Mesoscale Simulation Methods Ronojoy Adhikari The Institute of Mathematical Sciences Chennai
Outline What is mesoscale? Mesoscale statics and dynamics through coarse-graining. Coarse-grained equations for a binary fluid-fluid mixture. Numerical methods of solution. Example simulations.
Numbers and methods 7 Continuum methods CFD LB 6 5 4 Atomistic methods MD LD BD DPD 3 2 1 Ab initio methods DFT CPMD 0 Number of atoms (log) Simulation method Example of method
How lengths scale with numbers N = 1 L 1Å L N 1 3 N = 10 23 L 10 7 Å!! Try this for water Molar mass = 18 gm Molar volume = 18 ml!! Are other scalings possible? How does length grow with size for a polymer? Throughout, (!!) indicate exercises/derivations/points to ponder.
Length and time scales T ċ = D 2 c milli γṙ = U micro m R = U nano Hψ = Eψ pico nano micro milli L!! Are there systems with millimeter L but picosecond T?
How times scale with masses and lengths Harmonic Oscillator Wave Equation Diffusion Equation mẍ + kx = 0 ü = c 2 2 u u = D 2 u ω 2 0 = k m ω 0 = ±cq iω 0 = Dq 2 τ 0 m k τ 0 λ c τ 0 λ2 D
Thermal fluctuations Brownian motion : Einstein (1905) P (x x) = [ 1 (x x) 2 ] exp 2πDτ 2Dτ D = k BT 6πηa temperature size Q. Why does a smaller particle have a higher diffusion coefficient? A. The root N effect! The central limit theorem helps us understand this. Stokes-Einstein-Sutherland Relation!! Is it (central limit) theorem or central (limit theorem)?
Langevin theory damping reversible force inertia m v + γv F (x) = ξ(t) fluctuation deterministic stochastic mean behaviour root N fluctuations F = 0 Free Brownian particle regression to the mean equilibrium fluctuations v 2 = k B T
Mesoscale regime Mesoscale methods 7 6 Continuum } Coarse grained length scales. Coarse grained time scales. 5 4 Atomistic Retain thermal fluctuations. 3 Examples 2 1 0 Ab initio Brownian dynamics. Dissipative particle dynamics. Number of Simulation Time-dependent Ginzburg-Landau.
Coarse-graining in degrees of freedom What is the idea? Coarse-grained sum What is the distribution? x 1 } RV y = x 1 + x 2 P (x x 1, x 2 ) 2 P (y) P (y) = dx 1 dx 2 δ(y x 1 x 2 )P (x 1, x 2 ) P (y) contains less information about the system than P (x 1, x 2 ) Adequate if we are only interested in the sum variable.
General coarse-graining formula P (x 1, x 2,..., x N ) y = f(x 1, x 2,..., x N ) Microstate probability Mesoscale variable/order parameter P (y) = N i=1 dx i δ[y f(x 1,..., x N )]P (x 1,..., x N ) Mesostate probability
Coarse-grained Landau-Ginzburg functional Microscopic Hamiltonian H(q) Gibbs distribution P (q) = exp[ βh(q)] Z Order parameter ψ = f(q) Definition of Landau functional P (ψ) exp[ βl] Z ψ P (ψ) = N i=1 dq i δ[ψ f(q)] exp[ βh(q) Z exp[ βl] Z ψ
Explicit DOF coarse-graining... H(s 1, s 2,..., s N ) = J ij s i s j ψ λ = 1 N λ i λ s i Order parameter. Block spin. Coarse-grained magnetization.
Explicit DOF coarse-graining... H(s 1, s 2,..., s N ) = J ij s i s j ψ λ = 1 N λ i λ s i Order parameter. Block spin. Coarse-grained magnetization. P (ψ λ ) exp[ βl] Z ψ =? K. Binder, Z. Phys B, 43, 119 (1981)
Explicit DOF coarse-graining... H(s 1, s 2,..., s N ) = J ij s i s j ψ λ = 1 N λ i λ s i Order parameter. Block spin. Coarse-grained magnetization. P (ψ λ ) exp[ βl] Z ψ =? K. Binder, Z. Phys B, 43, 119 (1981)
Explicit DOF coarse-graining... H(s 1, s 2,..., s N ) = J ij s i s j ψ λ = 1 N λ i λ s i Order parameter. Block spin. Coarse-grained magnetization. Explicit DOF needs this! P (ψ λ ) exp[ βl] Z ψ =? K. Binder, Z. Phys B, 43, 119 (1981)
Or ansatz based on symmetry. L = A 2 ψ2 + B 4 ψ4 + K 2 ( ψ)2 All equal-time equilibrium properties are determined by this functional.!! Why is this called a free energy? Is it a thermodynamic free energy? P (s) a + δ(s 1) + a δ(s + 1) -J +J P (ψ) exp[ β( A 2 ψ2 + B 4 ψ4 )] K( ψ) 2 local part non-local part
What about dynamics? Temporal coarse-graining m v + γv = ξ(t) γ = 6πηa D = k BT 6πηa Why is this independent of particle mass? τ d τ d = m γ ẋ = v γẋ = ξ(t) Inertial time scale. Time the velocity remembers its initial condition. Overdamped equations of motion. Valid when τ τ d
Overdamped stochastic equations damping reversible force γẋ = U + ξ(t) fluctuation deterministic stochastic mean behaviour root N fluctuations
Overdamped coarse-grained equations of motion damping reversible force Model A equation ψ(r) = Γ δl δψ + ξ(r, t) fluctuation!! Zero-mean Gaussian noise ξ(r, t) = 0 Fluctuation Dissipation Relation. ξ(r, t)ξ(r, t ) = 2k B T Γδ(r r )δ(t t ) ψ(r) = Γ[Aψ + Bψ 3 K 2 ψ] + ξ(r, t) Stochastic partial differential equations with additive Gaussian noise
Conserved order parameters Ising spin Lattice gas Ising spins are usually non-conserved d dt ψ(r, t) 0 Lattice gas models are conserved d dt ψ(r, t) = 0
Conserved coarse-grained dynamics damping reversible force Model B equation ψ(r) = Γ 2 δl δψ + ξ(r, t) fluctuation!! Zero-mean vector ξ(r, t) = 0 Gaussian noise Fluctuation Dissipation Relation. ξ i (r, t)ξ j (r, t ) = 2k B T Γδ ij δ(r r )δ(t t ) ψ(r) = Γ 2 [Aψ + Bψ 3 K 2 ψ] + ξ(r, t)!! Derive this. Use the local conservation law ψ + j = 0
Numerical solution Stochastic PDE t ψ = D 2 xψ + ξ Spatial discretisation x ψ = ψ(x + h) ψ(x) h Stochastic ODE t ψ(i) = DL 2 ijψ(j) + ξ(i) Temporal integration ψ(i, t + t) ψ(i, t) = t+ t DL 2 ijψ(j) + ξ(i) Stochastic realisation!! Obtain the explicit form of the L matrix for a central difference Laplacian
Summary of TDGL mesoscale methods Microscopic Hamiltonian Symmetry + gradient expansion Explicit Coarse Graining Ginzburg-Landau Functional Overamped approximation Langevin equations Numerical solution
Conclusion Mesoscale methods are appropriate at length scales intermediate between the molecular and continuum scales. The continuum description must be supplemented by fluctuation terms. At the mesoscale, the number of particles is not so large that fluctuations can be neglected. Lengths and times must be coarse-grained. Intelligent coarse-graining improves computational efficiency, often by orders of magnitude, compared to direct MD. Mesoscale methods can be particle-based, for instance dissipative particle dynamics and Brownian dynamics, or field-based like time-dependent Ginzburg-Landau theory and fluctuating hydrodynamics. TDGL equations can be solved very efficiently on computers using matrix formulations.
Further reading Stochastic processes in Physics and Chemistry : van Kampen Handbook of stochastic processes : Gardiner Principles of Condensed Matter Physics : Chaikin and Lubensky Modern Theory of Critical Phenomena : Ma Reviews of Modern Physics article : Halperin and Hohenberg Numerical Solutions of Stochastic Differential Equations : Kloeden and Platen.
Example 0 : model A Amit Bhattacharjee, unpublished.
Example 1 : conserved binary mixture Pagonabarraga et al, New Journal of Physics.
Example 2 : Nematic liquid crystals Q ij = 3 2 S(n in j 1 3 δ ij) + T 2 (l il j m i m j ) Tensor order parameter B 6 4 2 0 biaxial nematic phase discotic phase superheating spinodal line I!N transition line supercooling spinodal line UN!BN transition line isotropic phase F = d 3 x[ 1 2 AT rq2 + 1 3 BT rq3 + 1 4 C(T rq2 ) 2 + E (T rq 3 ) 2 + 1 2 L 1( α Q βγ )( α Q βγ ) + 1 2 L 2( α Q αβ )( γ Q βγ )].!2!4 uniaxial nematic phase!6!6!4!2 0 2 4 6 A t Q αβ (x, t) = Γ αβµν δf δq µν Model A equation Γ αβµν = Γ[δ αµ δ βν + δ αν δ βµ 2 d δ αβδ µν ]
Spinodal decomposition into the nematic phase Amit Bhattacharjee, G. I. Menon, RA
Nucleation into the nematic phase with uniform nuclei Amit Bhattacharjee, G. I. Menon, RA
Nucleation into the nematic phase with non-uniform nuclei Amit Bhattacharjee, G. I. Menon, RA.
Thank you for your attention.