u x |i x x x x 2 u x x |i + (α − γ ) u x x x |i + (α + γ ) 2 6 x 3 u x x x x |i + O(x 4 ) + (α + γ ) (24) 12 We require three independent conditions to calculate the three unknowns α, β and γ . To determine these we impose that the expression (24) is consistent with increasing orders of accuracy. If the solution is constant, the left-hand side of (24) should be zero. This requires the coefficient of (1/x)u i to be zero and therefore α+β +γ = 0. If the solution is linear, we must have α−γ =1 to match u x |i . Finally whilst the first two conditions are necessary for consistency of the approximation in this case we are free to choose the third condition. We can therefore select the coefficient of (x/2) u x x |i to be zero to improve the accuracy, which means α + γ = 0. Solving these three equations we find the values α = 1/2, β = 0 and γ = −(1/2) and recover the second-order accurate centred formula u x |i =

u i+1 − u i−1 + O(x 2 ). 2x

Other approximations can be obtained by selecting a different set of points, for instance, we could have also chosen three points on the side of xi , e.g., u i , u i−1 , u i−2 . The corresponding approximation is known as a one-sided formula. This is sometimes useful to impose boundary conditions on u x at the ends of the mesh.

3.1.3. Higher-order derivatives In general, we can derive an approximation of the second derivative using the Taylor expansion 1 1 αu i+1 + βu i + γ u i−1 u x |i = (α + β + γ ) 2 u i + (α − γ ) 2 x x x 1 x u x x x |i + (α + γ ) u x x |i + (α − γ ) 2 6 x 2 u x x x x |i + O(x 4 ). + (α + γ ) 12

(25)

Numerical methods for partial differential equations

2427

Using similar arguments to those of the previous section we impose

α + β + γ = 0 α−γ =0 ⇒ α = γ = 1, β = −2. α+γ =2

(26)

The first and second conditions require that there are no u or u x terms on the right-hand side of Eq. (25) whilst the third conditon ensures that the righthand side approximates the left-hand side as x tens to zero. The solution of Eq. (26) lead us to the second-order centred approximation u i+1 − 2u i + u i−1 + O(x 2 ). (27) x 2 The last term in the Taylor expansion (α − γ )xu x x x |i /6 has the same coefficient as the u x terms and cancels out to make the approximation second-order accurate. This cancellation does not occur if the points in the mesh are not equally spaced. The derivation of a general three point finite difference approximation with unevenly spaced points can also be obtained through Taylor series. We leave this as an exercise for the reader and proceed in the next section to derive a general form using an alternative method. u x x |i =

3.1.4. Finite differences through polynomial interpolation In this section we seek to approximate the values of u(x) and its derivatives by a polynomial P(x) at a given point xi . As way of an example we will derive similar expressions to the centred differences presented previously by considering an approximation involving the set of points {xi−1 , xi , xi+1 } and the corresponding values {u i−1 , u i , u i+1 }. The polynomial of minimum degree that satisfies P(xi−1 ) = u i−1 , P(xi ) = u i and P(xi+1 ) = u i+1 is the quadratic Lagrange polynomial (x − xi )(x − xi+1 ) (x − xi−1 )(x − xi+1 ) + ui (xi−1 − xi )(xi−1 − xi+1 ) (xi − xi−1 )(xi − xi+1 ) (x − xi−1 )(x − xi ) . + u i+1 (xi+1 − xi−1 )(xi+1 − xi )

P(x) = u i−1

(28)

We can now obtain an approximation of the derivative, u x |i ≈ Px (xi ) as (xi − xi+1 ) (xi − xi−1 ) + (xi − xi+1 ) + ui (xi−1 − xi )(xi−1 − xi+1 ) (xi − xi−1 )(xi − xi+1 ) (xi − xi−1 ) . (29) + u i+1 (xi+1 − xi−1 )(xi+1 − xi )

Px (xi ) = u i−1

If we take xi − xi−1 = xi+1 − xi = x, we recover the second-order accurate finite difference approximation (14) which is consistent with a quadratic

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J. Peir´o and S. Sherwin

interpolation. Similarly, for the second derivative we have Px x (xi ) =

2u i−1 2u i + (xi−1 − xi )(xi−1 − xi+1 ) (xi − xi−1 )(xi − xi+1 ) 2u i+1 + (xi+1 − xi−1 )(xi+1 − xi )

(30)

and, again, this approximation leads to the second-order centred finite difference (27) for a constant x. This result is general and the approximation via finite differences can be interpreted as a form of Lagrangian polynomial interpolation. The order of the interpolated polynomial is also the order of accuracy of the finite diference approximation using the same set of points. This is also consistent with the interpretation of a Taylor expansion as an interpolating polynomial.

3.1.5. Finite difference solution of PDEs We consider the FDM approximation to the solution of the elliptic equation u x x = s(x) in the region = {x : 0 ≤ x ≤ 1}. Discretizing the region using N points with constant mesh spacing x = (1/N − 1) or xi = (i − 1/N − 1), we consider two cases with different sets of boundary conditions: 1. u(0) = α1 and u(1) = α2 , and 2. u(0) = α1 and u x (1) = g. In both cases we adopt a centred finite approximation in the interior points of the form u i+1 − 2u i + u i−1 = si ; x 2

i = 2, . . . , N − 1.

(31)

The treatment of the first case is straightforward as the boundary conditions are easily specified as u 1 = α1 and u N = α2 . These two conditions together with the N − 2 equations (31) result in the linear system of N equations with N unknowns represented by

1 0 ... 1 −2 1 0 ... 0 1 −2 1 0 ... .. .. .. . . . 0 ... 0 1 −2 1 0 ... 0 1 −2 0 ... 0

0 0 0

u1 u2 u3 .. .

u 0 N−2 1 u N−1

1

uN

α1 x 2 s2 x 2 s3 .. .

= x 2 s N−2 x 2 s N−1

α2

.

Numerical methods for partial differential equations

2429

This matrix system can be written in abridged form as Au = s. The matrix A is non-singular and admits a unique solution u. This is the case for most discretizations of well-posed elliptic equations. In the second case the boundary condition u(0) = α1 is treated in the same way by setting u 1 = α1 . The treatment of the Neumann boundary condition u x (1) = g requires a more careful consideration. One possibility is to use a one-sided approximation of u x (1) to obtain u x (1) ≈

u N − u N−1 = g. x

(32)

This expression is only first-order accurate and thus inconsistent with the approximation used at the interior points. Given that the PDE is elliptic, this error could potentially reduce the global accuracy of the solution. The alternative is to use a second-order centred approximation u x (1) ≈

u N+1 − u N−1 = g. x

(33)

Here the value u N+1 is not available since it is not part of our discrete set of values but we could use the finite difference approximation at x N given by u N+1 − 2u N + u N−1 = sN x 2 and include the Neumann boundary condition (33) to obtain 1 u N − u N−1 = (gx − s N x 2 ). 2

(34)

It is easy to verify that the introduction of either of the Neumann boundary conditions (32) or (34) leads to non-symmetric matrices.

3.2.

Time Integration

In this section we address the problem of solving time-dependent PDEs in which the solution is a function of space and time u(x, t). Consider for instance the heat equation u t − bu x x = s(x)

in

= {x, t : 0 ≤ x ≤ 1, 0 ≤ t ≤ T }

with an initial condition u(x, 0) = u 0 (x) and time-dependent boundary conditions u(0, t) = α1 (t) and u(1, t) = α2 (t), where α1 and α2 are known

2430

J. Peir´o and S. Sherwin

functions of t. Assume, as before, a mesh or spatial discretization of the domain {x1 , . . . , x N }.

3.2.1. Method of lines In this technique we assign to our mesh a set of values that are functions of time u i (t) = u(xi , t); i = 1, . . . , N . Applying a centred discretization to the spatial derivative of u leads to a system of ordinary differential equations (ODEs) in the variable t given by b du i {u i−1 (t) − 2u i (t) + u i+1 (t)} + si ; = dt x 2

i = 2, . . . , N − 1

with u 1 = α1 (t) and u N = α2 (t). This can be written as

u2

−2 1

u2 u3 .. . + u N−2 u N−1 1 −2

u3 1 −2 1 b d .. .. .. .. . = . . . x 2 dt u N−2 1 −2 1

u N−1

bα1 (t) s2 + x 2 s3 .. .

s N−2 bα2 (t)

s N−1 +

x 2

or in matrix form as du (t) = A u(t) + s(t). (35) dt Equation (35) is referred to as the semi-discrete form or the method of lines. This system can be solved by any method for the integration of initial-value problems [3]. The numerical stability of time integration schemes depends on the eigenvalues of the matrix A which results from the space discretization. For this example, the eigenvalues vary between 0 and −(4α/x 2 ) and this could make the system very stiff, i.e. with large differences in eigenvalues, as x → 0.

3.2.2. Finite differences in time The method of finite differences can be applied to time-dependent problems by considering an independent discretization of the solution u(x, t) in space and time. In addition to the spatial discretization {x1 , . . . , x N }, the discretization in time is represented by a sequence of times t 0 = 0 < · · · < t n < · · · < T . For simplicity we will assume constant intervals x and t in space and time, respectively. The discrete solution at a point will be represented by

Numerical methods for partial differential equations

2431

u ni ≈ u(xi , t n ) and the finite difference approximation of the time derivative follows the procedures previously described. For example, the forward difference in time is given by u t (x, t n ) ≈

u(x, t n+1 ) − u(x, t n ) t

and the backward difference in time is u t (x, t n+1 ) ≈

u(x, t n+1 ) − u(x, t n ) t

both of which are first-order accurate, i.e. T = O(t). Returning to the heat equation u t − bu x x = 0 and using a centred approximation in space, different schemes can be devised depending on the time at which the equation is discretized. For instance, the use of forward differences in time leads to − u ni u n+1 b n i u i−1 − 2u ni + u ni+1 . = 2 t x

(36)

This scheme is explicit as the values of the solution at time t n+1 are obtained directly from the corresponding (known) values at time t n . If we use backward differences in time, the resulting scheme is − u ni u n+1 b n+1 i n+1 n+1 = u − 2u + u i−1 i i+1 . t x 2

(37)

Here to obtain the values at t n+1 we must solve a tri-diagonal system of equations. This type of schemes are referred to as implicit schemes. The higher cost of the implicit schemes is compensated by a greater numerical stability with respect to the explicit schemes which are stable in general only for some combinations of x and t.

3.3.

Discretizations Based on the Integral Form

The FDM uses the strong or differential form of the governing equations. In the following, we introduce two alternative methods that use their integral form counterparts: the finite element and the finite volume methods. The use of integral formulations is advantageous as it provides a more natural treatment of Neumann boundary conditions as well as that of discontinuous source terms due to their reduced requirements on the regularity or smoothness of the solution. Moreover, they are better suited than the FDM to deal with complex geometries in multi-dimensional problems as the integral formulations do not rely in any special mesh structure.

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J. Peir´o and S. Sherwin

These methods use the integral form of the equation as the starting point of the discretization process. For example, if the strong form of the PDE is L(u) = s, the integral from is given by 1

L(u)w(x) dx =

0

1

sw(x) dx

(38)

0

where the choice of the weight function w(x) defines the type of scheme.

3.3.1. The finite element method (FEM) Here we discretize the region of interest = {x : 0 ≤ x ≤ 1} into N − 1 subdomains or elements i = {x : xi−1 ≤ x ≤ xi } and assume that the approximate solution is represented by u δ (x, t) =

N

u i (t)Ni (x)

i=1

where the set of functions Ni (x) is known as the expansion basis. Its support is defined as the set of points where Ni (x)=/ 0. If the support of Ni (x) is the whole interval, the method is called a spectral method. In the following we will use expansion bases with compact support which are piecewise continuous polynomials within each element as shown in Fig. 6. The global shape functions Ni (x) can be split within an element into two local contributions of the form shown in Fig. 7. These individual functions are referred to as the shape functions or trial functions.

3.3.2. Galerkin FEM In the Galerkin FEM method we set the weight function w(x) in Eq. (38) to be the same as the basis function Ni (x), i.e., w(x) = Ni (x). Consider again the elliptic equation L(u) = u x x = s(x) in the region with boundary conditions u(0) = α and u x (1) = g. Equation (38) becomes 1

w(x)u x x dx =

0

1

w(x)s(x) dx.

0

At this stage, it is convenient to integrate the left-hand side by parts to get the weak form −

1 0

wx u x dx + w(1) u x (1) − w(0) u x (0) =

1 0

w(x) s(x) dx.

(39)

Numerical methods for partial differential equations ui 1

u1

2433

ui ui 1 Ωi

x1

xi 1

xi

uN x i 1

xN

x

u1 x 1

x

.. . Ni (x)

ui x

1 x

.. . uN x

1 x

Figure 6. A piecewise linear approximation u δ (x, t) =

N

i=1 u i (t)Ni (x).

ui ui 1 Ωi

xi

ui

x i 1

Ni 1

Ni 1

xi

ui 1

x i 1

x

1

x i 1

Figure 7. Finite element expansion bases.

This is a common technique in the FEM because it reduces the smoothness requirements on u and it also makes the matrix of the discretized system symmetric. In two and three dimensions we would use Gauss’ divergence theorem to obtain a similar result. The application of the boundary conditions in the FEM deserves attention. The imposition of the Neumann boundary condition u x (1) = g is straightforward, we simply substitute the value in Eq. (39). This is a very natural way of imposing Neumann boundary conditions which also leads to symmetric

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J. Peir´o and S. Sherwin

matrices, unlike the FDM. The Dirichlet boundary condition u(0) = α can be applied by imposing u 1 = α and requiring that w(0) = 0. In general, we will impose that the weight functions w(x) are zero at the Dirichlet boundaries. N δ Letting u(x) ≈ u (x) = j =1 u j N j (x) and w(x) = Ni (x) then Eq. (39) becomes −

1

N dNi dN j uj (x) (x) dx = dx dx j =1

0

1

Ni (x) s(x) dx

(40)

0

for i =2, . . . , N . This represents a linear system of N − 1 equations with N − 1 unknowns: {u 2 , . . . , u N }. Let us proceed to calculate the integral terms corresponding to the i-th equation. We calculate the integrals in Eq. (40) as sums of integrals over the elements i . The basis functions have compact support, as shown in Fig. 6. Their value and their derivatives are different from zero only on the elements containing the node i, i.e., x − xi−1 xi−1 < x < xi x i−1 Ni (x) = xi+1 − x xi < x < xi+1 xi 1 xi−1 < x < xi x i−1

dNi (x) = dx −1 xi < x < xi+1 xi with xi−1 = xi − xi−1 and xi = xi+1 − xi . This means that the only integrals different from zero in (40) are xi

−

x i−1

dNi dNi−1 dNi + ui u i−1 dx dx dx

xi

=

Ni s dx +

x i−1

−

x i+1

xi

x i+1

Ni s dx

(41)

xi

The right-hand side of this equation expressed as xi

F= x i−1

x − xi−1 s(x) dx + xi−1

x i+1

xi

xi+1 − x s(x) dx xi

can be evaluated using a simple integration rule like the trapezium rule x i+1

xi

g(x) dx ≈

dNi dNi dNi+1 + u i+1 ui dx dx dx dx

g(xi ) + g(xi+1 ) xi 2

Numerical methods for partial differential equations and it becomes

F=

2435

xi xi−1 si . + 2 2

Performing the required operations in the left-hand side of Eq. (41) and including the calculated valued of F leads to the FEM discrete form of the equation as −

u i+1 − u i xi−1 + xi u i − u i−1 + = si . xi−1 xi 2

Here if we assume that xi−1 = xi = x then the equispaced approximation becomes u i+1 − 2u i + u i−1 = x si x which is identical to the finite difference formula. We note, however, that the general FE formulation did not require the assumption of an equispaced mesh. In general the evaluation of the integral terms in this formulation is more efficiently implemented by considering most operations in a standard element st = {−1 ≤ x ≤ 1} where a mapping is applied from the element i to the standard element st . For more details on the general formulation see Ref. [4].

3.3.3. Finite volume method (FVM) The integral form of the one-dimensional linear advection equation is given by Eq. (1) with f (u) = au and S = 0. Here the region of integration is taken to be a control volume i , associated with the point of coordinate xi , represented by xi− 1 ≤ x ≤ xi+ 1 , following the notation of Fig. 4, and the integral form is 2 2 written as x i+ 1

x i+ 1

2

u t dx +

x i− 1

2

f x (u) dx = 0.

(42)

x i− 1

2

2

This expression could also been obtained from the weighted residual form (4) by selecting a weight w(x) such that w(x) = 1 for xi− 1 ≤ x ≤ xi+ 1 and 2 2 w(x) = 0 elsewhere. The last term in Eq. (42) can be evaluated analytically to obtain x i+ 1

2

f x (u) dx = f u i+(1/2) − f u i−(1/2)

x i− 1 2

2436

J. Peir´o and S. Sherwin

and if we approximate the first integral using the mid-point rule we finally have the semi-discrete form

u t |i xi+ 1 − xi− 1 + f u i+ 1 − f u i− 1 = 0. 2

2

2

2

This approach produces a conservative scheme if the flux on the boundary of one cell equals the flux on the boundary of the adjacent cell. Conservative schemes are popular for the discretization of hyperbolic equations since, if they converge, they can be proven (Lax-Wendroff theorem) to converge to a weak solution of the conservation law.

3.3.4. Comparison of FVM and FDM To complete our comparison of the different techniques we consider the FVM discretization of the elliptic equation u x x = s. The FVM integral form of this equation over a control volume i = {xi− 1 ≤ x ≤ xi+ 1 } is 2

x i+ 1

2

x i+ 1

2

2

u x x dx = x i− 1

s dx. x i− 1

2

2

Evaluating exactly the left-hand side and approximating the right-hand side by the mid-point rule we obtain

u x xi+ 1 − u x xi− 1 = xi+ 1 − xi− 1 2

2

2

2

si .

(43)

If we approximate u(x) as a linear function between the mesh points i − 1 and i, we have u i − u i−1 u i+1 − u i , u x |i+ 1 ≈ , u x |i− 1 ≈ 2 2 xi − xi−1 xi+1 − xi and introducing these approximations into Eq. (43) we now have u i − u i−1 u i+1 − u i − = (xi+ 1 − xi− 1 ) si . 2 2 xi+1 − xi xi − xi−1 If the mesh is equispaced then this equation reduces to u i+1 − 2u i + u i−1 = x si , x which is the same as the FDM and FEM on an equispaced mesh. Once again we see the similarities that exist between these methods although some assumptions in the construction of the FVM have been made. FEM and FVM allow a more general approach to non-equispaced meshes (although this can also be done in the FDM). In two and three dimensions, curvature is more naturally dealt with in the FVM and FEM due to the integral nature of the equations used.

Numerical methods for partial differential equations

4.

2437

High Order Discretizations: Spectral Element/ p-Type Finite Elements

All of the approximations methods we have discussed this far have dealt with what is typically known as the h-type approximation. If h = x denotes the size of a finite difference spacing or finite elemental regions then convergence of the discrete approximation to the PDE is achieved by letting h → 0. An alternative method is to leave the mesh spacing fixed but to increase the polynomial order of the local approximation which is typically denoted by p or the p-type extension. We have already seen that higher order finite difference approximations can be derived by fitting polynomials through more grid points. The drawback of this approach is that the finite difference stencil gets larger as the order of the polynomial approximation increases. This can lead to difficulties when enforcing boundary conditions particularly in multiple dimensions. An alternative approach to deriving high-order finite differences is to use compact finite differences where a Pad´e approximation is used to approximate the derivatives. When using the finite element method in an integral formulation, it is possible to develop a compact high-order discretization by applying higher order polynomial expansions within every elemental region. So instead of using just a linear element in each piecewise approximation of Fig. 6 we can use a polynomial of order p. This technique is commonly known as p-type finite element in structural mechanics or the spectral element method in fluid mechanics. The choice of the polynomial has a strong influence on the numerical conditioning of the approximation and we note that the choice of an equi-spaced Lagrange polynomial is particularly bad for p > 5. The two most commonly used polynomial expansions are Lagrange polynomial based on the Gauss–Lobatto–Legendre quadratures points or the integral of the Legendre polynomials in combination with the linear finite element expansion. These two polynomial expansions are shown in Fig. 8. Although this method is more (a)

(b) 1

1

1

0

0

0

0

1

1

1

1

1

1

1

1

0

0

0

0

1

1

1

1

1

1

1

1

0

0

0

0

1

1

1

1

1

Figure 8. Shape of the fifth order ( p = 5) polynomial expansions typically used in (a) spectral element and (b) p-type finite element methods.

2438

J. Peir´o and S. Sherwin

involved to implement, the advantage is that for a smooth problem (i.e., one where the derivatives of the solution are well behaved) the computational cost increases algebraically whilst the error decreases exponentially fast. Further details on these methods can be found in Refs. [5, 6].

5.

Numerical Difficulties

The discretization of linear elliptic equations with either FD, FE or FV methods leads to non-singular systems of equations that can easily solved by standard methods of solution. This is not the case for time-dependent problems where numerical errors may grow unbounded for some discretization. This is perhaps better illustrated with some examples. Consider the parabolic problem represented by the diffusion equation u t − u x x = 0 with boundary conditions u(0) = u(1) = 0 solved using the scheme (36) with b = 1 and x = 0.1. The results obtained with t = 0.004 and 0.008 are depicted in Figs. 9(a) and (b), respectively. The numerical solution (b) corresponding to t = 0.008 is clearly unstable. A similar situation occurs in hyperbolic problems. Consider the onedimensional linear advection equation u t + au x = 0; with a > 0 and various explicit approximations, for instance the backward in space, or upwind, scheme is − u ni u n+1 u n − u ni−1 i +a i = 0 ⇒ u n+1 = (1 − σ )u ni + σ u ni−1 , i t x the forward in space, or downwind, scheme is u n − u ni − u ni u n+1 i + a i+1 =0 t x

⇒

(a)

u n+1 = (1 + σ )u ni − σ u ni+1 , i

(44)

(45)

(b)

0.3

0.3

t0.20 t0.24 t0.28 t0.32

0.2

t0.20 t0.24 t0.28 t0.32

0.2

0.1 u(x,t)

u(x,t)

0.1

0

0

0.1

0.1

0.2

0

0.2

0.4

0.6 x

0.8

1

0.2

0

0.2

0.4

0.6

0.8

1

x

Figure 9. Solution to the diffusion equation u t + u x x = 0 using a forward in time and centred in space finite difference discretization with x = 0.1 and (a) t = 0.004, and (b) t = 0.008. The numerical solution in (b) is clearly unstable.

Numerical methods for partial differential equations

2439

u(x,t)

0

u(x, 0) =

1 + 5x

1 − 5x

0

x ≤ −0.2 −0.2 ≤ x ≤ 0 0 ≤ x ≤ 0.2 x ≥ 0.2

a 1.0 0.0 0.2

0.2

x

Figure 10. A triangular wave as initial condition for the advection equation.

and, finally, the centred in space is given by u n − u ni−1 u n+1 − u ni i + a i+1 =0 t 2x

⇒

= u ni − u n+1 i

σ n (u − u ni−1 ) 2 i+1 (46)

where σ = (at/x) is known as the Courant number. We will see later that this number plays an important role in the stability of hyperbolic equations. Let us obtain the solution of u t + au x = 0 for all these schemes with the initial condition given in Fig. 10. As also indicated in Fig. 10, the exact solution is the propagation of this wave form to the right at a velocity a. Now we consider the solution of the three schemes at two different Courant numbers given by σ = 0.5 and 1.5. The results are presented in Fig. 11 and we observe that only the upwinded scheme when σ ≤ 1 gives a stable, although diffusive, solution. The centred scheme when σ = 0.5 appears almost stable but the oscillations grow in time leading to an unstable solution.

6.

Analysis of Numerical Schemes

We have seen that different parameters, such as the Courant number, can effect the stability of a numerical scheme. We would now like to set up a more rigorous framework to analyse a numerical scheme and we introduce the concepts of consistency, stability and Convergence of a numerical scheme.

6.1.

Consistency

A numerical scheme is consistent if the discrete numerical equation tends to the exact differential equation as the mesh size (represented by x and t) tends to zero.

2440

J. Peir´o and S. Sherwin 3

1 0.9

2 0.8 0.7

1

u(x,t)

u(x,t)

0.6 0.5

0

0.4 1

0.3 0.2

2

0.1 0

1

0.8

0.6

0.4

0.2

0

0.2

σ 0.5

0.4

0.6

0.8

3 1

1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

0.8

0.6

0.4

0.2

0 x

0.2

0.4

0.6

0.8

1

0.4

0.6

0.8

1

σ 1.5

30

20

2

10 1 0

u(x,t)

0 u(x,t)

10 1

20

2

3

30

1

0.8

0.6

0.4

0.2

0 x

0.2

0.4

0.6

0.8

1

40 1

σ 1.5

σ 0.5 3

1.2 1

2

0.8 1

0

0.4 u(x,t)

u(x,t)

0.6

0.2

1

0 2 0.2 3

0.4 0.6 1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

4 1

x

σ 0.5

0.8

0.6

0.4

0.2

0 x

0.2

σ 1.5

Figure 11. Numerical solution of the advection equation u t + au x = 0. Dashed lines: initial condition. Dotted lines: exact solution. Solid line: numerical solution.

Consider the centred in space and forward in time finite diference approximation to the linear advection equation u t + au x = 0 given by Eq. (46). Let us , u ni+1 and u ni−1 around (xi , t n ) as consider Taylor expansions of u n+1 i = u ni + t u t |ni + u n+1 i

t 2 u t t |ni + · · · 2

Numerical methods for partial differential equations

2441

x 2 x 3 u x x |ni + u x x x |ni + · · · 2 6 x 2 x 3 u ni−1 = u ni − x u x |ni + u x x |ni − u x x x |ni + · · · 2 6 Substituting these expansions into Eq. (46) and suitably re-arranging the terms we find that − u ni u n+1 u n − u ni−1 i + a i+1 − (u t + au x )|ni = T (47) t 2x where T is known as the truncation error of the approximation and is given by u ni+1 = u ni + x u x |ni +

t x 2 u t t |ni + au x x x |ni + O(t 2 , x 4 ). 2 6 The left-hand side of this equation will tend to zero as t and x tend to zero. This means that the numerical scheme (46) tends to the exact equation at point xi and time level t n and therefore this approximation is consistent. T =

6.2.

Stability

We have seen in the previous numerical examples that errors in numerical solutions can grow uncontrolled and render the solution meaningless. It is therefore sensible to require that the solution is stable, this is that the difference between the computed solution and the exact solution of the discrete equation should remain bounded as n → ∞ for a given x.

6.2.1. The Courant–Friedrichs–Lewy (CFL) condition This is a necessary condition for stability of explicit schemes devised by Courant, Friedrichs and Lewy in 1928. Recalling the theory of characteristics for hyperbolic systems, the domain of dependence of a PDE is the portion of the domain that influences the solution at a given point. For a scalar conservation law, it is the characteristic passing through the point, for instance, the line P Q in Fig. 12. The domain of dependence of a FD scheme is the set of points that affect the approximate solution at a given point. For the upwind scheme, the numerical domain of dependence is shown as a shaded region in Fig. 12. The CFL criterion states that a necessary condition for an explicit FD scheme to solve a hyperbolic PDE to be stable is that, for each mesh point, the domain of dependence of the FD approximation contains the domain of dependence of the PDE.

2442

J. Peir´o and S. Sherwin

(a)

(b) t

t

∆x

∆x

a∆t

Characteristic P

P

a∆t

∆t

∆t

x Q

x Q

Figure 12. Solution of the advection equation by the upwind scheme. Physical and numerical domains of dependence: (a) σ = (at/x) > 1, (b) σ ≤ 1.

For a Courant number σ = (at/x) greater than 1, changes at Q will affect values at P but the FD approximation cannot account for this. The CFL condition is necessary for stability of explicit schemes but it is not sufficient. For instance, in the previous schemes we have that the upwind FD scheme is stable if the CFL condition σ ≤ 1 is imposed. The downwind FD scheme does not satisfy the CFL condition and is unstable. However, the centred FD scheme is unstable even if σ ≤ 1.

6.2.2. Von Neumann (or Fourier) analysis of stability The stability of FD schemes for hyperbolic and parabolic PDEs can be analysed by the von Neumann or Fourier method. The idea behind the method is the following. As discussed previously the analytical solutions of the model diffusion equation u t − b u x x = 0 can be found in the form u(x, t) =

∞

eβm t e I km x

m=−∞

if βm + b km2 = 0. This solution involves a Fourier series in space and an expocomnential decay in time since βm ≤ 0 for b > 0. Here we have included the√ I km x = cos km x + I sin km x with I = −1, plex version of the Fourier series, e because this simplifies considerably later algebraic manipulations. To analyze the growth of different Fourier modes as they evolve under the numerical scheme we can consider each frequency separately, namely u(x, t) = eβm t e I km x .

Numerical methods for partial differential equations

2443

A discrete version of this equation is u ni = u(xi , t n ) = eβm t e I km xi . We can take, without loss of generality, xi = ix and t n = nt to obtain n

n

u ni = eβm nt e I km ix = eβm t e I km ix . The term e I km ix = cos(km ix) + I sin(km ix) is bounded and, therefore, any growth in the numerical solution will arise from the term G = eβm t , known as the amplification factor. Therefore the numerical method will be stable, or the numerical solution u ni bounded as n → ∞, if |G| ≤ 1 for solutions of the form u ni = G n e I km ix . We will now proceed to analyse, using the von Neummann method, the stability of some of the schemes discussed in the previous sections. Example 1 Consider the explicit scheme (36) for the diffusion equation u t − bu x x = 0 expressed here as u n+1 = λu ni−1 + (1 − 2λ)u ni + λu ni+1 ; i

λ=

bt . x 2

We assume u ni = G n e I km ix and substitute in the equation to get G = 1 + 2λ [cos(km x) − 1] . Stability requires |G| ≤ 1. Using −2 ≤ cos(km x) − 1 ≤ 0 we get 1 − 4λ ≤ G ≤ 1 and to satisfy the left inequality we impose −1 ≤ 1 − 4λ ≤ G

=⇒

1 λ≤ . 2

This means that for a given grid size x the maximum allowable timestep is t = (x 2 /2b). Example 2 Consider the implicit scheme (37) for the diffusion equation u t − bu x x = 0 expressed here as n+1 n + λu n+1 λu n+1 i−1 + −(1 + 2λ)u i i+1 = −u i ;

λ=

bt . x 2

The amplification factor is now G=

1 1 + λ(2 − cos βm )

and we have |G| < 1 for any βm if λ > 0. This scheme is therefore unconditionally stable for any x and t. This is obtained at the expense of solving a linear system of equations. However, there will still be restrictions on x

2444

J. Peir´o and S. Sherwin

and t based on the accuracy of the solution. The choice between an explicit or an implicit method is not always obvious and should be done based on the computer cost for achieving the required accuracy in a given problem. Example 3 Consider the upwind scheme for the linear advection equation u t + au x = 0 with a > 0 given by = (1 − σ )u ni + σ u ni−1 ; u n+1 i

σ=

at . x

Let us denote βm = km x and introduce the discrete Fourier expression in the upwind scheme to obtain G = (1 − σ ) + σ e−Iβm The stability condition requires |G| ≤ 1. Recall that G is a complex number G = ξ + I η so ξ = 1 − σ + σ cos βm ;

η = −σ sin βm

This represents a circle of radius σ centred at 1 − σ . The stability condition requires the locus of the points (ξ, η) to be interior to a unit circle ξ 2 + η2 ≤ 1. If σ < 0 the origin is outside the unit circle, 1 − σ > 1, and the scheme is unstable. If σ > 1 the back of the locus is outside the unit circle 1 − 2σ < 1 and it is also unstable. Therefore, for stability we require 0 ≤ σ ≤ 1, see Fig. 13. Example 4 The forward in time, centred in space scheme for the advection equation is given by = u ni − u n+1 i

σ n (u − u ni−1 ); 2 i+1

σ=

at . x

η

1 σ

1

G σ

ξ

Figure 13. Stability region of the upwind scheme.

Numerical methods for partial differential equations

2445

The introduction of the discrete Fourier solution leads to σ G = 1 − (e Iβm − e−Iβm ) = 1 − I σ sin βm 2 Here we have |G|2 = 1 + σ 2 sin2 βm > 1 always for σ =/ 0 and it is therefore unstable. We will require a different time integration scheme to make it stable.

6.3.

Convergence: Lax Equivalence Theorem

A scheme is said to be convergent if the difference between the computed solution and the exact solution of the PDE, i.e. the error E in = u ni − u(xi , t n ), vanishes as the mesh size is decreased. This is written as lim

x,t →0

|E in | = 0

for fixed values of xi and t n . This is the fundamental property to be sought from a numerical scheme but it is difficult to verify directly. On the other hand, consistency and stability are easily checked as shown in the previous sections. The main result that permits the assessment of the convergence of a scheme from the requirements of consistency and stability is the equivalence theorem of Lax stated here without proof: Stability is the necessary and sufficient condition for a consistent linear FD approximation to a well-posed linear initial-value problem to be convergent.

7.

Suggestions for Further Reading

The basics of the FDM are presented a very accessible form in Ref. [7]. More modern references are Refs. [8, 9]. An elementary introduction to the FVM can be consulted in the book by Versteeg and Malalasekera [10]. An in-depth treatment of the topic with an emphasis on hyperbolic problems can be found in the book by Leveque [2]. Two well established general references for the FEM are the books of Hughes [4] and Zienkiewicz and Taylor [11]. A presentation from the point of view of structural analysis can be consulted in Cook et al. [11] The application of p-type finite element for structural mechanics is dealt with in the book of Szabo and Babu˘ska [5]. The treatment of both p-type and spectral element methods in fluid mechanics can be found in the book by Karniadakis and Sherwin [6]. A comprehensive reference covering both FDM, FVM and FEM for fluid dynamics is the book by Hirsch [13]. These topics are also presented using a more mathematical perspective in the classical book by Quarteroni and Valli [14].

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References [1] J. Bonet and R. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge University Press, 1997. [2] R. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. [3] W. Cheney and D. Kincaid, Numerical Mathematics and Computing, 4th edn., Brooks/Cole Publishing Co., 1999. [4] T. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Publishers, 2000. [5] B. Szabo and I. Babu˘ska, Finite Element Analysis, Wiley, 1991. [6] G.E. Karniadakis and S. Sherwin, Spectral/hp Element Methods for CFD, Oxford University Press, 1999. [7] G. Smith, Numerical Solution of Partial Differential Equations: Finite Diference Methods, Oxford University Press, 1985. [8] K. Morton and D. Mayers, Numerical Solution of Partial Differential Equations, Cambridge University Press, 1994. [9] J. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Springer-Verlag, 1995. [10] H. Versteeg and W. Malalasekera, An Introduction to Computational Fluid Dynamics. The Finite Volume Method, Longman Scientific & Technical, 1995. [11] O. Zienkiewicz and R. Taylor, The Finite Element Method: The Basis, vol. 1, Butterworth and Heinemann, 2000. [12] R. Cook, D. Malkus, and M. Plesha, Concepts and Applications of Finite Element Analysis, Wiley, 2001. [13] C. Hirsch, Numerical Computation of Internal and External Flows, vol. 1, Wiley, 1988. [14] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, 1994.

8.3 MESHLESS METHODS FOR NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS Gang Li∗ , Xiaozhong Jin† , and N.R. Aluru‡ Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

A popular research topic in numerical methods recently has been the development of meshless methods as alternatives to the traditional finite element, finite volume, and finite difference methods. The traditional methods all require some connectivity knowledge a priori, such as the generation of a mesh, whereas the aim of meshless methods is to sprinkle only a set of points or nodes covering the computational domain, with no connectivity information required among the set of points. Multiphysics and multiscale analysis, which is a common requirement for microsystem technologies such as MEMS and Bio-MEMS, is radically simplified by meshless techniques as we deal with only nodes or points instead of a mesh. Meshless techniques are also appealing because of their potential in adaptive techniques, where a user can simply add more points in a particular region to obtain more accurate results. Extensive research has been conducted in the area of meshless methods in recent years (see [1–3] for an overview). Broadly defined, meshless methods contain two key steps: construction of meshless approximation functions and their derivatives and meshless discretization of the governing partial-differential equations. Least-squares [4–6, 8–13], kernel based [14–18] and radial basis function [19–23] approaches are three techniques that have gained considerable attention for construction of meshless approximation functions (see [26] for a detailed discussion on least-squares and kernel approximations). The meshless discretization of the partial-differential equations can be categorized into three classes: cell integration [5, 6, 12, 15, 16], local point integration [9, 24, 25], and point collocation [8, 10, 11, 17, 18, 20, 21]. Another class of important meshless methods are developed for boundaryonly analysis of partial differential equations. Boundary integral formulations 2447 S. Yip (ed.), Handbook of Materials Modeling, 2447–2474. c 2005 Springer. Printed in the Netherlands.

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[27], especially when combined with fast algorithms based on multipole expansions [28], Fast Fourier Transform (FFT) [29] and singular value decomposition (SVD) [30, 31], are powerful computational techniques for rapid analysis of exterior problems. Recently, several meshless methods for boundary-only analysis have been proposed in the literature. Some of the methods include the boundary node method [32, 33], the hybrid boundary node method [34] and the boundary knot method [35]. The boundary node method is a combined boundary integral/meshless approach for boundary only analysis of partial differential equations. A key difficulty in the boundary node method is the construction of interpolation functions using moving least-squares methods. For 2-D problems, where the boundary is 1-D, Cartesian coordinates cannot be used to construct interpolation functions (see [36] for a more detailed discussion). Instead, a cyclic coordinate is used in the moving least-squares approach to construct interpolation functions. For 3-D problems, where the boundary is 2-D, curvilinear coordinates are used to construct interpolation functions. The definition of these coordinates is not trivial for complex geometries. Recently, we have introduced a boundary cloud method (BCM) [36, 37], which is also a combined boundary-integral/scattered point approach for boundary only analysis of partial differential equations. The boundary cloud method employs a Hermite-type or a varying polynomial basis least-squares approach to construct interpolation functions to enable the direct use of Cartesian coordinates. Due to the length restriction, boundary-only methods are not discussed in this article. This paper summarizes the key developments in meshless methods and their implementation for interior problems. This material should serve as a starting point for the reader to venture into more advanced topics in meshless methods. The rest of the article is organized as follows: In Section 1, we introduce the general numerical procedures for solving partial differential equations. Meshless approximation and discretization approaches are discussed in Sections 2 and 3, respectively. Section 4 provides a brief summary of some existing meshless methods. The solution of an elasticity problem by using the finite cloud method is presented in Section 5. Section 6 concludes the article.

1.

Steps for Solving Partial Differential Equations: An Example

Typically, the physical behavior of an object or a system is described mathematically by partial differential equations. For example, as shown in Fig. 1, an irregular shaped 2-D plate is subjected to certain conditions of heat transfer: it has a temperature distribution of g(x, y) on the left part on its boundary (denoted as u ) and a heat flux distribution of h(x, y) on the remaining part of the boundary (denoted as q ). At steady state, the temperature at any point on

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u g(x,y)

Ω

Γu

∇2 u 0

Γq

u,n h(x,y) Figure 1. Heat conduction within a plate.

the plate is described by the steady-state heat conduction equation, i.e., ∇ 2u = 0

(1)

where u is the temperature. The temperature and the flux prescribed on the boundary are defined as boundary conditions. The prescribed temperature is called the Dirichlet or an essential boundary condition, i.e., u = g(x, y) on u

(2)

and the prescribed flux is called the Neumann or anatural boundary condition, i.e., ∂u = h(x, y) on q (3) ∂n where n is the outward normal to the boundary. The governing equations along with the Dirichlet and/or Neumann boundary conditions permit a unique temperature field on the plate. There are various numerical techniques available to solve the simple example considered above. Finite difference method (FDM) [38], finite element method (FEM) [39] and boundary element method (BEM) [27, 40] are the most popular methods for solving PDEs. Recently, meshless methods have been proposed and they have been successfully applied to solve many physical problems. Although the FDM, FEM, BEM and meshless methods are different in many aspects, all these methods contain three common steps: 1. Discretization of the domain 2. Approximation of the unknown function 3. Discretization of the governing equation and the boundary conditions.

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In the first step, a meshing process is often required for conventional methods such as finite element and boundary element methods. For objects with complex geometry, the meshing step could be complicated and time consuming. The key idea in meshless methods is to eliminate the meshing process to improve the efficiency. Many authors have shown that this can be done through meshless approximation and meshless discretization of the governing equation and the boundary conditions.

2.

Meshless Approximation

In meshless methods, as shown in Fig. 2, a physical domain is represented by a set of points. The points can be either structured or scattered as long as they cover the physical domain. An unknown function such as the temperature field in the domain is defined by the governing equation along with the appropriate boundary conditions. To obtain the solution numerically, one first needs to approximate the unknown function (e.g., temperature) at any location in the domain. There are several approaches for constructing the meshless approximation functions as will be discussed in the following sections.

2.1.

Weighted Least-squares Approximations

Assume we have a 2-D domain and denote the unknown function as u(x, y). In a weighted moving least-squares (MLS) approximation [41], the unknown function can be approximated by u a (x, y) =

m

a j (x, y) p j (x, y)

(4)

j =1

z

approximated unknown funtion x

weighting function

y support domain

Figure 2. Meshless approximation.

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where a j (x, y) are the unknown coefficients, p j (x, y) are the basis functions and m is the number of basis functions. Polynomials are often used as the basis functions. For example, typical 2-D basis functions are given by linear basis: p(x, y) = [1 x y]T qudratic basis: p(x, y) = [1 x y x 2 x y y 2 ]T cubic basis: p(x, y) = [1 x y x 2 x y y 2 x 3 x 2 y x y 2 y 3 ]T

m=3 m=6 m = 10

(5)

The basic idea in weighted least-squares method is to minimize the weighted error between the approximation and the exact function. The weighted error is defined as E(u) =

NP i=1

=

NP

wi (x, y) u a (xi , yi ) − u i

2

2 m wi (x, y) a j (x, y) p j (xi , yi ) − u i

i=1

(6)

j =1

where NP is the number of points, wi (x, y) is the weighting function centered at the point (x, y) and evaluated at the point (xi , yi ). If the weighting function is a constant, the weighted least-squares approach reduces to the classical least-squares approach. The weighting function is used in meshless methods for two reasons: first is to assign the relative importance of the error as a function of distance from the point (x, y); second, by choosing weighting functions whose value will vanish outside certain region, the approximation becomes local. The region where a weighting function has a non-zero value is called a support, a cloud or a domain of influence. The center point (x, y) is called a star point. As shown in Fig. 2, a typical weighting function is bellshaped. Several popular weighting functions used in meshless methods are listed below [1, 17, 42]: 2 3 2/3 − 4r + 4r

r ≤ 1/2 4/3 − 4r + 4r 2 − 4/3r 3 1/2 ≤ r ≤ 1 r >1 0 2 3 4 1 − 6r + 8r − 3r r ≤1 quartic spline: wi (r ) = r >1 0 2 2 e−(r/c) − e−(rmax /c) 0 ≤ r ≤ rmax Gaussian: wi (r) = 1 − e−(rmax /c)2 wi (r) 0 ≤ r ≤ rmax Modified Gaussian: wi (r) = 1 − wi (r) +

cubic spline:

wi (r) =

(7)

where r = r/rmax , r is the distance from the point (x, y) to the point (xi , yi ), i.e., r = |x − xi | = (x − xi )2 + (y − yi )2 and rmax is the radius of the support and c is called the dilation parameter which controls the sharpness of the weighting function. Typical value of c is between rmax /2 and rmax /3. The shape

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of the support, which defines the region where the weighting function is nonzero, can be arbitrary. The parameter in the modified Gaussian weighting is a small number to prevent the weighting function from being singular at the center. Multidimensional weighting functions can be constructed as products of one-dimensional weighting functions. For example, it is possible to define the 2-D weighting function as the product of two 1-D weighting functions in each direction, i.e., wi (x, y) = w(x − xi , y − yi ) = w(x − xi )w(y − yi )

(8)

In this case, the shape of the support/cloud is rectangular. The support size of the weighing function associated with a node i is selected to satisfy the following considerations [43]: 1. The support size should be large enough to cover a sufficient number of points and these points should occupy all the four quadrants of the star point (for boundary star points, the quadrants outside the domain are not considered). 2. The support size should be small enough to provide adequate local character to the approximation. Algorithm 1 gives a procedure for determining the support size for a given point i. Note that several other algorithms [8, 42, 44] are available for determining the support size. However, determining an “optimal” support size for a set of scattered points in meshless methods is still an open research topic. Algorithm 1 The implementation of determining support size rmax for a given point i 1: Select the nearest N E points in the domain (N E is typically several times of m). 2: For each selected point (x j , y j ), j = 1, 2, . . . , N E , compute the distance

3: 4: 5: 6:

from the point i, ρi j = (xi − x j )2 + (yi − y j )2 . Sort nodes in order of increasing ρi j and designate the first m nodes of the sort to a list. Draw a ray from the point i to each of the node in the list. If the angle between any two consecutive rays is greater than 90o , add the next node from the sort to the list and go to 4, if not, go to 6. Set rmax = Max(ρi j ) and multiply rmax by a scaling factor αs . The value of the scaling factor is provided by user.

Once the weighting function is selected, the unknown coefficients are computed by minimizing the weighted error (Eq. (6)) ∂E =0 ∂a j

j = 1, 2, . . . , m

(9)

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For a point (x, y), Eq. (9) leads to a linear system, which in matrix form is BW B T a = BW u

(10)

where a is the m × 1 coefficient vector, u is an NP × 1 unknown vector, B is an m × NP matrix,

B=

p1 (x1 , y1 ) p2 (x1 , y1 ) .. .

p1 (x2 , y2 ) p2 (x2 , y2 ) .. .

pm (x1 , y1 )

··· ··· .. .

pm (x2 , y2 ) · · ·

p1 (x NP , y NP ) p2 (x NP , y NP ) .. .

W =

0 .. .

0 ··· w(x − x2 , y − y2 ) · · · .. .. . .

0

0

···

,

(11)

pm (x NP , y NP )

W is an NP × NP diagonal matrix defined as w(x − x1 , y−y ) 1

0 .. . w(x − x NP ,

0

(12)

y − y NP )

Rewriting M(x, y) = BW B T

(13)

C(x, y) = BW

(14)

and where the matrix M(x, y) of size m × m is called the moment matrix and from Eqs. (10), (13), and (14), the unknown coefficients can be written as a = M −1 Cu

(15)

Therefore, the approximation of the unknown function is given by u a (x, y) = pT (M −1 C)u

(16)

One can write Eq. (16) in short form as u a (x, y) = N(x, y)u =

NP

Ni (x, y)u i

(17)

i=1

Note that typically u i =/ u a (xi , yi ). In the moving least-squares method, the unknown coefficients a(x, y) are functions of (x, y). The approximation of the first derivatives of the unknown function is given by

T −1 (M −1 C) + pT (M −1 C ,k ) u u a,k (x, y) = p,k ,k C + M

= N ,k (x, y)u

(18)

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where k = 1 is the x-derivative or k = 2 is the y-derivative. One alternative to the moving least-squares approximation is the fixed least-squares (FLS) approximation [10, 13]. In FLS, the unknown function u(x, y) is approximated by u a (x, y) =

m

a j p j (x, y)

(19)

j =1

Note that a j in Eq. (19) is not a function of (x, y), i.e., the coefficients a j , j = 1, 2, . . . , m are constants for a given support or cloud. The weighting matrix W in the fixed least-squares approximation is

w(x K − x1 , y −y ) 1 K

W =

0 .. . 0

0 ··· w(x K − x2 , y K − y2 ) · · · .. .. . . 0

···

0 .. . w(x K − x NP ,

0

(20)

y K − y NP )

where (x K , y K ) is the center of the weighting function. Note that (x K , y K ) can be arbitrary and consequently the interpolation functions can be multivalued (see [18] for details). A unique set of interpolation functions can be constructed by fixing (x K , y K ) at the center point (x, y), i.e., when computing Ni (x, y), i = 1, 2, . . . , NP and its derivatives, the center of the weighting function is always fixed at (x, y). Therefore, it is clear that the moment matrix M and matrix C are not functions of (x, y) and the derivatives of the function are given by T u a,k (x, y) = p,k (M −1 C)u

k ∈ {1, 2}

(21)

Comparing Eqs. (18) and (21), it is easily shown that the cost of computing the derivatives in FLS is much less than that in MLS. However, it is reported in literature [6] that the approximated derivatives obtained from FLS may be less accurate. Algorithm 2 gives the procedure for computing the moving leastsquares approximation. In Algorithm 2, N C is the number of points in a cloud.

2.2.

Kernel Approximations

Consider again an arbitary 2-D domain, as shown in Fig. (2), and assume the domain is discretized into NP points or nodes. Then, for each node an approximation function is generated by constructing a cloud about that node (also referred to as a star node). A support/cloud is constructed by centering a kernel (i.e., the weighting function in the case of weighted least-squares

Meshless methods for numerical solution

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Algorithm 2 The implementation of moving least-squares approximation 1: Discretize the domain into NP points to cover the entire domain and its boundary . 2: for each point in the domain, (x j , y j ), do 3: Center the weighting function at the point. 4: Search the nearby domain and determine the support size to get N C points in the cloud by using Algorithm 1. 5: Compute the matrices M, C and their derivatives. 6: Compute the approximation function Ni (x j , y j ), i = 1, 2, . . . , N C and its derivatives by using Eqs. (16,18). 7: end for approximation) about the star point. The kernel is non-zero at the star point and at few other nodes that are in the vicinity of the star point. Two types of the kernel approximations can be considered: the reproducing kernel [15] and the fixed kernel [18]. In a 2-D reproducing kernel approach, the approximation u a (x, y) to the unknown function u(x, y) is given by u (x, y) =

a

C (x, y, s, t)w(x − s, y − t)u(s, t)ds dt

(22)

where w is the kernel function centered at (x, y). Typical kernel functions are given by Eq. (7). C (x, y, s, t) is the correction function which is given by C (x, y, s, t) = pT (x − s, y − t)c(x, y)

(23)

pT ={p1 , p2 , . . . , pm } is an m ×1 vector of basis functions. In two dimensions, a quadratic polynomial basis vector is given by

pT = 1, x − s, y − t, (x − s)2 , (x − s)(y − t), (y − t)2

m = 6 (24)

c(x, y) is an m × 1 vector of unknown correction function coefficients. The correction function coefficients are computed by satisfying the consistency conditions, i.e.,

pT (x − s, y − t)c(x, y)w(x − s, y − t) pi (s, t)ds dt = pi (x, y)

i = 1, 2, . . . , m

(25)

In discrete form, Eq. (25) can be written as NP

pT (x − x I , y − y I )c(x, y)w(x − x I , y − y I ) pi (x I , y I )VI

I =1

= pi (x, y)

i = 1, 2, . . . , m

(26)

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where NP is the number of points in the domain and VI is the nodal volume of node I . Typically a unit nodal volume of the nodes is assumed (see [18] for a discussion on nodal volumes). Equation (26) can be written in a matrix form as M c(x, y) = p(x, y)

(27)

where M is the m × m moment matrix and is a function of (x, y). The entries in the moment matrix are given by Mij =

NP

p j (x − x I , y − y I )w(x − x I , y − y I ) pi (x I , y I )VI

(28)

I =1

From Eq. (27), the unknown correction function coefficients are computed as c(x, y) = M −1 (x, y) p(x, y)

(29)

Substituting the correction function coefficients into Eq. (23) and employing a discrete approximation for Eq. (22), we obtain u a (x, y) =

NP

pT (x, y)M −T (x, y) p(x − x I , y − y I )

I =1

×w(x − x I , y − y I )VI uˆ I =

NP

N I (x, y)uˆ I

(30)

I =1

where uˆ I is the nodal parameter for node I , and N I (x, y) is the reproducing kernel meshless interpolation function. The first derivatives of the correction function coefficients can be computed from Eq. (27) M ,k (x, y)c(x, y) + M(x, y)c,k (x, y) = p,k (x, y)

(31)

c,k = M −1 ( p,k − M ,k c)

(32)

where k = 1 (for x-derivative) or k = 2 (for y-derivative). Thus, the first derivatives of the approximation can be written as

u a (x, y)

,k

= =

NP

(cT ),k pw + cT p,k w + cT pw,k VI uˆ I

I =1 NP

N I,k (x, y)uˆ I

(33)

I =1

Similarly, the second derivatives of the correction function coefficients are given by M ,mn (x, y)c(x, y) + M ,m (x, y)c,n (x, y) + M ,n (x, y)c,m (x, y) + M(x, y)c,mn (x, y) = p,mn (x, y)

(34)

c,mn = M −1 ( p,mn − M ,mn c − M ,m c,n − M ,n c,m )

(35)

Meshless methods for numerical solution

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where m, n = x or y, and

u a (x, y)

,mn

=

NP

(cT ),mn pw + cT p,mn w + cT pw,mn + (cT ),m p,n w

I =1

+ (cT ),m pw,n + (cT ),n pw,m + (cT ),n p,m w

+ cT p,m w,n + cT p,n w,m VI uˆ I =

NP

N I,mn (x, y)uˆ I

(36)

I =1

The other major type of the kernel approximation is the fixed-kernel approximation. In a fixed-kernel approximation, the unknown function u(x, y) is approximated by

u (x, y) = C (x, y, x K − s, y K − t)w(x K − s, y K − t)u(s, t)ds dt (37) a

Note that in the fixed-kernel approximation, the center of the kernel is fixed at (x K , y K ) for a given cloud. Following the same procedure as in the reproducing kernel approximation, one can obtain the discrete form of the fixed kernel approximation u a (x, y) =

NP

pT (x, y)M −T (x K , y K ) p(x K − x I , y K − y I )

I =1

× w(x K − x I , y K − y I )VI uˆ I =

NP

N I (x, y)uˆ I

(38)

I =1

Since (x K , y K ) can be arbitrary in Eq. (38), the interpolation functions obtained by Eq. (38) are multivalued. A unique set of interpolation functions can be constructed by computing N I (x K , y K ), I = 1, 2, . . . , NP, when the kernel is centered at (x K , y K ) (see [18] for more details). Equation (38) shows that only the leading polynomial basis vector is a function of (x, y). Therefore, the derivatives of the interpolation functions can be computed simply by differentiating the polynomial basis vector in Eq. (38). For example, the first and second x derivatives are computed as:

N I ,x (x, y) = 0 1 0 2x y 0 M −T p(x K − x I , y K − y I ) ×w(x K − x I , y K − y I )VI

(39)

N I ,x x (x, y) = [0 0 0 2 0 0] M −T p(x K − x I , y K − y I ) ×w(x K − x I , y K − y I )VI

(40)

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It has been proved in [26] that, if the nodal volume is taken to be 1 for each node, the reproducing kernel approximation is mathematically equivalent to the moving least-squares approximation, and the fixed kernel approximation is equivalent to the fixed least-squares approximation. The algorithm to construct the approximation functions by using the fixed-kernel approximation method is given by Algorithm 3 The implementation of fixed-kernel approximation 1: Allocate NP points to cover the domain and its boundary . 2: for each point in the domain, (x j , y j ), do 3: Center the weighting function at the point. 4: Determine the support size to get N C points in the cloud by using Algorithm 1. 5: Compute the moment matrix M and the basis vector p(x, y). 6: Solve M c = p 7: Compute the approximation function N I (x j , y j ) I = 1, 2, . . . , N C and its derivatives by using Eqs. (38)–(40). 8: end for

2.3.

Radial Basis Approximation

In a radial basis meshless approximation, the approximation of an unknown function u(x, y) is written as a linear combination of NP radial functions [19], u a (x, y) =

NP

α j φ(x, y, x j , y j )

(41)

j =1

where NP is the number of points in the domain, φ is the radial basis function and α j , j = 1, 2, . . . , NP are the unknown coefficients. The unknown coefficients α1 , . . . , α NP can be computed by solving the governing equation by using either a collocation or a Galerkin method, which we will discuss in the following sections. The partial derivatives of the approximation function in a multidimensional space can be calculated as NP ∂ k φ(x, y, x j , y j ) ∂ k u a (x, y) = α j ∂ x a ∂ yb ∂ x a ∂ yb j =1

(42)

where a, b ∈ 0, 1, 2 and k = a + b. The multiquadrics [19–21] and thin-plate spline functions [45] are among the most popular radial basis functions. The multiquadrics radial basis function is given by φ(x, y, x j , y j ) = (x, x j ) = (r j ) = (r 2j + c2j )0.5

(43)

Meshless methods for numerical solution

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where r j = ||x − x j || is the Euclidian norm and c j is a constant. The value of c controls the shape of the basis function. The reciprocal multiquadrics radial basis function has the form (r) =

1 (r 2 + c2 )0.5

(44)

The thin-plate spline radial basis function is given by (r) = r 2m log r

(45)

where m is the order of the thin-plate spline. To avoid the singularity of the interpolation system, a polynomial function is often added to the approximation Eq. (41) [46]. The modified approximation is given by u a (x, y) =

NP

α j φ(x, y, x j , y j ) +

j =1

m

βi pi (x, y)

(46)

i=1

along with m additional constraints NP

α j pi (x j , y j ) = 0 i = 1, . . . , m

(47)

j =1

where βi , i = 1, 2, . . . , m are the unknown coefficients and p(x) are the polynomial basis functions as defined in Eq. (5). Equations (46) and (47) lead to a positive definite linear system which is gauranteed to be nonsingular. The radial basis function approximation shown above is global since the radial basis function are non-zero everywhere in the domain. It is required to solve a dense linear system to solve the unknown coefficients. The computational cost could be very high when the domain contains a large number of points. Recently, compactly supported radial basis functions have been proposed and applied to solve PDEs with largely reduced computational cost. For more details on compactly supported RBFs, please refer to [23].

3.

Discretization

As shown in Eqs. (17), (18), (30), (33), (36), (38) and (41), although each approximation method has a different way of computing the approximation functions, all the methods presented in previous sections represent u(x, y) in the same general form as u (x, y) = a

NP I =1

N I (x, y)uˆ I

(48)

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and the approximation of the derivtaives can also be written in the general form given by NP k ∂ N I (x, y) ∂ k u a (x, y) = uˆ I a b ∂x ∂y ∂ x a ∂ yb I =1

(49)

where a, b ∈ 0, 1, 2 and k = a + b. After the approximation functions are constructed, the next step is to compute the unknown coefficients in Eq. (48) by discretizing the governing equations. The meshless discretization techniques can be broadly classified into three categories: (1) point collocation; (2) cell integration and (3) local domain integration.

3.1.

Point Collocation

Point collocation is the simplest and the easiest way to discretize the governing equations. In a point collocation approach, the governing equations for a physical problem can be written in the following general form L (u(x, y)) = f (x, y) in G (u(x, y)) = g(x, y) on g H (u(x, y)) = h(x, y) on h

(50) (51) (52)

where is the domain, g is the portion of the boundary on which Dirichlet boundary conditions are specified, h is the portion of the boundary on which Neumann boundary conditions are specified and L , G and H are the differential, Dirichlet and Neumann operators, respectively. The boundary of the domain is given by = g ∪ h . After the meshless approximation functions are constructed, for each interior node, the point collocation technique simply substitutes the approximated unknown into the governing equations. For nodes with prescribed boundary conditions the approximate solution or the derivative of the approximate solution are substituted into the given Dirichlet and Neumann-type boundary conditions, respectively. Therefore, the discretized governing equations are given by L (u a ) = f (x, y) for points in G (u a ) = g(x, y) for points on g H (u a ) = h(x, y) for points on h

(53) (54) (55)

The point collocation approach gives rise to a linear system of equations of the form, K uˆ = F

(56)

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The solution of Eq. (56) provides the nodal parameters at the nodes. Once the nodal parameters are computed, the unknown solution at each node can be computed from Eq. (48). Let’s revisit the heat condution problem presented in Section 2 as an example. The governing equation is the steady-state heat conduction along with the appropriate boundary conditions stated in Eqs. (1)–(3). As shown in Fig. 3(a), the points are distributed over the domain and the boundary. Using the meshless approximation functions, the nodal temperature can be expressed by Eq. (48). If a node i is an interior node, the governing equation is satisfied, i.e.,

∇2

NP

N I (xi , yi )uˆ I

I =1

=

NP

(∇ 2 N I (xi , yi ))uˆ I = 0

(57)

I =1

If a node j is a boundary node with a Dirichlet boundary condition, we have NP

N I (x j , y j )uˆ I = g(x j , y j )

(58)

I =1

and if a node q is a boundary node with a Neumann boundary condition (heat flux at the boundary) ∂(

NP I =1

NP N I (xq , yq )uˆ I ) ∂(N I (xq , yq )) = uˆ I = h(xq , yq ) ∂n ∂n I =1

(59)

Assuming that there are ni interior points, nd Dirichlet boundary points, and nn Neumann boundary nodes (NP = ni + nd + nn) in the domain, the final

(a)

Governing equation

(b)

(c) background cells Ωs Γs

Dirichlet boundary condition

Neumann boundary condition

Ls Γsq

Figure 3. Meshlessdiscretization: (a) point collocation. (b) cell integration. (c) local domain integration.

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linear system takes the form

∇ 2 N1 (x1 ) ∇ 2 N1 (x2 ) . ..

∇ 2 N (x ) 1 ni N1 (xni+1 ) . . . N1 (xni+nd ) ∂(N (x 1 ni+nd+1 ) ∂n . .. ∂(N (x )) 1

∂n

NP

∇ 2 N2 (x1 ) ∇ 2 N2 (x2 ) .. .

∇ 2 N2 (xni ) N2 (xni+1 ) .. .

··· ··· .. . ··· ··· .. .

N2 (xni+nd ) · · · ∂(N2 (xni+nd+1 )) ··· ∂n .. .. . . ∂(N2 (x NP )) ··· ∂n

0 0 . . . 2 ∇ N NP (xni ) u ˆ 0 1 N NP (xni+1 ) g(x u ˆ ) 2 ni+1 .. . = . . . . . . N NP (xni+nd ) ) u ˆ g(x NP ni+nd ∂(N NP (xni+nd+1 )) ∂n ) h(x ni+nd+1 .. . . . . ∂(N NP (x NP ))

∇ 2 N NP (x1 ) ∇ 2 N NP (x2 ) .. .

∂n

h(x NP )

(60) where xni denotes the coordinates of node ni. Equation (60) can be solved ˆ The nodal temperature can be computed by to obtain the nodal parameters u. using Eq. (48). Algorithm 4 summarizes the key steps involved in the implementation of a point collocation method for linear problems. The point collocation steps are the same for nonlinear problems. However, a linear system such as Eq. (60) cannot be directly obtained by substituting the approximated unknown into the governing equation and the boundary conditions. A Newton’s method can be used to solve the discretized nonlinear system (please refer to [47] for detail). The point collocation method provides a simple, efficient and flexible meshless method for interior domain numerical analysis. Many meshless methods, such as the finite point method [10], the finite cloud method [18] and the h–p meshless cloud method [8], employ the point collocation technique to discretize the governing equation. However, there are several issues one needs to pay attention to improve the robustness of the point collocation method: 1. Ensuring the quality of clouds: We have found that, for scattered point distributions, the quality of the clouds is directly related to the numerical error in the solution. When the point distribution is highly scattered, it is likely that certain stability conditions, namely the positivity conditions (see [42] for details), could be violated for certain clouds. For this reason, the modified Gaussian, cubic or quartic inverse distance functions [42] are better choices for the kernel/weighting function in point collocation. In [42], we have proposed quantitative criteria to measure the cloud quality and approaches to ensure the satisfaction of the positivity conditions for 1-D and 2-D problems. However, for really bad point distributions, it could be difficult to satisfy the positivity conditions and modification of the point distribution may be necessary.

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Algorithm 4 Implementation of a point collocation technique for numerical solutions of PDEs 1: Compute the meshless approximations for the unknown solution 2: for each point in the domain do 3: if the node is in the interior of the domain then 4: substitute the approximation of the solution into the governing equation 5: else if the node is on the Dirichlet boundary then 6: substitute the approximation of the solution into the Dirichlet boundary condition 7: else if the node is on the Neumann boundary then 8: substitute the approximation of the solution into the Neumann boundary condition 9: end if 10: assemble the corresponding row of Eq. (60) 11: end for 12: Solve Eq. (60) to obtain the nodal parameters 13: Compute the solution by using Eq. (48)

2. Improving the accuracy for high aspect-ratio clouds: Like the conventional finite difference and finite element methods, large error could occur with the collocation meshless methods when the point distribution has a high aspect ratio (i.e. anisotropic cloud). Further investigation is needed to deal with the high aspect ratio problem.

3.2.

Cell Integration

Another approach to discretize the governing equation is the Galerkin method. The Galerkin approach is based on the weak form of the governing equations. The weak form can be obtained by minimizing the weighted residual of the governing equation. For the heat condution problem, a weak form of the governing equation can be written as

w ∇ u d + 2

v (u − g(x, y)) d = 0

(61)

u

where w and v are the test functions for the governing equation and the Dirichlet boundary condition, respectively. Note that the second integral in Eq. (61) is used to enforce the Dirichlet boundary condition. By applying the

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divergence theorem and imposing the natural boundary condition, Eq. (61) can b written as u

∂u wd + ∂n

q

∂u wd − ∂n

u ,i w,i d +

v (u − g(x, y)) d = 0

u

(62) The approximation for the unknown function is given by the meshless approximation (Eq. (48)) and the normal derivative of the unknown function can be computed by !

NP ∂ NI ∂ NI ∂u a = nx + n y uˆ I ∂n ∂x ∂y I =1

(63)

Denoting ∂ NI ∂ NI nx + ny ∂x ∂y

I =

(64)

The normal derivative of the unknown function can be rewritten as N ∂u = I uˆ I ∂n I =1

(65)

We choose the test functions w and v by w= v=

NP I =1 NP

N I uˆ I

(66)

I uˆ I

(67)

I =1

Subtituting the approximations into the weak form, we obtain NP I =1

" NP

uˆ I

N I,i N J,i duˆ J −

J =1

=

NP I =1

uˆ I

" q

NP J =1

u

N I h(x, y) d −

N I J d uˆ J − u

NP J =1

#

I g(x, y) d

#

I N J d uˆ J

u

(68)

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Equation (68) can be simplified as NP N I,i N J,i d − N I J d − I N J d uˆ J J =1

=

u

N I h(x, y) d −

q

u

I g(x, y) d

(69)

u

In matrix form

K − G − G T uˆ = h − g

(70)

where the entries of the coefficient matrix and the right hand side vector are given by

K IJ =

N I,i N J,i d

(71)

N I J d

(72)

N I h(x, y)d

(73)

I g(x, y)d

(74)

G IJ = u

hI = q

gI = u

As shown in Eqs. (71)–(74), the entries in the matrices and the right hand side vector are integrals over the domain or over the boundary. Since there is no mesh available to compute the various integrals, one approach is to use a background cell structure as shown in Fig. 3(b). The integrations are computed by appropriately summing over the cells and using Gauss quadrature in each cell. The implementation of cell integration is summarized in Algorithm 5. In a cell integration approach, the approximation order is reduced, i.e., for a second order PDE, there is no need to compute the second derivatives of the approximation functions. However, the cell integration approach requires background cells and the treatment of the boundary cells is not straightforward. Element-free Galerkin method [6], partition of unity finite element method [12], diffuse element method [5] and reproducing kernel particle method [15] are among the meshless methods using cell integration technique for discretizating the governing equation.

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Algorithm 5 Implementation of cell integration technique [48] 1: Compute the meshless approximations 2: Generate the background cells which cover the domain. 3: for each cell C i do 4: for each quadrature points x Q in the cell do 5: if the quadrature point is inside the physical domain then 6: Check all nodes in the cell Ci and surrounding cells to determine the nodes x I in the domain of influence of x Q 7: if x I − x Q does not intersect the boundary segment then 8: Compute the N I (x Q ) and N I,i (x Q ) at the quadrature point. 9: Evaluate contributions to the integrals. 10: Assemble contributions to the coefficient matrix. 11: end if 12: end if 13: end for 14: end for 15: Solve Eq. (77) to obtain the nodal parameters 16: Compute the solution by using Eq. (48)

3.3.

Local Domain Integration

Another method for discretizing the governing equation is based on the concept of local domain integration [9]. In the local domain integration method, the global domain is covered by local subdomains, as shown in Fig. 3(c). The local domains can be of arbitrary shape (typically circles or squares are convenient for integration) and can overlap with each other. In the heat conduction example, for a given node, a generalized local weak form over the node’s subdomain s can be written as s

v ∇ u d − α 2

v u − u b d = 0

(75)

su

where su = ∂s ∩ u is the intersection of the boundary of s and the global Dirichlet boundary. For nodes near or on the global boundary, ∂s = s + Łs . s is a part of the local domain boundary which is also located on the global boundary. Łs is the remaining part of the local boundary which is inside the global domain. α 1 is a penalty parameter used to impose the Dirichlet boundary conditions.

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By applying the divergence theorem and imposing the Neumann boundary condition, for any local domain s , we obtain the local weak form

∂u v d + ∂n

Ls

−

su

∂u v d + ∂n

u ,k v ,k d − α

s

h(x, y)v d sq

v (u − g(x, y)) d = 0

(76)

su

in which sq is the intersection of the boundary of s and the global Neumann boundary. For a sub-domain located entirely within the global domain, there is no intersection between ∂s and , the integrals over su and sq vanish. In order to simplify the above equation, one can deliberately select a test function v such that it vanishes over ∂s . This can be easily accomplished by using the weighting function in the meshless approximations as also the test function, with the support of the weighting function set to be the size of the corresponding local domain s . In this way, the test function vanishes on the boundary of the local domain. By substituting the test function and the meshless approximation of the unknown (Eq. (48)) into the local domain weak form (Eq. (76)), we obain the matrix form K uˆ = f

(77)

where

Ki j = si

N j,k v i,k d + α

sui

N j v i d −

N j,n v i d

(78)

sui

and

fi = sqi

h(x, y)v i d + α

g(x, y)v i d

(79)

sui

where si , sui and sqi are the domain and boundary for the local domain i. The integrations in Eqs. (78) and (79) can be computed within each local domain by using Gauss quadrature. The implementation of the local integration can be carried out as summarized in Algorithm 6. Meshless methods based on local domain integration include the meshless local Petrov–Galerkin method [9] and the method of finite spheres [24].

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Algorithm 6 Implementation of the local domain integration technique 1: Compute the meshless approximations for the unknown solution 2: for each node (x i , yi ) do 3: Determine the local sub-domain s and its corresponding local boundary ∂s 4: Determine Gaussian quadrature points x Q in s and on ∂s 5: for each quadrature points x Q in the local domain do 6: Compute the Ni (x Q ) and Ni, j (x Q ) at the quadrature point x Q . 7: Evaluate contributions to the integrals. 8: Assemble contributions to the coefficient matrix. 9: end for 10: end for 11: Solve Eq. (77) to obtain the nodal parameters 12: Compute the solution by using Eq. (48)

4.

Summary of Meshless Methods

In this paper, we have introduced several approaches to construct the meshless approximations and three approaches to discretize the governing equations. Many meshless methods published in the literature can be viewed as different combinations of the approximation and discretization approaches introduced in the previous sections. Table 1 lists the popular methods with their approximation and discretization components.

Table 1. The catagory of meshless methods Point collocation

Cell integration Galerkin

Local domain integration Galerkin

Moving leastSquares

Finite point method [10]

Element-free Galerkin method [6], partition of unity finite element method [12]

Meshless local Petrov-Galerkin method [3], method of finite spheres [24]

Fixed leastsquares

Geleralized finite difference method [7] h − p meshless cloud method [8], finite point method [10]

Diffuse element method [5]

Reproducing

Finite cloud method [18]

Repeoducing kernel

kernel

particle method [15]

Fixed kernel

Finite cloud method [18]

Radial basis

Many

Many

Meshless methods for numerical solution

5.

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Example: Finite Cloud Method for Solving Linear Elasticity Problems

As shown in Fig. 4, an elastic plate containing three holes and a notch is subjected to a uniform pressure at its right edge [49]. We solve this problem by using the finite cloud method to demonstrate the effectiveness of the meshless method. To show the accuracy of the solution, the problem is solved by both the finite element method by using ANSYS and the finite cloud method. We construct the FCM discretizations by employing the same set of FEM nodes. For two-dimensional elasticity, there are two unknowns associated with each node in the domain, namely the displacements in the x and y directions. The governing equations assuming zero body force, can be rewritten as the Navier–Cauchy equations of elasticity 1 ∂ ∇ 2u + 1 − 2ν ∂ x ∇ 2v +

∂ 1 1 − 2ν ∂ y

!

∂u ∂v + ∂x ∂y ∂u ∂v + ∂x ∂y

=0 (80)

!

=0

with

ν =

ν

ν 1+ν

for plane strain (81) for plane stress

where ν is the Poisson’s ratio. In this paper we consider the plane stress situation. In the finite cloud method, the first step is to construct the fixed kernel approximation for the displacements u and v by using Algorithm 3. In this example, the cloud size is set for each node to cover 25 neighboring nodes. 200

100

150 q

75

30

5

100

30

75

5

E 20

250 120

75 55

100

95

Thickness 1

115

Figure 4. Plate with holes.

υ 0.3 q 1.0

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The 2-D version of the modified Gaussian weighting function (Eq. (7)) is used as the kernel. After the approximation funcitons are computed, a point collocation approach is used for discretizing the governing equation and the boundary conditions by using Algorithm 4 to obtain the solution of the displacements. Figure 5 shows the deformed shape obtained by the FEM code ANSYS. The FEM mesh consists of 4474 nodes. All the 4474 ANSYS nodes are taken as the points in the FCM simulation. The deformed shapes obtained by FCM are shown in Fig. 6. The results obtained from the FEM and FCM agree with each other quite well and the difference of the maximum displacement is within 1%. Figure 7 shows a quantitative comparison of the computed σx x stress on the surfaces of the holes obtained from the two methods. The results

FEM solution

Figure 5. Deformed shape obtained by the finite element method.

400 FCM solution 350 300 250 200 150 100 50 0 0

100

200

300

400

500

Figure 6. Deformed shapeobtained by the finite cloud method.

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5 FEM (ANSYS) FCM 4

3

σxx

θ 2

1

0

1

0

10

20

30

40

50

60

70

80

90

100

θ (degree)

Figure 7. Results comparionfor σx x at the lower left circular boundary.

show very good agreement and demonstrate that the FCM approach provides accurate results for problems with complex geometries.

Remarks: 1. The construction of approximation functions is more expensive in meshless methods compared to the cost associated with construction of interpolation functions in FEM. The integration cost in Galerkin meshless methods is more expensive. Galerkin meshless methods can be a few times slower (typically more than five times) than FEM [25]. 2. Collocation meshless methods are much faster since no numerical integrations are involved. However, they may need more points and their robustness needs to be addressed [42]. 3. Meshless methods introduce a lot of flexibility. One needs to sprinkle only a set of points or nodes covering the computational domain as shown in Fig. 6, with no connectivity information required among the set of points. This property is very appealing because of its potential in adaptive techniques, where a user can simply add more points in a particular region to obtain more accurate results.

References [1] T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl, “Meshless methods: an overview and recent developments,” Comput. Methods Appl. Mech. Engrg., 139, 3–47, 1996.

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8.4 LATTICE BOLTZMANN METHODS FOR MULTISCALE FLUID PROBLEMS Sauro Succi1, Weinan E2 , and Efthimios Kaxiras3 1

Istituto Applicazioni Calcolo, National Research Council, viale del Policlinico, 137, 00161, Rome, Italy 2 Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA 3 Department of Physics, Harvard University, Cambridge, MA 02138, USA

1.

Introduction

Complex interdisciplinary phenomena, such as drug design, crackpropagation, heterogeneous catalysis, turbulent combustion and many others, raise a growing demand of simulational methods capable of handling the simultaneous interaction of multiple space and time scales. Computational schemes aimed at such type of complex applications often involve multiple levels of physical and mathematical description, and are consequently referred to as to multiphysics methods [1–3]. The opportunity for multiphysics methods arises whenever single-level methods, say molecular dynamics and partial differential equations of continuum mechanics, expand their range of scales to the point where overlap becomes possible. In order to realize this multiphysics potential specific efforts must be directed towards the development of robust and efficient interfaces dealing with “hand-shaking” regions where the exchange of information between the different schemes takes place. Two-level schemes combing atomistic and continuum methods for crack propagation in solids or strong shock fronts in rarefied gases have made their appearance in the early 90s. More recently, three-level schemes for crack dynamics, combining finite-element treatment of continuum mechanics far away from the crack with molecular dynamics treatment of atomic motion in the near-crack region and a quantum mechanical description of bond-snapping in the crack tip have been demonstrated. These methods represent concrete instances of composite algorithms which put in place seamless interfaces between the different mathematical models associated with different physical levels of description, say continuum and atomistic. An alternative approach is to explore methods 2475 S. Yip (ed.), Handbook of Materials Modeling, 2475–2486. c 2005 Springer. Printed in the Netherlands.

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that can host multiple levels of description, say atomistic, kinetic, and fluid, within the same mathematical framework. A potential candidate is the lattice Boltzmann equation (LBE) method. The LBE is a minimal form of Boltzmann kinetic equation in which all details of molecular motion are removed except those that are strictly needed to recover hydrodynamic behavior at the macroscopic scale (mass-momentum and energy conservation) [4, 5]. The result is an elegant and simple equation for the discrete distribution function f i ( x , t) describing the probability to find a particle at lattice site x at time t with speed v. LBE has potential to combine the power of continuum methods with the geometrical flexibility of atomistic methods. However, as multidisciplinary problems of increasing complexity are tackled, it is evident that significant upgrades are called for, both in terms of extending the range of scales accessible by LBE itself and in terms of coupling LBE downwards/upwards with micro/macroscopic methods. In the sequel, we shall offer a cursory view of both these research directions. Before proceeding further, a short review of the basic ideas behind LBE theory is in order.

2.

Lattice Boltzmann Scheme: Basic Theory

The lattice Boltzmann equation is based on the idea of moving pseudoparticles along prescribed directions on a discrete lattice (the discrete particle speeds define the lattice connectivity). At each lattice site, these pseudoparticles undergo collisional events designed in such a way as to conserve the basic mass, momentum and energy principles which lie at the heart of fluid behavior. Historically, LBE was generated in response to the major problems of its ancestor, the lattice gas cellular automaton, namely statistical noise, high viscosity, and exponential complexity of the collision operator with increasing number of speeds [6, 7]. A few years later, its mathematical connections with model kinetic equations of continuum theory have also been clarified [8]. The most popular, although not necessarily the most efficient, form of lattice Boltzmann equation (Lattice BGK, for Bhatnagar, Gross, Krook) reads as follows [9]

x + ci t, t + t) − f i ( x , t) − ωt f i − f ie ( x , t) + Fi t, f i ( →

(1)

where f i ( x , t) = f ( x , v = ci , t), i = 1,b, is the discrete one-body distribution function moving along the lattice direction defined by discrete speed ci. At the left hand side, we recognize the streaming operator of the Boltzmann equation, ∂t f + v · ∇ f, advanced in discrete time from t to t + t, along the characteristics xi = ci t. The right hand side represents the collisional operator in the form of single-time relaxation to the local equilibrium f ie · Finally, the effect of an external force, Fi , is also included. In order to recover fluid-dynamic

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behavior, the set of discrete speeds must guarantee the basic symmetries of fluid equations, namely mass, momentum and energy conservation, as well as rotational invariance. Only a limited subclass of lattices qualifies. A popular choice in three-dimensional space is the nineteen-speed lattice, consisting of one speed-zero (c = 0) particle sitting on the center of the cell, six speed-one (c = 1) particles√connecting to the face centers of the cell, and twelve particles with speed c = 2, connecting the center of the cell with edge centers. The local equilibrium is usually taken in the form of a quadratic expansion of a Maxwellian

uu · ci ci − cs2 I u · ci e , f i = ρωi 1 + 2 + cs 2cs4

(2)

i /ρ the flow speed. Here where ρ = i f i the fluid density, and u = = i fi c 2 cs is the lattice sound speed defined by the condition cs I = i ωi ci ci , where I denotes the unit tensor. Finally, ωi is a set of lattice-dependent weights normalized to unity. For athermal flows, the lattice sound speed is a constant of order one (cs2 = 1/3 for the 19-speed lattice of Fig. 1). Local equilibria obey the following conservation relations (mass made unity for convenience):

f ie = ρ,

(3)

f ie ci = ρ u,

(4)

i

i

f ie ci ci = ρ uu + cs2 I .

i

Figure 1. The D3Q19 lattice.

(5)

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Using linear transport theory, in the limit of long-wavelengths as compared to particle mean free path, (small-Knudsen number) and low fluid speed as compared to the sound speed (low-Mach number), the fluid density and speed are shown to obey the Navier-Stokes equations for a quasi-incompressible fluid (with no external force for simplicity) ∂t ρ + divρ u = 0,

(6)

u + ( u )T + λdiv uI , ∂t ρ u + divρ uu = − ∇ P + div µ(

(7)

where P = pcs2 is the fluid pressure, and µ = ρu is the dynamic viscosity, and λ is the bulk viscosity (this latter term can be neglected to all practical purposes since we deal with quasi-incompressible fluids). Note that, according to the above relation, the LBE fluid obeys an ideal equation of state, as it belongs to a system of molecules with no potential energy. Potential energy effects can be introduced via a self-consistent force Fi , but in this work we shall not deal with such non-ideal gas aspects. The kinematic viscosity of the LBE fluid turns out to be:

ν=

cs2

1 τ − t 2

x 2 . t

(8)

The term τ ≡ 1/ω is the relaxation time around local equilibria, while the factor –1/2 is a genuine lattice effect which stems from second order spatial derivatives in the Taylor expansion of the discrete streaming operator. It is fortunate that such a purely numerical effect can be reabsorbed into a physical (negative) viscosity. In particular, by choosing ωt = 2 − , very small viscosities of order O() (in lattice units) can be achieved, corresponding to the very challenging regime of fluid turbulence [10]. Main assets of LBE are: • • • •

mathematical simplicity; physical flexibility; easy implementation of complex boundary conditions; excellent amenability to parallel processing.

Mathematical simplicity is related to the fact that, at variance with the Navier-Stokes equations in which non-linearity and non-locality are lumped into a single term, u∇ u, in LBE the non-local term (streaming) is linear and the non-linear term (the local equilibrium) is local. This disentangling proves beneficial from both the analytical and computational point of views. Physical flexibility relates to opportunity of accomodating additional physics via generalizations of the local equilibria and/or the external source Fi , such as to include the effects of additional fields interacting with the fluid.

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Easy implementation of complex boundary conditions results from the fact that the most common hydrodynamic boundary conditions, such as prescribed speed at solid boundaries, or prescribed pressure at fluid oulets, can be imposed in terms of elementary mechanical operations on the discrete distributions. However, in the presence of curved boundaries, i.e., boundaries which do not fit into the lattice sites, the boundary procedure may become considerably more involved. This represents one of the most active research topic in the field. It must be pointed out that in addition to fluid density and pressure, LBE also carries along the momentum flux tensor, whose equilibrium part corresponds to the fluid pressure. As a result, LBE does not need to solve the Poisson problem to compute the pressure distribution corresponding to a given flow configuration. This is a significant advantage as compared to explicit finite-difference schemes for incompressible flows. The price to pay is an extra-amount of information as compared to a hydrodynamic approach. For instance, in two dimensions, the most popular LBE requires nine populations (one rest particle, four nearest-neighbors and four next-to-nearest neighbors) to be contrasted with only three hydrodynamic fields (density, two velocity components). On the other hand, since LBE populations always stream “upwind” (from x to x + ci t, only one time level needs to be stored, which saves a factor two over hydrodynamic representations. As per efficiency on parallel computers, the key is again the locality of the collision operator which can be advanced concurrently at each lattice site independently of all others. Owing to these highlights, LBE has been used for more than 10 years for the simulation of a large variety of flows, including flows in porous media, turbulence, and complex flows with phase transitions, to name but a few. Multiscale applications, on the other hand, have appeared only recently, as we shall discuss in the sequel.

3.

Multiscale Lattice Boltzmann

Multiscale versions of LBE were first proposed by Filippova and Haenel [11] in the form of a LBE working on locally embedded grids, namely regular grids in which the lattice spacing is locally refined or coarsened, typically in steps of two for practical purposes. The same option was available since even longer in commercial versions of LB methods [12]. In the sequel, we shall briefly outline the main elements of multiscale LBE theory on locally embedded cartesian grids.

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3.1.

Basics of the Multiscale LB Method

The starting point of multiscale LBE theory is the lattice BGK equation (1). Grid-refinement is performed by introducing an n-times finer grid with spacing: δx =

x t , δt = , n n

The kinematic viscosity on the coarse lattice is given by Eq. (8) from which we see that in order to achieve the same viscosity on both coarse and fine grids, the relaxation parameter in the fine grid has to be rescaled as follows

τn = nτ1 1 −

n − 1 t/2 , n τ1

(9)

where rn and τ1 ≡ τ are the relaxation parameters on the n times-refined and on the original coarse grids, respectively n = 2l after l levels of grid-refinement). Next, we need to set up the interface conditions controlling the exchange of information between the coarse and fine grids. The guiding requirement is the continuity of hydrodynamic quantities (density, flow speed) and of their fluxes. Since hydrodynamic quantities are microscopically conserved, the corresponding interface conditions simply consists in setting the local equilibria in the fine grid equal to those in the coarse one. The fluxes, however, do not correspond to any microscopic invariant, and consequently their continuity implies requirements on the non-equilibrium component of the discrete distribution function. Therefore, the first step of the interface procedure consists in splitting the discrete distribution function into an equilibrium and non-equilibrium components: f i = f ie + f ine .

(10)

Upon expanding the left hand side of the LBE equation (1) to first order in at, the non-equilibrium component reads as

f ine = −τ [∂t + cia ∂a ] f ie + O K n 2 ,

(11)

where the latin index a runs over spatial dimensions and repeated indices are summed upon. This is second-order accurate in the Knudsen number K n = x/L, where L is a typical macroscopic scale of the flow. In the low-frequency limit t/τ ∼ K n 2 , the time derivative can be neglected, and by combining the above relation with continuity of the hydrodynamic variables at the interface between the two grids, one obtains the following scaling relations between the coarse and fine grid populations

f i = F˜ie + F˜i − F˜ie −1 ,

Fi = f ie + ( f i − f ie ) ,

(12) (13)

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where capital means coarse-grid, prime means post-collision, and tilde stands for interpolation from the coarse grid. In the above,

=n

(τ1 − t) · (τn − t)

The basic one-step algorithm reads as follows: 1. Advance (Stream, and Collide) F on the coarse grain grid. 2. For all subcycles k = 0, l, . . . , n − 1 do: a. Interpolate F on the interface coarse-to-fine grid. b. Scale F to f via (12) on the interface coarse-to-fine grid. c. Advance (Stream and Collide) f on the fine-grain grid. 3. Scale back f to F via (13) on the interface of the fine-to-coarse grid. Step 1 applies to all nodes in the coarse grid, bulk and interface, Steps 2a and 2b apply to interface nodes which belong only to the fine grid, Step 2c applies to bulk nodes of the fine grid, and Step 3 applies to interface nodes which belong to both coarse and fine grids. It is noted that becomes singular at τn = t, corresponding to n = (t/2)/(τ1 − t/2) = cs2 t/2ν (see Eq. (8)). For high-Reynolds applications, in which v is of the order of ∼10−3 or less (in units of the original lattice), the above singularity is of no practical concern, for it would be met only after hundred levels refinement. For low-Reynolds flow applications, however, this flaw needs to be cured. To this purpose, a more general approach that avoids the singularity has been recently developed by Dupuis [13]. These authors show that by defining the scale transformations between the coarse and fine grain populations before they collide, the singularity disappears ( = n(τ1 )/(τn )). In practice, this means that, at variance with Filippova’s model, the collision operator is applied also to the interface nodes which belong to the fine grid only.

4.

Multiscale LBE Applications

To date, Multiscale LBEs have been applied mainly to macroscopic turbulent flows [14, 15]. Here, however, we focus our attention to microscale problems of more direct relevance to material science applications.

4.1.

Microscale Flows with Chemical Reactions

The LBE couples easily to finite difference/volume methods for continuum parial differential equations. Distinctive features of LBE in this context

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are: (1) Use of very-small time-steps, (2) Geometrical flexibility. Item (1) refers to the fact that since LBE is an explicit method ticking at the particle speed, not the fluid one, it advances in much smaller time-steps than usual fluid-dynamics methods, typically a factor ten. (The flip side, is that a large number of time-step is required in long-time evolutions.) As an example, take a millimetric flow with, say 100 grid points per side, yielding a mesh spacing dx =10 µm. Assuming a sound speed of the order of 300 m/s, we obtain a timestep of the order of dt= 30 ns. Such a small time-step permits to handle relatively fast reactions without going to implicit time stepping, thus avoiding the solution of large systems of algebraic equations. Item (2) is especially suited to heterogeneous catalysis since the simplicity of particle trajectories permits to describe fairly irregular geometries and boundary conditions. Because of these two points, LBE is currently being used to simulate reactive flows over microscopically corrugated surfaces, an application of great interest for the design of chemical traps, catalytic converters and related devices [16, 17] (Fig. 2). These problems are genuinely multiphysics, since they involve a series of hydrodynamic and chemical time-scales. The major control parameters are the Reynolds number Re = U d/ν, the Peclet number Pe = Ud/D, and the Damkohler number Da = d 2 /Dτc . In the above, U and d are typical flow speed and size, D is the mass diffusivity of the chemical species and τc is a typical chemical reaction time-scale. Depending on various physical and geometrical parameters, a wide separation of these time-scales can arise. In general, the LBE time-step is sufficiently small to resolve all the relevant time-scales.

Figure 2. A multiscale computation of a flow in a microscopic restriction of a catalytic converter. Local flow gradients may lead to significant enhancements of the fluid-wall mass transfer, with corresponding effects on the chemical reactivity of the device. Note that three levels of refinement are used.

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Whenever faster time-scales develop, e.g., fast chemical reactions, the chemical processes are sub-cycled, i.e., advanced in multiple steps each with the smallest time-scale, until completion of a single LBE step [18].

4.2.

Nanoscale Flows

When the size of the micro/nanoscopic flow becomes comparable to the molecular mean free path, the Knudsen number is no longer small, and the whole fluid picture becomes questionable. A fundamental question then arises as to whether LBE can be more than a “Navier-Stokes solver in disguise”, namely capture genuinely kinetic information not available at the fluid-dynamic level. Mathematically, this possibility stems from the fact that – as already observed – discrete populations f i consistently outnumber the set of hydrodynamic observables, so that the excess-variables are potentially available to carry non-hydrodynamic information. This would represent a very significant advance, for it would show that LBE can be used as a tool for computational kinetic theory, beyond fluid dynamics. Nonetheless, a few numerical simulations of LBE microflows in microscopic electro-mechanical systems (MEMS) seem to indicate that standard LBE can capture some genuinely kinetic features of rarefied gas dynamics, such as slip motion at solid walls [19]. LBE schemes for nanoflow applications will certainly require new types of boundary conditions. A simple way to accomodate slip motion within LBE is to allow a fraction of LBE particles to be elastically reflected at the wall. A typical slip-boundary condition for, say, southeast propagating molecules entering the fluid domain from the north wall, y = d, would read as follows (lattice spacing made unity for simplicity): f se (x, d) = (1 − r) fne (x − 1, d − 1) + r fnw (x + 1, d − 1). Here r is a bounce-back coefficient in the range 0 < r < 1, and subscripts se, ne stand for south-east and north-east propagation, respectively [20]. It is easily seen that the special case r = 1 corresponds to a complete bounceback along the incoming direction, a simple option to implement zero fluid speed at the wall. More general conditions, borrowed from “diffusive” boundary conditions used in rarefied gas dynamics for the solution of the “true” Boltzmann equation have also been developed [21]. Much remains to be done to show that existing LBE models, extended with appropriate boundary conditions, can solve non-hydrodynamic flow regimes. This is especially true if thermal effects must be taken into account, as it is often the case in nanoflows applications.

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Even if the use of LBE stand-alone turned out to be unviable, one could still think of coupling LBE with truly microscopic methods, such as direct simulation or kinetic Monte Carlo [22, 23]. A potential advantage of coupling LBE instead of Navier-Stokes solvers to atomistic, or kinetic Monte Carlo, descriptions of atomistic flows is that the shear tensor Sab =

ν(∂a u b + ∂b u a ) 2

(14)

can be computed locally as 1

Sab = µ ( f i − f ie )(cia cib − cs2 δab ) 2 i

(15)

with no need of taking spatial derivatives (a delicate, and often error-prone, task at solid interfaces). Moreover, while the expression (14) is only valid in the limit of small Knudsen number, no such restriction applies to the kinetic expression (15). Both aspects could significantly enhance the scope of sampling procedures converting fluid-kinetic information (the discrete populations) into atomistic information (the particles coordinates and momenta) and vice versa, at fluid–solid interfaces [24]. This type of coupling procedures represent one of the most exciting frontiers for multiscale LBE applications at the interface between fluid dynamics and material science [25].

5.

Future Prospects

LBE has already made proof of significant versatility in addressing a wide range of problems involving complex fluid motion at disparate scales. Much remains to be done to further boost the power of the LB method towards multiphysics applications of increasing complexity. Important topics for future research are: • robust interface conditions for strongly non-equilibrium flows; • locally adaptive LBEs on unstructured, possibly moving, grids; • acceleration strategies for long-time and steady-state calculations. Finally, the development of a solid mathematical framework identifying the general conditions for the validity (what can go wrong and why!) of multiscale LBE techniques is also in great demand [26]. There are good reasons to believe that further upgrades of the LBE technique, as indicated above, hopefully stimulated by enhanced communication with allied sectors of computational physics, will make multiphysics LBE applications flourish in the near future.

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References [1] M. Seel, “Modelling of solid rocket fuel: from quantum chemistry to fluid dynamic simulations,” Comput. Phys., 5, 460–469, 1991. [2] W. Hoover, A.J. de Groot, and C. Hoover, “Massively parallel computer simulation of plane-strain elastic–plastic flow via non-equilibrium molecular dynamics and Lagrangian continuum mechanics,” Comput. Phys., 6(2), 155–162, 1992. [3] F.F. Abraham, J. Broughton, N. Bernstein, and E. Kaxiras, “Spanning the length scales in dynamic simulation,” Comput. Phys., 12(6), 538–546, 1998. [4] R. Benzi, S. Succi, and M. Vergassola, “The lattice Boltzmann equation: theory and applications,” Phys. Rep., 222, 145–197, 1992. [5] S. Succi, “The lattice Boltzmann equation for fluid dynamics and beyond,” Oxford University Press, Oxford, 2001. [6] G. McNamara and G. Zanetti, “Use of the Boltzmann equation to simulate lattice gas automata,” Phys. Rev. Lett., 61, 2332–2335, 1988. [7] F. Higuera, S. Succi, and R. Benzi, “Lattice gas dynamics with enhanced collisions,” Europhys. Lett., 9, 345–349, 1989. [8] X. He and L.S. Luo, “A priori derivation of the lattice Boltzmann equation,” Phys. Rev. E, 55, R6333–R6336, 1997. [9] Y.H. Qian, D. d’Humieres, and P. Lallemand, “Lattice BGK models for the Navier– Stokes equation,” Europhys. Lett., 17, 479–484, 1992. [10] S. Succi, I.V. Karlin, and H. Chen, “Role of the H theorem in lattice Boltzmann hydrodynamic simulations,” Rev. Mod. Phys., 74, 1203–1220, 2002. [11] O. Filippova and D. H¨anel, “Grid-refinement for lattice BGK models,” J. Comput. Phys., 147, 219–228, 1998. [12] H. Chen, C. Teixeira, and K. Molvig, “Realization of fluid boundary conditions via discrete Boltzmann dynamic,” Int. J. Mod. Phys. C, 9, 1281–1292, 1998. [13] A. Dupuis, “From a lattice Boltzmann model to a parallel and reusable implementation of a virtual river,” PhD Thesis n. 3356, University of Geneva, 2002. [14] O. Fippova, S. Succi, F.D. Mazzocco, C. Arrighetti, G. Bella, and D. Haenel, “Multiscale lattice Boltzmann schemes with turbulence modeling,” J. Comp. Phys., 170, 812–829, 2001. [15] S. Chen, S. Kandasamy, S. Orszag, R. Shock, S. Succi, and V. Yakhot, “Extended Boltzmann kinetic equation for turbulent flows,” Science, 301, 633–636, 2003. [16] A. Gabrielli, S. Succi, and E. Kaxiras, “A lattice Boltzmann study of reactive microflows,” Comput. Phys. Commun., 147, 516–521, 2002. [17] S. Succi, G. Smith, O. Filippova, and E. Kaxiras, “Applying the Lattice Boltzmann equation to multiscale fluid problems,” Comput. Sci. Eng., 3(6), 26–37, 2001. [18] M. Adamo, M. Bernaschi, and S. Succi, “Multi-representation techniques for multiscale simulation: reactive microflows in a catalytic converter,” Mol. Simul., 25(1–2), 13–26, 2000. [19] X.B. Nie, S. Chen, and G. Doolen, “Lattice Boltzmann simulations of fluid flows in MEMS,” J. Stat. Phys., 107, 279–289, 2002. [20] S. Succi, “Mesoscopic modeling of slip motion at fluid–solid interfaces with heterogeneus catalysis,” Phys. Rev. Lett., 89(6), 064502, 2002. [21] S. Ansumali and I.V. Karlin, “Kinetic boundary conditions in the lattice Boltzmann method,” Phys. Rev. E, 66, 026311–17, 2002. [22] M. Silverberg, A. Ben-Shaul, and F. Rebentrost, “On the effects of adsorbate aggregation on the kinetics of surface-reactions,” J. Chem. Phys., 83, 6501–6513, 1985.

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[23] T.P. Schulze, P. Smereka, and Weinan E, “Coupling kinetic Monte Carlo and continuum models with application to epitaxial growth,” J. Comput. Phys., 189, 197–211, 2003. [24] W. Cai, M. de Koning, V.V. Bulatov, and S. Yip, “Minimizing boundary reflections in coupled-domain simulations,” Phys. Rev. Lett., 85, 3213–3216, 2000. [25] D. Raabe, “Overview of the lattice Boltzmann method for nano and microscale fluid dynamics in material science and engineering,” Model. Simul. Mat. Sci. Eng., 12(6), R13–R14, 2004. [26] W. E, B. Engquist, Z.Y. Huang, “Heterogeneous multiscale method: a general methodology for multiscale modeling,” Phys. Rev. B, 67(9), 092101, 2003.

8.5 DISCRETE SIMULATION AUTOMATA: MESOSCOPIC FLUID MODELS ENDOWED WITH THERMAL FLUCTUATIONS Tomonori Sakai1 and Peter V. Coveney2,∗ 1 Centre for Computational Science, Queen Mary, University of London, Mile End Road, London E1 4NS, UK 2 Centre for Computational Science, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK

1.

Introduction

Until recently, theoretical hydrodynamics has largely dealt with relatively simple fluids which admit or are assumed to have an explicit macroscopic description. It has been highly successful in describing the physics of such fluids by analyses based on the Navier-Stokes equations, the classical equations of fluid dynamics which describe the motion of fluids, and usually predicated on a continuum hypothesis, namely that matter is infinitely divisible [1]. On the other hand, many real fluids encountered in our daily lives, in industrial, biochemical, and other fields are complex fluids made of molecules whose individual structures are themselves complicated. Their behavior is characterized by the presence of several important length and time scales. It must surely be among the more important and exciting research topics of hydrodynamics in the 21st century to properly understand the physics of such complex fluids. Examples of complex fluids are widespread – surfactants, inks, paints, shampoos, milk, blood, liquid crystals, and so on. Typically, such fluids are comprised of molecules and/or supramolecular components which have a non-trivial internal structure. Such microscopic and/or mesoscopic structures lead to a rich variety of unique rheological characteristics which not only make the study of complex fluids interesting but in many cases also enhance our quality of life.

* Corresponding author: P.V. Coveney, Email address: P.V. [email protected]

2487 S. Yip (ed.), Handbook of Materials Modeling, 2487–2501. c 2005 Springer. Printed in the Netherlands.

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In order to investigate and model the behavior of complex fluids, conventional continuum fluid methods based on the governing macroscopic fluid dynamical equations are somewhat inadequate. The continuous, uniform, and isotropic assumptions on which the macroscopic equations depend are not guaranteed to hold in such fluids where complex and time-evolving mesoscopic structures, such as interfaces, are present. As noted above, complex fluids are ones in which several length and time scales may be of importance in governing the large scale dynamical properties, but these micro and mesoscales are completely omitted in macroscopic continuum fluid dynamics, where empirical constitutive relations are instead shoe-horned into the Navier–Stokes equations. On the other hand, fully atomistic approaches based on molecular dynamics [2], which are the exact antithesis of conventional continuum methods, are in most cases not viable due to their vast computational cost. Thus, simulations which provide us with physically meaningful hydrodynamic results are out of reach of present day molecular dynamics and will not be accessible within the near future. Mesoscopic models are good candidates for mitigating problems with both conventional continuum methods and fully atomistic approaches. Spatially and temporally discrete lattice gas automata (LGA)[3] and lattice Boltzmann (LB) [4–7] methods have proven to be of considerable applicability to complex fluids, including multi-phase [8, 9] and amphiphilic [8, 9] fluids, solid–fluid suspensions [10], and the effect of convection–diffusion on growth processes [11]. These methods have also been successfully applied to flow in complex geometries, in particular to flow in porous media, an outstanding contemporary scientific challenge that plays an essential role in many technological, environmental, and biological fields [12–16]. Another important advantage of LGA and LB is that they are ideally suited for high performance parallel computing due to the inherent spatial locality of the updating rules in their dynamical time-stepping algorithms [17]. However, lattice-based models have certain well-known disadvantages associated with their spatially discrete nature [4, 7]. Here, we describe another mesoscopic model worthy of study. The method, which we call discrete simulation automata (DSA), is a spatially continuous but still temporally discrete version of the conventional spatio-temporally discrete lattice gas method, whose prototype was proposed by Malevanets and Kapral [18]. Since the particles now move in continuous space, DSA has the advantage of eliminating the spatial anisotropy that plagues conventional lattice gases, while also providing conservation of energy which enables one to deal with thermohydrodynamic problems not easily accessible by conventional lattice methods. We have coined the name DSA by analogy with the direct simulation Monte Carlo (DSMC) method [24] to which it is closely related, as we discuss further in Section 2. Some authors have referred to this method as a “realcoded lattice gas” [19–22]. Others have used the terms “Malevanets–Kapral

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method” “stochastic rotation method” or “multiple particle collision dynamics”. We have proposed the term DSA which we hope will be widely adopted in order to avoid further confusion [23]. The remainder of our paper is structured as follows. Starting from a review of single-phase DSA in Section 2, Section 3 describes how DSA can deal with binary immiscible fluids. In Section 4, we describe the application of DSA to amphiphilic fluids. Two of the latest developments of DSA, flow in porous media and a parallel implementation, are discussed in Section 5. Section 6 concludes our paper with a summary of the method.

2.

The Basic DSA Model and its Physical Properties

DSA are based on a microscopic, bottom-up approach and are comprised of cartesian cells between which massive point particles with a certain mass move. For a single component DSA fluid, state variables evolve by a twostep dynamical process: particle propagation and multi-particle collision. Each particle changes its location in the propagation process r = r + v

(1)

and its velocity in a collision process v = V + σ (v − V ),

(2)

where V is the mean velocity of all particles within a cell in which the collision occurs and σ is a random rotation, the same for all particles in one cell but differing between cells. In these equations, primes denote post-collision values and the mass as of all the particles are set to unity for convenience. This collision operation is equivalent to that in the direct simulation Monte Carlo (DSMC) method [24], except that pairwise collisions in DSMC are replaced by multi-particle collisions. The loss of molecular detail is an unavoidable consequence of the DSA algorithms as with other mesoscale modeling methods; however, these details are not required in order to describe the universal properties of fluid flow. Evidently, the use of multi-particle collisions allows DSA to deal readily with phenomena on mesoscopic and macroscopic scales which would be much more costly to handle using DSMC. Mass, momentum and energy are locally and hence globally conserved during the collision process. The velocity distribution of DSA particles corresponds to that of a Maxwellian when the system has relaxed to an equilibrium state [18]. We can thus define a parameter which may be regarded as a measure of average kinetic energy of the particles; this is the temperature T . For example, T = 1.0 specifies a state when each cartesian velocity component for the particles is described by a Maxwell distribution, whose variance is equal to one lattice unit (i.e., one DSA cell length).

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The existence of an H-theorem has been established using a reduced one-particle distribution function [18]. By applying a Chapman–Enskog asymptotic expansion to the reduced distribution function, the Navier–Stokes equations can be derived, as in the case of LGA [3] and LB [5]. When σ rotates v − V (see Eq. (2)) by a random angle in each cell, the fluid viscosity in DSA is written as ν=

1 ρ + 1 − e−ρ +T , 12 2(ρ − 1 + e−ρ )

(3)

where ρ is the number density of particles.

3. 3.1.

DSA Models of Interacting Particles Binary Immiscible Fluids

DSA have been extended to model binary immiscible fluids by introducing the notion of “color”, in both two and three dimensions [20]. Individual particles are assigned color variables, e.g., red or blue, and “color charges” which act rather like electrostatic charges. This notion of “color” was first introduced by Rothman and Keller [8]. With the color charge Cn of the nth particle given by

Cn =

+1

red particle,

−1

blue particle,

(4)

there is an attractive force between particles of the same color and a repulsive force between particles of different colors. To quantify this interaction, we define the color flux vector N( r) Q(r) = Cn (vn − V (r)), (5) n=1

where the sum is over all particles, and the color field vector F(r) =

i

wi

N( r ) Ri i Cn , |Ri | n

(6)

where the first and the second sums are over all nearest neighbor cells and all particles, respectively. N (r) is the number of particles in the local cell, vn the velocity of the nth particle, and V (r) the mean velocity of particles in a cell. The weighting factors are defined as wi = 1/|Ri |, where Ri = r − r i and r i is the location of the centre of ith nearest neighbor cell. The range of the index i differs according to the definition of the neighbors. With two- and threedimensional Moore neighbors, for example, i would range from 0 to 7 and 0 to

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26, respectively. One can model the phase separation kinetics of an immiscible binary fluid by choosing a rotation angle for each collision process such that the color flux vector points in the same direction as the color field vector after the collision. The model exhibits complete phase separation in both two [20, 22] and three [22] dimensions and has been verified by investigating domain growth laws and the resultant surface tension between two immiscible fluids [21], see Figs. 1–3. Although the precise location of the spinodal temperature has not thus far been investigated within DSA, we have confirmed that all binary immiscible fluid simulations presented in this review operate below it.

Initial

50 steps

500 steps

1000 steps

Figure 1. Two-phase separation in a binary immiscible DSA simulation [22]. Randomly distributed particles of two different colors (dark grey for water, light grey for oil) in the initial state segregate from each other, until two macroscopic domains are formed. The system size is 32 × 32 × 32, and the number density of both water and oil particles is 5.0. 1.2 1.1

T ⫽ 2.0

1 0.9 0.8 ∆P

0.7 0.6

T ⫽ 1.0

0.5 0.4 0.3

T ⫽ 0.5

0.2 0.1 0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

1/ R

Figure 2. Verification of Laplace’s law for two-dimensional DSA [21]. The pressure difference between inside and outside of a droplet of radius R, P = Pin − Pout , was measured in a system of size 4R × 4R(R = 16, 32, 64, 128), averaged over 10 000 time-steps. The error bars are smaller than the symbols. T is the “temperature” which can be regarded as the indicator of averaged kinetic energy of particles and is defined by T = kT ∗ /m (k is Boltzmann’s constant, T ∗ the absolute temperature, and m the mass of the particles).

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3.6 ⴛ t⫺0.5

0.6

7.8 ⴛ t⫺0.67

0.5

0.4

0.3

0.2

100 Time step

200

300

Figure 3. Temporal evolution of the characteristic wave number [25] in two-dimensional DSA simulations of binary phase separation, averaged over seven independent runs [21]. The domain growth is characterized with two distinct rates, namely, a slow growth rate R ∼ t 1/2 in the initial stage and a fast growth rate R ∼ t 2/3 at later times.

3.2.

Ternary Amphiphilic Fluids

A typical surfactant molecule has a hydrophilic head and a hydrophobic tail. Within DSA this structure is described by introducing a dumbbell-shaped particle in both two [21] and three [22] dimensions. Figure 4 is a schematic description of the two-dimensional particle model. A and B correspond to the hydrophilic head and the hydrophobic tail. G is the centre of mass of the surfactant particle. Color charges Cphi and Cpho are assigned to A and B, respectively. If we take the other DSA particles to be water particles whose color charges are

l phi

F (r)

A C phi θ

l pho

G

x

B C pho Figure 4. The schematic description of the two-dimensional surfactant model. A and B with color charges Cphi and Cpho correspond to the hydrophilic head and the hydrophobic tail, respectively. The mass of the surfactant particle is assumed to be concentrated at G, the centre of mass of the dumbbell particle.

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positive, Cphi and Cpho should be set as Cphi > 0 and Cpho < 0. The attractive interaction between A and water particles and the repulsive interaction between A and oil particles and conversely for B are described in a similar way to those in the binary immiscible DSA. For simplicity, the mass of the surfactant particle is assumed to be concentrated at the centre of mass. This assumption provides the model with great simplicity especially in describing the rotational motion of surfactant particles, while adequately retaining the ability to reproduce essential properties of surfactant solutions. Since there is no need to consider the rotational motions of the surfactant particle explicitly, its degrees of freedom are reduced to only three, that is, its location, orientation angle, and translational velocity. Calculations of the color flux F(r) and the color field Q(r) resemble those in the binary immiscible DSA. For the calculation of F(r), we use Eq. (5), without taking the contributions of surfactant particles into account. Note that motions of A and B only result in suppressing the tendency of F(r) and Q(r) to overlap each other, because they would not influence the “non-color” momentum exchanges. Q(r) is determined by considering both the distribution and the structure of surfactant particles. When a surfactant particle is located at r G with an orientation angle θ (see Fig. 4), A and B ends of the particle are located at

rA =

rB =

r Ax r Ay

rBx rBy

=

=

rG x rGy rG x rGy

+

−

cos θ sin θ cos θ sin θ

· lphi ,

(7)

· lpho .

(8)

In these equations, lphi and lpho are the distance between G and the hydrophilic end (A in Fig. 4), and the distance between G and the hydrophobic end (B in Fig. 4), respectively. We then add the color charge Cphi and Cpho to cells located at r A and r B , which corresponds to modifying Eq. (4) into +1

red particle, −1 blue particle, Cn = hydrophilic head, C phi Cpho hydrophobic tail.

(9)

After calculating the color flux and the color field in each cell, a rotation angle is chosen using the same method as for binary immsicible DSA fluids, namely, the color flux vector overlaps the color field vector. Finally, the orientation angle θ of each surfactant particle, after the momentum exchange, is set in such a way that it overlaps with the color field, which can be expressed as:

cos θ sin θ

=

F(r) . |F(r)|

(10)

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Both two- and three-dimensional versions of this model have been derived in this way [21, 22]. Using this model, the formation of spherical micelles, water-in-oil and oil-in-water droplet microemulsion phases, and water/oil/ surfactant sponge phase in both two [21] and three [22] dimensions have been reported (see Figs. 5 and 6). Suppression of phase separation and resultant domain growth, the lowering of interfacial tension between two immiscible fluids, and the connection between the mesoscopic model parameters and the macroscopic surfactant phase behavior have been studied within the model in both two and three dimensions [21, 22]. These studies have been primarily qualitative in nature, and correspond to some of the early papers published on ternary amphiphilic fluids using LGA [26, 27] and LB [17, 28] methods. Much more extensive work on the

Initial

After 250 steps

After 2500 steps

After 10000 steps

Figure 5. A two-dimensional DSA simulation of a sponge phase in a ternary amphiphilic fluid starting from a random initial condition [21]. Surfactant is visible at the interface between oil (dark grey) and water (light grey) regions. The system size is 64×64, the number density of DS A particles 10, the concentration ratio of water/oil/surfactant 1 : 1 : 1, the temperature of the system 0.2, color charges for hydrophilic and hydrophobic end groups Cphi = 10.0, Cpho = − 10.0.

Figure 6. The formation of spherical micelles in aqueous solvent [22]. The system size is 32 × 32 × 32, the concentration of surfactant particles is 10%.

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quantitative aspects of self-assembly kinetics has already been published using these two techniques.

4.

Some Recent Developments and Applications

DSA is currently attracting growing attention; the most recent published works using the method include the modeling of colloids [29], a detailed quantitative analysis of single-phase fluid behavior [30], and studies on the theoretical and numerically determined viscosity [31, 32]. Here we describe our own latest developments, concerning flow in porous media and parallel implementation.

4.1.

Flow in Porous Media

Within DSA, updating the state in porous media simulations requires close attention to be paid to the propagation process. This is due to the fact that particles are allowed to assume velocities of arbitrary directions and magnitudes: it frequently happens that a particle penetrates unphysically through an obstacle and reaches a fluid area on another side. It is thus not enough to know only the information about the starting and ending sites of the moving particles, as is done in LGA and LB studies, but rather their entire trajectories need to be investigated. We detect whether a particle hits an obstacle or not in the following way. First, we look at the cell containing r =r +v. If the cell is inside an obstacle the particle move is rejected and bounce-back boundary conditions are applied to update the particle velocity in the cell. When the cell is within a pore region, we extract a rectangular set of cells where the cells including r and r face each other on the diagonal line, as shown in Fig. 7. From this set of cells we further extract cells which intersect the trajectory of the particle. In order to do this, every cell C j in the “box” shown in Fig. 7 except those containing r and r , is investigated by taking the cross product v × c j k , where c j k denotes the position vector of four points of a cell C j and k = 1, 2, 3 and 4. If the v × c j k s for all k have the same sign, this means that the whole of C j is located on either side of v, that is, it does not intersect v and there is no need to check whether the site is inside a pore or the solid matrix. Otherwise, C j intersects v and the move is rejected if the site is inside the solid matrix, see Fig. 7. Using this method we have simulated single phase and binary immiscible fluid flow in two-dimensional porous media [23]. Good linear force–flux relationships were observed in single phase fluid flows, as is expected from Darcy’s law. In binary immiscible fluid flows, our findings are in good

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r'

r

"box"

Solid matrices

Figure 7. Scheme for detecting particles’ collisions with obstacles within the discrete simulation automata model [23]. Assume a particle moves from r to r = r + v (v is the velocity of the particle) in the current time-step. This particle obviously collides with the obstacle which is colored gray. However, the collision cannot be detected if we only take into account the information on r which is within the pore region. The whole trajectory of the particle must be investigated to accurately detect the collisions. In order to do this, we first extract all – in this case twelve – cells comprising the “box”, a rectangular set of cells where the cells including r and r are aligned with each other on a diagonal line. Secondly, from within the box, we further extract cells which overlap with the trajectory of the particle. The six cells comprising the region bordered with slashed lines are such cells in this case. These cells except those which include r and r are finally checked to establish whether they are part of an obstacle or a pore region.

agreement with previous studies using LGA [12–14]: a well defined linear force–flux relationship was obtained only when the forcing exceeded specified thresholds. We also found a one-to-one correspondence between these thresholds and the interfacial tension between the two fluids, which supports the interpretation from previous LGA studies that the existence of these thresholds is due to the presence of capillary effects within the pore space. In the study [23], we assumed that the binary immiscible fluids are uncoupled. However, a more general force–flux relationship allows for the fluids to be coupled and there have been a few studies of two-phase flow taking such coupling into account [12–14, 33, 34]. Within LGA, using the gravitational

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j

i

υi Figure 8. Occluded particles [23]: some particles can be assigned to a pore completely surrounded by solid matrices at the initial state, like particle i. Other particles can be occluded in a depression on the surface of an obstacle, like particle j . By imposing the gravitational force on such particles, they will gain kinetic energy limitlessly because their energy cannot be dissipated through interactions with other particles.

forcing method, it is possible to apply the forcing to only one species of fluid and discuss similarities with the Onsager relations [12, 13]. In our DSA study, we have used pressure forcing [23] and thus have not been able to investigate the effect of the coupling of the two immiscible fluids. The difficulty in implementing gravitational forcing within DSA is partly due to the local heating effects caused by occluded particles which are trapped within pores and will gain kinetic energy in an unbounded manner by the gravitational force imposed on them at every time step; see Fig. 8.

4.2.

Parallel Implementation

For large scale simulations in three dimensions, the computational cost of DSA is high, as with LGA and LB methods. Due to the spatially local updating rules, however, all basic routines in DSA algorithms are parallelizable. Good

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computer performance can thus be expected given an efficient