Preface ........................................................ xv
Acknowledgements .............................................. xxi
Notes on Units, Scales and Conventions ....................... xxiv
Part one: Thinking About the Material World ..................... 1
1 Idealizing Material Response ............................... 3
1.1 A Material World ........................................... 3
1.1.1 Materials: A Databook Perspective ................... 3
1.1.2 The Structure-Properties Paradigm ................... 8
1.1.3 Controlling Structure: The World of Heat and Beat .. 12
1.2 Modeling of Materials ..................................... 14
1.2.1 The Case for Modeling .............................. 14
1.2.2 Modeling Defined: Contrasting Perspectives ......... 15
1.2.3 Case Studies in Modeling ........................... 18
1.2.4 Modeling and the Computer: Numerical Analysis vs
Simulation ......................................... 25
1.3 Further Reading ........................................... 26
2 Continuum Mechanics Revisited ............................. 29
2.1 Continuum Mechanics as an Effective Theory ................ 29
2.2 Kinematics: The Geometry of Deformation ................... 31
2.2.1 Deformation Mappings and Strain .................... 32
2.2.2 Geometry of Rigid Deformation ...................... 35
2.2.3 Geometry of Slip and Twinning ...................... 36
2.2.4 Geometry of Structural Transformations ............. 37
2.3 Forces and Balance Laws ................................... 39
2.3.1 Forces Within Continua: Stress Tensors ............. 39
2.3.2 Equations of Continuum Dynamics .................... 41
2.3.3 Configurational Forces and the Dynamics of
Defects ............................................ 44
2.4 Continuum Descriptions of Deformation and Failure ......... 51
2.4.1 Constitutive Modeling ............................ 51
2.4.2 Linear Elastic Response of Materials ............... 51
2.4.3 Plastic Response of Crystals and Polycrystals ...... 54
2.4.4 Continuum Picture of Fracture ...................... 60
2.5 Boundary Value Problems and Modeling ...................... 64
2.5.1 Principle of Minimum Potential Energy and
Reciprocal Theorem ................................. 64
2.5.2 Elastic Green Function ............................. 66
2.5.3 Method of Eigenstrains ............................. 69
2.5.4 Numerical Solutions: Finite Element Method ......... 72
2.6 Difficulties with the Continuum Approach .................. 75
2.7 Further Reading ........................................... 76
2.8 Problems .................................................. 78
3 Quantum and Statistical Mechanics Revisited ............... 81
3.1 Background ................................................ 81
3.2 Quantum Mechanics ......................................... 82
3.2.1 Background and Formalism ........................... 82
3.2.2 Catalog of Important Solutions ..................... 87
3.2.3 Finite Elements and Schrödinger .................... 94
3.2.4 Quantum Corrals: A Finite Element Analysis ........ 101
3.2.5 Metals and the Electron Gas ....................... 103
3.2.6 Quantum Mechanics of Bonding ...................... 109
3.3 Statistical Mechanics .................................... 115
3.3.1 Background ........................................ 115
3.3.2 Entropy of Mixing ................................. 119
3.3.3 The Canonical Distribution ........................ 122
3.3.4 Information Theoretic Approach to Statistical
Mechanics ......................................... 126
3.3.5 Statistical Mechanics Models for Materials ........ 129
3.3.6 Bounds and Inequalities: The Bogoliubov
Inequality ........................................ 135
3.3.7 Correlation Functions: The Kinematics of Order .... 137
3.3.8 Computational Statistical Mechanics ............... 139
3.4 Further Reading .......................................... 142
3.5 Problems ................................................. 144
Part two: Energetics of Crystalline Solids .................... 147
4 Energetic Description of Cohesion in Solids .............. 149
4.1 The Role of the Total Energy in Modeling Materials ....... 149
4.2 Conceptual Backdrop for Characterizing the Total
Energy ................................................... 152
4.2.1 Atomistic and Continuum Descriptions Contrasted ... 152
4.2.2 The Many-Particle Hamiltonian and Degree of
Freedom Reduction ................................. 154
4.3 Pair Potentials .......................................... 156
4.3.1 Generic Pair Potentials ........................... 156
4.3.2 Free Electron Pair Potentials ..................... 158
4.4 Potentials with Environmental and Angular Dependence ..... 164
4.4.1 Diagnostics for Evaluating Potentials ............. 164
4.4.2 Pair Functionals .................................. 165
4.4.3 Angular Forces: A First Look ...................... 172
4.5 Tight-Binding Calculations of the Total Energy ........... 176
4.5.1 The Tight-Binding Method .......................... 176
4.5.2 An Aside on Periodic Solids: k-space Methods ...... 184
4.5.3 Real Space Tight-Binding Methods .................. 189
4.6 First-Principles Calculations of the Total Energy ........ 197
4.6.1 Managing the Many-Particle Hamiltonian ............ 198
4.6.2 Total Energies in the Local Density
Approximation ..................................... 200
4.7 Choosing a Description of the Total Energy: Challenges
and Conundrums ........................................... 203
4.8 Further Reading .......................................... 204
4.9 Problems ................................................. 206
5 Thermal and Elastic Properties of Crystals ............... 210
5.1 Thermal and Elastic Material Response .................... 210
5.2 Mechanics of the Harmonic Solid .......................... 213
5.2.1 Total Energy of the Thermally Fluctuating Solid ... 214
5.2.2 Atomic Motion and Normal Modes .................... 216
5.2.3 Phonons ........................................... 228
5.2.4 Buckminsterfullerene and Nanotubes: A Case Study
in Vibration ...................................... 229
5.3 Thermodynamics of Solids ................................. 231
5.3.1 Harmonic Approximation ............................ 231
5.3.2 Beyond the Harmonic Approximation ................. 239
5.4 Modeling the Elastic Properties of Materials ............. 244
5.4.1 Linear Elastic Moduli ............................. 244
5.4.2 Nonlinear Elastic Material Response: Cauchy-Born
Elasticity ........................................ 248
5.5 Further Reading .......................................... 250
5.6 Problems ................................................. 251
6 Structural Energies and Phase Diagrams ................... 253
6.1 Structures in Solids ..................................... 253
6.2 Atomic-Level Geometry in Materials ....................... 254
6.3 Structural energies of solids ............................ 260
6.3.1 Pair Potentials and Structural Stability .......... 261
6.3.2 Structural Stability in Transition Metals ......... 264
6.3.3 Structural Stability Reconsidered: The Case of
Elemental Si ...................................... 265
6.4 Elemental Phase Diagrams ................................. 268
6.4.1 Free Energy of the Crystalline Solid .............. 268
6.4.2 Free Energy of the Liquid ......................... 275
6.4.3 Putting It All Together ........................... 277
6.4.4 An Einstein Model for Structural Change ........... 278
6.4.5 A Case Study in Elemental Mg ...................... 280
6.5 Alloy Phase Diagrams ..................................... 282
6.5.1 Constructing the Effective Energy: Cluster
Expansions ........................................ 283
6.5.2 Statistical Mechanics for the Effective
Hamiltonian ....................................... 291
6.5.3 The Effective Hamiltonian Revisited: Relaxations
and Vibrations .................................... 297
6.5.4 The Alloy Free Energy ............................. 299
6.5.5 Case Study: Oxygen Ordering in High TC
Superconductors ................................... 300
6.6 Summary .................................................. 304
6.7 Further Reading .......................................... 304
6.8 Problems ................................................. 305
Part three: Geometric Structures in Solids: Defects and
Microstructures ............................................... 309
7 Point Defects in Solids .................................. 311
7.1 Point Defects and Material Response ...................... 311
7.1.1 Material Properties Related to Point Disorder ..... 312
7.2 Diffusion ................................................ 318
7.2.1 Effective Theories of Diffusion ................... 318
7.3 Geometries and Energies of Point Defects ................. 326
7.3.1 Crystallographic Preliminaries .................... 327
7.3.2 A Continuum Perspective on Point Defects .......... 328
7.3.3 Microscopic Theories of Point Defects ............. 332
7.3.4 Point Defects in Si: A Case Study ................. 341
7.4 Point Defect Motions ..................................... 344
7.4.1 Material Parameters for Mass Transport ............ 345
7.4.2 Diffusion via Transition State Theory ............. 346
7.4.3 Diffusion via Molecular Dynamics .................. 351
7.4.4 A Case Study in Diffusion: Interstitials in Si .... 353
7.5 Defect Clustering ........................................ 356
7.6 Further Reading .......................................... 356
7.7 Problems ................................................. 359
8 Line Defects in Solids ................................... 362
8.1 Permanent Deformation of Materials ....................... 362
8.1.1 Yield and Hardening ............................... 363
8.1.2 Structural Consequences of Plastic Deformation .... 365
8.1.3 Single Crystal Slip and the Schmid Law ............ 367
8.2 The Ideal Strength Concept and the Need for
Dislocations ............................................. 369
8.3 Geometry of Slip ......................................... 371
8.3.1 Topological Signature of Dislocations ............. 372
8.3.2 Crystallography of Slip ........................... 375
8.4 Elastic Models of Single Dislocations .................... 382
8.4.1 The Screw Dislocation ............................. 382
8.4.2 The Volterra Formula .............................. 388
8.4.3 The Edge Dislocation .............................. 391
8.4.4 Mixed Dislocations ................................ 392
8.5 Interaction Energies and Forces .......................... 393
8.5.1 The Peach-Koehler Formula ......................... 395
8.5.2 Interactions and Images: Peach-Koehler Applied .... 398
8.5.3 The Line Tension Approximation .................... 402
8.6 Modeling the Dislocation Core: Beyond Linearity .......... 404
8.6.1 Dislocation Dissociation .......................... 404
8.6.2 The Peierls-Nabarro Model ......................... 406
8.6.3 Structural Details of the Dislocation Core ........ 412
8.7 Three-Dimensional Dislocation Configurations ............. 415
8.7.1 Dislocation Bow-Out ............................... 416
8.7.2 Kinks and Jogs .................................... 418
8.7.3 Cross Slip ........................................ 423
8.7.4 Dislocation Sources ............................... 426
8.7.5 Dislocation Junctions ............................. 430
8.8 Further Reading .......................................... 435
8.9 Problems ................................................. 437
9 Wall Defects in Solids ................................... 441
9.1 Interfaces in Materials .................................. 441
9.1.1 Interfacial Confinement ........................... 442
9.2 Free Surfaces ............................................ 446
9.2.1 Crystallography and Energetics of Ideal Surfaces .. 447
9.2.2 Reconstruction at Surfaces ........................ 452
9.2.3 Steps on Surfaces ................................. 474
9.3 Stacking Faults and Twins ................................ 476
9.3.1 Structure and Energetics of Stacking Faults ....... 477
9.3.2 Planar Faults and Phase Diagrams .................. 484
9.4 Grain Boundaries ......................................... 487
9.4.1 Bicrystal Geometry ................................ 489
9.4.2 Grain Boundaries in Polycrystals .................. 492
9.4.3 Energetic Description of Grain Boundaries ......... 494
9.4.4 Triple Junctions of Grain Boundaries .............. 500
9.5 Diffuse Interfaces ....................................... 501
9.6 Modeling Interfaces: A Retrospective ..................... 502
9.7 Further Reading .......................................... 503
9.8 Problems ................................................. 505
10 Microstructure and its Evolution ......................... 507
10.1 Microstructures in Materials ............................. 508
10.1.1 Microstructural Taxonomy .......................... 508
10.1.2 Microstructural Change ............................ 516
10.1.3 Models of Microstructure and its Evolution ........ 519
10.2 Inclusions as Microstructure ............................. 520
10.2.1 Eshelby and the Elastic Inclusion ................. 520
10.2.2 The Question of Equilibrium Shapes ................ 527
10.2.3 Precipitate Morphologies and Interfacial Energy ... 528
10.2.4 Equilibrium Shapes: Elastic and Interfacial
Energy ............................................ 529
10.2.5 A Case Study in Inclusions: Precipitate
Nucleation ........................................ 537
10.2.6 Temporal Evolution of Two-Phase Microstructures ... 540
10.3 Microstructure in Martensites ............................ 546
10.3.1 The Experimental Situation ........................ 547
10.3.2 Geometrical and Energetic Preliminaries ........... 551
10.3.3 Twinning and Compatibility ........................ 554
10.3.4 Fine-Phase Microstructures and Attainment ......... 560
10.3.5 The Austenite-Martensite Free Energy
Reconsidered ...................................... 565
10.4 Microstructural Evolution in Polycrystals ................ 566
10.4.1 Phenomenology of Grain Growth ..................... 567
10.4.2 Modeling Grain Growth ............................. 568
10.5 Microstructure and Materials ............................. 580
10.6 Further Reading .......................................... 580
10.7 Problems ................................................. 582
Part four: Facing the Multiscale Challenge of Real Material
Behavior ...................................................... 585
11 Points, Lines and Walls: Defect Interactions and
Material Response ........................................ 587
11.1 Defect Interactions and the Complexity of Real Material
Behavior ................................................. 587
11.2 Diffusion at Extended Defects ............................ 588
11.2.1 Background on Short-Circuit Diffusion ............. 588
11.2.2 Diffusion at Surfaces ............................. 589
11.3 Mass Transport Assisted Deformation ...................... 592
11.3.1 Phenomenology of Creep ............................ 593
11.3.2 Nabarro-Herring and Coble Creep ................... 595
11.4 Dislocations and Interfaces .............................. 599
11.4.1 Dislocation Models of Grain Boundaries ............ 600
11.4.2 Dislocation Pile-Ups and Slip Transmission ........ 604
11.5 Cracks and Dislocations .................................. 609
11.5.1 Variation on a Theme of Irwin ..................... 610
11.5.2 Dislocation Screening at a Crack Tip .............. 611
11.5.3 Dislocation Nucleation at a Crack Tip ............. 615
11.6 Dislocations and Obstacles: Strengthening ................ 620
11.6.1 Conceptual Overview of the Motion of Dislocations
Through a Field of Obstacles ...................... 622
11.6.2 The Force Between Dislocations and Glide
Obstacles ......................................... 625
11.6.3 The Question of Statistical Superposition ......... 628
11.6.4 Solution Hardening ................................ 633
11.6.5 Precipitate Hardening ............................. 636
11.6.6 Dislocation-Dislocation Interactions and Work
Hardening ......................................... 642
11.7 Further Reading .......................................... 644
11.8 Problems ............................................ 647
12 Bridging Scales: Effective Theory Construction ........... 649
12.1 Problems Involving Multiple Length and Time Scales ....... 651
12.1.1 Problems with Multiple Temporal Scales: The
Example of Diffusion .............................. 652
12.1.2 Problems with Multiple Spatial Scales: The
Example of Plasticity ............................. 653
12.1.3 Generalities on Modeling Problems Involving
Multiple Scales ................................... 655
12.2 Historic Examples of Multiscale Modeling ................. 658
12.3 Effective Theory Construction ............................ 668
12.3.1 Degree of Freedom Selection: State Variables,
Order Parameters and Configurational Coordinates .. 669
12.3.2 Dynamical Evolution of Relevant Variables:
Gradient Flow Dynamics and Variational
Principles ........................................ 674
12.3.3 Inhomogeneous Systems and the Role of Locality .... 685
12.3.4 Models with Internal Structure .................... 688
12.3.5 Effective Hamiltonians ............................ 697
12.4 Bridging Scales in Microstructural Evolution ............. 701
12.4.1 Hierarchical Treatment of Diffusive Processes ..... 701
12.4.2 From Surface Diffusion to Film Growth ............. 709
12.4.3 Solidification Microstructures .................... 711
12.4.4 Two-Phase Microstructures Revisited ............... 715
12.4.5 A Retrospective on Modeling Microstructural
Evolution ......................................... 718
12.5 Bridging Scales in Plasticity ............................ 719
12.5.1 Mesoscopic Dislocation Dynamics ................... 720
12.5.2 A Case Study in Dislocations and Plasticity:
Nanoindentation ................................... 728
12.5.3 A Retrospective on Modeling Plasticity Using
Dislocation Dynamics .............................. 731
12.6 Bridging Scales in Fracture .............................. 732
12.6.1 Atomic-Level Bond Breaking ........................ 732
12.6.2 Cohesive Surface Models ........................... 734
12.6.3 Cohesive Surface Description of Crack Tip
Dislocation Nucleation ............................ 735
12.7 Further Reading .......................................... 736
12.8 Problems ................................................. 738
13 Universality and Specificity in Materials ................ 742
13.1 Materials Observed ....................................... 743
13.1.1 What is a Material: Another Look .................. 743
13.1.2 Structural Observations ........................... 744
13.1.3 Concluding Observations on the Observations ....... 746
13.2 How Far Have We Come? .................................... 748
13.2.1 Universality in Materials ......................... 749
13.2.2 Specificity in Materials .......................... 750
13.2.3 The Program Criticized ............................ 751
13.3 Intriguing Open Questions ................................ 752
13.4 In Which the Author Takes His Leave ...................... 754
References .................................................... 757
Index ......................................................... 111
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