Coarse-graining molecular dynamics in crystalline solids

Transcription

1 in crystalline solids Xiantao Li Department of Mathematics Penn State University. July 19, 2009.

2 General framework Outline 1. Motivation and Problem setup 2. General methodology 3. The derivation of the effective equations 3.1 Static problems 3.2 Dynamic problems 4. Implementations

3 General framework Motivation System under consideration: Solid systems with crystal structures. Starting point: Full molecular dynamics model with empirical potential Observation: Very often, the interesting atoms are in a localized domain. Goal: Deriving an effective model in which the effect of the unimportant atoms are included implicitly.

4 General framework /statics Main issues, 1. Selecting coarse-grain variables 2. Deriving effective equations 3. Efficient computational methods Selecting coarse-grain variables near a crack.

5 General framework Problem setup As a boundary condition: 1. eliminating phonon reflection 2. model external loading 3. model heat bath to maintain system temperature Partition of the atoms

6 Coarse-graining molecular statics Boundary condition for molecular statics Full atomistic model: min V (u). u Partition of the displacement: u = (u I, u J ), dim(u I ) dim(u J ) u I : displacement of the atoms in the computational domain; u J : displacement of the atoms in the bath. Linearize the molecular statics model: i D i ju j = 0 Lattice Green s function: j D i j g j,k = I δ ik. Fourier representation of the Green s function: g j = 1 e ik R j D(k) 1 dk. B

7 Coarse-graining molecular statics Formulation based on the Green s functions Linearize molecular statics model in the bath j D ju i j = 0. For any atom in the bath u n = j = j g j k,n D k u j k J n,j u j i I n,i u i. A more convenient form: (Li 2009) u J = B JI u I, B JI = G JI G 1 II. The effective model min u I V (u I, B JI u I ).

8 w Coarse-graining molecular statics Example: an edge dislocation and an elliptic crack in Iron : Flexible BC 30 30: Fixed BC 60 60: Fixed BC 90 90: Fixed BC : Flexible BC 30 30: Fixed BC 60 60: Fixed BC 90 90: Fixed BC : Flexible BC 30 30: Fixed BC 60 60: Fixed BC 90 90: Fixed BC x

9 Coarse-graining molecular statics Coarse-graining molecular statics Quasicontinuum method: Tadmor et a Defining coarse-grained variables: q = Bu Eliminating the remaining degrees of freedom: min V (u), subject to q = Bu. The mechanical equilibrium given: q, u = Rq, R = D 1 B T (BD 1 B T ) 1. The effective model min V (Rq). q

10 Coarse-graining molecular statics Coarse-grained model at finite temperature Coarse-graining based on the free energy: F = k B T log Z, Z = e βv δ(q = Bu)du. LeSar et al 1989 Dupuy et al 2005 Blanc et al 2008

11 The dynamic problems Projection formalism Cf. Mori 1965, Zwanzig 1960, Ford, Kac and Mazur 1965, Chorin et al 1998, Hald and Kupferman 2004, etc. Dynamics of the entire system: mü i = ui V.

12 The dynamic problems Projection formalism Cf. Mori 1965, Zwanzig 1960, Ford, Kac and Mazur 1965, Chorin et al 1998, Hald and Kupferman 2004, etc. Dynamics of the entire system: mü i = ui V. Partition of the system: Retained variables Heat bath variables

13 The dynamic problems Mori-Zwanzig s formalism 1. Projection operator (conditional expectation), Pg = E(g retained variables), Q = I P. 2. For any function of the retained variables: ϕ, d dt ϕ(t) = etl Lϕ(0) = e tl PLϕ(0) + e tl QLϕ(0), 3. Dyson s formula, e tl = e tql + t 4. Generalized Langevin equation: d dt ϕ(t) = etl PLϕ(0) + 0 e (t s)l PLe sql ds. t 0 e (t s)l K(s)ds + F (t). F (t) = e tql QLϕ(0), K(t) = PLF (t).

14 Mori-Zwanzig s formalism for Molecular Dynamics Mori-Zwanzig s formalism for molecular dynamics Adelman and Doll 1974, 1976 Tully 1980 Sykes, 1976 Ciccotti and Ryckaert, 1981 Munakata, 1986 Curtarolo and Ceder 2002 Izvekov and Voth 2006 Lange and Grubmüller, 2006 Darve, Solomon and Kia 2009

15 Mori-Zwanzig s formalism for Molecular Dynamics Mori-Zwanzig s formalism for MD Partition of the system: u = (u I, u J ), v = (v I, v J ) The thermodynamic force: e tl PLv I (0) = W u I. The effective free energy: W (u I, T ) = k B T ln Z, Z = e V (u I,u J ) k B T du J. Coarse-graining MD based on the free energy (Rudd and Broughton 1998, 2005).

16 Mori-Zwanzig s formalism for Molecular Dynamics Mori-Zwanzig s formalism for MD boundary conditions Simplify the atomic interaction: V new = V ( ) 1( ) T ( ) u I, B JI u I + uj B JI u I DJJ uj B JI u I. 2 Harmonic approximation: 1. it is only made for the atoms in the bath 2. the force constants are consistent with the atomic potential 3. serves as the first approximation of the original Mori-Zwanzig model F (t) : stationary Gaussian process. F (t)f (s) T = k B T Θ(t s).

17 Mori-Zwanzig s formalism for Molecular Dynamics Mori-Zwanzig s formalism for MD the memory term The memory term: t Θ(τ) u I (t τ)dτ. 0 The generalized Langevin equation: (Li and E 2007, Li 2008) t mü I = ui V Θ(τ) u I (t τ)dτ + F (t) + f ex (t). 0

18 Mori-Zwanzig s formalism for Molecular Dynamics Example: molecular dynamics model for Iron

19 Mori-Zwanzig s formalism for Molecular Dynamics Example: fracture simulation anisotropic elastic solution (Sih and Liebowitz 1968). crack in a BCC system (< 110 >, < 1 10 >, < 001 >). generalized Langevin equations used at the boundary as the boundary condition. variational boundary condition

20 Mori-Zwanzig s formalism for Molecular Dynamics Example: finite temperature crack simulation (Li 2008) At zero temperature

21 Mori-Zwanzig s formalism for Molecular Dynamics Example: finite temperature crack simulation (Li 2008) At finite temperature 500K

22 Mori-Zwanzig s formalism for coarse-graining MD Mori-Zwanzig s formalism for coarse-graining MD Defining coarse-grained variables: q = Bu, p = Bv Projection matrices: P u = D 1 B T (BD 1 B T ) 1 B, P v = B T (BB T ) 1 B The generalized Langevin equation: q = p/m, Mṗ = R T V (Rq) t 0 Θ(τ)p(t τ)dτ + F (t) + f 0

23 Mori-Zwanzig s formalism for coarse-graining MD Energy dissipation The effective total energy: The energy flux: The energy flux: (Li 2009) E(t) = V (Rq) pt M 1 p. t J(t) = p T Θ(τ)p(t τ)dτ. 0 t E(t) E(0) = J(s)ds 0. 0

24 Mori-Zwanzig s formalism for coarse-graining MD Example: One-dimensional chain 2 K=4 K=8 K=16 K= K=4 K=8 K=16 K= U I U I Choose one atom out of every K atoms Choose the average of every K atoms

25 Mori-Zwanzig s formalism for coarse-graining MD Memory kernels for the one-dimensional chain The behavior depends on the selection of the CG variables: 1. A subset of atoms: the kernel converges to a Bessel function 2. Local averages: the kernel decays 1/K 3. Finite element nodal values: the kernel decays 1/K 2.

26 Implementation issues Computing the memory kernels For simple geometry, exact memory kernels can be obtained via Fourier/Laplace transform. Cf. Adelman and Doll 1974, Tully 1980, Berne et at 1990, Cai et al 2000, Wagner, Karpov, Liu, Park Other related work: Ciccotti and Ryckaert 1981, Tuckerman and Berne 1991, Izvekov and Voth nonlocal in both space and time numerical implementation is expensive premature truncation leads to large reflection

27 Implementation issues Local memory kernels (E and Huang, 2001; Li and E, ) the memory kernel is independent of the temperature at zero temperature, F (t) = 0 Basic principles: 1. Efficiency: local kernels 2. Stability: positive-definite kernels 3. Consistency: fluctuation-dissipation theorem Local Kernel Exact Kernel

28 Implementation issues Variational approach 1. Express Θ(t) in the form of, Θ(t) = Γ(t) is local: + Γ(s)Γ(t + s) T ds. Γ ij (t) = 0, if r i r j > r c, or t > t c. 2. Objective functions min Γ(t) e ( ω; Γ ) W (ω)dω. 3. Choose test functions, Phonon mode Green s functions

29 Sampling the random noise Sample the random noise The random noise F (t) is a stationary Gaussian process. The fluctuation-dissipation theorem is satisfied: F (t)f (0) T = k B T Θ(t). Let W (t) be white noise with variance k B TI, then, F (t) = Γ(s)W (t s)ds. k

30 Sampling the random noise Summary 1. Static problems: Green s function approach. 2. Dynamic problems: Mori-Zwanzig s viewpoint Reducing the dimension of the problem linearized interaction bath variables thermal equilibrium initially for the atoms in the continuum region, projection operator using conditional expectation, generalized Langevin equations. 3. Modeling the effective equations with GLEs a. compute the memory kernels Θ b. sample the random noise c. the fluctuation-dissipation theorem