Modeling & Simulation of Glass Structure VCG Lecture 21 John Kieffer Department of Materials Science and Engineering University of Michigan 1 Virtual Glass Course — 07 [email protected] Overview … Historical perspective Simulation methodologies Theoretical background Monte Carlo simulation Molecular dynamics simulation Reverse Monte Carlo Force fields Information retrieval Statistical mechanical formalisms Structural analyses Dynamics Application examples 2 Virtual Glass Course — 07 [email protected] Force Fields Types of interactions between two molecules Electrostatic interactions Between charges (if ions) and between permanent dipoles, quadrupoles and higher multipole moments – May be represented through partial charges Embedding terms (metals) Induction interactions Between, for example, a permanent dipole and an induced dipole Dispersion interactions Long-range attraction that has its origin in fluctuations of the charge distribution Valence (overlap) interactions Covalent bonds Repulsive interaction at short range which arises from exclusion principle Residual valence interactions Specific chemical forces giving rise to association and complex formation - e.g., hydrogen bonding Types of interactions within a molecule Bond stretch, bond-angle bending, torsional potential r q 3 Virtual Glass Course — 07 [email protected] Force Fields Non-bonded interactions are typically long-range and involve a large number of particles. Yet, contributions to the total energy and forces are considered as pair- wise additive Non-bonded (electrostatic) + - Non-bonded (van der Waals) + Johannes Diderik Van der Waals 1837-1923 4 Virtual Glass Course — 07 [email protected] Metals 0.2 Example: Morse force field (r) 0 Approximate physical behavior Computationally efficient -0.2 -0.4 Morse Harmonic 2 a r r0 a r r0 r D 0 e 2 e -0.6 a r r a r r D e 0 e 0 2 0 -0.8 -1 0.5 1 1.5 2 2.5 r 5 Virtual Glass Course — 07 [email protected] Van der Waals interactions The most popular model of the van der Waals non-bonded interactions between atoms is the Lennard-Jones potential. 12 6 U ( r ) 4 r r U(r) Collision typical cutoff diameter at 2.5 Sir John Edward minimum at r=1.122 Lennard-Jones 1894-1954 r = rij 6 Virtual Glass Course — 07 [email protected] Metals Embedded atom method 1 E F r t i ij 2 i i j i f r i ij j j 7 Virtual Glass Course — 07 [email protected] Metals Embedded atom method r re 1 f r f e e r re 1 r e e F E 1 ln 6 c e e e e 8 Virtual Glass Course — 07 [email protected] Embedded atom method 1 2 B E c Ec = cohesive energy Other factors are model parameters that are determined based on B = bulk modulus lattice parameter, bulk modulus, F= atomic E volume1 ln shear modulus, 6 cohesive energy, c e e e e vacancy formation energy, and average electron density 9 Virtual Glass Course — 07 [email protected] Formation of dislocations in metals V. Bulatov, LLNL 10 Virtual Glass Course — 07 [email protected] Simulation of Laser Ablation on Metal Surface QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. G.H. Gilmer, LLNL 3·106 atoms x 140 ps 11 Virtual Glass Course — 07 [email protected] Embedded atom method Simple closed form Computationally efficient Based on density functional theory concept Accurate for highly symmetric structures (FCC, BCC) Accurate for spherical electron orbitals Increasingly inaccurate for complex atomic electron configurations and low-symmetry structures 12 Virtual Glass Course — 07 [email protected] Range of interactions Short-range interactions: Long-range interactions: 1/2·N·NC force calculations 1/2·N·(N-1) force calculations 13 Virtual Glass Course — 07 [email protected] Ionic Materials Coulomb interactions are fundamental to atomistic systems Coulomb interactions have long-range effects N N 1 q iq j CC 4 r 0 i j i ij 14 Virtual Glass Course — 07 [email protected] Ionic Materials Degree of charge localization can vary Charge polarization effects can be accounted for analytically N N N N q CD 1 q i DC 1 j j r ij i r ij 4 r 3 4 r 3 0 i j i ij 0 i j i ij N N DD 1 3 i j r r 5 i ij j ij 3 4 r r 0 i j i ij ij 15 Virtual Glass Course — 07 [email protected] Ionic Materials Repulsion due to electron orbital overlap Repulsion due to interactions between nuclei … handled by empirical function N NC z BM z j i j rij ij A 1 i e ij n n i j 1 i j 16 Virtual Glass Course — 07 [email protected] The trouble with Coulomb interactions 5 CC (r ) ij 4 3 2 N N 1 q iq j CC 4 1 r 0 i j i ij 0 0 1 2 3 4 5 r ij 17 Virtual Glass Course — 07 [email protected] The trouble with Coulomb interactions Conditionally convergent … I.e., depends on how one adds it up … 18 Virtual Glass Course — 07 [email protected] Long-range interactions: Ewald summation method Total potential energy of one ion at the reference point. Summation decomposed into two parts, one to be evaluated in real space, the other in reciprocal space. … b 1 2 a 1 a b 2 3 2 r r l r r q e l l 2 19 Virtual Glass Course — 07 [email protected] Reciprocal space 2 a 4 iGr a c G e a G iGr G e G G 2 c e iGr 4 e iGr 2 G G c G 4 G G G G 20 Virtual Glass Course — 07 [email protected] Reciprocal space iGr G e G 3 2 2 r rl r r l q l e l l 21 Virtual Glass Course — 07 [email protected] Reciprocal space e iGr e iGr d r G G cell 2 iGr 3 2 r r iGr e d r q e l e d r l cell cell l 22 Virtual Glass Course — 07 [email protected] Reciprocal space e iGr e iGr d r G G cell 2 iGr 3 2 r r iGr e d r q e t e d r t cell space t 23 Virtual Glass Course — 07 [email protected] Reciprocal space 2 iGr iGr 3 2 r r iGr e e d r q e t e d r G G t cell space t 2 3 2 iGr iG q e t e d r G t t space 2 2 iGr t G 4 G 4 G q t e e S G e t 24 Virtual Glass Course — 07 [email protected] Reciprocal space 2 4 2 iG r G 4 a S G G e 4 2 2 G 4 S G G e b At r = 0: a Self-energy correction 1 2 4 r 2 r dr 2 q b t 0 2 4 2 G 4 1 2 1 S G G e 2 q t 25 Virtual Glass Course — 07 [email protected] Real space r l 1 1 r q r dr dr 2 l r r r l l l 0 rl q l erfc r 2 2 l r l l r l 26 Virtual Glass Course — 07 [email protected] Convergence in real and reciprocal space 4 2 1 2 q S G G 2 e G 4 2 q l erfc r i i l r l l 5 1 CC S(G) (r ) ij 4 0.5 3 0 2 -0.5 1 -1 0 0 5 10 15 20 25 0 1 2 3 4 5 G r ij 27 Virtual Glass Course — 07 [email protected] Molecular Simulation Massively parallel molecular dynamics Spatial domain decomposition - large systems Replicated data - long simulation times 28 Virtual Glass Course — 07 [email protected] Spatial Decomposition and Ewald Sum 1 2 > 1 Requires larger G 4 2 1 2 q S G G 2 e G 4 2 q l erfc r i i l G l rl 29 Virtual Glass Course — 07 [email protected] Particle-Mesh Ewald Method iG r i h x k y lz S G q e t q e t t t t t t t r r j j f ij m+1 f ij m r r i i f[u] interpolation function n n+1 iG u i h u k v lw S G f q e q e u t t t 30 Virtual Glass Course — 07 [email protected] Example: fracture in brittle materials Quic kTime™ and a GIF decompressor are needed to see this picture. 31 Virtual Glass Course — 07 [email protected] Overview … Historical perspective Simulation methodologies Theoretical background Monte Carlo simulation Molecular dynamics simulation Reverse Monte Carlo Force fields Information retrieval Statistical mechanical formalisms Structural analyses Dynamics Application examples 32 Virtual Glass Course — 07 [email protected] Ionic materials Fundamentally correct Accurate for strongly ionic compounds, including many ceramics and salts … Long-range interactions cause computational expense Increasingly inaccurate when partially covalent interactions and charge transfer occurs 33 Virtual Glass Course — 07 [email protected] Questions? 34 Virtual Glass Course — 07 [email protected] “Universal” force fields Goal is to develop force field that can be to describe intra- and inter-molecular interactions for arbitrary molecules UFF (Universal force field) A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard and W. M. Skiff, "UFF, a Full Periodic-Table Force-Field for Molecular Mechanics and Molecular-Dynamics Simulations," J.