Modeling and Simulation for Solid-State Dewetting Problems

Transcription

1 Modeling and Simulation for Solid-State Dewetting Problems Weizhu Bao Department of Mathematics National University of Singapore URL: Collaborators: Wei Jiang (Wuhan), David J. Srolovitz (U. Penn); Carl V. Thompson (MIT); Yan Wang (NUS); Quan Zhao (NUS)

2 Outline Motivation Sharp Interface models (SIMs) Isotropic surface energy Anisotropic (weak/strong) surface energy Numerical methods & results in D Phase field/diffuse interface model Mathematical model Numerical methods Results in 3D Conclusion & future works

3 Wetting / Dewetting in Fluid Mechanics Wetting spread of a liquid on a substrate Dewetting the rupture of a thin liquid film on a substrate

4 Solid-State Dewetting of Thin Films Most thin films are metastable in as-deposited state & dewet to form particles This occurs when the temperature is high enough for surface self diffusion, which can be well below the film s melting temperature γ = γ + γ cos( θ) -- Young VS FS FV γ γ γ VS FV FS : substrate free surface energy : film free surface energy : film-substracte interface energy

5 Dewetting on a flat substrate [1] E. Jiran & C. V. Thompson, Journal of Electronic Materials, 19 (1990), pp [] E. Jiran & C. V. Thompson, Thin Solid Films, 08 (1991), pp. 3-8.

6 Solid-State Dewetting Problems Solid-state dewetting Is driven by capillarity effects Occurs through surface diffusion controlled mass transport Belongs to capillarity-controlled interface/surface evolution problems Surface diffusion + contact line migration Applications of dewetting of thin films Play an improtant role in micorelectronics processing A common method to produce nanoparticles Catalyst for the growth of carbon nanotubes & semiconductor nanowires Recent experiments - [1] Geometric complexity, capillarity-driven instabilities, faceting Crystalline anisotropy, corner-induced instabilities, pinch-off,. Wetting/dewetting in fluids: TZ Qian, XP Wang&P Sheng; W. Ren&W E,... [1] J. Ye & C.V. Thompson, Acta Mater. 3 (011), 1567; 59 (011), 58; Appl. Phys. Lett. 97 (010),

7 Pinch-off

8 Dewetting Patterned Films [1] J. Ye & C.V. Thompson, Phys. Rev. B, 8 (010),

9 Effect of Size & Orientation of Pattern [1] J. Ye & C.V. Thompson, Adv. Mater., 3 (011), 1567.

10 Dewetting of Patterned Single-Crystal Films

11 Models and Methods for Dynamical Evolution Sharp interface model Isotropic case Model & power law --- Srolovitz & Safran JAP 86 Marker particle method ---- Wong, Voorheers & Miksis 00 ; Du etc JCP 10 ;. Anisotropic (weakly and strongly) case Model via thermodynamical variation Wang etc PRB 15 ; Jiang etc 16, Parametric finite element method (PFEM) -- Bao, Jiang, Wang &Zhao, 16 Kinetic Monte Carlo method Dufay & Pierre-Lious PRL 11 ; Pierre-Louis etc, EPL&PRL 09 Discrete surface chemical potential method Dornel etc. PRB 06 ; Klinger etc 1 Phase field model ---Jiang, Bao Thompson & Srolovitz, Acta Mater. 1 Crystalline formulation method Cahn & Taylor 94 ; Cater etc 95 ; Kim etc 13 ; Zucker etc 13 ; Roosen etc 94 &98 ;..

12 X(,) st t Sharp Interface Models (SIM) Γ : X( st, ) = ( xst (, ), yst (, )) s-- arclength Isotropic surface energy--dynamics: = Vn with V = B κ n n s X(,) st = ( xst (,), yst (,)) : moving front curve in D(surface in 3D) n : unit outward normal direction V n : normal velocity of the moving interface B : material constant s : surface Laplacian or Laplace-Beltrami operator κ : mean curvature of the surface [1] D.J. Srolovitz & S.A. Safran, J. Appl. Phys., 60 (1986), 55. [] H. Wong et al., Acta Mater., 48 (000), κ Vn = B (in D) s y x x y κ = + s s s s

13 SIM of Isotropic Surface Energy--Dynamics Γ : X( st, ) = ( xst (, ), yst (, )) s-- arclength Boundary conditions (D) Contact point condition (BC1) yx (,) t = 0 c Contact angle condition (BC) xc () t y ( x c,) t s = tan θi θi -- prescribed isotropic Young angle x ( x c,) t s γvs γ FS cos θi = : = Zero-mass flux condition (BC3) γ κ ( xc,) t = 0 s Total mass conservation of the film. No mass flux at the contact point-- no mass diffuses under the thin film at the film/substrate interface & any flux along the surface to the contact point simply moves the contact point along the substrate s.t. no flux remains x c FV σ

14 SIM for Isotropic Surface Energy Equilibrium State Total interfacial free energy W ( ω) = γ Σ + γ Σ + γ Σ FV FV FS FS VS VS γ FV -- interfacial energy (density) Σ -- interface length (D) or area (3D) FV Equilibrium configuration: min W ( ω) subject to ω dω = constant Wall Energy Wulff construction & Winterbottom construction

15 Existing Results for SIM of Isotropic Surface Energy /5 Asymptotic/mathematical results c For 3D linearized problem (small angle) [1]: For equilibrium static problem -- linearized stability analysis -- [] For dynamic nonlinear problem???? Front-track approaches (marker particle methods) [3]: Results: D with arclength works well 3D with marker particles too tedious and low accuracy!! Main difficulties Fourth-order derivatives along the surfaces -- accuracy Complex topological changes (pinch-off) complexity [1] D.J. Srolovitz & S.A. Safran, J. Appl. Phys., 60 (1986), 47; 60(1986), 55. [] H. Garcke, K. Ito & Y. Kohsaka, SIAM J. Math. Anal., 36 (005), [3] H. Wong, PW Voorhees, MJ Miksis, SH Davis, Acta Mater, 48 (000), x ( t) t t 1, 0 θ 1 i

16 SIM for Anisotropic Surface Energy (via thermodynamical variation) Γ : X( st, ) = ( xst (, ), yst (, )) s-- arclength Total interfacial free energy r l = γ FV ( θ) Γ+ ( γ FS γvs )( c c) W d x x Γ Anisotropic surface energy γ FV = γθ ( ) = γ0[1 + βcos( mθ)], m =,3,4,6 Thermo-dynamical variation: ψ( s) is arbitrary & ϕ ( s) ds = 0 ε ε ε Γ ( t) = ( x( st, ), y( st, )) =( xst (,) + εust (,), yst (,) + εvst (,)) L 0 ust (,) = x(,) stψ() s y(,) stϕ() s s vst (,) = x(,) stϕ() s + y(,) stψ() s s ε v(0,) t = vlt (,) = 0 & A( Γ ) A( Γ) C s s 0 ε

17 SIM for Anisotropic Surface Energy (via thermodynamical variation) Calculate first variation of the energy functional ε = 0 r l W = γ ( θ) dγ+ ( γ γ )( x x ) Γ FV FS VS c c ε ε r l W = γ ( θ ) dγ+ ( γ γ )[( x + εult (, )) ( x + εu(0, t))] Γ ε FV FS VS c c ε ε dw W W r l = lim = ( γ ( θ ) + γ ( θ )) κϕds + f ( θc) u( L, t) f ( θc) u(0, t) dε ε 0 ε δw δw δw µ : = = ( γθ ( ) + γ ( θ)) κ, = f( θ) r, = f( θ) r θ θc l δγ δx δx c 0 i = θ= θ c γ f ( θ) : = γ( θ)cos θ γ ( θ)sinθ γ cos θ, cos θ = σ : = [1] Wang, Jiang, Bao & Srolovitz, Phys. Rev. B, 015. L 0 i VS γ γ 0 FS l c

18 SIM for Weakly Anisotropic Surface Energy Γ : X( st, ) = ( xst (, ), yst (, )) s-- arclength The Model (Wang, Jiang, Bao, Srolovitz, PRB 15 ) X(,) st = Vn n, with Vn = B t µθ ( ) = B( γ( θ) + γ ( θ)) κ Boundary conditions Contact point condition (BC1): Relaxed contact angle condition (BC)[1]: Zero-mass flux condition (BC3): Anisotropic Young equation µ s η 0 r yx (,) t = 0 c r µ ( x,) t = 0 s r θ Dynamical contact angle c θ ----Isotropic Young contace angle i c γ( θ) γ γ( θ)cos θ γ ( θ)sinθ γ cosθ = 0 cosθ = cosθ [1] Ren, E, Phys. Fluid, 007&011; Ren, Hu & E, Phys Fluid, 010. i 0 r dxc () t δw r = η = ηf ( θc ), r dt δ x i c

19 Area (Mass) conservation Energy dissipation Dynamical Properties Lt () At () = y xds At () A(0) Lt () Lt () r l s= Lt () γ FV θ γ FS γ VS c c sµ µ sµ s= W () t = ( ) ds + ( )( x x ) W '() t = ( ) ds + r l Lt () r l dxc() t dxc() t dxc() t dxc() t + f ( θ) r f ( θ ) l = ( µ s) ds C 0 θ= θc θ= θc dt dt + dt dt 0 0 s Weakly anisotropic case: γθ ( ) + γ ( θ) > 0 0 β< m 1 1

20 Strongly Anisotropic Case Strongly anisotropic case β 1 > ( ) ( ) change sign m 1 γθ + γ θ ill-posedness Case I: Multiple solution of the anisotropic Young equation γ( θ)cos θ γ ( θ)sinθ γ0 cosθ i = 0 Regularize the total energy Case II: Surface energy is not smooth A typical example ( ) ( ) 1/ 1/ γθ ( ) = cosθ + sin θ γ( θ) = ε+ cos θ + ε+ sin θ ε Regularize (or smooth) the surface energy density γθ ( ) C γ( θ) C ε

21 Strongly Anisotropic Surface Energy Regularized interfacial free energy ε W = dγ+ x x + dγ ε r l γ FV ( θ) ( γ FS γvs )( c c) κ Γ Γ Calculate first variation of the energy functional 3 δwε κ µ ε : = = ( γθ ( ) + γ ( θ)) κ ε ssκ +, δγ δwε δwε γvs γ = f ( ) r, f ( ) l, cos i : r ε θ = θ θc l ε θ θ = σ = = θ= θc δx δx γ f Γ : X( st, ) = ( xst (, ), yst (, )) s-- arclength c c κθ ( ) θ γ θ θ γ θ θ γ θ ε θ κ θ θ ε ( ): = ( )cos ( )sin 0 cos i + cos s ( )sin [1] J. Lowengrub, A. Voigt,. [] Jiang, Wang, Zhao, Srolovitz & Bao, Script. Mater., FS

22 SIM for Strongly Anisotropic Surface Energy Γ : X( st, ) = ( xst (, ), yst (, )) s-- arclength The Model (Jiang, Wang, Zhao, Srolovitz, Bao, 16 ) X(,) st t = V n, with V = B n µ ε ( θ) B( γθ ( ) γ( θ)) κ ε ssκ Boundary conditions Contact point condition (BC1): Relaxed contact angle condition (BC): ε Zero-curvature condition (BC3): Zero-mass flux condition (BC4): n µ s 3 κ = + + r yx (,) t = 0 c κ ( x r,) t = 0 r µ ( x,) t = 0 s c c r θ Dynamical contact angle c θ ----Isotropic Young contace angle i r dxc () t δw r = η = ηf ( c ), r ε θ dt δ x c

23 Parametric Finite Element Method (PFEM) Weakly anisotropic case Mathematical model and variational form ( X ( s, t)) n = B µ ( X ( s, t)) n φ dp + B µ φdp=0, φ H t ss t s s C C Boundary conditions [ ] µθ ( ) = ( γ( θ) + γ ( θ)) κ µθ ( ) ( γ( θ) + γ ( θ)) κ ϕdp = 0, ϕ H C κn = ss X ( s, t) κn 1 ηdp + sx ( s, t) sηdp = 0, η ( H0) C Finite element discretization via piecewise polynomials Ref: [1] J.W. Barratt, H. Garcke & R. Nurnberg, J. Comput. Phys., 007; SISC (007); 01. [] W. Bao, W. Jiang & Q. Zhao, 015. C r r dxc () t r yx ( c, t) = 0, = ηf( θc) dt 1 0 1

24 Isotropic surface energy and short

25 Weakly anisotropic surface energy & short

26 Weakly anisotropic surface energy & long for pinch-off

27 Strongly anisotropic surface energy & short

28 Extension to Curved Substrate

29 Extension to Curved Substrate Γ () s = ((), xs ys ())with κ: = κ() s vs () C( σκ ) () s

30 Extension to Curved Substrate Experiment (Ye, et. Al, PRB, 10 ) Simulation

31 Phase Field/Diffuse Interface Model Phase field models --- S. Allen & J.W. Cahn ( ), J.W. Cahn & J. E. Hilliard (1958); G.J. Fix (1983), J.S. Langer (1986), L.Q. Chen (00); C.M. Elliott, X.F. Chen, J. Shen, Q. Du, J. Lowengrub, A. Voigt, Q. Wang, G. Forest, XB Feng, PW Zhang, A.A. Lee A. Munch & E. Suli, Introduce a phase function& write down the energy By variation Allen-Cahn or Cahn-Hilliard equations Applications in materials simulation: solidification; viscous fingering; fracture; solid-state nucleation; dislocation dynamics,. Advantages Interface/surface capturing method Eulerian coordinates Easy extension from D to 3D Naturally capture complex topological changes

32 Problem Set-up

33 Phase Field Model Introduce the phase function Total free energy-- isotropic surface energy ε ε ε W = W FV + W W Energy minimization problem: ε φ: = φ ( xt, ) [ 1,1] 1& (0,1] thin film phase φ = 1 & [ 1,0) vapor phase ε interface width = 0 interface/surface Film-vapor phase energy Wall energy = Ω+ Γ f d f d Γ FV W W W Ω Film-vapor energy density Wall energy density min W ε subject to BCs tanh ds ( x, Γ) ε

34 Phase Field Model Film/vapor phase energy Ginzburg-Landau free energy ε 1 ffv ( φ, φ) = λ φ + F( φ), F( φ) = ( φ 1) mixing constant ε film/vapor energy interface energy Convergence L. Modica & S. Mortola, Boll. Un. Mat. Ital. A, 14 (1977), If λ = γfv W ε FV γfv ΣFV 4 Wall energy must satisfy ' At homogeneous vapor phase φ = 1 fw ( φ) = γvs & fw ( φ) = 0 ' At homogeneous film phase φ = 1 fw ( φ) = γfs & fw ( φ) = 0 γvs + γfs φ(3 φ ) f W ( φ) = ( γ VS γ FS ) 4

35 Phase Field Model Cahn-Hilliard equation with function-dependent mobility φ = ( M µ ) t = 3 µ φ φ ε φ With BCs( Jiang, Bao, Thompson & Srolovitz, Acta Mater., 1 ) Γ W in Ω with M = (1 φ ) φ f ( φ) γ γ + = = On wall boundary ε 0 & cos θs n λ γfv φ µ ε + ( φ 1) cos θs = 0 & = 0 n n φ µ On other boundaries Γ0 Γ 1 = Ω \ ΓW = 0 & = 0 n n Recent debate on proper mobility for surface diffusion A.A. Lee, A. Munch & E Suli, APL 15 & 16 ; A. Voigt, APL 16,.. ' W VS FS

36 Numerical Method Stabilized semi-implicit method n+ 1 n φ φ n n+ 1 n+ 1 n n n = Aε ( φ φ ) + S ( φ φ ) + ( M µ ) τ n n 3 n n n n µ = ( φ ) φ ε φ, M = 1 ( φ ) A and S are two stabilizing constants x-direction by cosine pseudospectral method y-direction by finite difference/finite element 1 st order in time & can be upgraded to nd Can be solved efficiently at every step & Extension to 3D is easy [1] W. Jiang, W. Bao, C.V. Thompson &D. J. Srolovitz, Acta Mater. 60(01), [] J.Z. Zhu, L.Q. Chen, J Shen & V. Tikare, Phys. Rev. E, 60 (1999), [] J. Shen & X.F. Yang, SIAM J. Sci. Comput., 3 (010), 1159.

37 3D Results I -- Square θ s = 5 π /6

38 3D Results II Pinch-off θ s = 5 π /6

39 3D Results III Large-Thin Square

40 3D Results Comparison with Experiment Computational Results Experimental Results

41 Conclusion & Future Works Conclusion Propose sharp interface model (SIM) for anisotropic surface energy Propose phase field model Cahn-Hilliard Eq. with function-dependent mobility Design stable, efficient & accurate numerical methods Test parameters regimes & simulate D & 3D results Compare qualitatively with experimental / asymptotic results Future works Phase field model with anisotropic surface energy Sharp interface model in 3D and/or rough substrate Investigate role of different surface energy anisotropy & regularization Develop adaptive & parallel numerical method Mathematical analysis of the models Compare with experiments quantitatively & guide new experiments