String method for the Cahn-Hilliard dynamics Tiejun Li School of Mathematical Sciences Peking University tieli@pku.edu.cn Joint work with Wei Zhang and Pingwen Zhang
Outline Background Problem Set-up Algorithms and Results Summary
Background
Polymer Polymers or soft matters are extremely important for its softness, self-assembly and relatively easy response to fluctuations. Rich phase behavior (diblock copolymer) N A N B L G Composition f = N A /N, Flory-Huggins parameter χn C S
Triblock copolymer Richer phases than diblock case
Mathematical models Landau-Brazovski energy functional f[ϕ] = 1 V dr ξ 2 2 [( 2 + 1)ϕ(r)] 2 + τ 2 ϕ2 (r) γ 3! ϕ3 (r)+ 1 4! ϕ4 (r) Self-Consistent Field Theory H[w,w + ]= where dr 1 χn w2 w + V ln Q[w,w + ] Q = 1 drq(r, 1) V q(r, s) = 2 s rq(r, s) w(r, s)q(r, s), q(r, 0) = 1 w(r, s) =w + (r) w (r), s [0,f] w(r, s) =w + (r)+w (r), s [f,1]
Rare events and nucleation Metastable polymer phases can make transition because of the thermal noise Rare events: Wigner, Eyring, Kramers, Chandler, Jonsson, Elber, E-Ren-Vanden-Eijnden et al....
Some theory and methods Large deviation theory, potential theory, etc. Minimum action method, GMAM, string method, finite temperature string method, GAD method, TPT, NEB, dimer method, Max-flux, etc. Wentzell-Freidlin, Bovier, E-Ren-Vanden-Eijnden, Jonsson, Elber, Schutte,...
Key issues Numerical method to identify the transition path Locate the transition state Compute the transition rate
Nucleation study for diblock copolymer Landau-Brazovski model Lamella to cylinder Lin-Cheng-E-Shi-Zhang, JCP 2010 Self Consistent Field Theory Gyroid to cylinder Cheng-Lin-E-Zhang-Shi, PRL 2011
Large domain dynamics Multiple droplets: Nucleation and growth in large domain Slow nucleation and fast growth Heo-Zhang-Du-Cheng, Scipt. Mater. 2010
Problem: 1. What will happen if different dynamics are considered? 2. How to couple the slow and fast dynamics on the large domain?
Problem Set-up
Dynamics Allen-Cahn dynamics (E-Ren-Vanden-Eijnden, CPAM,2004) u t = δf (u) δu Cahn-Hilliard dynamics u t = δf (u) δu First consider the simplest case: 1D, double well potential, Cahn-Hilliard
Stochastic Cahn- Hilliard model Mathematical equations u t = 2 δf (u) x 2 δu Energy functional + 2ξ x [0, 1] F (u) = Noise 1 0 κ 2 2 u 2 + f(u) dx, x ξ = x η(x, t) f(u) = (u2 1) 2 4 η(x, t) Space-time Gaussian white noise.
Boundary conditions Neumann BC x u = xu 3 = 0 at x =0, 1, Periodic BC u, x u, xu 3 are periodic in R with period 1, η(x, t) =η(x +1,t). Both BCs can preserve the mass conservation 1 0 u(x, t)dx = 1 0 u(x, 0)dx = m
Analytical studies Stochastic PDE: well-posedness of the mild solution Da Prato-Debussche, Nonlinear Anal. 1996. Debussche-Zamboti, Ann. Prob. 2007 Large deviations and dynamical analysis Feng, Meth. Funct. Anal. Topol. 2003 Blomker-Gawron-Wanner, DCDS-A, 2010
Discretize to finite dimensions Define the grids x =1/n, x i =(i 1/2) x, i =1, 2,...,n and approximating function X i t u(x i,t) with centered difference to derivatives, we have dx t = A V (X t )dt + 2 x σdw t, where A 2 x 2, σ x
Neumann BC A = 1 x 2 σ = 1 x 1 1 0 0 0 1 2 1 0 0. 0........... 0 0 1 2 1 0 0 0 1 2 1 0 0 0 1 1 1 0 0 0 1 1 0 0. 0........ 0 0 1 1 0 0 0 1 1 0 0 0 1 n (n 1). We have A = σσ T
Periodic BC A = 1 x 2 σ = 1 x 2 1 0 0 1 1 2 1 0 0. 0........... 0 0 1 2 1 0 0 0 1 2 1 1 0 0 1 2 1 0 0 1 1 1 0 0. 0........ 0 0 1 1 0 0 0 1 1 n n. We have A = σσ T
Potential function The potential function with the form V (w) =F n (w)/ x F n (w) = n κ 2 i=2 2 wi w i 1 x 2 + f(wi ) + κ2 2 wn w 1 x 2 + f(w1 ) x for Periodic BC F n (w) = n κ 2 i=2 2 wi w i 1 x 2 + f(wi ) + f(w 1 ) x for Neumann BC
Meta-stable states Metastable states (homogeneous state, global minimum) and saddles Neumann BC Periodic BC Red: Homogeneous state. Black: Global minimum. Blue: Saddle points Bates-Fife, SIAM Appl Math 1993, Blomker-Gawron-Wanner, DCDS-A 2010
To be solved Numerical method to identify the transition path Locate the transition state Compute the transition rate
Algorithms and Results
Large deviation theory for Cahn-Hilliard dynamics Consider the transition between two metastable states a and b Define the considered path space B T = {u u(0) = a, u(t )=b, u P} we have from large deviation theory lim lnp T =limlnp (u B T )= min I(u) 0 0 u B T where the action functional I(u) = 1 2 I 0(u(0)) + and the negative norm T 0 1 4 u t δf δu 2 1dt, ρ 2 1 = ( ) 1 ρ, ρ
Minimal energy path The object is to minimize the action functional inf T inf I(u) u B T We have the relation u t δf δu 2 1dt = +4 T T u t + δf δu 2 1dt + df dt dt F (c) F (a) T u t δf δu 2 1dt If we take the MEP The minimum will be achieved. u ( ) =a, u (T )=c, u ( ) =b, u δf δu, <t<t t = δf δu, T <t<,
Minimal energy path We have the string equation for stochastic Cahn-Hilliard dynamics ϕ (s) δf δϕ (ϕ )
String method for Cahn-Hilliard dynamics Algorithm: Set up the initial string: ϕ 0 i,k =0 Update according to MEP equation with stepsize ϕ i = τ δf δu + ϕk i, i =0, 1, 2,,n, Reparameterize to get with equal arclength. τ ϕ k+1 i = L j+1 L i L j+1 L j ϕ j + L i L j L j+1 L j Iterate until the string converges. ϕ j+1
Transition path for the Neumann BC
Transition path for the periodic BC
Nucleation rates Focusing on the nucleation rates in the zero temperature limit, the finite temperature case is much more difficult The nucleation rates will involve the energy barrier, local geometry of the metastable state and the saddle point
Degeneracy of the saddle in periodic BC For periodic BC, if φ c (θ) S = is a saddle, then c := u c (x) φ c (θ, ) φ c (θ, ) =u c ( θ), θ [0, 1] are all saddles for any θ. We can obtain the degeneracy direction as v d = u c(x) u c(x) L 2.
Nucleation rate formula By asymptotic analysis of the mean first exit time in the zero temperature limit Neumann BC Periodic BC where k n = µ π k n = µ 2π (2π) 1 2 H means the Hessian confined in the space of mass conservation, nonzero eigenvalues. deth (a) Fn deth (c) e. deth (a) deth (c) u c L2 (R n ) e Fn. det means the product of
Computation of determinant ratio To compute the determinant ratio, we should modify the Hessian to SPD case. For Neumann BC, we take H 1 = H (c) 2λ 1 v 1 v T 1 For periodic BC, we take H 1 = H (c)+v d v T d 2λ 1 v 1 v T 1 We have deth 1 = deth (c)
Computation of determinant ratio Define U α (q) = 1 2 qt [αh 1 +(1 α)h (a)]q and then Z(α) = E 0 n dq exp =(2πβ 1 ) n 1 2 d dα lnz(α) = 1 Z(α) E 0 n βu α (q) det[αh 1 +(1 α)h (a)] 1/2. β dq 2 qt (H (a) H 1 )q exp β = 2 qt (H (a) H 1 )q α Q(α), βu α (q)
Computation of determinant ratio where α is the expectation w.r.t. π α (q) = 1 Z(α) exp βu α (q) We have the relation deth (a) deth 1 = Z(1) 2 =exp 2 Z(0) 1 0 Q(α)dα. Q(α) can be computed with infinite dimensional sampling algorithm.
Infinite dimensional sampling MCMC with Metropolis acceptancerejection strategy Random walk proposal MALA proposal q N(q k, σ 2 ni n ) q N(q k + σ2 n 2 logπ α(q k ), σ 2 ni n )
Nucleation rates: Neumann case MALA can significantly improve the RW proposal. When rate decreases. κ increases, the nucleation Beskos-Roberts-Stuart, Ann. Appl. Prob. 2009 Beskos-Pinski-Sanz Serna-Stuart, Stoch. Proc. Appl. 2011
Nucleation rates: Periodic case MALA can significantly improve the RW proposal. When rate decreases. κ increases, the nucleation
Nucleation rates Direct is the result by eigenvalues computation
Nucleation in the large domain Investigating the nucleation in the large domain corresponds to the multiple droplets formation With the rescaling x = x/l, t = t/l 2, κ = κ/l, = /L, m = m/l The equation reduces to with energy functional This corresponds to the small case in the zero temperature limit ũ t = 2 δ F (ũ) x 2 δũ + 2 ξ 1 κ 2 F (ũ) = 2 0 κ ũ 2 + f(ũ) d x x
Multiple droplets The saddle tends to have higher Morse index.
Compared with projection methods Previous projection method for computing the string actually corresponds to the SPDE u t = P δf δu + 2εη where P is the orthogonal projection to the space of mass conservation Cahn-Hilliard dynamics corresponds to some type of oblique projection
Summary
Summary String method for the stochastic Cahn-Hilliard dynamics is proposed and successfully applied to the 1D problem Infinite dimensional sampling methods are supposed to be useful for the high dimensional computations Further issues: higher dimensions, more complex models, algorithms on very large domain, finite temperature case... Thank you for your attention!