Multiscale Paradigms in Integrated Computational Materials Science and Engineering (eBook)

Preface 6
Contents 8
Contributors 9
1 Introduction 10
Abstract 10
1.1 Conceptual Framework in Theory, Modeling and Simulation 10
1.1.1 Theory 11
1.1.2 Model 13
1.1.3 Simulation 14
1.1.4 Additional Remarks 15
1.2 Multiscale Modeling and Simulation 16
1.2.1 Multiscale Approaches 16
1.2.2 Serial and Concurrent Multiscale Modeling and Simulation 17
1.2.3 Consistent Embedding 18
1.2.4 Multi-theory and Multi-model Approaches 18
Acknowledgments 19
References 19
2 Path Integral Molecular Dynamics Methods 21
Abstract 21
2.1 Introduction 22
2.2 Theoretical Framework and Model Development 24
2.2.1 Feynman Path Integral 24
2.2.1.1 Partition Function for a Single Particle 24
2.2.1.2 Systems of Interacting Particles Obeying Maxwell-Boltzmann Statistics 29
2.2.1.3 Two-Electron System 31
2.2.1.4 Many-Electron System 35
2.2.2 Path Integral with Non-local Exchange Using a Mean Field Approximation 39
2.2.3 Restricted Path Integral Method 41
2.2.4 Classical Isomorphism for Many-Body Fermion System 44
2.2.5 Path Integral with Non-local Pseudo-potential 46
2.2.5.1 Local and Nonlocal Pseudo-potential 46
2.2.5.2 One Electron System 47
2.2.5.3 Evaluation of the Non-local Density Matrix Elements 49
2.2.5.4 One Electron Partition Function 53
2.2.5.5 Model of a Valence Electron in a Sodium Atom 54
2.3 Path Integral Molecular Dynamics Simulation Method 56
2.3.1 Molecular Dynamics Method 56
2.3.2 Molecular Dynamics Classical Hamiltonian for N-Electron Plasma 58
2.3.3 Molecular Dynamics Hamiltonian for N-Alkali Atom Metal 60
2.3.4 Molecular Dynamics Hamiltonian for a Single Alkali Atom with Non-local Pseudo-potential 61
2.3.5 Periodic Boundary Conditions 62
2.3.5.1 Ewald Summation Method 62
2.3.5.2 Maintaining the Continuity of the Necklace Representation of Quantum Particles 65
2.3.5.3 Evaluation of { /det }/left( {E^{{/left( {p,q} /right)}} } /right) of the Effective Exchange Potential with PBC 67
2.3.6 Calculation of Forces 69
2.3.7 Isothermal Molecular Dynamics 70
2.3.8 Calculation of Properties 71
2.3.8.1 Time Averages 72
2.3.8.2 Energy Estimator 72
2.3.8.3 Pair Correlation Function 74
2.4 Applications of the Pimd Method 75
2.4.1 Electron Plasma 75
2.4.2 Alkali Metal 80
2.4.3 Expanded Body Centered Cubic (BCC) Hydrogenoid Crystal 90
2.4.4 Electron in Non-local Pseudo-potential 100
2.4.5 Conclusions 104
Appendix 1: Free Particle Propagator 106
Appendix 2: Exchange Force Calculation 107
Appendix 3: Derivation of the Force on an Electron in a Non-local Pseudo-potential 109
Appendix 4: Exchange Kinetic Energy Estimator for N-Electron System 110
Appendix 5: Energy Estimator for Electron in Non-local Pseudo-potential 111
References 112
3 Interatomic Potentials Including Chemistry 115
Abstract 115
3.1 Background 115
3.2 The Fragment View of Hamiltonians for Materials and Molecules 120
3.3 A Different Way to Decompose a System 123
3.4 Selected Many-Electron, Valence Basis States 129
3.4.1 The Fragment Hamiltonian Approach 134
3.4.2 New Variables for the Charge States of Each Fragment 136
3.4.3 Three-State Fragment Energies 139
3.4.4 Applications 142
3.4.4.1 Application I: H2 Molecule 143
3.4.4.2 Application II: Mulliken Electronegativity, Parr-Pearson Hardness, and Their FH Counterparts 144
3.4.4.3 Application III: Metallic Character in an Atomistic Model 148
3.4.4.4 Application IV: Charge-Flow Regulation in a Two-State, Two-Fragment Model 151
3.4.4.5 Application V: Chemical Potential Equalization 152
3.5 Informing the Embedded-Atom Method from FH: Two-Variable Embedding Energy 153
3.5.1 General Form of the Model Potential from FH 153
3.5.2 Many-Body Interactions 154
3.5.3 Structure of the FH Embedding Energy for Metals 157
3.5.4 Coordination Dependencies of Energy Scales 159
3.5.5 Background Densities 160
3.5.5.1 Generalized Energy Scales 161
3.5.6 FH Model for Elemental Ni 164
3.5.7 Metallic Character of Ni Structures 167
3.6 FH Model as a Variable Charge Model 169
3.6.1 Charge-Dependent Functional Forms as Embedding Energies 171
3.6.2 Embedded Atom (EAM), Modified Embedded Atom (MEAM), and N-Body Methods 175
3.7 FH View of Bonding 178
3.8 Environment-Dependent Dynamic Charge (EDD-Q) Model Potentials 178
3.8.1 EDD-Q Potential for Water 181
3.8.2 EDD-Q Potential for Silica 186
Appendix 193
References 199
4 Phase Field Methods 203
Abstract 203
4.1 Introduction 203
4.2 Methods 204
4.2.1 Conventional PF Methods 204
4.2.2 PFC and AE Method 206
4.3 Applications 208
4.3.1 Solid-State Phase Transformations 208
4.3.2 Ferroelectric Materials 213
4.3.3 Dislocation 214
4.3.4 Grain Growth 215
4.3.5 Dendrite Solidification 217
4.3.6 Fracture 221
4.3.7 Vesicle Morphology 221
4.4 Perspectives 222
References 223
5 Peridynamics 226
Abstract 226
5.1 Introduction 226
5.2 Integral Representation of Continuum Mechanics 227
5.3 Practical Implementation 230
5.4 Applications 231
5.4.1 Damage in a Ceramic Layer Due to Small Particle Impact 232
5.4.2 Fracture Patterns in Anodized Aluminum 239
5.4.3 Dynamic Fracture of Glass 250
References 254
6 Consistent Embedding Frameworks for Predictive Multi-theory Multiscale Simulations 255
Abstract 255
6.1 Introduction 255
6.2 Example 1: The Consistent Embedding Framework for Quantum-Classical Coupling 258
6.2.1 Transfer Hamiltonian 258
6.2.2 Classical Interatomic Potential 259
6.2.3 Pseudo Atoms 260
6.2.4 Quantum-Classical Multiscale Framework 261
6.3 The Compound Wavelet Matrix Method 265
6.3.1 Compounding Methodology for Coupling Scales: Forming the CWM 267
6.4 Example 2: Microstructural Evolution of Materials Using the Compound Wavelet Matrix Method 273
6.5 Example 3: The Dynamic-CWM (dCWM) Approach Applied to Reactive Flows 278
6.6 Example 4: A Cautionary Tale: Multiscale Models for Elastic Wave Propagation in Materials 291
References 302
Index 304