A sharp diffuse interface tracking method for approximating evolving interfaces

Transcription

1 A sharp diffuse interface tracking method for approximating evolving interfaces Vanessa Styles and Charlie Elliott University of Sussex

2 Overview Introduction Phase field models Double well and double obstacle potentials Applications of front tracking using the sharp diffuse interface method. Allen-Cahn equation (motion by mean curvature) Diffusion induced grain boundary motion (forced MBMC) Cahn-Hilliard equation (surface diffusion) Void electro migration Epitaxial growth Conclusions

3 Notation We are interested in tracking an evolving interface Γ t n Γ t evolving interface n unit normal V normal velocity κ mean curvature (negative when Γ t is a circle) s arc length d(x, t) signed distance function to Γ t (n = d, V = d t Γt, κ = d Γt )

4 Mathematical approaches Direct method parametric methods of differential geometry Γ t is parameterised by X(s, t) : [0, 1] [0, T ] IR 2 ( ) n = X s X s, V = X t X s X s, κ = 1 Xs X s X s Level set method Osher & Sethian 88 Γ t is taken to be the zero level set of a scalar ω. ( ) n = ω ω, V = ω t ω, κ = ω ω Phase field method Γ t is approximated by a diffuse transition layer in which an order parameter changes from 1 to +1. s

5 Phase field model To model motion by mean curvature V = κ Cahn & Allen 79 approximated Γ t by a diffuse transition layer. In this layer an order parameter ϕ changes from 1 to 1. The idea is based on the free energy ε F = 2 ϕ W (ϕ) (ε small) ε W ( ) is a double well potential with equal minima at ±1. The associated gradient flow of the free energy is εϕ t = ε ϕ + 1 ε W (ϕ) (Allen-Cahn equation)

6 Phase field model Setting P (ϕ) = εϕ t ε ϕ 1 ε W (ϕ) we have P (ψ(d/ε)) = ψ (d t d) 1 ε (ψ W (ψ)). Let ψ = ψ(z) be the unique solution of ψ + W (ψ) = 0 z IR, ψ(0) = 0, ψ (z) > 0, ψ(z) ±1 as z ± Then - (d t d) Γt = 0 - ψ ( d ε ) is only O(1) in an ε neighbourhood of Γ t - ψ ( d ε ) 0 away from Γ t So ψ( d ) is close to a solution of the Allen-Cahn equation. ε

7 Double Well Potential ϕ = 0 ϕ 1 ϕ W (ϕ) = 1 4 (1 ϕ2 ) 2 The diffuse interface is not well defined. Interface profile is ψ = tanh( d ε ) width O(ε log(ε) ).

8 Double Obstacle Potential ϕ < 1 ϕ = 1 ϕ = W (ϕ) = { 1 2 (1 ϕ2 ) ϕ 1 ϕ > 1 ϕ = 0 The diffuse interface is sharply defined. Interface profile is ψ = sin( d ε ), d ε < π 2 width O(ε).

9 Double Obstacle Potential Using the double obstacle potential W (ϕ) = { 1 2 (1 ϕ2 ) for ϕ 1 for ϕ > 1 the Allen-Cahn equation is now a parabolic variational inequality ε(ϕ t, η ϕ) + ε( ϕ, η ϕ) 1 (ϕ, η ϕ) ε η K = {η H 1 (Ω) : η 1}. Elliott & Chen 93, Nochetto, Paolini & Verdi 93 This problem approximates mean curvature flow.

10 Considering the gradient flow of the energy functional F = ε ϕ ε W (ϕ) + C Wfϕ Double Obstacle Potential yields ε(ϕ t, η ϕ) + ε( ϕ, η ϕ) ( ) ϕ ε C Wf, η ϕ η K C W = W (ϕ) = π 4 This converges to forced mean curvature flow V = κ + f. A finite element approximation of this model yields (for θ = 0, 1) ε t (ϕn ϕ n 1, χ ϕ n ) + ε( ϕ n θ, χ ϕ n ) ( 1 ε ϕn θ C W f n, χ ϕ n ) χ K h Convergence analysis Nochetto & Verdi 96, 97.

11 Double Obstacle Potential Phase field models can naturally deal with topological changes 1 t=0 1 t= t= f = 40 These computations and many others in this talk were obtained using the adaptive finite element toolbox ALBERT (Schmidt & Siebert 00).

12 Anisotropic Allen-Cahn with Forcing Considering the gradient flow of the energy functional F = εa( ϕ) + 1 W (ϕ) + fu ε yields an anisotropic Allen-Cahn equation with forcing ( ) ϕ ε(ϕ t, η ϕ) + ε(a ( ϕ), η ϕ) ε f, η ϕ η K A( ϕ) = γ2 2 (θ) ϕ 2 with tan θ = ϕ y ϕ x. Wheeler & McFadden 96, Bellettini & Paolini 96, Elliott & Schätzle 96 Asymptotics yield anisotropic mean curvature flow V = γ(γ + γ )κ + γf This model can be discretized using a standard explicit in time finite element method.

13 Anisotropic Allen-Cahn with Forcing γ = 1 γ = cos 4θ 1 t= t= f =

14 Diffusion Induced grain boundary motion (DIGM) When a film of Fe is immersed in a vapour containing Zn atoms the Zn diffuses into the Fe along the grain boundaries causing them to move. VAPOUR z y x A A+V B VAPOUR

15 Two dimensional reductions: Thick film VAPOUR Diffusion Induced grain boundary motion (DIGM) Assume y-independence A A+V B The grain boundaries span Ω VAPOUR Two dimensional reductions: Thin film C+V A A+V C A+V B Assume z-independence Rapid diffusion of solute into the film.

16 Thick film uni-directional motion Cahn, Fife & Penrose 97 proposed a phase field model for DIGM using the energy functional 1 F = 2ε c2 + ε ϕ W (ϕ) + ϕc2 ( ) ε c denotes the concentration of solute atoms in the material. The first term takes into account the chemical energy The second and third represent the interfacial energy The last term approximates the elastic energy caused by a difference in stress across the interface.

17 Thick film uni-directional motion Considering the gradient flow of the energy yields the Allen-Cahn equation with forcing ( ) 1 ε(ϕ t, η ϕ) + ε( ϕ, η ϕ) ε ϕ c2, η ϕ and a diffusion equation for c εc t = (D(ϕ) v). η K Here D(ϕ) = 1 ϕ 2 and v = c + εf ϕ. Deckelnick & Elliott 01 Existence with v = c. Deckelnick, Elliott & Styles 01 Finite element approximation, stability and convergence (v = c and the forcing term c 2 replaced by c). Fife, Cahn & Elliott 01 Formal asymptotics yield V = κ + c 2, c ss = V c

18 Thin film bi-directional motion C.M. Elliott & V. Styles Using the ideas of Fife & Wang 02, yields the following model for thin film bi-directional DIGM V = κ + [c] Γ, { 1 (x, t) s.t. x Γ( t) for some t t c(x, t) = 0 otherwise [c] Γ denotes the jump in concentration across Γ. We derived corresponding phase field and level set models with forcing terms that mimic the jump in concentration across the interfacial region. For the the phase field model we set F[c, ϕ] = c(x + δ + ϕ n, t) c(x δ ϕ n, t) x s.t. ϕ < 1. Garcke & Styles 04 Multi-order parameter phase field system for bi-directional DIGM in the presence of triple junctions.

19 Thin film bi-directional motion bi-directional motion bi-directional motion with triple junctions

20 DIGM with elasticity H. Garcke R. Nürnberg & V. Styles We study a phase field model for DIGM based on the free energy 1 F = + ε 2ε 2c2 2 ϕ ε W (ϕ)+ 1 (E(u) Ē : C(E(u) Ē)) 2 The last term models the elastic energy with u displacement vector E(u) = 1 2 ( u + ( u)t ) strain tensor (linear elasticity) Ē(x, ϕ, c) = Ē 1 (x)ϕ + Ē 2 (x)c + Ē 3 (x) stress free strain C = C (1 ϕ)c+ elasticity tensor (C ± constant tensors) C C +

21 DIGM with elasticity Considering the gradient flow of the free energy gives ( ) ϕ ε t, η ϕ ε ( ϕ, (η ϕ)) + 1 (ϕ, η ϕ) ε 1 2 (E(u) Ē : C ϕ (E(u) Ē)(η ϕ)) +(E(u) Ē : CĒ ϕ (η ϕ)) η K (E(u) Ē : CE(ξ)) = ( C(E(u) Ē), E(ξ) ) ξ (H 1 (Ω)) 2 The concentration c of solute satisfies c(x, t) = { 1 (x, t) s.t. ϕ(x, t) < 1 for some t t 0 otherwise We consider a finite element approximation of this model.

22 The following computations model DIGM in systems with cubic anisotropy. We consider two grains separated by a grain boundary. The two crystal lattices differ by a rotation. We set Ē = c(x) and C = C (1 ϕ)c+. We take C R to be a given elasticity tensor with cubic anisotropy with respect to a given reference orientation of the lattice. C ± are computed by setting for i, j, k, l = 1, 2 where C ± ijkl = 2 p,q,r,s=1 d ± ip d± jq d± kr d± ls CR pqrs ( ) cos θ sin θ D = sin θ cos θ This models a tilt boundary with rotation axis parallel to the x 2 -axis.

23 DIGM with elasticity θ = π 8 θ = 0 θ = π 8 θ = π 6

24 DIGM with elasticity t = 0, 4, 8, θ = π 8 θ = 0 θ = π 6 θ = 0

25 The Cahn-Hilliard equation The original motivation lay in the modelling of phase separation and spinodal decomposition in binary alloys. Order parameter u measuring the amount of one component satisfies the fourth order equation u t = ( ε u + 1 ε W (u)). Second order splitting with double obstacle potential, Blowey, Elliott 94 u t = w, ε( u, η u) (w + 1 u, η u) η K. ε chemical potential w is the functional derivative of F(u) := (ε u ε W (u)). Ω

26 Degenerate Cahn-Hilliard Introducing a concentration u dependent mobility M(u) in the diffusion equation yields:- εu t = (M(u) w), w = 4 π ( ε u + 1 ε W (u)) Taking M(u) := π 4 (1 u2 ) yields a degenerate system. Cahn, Elliott & Novick-Cohen 94 Formal asymptotics yield a geometric evolution law V = κ ss (surface diffusion), w approximates κ. Elliott & Garcke 96 Existence Barrett, Blowey & Garcke Finite element approximation, stability and convergence.

27 Void Electro Migration J.W. Barrett, R. Nürnberg & V.Styles The motion of voids in interconnect lines on microelectronic circuits is modelled by V = (κ + αϕ) ss ϕ is the electric potential associated with an applied electric field α > 0 is related to various physical constants. ϕ n = 0 ϕ n = 1 ϕ = 0 ϕ Γ = 0 ϕ n = 1 ϕ n = 0

28 A sharp diffuse interface approximation of this model is Void Electro Migration εu t = (M(u) (w + αϕ)) M(u) = 1 u 2 ε( u, χ u) (w + 1 u, χ u) ε χ K (c(u) ϕ) = 0 c(u) = 1 (1 u) 2 Barrett, Nürnberg & Styles 02 F.E.A., stability and convergence α = 0 Cahn-Hilliard

29 Void Electro Migration α = 40π α = 120π

30 Epitaxial growth Modern technology of growing single crystals that inherit atomic structure from substrates. It produces almost defect-free high quality materials. Consider the simple geometry of a single island Ω + on a terrace Ω. Set Γ t to be the boundary of the island. Ω Ω +

31 Epitaxial growth The following Burton-Cabrera-Frank type model was studied by Bansch, Hausser, Lakkis, Li & Voigt, 04 The adatom diffusion on the terrace and the island is described by t ρ± D ρ ± = F τ ρ ± ρ (ρ + ) denotes the adatom density on the terrace (island) D diffusion constant, F deposition flux rate, On the island boundary Γ(t) we have 1 τ desorbtion rate. q := D ρ n + V ρ = k (ρ κ) flux from terrace q + := D ρ + n V ρ + = k + (ρ + κ) flux from island k k + > 0 model the Ehrlich-Schwoebel effect. On Γ(t) we have V = κ ss + k + (ρ + κ) + k (ρ κ).

32 Epitaxial growth We study a sharp diffuse interface model of this problem εu t = (M(u) w) + 1 ε M(u)(k+ (ρ + w) + k (ρ w)) ε( u, χ u) (w + 1 u, χ u) ε χ K t t ( H (u)ρ ) (H (u) ρ ) + 1 ε M(u)k (ρ w) = H (u)(f τ ρ ) ( H + (u)ρ +) (H + (u) ρ + ) + 1 ε M(u)k+ (ρ + w) = H + (u)(f τ ρ + ) Here M(u) = 1 u 2, H + (u) = 1 2 (1 + u), H (u) = 1 (1 u). 2 We consider a finite element approximation to the weak formulation of this model.

33 Growth of a single circular island with no desorbtion. t=0, Ω 0 Ω

34 adatom density on the island ρ + adatom density on the terrace ρ

35 Conclusions We approximate interface evolution using phase field models with a double obstacle potential. The method is widely applicable and can be used for second and fourth order problems. It is frequently physically based as the gradient flow of a free energy functional. It can naturally deal with topological changes. Alternative methods used to study interface evolution are direct methods or level set methods.