Modeling Materials. Continuum, Atomistic and Multiscale Techniques. gg CAMBRIDGE ^0 TADMOR ELLAD B. HHHHM. University of Minnesota, USA

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1 HHHHM Modeling Materials Continuum, Atomistic and Multiscale Techniques ELLAD B. TADMOR University of Minnesota, USA RONALD E. MILLER Carleton University, Canada gg CAMBRIDGE ^0 UNIVERSITY PRESS

2 Preface A cknowledgments Notation page xiii xvi xxi 1 Introduction Multiple scales in crystalline materials Orowan's pocket watch Mechanisms of plasticity Perfect crystals Planar defects: surfaces Planar defects: grain boundaries Line defects: dislocations Point defects Large-scale defects: cracks, voids and inclusions Materials scales: taking stock 17 Further reading 18 Part I Continuum mechanics and thermodynamics 19 2 Essential continuum mechanics and thermodynamics Scalars, vectors, and tensors Tensor notation Vectors and higher-order tensors Tensor operations Properties of second-order tensors Tensor fields Kinematics of deformation The continuum particle The deformation mapping Material and spatial descriptions Description of local deformation Kinematic rates Mechanical conservation and balance laws Conservation of mass Balance of linear momentum Balance of angular momentum Material form of the momentum balance equations 59

3 mmmmamsm 2.4 Thermodynamics Macroscopic observables, thermodynamic equilibrium and state variables Thermal equilibrium and the zeroth law of thermodynamics Energy and the first law of thermodynamics Thermodynamic processes The second law of thermodynamics and the direction of time Continuum thermodynamics Constitutive relations Constraints on constitutive relations Local action and the second law of thermodynamics Material frame-indifference Material symmetry Linearized constitutive relations for anisotropic hyperelastic solids Boundary-value problems and the principle of minimum potential energy 105 Further reading 108 Exercises 109 Part II Atomistics Lattices and crystal structures Crystal history: continuum or corpuscular? The structure of ideal crystals Lattices Primitive lattice vectors and primitive unit cells 120 cell Voronoi tessellation and the Wigner-Seitz Conventional unit cells Crystal directions Crystal systems Point symmetry operations The seven crystal systems Bravais lattices Centering in the cubic system Centering in the triclinic system Centering in the monoclinic system Centering in the orthorhombic and tetragonal systems Centering in the hexagonal and trigonal systems Summary of the fourteen Bravais lattices Crystal structure Essential and nonessential descriptions of crystals Crystal structures of some common crystals Some additional lattice concepts Fourier series and the reciprocal lattice The first Brillouin zone Miller indices 149 Further reading 151 Exercises 151

4 4 Quantum mechanics of materials Introduction A brief and selective history of quantum mechanics The Hamiltonian formulation The quantum theory of bonding Dirac notation Electron wave functions Schrodinger's equation The time-independent Schrodinger equation The hydrogen atom The hydrogen molecule Summary of the quantum mechanics of bonding Density functional theory (DFT) Exact formulation Approximations necessary for computational progress The choice ofbasis functions Electrons in periodic systems The essential machinery of a plane-wave DFT code Energy minimization and dynamics: forces in DFT Semi-empirical quantum mechanics: tight-binding (TB) methods LCAO The Hamiltonian and overlap matrices Slater-Koster parameters for two-center integrals Summary of the TB formulation TB molecular dynamics From TB to empirical atomistic models 229 Further reading 235 Exercises Empirical atomistic models of materials Consequences of the Bom-Oppenheimer approximation (BOA) Treating atoms as classical particles Sensible functional forms Interatomic distances Requirement of translational, rotational and parity invariance The cutoff radius Cluster potentials Formally exact cluster potentials Pair potentials Modeling ionic crystals: the Born-Mayer potential Three-and four-body potentials Modeling organic molecules: CHARMM and AMBER Limitations of cluster potentials and the need for interatomic functionals Pair functionals The generic pair functional form: the glue-eam-emt~fs model Physical interpretations of the pair functional Fitting the pair functional model Comparing pair functionals to cluster potentials 266

5 5.6 Cluster functional Introduction to the bond order: the Tersoff potential Bond energy and bond order in TB ReaxFF The modified embedded atom method Atomistic models: what can they do? Speed and scaling: how many atoms over how much time? Transferability: predicting behavior outside the fit Classes of materials and our ability to model them Interatomic forces in empirical atomistic models Weak and strong laws of action and reaction Forces in conservative systems Atomic forces for some specific interatomic models Bond stiffnesses for some specific interatomic models The cutoff radius and interatomic forces 298 Further reading 299 Exercises Molecular statics The potential energy landscape Energy minimization Solving nonlinear problems: initial guesses The generic nonlinear minimization algorithm The steepest descent (SD) method Line minimization The conjugate gradient (CG) method The condition number The Newton-Raphson (NR) method Methods for rinding saddle points and transition paths The nudged elastic band (NEB) method Implementing molecular statics Neighbor lists Periodic boundary conditions (PBCs) Applying stress and pressure boundary conditions Boundary conditions on atoms Application to crystals and crystalline defects Cohesive energy of an infinite crystal The universal binding energy relation (UBER) Crystal defects: vacancies Crystal defects: surfaces and interfaces Crystal defects: dislocations The7-surface The Peierls-Nabarro model of a dislocation Dealing with temperature and dynamics 371 Further reading 371 Exercises 372

6 Part III Atomistic foundations of continuum concepts Classical equilibrium statistical mechanics Phase space: dynamics of a system of atoms Hamilton's equations Macroscopic translation and rotation Center of mass coordinates Phase space coordinates Trajectories through phase space Liouville's theorem Predicting macroscopic observables Time averages The ensemble viewpoint and distribution functions Why does the ensemble approach work? The microcanonical (NVE) ensemble The hypersurface and volume of an isolated Hamiltonian system The microcanonical distribution function Systems in weak interaction Internal energy, temperature and entropy Derivation of the ideal gas law Equipartition and virial theorems: microcanonical derivation The canonical (NVT) ensemble The canonical distribution function Internal energy and fluctuations Helmholtz free energy Equipartition theorem: canonical derivation Helmholtz free energy in the thermodynamic limit 432 Further reading 437 Exercises Microscopic expressions for continuum fields Stress and elasticity in a system in thermodynamic equilibrium Canonical transformations Microscopic stress tensor in a finite system at zero temperature Microscopic stress tensor at finite temperature: the virial stress Microscopic elasticity tensor Continuum fields as expectation values: nonequilibrium systems Rate of change of expectation values Definition of pointwise continuum fields Continuity equation Momentum balance and the pointwise stress tensor Spatial averaging and macroscopic fields Practical methods: the stress tensor The Hardy stress The virial stress tensor and atomic-level stresses The Tsai traction: a planar definition for stress Uniqueness of the stress tensor Hardy, virial and Tsai stress expressions: numerical considerations 488 Exercises 489

7 9 Molecular dynamics Brief historical introduction The essential MD algorithm The NVE ensemble: constant energy and constant strain Integrating the NVE ensemble: the velocity-verlet (VV) algorithm Quenched dynamics Temperature initialisation Equilibration time The NVT ensemble: constant temperature and constant strain Velocity rescaling Gauss' principle ofleast constraint and the isokinetic thermostat The Langevin thermostat The Nose-Hoover (NH) thermostat Liouville's equation for non-hamiltonian systems An alternative derivation of the NH thermostat Integrating the NVT ensemble The finite strain NcrE ensemble: applying stress A canonical transformation of variables The hydrostatic stress state The Parrinello-Rahman (PR) approximation The zero-temperature limit: applying stress in molecular statics The kinetic energy of the cell The NaT ensemble: applying stress at a constant temperature 533 Further reading 534 Exercises 534 Part IV Multiscale methods What is multiscale modeling? Multiscale modeling; what is in a name? Sequential multiscale models Concurrent multiscale models Hierarchical methods Partitioned-domain methods Spanning time scales 547 Further reading Atomistic constitutive relations for multilattice crystals Statistical mechanics of systems in metastable equilibrium Restricted ensembles Properties of a metastable state from a restricted canonical ensemble Relating mean positions to applied deformation: the Cauchy-Born rule Multilattice crystals and mean positions Cauchy-Born kinematics Centrosymmetric crystals and the Cauchy-Born rule Extensions and failures of the Cauchy-Born rule 562

8 11.3 Finite temperature constitutive relations for multilattice crystals Periodic supercell of a multilattice crystal Helmholtz free energy density of a multilattice crystal Determination of the reference configuration Uniform deformation and the macroscopic stress tensor Elasticity tensor Quasiharmonic approximation Quasiharmonic Helmholtz free energy Determination of the quasiharmonic reference configuration Quasiharmonic stress and elasticity tensors Strict harmonic approximation Zero-temperature constitutive relations General expressions for the stress and elasticity tensors Stress and elasticity tensors for some specific interatomic models Crystal symmetries and the Cauchy relations 595 Further reading 598 Exercises Atomistic-continuum coupling: static methods Finite elements and the Cauchy Born rule The essential components of a coupled model Energy-based formulations Total energy functional The quasi-continuum (QC) method The coupling of length scales (CLS) method The bridging domain (BD) method The bridging scale method (BSM) CACM: iterative minimization of two energy functionals Cluster-based quasicontinuum (CQC-E) Ghost forces in energy-based methods A one-dimensional Lennard-Jones chain ofatoms A continuum constitutive law for the Lennard-Jones chain Ghost forces in a generic energy-based model of the chain Ghost forces in the cluster-based quasicontinuum (CQC-E) Ghost force correction methods Force-based formulations Forces without an energy functional FEAtandCADD The hybrid simulation method (HSM) The atomistic-to-continuum (AtC) method Cluster-based quasicontinuum (CQC-F) Spurious forces in force-based methods Implementation and use of the static QC method A simple example: shearing a twin boundary Setting up the model Solution procedure Twin boundary migration Automatic model adaption 645

9 12.7 Quantitative comparison between the methods The test problem Comparing the accuracy of multiscale methods 650 of multiscale methods Quantifying the speed Summary of the relative accuracy and speed of multiscale methods 655 Exercises Atomistic-continuum coupling: finite temperature and dynamics Dynamic finite elements Equilibrium finite temperature multiscale methods Effective Hamiltonian for the atomistic region Finite temperature QC framework Hot-QC-static: atomistic dynamics embedded in a static continuum Hot-QC-dynamic: atomistic and continuum dynamics Demonstrative examples: thermal expansion and nanoindentation Nonequilibrium multiscale methods A naive starting point Wave reflections Generalized Langevin equations Damping bands Concluding remarks 689 Exercises 689 Appendix A Mathematical representation of interatomic potentials 690 A. 1 Interatomic distances and invariance 691 A.2 Distance geometry: constraints between interatomic distances 693 A.3 Continuously differentiable extensions of Vmt(s) 696 A.4 Alternative potential energy extensions and the effect on atomic forces 698 References Index

Structure and Dynamics : An Atomic View of Materials

Structure and Dynamics : An Atomic View of Materials MARTIN T. DOVE Department ofearth Sciences University of Cambridge OXFORD UNIVERSITY PRESS Contents 1 Introduction 1 1.1 Observations 1 1.1.1 Microscopic