Cahn-Hilliard Phase Decomposition and

Transcription

1 Decomposition and Roy H. Stogner 1 Graham F. Carey 1 1 Computational Fluid Dynamics Lab Institute for Computational Engineering and Sciences University of Texas at Austin 9th US National Congress on Computational Mechanics

2 Outline

3 Separation Phenomena Example Applications Tin-Lead solder aging Void lattice formation in irradiated semiconductors Self-assembly of thin film patterns

4 Patterned Separation Electric Voltage (Suo, Hong) Filler Particles (Lee, Douglas, Glotzer) Surface Monolayers (Böltau, Walheim, Mlynek, Krausch, Steiner; Karim, Douglas, Lee, Glotzer, Rogers, Jackman, Amis, Whitesides) Reacting Chemicals (Glotzer, Coniglio; Di Marzio, Muthukumar) Surface Stress (Lu, Suo; Yu, Lu)

5 Free Energy Formulation systems model material separation and interface evolution based on a weakly non-local free energy density. f (c, c) f 0 (c) + f γ ( c) f γ ( c) ɛ2 c c c 2 f 0 (c) NkT (c ln (c)+ (1 c) ln (1 c)) + Nωc(1 c) Bulk Free Energy Density Flory Huggins Free Energy Density Concentration NkT = 0.4 NkT = 0.5 NkT = 0.6 NkT = 0.7 NkT = 0.8 NkT = 0.9

6 A mobility coefficient M c defines the concentration flux J. For positive definite M c, the resulting equation gives globally non-increasing free energy. J = M c df dc = M c ( f 0(c) + f γ(c) ) c t = M c ( f 0(c) ɛ 2 c c )

7 Weak Taking a weighted residual and integrating by parts twice, ( c t, φ) Ω = ( M c f 0(c), φ ) Ω ( ɛ2 c c, M T c φ ) Ω + (( M c ( f 0(c) ɛ 2 c c )) n, φ ) Ω +ɛ 2 c ( c, M T c φ n ) Ω

8 C 1 Finite In the Galerkin formulation of transport, we find integrated products of second derivatives of trial and test functions. We use C 1 continuous,w 2,p conforming finite elements: Element Types Clough-Tocher macroelement triangles Bogner-Fox- Schmidt squares, analogous Hermite cubes

9 h Adaptivity Function space continuity requires constraining hanging node Degrees of Freedom in terms of DoFs on coarse elements. f k ( i u F = u C ) u F i φ F i = f k j u C j φ C j Both Hermite and Clough-Tocher elements are subdivision compatible. u i = (A ki ) 1 B kj u j

10 libmesh Finite Element Library Initial developers: Benjamin Kirk, John Peterson Key Features Mixed element geometries in unstructured grids Adaptive mesh h-refinement with hanging nodes Parallel system assembly and solution Integration w/ PETSc, ParMETIS, more Export/import to common data formats New Development C 1 macroelement, Hermite classes Fourth order problem calculations Parallel adaptivity for general element types Additional error estimators, adaptivity strategies New nonlinear solver, timestepping frameworks

11 Separation - Spinodal Decomposition Initial Evolution Initial homogeneous blend quenched below critical T Random perturbations rapidly segregate into two distinct phases, divided by a labyrinth of sharp interfaces Rapid anti-diffusionary process

12 Separation - Interfacial Flow Long-term Evolution Single-phase regions gradually coalesce Motion into curvature vector resembles surface tension Patterning may occur when additional stress, surface tropisms are applied

13 Mesh Refinement Coarse mesh bicubic elements t = t = Mesh resolution must capture equilibrium interface width Timesteps are limited by nonlinear solver

14 Mesh Refinement One Refinement bicubic elements t = t = Free energy decay limits most error growth Primary exception: interface topology changes

15 Mesh Refinement Two Refinements bicubic elements t = t = Finite Element error becomes negligible Uncertainty issues remain

16 Mesh Refinement L 2 error coarse medium fine Time Transient Error With sufficient resolution, error remains bounded Errors decay on moderately refined meshes

17 with AMR/C Interface Tracking Problem Coarsening in single-phase regions is traded for refinement in sharp layers Equivalent accuracy achieved here with 75% fewer degrees of freedom

18 with AMR/C Decomposition Problem Laplacian Jump error indicator keeps up with moving interfaces Coarsening strategy for spinodal decomposition raises new questions

19 Thin Film Patterning Material Self- Electrostatic or chemical surface treatment attracts one material component preferentially A patterned spatially varying bias is added to the configurational free energy

20 Film Thickness Effects Thin films rapidly become uniform in z direction Thick films show more connectivity, fewer defects

21 Effects of Bias Strength Low surface potential energy biases are overwhelmed by random noise

22 Effects of Bias Strength Higher surface potential energy biases form patterns with decreasing defect density

23 Correlation Lengths Correlation Length x 1y 3x 3y 5x 5y 7x 7y Time Correlation between c( x), c( x + y) averaged over all x Anisotropic effects quickly visible Correlation lengths in pattern direction diverge

24 Effects of Quench Temperature Defects Time Defect Count Reduction Lower T freezes random defects in place Higher T gives more diffuse interfaces, smoother/faster defect reduction Near-critical T prevents full phase decomposition

25 QuasiNewton, GMRES + Block Jacobi + ILU give efficient implicit parallel solve performance on a adaptive spline and macroelement meshes Realistic spinodal decomposition, pattern formation results with degrees of freedom Qualitative identification of key parameters for pattern formation

26 Ongoing Work Lyapunov free energy functional analysis Adaptive time discretization Monte Carlo simulation Continuing parameter studies Additional patterning physics