with deterministic and noise terms for a general non-homogeneous Cahn-Hilliard equation Modeling and Asymptotics

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1 Modeling and Asymptotics for a general non-homogeneous Cahn-Hilliard equation with deterministic and noise terms D.C. Antonopoulou (Joint with G. Karali and G. Kossioris) Department of Applied Mathematics, University of Crete, Heraklion, Greece Institute of Applied and Computational Mathematics, FO.R.T.H., Heraklion, Crete, Greece 1

2 1. The Problem Non-Homogeneous Cahn-Hilliard Equation tu = ( 2 u + f (u)) + F (x, t; ), x Ω R n, n 3, t > 0. Fourth order non-linear p.d.e., u = u(x, t) (mass concentration), > 0 (measure for the inner interfaces width). f (u) = uf(u) = u( 1 4 (u2 1) 2 ) = u(u 2 1). F smooth function on Ω (0, + ). f = 1 4 (u2 1) 2 double equal-well potential of minimum value 0 at u = ±1. 2

3 2. Model for concentration evolution The mass container: Ω bounded domain of R n, n 3. Mass concentration: u(x, t). Total mass in Ω: m(t) = Ω u(x, t)dx. Evolution process Reason: Non-Equilibrium state - Non-minimized Free Energy. (1) Concentration Evolution to equilibrium state lim t t eq u(x, t) = u eq(x). (2) Equilibrium (Free energy minimization) - Evolution stops u(x, t) = ueq(x), for any t teq. 3

4 Critical dynamics: Ginzburg-Landau model Hohenberg, Halperin, (Rev. Mod. Phys., 1977) Define: E := u 2 + f(u), for f(u) = 1 4 (u2 1) 2 double equal-well potential. System with no conservation laws: Model A (Allen-Cahn related) ut = Γ0Eu + Γ0h(x, t) + θ(x, t), h external field, θ Gaussian white noise source. System with conservation: Model B (Cahn-Hilliard related) in Model A replace Γ0 with λ0 and get ut = λ0 (Eu h(x, t)) + θ(x, t). 4

5 The generalized Cahn-Hilliard equation Gurtin, (Physica D, 1996) ut = ( 2 u + f (u) γ(x, t)) + m(x, t), γ external microforce, m external mass supply. Chemical potential: µ := 2 u + f (u) γ(x, t). Time dependent total free energy of concentration (1) Local energy of u: E(x, t; u) = Ek(x, t; u) + Ed(x, t; u), (2) Ek kinetic energy: u(x, t) (speed) concentration change in space. (3) Ed dynamical energy: Ed = F(x, t; u)du Due to a space-time force field F = F(x, t; u). Total Free Energy Functional: FE[u](t) := E(x, t; u)dx. Ω 5

6 Concentration Evolution Equation tu = K L, K const.> 0, L := ue(x, t; u) = uek(x, t; u) + F(x, t; u). Mass evolution: tm(t) = K Ω nl(s, t; u)ds. Mass conservation condition nl(x, t; u) = 0 on Ω n( uek(x, t; u) + F(x, t; u)) = 0 on Ω tm(t) = 0. Evolution Stages (a) Homogenization (Constant concentration) (b) Spinodal (c) Coarsening (Sharp Interface) (d) Equilibrium 6

7 Figure 1: (1) Homogenization Spinodal (2) Spinodal: Deterministic Model VS Stochastic Model 7

8 Figure 2: Coarsening 8

9 Figure 3: Spinodal Coarsening 9

10 3. Homogeneous Cahn-Hilliard equation Fourth order non-linear P.D.E. tu = K L, K := 1, L = uek + ued, with Ek(x, t; u) = u 2 Ed(x, t; u) = f (u)du = f(u) (double equal well potential) (1) Non-Linear force field considered F(x, t; u) = F(u) := f (u) = u(u 2 1) models the tendency of a binary alloy (two species) forced to homogenization to return in a two separated phases equilibrium. (2) The kinetic energy of the mass 2 models the material preference to be as uniform as possible. 10

11 Replace L and get Homogeneous Cahn-Hilliard equation tu = ( 2 u + f (u)), t > 0. Simple phase separation model of a binary alloy at a fixed temperature, > 0 measure for the inner interfaces width (phase transitions). B.C.: nl(x, t; u) = 0 on Ω mass conservation and free energy decreasing condition n( 2 u + f (u)) = 0 on Ω. Existence-Uniqueness of solution Homogeneous Cahn Hilliard with initial condition u0(x) H 2 (Ω), with nu0 = 0 on Ω and boundary conditions: n( 2 u + f (u)) = 0 on Ω, and nu = 0 on Ω, admits a unique solution, Elliott - Zheng, (Arch. Rat. Mech. Anal., 1986). 11

12 4. Non-homogeneous term F in Cahn-Hilliard: F := Fex + Fn Fex external field (source term, production rate in a reaction diffusion process A + B C). Fn Gaussian white noise source (thermal noise). Thermal noise= 0 source brings system to equilibrium. Source = deterministic + noise. Mean-field assumption: source noise = 0. The Equation - General Case Mass conservation condition: tu = ( 2 u + f (u)) + F (x, t; ), t > 0. Ω n( 2 u + f (u))ds = F (x, t; )dx. Ω 12

13 Case 1: Extra dynamics in free energy F := F2 = ψ(x, t; ) may model a free energy extra term (Gurtin-in chemical potential, Langer-ψ = ψ(x) (Annals of Physics, 1971)) tu = ( 2 u + [f (u) + ψ(x, t; )]) = ( 2 u + f (u)) + F2(x, t; ), t > 0. Ed(x, t; u) = f(u) + ψ(x, t; )u ued(x, t; u) = f (u) + ψ(x, t; ). Extra linear potential ψ(x, t; )u, extra space-time force field ψ (reaction - boundary interaction rate). Mass conservation b.c: n( 2 u + [f (u) + ψ(x, t; )]) = 0 on Ω. Case 2: Extra dynamics in free energy, free energy independent terms tu = ( 2 u + [f (u) + ψ(x, t; )]) + F1(x, t; ) = ( 2 u + f (u)) + F1(x, t; ) + F2(x, t; ), t > 0, F := F1 + F2. 13

14 5. Asymptotics 1. Multidimensional case, O() limit problem Alikakos-Bates-Chen (Arch. Rat. Mech. Anal., 1994), Pego (Proc. R. Soc. Lond., 1989) Homogeneous Cahn-Hilliard on R n, n 2 Hele-Shaw Inner Free Boundary Problem. Consider ut(x, t; ) = ( u + f (u) ), x Ω, t > 0, u(x, 0) = u0(x), nu = n( u + f (u) ) = 0 on Ω. 14

15 Set v = lim ). 0 + ( u f (u) Then v satisfies the sharp interface Hele-Shaw Problem: v = 0 in Ω\Γ(t), t > 0, nv = 0 on Ω, v = λh on Γ(t), V = 1 2 ( nv + nv ), Γ(0) = Γ0. Γ(t) closed n 1 dimensional hypersurface (sharp interface) H mean curvature of Γ, and V velocity of Γ Ω = Ω + (t) Ω (t) Γ(t), λ > 0. 15

16 Main result for the Non-Homogeneous case Consider G uniformly bounded as 0 +. tu(x, t; ) = ( u + f (u) ) + G(x, t; ), 1. Split G: G = G1 G2, G1(x, t; ), G2(x, t; ), uniformly bounded as Get: 3. Define: v() := u f (u) 4. Linearize: v() = v0 + v1 + O( 2 ). ut = ( u + f (u) + G2, get the system G2) + G1, ut = v + G1, v = u f (u) + G2. 16

17 Theorem. The order O() approximation of v (i.e. v0), satisfies the non-homogeneous problem v0 = G1 + O( 2 ) in Ω\Γ(t), t > 0, v0 = λh + G2 + O() on Γ(t), V = 1 2 ( Qz( ) Qz(+ )) + O( 2 ), Q(z) := v 1, z := d, d = d(x, t) := inf y Γ(t) x y. (1) Order of convergence to a homogeneous problem: = min{o(), O(G1), O(G2)}. (2) G2 ( Free energy): Local contribution while, G1: Global contribution. Assume that O(G1) = O( λ 1 ), O(G 2) = O( λ 2 ) then 1. If min{λ1, λ2} = 1 order of convergence to homogeneous Hele-Shaw remains the same. 2. If 0 < min{λ1, λ2} < 1 convergence is delayed. 3. If O(G1), O(G2) =const. Non-Homogeneous Cahn-Hilliard Non-Homogeneous Hele-Shaw. 17

18 Conclusion Assumption: Convergence to equilibrium admits that Homogeneous Hele-Shaw problem appears first. Non-Homogeneous terms may slow down equilibrium. Remarks: 1. Rogers, Elder, Desai, (Phys. Rev. B, 1988): Numerical experiments demonstrate that increasing of the noise strength leads to more diffuse interfaces and slower transitions. 2. Physical models: Noise in G. Split G On the limit: noise in G1, noise in G2. Free energy noise terms are local, as they appear at phase transitions Γ(t). Other noises (obtained by G1) influence globally evolution process. 18

19 Mass Conservation and minimal surface conditions Consider: Non-Homogeneous Hele-Shaw Problem v = G1 in Ω\Γ(t), t > 0, nv = 0 on Ω, v = λh + G2 on Γ(t), V = 1 2 ( nv + nv ), Γ(0) = Γ0. Get: d Per(Γ(t)) := dt Γ HV = 1 2λ d dt Vol(Ω (t)) := V = 1 Γ 2 Ω\Γ G1v + 1 λ G1 Ω\Γ Γ G2V 1 2λ Ω\Γ v 2 (1) d dt Per(Γ(t)) 0 minimal surface (2) d dt Vol(Ω (t)) = 0 mass conservation 19

20 7. One dimensional case, O( ) limit problem Ω R, 1 ξ(x, t ξ(x, t ) uniformly bounded as 0+. tu = ( u + f (u) ) 1 + ξ(x, t ), t > 0, ) deterministic free energy independent that follows white noise scale in t, Noise scale: Consider white noise Wt defined as the t derivative of an Itô integral ( L 2 (P )) Wt := d t dt 0 1dB t s, use 0 1dB s = Bt and get by Brownian scale in R Bt = B t = t 1dBs, 0 then set τ = t/ and get Wt = d dt B t = d dt t 0 1dBs = d dτ τ 0 1dBs = 1 Wτ. 20

21 Space and time scalings: x, z := x, and t, τ := t Linearize: u = u0(t, τ, x, z) + u1(t, τ, x, z) + u2(t, τ, x, z) + Outer expansion: u =1 + u 1/2 (τ, x) + v1(t, x)+ u1(τ, x) + 3/2 v 3/2 (t, x) + 3/2 u 3/2 (τ, x) + 2 v2(t, x) + 2 u2(τ, x) + Compute: tu = 1 τ u 1/2 + tv1 + τ u1 + 3/2 tv 3/2 + τ u 3/2 + 2 tv2 + τ u2 + 21

22 and u + f (u) = 1 f (1) + 1 f (1)u 1/2 + f (1)(v 1 + u 1 ) + 1 f (1)u 2 1/2 2 + ( u 1/2 (v 1 + u 1 )f (1)) + f (1)(u 3/2 + v 3/2 ) ) + ( f (1)(v 2 + u 2 ) (v 1 + u 1 ) 2 f (1) + f (1)u 1/2 (v 3/2 + u 3/2 ) ) + ( u 1/2 + f (1)u 1/2 (v 2 + u 2 ) + f (1)(v 1 + u 1 )(v 3/2 + u 3/2 ) ) + 2( v 1 u f (1)(v 3/2 + u 3/2 ) 2 + f (1)(v 1 + u 1 )(v 2 + u 2 ) ) + 5/2( v 3/2 u 3/2 + f (1)(v 3/2 + u 3/2 )(v 2 + u 2 ) ) + 3( v 2 u f (1)(v 2 + u 2 ) 2) + Collect the O( 1 ) terms to obtain τ u 1/2 = f (1) u 1/2 + ξ(x, τ) on Ω\Γ. 22

23 Ω R, expand d (distance from the interface): d(x, t, τ) = x ( x0(t) + y 1/2 (τ) + (x1(t) + y1(τ)) ) + Inner expansion: d(x, t, τ) u = m( ) + d(x, t, τ) m 1/2 ( ) + d(x, t, τ) u 1/2 (τ, x) + m1( ) +v1(t, x) + u1(τ, x) + 3/2 d(x, t, τ) m 3/2 ( ) + 3/2 v 3/2 (t, x) + 3/2 u 3/2 (τ, x) + 2 d(x, t, τ) m2( ) + 2 v2(t, x) + 2 u2(τ, x) +, m unique solution of Euler-Lagrange equation, for z := d m (z) + f ( m(z)) = 0, for z R lim m(z) = ±1, m(0) = 0. z ± 23

24 Expand f (u), use inner expansion, and seek: u + f (u) = 1 ( m + f ( m)) + 1 ( m 1/2 + f ( m)(m 1/2 + u 1/2 ) + ( m 1 + f ( m)(m 1 + v 1 + u 1 ) f ( m)(m 1/2 + u 1/2 ) 2 ) + ( (m 1/2 + u 1/2 )(m 1 + v 1 + u 1 )f ( m))+ ( m 3/2 + f ( m)m 3/2 ) +f ) ( m)(u 3/2 + v 3/2 ) }{{} L mm 3/2 ( + m 2 + f ( m)(m 2 + v 2 + u 2 ) + 1 (m 1 + v 1 + u 1 ) 2 f ( m) 2 +f ) ( m)(m 1/2 + u 1/2 )(m 3/2 + v 3/2 + u 3/2 ) + ( u 1/2 + f ( m)(m 1/2 + u 1/2 )(m 2 + v 2 + u 2 ) +f ) ( m)(m 1 + v 1 + u 1 )(m 3/2 + v 3/2 + u 3/2 ) + 2( v 1 u f ( m)(m 3/2 + v 3/2 + u 3/2 ) 2 2 +f ) ( m)(m 1 + v 1 + u 1 )(m 2 + v 2 + u 2 ) + 5/2( v 3/2 u 3/2 + f ( m)(m 3/2 + v 3/2 + u 3/2 )(m 2 + v 2 + u 2 ) ) + 3( v 2 u f ( m)(m 2 + v 2 + u 2 ) 2) + 24

25 Observe terms of the form: L mm 3/2 = m 3/2 + f ( m)m 3/2, L mm 1/2 = m 1/2 + f ( m)m 1/2 that are linearized operators around m. Apply Fredholm alternative solvability condition to get interface conditions: On the interface all the terms disappear and we end up to principal term Diffusion Equation: τ u 1/2 = f (1) u 1/2 + ξ(x, τ) on Ω\Γ. For ξ = 0, this diffusion is the linearization of the ill-posed problem τ u 1/2 (x, τ) = (f (u 1/2 )(x, τ)). Conclusions 1. Approximate problem Diffusion Equation. 2. No inner boundary conditions lower order approximation ( ) slower time scale. 3. ξ delays the phenomenon, and we capture evolution before coarsening. 25

26 REFERENCES REFERENCES References [1] N. D. Alikakos, P. W. Bates and X. Chen, Convergence of the Cahn-Hilliard Equation to the Hele-Shaw Model, Arch. Rat. Mech. Anal. 128 (1994), pp [2] T. Antal, M. Droz, J. Magnin and Z. Rácz, Formation of Liesengang Patterns: A Spinodal Decomposition Scenarion, Phys. Rev. Lett. 83 (1999), pp [3] J. W. Cahn, J. E. Hilliard, Free Energy of a Nonuniform System. I. Interfacial Free Energy, J. Chem. Phys. 28 (1958), pp [4] C. M. Elliott, S. Zheng, On the Cahn-Hilliard equation, Arch. Rat. Mech. Anal. 96 (1986), pp [5] P. C. Fife, Models for phase separation and their mathematics, El. Journ. Diff. E. 48 (2000), pp [6] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D 92 (1996), pp [7] P. C. Hohenberg and B.I. Halperin, Theory of dynamic critical phenomena, J. Rev. Mod. Phys. 49 (1977), pp

27 REFERENCES REFERENCES [8] J. S. Langer, Theory of Spinodal Decomposition in Alloys, Ann. of Phys. 65 (1971), pp [9] B. csendal, Stochastic Differential Equations, Springer, 2003, New York. [10] R. L. Pego, Front migration in the non-linear Cahn-Hilliard equation, Proc. R. Soc. Lond. A 422 (1989), pp [11] T. M. Rogers, K. R. Elder, and R. C. Desai, Numerical study of the late stages of spinodal decomposition, Phys. Rev. B 37 (1988), pp [12] C. Cardon-Weber, Cahn-Hilliard stochastic equation: existence of the solution and of its density, Bernoulli 7 (2001), pp