THREE-DIMENSIONAL NUMERICAL SIMULATIONS OF A PHASE-FIELD MODEL FOR ANISOTROPIC INTERFACIAL ENERGY

- Journal title : Communications of the Korean Mathematical Society
- Volume 22, Issue 3, 2007, pp.453-464
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/CKMS.2007.22.3.453

Title & Authors

THREE-DIMENSIONAL NUMERICAL SIMULATIONS OF A PHASE-FIELD MODEL FOR ANISOTROPIC INTERFACIAL ENERGY

Kim, Jun-Seok;

Kim, Jun-Seok;

Abstract

A computationally efficient numerical scheme is presented for the phase-field model of two-phase systems for anisotropic interfacial energy. The scheme is solved by using a nonlinear multigrid method. When the coefficient for the anisotropic interfacial energy is sufficiently high, the interface of the system shows corners or missing crystallographic orientations. Numerical simulations with high and low anisotropic coefficients show excellent agreement with exact equilibrium shapes. We also present spinodal decomposition, which shows the robustness of the pro-posed scheme.

Keywords

phase-field model;anisotropy;interfacial energy;Cahn-Hilliard equation;nonlinear multigrid method;

Language

English

Cited by

References

1.

W. K. Burton, N. Cabrera, and F. C. Frank, The growth of crystals and the equilibrium structure of their surfaces, Trans. R. Soc. Lond. A 243 (1951), 299-358

2.

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258-267

3.

J. J. Eggleston, G. B. McFadden, and P. W. Voorhees, A phase-field model for highly anisotropic interfacial energy, Physica D 150 (2001), 91-103

4.

D. J. Eyre, An unconditionally stable one-step scheme for gradient systems, Preprint, University of Utah, Salt Lake City, 1997

5.

F. C. Frank, Metal Surfaces, ASM, Cleveland, OH, 1963

6.

D. W. Hoffman and J. W. Cahn, A vector thermodynamics for anisotropic surfaces. I. Fundamentals and application to plane surface junctions, Surf. Sci. 31 (1972), 368-388

7.

J. S. Kim, A continuous surface tension force formulation for diffuse-interface models, J. Comput. Phys. 204 (2005), 784-804

8.

J. S. Kim and J. Sur, A hybrid method for higher-order nonlinear diffusion equations, Commun. Korean Math. Soc. 20 (2005), no. 1,179-193

9.

B. P. Vollmayr-Lee and A. D. Rutenberg, Fast and accurate coarsening simulation with an unconditionally stable time step, Phys Rev E, 68 (2003), 1-13

10.

W. W. Mullins, Proof that the two dimensional shape of minimum surface free energy is convex, J. Math. Phys. 3 (1962), 754-759

11.

M. Siegel, M. J. Miksis, and P. W. Voorhees, Evolution of material voids for highly anisotropic surface energy, J. Mech. Phys. Solids 52 (2004), 1319-1353

12.

T. Takaki, T. Hasebe, and Y. Tomita, Two-dimensional phase-field simulation of selfassembled quantum dot formation, J. Crystal Growth 287 (2006), 495-499

13.

U. Trottenberg, C. Oosterlee, and A. Schuller, MULTIGRID, Academic Press, 2001

14.

A. A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phase-field model for isothermal phase transitions in binary alloys, Phys. Rev. A 45 (1992), 7424-7439

15.

Y. Wang, L. Q. Chen, and A. G. Khachaturyan, Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap, Acta Metall. 41 (1993), 279-296