Multiscale Simulations

Transcription

1 Multiscale Simulations Prof. Yan Wang Woodruff School of Mechanical Engineering Georgia nstitute of Technology Atlanta, GA 30332, U.S.A.

2 DFT-MD coupling Topics First-Principles Molecular Dynamics DFT-KMC coupling on-the-fly KMC simulation MD-FEM coupling Quasicontinuum method Coarse-Grained Molecular Dynamics

3 Modeling & Simulation at Multiple Scales s ms μs ns pico-s femto-s Time Scales Finite Element Analysis Dislocation Dynamics Kinetic Monte Carlo Molecular Dynamics / Force Field Tight Binding Density Functional Theory Self-Consistent Field (Hartree-Fock) Quantum Monte Carlo Length Scales nm μm mm m Various methods used to simulate at different length and time scales

4 Zoo of Multiscale Simulation Methods First-principles MD (quantumatomistic coupling) Ehrenfest MD Born-Oppenheimer MD Car-Parrinello MD on-the-fly KMC (DFT-KMC coupling) QM/MM coupling Mathematical Homogenization Heterogeneous Multiscale Method Multiscale FEM quasi-continuum coarse-grained molecular dynamics variational multiscale method concurrent coupling coupled atomistic/discretedislocation adaptive multiscale modeling, bridging scale method bridging domain method DD/FEM coupling TB/MD/FEM coupling <new species born each year>

5 First-Principles Molecular Dynamics The major idea is to replace the predefined potentials in classical molecular dynamics (MD) by first-principles electronic structure calculation on-the-fly (i.e. keep electronic variables as active degrees of freedom in MD). The Algorithm: 1. solve the electronic structure problem for a set of ionic coordinates 2. evaluate forces 3. move atoms 4. repeat

6 Classical MD Based on Newtonian dynamics where approx M R () t = F () t = V { R ()} t e is a few-body-additive-interaction approximation of the true potential energy surface. The electrons follow adiabatically the classical nuclear motion and can be integrated out so that the nuclei evolve on a single global Born-Oppenheimer potential energy surface. A priori construction of the global potential energy surface suffers from the dimensionality bottleneck. Multiscale Systems Engineering Research Group ( ) N approx e J = 1 < J ( ) () 1 ( ) ( 2) { R ( )} = R ( ) + ( R ( ), R ( )) V t V t V t t ( R (), R (), R ()) + + ( 3) V t t t J K < J< K

7 Time-Dependent Schrödinger Equation where 2 H = 2M i Φ r R t = Φ r R t i i t 2 ({ },{ }; ) H ({ },{ }; ) ZZe e J i i 2me i< j r r < J R R J, i r R i j i = ({ },{ }) + V r R i n e i 2M i 2m e 2 2 = + H ({ r},{ R }) e i 2M Total wavefunction can be decomposed as t ( ) ( ) ( ) i Φ { r},{ R }; t Ψ { r}; t χ { R }; t exp dt' E ( t') i i e t0 with a phase factor E * * = d r d RΨ χ H Ψ χ e e Multiscale Systems Engineering Research Group Ze

8 Ehrenfest Molecular Dynamics Simultaneously solve E M R () t = V { ()} t e ( R ) = Ψ H Ψ e * = drψ H Ψ e i Ψ= t H e Ψ 2 2 = Ψ+ V ({ r},{ R }) Ψ i n e i i 2me

9 Born-Oppenheimer Molecular Dynamics n ground state BOMD, the time-independent electronic structure problem is solved self-consistently from each time for a given configuration of nuclei M R () t = min Ψ H Ψ Ψ 0 e 0 H Ψ = E Ψ e Electrons are explicitly set to be fully relaxed for a given configuration of nuclei, in contrast to Ehrenfest MD where electron relaxation is implicit by solving the timedependent Schrödinger equation.

10 BOMD with HF-SCF Assuming single Slater determinant Ψ = det{ ψ 0 j } M R () t = min Ψ H Ψ { ψ } 0 e 0 j Constrained (orthonormal orbitals) minimization min Ψ H Ψ st.. ψ ψ = δ { ψ } 0 e 0 i j ij j Define Lagrangian L = Ψ H Ψ + ( ) 0 e 0 Λ ψ ψ δ ij, ij i j ij The necessary condition of optimality L ψ j = 0 leads to Hartree-Fock equations HF H ψ = Λ ψ e j ij j i Then the new equations of motion are M R () t = min Ψ H Ψ { ψ } 0 0 j e H ψ = Λ ψ HF e j i ij j Multiscale Systems Engineering Research Group

11 BOMD with DFT H Based on the Hellmann-Feynman theorem, MD force is He F = Ψ H Ψ = Ψ H Ψ = Ψ Ψ 0 e 0 0 e R R Recall ( ) 2 e J { },{ } = r R + + e i i i 2 m e i< j r r < J R R J, i r R i j i The force is computed by DFT as ZZe He F = Ψ Ψ = 0 0 R ZZe Ze = + Ψ Ψ Ze 2 2 J R (,..., ) (,..., ) d... d 0 1 M 0 1 M 1 M J r r r r r r R R i r R J i 2 2 ZZe Ze J ρ() d J r r R R r R J = +

12 BOMD with DFT 1. Fix positions of nuclei {R 1,,R N }, solve DFT equations self-consistently; Ze ρ( ') ' r dr + + Vxc () r ψi () r = εψ i i () r 2 r R r r' ρ() r = f ψ () r 2. Find electrostatic force on each atom; ZZe Ze F r dr 2 2 J = () + ρ J R R r R J i 3. Perform a time step and find new positions of nuclei; i i 2 4. Repeat; M R () t = F

13 Drawbacks of BOMD The need to fully relax electronic subsystem while moving the atoms makes it computationally expensive. Full self-consistency at each MD step may not be necessary, especially when system is far from its equilibrium, since one simply needs a rough idea of the force field for a given atomic configuration

14 Car-Parrinello Molecular Dynamics R. Car and M. Parrinello (1985) Unified approach for molecular dynamics and density-functional theory. Phys. Rev. Lett. 55: Publication and citation analysis: : number of publications which appeared up to the year n that contain the keyword ab initio molecular dynamics (or synonyma first-principles MD, Car- Parrinello simulations etc.) in title, abstract or keyword list. : number of publications which appeared up to the year n that cite the 1985 paper by Car and Parrinello

15 Car-Parrinello Lagrangian L Car and Parrinello (1985) postulated the Lagrangian 1 1 = M R + μ ψ ψ Ψ H Ψ + Λ ( ψ ψ δ ) i 0 0 i ij 2 2 CP i i e ij i i ij Kinetic Energy Potential Energy Constraints where μ i s are fictitious masses for the dynamics of orbitals ψ i s.

16 Car-Parrinello Equations of Motion From Euler-Lagrange differential equations in classical mechanics (to ensure δ L ( R, R, ψ, ψ ; tdt ) = 0 ) CP i i d L L CP CP = dt R R d L L CP CP = dt ψ ψ Car-Parrinello equations of motion is derived as i M ( t) R = Ψ H Ψ + constraints({ ψ },{ }) R R i { R } 0 e 0 i δ δ μψ () t = Ψ H Ψ + constraints({ },{ }) δψ δψ { ψ R } i i 0 e 0 i i i

17 CPMD with DFT Car-Parrinello equations of motion M R E [{ ψ },{ R }] DFT i = R 2 Ze = ρ(,) r t dr r R ( t) δe [{ ψ },{ R }] DFT i μψ (,) r t = + ε ψ (,) r t i i ij j δψ (,) r t i j = H ψ (,) r t + ε ψ (,) r t DFT i ij j j

18 Verlet algorithm in CPMD Orbital Dynamics First, a verlet step ignoring orthogonality constraint Δt H ψ DFT i ψ ( t +Δ t) = 2ψ ( t) ψ ( t Δ t) + i i i μ Then, restore orthogonality ψ ( t +Δ t) = ψ ( t +Δ t) + ξ ψ ( t) i i i 2 ( ) ( ) Computationally dynamics is applied to c i s in the reciprocal space with Kohn-Sham orbitals 1 ψ (, rk) = c ( Gk, )exp i( + ) i i G k r Ω G i t

19 CPMD Code The CPMD code is a planewave implementation of DFT for first-principles molecular dynamics First version by Jürg Hutter at BM Zurich Research Lab with dozens of other contributors The code is copyrighted jointly by BM Corp and by Max Planck nstitute, Stuttgart t is distributed free of charge to non-profit organizations (

20 CPMD capabilities Wavefunction optimization: direct minimization and diagonalization Geometry optimization: local optimization and simulated annealing Molecular dynamics: NVE, NVT, NPT Path integral MD Response functions Excited states Time-dependent DFT (excitations, MD in excited states) Coarse-grained non-markovian metadynamics Wannier, EPR, Vibrational analysis QM/MM See on-line manual at:

21 On-the-fly KMC Searching saddle points on the potential energy surface (PES) on-the-fly while performing KMC simulation Search Saddle Points on PES Find Activation Energy Calculate Rate Constants (TST/hTST) Simulate Phase Transition by KMC

22 On-the-fly KMC 1. Start from a minimum configuration; 2. Randomly generate a set of configurations around the minimum and search the saddle points by the Dimer method; 3. Locate the saddle points connect to the minimum; 4. nsert the new events and propensities in the event table in KMC and simulate one step; 5. Repeat;

23 Diffusion of adatom on Al(100) Multiscale Systems Engineering Research Group

24 ΔR V V 1 V 0 2 F 1 F F R 2 Dimer energy: E = V1+ V2 Curvature along Dimer: ˆN C Dimer Method [Henkelman & Jónsson 1999] = E (2 V ) ( ΔR) Estimate: F1 = V1 F2 = V2 E ΔR V ˆ 0 = + ( F1 F2) in 2 4 F R = ( F 1 + F 2) 2 2. Dimer rotates to find the lowest curvature mode of PES, i.e. minimize Dimer energy 3. Translate Dimer towards uphill according to 4. Repeat = 1 2 = F F F F ( F F )/2 F F if C > 0 = FR 2F if C < 0

25 Quasicontinuum (QC) Method Based on the full atomistic model, use mesh to reduce the degrees of freedom representative atoms (repatoms)

26 QC Method Displacement of atom i: Displacement of N atoms: {u 1,u 2,,,u N } Empirically the total energy is the sum of site energy of each atom the Stillinger-Weber (SW) type site energy for atom i is with two-body potential and three-body potential where

27 QC Method The total potential energy of the system (atoms + external loads) is where f i u i is the potential energy of the applied load f i on atom i The goal of the static QC method is to find the atomic displacements that minimize the total potential such that the number of degrees of freedom is substantially reduced from 3N; the computation of the total potential is accurately approximated without the need to explicitly compute the site energy of all the atoms; the critical regions can evolve with the deformation by addition/removal of repatoms.

28 QC Method Reduce DoF Any atom not chosen as a repatom is constrained to move according to the interpolated displacements This first approximation of the QC then, it to replace the energy E tot by E tot,h : with continuum displacement field where S α is the interpolation function with local support

29 QC Method Local Energy The Cauchy-Born rule assumes that a uniform deformation gradient at the macro-scale can be mapped directly to the same uniform deformation on the micro-scale. Thus, every atom in a region subject to a uniform deformation gradient will be energetically equivalent. The energy within an element in crystals can be estimated by computing the energy of one atom in the deformed state and multiplying by the number of atoms in the element. Energy density for each element is where Ω 0 is the unit cell volume and E 0 is the energy of the unit cell when its lattice vectors are distorted according to F The total energy of an element is Multiscale Systems Engineering Research Group

30 QC Method Nonlocal Energy Local energy does not approximate well where deformation of crystal is non-uniform (e.g. surfaces and interfaces) and shorter than the cutoff radius of inter-atomic potential. Energy-based formulation: Nonlocal energy is weighted sum of those of repatoms as where Force-based formulation: The force on repatom β is determined by its neighborhood C β (α,β, for repatom) Atomic-level force in neighborhood c of repatom α weights of the atomic forces Multiscale Systems Engineering Research Group

31 QC Method detailed issues Local-nonlocal energy coupling where N loc +N nl =N rep Local/nonlocal criterion: whether a repatom should be local or nonlocal? whether there is significant variantion of the deformation gradient Effects of local-nonlocal interface: Polycrystals Elastic/plastic deformation decomposition

32 QC Applications nanoindentation (Smith et al. 2001) Silicone Phase transformation and dislocation nucleation observed Different phases observed

33 QC Applications dislocation behavior (Rodney & Phillips 1999) Dislocation junction under shear stress

34 Coarse Grained (CG) Molecular Dynamics displacement of mesh node j (does not have to coincident with an atom) u = j μ fj μ u μ, is a weighted average of displacements of atoms μ s. j Displacement field u( x) = N ( x) u j j j μ Coarse grained energy is the average of the canonical ensemble of the atomistic Hamiltonian on the constrained phase space {(x,p)} ( ) u, MD E u, u = H = d x d p H e Δ/ Z where β=1/(kt), Z d d e βh MD = x p Δ is partition function, and μ μ Δ= δ ( u f ) ( f / m ) j j u δ u μ μ jμ j p μ μ jμ μ enforces constraints. The atomistic Hamiltonian MD k k MD u μ μ MD k k βh p p H = + E + D T μ μ coh 1 T u u μ μ μν ν μ 2m μ μν, 2 Cohesive energy of atoms

35 CGMD Partition function Z = Z kin Z pot Potential part of partition function is CG potential energy then is w/ stiffness Computationally, the full CG energy is where T ( N N ) u K 2 2 Z ( u, β) = CC β e β pot k atom node j jk k 3 1 T E ( u ) = log Z = ( N N ) kt + u K u β 2 2 u pot k pot atom node j jk k ( ) int Thermal 1 E u, u = U + ( M u u + u K u ) T T k k jk j k j jk k 2 j, k coh atom atom node U = N E + 3( N N ) kt int K = ( f D f ) 1 1 jk μν, jμ μν kν Harmonic is internal energy and x x K f m f m N N 1 1 = ( ) = ( ) ( ) jk μ jμ μ kμ μ μ j μ k μ is CG mass matrix

36 CGMD Applications MEMS/NEMS NEMS silicon microresonator The coarse grained (CG) region comprises most of the volume The molecular dynamics (MD) region contains most of the simulated degrees of freedom the CG mesh is refined to the atomic scale where it joins with the MD lattice.

37 Summary First-Principles MD On-the-fly KMC Quasicontinuum method Coarse-Grain Molecular Dynamics

38 Further Readings First-Principles Molecular Dynamics Marx D. and Hutter J. (2000) Ab nitio Moleular Dynamics: Theory and mplementation. n: J. Grotendorst (Ed.) Modern Methods and Algorithms of Quantum Chemistry (Jülich: John von Neumann nstitute for Computing, SBN ), Vol.3, pp On-the-fly KMC Mei, D., Ge, Q., Neurock, M., Kieken, L., and Lerou, J. (2004) First-principles-based kinetic Monte Carlo simulation of nitric oxide decomposition over Pt and Rh surfaces under lean-burn conditions. Molecular Physics, 102(4): Kratzer P. and Scheffler M. (2002) Reaction-limited island nucleation in molecular beam epitaxy of compound semiconductors. Physical Review Letters, 88(3): (1-4) Battaile C.C., Srolovitz D.J., Oleinik.., Pettifor D.G., Sutton A.P., Harris S.J., and Butler J.E. (1999) Etching effects during the chemical vapor deposition of (100) diamond. Journal of Chemical Physics, 111(9): Henkelman, G. and Jónsson, H. (2001) Long time scale kinetic Monte Carlo simulations without lattice approximation and predefined event table. Journal of Chemical Physics, 115(21): Trushin, O., Karim, A., Kara, A. and Rahman, T.S. (2005) Self-learning kinetic Monte Carlo method: Application to Cu(111). Physical Review B, 72: (1-9). Xu, L. and Henkelman, G. (2008) Adaptive kinetic Monte Carlo for first-principles accelerated dynamics. Journal Chemical Physics, 129: (1-9). Mei D., Xu L., and Henkelman G. (2009) Potential energy surface of Methanol decomposition on Cu(110). Journal of Physical Chemistry C, 113:

39 Further Readings Quasicontinuum Tadmor E.B., Ortiz M., and Phillips R. (1996) Quansicontinuum analysis of defects in solids. Philosophical Magazine A, 73(6): Shenoy V.B., Miller R., Tadmor E.B., Rodney D., Phillips R., and Ortiz M. (1998) An adaptive methodology for atomic scale mechanics: the quasicontinuum method. J. Mech. Phys. Sol., 47: Knap J. and Ortiz M. (2001) Ana analysis of the quasicontinuum method. J. Mech. Phys. Sol., 49: Smith G.S., Tadmor E.B., Bernstein N. and Kaxiras E. (2001) Multiscale simulations of Silicon Nanoindentation. Acta Mater., 49: Rodney D. and Phillips R. (1999) Structure and strength of dislocation juncitons: An atomic level analysis. Physical Review Letters, 82(8): Coarse-grained molecular dynamics Rudd R.E. and Broughton J.Q. (1998) Coarse-grained molecular dynamics and the atomic limit of finite elements. Physical Review B, 58(10):R5893-R5896 Rudd R.E. and Broughton J.Q. (2005) Coarse-grained molecular dynamics: Nonlinear finite elements and finite temperature. Physical Review B, 72(14): Variational multiscale method Hughes T.J.R., Feijóo G.R., Mazzei L., and Quincy J.-B. (1998) The variational multiscale method a paradigm for computational mechanics. Computational Methods in Applied Mechanics & Engineering, 166(1-2):3-24

40 Further Readings Concurrent coupling Broughton J.Q., Abraham F.F., Bernstein N., Kaxiras E. (1999) Concurrent coupling of length scales: methodology and application. Physical Review B, 60(4): Bridging domain method Xiao S.P. and Belytschko T. (2004) A bridging domain method for coupling continua with molecular dynamics. Computational Methods in Applied Mechanics & Engineering, 193(17-20): Bridging scale method Liu W.K., Park H.S., Qian D., Karpov E.G., Kadowaki H., Wagner G.J. (2006) Bridging scale methods for nanomechanics and materials. Computational Methods in Applied Mechanics & Engineering, 195(13-16): Wagner G.J., Liu W.K. (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. Journal of Computational Physics, 190(1): Park H.S., Karpov E.G., Liu W.K., Klein P.A. (2005) The bridging scale for twodimensional atomistic/continuum coupling. Philosophical Magazine, 85(1):