Introduction to Computational Materials Science

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1 Introduction to Computational Materials Science Fundamentals to Applications RICHARD LESAR lowa State University.CAMBRIDGE ::: UNIVERSITY PRESS

2 CONTENTS Preface 1 Introduction to materials modeling and simulation 1.1 Modeling and simulation 1.2 What is meant by computational materials science and engineering? 1.3 Scales in materials structure and behavior 1.4 How to develop models 1.5 Summary PART ONE SOME BASICS 2 The random-walk model 2.1 Random-walk model of diffusion 2.2 Connection to the diffusion coefficient 2.3 Bulk diffusion 2.4 A random-walk simulation 2.5 Random-walk models for materials 2.6 Summary 3 Simulation of finite systems 3.1 Sums of interacting pairs of objects 3.2 Perfect crystals 3.3 Cutoffs 3.4 Periodic boundary conditions 3.5 Implementation 3.6 Long-ranged potentials 3.7 Summary 3.8 Appendix PART TWO ATOMS AND MOLECULES 4 Electronic structure methods 4.1 Quantum mechanics of multielectron systems 4.2 Early density functional theories

3 vi Contents 4.3 The Hohenberg-Kohn theorem 4.4 Kohn-Sham method 4.5 The exchange-correlation functional 4.6 Wave functions 4.7 Pseudopotentials 4.8 Use of density functional theory 4.9 Summary 5 Interatomic potentials 5.1 The cohesive energy 5.2 lnteratomic potentials 5.3 Pair potentials 5.4 lonic materials 5.5 Metals 5.6 Covalent solids 5.7 Systems with mixed bonding 5.8 What we can simulate 5.9 Determining parameters in potentials 5.10 Summary 5.11 Appendix 6 Molecular dynamics 6.1 Basics of molecular dynamics for atomic systems 6.2 An example calculation 6.3 Velocity rescaling 6.4 Molecular dynamics in other ensembles 6.5 Accelerated dynamics 6.6 Limitations of molecular dynamics 6.7 Molecular dynamics in materials research 6.8 Summary 6.9 Appendix 7 "rhe Monte (arlo method 7.1 lntroduction 7.2 Ensemble averages 7.3 The Metropolis algorithm 7.4 The lsing model 7.5 Monte Carlo for atomic systems 7.6 Other ensembles 7.7 Time in a Monte Carlo simulation 7.8 Assessment of the Monte Carlo method 7.9 Uses of the Monte Carlo method in materials research

4 Contents I vii 7.10 Summary 7.11 Appendix 8 Molecular and macromolecular systems 8.1 Introduction 8.2 Random-walk models of polymers 8.3 Atomistic simulations of macromo1ecules 8.4 Coarse-grained methods 8.5 Lattice models for polymers and biomo1ecules 8.6 Simulations of molecular and macromolecular materials 8.7 Summary 8.8 Appendix PART THREE MESOSCOPIC METHODS 9 Kinetic Monte Carlo The kinetic Monte Carlo method Time in the kinetic Monte Carl0 method Kinetic Monte Carlo calculations Applications Summary Monte Carlo methods at the mesoscale Modeling Grain Growth The Monte Carl0 Potts model The N-fold way Example applications of the Potts model Applications in materials science and engineering Summary Cellular automata Basics of cellular automata Examp1es of cellular automata in two dimensions Lattice-gas methods Examples of cellular automata in materials research Relation to Monte Carl Summary Phase-field methods Conserved and non-conserved order parameters Governing equations 230

5 viii Contents 12.3 A one-dimensional phase-field calculation Free energy of an interface Local free-energy functions Two examples Other applications in materials research Summary Appendix Mesoscale dynamics Damped dynamics Langevin dynamics Simulation "entities" at the mesoscale Dynamic models of grain growth Discrete dislocation dynamics simulations Summary 265 PART FOUR SOME FINAL WORDS 14 Materials selection and design Integrated computational materials engineering Concurrent materials design Methods Materials informatics Summary 278 PART FIVE APPENDICES A Energy units, fundamental constants, and conversions 281 A.l Fundamental constants 281 A.2 Units and energy conversions 281 B Abrief introduction to materials 283 B.l Introduction 283 B.2 Crystallography 284 B.3 Defects 291 BA Point defects 291 B.5 Dislocations 292 B.6 Polycrystalline materials 302 B.7 Diffusion 306

6 Contents ix C D E F G Mathematical backgound C.l Vectors and tensors C.2 Taylor series C.3 Complex numbers CA Probability C.5 Common functions C.6 Functionals A brief summary of c1assical mechanics D.l Newton's equations D.2 The Hamiltonian D.3 Example: the harmonie oscillator DA Central-force potentials Electrostatics E. 1 The force E.2 Electrostatic potentials and energies E.3 Distribution of charges: the multipole expansion Elements of quantum mechanics EI History E2 Wave functions E3 The Schrödinger equation FA Observables ES Some solved problems F.6 Atoms with more than one electron F.7 Eigenvalues and eigenvectors E8 Multielectron systems E9 Quantum mechanics of periodic systems ElO Summary Statistical thermodynamics and kinetics G.l Basic thermodynamic quantities G.2 Introduction to statistical thermodynamics G.3 Macrostates versus microstates GA Phase space and time averages G.5 Ensembles G.6 Fluctuations G.7 Correlation functions G.8 Kinetic rate theory G.9 Summary

7 x Contents H linear elasticity 375 H.I Stress and strain 375 H.2 Elastic constants 378 H.3 Engineering stress and strain 378 HA Isotropie solids 379 H.5 Plastic strain 381 Introduction to (Omputation Some basie concepts Random-number generators Binning Numerical derivatives Summary 391 References 392 Index 409

Introduction to Computational Materials Science

Introduction to Computational Materials Science Emphasizing essential methods and universal principles, this textbook provides everything students need to understand the basics of simulating materials