Handbook of Materials Modeling

HANDBOOK OF MATERIALS MODELING HANDBOOK OF MATERIALS MODELING Part B. Models Editor Sidney Yip, Massachusetts Institu...

Author: Sidney Yip

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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, Universitá 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. Blöhl 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, Université 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, Unité mixte de recherche CNRS/UHP 7565, Institut nancéen de chimie moléculaire, Université Henri Poincaré, BP 239, 54506 Vanduvre-lès-Nancy cedex, France 2.26 Giovanni Ciccotti INFM and Dipartimento di Fisica, Università “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 Gábor Csányi 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. Först 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 Educación 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 München, 85747 Garching, Germany [email protected] 5.10 Peter Gumbsch Institut für Zuverlässigkeit von Bauteilen und Systemen izbs, Universität Karlsruhe (TH), Kaiserstr. 12, 76131Karlsruhe, Germany and Fraunhofer Institut für Werkstoffmechanik IWM, Wöhlerstr. 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


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 Matiére Condensée et des Nanostructures, CNRS and Université Claude Bernard Lyon-1, 69622 Villeurbanne Cédex, 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 João F. Justo Escola Politécnica, Universidade de São Paulo, São 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 Kästner 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, Université 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 São Paulo, São 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. Lignères 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


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 Fédération, 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, Università 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 Ordejón 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 Peiró 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 für Eisenforschung, Max-Planck-Str. 1, D-40237 Düsseldorf, 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 Sánchez-Portal Donostia International Physics Center, Donostia, Spain [email protected] 1.5 Joachim Sauer Institut für Chemie, Humboldt-Universität 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 für Physikalische Chemie, Lehrstuhl für Theoretische Chemie, Universität 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 Frédéric Soisson CEA Saclay, DMN-SRMP, 91191 Gif-sur-Yuette, France [email protected] 7.9 José M. Soler Universidad Autónoma 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 Université 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


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, Unité mixte de recherche CNRS/UHP 7565, Institut nancéien de chimie moléculaire, Université Henri Poincaré, BP 239, 54506 Vanduvre-lès-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 Göran Wahnström Chalmers University of Technology and Göteborg University Materials and Surface Theory, SE-412 96 Göteborg, Sweden [email protected] 5.14

xxvi


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 Ordejón, Deniel Sánchez-Portal, and José M. Soler 1.6 Electronic Structure Methods: Augmented Waves, Pseudopotentials and the Projector Augmented Wave Method Peter E. Blöchl, Johannes Kästner, and Clemens J. Först 1.7 Electronic Scale James R. Chelikowsky 1.8 An Introduction to Orbital-Free Density Functional Theory Vincent L. Lignères 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 João 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


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 Göran Wahnström 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 Frédéric 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


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 Peiró 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 Español 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 Müller 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


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, Gábor Csányi, 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


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 Jónsson [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. Hänggi, 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. Jónsson, “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. Jónsson, “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. Cárdenas, Long time dynamics of complex systems, Acc. Chem. Res., 35, 396, 2002. [3] H. Jónsson, 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, Università “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

Simulating reactions that occur once in a blue moon

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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

Phase-field modeling

2093

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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A. Karma

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.


4.

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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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A. Karma

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


2101

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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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)

2106

S.G. Kim and W.T. Kim

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

Phase-field modeling of solidification

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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

2112


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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C. Shen and Y. Wang

(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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C. Shen and Y. Wang

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

Coherent precipitation – phase field method

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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)


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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.


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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. Fährmann). (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


(b)

t* = 20

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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


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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.


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.

References [1] L. Wang, D.E. Laughlin et al., “Magnetic domain structure of Fe-55 at %Pd alloy at different stages of atomic ordering,” J. Appl. Phys., 93, 7984–7986, 2003. [2] V.I. Syutkina and E.S. Jakovleva, Phys. Stat. Sol., 21, 465, 1967. [3] Y. Le Bouar and A. Loiseau, “Origin of the chessboard-like structures in decomposing alloys: Theoretical model and computer simulation,” Acta Mater., 46, 2777, 1998. [4] C. Manolikas and S. Amelinckx, “Phase-transitions in ferroelastic lead orthovanadate as observed by means of electron-microscopy and electron-diffraction 1. Static observations,” Phys. Stat. Sol., A60(2), 607–617, 1980. [5] K.C. Russell, “Introduction to: Coherent fluctuations and nucleation in isotropic solids by John W. Cahn,” In: W. Craig Carter and William C. Johnson (eds.), The selected works of John W. Cahn, Warrendale, Pennsylvania, The Minerals, Metals & Materials Society, 105–106, 1998. [6] Y. Wang, L.Q. Chen et al., “Computer simulation of microstructure evolution in coherent solids,” Solid → Solid Phase Transformations, Warrendale, PA, TMS, 1994.

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[7] Y. Wang and L. Chen, “Simulation of microstructural evolution using the phase field method,” In: E.N. Kaufman (Editor in chief) Methods in materials research, a current protocols, Unit 2a.3, John Wiley & Sons, Inc., 2000 [8] L.Q. Chen, “Phase field models for microstructure evolution,” Annu. Rev. Mater. Res., 32, 113–140, 2002. [9] Y. Wang, H.Y. Wang et al., “Microstructural development of coherent tetragonal precipitates in Mg-partially stabilized zirconia: a computer simulation,” J. Am. Ceram. Soc., 78, 657, 1995. [10] Y. Wang and A.G. Khachaturyan, “Three-dimensional field model and computer modeling of martensitic transformation,” Acta Metall. Mater., 45, 759, 1997. [11] Y. Wang, L.Q. Chen et al., “Particle translational motion and reverse coarsening phenomena in multiparticle systems induced by a long-range elastic interaction,” Phys. Rev. B, 46, 11194, 1992. [12] Y. Wang, L.Q. Chen et al., “Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap,” Acta Metall. Mater., 41, 279, 1993. [13] D.Y. Li and L.Q. Chen, “Shape evolution and splitting of coherent particles under applied stresses,” Acta Mater., 47(1), 247–257, 1998. [14] J.D. Zhang, D.Y. Li et al., “Shape evolution and splitting of a single coherent precipitate,” Materials Research Society Symposium Proceedings, 1998. [15] M. Doi, “Coarsening behavior of coherent precipitates in elastically constrained systems – with particular emphasis on gamma-prime precipitates in nickel-base alloys,” Mater. Trans. Japan. Inst. Metals, 33, 637, 1992. [16] A.G. Khachaturyan, Theory of Structural Transformations in Solids, John Wiley & Sons, New York, 1983. [17] 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. [18] J.W. Cahn, “Coherent fluctuations and nucleation in isotropic solids,” Acta Met., 10, 907–913, 1962. [19] J.W. Cahn, “On spinodal decomposition in cubic solids,” Acta Met., 10, 179, 1962. [20] J.W. Cahn, “Coherent two-phase equilibrium,” Acta Met., 14, 83, 1966. [21] J.W. Cahn, “Coherent stress in elastically anisotropic crystals and its effect on diffusional proecesses,” In: The Mechanism of Phase Transformations in Crystalline Solids, The Institute of Metals, London, 1, 1969. [22] J.W. Cahn and J.E. Hilliard, “Free energy of a nonuniform system. I. Interfacial free energy,” J. Chem. Phys., 28(2), 258–267, 1958. [23] J.W. Christian, The Theory of Transformations in Metals and Alloys, Pergamon Press, Oxford, 1975. [24] A.J. Ardell, “The Ni-Ni3 Al phase diagram: thermodynamic modelling and the requirements of coherent equilibrium,” Modell. Simul. Mater. Sci. Eng., 8, 277–286, 2000. [25] K.C. Russell, “Nucleation in solids,” In: Phase Transformations, ASM, Materials Park, OH, 219–268, 1970. [26] H.I. Aaronson and J.K. Lee, “The kinetic equations of solid → solid nucleation theory and comparisons with experimental observations,” In: H.I. Aaronson (ed.), Lectures on the Theory of Phase Transformation, TMS, 165–229, 1999. [27] Y.H. Wen, Y. Wang et al., “Phase-field simulation of domain structure evolution during a coherent hexagonal-to-orthorhombic transformation,” Phil. Mag. A, 80(9), 1967–1982, 2000.


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[28] J.P. Simmons, C. Shen et al., “Phase field modeling of simultaneous nucleation and growth by explicit incorporating nucleation events,” Scripta Mater., 43, 935–942, 2000. [29] C. Shen, J.P. Simmons et al., “Modeling nucleation during coherent transformations in crystalline solids,” (to be submitted), 2004. [30] Y.H. Wen and J.P. Simmons et al., “Phase-field modeling of bimodal particle size distributions during continuous cooling,” Acta Mater., 51(4), 1123–1132, 2003. [31] W.C. Johnson and P.W. Voorhees, Solid State Phenomena, 23–24, 87, 1992. [32] Y.S. Yoo, Ph.D. dissertation, Korea Advanced Institute of Science and Technology, Taejon, Korea, 1993. [33] A.G. Khachaturyan, “Some questions concerning the theory of phase transformations in solids,” Sov. Phys. Solid State, 8, 2163, 1967. [34] A.G. Khachaturyan and G.A. Shatalov, “Elastic interaction potential of defects in a crystal,” Sov. Phys. Solid State, 11, 118, 1969. [35] J.D. Eshelby, “The determination of the elastic field of an ellipsoidal inclusion, and related problems,” Proc. R. Soc. A, 241, 376–396, 1957. [36] J.D. Eshelby, “The elastic field outside an ellipsoidal inclusion,” Proc. R. Soc. A, 252, 561, 1959. [37] L.E. Malvern, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Englewood Cliffs, 1969. [38] D.Y. Li and L.Q. Chen, “Shape of a rhombohedral coherent Ti11Ni14 precipitate in a cubic matrix and its growth and dissolution during constrained aging,” Acta Mater., 45(6), 2435–2442, 1997. [39] Y.U. Wang, Y.M. Jin et al., “Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid,” J. Appl. Phys., 92(3), 1351–1360, 2002. [40] Y. Wang, D. Banerjee et al., “Field kinetic model and computer simulation of precipitation of L12 ordered intermetallics from fcc solid solution,” Acta Mater., 46(9), 2983–3001, 1998. [41] L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon Press, Oxford, New York, 1980. [42] P. Tolèdano and V. Dimitriev, Reconstructive Phase Transitions : In Crystals and Quasicrystals, World Scientific, Singapore, River Edge, NJ, 1996. [43] H. Pottebohm, G. Neitze et al., “Elastic properties (the stiffness constants, the shear modulus and the dislocation line energy and tension) of Ni-Al solid-solutions and of the nimonic alloy pe16,” Mat. Sci. Eng., 60, 189, 1983. [44] J. Vicens and P. Delavignette, Phys. Stat. Sol., A33, 497, 1976. [45] R. Sinclair and J. Dutkiewicz, Acta Met., 25, 235, 1977. [46] L.A. Bendersky and W.J. Boettinger, “Transformation of bcc and B2 hightemperature phases to hcp and orthorhombic structures in the ti-al-nb system 2. Experimental tem study of microstructures,” J. Res. Natl. Inst. Stand. Technol., 98(5), 585–606, 1993. [47] Y.H. Wen, Y. Wang et al., “Coarsening dynamics of self-accommodating coherent patters,” Acta Mater., 50, 13–21, 2002. [48] Y.M. Jin, A. Artemev et al., “Three-dimensional phase field model of low-symmetry martensitic transformation in polycrystal: simulation of ζ2 martensite in aucd alloys,” Acta Mater., 49, 2309–2320, 2001. [49] S. Aoki, K. Morii et al., “Self-accommodation of ζ2 martensite in a Au-49.5%Cd alloy. Solid → Solid Phase Transformations, Warrendale, PA, TMS, 1994.

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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.

Ferroic domain structures using Ginzburg–Landau methods

1.

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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. Castán, 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. Tolédano 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. Tolédano and P. Tolédano, 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. Castán, 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

Phase-field modeling of grain growth (a)

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1.4 1.0 0.8 0.6 800˚C 750˚C 700˚C

0.4 0.2

f (R /, t )

(mm)

2.5 min 5 min 12 min Hillert (3D)

1.0

1.2

0.8 0.6 0.4 0.2 0.0

0.0 0

50

100

150

Annealing time (min)

200

0.0

0.5

1.0

1.5

2.0

2.5

R /

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

Phase-field modeling of grain growth

2161

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

φ2

0

φ3

(0, 0, 1, 0)

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

(c)

(d) φ1 θ 10˚

φ1 θ 18˚

φ1 θ 46˚ φ1 θ 72˚ φ1 θ 34˚

φ1 θ 27˚

crystal 50˚

0˚ 1

boundary

crystal θ

φ

0

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

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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)

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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]

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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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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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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. Göbel, “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

Modeling coarsening dynamics using interface tracking methods

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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.


2209

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


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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,


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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


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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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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

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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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J. Lowengrub

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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J. Lowengrub

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 Frédéric Soisson2 1

´ Commissariat a` l’Energie Atomique, Cab. H.C., 33 rue de la Fédération, 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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G. Martin and F. Soisson

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

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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


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.


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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

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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


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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

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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


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


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 ,


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


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 Athènes 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


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, Athènes’ 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


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


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.

2240


– 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].


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 Athènes [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


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].


2243

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”

2244


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].


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;

2246


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 ) .

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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, Cinétiques de Précipitation du Carbure de Niobium dans la ferrite, CEA Report, 0429–3460, 2001. [29] M. Athènes, 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. Athènes and P. Bellon, “Antisite diffusion in the L12 ordered structure studied by Monte Carlo simulations,” Philos. Mag. A, 79, 2243–2257, 1999. [31] A. Athènes, 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. Athènes, 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.

2250

I.R. Pankratov and V.G. Vaks

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

2252


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,


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].

2254


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)


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)

2256


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


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

2258


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.


2259

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)

2260


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.


2261

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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I.R. Pankratov and V.G. Vaks a

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].


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,

2264


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.


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

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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)


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,

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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)


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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 )

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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


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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.

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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)


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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.

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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


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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

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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


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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].

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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. Hähner, 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

Dislocation dynamics – phase field

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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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Y.U. Wang et al.

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. Cuitiño, 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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Y. Xiang and D.J. Srolovitz

[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.

2309

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)

2310


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)


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).

2312


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


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,

2314


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).


2315

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.

2316


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.


2317

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.

2318


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

y

0 ⫺0.2 ⫺0.4 ⫺0.6 ⫺0.8 ⫺1 ⫺1 ⫺0.8 ⫺0.6 ⫺0.4 ⫺0.2

0

0.2

0.4

0.6

0.8

1

x

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)

(b)

(d)

(e)

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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.


2319

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

z

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.

2320


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

3

3

3

2

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Figure 7. An edge dislocation bypassing a linear array of impenetrable particles, leaving Orowan loops [37] around the particles behind. The Burgers vector b is in the x direction. The applied stress σx z =/ 0, where the z direction is pointing out of the paper.

z b y

x

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).


2321

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.

0.4

Critical stress (Gb/L)

0.35

0.3

0.25 slope⫽1/2π 0.2

0.15

0.1 0.1

0.2

0.3

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0.5

0.6

0.7

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0.9

1

log(D /r ) 1 0

Figure 10. The critical stress for an edge dislocation to bypass co-planar impenetrable particles by the Orowan mechanism. The stress is plotted in the unit Gb/L against log(D1 /r0 ).

2322


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.

2326

J.M. Rickman and R. LeSar

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

2328


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 Kröner’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)


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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3.


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


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


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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. Kröner, 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

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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.

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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)


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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.

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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).


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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

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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.


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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

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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

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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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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.


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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. Schöll, “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.


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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

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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)

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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


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)

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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


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)


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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D.D. Vvedensky

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. Godrèche (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. Barabási 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

Monte Carlo methods for simulating thin film deposition

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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


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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)


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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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C. Battaile

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


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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).


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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


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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. Schröder, “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. Gué, M. Sahlaoui, and D. Estève, “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

Microstructure optimization

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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.


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

Mesoscale models of fluid dynamics

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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. Español, “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 Peiró 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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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)


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. Peiró 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.


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. Peiró 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.


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.

2424


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].


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)

2426


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)


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

2428


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

      .     


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


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


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.

2432


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

2434


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


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.


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 Padé 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


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.


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. Peiró 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


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


(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 .


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


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.


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].

2446


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.

2448

G. Li et al.

[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

Meshless methods for numerical solution

2449

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


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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)


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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)


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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


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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[2] S. Li and W.K. Liu, “Meshfree and particle methods and their applications,” Appl. Mech. Rev., 55, 1–34, 2002. [3] S.N. Atluri, The Meshless Local Petrov–Galerkin (MLPG) Method, Tech Science Press, 2002. [4] P. Lancaster and K. Salkauskas, “Surface generated by moving least squares methods,” Math. Comput., 37, 141–158, 1981. [5] B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element method: diffuse approximation and diffuse elements,” Comput. Mech., 10, 307–318, 1992. [6] T. Belytschko, Y.Y. Lu, and L. Gu, “Element free galerkin methods,” Int. J. Numer. Methods Eng., 37, 229–256, 1994. [7] T.J. Liszka and J. Orkisz, “The finite difference method at arbitrary irregular grids and its application in applied mechanics,” Comput. Struct., 11, 83–95, 1980. [8] T.J. Liszka, C.A. Duarte, and W.W. Tworzydlo, “hp-meshless cloud method,” Comput. Methods Appl. Mech. Eng., 139, 263–288, 1996. [9] S.N. Atluri and T. Zhu, “A new meshless local Petrov–Galerkin (MLPG) approach in computational mechanics,” Comput. Mech., 22, 117–127, 1998. [10] E. Oñate, S. Idelsohn, O.C. Zienkiewicz, and R.L. Taylor, “A finite point method in computational mechanics. Applications to convective transport and fluid flow,” Int. J. Numer. Methods Eng., 39, 3839–3866, 1996. [11] E. Oñate, S. Idelsohn, O.C. Zienkiewicz, R.L. Taylor, and C. Sacco, “A stabilized finite point method for analysis of fluid mechanics problems,” Comput. Methods Appl. Mech. Eng., 139, 315–346, 1996. [12] I. Babuska and J.M. Melenk, “The partition of unity method,” Int. J. Numer. Meth. Eng., 40, 727–758, 1997. [13] P. Breitkopf, A. Rassineux, G. Touzot, and P. Villon, “Explicit form and efficient computation of MLS shape functions and their derivatives,” Int. J. Numer. Methods Eng., 48(3), 451–466, 2000. [14] J.J. Monaghan, “Smoothed particle hydrodynamics,”Annu. Rev. Astron. Astrophys., 30, 543–574, 1992. [15] W.K. Liu, S. Jun, S. Li, J. Adee, and T. Belytschko, “Reproducing Kernel particle methods for structural dynamics,” Int. J. Numer. Methods Eng., 38, 1655–1679, 1995. [16] J.-S. Chen, C. Pan, C. Wu, and W.K. Liu, “Reproducing Kernel particle methods for large deformation analysis of non-linear structures,” Comput. Methods Appl. Mech. Eng., 139, 195–227, 1996. [17] N.R. Aluru, “A point collocation method based on reproducing Kernel approximations,” Int. J. Numer. Methods Eng., 47, 1083–1121, 2000. [18] N.R. Aluru and G. Li, “Finite cloud method: a true meshless technique based on a fixed reproducing Kernel approximation,” Int. J. Numer. Methods Eng., 50(10), 10, 2373–2410, 2001. [19] R.L. Hardy, “Multiquadric equations for topography and other irregular surfaces,” J. Geophys. Res., 176, 1905–1915, 1971. [20] E.J. Kansa, “Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics – I, surface approximations and partial derivative estimates,” Comp. Math. Appl., 19, 127–145, 1990. [21] E.J. Kansa, “Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics – II, solutions to parabolic, hyperbolic and elliptic partial differential equations,” Comp. Math. Appl., 19, 147–161, 1990. [22] M.A. Golberg and C.S. Chen, “A bibliography on radial basis function approximation,” Boundary Elements Comm., 7, 155–163, 1996.


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[23] H. Wendland, “Piecewise polynomial, positive definite and compactly supported radial functions of minial degree,” Adv. Comput. Math., 4, 389–396, 1995. [24] S. De and K.J. Bathe, “The method of finite spheres,” Comput. Mech., 25, 329–345, 2000. [25] S. De and K.J. Bathe, “Towards an efficient meshless computational technique: the method of finite spheres,” Eng. Comput., 18, 170–192, 2001. [26] X. Jin, G. Li, and N.R. Aluru, “On the equivalence between least-squares and Kernel approximation in meshless methods,” CMES: Comput. Model. Eng. Sci., 2(4), 447– 462, 2001. [27] J.H. Kane, Boundary Element Analysis in Engineering Continuum Mechanics, Prentice-Hall, 1994. [28] L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys., 73(2), 325–348, 1987. [29] J.R. Phillips and J.K. White, “A precorrected-FFT method for electrostatic analysis of complicated 3-D structures,” IEEE Transact. on Comput.-Aided Des. of Integrated Circuits Sys., 16(10), 1059–1072, 1997. [30] S. Kapur and D.E. Long, “I E S 3 : a fast integral equation solver for efficient 3-dimensional extraction,” IEEE Computer Aided Design, 1997, Digest of Technical Papers 1997, IEE/ACM International Conference, 448–455, 1997. [31] V. Shrivastava and N.R. Aluru, “A fast boundary cloud method for exterior 2-D electrostatics,” Int. J. Numer. Methods Eng., 56(2), 239–260, 2003. [32] Y.X. Mukherjee and S. Mukherjee, “The boundary node method for potential problems,” Int. J. Numer. Methods Eng., 40, 797–815, 1997. [33] M.K. Chati and S. Mukherjee, “The boundary node method for three-dimensional problems in potential theory,” Int. J. Numer. Methods Eng., 47, 1523–1547, 2000. [34] J. Zhang, Z. Yao, and H. Li, “A hybrid boundary node method,” Int. J. Numer. Methods Eng., 53(4), 751–763, 2002. [35] W. Chen, “Symmetric boundary knot method,” Eng. Anal. Boundary Elements, 26(6), 489–494, 2002. [36] G. Li and N.R. Aluru, “Boundary cloud method: a combined scattered point/boundary integral approach for boundary-only analysis,” Comput. Methods Appl. Mech. Eng., 191, (21–22), 2337–2370, 2002. [37] G. Li and N.R. Aluru, “A boundary cloud method with a cloud-by-cloud polynomial basis,” Eng. Anal. Boundary Elements, 27(1), 57–71, 2003. [38] G.E. Forsythe and W.R. Wasow, Finite Difference Methods for Partial Differential Equations, Wiley, 1960. [39] T.J.R. Hughes, The Finite Element Method, Prentice-Hall, 1987. [40] C.A. Brebbia and J. Dominguez, Boundary Elements An Introductory Course, McGraw-Hill, 1989. [41] K. Salkauskas and P. Lancaster, Curve and Surface Fitting, Elsevier, 1986. [42] X. Jin, G. Li, and N.R. Aluru, “Positivity conditions in meshless collocation methods,” Comput. Methods Appl. Mech. Eng., 193, 1171–1202, 2004. [43] W.K. Liu, S. Li, and T. Belytschko, “Moving least-square reproducing kernel methods (I) methodology and convergence,” Comput. Methods Appl. Mech. Eng., 143, 113– 154, 1997. [44] P.S. Jensen, “Finite difference techniques for variable grids,” Comput. Struct., 2, 17– 29, 1972. [45] G.E. Fasshauer, “Solving differential equations with radial basis functions: multilevel methods and smoothing,” Adv. Comput. Math., 11, 139–159, 1999.

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[46] M. Zerroukat, H. Power, and C.S. Chen, “A numerical method for heat transfer problems using collocation and radial basis functions,” Int. J. Numer. Methods Eng., 42, 1263–1278, 1998. [47] M.T. Heath, Scientific Computing: An Introductory Survey, McGraw-Hill, 1997. [48] Y.Y. Lu, T. Belytschko, and L. Gu, “A new implementation of the element free galerkin method,” Comput. Methods Appl. Mech. Eng., 113, 397–414, 1994. [49] G. Li, G.H. Paulino, and N.R. Aluru, “Coupling of the meshfree finite cloud method with the boundary element method: a collocation approach,” Comput. Meth. Appl. Mech. Eng., 192(20–21), 2355–2375, 2003.

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

Lattice Boltzmann methods for multiscale fluid problems

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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. Hänel, “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]


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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

Discrete simulation automata

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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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T. Sakai and P.V. Coveney DSA cells

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


2497 υj

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 parallel implementation. We have parallelized our DSA codes in two and three dimensions, written in C++ and named DSA2D and DSA3D, respectively, by spatially decomposing the system and implementing the MPI libraries [35]. It is in the propagation process that the MPI library functions are mainly used. There are two key features which are worth pointing out here. First, in the propagation process, information on the particles which exit each domain is stored in arrays which are then handed over to a send function MPI_Isend. The size of the arrays depends on temperature and the direction of the target domain. Second, as the number of particles within a domain fluctuates, 10 – 20% of the memory allocated for particles in the domain is used as an absorber. (Particles are allocated at an initial stage up to 80 – 90% of the total capacity.) Figures 9 and 10 show the parallel performance in two and three dimensions, respectively. Although DSA2D scales superlinearly across all processor counts, DSA3D scales well only with a large number of CPU (DSA3D’s propagation() routine even slows down with increasing processor counts for certain sets of parameters). The difference here is due to the way the system is spatially decomposed: DSA2D has been domain decomposed in one direction whereas DSA3D has been decomposed in three directions. In order to realise good scalability in three dimensions for our current parallel

50 45 collision()

40 35 Performance

TOTAL

30 IDEAL

25 propagation()

20 15 10 5 0

0

5

10

15

20

25

30

35

Number of CPUs

Figure 9. Scalability of two-dimensional DSA (DSA2D) for single-phase fluids on SGI Origin 3000 (400 MHz MIPS R12000) processors. “Performance” (vertical axis) means “speed-up”, which is relative to the number of processors. The overall performance is indeed superlinear.


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(a) 60 IDEAL

50 Performance

collision() 40 30

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10 0

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collision()

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(c) 60 IDEAL

Performance

50

collision()

40 TOTAL 30 propagation() 20 10 0

0

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20

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Figure 10. Parallel performance of DSA3D for single-phase fluids of varying system sizes: (A) 643 ; (B) 1283 ; (C) 2563 , on SGI Origin 3000 (400 MHz MIPS R12000) processors. “Performance” (vertical axis) means “speed-up”, which is relative to the number of processors. For 643 and 1283 systems the performance of the propagation process actually decreases when the number of CPUs becomes large.

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implementation, a large system and a large number of CPUs are required. The present parallel implementation should be regarded only as preliminary; further optimization may be expected to result in better overall performance.

5.

Summary

Discrete simulation automata (DSA) represent a mesoscopic fluid simulation method which, in common with lattice gas automata (LGA) and the lattice Boltzmann (LB) methods, has several advantages over conventional continuum fluid dynamics. Beyond LGA and LB’s beneficial aspects, DSA’s most eminent characteristic is that a temperature can be defined very naturally. It is thus a promising candidate to deal with complex fluids where fluctuations can often play an essential role in determining macroscopic behavior. There remain, however, some drawbacks to the DSA technique. The existence of particles with continuously valued velocities coupled to the intrinsic temporal discreteness of the model leads to some problems in handling wall boundary collisions, including trapping of particles with increasing energy in certain flow regimes, which do not arise with LGA and LB methods. Nonetheless, DSA appears to be a promising technique for the study of numerous complex fluids. We have reviewed a few examples here, including immiscible fluids, amphiphilic fluids, and flow in porous media. Most of these studies have so far not reached an equivalent maturity and quantitative level to that of LGA and LB publications. DSA is amenable to fairly straightforward parallel implementation. We therefore expect to see further fruitful explorations of complex fluid dynamics using DSA in the future.

References [1] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967. [2] D.C. Rapaport, The Art of Molecular Dynamics Simulation, Cambridge University Press, Cambridge, 1995. [3] U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett., 56, 1505, 1986. [4] S. Succi, The Lattice Boltzmann Equation, Oxford University Press, Oxford, 2001. [5] R. Benzi, S. Succy, and M. Vergassola, Phys. Rep., 222, 145, 1992. [6] S. Chen, Z. Wang, X. Shan, and G. Doolen, J. Stat. Phys., 68, 379, 1992. [7] D.H. Rothman and S. Zaleski, Lattice Gas Cellular Automata, Cambridge University Press, Cambridge, 1997. [8] D.H. Rothman, and J. Keller, J. Stat. Phys., 52, 1119, 1988. [9] D. Grunau, S. Chen, and K. Eggert, Phys. Fluids A, 5, 2557, 1993. [10] A.J.C. Ladd, J. Fluid Mech., 271, 285, 1994. [11] J.A. Kaandorp, C. Lowe, D. Frenkel, and P.M.A. Sloot, Phys. Rev. Lett., 77, 2328, 1996.


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[12] P.V. Coveney, J.-B. Maillet, J.L. Wilson, P.W. Fowler, O. Al-Mushadani, and B.M. Boghosian, Int. J. Mod. Phys. C, 9, 1479, 1998. [13] J.-B. Maillet and P.V. Coveney, Phys. Rev. E, 62, 2898, 2000. [14] P.J. Love, J.-B. Maillet, and P.V. Coveney, Phys. Rev. E, 64, 061302, 2001. [15] N.S. Martys and H. Chen, Phys. Rev. E, 53, 743, 1996. [16] A. Koponen, D. Kandhai, E. Hellen, M. Alava, A. Hoekstra, M. Kataja, K. Niskanen, P. Sloot, and J. Timonen, Phys. Rev. Lett., 80, 716, 1998. [17] P.J. Love, M. Nekovee, P.V. Coveney, J. Chin, N. Gonzalez-Segredo and J.M.R. Martin, Comput. Phys. Commun., 153, 340, 2003. [18] A. Malevanets and R. Kapral, J. Chem. Phys., 110, 8605, 1999. [19] Y. Hashimoto, Y. Chen, and H. Ohashi, Int. J. Mod. Phys. C, 9(8), 1479, 1998. [20] Y. Hashimoto, Y. Chen, and H. Ohashi, Comput. Phys. Commun., 129, 56, 2000. [21] T. Sakai, Y. Chen, and H. Ohashi, Phys. Rev. E, 65, 031503, 2002. [22] T. Sakai, Y. Chen, and H. Ohashi, J. Coll. and Surf., 201, 297, 2002. [23] T. Sakai and P.V. Coveney, “Single phase and binary immiscible fluid flow in two-dimensional porous media using discrete simulation automata,” 2002 (preprint). [24] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, Oxford, 1994. [25] T. Kawakatsu, K. Kawasaki, M. Furusaka, H. Okabayashi and T. Kanaya, J. Chem. Phys., 99, 8200, 1993. [26] B.M. Boghosian, P.V. Coveney, and A.N. Emerton, Proc. R. Soc. A, 452, 1221, 1996. [27] B.M. Boghosian, P.V. Coveney, and P.J. Love, Proc. R. Soc. A, 456, 1431, 2000. [28] H. Chen, B.M. Boghosian, P.V. Coveney, and M. Nekovee, Proc. R. Soc. A, 456, 2043, 2000. [29] S.H. Lee and R. Kapral, Physica A, 298, 56, 2001. [30] A. Lamura and G. Gompper, Eur. Phys. J.E, 9, 477, 2002. [31] T. Ihle and D.M. Kroll, Phys. Rev. E, 63, 020201(R), 2001. [32] A. Lamura, G. Gompper, T. Ihle, and D. M. Kroll, Europhys. Lett., 56, 319, 2001. [33] C. Zarcone and R. Lenormand, C.R. Acad. Sci. Paris, 318, 1429, 1994. [34] J.F. Olson and D.H. Rothman, J. Fluid Mech., 341, 343, 1997. [35] http://www-unix.mcs.anl.gov/mpi/index.html

8.6 DISSIPATIVE PARTICLE DYNAMICS Pep Español Dept. Física Fundamental, Universidad Nacional de Educación a Distancia, Aptdo. 60141, E-28080 Madrid, Spain

1.

The Original DPD Model

In order to simulate a complex fluid like a polymeric or colloidal fluid, a molecular dynamics simulation is not very useful. The long time and space scales involved in the mesoscopic dynamics of large macromolecules or colloidal particles as compared with molecular scales imply to follow an exceedingly large number of molecules during exceedingly large times. On the other hand, at these long scales, molecular details only show up in a rather coarse form, and the question arises if it is possible to deal with coarse-grained entities that reproduce the mesoscopic dynamics correctly. Dissipative particle dynamics (DPD) is a fruitful modeling attempt in that direction. DPD is a stochastic particle model that was introduced originally as an off-lattice version of Lattice gas automata (LGA) in order to avoid its lattice artifacts [1]. The method was put in a proper statistical mechanics context a few years later [2] and the number of applications since then is growing steadily. The original DPD model consists of a collection of soft repelling frictional and noisy balls. From a physical point of view, each dissipative particle is regarded not as a single molecule of the fluid but rather as a collection of molecules that move in a coherent fashion. In that respect, DPD can be understood as a coarse-graining of molecular dynamics. There are three types of forces between dissipative particles. The first type is a conservative force deriving from a soft potential that tries to capture the effects of the “pressure” between different particles. The second type of force is a friction force between the particles that wants to describe the viscous resistance in a real fluid. This force tries to reduce velocity differences between dissipative particles. Finally, there is a stochastic force that describe the degrees of freedom that have been eliminated from the description in the coarse-graining process. 2503 S. Yip (ed.), Handbook of Materials Modeling, 2503–2512. c 2005 Springer. Printed in the Netherlands.

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This stochastic force will be responsible for the Brownian motion of polymer and colloidal particles simulated with DPD. The postulated stochastic differential equations (SDEs) that define the DPD model are [2] dri = vi dt FCi j (ri j )dt − γ ω(ri j )(ei j ·vi j )ei j dt m i dvi = j= /i

+σ

(1)

j= /i

ω

1/2

(ri j )ei j dWi j

j= /i

Here, ri , vi are the position and velocity of the dissipative particles, m i is the mass of particle i, FCi j is the conservative repulsive force between dissipative particles i, j , ri j = ri −r j , vi j = vi −v j , and the unit vector from the j th particle to the ith particle is ei j = (ri − r j )/ri j with ri j = |ri − r j |. The friction coefficient γ governs the overall magnitude of the dissipative force, and σ is a noise amplitude that governs the intensity of the stochastic forces. The weight function ω(r) provides the range of interaction for the dissipative particles and renders the model local in the sense that the particles interact only with their neighbors. A usual selection for the weight function in the DPD literature is a linear function with the shape of a Mexican hat, but there is no special reason for such a selection. Finally, dWi j = dW j i are independent increments of the Wiener process that satisfy the Itô calculus rule dWi j dWi j = (δii δ j j + δi j δ j i ) dt. There are several remarkable features of the above SDEs. They are translationally, rotationallyand Galilean invariant. Most importantly, total momentum is conserved, d( i pi )/dt = 0, because the three types of forces satisfy Newton’s Third Law. Therefore, the DPD model captures the essentials of mass and momentum conservation which are responsible for the hydrodynamic behavior of a fluid at large scales [3, 4]. Despite its appearance as Langevin equations, Eq. (2) is quite different from the ones used in Brownian Dynamics simulations. In the Brownian Dynamics method, total momentum of the particles is not conserved and only mass diffusion can be studied. The above SDE are mathematically equivalent to a Fokker–Planck equation (FPE) that governs the time-dependent probability distribution ρ(r, v; t) of positions and velocities of the particles. The explicit form of the FPE can be found in Ref. [2]. Under the assumption that the noise amplitude and the friction coefficient are related by the fluctuation–dissipation relation σ = (2kB T γ )1/2, the equilibrium distribution ρ eq of the FPE has the familiar form

1 1 ρ (r, v) = exp − Z kB T eq

m i v i2 i

2

+ V (r)

(2)

where V is the potential function that gives rise to the conservative forces FC , kB is Boltzmann’s constant, T is the equilibrium temperature and Z is the normalizing partition function.

Dissipative particle dynamics

2.

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DPD Simulations of Complex Fluids

One of the most attractive features of the model is its enormous versatility in order to construct simple models for complex fluids. In DPD, the Newtonian fluid is made “complex” by adding additional interactions between the fluid particles. Just by changing the conservative interactions between the fluid particles, one can easily construct polymers, colloids, amphiphiles, and mixtures. Given the simplicity of modeling of mesostructures, DPD appears as a competitive technique in the field of complex fluids. We review now some of the applications of DPD to the simulation of different complex fluids systems (see also Ref. [5]). Colloidal particles are constructed by freezing fluid particles inside certain region, typically spheres or ellipsoids, and moving those particles as a rigid body. The idea was pioneered by Koelman and Hoogerbrugge [6] and has been explored in more detail by Boek et al. [7]. The simulation results for shear thinning curves of spherical particles compare very well with experimental results for volume fractions below 30%. At higher volume fractions somewhat inconsistent results are obtained, which can be attributed to several factors. The colloidal particles modeled in this way are to certain degree “soft balls” that can interpenetrate leading to unphysical interactions. At high volume fractions solvent particles are expelled from the region in between two colloidal particles. Again, this misrepresents the hydrodynamic interaction, which is mostly due to lubrication forces [8]. Depletion forces appear [9, 10] which are unphysical and due solely to the discrete representation of the continuum solvent. It seems that a judicious selection of lubrication forces that would take into account the effects of the solvent when no dissipative particle exist in between two colloidal particles can eventually solve this problem. Finally, we note that DPD can resolve the time scales of fluid momentum transport on the length scale of the colloidal particles or their typical interdistances. These scales are probed experimentaly by diffusive wave spectroscopy [11]. Polymer molecules are constructed in DPD through the linkage of several dissipative particles with springs (either Hookean or FENE [12]). Dilute polymer solutions are modeled by a set of polymer molecules interacting with a sea of fluid particles. The solvent quality can be varied by fine tuning the solvent–solvent and solvent–monomer conservative interactions. In this way, a collapse transition has been observed in passing from a good solvent to a poor solvent [13]. Static scaling results for the radius of gyration and relaxation time with the number of beads are consistent with the Rouse/Zimm models [14]. The model displays hydrodynamic interactions and excluded volume interactions, depending on solvent quality. Rheological properties have been also studied showing a good agreement with known kinetic theory results [15, 16]. Polymer solutions confined between walls have also been modeled showing anisotropic relaxation in nanoscale gaps [17]. Polymer melts have been

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simulated showing that the static scaling and rheology correspond to the Rouse theory, as a result of screening of hydrodynamic and excluded volume interactions in the melt [14]. The model is unable to simulate entanglements due to the soft interactions between beads that allow polymer crossing [14], although this effect can be partially controlled by suitably adjusting the length and intensity of the springs. At this point, DPD appears as a well benchmarked model for the simulation of polymer systems. Nevertheless, there is still not a direct connection between the model parameters used in DPD and actual molecular parameters like molecular weight, torsion potentials, etc. Immiscible fluid mixtures are modeled in DPD by assuming two types of particles [18]. Unequal particles repel each other more strongly than equal particles thus favoring phase separation. Starting from random initial conditions representing a high temperature miscible phase suddenly quenched, the domain growth has been investigated [19, 20]. Although lattice Boltzmann simulations allow to explore larger time scales than DPD [21], the simplicity of DPD modeling allows one to generalize easily to more complex systems in a way that lattice Boltzmann cannot. For example, mixtures of homopolymer melts have been modeled with DPD [22]. Surface tension measurements allow for a mapping of the model to the Flory–Huggins theory [22]. In this way, thermodynamic information has been used to fix the model parameters of DPD. A recent more detailed analysis of this procedure has been presented in Refs. [23, 24], where a calculation of the phase diagram of monomer and polymer mixtures of DPD particles allowed to discuss the connection of the repulsion parameter difference and the Flory–Huggins parameter χ. Another successful application of DPD has been the simulation of microphase separation of diblock copolymers [25], that has allowed to discuss the pathway to equilibrium. This pathway is strongly affected by hydrodynamics [26]. In a similar way, simulations of rigid DPD dimers in a solution of solvent monomers has allowed to study the growth of amphiphilic mesophases and its behavior under shear [27] and the self-assembly of model membranes [28]. DPD has also been applied to other complex situations like the dynamics of a drop at a liquid–solid interface [29], flow and rheology in the presence of polymers grafted to walls [30], vesicle formation of amphiphilic molecules [31] and polyelectrolytes [32].

3.

Thermodynamically Consistent DPD Model

Despite its successes, the DPD model suffers from several conceptual shortcomings that originate from the oversimplification of the so-formulated


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dissipative particles as representing mesoscopic portions of fluid. There are several issues in the original model that are unsatisfactory. For example, even though the macroscopic behavior of the model is hydrodynamic [3], it is not possible to relate in a simple direct way the viscosity of the fluid with the model parameters. Only after a recourse to the methods of kinetic theory can one estimate what input values for the friction coefficient should be imposed to obtain a given viscosity [4]. Another problem with the original model is that the conservative forces fix the thermodynamic behavior of the fluid [22]. The pressure equation of state, for example, is an outcome of the simulation, not an input. The model is isothermal and not able to study energy transport. There are no rules for specifying the range and shape of the weight functions that affect both, thermodynamic and transport properties. Perhaps the biggest problem of the model is the unclear physical length and time scales that are actually simulated. How big is a dissipative particle is not known from the model parameters. DPD appeared as a quick way of getting hydrodynamics suitable for “mesoscales”. Of course, the fact that there exists a well-defined Hamiltonian with a proper equilibrium ensemble, still makes the DPD model useful, at least as a thermostating device that respect hydrodynamics. In particular, when considering models of coarse-grained complex molecules (like amphiphiles or macromolecules) DPD as it was originally formulated can be very useful, despite the fact that an explicit correspondence between molecular parameters and DPD parameters are not known. However, the above-mentioned problems render DPD as a poor tool for the simulation of Newtonian fluids at mesoscopic scales. One needs to simulate a Newtonian fluid when dealing with colloidal suspension, dilute polymeric suspensions or mixtures of Newtonian fluids. In these cases, one should use better models that are thermodynamically consistent. These models consider each dissipative particle as a fluid particle, this is, a small moving thermodynamic system with proper thermodynamic variables. The idea of introducing an internal energy variable in the DPD model was developed in Refs. [33, 34] in order to obtain an energy conserving DPD model. Yet, it is necessary to introduce a second thermodynamic variable to have a full thermodynamic description. This variable is the volume of the fluid particles. There have been also attempts to introduce a volume variable in the isothermal DPD model [35, 36], but a full non-isothermal and thermodynamically consistent model has only appeared recently [37]. One way to define the volume is with the help of a bell-shaped weight function W (r) of finite range h normalized to unity.We introduce the density of every fluid particle through the relation di = j W (ri j ). Clearly, if around particle i there are many particles j , the density di defined above will be large. One associates a volume V i = di−1 to the fluid particle. Another possibility for defining the volume of each fluid particle relies on the Voronoi tessellation [38–40].

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The equations for the evolution of the position, velocity, and entropy of each fluid particle in the thermodynamically consistent DPD model are [37] r˙ i = vi m v˙ i =

j

Ti S˙i = −2κ

Pj 5η Fi j Pi + F r − vi j + ei j ei j ·vi j + F˜ i i j i j 2 2 3 j di d j di dj

Fi j j

di d j

Ti j +

5η Fi j 2 vi j + (ei j ·vi j )2 + Ti J˜i 6 j di d j

(3)

Here, Pi , Ti are the pressure and temperature of the fluid particle i, given in terms of equations of state, and Ti j = Ti − T j . We have introduced the function F(r) through ∇W (r) = −rF(r) and F˜ i , J˜i are suitable stochastic forces that obey the fluctuation–dissipation theorem [37]. Some small terms have been neglected in Eq. (3) for the sake of presentation. It can be shown that the above model conserves mass, momentum and energy and that the total entropy is a non-decreasing function of time, rendering the model consistent with the Laws of Thermodynamics. What are the similarities and differences between the thermodynamically consistent DPD model in Eq. (3) and the original DPD model of Eq. (2)? As in DPD, now particles of constant mass m move according to their velocities and exert forces of finite range to each other of different nature. The conservative forces of DPD are now replaced by a repulsive force directed along the line joining the particles that has a magnitude given by the pressure Pi and densities of the particles. Because the pressure Pi depends on the density, these type of force is not pair-wise but multibody [35]. The friction forces still depend on velocity differences between neighbor particles, but there is an additional term directly proportional to vi j . This new bit is necessary in order to have a faithful representation of the second space derivative terms that appear in the continuum equations of hydrodynamics [41]. In other words, it can be shown that, when thermal fluctuations can be neglected, Eq. (3) is a Lagrangian discretization of the continuum equations for hydrodynamics. Note that the friction coefficient is now given by the actual viscosity η of the fluid to be modeled and not an arbitrary tuning parameter. Finally, there is an additional dynamic equation for the entropy Si of the fluid particles. The terms in the entropy equation have a simple meaning as heat conduction and viscous heating. The heat conduction term tries to reduce temperature differences between particles by suitable energy exchange [42], whereas the viscous heating term proportional to the square of the velocities ensures that the kinetic energy dissipated by the friction forces is transformed into internal energy of the fluid particles. The model solves all the conceptual problems of DPD mentioned in the beginning of this section. In particular, the pressure and any other thermodynamic information is introduced as an input. The conservative forces of the


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original model become physically sounded pressure forces. Arbitrary equations of state and, in particular, of the van der Waals type can be used to study liquid–vapor coexistence in dynamic situations. Energy is conserved and we can study transport of energy in the system. The Second Law is satisfied. The transport coefficients are input of the model. The range functions of DPD enter in a very specific form, both in the conservative part of the dynamics through the density and pressure and in the dissipative part through the function Fi j . The particles have a physical size given by its physical volume and it is possible to specify the physical scale being simulated. The concept of resolution enters into play in the sense that one has to use many fluid particles per relevant length scale in order to recover the continuum results. Therefore, for resolving micron-sized objects one has to use very small fluid particles, whereas for resolving meter-sized objects large fluid particles are sufficient. In the model, it turns out that the amplitude of the thermal fluctuations scales with the square root of the volume of the fluid particles, in accordance with the usual notions of equilibrium statistical mechanics. Therefore, we expect that thermal fluctuations can be neglected in a simulation of meter-sized objects, but they are essential in the simulation of colloidal particles. This natural switching off thermal fluctuations with size is absent in the original DPD model. The model in Eq. (3) (without thermal fluctuations) is actually a version of the smoothed particle hydrodynamics (SPH) model, which is a Lagrangian particle method introduced by Lucy [43] and Monaghan [44] in the 70s in order to solve hydrodynamic problems in astrophysical contexts. Generalizations of SPH in order to include viscosity and thermal conduction and address laboratory scale situations like viscous flow and thermal convection have been presented only quite recently [42, 45, 46]. In order to formulate the thermodynamically consistent DPD model in Eq. (3), we have resorted to the GENERIC framework, which is a very elegant and useful way of writing dynamic equations that, by structure, are thermodynamically consistent [47]. It is possible to derive new fluid particle models based on both, the SPH methodology for discretizing continuum equations, and the GENERIC framework to ensure thermodynamic consistency. Continuum models for complex fluids typically involve additional structural or internal variables that are coupled with the conventional hydrodynamic variables. The coupling renders the behavior of the fluid non-Newtonian and complex. For example, polymer melts are characterized by additional conformation tensors, colloidal suspensions can be described by further concentration fields, mixtures are characterized by several density fields (one for each chemical species), emulsions are described with the amount and orientation of interface, etc. All these continuum models rely on the hypothesis of local equilibrium and, therefore, the fluid particles are regarded as thermodynamic subsystems. The physical picture that emerges from these fluid particles is that they represent “large” portions of the fluid and therefore, the scale of these

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fluid particles is supramolecular. This allows one to study large time scales. The price, of course, is the need for a deep understanding of the physics at this more coarse-grained level. In order to model polymer solutions, for example, ten Bosch [48] has associated to each dissipative particle an elongation vector representing the average elongation of polymer molecules. Although the ten Bosch model has all the problems of the original DPD model, it can be cast into a thermodynamically consistent model for non-isothermal dilute polymer solutions [49]. Another example where the strategy of internal variables can be successful is in the simulation of chemically reacting mixtures. Chemically reacting mixtures are not easily implemented with the usual approach taken by DPD in order to model mixtures. In DPD, mixtures are represented by “red” and “blue” particles. It is not trivial to specify a chemical reaction in which, for example, two red particles react with a blue particle to form a “green” particle. In this case, it is better to start from the well-established continuum equations for chemical reactions [41]. The fluid particles in the model have as additional variable the fraction of component red and blue inside the fluid particle. These two examples show how one can address viscoelastic flow problems and chemical reacting fluids with a simple methodology that involves fluid particles with internal variables. The idea can, of course, be applied to other complex fluids where the continuum equations are known.

Acknowledgments This work has been partially supported by the project BFM2001-0290 of the Spanish Ministerio de Ciencia y Tecnología.

References [1] P.J. Hoogerbrugge and J.M.V.A. Koelman, “Simulating microscopic hydrodynamics phenomena with dissipative particle dynamics,” Europhys. Lett., 19(3), 155–160, 1992. [2] P. Español and P. Warren, “Statistical mechanics of dissipative particle dynamics,” Europhys. Lett., 30, 191, 1995. [3] P. Español, “Hydrodynamics from dissipative particle dynamics,” Phys. Rev. E, 52, 1734, 1995. [4] C. Marsh, G. Backx, and M.H. Ernst, “Static and dynamic properties of dissipative particle dynamics,” Phys. Rev. E, 56, 1976, 1997. [5] P.B. Warren, “Dissipative particle dynamics,” Curr. Opinion Colloid Interface Sci., 3, 620, 1998. [6] J.M.V.A. Koelman and P.J. Hoogerbrugge, “Dynamic simulations of hard-sphere suspensions under steady shear,” Europhys. Lett., 21, 363–368, 1993.


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[7] E.S. Boek, P.V. Coveney, H.N.W. Lekkerkerker, and P. van der Schoot, “Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics,” Phys. Rev. E, 55(3), 3124–3133, 1997. [8] J.R. Melrose, J.H. van Vliet, and R.C. Ball, “Continuous shear thickening and colloid surfaces,” Phys. Rev. Lett., 77, 4660, 1996. [9] E.S. Boek and P. van der Schoot, “Resolution effects in dissipative particle dynamics simulations,” Int. J. Mod. Phys. C, 9, 1307, 1997. [10] M. Whittle and E. Dickinson, “On simulating colloids by dissipative particle dynamics: issues and complications,” J. Colloid Interface Sci., 242, 106, 2001. [11] M. Kao, A. Yodh, and D.J. Pine, “Observation of brownian motion on the time scale of hydrodynamic interactions,” Phys. Rev. Lett., 70, 242, 1993. [12] A.G. Schlijper, P.J. Hoogerbrugge, and C.W. Manke, “Computer simulation of dilute polymer solutions with dissipative particle dynamics,” J. Rheol., 39(3), 567–579, 1995. [13] Y. Kong, C.W. Manke, W.G. Madden, and A.G. Schlijper, “Effect of solvent qualityon the conformation and relaxation of polymers via dissipative particle dynamics,” J. Chem. Phys., 107, 592, 1997. [14] N.A. Spenley, “Scaling laws for polymers in dissipative particle dynamics,” Mol. Simul., 49, 534, 2000. [15] Y. Kong, C.W. Manke, W.G. Madden, and A.G. Schlijper, “Modeling the rheology of polymer solutions by dissipative particle dynamics,” Tribol. Lett., 3, 133, 1997. [16] A.G. Schlijper, C.W. Manke, W. GH, and Y. Kong, “Computer simulation of non-Newtonian fluid rheology,” Int. J. Mod. Phys. C, 8(4), 919–929, 1997. [17] Y. Kong, C.W. Manke, W.G. Madden, and A.G. Schlijper, “Simulation of a confined polymer on solution using the dissipative particle dynamics method,” Int. J. Thermophys., 15, 1093, 1994. [18] P.V. Coveney and K. Novik, “Computer simulations of domain growth and phase separation in two-dimensional binary immiscible fluids using dissipative particle dynamics,” Phys. Rev. E, 54, 5134, 1996. [19] S.I. Jury, P. Bladon, S. Krishna, and M.E. Cates, “Test of dynamical scaling in threedimensional spinodal decomposition,” Phys. Rev. E, 59, R2535, 1999. [20] K.E. Novik and P.V. Coveney, “Spinodal decomposition off of-critical quenches with a viscous phase using dissipative particle dynamics in two and three spatial dimensions,” Phys. Rev. E, 61, 435, 2000. [21] V.M. Kendon, J.-C. Desplat, P. Bladon, and M.E. Cates, “Test of dynamical scaling in three-dimensional spinodal decomposition,” Phys. Rev. Lett., 83, 576, 1999. [22] R.D. Groot and P.B. Warren, “Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation,” J. Chem. Phys., 107, 4423, 1997. [23] S.M. Willemsen, T.J.H. Vlugt, H.C.J. Hoefsloot, and B. Smit, “Combining dissipative particle dynamics and Monte Carlo techniques,” J. Comput. Phys., 147, 50, 1998. [24] C.M. Wijmans, B. Smit, and R.D. Groot, “Phase behavior of monomeric mixtures and polymer solutions with soft interaction potential,” J. Chem. Phys., 114, 7644, 2001. [25] R.D. Groot and T.J. Madden, “Dynamic simulation of diblock copolymer microphase separation,” J. Chem. Phys., 108, 8713, 1997. [26] R.D. Groot, T.J. Madden, and D.J. Tildesley, “On the role of hydrodynamic interactions in block copolymer microphase separation,” J. Chem. Phys., 110, 9739, 1999.

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[27] S. Jury, P. Bladon, M. Cates, S. Krishna, M. Hagen, N. Ruddock, and P.B. Warren, “Simulation of amphiphilic mesophases using dissipative particle dynamics,” Phys. Chem. Chem. Phys., 1, 2051, 1999. [28] M. Venturoli and B. Smit, “Simulating the self-assembly of model membranes,” Phys. Chem. Commun., 10, 1, 1999. [29] J.L. Jones, M. Lal, N. Ruddock, and N.A. Spenley, “Dynamics of a drop at a liquid/solid interface in simple shear fields: a mesoscopic simulation study,” Faraday Discuss., 112, 129, 1999. [30] P. Malfreyt and D.J. Tildesley, “Dissipative particle dynamics of grafted polymer chains between two walls,” Langmuir, 16, 4732, 2000. [31] S. Ymamoto, Y. Maruyama, and S. Hyodo, “Dissipative particle dynamics study of spontaneous vesicle formation of amphiphilic molecules,” J. Chem. Phys., 116(13), 5842, 2003. [32] R.D. Groot, “Electrostatic interactions in dissipative particle dynamics – simulation of polyelectrlytes and anionic surfactants,” J. Chem. Phys., 118, 11265, 2003. [33] J. Bonet-Avalós and A.D. Mackie, “Dissipative particle dynamics with energy conservation,” Europhys. Lett., 40, 141, 1997. [34] P. Español, “Dissipative particle dynamics with energy conservation,” Europhys. Lett., 40, 631, 1997. [35] I. Pagonabarraga and D. Frenkel, “Dissipative particle dynamics for interacting systems,” J. Chem. Phys., 115, 5015, 2001. [36] S.Y. Trofimov, E.L.F. Nies, and M.A.J. Michels, “Thermodynamic consistency in dissipative particle dynamics simulations of strongly nonideal liquids and liquid mixtures,” J. Chem. Phys., 117, 9383, 2002. [37] P. Español and M. Revenga, “Smoothed dissipative particle dynamics,” Phys. Rev. E, 67, 026705, 2003. [38] E.G. Flekkøy, P.V. Coveney, and G. DeFabritiis, “Foundations of dissipative particle dynamics,” Phys. Rev. E, 62, 2140, 2000. [39] M. Serrano and P. Español, “Thermodynamically consistent mesoscopic fluid particle model,” Phys. Rev. E, 64, 046115, 2001. [40] M. Serrano, G. DeFabritiis, P. Español, E.G. Flekkoy, and P.V. Coveney, “Mesoscopic dynamics of voronoi fluid particles,” J. Phys. A: Math. Gen., 35, 1605–1625, 2002. [41] S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics, North Holland, Amsterdam, 1964. [42] P.W. Cleary and J.J. Monaghan, “Conduction modelling using smoothed particle hydrodynamics,” J. Comput. Phys., 148, 227, 1999. [43] L.B. Lucy, “A numerical testing of the fission hypothesis,” Astron. J., 82, 1013, 1977. [44] J.J. Monaghan, “Smoothed particle hydrodynamics,” Annu. Rev. Astron. Astrophys., 30, 543–574, 1992. [45] H. Takeda, S.M. Miyama, and M. Sekiya, “Numerical simulation of viscous flow by smoothed particle hydrodynamics,” Prog. Theor. Phys., 92, 939, 1994. [46] O. Kum, W.G. Hoover, and H.A. Posch, “Viscous conducting flows with smoothparticle applied mechanics,” Phys. Rev. E, 52, 4899, 1995. ¨ [47] H.C. Ottinger and M. Grmela, “Dynamics and thermodynamics of complex fluids. II. Ilustrations of a general formalism,” Phys. Rev. E, 56, 6633, 1997. [48] B.I.M. ten Bosch, “On an extension of dissipative particle dynamics for viscoelastic flow modelling,” J. Non-Newtonian Fluid Mech., 83, 231, 1999. [49] M. Ellero, P. Español, and E.G. Flekkøy, “Thermodynamically consistent fluid particle model for viscoelastic flows,” Phys. Rev. E, 68, 041504, 2003.

8.7 THE DIRECT SIMULATION MONTE CARLO METHOD: GOING BEYOND CONTINUUM HYDRODYNAMICS Francis J. Alexander Los Alamos National Laboratory, Los Alamos, NM, USA

The Direct Simulation Monte Carlo method is a stochastic, particle-based algorithm for solving kinetic theory’s Boltzmann equation. Materials can be modeled at a variety of scales. At the quantum level, for example, time-dependent density functional theory or quantum Monte Carlo may be used. At the atomistic level, typically molecular dynamics is used, while at the continuum level, partial differential equations describe the evolution of conserved quantities and slow variables. Between the atomistic and continuum descriptions lives is the kinetic level. The ability to model at this level is crucial for electron and phonon transport in materials. For classical fluids, especially gases in certain regimes, modeling at this level is required. This article addresses computer simulations at the kinetic level.

1.

Direct Simulation Monte Carlo

The equations of continuum hydrodynamics, such as Euler and NavierStokes, model fluids under a variety of conditions. From capillary flow, to river flow, to the flow of galactic matter, these equations describe the dynamics of fluids over a wide range of space and time scales. However, these equations do not apply in important situations such as gas flow in nanoscale channels and flight in rarefied atmospheric conditions. Because these flows may be collisionless, or nonequilibrium or have sharp gradients, they require a finer-grained description than that provided by hydrodynamics. In these situations, the single particle distribution function, f (r, v, t) is used. Here, f is the number density of atoms or molecules in an infinitesimal six-dimensional volume of phase space, centered at location r and with 2513 S. Yip (ed.), Handbook of Materials Modeling, 2513–2522. c 2005 Springer. Printed in the Netherlands.

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velocity v. For dilute gases, Boltzmann was the first to determine how this distribution changes in time. His insight led to the equation that bears his name [1]: ∂ f (r, v, t) ∂t

=

+v·

dv1

∂ f (r, v, t) ∂r

+

F ∂ f (r, v, t) · m ∂v

d ( f (r, v , t) f (r, v1 , t) − f (r, v, t) f (r, v1 , t))|v − v1 |σ (v − v1 ).

(1)

The Boltzmann equation for hard spheres (1) accounts for all of the processes which change the particle distribution function. The advection term, v · (∂ f /∂r), accounts for the change in f due to particles’ velocities carrying them into and out of a given region of space around r. The force term, (F/m) · (∂ f /∂v) accounts for the change in f due to forces acting on particles of mass m to carry them into and out of a given region of velocity space around v. Terms on the right hand side represent the changes due to collisions. The first term on the right accounts for particles at r, with velocities v1 and v1 which, upon collision, are scattered into a small volume of velocity phase-space around v. The second term accounts for particles at r which, upon collision, are scattered out of this region of velocity space. The collision rate is given by σ and is a function of relative velocities. Though it provides the level of detail necessary to describe many important flows, the Boltzmann equation (1) has several features which make solving it extremely difficult. First, it is a nonlinear integro-differential equation. Only in special cases has it been amenable to exact analytic solution. Second, the Boltzmann equation lives in infinite dimensional phase space. Thus, the methods which work so well for partial differential equations cannot be used. As a result, approximate numerical methods are required. Monte Carlo methods are ideally suited for such high dimensional problems. In the early 1960s, Graeme Bird developed a Monte Carlo technique to solve the Boltzmann equation. This method, now known as Direct Simulation Monte Carlo (DSMC), has been extraordinarily successful in aerospace applications and is also gaining popularity with computational scientists in many fields. A brief outline of DSMC is given here. For more comprehensive descriptions, see Refs. [2–4]. The DSMC method solves the Boltzmann equation by using a representative sample of particles drawn from the actual single particle distribution function. Each DSMC particle represents Ne molecules in the original physical system. For flows of practical interest, typically Ne 1. This approach allows the modeling of extremely large systems while using a computationally tractable number of particles Ntot ≤ 108 , instead of a macroscopic number, 1023 .

The direct simulation monte carlo method

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A DSMC simulation is set up in the following way. First the spatial domain, boundary conditions and initial conditions of the simulation are specified. The domain is then partitioned into cells, typically, though not always, of uniform size. These cells are later used in the collision phase of the algorithm. Particles are placed according to a density distribution specified by the initial conditions. To guarantee accuracy, the number of particles used in the simulation should not be too small, i.e., not fewer than about 20 particles per cell [5]. Along with its spatial location ri , each particle is also initialized with a velocity vi . If the system is in equilibrium, this velocity distribution is Maxwellian. However, the velocity distribution can be set to accomodate any flow. The state of the DSMC system is given by the positions and velocities of the particles, {ri , vi }, for i = 1, . . . , N . The DSMC method simulates the dynamics of the single particle distribution using a two-step, splitting algorithm. These steps are advection and collision and model two physical processes at work in the Boltzmann equation. Advection models the free streaming between collisions, and the collision step models the two-body collisions. Each advection–collision step simulates a time t.

2.

Advection Phase

During the advection phase, all particles’ positions are changed from ri to ri + vi t. When a particle strikes a boundary or interface, it responds according to the appropriate boundary condition. The time of the collision is then determined by tracing the straight line trajectory from the initial location ri to the point of impact, rw . The time of flight from the particle’s initial position ˆ i · n), ˆ where nˆ is the unit norto the point of impact is tw = (rw − ri ) · n/(v mal to the surface. After striking the surface, the particle rebounds with a new velocity. This velocity depends on the boundary conditions. The particle then propagates freely for the remaining time t − tw . If, in the remaining time, the same particle again strikes a wall, this process is repeated until all of the time in that step has been exhausted. DSMC can model several types of boundaries (for example, specular surfaces, periodic boundaries, and thermal walls). Upon striking a specular surface, the component of a particle’s velocity normal to the surface is reversed. If a particle should strike a perfect thermal wall at temperature Tw , then all three components of the velocity are reset according to a biasedMaxwellian distribution. The resulting component normal to the wall is distributed as m 2 P⊥ (v ⊥ ) = v ⊥ e−mv ⊥ /2kTw . (2) kTw

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The individual parallel components are distributed as

P (v ) =

2 m e−mv /2kTw , 2πkTw

(3)

where Tw is the wall temperature, m is the particle’s mass and k is Boltzmann’s constant. Along with the tangential velocity component generated by thermal equilibration with the wall, an additional velocity is required to account for any translational motion of the wall. The distribution (3) is given in the rest frame of the wall. Assume the x and y axes are parallel to the wall. If the wall is moving in the lab frame, for example in the x-direction with velocity u w , then u w is added to the x-component of velocity for particles scattering off the wall. The components of the velocity of a particle leaving a thermal wall are then

vx =

vy =

v⊥ =

kTw RG + u w m

(4)

kTw R m G

(5)

−

2kTw ln R m

(6)

where R is a uniformly distributed random number in [0,1) and RG , RG are Gaussian distributed random numbers with zero mean and unit variance. For most engineering applications, gas-surface scattering is far more complicated. Nevertheless, these scattering rates can usually be effectively modeled in the gas-surface scattering part of the algorithm [6].

3.

Collision Phase

Interparticle collisions are addressed independently from the advection phase. For this to be accurate, the interaction potential between molecules must be short-range. While many short-range interaction models exist, the DSMC algorithm is formulated in this article for a dilute gas of hard sphere particles with diameter σ . When required for specific engineering applications, more realistic representations of the molecular interaction may be used [2, 8]. During the collision phase, some of the particles are selected at random to undergo collisions with each other. The selection process is determined by classical kinetic theory. While there are many ways to accomplish this, a simple and effective method is to sort the particles into spatial cells, the size of which should be less than a mean free path. Only particles in the same cell


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are allowed to collide. As with the particles themselves, the collisons are only statistical surrogates of the actual collisions that would occur in the system. At each time-step, and within each cell, sets of collisions then are generated. All pairs of particles in a cell are eligible to become collision partners. This eligibility is independent of the particles’ positions within the cell. Only the magnitude of the relative velocity between particles is used to determine their collision probability. Even particles that are moving away from each other may collide. The collision probability for a pair of hard spheres, i and j , is given by Pcoll (i, j ) =

2|vi − v j | Nc (Nc − 1)|v rel |

(7)

where |v rel | is the mean magnitude of the relative velocities of all pairs of particles in the cell, and Nc is the number of particles in the cell. To implement this in an efficient manner, a pair of potential collision partners, i and j , is selected at random from the particles within the cell. The pair collides if |vi − v j | > r, v r,max

(8)

where v r,max is the maximum relative speed in the cell and r is a uniform random variable chosen from the interval [0, 1). (Rather than determining v r,max exactly each time step, it is sufficient to simply update it everytime a relative velocity is actually calculated.) If the pair does not collide,then another pair is selected and the process repeats until the required number of pairs Mcoll (explained below) in the cell have been handled. If the pair does collide, then the new velocities of the particles are determined by the following procedure, and the process repeats until all collisions have been processed. In an elastic hard sphere collision, linear momentum and energy are conserved. These conserved quantities fix the magnitude of the relative velocity and center of mass velocity v r = |vi − v j | = |vi − vj | = v r ,

(9)

and vcm = 12 (vi + v j ) = 12 (vi + vj ) = vcm ,

(10)

where vi and vj are the post-collision velocities. In three dimensions, Eqs. (9) and (10) constrain four of the six degrees of freedom. The two remaining degrees of freedom are chosen at random. These correspond to the azimuthal and polar angles, θ and φ, for the post-collision relative velocity. vr = v r [(sin θ cos φ) xˆ + (sin θ sin φ) yˆ + cos θ zˆ ].

(11)

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For the hard sphere model, these angles are uniformly distributed over the unit sphere. Specifically, the azimuthal angle φ is uniformly distributed between 0 and 2π, and the angle θ has the following distribution P(θ) dθ = 12 sin θ dθ.

(12)

Since only sin θ and cos θ are required, it is convenient to change variables from θ to ζ = cos θ.Then ζ is chosen uniformly from [−1, 1], and setting cos θ = ζ and sin θ = 1 − ζ 2 . These values are used in (11). The post-collision velocities are then given by vi = vcm + 12 vr , vj = vcm − 12 vr .

(13)

The mean number of collisions that take place in a cell during a time-step is given by Nc (Nc − 1)πσ 2 v r Ne t , (14) 2Vc where Vc is the volume of the cell, and v r is the average relative velocity in the cell. To avoid the costly computation of v r , and since the ratio of total accepted to total candidates is v r Mcoll = . (15) Mcand v r,max Using (14) and (15) Mcoll =

Nc (Nc − 1)πσ 2 v r,max Ne t , (16) 2Vc produces the number of candidate pairs to select over a time step t. Note that Mcoll will, on average, equal the acceptance probability (8) multiplied by (16) and is independent of v r,max . Setting v r,max too high still processes the same number of collisions on the average, but the program is inefficient because the acceptance probability is low. This procedure selects collision pairs according to (7). Even if the value of v r,max is overestimated, the method is still correct, though less efficient because too many potential collisions are rejected. A better option is to make a guess which slightly overestimates v r,max [7]. To maintain accuracy while using the two-step, advection–collision algorithm, t should only be a fraction of the mean free time. If too large a time-step is used, then particles move too far between collisions. On the other hand, if the spatial cells are too large, then collisions can occur between particles which are “too far” from each other. Time steps beyond a mean free time and spatial cells larger than a mean free path have the effect of artificially enhancing transport coefficients such as viscosity and thermal conductivity [17, 18]. Mcand =


4.

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Data Analysis

In DSMC, as with other stochastic methods, most quantities of interest are computed as averages. For example, the instantaneous, fluctuating mass ˜ t), and energy density e(r, density, ρ(r, ˜ t), momentum density, p(r, ˜ t) are given by   ˜ t)   ρ(r,





1  m  ˜ t) = . p(r, mvi   Vs 1 2 i e(r, ˜ t) m|vi | 2

(17)

The sum is over particles over a volume of space surrounding r. Because it contains details of the single particle distribution function, DSMC can provide far more information than what is contained in the hydrodynamic variables above. However, this extra information comes at a price. As with other Monte Carlo-based methods, DSMC suffers from errors√due to the finite number of particles used. Convergence is typically O(1/ N ). These errors can be reduced by using more particles in the simulation, but for some systems, that can be prohibitive. For a detailed discussion on the statistical errors in DSMC and the techniques to estimate them in a variety of flow situations, refer to the recent work of Hadjiconstantinou et al. [9]. To reduce the fluctuations in the average quantities, a large number of particles is used, or, in the case of time-independent flows, statistics are gathered over a long run after the system has reached its steady state. For timedependent problems, a statistical ensemble of realizations of the simulation is used. Physical quantities of interest can be obtained from these averages. From the description of the algorithm above it should be clear that DSMC is computationally very expensive and should not be used in situations when Navier-Stokes or Euler PDE solvers apply. To check if DSMC is necessary, one should determine the Knudsen number K n. This dimensionless parameter is defined as K n = λ/L, where L is the characteristic length scale of the physical system, and λ is the molecular mean free path (i.e., the average distance between successive collisions of a given molecule). While there is no clear dividing line, a useful rule of thumb is that DSMC should be used when K n > 1/10. For a dilute gas, the mean free path λ is given by λ= √

1 2 πσ 2 n

,

(18)

where n is the number density, and σ is the effective diameter of the molecule. Air at atmospheric pressure has λ ≈ 50 nm. In the upper atmosphere, however, (e.g., > 100 km altitude), the mean free path is several meters. The Knudsen number for air flow through a nanoscale channel or around a meter scale space vehicle can therefore easily exceed K n ≈ 1. For these cases

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continuum hydrodynamics is not an option and DSMC should be used. Other, more detailed criteria can also be used [7].

5.

Discussion

Despite obvious similarities, key differences exist between DSMC and molecular dynamics. In molecular dynamics, the trajectory of every particle in the gas is computed from Newton’s equations, given an empirically determined interparticle potential. Each MD particle represents one atom or molecule. In DSMC, each particle represents Ne atoms or molecules, where Ne is on the order of 1/20 of the number of atoms/molecules in a cubic mean free path. Using MD to simulate one cubic micron of air at standard temperature and pressure MD requires integrating Newton’s equations for approximately 1010 molecules for 104 time steps to model one mean free time. With DSMC, only 105 particles and approximately 10 time steps are required. The DSMC method is therefore an efficient alternative for simulating a dilute gas. The method can be viewed as a simplified molecular dynamics (though DSMC is several orders of magnitude faster). DSMC can also be considered a Monte Carlo method for solving the time-dependent nonlinear Boltzmann equation. Instead of exactly calculating collisions as in molecular dynamics, the DSMC method generates collisions stochastically with scattering rates and post-collision velocity distributions determined from the kinetic theory of a dilute gas. Although DSMC simulations are not correct at the length scale of an atomic diameter, they are accurate at scales smaller than a mean free path. However, if more detail is required, then MD is the best option.

6.

Outlook

Though it originated in the aerospace community, since the mid-1980s DSMC has been used in a variety of other areas which demand a kinetic level formulation. These include the study of nonequilibrium fluctuations [10], nanoscale fluid dynamics [11] and granular gases [13]. Originally, DSMC was confined to to dilute gases. Several advances, however, such as the consistent Boltzmann algorithm (CBA) [8] and Enskog simulation Monte Carlo (ESMC) [12] have extended DSMC’s reach to nonideal, dense gases. Among other areas, CBA has found applications in heavy ion dynamics [14]. Similar methods also are used in transport theories of condensed matter physics [15]. While the DSMC method has been quite successful in these applications, only within the last decade has it been put on a firm mathematical foundation.


2521

Wagner [16], for example, proved that the method, in the limit of infinite particle number, has a deterministic evolution which solves an equation “close” to the Boltzmann equation. Subsequent work has shown that DSMC and its variants converge to a variety of kinetic equations. Other analytical work has determined the error incurred in DSMC by the use of a space and time discretization [17, 18]. Efforts have been made to improve the computational efficiency of DSMC for flows in which some spatial regions are hydrodynamic and others kinetic. Pareschi and Caflisch [19] have developed an implicit DSMC method which seamlessly interpolates between the kinetic and hydrodynamic scales. Another hybrid approach optimizes performance by using DSMC where required and then using Navier-Stokes or Euler in regions where allowed. The two methods are then coupled across an interface to provide information to each other [20, 21]. This is currently a rapidly growing field.

Acknowledgments This document was prepared at LANL under the auspices of the Department of Energy LA-UR 03-7358.

References [1] C. Cercignani, The Boltzmann Equation and its Applications, Springer, New York, 1988. [2] G.A. Bird, Molecular Gas Dynamics, Clarendon, Oxford, 1976; G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon, Oxford, 1994. [3] A.L. Garcia, Numerical Methods for Physics, Prentice Hall, Englewood Cliffs, 1994. [4] E.S. Oran, C.K. Oh, and B.Z. Cybyk, Annu. Rev. Fluid Mech., 30, 403, 1998. [5] M. Fallavollita, D. Baganoff, and J. McDonald, J. Comput. Phys., 109, 30, 1993; G. Chen and I. Boyd, J. Comput. Phys., 126, 434, 1996. [6] A.L. Garcia and F. Baras, Proceedings of the Third Workshop on Modeling of Chemical Reaction Systems, Heidelberg, 1997 (CD-ROM only). [7] I. Boyd, G. Chen, and G. Candler, Phys. Fluids, 7, 210, 1995. [8] F. Alexander, A.L. Garcia, and B. Alder, Phys. Rev. Lett., 74, 5212, 1995; F. Alexander, A.L. Garcia, and B. Alder, in 25 Years of Non-Equilibrium Statistical Mechanics, J.J. Brey, J. Marco, J.M. Rubi, and M. San Miguel (eds.), Springer, Berlin, 1995; A. Frezzotti, A particle scheme for the numerical solution of the Enskog equation, Phys. Fluids, 9(5), 1329–1335, 1997. [9] N. Hadjiconstantinou, A. Garcia, M. Bazant, and G. He, J. Comput. Phys., 187, 274–297, 2003. [10] F. Baras, M.M. Mansour, A.L. Garcia, and M. Mareschal, J. Comput. Phys., 119, 94, 1995. [11] F.J. Alexander, A.L. Garcia, and B.J. Alder, Phys. Fluids, 6, 3854, 1994. [12] J.M. Montanero and A. Santos, Phys. Rev. E, 54, 438, 1996; J. M. Montanero and A. Santos, Phys. Fluids, 9, 2057, 1997.

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F.J. Alexander

[13] H.J. Herrmann and S. Luding, Continuum Mechanics and Thermodynamics, 10, 189, 1998; J. Javier Brey, F. Moreno, R. García-Rojo, and M.J. Ruiz-Montero, “Hydrodynamic Maxwell demon in granular systems,” Phys. Rev. E, 65, 011305, 2002. [14] G. Kortemeyer, F. Daffin, and W. Bauer, Phys. Lett. B, 374, 25, 1996. [15] C. Jacoboni and L. Reggiani, Rev. Mod. Phys., 55, 645, 1983. [16] W. Wagner, J. Stat. Phys., 66, 1011, 1992. [17] F.J. Alexander, A.L. Garcia, and B.J. Alder, Phys. Fluids, 10, 1540, 1998; Phys. Fluids, 12, 731, 2000. [18] N.G. Hadjiconstantinou, Phys. Fluids, 12, 2634, 2000. [19] L. Pareschi and R.E. Caflisch, J. Comput. Phys., 154, 90, 1999. [20] H.S. Wijesinghe and N.G. Hadjiconstantinou, “Hybrid atomistic-continuum formulations for multiscale hydrodynamics,” Article 8.8, this volume. [21] A.L. Garcia, J.B. Bell, W.Y. Crutchfield, and B.J. Alder, J. Comput. Phys., 154, 134, 1999.

8.8 HYBRID ATOMISTIC–CONTINUUM FORMULATIONS FOR MULTISCALE HYDRODYNAMICS Hettithanthrige S. Wijesinghe and Nicolas G. Hadjiconstantinou Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Hybrid atomistic-continuum formulations allow the simulation of complex hydrodynamic phenomena at the nano and micro scales without the prohibitive cost of a fully atomistic approach. Hybrid formulations typically employ a domain decomposition strategy whereby the atomistic model is limited to regions of the flow field where required and the continuum model is implemented side-by-side in the remainder of the domain within a single computational framework. This strategy assumes that non-continuum phenomena are localized and that coupling of the two descriptions can be achieved in a spatial region where both formulations are valid. In this article we review hybrid atomistic-continuum methods for multiscale hydrodynamic applications. Both liquid and gas formulations are considered. The choice of coupling method and its relation to the fluid physics as well as the differences between incompressible and compressible hybrid methods are discussed using illustrative examples.

1.

Background

While the fabrication of MEMS devices has received much attention, transport mechanisms at the nano and micro scale environment are currently poorly understood. Furthermore, efficient and accurate design capabilities for nano and micro engineering components are also somewhat limited since design tools based on continuum formulations are increasingly reaching their limit of applicability. 2523 S. Yip (ed.), Handbook of Materials Modeling, 2523–2551. c 2005 Springer. Printed in the Netherlands.

2524

H.S. Wijesinghe and N.G. Hadjiconstantinou

For gases, deviation from the classical Navier–Stokes behavior is typically quantified by the Knudsen number, K n = λ/L where λ is the atomistic mean free path ( = 4.9 × 10−8 m for air) and L is a characteristic dimension. The Navier–Stokes formulation is found to be invalid for K n 0.1. Ducts of width 100 nm or less which are common in N/MEMS correspond to Knudsen numbers of order 1 or above [1]. The Knudsen number for Helium leak detection devices and mass spectrometers can reach values of up to 200 [2]. Also material processing applications such as chemical vapor deposition and molecular beam epitaxy involve high Knudsen number flow regimes [3]. The Navier–Stokes description also deteriorates in the presence of sharp gradients. One example comes from Navier–Stokes formulations for high Mach number shock waves which are known to generate spurious post-shock oscillations [4, 5]. In such cases, a Knudsen number can be defined using the characteristic length scale of the gradient. A significant challenge therefore exists to develop accurate and efficient design tools for flow modeling at the nano and micro scales. Liquids in nanoscale geometries or under high stress and liquids at material interfaces may also exhibit deviation from Navier–Stokes behavior [6]. Examples of problems which require modeling at the atomistic scale include the moving contact-line problem between two immiscible liquids [6], corner singularities, the breakup and merging of droplets [7], dynamic melting processes [8], crystal growth from a liquid phase and polymer/colloid wetting near surfaces. Accurate modeling of wetting phenomena is of particular concern in predicting microchannel flows. While great accuracy can be obtained by an atomistic formulation over a broader range of length scales, a substantial computational overhead is associated with this approach. To mitigate this cost, “hybrid” atomisticcontinuum simulations have been proposed as a novel approach to model hydrodynamic flows across multiple length and time scales. These hybrid approaches limit atomistic models to regions of the flow field where needed, and allow continuum models to be implemented in the remainder of the domain within a single computational framework. A hybrid method therefore allows the simulation of complex hydrodynamic phenomena which require modeling at the microscale without the prohibitive cost of a fully atomistic calculation. In what follows we provide an overview of this rapidly expanding field and discuss recent developments. We begin by discussing the challenges associated with hybrid formulations, namely the choice of the coupling method and the imposition of boundary conditions on atomistic simulations. We then illustrate hybrid methods for incompressible and compressible flows by describing recent archetypal approaches. Finally we discuss the effect of statistical fluctuations in the context of developing robust criteria for adaptive placement of the atomistic description.

Hybrid atomistic–continuum formulations

2.

2525

Challenges

Over the years a fair number of hybrid simulation frameworks have been proposed leading to some confusion over the relative merits and applicability of each approach. Original hybrid methods focused on dilute gases [9–12], which are arguably easier to deal with within a hybrid framework than dense fluids, mainly because boundary condition imposition is significantly easier in gases. The first hybrid methods for dense fluids appeared a few years later [13–16]. These initial attempts have led to a better understanding of the challenges associated with hybrid methods. Coupling the continuum and atomistic formulations requires a region of space where information exchange takes place between the two descriptions. This information exchange between the two subdomains is typically in the form of state variables and/or hydrodynamic fluxes, with the latter typically measured across the matching interface. This process may be viewed as a boundary condition exchange between subdomains. In some cases transfer of information is facilitated by an overlap region. The transfer of information from the atomistic subdomain to the continuum subdomain is fairly straightforward. A hydrodynamic field can be obtained from atomistic simulation data through averaging (see for example the article “The Direct Simulation Monte Carlo: going beyond continuum hydrodynamics” in the Handbook). The relative error due to statistical sampling in atomistic hydrodynamic formulations was also recently characterized [17]. Imposition of the latter data as boundary conditions on the continuum method is also well understood and depends on the numerical method used (see article “Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations” of the Handbook). As discussed below, the most challenging aspect of the information exchange lies in imposing the hydrodynamic field obtained from the continuum subdomain onto the atomistic description, a process which is not well defined in the absence of the complete distribution function (hydrodynamic fields correspond to the first few moments of the distribution function). Thus to a large extent, the two major issues in developing a hybrid method is the choice of a coupling method and the imposition of boundary conditions on the atomistic simulation. Generally speaking, these two can be viewed as decoupled. The coupling technique can be developed on the basis of matching two compatible and equivalent over some region of space hydrodynamic descriptions and can thus be borrowed from the already existing and extensive continuum-based numerical methods literature. Boundary condition imposition can be posed as a general problem of imposing “macroscopic” boundary conditions on an atomistic simulation. In our opinion this is a very challenging problem that has not been, in general, resolved to date completely satisfactorily. Boundary condition imposition on the atomistic subdomain is discussed shortly.

2526

3. 3.1.


Developing a Hybrid Method The Choice of Coupling Method

Coupling a continuum to an atomistic description is meaningful in a region where both can be presumed valid. In choosing a coupling method it is therefore convenient to draw upon the wealth of experience and large cadre of coupling methods nearly 50 years of continuum computational fluid dynamics have brought us. Coupling methods for the compressible and incompressible formulations generally differ, since the two correspond to two different physical and mathematical limits. Faithful to their mathematical formulations, the compressible formulation lends itself naturally to time-explicit flux-based coupling while incompressible formulations are typically coupled using either state properties (Dirichlet) or gradient information (Neumann). Given that the two formulations have different limits of applicability and physical regimes in which each is significantly more efficient than the other, care must be exercised when selecting the ingredients of the hybrid method. In other words, the choice of a coupling method and continuum subdomain formulation needs to be based on the degree to which compressibility effects are important in the problem of interest, and not on a preset notion that a particular coupling method is more appropriate than all others. The latter approach was recently pursued in a variety of studies which enforce the use of the compressible formulation to steady and essentially incompressible problems to achieve coupling by time-explicit flux matching. This approach is not recommended. On the contrary, for an efficient simulation method, similarly to the case of continuum solution methods, it is important to allow the flow physics to dictate the appropriate formulation, while the numerical implementation is chosen to cater to the particular requirements of the latter. Below, we expand on some of the considerations which influence the choice of coupling method under the assumption that the hybrid method is applied to problems of practical interest and therefore the continuum subdomain is appropriately large. Our discussion focuses on timescale considerations that are more complex but equally important to limitations resulting from lengthscale considerations, such as the size of the atomistic region(s). It is well known that the timestep for explicit integration of the compressible Navier–Stokes formulation, τc , scales with the physical timestep of the problem, τx , according to [18] M τx (1) 1+ M where M is the Mach number. As the Mach number becomes small, we are faced with the classical stiffness problem whereby the numerical efficiency of the solution method suffers [18] due to disparity of the time scales in the τc ≤


2527

system of governing equations. For this reason, when the Mach number is small, the incompressible formulation is used which allows integration at the physical timestep τx . In the hybrid case matters are complicated by the introduction of the atomistic integration timestep, τm , which is at most of the order of τc in gases (if the discretization scale is O(λ)) and in most cases significantly smaller. Thus as the global domain of interest grows, the total integration time grows, and transient calculations in which the atomistic subdomain is explicitly integrated in time become more computationally expensive and eventually infeasible. The severity of this problem increases with decreasing Mach number and makes unsteady incompressible problems very computationally expensive. New integrative frameworks which coarse grain the time integration of the atomistic subdomain are therefore required. Fortunately, for low speed steady problems implicit (iterative) methods exist which provide solutions without the need for explicit integration of the atomistic subdomain to the global problem steady state. One such implicit method is discussed in this review; it is known as the Schwarz method. This method decouples the global evolution timescale from the atomistic evolution timescale (and timestep) by achieving convergence to the global problem steady state through an iteration between steady state solutions of the continuum and atomistic subdomains. Since the atomistic subdomain is small, explicit integration to its steady state is feasible. Although the steady assumption may appear restrictive, it is interesting to note that the majority of both compressible and incompressible test problems solved to date have been steady. A variety of other iterative methods may be suitable if they provide for timescale decoupling. The choice of the Schwarz coupling method using state variables versus a flux matching approach was motivated by the fact (as explained below) that state variables suffer from smaller statistical noise and are thus easier to prescribe on a continuum formulation. The above observations do not preclude the use of the compressible formulation in the continuum subdomain for low speed flows. In fact, preconditioning techniques which allow the use of the compressible formulation at very low Mach numbers have been developed [18]. Such a formulation can, in principle, be used to solve for the continuum subdomain while this is being coupled to the atomistic subdomain via an implicit (eg., Schwarz) iteration. What should be avoided is a time-explicit compressible flux-matching coupling procedure for solving essentially incompressible steady state problems. The issues discussed above have not been very apparent to date because in typical test problems published so far, the continuum and atomistic subdomains are of the same size (and, of course, small). In this case the large cost of the atomistic subdomain masks the cost of the continuum subdomain and also typical evolution timescales (or times to steady state) are small. It should not be forgotten, however, that hybrid methods make sense when the continuum subdomain is significantly larger than the atomistic subdomain.

2528


The stiffness resulting from a small timestep in the atomistic subdomain may be remedied by implicit timestepping methods [19]. However, flux-based coupling additionally suffers from adverse signal to noise ratios in connection with the averaging required for imposition of boundary conditions from the atomistic subdomain to the continuum subdomain. In the case of an ideal gas it has been shown for low speed flows [17] that for the same number of samples, flux (shear stress, heat flux) averaging exhibits relative noise, E f , which scales as Ef ∝

E sv Kn

(2)

where E sv is the relative noise in √ the corresponding state variable (velocity, temperature) which varies as 1/ (number of samples). Here K n is based on the characteristic lengthscale of the transport gradients. Since, by assumption, in the matching region a continuuum description is appropriate, we expect K n = λ/L 1. It thus appears that flux coupling will be significantly disadvantaged in this case, since 1/K n 2 times the number of samples required by state-variable averaging is required to achieve comparable noise levels in the matching region. Statistical noise has important implications on hybrid methods which will be discussed throughout this paper. The effect of statistical noise becomes of critical importance in unsteady incompressible flows which are discussed later.

4.

Boundary Condition Imposition

Consider the boundary, ∂ of the atomistic region on which we wish to impose a set of hydrodynamic (macroscopic) boundary conditions. Typical implementations require the use of particle reservoirs R (see Fig. 1) in which particle dynamics may be altered in such a way that the desired boundary conditions appear on ∂; the hope is that the influence of the perturbed dynamics in the reservoir regions decays sufficiently fast and does not propagate into the region of interest, that is, the relaxation distance both for the velocity distribution function and the fluid structure is small compared to the characteristic scale of . Since ∂ represents the boundary with the continuum region, R extends into the continuum subdomain. Knowledge of the continuum solution in R is typically used to aid imposition of the above on ∂. In a dilute gas, the non-equilibrium distribution function in the continuum limit has been characterized [20] and is known as the Chapman–Enskog distribution. Use of this distribution to impose boundary conditions on atomistic simulations of dilute gases results in a robust, accurate and theoretically elegant approach. Typical implementations [21] require the use of particle generation and initialization within R. Particles that move into within the


2529 ∂Ω

Ω

R

Figure 1. Continuum to atomistic boundary condition imposition using reservoirs.

simulation timestep are added to the simulation whereas particles remaining in R are discarded. For liquids, both the particle velocity and the fluid structure distribution functions are important and need to be imposed. Unfortunately no theoretical results for these distributions exist. A related issue is that of domain termination; due to particle interactions, , or in the presence of a reservoir R, needs to be terminated in a way that does not have significant effect on the fluid state inside of . As a result, researchers have experimented with possible methods to impose boundary conditions. It is now known that similarly to a dilute gas, use of a Maxwell–Boltzmann distribution for particle velocities leads to slip [14]. In [22] a Chapman–Enskog distribution is used to impose boundary conditions to generate a liquid shear flow. In this approach, particles crossing ∂ acquire velocities that are drawn from a Chapman–Enskog distribution parametrized by the local values of the required velocity and stress boundary condition. Although this approach was only tested for a Couette flow, it appears to give reasonable results (within atomistic fluctuations). Since in Couette flow no flow normal to ∂ exists, ∂ can be used as symmetry boundary separating two back-to-back shear flows; this sidesteps the issue of domain termination. Boundary conditions on MD simulations can also be imposed through the method of constraint dynamics [13]. Although the approach in [13] did not allow hydrodynamic fluxes across the matching interface, the latter feature can be integrated into this approach with a suitable domain termination. In a different approach [16] external forces are used to impose boundary conditions. More specifically, the authors apply an external field with a magnitude such that the total force on the fluid particles in R is the one required by momentum conservation as required by the coupling procedure.

2530


The outer boundary of the reservoir region is terminated by using a force preventing particles from leaving the domain and an ad-hoc weighting factor for the force distribution on particles. This weighting factor diverges as particles approach the edge of R. This prevents their escape and also ensures new particles introduced in R move towards . Particles introduced into the reservoir are given velocities drawn from a Maxwell–Boltzmann distribution, while a Langevin thermostat keeps the temperature constant. The method appears to be successful although the non-unique choice of force fields and Maxwell– Boltzmann distribution makes it not very theoretically pleasing. It is also not clear what the effect of these forces are on the local fluid state (it is well known that even in a dilute gas gravity driven flow [23] exhibits significant deviations from Navier–Stokes behavior) but this effect is probably small since force fields are only acting in the reservoir region. The above approach was refined [24] by using a version of the Usher algorithm to insert particles in the energy landscape such that they have the desired specific energy, which is beneficial to imposing a desired energy current while eliminating the risk of particle overlap at some computational cost. This approach uses a Maxwell– Boltzmann distribution, however, for the initial velocities of the inserted particles. Temperature gradients are imposed by a small number of thermostats placed in the direction of the gradient. Although no proof exists that the disturbance to the particle dynamics is small, it appears that this technique is successful at imposing boundary conditions with moderate error [24]. A method for terminating incompressible molecular dynamics simulations with small effect on particle dynamics has been suggested and used [14]. This simply involves making the reservoir region fully periodic. In this manner, the boundary conditions on ∂ also impose a boundary value problem on R, where the inflow to is the outflow from R. As R becomes bigger, the gradients in R become smaller and thus the flowfield in R will have a small effect on the solution in . The disadvantage of this method is the number of particles that are needed to fill R as this grows, especially in high dimensions. We believe that significant contributions can still be made by developing methods to impose boundary conditions in hydrodynamically consistent and, most importantly, rigorous approaches.

4.1.

Particle Generation in Dilute Gases Using the Chapman–Enskog Velocity Distribution Function

In the case of dilute gases the atomistic structure is not important and the gas is characterized by the single-particle distribution function. This relative simplicity has led to solutions of the governing Boltzmann equation [25, 26] in the Navier–Stokes limit. The resultant Chapman–Enskog solution [20, 25] can be used to impose boundary conditions in a robust and rigorous manner.


2531

In what follows we illustrate this procedure using the direct simulation Monte Carlo (DSMC) as our dilute gas simulation method. DSMC is an efficient method for the simulation of dilute gases which solves the Boltzmann equation [27] using a splitting approach. The time evolution of the system is approximated by a sequence of discrete timesteps, t, in which particles undergo, successively, collisionless advection and collisions. An appropriate number of collisions are performed between randomly chosen particle pairs within small cells of linear size x. DSMC is discussed further in the article, “The Direct Simulation Monte Carlo Method: going beyond continuum hydrodynamics” of the Handbook. In a typical hybrid implementation, particles are created in a reservoir region in which the continuum field to be imposed is known. Correct imposition of boundary conditions requires generation of particles with the correct single particle distribution function which includes the local value of the number density [21]. Current implementations [21, 28, 29] show that linear interpolation of the density gradient within the reservoir region provides sufficient accuracy. Generation of particles according to a linear density gradient can be achieved with a variety of methods, including acceptance-rejection schemes. In the next section we outline an approach for generation of particle velocities from a Chapman–Enskog distribution parametrized by the required flow variables. After particles are created in the reservoir they move for a single DSMC timestep. Particles that enter the atomistic region are incorporated into the standard convection/collision routines of the DSMC algorithm. Particles that remain in the reservoir are discarded. Particles that leave the atomistic region are also deleted from the computation.

4.2.

Generation of Particle Velocities Using the Chapman–Enskog Velocity Distribution

The Chapman–Enskog velocity distribution function f (C) can be written as [30], f (C) = f 0 (C)(C)

(3)

where, C = C/(2kT /m) 1 ∈ f 0 (C) = 3/2 e−C π and,

1/2

is the normalized thermal velocity, (4)

2 2 C −1 (C) = 1 + (qx Cx + q y C y + qz Cz ) 5 − 2(τx y Cx C y + τx z Cx Cz + τ yz C y Cz ) − τx x (Cx2 − Cz2 ) − τ yy (C y2 − Cz2 )

(5)

2532


with,

2m 1/2 ∂ T kT ∂ xi µ ∂v i ∂v j 2 ∂v k τi j = + − δi, j P ∂x j ∂ xi 3 ∂ xk qi = −

κ P

(6) (7)

where qi and τi j are the dimensionless heat flux vector and stress tensor respectively with µ, κ, P and v = (v x , v y , v z ) being the viscosity, thermal conductivity, pressure and mean fluid velocity. An “Acceptance–Rejection” scheme [30, 31] can be utilized to generate Chapman–Enskog distribution velocities. In this scheme an amplitude parameter A = 1 + 30B is first chosen where B = max(|τi j |, |qi |). Next a trial velocity Ctry is drawn from the Maxwell–Boltzmann equilibrium distribution function f 0 given by Eq. (4). Note f 0 is a normal (Gaussian) distribution that can be generated using standard numerical techniques [32]. The trial velocity Ctry is accepted if it satisfies AR ≤ (Ctry ) where R is a uniform deviate in [0, 1). Otherwise a new trial velocity Ctry is drawn. The final particle velocity is given by C = (2kT /m)1/2 Ctry + v

5.

(8)

Incompressible Formulations

Although in some cases compressibility may be important, a large number of applications are typically characterized by flows where use of the incompressible formulation results in a significantly more efficient approach [18]. As explained earlier, our definition of incompressible formulation is based on the flow physics and not on the numerical method used. Although in our example implementation below we have used a finite element discretization based on the incompressible formulation, we believe that a preconditioned compressible formulation [18] could also be used to solve the continuum subdomain problem if it could be successfully matched to the atomistic solution through a coupling method which takes into account the incompressible nature of the (low speed) problem to provide solution matching consistent with the flow physics. From the variety of methods proposed to date, it is becoming increasingly clear that almost any continuum–continuum coupling method can be used so long as it is properly extended to account for boundary condition imposition. The challenge thus lies more in choosing a method that best matches the physics of the problem of interest (as explained above) rather than developing general methods for large classes of problems. Below we illustrate a hybrid implementation appropriate for incompressible steady flow using the Schwarz alternating coupling method.


2533

Before we proceed with our example, a subtle numerical issue associated with the incompressible formulation should be discussed. Due to inherent statistical fluctuations, boundary conditions obtained from the atomistic subdomain may lead to mass conservation discrepancies. Although this phenomenon is an artifact of the finite sampling, in the sense that if a sufficiently large (infinite) number of samples are taken the mean field obtained from the atomistic simulation should be appropriately incompressible, it is sufficient to cause a numerical instability in the continuum calculation. A simple correction that can be applied consists of removing the discrepancy in mass flux equally across all normal velocity components of the atomistic boundary data that are to be imposed on the continuum subdomain. If 1 is the portion of the continuum subdomain φ that receives boundary data from the atomistic subdomain (φ ⊇ 1 ), n is the unit outward normal vector to φ, and d S is a differential element of φ, the correction to the atomistic data on 1 , v1 , can be written as

(v1 .n)corrected = v1 .n −

φ

vφ .ndS

1

dS

(9)

Tests with various problems [14, 15, 28] indicate that this simple approach is successful at removing the numerical instability.

5.1.

The Schwarz Alternating Method for Steady Flows

The Schwarz method was originally proposed for molecular dynamicscontinuum methods [14, 15], but it is equally applicable to DSMC-continuum hybrid methods [28, 33]. This approach was chosen because of its ability to couple different descriptions through Dirichlet boundary conditions (easier to impose on liquid atomistic simulations compared to flux conditions, because fluxes are non-local in liquid systems), and its ability to reach the solution steady state in an implicit manner which requires only steady solutions from each subdomain. The importance of the latter characteristic cannot be overemphasized; the implicit convergence in time through steady solutions guarantees timescale decoupling that is necessary for the solution of macroscopic problems; the integration of atomistic trajectories at the atomistic timestep for total times corresponding to macroscopic evolution times is, and will for a long time be, infeasible, while integration of the small molecular region to its steady state solution is feasible. A continuum–continuum domain decomposition can be used to illustrate the Schwarz alternating method as shown graphically in Figs. 2–4 (adapted from [34]) to solve for the velocity in a simple, one-dimensional problem, a pressure driven Poiseuille flow. Starting with a zero guess for the solution in domain 2, the first steady solution in domain 1 can be obtained. This provides the first boundary condition for a steady solution in domain 2 (Fig. 2). The

2534


Domain 1 Domain 2 Overlap Region

Wall

Wall First BC for domain 2 First BC for domain 1

Domain 1 first iteration a x

b L

Figure 2. Schematic illustrating the Schwarz alternating method for Poiseuille flow. Solution at the first Schwarz iteration. Adapted from [34].

Figure 3. Schematic illustrating the Schwarz alternating method for Poiseuille flow. Solution at the second Schwarz iteration. Adapted from [34].


2535

Figure 4. Schematic illustrating the Schwarz alternating method for Poiseuille flow. Solution at the third Schwarz iteration. Adapted from [34].

new solution in domain 2 provides an updated second boundary condition for domain 1 (Fig. 3). This process is repeated until the two solutions are identical in the overlap region. As seen in Fig. 4 the solution across the complete domain rapidly approaches the steady state solution. This method is guaranteed to converge for elliptic problems [35]. The Schwarz method was recently applied [33] to the simulation of flow through micromachined filters. These filters have passages that are sufficiently small that require an atomistic description for the simulation of the flow through them. Depending on the geometry and number of filter stages the authors have reported computational savings ranging from 2 to 100.

5.2.

Driven Cavity Test Problem

In this section we solve the steady driven cavity problem using the Schwarz alternating method. The driven cavity problem is used here as a test problem for verification and illustration purposes. In fact, although wall effects might be important in small scale flows, and a hybrid method which treats only the regions close to the walls using the molecular approach may be an interesting problem, the formulation chosen here is such that no molecular effects are present. This is achieved by placing the molecular description in the center of the computational domain such that it is not in contact with any of the system

2536


walls (see Fig. 5). The rationale is that the hybrid solution of this problem should reproduce the full Navier–Stokes solution and thus the latter can be used as a benchmark result. In our formulation the continuum subdomain is described by the Navier– Stokes equations solved by finite element discretization. Standard Dirichlet velocity boundary conditions for a driven cavity problem were applied on the system walls which in this implemetation are captured by the continuum subdomain; the horizontal velocity component on the left, right and lower walls were held at zero, while on the upper wall it was set to 50 m/s. The vertical velocity component on all boundaries was set to zero. Boundary conditions from the atomistic domain are imposed on nodes that have been centered on DSMC cells (see Fig. 6). The pressure is scaled by setting the middle node on the lower boundary at atmospheric pressure (1.013×105 Pa). Despite the relatively high flow velocity, the flow is essentially incompressible and isothermal.

Figure 5. Continuum and atomistic subdomains for Schwarz coupling for the twodimensional driven cavity problem.


2537

Figure 6. Boundary condition exchange. Only the bottom left corner of the matching region is shown for clarity. Particles are created with probability density proportional to the local number density.

The imposition of boundary conditions on the atomistic subdomain is facilitated by a particle reservoir as shown in Fig. 6. Note that in this particular implementation the reservoir region serves also as part of the overlap region, thus reducing the overall cost of the molecular description. Particles are created at locations x, y within the reservoir with velocities C x , C y drawn from a Chapman–Enskog velocity distribution. The Chapman Enskog distribution is generated, as explained above, by using the mean and gradient of velocities from the continuum solution; the number and spatial distribution of particles in the reservoir are chosen according to the overlying continuum cell mean density and density gradients. The rapid convergence of the Schwarz approach is demonstrated in Fig. 7. The continuum numerical solution is reached to within ±10% at the 3rd Schwarz iteration and to within ±2% at the 10th Schwarz iteration. The error estimate which includes the effects of statistical noise [17] and discretization error due to finite timestep and cell size is approximately 2.5%. Similar convergence is also observed for the velocity field in the vertical direction. The close agreement with the fully continuum results indicates that the Chapman–Enskog procedure is not only theoretically appropriate but also robust. Despite a Reynolds number of Re ≈ 1, the Schwarz method converges

2538


Figure 7. Convergence of the horizontal velocity component along the Y = 0.425 × 10−6 m plane with successive Schwarz iterations.

with negligible error. This is in agreement with findings [36] which have recently shown that the Schwarz method is expected to converge for Re ∼ O(1).

5.3.

Unsteady Formulations

Unsteady incompressible calculations are particularly challenging for two reasons. First, due to the low flow speeds associated with them and the associated large number of samples required, the computational cost of the atomistic subdomain simulation rises sharply. Second, because of Eq. (1) and the fact that τm τc (typically), explicit time integration to the time of interest is very expensive. Approaches which use explicit time coupling based on compressible fluxmatching schemes have been proposed for these flows but it is not at all clear that these approaches provide the best solution. First, they suffer from signal to noise problems more than state-variable based methods. Second, integration of the continuum subdomain using the compressible formulation for an incompressible problem becomes both expensive and inaccurate [18]. On the other hand, iterative methods require a number of re-evaluations of the molecular


2539

solution to achieve convergence. This is an additional computational cost that is not shared by the time-explicit coupling and leads to a situation whereby (for incompressible unsteady problems) the choice between a time-explicit fluxmatching coupling formulation or an iterative (Schwarz-type) coupling formulation is not clear and may be problem dependent. An alternative approach would be the adaptation of non-iterative continuum-continuum coupling techniques which take into account the incompressible nature of the problem and avoid the use of flux matching, such as the coupling approach presented in O’Connell and Thompson [13]. We should also recall that from Eq. (1), unless time coarse-graining techniques are developed, large, low-speed, unsteady problems are currently too expensive to be feasible by any method.

6.

Compressible Formulations

As discussed above, consideration of the compressible equations of motion leads to hybrid methods which differ significantly from their incompressible counterparts. The hyperbolic nature of compressible flows means that steady state formulations typically do not offer a significant computational advantage, and as a result, explicit time integration is the preferred solution approach and flux matching is the preferred coupling method. Given that the characteristic evolution time, τh , scales with the system size, the largest problem that can be captured by a hybrid method is limited by the separation of scales between the atomistic integration time and τh . Local mesh refinement techniques [21, 29] minimize the regions of space that need to be integrated at small CFL timesteps (due to a fine mesh), such as the regions adjoining the atomistic subdomain. Implicit timestepping methods [19] can also be used to speed up the time integration of the continuum subdomain. Unfortunately, although both approaches enhance the computational efficiency of the continuum sub-problem, they do not alleviate the issues arising from the disparity between the atomistic timestep and the total integration time. Compressible hybrid continuum-DSMC approaches are popular because compressible behavior is often observed in gases. In these methods, locally refining the continuum solution cells to the size of DSMC cells leads to a particularly seamless hybrid formulation in which DSMC cells differ from the neighboring continuum cells only by the fact that they are inherently fluctuating. The DSMC timestep required for accurate solutions [37–39] is very similar to the CFL timestep of a compressible formulation, and thus a finite volume formulation can be used to couple the two descriptions (for finite volume methods see the article, “Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations” in the Handbook). In such a method [9, 10, 40] the flux of mass, momentum and energy from DSMC

2540


to the continuum domain can be used directly for finite volume integration. Going from the continuum solution to DSMC requires the use of reservoirs. A DSMC reservoir extending into the continuum subdomain is attached at the end of the DSMC subdomain and initialized using the overlying continuum field properties. Flux of mass, momentum and energy is then provided by the particles entering the DSMC subdomain from the reservoir. The particles leaving the DSMC subdomain to the reservoir are discarded (after their contribution to mass, momentum and energy flux to the continuum is recorded). Another characteristic inherent to compressible formulations is the possibility of describing parts of the domain by the Euler equations of motion [29]. In that case, consistent coupling to the atomistic formulation can be performed using a Maxwell–Boltzmann distribution [21]. It has been shown [41] that explicit time-dependent flux-based formulations preserve the fluctuating nature of the atomistic description within the atomistic regions but the fluctuation amplitude decays rapidly within the continuum regions; correct fluctuation spectra can be obtained in the entire domain by solving a fluctuating hydrodynamics formulation [42] in the continuum subdomain. Below we discuss a particular hybrid implementation to illustrate atomisticcontinuum coupling in the compressible limit. We would like to emphasize again that a variety of methods can be used, although the compressible formulation is particularly well suited to flux matching. The method illustrated here is an extended version of the original Adaptive Mesh and Algorithm Refinement (AMAR) method [21]. This method was chosen since it is both the current state of the art in compressible fully adaptive hybrid methods and since it also illustrates how existing continuum multiscale techniques can be used directly for atomistic-continuum coupling with minimum modification.

6.1.

Fully Adaptive Mesh and Algorithm Refinement for a Dilute Gas

The compressible adaptive mesh and algorithm refinement formulation of Garcia et al., [21], referred to as AMAR, pioneered the use of mesh refinement as a natural framework for the introduction of the atomistic description in a hybrid formulation. In AMAR the typical continuum mesh refinement capabilities are supplemented by an algorithmic refinement (continuum to atomistic) based on continuum breakdown criteria. This seamless transition is both theoretically and practically very appealing. In what follows we briefly discuss a recently developed [29] fully adaptive AMAR method. In this method DSMC provides an atomistic description of the


2541

flow while the compressible two-fluid Euler equations serve as the continuumscale model. Consider the Euler equations in conservative integral form d dt where

U dV +

φ

F · nˆ dS = 0

(10)

∂φ



    U=   

ρ px py pz e ρc





    ;   

    x F =   

ρu x ρu 2x + P ρu x u y ρu x u z (e + P)u x ρcu x

        

(11)

Only the x-direction component of the flux terms are listed here; other directions are similar. A two-species gas is assumed with the mass concentrations being c and (1 − c). Discrete time integration is achieved by using a finite volume approximation to Eq. (10). This yields a conservative finite difference expression with Unij k appearing as a cell-centered quantity at each x,n+1/2 time level and Fi+1/2, j,k located at faces between cells at half-time levels. A second-order version of an unsplit Godunov scheme is used to approximate the fluxes [43–45]. Time stepping on an AMR grid hierarchy involves interleaving time steps on individual levels [46]. Each level has its own spatial grid resolution and timestep (typically constrained by a CFL condition). The key to achieving a conservative AMR algorithm is to define a discretization for Eq. (10) that holds on every region of the grid hierarchy. In particular, the discrete cell volume integrals of U and the discrete cell face integrals of F must match on the locally-refined AMR grid. Thus, integration of a level involves two steps: solution advance and solution synchronization with other levels. Synchronizing the solution across levels assumes that fine grid values are more accurate than coarse grid values. So, coarse values of U are replaced by suitable cell volume averages of finer U data where levels overlap, and discrete fine flux integrals replace coarse fluxes at coarse-fine grid boundaries. Although the solution is computed differently in overlapping cells on different levels as each level is advanced initially, the synchronization procedure enforces conservation over the entire AMR grid hierarchy.

6.2.

Details of Coupling

During time integration of continuum grid levels, fluxes computed at each cell face are used to advance the solution U (Fig. 8b). Continuum values are

2542


(a)

(b)

(c)

(d)

(e)

(f)

Figure 8. Outline of AMAR hybrid: (a) Beginning of a time step; (b) Advance the continuum grid; (c) Create buffer particles; (d) Advance DSMC particles; (e) Refluxing; (f) Reset overlying continuum grid. Adapted from [29].

advanced using a time increment tc appropriate for each level, including those that overlay the DSMC region. When the particle level is integrated, it is advanced to the new time on the finest continuum level using a sequence of particle time steps, tp . The relative magnitude of tp to the finest continuum grid tc depends on the finest continuum grid spacing x (typically a few λ) and the particle mean collision time. Euler solution information is passed to the particles via buffer (reservoir) cells surrounding the DSMC region. At the beginning of each DSMC integration step, particles are created in the buffer cells using the continuum hydrodynamic values (ρ, u, T ) and their gradients (Fig. 8c) in a manner analogous to the incompressible case discussed above and the guidelines of the section on particle generation in dilute gases. Since the continuum solution is advanced first, these values are time interpolated between continuum time steps for the sequence of DSMC time steps needed to reach the new continuum solution time. DSMC buffer cells are one mean free path wide; thus, the time step tp is constrained so that it is extremely improbable that a particle will travel further than one mean free path in a single time step. The particle velocities are drawn from an appropriate distribution for the continuum solver, such as the Chapman–Enskog distribution when coupling to a Navier–Stokes description and a Maxwell–Boltzmann when coupling to an Euler description. During each DSMC time integration step, all particles are moved, including those in the buffer regions (Fig. 8d). A particle that crosses the interface


2543

between continuum and DSMC regions will eventually contribute to the flux at the corresponding continuum cell face during the synchronization of the DSMC level with the finest continuum level. After moving particles, those residing in buffer regions are discarded. Then, collisions among the remaining particles are evaluated and new particle velocities are computed. After the DSMC region has advanced over an entire continuum grid timestep, the continuum and DSMC solutions are synchronized in a manner analogous to the AMR level synchronization process described earlier. First, the continuum values in each cell overlaying the DSMC region interior are set to the conservative averages of data from the particles within the continuum grid cell region (Fig. 8e). Second, the continuum solution in cells adjacent to the DSMC region is recomputed using a “refluxing” process (Fig. 8f). That is, a flux correction is computed using a space and time integral of particle flux data, δF = −AFn+(1/2) +

Fp.

(12)

particles

The sum represents the flux of the conserved quantities carried by particles passing through the continuum cell face during the DSMC updates. Finally, Un+1 = Un+1 +

tc δF xyz

(13)

is used to update the conserved quantities on the continuum grid where Un+1 is the coarse grid solution before computing the flux correction. Note, multiple DSMC parallelepiped regions (i.e., patches) are coupled by copying particles from patch interiors to buffer regions of adjacent DSMC patches (see Fig. 9). That is, particles in the interior of one patch supply boundary values (by acting as a reservoir) for adjacent particle patches. After copying particles into buffer regions, each DSMC patch may be integrated independently, in the same fashion that different patches in a conventional AMR problems are treated after exchanging boundary data. In summary, the coupling between the continuum and DSMC methods is performed in three operations. First, continuum solution values are interpolated to create particles in DSMC buffer cells before each DSMC step. Second, conserved quantities in each continuum cell overlaying the DSMC region are replaced by averages over particles in the same region. Third, fluxes recorded when particles cross the DSMC interface are used to correct the continuum solution in cells adjacent to the DSMC region. This coupling procedure makes the DSMC region appear as any other level in the AMR grid hierarchy. Figure 10 shows the adaptive tracking of a shockwave of Mach number 10 used as a validation test for this method. Density gradient based mesh refinement ensures the DSMC region tracks the shock front accurately. Furthermore, as shown in Fig. 11 the density profile of the shock wave remains smooth and

2544


Figure 9. Multiple DSMC regions are coupled by copying particles from one DSMC region (upper left) to the buffer region of an adjacent DSMC region (lower right). After copying, regions are integrated independently over the same time increment. Adapted from Wijesinghe et al. [29].

Figure 10. Moving Mach 10 shock wave though Argon. The AMAR algorithm tracks the shock by adaptively moving the DSMC region with the shock front. Note, dark Euler region shading corresponds to density = 0.00178 g/cm3 , light Euler region shading corresponds to density = 0.00691 g/cm3 .


2545

Figure 11. Moving Mach 10 shock wave though Argon. The AMAR profile (dots) is compared with the analytical time evolution of the initial discontinuity (lines). τm is the mean collision time.

is devoid of oscillations that are known to plague traditional shock capturing schemes [4, 5]. Further details of the implementation using the Structured Adaptive Mesh Refinement Application Infrastructure (SAMRAI) developed at Lawrence Livermore National Laboratory [47] can be found in [29].

7.

Refinement Criteria

The AMAR scheme discussed above allows grid and algorithm refinement based on any combination of flow variables and their gradients. Density gradient based refinement has has been found to be generally robust and reliable. However, refinement may be triggered by any number of user defined criteria. For example, concentration gradients or concentration values within some interval are also effective refinement criteria especially for multispecies flows. In the AMAR formulation, refinement is triggered by spatial gradients exceeding user defined tolerances. This approach follows from the continuum breakdown parameter method [48]. Due to spontaneous stochastic fluctuations in atomistic computations, it is important to track gradients in a manner that does not allow the fluctuations

2546


to trigger unnecessary refinement and excessively large atomistic regions. Let us consider a dilute gas for simplicity and the gas density as an example. For an ideal gas under equilibrium conditions, the number of particles in a given volume is Poisson distributed; the standard deviation in the normalized density gradient perceived by the calculation at cell i is dρ/dx 2 ≈

ρ

Ni+1 − Ni 2 =

x Ni

√

2 √ x N

(14)

where N is the number of particles in a cell where macroscopic properties are defined. The use of equilibrium fluctuations is sufficiently accurate as long as the deviation from equilibrium is not too large [17]. The fluid density fluctuation can thus only be reduced by increasing the number of simulation particles. This has consequences for the use of density gradient tolerances Rρ , the value of which, as a result, must be based on the number of particles used in the atomistic subdomain. Let us illustrate this through an example. Consider the domain geometry shown in Fig. 12 where an atomistic region is in contact with a continuum region. Let the gas be in equilibrium. As stated above, the effect of nonequilibrium on fluctuations is small. In this problem, grid refinement occurs when the density gradient at the interface between two descriptions exceeds a normalized threshold,

2λ dρ Rρ < ρ dx

(15)

After such a “trigger” event the atomistic region grows by a single continuum cell width. Lets us assume that we would like to estimate the value of

Figure 12. 3D AMAR computational domain for investigation of tolerance parameter variation with number of particles in DSMC cells. From [29].


2547

refinement threshold such that a given trigger rate, say 5–10%, is achieved. The interpretation of this trigger rate, is that there is a probability of 5–10% of observing spurious growth of the atomistic subdomain due to density fluctuations. Following [29] we show how the trigger rate can be related to the number of particles per cell used in the calculation. For the geometry considered in the above test problem, each continuum cell consists of 8 DSMC cells and hence effectively the contribution of 8 × N particles is averaged to determine the density gradient between continuum cells. Applying Eq. (14) to these continuum cells we obtain, dρ/dx 2 ≈ σ=

ρ

c

1 √

2x N

(16)

Note that we are assuming that the fluctuation of the continuum cells across from the atomistic-continuum interface is approximately the same as that in the atomistic region. This was shown to be the case for the diffusion equation and a random walk model [41], and has been verified for the Euler–DSMC system [29] (see Fig. 14). This allows the use of Eq. (14) that was derived assuming 2 atomistic cells. Note that the observed trigger event is a composite of a large number of possible density gradient fluctuations that could exceed Rρ ; gradients across all possible nearest neighbor cells, next-to-nearest neighbor cells and diagonally-nearest neighbor cells are all individually evaluated by the refinement routines and checked against Rρ . For a 10% trigger rate (or equivalent probability of trigger) the probability of an individual cell having a density fluctuation exceeding Rρ can be estimated as O(0.1/100) by observing that, 1. since the trigger event is rare, probabilities can be approximated as additive, 2. for the geometry considered, there are ≈ 300 nearest neighbor, nextnearest neighbor and diagonal cells that can trigger refinement and 3. the rapid decay of the Gaussian distribution ensures the decreasing probability (O(0.1/100) ∼ O(0.001)) of a single event does not significantly alter the corresponding confidence interval and thus an exact enumeration of all possible trigger pairs with correct weighting factors is not necessary. Our probability estimate at O(0.001) suggests that our confidence interval is 3σ − 4σ . This is verified in Fig. 13. Smaller trigger rates can be achieved by increasing Rρ , that is, by increasing the number of particles per cell.

2548

Figure 13.


Variation of density gradient tolerance with number of DSMC particles. From [29].

Figure 14. Average density for stationary fluid Euler–DSMC hybrid simulation with 80 particles per cubic mean free path. Errorbars give one standard deviation over 10 samples. From [29].


8.

2549

Outlook

Although hybrid methods provide significant savings by limiting atomistic solutions only to the regions where they are needed, solution of timeevolving problems which span a large range of timescales is still not possible if the atomistic subdomain, however small, needs to be integrated for the total time of interest. New frameworks are therefore required which allow timescale decoupling or coarse grained time evolution of atomistic simulations. Significant computational savings can be obtained by using the incompressible formulation, when appropriate, for steady problems. Neglect of these simplifications can lead to a problem that is simply intractable when the continuum subdomain is appropriately large. It is interesting to note that, when a hybrid method was used to solve a problem of practical interest [33] while providing computational savings, the Schwarz method was preferred because it provides a steady solution framework with timescale decoupling. For dilute gases the Chapman–Enskog distribution provides a robust and accurate method for imposing boundary conditions. Further work is required for the development of similar frameworks for liquids.

Acknowledgments The authors wish to thank R. Hornung and A.L. Garcia for help with the computations and valuable comments and discussions, and A.T. Patera and B.J. Alder for helpful comments and discussions. This work was supported in part by the Center for Computational Engineering, and the Center for Advanced Scientific Computing, Lawrence Livermore National Laboratory, US Department of Energy, W-7405-ENG-48. The authors also acknowledge the financial support from the University of Singapore through the Singapore-MIT alliance.

References [1] A. Beskok and G.E. Karniadakis, “A model for flows in channels, pipes and ducts at micro and nano scales,” Microscale Thermophys. Eng., 3, 43–77, 1999. [2] S.A. Tison, “Experimental data and theoretical modeling of gas flows through metal capillary leaks,” Vacuum, 44, 1171–1175, 1993. [3] D.G. Coronell and K.F. Jensen, “Analysis of transition regime flows in low pressure CVD reactors using the direct simulation Monte Carlo method,” J. Electrochem. Soc., 139, 2264–2273, 1992. [4] M. Arora and P. L. Roe, “On postshock oscillations due to shock capturing schemes in unsteady flows,” J. Comput. Phys., 130, 25, 1997. [5] P.R. Woodward and P. Colella, “The numerical simulation of two-dimensional fluid flow with strong shocks,” J. Comput. Phys., 54, 115, 1984.

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[6] J. Koplik and J.R. Banavar, “Continuum deductions from molecular hydrodynamics,” Annu. Rev. Fluid Mech., 27, 257–292, 1995. [7] M.P. Brenner, X.D. Shi, and S.R. Nagel, “Iterated instabilities during droplet fission,” Phys. Rev. Lett., 73, 3391–3394, 1994. [8] P.A. Thompson and M.O. Robbins, “Origin of stick–slip motion in boundary lubrication,” Science, 250, 792–794, 1990. [9] D.C. Wadsworth and D.A. Erwin, “One-dimensional hybrid continuum/particle simulation approach for rarefied hypersonic flows,” AIAA Paper 90-1690, 1990. [10] D.C. Wadsworth and D.A. Erwin, “Two-dimensional hybrid continuum/particle simulation approach for rarefied hypersonic flows,” AIAA Paper 92-2975, 1992. [11] J. Eggers and A. Beylich, “New algorithms for application in the direct simulation Monte Carlo method,” Prog. Astronaut. Aeron., 159, 166–173, 1994. [12] D. Hash and H. Hassan, “A hybrid DSMC–Navier Stokes solver,” AIAA Paper 95-0410, 1995. [13] S.T. O’Connell and P. Thompson, “Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows,” Phys. Rev. E, 52, R5792–R5795, 1995. [14] N.G. Hadjiconstantinou and A.T. Patera, “Heterogeneous atomistic-continuum representations for dense fluid systems,” Int. J. Mod. Phys. C, 8, 967–976, 1997. [15] N.G. Hadjiconstantinou, “Hybrid atomistic-continuum formulations and the moving contact-line problem,” J. Comput. Phys., 154, 245–265, 1999. [16] E.G. Flekkoy, G. Wagner, and J. Feder, “Hybrid model for combined particle and continuum dynamics,” Europhys. Lett., 52, 271–276, 2000. [17] N.G. Hadjiconstantinou, A.L. Garcia, M.Z. Bazant, and G.He, “Statistical error in particle simulations of hydrodynamic phenomena,” J. Comput. Phys., 187, 274–297, 2003. [18] P. Wesseling, Principles of Computational Fluid Dynamics, Springer, 2001. [19] X. Yuan and H. Daiguji, “A specially combined lower-upper factored implicit scheme for three dimensional compressible Navier-Stokes equations,” Comput. Fluids, 30, 339–363, 2001. [20] S. Chapman and T.G. Cowling, The Mathematical Theory of Non-uniform Gases, Cambridge University Press, 1970. [21] A.L. Garcia, J.B. Bell, W.Y. Crutchfield et al., “Adaptive mesh and algorithm refinement using direct simulation Monte Carlo,” J. Comput. Phys., 54, 134, 1999. [22] J. Li, D. Liao and S. Yip, “Nearly exact solution for coupled continuum/MD fluid simulation,” J. Comput. Aided Mater. Design, 6, 95–102, 1999. [23] M.M. Mansour, F. Baras, and A.L. Garcia, “On the validity of hydrodynamics in plane poiseuille flows,” Physica A, 240, 255–267, 1997. [24] R. Delgado–Buscalioni and P.V. Coveney, “Continuum–particle hybrid coupling for mass, momentum and energy transfers in unsteady fluid flow,” Phys. Rev. E, 67(4), 2003. [25] C. Cercignani, The Boltzmann Equation and its Applications, Springer-Verlag, New York, 1988. [26] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1994. [27] W. Wagner, “A convergence proof for bird’s direct simulation Monte Carlo method for the Boltzmann equation,” J. Statist. Phys., 66, 1011, 1992. [28] H.S. Wijesinghe and N.G. Hadjiconstantinou, “A hybrid continuum-atomistic scheme for viscous incompressible flow,” In: Proceedings of the 23th International Symposium on Rarefied Gas Dynamics, 907–914, Whistler, British Columbia, 2002.


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[29] H.S. Wijesinghe, R. Hornung, A.L. Garcia et al., “Three–dimensional hybrid continuum–atomistic simulations for multiscale hydrodynamics,” ASME J. Fluids Eng., 126, 768–777, 2004. [30] A.L. Garcia and B.J. Alder, “Generation of the Chapman Enskog distribution,” J. Comput. Phys., 140, 66, 1998. [31] L. Devroye, “Non-uniform random variate generation,” In: A.L. Garcia (ed.), Numerical Methods for Physics, Prentice Hall, New Jersey, 1986. [32] W.H. Press, B.P. Flannery, S.A. Teukolsky, and W.A. Vetterling, Numerical Recipes in Fortran, Cambridge University Press, 1992. [33] O. Aktas and N.R. Aluru, “A combined continuum/DSMC Technique for multiscale analysis of microfluidic filters,” J. Comput. Phys., 178, 342–372, 2002. [34] N. G. Hadjiconstantinou, Hybrid Atomistic-Continuum Formulations and the Moving Contact Line Problem, Phd Thesis edn., Mechanical Engineering Department, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1998. [35] P.L. Lions, “On the Schwarz alternating method,” I. In: R. Glowinski, G. Golub, G. Meurant, and J. Periaux (eds.), First International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp. 1–42, SIAM, Philadelphia, 1988. [36] S.H. Liu, “On Schwarz alternating methods for the incompressible Navier–Stokes equations,” SIAM J. Sci. Comput., 22(6), 1974–1986, 2001. [37] F.J. Alexander, A.L. Garcia, and B.J. Alder, “Cell size dependence of transport coefficients in stochastic particle algorithms,” Phys. Fluids, 10, 1540, 1998. [38] N.G. Hadjiconstantinou, “Analysis of discretization in the direct simulation Monte Carlo,” Phys. Fluids, 12, 2634–2638, 2000. [39] A.L. Garcia and W. Wagner, “Time step truncation error in direct simulation Monte Carlo,” Phys. Fluids, 12, 2621–2633, 2000. [40] R. Roveda, D.B. Goldstein, and P.L. Varghese, “Hybrid Euler/direct simulation Monte Carlo calculation of unsteady slit flow,” J. Spacecraft and Rockets, 37(6), 753–760, 2000. [41] F.J. Alexander, A.L. Garcia, and D. Tartakovsky, “Algorithm refinement for stochastic partial diffential equations: I. Linear diffusion,” J. Comput. Phys., 182(1), 47–66, 2002. [42] L.D. Landau and E.M. Lifshitz, Statistical Mechanics Part 2, Pergamon Press, Oxford, 1980. [43] P. Colella, “A direct Eulerian (MUSCL) scheme for gas dynamics,” SIAM J. Sci. Statist. Comput., 6, 104–117, 1985. [44] P. Colella and H.M. Glaz, “Efficient solution algorithms for the riemann problem for real gases,” J. Comput. Phys., 59, 264–289, 1985. [45] J. Saltzman, “An unsplit 3D upwind method for hyperbolic conservation laws,” J. Comput. Phys., 115, 153, 1994. [46] M. Berger and P. Colella, “Local adaptive mesh refinement for shock hydrodynamics,” J. Comput. Phys., 82, 64, 1989. [47] CASC, “Structured adaptive mesh refinement application infrastructure,” http://www.llnl.gov/CASC/, 2000. [48] G.A. Bird, “Breakdown of translational and rotational equilibrium in gaseous expansions,” Am. Inst. Aeronautics and Astronaut. J., 8, 1998, 1970.

9.1 POLYMERS AND SOFT MATTER L. Mahadevan1 and Gregory C. Rutledge2 1

Division of Engineering and Applied Sciences, Department of Organismic and Evolutionary Biology, Department of Systems Biology, Harvard University Cambridge, MA 02138, USA 2 Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

1.

Introduction

Within the context of this Handbook, the combined areas ofpolymers and soft matter encompasses a vast range of complex materials, including both synthetic and natural polymers, many biological materials, and complex fluids such as colloids and viscoelastic media. What distinguishes these materials from most of those considered in other chapters of this Handbook is the macromolecular or supermolecular nature of the basic components of the material. In addition to the usual atomic level interactions responsible for chemically specific material behavior, as is found in all materials, these macromolecular and supermolecular objects exhibit topological features that lead to new, larger scale, collective nonlinear and nonequilibrium behaviors that are not seen in the constituents. As a consequence, these materials are typically characterized by a broad range of both length and time scales over which phenomena of both scientific and engineering interest can arise. In polymers, for instance, the organic nature of the molecules is responsible for both strong (intramolecular, covalent) and weak (intermolecular, van der Waals) interactions, as well as interactions of intermediate strength such as hydrogen bonds that are common in macromolecules of biological interest. In addition, however, the long chain nature of the molecule introduces a distinction between dynamics that occur along the chain or normal to it; one consequence of this is the observation of certain generic behaviors such as the “slithering snake” motion, or reptation, in polymer dynamics. It is often the very ability of polymers and soft matter to exhibit both atomic (or molecular) and macro- (or super-) molecular behavior that makes them so interesting and powerful as a class of materials and as building blocks for living systems. 2555 S. Yip (ed.), Handbook of Materials Modeling, 2555–2559. c 2005 Springer. Printed in the Netherlands.

2556

L. Mahadevan and G.C. Rutledge

Nevertheless, polymers and soft matter are, at their most basic level, collections of atomic and subatomic particles, like any other class of materials. They exhibit both liquid-like and crystalline (or at least semi-crystalline) order in their condensed forms. For polymers, vitrification and the glassy state are particularly important, as both the vitrification temperature and the kinetics of vitrification are strong functions of the inverse of molecular weight. For the most part, the methods developed for atomic and electronic level modeling described in the earlier chapters of this Handbook are equally applicable, at least in principle, to the descriptive modeling of polymers and soft matter. Electronic structure calculations, atomistic scale molecular dynamics and Monte Carlo simulations, coarse-grained and mesoscale models such as Lattice Boltzmann and Dissipative Particle Dynamics all have a role to play in modeling of polymers and soft matter. As materials, these interesting solids and fluids exhibit crystal plasticity, amorphous component viscoelasticity, rugged energy landscapes, and fascinating phase transitions. Indeed, block copolymers consisting of two or more covalently-joined but relatively incompatible chemical segments, and the competition they represent between intermolecular interactions and topological constraints, give rise to the rich field of microphase separation, with all its associated issues and opportunities regarding manipulation of microstructure, size and symmetry. It has not been our objective in assembling the contributions to this chapter to repeat any of the basic elements of modeling that have been developed to describe materials at any of these particular length and time scales, or strategies for generating thermodynamics information relevant to ensembles, phase transitions, etc. Rather, in recognition of those features which make polymers and soft matter distinct and novel with respect to their atomic or monomolecular counterparts, we have attempted to assemble a collection of contributions which highlight these features, and which describe methods developed specifically to handle the particular problems and complexities of dimensionality, time and length scale which are unique to this class of materials. With this in mind, the following sections in this chapter should be understood as extensions and revisions of what has gone before. We begin with a discussion of interatomic potentials specific to organic materials typical of synthetic and natural polymers and other soft matter. Accurate force fields lie at the heart of any molecular simulation intended to describe a particular material. Over the years, numerous apparently dissimilar force fields for organic materials have been proposed. However, certain motifs consistently reappear in such force fields, and common pitfalls in parameterization and guidelines for application of such force fields can be identified. These are discussed in the contribution by Smith. The recognition that one of the defining features of macromolecules is their very large conformation space motivated the relatively early development by Volkenstein in the late 1950s of the concept of rotational isomeric

Polymers and soft matters

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states for each of the rotatable bonds along the backbone (i.e., the topologically connected dimension) of molecular chains. This essential discretization of conformation space allowed the development by Flory and others of what is now known as the rotational isomeric states (RIS) method, discussed in the section by Mattice. This method for evaluation of conformational averages is unique to polymers and provides an important alternative to the sampling strategies embodied by molecular dynamics and Monte Carlo simulation. What RIS gives up in assuming a simplified, discrete set of allowed rotational states for each bond, it more than makes up for in its computational efficiency and rigorous representation of contributions from all allowed conformers to the partition function and resulting conformational averages. The issues in sampling of phase space using molecular dynamics or Monte Carlo simulations for chain models are discussed by Mavrantzas. Molecular dynamics is of course applicable to the study of polymers and soft matter, but the broad range of length and, in particular, time scales alluded to earlier as being a consequence of the macromolecular and/or supermolecular nature of such matter, render this method of limited utility for many of the most interesting and unique behaviors in this class of materials. For this reason, Monte Carlo simulation has come to play a particularly important role in the modeling of polymers and soft matter. At the expense of detailed dynamics, the state of the art in Monte Carlo simulations of chain molecules and aggregates has advanced through the development of new sampling schemes that permit drastic, sometimes seemingly unphysical, moves through phase space. These moves are designed with both intermolecular interactions and intramolecular topology in mind. Without them, full equilibration and accurate simulation of complex materials are all but impossible. An alternative approach to accessing the long length and time scales of interest in polymers and soft matter is to coarse-grain the model description, gaining computational efficiency at the price of atomic scale detail. Such methods are useful for studying the generic, or universal, properties of polymers and aggregates. In the field of polymers and soft matter, lattice models have long been employed for rendering such coarse-grained models. The Bond Fluctuation Model, in particular, is typical of this class of methods and has enjoyed widespread application, due at least in part to the delicate compromise it achieves between the complexity of conformation space and the simplification inherent in rendering on a lattice. Importantly, it does so while retaining the essential topological connectivity. These methods are discussed by Müller and provide a link to continuum-based methods. Continuum based methods start to become relevant when the number of particles involved is very large and one is interested in long wavelength, long time modes, as is typical of hydrodynamics. The dimensionalities of both the “material” component and the embedding component, or matrix, play important roles in determining the behavior of mesophases such as suspensions, colloids

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and membranes. The article by Sierou provides an introduction to Stokesian dynamics, a molecular dynamics-like method for simulating the multi-phase behavior of particles suspended in a fluid. The particles are treated in a discrete sense, while the surrounding fluid is treated using a continuum approximation and is thus valid when the particle size is much larger than that of the molecules of the solvent. By accounting for Brownian motion, Stokesian dynamics provides a generalization of Brownian dynamics, treated by Doyle and Underhill in the next section, wherein the many-body contribution from hydrodynamics is accounted for properly. It thus paves the road for a study of the equilibrium and non-equilibrium rheology of colloids and other complex multiphase fluids. Moving up in dimensionality from particles to chains, the section by Doyle and Underhill discusses Brownian dynamics simulation of long chain polymers. The topological connectivity of these polymers implies a separation in time and energy scales for deformations tangential to and normal to the backbone. Coarse-grained models that account for this separation of scales range from bead-spring models to continuum semi-flexible polymers. While these models have been corroborated with each other and with simple experiments involving single molecules, the next frontier is clearly the use of these dynamical methods to probe the behavior of polymer solutions, a subject that still merits much attention. Next, Powers looks at the 2D generalization of polymers, i.e., membranes, which are assemblies of lipid molecules that are fluid-like in the plane but have an elastic response to bending out of the plane. In contrast to the previous sections, the focus here is on the continuum and statistical mechanics of these membranes using analytical tools via a coarse-grained free energy written in terms of the basic broken-symmetries of the system. Once again the role of non-equilibrium dynamics comes up in the example of active membranes. The last section in this chapter offers a union of the molecular and continuum perspectives, in some sense, to address problems such as molecular structure-mediated microphase formation. Here again continuum models based on density fields and free energy functionals are most appropriate. It is a relatively recent development, however, that such models have been used as a starting point for computer simulations. The Field Theoretic Simulation method developed by Frederickson and co-workers does just this, and is discussed by Ganesan and Frederickson in this chapter. They provide a prescription by which a molecular model can be recast as a density field with its projected Hamiltonian, and then present appropriate methods for discretizing and sampling phase space during the simulation. Thus, polymers and soft matter are in some sense no different than hard matter, in that their constituents are atomic in nature. Yet they are distinguished by the predominance of weak interactions comparable to the thermal fluctuations, which makes them amenable to change. Looking to the future, the wide

Polymers and soft matters

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variety of phases and broken symmetries that they embody is nowhere more abundant than in living systems that operate far from equilibrium and are eternally mutable. From a materials perspective, polymers and soft matter offer opportunities to mimic and understand nature in ways that we are only just beginning to appreciate. It is our hope that the sections in this chapter offer a glimpse of the techniques that one may use and the questions that motivate them.

9.2 ATOMISTIC POTENTIALS FOR POLYMERS AND ORGANIC MATERIALS Grant D. Smith Department of Materials Science and Engineering, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, USA

Accurate representation of the potential energy lies at the heart of all simulations of real materials. Accurate potentials are required for molecular simulations to accurately predict the behavior and properties of materials, and even qualitative conclusions drawn from simulations employing inaccurate or unvalidated potentials are problematic. Various forms of classical potentials (force fields) for polymers and organic materials can be found in the literature [1–3]. The most appropriate form of the potential depends largely upon the properties of interest to the simulator. When interest lies in reproducing the static, thermodynamic and dynamic (transport and relaxational) properties of non-reactive organic materials, the potential must accurately represent the molecular geometry, nonbonded interactions, and conformational energetics of the materials of interest. The relatively simple representation of the classical potential energy discussed below has been found to work remarkable well for these properties. More complicated potentials that can handle chemical reactions [4] or are designed to very accurately reproduce vibrational spectra [5] can be found in the literature. The form of the force field considered here has the advantages of being more easily parameterized than more complicated forms. Parameterization of even simple potentials is a challenging task, however, as discussed below.

1.

Form of the Potential

The classical force field represents the total potential energy of an ensemble of atoms V ( r ) with positions given by the vector r as a sum of nonbonded


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G.D. Smith

interactions V N B ( r ) and energy contributions due to all bond, valence bend, and dihedral interactions: V ( r ) = V nb ( r) +

V bond(ri j ) +

bonds

V bend (θi j k ) +

bends

V tors (ϕi j kl )

dihedrals

(1) The various interactions are illustrated in Fig. 1. The dihedral term also includes four-center improper torsion or out-of-plane bending interactions that occur at sp2 hybridized centers. r ) consists of a sum of the twoCommonly, the nonbonded energy V N B ( body repulsion and dispersion energy terms between atoms i and j represented by the Buckingham (exponential-6) potential, the energy due to the interactions between fixed partial atomic or ionic charges (Coulomb interaction), and the energy due to many-body polarization effects: r ) = V pol ( r) + V nb (

N 1 Ci j qi q j Ai j exp(−Bi j ri j ) − 6 + 2 i, j =1 4π ε0ri j ri j

(2)

The generic behavior of the dispersion/repulsion energy for an atomic pair is shown in Fig. 2. The dispersion interactions are weak compared to repulsion, but are longer range, resulting in an attractive well with well depth ε at an interatomic separation of σ ∗ . The separation where the net potential is zero, σ , is often used to define the atomic diameter. In addition to the exponential-6 dihedral twist

intramolecular nonbonded intermolecular nonbonded

bond stretch

valence angle bend

Figure 1. Schematic representation of intramolecular bonded and nonbonded (intramolecular and intermolecular) interactions in a typical polymer.

V DIS-REP(r)

Atomistic potentials for polymers and organic materials

0

2563

ε σ σ∗

r Figure 2. Schematic representation of the dispersion/repulsion potential between two atoms as a function of separation.

form, the Lennard–Jones form of the dispersion–repulsion interaction, 

Ai j Ci j σ V D I S−R E P (ri j ) = 12 − 6 = 4ε  ri j ri j ri j

12

 12 6  σ ∗ σ ∗  = ε −2

ri j

ri j

−

σ ri j

6  

(3)

is commonly used, although this form tends to yield a poorer (too stiff) description of repulsion. The relationship between the well depth and atomic diameter and the dispersion–repulsion parameters is particularly simple for the Lennard–Jones potential (ε = C 2 /4A, σ = (A/C)1/6 , σ ∗ = 21/6 σ ), allowing the dispersion–repulsion interaction to be expressed in terms of these parameters, as shown in Eq. (3). Nonbonded interactions are typically included between all atoms of different molecules and between atoms of the same molecule separated by more than two bonds (see Fig. 1). It is not uncommon, however, to scale intramolecular nonbonded interactions between atoms separated by three bonds. Care must therefore be taken in implimenting a potential that the 1–4 intramolecular nonbonded interactions are correctly treated. Repulsion parameters have the shortest range and typically become negligible at 1.5 σ. Dispersion parameters are longer range than the repulsion parameters requiring cutoff distances of 2.5 σ. The Coulomb term is long-range, necessitating use of special summing methods [6, 7]. While dispersion interactions are typically weaker and are shorter range than Coulomb interactions, they are always attractive, independent of the configuration of the molecules, and typically make the dominate contribution to the cohesive energy even in highly polar polymers and organic materials.

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A further complication arises in cases where many-body dipole polarization needs to be taken into account explicitly. The potential energy due to dipole polarization is not pair-wise additive and is given by a sum of the interaction energy between the induced dipoles µi and the electric field Ei0 at atom i generated by the permanent charges in the system (qi ), the interaction energy between the induced dipoles and the energy required to induce the dipole moments [7] V

pol

(r) = −

N i=1

N N 1 µ i • µ i 0 µ • Ei − µ i • T ij • µ j + 2 i, j 2α i i=1

(4)

tot where µ i = αi E tot i , αi is the isotropic atomic polarizability, E i is the total electrostatic field at the atomic site i due to permanent charges and induced dipoles, and the second order dipole tensor is given by 1 1 T i j = ∇i ∇ j = 4π ε0ri j 4π ε0ri3j

3 ri j ri j −1 ri2j

(5)

where ri j is the vector from atom i to atom j . Because of the expense involved in simulations with explicit inclusion of many-body dipole polarization, it may be desirable to utilize a two-body approximation for these interactions [8]. The contributions due to bonded interactions are represented as

ri j − ri0j V bond (ri j ) = 12 kibond j

2

θi j k − θi0j k V bend (θi j k ) = 12 kibend jk V tors(ϕi j kl ) = V tors(ϕi j kl ) =

1 2

(6) 2

= 12 k bend cos θi j k − cos θi0j k ijk

kitors j kl (n) 1 − cos nϕi j kl

n 1 oop k 2 i j kl

φi j k

2

2

(7)

or (8)

Here, ri0j is an equilibrium bond length and θi0j l is an equilibrium valence bend oop bend tors angle while kibond j , ki j k , ki j kl (n) and ki j kl are the bond, bend, torsion and outof-plane bending force constants, respectively. Note that the forms given for the valence bend interaction are entirely equivalent for sp2 and sp3 bonding geometries for reasonably stiff bends at reasonable temperatures, with k = k/sin2 θ 0 . The indices indicate which (bonded) atoms that are involved in the interaction. These geometric parameters and force constants, combined with the nonbonded parameters qi , αi , Ai j , Bi j and Ci j , constitute the classical force field for a particular material. In contrasting the form of the potential for polymers and organics with potentials for other materials, the nature of bonding in organic materials becomes manifestly apparent. In organic materials the relatively strong covalent bonds and valence bends serve primarily to define the geometry of the


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molecule. Much weaker/softer intramolecular degrees of freedom, namely torsions, and intermolecular nonbonded interactions, primarily determine the thermodynamic and dynamic properties of polymers and large organic molecules. Hence relatively weak (and consequently difficult to parameterize) torsional and repulsion/dispersion parameters must be determined with great accuracy in potentials for polymers and organics. However, this separation of scales of interaction strengths (strong intramolecular covalent bonding, weak intermolecular bonding) has the advantage of allowing many-body interactions, which often must be treated through explicit many-body nonbonded terms in simulations of other classes of materials, to be treated much more efficiently as separate intramolecular bonded interactions in organic materials.

2.

Existing Potentials

By far the most convenient way to obtain a force field is to utilize an extant one. In general, force fields can be divided into three categories: (a) force fields parametrized based upon a broad training set of molecules such as small organic molecules, peptides, or amino acids including AMBER [1], COMPASS [9], OPLS-AA [3] and CHARMM [10]; (b) generic potentials such as DREIDING [11] and UNIVERSAL [12] that are not parameterized to reproduce properties of any particular set of molecules; and (c) specialized force fields carefully parametrized to reproduce properties of a specific compound. A procedure for parameterizing the latter class of potential is described below. A summary of the data used in the parametrization of some of the most common force fields is presented in Table 1. Parametrized force fields (AMBER, OPLS and CHARMM) can work well within the class of molecules they have been parametrized upon. However, when the force field parameters are utilized for compounds similar to those in the original training set but not contained in the training set significant errors can appear and the quality of force field predictions is often no better than that of the generic force fields [13]. Similar behavior is expected when parameterized force fields transferred to new classes of compounds. Therefore, in choosing a potential, both the quality of the potential and the transferability of the potential need to be considered. The quality of a potential can be estimated by examining the quality and quantity of data used in its parameterization. For example, AMBER ff99 (Table 1) uses a much higher level of quantum chemistry calculation for determination of dihedral parameters than the early AMBER ff94. The ability of the force fields to describe the molecular and condensed-phase properties of the training set is another indicator of the force field quality. The issue of transferability of a potential is faced when a high-quality force field, adequately validated for compounds similar to the one of interest, is used in modeling

[N/A, N/A, experiment]

X-ray structure, IR, Raman

various experimental sources, QC

peptides, nucleic acids, organics

polarization

bond/bend

torsion

training set

organic liquids

QC

AMBER[ff94] with some values from CHARMM

peptides

Microwave and electron diffraction, QC

IR, Raman, microwave and electron diffraction, X-ray crystal data, QC

N/A

QC, experimental dipoles

PVT, H vap

QC

electrostatic N/A

PVT, H vap , crystal structures, QC

PVT, H vap

PVT, H vap

repulsion/dispersion

CHARMM

OPLS-AA

AMBER[ff94, ff99, ff02]

Interactions

Table 1. Summary of the primary data used in parameterization of popular force fields

generic

Generic based on hybridization

Generic

N/A

predictive method

Crystal structures and sublimation energies

DREIDING

2566 G.D. Smith


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related compounds not in the training set, or in modeling entirely new classes of materials. Transferability varies tremendously upon the potential function parameter, with some parameters being in general quite transferable between similar compounds and others being much less so.

3.

Sources of Data for Force Field Parametrization

In order to judge the quality of existing force fields for a compound of interest, or to undertake the demanding but often inevitable task of parameterizing or partially parameterizing a new force field, one requires data against which the force field parameters (or subset thereof) can be tested and if necessary, fit. As can be seen in Table 1, there are two primary sources for such data: experiment and ab initio quantum chemistry calculations. Experimentally measured structural, thermodynamic and dynamic data for condensed phases (liquid and/or crystal) of the material of interest or closely related compounds are particularly useful in force field parameterization and validation. Highlevel quantum chemistry calculations are the best source of molecular level information for force field parameterization. While such calculations are not yet possible on high polymers and very large organic molecules, they are feasible on small molecules representative of polymer repeat units and oligomers, fragments of large molecules, as well as molecular clusters that reproduce interactions between segments of polymers or organic molecules or the interaction of a these with surfaces, solvents, ions, etc. These calculations can provide the molecular geometries, partial charges, polarizabilities, conformational energy surface, and intermolecular nonbonded interactions critical for accurate prediction of structural, thermodynamic and dynamic properties of polymers. Of key importance in utilizing quantum chemistry calculations for force field parameterization is use of an adequate level of theory and the choice of the basis set. As a rule of thumb, augmented correlation-consistent polarizable basis sets (e.g., aug-cc-pVDZ) utilizing DFT geometries (e.g., B3LYP) and correlated (MP2) energies work quite well, often providing molecular dipole moments within a few percent of experimental values, conformer energies within ±0.3 kcal/mol, rotational energy barriers between conformations within ± 0.5 kcal/mol, and intermolecular binding energies after basis set superposition error (BSSE) correction within 0.1–1 kcal/mol. However, whenever force field parameterization for any new class of molecule for which extensive quantum chemistry studies do not exist is undertaken, a comprehensive study of the influence of basis set and electron correlation on molecular geometries, conformational energies, cluster energies, dipole moments, molecular polarizabilities and electrostatic potential is warranted.

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4.

G.D. Smith

Determining Potential Function Needs

In examining candidate potentials for a material, one should ascertain whether they have been parameterized for the material of interest or for closely related materials. One should also determine what data (quantum chemistry and experimental) were used in the parametrization, the quality of the data employed, and how well the potential reproduces the “training set” data. Finally, what if any validation steps that have been carried by the originators of the potential or by others who have utilized the potential should be determined. Next, one should determine what force field parameters are missing or may need reparameterization for the material of interest. The parameters that have most limited transferability from the training set to related compounds and hence are most likely to need parameterization are partial charges and dihedral parameters. Other parameters that may need parameterization in order of decreasing probability (increasing transferability) are equilibrium bond lengths and angles, bond, bend and improper torsion force constants, dispersion/repulsion parameters and atomic polarizabilities (for many-body polarizable potentials). A general procedure for systematic parameterization and validation of potential functions suitable for any polymer, organic compound or solution is provide below. Detailed derivations of quantum-chemistry based potentials for organic compounds and polymers can be found in the literature [9, 14].

5.

Establishing the Quantum Chemistry Data Set

Once it has been determined that parameterization or partial parameterization of a potential function is needed, it is necessary to determine the set of model molecules to be utilized in the potential function parameterization. If dispersion/repulsion parameters are needed, this may include molecular complexes containing the intermolecular interactions of interest. For smaller organic molecules, the entire molecule should be included in the data set. For polymers and larger organic molecules, oligomers/fragments containing all single conformations and conformational pairs extant in the polymer/large organic should be included. A search for existing quantum chemistry studies of these and related molecules should be conducted before beginning quantum chemistry calculations. When a new class of material (one for which extensive quantum chemistry studies have not yet been conducted) is being investigated, the influence of basis set and level of theory should be systematically investigated. Comparison with experiment (binding energies, molecular geometry, conformational energies, etc.) can help establish what level of theory is adequate.


2569

Once the level of theory is established, all important conformers and rotational energy barriers for the model molecule(s) in the data set should be found, as well as dipole moments and electrostatic potential for the lowest energy conformers. BSSE corrected binding energies for important configurations of molecular clusters should also be determined if parameterization of dispersion/repulsion interactions is required. These data provide the basis for parameterization of the potential as described briefly below.

6. 6.1.

Potential Function Parameterization and Validation Partial Charges

Most organic molecules are sufficiently polar that Coulomb interactions must be accurately represented. Often it is sufficient to treat Coulomb interactions with fixed partial atomic charges (Eq. (2)) and neglect explicit inclusion of many-body dipolar polarizability. The primary exception occurs when small ionic species are present. In such cases the force field needs to be augmented with additional terms describing polarization of a molecule (Eq. (4)). When needed, atomic polarizabilities can be determined straightforwardly from quantum chemistry [14, 15]. In parameterization of partial atomic charges, one attempts to reproduce the molecular dipole moment and electrostatic potential in the vicinity of model molecules as determined from high-level quantum chemistry calculations with a set of partial charges of the various atoms. Fig. 3 illustrates the quality of agreement that can be achieved in representing the electrostatic potential with partial atomic charges.

6.2.

Dispersion and Repulsion Interactions

Carrying out quantum chemistry studies of molecular clusters of sufficient accuracy to allow for final determination of dispersion parameters is very computationally expensive. Fortunately repulsion and dispersion parameters are highly transferable. Therefore, it is expedient to utilize literature values for repulsion and dispersion parameters where high-quality, validated values exist. Where necessary BSSE corrected Hartree–Fock binding energies of molecular clusters can be used establish repulsion parameters and initial values for dispersion parameters can be determined from fitting to correlated binding energies [14, 15]. Regardless of the source of data utilized to parameterize dispersion interactions (experimental thermodynamic or structural data, quantum chemistry data on molecular clusters, or direct use of existing parameters)

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G.D. Smith

5 5

-30 -20 0 0

-10

Figure 3. Electrostatic potential in the plane of a 1,2-dimethoxyethane molecule from ab initio electronic structure calculations (QC) and from partial atomic charges (FF) parameterized to reproduce the potential. Energy contours are in kcal/mol.

it may be necessary to make (hopefully) minor empirical adjustments (as large as ±10%) to the dispersion parameters so as to yield highly accurate thermodynamic properties for the material of interest. This can be accomplished by carrying out simulations of model molecules and comparing predicted thermodynamic properties (density, heat of vaporization, thermal expansion, compressibility) with experiment and adjusting dispersion parameters as needed to improve agreement.

6.3.

Bond and Bend Interactions

The covalent bond and valence bend force constants are also highly transferable between related compounds. As long as the dihedral potential (see


2571

below) is parameterized with the chosen bond and bend force constants, the particular (reasonable) values of the force constants will not strongly influence structural, thermodynamic, or dynamic properties of the material. It is therefore recommended that stretching bending force constants be taken from the literature where available. When not available, stretching and bending force constants can be taken directly from quantum chemistry normal mode frequencies determined for representative model molecules with appropriate scaling of the force constants.

6.4.

Molecular Geometry

The molecular geometry can strongly influence static, thermodynamic and dynamic properties and needs to be accurately reproduced. Therefore, accurate representation of bond lengths and angles is important. Equilibrium bond lengths and bond angles can be adjusted so as to accurate reproduce the bond lengths and bond angles of model compounds determined from high-level quantum chemistry.

6.5.

Dihedral Potential

It is crucial that the conformational energies, specifically the relative energies of important conformations and the rotational energy barriers between them, be accurately represented for polymers and conformationally flexible organic compounds. As a minimum a force field must be able to reproduce the relative energies of the important conformations of single dihedrals and dihedral pairs (dyad) in model molecules. The conformational energies and rotational energy barriers obtained from quantum chemistry for model molecules are quite sensitive to the level of theory utilized, both basis set size and electron correlation. Fortunately, it is typically not necessary to conduct geometry optimizations with electron correlation—for many compounds SCF or DFT geometries are sufficient. Unfortunately, relative conformational energies and rotational energy barriers obtained at the SCF and DFT level are usually not sufficient accurate, necessitating the calculation of MP2 energies at SCF or DFT geometries. In fitting the dihedral potential, it is sometimes possible to utilize only 1, 2 and 3-fold dihedral terms (n = 1–3 in Eq. (8)). However, it is often necessary to up to 6-fold dihedral terms to obtain a good representation of the conformational energy surface. One must be cognizant of possible artifacts (e.g., spurious minima and conformational energy barriers) that can be introduced into the conformational energy surface when higher-fold terms (n > 3) with large amplitudes are utilized. Fig. 4 show the quality of agreement for conformational energies between quantum chemistry and molecule

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G.D. Smith 7

conformational energy (kcal/mol)

QC

6

FF

5 4 3 2 1 0 0

60

120

180

240

300

360

β dihedral angle Figure 4. The relative conformational energy for rotation about the β-dihedral in 1,5hexadiene from ab initio electronic structure calculations (QC) and a force field parameterized to reproduce the conformational energy surface (FF).

mechanics that is possible with a 1-3 fold potential for model molecules for poly(butadiene).

6.6.

Validation of the Potential

As a final step, the potential, regardless of its source, should be validated through extensive comparison of structural, thermodynamic and dynamic properties obtained from simulations of the material of interest, closely related materials, and model compounds used in the parameterization, with available experimental data. The importance of potential function validation in simulation of real materials cannot be overemphasized.

References [1] W.D. Cornell et al., “A second generation force field for simulations of proteins, nucleic acids, and organic molecules,” J. Am. Chem. Soc., 117, 5179–5197, 1995. [2] J.W. Ponder and D.A. Case, “Force fields for protein simulation,” Adv. Prot. Chem., 66, 27–85, 2003. [3] W.L. Jorgensen, D.S. Maxwell, and J. Tirado-Rives, “Development and testing of the OPLS all-atom force field on conformational energetics and properties of organic luquids,” J. Am. Chem. Soc., 118, 11225–11236, 1996. [4] D.W. Brenner, “Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films,” Phys. Rev. B, 42, 9458–9471, 1990.


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[5] T.A. Halgren, “Merck molecular force field. III. Molecular geometries and vibrational frequencies for MMFF94,” J. Comput. Chem., 17, 553–586, 1996. [6] A. Toukmaji, C. Sagui, J. Board, and T.Darden, “Efficient particle-mesh ewald based approach to fixed and induced dipolar interactions,” J. Chem. Phys., 113, 10912– 10927, 2000. [7] T.M. Nymand and P. Linse, “Ewald summation and reaction field methods for potentials with atomic charges, dipoles, and polarizabilities,” J. Chem. Phys., 112, 6152–6160, 2000. [8] O. Borodin, G.D. Smith, and R. Douglas, “Force field development and MD simulations of poly(ethylene oxide)/LiBF4 polymer electrolytes,” J. Phys. Chem. B, 108, 6824–6837, 2003. [9] H. Sun, “COMPASS: An ab initio force-field optimized for condensed-phase applications-overview with details on alkane and benzene compounds,” J. Phys. Chem. B, 102, 7338–7364, 1998. [10] A.D. MacKerell et al., “All-atom empirical potential for molecular modeling and dynamics studies of proteins,” J. Phys. Chem. B, 102, 3586–3616, 1998. [11] S.L. Mayo, B.D. Olafson, and W.A. Goddard, III, “DREIDING: A generic force field for molecular simulations,” J. Phys. Chem., 94, 8897–8909, 1990. [12] A.K. Rappé, C.J. Casewit, K.S. Colwell, W.A. Goddard, and W.M. Skiff, “UFF, a full periodic table force field for molecular mechanics and molecular dynamics simulations,” J. Am. Chem. Soc., 114, 10024–10035, 1992. [13] F. Sato, S. Hojo, and H. Sun, “On the transferability of force field parameters-with an ab initio force field developed for sulfonamides,” J. Phys. Chem. A., 107, 248–257, 2003. [14] O. Borodin and G.D. Smith, “Molecular modeling of poly(ethylene oxide) melts and poly(ethylene oxide)-based polymer electrolytes,” In: L. Curtiss and M. Gordon, (eds.), Methods and Applications in Computational Materials Chemistry, Kluwer Academic Publishers, 35–90, 2004. [15] O. Borodin and G.D. Smith, “Development of the quantum chemistry force fields for poly(ethylene oxide) with many-body polarization interactions,” J. Phys. Chem. B, 108, 6801–6812, 2003.

9.3 ROTATIONAL ISOMERIC STATE METHODS Wayne L. Mattice Department of Polymer Science, The University of Akron, Akron, OH 44325-3909

At very small degree of polymerization, x, the conformation-dependent physical properties of a chain are easily evaluated by discrete enumeration of all allowed conformations. Each conformation can be characterized in terms of bond lengths, l, bond angles, θ, torsion angles, φ, and conformational energy, E. The rapid increase in conformations as x → ∞ prohibits discrete enumeration when the chain reaches a degree of polymerization associated with a high polymer. This difficulty is overcome with the rotational isomeric state (RIS) model. This model provides a tractable method for computation of average conformation-dependent physical properties of polymers, based on the knowledge of the properties of the members of the homologous series with very small values of x. The physical property most commonly computed with the RIS method is the mean square unperturbed end-to-end distance, r 2 0 . Zero as a subscript denotes the unperturbed state, where the properties of the chain are controlled completely by the short-range interactions that are present at very small values of x. This assumption is appropriate for the polymer in its melt, which is a condition of immense importance both for modeling studies and for the use of polymers in reality. The assumption also applies in dilute solution in a solvent, where the excluded volume effect is nil [1]. The second virial coefficient for the osmotic pressure is zero in this special solvent. In good solvents, where the second virial coefficient is positive, the mean square end-to-end distance is larger than r 2 0 , due to the expansion of the chain produced by the excluded volume effect. The excluded volume effect is not incorporated in the usual applications of the RIS model. The first use of the RIS method was reported over five decades ago, well before the widespread availability of fast computers [2]. Given this date of origin of the method, it is not surprising that the correct numerical evaluation of a RIS model requires very little computer time, in comparison with newer 2575 S. Yip (ed.), Handbook of Materials Modeling, 2575–2582. c 2005 Springer. Printed in the Netherlands.

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W.L. Mattice

simulation methods that were developed after fast computers populated nearly every desktop.

1.

Information Required for Calculation of r 2 0

The essential features of the RIS method are well illustrated by the classic calculation of r 2 0 for a long unperturbed polyethylene chain, using as input the properties of n-butane and n-pentane [3]. This illustration identifies the information that is required from the small molecules, and shows how that information is incorporated into the model in order to calculate r 2 0 for a very long chain. The information required for a successful RIS treatment of polyethylene is summarized in Table 1. From n-butane we obtain the values for the length of the C–C bond, l = 0.154 nm, and the C–C–C bond angle, 112◦ . The internal C–C bond is subject to a symmetric torsion potential with three preferred conformations, ν = 3, denoted by trans(t), gauche+ (g + ), and gauche− (g − ). When φ is defined to be zero in the cis state, the torsion angles are 180◦ and ± (60◦ + φ), with the value of φ being about 7.5◦ . The g states are higher in energy that the t state by E σ = E g − E t = 2.1 kJ/mol. This first-order (dependence on a single torsion angle) interaction energy specifies a temperature-dependent statistical weight of σ = exp (−E σ /RT) for a g state relative to a t state. The input from n-butane would be sufficient for the RIS model if the bonds in polyethylene were independent of one another. However, independence of bonds in not observed in polyethylene or in most other polymers. Information about the pair-wise interdependence of the bonds comes from the next higher alkane in the homologous series. Specifically it is from the examination of the energies of the four conformations of n-pentane in which both internal C–C bonds adopt g states. If the two bonds were independent, the four gg states would have the same conformational energy, and that energy would be higher Table 1. Input from small n-alkanes to the RIS model for polyethylene Alkane Butane

Pentane

Information

Symbol

Value for polyethylene

C–C bond length C–C–C bond angle Number of rotational isomeric states Torsion angles

l θ

0.154 nm 112◦

ν φ

First-order interaction energy Second-order interaction energy

Eσ = E g – Et

3 180◦ and ± (60◦ + φ), φ = 7.5◦ 2.1 kJ/mol

E ω = E g + g− − E g+g+

8.4 kJ/mol

Rotational isomeric state methods

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by 2Eσ than the conformational energy in the tt state. This expectation is realized if both g states are of the same sign. However, if they are of opposite sign, a strong repulsive interaction of the pendant methyl groups causes the energy to be higher than the energy of the tt conformation by 2Eσ + 8.4 kJ/mol. This important extra energy, denoted E ω , is termed a second-order interaction because it depends on two torsion angles. Examination of the remaining conformations of n-pentane reveals no other important second-order interactions. Third- and higher-order interactions can be incorporated in the model, but often they are unnecessary. Polyethylene is an example of a chain where the performance of the model is not improved by the incorporation of thirdorder interactions. Third-order interactions occur between the methyl groups in n-hexane. Their interaction is prohibitively repulsive when the intervening C–C bonds are all in g states that alternate in sign. However, the g + g − g + conformation of n-hexane is severely penalized by the second-order interactions described in the previous paragraph. Penalizing it further by specifically incorporating the third-order interaction has a trivial effect on numerical results calculated from the model. Therefore the simpler approach, based on first- and second-order interactions only, is the one usually adopted. All of the information in Table 1 is used in the calculation of r 2 0 for a long unperturbed polyethylene chain via the RIS method. Initially the thermodynamic (or energetic) and structural (bond lengths, bond angles, torsion angles) contributions are considered separately. Then these two pieces of the problem are combined for the final answer.

2.

Thermodynamic (energetic) Information: The Conformational Partition Function

The thermodynamic information appears in the conformational partition function, Z, which is the sum of the statistical weights for all ν (n−2) conformations for an unperturbed chain with n bonds. The first- and second-order interactions from Table 1 are counted correctly in an expression for Z that uses a statistical weight matrix, Ui , for each bond. Z = U1 U2 . . . Un

(1)

For internal bonds, Ui is a ν × ν matrix, with rows indexed by the state at bond i − 1, and columns indexed in the same order by the state at bond i. Each column contains the first-order statistical weight appropriate for the conformation that indexes that column, and each element contains the second-order statistical weight appropriate for the pair of states defined by that row and

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column. If the order of indexing is t, g + , g − , Ui is specified by Eq. (2) for 1 < i < n. 



1 σ σ Ui =  1 σ σ ω , 1 σω σ

1 0) Pr(

(3)

A direct check of the validity of the transient FT itself was recently performed for a simpler experiment. In this experiment, an equilibrium ensemble of trapped particles was subject to a step function change in the strength of the optical trap. In this case the dissipation function appearing in the transient FT is not directly related to entropy production. Instead the dissipation function ¯ t is ¯ t = (tkB T )−1 (k0 − k1 )

0

t

•

ds r(s) • r(s)

= (2tkB T )−1 (k0 − k1 )(r(t)2 − r(0)2 )

(4)

where k0 , k1 are the initial and final values of the spring constant for the optical trap and and r(t) is the vector position of the colloid particle relative to the centre of the optical trap. From its definition (4) we can see the dissipation function is like entropy production in that its ensemble average is always expected to be positive. For this system the transient FT states that, ¯ t = A) Pr( = exp( At). ¯ Pr(t = −A)

(5)

This prediction was confirmed in the experiments of Carberry, et al., 2004 [5]. A recent review of the theoretical status of the FT has been published by Evans and Searles 2002 [4]. Fluctuation Theorems are extraordinarily general. There are stochastic versions of the FT. The FT is completely consistent with Langevin dynamics [6]. There are quantum versions of the FT [7, 8]. This theorem is obviously very general. It can be used to derive Green–Kubo relations for linear transport coefficients and the Fluctuation Dissipation Theorem [9]. However, it is more general than either of these two relations since the FT applies to the nonlinear regime far from equilibrium where Green– Kubo and Fluctuation Dissipation relations fail. The transient FT can be applied to nonequilibrium paths that connect two equilibrium states. When this is done the FT can be used to derive new expressions for free energy differences between equilibrium states in terms of sums over all nonequilibrium path integrals which connect those two equilibrium states [10–14]. We expect that over the next decade further extensions of the Fluctuation Theorem will be explored. The practical utility of these nonequilibrium free energy expressions is yet to be properly assessed. However the mere fact that equilibrium free energy differences can be related to nonequilibrium path integrals is quite surprising.

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So one hundred years after Boltzmann, I think we can finally say to our students that we understand how macroscopic irreversibility arises from reversible microscopic dynamics.

References [1] D.J. Evans, E.G.D. Cohen, and G.P. Morriss, “Probability of second law violations in shearing steady states,” Phys. Rev. Lett., 71, 2401–2402, 1993. [2] D.J. Evans and D.J. Searles, “Equilibrium microstates which generate second law violating steady states,” Phys. Rev. E, 50, 1645–1648, 1994. [3] G.M. Wang, E.M. Sevick, E. Mittag et al., “Experimental demonstration of violations of the second law of thermodynamics for small systems and short time scales,” Phys. Rev. Lett., 89, 050601–4, 2002. [4] D.J. Evans and D.J. Searles, “The Fluctuation theorem,” Adv. In Phys., 51, 1529– 1585, 2002. [5] D.M. Carberry et al., “Fluctuations and irreversibility: An experimental demonstration of a second-law-like theorem using a colloidal particle held in an optical trap,” Phys. Rev. Lett., 92, 140601–4, 2004. [6] J.C. Reid et al., “Reversibility in non-equilibrium trajectories of an optically trapped particle,” Phys. Rev. E, 70, 016111–9, 2004. [7] J. Kurchan, “A quantum fluctuation theorem,” arXiv:cond-mat/0007360, 1–8, 2001. [8] T. Monnai and S. Tasaki, “Quantum correction of fluctuation theorem,” J.Phys.A Math.Gen., 37, L75–L79, 2004. [9] D.J. Evans, D.J. Searles, and L. Rondoni, “On the application of the gallavotti-cohen fluctuation relation to thermostatted steady states near equilibrium,” arXiv:condmat/0312353, 1–40, 2004. [10] C. Jarzynski, “Nonequilibrium equality for free energy differences,” Phys. Rev. Lett., 78, 2690–2693, 1997a. [11] C. Jarzynski, “Equilibrium free energy differences from nonequilibrium measurements: A master equation approach,” Phys. Rev. E, 56, 5018–5035, 1997b. [12] G.E. Crooks, “Entropy production fluctuation theorem and nonequilibrium work relation for free energy differences,” Phys. Rev. E, 60, 2721–2726, 1999. [13] G.E. Crooks, “Path-ensemble averages in systems driven far from equilibrium,” Phys. Rev. E, 61, 2361–2366, 2000. [14] D.J. Evans, “A non-equilibrium free energy theorem for deterministic systems,” Mol. Phys., 101, 1551–1554, 2003.

Perspective 18 THE LIMITS OF STRENGTH J.W. Morris, Jr. Department of Materials Science and Engineering, University of California, Berkeley

In the usual case the strength of a crystalline material is determined by the motion of defects such as dislocations or cracks that are present within it. Materials scientists control strength by modifying the microstructure of the material to eliminate defects or flaws and inhibit the motion of dislocations. There is, however, an ultimate limit to the strength that can be obtained in this way. The mechanical stresses that are not relieved by plastic deformation or fracture are supported by elastic deformation, which is, essentially, the stretching of the interatomic bonds. These bonds have finite strength. There is a value of the stress at which bonding itself becomes unstable and the material must fracture or deform, whatever its microstructure. This elastic instability sets an upper bound on mechanical strength that cannot be exceeded, however creative a scientist may be. There are several practical reasons to be interested in the limit of elastic stability [1]. First, elastic instability defines the ideal strength [2–4], and it is useful to know the highest strength a particular material could possibly have. This is particularly true in times when new concepts in materials, such as nanomaterials, have led to optimistic predictions that have not always been vetted against the limits nature has set. Second, the elastic limit is reached, or, at least, closely approached in a number of experimental situations. A familiar example is deformation via stress-induced phase transformations, as in certain austenitic steels. However, even normal, ductile metals seem to approach the limit of strength in nanoindentation experiments, and stronger alloys may also do so in the region of stress concentration ahead of a sharp crack. Third, elastic instability is one of the few problems in solid mechanics that can actually be solved ab initio. Existing pseudopotential codes are capable of following elastic deformation to the point of instability with reasonable accuracy, and a number of calculations have been done [1, 5–9]. Fourth, as we shall see, the 2777 S. Yip (ed.), Handbook of Materials Modeling, 2777–2785. c 2005 Springer. Printed in the Netherlands.

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theoretical study of behavior at the limit of strength can provide new insight into mechanical behavior in more common situations. From the perspective of understanding deformation, it is interesting that a wide variety of mechanical phenomena that are ordinarily attributed to the specific behavior of dislocations would also be found in a defect-free world. In the limited space available here we discuss three examples. (1) In a defectfree world, the common bcc metals would cleave on {100} and exhibit “pencil glide” in 111. Most would be brittle at low temperature, as they are. (2) The common fcc metals would glide in {111} and would not cleave under simple tensile loads. They would be ductile at low temperature, as most of them are. (3) The maximum values of the nanohardness of simple metals would be very nearly what they are.

1.

Methodology

The calculation of ideal strength is based on modern methods that make it possible to compute the energy of a crystalline configuration of atoms with rather good accuracy. A number of computational techniques are available, almost all of them based on density functional theory. The Vienna Ab-Initio Simulation Package (VASP) [10] provides a set of readily available tools to accomplish these calculations, and other packages are also available. In most cases the local density approximation (LDA) to density functional theory yields good results. In certain circumstances (e.g., Fe) the generalized gradient approximation (GGA) gives significantly more accurate results [11]. The relative quality of the two approximations can be judged by comparing experimental data to theoretical predictions. In these methods Schrodinger’s equation is solved within a single particle approximation. It is usually sufficient to employ pseudopotentials for the atomic cores and a plane wave expansion of the wave functions (this is the approach used in VASP). This technique works reasonably well in situations for which the core states are not strongly affected by the imposed strains. However, under severe compressive stresses, for example, all electron methods, such as that employed in WIEN97, provide a better description of the solid’s properties. The elastic stress within a solid is related to the derivative of the energy with respect to the elastic strain. While there is some subtlety involved in doing this (there is no unique definition of the strain), it is usually sufficient to define the strain from the displacements of the crystal lattice points from a reference configuration [11, 12]. One can then compute the energy as a function of strain by incrementally displacing the atom positions within the unit cell to increment the strain along the desired path, and evaluate the associated stresses from the energy increment. The ultimate strength is associated with

The limits of strength

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the maximum in the stress, which occurs at (or near) the inflection point of a plot of energy against strain. The precise way in which the stress-strain relation is evaluated depends on the specific problem one is trying to solve. For example, one is often interested in the ideal strength under simple tension along a particular crystal axis, or in simple shear on a particular crystallographic plane. In these cases the only nonzero stress is the stress that is conjugate to the strain of interest: the uniaxial tensile stress in the case of simple tension, the conjugate shear stress in the case of simple shear. In these cases the atom positions must be adjusted after each increment to the strain so that all other stresses are relaxed to zero. Methods for doing this in a variety of load geometries and crystal types are described in the references cited at the end of this article. We shall now describe the results of some of these calculations with an emphasis on the physical insight they provide.

2. 2.1.

BCC Metals Ideal Strength in Tension

Computations of the ideal tensile strengths of unconstrained bcc metals show that they are weakest when pulled in a 100 direction. The majority of those we have investigated fail in tension (cleave) on {100} planes, both in theory and in experiment. Ab initio calculations [1, 11–15] give an ideal tensile strength of about 30 GPa for W (0.07E100 ), 29 GPa for Mo (0. 078E100 ) and 13 GPa for Fe (0.087E100 ) where E100 is the tensile modulus in the 100 direction, and the stresses are computed at 0K. There is a simple crystallographic argument that explains both the cleavage plane and the ideal tensile strength (Fig. 1). A relaxed tensile strain along

Figure 1. The Bain strain connecting the bcc and fcc structures. If bcc is pulled in tension on [001] while contracting along [100] and [010] it generates an fcc crystal as shown.

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100 carries the bcc structure into an fcc structure with the same volume at a tensile strain of 0.26 (the “Bain strain”). By symmetry, both structures are unstressed, so the tensile stress must pass through at least one maximum along the transformation path, at a critical stain much less than 0.26. If we fit the stress-strain curve to a sinusoid that has the correct modulus at low strain, the tensile strength in 100 is given by

σm ∼

eB E100 = 0.08E100 π

(1)

where eB is the Bain strain and E100 is Young’s modulus for 100 tension. The same reasoning explains why the ideal strength increases when the normal tension is supplemented by a hydrostatic tension, as it is, for example, near the tip of a crack. Hydrostatic tension expands the unit cell, which increases the Bain strain and raises the stress at instability. Ab initio calculations for Fe [16] show that the ideal tensile strength increases by almost 50% when the tensile stress is supplemented by a hydrostatic tension that is equal in magnitude. The element Nb is anomalous among the bcc metals we have studied [13, 17]. While the ideal tensile strength of Nb is lowest for 100 loading, as in the other bcc’s, the failure mode is not in tension across the {100} planes, but in shear on the 111{112} system. After some significant tensile strain, Nb deviates from the tetragonal, Bain strain path onto an orthorhombic strain path that is characterized by unequal contractions in the 100 directions perpendicular to the axis of load. The eventual failure is in shear, rather than tension. This preference for shear failure is preserved, though with a smaller margin, when hydrostatic tension is superiomposed. The results suggest that Nb may not exhibit the ductile-brittle transition that is typical of bcc metals. The experimental evidence is unclear [13].

2.2.

Ideal Strength in Shear

The ideal shear strengths of bcc metals also reflects their symmetry. Calculations of the ideal strength of bcc W [12] in relaxed shear in the 111 direction on the {110}, {112} and {123} planes give almost identical values, τm ∼ 17.7 GPa ∼ 0.11G111 , where G111 is the shear modulus for shear in the 111 direction. In all three cases the shear strain at instability is about 0.17. Calculations for Fe [11] and Mo [13] give very similar results, with τm ∼ 7.2 GPa (0.11G111 ) for Fe, ∼15.8 GPa ( 0.12G111 for Mo). Nb is, again, unusual; the ideal shear strength in the 111{112} system is anomonously large, 6.4 GPa (0.15G111 ), and is still larger for the common alternative systems [13].


2781

The symmetry rule that governs the shear strength of typical bcc’s is illustrated in Fig. 2 [1, 12]. Essentially, a shear in the 111 direction tilts planes that are perpendicular to the 111 axis. If we allow relaxation of the atom positions in these planes, they come into an atomic registry that changes the crystal symmetry at a shear strain of ∼ 0.34, irrespective of the plane of tilt. This common stress-free, saddle-point structure is body-centered tetragonal. There is a maximum in the shear strength at about half the saddle-point shear, at e ∼ 0.17. If we fit the stress-strain relation with a sinusoid that gives the correct modulus in the elastic limit, we obtain

τm ∼ 0.11G111

(2)

in good agreement with the ab initio calculations. A number of bcc metals have similar strengths for slip in the 111 direction on various planes, a phenomenon that is known as “pencil glide” and is attributed to the peculiarities of dislocation glide in bcc. These calculations show that defect-free bcc crystals would tend to behave in a very similar way. The balance between the shear and tensile strengths suggested by Eqs. (1) and (2) is such that, in a defect-free world, the common bcc metals would cleave if loaded along 100, but not if loaded in other directions. Taking W as an example, a uniaxial load along 100 would reach the ideal cleavage strength, ∼ 30 GPa, when the shear stress in the most favorable slip system was only around 14 GPa, below the ideal shear strength. However, a uniaxial load along 111 or 110 would cause the shear strength to be exceeded before the tensile stress in 100 reached the ideal value. A single crystal would be ductile or brittle, depending on the direction of the load.

Figure 2. Instability in shear in [111], [111] shear tilts planes of atoms (equilateral triangles perpendicular to [111]) until they come into registry, as at right, creating a new symmetry.

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FCC Metals

3.1.

Ideal Strength in Tension

The tensile strength of defect-free fcc crystals could, in theory, also be governed by the Bain strain [18]. As illustrated in Fig. 3, an fcc crystal can be converted into bcc by straining in tension in the [110] direction. The tensile strain required to reach bcc is, in fact, relatively small, so the estimated strength would be small as well. However, reaching bcc from fcc with a [110] pull requires very substantial relaxations in the perpendicular directions. The ¯ crystal must expand along [110] and contract to an even greater degree along [001]. These large relaxations are inconsistent with the Poisson contractions of typical fcc metals in the linear elastic limit. Fcc metals do not start out along this deformation path when pulled along [110] and, apparently, never find it. Nonetheless, the 110 directions are the weak directions for tension in all of the fcc metals that have been studied to date: Al, Cu, Ir and Pd [19]. If the fcc crystal is pulled quasistatically to failure under uniaxial tension, the failure mode at the elastic limit is not a tensile failure across the perpendicular {110} plane, but rather a shear failure (the “flip strain”). In this deformation mode the 110 tensile direction is stretched while the perpendicular 100 direction contracts, with the ultimate consequence that the two directions are interchanged. It can be shown that the failure mode is, in fact, a failure in shear in the 112{111} system, which is the normal mode of shear failure in an fcc crystal. This result suggests a simple explanation for the fact that fcc crystals do not exhibit a conventional ductile-brittle transition on cooling; the inherent failure mechanism is in shear. Those fcc metals that do become brittle at low temperature, such as Ir and nitrided austenitic steels, do so only after [110] [1⫺10]

[001]

[110]

[1⫺10]

[001]

Figure 3. Bain strain of an fcc lattice through tension along [110]. The crystal must expand ¯ equally along [110] and contract dramatically along [001]. The alternative is the “flip strain” shown at right.


2783

significant plastic deformation; their “cleavage” is by decohesion on slip or twin planes. If we assume deformation at constant volume with a sinusoidal stress-strain curve, the “flip” instability occurs at an engineering strain of 0.08 and a stress of σm = 0.05E110 , where E110 is Young’s modulus for tension in the 110 direction. The most recent ab initio calculations of the ideal tensile strengths of fcc metals under quasistatic loading give the following values: for Al, σm = 5.2 GPa = 0.07E110 , for Cu, σm = 6.2 GPa = 0.05E110 , for Pd, σm = 5.2 GPa = 0.06 E110 , and for Ir, σm = 36 GPa = 0.06E110 [19]. Note that the exceptional strength of Ir is a consequence of its large elastic modulus, and is in no way anomalous. The anomaly is the high dimensionless strength of Al. However, recent research has shown that the actual strength of Al is determined by a phonon instability that intrudes slightly before the elastic instability [20], decreasing the ideal strength. No similar instability has been found in other fcc’s.

3.2.

The Ideal Strength in Shear

The mode of failure of fcc crystals in shear is conventional, though the behavior of at least some fcc’s, Al in particular, is not. The weak directions in shear are 112 directions in {111} planes, as one would expect from a rigidball model of the close-packed fcc structure. A sinusoidal model of that failure mode predicts a shear strength, τm ∼ 0.085G111 . The ideal shear behavior of Al and Cu [20, 21] make an interesting comparison. While the deformation of Cu remains nearly planar in the {111} shear plane as the instability is approached, Al expands significantly perpendicular to the shear plane. The consequence is that while Cu has an ideal strength near the estimate, τm = 2.7 GPa ∼ 0.11G 111 , the calculated shear strength of Al is much larger, both in absolute magnitude and in dimensionless terms: τm ∼ 3.4 GPa = 0.15G 111 . (This number is a bit of an overestimate, since Al experiences a phonon instability before reaching peak strength [20], but is qualitatively correct. It is significantly higher than the number reported in earlier work [8], which used a less accurate pseudopotential.) Comparing the ideal tensile and shear strengths of Al and Cu leads to a curious result: at the limit of strength, Cu is stronger than Al in tension, though weaker in shear. This is true despite the fact that the failure mode is precisely the same in the two cases: a shear instability in the system 112{111}. The reason is the perpendicular expansion of Al during shear. Tension in the 110 direction imposes a tension across the {111} shear plane, assisting the normal displacement of {111}planes and lowering the stress required for Al to fail in shear.

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Nanoindentation

A nanoindentation test is, essentially, a microhardness test done with a nanotipped indenter. Until the substrate yields, the deformation field of the indenter should be approximately Hertzian, which makes it possible to use the data to infer the stresses and strains at which yielding occurred. Moreover, since the maximum shear in a Hertzian strain field is well beneath the surface, nanoindentation tests can sample defect-free volumes, and may, therefore, test the ideal strength. Surprisingly, the shear strengths inferred from recent nanoindentation tests substantially exceed the computed ideal strengths. Thus, Bahr, et al. [22] report data showing shear stresses as high as 28 GPa in W prior to yielding, well beyond the value (18 GPa) that corresponds to the ideal strength on any of the common slip planes. Nix [23] reported preliminary Mo data giving a maximum strength of 23 GPa, compared to the theoretical shear strength of 15.6 GPa. The discrepancy between these values is almost entirely removed if one makes two corrections [24]. First, the Hertzian stress field is modified by nonlinearity as the ideal strength is approached. Finite-element calculations using a sinusoidal stress-strain relation show that the Hertzian stress field is correct except in the immediate vicinity of the maximum shear stress, even when the maximum shear stress approaches the ideal strength. However, the value of the maximum shear stress is significantly decreased, to τm ∼ 0.69 τH , where τH is the Hertzian value. Second, the triaxiality of the stress field near the point of maximum shear increases the ideal shear strength. When these (and a couple of other, minor corrections) are made the maximum shear strengths that can be inferred from nanoindentation experiments on W (22.8–24.0 GPa) and Mo (16.0–16.8 GPa) are very close to the theoretical values of the ideal strength (W = 22.1–23.3 GPa; Mo = 17.6–18.8 GPa), as they should be. This result suggests that nanoindentation may provide a viable means for measuring ideal strength.

References [1] J.W. Morris, Jr., C.R. Krenn, D. Roundy et al., In: P.E. Turchi and A. Gonis (eds.), Phase Transformations and Evolution in Materials, TMS, Warrendale, PA, pp. 187– 208, 2000. [2] R. Hill and F. Milstein, Phys. Rev. B, 15, 3087–3097, 1977. [3] J. Wang, J. Li, S. Yip et al., Phys. Rev. B, 52, 12, 627–635, 1995. [4] J.W. Morris, Jr. and C.R. Krenn, Phil. Mag. A, 80, 2827–2840, 2000. [5] A.T. Paxton, P. Gumbsch, and M. Methfessel, Phil. Mag. Lett., 63, 267–274, 1991. [6] W. Xu and J.A. Moriarty, Phys. Rev. B, 54, 6941–6951, 1996. [7] M. Sob, L.G. Wang, and V. Vitek, Mat. Sci. Eng., A234–236, 1075-1078, 1997.

The limits of strength [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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D. Roundy, C.R. Krenn, M.L. Cohen et al., Phys. Rev. Lett., 82, 2713–2716 1999. S. Ogata, J. Li, and S. Yip, Phys. Rev. B, in press, 2004. G. Kresse and J. Hafner, J. Phys. Condens. Matter, 6, 8245, 1994. D.M. Clatterbuck, D.C. Chrzan, and J.W. Morris, Jr., Acta Mater., 51, 2271–2283, 2003. D. Roundy, C.R. Krenn, M.L. Cohen et al., Phil. Mag. A, 81, 1725–1747, 2001. W. Luo, D. Roundy, M. L. Cohen et al., Phys. Rev. B, 66, 94110, 2002. D.M. Clatterbuck, D.C. Chrzan, and J.W. Morris, Jr., Phil. Mag. Lett., 82, 141–147, 2002. M. Friak, M. Sob, and V. Vitek, Proc. Int. Conf. Juniormat 2000, Brno Univ. Technology, Brno, 2001. D.M. Clatterbuck, D.C. Chrzan, and J.W. Morris, Jr., Scripta Mat., 49, 1007, 2003. C.R. Krenn, D. Roundy, J.W. Morris, Jr. et al., Mat. Sci. Eng. A, A319–321, 111–114, 2001. J.W. Morris, Jr., C.R. Krenn, D. Roundy, and M.L. Cohen, Mat. Sci. Eng. A, 309– 310, 121–124, 2001. J.W. Morris, Jr., D.M. Clatterbuck, D.C. Chrzan et al., Mat. Sci. Forum, 426–432, 4429–4434, 2003. D.M. Clatterbuck, C.R. Krenn, M.L. Cohen et al., Phys. Rev Lett., 91, 135501, 2003. S. Ogata, J. Li, and S. Yip, Science, 298, 807, 2002. D.F. Bahr, D.E. Kramer, and W.W. Gerberich, Acta Mater., 46, 3605–3617, 1998. W.D. Nix, Dept. Materials Science, Stanford Univ., Private Communication, 1999. C.R. Krenn, D. Roundy, M.L. Cohen et al., Phys. Rev. B, 65, 13411–13416, 2002.

Perspective 19 SIMULATIONS OF INTERFACES BETWEEN COEXISTING PHASES: WHAT DO THEY TELL US? Kurt Binder Institut fuer Physik, Johannes Gutenberg Universitaet Mainz, Staudinger Weg 7, 55099 Mainz

Interfaces between coexisting phases are ubiquitous in the physics and chemistry of condensed matter: Bloch walls in ferromagnets; antiphase domain boundaries in ordered binary (AB) or ternary alloys; the surface of a liquid droplet against its vapor; boundaries between A-rich regions and B-rich regions in fluid binary mixtures, etc. These interfaces control material properties in many ways (e.g., when a fluid polymer mixture is frozen-in to form an amorphous material, the mechanical strength of this macromolecular glass is controlled by the extent that A-polymers are entangled with B-polymers across the interface, etc.). For a detailed understanding of the properties of such interfaces, one must consider their structure from the scale of “chemical bonds” between atoms in the interfacial region up to much larger, mesoscopic, scales (e.g., when one tries to measure a concentration profile across an interface between coexisting phases in a partially unmixed polymer blend by suitable depth profiling methods, typical results for the interfacial width are of the order of 50 nm [1]). So from the point of view of simulations, one deals with a multiscale problem [2]. However, the situation is even worse: interfaces may exhibit longwavelength excitations, such as capillary waves [3], which lead to the effect that many properties associated with interfaces do depend on the geometry used for the experiment [1] or the simulation [4]. This fact causes a strong danger so that the properties observed in the simulation (or experiment, respectively) are mis-interpreted [2]. For simplicity, we shall consider in the following only interfaces in fluid systems (the gas liquid-interface or the interface in a fluid binary mixture, respectively); although many considerations can be carried over immediately 2787 S. Yip (ed.), Handbook of Materials Modeling, 2787–2791. c 2005 Springer. Printed in the Netherlands.

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to interfaces in solid phases (e.g., Bloch walls in antiferromagnets, antiphase domain boundaries in ordered alloys with negligible mismatch between the lattice parameters of the constituent atomic species, etc.), there are many cases in solids where additional complications arise due to long-range elastic interactions, and it is clear that the latter effects are very important (consider e.g., “coherent” vs. “incoherent” precipitation: in the latter case elastic strains destroy the coherence of the lattice structure between a precipitated grain and the surrounding matirx). These complications will stay beyond the considerations of the present note, however. Now the standard concept to describe an interfacial profile (i.e., a density or concentration variable c(z) observed as a function of the coordinate z across the interface) is the concept of an “intrinsic interfacial profile”. This concept dates back to van der Waals in the 19th century (and to Cahn and Hilliard in the late 50s of the 20th century, as far as ordinary binary mixtures are concerned, and to deGennes, Joanny and Leibler in the late 70s for polymer blends). The result for the interfacial profile near the critical point of the fluid (or fluid binary mixture, respectively) can be cast into the familiar tanh-form

c(z) =

z − z0 1 c1 + c2 + (c2 − c1 ) tanh 2 w0

.

(1)

Here c1 , c2 are the concentrations (or densities, respectively) of the two coexisting phases very far away from the interface that is located at z = z 0 . The variation from c1 to c2 extends essentially over the distance z = w0 around z 0 , the “intrinsic width” w0 . Near the critical point, w0 is just twice the correlation length of the order parameter fluctuations (remember that the order parameter is the concentration (density) difference c2 − c1 , for the considered systems). Near the critical point, the correlation length is large, particularly for polymer blends (where the scale for this length is set by the gyration radius of the coils, see e.g., [5]). Also away from the critical point, Eq. (1) often is obtained, at least as a good approximation, though the required theories are more complicated than the Ginzburg–Landau type mean field theory required to derive Eq. (1) near the critical point [5]. However, these theories (such as the density functional theories for fluids, or the self-consistent field theories for polymeric systems in a sense also have a mean field character, and all ignore lateral fluctuations in the (x, y) directions parallel to the interface. For very large length scales of an interface in a fluid system, these lateral fluctuations are nothing but the well-known capillary waves [3] (for simplicity, effects of gravity are ignored here: this is anyway a very good approximation for polymer mixtures, but also reasonable for many after systems). One can show from the equipartition theorem that the thermally averaged mean square amplitude of interfacial height fluctuations due to the capillary waves with wavelenghths 2 /q scales like the inverse of the square of the wavenumber q. This causes a long wavelength instability of the interface: on a lateral length

Simulations of interfaces between coexisting phases

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scale L parallel to the interface the mean square height scales proportional to the logarithm of L, ln (L/B), where the length B is a short-wavelength cutoff needed in the theory. In fact, on scales of the order of atomic distances (and near the critical point, already on scales of the order of the correlation length), the picture of a sharp interface locally displaced by capillary waves makes no sense anymore. However, a good understanding is lacking of what cutoff length B is [2–4, 6]. And when one considers the convolution of the intrinsic profile with this capillary wave broadening, there is no unique way to separate in the resulting dependence of the total width w the “intrinsic width” w0 from the term involving ln (B) [2, 4, 6]. These considerations have indeed dramatic consequences on simulations (and experiments [1]), as shown in [4]. Simulations intended to study interfacial profiles typically use one of the following two geometries: 1. A single interface can be stabilized in the geometry of a thin film of thickness D between two walls (of lateral linear dimensions L ). The nature of the walls is chosen such that they distinguish between the phases: in a binary mixture, one wall prefers A, the other wall prefers B; for a gas-liquid interface, one wall has a purely repulsive interaction between the wall and the fluid particles (prefering the gas phase), the other wall exerts an attractive interaction (prefering the liquid phase). In the lateral directions, periodic boundary conditions are used. The disadvantage of this geometry, of course, is that D must be rather large, to avoid a too strong perturbance of the interfacial profile due to the forces from the walls (there must be room on both sides of the interface for the bulk phases unperturbed by the walls). 2. One chooses periodic boundary conditions in all directions, and D considerably larger than L, and creates via suitable initial conditions a slab configuration, e.g., a liquid separated by two parallel interfaces from the gas (e.g., [7]). D must be large enough, so that any interactions between the two interfaces safely are avoided. In addition, L should not be too small either, because otherwise the average orientation of the interfaces around the z-directions also would fluctuate, and hence typically only part of the capillary wave spectrum is suppressed by the periodic boundary condition. In view of these problems, it is not clear to what extent observations of interfacial widths [7] should be compared with experiments, in particular when a careful study of the effects of varying both linear dimensions has not been made. A caveat needs also to be made with respect to the interfacial free energy f, which usually is computed from the profile of the anisotropy of the pressure tensor across the interface [3], since this profile is size-dependent as well. A variant of this geometry is used for symmetric systems (e.g., Isinglattice gas models, where there is a symmetry between particles and holes or

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“spin up” and “spin down” phases), there one can simulate a single interface but dispense of walls in favour of an “antiperiodic” boundary condition (crossing the boundaries in z-direction the sign of the spin is flipped) [8]. For a symmetric binary mixture, this means A turns into B (and vice versa) when a boundary in z-direction is crossed [6]. For this geometry (as well as for the above alluded “slab geometry” with two parallel interfaces) one has the problem already alluded to above, that the mean squared interfacial width varies with ln (L), and a unique identification of the intrinsic width (and intrinsic profile) is not possible. However, for the geometry with real walls (above choice No. 1) there is an additional very strong effect of the linear dimension D : for L , and shortrange forces due to the walls, the mean square width of the interface varies linearly with D (while it varies only logarithmically with D for long-range wall forces that decay with a power law with the distance from the wall and hence stabilize better the position of the interface in the middle of the thin film). This very strong size effect results from a kind of “soft mode”, the local position of the interface can be pushed away from the center of the thin film with very little energy cost, and in thermal equilibrium these “cheap” fluctuations cause a very strong broadening of the interfacial profile. These effects have been seen in simulations of Ising models and of polymer mixtures [2] as well as in experiments [1]. We conclude these comments by noting that finite size effects, on the other hand, are useful for the estimation of interfacial free energies f {e.g., [8,9]}. Choosing a geometry with periodic boundary conditions in all spatial directions and the (semi-) grandcanonical ensemble, the distribution of the order parameter is sampled. While its maxima correspond to the pure phases, its minimum corresponds to the slab configuration mentioned above. To be able to sample the minimum accurately, “multicanonical Monte Carlo” or “umbrella sampling techniques” are needed. From the ratio of the logarithms of the probabilities in the maximum and in the minimum, one can extract the free energy excess due to the interfaces, which is 2 times the area of a single interface times f . Checking that the result indeed scales linearly with the interface area proves that the asymptotic regime has indeed been reached. This “first principles” method avoids the need for taking any “measurements” at the interfaces or even to observe them explicitly in the simulations! There is now ample evidence for the reliability of this approach [6].

References [1] T. Kerle, J. Klein, and K. Binder, “Effects of finite thickness on interfacial widths in confined thin films of coexisting phases,” Eu. Phys. J. B7, 401–410, 1999.

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[2] K. Binder, “Simulations of interfaces between coexisting phases: multiscale aspects,” In: Multiscale Computational Methods in Chemistry and Physics, pp. 207–220, IOS Press, Amsterdam, 2001. [3] J.S. Rowlinson and B. Widom, “Molecular theory of capillarity,” Clarendon, Oxford, 1982. [4] A. Werner, F. Schmid, M. Mueller, and K. Binder, “Anomalous size-dependence of interfacial profiles between coexisting phases of polymer mixtures in thin film geometry: a Monte Carlo simulation,” J. Chem. Phys., 107, 8175–8188, 1997. [5] K. Binder, “Phase transitions in polymer blends and block copolymer melts: some recent developments,” Adv. Polym. Sci., 112, 181–299, 1994. [6] A. Werner, F. Schmid, M. Mueller, and K. Binder, “Intrinsic profiles and capillary waves at homopolymer interfaces: a Monte Carlo study,” Phys. Rev. E, 59, 728–738, 1999. [7] J. Alejandre, D.J. Tildesley, and G.A. Chapela, “Molecular dynamics simulation of the orthobaric densities and surface tension of water,” J. Chem. Phys., 102, 4754– 4583, 1995. [8] K. Binder, “Monte Carlo simulations of surfaces and interfaces in materials,” In: A. Gonis, P.A. Turchi, and J. Kudrnovsky (eds.), Stab. Mater., pp. 3–37, Plenum, New York, 1996. [9] M. Mueller, K. Binder, and W. Oed, “Structural and thermodynamic properties of interfaces between coexisting phases in polymer blends: a Monte Carlo investigation,” J. Chem. Soc. Faraday Trans., 91, 2369–2379, 1995.

Perspective 20 HOW FAST CAN CRACKS MOVE? Farid F. Abraham IBM Almaden Research Center, San Jose, California

1.

Molecular Dynamics Experiments

With the present-day supercomputers, simulation is becoming a very powerful tool for providing important insights into the nature of materials failure. Atomistic simulations yield “ab initio” information about materials deformation at length and time scales unattainable by experimental measurement and unpredictable by continuum elasticity theory. Using our “computational microscope,” we can see what is happening at the atomic scale. Our simulation tool is computational molecular dynamics [2], and it is very easy to describe. Molecular dynamics predicts the motion of a large number of atoms governed by their mutual interatomic interaction, and it requires the numerical integration of the equations of motion, “force equals mass times acceleration or F = ma.” We learn in beginning physics that the dynamics of two atoms can be solved exactly. Beyond two atoms, this is impossible except for a few very special cases, and we must resort to numerical methods. A simulation study is defined by a model created to incorporate the important features of the physical system of interest. These features may be external forces, initial conditions, boundary conditions, and the choice of the interatomic force law. In the present simulations, we adopt simple interatomic force laws since we wish to investigate the generic features of a particular many-body problem common to a large class of real physical systems and not governed by the particular complexities of a unique molecular interaction. It is very important to emphasize that this is a conscious choice since it is not uncommon to hear others object that one is not studying “real” materials when using simple potentials. Feynman summarized this viewpoint well on page two, volume I of his famous three volume series Feynman’s Lectures In Physics [3]: “If in some cataclysm all scientific knowledge were to be destroyed and only one sentence passed on to the next generation of creatures, what statement would contain the most information in the fewest words? I believe it is 2793 S. Yip (ed.), Handbook of Materials Modeling, 2793–2804. c 2005 Springer. Printed in the Netherlands.

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the atomic hypothesis that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see there is an enormous amount of information about the world, if just a little imagination and thinking are applied”. A simple interatomic potential may be thought of as a “model potential,” and the model potentials for the present studies are harmonic and anharmonic springs and the Lennard–Jones 12:6 potential. Complaints of model approximations are not new. In his book entitled The New Science of Strong Materials, Gordon comments on Griffith’s desire to have a simpler experimental material that would have an uncomplicated brittle fracture [4]. He writes, “In those days, models were all very well in the wind tunnel for aerodynamic experiments but, damn it, who ever heard of a model material?” In the mid 1960s, a few hundred atoms could be treated. In 1984, we reached 100,000 atoms. Before that time, computational scientists were concerned that the speed of scientific computers could not go much beyond 4 Gigaflops, or 4 billion arithmetic operations per second and that this plateau would be reached by the year 2000! That became forgotten history with the introduction of concurrent computing. A modern parallel computer is made up of several (tens, hundreds or thousands) small computers working simultaneously on different portions of the same problem and sharing information by communicating with one another. The communication is done through message passing procedures. The present record is well over a few tens of Teraflops for optimized performance. Moore’s Law states that computer speed doubles every one and one-half years. For 35 years, that translates into a computer speed increase of ten million. This is exactly the increase in the number of atoms that we could simulate over the last 35 years.

2.

Supersonic Crack Propagation in Brittle Fracture

Our simulation study addresses the important question “how fast can cracks propagate?” In this study, we used system sizes of about twenty million atoms, very large by present-day standards but modest compared to our second study on ductile failure, which will follow [1]. Based on our current simulation model, we develop our earlier studies on transonic crack propagation in linear materials and supersonic crack propagation in nonlinear solids. Our new finding centers on a bilayer solid which behaves under large strain like an interface crack between a soft (linear) material and a stiff (nonlinear) material. In this mixed case, we observe that the initial mother crack propagating at the Rayleigh sound speed gives birth to a transonic daughter crack. Then, quite unexpectedly, we observe the birth of a supersonic granddaughter crack.

How fast can cracks move?

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We verify that the crack behavior is dominated by the local (nonlinear) wave speeds, which can be in excess of the conventional sound speeds of a solid. In this problem, there are three important wave speeds in the solid. In order of increasing magnitude, they are the Rayleigh wave speed, or the speed of sound on a solid surface, the shear (transverse) wave speed and the longitudinal wave speed. Predictions of continuum mechanics [5] suggest that a brittle crack cannot propagate faster than the Rayleigh wave speed. For a mode I (tensile) crack, the energy release rate vanishes for all crack velocities in excess of the Rayleigh wave speed, implying that the crack cannot propagate at a velocity greater than the Rayleigh wave speed. A mode II (shear) crack behaves similarly to a mode I crack in the subsonic velocity range; i.e., the energy release rate monotonically decreases to zero at the Rayleigh wave speed and remains zero between the Rayleigh and shear wave speeds. However, the predictions for the two loading modes differ for crack velocities greater than the shear wave speed. While the energy release rate remains zero for a mode I crack, it is positive for a mode II crack over the entire range of intersonic velocities. From these theoretical solutions, it has been concluded that a mode I crack’s limiting speed is clearly the Rayleigh speed. The same conclusion has also been suggested for a mode II crack’s limiting speed because the “forbidden velocity zone” between the Rayleigh and shear wave speeds acts as an impenetrable barrier for the shear crack to go beyond the Rayleigh wave speed. The first direct experimental observation of cracks moving faster than the shear wave speed has been reported by Rosakis et al., [6]. They investigated shear dominated crack growth along weak planes in a brittle polyester resin under dynamic loading. Around the same time, we performed 2D molecular dynamics simulations of crack propagation along a weak interface joining two strong crystals [7]. We assumed that the interatomic forces are harmonic except for those pairs of atoms with a separation cutting the centerline of the simulation slab. For these pairs, the interatomic potential is taken to be a simple model potential that allows the atomic bonds to break. Our simulations demonstrated intersonic crack propagation and the existence of a mother-daughter crack mechanism for a subsonic shear crack to jump over the forbidden velocity zone. We have since discovered that a crack cannot only travel supersonically [8] but that there exists a mother-daughter-granddaughter crack mechanism in bilayer slabs. The classical theories of fracture [5] are largely based on linear elastic solutions to the stress fields near cracks. An implicit assumption in such theories is that the dynamic behavior of cracks is determined by the linear elastic properties of a material. We have found [9] that the MD simulation results for harmonic atomic forces are indeed well interpreted by elasticity theories. However, the effects of anharmonic material properties on dynamic behaviours of cracks are not clearly understood. This is partly due to the

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general difficulties in obtaining non-linear elastic solutions to dynamic crack problems. Molecular dynamics (MD) simulations can be easily adapted to the anharmonic case so that non-linear effects can be thoroughly investigated. We now discuss the anharmonic simulations.

3.

The Computer Experiments Setup

We consider a strongly non-linear elastic solid described by a tethered Lennard–Jones potential where the compressive part of this potential is identical to that of the usual Lennard–Jones 12:6 function and the tensile part is the reflection of the compressive part with respect to the potential minimum. An FCC crystal formed by this potential exhibits a strongly non-linear stress-strain behaviour resulting in elastic stiffening and an increase of the elastic modulus with strain, as shown in Fig. 1. Note that the elastic modulus increases by a factor of 10 at 13% of elastic strain, indicating that the material properties of such a solid is strongly non-linear in the hyperelastic regime. We performed 3D MD simulations of two face-center-cubic (fcc) crystals joined by a weak interface. In this study, we used system sizes of about twenty million atoms, though a billion atoms were used in a preliminary simulation [8]. For comparison, we consider the anharmonic tethered potential together with the harmonic potential where the spring constant is equal to the

3500 anhrarmonic potential modulus

2500

1500

500 0

0.05

0.1 strain

0.15

0.2

Figure 1. Variation of elastic modulus as a function of strain under uniaxial stretching in the [110] direction of a fcc crystal formed by the anharmonic (tethered repulsive LJ) potential.


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tethered potential at equilibrium. The interatomic force bonding the two crystals is given by the Lennard–Jones 12:6 potential. The simulation results are expressed in reduced units: lengths are scaled by the value of the interatomic separation where the LJ potential is zero and energies are scaled by the depth of the minimum of the LJ potential. Atoms bond only with their original nearest neighbors. Hence, rebonding of displaced atoms due to applied loading does not occur. For the adjoining two crystal slabs, we consider the following three simulation cases: (1) Harmonic case: both crystal slabs are characterized by the harmonic potential. This is used as a control for the anharmonic studies; (2) Anharmonic case: both crystal slabs are characterized by the anharmonic potential; (3) Mixed case: one crystal slab is characterized by the anharmonic potential, and the other is characterized by the harmonic potential. In each case, a shear crack lies along the (110) plane and oriented toward the [110] direction. The crack front is parallel to the [001] direction. The applied loading is dominated by shear. In order to interpret the results of MD simulation, we need the following wave speeds in the fcc crystals formed by harmonic and/or anharmonic potentials: the conventional longitudinal and shear wave speeds in the harmonic and anharmonic crystals in the direction of crack propagation; the longitudinal and shear wave speeds under applied loading in the anharmonic crystal in the direction of crack propagation; the local longitudinal and shear wave speeds near the crack tip in the anharmonic crystal in the direction of crack propagation. The calculations of these wave speeds will be presented in detail in a forthcoming paper. We give the results in Table 1. We add some comments with regard to the concept of local wave speeds near the crack tip, which play a critically important role in explaining our simulation results. Conceptually, it is clear that the fracture process in brittle solids involves breaking of atomic bonds and is intrinsically a highly non-linear process. The anharmonic material properties of solids near the cohesive strength of atomic bonds would in general be quite different from the harmonic properties. In an earlier attempt to explain why the highest crack velocities recorded for mode I crack propagation in a homogeneous body are significantly lower than the Rayleigh wave speed, Broberg [5, 10] has suggested that the reason could be due to some kind of a “local” Rayleigh wave speed in the highly strained region near the crack tip, rather than the Rayleigh wave speed in the

Table 1. Calculated wave speeds for the harmonic wave speeds, bulk anharmonic wave speeds due to applied loading and local wave speeds Longitudinal wave Shear wave

Bulk harmonic

Bulk anharmonic

Local

9.49 4.24

10.36 5.95

13.4 10.4

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undisturbed material. Since the local strain is an extremely strong function of position from the crack tip, one might think that the local wave speed should depend on distance from the crack tip and cannot have one single value. It was not until 30 years after Broberg’s suggestion that this issue was finally quantitatively studied by Gao [11] who made use of the Barenblatt cohesive model of a mode I crack and, for the first time, defined the local wave speed unambiguously as the wave speed at the location where stress is exactly equal to the cohesive strength of the material, i.e., the true point of “fracture nucleation”. The local wave speed characterizes how fast elastic energy is transported near the region of bond breaking in front of a crack tip. For example, for a mode I crack√in a homogeneous isotropic elastic, the local wave speed is calculated to be σmax /ρ [11] where σmax is the cohesive strength and ρ is the density of the undisturbed materials. The cohesive strength is typically around 1/10 of the shear modulus, suggesting that the local wave speed for a mode I crack is approximately 1/3 of the shear wave speed. Interestingly, experiments [12] and MD simulations [13] show that mode I cracks exhibit a dynamic instability at 30% of the shear wave speed which suggests a possible dependence on the local wave speed [11]. We note that the local wave speeds differ from the conventional wave speeds both qualitatively and quantitatively. In the present study, the focus is shifted to study the effect of stiffening anharmonic behaviors of materials (as in many polymers) on a mode II crack propagating along a weak interface described by the Lennard–Jones potential. The solid itself is described by the tethered Lennard–Jones potential. We follow the same approach used in [11] and define the local wave speed as the wave speed of the solid at the location adjacent to the interface where the shear stress is exactly equal to the cohesive strength of the interface. Note that in this case the local wave speed depends on both the interface cohesive strength and the non-linear elastic properties of the solid.

4.

The Computer Experiments Results & Discussion

Figure 2 presents distance-time histories of a crack moving in the three different slab configurations: two weakly bonded harmonic crystals (designated “harmonic”); two weakly bonded anharmonic crystals (designated “anharmonic”); and a harmonic crystal weakly bonded to an anharmonic crystal (designated “mixed”). The cracks begin their motion when the Griffith criterion is satisfied. The respective dip-spike regions for each history represent the birth of a new crack. For the harmonic and anharmonic simulations (see Figs. 3 and 4), we observe one such region representing the birth of a daughter crack, the former traveling at the longitudinal sound speed for the harmonic solid and the latter achieving a supersonic sound speed of Mach 1.6. For the mixed simulation (see Fig. 5), we see two such dip-spike regions where a daughter crack


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speed

30

20 anharmonic 10 harmonic 100

mixed 200

time Figure 2. The space-time history of the crack tip for the three different simulations described in the text (reduced units are used).

Figure 3. Snapshot pictures of a crack traveling in the harmonic slab, where the progression in time is from the top to bottom. The boxed snapshot pictures represent a progression in time from the top to bottom.

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Figure 4. Snapshot pictures of a crack traveling in the anharmonic slab, where the progression in time is from the top to bottom. The boxed snapshot pictures represent a progression in time from the top to bottom.

is born from the mother crack, which then gives birth to a granddaughter crack at a later time. In Figs. 3 and 4, the harmonic and anharmonic simulations are shown, respectively. The boxed snapshot pictures in each figure represent a progression in time from the top to bottom. In the top images, the mode II daughter cracks are born. The early-time occurrence of the Mach cone attached to the crack tip is evident in the anharmonic slab; In middle image of Fig. 3, the crack in the harmonic slab has a single Mach cone and a circular stress-wave halo, which indicates a crack speed equal to the longitudinal sound speed of the linear solid. This is in contrast to the middle image of Fig. 4 where the two Mach cones for the crack in the anharmonic slab signify that the crack is traveling supersonically. For the bottom images of the two figures, we conclude that the crack in the anharmonic slab wins. In Fig. 5, snapshot pictures of the crack traveling in the mixed slab are presented, where the progression in time is from the top to bottom. The material properties for the harmonic and anharmonic regions are labeled, and the different sound waves associated with the crack’s dynamics are denoted. The sequence shows the following progression of events: (1) the daughter


2801

Figure 5. Snapshot pictures of a crack traveling in the mixed slab, where the progression in time is from the top to bottom. The sound waves associated with the crack’s dynamics and the material properties of the harmonic and anharmonic regions of the mixed slab are labeled.

crack is born; (2) the daughter crack is traveling at the longitudinal sound speed of the harmonic slab; (3) the granddaughter crack is born; (4) the granddaughter crack speeds ahead to Mach 1.6, matching the crack speed in the anharmonic slab. We have shown that the behavior of the crack in the harmonic crystal is controlled by the conventional elastic wave speeds [7]. In contrast, the crack behavior in the anharmonic crystal is controlled by the local wave speeds, which play an important role in the dynamic behavior of crack propagation. A similar dependence has been identified in a related phenomenon where hyperelasticity plays an important role in whether a solid will undergo brittle fracture or ductile failure at the crack tip [14]. The local wave speed represents the non-linear, hyperelastic, material properties near cohesive failure of atomic bonds while the conventional elastic (shear and longitudinal) wave speeds represent the material properties under infinitesimal deformation and can differ significantly from the harmonic wave speeds. The crack propagation speeds observed in molecular dynamics simulations are tabulated in Table 2 for comparison.

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F.F. Abraham Table 2. Observed speeds at which a daughter crack or a granddaughter crack nucleates and the limiting speeds for the harmonic, anharmonic and mixed simulation cases Daughter crack Grand daughter Limiting speed

Harmonic

Anharmonic

Mixed

3.5 – 9.4

4.1 – 15

4.1 10.35 15

Comparing the MD simulation results with the calculated wave speeds for harmonic and anharmonic crystals, we reach the following conclusions. The harmonic case is consistent with the linear elastic theory of intersonic crack propagation. We have previously discussed this case in detail for the 2D harmonic MD simulation [7]. The same theory is found to apply for the present 3D case. Essentially, the initial crack starts to propagate when the Griffith criterion is satisfied. Near the Rayleigh wave speed, the crack encounters a velocity barrier and a vanishing of stress singularity at the crack tip; i.e., both stress intensity factor and energy release rate vanish at the Rayleigh wave speed. This velocity barrier is overcome by the nucleation of a daughter crack at a distance ahead of the mother crack. This distance corresponds to the shear wave front at which a peak of shear stress increases to a critical magnitude to cause cohesive failure of the interface. The daughter crack’s speed is only limited by the longitudinal wave speed. Comparing Tables 1 and 2, we see that the daughter crack indeed nucleates at the Rayleigh wave speed and the limiting speed agrees very well with the longitudinal wave speed for the harmonic crystal. The mother-daughter mechanism described above is consistent with the Burridge–Andrew model of intersonic crack propagation [15]. In the anharmonic case, the nucleation of the daughter crack is consistent with linear elastic theory. The mother crack initiates according to the Griffith criterion and achieves a limiting velocity equal to the Rayleigh wave speed. At this point, it is necessary to nucleate a daughter crack to break the velocity barrier. The limiting speed of the daughter crack is more than 50% higher than the longitudinal wave speed and cannot be explained by the linear theory of intersonic fracture. In comparison, the calculated local wave speed is approximately equal to (only 10% lower than) the observed limiting speed. In calculating the local wave speeds, we have ignored the large gradient of deformation field near the crack tip. In view of this simplification, we conclude that the local wave speed provides a reasonable explanation of the observed limited crack speed. In the mixed case, the nucleation of the daughter crack still occurs at the Rayleigh wave speed for reasons discussed above. The daughter crack breaks the velocity barrier at the Rayleigh speed and propagates near the longitudinal wave speed. This behavior is similar to the harmonic case. However, at a


2803

velocity of 10.35, we observe a granddaughter crack forming ahead of the daughter crack. The critical speed at which this transition occurs is very close to the local shear wave speed in the anharmonic crystal. The granddaughter crack rapidly accelerates toward the local longitudinal wave speed of the stretched non-linear solid. It is tempting to conclude that the nucleation of the granddaughter crack is controlled by the local Rayleigh wave speed magnified by the local stress concentration, although this issue is worth further investigation. In summary, we conclude that the local wave speeds play a dominant role in the behavior of cracks in the anharmonic crystals. The mixed case behaves somewhat like an interface crack between a soft material and a stiff material. Although the harmonic and anharmonic crystals have identical material properties under infinitesimal deformation, the local material properties near the crack tip are resembled by a bimaterial. Rosakis et al. [16] have previously studied crack propagation along an interface between PMMA (soft) and Al (hard) and found that the crack speed can significantly exceed the longitudinal wave speed of PMMA. Our present study shows that the crack behavior is dominated by the local (nonlinear) wave speeds. This is not only of theoretical interest, but also of practical importance. It is known that many polymeric materials, especially rubbers, increase their modulus significantly when stretched. The underlying physical mechanism is that initial elasticity in rubbers is due to entropic effects. When stretched to large deformation, the polymeric chains are straightened and covalent atomic bonds eventually dominate their hyperelastic response. In such solids, the elastic modulus increases with strain and the local wave speeds near a crack tip would be larger than the linear elastic wave speeds. The dynamic behavior of cracks in such solids should propagate at a speed exceeding the conventional wave speeds. Another point worth commenting here is the remarkable success of continuum theories in predicting the behaviors obtained by atomistic simulations. We have found previously [9] that the MD simulation results for shear crack propagation along a weak interface in harmonic solids are well interpreted by elasticity theories. In particular, calculations based on linear elasticity were able to predict the time and location of the daughter crack as well as the initiation time of the mother crack. Our atomistic simulation of the mother-daughter crack mechanism for intersonic shear crack is consistent with the continuum mechanics based discovery made earlier by Burridge and Andrew [15]. An important message of the present study is that the nonlinear continuum theory of local wave speeds is capable of predicting crack velocities in strongly nonlinear solids. Indeed, it is quite remarkable that the dynamic behavior of cracks may retain its basic nature over such a wide range of length scales, from atomistic calculations using interatomic potentials all the way up to macroscopic laboratory experiments and continuum elasticity treatments. This bridging of length scales in dynamic materials failure should be of great interest to the

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general scientific audience since it points out the fantastic power of continuum mechanics.

References [1] F.F. Abraham, R. Walkup, H. Gao et al., Proc. Na. Acad. Sci., 99, 5777, 2002. [2] M.P. Allen and D.J. Tildesley, “Computer simulation of liquids,” Clarendon Press, Oxford, 1987. [3] R. Feynman, R. Leighton, and M. Sands, “The Feynman lectures on physics,” Addison–Wesley, Redwood City, 1963. [4] J.E. Gordon, “The new science of strong materials or why you don’t fall though the floor,” Princeton University Press, Princeton, 1988. [5] L.B. Freund, “Dynamical fracture mechanics,” Cambridge Univ. Press, New York, 1990; B. Broberg, Cracks and Fracture, Academic Press, San Diego, 1999. [6] A.J. Rosakis, O. Samudrala, and D. Coker, Science, 284, 1337, 1999. [7] F.F. Abraham and H. Gao, Phys. Rev. Lett., 84, 3113, 2000. [8] F.F. Abraham, J. Mech. Phys. Solids, 49, 2095, 2001. [9] H. Gao, Y. Huang, and F.F. Abraham, J. Mech. Phys. Solids, 49, 2113, 2001. [10] B. Broberg, J. Appl. Mech., 546, 1964. [11] H. Gao, J. Mech. Phys. Solids, 44, 1453, 1996. [12] J. Fineberg, S.P. Gross, M. Marder, and H.L. Swinney, Phys. Rev. Lett., 67, 457, 1991; Phys. Rev. B, 45, 5146, 1992, The dynamic instability is generally known as the “mirror-mist-hackle” instability. A quantitative study of the onset of this instability was not done previously. [13] F.F. Abraham, D. Brodbeck, R. Rafey et al., Phys. Rev. Lett., 73, 272, 1994; J. Mech. Phys. Solids, 45, 1595, 1997. [14] F.F. Abraham, Phys. Rev. Lett., 77, 869, 1996; F.F. Abraham, D. Schneider, B. Land et al., J. Mech. Phys. Solids, 45, 1461, 1997. [15] R. Burridge, Geophys. J. Roy. Astron. Soc., 35, 439, 1973; D.J. Andrews, J. Geophys. Res., 81, 5679, 1976. [16] A.J. Rosakis, O. Samudrala, R.P. Singh et al., J. Mech. Phys. Solids, 46, 1789, 1998.

Perspective 21 LATTICE GAS AUTOMATON METHODS Jean Pierre Boon Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, 1050-Bruxelles, Belgium

When one is interested in studying the dynamical behavior of fluid systems starting at the microscopic level, a logical approach is to begin with a molecular dynamics description of the interactions between the constituting particles. This approach quite often turns into a formidable task when the fluid evolves into a non-linear regime where chaos, turbulence, or reactive processes take place. But one may question whether a ‘realistic’ description of the microscopic dynamics is indispensable to gain insight on the underlying mechanisms of large scale non-linear phenomena. Around 1985, a considerable simplification was introduced [1] when pioneering studies established theoretically and computationally the feasibility of simulating fluid dynamics via a microscopic approach based on a new paradigm: a virtual simplified micro-world is constructed as an automaton universe based not on a realistic description of interacting particles, but merely on the laws of symmetry and of invariance of macroscopic physics. Suppose we implement point-like particles on a regular lattice where they move from node to node at each time step and undergo collisions when their trajectories meet at the same node. As the system evolves, we observe its collective dynamics by looking at the lattice from a distance. And the remarkable fact is that, if the collisions occur according to some simple logical rules (satisfying fundamental conservations) and if the lattice has the proper symmetry, this Lattice Gas Automaton shows global behavior very similar to that of a real fluid. So we can infer that, despite its simplicity at the microscopic scale, the lattice gas automaton (LGA) should contain, at the elementary level, the essentials that are responsible for the emergence of complex behavior, and thereby can help us understand the basic mechanisms where from complexity builds up. The LGA consists of a set of particles moving on a regular d-dimensional lattice L at discrete time steps, t = nt, with n an integer. The lattice is 2805 S. Yip (ed.), Handbook of Materials Modeling, 2805–2809. c 2005 Springer. Printed in the Netherlands.

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composed of V nodes labeled by the d-dimensional position vectors r ∈ L. Associated to each node there are b channels (labeled by indices i, j, . . . , running from 1 to b). At a given time t, a channel can be either occupied by one particle or empty, so that the occupation variable n i (r, t) = 1 or 0. When channel i at node r is occupied, then the particle at the specified node r has velocity ci . The set of allowed velocities is such that the condition r + ci t ∈ L is fulfilled. The “exclusion principle” requirement that the maximum occupation be of one particle per channel allows for a representation of the automaton configuration in terms of a set of bits {n i (r, t)}; r ∈ L, i = {1, b}. The evolution rules are thus simply logical operations over sets of bits. The time-evolution of the automaton takes place in two stages: propagation and collision. In the propagation phase, particles are moved according to their velocity vector, and in the (local) collision phase, the particles occupying a given node are redistributed amongst the channels associated to that node. So the microscopic evolution equation of the LGA reads n i (r + ci t, t + t) = n i (r, t) + i ({n j (r, t)}),

(1)

where i ({n j }) represents the colision term which depends on all channel occupations at node r. By performing an ensemble average (denoted by angular brackets) over an arbitrary distribution of initial occupations, one obtains a hierarchy of coupled equations for the successive n-body distribution functions. This hierarchy can be truncated to yield the Lattice Boltzmann equation for the single particle distribution function f i (r, t) = n i (r, t): ({ f j (r, t)}), f i (r + ci t, t + t) − f i (r, t) = Boltz i

(2)

The l.h.s. can easily be recognized as the discrete version of the l.h.s. of the classical Boltzmann equation for continuous sytems, and the r.h.s. denotes the collision term where the precollisional uncorrelated state ansatz has been used to factorize the b-particle distribution function. The Lattice Boltzmann Eq. (2) is one of the most important results in LGA theory. It can be used as the starting point for the derivation (via multi-scale analysis) of the macroscopic equations describing the long wave-length behavior of the lattice gas. The LGA macroscopic equations are found to exhibit the same structure as the classical hydrodynamic equations, and under the incompressibility condition, one retrieves the Navier–Stokes equations for non-thermal fluids. Another important feature of the Lattice Boltzmann equation is that it can be used as an efficient and powerful simulation algorihtm. In practice one usually prefers to use a simplified equation where the collision term is approximated by a single relaxation time process inspired by the Bhatnagar–Gross– Krook model, known in its lattice version as the LBGK equation: f i (r + ci t, t + t) − f i (r, t) = −

1 leq f i (r, t) − f i (r, t) , τ

(3)

Lattice gas automaton methods

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where the r.h.s. is proportional to the deviation from the local equilibrium distribution function. There is a wealth of applications of the lattice gas methods which have established their validity and their usefulness. LGA simulations, based on Eq. (1), are most valuable for fundamental problems in statistical mechanics such as for instance the study of fluctuation correlations in equilibrium and non-equilibrium sytems [2, 3]. As an example, Fig. 1 shows the trajectories of tracer particles suspended in a Kolmogorov flow (above the critical Reynolds number) produced by a lattice gas automaton and where from turbulent diffusion was analyzed [4]. Simulations of more direct practical interest such as for instance profile optimization in car design or turbulent drag problems are most efficiently treated with the lattice Boltzmann method, in particular using the LBGK model. The examples given in Figs. 2 and 3 illustrate the method for the study of viscous fingering in Hele–Shaw geometry, showing the effect of reactivity between the two fluids as a determinant factor in the dynamics of the moving interface [5]. Applications of the LGA approach and of the lattice Boltzmann equation cover a wide variety of theoretical and practical problems ranging from the

Figure 1. Lattice gas simulation of the Kolmogorov flow: the tracer trajectories reflect the topology of the A BC flow in the regime beyond the critical Reynolds number (Re = 2.5× Rec ).

Figure 2. Lattice Boltzmann (LBGK) simulation of viscous fingering in miscible fluids.

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Figure 3. Lattice Boltzmann (LBGK) simulation of viscous fingering showing the interface sharpening effect of a reactive process between the two fluids (compare with Fig. 2).

dynamics of thermal fluctuations and quantum lattice gas automata to multiphase flow, complex fluids, reactive sytems and inhomogeneous turbulence. The list of selected references given below should guide the reader interested in specific topics related to the subject: Statistical Mechanics of Lattice Gas Automata [2], Lattice Gas Automaton applications [3], Reactive Lattice Gas Automata [6], Quantum Lattice Gas Automata [7], Complex fluids [8], Lattice Boltzmann method [9], and recent developments [10].

References [1] U. Frisch, B. Haslacher, and Y. Pomeau, “Lattice gas automata for the Navier–Stokes equation,” Phys. Rev. Lett., 56, 1505–1508, 1986. [2] J.P. Rivet and J.P. Boon, Lattice Gas Hydrodynamics, Cambridge, Cambridge University Press, 2001. [3] D. Rothman and S. Zaleski, Lattice Gas Cellular Automata, Cambridge, Cambridge University Press, 1997. [4] J.P. Boon, D. Hanon, and E. Vanden Eijnden, “Lattice gas automaton approach to turbulent diffusion,” Chaos, Solitons and Fractals, 11, 187–192, 2000. [5] P. Grosfils and J.P. Boon, “Viscous fingering in miscible, immiscible, and reactive fluids,” J. Modern Phys. B, (to appear), 2002. [6] J.P. Boon, D. Dab, R. Kapral, and A. Lawniczak, “Lattice gas automata for reactive systems,” Phys. Rep., 273(2), 55–148, 1996. [7] D. Meyer, “From quantum cellular automata to quantum lattice gases,” J. Statist. Phys., 85, 551–574, 1996.

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[8] B.M. Boghosian, P.V. Coveney, and A.N. Emerton, “A lattice gas model of microemulsions,” Proc. Soc., A452, 1221–1250, 1996. [9] S. Succi, The Lattice Boltzmann Equation, Oxford, Clarendon Press, 2001. [10] P.V. Coveney and S. Succi (eds.), “Discrete modeling and simulation of fluid dynamics,” Philos. Trans. R. Soci., 360, 291–573, 2002.

Perspective 22 MULTI-SCALE MODELING OF HYPERSONIC GAS FLOW Iain D. Boyd University of Michigan, Ann Arbor, MI, USA

On March 27, 2004, NASA successfully flew the X-43A hypersonic test flight vehicle at a velocity of 5000 mph to break the aeronautics speed record that had stood for over 35 years. The final flight of the X-43A on November 16, 2004 further increased the speed record to 6,600 mph which is almost ten times the speed of sound. The very high speed attainable by hypersonic airplanes could revolutionize air travel by dramatically reducing inter-continental flight times. For example, a hypersonic flight from New York to Sydney, Australia, a distance of 10,000 miles, would take less than 2 h. Reusable hypersonic vehicles are also being researched to significantly reduce the cost of access to space. Computer modeling of the gas flows around hypersonic vehicles will play a critical part in their development. This article discusses the conditions that can prevail in certain hypersonic gas flows that require a multi-scale modeling approach.

1.

Hypersonic Flight

Hypersonic flight is one in which the vehicle velocity is much greater than the speed of sound. For the remainder of this article, we will use the Galilean transformation that allows us to consider the hypersonic flow of air around a fixed vehicle as being the same as when a hypersonic vehicle flies through stationary air. In gas dynamics, the Mach number is the ratio of the flow velocity to the speed of sound. Although there is not a fixed definition, hypersonic flow is generally considered to involve a Mach number greater than five. The fastest commercial passenger airplane ever developed is the Concorde that cruised at a Mach number of about two. The fastest military aircraft ever developed is the SR-71 Blackbird that cruised at a Mach number of three. Important examples of truly hypersonic vehicles include the X-15 powered by rockets and flown 2811 S. Yip (ed.), Handbook of Materials Modeling, 2811–2818. c 2005 Springer. Printed in the Netherlands.

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in the 1960s, the recently flown X-43A powered by a scramjet that reached a peak Mach number of about ten, and the space shuttle that re-enters the atmosphere at about Mach 25. Note, however, that the space shuttle during its hypersonic re-entry is a glider and does not fly under its own power. Flights of hypersonic passenger aircraft will not happen any time in the near future as there are many difficult technology issues faced in the development of such vehicles. Some of the most important of these issues concern the basic aerodynamics of the vehicle. Because hypersonic vehicles fly at very high speed, they generate a lot of aerodynamic drag. For cruise conditions, the drag must be overcome by the propulsion system and so directly affects the efficiency of the vehicle. Note that lack of economic viability for flight at Mach 2 for the Concorde was the primary reason for the end of commercial service of that aircraft. On hypersonic vehicles, the problem is even more acute calling for minimization of vehicle drag to the greatest extent possible. Also, at hypersonic speed, the high velocity is converted into high gas temperature at the vehicle surface requiring development of a thermal protection system (TPS) that may consist of special surface materials and active cooling strategies.

2.

Need for Computer Simulation of Hypersonic Flows

The development of hypersonic vehicles relies heavily on computational modeling. This is in part because laboratory experiments are both technically challenging and cost a lot of money. In order to generate in the laboratory an air flow at Mach 12 representative of a hypersonic flight condition, the air must be compressed to about 100 times atmospheric pressure and heated to a temperature of 7500 K. Expanding this hot, compressed gas through the hypersonic nozzle of a wind-tunnel will typically produce test times of less than one second making measurements difficult. Flight development programs for hypersonic vehicles such as the X-43A are very rare due to the orders of magnitude increase in cost compared to laboratory testing. By comparison, computer simulation techniques offer the potential to provide useful results to aid in the design of hypersonic vehicles in a fraction of the time and money required for laboratory and flight investigations. However, computer simulations are only useful if they are demonstrated to provide accurate results through direct comparison with measured data. Thus, laboratory testing remains an extremely important component in hypersonic gas dynamics research. Finally, similar to the development of any air vehicle, flight tests will always be required to address issues that cannot be foreseen in simulation and laboratory investigations.

Multi-scale modeling of hypersonic gas flow

3.

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Modeling Hypersonic Gas Dynamics

The hypersonic flow of air over a vehicle shape involves a number of complex gas dynamic phenomena. The air becomes compressed as it is decelerated by the presence of the vehicle surfaces. The hypersonic character of the free stream results in an extremely compact region of strong compression called a shock wave. For hypersonic vehicle flight conditions, the shock wave typically has a thickness much less than 1 mm over which gas properties such as pressure, temperature, and density are increased by substantial factors. Downstream of the shock wave, the high pressure and high temperature gas may undergo excitation of vibrational energy modes of the air molecules and the molecules may even begin to react chemically. As the gas finally approaches the surface of the vehicle, it encounters a relatively cold surface that causes the density to rise further in another thin layer of fluid called the boundary layer. It is the properties of the gas in this final layer of fluid immediately adjacent to the vehicle surface that determine the drag force and heat transfer to the vehicle. The most fundamental model of dilute gas dynamics is the Boltzmann equation that describes the evolution of the velocity distribution function of molecules. By taking moments of the Boltzmann equation, Maxwell derived the equation of change [1]: ∂ (n W ) ∂ (n ci W ) = [W ] + ∂t ∂ xi

(1)

in which W (ci ) is some molecular property that depends on the random velocity ci in the xi direction, indicates the average over the velocity distribution function otherwise called the moment, and [ ] indicates the change due to collisions. For example, if W is the mass of a molecule, then in the absence of chemical reactions the mass of the particle is conserved in a collision, and Eq. (1) becomes the well-known continuity equation of continuum gas dynamics. In addition, if equilibrium is assumed, and W is set to the momentum vector and the energy of a molecule, Eq. (1) provides a set of five continuum transport equations that is often called the Euler equations. This equation set corresponds to the case where the velocity distribution function everywhere in the flow is of the equilibrium Maxwellian form. To derive higher order sets of transport equations from Maxwell’s equation of change, it must be noted from Eq. (1) that the temporal derivative of any moment W depends on the divergence of the next higher velocity moment. This problem is addressed by one of two methods. In the Chapman–Enskog approach, a specific form of the velocity distribution function is assumed for flows perturbed slightly from the equilibrium state. In Grad’s method of moments, specific relations are assumed between the second and fourth order velocity moments. Setting W = mci c j and W = mci c j ck in Eq. (1), using either the Chapman–Enskog or

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the Grad methods, leads to a set of 20 equations (the 20-moment equations) consisting of the five Euler equations, five further equations involving the shear stress tensor τ i j , and ten further equations involving the symmetric heat flux tensor Q i j k . For modeling hypersonic gas flow (and many other flows involving viscosity and thermal conductivity effects), a set of transport equations called the Navier–Stokes (NS) equations is widely employed. The NS equations are obtained from the 20-moment equations by replacing the heat flux tensor with a heat flux vector, qi . As discussed above, one of the methods for deriving transport equation sets from Maxwell’s equation of change is the Chapman–Enskog approach in which the following form for the velocity distribution function is assumed explicitly:

f (C) = f 0 (C) (C),

f 0 (C)dC =

1

c (2kT /m)1/2

(2)

e−C dC 2

π 3/2

C=

(3)

(C) = 1 + qi∗ Ci ( 25 C 2 − 1) − τi∗j Ci C j qi∗

κ =− P

2m kT

1/2

∂T , ∂ xi

τi∗j

µ = P

∂ci ∂c j 2 ∂ck + − δi j ∂x j ∂ xi 3 ∂ xk

(4)

(5)

where f o is the equilibrium Maxwellian form of the distribution. When the Chapman–Enskog parameter, , is slightly perturbed from unity, then the 20moment (or perhaps the Navier–Stokes) equations are valid. When is sufficiently far from unity then these equation sets can be expected to fail, and a more detailed approach is required. Examination of Eqs. (4) and (5) indicates that will become large when gradients in temperature and/or velocity become significant. This is exactly the situation in the critical regions of the shock waves and boundary layers created around hypersonic vehicles. In these regions, the characteristic length scales of the flow based on flow field gradients are so steep that the Chapman–Enskog distribution function cannot accurately describe the strong non-equilibrium behavior. Thus, higher order terms must be included in the perturbation to the Maxwellian distribution. In a continuum sense, these additional terms introduce higher order derivatives into the conservation equations. The next higher set of conservation equations produced in this manner is called the Burnett equations. Research on numerical solution of the Burnett equations has met with mixed success. The mathematical properties of the full set of Burnett equations presents several difficulties in terms of numerical solution. Attempts to simplify the equations by eliminating “difficult” terms have resulted in unfortunate side effects such as a failure to obey the second law of thermodynamics. In addition, development of the


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associated boundary conditions required for these higher order equation sets presents problems of its own. An alternative approach to the numerical simulation of the strongly nonequilibrium conditions experienced in hypersonic shock waves and boundary layers is the direct simulation Monte Carlo (DSMC) method [2]. In the DSMC technique, model particles are used to simulate the motions and collisions of real molecules. In a single iteration of the DSMC technique, the model particles are first moved over the distance given by the product of their velocity vector and a time-step t that is smaller than the mean free time, the average time that elapses between successive collisions of a molecule. After all particles are moved, any boundary interactions are processed, such as collision of particles with a solid wall. Next, the particles are binned into computational cells. The size of each cell should be less than the local mean free path, λ, that is the average distance a molecule travels between successive collisions. Within each of the cells, a number of particles are paired together, and collision probabilities determined for each pair to determine which particles actually collision. For the pairs that collide, collision mechanics is applied that conserve linear momentum and energy. Finally, the particle properties in each cell are sampled to determine time averaged, macroscopic properties such as density, flow velocity, temperature, and pressure. The DSMC technique has been successfully applied to a variety of nonequilibrium gas flow systems including hypersonic aerothermodynamics, spacecraft propulsion systems, materials processing, micro-scale gas flows, and space physics. Since the DSMC technique requires that the dimensions of its computational cells are of the order of a local mean free path (less than 1 mm under hypersonic vehicle flight conditions), the method becomes prohibitively expensive for continuum flows.

4.

The Multi-scale Hybrid Simulation Approach

The above discussion indicates that, in aiming to develop a computer model of the gas flow around hypersonic vehicles, we are faced with a dilemma. Solution of the sets of continuum conservation equations (such as the Navier– Stokes equations) using standard methods from computational fluid dynamics (CFD) [3], is relatively efficient numerically but may be physically inaccurate in the important regions of the shock waves and boundary layers where flow field gradients become very steep. The direct simulation Monte Carlo method is physically valid for the entire flow field, but is so numerically expensive as to make this approach impossible. One way of considering this problem is from a multi-scale perspective. Specifically, the continuum CFD approach is physically valid except in small thin regions where the scale length changes so

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dramatically that sub-scale physical processes become important. Since these sub-scale, molecular level processes can be accurately simulated using the DSMC technique, a multi-algorithm approach to the problem presents itself as a natural way to proceed. This is a common solution technique in many areas of physics and engineering for multi-scale phenomena. Thus, a hybrid method that uses a CFD approach for most of the flow field, and the DSMC technique only in the high-gradient, sub-scale regions, appears to offer the potential to combine the physical accuracy and numerical efficiency of each of the component methods. There are two key issues in the development of a hybrid CFD-DSMC method: (1) how to determine the location of the domain boundaries between the CFD and DSMC regions; and (2) how to communicate information back and forward between continuum and discrete representations of the same gas flow. The first issue is resolved using continuum breakdown parameters that assess the physical accuracy of the continuum conservation equations. There are a number of such parameters in the literature [4, 5], and the most successful of these are defined in terms of flow field gradients similar to those that appear in the Chapman–Enskog distribution. The second issue is usually resolved by computing fluxes of conserved properties (mass, momentum, and energy) across the CFD/DSMC domain interfaces. This can either be achieved directly using a finite volume CFD formulation, or by evaluating the fluxes from primitive variables such as density, velocity, and temperature.

5.

Illustrative Results

The primary objective of a hybrid CFD-DSMC simulation is to improve the physical accuracy of a pure CFD solution without paying the enormous computational penalty of a pure DSMC solution. A common approach is to first obtain a pure CFD solution and to then use it to initialise a hybrid simulation. Figures 1 and 2 show some illustrative results for Mach 12 flow of molecular nitrogen over a blunt cone from a hybrid simulation performed in this manner. In Fig. 1, the decomposition of the flow domain between the CFD and DSMC methods is shown. A breakdown parameter based on flow field gradients is employed [4] and shows, as expected, that the DSMC technique is required in the bow shock wave and in the boundary layer next to the body surface. Figure 2 shows comparisons of three different numerical solutions along the axis of the flow (the stagnation streamline). Assuming that the pure DSMC solution is more physically accurate than the pure CFD solution, what we would like to see from the hybrid simulation is that it starts from the initial CFD solution, and moves it to the pure DSMC solution. This is exactly the behavior achieved in the results shown in Fig. 2.


2817

Figure 1. CFD (white) and DSMC (grey) domains for Mach 12 flow over a cone.

Figure 2. Numerical solutions of temperature along the flow axis.

6.

Outlook

The results shown in the figures represent a solid foundation for the further development of hybrid CFD-DSMC methods for hypersonic gas flows. The physical modeling capabilities of the hybrid methods need to be extended beyond the perfect gas description that is presently employed. Specifically,

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models for vibrational excitation and chemical reactions must be developed. In addition, the numerical performance of the hybrid scheme must be improved dramatically. In the results shown here, the cost of the hybrid simulation was greater than that of the pure DSMC solution! Therefore, it is expected that refinement of hybrid CFD-DSMC hybrid methods will continue for several years. The development of such methods is intellectually challenging, and the need for accurate simulations of these complex hypersonic flows provides plenty of practical motivation.

References [1] T.I. Gombosi, Gaskinetic Theory, Cambridge University Press, Cambridge, 1994. [2] G.A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, New York, 1994. [3] J.C. Tannehill, D.A. Anderson, and R.H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Taylor and Francis, Philadelphia, 1997. [4] W.-L. Wang and I.D. Boyd, “Predicting continuum breakdown in hypersonic viscous flows,” Phys. Fluids, 15, 91–100, 2003. [5] A.L. Garcia, J.B. Bell, W.Y. Crutchfield, and B.J. Alder, “Adaptive mesh and algorithm refinement using direct simulation Monte Carlo,” J. Comp. Phys., 154, 134– 155, 1999.

Perspective 23 COMMENTARY ON LIQUID SIMULATIONS AND INDUSTRIAL APPLICATIONS Raymond D. Mountain Physical and Chemical Properties Division, Chemical Science and Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899-8380, USA

Molecular dynamics and Monte Carlo simulations have proven to be invaluable tools in the study of liquids. Statistical mechanics tells us what to calculate to obtain liquid state properties (equation of state, specific heat, and transport coefficients for example) from a model of the interaction between molecules. Simulations are the tools needed to make those calculations for physically realistic models. While the impact of simulations on the investigation of fluid properties and structure cannot be overstated, there has been very little success in the use of these methods to predict liquid properties outside the range where model potentials have been fit to some property. One result of this is that industrial modelers, who are called upon to estimate properties for various compositions, temperatures, and pressures of liquids, have not found simulation methods to be of much value for their work. Some of the reasons for the lack of use of molecular level simulations in the industrial sector will be examined here. These reasons point to opportunities and challenges for research that could lead to industrially useful molecular simulation methods and practices. A situation where simulations could be industrially useful involve conditions where data are scarce and measurements would be difficult, expensive, or hazardous. The solubility of oxygen in a combustible liquid at elevated temperature is an example. Another would involve compounds that are expensive to prepare. Specific properties that have industrial interest and where simulations could be useful are the vapor pressure, solubility of gases in liquids, and the thermal conductivity and shear viscosity of liquids. Simulation methods exist for determining these properties, although they are not easily used by non-experts.


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The ingredients that go into a simulation are model interaction potentials and state conditions. For this discussion, boundary conditions, initial conditions, and simulation software will not be discussed. At present, almost all model intermolecular potential functions are empirical or semiempirical constructs that have parameters adjusted so that some set of calculated properties match the experimental values of those properties over some range of state conditions. The accuracy of property predictions outside the range where the fit was made is unpredictable. This is not a satisfactory state of affairs as uncertainty estimates are essential for industrial purposes. The way a molecule–molecule interaction is represented can be quite artibrary. For example, interaction sites may be placed on atomic sites or are they may be placed elsewhere. Often, the location of “atomic sites” differs from the location of the sites on the isolated molecule. A related issue is the placement of partial charges to mimic the charge distribution in a molecule. When a mean-field approach to induced polarization is employed, the criteria for the placement and magnitude of the charges is not governed by unambiguous physical considerations. Since the electrostatic interactions are generally much stronger than dispersion interactions, small changes in the magnitude and/or position of charges can have significant consequences. How to improve the predictive ability of molecular simulations for thermodynamic and transport properties is an open topic. At the root of the problem is the quantum mechanical nature of the potentials that are the result of effectively replacing electronic degrees of freedom by potentials. This suggests that at least some of the electronic degrees of freedom should be explicitly included in the simulation. How to do this in a way that is computationally tractable for system sizes of interest, for simulation intervals of sufficient length that statistically adequate samples are obtained, and that include the necessary physics so that the arbitrariness is reduced to acceptable levels is not yet understood, although there are continuing research efforts. One approach is to use high level quantum chemistry methods to calculate directly the energy of interaction between pairs of molecules and then to map the results onto functions that represent the energy of interaction for various separations and orientations of a pair of molecules. This can result in an accurate representation of the energy of a pair of molecules, but may be quite complicated and may require considerable computational effort if used in a molecular dynamics or Monte Carlo simulation. Also, significant many body effects associated with induced polarization require separate calculations. Extending this approach to explicitly consider three or more molecules simultaneously is beyond current capabilities for all but very simple molecules. A second scheme is to use quantum mechanics to calculate energies and forces as needed during a simulation. These methods are currently limited to using low level methods such as density functional based quantum methods. Even so, the computational load is heavy, the number of molecules is restricted

Commentary on liquid simulations and industrial applications

2821

to 100 or so, and and accuracy of predicted properties relative to experimental values is often no better than that obtained from empirical models. The challenge and opportunity is to find and develop ways to incorporate the relevant quantum mechanical interactions into forms that are computationally manageable for molecular simulations. A second reason why industrial modelers have not embraced molecular simulations is that the computational time required to obtain statistically significant results, particularly for the viscosity and the thermal conductivity, can be quite long. This is the case when either the equilibrium time correlation function/Einstein relation approach is used or a nonequilibrium externally applied gradient is imposed on the fluid. The challenge and opportunity is to develop alternative, efficient ways to extract transport coefficients with known uncertainties from simulations.

Perspective 24 COMPUTER SIMULATIONS OF SUPERCOOLED LIQUIDS AND GLASSES Walter Kob Laboratoire des Verres, Université Montpellier 2, 34095 Montpellier, France

Glasses are materials that are ubiquitous in our daily life. We find them in such diverse items as window pans, optical fibers, computer chips, ceramics, all of which are oxide glasses, as well as in food, foams, polymers, gels, which are mainly of organic nature. Roughly speaking glasses are solid materials that have no translational or orientational order on the scale beyond O(10) diameters of the constituent particles (atoms, colloids, . . .) [1]. Note that these materials are not necessarily homogeneous since, e.g., alkali-glasses such as Na2 O-SiO2 show (disordered!) structural features on the length scale of 6–10 Å (compare to the interatomic distance of 1–2 Å) and gels can have structural inhomogeneities that extend up to macroscopic length scales. Apart from their relevance as a technological material, glasses, or more general glass-forming liquids, are also an important subject of fundamental research since many of their unique properties are not understood well (or at all) from a microscopic point of view and it is a great challenge to close this gap in our knowledge. For example, why does the addition of 100 ppm of water change the viscosity of silica, SiO2 , by 2–3 orders of magnitude, a question that is important, e.g., to understand the flow of magmatic material? What are the mechanisms that make a glass “age”, i.e., lead to a change in its properties with time, a phenomenon that is highly important to understand material properties on long time scales? What are the “best” compositions to obtain metallic glasses with a prescribed mechanical/electric/magnetic/. . . property? The most pertinent question is however an embarrassingly simple one: What is a glass? To understand this question we can consider the temperature dependence of the viscosity η of a glass-forming liquid, i.e., of a liquid that can be supercooled by a significant fraction of its melting temperature. 2823 S. Yip (ed.), Handbook of Materials Modeling, 2823–2828. c 2005 Springer. Printed in the Netherlands.

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Experimentally it is found that η shows a very strong T -dependence in that it increases from values on the order of 10−2 Pa · s– 1014 Pa · s (these values are typical for atomic and molecular liquids). This is demonstrated in Fig. 1 where we show an Arrhenius plot of η(T ) for various glass-forming liquids. Note that in order to take into account the different nature of the liquids, and hence the different relevant temperature scales, we have normalized the temperature axis by Tg , the temperature at which the viscosity of the liquid takes the value 1012 Pa · s, i.e., a value that corresponds to a typical relaxation time of 1–100 s. From this plot, we thus see that a change in temperature by a factor of two leads to an increase of the viscosity by about 10–15 decades! If the temperature is lowered a bit further, the viscosity, and hence the relaxation times, increases even more and the material does not flow anymore on human time scales, i.e., it has become a glass. The reason for the dramatic slowing down of the dynamics of glass-forming liquids is presently not really known. Although there are reliable theoretical approaches, such as the

Figure 1. Temperature dependence of the viscosity of various glass-forming liquids as a function of Tg /T , where Tg is the glass transition temperature. Adapted from Angell et al. [2] (reproduced with permission).

Computer simulations of supercooled liquids and glasses

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so-called “mode-coupling theory of the glass transition” [3], that are able to rationalize in a semi-quantitative and sometimes even quantitative way the observed slowing down, these approaches usually work only in a relatively narrow temperature range, i.e., in a T -window in which η is on the order of 10−1 −104 Pa · s and thus changes by “only” 3–5 decades. Finding a reliable theoretical description that is able to describe the slowing-down in the whole T -range is one of the most important issues in the field. Despite the many experiments that have been made to determine the properties of glass-forming systems [4] and hence to shed some light on the reason for the occurrence of the glass transition, one is currently still far away to have an answer. The problem is that in experiments, the microscopic information that is needed to identify the mechanism for the slowing down, namely the trajectories of the particles as well as their statistical properties, is not really available. Therefore, also in the last twenty years there have been large efforts to use computer simulations to study glass-forming systems [5]. Roughly speaking these numerical efforts can be divided into two large, not completely disjoint, groups. Simulations that aim to increase our understanding of a (real) specific material (or a narrow class of materials) such as silica, ortho-terphenyl, polyethylene, etc. Since the properties of the materials are often very specific, it is necessary to use force fields that are highly accurate. Although for the case of crystalline systems accurate effective force fields are often available, this is unfortunately not the case for supercooled liquids and therefore the results obtained with these types of interactions must always be looked at with some caution. The alternative is to use ab initio calculations in which the forces between the ions are determined directly from the electronic degrees of freedom [6]. Although the so obtained interactions are very accurate, the price one has to pay is an increase of about a factor 106 in the computational burden. Therefore the current state of the art ab initio simulations are done on systems with only a few hundred particles and cover a time scale of just a few tens of a pico-second. The second large group of simulations of glass-forming materials are numerical investigations aimed to understand the more universal aspects of these systems. Therefore one uses models that are relatively simple, such as Lennard–Jones particles, lattice gases, or spin models, i.e., systems in which the positions of the “particles” are frozen on a lattice and only their orientational degrees of freedom are relevant. Due to their simplicity these models are very useful to obtain results with a good statistics and to study the dynamical behavior of the system at relatively long times, which presently means O(109 ) time steps. Although making 109 time steps for a system size of O(1000) particles is presently still a huge numerical effort, it is still significantly less than the number of steps that one needs in order to reach macroscopic time scales. For example, in the case of silica a typical time step is around 1 fs and hence 109 steps correspond to only 1 µs! Thus the last 6–8 decades in η of the curves

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shown in Fig. 1 are presently not accessible in (equilibrium!) simulations and thus this technique is presently not able to make a significant contribution to our understanding of these highly supercooled liquids. Note that here we emphasize the notion of “equilibrium”. Of course it is possible to quench the system rapidly to a relatively low temperature, e.g., by coupling it to a heat bath, and then to start a microcanonical or canonical simulation. However, since the system is now out of equilibrium the measured properties might be, and usually are, very different from the ones found for the same system at this temperature but in equilibrium Bouchaud 1998 [10]. Note that in principle one is of course not forced to use these 109 time steps to do a “realistic dynamics”, i.e., one in which Newton’s equations of motions are solved by means of the standard algorithm, such as the one of Verlet. Instead it would be perfectly reasonable to use a cleverly designed Monte Carlo scheme which allows to equilibrate the system at relatively low temperatures and then to use the so generated equilibrium configurations as a starting point for a “conventional” simulation, i.e., a dynamics in which the particles follow Newton’s equations of motion. This approach is, e.g., used very successfully in the context of second order phase transitions, another physical situation where the dynamics becomes exceedingly slow (critical slowing down). However, whereas in this latter case there are indeed efficient algorithms, such as the one by Swendsen and Wang [7] that allows to equilibrate the system even very close to the critical point, this is not the case for glass-forming liquids, despite strong efforts to find such accelerating schemes. Nevertheless, some progress has already been made [8] and therefore it can be hoped that in the not too distant future it will indeed be possible to equilibrate glass-forming systems even at very low temperatures. (Note that since from a mathematical point of view equilibrating a glass-forming liquid has many similarities with more formal optimization problems, such as the traveling salesman problem (in both cases one searches for low lying minima of a complicated function, the potential energy/cost function), in recent years, there have been quite a few very fruitful exchanges between these two communities that have lead to an improvement of the algorithms [8].) Once such a method has been found, it will be possible to investigate on the microscopic level the vibrational and relaxation dynamics of such systems, and hence to make a much closer connection to experimental data and thus to allow to interpret it in a more reliable way and to make new predictions on the structure and dynamics of glass-forming systems. On the other hand such simulations will also give the possibility to obtain a better theoretical (analytical) description of the dynamics of supercooled liquids. As a possible example we return to the above mentioned mode-coupling theory. In this theory, one discusses the temperature dependence of the so-called intermediate scattering function F(q, t), a space-time correlation function that is directly measurable

Computer simulations of supercooled liquids and glasses

2827

in light or neutron scattering experiments and hence is of great theoretical and practical importance [9]. This function is defined given by F(q, t) =

N N 1 exp(iq · (r j (t) − rk (0)), N j =1 k=1

(1)

where r j (t) is the location of particle j at time t and q is a wave-vector. Using a procedure that is called “Zwanzig-Mori projection operator formalism” it is possible to derive exact equations of motion for F(q, t) and one finds that they have the form [9]: ¨ F(q, t) +

2q F(q, t)

+

2q

t

M(q, t − t )F(q, t )dt = 0.

(2)

0

Here the (squared) frequency 2q is given by q 2 k B T /(m S(q)), where m is the mass of the particles and S(q) is the static structure factor. The mode-coupling theory now makes the approximation that the so-called “memory function” M(q, t) is a bi-linear product of F(q , t) with coefficients that depend only on S(q ) [3]. As mentioned above, the theory works very well at temperatures corresponding to low and intermediate values of the viscosity. However, at lower temperatures the approximation seems not to be accurate any more but for the moment it is not clear at all how it can be improved. Thus to advance, it will be necessary to determine the memory function and to see why the approximation is no longer good. This can be done, e.g., by making large scale molecular dynamics simulations in order to determine F(q, t) with high precision. Using a simple Laplace transform it is then possible [9] to invert Eq. (2) in order to express M(q, t) as a function of F(q, t) and hence to see how the mode-coupling approximations can be improved. This procedure would thus allow to make finally one important step forward to our understanding of the relaxation dynamics of deeply supercooled liquids, a field that has not seen much theoretical progress since about 20 years, i.e., after the mode-coupling theory has been proposed. Thus it is evident that for the next few years the big challenges in simulations of glass-forming liquids and glasses are to obtain potentials that are sufficiently accurate to describe these disordered structures on a quantitative level and to develop new simulation algorithms that allow to equilibrate these systems also at low temperatures. Once these two goals are attained it will be possible to make on the one hand qualitative and quantitative predictions on the properties of glass-forming materials and on the other hand allow to make an important step forward in our understanding of the nature of the glassy state.

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References [1] J. Zarzycki (ed.), Materials Science and Technology, Vol. 9: Glasses and Amorphous Materials, VCH Publ., Weinheim, 1991. [2] C.A. Angell, P.H. Poole, and J. Shao, “Glass-forming liquids, anomalous liquids, and polyamorphism in liquids and biopolymers,” Nuovo Cim. D, 16, 993–1025, 1994. [3] W. Götze and L. Sjögren, “Relaxation processes in supercooled liquids,” Rep. Prog. Phys., 55, 241–376, 1992. [4] K.L. Ngai, G. Floudas, A.K. Rizos, and E. Riande (eds.), “Relaxation in complex systems,” J. Non-Cryst. Solids, 307–310, 1–1080, 2002. [5] W. Kob, “Supercooled liquids, the glass transition, and computer simulations,” In: J.-L. Barrat, M. Feigelman, J. Kurchan, and J. Dalibard (eds.), Lecture Notes for Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter, Les Houches July, 1–25, 2002; Les Houches Session LXXVII. Springer, Berlin, pp. 199–270, 2003. [6] M.E. Tuckerman, “Ab Initio, molecular dynamics: basic concepts, current trends and novel applications,” J. Phys. Condens. Matter., 14, R1297–R1355, 2002. [7] D.P. Landau and K. Binder, Monte Carlo Simulations in Statistical Physics, Cambridge University, Cambridge, 2000. [8] Y. Okamoto, “Generalized-ensemble algorithms: enhanced sampling techniques for Monte Carlo and molecular dynamics simulations,” J. Mol. Graphics Modelling, 22, 425–439, 2004. [9] U. Balucani and M. Zoppi, Dynamics of the Liquid State, Oxford University Press, Oxford, 1994. [10] J.-P. Bouchaud, L.F. Cugliandolo, J. Kurchan, and M. Mézard, “Out of equilibrium dynamics in spin glasses and other glassy systems,” In: A.P. Young (ed.), Spin Glasses and Random Fields, World Scientific, Singapore, pp. 161–224, 1998.

Perspective 25 INTERPLAY BETWEEN MATERIALS THEORY AND HIGH-PRESSURE EXPERIMENTS Raymond Jeanloz University of California, Berkeley, CA, USA

High-pressure experiments have played an important role in materials physics, both as a route for synthesizing new materials and as a means of validating theory. These roles are complementary, and have proven to be remarkably synergistic. Perhaps the most famous example of materials synthesis under pressure is that of diamond, the high-pressure form of carbon that is produced in the Earth’s mantle and – since the early 1950s – in the laboratory; industrial diamonds have subsequently found a wide range of applications [1, 2]. Focusing on a property rather than a particular material, superconductivity illustrates an important characteristic that has been systematically pursued in the laboratory. Not only do the examples discovered under pressure almost double the number of superconductors documented among elements (Fig. 1)[3], but the highesttemperature superconducting transition found to date has been measured at high pressure [4]. Pressure is also significant for condensed-matter theory, which is founded on quantum mechanics but involves important approximations in actual implementation (e.g., of exchange and correlation effects; applicability of one-electron and density-functional approaches; separation of electronic and vibrational degrees of freedom; and considerations of electron-spin and of relativistic effects). The best way to evaluate the reliability of theory is to have it predict the stability and properties of new materials, or of known materials under new conditions. If one changes composition to form a new material, that also changes the approximations embedded in the potential term of the Schrödinger equation, however; if one changes temperature, new complications arise as thermal excitations have to be reliably accounted for in the theory. 2829 S. Yip (ed.), Handbook of Materials Modeling, 2829–2835. c 2005 Springer. Printed in the Netherlands.

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R. Jeanloz Superconductivity of Elements He

H

Li

Be

17

0.03

Na

Mg

K Rb

T

P⫽0

TC in K for bulk sample

B

C

N

T

P>0

maximum reported TC in K

Ca

Sc

Ti

V

Cr

15

0.3

0.5

5.4

3

Sr

Y

Zr

Nb

Mo

Tc

Ru

4

3

0.6

9.3

0.9

8.2

0.5 10

Mn

Fe

Co

Ni

Cu

2 Rh

Pd

Ag

⫺6

Cs

Ba

Hf

Ta

W

Re

Os

Ir

1.7

5

0.4

4.5

0.01

1.7

0.7

0.1

Fr

Ra

Rf

Db

Sg

Bh

Hs

Mt

Ds

Rg

La

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

6

1.8

Ac

Th 1.4

Pt

Au

O

F

Ne

Cl

Ar Kr

0.6

11 Al

Si

P

S

1.2

8

20

17

Zn

Ga

Ge

As

Se

Br

0.9

1.1

5.5

2.7

11

1.5

Cd

In

Sn

Sb

Te

I

0.5

3.4

3.7

3.6

7.5

1.2

Po

At

Hg

Tl

Pb

Bi

4.2

2.4

7.2

7

Dy

Ho

Er

Tm

Yb

Xe

Rn

Lu 1.2

Pa 1.4

U 1.3

Np

Pu

Am 0.6

Cm

Bk

Cf

Es

Fm

Md

No

Lr

Figure 1. Periodic table identifying elements that have been found to be superconductors at high pressures (shaded), and those elements known as superconductors at zero pressure, with transition temperatures shown for bulk samples at zero applied magnetic field [3].

These and other means of perturbing the system can be experimentally applied and theoretically modeled, but none is as straightforward for theory as considering the effect of pressure. Quite simply, the inter-atomic spacing is changed, and the quantum mechanical problem is now solved (and optimized) again in order to predict the properties of matter under new conditions. Experiments then verify the predictions, or not.

1.

Effect of Pressure on Materials Chemistry

As shown by many theoretical studies, pressure can dramatically change the properties of materials. One of the earliest examples is the 1935 prediction by E. Wigner and H. B. Huntington that hydrogen – normally a transparent, electrically insulating gas at ambient conditions – would become a metal at high pressures [5]. Ironically, this example has become notorious because, though there is no doubt that hydrogen becomes metallic at high enough pressures and temperatures, the conditions of metallization and the detailed processes involved remain controversial. It is unclear whether the metallization of

Interplay between materials theory and high-pressure experiments

2831

hydrogen can be “exactly solved” even with present-day quantum mechanical methods. That the periodic table of the elements is fundamentally altered under pressure is suggested by the following reasoning. The PdV pressure–volume work induced by compression to 1 million times atmospheric pressure (1 Mbar or 100 GPa), conditions readily achieved using either static (diamond-anvil cell) or dynamic (shock-wave) methods, amounts to an internal-energy change of order eV (∼105 J/mol of atoms). This is comparable to the energies of valence-shell electrons, and thus to chemical-bonding energies [6, 7]. In other words, the thermodynamic perturbation caused by Megabar pressures is enough to alter chemical bonding. Numerous experiments document major changes in chemical properties under pressure. Oxygen becomes chalcogenide-like, for example, transforming from a transparent, electrically insulating gas to an opaque metal by about

300

250

Pressure (GPa)

CsI-Xe CONVERGENCE 200

150

Insulator-Metal Transition 100

50

Structural Transitions 0 0

10

20

30

40

50

60

Volume (A3/atom) Figure 2. Crystalline phases and equations of state of Xe (open symbols; face-centered cubic structure on extreme right) and CsI (closed symbols; CsCl structure on right) at low pressures converge at pressures above 50 GPa (hexagonal close-packed structure on left), and both materials are metallic at pressures above 120 GPa [7].

2832

R. Jeanloz

100 GPa (and a superconductor at somewhat higher pressures) [7]. Perhaps more remarkable, the “inert” gas xenon becomes a close-packed metal by 100 GPa, with a crystal structure and equation of state matching those of the iso-electronic compound CsI (Fig. 2). Indeed, the salt (with a gap between valence– and conduction–electron energy bands exceeding 6 eV at zero pressure) becomes metallic by 100 GPa, and then superconducting by 200 GPa. Could it be that xenon will also become a superconducting metal at high pressures? Experiments have, so far, failed to confirm this.∗

2.

Comparison of Theory with Experiment

One of the earliest high-pressure successes of modern electronic bandstructure calculations was the prediction that potassium would transform from an alkali to a transition metal at high pressure [8], and could therefore alloy with iron under compression.† Spectroscopic and other measurements confirmed that K does take on a transition-metal character by 25 GPa [9], and potassium has subsequently been alloyed with nickel and with iron at pressures above 25–30 GPa [10]. Quantitative predictions of structural transitions are notoriously difficult because the energy differences between bulk crystalline phases are often quite small: comparable, in many cases, to the energy resolution of calculations until the 1970s. By the 1980s, however, the feasibility of theoretically deriving transition pressures was demonstrated through studies on Si. Superconductivity was then predicted for some of the high-pressure phases of silicon, and experimentally confirmed shortly thereafter [11]. The role of theory relative to high-pressure experiment underwent a quiet but significant change in the 1990s, with predictions of phase stability being made well before they were experimentally studied, and not just for elements but also for compounds exhibiting more complex structures and variations in bonding. A subtle structural transition (from corundum to Rh2 O3 II structures) was predicted for Al2 O3 in 1987–1994 for example, and only experimentally confirmed in 1997 [12].

∗ The failure to find superconductivity in xenon is plausibly due to the difficulty of experiments in the

“multi-Megabar” pressure range, but might also be due to subtle effects being unexpectedly important. For example, the slightly different masses of Cs and I result in the compound having twice as many vibrational mode-branches as the element, with optic as well as acoustic modes, and the possible effect of this in xenon could be explored through studies of a solid solution consisting of a crystallographically ordered 50:50 mixture of 126 and 132 isotopes of Xe. † The most abundant naturally occurring radioactive isotope with a half life in the billion-year range, 40 K, is an important source of heat for planets. If it alloys with iron at high-pressures, it could also be an important source of heat for the Earth’s iron-rich core and its dynamo-generated magnetic field.


2833

At the same time, molecular dynamics, with interatomic forces derived from first-principles (quantum mechanical) calculations, was being developed to make predictions at high temperatures as well as high pressures. That crystalline CH4 methane would polymerize and ultimately form diamond at elevated pressures and temperatures was thus predicted, and subsequently observed experimentally [13]. By the end of the 20th century, it was becoming evident to high-pressure experimentalists that they had to pay attention to theory, not only to help identify interesting problems (determine which combination of elements to study, at what conditions and with what transformations or other phenomena to expect), but also to help interpret and go beyond the experimental measurements. At the highest pressures, many observations are indirect or require considerable interpretation, and theory can play a crucial role in validating and quantifying these interpretations. Also, to the degree that measured properties agree with theoretical predictions, one can have some confidence in the theoretical values for properties that cannot be measured (for example, elastic shear moduli are generally more difficult to measure under pressure than compressional moduli, so theory can be used to estimate the former if it has been shown to yield good estimates for the latter). Although it is useful to validate theory by demonstrating good agreement with measurement, comparisons between the two are especially powerful when disagreements are uncovered, because this is when the limitations in current modeling approaches or approximations are revealed. For example, the molecular-dynamics prediction that pressures of 100–300 GPa would polymerize methane and then transform it to diamond is experimentally found to be too high by approximately one order of magnitude. Such quantitative discrepancies provide important information about how theory can be improved.

3.

Future Directions for Materials Modeling

Theoretical modeling is pursued in order to develop a fundamental understanding of material properties, including what controls those properties and how those properties can be tuned in practice. An important goal of theory is thus to be able to take a desired property, such as superconductivity or high elastic modulus, and deduce the range of materials that would have these properties as well as predicting the conditions under which the relevant phases are thermodynamically stable. Extension beyond classical thermodynamic equilibrium (the material’s ground-state) is important in at least two regards. First, non-equilibrium properties such as strength and absorption or reflectance spectra (i.e., excited electronic or vibrational states) are of great practical interest. There is again much

2834

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opportunity for comparing theoretical predictions against experimental observations, in particular with pressure as a variable over which to make the comparisons (as a function of compression for a given phase, as well as across pressure-induced phase transformations). Second, once materials having particular desirable properties have been identified, it becomes all-important to consider synthesis routes going beyond the bulk equilibrium of classical thermodynamics. After all, the value of a material is directly proportional both to the nature of its properties and to the ease with which it is synthesized. Carbon is a case in point, with formation by chemical vapor deposition (CVD) having revolutionized the utility of synthetic diamond. High-pressure experiments thus helped establish the equilibrium properties and conditions for synthesis of diamond, but it is the low-pressure method of growth that has taken over as one of the most effective means of making high-quality samples [14]. Indeed, the toughness properties of CVD diamond can be tuned, and this in turn will prove useful for future high-pressure experiments as the diamonds are used as anvils. The ultimate hope is that by fine-tuning theoretical modeling of materials, it will become possible to predict optimal routes for synthesis. Understanding the microscopic details of how materials are nucleated or can grow, and not only under conditions of bulk thermodynamic equilibrium, can then provide the practical answer to those needing a particular set of material properties for a given application: from a prediction of materials that have the required properties to a recipe for the optimal synthesis route. At that point, high-pressure experiments may no longer have their current utility for the study of materials.

References [1] K. Nassau and J. Nassau, J. Crystal Growth, 46, 157–172; F.P. Bundy and R.C. DeVries, “Diamond: high-pressure synthesis,” In: Encyclopedia of Materials: Science and Technology, Elsevier, Amsterdam, 2001. [2] R.M. Hazen, The Diamond Makers, Cambridge University Press, Cambridge, 1999. [3] C. Buzea and K. Robbie, Supercond. Sci. Technol., 18, R1–R8, 2005. [4] L. Gao, Y.Y. Xue, F. Chen, Q. Xiong, R.L. Meng, D. Ramirez, C.W. Chu, J.H. Eggert, and H.K. Mao, Phys. Rev. B, 50, 4260–4263, 1994. [5] E. Wigner and H.B. Huntington, J. Chem. Phys., 3, 764–770, 1935. [6] R. Jeanloz, Ann. Rev. Phys. Chem., 40, 237–259, 1989. [7] R.J. Hemley and N.W. Aschroft, Phys. Today, 51, 26, 1998; R.J. Hemley, Ann. Rev. Phys. Chem., 51, 763–800, 2000. [8] M.S.T. Bukowinski, Geophys. Res. Lett., 3, 491–503, 1976. [9] K. Takemura and K. Syassen, Phys. Rev. B, 28, 1193–1196, 1983. [10] L.J. Parker, T. Atou, and J.V. Badding, Science, 273, 95–97, 1996; L.J. Parker, M. Hasegawa, T. Atou, and J.V. Badding, Eur. J. Solid-State Inorg. Chem., 34, 693–704, 1997; K.K.M. Lee and R. Jeanloz, Geophys. Res. Lett., 30,


[11] [12]

[13]

[14]

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doi:10.1029/2003GL018515, 2003; K.K.M. Lee, G. Steinle-Neumann, and R. Jeanloz, Geophys. Res. Lett., 31, doi:10.1029/2004GL019839, 2004. D. Erskine, P.Y. Yu, K.J. Chang, and M.L. Cohen, Phys. Rev. Lett., 57, 2741–2744, 1986. R.E. Cohen, Geophys. Res. Lett., 14, 37, 1987; H. Cynn, D.G. Isaak, R.E. Cohen, M.F. Nicol, and O.L. Anderson, Am. Mineral., 75, 439, 1990; F.C. Marton and R.E. Cohen, Am. Mineral., 79, 789, 1994; K.T. Thompson, R.M. Wentzcovitch, and M.S.T. Bukowinski, Science, 274, 1880, 1996; N. Funamori and R. Jeanloz, Science, 278, 1109–1111, 1997. F. Ancilotto, G.L. Chiarotti, S. Scandolo, and E. Tosatti, Science, 275, 1288–1290, 1997; L.R. Benedetti, J.H. Nguyen, W.A. Caldwell, H. Liu, M. Kruger, and R. Jeanloz, Science, 286, 100–102, 1999. C.S. Yan, K.K. Mao, W. Li, J. Qian, Y. Zhao, and R.J. Hemley, Phys. Sta. Sol.(a), 201, R25–R27, 2004.

Perspective 27 ATOMISTIC SIMULATION OF FERROELECTRIC DOMAIN WALLS I-Wei Chen Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6282, USA

1.

Introduction

Atomistic simulation of moving discommensurations is most useful when atomic details have a strong influence on the outcome. A classical example is the problem of dislocation core and movement in which the nature of atomic bonding has a direct effect on the configuration of the dislocation core, which in turn affects the manner the dislocation moves [1]. The final macroscopic characteristics of plastic deformation are intimately related to these details, of which a complete understanding can only come from the atomistic simulation of both the static and the (activated) non-equilibrium configurations of dislocations. The extension to the interface problem is encountered in martensitic transformations, involving either the parent/product interface or the product/product interface, which we call variant interface [2]. Variant interfaces are mobile and may be regarded as a group of dislocations, and like dislocations they move under a stress. The problem is relevant to shape memory alloys in which the stress and rate dependence of shape change is ultimately controlled by the atomic configurations at variant interfaces during their movement, even though the final (presumably equilibrium) multi-variant configurations are dictated by crystallography and elastic energy minimization. As a further extension to the interface problem, domain walls in ferroelectric crystals are like variant interfaces in that, crystallographically, both belong to the class of twin boundaries, but domain walls are complicated by electrostatic considerations [3, 4]. Since the origin of ferroelectricity lies in the instability of covalent bonding, the atomic configurations of domain walls are necessarily sensitive to both covalent bonding and long-range elastic and electrostatic interactions. Such complexity makes the problem a good candidate for atomistic simulation studies from which one may hope to better understand domain switching 2843 S. Yip (ed.), Handbook of Materials Modeling, 2843–2847. c 2005 Springer. Printed in the Netherlands.

2844

I-W. Chen

characteristics such as coercivity and fatigue, which are important for ferroelectric applications. To date, however, there have been very few atomistic simulation studies on domain walls, none being relevant to domain wall movement. Nevertheless, the recent development of phenomenological but realistic potentials for Pb(Zr,Ti)O3 (PZT) [5] makes such studies entirely feasible within today’s computational resources. The purpose of this commentary is to outline the basic domain wall problem to stimulate the community to undertake atomistic simulations.

2.

Synopsis

Some general considerations based on our theoretical understanding of dislocation and ferroelectricity provide the following synopsis. A domain wall separates two domains of different polarization vectors. Like twinning, ferroelectric distortion causes internal strains that vary from domain to domain, albeit the net shear component of such strains can be greatly reduced by alternating the shear direction. Therefore, the simplest domain wall (called 180◦ domain wall) has opposite polarization on the two sides, and it has no elastic distortion at all. All the non-180◦ domain walls, however, are like twin boundaries and have some short-range strains associated with them. Therefore, they interact with stresses, including self-stress of the walls and internal stresses due to impurities and heterogeneities. A stress-free domain wall, in turn, is electrically neutral, which requires the normal component of the polarization vector to be continuous across the domain wall. This is illustrated for a tetragonal crystal (e.g., BaTiO3 ) for both 180◦ (BB’ in Fig. 1) and 90◦ (AA’ in Fig. 1) domain walls. This neutrality condition is not satisfied if the domain wall is aligned in other directions. (For example, BB” in Fig. 1 has a A'

B'

B"

A' D C

A

B

A Figure 1.

B

Atomistic simulation of ferroelectric domain walls

2845

negatively charged kink at CD.) Therefore, curved or kinked domain walls are generally charged. Such is the case at the edge of a lens-shaped domain where two domain walls bend to meet each other. Also, when a domain wall moves, it prefers to first form charged kinks (or ledges in three dimensions) as shown in Fig. 1, then propagate the kinks sidewise, rather than moving the entire domain wall forward all at once. (The electrostatic forces on the charged kinks always move them in a way to enlarge the side where the polarization vector is electrically favored.) These kinks and ledges are structurally similar to those found on moving dislocations, twin boundaries and variant interfaces, and their formation determines the activation energy for the wall movement. Since all moving walls are charged and all stationary non-180◦ walls are strained, they will attract or repel defects either electrically or elastically. Oxygen vacancies, which are positively charged and elastically soft, are known to be a potent defect that strongly interacts with domain walls and greatly influences the ferroelectric behavior of BaTiO3 and PZT. Second phases can similarly interact with domain walls and modify ferroelectric behavior.

3.

Stationary Domain Wall and Wulff Plot

The above synopsis makes evident the path to follow in the atomistic simulation of stationary and moving domain walls. In the following, we outline the case primarily for (strain-free) 180◦ domain walls to focus on the electrostatic aspect that is unique to the domain wall problem. To begin with, we recognize that there is a strong anisotropy due to both crystallography and charge. Take the case of BaTiO3 for example, in which the polarization is along 001. Then all the (hk0) domain walls are neutral, so their different domain wall energies are due to crystallographic anisotropy. On the other hand, the (hkl) domain walls are charged, so with increasing l there is a rapid rise in energy from the electrostatic contribution. The Wulff plot of the domain wall is therefore a highly elongated rod around the four-fold symmetry axis 001. Also of interest is the second derivative of the domain wall energy which determines the torque on a curved domain wall. These aspects, as well as the atomic positions and polarization distribution across the domain wall, can be readily studied by atomistic simulation. Our current understanding is that domain walls are relatively sharp, across which a near discontinuity of polarization vector exists. In this respect, ferroelectric domain walls are different from magnetic domain walls which can be quite diffuse. The sharpness of ferroelectric domain walls is due to the strong preference for polarization to align in certain crystallographic orientations and with a constant magnitude, given by the minimum of the “doublewell”. Ionic displacements at intermediate positions and with intermediate orientations may simply have too high an energy to be allowed even in the

2846

I-W. Chen

transition layer at the domain wall. This should be verified by atomistic simulation, however, and any deviation from the double-well minimum will modify the domain wall energy due to the so-called “gradient energy” term in the Ginsburg Landau equation. Moreover, the possibility of a more extended domain wall cannot be ruled out in some ferroelectric crystals. In BCC metals, extended dislocation cores with three-fold symmetry have been seen in both electron microscopy and atomistic simulations.

4.

Activation and Domain Wall Movement

As a first step to understand domain wall movement, we consider a translation that brings a domain wall initially to a higher energy state, then passing a maximum, before finally returning to the equilibrium state after one unit cell displacement. The periodic energy landscape along such translation can be studied by atomistic simulation, and its maximum slope gives the theoretical coercive field (at 0 K) of a flat domain wall. This energy landscape corresponds to the Peierls (lattice) energy for dislocations, and the theoretical coercive field corresponds to the Peierls stress. Lattice periodicity similarly manifests itself in energy profiles of a kinked/ledged domain wall when the entire wall is translated forward or when the kink/ledge is translated sideway. Polarization distribution across the translated wall/kink/ledge is again of interest. Both an electric and a stress field can motivate domain wall movement, but the energetics of domain wall movement can be studied without applying an external field. For a flat wall, the energy barrier for first forming a bulge, then enlarging the bulge sideway, is lower than the “Perierls” energy for translation. This bulge has a thickness of a unit cell, is electrically neutral as a whole, but is locally charged on some portions of its perimeter. The total energy of a bulged domain wall as a function of the size and shape of the bulge can be studied by atomistic simulation, which determines the activation barrier (and the coercive field at 0 K) for the flat wall. We expect the shape of the bulge to be highly anisotropic in order to minimize the domain wall energy of the charged perimeter and to keep the oppositely charged segments as far apart as possible. For the latter reason, the shape optimization of a bulge (of a constant area) involves non-local interaction and does not immediately follows from the Wulff plot. In addition, the incline of the perimeter may not be vertical but, instead, may be quite gradual especially when the wall energy of the segment is low. Therefore, atomistic simulation is again useful for elucidating these details. Of course, the simulation can also be repeated under an applied field. Since the activation nature implies that the wall movement is a dissipative, frictional process even at 0 K, a finite wall velocity obtains under an applied field once it exceeds the coercive field.

Atomistic simulation of ferroelectric domain walls

5.

2847

Extension to Finite Temperature and non-180◦ Domain Walls

Clearly, the simulation studies outline above can be extended to finite temperature. Information of domain wall entropy, roughening, and thermally activated movement can then be analyzed. Because of the non-local electrostatic interaction (and more generally, elastic interaction for non-180◦ walls) and the strong anisotropy of domain wall energy, the size of the simulated volume should be sufficiently large to avoid artifacts that either suppress or promote a particular mode of fluctuation. This size limitation is the main concern for atomistic domain wall simulations but it should be within reach of today’s computational resources. We have thus far emphasized the electrostatic aspect which is unique to the domain wall problem and is appropriate for the 180◦ domain walls, which only respond to the electrical field. The insensitivity to the stress field makes 180◦ domain walls less likely trapped and more mobile. In a real ferroelectric crystal, however, other domain walls are also present and they may well control the ferroelectric properties. For these domain walls, incorporation of both electrostatic and elastic interactions is obviously needed. Two sets of scaling parameters are therefore necessary. For electrostatic interaction, it is polarization; for elastic interaction, it is shear modulus and strain, which scales as (polarization)2 . Since the polarization vanishes at the Curie temperature whereas the shear modulus thermally softens only gradually, the temperature dependence in the domain wall problem is much stronger than in the dislocation problem, where only the shear modulus matters while strain remains constant. The additional influence of crystal chemistry and crystallography also need to be considered. This can be best incorporated into the simulation by parameterizing the semi-empirical potentials employed to allow simple interpretation in terms of charge, covalency, elasticity and polarization. No additional computational difficulty is foreseen with these studies.

References [1] J.P. Hirth and J. Lothe, Theory of Dislocations, 2nd edn., John Wiley & Sons, New York, 1982. [2] Y-H. Chiao and I-W. Chen, “Martensitic growth in ZrO2 –an in situ, small particle TEM study of a single-interface transformation,” Acta Metall., 38, 1163–1174, 1990. [3] F. Jona and G. Shirane, Ferroelectric Crystals, Pergamon Press, New York, 1962. [4] M.E. Lines and A.M. Glass, Principles and Applications of Ferroelectrics and Related Materials, Clarendon, Oxford, 1977. [5] I, Grinberg, V.R. Cooper, and A.M. Rappe, “Relationship between local structure and phase transitions of a disordered solid solution,” Nature, 419, 6910, 909–911, 2002.

Perspective 28 MEASUREMENTS OF INTERFACIAL CURVATURES AND CHARACTERIZATION OF BICONTINUOUS MORPHOLOGIES Sow-Hsin Chen Department of Nuclear Engineering, MIT, Cambridge, MA 02139, USA

1.

Introduction

Studies of bicontinuous and interpenetrating domain structures developed in the process of spinodal decomposition (SD) of binary mixtures of molecular fluids, binary alloys, and polymer blends have been an attractive research theme over the past several decades [1]. Complex structures can be observed in the phase separation process induced by a temperature quench from the stable one-phase region into the unstable two-phase spinodal region. The fact that the dynamical processes in complex fluids such as polymer blends, glasses, and gels are extremely slow due to their long characteristic relaxation times allows one to study their morphologies with better accuracy than those of simple liquid mixtures [2]. Similar kinds of bicontinuous structures have been observed in the water/oil/ surfactant three-component microemulsion system in the one-phase region close to the three-phase boundary (often called the “fish tail” in microemulsion literature because of the similarity in shape of the phase boundary of this region to a fish tail) and in the vicinity of hydrophile-lipophile balance temperature [3, 4]. It is interesting and physically relevant to examine the common and universal features of these bicontinous structures observed in phase-separated polymer blends [5] and in microemulsions [6]. During the course of the past three decades our comprehension of amphiphile solutions has been profoundly transformed [7]. A major conceptual force behind these developments has been the realization that the bulk phases of amphiphiles in solution could be visualized as systems of interfacial films and 2849 S. Yip (ed.), Handbook of Materials Modeling, 2849–2863. c 2005 Springer. Printed in the Netherlands.

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S.H. Chen

could therefore be successfully described in terms of their properties. Thus, the structural and thermodynamic features of micellar or vesicular solutions, of microemulsions, and of lyotropic liquid crystals, began to be progressively rationalized in terms of the properties of fluctuating surfaces [8]. Because interfaces formed by amphiphiles have often very small or vanishing interfacial tensions the dominant contribution to their free energy is its bending elastic energy. The latter, in turn, is given by the following expression,

FH =

1 dS κ (H − c0 )2 + κ K 2

(1)

where dS is the element of the surface area, H and K, are respectively, the mean and the Gaussian curvatures of the surface that represents the interface, and c0 , κ and κ are, respectively, the spontaneous curvature, the bending rigidity and the saddle-splay constants. FH is known as the Helfrich free energy [9] in the literature. Thus it is seen from the above formula, that the basic physical variables describing the elastic properties of an interface are the mean and Gaussian curvatures, much the same as those describing the mechanical properties of an interacting particle system are the kinetic and the potential energies. It follows that a statistical mechanical description of an interface must involve the average mean and Gaussian curvatures over the whole interface. It is then clear that to devise a method to measure and compute these average curvatures for physically relevant interfacial systems is utmost important. At any point on the oil–water interface (with the surfactant mono-layer in-between), the interface may be characterized by its two principal radii of curvature R1 and R2 , which by convention are positive when the interface is curved towards the oil phase and are negative when it is curved towards the water phase. One then defines a mean curvature by H =1/2 ((1/R1 ) + (1/R2 )), and a Gaussian curvature by K = 1/(R1 R2 ) at that point. The geometry of the surface is completely specified by giving a pair of values (H, K) at every point on the surface. The first practical method for measuring the average mean curvature H was proposed and implemented by Lee and Chen [10]. The system they studied was a three-component non-ionic microemulsion system composed of water/octane/ C10 E4 (tetraethylene glycol monodecyl ether) with equal volume fractions of water and oil. The phase diagram in the temperature- φs (volume fraction of the surfactant) plane shows that the hydrophile–lipophile balance (HLB) temperature is about 24 ◦ C and the one-phase microemulsion begins to form at the minimum volume fraction, φs = 0.10. Hence at φs = 0.20, if one varies the temperature from 15◦ to 30 ◦ C, the one-phase microemulsion should transform from a water-in-oil droplet microemulsion to an oil-in-water microemulsion through an intermediate state of lamellar microemulsion. In this process the average mean curvature H should switch sign from an initial

Measurements and characterization of bicontinuous morphologies

2851

positive value to a final negative value through an intermediate value of zero at the HLB temperature. They indeed found that the average mean curvature varies linearly in this temperature interval starting with a value H = 0.008 Å−1 at 15 ◦ C and ends up with a value H = −0.006 Å−1 at 30 ◦ C. The method was based on the measurements of three separate specific interfacial areas using a contrast variation technique in a small-angle neutron scattering (SANS) experiment. Because the surfactant layer in-between the water and oil phases of a microemulsion has a finite thickness d, the water–surfactant interfacial area AW and the oil–surfactant interfacial area AO are generally not equal if the film is curved. This can most easily seen by considering the case of a spherical shell; the inner surface has less area than the outer surface and this difference is related to the thickness of the shell and the radius of the sphere. The exact geometrical relationships between AW , AO and the surface area AS measured at the midpoints of the surfactant molecules to the average curvatures H and K are given by:

Aw = As

d2 K , 1 + d H + 4

Ao = As

d2 K 1 − d H + 4

(2) The average mean curvature can thus be calculated from thetwo measured areas AW and AO by a formulaH = ( Aw − AO ) / ( Aw + Ao ) /d. With this method, errors involved in the specific interfacial areas measurements were large enough that the average Gaussian curvature could not be deduced with enough accuracy. In order to measure the average Gaussian curvature Chen et al. proposed another method. This is called a Clipped Random Wave (CRW) model [6]. Cahn in 1965 proposed a scheme for generating a three-dimensional morphology of a phase separated A–B alloy by clipping a Gaussian random field generated by superposing many isotropically propagating sinusoidal waves with random phases [11]. The Gaussian random field can be normalized in such a way that it fluctuates continuously between −1 and +1. One can then realizes the bicontinuous two-phase morphology by clipping the continuous random process, namely, by assigning say 0 to all negative signals representing A, and +1 to all positive signals representing B. In this scheme, the essential features of the morphology depends only on the “spectral density function” (SDF), which is the three-dimensional Fourier transform of the twopoint correlation function g(r) of the Gaussian random field. The SDF gives the distribution of the magnitudes of the propagation wave vectors of the sinusoidal waves. Berk in 1987 further developed the idea of Cahn mathematically for the purpose of analyzing scattering data [12]. In particular, he derived an important relation connecting the two-point correlation function and the Debye correlation function which determines the scattering intensity. In his

2852

S.H. Chen

original paper, however, Berk discussed only a SDF which is a delta function. This resulted in generating a morphology which is only partially disordered and is not realistic. Chen, Chang and Strey [13] later point out that a broader but peaked SDF is necessary to generate a more disordered morphology which will reproduce small-angle neutron scattering (SANS) intensity distributions. In particular, it was recognized that the SDF should be chosen in such a way that it has the correct second moment to ensure the agreement with the measured finite interfacial area per unit volume (the specific interface area S/V).

2.

Theory of Scattering From Bicontinuous Porous Materials in Bulk Contrast

The intensity distribution of SANS from an isotropic, disordered twocomponent porous material can be calculated generally from a Debye correlation function (r) by the following formula [14]: I (Q)= < η >

∞

2

dr4πr 2 j0 (Qr)(r)

(3)

0

whereη2 = ϕ1 ϕ2 (ρ1 − ρ2 )2 , is the mean square fluctuation of the local scattering length density, ϕ1 and ϕ2 the volume fractions of components 1 and 2, ρ1 and ρ2 the corresponding scattering length densities. There are two physical boundary conditions that the Debye correlation function, a function of a scalar distance between the two points under consideration, must satisfy: it is normalized to unity at the origin, r = 0, and it should decay to zero at infinity. The most important property of the Debye correlation function for the case with a sharp boundary between two regions having different scattering length densities, ρ1 and ρ2 , is that it has a linear and a cubic terms in the small r expansion of the form: (r → 0) = 1 − ar + br 3 + . . . 1 S b = 1− r 1 − r2 + . . . 4ϕ1 ϕ2 V a

(4)

where a = (S/ V )/4ϕ1 ϕ2 is a factor proportional to the total interfacial area per unit volume of the sample and the ratio of the coefficient of the cubic term to the linear term has been given by Kirste and Porod [15] in terms of curvatures as: b 1 2 1 K . = H − a 8 24

(5)


3.

2853

Clipped Random Wave (CRW) Model for Computation of Interfacial Curvatures

In the clipped random wave model of Berk [12], an order parameter field ψ( r ), which for example, can be a quantity proportional to the volume fraction of the majority phase in the system, is constructed by superposition of a large number N of cosine waves with random phases:

ψ( r) =

N 2 cos(ki · r + ϕi ) N i=1

(6)

where directions of the wave vector ki are assumed to be distributed isotropically over a unit sphere and the phase, ϕi distributed randomly over an interval (0, 2π ). In constructing the sum, the magnitude of the ki vector is sampled from a scalar distribution function f (k) called the “spectral density function”. This order parameter field is certainly a Gaussian random field due to the central limit theorem in statistics. The statistical properties of a Gaussian random field is completely characterized by giving its spectral density function f (k) or the corresponding two-point correlation function g(r). We define the two-point correlation funcri )ψ( r2 ) and the associated SDF f (k) by a Fourier tion by g(|ri − r2 |) = ψ( transform relation: g(|ri − r2 |) =

∞

4π k 2 j0 (k | ri − r2 |) f (k) dk

(7)

0

This continuous random process ψ( r ), varying between +1 and −1, having a mean square value of unity (by its definition Eq. 6), is then clipped and transformed into a discrete, two-state discrete random process. By clipping we mean assigning a constant value +1 to the function whenever the Gaussian random field at that point is above a certain “clipping level” called α, and a constant 0 whenever its value is below α. This transformation can be defined as follows: 1, when ψ( r) ≥ α r )) = (8) ζ( r ) = α (ψ( 0, otherwise where α is a step function. Then the Debye correlation function can be written as a function of the discrete random variable ς (r) in the form: ζ(0)ζ( r ) − ζ 2 (9) ζ − ζ 2 where the quantities on the right-hand side are given by Teubner [16] as: ( r) =

1 1 ζ = − √ 2 2π

0 α

exp(−x 2 /2)dx

(10)

2854

S.H. Chen

1 ζ(0)ζ( r ) = ζ − 2π

cos−1 (g(r))

α2 exp − 1 + cos θ

0

dθ.

(11)

The average value of the clipped Gaussian random filed, ζ , and 1 − ζ can be interpreted as volume fractions of the majority and minority phases, ϕ1 and ϕ2 , respectively. Using (10) and ζ = ϕ1 , Eq. (9) can be rewritten as 1 ( r) = 1 − 2π ϕ1 (1 − ϕ1 )

cos−1 (g(r))

0

α2 exp − dθ 1 + cos θ

(12)

For a small α, meaning a slight deviation from an isometric case, Eq. (12) can be approximated as:

( r) ∼ = 1−

cos−1 (g(r)) 1 cos−1 (g(r)) − α 2 tan 2π ϕ1 (1 − ϕ1 ) 2

(13)

where the volume fraction ϕ1 can be approximated as α 1 ϕ1 ∼ = −√ . 2 2π

(14)

For an isometric (ϕ1 = ϕ2 = 1/2) microemulsion α = 0, and Eq. (12) reduces to a simple form 2 sin−1 (g(r)) (15) π which is a well-known result given by Berk [12]. From the relation in Eq. (7), one has the small r expansion of g(r) of the form ( r) =

∞

g(r) = 0

1 1 4 4 k r + . . . f (k)dk 4π k 2 1 − k 2r 2 + 6 120

=1−

1 4 4 1 2 2 k r + k r + ... 6 120

(16)

where we used the normalization condition g(0) = 1, and k 2 and k 4 denote the second and fourth moment of the spectral density function. Note that this expansion has a quadratic term followed by a quartic term. Using the result of Eq. (16) in Eq. (12), we obtain a small r expansion of the Debye correlation function the form: 1/2 1 2 √ k2 e−α /2r 2π 3ϕ1 ϕ2 1 2 2 1 k4 − k (α − 1) r 2 × 1− 40 k 2 72

B (r → 0) = 1 −

(17)


2855

Comparing this with Eq. (4), we arrive at a useful relation connecting the second moment of the spectral density function to the fundamental quantity of a porous material, the interfacial area per unit volume, 2 2 1/2 −α 2 /2 S = √ k e . V π 3

(18)

This relation also implies that one of the basic requirements for the physically acceptable spectral function is that the second moment be finite. We next consider a random surface generated by the level set ψ( r ) = ψ(x, y, z) = α

(19)

where the function ψ(x, y, z) is defined by Eq. (6). Teubner [16] has proved a remarkable theorem that for this random surface the average mean, Gaussian and mean square curvatures are given by:

K =

(20)

1 2 2 k α −1 6

(21)

H2 =

where

π 2 k 6

α H = 2

1 2 2 k α + v2 6

(22)

6 k4 v = 2 − 1 5 k2 2

(23)

So the second requirement of the physically acceptable spectral function is that it has a finite fourth moment also. Since for a bicontinuous microemulsion the level surface defined by Eq. (19) is approximately the mid-plane passing through the surfactant monolayer in a bulk contrast experiment, the average mean, Gaussian and square mean curvatures of the surfactant monolayer can be computed once a physically acceptable SDF can be found. Choice of the SDF can be based on a criterion that when it is substituted into equations Eqs. 3, 7 and 12, it would give an intensity distribution which agrees with SANS data in an absolute scale. A suitable form of the SDF has been proposed by Chen and Choi [17], which is an inverse eighth order polynomial in k and which contains three parameters a, b, and c. The first two parameters a and b have their approximate correspondences in the approximate theory of Teubner and Strey [18]. In the T-S theory, the Debye correlation function is given by: TS (r) = e−r/ξ

sin(2πr/d) (2πr/d)

(24)

2856

S.H. Chen

The correspondences are: a ≈ 2π/d and b ≈ 1/ξ , where d is the interdomain (water–water or oil–oil) repeat distance and ξ the decay length of the local order [19, 20]. The parameter c controls the transition from the main Q peak to the large Q behavior of the scattering intensity distribution, the existence of which is essential for the good agreement of the theory and experiment for the entire range of Q. This agreement is important if the correct surface to volume ratio of the system is to be incorporated into the theory. We thus choose [17] a SDF for which both the second and fourth moments exist. This function is an inverse eighth order polynomial containing three parameters, which are the minimal set for the physical situation under study: f (k) =

bc(a 2 + (b + c)2 )2 /(b + c)π 2 (k 2 + c2 )2 (k 4 + 2(b2 − a 2 )k 2 + (a 2 + b2 )2 )

(25)

√ This spectral density function has a peak approximately at kmax ≈ a 2 −b2 . From Eq. (25), the second and fourth moments of the spectral density function are given by:

c(a 2 + b2 + bc) (b + c)

c(a 4 + 2a 2 b2 + b4 + 4a 2 bc + 4b3 c + 4b2 c2 + bc3 ) k4 = (b + c) k2 =

(26)

The two-point correlation function g(r) can be given in an analytical form as well, which has only even powers of r in the small r expansion. From this two-point correlation function, the Debye correlation function can be calculated analytically using Eq. (12), once the clipping level (i.e., the volume fraction ϕ1 ) is specified. Explicit expressions for the curvatures are given in terms of the three parameters and the clipping level α (determined by the volume fraction of the majority phase as calculated by Eq. (10)) as:

π c(a 2 + b2 + bc) 6 (b + c) 2 2 1 c(a + b + bc) K = − (1 − α 2 ) 6 (b + c)

a 4 + 2a 2 b2 + b4 + 4a 2 bc + 4b3 c + 4b2 c2 + bc3 H 2 = K + 5(a 2 + b2 + bc) α H = 2

(27) (28) (29)

This model can be used to fit all the bulk contrast data nicely to obtain parameters a, b and c. One can then use Eqs. (27)–(29) to calculate the respective curvatures.


4.

2857

Examples of Small-angle Neutron Scattering (SANS) Data Analysis Using CRW Model

Figure 1a shows a series of scattering intensities I (Q) measured from an AOT microemulsion system at points in the one-phase region close to the lamellar phase boundary. The symbols are experimental data and solid lines are theoretical fits using Eq. (7), (15) and (25) in Eq. (3) plus an incoherent background. These curves show good agreement between the theory and experimental data over the entire range of Q. The fitted parameters a, b, and c are listed in Table 1. As the surfactant volume fraction increases a and c increase rapidly while b increases slowly. Considering the relations d ≈ 2π/a and ξ ≈ 1/b, the inter-domain distance d and the decay length ξ decreases with the surfactant volume fraction. This makes sense because we create more surfaces per unit volumes as the number of surfactant molecules increase and thus the inter-domain distance should decrease. Considering the ratio ξ /d, one sees that a bicontinuous microemulsion becomes more disordered at smaller surfactant volume fractions. The average Gaussian and square mean curvatures calculated by using Eqs. (28) and (29), respectively, are given in Fig. 2a. The solid line is a fit using a phenomenological parabolic equation K = co + c1 (φs − φo )2 (a)

(30)

(b) 104

103

102

102 101

Theory

100

0.08 0.11 0.14 0.17 0.20

I(Q) (cm⫺1)

I(Q) (cm⫺1)

103

φs

10⫺1 0.01

Theory octane decane dodecane tetradecane

100 0.1

Q (Å⫺1)

101

10⫺2

10⫺1

100

Q (Å⫺1)

Figure 1. (a) Analyses of the bulk contrast scattering intensities of isometric AOT-based microemulsions as a function ofsurfactant volume fraction. The scattering intensities were measured at points close to the lamellar phase boundary, where the average mean curvature is nearly zero; (b) analyses of the bulk contrast scattering intensities of isometric Ci E j -based microemulsion systems taken at the fish tail of its phase diagram.

2858

S.H. Chen

Table 1. Fitted parameters a, b, c and the calculated interfacial curvatures for an isometric (equal oil and water volume fractions,H = 0) AOT/decane/water (NaCl) microemulsion system in the one-phase region along the lamellar phase boundary. ϕs is volume fraction of the surfactant AOT φs

a (Å−1 )

b (Å−1 )

c (Å−1 )

0.05 0.08 0.11 0.14 0.17 0.20

0.00613 0.01317 0.02139 0.03005 0.03939 0.04968

0.01104 0.01198 0.01120 0.01236 0.01346 0.01459

0.01710 0.04682 0.09114 0.1096 0.1530 0.2096

2 η

backgrd

25.92 13.07 10.86 9.33 9.30 9.00

0.385 0.383 0.393 0.382 0.352 0.321

K

H 2

NaCl

–0.353 –1.165 –2.382 –3.610 –5809 –8.943

1.62 6.35 16.78 21.65 37.67 64.85

0.49 0.46 0.42 0.36 0.32 0.26

(1020 cm −4 ) (cm −1 ) (10−4 Å−2 ) (10−4 Å−2 ) (wt%)

and the dashed line is a fit using a similar equation

H 2 = c0 + c1 (φs − φo )2 .

(31)

Results of the fit are c0 = −0.434 × 10−4 Å−2 ; c1 = −310.2 × 10−4 Å−2 ; φ0 = 0.036 and c0 = 2.35 × 10−4 Å−2 ; c1 = 2530 × 10−4 Å−2 ; φ0 = 0.046. In both cases, φ0 turns out to be very close to the volume fraction at the fish tail of the phase diagram.We can say that c0 and c0 are the average Gaussian curvature and the average square mean curvature at the fish tail which is the lowest volume fraction of the surfactant where the one-phase microemulsion first forms. Magnitudes of average Gaussian curvatures obtained in this experiment are comparable to that obtained for a C10 E 4 /D2 O/Octane microemulsion near the fish tail before [10]. The quadratic dependence of the average Gaussian curvature on the volume fraction of surfactant is also understandable. According to Eqs. (18) and (21), the magnitude of the average Gaussian curvature is proportional to the square of the interfacial area per unit volume. The interfacial area is in turn proportional to the amount of surfactant added to the system. Since the average mean curvature is expected to be zero for microemulsions that we studied at these phase points, the average square mean curvature is the variance of the fluctuation of the mean curvature. One also observes that the variance of the fluctuation increases quadratically as the volume fraction increases. This is reasonable because as more surfactant is added to the system, more interfacial area is created. The surfactant film has to bend around to accommodate itself in the liquid. Figure 1b shows scattering intensities and their analyses for a series of isometric microemulsion systems composed of C8 E 3 /D2 O/n-alkane at their respective fish tail in the phase diagram. The theory again agrees with experiments uniformly well and the extracted curvatures are plotted as symbols in Fig. 2b. The fish tail is a very special point in the phase diagram where the system has the lowest specific internal surface area. Figure 2b shows results


⫺2

60

⫺4

50

40

⫺6

30

⫺8 ⫺10 0

0.05

0.1

φs

(a)

0.15

0.2

120 100 80

⫺6

⫺8

60

⫺10

20

⫺12

10

⫺14

0

⫺16

40

(10⫺4 Å⫺2)

⫺4

0

70

(10⫺4 Å⫺2)

(10⫺4 Å⫺2)

⫺2

80

(10⫺4 Å⫺2)

0

2859

20

0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

φs

(b)

Figure 2. (a) Average Gaussian and square mean curvatures as a function of surfactant volume fraction obtained from analyses of a series of isometric AOT-based microemulsions. The corresponding SANS intensities are those shown in Fig. 1a. Solid lines are parabolic fits based on Eqs. (30) and (31). Average Gaussian and square mean curvatures as a function of surfactant volume fraction at the fish tail obtained from analyses of a series of isometric Ci E j -based microemulsions. The corresponding SANS intensities are those shown in Fig. 1b. Solid lines are the parabolic fits.

of curvatures we extracted from 18 generic non-ionic microemulsion systems [21]. Again, the parabolic dependence of the curvatures on the surfactant volume fraction is evident. Figure 3 shows the level surface as defined by Eq. (19) for the system C8 E3 /D2 O/tetradecane at its fish tail. The three parameters in the spectral density function needed for the three-dimensional reconstruction comes from analysis of the fourth intensity curve from the top shown in Fig. 1b. It is clear from the figure the geometry of the level surface is locally hyperbolic giving rise to a negative Gaussian curvature. In Fig. 4 we show scattering intensity distributions and their analyses for three non-isometric microemulsions. This new microemulsion system is composed of F8 H16 /H2 O/perfluorooctane [22]. We use an abbreviation F8 H16 ≡ CF3 (CF2 )7 −COO(CH2 CH2 O)7.2 CH3 . The analyses are successful and we are able to extract both the average mean and Gaussian curvatures given in Table 2. It is interesting to ask the following question at this point: is CRW method of analysis applicable to bicontinuous structures generated by a late stage spinodal decomposition of an isometric system such as a binary alloy or a polymer blend? Answer is partially yes. In Fig. 5 we show a light scattering intensity distribution [5] of a 50–50 polymer blend of near critical mixture of perdeuterated polybutadiene (dPB) and polyisoprene (PI) at the late stage spinodal decomposition. The analysis result is shown as the solid line. The plot

2860

S.H. Chen

Figure 3. ‘The level surface as defined by Eq. (19) for the system C8 E3 /D2 O/tetradecane at its fish tail. It is clear from the picture that the surface is locally hyperbolic having a negative average Gaussian curvature. The size of cube is 240 × 240 × 240µm3 .

I(Q) (cm -1 )

102

101

The ory 100 α⫽0.0530 α⫽0.2534 ⫺1

10

10⫺2

α⫽0.4438

10⫺2

10⫺2 ⫺1

Q (Å )

Figure 4. Analyses of scattering intensity distributions for threenon-isometric microemulsions composed of F8 H16 /H2 O/perfluorooctane [22]. The curvatures obtained are given in Table 2.


2861

Table 2. Average mean and Gaussian curvatures extracted from non-isometric microemulsions α (clipping level) 0.0530 0.2544 0.4438

φ1 (volume fraction)

H in 10−3 Å−1

K in 10−4 Å−2

0.48 0.40 0.33

0.86 4.80 9.90

−3.35 −4.27 −5.05

is presented in a scaling form because the characteristic length scale of the microstructure is of the order of 10 µm. Agreement between the experiment and the theory is excellent for large Q portion of the curve but is only moderately satisfactory for smaller Q region. The following results are obtained:

α = 0, η2 = 6.836 × 1012 cm−4 a = 5.36 × 10−5 Å−1 ; b = 1.04 × 10−5 Å−1 ; K = −1.99 × 10−9 Å−2 ≈ 0.2 µm−2

c = 87.2 × 10−5 Å−1 (32)

101

Q3maxI(Q)/

100

10⫺1

10⫺2

Experiment Theory

10⫺3

100

101

Q /Qm ax Figure 5. Analysis of a light scattering intensity distribution [5] of a 50–50 (isometric) polymer blend of near critical mixture of perdeuterated polybutadiene (dPB) and polyisoprene (PI) at its late stage spinodal decomposition. The analysis results are given in Eq. (32).

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Conclusion

We have outlined a method, called the Clipped Random Wave model, by which the interfacial curvatures of a porous material can be measured through a scattering experiment. This method also allows one to reconstruct a three dimensional interfacial structure of the porous medium from which the morphology of the two-phase bicontinuous microstructure can be visualized. We illustrate this method using SANS data taken from bicontinuous microemulsion samples having bulk contrast. Specifically, we can extract the average mean, Gaussian and square mean curvatures of the internal interfaces. The method relies on a choice of a physically reasonable SDF of the underlying Gaussian random field which upon clipping generates a bicontinuous structure that produces the given scattering intensity pattern. A physically reasonable SDF of the Gaussian random field should have finite lower order moments to ensure a finite internal surface area per unit volume and a finite average square mean curvature. The method has been put to test in three cases: a nearly isometric AOT/D2 O(NaCl)/decane microemulsion system as a function of surfactant volume fraction; Several isometric microemulsions made of Ci E j /D2 O/alkane at their respective fish tails in the phase diagrams; and several non-isometric microemulsions made of F8 H16 /H2 O/perfluorooctane. Theoretical intensities calculated by the CRW model can be made to agree with the scattering data in an absolute scale by adjusting the three length scale parameters, 1/a,1/b,1/c, in the SDF. The commonly used two-parameter Teubner–Strey model for the Debye correlation function can also be used to fit the scattering data in the small Q region. But the fit is distinctly worse compared to our three parameter theory for Q region after the peak.The parameter a can be approximately identified as a = 2π/d and b = 1/ξ of the corresponding parameters in the T-S theory. The third parameter c is necessary for ensuring a smooth transition of scattering intensity from small Q behavior near the peak to large Q behavior dominated by the Porod’s law. We can thus say that three length scales are necessary for a complete description of the microstructure of bicontinuous microemulsions. The CRW model with an appropriate choice of the SDF is also shown to be a useful tool for quantitative analyses of small angle scattering data from microphase-separated system such as a symmetric binary mixture of polymers at late stage of spinodal decomposition. It not only gives information on curvatures of the interface, which cannot be obtained by other means, but can also generate the morphology of 3-d microstructure as shown in Fig. 3.

Acknowledgment This research is supported by a grant from Materials Science Program of US Department of Energy. We are grateful to the Intense Pulse Neutron Source


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Division of Argonne National Laboratory for neutron beam time at the SAND Low-Angle Diffractometer.

References [1] J.D. Gunton, M. San Miguel and P. Sahni, In: C. Domb and J.L. Lebowitz (eds.), Phase Transitions and Critical Phenomena, Academic Press, New York, 1983, p. 269. [2] T. Hashimoto, Phase Transitions, 1988, 12, 47; Materials Science and Technology, Vol 12, Structure and Properties of Polymers, VCH, Weinheim, 1993. [3] W. Jahn and R. Strey, J. Phys. Chem., 92, 2294, 1988. [4] S.H. Chen, S.L. Chang, and R. Strey, J. Chem. Phys., 93, 1907, 1990. [5] H. Jinnai, T. Hashimoto, D.D. Lee, and S.H. Chen, Macromolecules, 30, 130, 1997. [6] S.H. Chen, D.D. Lee, K. Kimishima, H. Jinnai, and T. Hashimoto, Phys. Rev. E, 54, 6526, 1996. [7] W.M. Gelbart, A. Ben-Shaul and D. Roux (eds.), Micelles, Membranes, Microemulsions, and Monolayers, Springer, New York, 1994. [8] S.A. Safran, Statistical Thermodynamics of Surfaces, Interfaces, and Membranes, Addison-Wesley, 1994. [9] W. Helfrich and Z. Naturforsch, 28c, 693, 1973. [10] D.D. Lee and S.H. Chen, Phys. Rev. Lett., 73, 106, 1994. [11] J.W. Cahn, J. Chem. Phys., 42, 93, 1965. [12] N.F. Berk, Phys. Rev. Lett., 73, 106, 1987. [13] S.H. Chen, S. L. Chang, and R. Strey, J. Appl. Crystallogr., 24, 721, 1991. [14] P. Debye, H.R. Anderson, Jr. and H. Brumberger, J. Appl. Phys., 28, 679–683, 1957. [15] R. Kirste, Von & G. Porod, Kolloid-Z&Z.f.Polym., 184, 1–7, 1962. [16] M. Teubner, Europhys. Lett., 14(5), 403–408, 1991. [17] S.H. Chen and S.M. Choi, J. Appl. Cryst., 30, 755–760, 1997. [18] M. Teubner and R. Strey, J. Chem. Phys., 87, 3195–3200, 1987. [19] S.H. Chen, S.L. Chang, and R. Strey, Prog. Colloid Polymer. Sci., 81, 30–35, 1990. [20] S.H. Chen and S.L. Chang, and R. Strey, J. Appl. Cryst., 24, 721–731, 1991. [21] S.-M Choi, S.H. Chen, T. Sottmann, and R. Strey, “The existence of three length scales and their relation to the interfacial curvatures in bicontinuous microemulsions,” Physica A, 304, 85–92, 2002. [22] P. LoNostro, S.M. Choi, C.Y. Ku, and S.H. Chen “Fluorinated microemulsions–a study of the phase behavior and structure by SANS,” J. Phys. Chem., 103, 5347– 5352, 1999.

Perspective 29 PLASTICITY AT THE ATOMIC SCALE: PARAMETRIC, ATOMISTIC, AND ELECTRONIC STRUCTURE METHODS Christopher Woodward Northwestern University, Evanston, Illinois, USA

Over the last hundred years our evolving comprehension of deformation has been based on the discovery and understanding of the line defects (dislocations) that control plasticity. While our ability to directly model various defects has improved over this time, a great deal has been learned about deformation processes using parametric approaches. A natural extension of analytic models, parametric studies are sometimes overlooked in our search for the most accurate computational representation of the mechanism that controls a given materials property. However, parametric strategies has been broadly employed in the materials community. For example, we have used parametric approaches to study the influence of micro-structural properties on the strengthening mechanisms in model Ni-based super alloys and the influence of chemistry on high temperature strengthening in Ti–Al alloys [1, 2]. Also, current dislocation dynamics calculations can be viewed as a template for parametric studies of the role of various defect-defect interactions and how they control macroscopic behavior. The deformation behavior of the bcc transition metals provides an excellent historical example of this type of work and how it influenced our understanding of these materials. In the early 1920s as materials scientists were exploring the deformation behavior of various simple metals. Schmid proposed the seemingly reasonable premise that: Plastic flow will occur when the shear stress resolved on a particular slip system reaches a critical value, the critical resolved shear stress (CRSS), which will be independent of slip system and the sense of slip. Further, the CRSS should not be influenced by other components of the applied stress tensor. Since that time many examples of violations to Schmid’s law have been documented. In fact for materials slated for high-temperature structural 2865 S. Yip (ed.), Handbook of Materials Modeling, 2865–2869. c 2005 Springer. Printed in the Netherlands.

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applications (refractory metals and intermetallics) violations to Schmid’s law seem to be the rule rather than the exception. In these materials edge dislocations tend to be very mobile and deformation is limited by the lattice friction stress (Peierls stress) of screw dislocations. These materials also exhibit large Peierls stresses compared to the simple fcc metals, particularly at low temperatures. This and the significant deviations from Schmids law led Mitchel and co-workers [3] to propose that in the bcc transition metals this behavior was produced by dislocations with non-planar dislocation centers (cores) that made planar glide difficult. Realizing that the differences in geometry for fcc and bcc metals may explain the gross differences in the plasticity of these materials, several groups in the early 1960s developed parametric atomistic potentials for the bcc metals. One set of parametric potentials were based on variations of a generalized stacking fault energy [4, 5]. This is defined as the energy as a function of shear for two blocks of material along the screw direction where the plane of shear is on the assumed glide plane. Structural parameters (lattice and elastic constants) were held fixed for the suite of potentials, but no effort was made to model a particular material. These simple studies showed that a range of non-planar dislocation cores could be obtained for screw dislocations in the bcc metals. Even though these studies were hampered by the simple forms of inter-atomic potentials for many years the qualitative parametric description of the screw dislocations was found to be satisfactory. The large lattice friction stresses, and some of the deviations from Schmids law could be understood in terms of the derived non-planar dislocation core structures. In the last 15 years, as confidence grew in new atomistic potential schemes such as the Embedded Atom Method, and the Model Generalized Pseudopotential Theory several groups produced carefully designed atomistic potentials for specific bcc transition metals [6–9]. These methods produced core structures consistent with previous, parametric studies, lending credence to the idea that the local geometry determines the shape and mobility of dislocations on the atomic scale. Specifically the studies showed that the predicted dislocation cores for the group V and VI bcc transition metals spread into three {110} planes and the shapes of the cores fell into two distinct classes (Fig. 1). The group V metals exhibited a core spread symmetrically about a central point (Fig. 1a) while the core for the group VI metals spreads asymmetrically about this central point (Fig. 1b). Differences in the macroscopic plasticity of the group V and VI transition metals, for example higher sensitivity to non-glide stresses propensity in the group VI metals, were linked to the differences in the equilibrium core structures. Also, atomistic studies showed that details of the generalized stacking fault determined the differences in equilibrium shape of the dislocations [10]. Only recently has it been possible to actually model these dislocation cores using electronic structure methods based on Density Functional Theory (DFT).

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(a)

(b)

Figure 1. Schematic of the strain field near the dislocation cores found using atomistic potentials (before 2001) for the group V and group VI bcc transition metals.

The geometry of an isolated dislocation is problematic for current large-scale electronic structure methods, so two approaches have been taken to accommodate these boundary conditions. The group at Wright Patterson Air Force Base employed a flexible boundary condition method that self-consistently couples the local strain field produced by the dislocation core to the long range elastic field [11]. This technique is an extension of a method originally proposed by Sinclair and allows the dislocation to be contained in a very small simulation cell [12, 13]. Other groups have used simulation cells with dislocation dipoles carefully arranged to minimize the total stress field produced by the dislocation array [14, 15]. Surprisingly all the electronic structure calculations show Ta (group V) and Mo (group VI) transition metals to have a dislocation core consistent with Fig. 1a. Frederiksen and Jacobsen also find the same result for the screw dislocation in bcc Fe (group VIII). They also find that the generalized stacking fault calculated using DFT for all the group V and VI bcc metals are consistent with a core spread symmetrically about a central point (Fig. 1a). Our calculations also show the atomic scale response of the screw dislocations to applied stress in Mo and Ta is similar to that found using atomistic potentials. Thus the differences in macroscopic plasticity of these two materials is probably not linked to the shape of the equilibrium cores, but is more likely tied to anisotropies in the local bonding of the dislocation core. While the atomistic methods are correctly incorporating a great deal of this information content, predicting differences between elemental metals may require improved interaction models or ab-initio methods.

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Electronic structure calculations indicate that one extreme of the parametric model (Fig. 1b) is not realized in the bcc transition metals. This is not a failure of the previous methods; it just indicates that this level of detail resides at the electronic level and must be modeled appropriately if we are to understand the behavior of a particular material. The question remains; can we find examples of the core shown in Fig. 1b. in nature and how will this configuration affect plasticity. Thus the original parametric investigations, based on the geometry and reasonable values for the physical parameters, had great value in breaking into a new area of materials science. These configurations may be realized in other bcc metals, or perhaps in studies of interstitial-dislocation interactions. Simple parametric-models will continue to play and important role extending our understanding of materials behavior, don’t overlook this strategy when approaching a new area in computational materials science.

References [1] S.I. Rao, T.A. Parthasarathy, D.M. Dimiduk, and P.M. Hazzledine, “Discrete dislocation simulation of precipitate hardening in superalloys,” Phil. Mag., in press, 2004. [2] C. Woodward, and J.M. MacLaren, “Planar fault energies and sessile dislocation configurations in substitutionally disordered Ti-Al with Nb and Cr ternary additions,” Phil. Mag., A74, 337, 1996. [3] T.E. Mitchell, R.A. Foxall, and P.B. Hirsch, “Work hardening in Niobium Single Crystals,” Phil. Mag., 8, 1895, 1963. [4] V. Vitek, R.C. Perrin, and D.K. Bowen, “The core structure of a/2111 screw dislocations in BCC crystals,” Phil. Mag., 21, 1049, 1970. [5] M.S. Duesbery, V. Vitek, and D.K. Bowen, “The effect of shear stress on the screw dislocation core structure in BCC cubic lattices,” Proc. Roy. Soc. Lond., A332, 85– 111, 1973. [6] G.J. Ackland and V. Vitek, “Many body potentials and atomic scale relaxations in nobel metals alloys,” Phys. Rev., B 41, 10324, 1990. [7] D. Farkas and P.L. Rodriguez, “Embedded atom study of dislocation cores structure in Fe,” Scripta Metall. Mater., 30, 921, 1994. [8] W. Xu and J.A. Moriarty, “Atomistic simulation of ideal shear strength, point defects, and screw dislocations in BCC transition metals: Mo and a prototype,” Phys. Rev., B 54, 6941, 1996. [9] L.H. Yang, P. Soderlind, and J.A. Moriarty, “Accurate atomistic simulation of (a/2)111, screw dislocations and other defects in BCC tantalum” Phil. Mag., A81, 1355–85, 2001. [10] M.S. Duesbery and V. Vitek, “Plastic anisotrophy in BCC transition metals,” Acta Mater., 46, 1481–1492, 1998. [11] C. Woodward and S.I. Rao, “Flexible ab-initio boundary conditions: Simulating isolated dislocations in BCC Mo and Ta,” Phys. Rev. Lett., 88, 216402-1-4, 2002. [12] J.E. Sinclair, P.C. Gehlen, R.G. Hoagland et al., “Flexible boundary conditions and nonlinear geometric effects an atomic dislocation modeling,” J. Appl. Phys., 3890– 3897, 1978.

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[13] S.I. Rao, C. Hernandez, J.P. Simmons et al., “Green function boundary conditions in 2-d and 3-d atomistic simulations of dislocations,” Philos. Mag. A77, 231, 1998. [14] S. Ismail-Beigi and T.A. Arias, “Ab-initio study of screw dislocations in Mo and Ta: A new picture of plasticity in BCC transition metals,”Phys. Rev. Lett., 84, 1499– 1503, 2000. [15] S.L. Frederiksen and K. Jacobsen, “Density functional theory studies of screw dislocation core structures in BCC metals,” Phil. Mag., 83, 365–375, 2003.

Perspective 30 A PERSPECTIVE ON DISLOCATION DYNAMICS Nasr M. Ghoniem Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA 90095-1597, USA

A fundamental description of plastic deformation has been recently pursued in many parts of the world as a result of dissatisfaction with the limitations of continuum plasticity theory. Although continuum models of plastic deformation are extensively used in engineering practice, their range of application is limited by the underlying database. 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. Although homogenization methods have played a significant role in determining the elastic properties of new materials from their constituents (e.g., composite materials), the same methods have failed to describe plasticity. 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 heterogeneous at all length-scales. Depending on the deformation mode, heterogeneous dislocation structures appear with definitive wavelengths. A satisfactory description of realistic dislocation patterning and strain localization has been rather elusive. Attempts aimed at this question have been based on statistical mechanics, reaction-diffusion dynamics, or the theory of phase transitions. Much of the efforts have aimed at clarifying the fundamental origins of inhomogeneous plastic deformation. On the other hand, engineering descriptions of plasticity have relied on experimentally verified constitutive equations. At the macroscopic level, shear bands are known to localize plastic strain, leading to material failure. At smaller length scales, dislocation distributions are mostly heterogeneous in deformed materials, leading to the formation of 2871 S. Yip (ed.), Handbook of Materials Modeling, 2871–2877. c 2005 Springer. Printed in the Netherlands.

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a number of strain patterns. Generally, dislocation patterns are thought to be associated with energy minimization of the deforming material, and manifest themselves as regions of high dislocation density separated by zones of virtually undeformed material. Dislocation-rich regions are zones of facilitated deformation, while dislocation poor regions are hard spots in the material, where plastic deformation does not occur. Dislocation structures, such as Persistent slip Bands (PSB’s), planar arrays, dislocation cells, and subgrains, are experimentally observed in metals under both cyclic and steady deformation conditions. Persistent slip bands are formed under cyclic deformation conditions, and have been mostly observed in copper and copper alloys. They appear as sets of parallel walls composed of dislocation dipoles, separated by dislocation-free regions. The length dimension of the wall is orthogonal to the direction of dislocation glide. Dislocation planar arrays are formed under monotonic stress deformation conditions, and are composed of parallel sets of dislocation dipoles. While PSB’s are found to be aligned in planes with normal parallel to the direction of the critical resolved shear stress, planar arrays are aligned in the perpendicular direction. Dislocation cell structures, on the other hand, are honeycomb configurations in which the walls have high dislocation density, while the cell interiors have low dislocation density. Cells can be formed under both monotonic and cyclic deformation conditions. However, dislocation cells under cyclic deformation tend to appear after many cycles. Direct experimental observations of these structures have been reported for many materials. Two of the most fascinating features of micro-scale plasticity are the spontaneous formation of dislocation patterns, and the highly intermittent and spatially localized nature of plastic flow. Dislocation patterns consist of alternating dislocation rich and dislocation poor regions usually in the µm range (e.g., dislocation cells, sub-grains, bundles, veins, walls, and channels). On the other hand, the local values of strain rates associated with intermittent dislocation avalanches are estimated to be on the order of 1–10 million times greater than externally imposed strain rates. Understanding the collective behavior of defects is important because it provides a fundamental understanding of failure phenomena (e.g., fatigue and fracture). It will also shed light on the physics of elf-organization and the behavior of critical-state systems (e.g., avalanches, percolation, etc.) Because the internal geometry of deforming crystals is very complex, a physically-based description of plastic deformation can be very challenging. The topological complexity is manifest in the existence of dislocation structures within otherwise perfect atomic arrangements. Dislocation loops delineate regions where large atomic displacements are encountered. As a result, longrange elastic fields are set up in response to such large, localized atomic displacements. As the external load is maintained, the material deforms

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plastically by generating more dislocations. Thus, macroscopically observed plastic deformation is a consequence of dislocation generation and motion. A closer examination of atomic positions associated with dislocations shows that large displacements are confined only to a small region around the dislocation line (i.e., the dislocation core). The majority of the displacement field can be conveniently described as elastic deformation. Even though one utilizes the concept of dislocation distributions to account for large displacements close to dislocation lines, a physically-based plasticity theory can paradoxically be based on the theory of elasticity! 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. When the mechanical behavior is dominated by microstructure heterogeneity, the mechanics problem can be greatly simplified if all atomic degrees of freedom were adiabatically eliminated, and only those associated with defects are retained. Because the motion of all atoms in the material is not relevant, and only atoms around defects determine the mechanical properties, one can just follow material regions around defects. Since the density of defects is many orders of magnitude smaller than the atomic density, two useful results emerge. First, defect interactions can be accurately described by long-range elastic forces transmitted through the atomic lattice. Second, the number of degrees of freedom required to describe their topological evolution is many orders of magnitude smaller than those associated with atoms. These observations have been instrumental in the emergence of meso-mechanics on the basis of defect interactions by Eshelby, Kröner, Kossevich, Mura and others. Thanks to many computational advances during the past two decades, the field has steadily moved from conceptual theory to practical applications. While early research in defect mechanics focused on the nature of the elastic field arising from defects in materials, recent computational modelling has shifted the emphasis on defect ensemble evolution. Although the theoretical foundations of dislocation theory are wellestablished, efficient computational methods are still in a state of development. Other than a few cases of perfect symmetry and special conditions, the elastic field of 3-D dislocations of arbitrary geometry is not analytically available. The field of dislocation ensembles is likewise analytically unattainable. A relatively recent approach to investigating the fundamental aspects of plastic deformation is based on direct numerical simulation of the interaction and motion of dislocations. This approach, which is commonly known as dislocation dynamics (DD), was first introduced for 2-D straight, infinitely long dislocation distributions, and then later for complex 3-D microstructure. In DD simulations of plastic deformation, the computational effort per time-step is proportional to the square of the number of interacting segments, because

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of the long-range stress field associated with dislocation lines. The computational requirements for 3-D simulations of plastic deformation of even single crystals are thus very challenging. The study of dislocation configurations at short-range can be quite complex, because of large deformations and reconfiguration of dislocation lines during their interaction. Thus, adaptive grid generation methods and more refined treatments of self-forces have been found to be necessary. In some special cases, however, simpler topological configurations are encountered. For example, long straight dislocation segments are experimentally observed in materials with high Peierls potential barriers (e.g., covalent materials), or when large mobility differences between screw and edge components exist (e.g., some bcc crystals at low temperature). Under conditions conducive to glide of small prismatic loops on glide cylinders, or the uniform expansion of nearly circular loops, changes in the loop shape is nearly minimal during its motion. Also, helical loops of nearly constant radius are sometimes observed in quenched or irradiated materials under the influence of point defect fluxes. It is clear that, depending on the particular application and physical situation, one would be interested in a flexible method which can capture the essential physics at a reasonable computational cost. A consequence of the longrange nature of the dislocation elastic field is that the computational effort per time step is proportional to the square of the number of interacting segments. It is therefore advantageous to reduce the number of interacting segments within a given computer simulation, or to develop more efficient approaches to computations of the long range field. While continuum approaches to constitutive models are limited to the underlying experimental data-base, DD methods offer new directions for modeling microstructure evolution from fundamental principles. The limitation to the method presented here is mainly computational, and much effort is needed to overcome several difficulties. First, the length and time scales represented by the present method are still short of many experimental observations, and methods of rigorous extensions are still needed. Second, the boundary conditions of real crystals are more complicated, especially when external and internal surfaces are to be accounted for. Thus, the present approach does not take into account large lattice rotations, and finite deformation of the underlying crystal, which may be important for explanation of certain scale effects on plastic deformation. And finally, a much expanded effort is needed to bridge the gap between atomistic calculations of dislocation properties on the one hand, and continuum mechanics formulations on the other. Nevertheless, with all of these limitations, the DD approach is worth pursuing, because it opens up new possibilities for linking the fundamental nature of the microstructure with realistic material deformation conditions. It can thus provide an additional tool to both theoretical and experimental investigations of plasticity and failure of materials.


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Two main approaches have been advanced to model the mechanical behavior in this meso length scale. The first is based on statistical mechanics methods. 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. The main challenge in this regard is that, unlike the situation encountered in the development of the kinetic theory of gases, the topology of interacting dislocations within the system must be included. The second approach, commonly known as Dislocation Dynamics (DD), was initially motivated by the need to understand the origins of heterogeneous plasticity and pattern formation. An early variant of this approach (the cellular automata) was first developed by Lepinoux and Kubin [1], and that was followed by the proposal of DD [2–4]. In these early efforts, dislocation ensembles were modelled as infinitely long and straight in an isotropic infinite elastic medium. The method was further expanded by a number of researchers, with applications demonstrating simplified features of deformation microstructure. Since it was first introduced in the mid-eighties independently by Lepinoux and Kubin, and by Ghoniem and Amodeo, Dislocation Dynamics (DD) has now become an important computer simulation tool for the description of plastic deformation at the micro- and meso-scales (i.e., the size range of a fraction of a micron to tens of microns). The method is based on a hierarchy of approximations that enable the solution of relevant problems with today’s computational resources. In its early versions, the collective behavior of dislocation ensembles was determined by direct numerical simulations of the interactions between infinitely long, straight dislocations [5]. Recently, several research groups extended the DD methodology to the more physical, yet considerably more complex 3-D simulations. The method can be traced back to the concepts of internal stress fields and configurational forces. The more recent development of 3-D lattice dislocation dynamics by Kubin and co-workers has resulted in greater confidence in the ability of DD to simulate more complex deformation microstructure [6–8]. More rigorous formulations of 3-D DD have contributed to its rapid development and applications in many systems [9–15]. We can classify the computational methods of DD into the following categories: 1. The Parametric Method: 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. Other approximation methods have been developed by a number of groups. These approaches differ mainly in the representation of dislocation loop geometry, the manner by which the elastic field and self energies are calculated, and some additional details

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3.

4.

5.

N.M. Ghoniem related to how boundary and interface conditions are handled. The suitability of each method is determined by the required level of accuracy and resolution in a given application. dislocation loops are divided into contiguous segments represented by parametric space curves. The Lattice Method: 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. The method is computationally fast, but gives coarse resolution of dislocation interactions. The Force Method: Straight dislocation segments of mixed character in the are moved in a rigid body fashion along the normal to their midpoints, 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: This 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 Phase Field Microelasticity Method: This method is based on the 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 [17].

References [1] J. Lepinoux and L.P. Kubin, “The dynamic organization of dislocation structures: a simulations,” Scripta Met., 21(6), 833, 1987. [2] N.M. Ghoniem and R.J. Amodeo, “Computer simulation of dislocation pattern formation,” Solid State Phenomena, 3 & 4, 377, 1988. [3] R.J. Amodeo and N.M. Ghoniem, “Dislocation dynamics I: a proposed methodology for deformation micromechanics,” Phys. Rev., 41, 6958, 1990b. [4] R.J. Amodeo and N.M. Ghoniem, “Dislocation dynamics II: applications to the formation of persistent slip bands, planar arrays, and dislocation cells,” Phys. Rev., 41, 6968, 1990a. [5] H.Y. Wang and R. LeSar, “O(N) algorithm for dislocation dynamics,” Phil. Mag. A, 71, 1, 149, 1995. [6] L.P. Kubin, G. Canova, M. Condat, B. Devincre, V. Pontikis, and Y. Brechet, “Dislocation microstructures and plastic flow: a 3D simulation,” Diffussion and Defect Data – Solid State Data, Part B (Solid State Phenomena), 23–24, 455, 1992.


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[7] G. Canova, Y. Brechet, and L.P. Kubin, “3D dislocation simulation of plastic instabilities by work-softening in alloys,” In: S.I. Anderson et al., Modeling of Plastic Deformation and its Engineering Applications, RISØ National Laboratory, Roskilde, Denmark, 1992. [8] B. Devincre and L.P. Kubin, “Simulations of forest interactions and strain hardening in fcc crystals,” Mod. Sim. Mater. Sci. Eng., 2(3A), 559, 1994. [9] J.P. Hirth, M. Rhee, and H. Zbib, “Modeling of deformation by a 3D simulation of multipole, curved dislocations,” J. Comp.-Aided Mat. Design, 3, 164, 1996. [10] K.V. Schwarz and J. Tersoff, “Interaction of threading and misfit dislocations in a strained epitaxial layer,” App. Phys. Lett., 69(9), 1220, 1996. [11] R.M. Zbib, M. Rhee, and J.P. Hirth, “On plastic deformation and the dynamics of 3D dislocations,” In: J. Mech. Sci., 40(2–3), 113, 1998. [12] M. Rhee, H.M. Zbib, J.P. Hirth, H. Huang, and T. de la Rubia, “Models for long/ short-range interactions and cross slip in 3D dislocation simulation of bcc single crystals,” Mod. Sim. Mater. Sci. Eng., 6(4), 467, 1998. [13] N.M. Ghoniem and L.Z. Sun, “Fast sum method for the elastic field of 3-D dislocation ensembles,” Phy. Rev. B, 60(1), 128–140, 1999. [14] N.M. Ghoniem, S.-H. Tong, and L.Z. Sun, “Parametric dislocation dynamics: a thermodynamics-based approach to investiagations of mesoscopic plastic deformation,” Phys. Rev., 61(2), 913–927. [15] 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. [16] L.M. Brown, “A proof of Lothe’s theorem,” Phil. Mag., 15, 363–370, 1967. [17] Y. Wang, Y. Jin, A. Cuitino, and A.G. Khachaturyan, “Nanoscale phase field microelasticity theory of dislocations: model and 3D simulations,” Acta Mat., 49, 1847, 2001.

Perspective 31 DISLOCATION-PRESSURE INTERACTIONS J.P. Hirth Ohio State and Washington State Universities, 114 E. Ramsey Canyon Rd., Hereford, AZ 85615, USA

1.

Introduction

There are various ways in which a dislocation can interact with isostatic stress, with most effects being important at elevated stress levels. Some of these that have received extensive attention are the effects of isostatic stress on elastic constants [1–3]; the direct influence of pressure, or the indirect effect via the stress normal to the glide plane, on the Peierls stress, or more specifically the kink formation energy in many metals and alloys [4, 5]; the influence of the degree and symmetry of core splitting of screw dislocations in bcc metals [6–8] as well as some semiconducting crystals and intermetallic compounds [8, 9]; the implementation of core-splitting effects in terms of the general stress tensor [10, 11]; the indirect effect on operative slip systems that modifies geometric hardening [12]; and weak effects on factors such as lattice parameter, vibrational frequency and diffusivity. However, little or no consideration has been given to the coupling of isostatic stress with the nonlinear, long-range, elastic field of the dislocation. Here, we briefly treat this effect and indicate how it can be incorporated into constitutive models for dislocation motion.

2.

Origin of the Coupling

The nonlinear elastic theory of dislocations has been developed, in the perturbation sense of including third and fourth order elastic constants, for both the isotropic and anisotropic elastic cases [1, 13, 14]. These treatments predict, for example, that there is a biaxial dilatation associated with a dislocation, and X-ray diffraction verifies the presence of the dilatation [1]: of course, in low symmetry crystals such as triclinic, there is a dilatational response to 2879 S. Yip (ed.), Handbook of Materials Modeling, 2879–2882. c 2005 Springer. Printed in the Netherlands.

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any applied stress so local dilatation should exist near a dislocation.. However, the perturbation models are not accurate in predicting the magnitude of the dilatation, although they are useful in giving the dominant terms for the near core region once the core fields are known. The dilatation is associated with highly nonlinear core effects and can be estimated from atomistic simulations of the near-core region [15] or from an atomistic simulation involving the variation of strain energy with stress in a local near-core region [16]. The core field entails sets of line-force dipoles without moment [15], but at long-range converges to a biaxial dilatation, so a consideration of the latter suffices at long range. A list of typical biaxial dilatations determined in such a manner is given by Puls [17]. Typically, the biaxial dilatational area, δa, including image relaxation at free surfaces, is of the order of 1 to 1.5 atomic areas per plane cut by the dislocation. This corresponds to a volume change per unit length δv/L = δa.

3.

Influence on Dislocation Motion

We consider local coordinates xi fixed on the dislocation with x3 // ξ , the sense vector of the dislocation. The dilatation will then interact with the biaxial stress σ = (σ11 +σ22 )/2. In most practical situations σ will correspond to the isoaxial stress σ I = − p = (σ11 + σ22 + σ33 )/3, where p is the isostatic pressure. We consider this correspondence to apply, although the more specific σ case can easily be introduced into the results. The local coupling will then produce a positive interaction energy per unit length given by W p = pδa.

4.

Systems Deforming by Kink Motion

As a first application, we treat kink pair formation on screw dislocations in bcc metals. For Fe a typical kink formation energy is 0.8 eV [7], δa = 0.62 b2 [18], and kinks should be inclined at an angle φ ∼45◦ to the Peierls valley. Here b is the length of the Burgers vector of a 12 [111] dislocation. These values lead to an energy contribution from the pressure interaction of W p λ=7.7 ( p/µ) eV, normalized to the shear modulus µ, where λ is the kink length. Thus at ( p/µ) = 0.01, the contribution of W p λ is about ten percent of Wk . The value ( p/µ) = 0.01 is readily attained in shock loading and it is of the order of pressures achieved in constrained high pressure tensile testing devices. This value is also typical of the isostatic stress σI = − p achieved in uniaxial tensile tests of heavily drawn steel wire [19]. In general, the kink formation energy can be written W f = Wk + Wh + W p λ +

Wσ 2

(1)

Dislocation-pressure interactions

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Entropy terms would also enter the kink free energy [20]. Here, Wk is the kink formation energy in the absence of stress, Wh is the contribution arising from core splitting and the influence of the applied stress tensor [10], and Wσ is the work done by the effective shear stress resolved on the glide plane as a kink pair is created. The numerical example indicates that W p λ is significant and should be included in the analysis of kink formation at high stress levels. Less obviously, there is an indirect effect of pressure on kink energy. When W p is appreciable, there is a tendency to increase φ from its zero-stress value in order to minimize the length λ that interacts with p: that is the minimum free energy kink will become sharper. This means that Wk will increase relative to the zero-stress value. The effect is second-order relative to the direct effect just discussed, but it can be important at very high stress levels. For fcc metals, the isostatic stress interaction term is roughly W p λ ≈ 5 ( p/µ) eV. This is similar to the value for bcc metals. Thus for fcc metals, the kink mechanism is only operative at temperatures typically well below room temperature, and there the interaction again can be important under shock loading. The effect is negligible for bulk metals under uniaxial tension because of the relatively low values of the flow stress for fcc metals. However, it could also be important for in-situ composites [21] and thin multiplayer structures [22], where stresses also reach values where ( p/µ) ∼0.01. When the interaction is important, the influence on φ should be marked because of its initially low value ∼ 5◦ [20]. Other systems with large kink formation energies should behave analogously to the bcc screw case. Intermetallic compounds [9] and ceramic crystals such as alumina [23] should exhibit such behavior.

5.

Systems Deforming by Dislocation Bowout and Breakaway

In fcc metals at room temperature and in most bcc metals above a critical temperature that exceeds room temperature, deformation involves locally bowed out segments that breakaway from obstacles by cutting or bypassing them. The barrier to bowout is the increase in line length of the bowing dislocation. The exact line energy is a complicated function of segment length, dislocation character and local surrounding segment configurations [20]. For our purposes, it suffices to use the simple line tension approximation. The increase in energy per unit length is then W = S L L = (µb2 /2)L where S L is the constant line tension and L is the increase in line length. With the pressure interaction present, W p directly adds to S L . Thus, the total line tension is S = S L + W p = µb2 /2 + pδa

(2)

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And the increase in energy with an increase of line length is W = SL. The importance of the pressure interaction term can be determined by direct substitution.

6.

Summary

Dislocations have a long-range dilatational strain field arising from large nonlinearities in the core. The field interacts with isostatic stresses. The resultant interaction energy can be an important contribution to kink formation energy and to dislocation line tension at high pressures.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

A. Seeger and P. Haasen, Philos. Mag., 3, 470, 1958. D.J. Steinberg, S.G. Cochran, and J. Guinan, J. Appl. Phys., 51, 1498, 1980. L.H. Yang, P. Söderlind, and J.A. Moriarty, Philos. Mag., A81, 1355, 2001. J.A. Moriarty, Phys. Rev., B49, 12431, 1994. J.A. Moriarty, V. Vitek, V.V. Bulatov et al., J. Computer-Aid. Mater. Des., 9, 99, 2002. V. Vitek, Cryst. Lattice Defects, 5, 1, 1974. M.S. Duesbery, Acta Metall., 31, 1747, 1983. W. Cai, V.V. Bulatov, J. Chang et al., In: F.R.N. Nabarro and J.P. Hirth (eds.), Dislocations in Solids, vol. 12, N. Holland, Amsterdam, 2004. V. Paidar, D.P. Pope, and V. Vitek, Acta Metall., 32, 435, 1984. Q. Qin and J.L. Bassani, J. Mech. Phys. Solids, 40, 835, 1992. M.S. Duesbery and V. Vitek, Acta Metall. Mater., 46, 1481, 1998. R.J. Asaro and J.R. Rice, J. Mech. Phys. Solids, 25, 309, 1977. J.R. Willis, Int. J. Engin. Sci., 5, 171, 1967. C. Teodosiu, Elastic Models of Crystal Defects, Springer-Verlag, Berlin, p. 208, 1982. R.G. Hoagland, J.P. Hirth, and P.C. Gehlen, Philos. Mag., 34, 413, 1976. R.G. Hoagland, M.S. Daw, and J.P. Hirth, J. Mater. Sci., 6, 2565, 1991. M.P. Puls, Dislocation Modeling of Physical Systems, M.F. Ashby et al. (eds.), Pergamon, Oxford, p. 249, 1981. B.L. Adams, J.P. Hirth, P.C. Gehlen, et al., J. Phys. F, 7, 2021, 1977. J.D. Embury, A.S. Keh, and R.M. Fisher, Metall. Trans., 12, 478, 1967. J.P. Hirth and J. Lothe, Theory of dislocations, Kliewer, Melbourne, FL, 1992. K. Han, J.D. Embury, J.J. Petrovic et al., Acta Mater., 46, 4691, 1998. A. Misra and H. Kung, Adv. Engin. Mater., 3, 217, 2001. T.E. Mitchell, P. Peralta, and J.P. Hirth, Acta Mater., 47, 3687, 1999.

Perspective 32 DISLOCATION CORES AND UNCONVENTIONAL PROPERTIES OF PLASTIC BEHAVIOR V. Vitek Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104

1.

Concept of Dislocations

Dislocations are line defects found in all crystalline materials and their motion produces plastic flow. The notion of dislocations has two starting points. First, the dislocation was introduced as an elastic singularity by considering the deformation of a body occupying a multiply connected region of space. Secondly, dislocations were introduced into crystal physics when analyzing the large discrepancy between the theoretical and experimental strength of crystals. These two approaches are intertwined since the crystal dislocations are sources of long-ranged elastic stresses and strains that can be examined in the continuum framework. In fact, the bulk of the dislocation theory employs the continuum elasticity when analyzing a broad variety of dislocation phenomena encountered in plastically deforming crystals [1–4]. From the continuum point of view dislocations are line singularities invoking long-ranged elastic stress and strain fields that decrease as d −1 , where d is the distance from the dislocation line. Formally, both the stress and strain diverge at the dislocation line and the corresponding strain energy would also diverge. However, physically this means that there is a region, centered at the dislocation line, in which the linear elasticity does not apply. This region, the dimensions of which are of the order of the lattice spacing, is called the dislocation core. The properties of the core region and its impact on dislocation motion and thus on plastic yielding, can only be fully understood when the atomic structure is adequately accounted for. In general, when a dislocation glides its core undergoes changes that are the source of an intrinsic lattice friction. This 2883 S. Yip (ed.), Handbook of Materials Modeling, 2883–2896. c 2005 Springer. Printed in the Netherlands.

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friction is periodic with the period of the crystallographic direction in which the dislocation moves. The applied stress needed to overcome this friction at 0 K temperature is called the Peierls stress and the corresponding periodic energy barrier is called the Peierls barrier. In this Commentary we first focus on the dominant aspects of dislocation cores, in particular their symmetries and the form of spreading of atomic displacements in the core region. Specifically, the core displacements can be either confined to a given crystallographic plane or extended three-dimensionally. The latter cores, encountered in a broad variety of materials, are responsible for unconventional aspects of the plastic deformation such as the break-down of the Schmid law, anomalous dependence of the yield stress on temperature, unusually strong orientation and temperature dependence of the yield and flow stress, orientation dependence of the ductility and brittleness, strong strain rate sensitivity etc., [5–10]. Using body-centered-cubic (bcc) metals as an example, we discuss the core features that have to be captured in any theoretical description of the dislocation motion and related plastic properties. Finally, we demonstrate by a brief overview that analogous core phenomena are encountered in many other materials than bcc metals. However, prior to the discussion of atomic structures of dislocations, we reflect on the most important “core phenomenon”, dislocation dissociation into partial dislocations separated by metastable stacking-fault like planar defects.

2.

Dislocation Dissociation and Stacking Faults

A vital characteristic of dislocations in crystalline materials is their possible dissociation into partial dislocations with Burgers vectors smaller than the lattice vector. Such dislocation splitting can occur if the displacements corresponding to the Burgers vectors of the partials lead to the formation of metastable planar faults which then connect the partials. The reason for the splitting is, of course, the decrease of the dislocation energy when it is divided into dislocations with smaller Burgers vectors. A well-known example is splitting of 1/2110 dislocations in fcc materials into two Shockley partials with the Burgers vectors of the type 1/6112 on {111} planes. The planar fault formed by the 1/6112 displacement is an intrinsic stacking fault. In general, stacking-fault-like defects that include not only stacking faults but also other planar defects, such as antiphase domain boundaries and complex stacking faults encountered in ordered alloys and compounds, can be very conveniently analyzed using the notion of γ-surfaces, first employed in investigation of possible stacking faults in bcc metals [11]. To introduce the idea of a γ-surface, we first define a generalized stacking-fault: Imagine that the crystal is cut along a given crystallographic plane and the upper part displaced , parallel to the plane of the cut, as with respect to the lower part by a vector u

Dislocation cores and unconventional properties of plastic behavior

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shown in Fig. 1. The fault created in this way is called the generalized stackingfault and it is not in general metastable. The energy of such fault, γ ( u), can be evaluated using atomistic and/or density functional theory (DFT) methods; relaxation perpendicular to the fault has to be carried in such calculations. Re within the repeat cell of the given peating this procedure for various vectors u crystal plane, an energy-displacement surface can be constructed, commonly called the γ-surface. The local minima on this surface determine the displacement vectors of all possible metastable stacking-fault-like defects, and the values of γ at these minima are the energies of these faults. Many such calculations were performed employing a broad variety of descriptions of interatomic interactions. They became particularly popular with the advent of the DFT based methods since γ-surface calculations are much less computation intensive than studies of dislocations while providing information that is often sufficient for understanding the atomic level properties of dislocations. Examples are recent calculation of γ-surfaces in bcc transition metals [12], aluminum [13], silicon [13, 14], graphite [15], TiAl [16, 17] and MoSi2 [18, 19]. Symmetry arguments can be utilized to assess the general shape of γ-surfaces. If a mirror plane of the perfect lattice perpendicular to the plane of a generalized stacking-fault passes through the point corresponding to a , the first derivative of the γ-surface along the normal to this displacement u mirror plane vanishes owing to the mirror symmetry. This implies that the

Upper part of the crystal

u Generalized stacking fault

Lower part of the crystal Figure 1. Formation of the generalized stacking fault.

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γ-surface will possess extrema (minima, maxima or inflexions) for those displacements, for which there are at least two non-parallel mirror planes of the perfect lattice perpendicular to the fault. Whether any of the extrema correspond to minima, and thus to metastable faults, can often be determined by considering the change in the nearest neighbor configuration produced by the corresponding displacement. Hence, the symmetry-dictated metastable stacking-fault-like defects can be ascertained on a crystal plane by analyzing its symmetry. Such faults are then common to all materials with a given crystal structure. The intrinsic stacking faults in fcc crystals are symmetry dictated since three mirror planes of the {101} type pass through the points that correspond to the displacements 1/6112. However, other minima than those associated with symmetry-dictated extrema may exist in any particular material. These can not be anticipated on crystallographic grounds but their existence depends on the details of atomic interactions and they can only be revealed by calculations of the γ-surface. The primary significance of the dislocation dissociation is that it determines uniquely the slip planes, identified with the planes of splitting and corresponding stacking-fault-like defects and, consequently, the operative slip systems. The fact that {111} planes are the slip planes in fcc materials is a typical example. However, the core of individual partial dislocations may still spread spatially and introduce effects similar to those found in undissociated dislocations with non-planar cores. In summary, when analyzing dislocation core structure, possible splitting into well-defined partials separated by metastable stacking fault-like defects has to be considered first. If such splitting cannot occur, either because no metastable stacking fault-like defects exist or their energy is so high that the splitting is not favored, the core of total dislocations has to be studied. In the opposite case, investigation of the cores of the corresponding partials needs to be performed.

3.

Dislocation Cores

In general, the dislocation core structure can be described in terms of the relative displacements of atoms in the core region. These displacements are usually not distributed isotropically but are confined to certain crystallographic planes. Two distinct types of dislocation cores have been found, depending on the mode of the distribution of significant atomic displacements. When the core displacements are confined to a single crystallographic plane the core is planar∗ . In metallic materials dislocations with such cores usually glide

∗ A more complex planar core, called zonal, may be spread into several adjacent parallel crystallographic

planes of the same type [20].


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easily in the planes of the core spreading and their Peierls stress is commonly low. However, the Peierls stress can be high even for planar cores in covalently bonded solids owing to the need to break the covalent bonds during the dislocation glide. In contrast, if the core displacements spread into several nonparallel planes of the zone of the dislocation line, the core is non-planar. The glide planes of dislocations with such cores are often not defined uniquely, the corresponding Peierls stress is high and well below the melting temperature the dislocation glide is enabled by thermal activation over the Peierls barriers.

3.1.

Planar Cores

Only full-scale atomistic calculations can reveal all the details of the core structure for both planar and non-planar cores. However, for planar cores semiatomistic models of the Peierls-Nabarro type are capable to describe the cores with high precision [3]. In such models the core is regarded as a continuous distribution of dislocations in the plane of the core spreading. If we choose the coordinate system in the plane of the core spreading such that the axes x1 and x2 are parallel and perpendicular to the dislocation line, respectively, the corresponding density of the continuously distributed dislocations has two components ρα = ∂ u α /.∂ x2 (α = 1, 2), where u α is the α component of the dis in the x1 , x2 plane. The displacement u increases graduplacement vector u +∞ ally in the direction x2 from zero to the Burgers vector so that −∞ ρα dx2 = bα , where bα is the corresponding component of the Burgers vector. In the continuum approximation the elastic energy of such dislocation distribution can be expressed as the interaction energy of the dislocations within this distribution: E el =

+∞ +∞ 2

K αβ ρα ρβ ln x2 − x2 dx2 dx2

(1)

α,β=1−∞ −∞

where K αβ are constants depending on the elastic moduli and orientation of the dislocation line. On the atomic scale the displacement across the plane of the core spreading causes a disregistry that leads to an energy increase. The produces locally a generalized stacking-fault and in the local displacement u approximation the energy associated with the disregistry can be approximated as +∞

γ ( u) dx2

Eγ =

(2)

−∞

where γ ( u) is the energy of the corresponding γ -surface for the displacement . The continuous distribution of dislocations describing the core structure is u

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then found by the functional minimization of the total energy E tot = E el + E γ [21]∗ . with respect to the displacement u The salient feature of this model is that the energy E tot does not depend on the position of the core and thus when the dislocation moves no energy change would be found. Nevertheless, the Peierls stress can be evaluated by re-introducing the discrete structure of the lattice by placing the atoms of this obtained from lattice into the positions determined by the displacement field u the model. The Peierls stress and the Peierls barrier can then be evaluated by gradually moving this displacement distribution along the slip plane by one period along the glide plane. This is the approach adopted originally by Nabarro [22]. The result is a rather low Peierls stress in the case of planar cores. This is seen from the analytical expression obtained for the original model with sinusoidal restoring force when the Peierls stress is τ P = 2µ/α exp(−4π ζ /b); µ is the shear modulus and α a factor of the order of one and ζ is the width of the core. This stress is very small even when the core is very narrow; for example, if ζ = b, τ P ≈ 10−5 µ. Comparisons with full-scale atomistic calculations have shown a good agreement with this approximate approach in a number of cases, in particular when the cores are wide. A variety of improvements have been suggested recently in which the atomic structure is taken into account explicitly in the model rather than a posteriori [13, 23, 24]. However, the model generally performs poorly for cores the width of which is comparable with the lattice spacing and when covalent bonds dominate crystal bonding. In the former case the continuum description of the core is poor since it is applied to distances smaller than the lattice spacing and in the latter case possible breaking and/or formation of covalent bonds is not included into the model.

3.2.

Non-planar Cores

The non-planar cores can be divided into two classes: cross slip and climb cores. In the former case the core displacements lie in the planes of the core spreading while in the latter case they possess components perpendicular to these planes. Climb cores are less common and are usually formed at high temperatures by a climb process. The best known example of the cross-slip core is the core of 1/2111 screw dislocations in bcc metals. Atomistic studies of dislocations are the principal source of our understanding of the structure of such cores since direct observations are mostly outside the limits of experimental techniques. Two alternate structures of the core of the 1/2[111] ∗ If the displacement vector is all the time parallel to the Burgers vector, so that only the component u in ∂γ / 0 for any finite value of u, the Euler equation corresponding this direction needs to be considered, and ∂u = to the condition δ E tot = 0 leads to the well-known Peierls equation [21].


(a)

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(b)

Figure 2. Two alternate core structures of the 1/2[111] screw dislocation depicted using differential displacement maps. (a) Calculated using Finnis–Sinclair type central force potential for Mo [25], (b) Calculated using a bond-order potential for Mo [27]. The atomic arrangement is shown in the projection perpendicular to the direction of the dislocation line ([111]) and circles represent atoms within one period. An arrow between them represents the [111] (screw) component of the relative displacement of the neighboring atoms produced by the dislocation. The length of the arrows is proportional to the magnitude of these components. The arrows, which indicate out-of-plane displacements, are always drawn along the line connecting neighboring atoms and their length is normalized such that it is equal to the separation of these atoms in the projection when the magnitude of their relative displacement is equal to |1/6 [111]|.

screw dislocation found by atomistic modeling are presented in Fig. 2. The core ¯ ¯ and (110) ¯ in Fig. 2a is spread asymmetrically into the (101), (011) planes that ¯ diad; belong to the [111] zone and is not invariant with respect to the [101] another energetically equivalent configuration related by this symmetry operation exists and this core is called degenerate or polarized. The core in Fig. 2b ¯ diad and it is called non-degenerate or is invariant with respect to the [101] non-polarized. The core structures shown in Figs. 2a and b were found by atomistic calculations employing central-force many-body potentials [25] and a tight binding and/or density functional theory based approaches [26, 27], respectively. This example demonstrates that the structure of the core is not determined solely by the crystal structure but may vary from material to material with the same crystal structure. The non-planar dislocation cores are the more common the more complex is the crystal structure and for this reason these cores are more prevalent than planar cores. In this respect, fcc materials (and also hcp materials with basal slip) in which the dislocations possess planar cores, are a special case rather than a prototype for more complex structures [5]∗ . ∗ Additional complexities of the dislocation core structures arise in covalent crystals where the breaking and/or readjustment of the bonds in the core region may be responsible for a high lattice friction stress, and in ionic solids where the cores can be charged which then strongly affects the dislocation mobility. Such dislocation cores affect not only the plastic behavior but also electronic and/or optical properties of covalently bonded semiconductors and ionically bonded ceramic materials.

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3.3.

V. Vitek

Glide of Dislocations with Non-planar Cores and Breakdown of the Schmid Law

The Peierls stress of dislocations with non-planar cores is typically at least an order of magnitude higher than that of dislocations with planar cores. Furthermore, the movement of such dislocations is frequently affected not just by the shear stress parallel to the Burgers vector in the slip plane but by other stress components. Thus the deformation behavior of materials with non-planar dislocation cores may be very complex, often displaying unusual orientation dependencies and breakdown of the Schmid law. It has been known ever since the first studies of the plastic deformation of bcc metals that the Schmid law does not apply [7, 28] and, indeed, already the early atomistic calculations showed that the non-planar core has to transform prior to the dislocation motion and this transformation may be influenced by shear stresses in the slip direction acting in planes other than the slip plane [29]. Furthermore, more recent calculations revealed that such transformation may also be affected by shear stresses in the direction perpendicular to the Burgers vector [25, 30, 31]. This is well demonstrated by calculating the Peierls stress as a function of the shear stresses perpendicular to the Burgers vector while keeping the plane of the maximum resolved shear stress parallel to the Burgers vector (MRSSP) fixed. As suggested by Ito and Vitek [25], this can be achieved by applying the stress tensor with the components σ11 = −τ2 , σ22 = τ2 , σ33 = σ12 = σ13 = σ23 = 0 in the right-handed coordinate system with the x1 axis in the MRSSP, x2 axis perpendicular to the MRSSP and x3 axis parallel to [111], together with the shear stress parallel to the Burgers vector. Figure 3 shows the calculated dependence of the Peierls ¯ stress, τM , for the core shown in Fig. 2b on τ2 when the MRSSP is the (101) plane. It is important to note that changing the sign of τ2 corresponds to the rotation of the coordinates by 180◦ around the [111] axis. However, the core structure is not invariant with respect to this transformation and thus the effect of τ2 upon the dislocation behavior will, in general, be different for positive and negative values, as observed. The values of τM corresponding to tensile and compressive loadings along ¯ [238] and [012] axes are also shown in Fig. 3; for each uniaxial loading the value of τ2 is uniquely related to the loading stress at which the disloca¯ tion started to move. For these axes the (101) plane is the MRSSP and if there were no effect of shear stresses perpendicular to the Burgers vector the value of τM would be the same for both axes as well as for tension and compression. Obviously, this is not the case and the magnitude of τM depends on the shear stresses perpendicular to the slip direction, described by τ2 , that are different for different uniaxial loading. Moreover, for large negative τ2 the ¯ ¯ plane although the shear slip plane changes from the (101) plane to (011)


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τM C44

τ2/C44 Figure 3. Dependence of the Peierls stress, τM , on τ2 that defines the shear stress perpendic¯ ular to the Burgers vector, when the MRSSP is the (101) plane.

stress in the direction of the Burgers vector is lower in this plane. Apparently, the shear stress perpendicular to the Burgers vector alters the core such that ¯ plane is preferred. transformation into the (011) It should be emphasized at this point that the phenomena discussed above have been observed for both polarized and non-polarized cores [25]; similarly the polarity has not been found to influence significantly the magnitude of the Peierls stress. This is in contrast with the recent suggestions that the polarity plays a considerable role in the glide of dislocations with non-planar cores [32]. The reason is that when a stress is applied the symmetry of the nonpolarized core is broken and becomes the same as that of the polarized one. This has recently been discussed in detail by Vitek [33]. The results of atomistic calculations suggest that the motion of the 1/2[111] screw dislocation is governed by four distinct shear stress components when ¯ it glides in the (101) plane. First two are the Schmid stress, i.e., the stress in ¯ the direction of the Burgers vector in the slip plane (101), and the shear stress parallel to the Burgers vector in another {110} plane of the [111] zone. The other two are shear stresses perpendicular to the Burgers vector acting in two different {110} planes of the [111] zone. As discussed in detail in [34, 35], the effects of non-glide stresses may enter the flow rules for a single crystal

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loaded by a stress tensor σ by introducing for a slip system α an effective yield stress, τ ∗α = τ α +

3 η=1

aηα τηα

(3)

Here τ α = nα ·σ·mα is the Schmid stress; nα is the slip plane normal and mα the slip direction. τηα = nαη ·σ·mαη are the non-glide shear stresses; nαη is the normal to the {110} plane η, and mαη is the direction in this plane, either parallel or perpendicular to the Burgers vector, depending on which type of the shear stress is considered. At the onset of the plastic flow this effective yield stress has to attain a critical value τcr∗α . The coefficients aηα and the value of τcr∗α are determined so as to fit the results of atomistic calculations of the dislocation motion. This criterion reduces to the Schmid law if there is no influence of non-glide stresses and all aηα = 0. All the above results of atomistic studies relate to the glide of straight dislocations at 0 K but at finite temperatures the dislocation motion is aided by thermal activation and a generally accepted mechanism of the dislocation motion involves formation and extension of pairs of kinks. Assuming the usual rate theory, the dislocation velocity is then

U − W v ∝ exp − kB T

(4)

where kB is the Boltzman constant, T temperature, U = Uactivated − Uground the difference in the energy between the activated state and the state prior to the activation and W = τ α · b · A is the work done by the shear stress parallel to the Burgers vector (Schmid stress) during the activation process; b is the magnitude of the Burgers vector and A the area swept by the dislocation segment during activation. This mechanism of the thermally activated overcoming of the Peierls barriers has been employed in the framework of the dislocation theory by a number of authors, starting with the classical paper of Dorn and Rajnak [36] [37–39]. In these developments U had either been taken as constant or considered as a function of the Schmid stress τ α for a given slip system α. The same applies in the recent atomistic studies of kinks [40–43]. However, to include fully the effects of non-glide stresses at finite temperatures it has to be recognized that both Uactivated and Uground are functions of the applied stress tensor, so that U = f (σi j ), where σi j includes both glide and non-glide stress components affecting the dislocation motion. In principle, this could be achieved by molecular dynamics simulations of the formation of kink pairs at finite temperatures with applied stresses involving both glide and non-glide components. However, such calculations are feasible only for very


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high stresses and thus very large strain rates even when using the most powerful supercomputers [44]. The reason is that at usual strain rates the frequency of the formation of kink pairs is many orders of magnitude lower than the frequency of atomic vibration. Hence, the yet unsolved task is to develop a mesoscopic theory of the formation of kink pairs that will utilize the results of atomistic studies of the motion of straight dislocations to establish the dependence of U on all components of the stress tensor that govern the dislocation glide. A possible approach is to consider that U is a function of the effective yield stress τ ∗α such that U = 0 when τcr∗α = τcr∗α .

4.

Conclusions: Generality of the Core Effects

The non-planar dislocation cores, the example of which is the 1/2111 screw dislocation in bcc metals, are common in many materials. In hexagonal crystals such cores are not found when the slip is confined to basal planes but they are common in the case of prismatic and/or pyramidal slip [45] where they are responsible for strong temperature dependence of the yield stress. Similarly, non-planar dislocation cores have been identified in many intermetallic compounds. Examples are screw dislocations in NiAl [46], TiAl [47, 48] and MoSi2 [19, 49] where they are, presumably, responsible for an unusually large orientation dependence of the yield stress [50]. The transformation of 110 superdislocations in Ni3 Al from the planar glissile form to non-planar sessile form is responsible for the anomalous increase of the flow stress with temperature [10, 51]. However, non-planar cores leading to unusual temperature, strain rate and orientation dependencies have not been found only in metallic materials but also in materials such as olivine [52], sapphire [53] and anthracene [54]. Recently, an unusual ductile-to-brittle transition has been observed in the perovskite SrTiO3 that deforms plastically at room temperature but becomes brittle at about 1000 K [55]. This unusual behavior is most likely associated with formation of non-planar dislocation cores at high temperatures associated with local changes in stoichiometry [56]. In covalently bonded semiconductors it is both the atomic and electronic effects which affect significantly the Peierls barrier [57] that may lead to very complex mechanisms of formation of kinks [58]. In all these cases atomic level modeling is capable to identify the essential features of dislocation cores and identify the stress components governing the dislocation motion. The development of mesoscopic models of the thermally activated dislocation motion that incorporate the results of atomic level calculations is then the next essential step. However, at present such studies are either very rudimental or non-existent and this is, therefore, an open avenue for further research linking atomic level, nano-scale and macroscale deformation properties.

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References [1] [2] [3] [4] [5] [6]

[7] [8] [9] [10]

[11] [12] [13]

[14]

[15] [16] [17]

[18] [19]

[20] [21] [22] [23]

J. Friedel, Dislocations, Pergamon Press: Oxford, 1964. F.R.N. Nabarro, Theory of Crystal Dislocations, Clarendon Press: Oxford, 1967. J.P. Hirth and J. Lothe, Theory of Dislocations, Wiley-Interscience: New York, 1982. D. Hull, and D.J. Bacon, Introduction to Dislocations, Butterworth–Heinemannn, Oxford, 2001. A. Cottrell, “Closing Remarks,” Dislocations and Properties of Real Materials, M. Lorretto (ed.), London: The Institute of Metals, 378–381, 1985. V. Vitek, “Effect of dislocation core structure on the plastic properties of metallic materials,” M. Lorretto (ed.), Dislocations and Properties of Real Materials, London: The Institute of Metals, 30–50, 1985. M.S. Duesbery, “The dislocation core and plasticity,” In: F.R.N. Nabarro (ed.), Dislocations in Solids, Amsterdam: North Holland, vol. 8, p. 67, 1989. M.S. Duesbery and G.Y. Richardson, “The dislocation core in crystalline materials,” CRC Critical Reviews in Solid State and Materials Science, 17, 1, 1991. V. Vitek, “Structure of dislocation cores in metallic materials and its impact on their plastic behaviour,” Prog. Mater. Sci., 36, 1–27, 1992. V. Vitek, D.P. Pope, and J.S. Bassani, “Anomalous yield behaviour of compounds with L1(2) structure,” In: F.R.N. Nabarro (ed.), Dislocations in Solids, Amsterdam: North Holland, vol. 10, 135–185, 1995. V. Vitek, “Intrinsic stacking faults in bcc crystals,” Philos. Mag. A, 18, 773, 1968. S.L. Frederiksen and K.W. Jacobsen, “Density functional theory studies of screw dislocation core structures in bcc metals,” Philos. Mag., 83, 365–375, 2003. G. Lu, N. Kioussis, V.V. Bulatov, and E. Kaxiras, “Generalized-stacking-fault energy surface and dislocation properties of aluminum,” Phys. Rev. B, 62, 3099–3108, 2000a; “The Peierls-Nabarro model revisited,” Philos. Mag. Lett., 80, 675–682, 2000b. Y.M. Juan and E. Kaxiras, “Generalized stacking fault energy surfaces and dislocation properties of silicon: a first-principles theoretical study,” Philos. Mag. A, 74, 1367–1384, 1996. R.H. Telling and M.I. Heggie, “Stacking fault and dislocation glide on the basal plane of graphite,” Philos. Mag. Lett., 83, 411–421, 2003. J. Ehmann and M. Fähnle, “Generalized stacking-fault energies for TiAl: mechanical instability of the (111) antiphase boundary,” Philos. Mag. A, 77, 701–714, 1998. S. Znam, D. Nguyen-Manh, D.G. Pettifor et al., “Atomistic modelling of TiAl. I. Bond-order potentials with environmental dependence,” Philos. Mag., 83, 415–438, 2003. U.V. Waghmare, V. Bulatov, E. Kaxiras et al., “331 slip on {013} planes in molybdenum disilicide,” Philos. Mag. A, 79, 655–663, 1999. T.E. Mitchell, M.I. Baskes, R.G. Hoagland, and A. Misra, “Dislocation core structures and yield stress anomalies in molybdenum disilicide,” Intermetallics, 9, 849–856, 2001. M.L. Kronberg, “Zonal dislocations in hcp crystals,” Acta Metall., 9, 970, 1961. J.W. Christian and V. Vitek, “Dislocations and stacking faults,” Rep. Prog. Phys., 33, 307, 1970. F.R.N. Nabarro, “Dislocations in simple cubic lattice,” Proc. Phys. Soc. London B, 59, 256, 1947. K. Ohsawa, H. Koizumi, H.O.K. Kirchner, and T. Suzuki, “The critical stress in a discrete Peierls-Nabarro model,” Philos. Mag. A, 69, 171–181, 1994.


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[24] G. Schoeck, “The Peierls energy revisited,” Philos. Mag. A, 79, 2629–2636, 1999. [25] K. Ito and V. Vitek, “Atomistic study of non-Schmid effects in the plastic yielding of bcc metals,” Philos. Mag. A, 81, 1387–1407, 2001. [26] C. Woodward and S.I. Rao, “Ab initio simulation of isolated screw dislocations in bcc Mo and Ta,” Philos. Mag. A, 81, 1305–1316, 2001. [27] M. Mrovec, D. Nguyen-Manh, D.G. Pettifor, and V. Vitek, “Bond-order potential for molybdenum: application to dislocation behavior,” Phys. Rev. B, 69, 094115, 2004. [28] J.W. Christian, “Some surprising features of the plastic deformation of body-centered cubic metals and alloys,” Metall. Trans. A, 14, 1237, 1983. [29] M.S. Duesbery, V. Vitek, and D.K. Bowen, “Screw dislocations in bcc metals under stress,” Proc. Roy. Soc. London A, 332, 85, 1973. [30] M.S. Duesbery, “On non-glide stresses and their influence on the screw dislocation core in bcc metals I: the Peierls stress,” Proc. Roy. Soc. London A, 392, 145–173, 1984. [31] M.S. Duesbery and V. Vitek, “Plastic anisotropy in bcc transition metals,” Acta Mater., 46, 1481–1492, 1998. [32] G.F. Wang, A. Strachan, T. Cagin et al., “Role of core polarization curvature of screw dislocations in determining the Peierls stress in bcc Ta: a criterion for designing highperformance materials,” Phys. Rev. B, 6714, 101, 2003. [33] V. Vitek, “Core structure of dislocations in body-centred cubic metals: relation to symmetry and interatomic bonding,” Philos. Mag., 84, 415–428, 2004. [34] J.L. Bassani, K. Ito, and V. Vitek, “Complex macroscopic plastic flow arising from non-planar dislocation core structures,” Mat. Sci. Eng. A, 319, 97–101, 2001. [35] V. Vitek, M. Mrovec, and J.L. Bassani, “Influence of non-glide stresses on plastic flow: from atomistic to continuum modeling,” Mater. Sci. Eng. A, 365, 31–37, 2004. [36] J.E. Dorn and S. Rajnak, “Nucleation of kink pairs and the Peierls mechanism of plastic deformation,” Trans. TMS-AIME, 230, 1052, 1964. [37] M.S. Duesbery, “Dislocation motion in Nb,” Philos. Mag., 19, 501, 1969. [38] C.H. Woo and M.P. Puls, “The Peierls mechanism in MgO,” Philos. Mag., 35, 1641– 1652, 1977. [39] A. Seeger, “Peierls barriers, kinks, and flow stress: recent progress,” Z Metallk, 93, 760–777, 2002. [40] A.H.W. Ngan and M. Wen, “Dislocation kink-pair energetics and pencil glide in body-centered-cubic crystals,” Phys. Rev. Lett., 8707, 5505, 2001. [41] S.I. Rao and C. Woodward, “Atomistic simulations of (a/2)111 screw dislocations in bcc Mo using a modified generalized pseudopotential theory potential,” Philos. Mag. A, 81, 1317–1327, 2001. [42] L.H. Yang, and J.A. Moriarty, “Kink-pair mechanisms for a/2 111 screw dislocation motion in bcc tantalum,” Mater, Sci, Eng, A Struct Mater., 319, 124–129, 2001. [43] L.H. Yang, P. Soderlind, and J.A. Moriarty, “Accurate atomistic simulation of (a/2) 111 screw dislocations and other defects in bcc tantalum,” Philos. Mag. A, 81, 1355–1385, 2001. [44] J. Marian, W. Cai, and V.V. Bulatov, “Dynamic transitions from smooth to rough to twinning in dislocation motion,” Nature Materials, 3, 158–163, 2004. [45] D.J. Bacon, and V. Vitek, “Atomic-scale modeling of dislocations and related properties in the hexagonal-close-packed metals,” Metal. Mater. Trans. A, 33, 721–733, 2002. [46] R. Schroll, V. Vitek, and P. Gumbsch, “Core properties and motion of dislocations in NiAl,” Acta Mater., 46, 903–918, 1998.

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[47] R. Porizek, S. Znam, D. Nguyen-Manh, V. Vitek, and D.G. Pettifor, “Atomistic studies of dislocation glide in y-TiAl,” Defect Properties and Related Phenomena in Intermetallic Alloys, E.P. George, H. Inui, M.J. Mills, and G. Eggeler (eds.), Pittsburgh: Materials Research Society, vol. 753, p. BB4.3.1–BB4.3.6, 2003. [48] C. Woodward and S.I. Rao, “Ab initio simulation of a/2110 screw dislocations in γ -TiAl,” Philos. Mag., 84, 401–414, 2003. [49] M.I. Baskes and R.G. Hoagland, “Dislocation core structures and mobilities in MoSi2 ,” Acta Mater., 49, 2357–2364, 2001. [50] K. Ito, H. Inui, Y. Shirai, and M. Yamaguchi, “Plastic deformation of MoSi2 single crystals,” Philos. Mag. A, 72, 1075–1097, 1995. [51] T. Kruml, E. Conforto, B. LoPiccolo, D. Caillard, and J.L. Martin, “From dislocation cores to strength and work-hardening: A study of binary Ni3 Al,” Acta Mater., 50, 5091–5101, 2002. [52] J.-P. Poirier, and B. Vergobbi, “Unusual deformation properties of olivire,” Physics of the Earth and Planetary Interiors, 16, 370–382, 1978. [53] J. Chang, C.T. Bodur, and A.S. Argon, “Pyramidal edge dislocation cores in sapphire,” Philos. Mag. Lett., 83, 659–666, 2003. [54] N. Ide, I. Okada, and K. Kojima, “Computer simulation of core structure and Peierls stress of dislocations in anthracene crystals,” J. Phys.: Condens. Matter, 5, 3151–3162, 1993. [55] P. Gumbsch, S. TaeriBaghbadrani, D. Brunner et al., “Plasticity and an inverse brittle-to-ductile transition in strontium titanate,” Phys. Rev. Lett., 8708, 5505+, 2001. [56] Z.L. Zhang, W. Sigle, W. Kurtz et al., “Electronic and atomic structure of a dissociated dislocation in SrTiO3 ,” Phys. Rev. B, 6621, 4112–4118, 2002a; “Atomic and electronic characterization of the a[100] dislocation core in SrTiO3 ,” Phys. Rev. B, 6609, 4108–4115, 2002b. [57] J.F. Justo, A. Antonelli, and A. Fazzio, “The energetics of dislocation cores in semiconductors and their role in dislocation mobility,” Physica B, 302, 398–402, 2001. [58] V.V. Bulatov, J.F. Justo, W. Cai et al., “Parameter-free modelling of dislocation motion: the case of silicon,” Philos. Mag. A, 81, 1257–1281, 2001.

Perspective 33 3-D MESOSCALE PLASTICITY AND ITS CONNECTIONS TO OTHER SCALES Ladislas P. Kubin LEM, CNRS-ONERA, 29 Av. de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France

1.

Multiscale Analysis

There is a dislocation theory but there is no dislocation theory of plasticity. The reasons for this situation are multiple. By definition, a dislocation is a linear defect that ensures compatibility between slipped and unslipped parts of a crystal. The motion of this defect propagates microscopic shears and is responsible for the plastic (i.e., permanent) deformation of crystalline materials. A dislocation line can be viewed in two different manners. From an atomistic viewpoint, it consists of a highly distorted region, the core, which surrounds the geometrical line of the defect and has a diameter of a few lattice spacings. In the continuum, a dislocation consists of a singularity line to which are associated long range stress and strain fields. The line energy of a dislocation is mainly located outside the core region and can conveniently be calculated by elasticity theory. This allows treating all the elementary dislocation properties that derive from their self and interaction energies to any desired degree of accuracy. However, the properties of dislocation cores also govern important properties, like dislocation mobility or the selection of the dislocation slip planes. Although our current knowledge of core properties has significantly improved in the past years, it is still far from being just satisfactory. Thus, dislocation theory has not yet been able to bridge the gap between atomic scale properties and mesoscale properties (the mesoscale is understood here as the scale of the defect microstructure). At the mesoscale, the treatment of dislocation ensembles and the spontaneous emergence of dislocation patterns poses several types of problems. It is now understood that plasticity is a dissipative process far from thermal equilibrium. In such conditions, which are beyond the reach of thermodynamics, dislocation patterning is now modeled as a purely dynamic phenomenon. 2897 S. Yip (ed.), Handbook of Materials Modeling, 2897–2901. c 2005 Springer. Printed in the Netherlands.

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Nevertheless, there is no common agreement on the mechanisms that trigger the formation of self-organized dislocation microstructures, or on the way the latter influence the mechanical response. Finally, in many important situations, of fundamental or applied relevance, a connection has to be established with continuum formulations. Indeed, only the latter can compute, through a proper treatment of boundary conditions, the complex field of a dislocated crystal in a specimen with finite dimensions. The objective of what is called multiscale analysis is to reach, through a combination of simulation and modeling, a physically-based picture of the plasticity of crystalline solids. Within this goal, 3-D dislocation dynamics (DD) simulations have been developed in the last decade as a complement to more ancient and well established simulation methods that exist in the domains of solid state physics or solid mechanics.

2.

Connection to the Continuum

At the mesoscale, dislocation core properties can only be incorporated by imposing “local rules”, which translate atomic scale properties into a continuum formulation. Indeed, in the absence of these rules, DD simulations make little sense, especially if they cannot treat thermally activated processes. Since atomistic simulations provide everything but analytical models, one has to make use of intermediate models or approaches to implement this connection. The most important core mechanisms are related to changes in core structure and energy under stress. They involve small energy changes, in the eV range, and are, therefore, assisted by thermal fluctuations. Thus, they can be described with the help of rate equations, for instance semi-empirical Arrhenius forms, forms derived from elastic models for the core structure or Monte– Carlo simulations. This information passing procedure between the atomic and mesoscopic scales presents a major advantage, that of decoupling the time and length scales between the two types of simulations. As thermally activated events are slow events, with a time scale of the order of a fraction of a second in conditions of conventional deformation testing, they cannot be fully treated by molecular dynamics (MD) simulations. The search for saddle point configurations is then performed with the help of various types of energy minimization techniques. In this domain, progress has been slow but continuous, taking advantage of the now possible comparison between ab initio calculations and quantum-based potentials. The most important results recorded in the past years are concerned with the lattice friction (also called Peierls stress), especially in bcc metals. In silicon, the absence of consistency between experiment, continuum modeling and atomistic modeling is still a bit confusing. Studies of the cross-slip mechanism in fcc crystals have confirmed the physical picture yielded by early

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elastic models, but have stopped short of providing stress-dependent activation energies and scaling laws for fcc metals. The question of solute hardening, notably in the presence of lattice friction, is still an open domain. Studies on fast moving dislocations by MD simulations have a potential in relation to dynamic deformation tests (i.e., shock loading). A critical area of investigation is concerned with the study of athermal or weakly thermally assisted events, of which the most important one is dislocation generation in dislocation-free volumes, through either homogeneous or heterogeneous processes (occurring at surfaces, interfaces, voids, crack tips. . .). At present, the lack of local rules in this domain constitutes a serious handicap to mesoscale simulations. The investigation of fundamental dislocation mechanisms at atomic scale and the completion of dislocation theory is, indeed, a challenging task. Although this exercise may seem less seducing than the production of colorful but complex events, it is critical for the progress of multiscale analyses.

3.

DD Simulations: Limitations and Prespective

As exemplified by the variety of solutions adopted by the groups working in the field, implementing the elastic properties of dislocations in DD simulations is no longer a challenge. The microstructure may include defects other than dislocations, e.g., small clusters, precipitates, or grain boundaries. This entails the definition of additional local rules of topological nature, for instance telling dislocations in which conditions they can penetrate precipitates or grain boundaries. However, the elastic theory of dislocations is not providing tractable solutions for the treatment of complex stress fields originating from the conditions of compatible deformation at interfaces. As a result, DD simulations are essentially suited to studies on dislocation dynamics and collective behavior under uniform applied stresses, in crystals of infinite dimensions. Several technical factors (time step, number of interacting segments, dimension of the simulated volume, applied strain rate, available computing power) contribute to the maximum strain that can be achieved by DD simulations and, most of all, to the accuracy of the results. In spite of these limitations, the domain accessible to DD simulations is enormous. Globally, these simulations can be divided into “mass” simulations and “model” simulations, dedicated to the study of one or a few elementary mechanisms. Both are complementary and, as always, cannot be carried out without reference to experiment and current theoretical modeling. A nonexhaustive list of the main topics under investigation or that can be treated in the near future is as follows. – Single crystal studies, particularly in the absence of lattice friction. Such investigations can be very useful to clear up a number of controversies

2900

–

–

–

–

–

4.

L.P. Kubin accumulated over the years on dislocation pattern formation and strain hardening in monotonic and cyclic deformation. In materials with high lattice friction, especially covalent or iono-covalent compounds, similar investigations are feasible provided that enough information is available to construct local rules. Several models, deterministic or stochastic, some of them constructed along the lines of continuum dislocation theory, describe the collective behavior of dislocations in crystals. These models can benefit from a comparison with simulation data. Mechanical properties at medium temperatures (T > 0.3−0.4 Tm , where Tm is the melting temperature) have not been studied up to now by DD simulations. Any attempt to go beyond the current phenomenology of creep mechanisms at intermediate temperatures would be highly welcome. Another topic of practical interest, which has so far been treated only in 2-D, is the strength of alloys containing coherent or semi-coherent particles in the presence of cross-slip or climb. A major unknown in plasticity theory is concerned with the composition rules for obstacles of different strengths and of same or different nature. Only semi-empirical estimates exist in this domain, and mesoscale simulations could be of a great help to initiate modeling in a few simple cases. The interaction of dislocations with small clusters, in practice small prismatic loops, is involved in two important traditional domains: one is the instabilities and strain localizations observed during the plastic deformation of irradiated materials. The other is dislocation patterning in cyclic deformation. The investigation of size effects is another very seducing topic, although it has to be approached with care in the absence of a rigorous solution for the local fields. The influence of reduced dimensionality on line tension effects and dislocation mean-free paths is now being investigated. It implies defining new local rules, for instance for the interactions between dislocations and interfaces.

Connection to Continuum Mechanical Aspects

Simulations that simultaneously treat discrete dislocations and solve the boundary value problem have an enormous potential. In fundamental terms, they allow introducing length scales into the continuum framework. They are particularly suited for the study of size effects in nanostructured materials, of plasticity under strain gradients, of materials with complex microstructure, of complex modes of loading, static or dynamic, and fracture processes.

3-D mesoscale plasticity and its connections to other scales

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Two methods are being developed to establish this connection. One is based on the combination of DD simulations, which treat the plastic strains resulting from dislocation motion, with finite element (FE) codes, which solve the elastic fields. The existing solutions only differ in the distribution of tasks between the two codes. A variant consists in applying the boundary conditions via a Green function methods when the latter can be computed. The phase field method, which was initially developed to simultaneously treat stress and concentration fields, is now being applied to dislocation problems. At present, it is perhaps too early to assess the potential of this method. Its major advantage resides in the fact that it incorporates a “chemical” dimension that was up to now missing in mesoscale simulations. Another type of connection is possible, which is much less heavy. It simply consists in constructing physically based constitutive formulations for the evolution of dislocation densities and implementing them in a FE code that cares of the boundary value problem. Although this procedure is not really new, it takes advantage of the recent development of mesoscale simulations, for checking or developing existing models, and of continuum crystal plasticity codes, i.e., codes that include the slip geometry. This methodology is also very attractive for two reasons: it allows making use of a substantial existing body of knowledge on constitutive formulations, and it yields access to large strain behavior. How far it can be developed will depend on its ability to account for the spatial organization of the microstructures.

5.

Conclusion

At present, the development of DD simulations is governed in the one hand by the input information generated at atomic scale and, on the other hand, by the strain limit fixed by the available computing power. Within these bounds, DD simulations are still far from having exhausted their potential. The connection to continuum mechanical aspects constitutes the final step toward a physically-based modeling of plasticity. It will probably take a major importance in the coming years. Indeed, it is essential to keep in mind that this has been from the beginning the main objective of dislocation theory. Mesoscale simulations perhaps still have to confirm that they can go beyond providing textbook illustrations and are able to bring major useful information to materials scientists. This objective will, however, not be reached without a strong synergy with theoretical modeling. Current simulations are now able to reproduce known patterns of behavior, but reproducing does not necessarily means understanding.

Perspective 34 SIMULATING FLUID AND SOLID PARTICLES AND CONTINUA WITH SPH AND SPAM Wm.G. Hoover Department of Applied Science, University of California at Davis/Livermore and Lawrence Livermore National Laboratory, Livermore, California, 94551-7808

1.

SPH and SPAM

In my own research career I have been primarily interested in the statistical mechanics of nonequilibrium systems of particles, stressing new techniques for undertaking and understanding computer simulations [1, 2]. The blind alleys toward which molecular dynamics naturally leads provide a compensating appreciation of continuum mechanics, with its length scale more appropriate to everyday experiences. For me, it was a pleasure to learn that the two approaches, microsopic and macroscopic, can be usefully combined, using ideas due to Lucy and Monaghan. About 25 years ago these men independently introduced a new numerical method for simulating continuum problems. They used particles to represent extended macroscopic material elements. Though their method is not at all restricted to fluids, and certainly not to water, Monaghan has consistently adopted the name “smooth-particle hydrodynamics” for it. In the work I have carried out during the last decade I have used instead the alternative “Smooth particle applied mechanics,” (SPAM) for short, thinking it more apt, particularly for solid-phase applications. A Google search of the internet turns up hundreds of references to sph and SPAM. The interested reader could begin with my website at http:// williamhoover.info or could work backward from Carol’s and my review [3]. SPAM defines the local average value (at any location r) of any particle quantity Fi as a quotient of weighted sums:

F(r) ≡ i

Fi m i w(|r − ri| ) . i m i w(|r − ri| ) 2903

S. Yip (ed.), Handbook of Materials Modeling, 2903–2906. c 2005 Springer. Printed in the Netherlands.

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All particles lying within a distance h of the location r (h is the “range” of w) are to be included in the sums. The “weight function” w(|r| < h) is short-ranged and must be continuously twice differentiable, resembling a foreshortened Gaussian function. Lucy’s choice for w is the simplest polynomial satisfying these conditions: w(x < 1) ∝ (1 − x)3 (1 + 3x) = 1 − 6x 2 + 8x 3 − 3x 4 ;

x ≡ |r|/ h.

The implied multiplicative proportionality constant depends upon the dimensionality of the problem (1, 2, or 3) and is to be chosen such that the spatial integral of w is unity. In applying SPAM tofluids and solids a fixed mass m i is associated with the ith particle so that i m i w(|r − ri |) is the mass density ρ(r) while the density at the location of the ith particle is a sum over nearby particle pairs (and including the term i = j ): ρi ≡

m j w(|ri − r j |).

j

The related identities, F(r)ρ(r) ≡

Fi m i w(|r − ri |) −→

i

∇r [F(r)ρ(r)] ≡

Fi m i ∇r w(|r − ri |),

i

show off the method’s key advantage. First-order or second-order spatial derivatives (of velocity, temperature, stress, . . . ) can all be evaluated from simple sums involving w and w . Only the polynomial w is affected by the gradient operation. A consistent application of these ideas leads to new particle forms for the continuum equations of motion,

rï = υi = −m

(P/ρ )i + (P/ρ ) j · ∇i wi j , 2

2

i

and for the particle representations of the continuum energy equation,

e˙i ≡ −

(m/2)[(P/ρ 2 )i + (P/ρ 2 ) j ] :(υ j − υ˙ i ) ∇i wi j

j

−

m[(Q/ρ )i + (Q/ρ ) j ]·∇i wi j . 2

2

j

Note that the special case P ∝ ρ 2 gives particle trajectories isomorphic to those found with molecular dynamics (with w playing the rôle of a pair potential).

Simulating fluid and solid particles and continua with SPH and SPAM 2905 In a SPAM particle simulation the pressure tensor P and the heat-flux vector Q follow from the underlying continuum constitutive relations (such as Newtonian viscosity and Fourier conductivity). A SPAM simulation proceeds by solving the differential equations for all the particles’ {˙r , v, ˙ e} ˙ subject to assumed initial and boundary conditions. (It must be emphasized that developing properly-formed algorithms for boundary conditions is a high art form.) There is a tremendous literature on particular fluid and solid applications, including the effects of analogs of the artificial viscosity and artificial heat conductivity encountered in the more usual simulations using regular meshes. An interesting and simple problem, with philosophical connections to fundamental statistical mechanics, is Gibbs’ (time-reversible!) expansion problem, in which a fluid expands (irreversibly!) to fill a larger container [4, 5]. See Fig. 1. Smooth particles have an advantage, for multiscale physics, in that their length scale h is arbitrary (it can also be a function of time or direction in space), so that “big” zones can be linked to “smaller” ones, even to zones of molecular dimensions. The particles also promote rezoning in a natural way Because their basic attributes include mass, momentum, and energy, particles

Figure 1. Fourfold free expansion of a 2D gas using smooth-particle applied mechanics. The particles themselves are shown, as well as are contours of density and kinetic energy density, computed using the smooth-particle weight functions. The separation between the white and black regions in the density and kinetic energy plots is the contour characterizing the corresponding mean value. The simulations show rapid equilibration, on a timescale of a few sound traversal times. In the Figure τ is the sound-traversal time for the (spatially-periodic) system.

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can easily be combined or subdivided in such a way as to conserve all three variables. Hybrid atomistic-continuum simulations based on these ideas are another useful application area.

References [1] W.G. Hoover, “Computational statistical mechanics,” Elsevier, New York, 1991. [2] Wm.G. Hoover, “Time reversibility, computer simulation, and chaos,” World Scientific, Singapore, 1999 and 2001. [3] Wm.G. Hoover and C.G. Hoover, “Links between microscopic and macroscopic fluid mechanics,” Mol. Phys., 101, 1559–1573, 2003. [4] Wm.G. Hoover and H.A. Posch, “Entropy increase in confined free expansions via molecular dynamics and smooth particle applied mechanics,” Phys. Rev. E, 59, 1770–1776, 1999. [5] Wm.G. Hoover, H.A. Posch, V.M. Castillo, and C.G. Hoover, “Computer simulation of irrversible expansions via molecular dynamics, smooth particle applied mechanics, eulerian, and Lagrangian continuum mechanics,” J. Stat. Phys., 100, 313–326, 2000.

Perspective 35 MODELING OF COMPLEX POLYMERS AND PROCESSES Tadeusz Pakula Max Planck Institute for Polymer Research, Mainz, Germany and Department of Molecular Physics, Technical University, Lodz, Poland

1.

Can Complex Macromolecular Architectures Lead to New Properties?

Creating new macromolecular architectures with increasing complexity can constitute a challenge for synthetic chemists but can additionally be justified if it would result in new material properties. For example, joining monomeric units into linear polymer chains, results in a dramatic change of properties of products with respect to properties of substrates. Whereas, a monomer in bulk can usually be only liquid-like or solid (e.g., glassy), the polymer can additionally exhibit a rubbery state with properties, which make these materials extraordinary for a large number of applications. This new state is due to the very slow relaxation of polymer chains in comparison with a fast motion of the monomers, especially, when the chains become so long that they can entangle in a bulk melt. The dynamic mechanical characteristics indicate a single relaxation in the monomer system (Fig. 1a) in contrast to the two characteristic relaxations in the polymer (Fig. 1b). The rubbery state of the polymer extends in the time scale between the segmental (monomer) and the chain relaxation times and is controlled by a number of parameters related to the polymer structure. The most important among these parameters is the chain length determining the ratio of the two relaxation rates. In the rubbery state, the material is much softer than in the solid state. If expressed by the real part of the modulus, the typical glassy state elasticity is of the order of 109 Pa and higher, whereas the rubber like elasticity in bulk polymers is of the order of 105 –106 Pa. It has been demonstrated, recently, that some highly branched macromolecular structures can lead to considerably different properties than these obtained 2907 S. Yip (ed.), Handbook of Materials Modeling, 2907–2915. c 2005 Springer. Printed in the Netherlands.

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(a)

109

107

liquid

106

glassy state

105

T = 233K

(c)

104 103 102 108

109

1010 1011 1012 1013 1014

ω [rad/s.]

(b)

109

G', G" [Pa]

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Figure 1. A comparison of mechanical behavior of bulk systems with various molecular complexity: (a) low molecular liquid, (b) linear entangled polymer and (c) molecular brush with PnBA side chains.

by linking monomeric units into linear chains [1, 2]. Examples consider dynamic behavior of multiarm stars in the melt [3], the melts of brush-like macromolecules [2] and hairy micelles dispersed in linear polymer matrices. In all these systems, the more complex structures extend the spectrum of the relaxations by a third well distinguishable process with the longest relaxation time, which is interpreted as related to slow structural rearrangements in the structured system. When the new process is slow enough, it can create a new elastic plateau with the plateau modulus (102 –103 Pa) by orders of magnitude lower than that characteristic for the conventional polymeric rubbery state (Fig. 1c). Appropriate cross-linking of such structured systems based on macromolecules with complex architectures can lead to bulk super soft elastomers for which we expect a broad range of application possibilities [2].

2.

Where are the Main Obstacles? What is Predictable?

In contrast to the situation we have in the case of linear polymers, an understanding, a theoretical description and consequently a predictability of

Modeling of complex polymers and processes

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behavior of the bulk systems with macromolecules having complex architecture are in a stage in which the problems are more or less recognized but because of the increased complexity the methods and solution are missing. Tackling the problems of complexity in macromolecular systems in order to provide new ways advancing our possibilities in predicting controlling and analyzing the complex macromolecular systems is, therefore, required. This concerns both natural and synthetic systems and consideration of the later along the whole pathway from substrates to products i.e., from monomers to bulk materials. The macromolecular chemistry is receiving recently a special attention because of development of new synthetic methods and skills which make possible creation of macromolecules with a complexity and precision comparable with that met in biological systems. This includes macromolecules having complex architectures such as multiarm stars, hyperbranched polymers, dendrimers, comb-like or dendronized polymers, rings, catenanes as well as various variants of these topologies made additionally more complex by various distributions of interacting fragments such as functional groups, incompatible blocks or intramolecular composition distributions of chemically different units (Fig. 2). Synthesis of such complex macromolecules requires well controlled processes for which a detailed understanding and possibility of description or modeling is of crucial importance. Existing theoretical methods of description of synthetic processes in macromolecular systems are not sufficiently developed. The problems result from neglecting that for bond formation a presence homopolymers linear

cyclic

star

comb

microgel

network

block copolymers

Figure 2. Some examples of complex macromolecular architectures differing by topology and intramolecular composition distributions.

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of reacting sites and catalysts within the reaction volume is necessary. This condition can only be reasonably considered in spatial models of reacting systems in which substrates and products coexist in space interacting mutually under reaction progress dependent conditions. Therefore, one of the important challenges for the future should be a development of effective tools for modeling of complex synthetic processes in 3D space under controlled conditions. A lot of systems have been developed in which a specific molecular architecture leads to self-organization of molecules to various supramolecular structures. In spite of a considerable progress in this field still many questions such as, what and how to synthesize in order to achieve desired functions or properties, remain open. Also in natural systems the mechanisms of adopting certain structural states and fulfilling certain functions are for most systems not understood. In all these systems, complexity and related physical effects play an important role and can not be considered within the existing theoretical tools. Experimental analysis of such processes and of products is difficult and in many cases leads to misleading results because of unknown effects of the complex macromolecules on the measured quantities (e.g., size exclusion chromatography). On the other hand, it can be expected that appropriate modeling can provide information concerning kinetics of such processes and properties of the products. They can be characterized, for example, by molecular weights, molecular weight distributions, compositions in the case of copolymers as well as composition heterogeneities and distributions, etc. When the complex molecules are obtained, the questions arise: which properties they have, which functions they can fulfill or what they are good for? The further aim of our efforts should, therefore, concern an analysis of structure and dynamics of the complex macromolecules in order to answer the above questions. The analysis should consider single complex macromolecules, spatially confined aggregates, as well as, bulk systems in which specific spatial arrangements, self-organization and formation of heterophases influence the behavior. The later structures constitute usually hierarchical supramolecular architectures in which the structural and dynamic complexity requires new methodological developments in order to be analyzed. Such structures can extend over many orders of magnitude in the size scale. The broad size range involves a broad variety of related relaxations which contribute to the dynamics extending over an extremely broad time range. The dynamic spectrum of such materials becomes very important for understanding the correlation between parameters of the molecular and supramolecular structures on one hand, and their specific functions or macroscopic properties on the other hand [2]. Analysis of these correlations appears to be extremely difficult because of complexity of the systems. Experimentally, it usually requires application of many techniques for characterization of both the structure


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and the dynamics. It often, however, leaves many questions open because of limitations in resolution, accuracy, sensitivity or selectivity of the experimental methods. The systems are usually to complex to be considered by existing theoretical tools. A hope is in computer modeling which due to rapid software and hardware developments becomes a valuable research tool providing a lot of details concerning structure and dynamics in such complex macromolecular systems [1].

3.

Some Perspectives for a Progress

The other important scientific objective of the future efforts should, therefore, be a methodological improvement in solving problems which appear in complex systems of a large number of molecular units interacting strongly or subjected to strong external constraints. Two aspects are considered as crucial for understanding such systems: hierarchy and cooperativity. We can take advantage of recent developments made in modeling of dynamics in complex molecular systems. A new liquid model – the Dynamic Lattice Liquid model [4] – could be used. The model seems to solve the long standing problem of cooperativity in dense complex molecular or macromolecular systems. It provides a microscopic picture of cooperative molecular rearrangements (cooperative loops) resulting from system continuity under conditions of excluded volume and dense packing of molecules (Fig. 3). The rearrangements are considered as taking place in systems with fluctuating density and with rates dependent on thermal activation barriers which depend on and fluctuate with the local density (intermolecular distances). Dependencies of this kind

single molecule trajectory

collective rearrangement

Figure 3. Illustration of a dynamic simplification of a rearrangement of beads in a dense system with all lattice sites occupied. The rearrangement consists in cooperative displacement of system elements along closed trajectories within which each element replaces one nearest neighbor.

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may include chemical specificity of systems. It has been shown, that the model is able to reproduce the extreme cases of behavior of so called fragile and strong liquids, as well as, various cases filling the gap between these extremes [4]. It reproduces also effects of pressure on the molecular dynamics. The model considering a detailed microscopic behavior become a basis for parallel computer simulation algorithms and in this form can be applied for both liquids and polymers of various complexities [1, 4, 5]. It constitutes new chances for very efficient molecular modeling. Computer modeling is nowadays one of the most important sources of information about details concerning behavior of complex molecular and macromolecular systems at the molecular or atomic scale. Main obstacles in exploring possibilities in this field concern limitations in computational speed and in related limits of spatial resolution of considered models. Possibilities to overcome these problems are both in development of appropriate efficient software, as well as, in development of suitable super fast hardware. Recent modeling techniques allow simulating systems over time intervals not exceeding six orders of magnitude and with spatial resolution still far from reaching two orders of magnitude in the 1D size scale. In most problems concerning material development on nano- and micro-meter size scales, as well as, in biological systems on the size scale of cells, the requirements for the spatial resolution of models are higher than the resolutions accessible recently in modeling of molecular or macromolecular dynamics and organization. Tests of the models based on cooperative rearrangements on the sequentially working computing hardware indicted already a high potential of this type of models for predicting behavior of the complex molecular systems (CMA algorithms) [1]. Based on this experience, we suggest a development and construction of a multipurpose fast logical unit which can serve as a basic element of a parallel computing system realizing the 3D architecture and the DLL dynamics. A building block of such DLL system will be a single chip that integrates multiple processors, each representing single site in the model, and a connectivity and communication logic corresponding to the DLL architecture. This should allow to build a full system by replicating such a chip. A prototype of such a system should allow estimations concerning finite size limits of the DLL machine and its computational efficiency, as well as the technological problems related to energy supply, thermal stabilization and spatial requirements and so on. Schematic illustration of the pathway between the DLL model and the very innovative and technologically pretentious computing system is shown in Fig. 4. Realization of such a system is of a great challenge and when successful would constitute a unique example of a modeling system having the architecture directly corresponding to the structure of modeled system. This would allow using of highest level of parallelism of computation and logic in a similar way as in real systems. To our knowledge such computing systems do not


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3D network of logical units corresponding to structure of modeled system CDLL

Basic Cell of the DLL

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Figure 4. Illustration of a pathway between the DLL model and a DLL massively parallel computing system.

exist yet. It is worth while to mention that the systems will constitute a 3D network with 12 coordinated knots (sites) which probably will make the system suitable for modeling of complex networks such as brain system as well. We postulate future research to be related both to computational modeling of processes in complex macromolecular systems but embedded in, supported by and confronted with experimental investigations of related systems. Many, more and more complex macromolecular systems are synthesized nowadays with arguments that they potentially can show interesting properties. This, however, can only be tested after often very difficult and time consuming synthetic and characterization efforts. Some modeling examples demonstrate, however, another access to this kind of tests providing molecular modeling tools having predictive potential both in the synthetic and in characterization areas [1, 2]. The main properties towards which the predictive recognition postulated here is directed are the mechanical properties dependent on the macromolecular dynamics specific for the complex macromolecular systems. It has been recently discovered (MPG and CMU) that such systems can exhibit super soft elastic states, the presence of which is controlled by the details of the complex macromolecular architecture and related dynamics [2]. An improvement of modeling methods should provide answers to questions which structures

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under which conditions can lead to such states which can become a basis of useful applications. On the other hand, the development discussed should be considered as giving bases for much broader further investigations of other properties, other systems and other processes.

4.

Conclusions

The models based on the cooperative dynamics allow to expect that a real qualitative improvement in the field of macromolecular modeling can be achieved. Main advantages of these methods are: physically reasonable simplifications of the dynamics and structures, applicability to dense systems, flexibility in representation of various molecular topologies and high computational efficiency [1]. The cooperativity understood as concerted action of a number of system elements which results in realization of effects remaining strongly restricted or impossible to achieve in actions of individual elements is responsible for the high efficiency of the related computational methods. The simplicity of the cooperative phenomena in the suggested models lies in recognition of the essential conditions under which in a many body system the cooperative actions can take place [4]. The concepts of the DLL model can directly serve as basis for a parallel special purpose computer system realizing super fast modeling which can open new possibilities to improve spatial resolution of models by decoupling it from limitations imposed by computational speed. The building block of such a system will be a chip that integrates multiple processors each corresponding to the elemental site and the communication logic corresponding to the architecture of the DLL system. We consider it as a perspective which might be very helpful from the point of view of the material science and technology and which may allow in future simulations of systems comparable in sizes and complexity with biological cells, still represented with molecular resolution.

References [1] T. Pakula, “Simulations on the completely occupied lattice,” In: Simulation Methods for Polymers, M.J. Kotelyanskii and D.N. Theodorou, Marcel-Dekker, New York, Ch. 5. pp. 147–176, 2004. [2] T. Pakula, P. Minkin, and K. Matyjaszewski, “Polymers, particles, and surfaces with hairy coatings: Synthesis, structure, dynamics, and resulting properties,” In: K. Matyjaszewski (ed.), Advances in Controlled/Living Radical Polymerization, ACS Symp. Series 854, pp. 366–382, 2003. [3] T. Pakula, D. Vlassopoulos, G. Fytas, and J. Roovers, “Structure and dynamics of melts of multiarm polymer stars,” Macromolecules, 31, 8931–8940, 1998.


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[4] T. Pakula, “Collective dynamics in simple and supercooled polymer liquids,” Molecular Liquids, 86, 109–121, 2000. [5] P. Polanowski and T. Pakula, “Studies of mobility, interdiffusion, and self-diffusion in two-component mixtures using the dynamic lattice liquid model,” J. Chem. Phys., 118, 11139–11146, 2003.

Perspective 36 LIQUID AND GLASSY WATER: TWO MATERIALS OF INTERDISCIPLINARY INTEREST H. Eugene Stanley Center for Polymer Studies and Department of Physics Boston, University, Boston, MA 02215, USA

1.

Puzzling Behavior of Liquid Water

We can superheat water above its boiling temperature and supercool it below its freezing temperature, down to approximately −40 ◦ C, below which water inevitably crystallizes. In this deeply supercooled region, strange things happen: response functions and transport functions appear as if they might diverge to infinity at a temperature of about −45 ◦ C. These experiments were pioneered by Angell and co-workers over the past 30 years [1–4]. Down in the glassy region of water, additional strange things happen, e.g., there is not just one glassy phase [1]. Rather, just as there is more than one polymorph of crystalline water, so also there appears to be more than one polyamorph of glassy water. The first clear indication of this was a discovery of Mishima in 1985: at low pressure there is one form, called low-density amorphous (LDA) ice [5], while at high pressure Mishima discovered a new form, called highdensity amorphous (HDA) ice [6]. The volume discontinuity separating these two phases is comparable to the volume discontinuity separating low-density and high-density polymorphs of crystalline ice, 25–35 percent [7, 8]. In 1992, Poole and co-workers hypothesized that the first-order transition line separating two glassy states of water does not terminate when it reaches the no-man’s land (the region of the phase diagram where only crystalline ice is found experimentally), but extends into it [9]. If experiments could avoid the no-man’s land connecting the supercooled liquid with the glass, then the LDA–HDA first order transition line would continue into the liquid phase. This first-order liquid-liquid (LL) phase transition line separates two phases of liquid—high-density liquid (HDL) and low-density liquid (LDL)—which 2917 S. Yip (ed.), Handbook of Materials Modeling, 2917–2922. c 2005 Springer. Printed in the Netherlands.

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are the precise analogs of the two amorphous solids LDA and HDA. Like essentially all first-order transition lines, the LL transition line between noncrystalline phases must terminate in a critical point. Above the critical point is an analytic extension of the LL phase transition line. Called the Widom line, this extension exhibits apparent singularities—i.e., if the system approaches the Widom line, then thermodynamic response functions appear to diverge to infinity until the system is extremely close, when the functions will round off and ultimately remain finite—as seen in adiabatic compressibility data [10].

2.

Plausibility Arguments

That a LL phase transition exists is at least plausible. Liquid water is a tetra-hedral liquid, and two water tetrahedra can approach each other together in many different ways. One way is coplanar, as in ordinary hexagonal ice Ih —creating a “static heterogeneity” with a local density not far from that of ordinary ice, about 0.9 g/cm3 . A second way is altogether different: one of the two tetrahedra is rotated 90◦ , resulting in a closer distance where the minimum of potential energy occurs, and hence a static heterogeneity with a local density substantially larger (by about 30%) than that of ordinary ice [11]. In fact, this rotated configuration occurs in solid crystalline water (“ice VI”), which occurs at very high pressure. In liquids close to the freezing temperature, there are heterogeneities with local order resembling that of the nearby crystalline phases. Not surprisingly, then, in water at low pressure, there are more heterogeneities that have icelike entropy (local order) and specific volume, while at high pressure there are more heterogeneities that have a Ice VI-like entropy and specific volume. The potential that represents the relative orientations of two water tetrahedra has two wells: a deeper, “high-volume, low-entropy” well corresponding to LDL and a shallower, “low-volume, high-entropy” well corresponding to HDL. Note that LDL has a higher specific volume and a lower entropy. Therefore when water cools, each molecule must decide how to partition itself between these two minima. The specific volume fluctuations increase because of these two possibilities. The entropy fluctuations also increase, and the cross-fluctuations of volume and entropy have a negative contribution—i.e., high volume corresponds to low entropy so that the coefficient of thermal expansion, proportional to these cross fluctuations, can become negative. The possibility that these static heterogeneities gradually shift their balance between low-density and high-density as pressure increases is plausible, but need not correspond to a genuine phase transition. There is no inherent reason why these heterogeneities need to “condense” into a phase, and the first guess might be that they do not condense – what is now called

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the singularity free hypothesis [12, 13]. However if we reason by analogy with the gas-liquid transition, then there is one reason to believe that they will condense. This is related to the fact that a permanent gas is impossible so long as there is a weak attraction, no matter how weak. If such a weak attraction has an energy scale e, at low enough temperature T the ratio /T will become large enough to influence the Boltzmann factor sufficiently that the system will condense. For example, if a a lattice-gas fluid (ex: single-well potential) with = 1 condenses below Tc = 1, then if = 0.001 one anticipates condensation below Tc ≈ 0.001. For this reason, one anticipates that at low enough temperature the one-component liquid would condense into a low density liquid corresponding to a deeper potential well.

3.

Simulations

That the static heterogeneities should condense at sufficiently low temperature is found in simulations using a wide range of molecular potentials, ranging from “overstructured” potentials such as ST2 to “understructured” potentials like SPC/E. Recent work has focused on the newest of all water potentials, Tip5p [14]. Regardless of potential used, all results seem to be consistent with the LL phase transition hypothesis [2].

4.

Experiments

Experimental data are consistent with the LL phase transition hypothesis. The volume fluctuations are proportional to the compressibility, and this compressibility is a spectacularly anomalous function. Below 46 ◦ C, the compressibility start to increase as the temperature is lowered. This phenomenon is no longer counterintuitive if the double-well potential is correct. Similarly, below 35 ◦ C the entropy fluctuations, which correspond to the specific heat, start to increase. Finally, consider the coefficient of thermal expansion, which is proportional to the product of the entropy and volume fluctuations. This is positive in a typical liquid because large entropy and large volume go together, but for water this cross-correlation function has a negative contribution—and as we lower the temperature this contribution gets larger and larger until we reach 4 ◦ C, at which point the coefficient of thermal expansion passes through zero. The experimental work of Angell and collaborators shows apparent singularities when experimental data are extrapolated into the No-Man’s Land. Mishima measured the metastable phase transition lines of ice polymorphs and found that the slopes of these lines exhibited sharp kinks in the vicinity of the hypothesized liquid-liquid phase transition line as predicted by extrapolation

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[15, 16]. The nature of these kinks can be explained if we take into account that an ice polymorph must melt into a metastable liquid before it can recrystallize into a different polymorph. By the Clausius–Clapeyron relation the slope of that first-order metastable melting line must be equal to the ratio of the entropy change divided by the volume change of the two phases that coexist. In one phase the coexistence is always the high-pressure polymorph of ice. The other phase is either HDL or LDL. The volumes and entropies of those two liquids are different, and therefore as the first-order solid-liquid phase transition line crosses the hypothesized LL phase transition line, the slope changes. The Gibbs potential of two phases that coexist along a first-order transition line must be equal. We already know the Gibbs potential of all the polymorphs of ice, so we know, experimentally, the Gibbs potential of the LDL and the HDL. From the Gibbs potential of any substance one can obtain, by differentiation, the volume. Thus, if we know the Gibbs potential as a function of temperature and pressure, we know the volume as a function of temperature and pressure—which is called the equation of state. In this way Mishima and Stanley were able to find an experimental equation of state for water deep inside the no-man’s land. This is, of course, not quite the same as actually measuring the densities of two liquids coexisting at the LL phase transition line since the Mishima experiments concerned metastable melting lines in which the Gibbs potentials of the two phases are not necessarily equal to each other. Very recently, Reichert and collaborators [17] discovered experimentally a HDL under conditions outside the no-man’s land. They studied the thin, quasi-liquid layer between ice Ih and a solid substrate (amorphous SiO2 ). Using a clever experimental technique at Grenoble, they were able to measure the density and found 1.17 g/cm3 , the density of HDA.

5.

Discussion

The LL phase transition hypothesis does not fully answer the question “what matters?”—i.e., it does not tell us which liquids should exhibit LL phase transitions and which should not. It has been conjectured that it is local tetrahedral geometry of water that matters, since a tetrahedral local geometry leads to static heterogeneities which lead to a LL phase transition [18]. But then what about other tetrahedral liquids? Phosphorus [19] and SiO2 [20] have a local tetrahedral geometry, and experimental evidence supports the LL phase transition hypothesis. There is recent evidence for a LL phase transition in silicon, which is also a tetrahedral liquid [21]. It has been argued that LL phase transitions are associated with liquids possessing a line in the T-P phase diagram at which the density achieves a maximum [22–24].

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In summary, the presence of a local tetrahedral geometry leads to two distinct forms of static heterogeneities (or “local order”) differing in specific volume and entropy, with the specific volume and entropy anticorrelated. This fact gives rise to anomalous fluctuations in compressibility, specific heat, and the coefficient of thermal expansion. The hypothesis that at low enough temperatures these small regions of local order condense into two separate phases (LDA and HDA) is supported by simulations, but remains an open question experimentally.

References [1] C.A. Angell, “Amorphous water,” Ann. Rev. Chem., 55, 559–583, 2004. [2] P.G. Debenedetti, “Supercooled and glassy water,” J. Phys.: Condens. Matter, 15, R1669–R1726, 2003. [3] P.G. Debenedetti and H.E. Stanley, “The physics of supercooled and glassy water,” Physics Today, 56[6], 40–46, 2003. [4] O. Mishima and H.E. Stanley, “The relationship between liquid, supercooled, and glassy water,” Nature, 396, 329–335, 1998. [5] P. Briigeller and E. Mayer, “Complete vitrification in pure liquid water and dilute aqueous solutions,” Nature, 288, 569–571, 1980. [6] O. Mishima, L.D. Calvert, and E. Whalley, “An apparently first-order transition between two amorphous phases of ice induced by pressure,” Nature, 314, 76–78, 1985. [7] O. Mishima, “Reversible first-order transition between two H2 O amorphs at −0.2 GPa and 135 K,” J. Chem. Phys., 100, 5910–5912, 1994. [8] O. Mishima, “Relationship between melting and amorphization of ice,” Nature, 384, 546–549, 1996. [9] P.H. Poole, F. Sciortino, U. Essmann, and H.E. Stanley, “Phase behaviour of metastable water,” Nature, 360, 324–328, 1992. [10] E. Trinh and R.E. Apfel, “Sound velocity of supercooled water down to −33 ◦ C using acoustic levitation,” J. Chem. Phys., 72, 6731–6735, 1980. [11] M. Canpolat, F.W. Starr, M.R. Sadr-Lahijany et al., “Local structural heterogeneities in liquid water under pressure,” Chem. Phys. Lett., 294, 9–12, 1998. [12] H.E. Stanley and J. Teixeira, “Interpretation of the unusual behavior of H2 O and D2 O at low temperatures: tests of a percolation model,” J. Chem. Phys., 73, 3404– 3422, 1980. [13] S. Sastry, P. Debenedetti, F. Sciortino, and H.E. Stanley, “Singularity-free interpretation of the thermodynamics of supercooled water,” Phys. Rev. E, 53, 6144–6154, 1996. [14] M. Yamada, S. Mossa, H.E. Stanley, and F. Sciortino, “Interplay between timetemperature-transformation and the liquid-liquid phase transition in water,” Phys. Rev. Lett., 88, 195701, 2002. [15] O. Mishima and H.E. Stanley, “Decompression-induced melting of ice IV and the liquid-liquid transition in water,” Nature, 392, 164–168, 1998. [16] O. Mishima and Y. Suzuki, “Vitrification of emulsified liquid water under pressure,” J. Chem. Phys., 115, 4199–4202, 2001. [17] S. Engemann, H. Reichert, H. Dosch, J. Bilgram, V. Honkimaki, and A. Snigirev, “Interfacial melting of ice in contact with SiO2 , Phys. Rev. Lett., 92, 205 701, 2004.

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[18] H.E. Stanley, S.V. Buldyrev, N. Giovambattista, E. La Nave, A. Scala, F. Sciortino, and F.W. Starr, “Statistical physics and liquid water: ‘what matters’,” Physica A, 306, 230–242, 2002. [19] Y. Katayama, T. Mizutani, W. Utsumi, O. Shimomura, M. Yamakata, and K.-i. Funakoshi, “A first-order liquid-liquid phase transition in phosphorus,” Nature, 403, 170–173, 2000. [20] P.H. Poole, M. Hemmati, and C.A. Angell, “Comparison of thermodynamic properties of simulated liquid silica and water,” Phys. Rev. Lett., 79, 2281–2284, 1997. [21] S. Sastry and C.A. Angell, “Liquid–liquid phase transition in supercooled silicon,” Nature Materials, 2, 739–743, 2003. [22] F. Sciortino, E. La Nave, and P. Tartaglia, “Physics of the liquid–liquid critical point,” Phys. Rev. Lett., 91, 155701, 2003. [23] G. Franzese and H.E. Stanley, “A theory for discriminating the mechanism responsible for the water density anomaly,” Physica A, 314, 508–513, 2002. [24] G. Franzese, G. Malescio, A. Skibinsky, S.V. Buldyrev, and H.E. Stanley, “Generic mechanism for generating a liquid-liquid phase transition,” Nature, 409, 692–695, 2001.

Perspective 37 MATERIAL SCIENCE OF CARBON Wesley P. Hoffman Air Force Research Laboratory, Edwards, CA, USA

Carbon is a ubiquitous material that is essential for the functioning of modern society. Because carbon can exist in a multitude of forms, it can be tailored to possess practically any property that might be required for a specific application. The list of applications is very extensive and includes: aircraft brakes, electrodes, high temperature molds, rocket nozzles and exit cones, tires, ink, nuclear reactors and fuel particles, filters, prosthetics, batteries and fuel cells, airplanes, and sporting equipment. The different forms of carbon arise from the fact that carbon exists in three very different crystalline forms (allotropes) with a variety of crystallite sizes, different degrees of purity and density, as well as various degrees of crystalline perfection. These allotropes are possible because carbon has four valence electrons and is able to form different kinds of bonds with other carbon atoms. For example, diamond with a covalently-bonded face-centered cubic structure can exist as a naturally–formed single crystal as large as 200 g. Diamond, the hardest material known to man, is also able to be made synthetically by a variety of processes. For example, high pressure anvils can be utilized to produce relatively small single crystals while a vapor-phase process such as chemical vapor deposition (CVD) is employed to deposit crystalline and amorphous coatings having grain sizes on the micron scale, and a variety of degrees of crystallite orientations. Synthetic diamonds in all forms are used as hard scratch-resistant coatings and tool coatings for grinding, cutting, drilling and wire drawing. Other applications include heat sinks and optical windows among others. The most abundant forms of carbon exist as various forms of the allotrope hexagonal graphite. The perfect crystalline structure of graphite is a hexagonal layered structure in which the atoms in each layer are covalently-bonded while the graphene layers are held together by weaker Van Der Waals forces. This difference in bonding is what is responsible for the great anisotropy in mechanical, thermal, electrical and electronic properties. 2923 S. Yip (ed.), Handbook of Materials Modeling, 2923–2928. c 2005 Springer. Printed in the Netherlands.

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W.P. Hoffman 0.1415 nm

0.2456 nm

0.3354 nm

Diamond Structure

Hexagonal Graphite Structure

With the relative recent discovery of the nano-forms of carbon in the last two decades, the range of properties that carbon can possess and the gamut of potential applications has greatly increased. The linear portions of these molecules are simply rolled up graphene layers, while the curved portions consist of graphite hexagons in contact with pentagons. Although commercial applications for both bucky balls and carbon nanotubes are not well defined at the time of this writing, the high aspect ratio of nanotubes along with the fact that they are stronger ( 63 GPa ) and stiffer (∼1000 GPa ) than any other known material, means that the potential for these materials is great.

BUCKYBALL

NANOTUBE

Adding to the complexity of understanding and modeling carbon is the fact that, in its various forms, it rarely exists in a perfect single crystalline state. For example, perfect graphitic structure only exists in the various forms of natural graphite flakes and graphitizable carbons, which are carbons formed from a gas or liquid phase process. An example of a graphitizable carbon would be highly oriented pyrolytic graphite (HOPG), which is formed by depositing one atom at a time on a surface utilizing the pyrolysis of a hydrocarbon, such as methane or propylene. This deposited material is then graphitized employing both thermal and mechanical stress. In the overwhelming number of applications single crystal graphite is not employed. Rather a carbon with some degree of graphite structure is utilized. Excluding crystallite imperfection, such as, vacancies, interstitials,

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substitutions, twin planes, etc., the form of carbon that most closely approaches single crystal graphite is turbostratic graphite. This form of carbon looks very similar to graphite except that, although there may be some degree of perfection within the planes, the adjacent planes are out of registry with one another. That is, in the hexagonal graphite structure, there is an atom in each adjacent plane that sits directly over the center of the hexagonal ring. In turbostratic graphite, the adjacent planes are shifted with respect to one another and are out of registry. This results in an increase in the interlayer spacing, which can increase from 0.3354 nm to more than 0.345 nm. If the value exceeds this, the structure exfoliates. Heating to temperatures in excess of 2800 K provides energy for mobility and can convert turbostratic graphite to single crystal graphite in a process called graphitization. As stated above, a carbon material that goes through either a gas phase or liquid phase process in its conversion to carbon (carbonization process) can be convert to a graphitic structure employing time and temperature in the range from 2000–2800 K. This means that materials formed in the gas phase like carbon black and pyrolytic carbon can be converted to a graphitic structure. Cokes and carbon fibers fabricated from petroleum, coal tar, or mesophase pitch are also graphitizable. On the other hand, chars formed directly from organic materials, such as wood and bone used for activated carbon, PAN fibers formed from polyacrylonitrile, and vitreous carbons formed from polymers, such as, phenolic or phenyl-formaldehyde, are amorphous and not graphitizable because they maintain the same rigid non-aligned structure that they possessed before carbonization. In addition to the degree of crystalline order, the properties of carbons are also determined by the crystallite size and orientation of polycrystalline carbons. The largest carbon parts that are manufactured are electrodes for the steel and aluminum industries. These electrodes can weigh more than 3 tons and are fabricated by extruding a mixture of fine petroleum coke and coal tar pitch. The extrusion process causes some preferential alignment of the crystallites and baking to 2800 K produces a polycrystalline graphite part that has high strength and conductivity. To make isotropic graphite, fine grain coke and petroleum pitch are isostatically pressed. By proper selection of precursor material and processing condition, a carbon with practically any property can be produced. For example carbons can be hard (chars) or soft (blacks), strong (PAN fibers) or weak ( aerogel), r as well as anisotropic (HOPG) or stiff (pitch fibers) or flexible (Graphoil), isotropic (polycrystalline graphite). In addition, porosity, lubricity, hydrophobicity, hydrophilicity, thermal conductivity and surface area can be varied over a wide range. For example surface area can vary from 0.5 m2 /g for a fiber to >2000 m2 /g for an activated carbon while thermal conductivity can range from 0.001 W/(m-K) for an amorphous carbon foam to 1100 W/(m-K) for a pitch fiber, and 3000 W/(m-K) for diamond.

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W.P. Hoffman

Carbon materials have been studied and produced for thousands of years. (The Chinese used lampblack 5000 years ago.) While much is understood about carbon, there are some very important areas in which there is still a lack of understanding. These areas fall generally into the production of graphitic material and the oxidation protection of carbon and graphite materials. A better understanding of the science of carbon formation will allow increased performance at reduced cost, while effective oxidation protection at ultra high temperature will enable a whole range of new technology. Currently, the highest performance and highest cost form of carbon are carbon-carbon composites, which are truly a unique class of materials. These composites, which are stronger and stiffer than steel as well as being lighter than aluminum, are currently used principally in high performancehigh value applications in the aerospace and astronautics industries. The highest volume of carbon-carbon is used as brake rotors and stators for military and commercial aircraft because it has high thermal conductivity, good frictional properties, and low wear. Astronautic applications include, rocket nozzles, exit cones, and nose tips for solid rocket boosters as well as leading edges and engine inlets for hypersonic vehicles. In these applications carbon-carbon’s high strength and stiffness as well as its thermal shock resistant are keys to its success. For re-useable hypersonic vehicles, the fact that carbon does not go through phase changes like some ceramics and in fact its mechanical properties actually increase with temperature make this a very valuable material. For satellite applications carbon-carbon’s high specific strength and stiffness as well as its near zero thermal expansion make it an ideal material for large structures that require dimensional stability as they circle the earth. Carbon–carbon composites are fabricated through a multi-step process. First, carbon fibers, which carry the mechanical load, are woven, braided, felted, or filament wound into a preform which has the shape of the desired part. The perform is then densified with a carbon matrix, which fills the space between the fibers and distributes the load among the fibers. It is this densification process that is not well understood. Unlike the manufacture of fiberglass, in the formation of carbon-carbon composites the matrix precursor is not just cured but must be converted into carbon. The conversion of polymers to produce a char, as well as the conversion of a hydrocarbon gas to produce a graphitizable matrix, is fairly well understood. What is not clear is the process of converting a pitch-based material to a high quality graphitizable matrix. That is, for example, starting with a petroleum or coal tar pitch, heat soaking converts this isotropic mixture of polyaromatics into anisotropic liquid crystalline spherical droplets called mesophase spherules. These spherules ultimately coalesce to form a continuous second phase which ultimately pyrolyzes to form a graphitic structure. Although this process has been observed microscopically, little is known about the process, or even exactly what mesophase is or what molecular precursors are needed to form

Material science of carbon

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mesophase. Little is known because it has proved impossible to accurate analyze mesophase precursors, mesophase itself, or the intermediates between mesophase and graphite. What is needed is a model that can predict the growth of single aromatic rings through the mesophase intermediate and into an ordered graphitic structure. Moreover, it is well known that during pyrolysis, mesophase converts into a matrix that is very anisotropic. The formation of onion-like “sheaths” takes place on the surface of individual carbon fibers in the carbon composite. As these sheaths grow outward from the fiber surfaces, they ultimately collide forming point defects called disclinations. This behavior has a pronounced effect on both the chemical and physical properties of the carbon-carbon composite. A model that describes this matrix contribution to the composite’s properties would be of great benefit both for understanding experimental data, such as thermal conductivity and mechanical properties, as well as for prediction of composite behavior. Although carbon–carbon composites possess high strength, stiffness, and thermal shock resistance making them an excellent high temperature structural material, their Achilles heel is oxidation. Above 670 K, carbon oxidizes. This means that if it is unprotected, it can not be used for long term high temperature applications. Today it is used for short term applications, such as, aircraft brakes as well as rocket nozzles, exit cone, and nose tips. To be used in long term applications such as reusable hypersonic vehicles, it must be protected. Currently, an adequate protection system at ultra high temperatures (>2700 K) does not exist. There is a tremendous need and payoff for a non-structural barrier that can keep oxygen from reaching a carbon surface at 2700 K. Efforts to fabricate these coatings have been unsuccessful. What is probably required for success is some sort of novel functionally graded coating which will require modeling material properties as well as thermal stresses. Finally, the rather recent discovery of Buckyballs and nanotubes, which are 3D analogs of hexagonal graphite, have also rekindled the need for models of the intercalation of graphite. This process occurs when various elements, such as lithium, sodium, or bromine, “sandwich” themselves between graphene planes. This greatly alters the thermal and electrical properties of the graphite. Similar behavior is beginning to be demonstrated in Buckyballs and nanotubes. The need for models based on computational chemistry would be of great help in this area. For additional information on carbon, there are many good reference works. Most focus on specific forms of carbon such as carbon blacks, active carbons, fiber, composites, intercalation compounds, nuclear graphites, etc. For general topics the handbook by Pierson [1] is a good text. A reference work covering scores of subjects in great detail is the 28 Volume Chemistry and Physics of Carbon [2] which has had three Editors over the last 40 years.

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References [1] H.O. Pierson, “Handbook of carbon, graphite, diamond and fullerenes – properties, processing and applications,” Noyes Punlications, Park Ridge, 1993. [2] P.L. Walker Jr., P. Thrower, and L.R. Radovic (eds.), “Chemistry and physics of carbon,” vol. 1–28, 1965–2004, Marcel Decker, New York, 2001.

Perspective 38 CONCURRENT LIFETIME-DESIGN OF EMERGING HIGH TEMPERATURE MATERIALS AND COMPONENTS Ronald J. Kerans Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, Ohio, USA

We ask a great deal of structural components used at high temperatures. In addition to performing their primary load bearing function, most of them are subjected to sizable thermal stresses and aggressive atmospheres. Their microstructures and phase compositions, and hence their properties, evolve throughout their lifetime. Many are used in regimes where significant local creep deformation accumulates, so their shapes and residual stress states change also. Finally, many of them are used in situations where failure is extremely undesirable. Failure of a power generation turbine or an aircraft engine carries substantial fiscal costs, and the latter has potential for tragic human costs. The customers – all of us – have very low tolerance for other than extreme reliability. This is a challenging environment in which to introduce new materials. It is an equally challenging environment in which to substitute computation for time-proven testing and design techniques. The penalties for mistakes are extreme and the development costs are correspondingly large. On the other hand, the high stakes involved provide strong motivation to use computation to improve the ability to predict component end-of-life and thereby avoid both catastrophic failures and expensive premature retirement. Likewise, the long and costly development / certification cycles for introducing even evolutionary changes in materials provide a substantial benefit for substituting computation for experiment. Fortunately the fundamental understanding and many of the multi-scale modeling tools will be similar for both tasks.


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R.J. Kerans

Current Efforts

In many ways, the need for computational infrastructure for both design and life management is most pronounced for new materials systems for which both design and life limiting behavior differ from materials now in service. However, the breadth and depth of understanding of behavior, substantial existing databases, and cost benefit associated with fielded systems substantially argues for application to existing and evolutionary systems. The computational tools required for either task are numerous, as are the materials science aspects that will require enhanced understanding; hence a comprehensive, coordinated attack on either is a significant undertaking. In fact, a number of efforts have been coordinated under the umbrellas of three substantial activities which have begun addressing both design [1–3] and life management [4, 5]. The long-range scope of the design-related modeling effort extends from atomic level to component design via finite element analysis (FEA). The life management effort covers a broad span in degradation and failure behavior, plus goals of comparable complexity in logging and predicting the effects of actual use history and in sensing the actual state of the system in situ.

2.

Emerging Materials

While current efforts boldly attempt to devise entirely new approaches to design and life management, the focus is quite logically on current or evolutionary variations of current materials. However, the real impact of the approach may be most profound when applied to the introduction of new materials, such as low-ductility intermetallics, ceramic composites and perhaps hybrids combining two or more materials systems. These systems almost “demand” a more robust computational approach to design. Consider that while developmental ceramic composites are made in sheet form with uniform properties, many actual components will have fiber and void volume fractions that vary strongly with location in the part. The magnitude and anisotropy of the properties of the material will be a local feature of the component. In this sense, it is almost impossible to separate the material from the component; optimal design will require that the material and component be dealt with concurrently. While these issues apply to all fibrous composites, they will apply especially to ceramic composites. Organic-matrix composites are often used for large sheet-like components such as wing skins, whereas ceramic composites will largely be used for the smaller, more complex shapes of high temperature hardware. In almost all aspects, ceramic composites represent a significant departure from conventional practice. Optimizing the distributed damage mechanisms

Design of emerging high temperature materials and components

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that provide macroscopic fracture toughness is an exercise in carefully designed fracture processes. In conventional materials, the design work is mostly oriented towards avoiding fracture entirely, or at least designing for minimum growth rates. The task of promoting local fracture and designing crack paths is relatively new territory [6]. A consequent significant difference between ceramic composites and other systems is the relationship between properties and damage. Local deformation in metals, for example, does not radically change properties whereas comparable deformation in a composite may significantly change the local properties with the largest changes possibly being in an unloaded direction. The local nature of properties both imposes obligation and provides opportunity to the designer. The idea is to design the local material/properties to best suit the requirements of the component. In many cases this will be best achieved not by fastening sub components together, but by way of designed architectures and blending materials systems within the component [7, 8]. Such hybrids might comprise oxide composites at the hot surface blending to nonoxide reinforcements in mid-temperature regions, blending to metal matrices in warm regions and finally to monolithic metals for cool attachments. The idea of hybrids is a natural extension of local design.

3.

Processing

Further complicating the local property situation is the certainty that the properties will also be a consequence of local processing. For most materials systems, processes will need to vary with the component. For example, the microstructures of both wrought and cast metal parts vary with location and depend on the geometry of the part. Ceramic composites are by their very nature particularly difficult to process. The basic goal is to consolidate a very refractory matrix material that is constrained by an equally refractory fiber structure, without damaging the fibers. The degree of densification tends to vary with local fiber fraction and most processes will always yield significant void space that varies with location. If the void space can’t be eliminated, then the objective should be to control it to be distributed in the most benign arrangement. Uniform fine porosity is typically more desirable than large pores, which are more desirable than large cracks. The challenge is three-fold; to design what can be processed reliably, to process what was designed, and to model the performance and life of what actually exists. This will require the ability to know and model the effects of such things as actual fiber locations in real preforms and components. Irrespective of the material system, modeling that represents real processes will be essential.

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R.J. Kerans

Lifetime Design

Materials that operate at very high temperatures will evolve over time; no component in use is in a truly stable state. While this fact is not currently neglected entirely, there is opportunity to develop a much higher level of sophistication. The long-range goal should be to establish materials design and modeling capabilities that allow design of materials and components to achieve an optimum blend of properties over their lifetimes and reliable lifetimes in the specific application. This optimization will require: in-depth knowledge of, and the ability to simulate, the service environments; and the ability to predict the evolution of microstructures and damage and the consequent effects on properties and life limiting behavior. The ability to use this knowledge to predict the correct initial state of the materials for optimum lifetime performance implies a very high level of sophistication. The ability to achieve the goal of concurrent lifetime-designed, life-managed materials/components is completely dependent upon conquering what may be an equally challenging task: acquiring knowledge of the actual use environment to a far greater depth than is currently possible. It requires knowing well the actual conditions, not just nominal conditions, and the impact that has on each particular component, which may be as challenging as the preceding items.

5.

Models

The highest-level models may be similar to current design Finite Element Analysis. Much of the design portion of the work might best be done through inverse analysis. That is, the design analysis would determine what the component “wants” for properties and design the material to best suit that. In any event, the more basic analysis will need to be integrated such that it is invisible to the designer [3]. The shortest-scale models will depend somewhat on the materials systems and the nature of the component. In at least some cases, there will be a continuing need to deal with quantum-level modeling. An example of an atomic level issue in a metal alloy might be the introduction of a minor alloying element for the purpose of slowing diffusive microstructural changes. The modification could easily introduce unintended effects on boundary strength, or perhaps subtly affect stability of undesirable minor phases that could be problematic late in service life. Evaluation for all such effects requires very basic modeling, or, as in the current system, extensive prior knowledge. For ceramic composites, it has thus far seemed unnecessary to consider scales below the micromechanics level for design purposes; that is, the smallest dimension would be the approximate 200 nm thickness of the fiber coating.

Design of emerging high temperature materials and components

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It has been assumed that all constituents remain perfectly elastic (brittle) even at quite high temperatures, and that composite performance is not directly affected by subtleties at scales below that of the smallest constituent. However, recent basic work on deformation of monazite and scheelite fiber-coating materials indicates a large role of plasticity in the function of the coatings [9–11]. Further evidence confirms that the extent of plasticity will be temperature dependent, though considerably less so than for most metals. This will add a significant temperature dependent component to composite behavior, and a high level of modeling fidelity may require dealing with ductility via dislocations and deformation twins in the 200 nm coating. Moreover, it is clear that there are significant corrosion effects on fibers that remain uncharacterized, much less understood [12, 13]. These will have profound effects on lifetime properties, and surely other such effects lurk in the details. The degree to which we see simplicity in the modeling of ceramic composites is probably more a reflection of our depth of understanding of the materials than of their actual complexity in long-term service. In any event, very basic modeling will be necessary, but not nearly sufficient. Computational concurrent design will require integrated modeling on a wide variety of time and length scales, integration of materials disciplines, and integration of materials and design disciplines. An elegant and readily usable solution to multi-scale integration will be essential to an infrastructure that has real impact [3].

6.

Outlook

The ultimate goal could be: (1). concurrent material/component design for the best balance of properties over the lifetime of the component; (2). a reliable lifetime with graceful, detectable failure processes; (3). the ability to determine the current state of the component and predict its remaining life; (4). the power to do much of the preceding computationally with an economical set of physical experiments to confirm the results to high reliability. The achievement of this goal is dependent as much upon progress in the science of materials as upon progress in the science of modeling. It is probably safe to say that all current models are situation specific. They do not describe the actual physics; they describe an approximation of the actual physics. Each one describes a facet of the situation wonderfully well in certain circumstances and is completely wrong in others. Integrating them and their successors into an infrastructure that will apply them properly to this nearly infinitely faceted, multi-scale problem is a formidable challenge. There is the attendant danger that each implementation will be so material-system specific that applying the approach to a new system will require rebuilding the entire infrastructure. While such an outcome might still yield significant benefit, much of the

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savings would be lost. The resources currently spent building temporarily useful databases would largely be spent building temporarily useful models. Nevertheless, it seems self-evident that it is the logical goal of materials and computational science. It is a goal with tremendous benefit in both monetary and human terms and that is wonderfully engaging in addition. Though when it will be achieved is uncertain, we can look forward to the day when intrinsic failures are an artifact of the past.

References [1] D.M. Dimiduk and P.L. Martin, et al., “Accelerated insertion of materials: the challenges of gamma alloys are really not unique,” Gamma Titanium Alumindes, H.C.Y-W. Kim and A.H. Rosenberger, TMS, Warrendale, PA, 15–28, 2003. [2] D.M. Dimiduk and T.A. Parthasarathy et al., “Structural alloy performance prediction for accelerated use: evolving computational materials science & multiscale modeling,” 2nd International Conference on Multiscale Materials Modeling, University of California Los Angeles, Los Angeles, CA USA, American Scientific Publishers, 2004. [3] D.M. Dimiduk and T.A. Parthasarathy et al., “Predicting the microstructuredependent mechanical performance of materials for early-stage design,” NUMIFORM 2004, Columbus, OH, USA, 2004. [4] J.M. Larsen and B. Rasmussen et al., “The engine rotor life extension (ERLE) initiative and its contributions to increased life and reduced maintenance cost,” 6th National Turbine Engine High Cycle Fatigue (HCF) Conference, Jacksonville, FL, USA, 2001. [5] L. Christodoulou and J.M. Larsen, “Using materials prognosis to maximize the utilization potential of complex mechanical systems,” JOM(March), 15–19, 2004. [6] R.J. Kerans and R.S. Hay et al., “Interface design for oxidation resistant ceramic composites,” J. Am. Ceram. Soc., 85(11), 2599–632, 2002. [7] G. Jefferson and T.A. Parthasarathy et al., “Materials design of hybrid ceramic composites for hot structures,” Proceedings of the 35th International SAMPE Technical Conference, Dayton, OH, SAMPE, 2003. [8] T.A. Parthasarathy and R.J. Kerans et al., “Reduction of thermal gradient-induced stresses in composites using mixed fibers,” J. Amer. Cer. Soc., 87(4), 617–625, 2004. [9] R.S. Hay, “(120) and (122) monazite deformation twins,” Acta Mater., 51(18), 5255– 5262, 2003. [10] R.S. Hay and D.B. Marshall, “Monazite deformation twins,” Acta Mater., 51(18), 5235–5254, 2003. [11] R.S. Hay, “Climb dissociation of dislocations in monazite at low temperature,” J. Am. Ceram. Soc., 87(6), 1149–1152, 2004. [12] E.E. Boakye, R.S. Hay et al., “Monazite coatings on fibers: II, coating without strength degradation,” J. Am. Ceram. Soc., 84(12), 2793–2801, 2001. [13] R.S. Hay and E.E. Boakye, “Monazite coatings on fibers: I, effect of temperature and alumina-doping on coated fiber tensile strength,” J. Am. Ceram. Soc., 84(12), 2783–2792, 2001.

Perspective 39 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 Department of Physics, University of Louisville, Louisville, KY 40292

The construction of semi-empirical Hamiltonians for materials that have the predictive power is an urgent task in materials simulation. This task is necessitated by the bottleneck encountered in using density functional theory (DFT)-based molecular dynamics (MD) schemes for the determination of structural properties of materials. Although DFT/MD schemes are expected to have predictive power, they can only be applied to systems of about a few hundreds of atoms at the moment. MD schemes based on tight-binding (TB) Hamiltonians, on the other hand, are much faster and applicable to larger systems. However, the conventional TB Hamiltonians include only two-center interactions and they do not have the framework to allow the self-consistent determination of the charge redistribution. Therefore, in the strictest sense, they can only be used to provide explanation for system-specific experimental results. Specifically, their transferability is limited and they do not have predictive power. To overcome the size limitation of DFT/MD schemes on the one hand and the lack of transferability of the conventional two-center TB Hamiltonians on the other, there exists an urgent need for the development of semi-empirical Hamiltonians for materials that are transferable and hence, have predictive power. The key ingredient to the development of semi-empirical Hamiltonians for materials that have predictive power is a reliable and efficient scheme to mimic the effect of screening by electrons when atoms are brought together to form a stable aggregate. Such an ingredient requires the construction of


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the semi-empirical Hamiltonian based on a framework that allows a coherent treatment of the self-consistent (SC) determination of charge redistribution and environment-dependent (ED) multi-center interactions. Various schemes can be found in the literature in recent years that are designed to improve the transferability of TB Hamiltonians by including the self-consistency and/or the environment-dependency. Among these are methods that can also be conveniently implemented in MD schemes because the atomic forces can be readily calculated. They include methods whose emphasis is placed on a phenomenological description of the environment-dependency [1, 2] and two similar methods whose frameworks take into account the self-consistency as well as the environment-dependency [3, 4]. The latter two approaches can be construed as the expansion of the DFT-total energy in terms of the charge density fluctuations about some reference density. To the second order in the density fluctuations, the total energy is approximated as the sum of a band structure term, a short-range repulsive term akin to that in the conventional two-center TB Hamiltonian, and a term representing the Coulomb interaction between charge fluctuations. The charge fluctuations in this approach are self-consistently determined by solving an eigenvalue equation with the two-center Hamiltonian modified by a term that depends on the charge redistribution. In this framework, the Hamiltonian does contain the features of self-consistency in the charge redistribution and the environment-dependency for systems with charge fluctuations. The environment-dependent feature, however, disappears when systems under consideration do not involve charge fluctuations, e.g., periodic systems with one atomic species per primitive unit cell. But the environment-dependent multi-center interactions are key features in a realistic modeling of the screening effect of the electrons in an aggregate of atoms, including extended periodic systems. This deficiency in properly mimicking the screening of the electrons can be critical in the development of a truly transferable Hamiltonian. Thus the development of semi-empirical Hamiltonians for materials with predictive power requires the treatment of the self-consistency as well as the environment-dependency on equal footing. We have recently developed a scheme for the construction of semi-empirical Hamiltonians for materials within the framework of linear combination of atomic orbitals (LCAO) that allows a coherent treatment of the SC determination of the charge redistribution and the environment-dependent (ED) multi-center interactions in a transparent manner [5]. In this scheme, we set up the framework of the semi-empirical Hamiltonian in accordance with the Hamiltonian of the many-atom aggregate. The salient feature of the resulting semi-empirical Hamiltonian, referred as the SCED/LCAO Hamiltonian, is that it has the flexibility to allow the database to provide the necessary ingredients for fitting parameters to capture the effect of electron screening.

Semi-empirical Hamiltonian for materials simulation

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The Hamiltonian of a many-atom aggregate may be written as H =−

h¯ 2 l

2m

∇l2 +

ν( rl − Ri ) +

l,i

l,l

Z i Z j e2 e2 + 4π ε0rll 4π ε0 Ri j i, j

(1)

energy where ν( rl − Ri ) is the potential between an electron at rl and the ion at Ri , rll = rl − rl , Ri j = Ri − R j , and Z i corresponds to the number of valence electrons associated with the ion at site Ri . Within the one-particle approximation in the framework of LCAO, the on-site (diagonal) element of the Hamiltonian can be written as 0 inter + u intra Hiα,iα = εiα iα + u iα + v iα

(2)

0 denotes the sum of the kinetic energy and the energy of interaction where εiα with its own ionic core of an electron in the orbital iα. The terms u intra iα and inter u iα are the energies of interaction of the electron in orbital iα with other electrons associated with the same site i and with other electrons in orbital jβ ( j =/ i), respectively. The term v iα represents the interaction energy between the electron in orbital α at site i and the ions at the other sites. In our scheme, the terms in Eq. (2) are represented by 0 = εiα − Z i U εiα

(3)

u intra iα = Ni U

(4)

and u inter iα + v iα =

[Nk VN (Rik ) − Z k VZ (Rik )]

(5)

k= /i

where εiα may be construed as the energy of the orbital α for the isolated atom at i, Z i the number of positive charges carried by the ion at i (also the number of valence electrons associated with the isolated atom at i), Ni the number of valence electrons associated with the atom at i when the atom is in the aggregate, U , a Hubbard-like term, the effective energy of electronelectron interaction for electrons associated with the atom at site i, VN (Rik ) the effective energy of electron-electron interaction for electrons associated with different atoms (atoms i and k), and Z k VZ (Rik ) the effective energy of interaction between an electron associated with an atom at i and an ion at site k.

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Following the same reasoning, we can set up the off-diagonal matrix element Hiα, jβ ( j =/ i) as

Hiα, jβ

1 = K (Ri j )(ε iα + εjβ ) + [(Ni − Z i ) + (N j − Z j )]U 2 +

(Nk VN (Rik ) − Z k VZ (Rik ))

k= /i

+

(Nk VN (R j k ) − Z k VZ (R j k ))

Siα, jβ (Ri j )

(6)

k= /j

Thus, in addition to the conventional two-center hopping-like first term, Eq. (6) also includes both intra- and inter-electron-electron interaction terms as well as environment-dependent multi-center (three-center explicitly and four-center implicitly) interactions. In its broadest sense, the first term in Eq. (6) corresponds to the Wolfsberg– Helmholtz relation in the extended Hückel theory. In our approach, K is treated as a function of Ri j rather than a constant parameter to ensure a reliable description of the dependence of the two-center term on Ri j in the off-diagonal Hamiltonian matrix element. The overlap matrix elements Siα, jβ (Ri j ) are expressed in terms of Si j,τ , with τ denoting, for example, molecular orbitals ssσ, spσ, ppσ, and ppπ in a sp3 configuration. They are expected to be shortranged function of Ri j . Equations (2) through (6) completely define the recipe for constructing semi-empirical SCED-LCAO Hamiltonians for materials in terms of parameters and parameterized functions. An examination of Eqs. (2)–(6) clearly indicates that the presence of Ni , the charge distribution at site i, in the Hamiltonian provides the framework for a self-consistent determination of the charge distribution. From Eqs. (5) and (6), it can be seen that the environmentdependent multi-center interactions are critically dependent on VN (Rik ) and VZ (Rik ), in particular their difference VN (R ik ) = VN (Rik ) − VZ (Rik ). As both VN (Rik ) and VZ (Rik ) must approach E 0 Rik for Rik beyond a few nearest neighbor separations, VN (Rik ) is expected to be a short ranged function of Rik . The parameters, including those characterizing the parameterized functions, are to be optimized with respect to a judiciously chosen database for a particular material. In our approach, εiα may be chosen according to its estimated value based on the orbital iα, or treated as a parameter of optimization. The quantity U will be treated as a parameter of optimization while VZ (Rik ) and VN (Rik ) will be treated as parameterized functions to be optimized. The parameterized function VZ (Rik ) is modeled as the energy of the effective interaction per ionic charge between an ion at site k and an electron associated with the atom at site i. VN (Rik ) is then modeled in terms of VZ (Rik ) and the short-range function VN .


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The recognition of the difference between VN (Rik ) and VZ (Rik ) in the SCED/LCAO Hamiltonian assures that the environment-dependent feature will not disappear even for systems with no charge redistribution. The presence of the environment-dependent terms in the SCED/LCAO Hamiltonian for systems with no on-site charge redistribution affects the distribution of the electrons among the orbitals even though the total charge associated with a given site is not changed. Therefore, the effect of the environment-dependency will be reflected in the band structure energy through the solution to the general eigenvalue equation corresponding to the SCED/LCAO Hamiltonian as well as the total energy. This feature, together with the self-consistency in the determination of the charge redistribution, provides the flexibility for the SCED/LCAO Hamiltonian to mimic the effect of electron screening. According to the strategy given above, the framework of the proposed semi-empirical SCED-LCAO Hamiltonian will allow the self-consistent determination of the electron distribution at site i. The inclusion of environmentdependent multi-center interactions will provide the proposed Hamiltonian with the flexibility of treating the screening effect associated with electrons which is important for the structural stability of narrow band solids such as d-band transition metals, while at the same time, handling the effect of charge redistribution for systems with reduced symmetry on equal footing. Furthermore, as described above, the Hamiltonian is set up in such a way that the physics underlying each term in the Hamiltonian is transparent. Therefore, it will be convenient to trace the underlying physics for properties of a system under consideration when such a Hamiltonian is used to investigate a manyatom aggregate and predict its properties. The salient feature of our strategy is that, with the incorporation of all the relevant terms discussed previously, there is no intrinsic bias towards ionic, covalent, or metallic bonding for the proposed Hamiltonian. The construction of the SCED/LCAO Hamiltonian depends critically on the database. If one can judiciously compile a systematic and reliable database, the scheme has the flexibility to allow the database to properly model the screening effect of the electrons in an atomic aggregate. Thus the strategy represents an approach that provides the appropriate conceptual framework to allow the chemical trend in a given atomic aggregate to determine the structural as well as electronic properties of condensed matter systems. The total energy of the system consistent with the Hamiltonian described by Eqs. (2)–(6) is given by E tot = E B S − E corr + E ion−ion

(7)

where E B S is the band-structure energy and is obtained by solving the general eigenvalue equation corresponding to the SCED/LCAO Hamiltonian, E corr is the correction to the double counting of the electron-electron interactions between the valence electrons in the band-structure energy calculation, and

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E ion−ion is the repulsive interaction between ions. Based on Eqs. (2)–(6), Eq. (7) can be rewritten as E tot = E B S + +

1 2 1 (Z i − Ni2 )U − Ni Nk VN (Rik ) 2 i 2 i,k(i =/ k)

1 Z i Z k VC 2 i,k(i =/ k)

(8)

with VC =

e2 E0 = 4π ε0 Rik Rik

(9)

For the MD simulation, the forces acting on the atoms in the atomic aggregate must be calculated at each MD step. The calculation of the band structure contribution to atomic forces can be carried out by the Hellmann–Feynman theory. With the presence of terms involving Ni and Nk in the SCED-LCAO Hamiltonian (see Eqs. (5) and (6)), terms such as ∇k Ni where ∇k refers to the gradient with respect to Rk will appear in the electronic contribution to the atomic forces. However, these terms are canceled exactly by terms arising from the gradients of the second and the third terms in the total energy expression (Eq. (8)). Thus terms involving ∇k Ni will not contribute to the calculation of atomic forces. This fact greatly simplifies the calculation of atomic forces needed in the MD simulations. In other words, if one disregards the extra time due to the self-consistency requirement, the calculation of atomic forces based on the SCED–LCAO Hamiltonian is not anymore difficult compared with conventional TB approaches. We have tested the SCED/LCAO Hamiltonian by investigating a variety of different structures of silicon (Si), including the bulk phase diagrams of Si, the equilibrium structure of an intermediate-size Si71 cluster, the reconstruction of the Si(100) surface, and the energy landscape for a Si monomer adsorbed on the reconstructed Si(111)-7×7 surface [5]. In all the cases studied, the results have demonstrated the robustness of the SCED/LCAO Hamiltonian. For example, results showing the binding energy vs relative atomic volume curves for the diamond, the simple cubic (sc), the body centered cubic (bcc), and the face centered cubic (fcc) phases of silicon, obtained by using the SCED–LCAO Hamiltonian constructed for Si with our scheme, are presented in Fig. 1. Also shown in Fig. 1 are the corresponding curves obtained using three existing traditional (two-center and non-self consistent) non-orthogonal tight binding (NOTB) Hamiltonians [6–8], and two more recently developed non-self consistent but environment-dependent Hamiltonians [1, 2]. All the curves (solid) are compared with the results obtained by DFT–LDA calculations [9].


2941

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fcc DFT LCAO

bcc

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sc

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Wang et al.

Frauenheim et al.

binding energy per atom

0.6

Bernstein & Kaxiras

Menon & Subbaswamy

SCED

0.8

0.4

0.2 0 0.4

1.4

relative atomic volume Figure 1. The binding energy versus relative atomic volume curves for the diamond (cdia), the simple cubic (sc), the body centerd cubic (bcc), and the face centered cubic (fcc) phases of silicon. Top-left panel: SCED-LCAO Hamiltonian. Top-central panel: [7]; Top-right panel: [8]; Bottom-left panel: [6]; Bottom-central panel: [1]; Bottom-right panel: [2]. All the curves (solid) are compared with the result obtained by a DFT–LDA calculation [9].

It can be seen that while the results obtained by all the existing Hamiltonians fail for the high pressure phases, those obtained using Hamiltonians with environment-dependent terms give much better agreement for those phases. This is an indication of the importance of the inclusion of the environmentdependent effects in the Hamiltonian, even for single-element extended crystalline phases where there is no charge redistribution. However, the most striking message conveyed by Fig. 1 is how well our result compares with the DFT–LDA results for all the extended crystalline phases, both at low as well as high pressures. It indicates that the SCED/LCAO Hamiltonian has the capacity and the flexibility of capturing the environment-dependent screening effect under various local configurations. The framework of the SCED/LCAO Hamiltonian outlined in Eqs. (2)–(6) is very flexible. It can be conveniently extended to include the spin-polarized effect and to construct SCED/LCAO Hamiltonians for heterogeneous systems in terms of parameters of SCED/LCAO Hamiltonians of their constituent elemental systems. Work along these lines is in progress.

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Acknowledgment This work was supported by NSF (DMR-0112824 and ECS-0224114) and DOE (DE-FG02-00ER4582).

References [1] C.Z. Wang, B.C. Pan, and K.M. Ho, “An environment-dependent tight-binding potential for Si,” J. Phys.: Condens. Matter., 11, 2043–2049, 1999. [2] D.A. Papaconstantopoulos, M.J. Mehl, S.C. Erwin et al., “Tight-binding Hamiltonians for carbon and silicon,” In: P.E.A. Turchi, A. Gonis and L. Colombo (eds.), TightBinding Approach to Computational Materials Science. MRS Symposia Proceedings No. 491, Materials Research Society, Pittsburg, pp. 221–230, 1998. [3] K. Esfarjani and Y. Kawazoe, “Self-consistent tight-binding formalism for charged systems,” J. Phys.: Condens. Matter., 10, 8257–8267, 1998. [4] Th. Frauenheim, M. Seifert, M. Elsterner et al., “A self-consistent charge densityfunctional based tight-binding method for predictive materials simulations in physics, chemistry, and biology,” Phys. Stat. Sol. (b), 217, 41–62, 2000. [5] C. Leahy, M. Yu, C.S. Jayanthi, and S.Y. Wu, “Self-consistent and environmentdependent Hamiltonians for materials simulations: case studies on Si structures,” http://xxx.lanl.gov/list/cond-mat/0402544, 2004. [6] Th. Frauenheim, F. Weich, Th. Köhler et al., “Density-functional-based construction of transferable nonorthogonal tight-binding potentials for Si and SiH,” Phys. Rev. B, 52, 11492–11501, 1995. [7] M. Menon and K.R. Subbaswamy, “Nonorthogonal tight-binding moleculardynamics scheme for Si with improved transferability,” Phys. Rev. B, 55, 9231–9234, 1997. [8] N. Bernstein and E. Kaxiras, “Nonorthogonal tight-binding Hamiltonians for defects and interfaces,” Phys. Rev. B, 56, 10488–10496, 1997. [9] M.T. Yin and M.L. Cohen, “Theory of static structural properties, crystal stability, and phase transformations: applications to Si and Ge,” Phys. Rev. B, 26, 5668–5687, 1982.

INDEX OF CONTRIBUTORS (Article/Perspective numbers are given in bold.) Abraham, F.F. P20, 2793 Alexander, F.J. 8.7, 2513 Aluru, N.R. 8.3, 2447 Angelis, F. de 4, 59 Artacho, E. 1.5, 77 Asta, M. 1.16, 349 Averback, R. 6.2, 1855 Bammann, D.J. 3.2, 1077 Barmak, K. 7.19, 2397 Baroni, S. 1.1, 195 Bartlett, R.J. 1.3, 27 Battaile, C. 7.17, 2363 Bazant, M.Z. 4.1, 1217; 4.1, 1417 Bernstein, N. 2.24, 855 Binder, K. P19, 2787 Blöchl, P.E. 1.6, 93 Boghosian, B.M. 8.1, 2411 Boon, J P. P21, 2805 Boyd, I.D. P22, 2811 Bulatov, V.V. P7, 2695 Caflisch, R. 7.15, 2337 Cai, W. 2.21, 813 Car, R. 4, 59 Carloni, P. 1.13, 259 Carter, E.A. 1.8, 137 Catlow, C.R.A. 6.1, 1851; 2.7, 547 Ceder, G. 1.17, 367; 1.18, 395 Chadwick, A.V. 6.5, 1901 Chan, H S. 5.16, 1823 Chelikowsky, J.R. 1.7, 121 Chen, I-W. P27, 2843 Chen, L-Q. 7.1, 2083 Chen, S-H. P28, 2849 Chipot, C. 2.26, 929 Ciccotti, G. 2.17, 745; 5.4, 1597 Cohen, M.L. 1.2, 13 Corish, J. 6.4, 1889 Coveney, P.V. 8.5, 2487 Crocombette, J-P. 2.28, 987 Crowdy, D. 4.1, 1417

Csányi, G. P16, 2763 Cuong, N.N. 4.15, 1529 de Koning, M. 2.15, 707 De Vita, A. P16, 2763 Dellago, C. 5.3, 1585 Doll, J.D. 5.2, 1573 Doyle, P.S. 9.7, 2619 Eggers, J. 4.9, 1403 Español, P. 8.6, 2503 Evans, D.J. P17, 2773 Evans, J.W. 5.12, 1753 Falk, M.L. 4.3, 1281 Farkas, D. 2.23, 839 Först, C.J. 1.6, 93 Fredrickson, G H. 9.9, 2645 Frenkel, D. 2.14, 683 Gale, J.D. 2.3, 479; 1.5, 77 Galli, G. P8, 2701 Ganesan, V. 9.9, 2645 García, A. 1.5, 77 Gear, C.W. 4.11, 1453 Germann, T.C. 2.11, 629 Geva, E. 5.9, 1691 Ghoniem, N M. 7.11, 2269; P11, 2719; P30, 2871 Giannozzi, P. 1.1, 195; 4, 59 Gillespie, D.T. 5.11, 1735 Gilmer, G. 2.1, 613 Goddard, W.A. III. P9, 2707 Gro, A. 5.1, 1713 Gumbsch, P. P10, 2713 Gygi, F. P8, 2701 Hadjiconstantinou, N.G. 8.1, 2411; 8.8, 2523 Hirth, J.P. P31, 2879 Ho, K.M. 1.15, 307 Hoffman, W.P. P37, 2923 Hoover, W.G. P34, 2903 Horstemeyer, M.F. 3.1, 1071; 3.5, 1133 Hou, T.Y. 4.14, 1507 Huang, H. 2.3, 1039

2943

2944 Hummer, G. 4.11, 1453 Islam, M.S. 6.6, 1915 Jang, S. 5.9, 1691 Jayanthi, C.S. P39, 2935 Jeanloz, R. P25, 2829 Jensen, P. 5.13, 1769 Jin, X. 8.3, 2447 Jin, Y.M. 7.12, 2287 Joannopoulos, J.D. P4, 2671 Junquera, J. 1.5, 77 Justo, J.F. 2.4, 499 Kaburaki, H. 2.18, 763 Kalia, R.K. 2.25, 875 Kapral, R. 2.17, 745; 5.4, 1597 Karma, A. 7.2, 2087 Kästner, J. 1.6, 93 Katsoulakis, M.A. 4.12, 1477 Kaxiras, E. 2.1, 451; 8.4, 2475 Kerans, R.J. P38, 2929 Kevrekidis, I.G. 4.11, 1453 Khachaturyan, A.G. 7.12, 2287 Khraishi, T.A. 3.3, 1097 Kim, S G. 7.3, 2105 Kim, W.T. 7.3, 2105 Klein, M.L. 2.26, 929 Kob, W. P24, 2823 Kofke, D.A. 2.14, 683 Korkin, A. 1.3, 27 Kremer, K. P5, 2675 Krill, C.E., III. 7.6, 2157 Kubin, L.P. P33, 2897 Landau, D.P. P2, 2663 Langer, J.S. 4.3, 1281; P14, 2749 Leahy, C. P39, 2935 LeSar, R. 7.14, 2325 Li, G. 8.3, 2447 Li, J. 2.8, 565; 2.19, 773; 2.31, 1051 Li, X. 4.13, 1491 Lignères, V.L. 1.8, 137 Lookman, T. 7.5, 2143 Louie, S.G. 1.11, 215 Lowengrub, J. 7.8, 2205 Lu, G. 2.2, 793 MacKerell, A.D., Jr. 2.5, 509 Magistrato, A. 1.13, 259 Mahadevan, L. 9.1, 2555 Margetis, D. 4.8, 1389 Marin, E.B. 3.5, 1133 Maroudas, D. 4.1, 1217 Martin, G. 7.9, 2223 Martin, R.M. 1.5, 77 Marzari, N. 1.1, 9; 1.4, 59 Mattice, W.L. 9.3, 2575 Mavrantzas, V.G. 9.4, 2583 McDowell, D.L. 3.6, 1151; 3.9, 1193

Index of contributors Mehl, M.J. 1.14, 275 Metiu, H. 5.1, 1567 Miller, R.E. 2.13, 663 Milstein, F. 4.2, 1223 Mishin, Y. 2.2, 459 Montalenti, F. 2.11, 629 Morgan, D. 1.18, 395 Moriarty, J.A. P13, 2737 Morris, J.W., Jr. P18, 2777 Mountain, R.D. P23, 2819 Müller, M. 9.5, 2599 Nakano, A. 2.25, 875 Needleman, A. 3.4, 1115 Nitzan, A. 5.7, 1635 Nordlund, K. 6.2, 1855 Odette, R.G. 2.29, 999 Ogata, S. 1.2, 439 Olson, G.B. P3, 2667 Ordejón, P. 1.5, 77 Pakula, T. P35, 2907 Pande, V. 5.17, 1837 Pankratov, I.R. 7.1, 2249 Papaconstantopoulos, D.A. 1.14, 275 Pask, J.E. 1.19, 423 Patera, A.T. 4.15, 1529 Payne, M. P16, 2763 Pechenik, L. 4.3, 1281 Peiró, J. 8.2, 2415 Phillpot, S.R. 2.6, 527; 6.11, 2009 Potirniche, G.P. 3.5, 1133 Powers, T.R. 9.8, 2631 Raabe, D. 7.7, 2173 Radhakrishnan, R. 5.5, 1613 Ratsch, C. 7.15, 2337 Ray, J.R. 2.16, 729 Reinhard, W.P. 2.15, 707 Reuter, K. 1.9, 149 Rickman, J.M. 7.14, 2325; 7.19, 2397 Rubio, A. 1.11, 215 Rudd, R.E. 2.12, 649 Rutledge, G.C. 9.1, 2555 Sakai, T. 8.5, 2487 Sánchez-Portal, D. 1.5, 77 Sauer, J. 1.12, 241 Saxena, A. 7.5, 2143 Scheffler, M. 1.9, 149 Schulten, K. 5.15, 1797 Schwartz, S.D. 5.8, 1673 Selinger, R.L.B. 2.23, 839 Sepliarsky, M. 2.6, 527 Sergi, A. 2.17, 745; 5.4, 1597 Sethian, J.A. 4.6, 1359 Shelley, M.J. 4.7, 1371 Shen, C. 7.4, 2117 Sherwin, S. 8.2, 2415

Index of contributors Sierka, M. 1.12, 241 Sierou, A. 9.6, 2607 Smith, G.D. 9.2, 2561 Soisson, F. 7.9, 2223 Soler, J.M. 1.5, 77 Sornette, D. 4.4, 1313 Srolovitz, D.J. 7.1, 2083; 7.13, 2307 Stachiotti, M.G. 2.6, 527 Stampfl, C. 1.9, 149 Stanley, E.H. P36, 2917 Sterne, P.A. 1.19, 423 Stone, H.A. 4.8, 1389 Stoneham, M. P12, 2731 Succi, S. 8.4, 2475 Tadmor, E.B. 2.13, 663 Tajkhorshid, E. 5.15, 1797 Tang, M. 2.22, 827 Tarek, M. 2.26, 929 Taylor, DeCarlos E. 1.3, 27 Theodorou, D.N. P15, 2757 Thompson, C.V. P26, 2837 Tornberg, A.K. 4.7, 1371 Torquato, S. 4.5, 1333; 7.18, 2379 Trout, B.L. 5.5, 1613 Tuckerman, M.E. 2.9, 589 Uberuaga, B.P. 5.6, 1627; 2.11, 629 Underhill, P.T. 9.7, 2619 Vaks, V.G. 7.1, 2249 Van de Walle, A. 1.16, 349 Van de Walle, C.G. 6.3, 1877

2945 Van der Giessen, E. 3.4, 1115 Van der Ven, A. 1.17, 367 Vashishta, P. 2.25, 875 Veroy, K. 4.15, 1529 Vitek, V. P32, 2883 Vlachos, D.G. 4.12, 1477 Voter, A.F. 2.11, 629; 5.6, 1627 Voth, G.A. 5.9, 1691 Voyiadjis, G.Z. 3.8, 1183 Vvedensky, D.D. 7.16, 2351 Wahnström, G. 5.14, 1787 Wallace, D.C. P1, 2659 Wang, C.Z. 1.15, 307 Wang, Y. 7.4, 2117 Wang, Y.U. 7.12, 2287 Weinan, E. 4.13, 1491; 8.4, 2475 Wijesinghe, H S. 8.8, 2523 Wirth, B.D. 2.29, 999 Wolf, D. 6.7, 1925; 6.9, 1953; 6.1, 1985; 6.11, 2009; 6.12, 2025; 6.13, 2055 Woo, C.H. 2.27, 959 Woodward, C. P29, 2865 Wu, S.Y. P39, 2935 Xiang, Y. 7.13, 2307 Yip, S. 2.1, 451, 613; 6.7, 1925; 6.8, 1931; 6.11, 2009 Yu, M. P39, 2935 Zbib, H.M. 3.3, 1097 Zhu, F. 5.15, 1797 Zikry, M. 3.7, 1171

INDEX OF KEYWORDS ab-initio 1877, 2671, 2687, 2865 ab initio calculations 13, 349, 423, 2823 ab initio molecular dynamics 9, 59, 93, 195, 259, 349, 2701 ab initio potentials 1901 ab initio pseudopotential 121 abnormal grain growth 2157 accelerated molecular dynamics 629 acceptors 1877 acoustic emissions 1313 activated processes 1613 activation barrier 1281, 2223 activation energy 773, 1985, 2055 activation free energy 259 activation volume 1281 active sites 241 adaptive simulation 2675 adatom diffusion 613, 2337 adiabatic energy surfaces 2731 adsorption resonances 1713 aggregation 1769 AgI 1901 AIM 2667 Al(Zr,Sc) 2223 alkali metals 1223 all-atom-models 2675 all-electron potential 121 Allen-Cahn equation 2157 alumina 479 aluminum in melting 2009 Alzheimer’s disease 259 AMBER 509, 2561 amorphization 987, 2009 amorphous 1953, 1985 amorphous carbon 307 amorphous cement model 1953, 1985 amorphous GaAs static structure factor 875 amorphous polymers 1281 amorphous solid water 2917 amorphous solids 1901, 2055, 2749 amorphous structure of high-energy grain boundaries 1953, 1985

amphiphilic fluids 2411, 2487 Andrade law 1313 angular-dependent forces 459 angular-force multi-ion potential 2737 anharmonic 349 anharmonic correction 1877 anisotropic diffusion 959 anisotropic grain growth 2157 anisotropic long-range interaction 2143 anisotropic media 729 anisotropy 2173 annealing stages 1855 annealing temperature 613 anticancer drugs 259 antifreeze proteins 1613 antiphase domain boundaries 2787 antisymmetrically coupled environmental modes 1673 AOT microemulsion 2849 a posteriori error estimation 1529 aquaporins (AQPs) 1797 argon 1635 Arrhenius 1635 Arrhenius dynamics 1477 Arrhenius factor 1735 Arrhenius plot 1985 a-SiO2 structure factor 875 asymmetric GB 1953 asymmetric tilt GB (ATGB) 1953 asymmetrical grain boundaries 1931 asymmetry parameter 2223 ATGB cusp 1953 atomic configuration 613 atomic displacement 349 atomic displacements at interfaces 1931, 2055 atomic force field 1837 atomic hypothesis 451 atomic jump 2223 atomic jump frequency 1787 atomic-level geometry of grain boundaries 1953

2947

2948 atomic mechanism [of diffusion] 1787 atomic misfit energy 2117 atomic pseudopotential wave functions 121 atomic simulations 929 atomic structure 1953, 1985 atomic transport 1851 atomistics 649 atomistic/continuum coupling 663 atomistic description 2523 atomistic (modeling) 2757, 2763 atomistic picture 959 atomistic simulation 451, 793, 1837, 1931, 2737 atomistic simulation of interfaces 1925 atomistic simulation of melting 2009 atomistic thermodynamics 149 automatic model adaptation 663 Bain strain 2777 Bain transformation 1223 ballistic jumps 2223 bands 13 bandgap problem 215 barrier crossing 1635 basin constrained molecular dynamics 629 basis function 2447 basis-sets 93 bcc lattice 827, 1953 bcc metals 2777, 2865, 2883 Beeman algorithm 565 Beevers–Ross site 1901 bending elasticity 2631 Bethe–Salpeter equation 215 Bhatnagar–Gross–Krook model 2805 bias potential 629 bias sampling 613, 1613 biased random walk 1635 bicontinuous morphologies 2849 bicrystal 1953, 1985 bicrystal melting 2009 bifurcation 1223 biharmonic functions 1417 bilayers 2631 bimolecular modeling 259 bimolecular reactions 1735 binary collision approximation 987 binary mixture 2787 binding energy 1877, 2659 biomolecules 1613 biosystems 2731 Bloch walls 2787 Bloch-periodic boundary conditions 423 Bloch’s theorem 423 block copolymers 2599 Blue Moon ensemble 1597

Index of keywords boiling point 395 Boltzmann equation 2411, 2513 Boltzmann factor 613 Boltzmann H theorem 2487 bonds 13 bond-angle distribution function 1985 bond breaking 2763 bond fluctuation model 2599 bond-order 499 bond-order potentials 2737 bond orientational order parameter 1613 border conditions 1931 Born effective charges 195, 479 Born stability criteria 2009 Born-Oppenheimer approximation 59, 195 Born–von Kármán model 349 Bortz algorithm 1753 boundary 1491 boundary conditions 1931, 2475 boundary integral methods 2205 boundary layer 1389 boundary-tracking 2157 Bravais lattice 1953 Bredig transition 1901 brittle 855 brittle fracture 1417 brittle-to-ductile transition 839, 773, 855 broken-bond model 1953 Brønsted acidic sites 241 Brownian dynamics 649, 2757, 2607, 2619 buckyball 2923 bulk free surface 2025 bulk interfaces 1925, 2025 bulk melting point 1985 bulk modulus 855, 2009 Burgers vector 2307 C60 307, 1627 calorimetric 1823 canonical d-bands 2737 canonical ensemble 613 capillary waves 2787 carbon 855, 2923 carbon nanotubes 215 carbon-carbon composite 2923 carboxylate bridged binuclear motif 259, 1877 Car–Parrinello molecular dynamics 59, 259, 1877 cascade 987 CASCADE 1889, 1901 cascade damage 959 CASTEP 1901 catalysis 241, 395, 1753 Cauchy relation 2025

Index of keywords Cauchy–Born rule 663 cellular automata 2351 cellular automata 2687 central limit theorem 1635 central symmetry parameter 1051 centroid molecular dynamics 1691 ceramic composites 527, 875, 2929 certification 1529 CGMD 649 chain entropy 2675 Chapman Kolmogorov equation 1635 Chapman–Enskog analysis 2475, 2487 charge localisation 2731 charge transfer 2731 charged state 1877 CHARMM 509, 2561 chemical-bonding changes 2829 chemical Fokker–Planck equation 1735 chemical Langevin equation 1735 chemical master equation 1735 chemical potential 349, 407, 707, 1389, 1877, 2645 chemical rates 59, 149, 1573, 1585 chemically reacting flows 2475 chemically synthesized quantum dot structures 875 cisplatin-DNA adduct 259 classical limit 349 classical molecular dynamics 59, 349, 451 classical potentials – DFT comparisons 27 clathrate hydrates 1613 cleavage anisotropy 855 climb 2307 clipped random wave model 2849 closure 1477 closure on demand 1453 cluster 241, 349, 1851, 1877 cluster dynamics 2223 cluster expansion 349, 367 cluster migration 241, 2223 coarse bifurcation (algorithms/computations) 1453 coarse grained models 2675 coarse projective integration 1453 coarse time stepper(s) 1453 coarse-grain models 929 coarse-grained molecular dynamics 649 coarse-grained Monte Carlo 1477 coarse-grained polycrystal 2055 coarse-grained statistical model 349 coarse-grained stochastic models 1477 coarse-graining 649, 1477, 1613, 2083, 2325, 2351 2503, 2599, 2645, 2687, 2757 coarsening 2117, 2205 coherency strain energy 2117

2949 coherent 2025 coherent interfaces 1925 coherent phase diagram 2117 coherent potential approximation 349 coherent precipitation 2117 coherent treatment 2935 coherent-twin boundary 1953, 2055 cohesive zone model 855 coincident-site lattice (CSL) 1953 collective damage 1313 collective diffusion model 1797 colloidal fluids 2503, 2607, 2645 combined quantum mechanics–interatomic potential functions method 241 commensurate GB 1953 commensurate interfaces 1925, 2025 committor(s) 1585 COMPASS 2561 compensation 1877 complex fluids 1371, 2411, 2487, 2503, 2675 complex Langevin sampling 2645 complex systems 1217, 1453, 1475 component simulation 2713 composite 2173, 2379 composition-modulated superlattice 2025 compressibility 745 compressibility in melting 2009 computational efficiency 2907 computational fluid dynamics 2811 computational materials design facility 2707 computational nanoscience 2701 computer-aided analysis 1453 concerted rotation 2583 concerted rotation Monte Carlo 2757 concurrent 2929 concurrent material/component design 2929 concurrent multiscale simulation 649 condensed matter 137, 2659 condensed phase systems 1597 conditional convergence 813 conditional probability 1635 conditional reaction probability 1735 conductivity 1333, 1877, 1901 configuration 349 configuration coordinate diagram 1877 configuration phase space 613 configurational and displacive states 349 configurational arrangement 349 configurational bias 2583 configurational bias Monte Carlo 2757 configurational disorder 367 confined liquid 1985 conformal map 1417 conformational partition function 2575

2950 conjugate gradients 2671 conjugated polymers 215 conservation laws 1491 consistency 2415 constant pressure molecular dynamics 589 constant-coverage algorithms 1753 constitutive equations 1071 constitutive formulations 2897 constitutive law 745 constitutive theories 1281 constrained equations 745 constrained equilibrium 149 constrained phase space flow 745 continuity equation 745 continuous Markov process 1735 continuous-orientation model 2157 continuous-time random-walk (CTRW) model 1797 continuum 2173 continuum approach [to modeling diffusion] 1787 continuum description 2523 continuum limit 2351 continuum mechanics 1529, 2903 convergence 2415 convex hull 349 cooperative rearrangement 2907 cooperativity 1823, 2907 coordinate scaling 707 coordination number 1051 copolymers 2645 copper 629, 1223 copper100 629 core energy 813 core properties 793 core-shell nanoparticles 875 correlated events 629 correlation factor 1855 correlation function 1333, 1635, 1673, correlation length 2787 correlation time 1635 correspondence relation 649 coupling method 2523 covalent bonding 499 covalent solids 451 cracks 839, 2287 crack dynamics 2475 crack propagation 2087, 2763 crack propagation in amorphous SiO2 875 crack tip 773 crack tip plasticity 839 creep 959, 1313, 2719 critical behavior 1613 critical point 2787

Index of keywords criticality 1313 cross-slip 793, 2307, 2897 cross-validation 395 crowdion 1855 crown thioether complexes of rhenium and technetium 259 crystal growth 613, 629 crystal interface 1925 crystal plasticity code 2897 crystal structure 395, 423, 2829 crystal symmetry 1223 crystal-growth simulation 2055 crystallography 2173 crystal-to-amorphous transition 2009 C-S bond cleavage 259 CSL misorientation 1953 curvature 1039, 2837 curvature-driven grain growth 2157 cusped orientation 1953 CVD diamond 2829 cyclic plasticity 1193 DAD effect 959 damage 1071, 1183 Damkohler number 2475 data management and mining 875 dislocation dynamics simulations 2897 deposition 1039 dissipative particle dynamics (DPD) 2503 de novo 2707 deacylation of peptide 259 Debye correlation function 2849 decoherence 2731 defects 241,1855, 1877, 1915, 2269, 2737, 2871 defect annealing 1855 defect calculations 547 defect concentration 1855 defect-dependent properties 1851 defect diffusion 613, 1855 defect entropies 1889 defect-induced melting 2009 defects in amorphous materials 1855 defects in intermetallics 1855 deformation 1281, 2173 deformation gradient tensor 439, 1133 deformed materials 1217 degrees of freedom (DOF) of a grain boundary 1953 deNOx process 241 density field 2645 density functional theory (DFT) 9, 59, 93, 121, 137, 149, 195, 349, 423, 439, 451, 855, 1613, 1877, 1889, 2737

Index of keywords density of CSL sites 1953 density of states 683, 1823 deposition 2363 deprotonation energy 241 design 2667 detailed balance 613, 1477, 1585, 1753 deterministic dynamics 613 diamond 2923 diamond-anvil cell 2829 diamond nanoparticle 307 diamond structure 855 diamond surface 307 diamond-to-graphite transition 307 dielectric breakdown 1417 dielectric tensor 195 differential displacement map 773 diffraction 1713 diffuse interface 2087, 2117 diffusion 629, 1627, 2117 diffusion coefficient 1635, 1691, 1901, 1931, 2055 diffusion controlled reactions 1635 diffusion equation 1635, 1787 diffusion mechanism 2223 diffusion permeability 1797 diffusional phase-transition 2205 diffusional width 1985 diffusion-limited aggregation 1417 diiron proteins 259 dimanganese proteins 259 dimensional analysis 565 dimer method 629, 1627 dimer-TAD 629, 1627 diomimetic compound of methane monooxygenase 259 direct correlation function 1613 direct force method 349 direct simulation Monte Carlo (DSMC) 2411, 2811, 2487, 2523 directed end-bridging 2583 directed internal bridging 2583 disclinations 2923 discrete simulation automata (DSA) 2411, 2487 discrete-orientation model 2157 discretization 2415 dislocation 793, 813, 839, 855, 1077, 1098, 1115, 2083, 2269, 2843, 2865, 2871 dislocation core energy 773 dislocation density tensor 2325 dislocation dipole 2325 dislocation dynamics 813, 827, 2083, 2269, 2307, 2325, 2871 dislocation dynamics simulation, 3D 2695 dislocation GB 1953, 1985

2951 dislocation generation 2897 dislocation microstructure 827 dislocation nucleation 773 dislocation patterns 2897 dislocation theory 2695 dislocation-pressure interaction 2879 disorder at interfaces 1931 disorder temperature 2749 disordered crystalline alloy 349 disordered materials 349, 1217 dissipation 1151, 1281, 1635, 2773 dissipative particle dynamics (DPD) 2411, 2757 dissociation 1713 distributed computing 1837 distributed damage 2929 distribution of grain sizes 2837 distribution of nucleation sites 2397 dividing surface 629 DL POLY code 1901 DNA 77, 2619 DNA binding 259 domain 2083, 2843 domain wall, 90◦ 2843 domain wall, 180◦ 2843 doping 1877 double bridging 2583 double bridging Monte Carlo 2757 DREIDING 2561 drift 1635 drift-diffusion process 1635 driven alloys 2223 DSC lattice 1931 d-state directional bonding 2737 ductile 855 due ferro 1 (DF1) 259 dumbbell 1855 dynamic density functional theories 2757 dynamic fracture in n-Si3 N4 875 dynamic heterogeneities 2917 dynamic lattice liquid 2907 dynamical matrix 195, 349, 459, 649 dynamical scaling hypoethesis 2351 dynamics 1491, 2083 EAM potential 1931, 2055 Earth’s interior 2829 edge 1098, 1115, 2307 edge diffusion 2337 EDTB potential for molybdenum 307 EDTB potential for Si 307 Edwards–Wilkinson equation 2351 effective cluster interaction 349 effective Hamiltonian 349, 2645 effective lifetime 959

2952 effective medium theory 2737 effective potential 499, 1823 effective temperature 1281, 2749 eigenvalue 349 eigenvector following [method] 1573 Einstein crystal 349 elastic anomalies at interfaces 2025 elastic compatibility 2143 elastic constants 275, 459, 479, 729, 773, 1223, 1333, 2025 elastic effect 349 elastic energy 2117 elastic instability 2009, 2777 elastic stability 1223 elastic stiffness 773 elastic wave 649 elastically-mediated interaction 349 elasticity 649, 1371, 1529 elastodiffusion 959 electro-migration 1417 electronegativity equalisation 479 electron-hole interaction 215 electronic 2829 electronic band energy 349 electronic chemical potential 349 electronic density of states 349 electronic entropy 307, 349 electronic excitation 349 electronic excited states 2731 electronic free energy 349 electronic structure 13, 93, 137, 149, 349, 423, 1889 electronic structure, coarse grained 2737 electrostatic effect 349, 423, 479 element modeling 649 elliptic equations 2415 embedded atom method (EAM) 459, 1223, 1953, 1985, 2025, 2737 embrittlement 2719 empirical potential 499, 509, 855 empirical valence bond (EVB) method 241 end bridging Monte Carlo 2757 end-bridging 2583 end-of-life 2929 end-to-end distance 2575 end-to-end vector 2575 energy barrier 1585 energy cusp 1953, 2055 energy density functionals 137 energy function 509 energy localisation 2731 energy minimization 855, 1931 energy release rate 855 energy transfer 1713, 2731 ensemble 729, 2645

Index of keywords ensemble average 349, 707, 763 entropic elasticity 2631 entropy 349, 707, 1931 entropy production 707 environment dependence 2935 environmental effects on fracture 875 environment-dependent tight-binding (EDTB) potential 307 enzyme 1613 epitaxial growth 2337 equation of continuity 763 equation of state 2599 equation-free modeling 1453 equilibrium ensembles 589 equilibrium Monte Carlo 613 equilibrium properties 613 equilibrium volume 349 ergodic hypothesis 613 ergodic systems 1931 error bounds 1529 Eshelby’s virtual procedures 2117 Euler–Lagrange equation 649 evaporation 1769 evaporation–condensation 1389 evolution 1823, 2083 Ewald method 479, 1889 exact enumeration 1823 EXAFS 1901 excess energy 1985 excess energy profile 1985 exchange process 1627 exchange-correlation potential 1877 exciton 2731 excitonic effects 215 explicit solvent model 1837 explicit tau-leaping simulation procedure 1735 external free surface 1953 facets 1389 fast ion conductors 1901 fast marching methods 1359 fast multipole method 875 fast sweeping method 2307 fatigue 1071, 1193 fcc lattice 1953 fcc metals 2777 Fe(Cu) 2223 Fe(Nb,C) 2223 f-electron metals 2737 femtosecond laser pulse 307 FENE 2619 Fermi distribution 349 Fermi level 349 fermi surface 275

Index of keywords ferroelectrics 349, 527, 2843 Feynman propagators 1673 fiber-bundle models 1313 film growth 629 finite difference 2415 finite difference method 423 finite difference time domain 2671 finite element method 121, 423, 649, 663, 1529, 2173, 2415, 2663 finite size effects 2787 finite temperature 349, 1491 finite volume 2415 finite-time singularity 1313 Finnis–Sinclair model 2737 first generation radiopharmaceutical compounds 259 first principles 149, 349, 367, 423, 1877, 2707, 2865 first wall 2719 first-principles density functional theory 275 fitting 479 ‘flip’ instability 2777 Flory–Huggins theory 2599 flow defect 2749 flow in porous media 1507, 2487 fluctuation theorem 2773 fluctuation-dissipation theorem 1635, 2411 fluctuations 1635, 1735 fluid binary mixture 2787 fluid dynamics 2411 fluid flow 1507, 1529 fluid permeability 1333 fluorite structure 1901 flux 1635, 1787 flux autocorrelation function 1673 Fokker–Planck equation 1635, 2503 folding temperature 1823 force constant 349, 1877 force field 2561, 2707 forced oscillator model 1713 formation energies 1877 formation enthalpy 1855 formation entropy 1855 formation volume 1855 Fourier transforms 121, 423 Fourier’s law of thermal conduction 763 fractal 1417 fracture 839, 855, 1071, 1171, 2793 Frank–Read source 827, 2307 free energy 613, 683, 707, 1585, 1613 free volume 1281, 1953, 1985 free-energy perturbation 683 freezing 1613 Frenkel defects 1901 friction coefficient 1635

2953 front-tracking 2837 fuel cells 1915 functional integral 2645 funnel sampling 683 fusion energy 2719 fusion reactors 999 GaAs phonon dispersion 875 gamma-surface 855, 2883 Gaussians 121 Gaussian (“normal”) distribution 1635 Gaussian chains 2599 Gaussian curvature 2849 Gaussian Markovian stochastic process 1635 Gaussian random field 2849 Gaussian random force 1635 Gaussian stochastic process(es) 1635 Gaussian variable 1635 Gaussian white noise 1735 general grain boundary 1953, 2055 generalized configurational bias 2583 generalized gradient approximation 439, 1877 generalized Langevin equation 1673 generalized pseudopotential theory 2737 generalized stacking fault (GSF) energy 793 GENERIC framework 2503 genetic algorithm 307, 547, 1613 geometric glide plane 2695 germanium 855 grain boundary sliding and migration 1931 ghost forces 663 Gibbs–Duhem integration 683 Gibbs ensemble 683 Gibbs–Feynman identity 707 Gibbs free energy 2009 Ginzburg-Landau framework 2143 glass transition 1281, 1985 glass transition temperature 1823 glasses 2823 glide 1098, 1115, 2307 global climatic changes 1613 global minimum 613 global structure optimization 307 go model / go-like 1823 GPT 2737 gradient thermodynamics 2117 grain area distributions 2397 grain boundary (GB) 547, 1039, 1925, 1931, 1953, 1985, 2025, 2055, 2083 grain boundary area 2157 grain boundary diffusion 1953, 1985, 2055 grain boundary diffusion creep 1985, 2055

2954 grain boundary energy 1953, 1985, 2025, 2055, 2157 grain boundary fracture 1953 grain boundary inclination 2157 grain boundary migration 1985 grain boundary misorientation 2157 grain boundary mobility 1985, 2157 grain boundary self-diffusion 1985 grain boundary sliding 2055 grain boundary superlattice (GBSL) 2025 grain boundary width 2157 grain growth 1985, 2837 grain growth exponent 2157 grain junction 1953, 1985, 2055 grain shape 2055 grain size 2055, 2157 Granato model 1855 grand canonical ensemble 1931 graphite 2923 gray scale operations and filters 2397 Green’s function 1877 Green-Kubo formula 745, 763 Green’s function 2117 grid 2415 grid computing 875 grid methods 121 Griffith criterion 855 GROMOS 509 Grotthus mechanism 1901 ground state 349 growth 1769, 2117 Grüneisen parameter 349 GULP 1889, 1901 GW approximation 215 GW bandgap 215 GW-BSE approach 215 HADES 1889, 1901 Hamiltonian 2935 Hamiltonian matrix 121, 307 Hamilton’s equations of motion 763 handshake regions 875 hardening 1281 hardness 855 hard-sphere model 1953 harmonic approximation 349 harmonic functions 1417 harmonic transition state theory 629 Hartree potential 121 Hartree–Fock 1877 heat capacity 1823 heat flux 763 heat of formation 275 Hele–Shaw 1417 Helfrich free energy 2849

Index of keywords Hellmann–Feynman theorem 59, 195 Helmholtz 1529 Helmholtz inequality 707 Herring relation 2055 heterogeneous and homogeneous melting 2009 heterogeneous catalysis 149 heterogeneous materials 1217, 1333 heterogeneous microstructure development 959 heterogeneous multiscale method 1491 heteropolymer 1823 hexatic phase 1613 hierarchical modeling 649 hierarchical processes 2731 high friction limit 1635 high pressure 2737, 2829 high temperature limit 349 high-angle GB 1953, 1985, 2055 high-energy GB 1953, 1985, 2055 higher order finite difference expansions 121 high-temperature relaxed structure 1953, 1985 Hillert model 2157 histogram reweighting 683 histograms 1613 holonomic constraints 745 homogeneous 1613 homogeneous system 1635, 2025 homogenization 1333, 2325, 2379 HP model 1823 hybrids 2929 hybrid algorithms 1753 hybrid Car–Parrinello/MM molecular dynamics 259 hybrid energy 349 hybrid FE/MD scheme 875 hybrid FE/MD/QM scheme 875 hybrid formulation 2523 hybrid MD/QM scheme 875 hybrid multiscale modeling 649, 2763 hybrid quantum mechanics/molecular mechanics methods 241 hydrocarbons conversion 241 hydrodynamic interaction 2607, 2619 hydrodynamics 1403, 2513, 2523 hydrogen 1877 hydrogen bonding 1613 hydrogen tunneling 259 hydrophobic 1613 hydrostatic (compression/expansion) 1223, 2009 hyperbolic equations 2415 hyperdynamics 629

Index of keywords hyperfine parameters 1877 hypersonic flows 2811 ideal strength 439, 855, 2777 image analysis 2397 image force effect 2287 immiscible fluids 2487, 2503 implicit solvent model 1837 importance sampling 349, 613, 2583 impurities 1877 In melts 1985 InAs/GaAs square nanomesa 875 incoherent 2025 incubation 1193 incubation time for nucleation 2223 indented surface of Si3 N4 875 industrial applications 2819 industrial applications of materials modeling 2713 information theory 1477 infrared spectroscopy 195 infrequent event system 629, 1627 inherent structure(s) 1573 inhomogeneous 1613, 2025 instability 1371 integral materials simulation 2713 interatomic force constants 195 interatomic potential 241, 451, 459, 479, 499, 1901, 1931, 2731, 2763 interconversion of atoms 349 inter-diffusion 367 interface 1953 interface capturing 2205 interface fluctuations 2351 interface-plane method 1953 interface tracking 2205 interfacial 2025 interfacial curvatures 2849 interfacial dynamics 1217 interfacial energy 2157 interfacial fracture and dislocation emission 875 interfacial free energies 2787 interfacial profile 2787 interfacial width 2787 intergranular fracture 1931 interionic potentials 1889 intermediate scattering function 2823 intermetallic compounds and alloys 2737 intermetallics 2883 internal interface 1953 internal state variables 1077, 1183 internal variables 1151 interplanar spacing 1953 interstitial 2223

2955 interstitial diffusion 367 interstitial position 1877 interstitialcy 2223 intrinsic interfacial profile 2787 intrinsic profile 2787 intrinsic width 2787 invariance 1223 invariant volume element 589 inversion symmetry 1953 ion core 121, 613 ion implantation 649 ion polarizability 1901 ion transport mechanism 1901 ionic materials 479 ionization energy 1877 ion-pair potentials 241 IR spectra 259 irradiation 987 irradiation growth 959 irreversibility paradox 2773 irreversible aggregation 2337 irreversible process 707 irreversible work 707 Ising 2687 Ising models 2787 Ising-type models 1477 island dynamics 2337 island number density 2337 isostress molecular dynamics 1223 iterative diagonalization methods 121 IVR 1673 Jahn–Teller effect 2731 jammed state 1281 Jarzynski average 707 jump frequency 2223 jump Markov process 1735 JWKB approximation 1673 kinematics 1183 kinetic 137 kinetic equation 2249 kinetic Ising model 2223 kinetic Monte Carlo 149, 613, 629, 1627, 1753, 1769, 2083, 2223, 2351, 2363, 2757 kinetic pathway 2223 kinetic theory 2411, 2475, 2513 kinetics 959, 2083 kinetics (of reactions) 1585 kinetics of melting 2009 kink-pair mechanism 827 kink-pressure 2879 Kleinman–Bylander form 121 Knudsen number 2475, 2513, 2523

2956 Kohn–Sham eigenvalues 215 Kohn–Sham equation 121 Kolmogorov flow 2805 Kosevich energy functional 2325 k-point sampling 439 Kramer–Moyal–van Kampen expansion 2351 Kroger–Vink notation 1889 Kubo–Green 367 Lagrange multipliers 745 Lagrangian formulations 1281 Landau free energy 1613 Landau, Ginzburg, De Gennes 1613 Landau-type coarse-grained “chemical” free energy 2287 Langevin equation 649, 2351, 2411, 1635 Langevin leaping 1735 Langevin noise 2117 Langevin update formula 1735 LAPW 93 large strain mechanical response 1223 large-scale deformation 2749 laser ablation 307 latent heat and volume 2009 lateral interactions 149 lattice Boltzmann equation 2411, 2475, 2805 lattice dynamics 649, 1931, 2025, 2055 lattice friction 2897 lattice gas 1753, 2351 lattice gas automata 2805 lattice gas Hamiltonians 149 lattice Green’s function 649 lattice model 2599 lattice Monte Carlo 613 lattice protein 1823 lattice site 349 lattice statics 1223, 1931, 2025 lattice strain in melting 2009 lattice symmetry 349 lattice trapping 855 lattice vibration 349 layer-by-layer growth 2337 layered and tunnel structures 1901 leap condition 1735 leap-frog algorithm 565 Lee–Edwards boundary condition 745 legacy codes 1453 length and time scales 2663 length scales 1077 Lennard Jones potential 565, 1953, 1985, 2025 level set methods 1359, 2337, 2083, 2307 level surface 2849

Index of keywords life management 2929 life-managed 2929 limitations of atomistic simulations 451 Lindemann criterion 1985 linear inequality 349 linear programming 349 linear regression 395 linear response theory 195, 349, 707, 745 linear scaling 77, 649 line–tension–pressure interaction 2879 link atoms 241 Liouville equation 745, 763 Liouville operator 589 lipid bilayers 929 lipid membrane structure 929 liquid 349 liquid simulations 2819 liquid–liquid phase transition 2917 liquid–solid interface 2009 LMTO 93 load balancing 875 local density approximation 121, 439, 1877 local expansion at interfaces 1931 local processing 2929 local property 2929 local pseudopotentials 137 local rules 2897 localization 1713 localized basis sets 423 localized orbitals 77 localized vibrational modes 1877 lognormal 2397 long range elastic fields 2763 long-range interactions 241 long-range interatomic interaction 349 long-range order parameter 2117 long-range structural order 1953, 1985 look-up tables 1753 low viscosity simulation 1837 low-angle GB 1953, 1985 low-dimensional continuum models 2631 low energy 1953, 2055 low-energy GB 1985 lubrication forces 2607 macromolecular architectures 2907 macroscale 1071 macroscopic 1953 macroscopic (modeling) 2757 magnetism 275 manganese catalase 259 many-body expansion 499 many-body perturbation theory 215 many-electron Green’s function 215 mapping 1039

Index of keywords Markov chain 2583 Markov process 1477, 1735 Markovian stochastic processes 1635 martensitic phase transformation 349, 2117 mass matrix 649 mass-action equations 1735 master equation 613, 1281, 1635, 1753, 2351 materials by design 2667 materials failure 2793 materials for fission and 999 materials modeling 1217, 2707 materials processing 1529 materials simulation 2935 mathematical methods 1217 matrix-free (numerical methods) 1453 Maxwell’s equations 2671 mean curvature 2849 mean field theory 2787 mean free path 2513 mean occupation 2249 mean square curvatures 2849 mean-field methods 349 mean-squared displacement 1931, 1985, 2055 measure 745 mechanical deformation 439 mechanical embedding 241 mechanical properties 2173, 2737 mechanism 1613, 1985, 2687 melting 2009 melting at interfaces 1931 melting point 349 membrane dynamics 929 membrane transport 929 membranes 2631, 2675 memory kernel 649 memory time 1635 Mermin order parameter 1613 mesh 2415 meshless method 2447 mesophase 2923 mesoscale 1071 mesoscale phenomena 2411, 2487 mesoscopic (modeling) 2731, 2749, 2757 metallic glasses 1281, 2749 metallic hydrogen 2829 metallization 2829 metals 451, 2173 metastable phase 2009 metastable phase diagram 349 metastable states 1613 methane monooxygenase 259 Metropolis 2363

2957 Metropolis algorithm 349, 613, 1585, 1823, 1931, 2583 Metropolis-type dynamics 1477 MgO 1627 MGPT 2737 microemulsion 2645, 2849 micro–macro hybrid methods 2903 microphotonics 2671 microporous materials 547 microscopic 1953 microscopic electro-mechanical systems (MEMS) 2475 microscopic reversibility 613 microscopic state 349 microstructural evolution 2087 microstructurally small crack 1193 microstructure 999, 1193, 1333, 1371, 2083, 2117, 2269, 2379, 2687, 2871 microstructure evolution 2157, 2249 microstructure, ferroic 2143 migration barrier 1877 migration enthalpy 1855 migration entropy 1855 minimum energy path 773 mirror symmetry 1953 misfit 2307 misfit energy 793 misfit localization 1953 misfit strain 2287 mixed basis 2737 mixed quantum classical [time evolution methodologies] 1673 MMFF 509 mobility 1877, 2307 mobility constant 745 mode-coupling theory 2823 model order reduction 1529 modeling 2173, 2269, 2871 modeling of excited states 259 modeling of interfaces 1925 modified embedded atom method 459, 855 modulation wavelength 2025 molecular assemblies 2675 molecular complexity 2907 molecular dynamics (MD) 275, 451, 527, 565, 629, 649, 707, 729, 813, 839, 855, 1039, 1491, 1613, 1627, 1735, 1787, 1837, 1877, 1889, 1901, 1931, 2523, 2663, 2737, 2793, 2411, 2475, 2487, 2503, 2707 molecular dynamics and Monte Carlo comparison 613 molecular dynamics simulation 349, 509, 929, 1585, 1597, 1797

2958 molecular dynamics simulation of melting 2009 molecular mechanics 2763 molecular mechanics force fields 241 molecular modeling 509 molecular statics 839 molten sub-lattice model 1901 monazite 2929 mon-equilibrium molecular dynamics 745 Monte Carlo 451, 707, 729, 1039, 1931, 2663, 2687 Monte Carlo [path] 1613 Monte Carlo [procedure] 1585 Monte Carlo simulation 349, 1901, 2583, 2599 morphological and microstructural evolution 1217 morphological evolution 2351 morphology 2083 Morse model 1223 Mott–Littleton 1889 Mott–Littleton procedure 1901 multicanonical Monte Carlo 2787 multicomponent alloy 349 multifunctionality 2379 multilayer growth 2337 multimillion atom molecular dynamics simulations 875 multi-paradigm 2707 multiphysics 2475 multiplicity 349 multipole expansion 2325 multiscale 2523, 2707 multiscale analysis 1491, 1507 multiscale computation 1507 multiscale gas dynamics 2811 multiscale modeling 149, 793, 999, 1217, 2737, 2657 multiscale modeling of polymers 2757 multiscale visualization 875 Nabarro–Herring creep 2055 nanoclusters 547 nanocrystalline 839 nanocrystalline material 1985, 2055 nanocrystals 1925 nanodiamond 307 nano-forms of carbon 2923 nanoindentation 2777 nanoindentation of silicon nitride 875 nanomechanics 649 nanophase SiC 875 nanophotonics 2671 nanostructured a-SiO2 875 nanostructured ceramics 875

Index of keywords nanostructured materials 875 nanostructured Si3 N4 875 nanostructures 215 nanostructures in solution 2701 nanotube 307, 2923, 1797 native point defects 1877 native-centric 1823 natural convection 1529 Navier–Stokes 1529 Navier–Stokes equations 2411, 2475, 2805 NDDO 27 neighbor distribution 2397 neighbor list 565 nematic order parameter 1613 NEMS 649 neural network 395 neutral net 1823 Newtonian equations of motion 1931 Newtonian fluids 2503 n-fold way 2363 Ni(Cr,Al) 2223 nickel 1223 NMR 1901 NMR spectra 259 nonbonded interaction 2561 non-collinear magnetization 275 non-Debye relaxation 2731 non-equilibrium alloy 2249 non-equilibrium free-energy calculation 707 non-equilibrium processes 1753 non-equilibrium states 745 non-equilibrium work 683 non-Hamiltonian dynamical systems 589 non-Hamiltonian system 745 non-linear perturbations 745 nonlocal pseudopotentials 423 nonlocality 121 non-Markovian 1635 non-Markovian dynamics 1635 non-Newtonian fluids 2503 nonorthogonal tight-binding 307 nonperiodic systems 121 non-planar core 2883 non-reactive collisions 1735 nonstoichiometry 1851 norm conserving 121 normal modes 195, 349, 1635 normal random variable 1735 Nose–Hoover chains 589 nucleation 1098, 1115, 1613, 2117, 2223, 2337 nucleation and growth 2397 nucleation barrier 2223 nucleation of voids and dislocation loops 959

Index of keywords nucleation rate 2223 nucleation sites in melting 2009 nudged elastic band 773 numerical analysis 1453 O(N) multiresolution algorithms 875 observable variables 1151 occupation variable 349 offline–online procedures 1529 Omori law 1313 one-dimensional diffusion 959 ONIOM method 241 Onsager’s regression hypothesis 707, 745 on-the-fly algorithms 1753 on-the-fly kinetic Monte Carlo 1627 open systems 2731 operator resummation 1673 OPLS 509, 2561 optical properties 215 optical properties of silicon dots 2701 optimization 613, 2083, 2379 orbital-free density functional theory 137 order parameter 683, 1585, 1613 order-disorder transition 2645 ordered phase 349 ordering 2223, 2249 order-N 77, 565 orientation variants 2117 Orowan looping 2307 orthogonal tight-binding 307 osmotic permeability 1797 Ostwald ripening 2337 output bounds 1529 overdamped limit 1635 overlap matrix 307 oxidation of alkanes 259 oxidation of aluminium nanoparticles 875 oxygen diffusion 1915 oxygen molecule 121 pair correlation function 2325, 2397, 2583 pair potentials 499 parabolic equations 2415 parallel (multiscale modeling approaches) 2757 parallel calculations 423, 2487 parallel molecular dynamics 875 parallel replica dynamics 629, 1627 parallel tempering 1613 parallel-replica 629, 1627 parametrized problems 1529 parent lattice 349 partial charge 2561 partial differential equations 1529, 2415, 2447

2959 partial least squares 395 partial occupation 349 particle and radiation transport 613 particle-based 2513 particle bypass 2307 particle-laden flows 2607 particle tracking 451, 613 particle trajectories 613 partition function 349, 707, 2645 path integral molecular dynamics 259 path-integral propagator 1691 path integral quantum transition state theory 1713 path sampling 1837 pattern formation 2117 pattern recognition 1613 Peach–Koehler force 1098, 1115 Peclet number 2475, 2607 Peierls stress 793, 813, 2865, 2883 Peierls–Nabarro model 793, 2287, 2695 ‘pencil glide’ 2777 peptide bundle, 4-α-helix 259 perfect crystal 2025 periodic boundary condition 241, 423, 565, 773, 813, 1051, 2695 perovskite oxides 527, 1915 Pettifor map 395 phase behavior 527 phase coexistence 2009 phase diagram 149, 349 phase equilibria 683 phase field 2083, 2105, 2117, 2157, 2287, 2837 phase-field models 2087 phase-field simulation 2157 phase separation 2249 phase-separated polymer blends 2849 phase space 349, 683, 1635 phase space flow 745 phase space sampling 613 phase stability 349 phase transformation 1223, 2117 phase transitions 149, 1613, 2829 phonons 195, 1713 phonon density of states 349 phonon dispersion 459 phonon frequencies 275, 349 Photobacterium leiognathi 1597 photonic band gap 2671 photonic crystals 2671 physical properties 2379 π-back donation 259 planar core 2883 planar stacking 1953 planar structure factor 1953, 1985

2960 plane waves 59, 121, 195 plane-wave basis 93 plane-wave method 423 plastic deformation 663, 793, 1985, 2055, 2713 plastic spin 1133 plasticity 1071, 1151, 1281, 2173, 2269, 2749, 2793, 2865, 2871, 2929 plateau value 1573 point charge model 1901 point collocation 2447 point defects 1851, 1855 Poisson–Boltzmann 1837 Poisson effect 2025 Poisson process 1797 Poisson random numbers 1735 Poisson random variable 1735 polarization 479, 2561 polycarbonate 2675 polycrystals 1953, 1985 polycrystalline 839, 1039, 2837 polydispersity 2583 polyethylene 2575 polyhedral unit model 1953 polymer blends 2599, 2645, 2787 polymer glass 2599 polymer melts 2599 polymer mixtures 2787 polymer solutions 2599, 2645 polymeric fluids 2503 polymers 2173, 2675 potential cutoff 565 potential energy function 509, 2561 potential energy surface 149, 1713 potential of mean force 2645, 2757 Potts model 2687, 2837 powder metallurgical processing 2713 power laws 1313 precipitation 2117 predictive simulations 27 predictor–corrector algorithm 565 premelting 1985 premelting at surface 2009 pressure distribution in a Si/Si3 N4 nanopixel 875 principal component analysis 395 probabilistic clustering 1573 problem posing 2731 process simulation 2713 processing 2173 processing–structure–property relationships 395 production bias 959 projection strategies (in coarse-graining) 2757

Index of keywords projector augmented wave method (PAW) 93 propensity function 1735 protein data bank 1051 protein folding 1837 protein-based nanostructures 875 proton conductors 1901 proton exclusion 1797 proton mobility in zeolites 241 proton transport 1915 prototype sequence 1823 pseudopotential approximation 439 pseudopotential model 1223 pseudopotential perturbation theory 2737 pseudopotentials 13, 59, 93, 121, 195, 423, 1877 Pt chemical shift 259 Pulay forces 59, 121, 195 pyrolsis 2923 pyrosilicic acid 27 QM/MM 259, 241, 2675, 2763 QM-Pot method 241 quadrature domains 1417 quantitative structure-activity relationships 395 quantitative structure–property relationships 395 quantum-based interatomic potentials 2737 quantum dots 121, 649 quantum effects 1713 quantum entanglement 2731 quantum information processing 2731 quantum Kramers [approach] 1673 quantum mechanical rate constants 1691 quantum mechanics 27, 349, 2707, 2763 quantum molecular dynamics 2737 quantum Monte Carlo 2701 quantum reflection 1713 quantum rods and tetrapods 875 quantum simulations 451, 2701 quantum statistics 2731 quantum tunnelling 2731 quasi-atomic minimal-basis-sets orbitals (QUAMBOs) 307 quasicontinuum 649, 663, 1491 quasi-elastic neutron scattering 1787 quasiharmonic approximation 349, 1931 quasiparticle calculations 1877 quasiparticle excitations 215 quasiparticle lifetime 215

Index of keywords radial distribution function 1901, 1953, 1985 radial distribution function at interfaces 1925, 2025 radiation damage 613, 649, 999, 1627 radiation effects 999, 2719 radiopharmaceutical compounds 259 Raman spectra 195, 259 random deposition 2351 random-dissipative forces 649 random force 1635, 1673 random numbers 613, 1635 random sampling 613 random walk 1585, 1635, 1787 rare event 1585 rare reactive events 1597 rare transitions 1613 ratcheting 1193 rate catalog 1627 rate dependent 1133 rate equations 2337 rate independent 1133 rate table 1627 Rayleigh and Gamma probability densities 2397 Rayleigh scattering 649 Rayleigh–Taylor 1417 Rayleigh wave speed 855 reaction coordinates 1585, 1635, 1597 reaction force field 241 reaction mechanism 1585 reaction of H2 O molecules at a silicon crack tip 875 reaction rate constants 1585, 1691, 1735 reaction rate equations 1735 reaction time 1585 reactive collisions 1735 reactive flux correlation function 1597 reactive force fields (ReaxFF) 2707 Read–Shockley model 1953 real space 121 real space methods 121, 423 real-time computation 1529 realizations (stochastic trajectories) 1635 reciprocal space 121 reduced basis 1529 reduced coordinates 1051 reduced descriptions 1635 reduced dimensional systems 215 reduced dynamics 1635 reduced unit system 565 re-emission 1359 reference system 349 refinement 2447, 2523 reflection coefficient 649

2961 refractory metals 2865 relative fluctuations 1735 relaxation 1953 relaxation time 1635 relaxation volume 1855 renormalization 2687 renormalization group 1613, 2351 reptation 1901, 2583, 2599, 2675 reversible aggregation 2337 reversible glass transition in high-energy grain boundaries 1985 reversible process 707 reversible work 707, 1585 Reynolds number 2475 rezoning 2903 rheology 2607 Rice criterion 855 rigid-body translation 1953, 2055 ROCKS 259 rolling 2173 rotation 1953 rotational excitation 1713 rotational isomeric state 2575 roughening transition 1389 Rouse dynamics 2599 rupture prediction 1313 saddle points 1585, 1877, 2223 sampling strategies (in coarse-graining) 2757 scalable data-compression scheme 875 scalable simulation 875 scale-bridging 2687 scale-bridging simulation 2675 scaling 2687 scaling properties of fracture surfaces 875 scattering 649, 1713 scheelite 2929 Schmid law 2883 screened Coulomb interaction 215 screw dislocation 1098, 1115, 2307 screw dislocation in bcc molybdenum 307 second law 2773 sedimentation 2607 segregation 1931 self-avoiding walks 2599 self-consistency 2935 self-consistency cycle 121 self-consistent field (SCF) 2645, 2757 self-diffusion 367, 1985 self end-bridging 2583 self-energy operator 215 self-limiting growth 875 self-organization 2117 self-similarity 1403

2962 self-trapping 2731 semiclassical mechanics 1673 semiconductor etching and deposition 1359 semiconductor growth 149 semiconductor nanostructures 2701 semiconductors 855, 1877, 2737 semi-discrete variational Peierls–Nabarro model 793 semi-empirical 2935 semi-empirical potentials 839 semi-empirical quantum chemistry 27 sequential (multiscale modeling approaches) 2757 sequential multiscale modeling 649 shape function 649 shape selectivity 241 shape transition 2117 shear deformation 2009 shear failure 2777 shear instability 2009 shear localization 2749 shear modulus 855, 2009 shear-rate dependence 745 shear softening 1281 shear strength 2777 shear stress 1223 shear-transformation zone 2749 shear viscosity 745 shell model 527, 1889 shock waves 2829 shooting and shifting [algorithms] 1585 short-range interactions 2575 short-range order 349, 1953, 1985 Si/Si3 N4 interface 875 Si3 N4 phonon density of states 875 SIESTA 77 silica stress and strain curves 27 silicon 499, 613, 855, 2009 silicon (110) 855 silicon (111) 855 silicon cluster 307 silicon nitride 855 silver 629 simple ionic solids 349, 547, 613, 1889 simulated annealing 613 simulations 275, 729, 2173, 2363 simulations of nanostructured materials 875 single crystal plasticity 827 single exponential kinetics 1837 single molecule spectroscopy 1635 single particle distribution function 2513 single-file water transport 1797 singularities 1403 sink competition 959 sintering of n-SiC 875

Index of keywords sintering of silicon nitride nanoclusters 875 size effects 2897 Slater–Koster theory 307 slip 793 Smoluchowski equation 1635 smooth particles 2903 smoothed particle hydrodynamics 2503 soft matter 2675 soft matter properties 2675 soft mode 2787 solid 349 solid electrolytes 1901 solid oxide fuel cells 1901 solid polymer electrolytes 1901 solid with a phase transition 1901 solidification 2087, 2105 solid–liquid interface 2009 solid–liquid phase boundary 349 solid-on-solid model 2337 solid-state amorphization 2055 solubility 1877 solute 349 solvent 349, 1613 solvent model 1837 Sommerfeld model 349 SPAM 2903 spatial decomposition 875 special GB 1953 special grain-boundary plane 1953 specific reaction probability rate constant 1735 spectral density 1673 spectral density function 2849 spectroscopic properties 1851 sph 2903 spherical chicken 2083 spinodal decomposition 2487 splitting algorithm 2513 static structure factor 1931 stability 2415 stacking fault 1953 stacking period 1953 stacking sequence 1953 standard model 13 state-change vector 1735 static heterogeneities 2917 static lattice 547 static lattice calculations 1889, 1901 stationary 1635 statistical mechanics 149, 729 statistical sampling 613 statistical weight matrix 2575 statistics of point processes 2397 steepest descent sampling 2645 Steinhardt order parameter 1613

Index of keywords steps 1389 STGB cusp 1953 sticking 1713 stiffness 1735 stiffness matrix 649 Stillinger–Weber potential 499, 855 stochastic 1635, 1735, 2513 stochastic differential equations 2503, 2619 stochastic dynamics 613 stochastic effects 959 stochastic equations of motion 1635, 2083 stochastic mesoscopic models 1477 stochastic PDE 1477 stochastic simulation algorithm 1735 stochastic time evolution 1635 stochastic trajectory 1635 stochastic variable 1635 Stokesian dynamics 2607 strain 773 strain faceting 2337 strength 2777 stress 439, 773, 2205 stress-free transformation strain 2117 stress intensity factor 839 stress–strain curve 827 stress tensor 459, 2619 structural 2737 structural components 2929 structural disorder 1953, 1985 structural instability 2009 structural materials 2719 structural phase transitions 195, 1985, 2009 structural properties 2737 structural relaxation 1931 structural transformation in GaAs nanocrystals 875 structural width 1985 structure 13, 349 structure map 395 structure–property correlations 1925, 1931, 1953, 2025, 2055, 2675 submonolayer growth 2337 substitutional diffusion 367 subtraction scheme 241 subtraction technique 745 super soft elastomers 2907 supercell 565, 813, 1051, 1877, 1889 superconductivity 2829 supercooled liquids 2009, 2823 supercooled melt 1985 supercooled water 2917 superfunnel 1823 superhard materials 2829 superheating 2009 superheating limit 1985

2963 superionic conductors 1901 superlattice 2025 super-plasticity 1281 surface 855, 1953 surface chemistry 2337 surface diffusion 629, 1389, 1627 surface exchange mechanism 1627 surface growth 629, 2351 surface hopping 1673 surface roughness 2337, 2351 surface simulations 547 surface steps 1953 surface structure 149 surface tension 1403 surface-state excitons 215 suspensions 1371, 2607 Suzuki–Yoshida factorization scheme swelling 2719 switched Hamiltonian 349 switching parameter 707 symmetric GB 1953 symmetric tilt GB 1953 symmetry breaking 2009 symplectic integration 565, 589 system instability 959 systems theory 1453

589

tau leaping 1735 temperature accelerated dynamics 629, 1627 temperature-dependent elastic constants and moduli 2025 temperature-dependent potentials 2737 tensile strength 2777 tensor 1183 Tersoff model 499 tertiary creep 1313 tetrahedral order parameter 1613 texture 1039, 1133, 2173, 2837 theoretical strength 1223 thermal activation 1223 thermal conduction 763 thermal conductivity 2819 thermal defect 349 thermal expansion 349, 459, 2659 thermal properties 707, 2659 thermally activated processes 2897 thermodynamics 1151, 2083, 2773 thermodynamic and mechanical melting 2009 thermodynamic driving force 707 thermodynamic factor 367 thermodynamic integration 259, 349, 683, 707, 1855 thermodynamic limit 1735

2964 thermodynamic path 707 thermodynamic phase diagram 2009 thermodynamic stability 349 thermo-elastic behavior of interfaces 2025 thermo-mechanical behavior 2055 thermostatted molecular dynamics 589 theta state 2575 thin films 1039, 2363, 2837 thin films, structure and elastic behavior of 2025 thin film growth 1753, 2337 thin-film interfaces 1925 threshold displacement energy 987 threshold effects 1713 tight-binding 275, 855, 1877, 2737 tight-binding Hamiltonian 307, 451 tight-binding model for carbon 307 tight-binding molecular dynamics 307 tilt 1953 tilt axis 1953 tilt boundary structures in Si 307 tilt GB 1953, 2055 time average 763 time correlation function 763, 1635 time dependent DFT (TDDFT) 259 time evolution of dislocation motion 875 time integration 2415 time-dependent Ginzbur–Landau equation 2287 time-dependent rate coefficient 1597 time–temperature–transformation (TTT) diagrams 395 topology 2157 total energy 423 total-energy calculation 349 total energy functional 93 total energy surface 1877 tracer 1901 tracer (self-diffusion) coefficient 1787 trajectory in phase space 763 transfer Hamiltonian 27 transitions 2143 transition level 1877 transition metals 2737 transition metals containing zeolites 241 transition path ensemble 1585 transition path sampling 1585 transition rates 613, 1635 transition state 1585, 1613, 1635 transition state ensemble 1585 transition state rate 149,1635 transition-state theory 629, 1573, 1627, 1635, 1673, 2757 translational 1953 translational order parameter 1613

Index of keywords transport coefficient 745, 763 trapping constant 1333 trimolecular reactions 1735 triple junctions 2055, 2157 triple lines 2055 Trotter theorem 589 Tsallis statistics 1613 TVD Runge–Kutta 2307 twin boundary 2843 twinning plane 1953 twist 1953 twist GB 1953, 2055 two-center approximation 307 two-phase flow 1403 two-phase microstructure 2205 two-phase region 1985 two-state model 2025 two-state systems 1281 ultrasoft pseudopotential 59, 195 umbrella sampling 613, 683, 1613, 2787 uniaxial loading 1223 unimolecular reactions 1735 unique properties of molecular dynamics and Monte Carlo 451 unit-cell area 1953 unit-cell volume 1953 UNIVERSAL 2561 unmixing 2223 unperturbed state 2575 unstable mode 349 unstable stacking energy 855 unstructured mesh 649 upscaling 1507 vacancies 1901, 1985, 2223 vacancy formation energy 275 vacancy source, sink 2223 valence interaction 2561 variable charge MD simulation 875 variable connectivity Monte Carlo 2583 variance 1635 variance reduction 2619 variational transition state theory 1635 velocity gap 855 velocity Verlet algorithm 565 velocity Verlet time integration 649 Verlet algorithm 565 vertex tracking 2837 vesicles 2631 vibrational energy relaxation rate constants 1691 vibrational entropy 349 vibrational free energy 349 vicinal GB 1953

Index of keywords vicinal surface 1953 virtual environment 875 viscosity 2675, 2815 viscous fingering 1417, 2805 viscous sintering 1417 visibility 1359 visualization 451, 1051 visualization algorithms 875 void coalescence 1171 void growth 1171 void nucleation 1171 volumetric relaxation 2157 von Mises strain 1051 Voronoi construction 839 Voronoi diagram 2503 waiting-time distribution 2397 water 1613, 2917 water channels 1797 water nucleophilic attack 259 water/silica interactions 27

2965 water/water interactions 27 wave reflection 649 weight function 2415 weighting function 2447 WENO 2307 width 1953, 1985 Wolf–Villain model 2351 wormlike chain 2619 wrapper, computational 1453 yield stress 827, 1281, 2749 yield surface 1077, 2173 Zeldovich factor 2223 zeolite catalysts 241 zero-point vibrations 1931 zero-temperature relaxed structure 1985 Zwanzig Hamiltonian 1673

1953,

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