MATHEMATICAL RESEARCH IN MATERIALS SCIENCE

Opportunities and Perspectives

Committee on the Mathematical Sciences Applied to Materials Science

Board on Mathematical Sciences

Commission on Physical Sciences, Mathematics, and Applications

National Research Council

National Academy Press
Washington, D.C. 1993



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Mathematical Research in Materials Science: Opportunities and Perspectives MATHEMATICAL RESEARCH IN MATERIALS SCIENCE Opportunities and Perspectives Committee on the Mathematical Sciences Applied to Materials Science Board on Mathematical Sciences Commission on Physical Sciences, Mathematics, and Applications National Research Council National Academy Press Washington, D.C. 1993

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Mathematical Research in Materials Science: Opportunities and Perspectives NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance. This report has been reviewed by a group other than the authors according to procedures approved by a Report Review Committee consisting of members of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and technical matters. Dr. Bruce Alberts is president of the National Academy of Sciences. The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as a parallel organization of outstanding engineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers. Dr. Robert M. White is president of the National Academy of Engineering. The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, upon its own initiative, to identify issues of medical care, research, and education. Dr. Kenneth I. Shine is president of the Institute of Medicine. The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy’s purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of Medicine. Dr. Bruce Alberts and Dr. Robert M. White are chairman and vice chairman, respectively, of the National Research Council. Support for this project was provided by the Department of the Army, Army Research Office, and the National Science Foundation. The content of this report does not necessarily reflect the position or the policy of the government, and no official endorsement should be inferred. Library of Congress Catalog Card Number 93-84439 International Standard Book Number 0-309-04930-X Copyright 1993 by the National Academy of Sciences. All rights reserved. Additional copies of this report are available from: National Academy Press 2101 Constitution Avenue, N.W. Washington, D.C. 20418 B-157 Printed in the United States of America

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Mathematical Research in Materials Science: Opportunities and Perspectives COMMITTEE ON THE MATHEMATICAL SCIENCES APPLIED TO MATERIALS SCIENCE AVNER FRIEDMAN, University of Minnesota, Chair I.-WEI CHEN, University of Michigan MORTON M. DENN, University of California at Berkeley KARL F. FREED, University of Chicago JAMES E. GUBERNATIS, Los Alamos National Laboratory RICHARD D. JAMES, University of Minnesota ALEXANDER KAPLAN, Johns Hopkins University WILLIAM W. MULLINS, Carnegie Mellon University SOKRATES T. PANTELIDES, IBM T.J. Watson Research Center FRANK STILLINGER, AT&T Bell Laboratories JEAN E. TAYLOR, Rutgers University Staff JOHN R. TUCKER, Senior Program Officer

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Mathematical Research in Materials Science: Opportunities and Perspectives BOARD ON MATHEMATICAL SCIENCES SHMUEL WINOGRAD, IBM T.J. Watson Research Center, Chair JEROME SACKS, National Institute of Statistical Sciences, Vice Chair LOUIS AUSLANDER, City University of New York System HYMAN BASS, Columbia University LAWRENCE D. BROWN, Cornell University AVNER FRIEDMAN, University of Minnesota JOHN F. GEWEKE, University of Minnesota JAMES GLIMM, State University of New York at Stony Brook GERALD J. LIEBERMAN, Stanford University PAUL S. MUHLY, University of Iowa RONALD F. PEIERLS, Brookhaven National Laboratory DONALD ST. P. RICHARDS, University of Virginia KAREN K. UHLENBECK, University of Texas at Austin MARY F. WHEELER, Rice University ROBERT J. ZIMMER, University of Chicago Ex Officio Member JON R. KETTENRING, Bell Communications Research Chair, Committee on Applied and Theoretical Statistics Staff JOHN E. LAVERY, Director RUTH E. O'BRIEN, Staff Associate JOHN R. TUCKER, Senior Program Officer BARBARA WRIGHT, Administrative Assistant

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Mathematical Research in Materials Science: Opportunities and Perspectives COMMISSION ON PHYSICAL SCIENCES, MATHEMATICS, AND APPLICATIONS RICHARD N. ZARE, Stanford University, Chair RICHARD S. NICHOLSON, American Association for the Advancement of Science, Vice Chair JOHN A. ARMSTRONG, IBM Corporation (retired) SYLVIA T. CEYER, Massachusetts Institute of Technology GEORGE W. CLARK, Massachusetts Institute of Technology AVNER FRIEDMAN, University of Minnesota SUSAN L. GRAHAM, University of California at Berkeley ROBERT J. HERMANN, United Technologies Corporation NEAL F. LANE, Rice University HANS MARK, University of Texas at Austin CLAIRE E. MAX, Lawrence Livermore National Laboratory CHRISTOPHER F. MCKEE, University of California at Berkeley JAMES W. MITCHELL, AT&T Bell Laboratories JEROME SACKS, National Institute of Statistical Sciences A. RICHARD SEEBASS III, University of Colorado at Boulder CHARLES P. SLICHTER, University of Illinois at Urbana-Champaign ALVIN W. TRIVELPIECE, Oak Ridge National Laboratory NORMAN METZGER, Executive Director

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Mathematical Research in Materials Science: Opportunities and Perspectives PREFACE This report is the product of the second phase of a two-phase study by the Committee on the Mathematical Sciences Applied to Materials Science, a committee convened by the Board on Mathematical Sciences (BMS). It builds on the committee's short phase-one survey,1 which (along with a briefing) was produced in response to a National Science Foundation (NSF) request. That report briefly described general mathematical theory and techniques that have been or show promise of being fruitful for ongoing and future materials science research. It was primarily aimed at and distributed to federal agencies that fund mathematical sciences and materials science research. This more comprehensive technical report documents and presents technical details of fruitful past collaborations between the mathematical sciences and materials science, and it indicates which particular areas of mathematical sciences research hold the most promise for advancing materials science. Materials research is now undergoing a transformation into a quantitative science.2 Although interaction between the mathematical sciences and materials science is increasing, many researchers in both communities are unaware that fruitful collaborations are possible and that a broad mathematical theory of materials is already being developed. However, materials science has been a prominent theme of several recent mathematics professional society meetings. Also, materials and processing have become the focus of a major cross-government initiative3 because they are critical to the success of industries such as the aerospace, automotive, biomaterials, chemical, electronics, energy, metals, and telecommunications industries. In light of the subject's timeliness, and to follow up and build on the brief survey prepared for NSF, the BMS chose materials science as the focus for a BMS cross-disciplinary report. This is one of a series of BMS reports that highlight areas on the interface between the mathematical sciences and other fields. The purpose of this report is not only to focus on directions for potentially promising collaboration between materials scientists and mathematical scientists, but also to encourage both communities to increase such collaborations. It is written primarily for mathematical and materials science researchers with an interest in advancing research at this interface, as well as for federal and state agency representatives interested in encouraging such collaborations. The opening and closing chapters (1 and 9) are intended for any persons wanting general information on how such cross-disciplinary, collaborative efforts can be successfully accomplished. To articulate the many mathematical challenges faced by materials scientists, the committee asked a large number of researchers (see appendix) to provide short write-ups briefly describing materials science research areas and identifying mathematical challenges in those areas. The committee incorporated the information received into the committee's descriptions and perspectives presented here. This report emphasizes that both the mathematical sciences and materials science communities have much to gain from an increase in cross-disciplinary collaborations, and it presents the committee's recommendations for facilitating mathematical sciences research that bears on important issues in materials science, including recommendations on how to attract students and young

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Mathematical Research in Materials Science: Opportunities and Perspectives researchers to this area. These recommendations are general and are not intended to be a detailed "blueprint" for action. It is hoped that this report will encourage research directions in the mathematical sciences that complement vital materials science research, as well as raise awareness of the value of quantitative methods in materials science. The committee is very grateful to the anonymous reviewers who provided excellent feedback in a short time, and to the many individuals who contributed information at the request of the committee. These colleagues strengthened this report significantly. NOTES 1.   National Research Council. 1991. Applications of the Mathematical Sciences to Materials Science. Board on Mathematical Sciences. Washington, D.C.: National Academy Press. 36 pp. 2.   See, for example, National Research Council, 1989, Materials Science and Engineering for the 1990s, Board on Physics and Astronomy, and National Materials Advisory Board, Washington, D.C.: National Academy Press. 3.   Federal Coordinating Council for Science, Engineering and Technology. 1992. Advanced Materials and Processing: The Fiscal Year 1993 Program. Committee on Industry and Technology. Washington, D.C.: Office of Science and Technology Policy.

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Mathematical Research in Materials Science: Opportunities and Perspectives CONTENTS 1   SUMMARY AND OVERVIEW   1 2   ATOMIC SCALE   5 3   MACROMOLECULAR STRUCTURES   11     Introduction   11     Single-Chain Conformations   12     Modeling Protein Structure and Dynamics   15     Entanglements, Reptation, and Elasticity   18     Constitutive Equations   18     Existence, Well-Posedness   20     Numerical Methods and Singularities   21     Sharkskin and Spurt Flow   22     Flow Instabilities   23     Micromechanics   24     Theory of the Liquid State of Polymers   24     Interfaces in Polymer Systems   27     Block Copolymers   30     Stiff Polymers and Liquid Crystals   32     Other Problems   33 4   EVOLUTION OF MICROSTRUCTURES   34     Introduction   34     Spinodal Decomposition and Nucleation   34     Grain Growth and Other Interface Motion Controlled by Interface Kinetics   35     Computer Algorithms   38     Shape Evolution Controlled by Surface Diffusion   38     Morphological Stability   39     Phase Transformations and Pattern Formation   40     Dendritic Growth   42     Mushy Zones   43     Precipitation and Coarsening   43     Evolution of Microstructures; Stress and Current Effects   45     Martensite and Shape-Memory Materials   45     Magnetic Materials   48     Superconductivity   49

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Mathematical Research in Materials Science: Opportunities and Perspectives 5   DEFECTS, DEFORMATION, AND INTERFACES   52     Introduction   52     Development of Mesoscale Statistical Mechanics of Solids   54     Mechanics of Defects and Interfaces   54     Plasticity and Fracture   55     Large Local Field-Induced Instability in Random Systems   57     Dynamic Fracture   58     Liquid Crystals   59     Equilibrium and Nonequilibrium Surface Structure   62     Development of a Lattice Model of Microemulsions   63     Grain Boundaries   64     Statistical Issues   64     Statistical Mechanics Models   65     Computer Simulation   65 6   AGGREGATES AND DISORDERED MATERIAL   66     Introduction   66     Colloidal Suspensions   66     Stokesian Dynamics   67     Computational Microhydrodynamics   68     Nonequilibrium Statistical Mechanics   68     Variational Techniques   69     Self-Consistent Field Theories   69     Equilibrium Structure   69     Effective Moduli of Composites   70     Future Directions   72     Optimal Composites   74     Glasses and Other Amorphous Solids   75 7   PROCESSING, FABRICATION, AND EVALUATION   77     Introduction   77     Processing of Semiconductor Chips   77     Amorphous Semiconductors   79     Casting   79     Polymer Processing   80     Other Processing   80     Mixing   80     Mathematical Modeling in Quantitative Nondestructive Evaluation   81     Functionally Gradient Materials   82     Nonlinear Optical Materials   83

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MATHEMATICAL RESEARCH IN MATERIALS SCIENCE: OPPORTUNITIES AND PERSPECTIVES 8   Mathematical And Numerical Methods   88     Introduction   88     Microscopic Scale   89     Macroscopic Scale   91     Mesoscopic Scale   93     Potentially Applicable Mathematical Sciences Developments   94 9   RECOMMENDATIONS   97     Introduction   97     Acknowledging Obstacles to Collaboration   97     Fostering Increased Collaboration   98     Recommendations   99     Universities   99     Federal and State Government   100     Industry   101     Universities, Government, and Industry Together   101     Professional Societies   102     BIBLIOGRAPHY   103     APPENDIX   127

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