Modeling materials : continuum, atomistic, and multiscale techniques

Material properties emerge from phenomena on scales ranging from Angstroms to millimeters, and only a multiscale treatment can provide a complete understanding. Materials researchers must therefore understand fundamental concepts and techniques from different fields, and these are presented in a comprehensive and integrated fashion for the first time in this book. Incorporating continuum mechanics, quantum mechanics, statistical mechanics, atomistic simulations and multiscale techniques, the book explains many of the key theoretical ideas behind multiscale modeling. Classical topics are blended with new techniques to demonstrate the connections between different fields and highlight current research trends. Example applications drawn from modern research on the thermo-mechanical properties of crystalline solids are used as a unifying focus throughout the text. Together with its companion book, Continuum Mechanics and Thermodynamics (Cambridge University Press, 2011), this work presents the complete fundamentals of materials modeling for graduate students and researchers in physics, materials science, chemistry and engineering. More information on this book is available at the following website maintained by the authors: http://www.modelingmaterials.org Modeling Meterials......Page 2 Title......Page 4 Copyright......Page 5 Contents......Page 6 Preface......Page 14 Acknowledgments......Page 17 Notation......Page 22 1.1.1 Orowan's pocket watch......Page 30 1.1.2 Mechanisms of plasticity......Page 32 1.1.3 Perfect crystals......Page 33 1.1.4 Planar defects: surfaces......Page 36 1.1.5 Planar defects: grain boundaries......Page 39 1.1.6 Line defects: dislocations......Page 41 1.1.7 Point defects......Page 44 1.1.8 Large-scale defects: cracks, voids and inclusions......Page 45 1.2 Materials scales: taking stock......Page 46 Further reading......Page 47 PART I CONTINUUM MECHANICS AND THERMODYNAMICS......Page 48 2 Essential continuum mechanics and thermodynamics......Page 50 2.1.1 Tensor notation......Page 51 2.1.2 Vectors and higher-order tensors......Page 55 2.1.3 Tensor operations......Page 62 2.1.4 Properties of second-order tensors......Page 66 2.1.5 Tensor fields......Page 68 2.2.1 The continuum particle......Page 71 2.2.2 The deformation mapping......Page 72 2.2.3 Material and spatial descriptions......Page 73 2.2.4 Description of local deformation......Page 75 2.2.5 Kinematic rates......Page 78 2.3.1 Conservation of mass......Page 80 2.3.2 Balance of linear momentum......Page 82 2.3.3 Balance of angular momentum......Page 87 2.3.4 Material form of the momentum balance equations......Page 88 2.4.1 Macroscopic observables, thermodynamic equilibrium and state variables......Page 90 2.4.2 Thermal equilibrium and the zeroth law of thermodynamics......Page 94 2.4.3 Energy and the first law of thermodynamics......Page 96 2.4.4 Thermodynamic processes......Page 100 2.4.5 The second law of thermodynamics and the direction of time......Page 101 2.4.6 Continuum thermodynamics......Page 112 2.5 Constitutive relations......Page 119 2.5.1 Constraints on constitutive relations......Page 120 2.5.2 Local action and the second law of thermodynamics......Page 121 2.5.3 Material frame-indifference......Page 126 2.5.4 Material symmetry......Page 128 2.5.5 Linearized constitutive relations for anisotropic hyperelastic solids......Page 130 2.6 Boundary-value problems and the principle of minimum potential energy......Page 134 Further reading......Page 137 Exercises......Page 138 Part II ATOMISTICS......Page 142 3.1 Crystal history: continuum or corpuscular?......Page 144 3.3 Lattices......Page 148 3.3.1 Primitive lattice vectors and primitive unit cells......Page 149 3.3.2 Voronoi tessellation and the Wigner-Seitz cell......Page 151 3.3.3 Conventional unit cells......Page 152 3.3.4 Crystal directions......Page 153 3.4.1 Point symmetry operations......Page 154 3.4.2 The seven crystal systems......Page 158 3.5.1 Centering in the cubic system......Page 163 3.5.3 Centering in the monoclinic system......Page 166 3.5.5 Centering in the hexagonal and trigonal systems......Page 167 3.6 Crystal structure......Page 168 3.6.2 Crystal structures of some common crystals......Page 171 3.7.1 Fourier series and the reciprocal lattice......Page 175 3.7.2 The first Brillouin zone......Page 177 3.7.3 Miller indices......Page 178 Exercises......Page 180 4.1 Introduction......Page 182 4.2 A brief and selective history of quantum mechanics......Page 183 4.2.1 The Hamiltonian formulation......Page 186 4.3.1 Dirac notation......Page 189 4.3.2 Electron wave functions......Page 192 4.3.3 Schrodinger's equation......Page 197 4.3.4 The time-independent Schrödinger equation......Page 200 4.3.5 The hydrogen atom......Page 201 4.3.6 The hydrogen molecule......Page 208 4.3.7 Summary of the quantum mechanics of bonding......Page 216 4.4.1 Exact formulation......Page 217 4.4.2 Approximations necessary for computational progress......Page 225 4.4.3 The choice of basis functions......Page 228 4.4.4 Electrons in periodic systems......Page 229 4.4.5 The essential machinery of a plane-wave DFT code......Page 239 4.4.6 Energy minimization and dynamics: forces in DFT......Page 250 4.5.1 LCAO......Page 252 4.5.2 The Hamiltonian and overlap matrices......Page 253 4.5.3 Slater-Koster parameters for two-center integrals......Page 256 4.5.5 TB molecular dynamics......Page 257 4.5.6 From TB to empirical atomistic models......Page 258 Exercises......Page 264 5 Empirical atomistic models of materials......Page 266 5.1 Consequences of the Born-Oppenheimer approximation (BOA)......Page 267 5.2 Treating atoms as classical particles......Page 269 5.3 Sensible functional forms......Page 270 5.3.1 Interatomic distances......Page 271 5.3.2 Requirement of translational, rotational and parity invariance......Page 272 5.3.3 The cutoff radius......Page 274 5.4 Cluster potentials......Page 275 5.4.1 Formally exact cluster potentials......Page 276 5.4.2 Pair potentials......Page 280 5.4.3 Modeling ionic crystals: the Born-Mayer potential......Page 285 5.4.4 Three- and four-body potentials......Page 286 5.4.5 Modeling organic molecules: CHARMM and AMBER......Page 288 5.4.6 Limitations of cluster potentials and the need for interatomic functionals......Page 290 5.5 Pair functionals......Page 291 5.5.1 The generic pair functional form: the glue-EAM-EMT-FS model......Page 292 5.5.2 Physical interpretations of the pair functional......Page 293 5.5.3 Fitting the pair functional model......Page 294 5.5.4 Comparing pair functionals to cluster potentials......Page 295 5.6.1 Introduction to the bond order: the Tersoff potential......Page 297 5.6.2 Bond energy and bond order in TB......Page 300 5.6.3 ReaxFF......Page 303 5.6.4 The modified embedded atom method......Page 305 5.7.1 Speed and scaling: how many atoms over how much time?......Page 308 5.7.2 Transferability: predicting behavior outside the fit......Page 311 5.7.3 Classes of materials and our ability to model them......Page 314 5.8.1 Weak and strong laws of action and reaction41......Page 317 5.8.2 Forces in conservative systems......Page 320 5.8.3 Atomic forces for some specific interatomic models......Page 323 5.8.4 Bond stiffnesses for some specific interatomic models......Page 326 5.8.5 The cutoff radius and interatomic forces......Page 327 Further reading......Page 328 Exercises......Page 329 6.1 The potential energy landscape......Page 333 6.2.1 Solving nonlinear problems: initial guesses......Page 335 6.2.2 The generic nonlinear minimization algorithm......Page 336 6.2.3 The steepest descent (SD) method......Page 337 6.2.4 Line minimization......Page 339 6.2.5 The conjugate gradient (CG) method......Page 340 6.2.6 The condition number......Page 341 6.2.7 The Newton–Raphson (NR) method......Page 342 6.3 Methods for finding saddle points and transition paths......Page 344 6.3.1 The nudged elastic band (NEB) method......Page 345 6.4.1 Neighbor lists......Page 350 6.4.2 Periodic boundary conditions (PBCs)......Page 354 6.4.3 Applying stress and pressure boundary conditions......Page 357 6.4.4 Boundary conditions on atoms......Page 359 6.5 Application to crystals and crystalline defects......Page 360 6.5.1 Cohesive energy of an infinite crystal......Page 361 6.5.2 The universal binding energy relation (UBER)......Page 363 6.5.3 Crystal defects: vacancies......Page 367 6.5.4 Crystal defects: surfaces and interfaces......Page 368 6.5.5 Crystal defects: dislocations......Page 376 6.5.6 The γ-surface......Page 386 6.5.7 The Peierls–Nabarro model of a dislocation......Page 389 Further reading......Page 400 Exercises......Page 401 PART III ATOMISTIC FOUNDATIONS OF CONTINUUM CONCEPTS......Page 404 7 Classical equilibrium statistical mechanics......Page 406 7.1.1 Hamilton's equations......Page 407 7.1.2 Macroscopic translation and rotation......Page 408 7.1.3 Center of mass coordinates......Page 409 7.1.4 Phase space coordinates......Page 410 7.1.5 Trajectories through phase space......Page 411 7.1.6 Liouville's theorem......Page 413 7.2.1 Time averages......Page 416 7.2.2 The ensemble viewpoint and distribution functions......Page 418 7.2.3 Why does the ensemble approach work?......Page 421 7.3.1 The hypersurface and volume of an isolated Hamiltonian system......Page 432 7.3.2 The microcanonical distribution function......Page 435 7.3.3 Systems in weak interaction......Page 438 7.3.4 Internal energy, temperature and entropy......Page 441 7.3.5 Derivation of the ideal gas law......Page 447 7.3.6 Equipartition and virial theorems: microcanonical derivation......Page 449 7.4 The canonical (NVT) ensemble......Page 452 7.4.1 The canonical distribution function......Page 453 7.4.2 Internal energy and fluctuations......Page 457 7.4.3 Helmholtz free energy......Page 458 7.4.4 Equipartition theorem: canonical derivation......Page 460 7.4.5 Helmholtz free energy in the thermodynamic limit......Page 461 Further reading......Page 466 Exercises......Page 467 8 Microscopic expressions for continuum fields......Page 469 8.1.1 Canonical transformations......Page 471 8.1.2 Microscopic stress tensor in a finite system at zero temperature......Page 476 8.1.3 Microscopic stress tensor at finite temperature: the virial stress......Page 479 8.1.4 Microscopic elasticity tensor......Page 489 8.2 Continuum fields as expectation values: nonequilibrium systems......Page 494 8.2.1 Rate of change of expectation values......Page 495 8.2.2 Definition of pointwise continuum fields......Page 496 8.2.4 Momentum balance and the pointwise stress tensor......Page 498 8.2.5 Spatial averaging and macroscopic fields......Page 504 8.3 Practical methods: the stress tensor......Page 508 8.3.1 The Hardy stress......Page 509 8.3.2 The virial stress tensor and atomic-level stresses......Page 510 8.3.3 The Tsai traction: a planar definition for stress......Page 511 8.3.4 Uniqueness of the stress tensor......Page 516 8.3.5 Hardy, virial and Tsai stress expressions: numerical considerations......Page 517 Exercises......Page 518 9.1 Brief historical introduction......Page 521 9.2 The essential MD algorithm......Page 524 9.3.1 Integrating the NVE ensemble: the velocity-Verlet (VV) algorithm......Page 526 9.3.3 Temperature initialization......Page 533 9.4 The NVT ensemble: constant temperature and constant strain......Page 536 9.4.1 Velocity rescaling......Page 537 9.4.2 Gauss ́ principle of least constraint and the isokinetic thermostat......Page 538 9.4.3 The Langevin thermostat......Page 540 9.4.4 The Nosé-Hoover (NH) thermostat......Page 542 9.4.5 Liouville’s equation for non-Hamiltonian systems......Page 545 9.4.6 An alternative derivation of the NH thermostat......Page 546 9.4.7 Integrating the NVT ensemble......Page 547 9.5 The finite strain NσE ensemble: applying stress......Page 549 9.5.1 A canonical transformation of variables......Page 550 9.5.2 The hydrostatic stress state......Page 556 9.5.3 The Parrinello-Rahman (PR) approximation......Page 557 9.5.4 The zero-temperature limit: applying stress in molecular statics......Page 559 9.6 The NσT ensemble: applying stress at a constant temperature......Page 562 Exercises......Page 563 PART IV MULTISCALE METHODS......Page 566 10.1 Multiscale modeling: what is in a name?......Page 568 10.2 Sequential multiscale models......Page 570 10.3 Concurrent multiscale models......Page 572 10.3.1 Hierarchical methods......Page 573 10.3.2 Partitioned-domain methods......Page 575 10.4 Spanning time scales......Page 576 Further reading......Page 578 11 Atomistic constitutive relations for multilattice crystals......Page 579 11.1.1 Restricted ensembles......Page 583 11.1.2 Properties of a metastable state from a restricted canonical ensemble......Page 585 11.2.1 Multilattice crystals and mean positions......Page 587 11.2.2 Cauchy-Born kinematics......Page 588 11.2.3 Centrosymmetric crystals and the Cauchy-Born rule......Page 590 11.2.4 Extensions and failures of the Cauchy-Born rule......Page 591 11.3.1 Periodic supercell of a multilattice crystal......Page 592 11.3.2 Helmholtz free energy density of a multilattice crystal......Page 595 11.3.3 Determination of the reference configuration......Page 596 11.3.4 Uniform deformation and the macroscopic stress tensor......Page 599 11.3.5 Elasticity tensor......Page 604 11.4.1 Quasiharmonic Helmholtz free energy......Page 607 11.4.2 Determination of the quasiharmonic reference configuration......Page 611 11.4.3 Quasiharmonic stress and elasticity tensors......Page 615 11.4.4 Strict harmonic approximation......Page 619 11.5.1 General expressions for the stress and elasticity tensors......Page 621 11.5.2 Stress and elasticity tensors for some specific interatomic models......Page 622 11.5.3 Crystal symmetries and the Cauchy relations......Page 624 Exercises......Page 627 12.1 Finite elements and the Cauchy–Born rule......Page 630 12.2 The essential components of a coupled model......Page 633 12.3.1 Total energy functional......Page 637 12.3.2 The quasi-continuum (QC) method......Page 639 12.3.3 The coupling of length scales (CLS) method......Page 642 12.3.4 The bridging domain (BD) method......Page 643 12.3.5 The bridging scale method (BSM)......Page 645 12.3.6 CACM: iterative minimization of two energy functionals......Page 646 12.3.7 Cluster-based quasicontinuum (CQC-E)......Page 647 12.4 Ghost forces in energy-based methods......Page 649 12.4.1 A one-dimensional Lennard-Jones chain of atoms......Page 651 12.4.3 Ghost forces in a generic energy-based model of the chain......Page 652 12.4.4 Ghost forces in the cluster-based quasicontinuum (CQC-E)......Page 656 12.4.5 Ghost force correction methods......Page 659 12.5.1 Forces without an energy functional......Page 660 12.5.2 FEAt and CADD......Page 662 12.5.4 The atomistic-to-continuum (AtC) method......Page 663 12.5.6 Spurious forces in force-based methods......Page 665 12.6.1 A simple example:shearing a twin boundary......Page 667 12.6.2 Setting up the model......Page 669 12.6.3 Solution procedure......Page 671 12.6.4 Twin boundary migration......Page 673 12.6.5 Automatic model adaption......Page 674 12.7 Quantitative comparison between the methods......Page 676 12.7.1 The test problem......Page 677 12.7.2 Comparing the accuracy of multiscale methods......Page 679 12.7.3 Quantifying the speed of multiscale methods......Page 683 12.7.4 Summary of the relative accuracy and speed of multiscale methods......Page 684 Exercises......Page 685 13 Atomistic–continuum coupling: finite temperature and dynamics......Page 687 13.1 Dynamic finite elements......Page 688 13.2 Equilibrium finite temperature multiscale methods......Page 690 13.2.1 Effective Hamiltonian for the atomistic region......Page 691 13.2.2 Finite temperature QC framework......Page 696 13.2.3 Hot-QC-static:atomistic dynamics embedded in a static continuum......Page 699 13.2.4 Hot-QC-dynamic: atomistic and continuum dynamics......Page 701 13.2.5 Demonstrative examples: thermal expansion and nanoindentation......Page 704 13.3 Nonequilibrium multiscale methods......Page 706 13.3.2 Wave reflections......Page 707 13.3.3 Generalized Langevin equations......Page 712 13.3.4 Damping bands......Page 716 Exercises......Page 718 A: Mathematical representation of interatomic potentials......Page 719 A.1 Interatomic distances and invariance......Page 720 A.2 Distance geometry: constraints between interatomic distances......Page 722 A.3 Continuously differentiable extensions of Vint(s)......Page 725 A.4 Alternative potential energy extensions and the effect on atomic forces......Page 727 References......Page 731 Index......Page 775 Material Properties Emerge From Phenomena On Scales Ranging From Angstroms To Millimeters, And Only A Multiscale Treatment Can Provide A Complete Understanding. Materials Researchers Must Therefore Understand Fundamental Concepts And Techniques From Different Fields, And These Are Presented In A Comprehensive And Integrated Fashion For The First Time In This Book. Incorporating Continuum Mechanics, Quantum Mechanics, Statistical Mechanics, Atomistic Simulations And Multiscale Techniques, The Book Explains Many Of The Key Theoretical Ideas Behind Multiscale Modeling. Classical Topics Are Blended With New Techniques To Demonstrate The Connections Between Different Fields And Highlight Current Research Trends. Example Applications Drawn From Modern Research On The Thermo-mechanical Properties Of Crystalline Solids Are Used As A Unifying Focus Throughout The Text. Together With Its Companion Book, Continuum Mechanics And Thermodynamics (cambridge University Press, 2011), This Work Presents The Complete Fundamentals Of Materials Modeling For Graduate Students And Researchers In Physics, Materials Science, Chemistry And Engineering--provided By Publisher. Machine Generated Contents Note: 1. Introduction; Part I. Continuum Mechanics And Thermodynamics: 2. Essential Continuum Mechanics And Thermodynamics; Part Ii. Atomistics: 3. Lattices And Crystal Structures; 4. Quantum Mechanics Of Materials; 5. Empirical Atomistic Models Of Materials; 6. Molecular Statics; Part Iii. Atomistic Foundations Of Continuum Concepts: 7. Classical Equilibrium Statistical Mechanics; 8. Microscopic Expressions For Continuum Fields; 9. Molecular Dynamics; Part Iv. Multiscale Methods: 10. What Is Multiscale Modeling?; 11. Atomistic Constitutive Relations For Multilattice Crystals; 12. Atomistic/continuum Coupling: Static Methods; 13. Atomistic/continuum Coupling: Finite Temperature And Dynamics; Appendix; References; Index. Ellad B. Tadmor, Ronald E. Miller. Includes Bibliographical References And Index. "Material properties emerge from phenomena on scales ranging from Angstroms to millimeters, and only a multiscale treatment can provide a complete understanding. Materials researchers must therefore understand fundamental concepts and techniques from different fields, and these are presented in a comprehensive and integrated fashion for the first time in this book. Incorporating continuum mechanics, quantum mechanics, statistical mechanics, atomistic simulations and multiscale techniques, the book explains many of the key theoretical ideas behind multiscale modeling. Classical topics are blended with new techniques to demonstrate the connections between different fields and highlight current research trends. Example applications drawn from modern research on the thermo-mechanical properties of crystalline solids are used as a unifying focus throughout the text. Together with its companion book, Continuum Mechanics and Thermodynamics (Cambridge University Press, 2011), this work presents the complete fundamentals of materials modeling for graduate students and researchers in physics, materials science, chemistry and engineering"-- Provided by publisher