A Flux-Corrected Phase-Field Method for Surface Diffusion

Phase-field methods with a degenerate mobility have been widely used to simulate surface diffusion motion. However, apart from the motion induced by surface diffusion, adverse effects such as shrinkage, coarsening and false merging have been observed in the results obtained from the current phase-field methods, which largely affect the accuracy and numerical stability of these methods. In this paper, a fluxcorrected phase-field method is proposed to improve the performance of phase-field methods for simulating surface diffusion. The three effects were numerically studied for the proposed method and compared with those observed in the two existing methods, the original phase-field method and the profile-corrected phase-field method. Results show that compared to the original phase-field method, the shrinkage effect in the profile-corrected phase-field method has been significantly reduced. However, coarsening and false merging effects still present and can be significant in some cases. The flux-corrected phase field performs the best in terms of eliminating the shrinkage and coarsening effects. The false merging effect still exists when the diffuse regions of different interfaces overlap with each other. But it has been much reduced as compared to that in the other two methods. AMS subject classifications: 82C24

[1]  Axel Voigt,et al.  A new phase-field model for strongly anisotropic systems , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[2]  John S. Lowengrub,et al.  An improved geometry-aware curvature discretization for level set methods: Application to tumor growth , 2006, J. Comput. Phys..

[3]  Chunfeng Zhou,et al.  Spontaneous shrinkage of drops and mass conservation in phase-field simulations , 2007, J. Comput. Phys..

[4]  Wei Lu,et al.  A Local Semi-Implicit Level-Set Method for Interface Motion , 2008, J. Sci. Comput..

[5]  Bruce T. Murray,et al.  Computation of Dendrites Using a Phase Field Model , 2017 .

[6]  J. Taylor,et al.  Overview no. 113 surface motion by surface diffusion , 1994 .

[7]  Jie Shen,et al.  A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method , 2003 .

[8]  A. Karma,et al.  Phase-Field Simulation of Solidification , 2002 .

[9]  Åsmund Ervik,et al.  A robust method for calculating interface curvature and normal vectors using an extracted local level set , 2014, J. Comput. Phys..

[10]  A. Karma,et al.  Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[11]  Wei Jiang,et al.  Phase field approach for simulating solid-state dewetting problems , 2012 .

[12]  R. M. Bradley,et al.  Phase field model of surface electromigration in single crystal metal thin films , 1999 .

[13]  Jung-Il Choi,et al.  A phase-field fluid modeling and computation with interfacial profile correction term , 2016, Commun. Nonlinear Sci. Numer. Simul..

[14]  Long-Qing Chen.  Phase-Field Models for Microstructure Evolution , 2002 .

[15]  Martin Grant,et al.  Phase field model of stress-induced surface instabilities: Surface diffusion , 2006 .

[16]  I. Steinbach,et al.  A phase field concept for multiphase systems , 1996 .

[17]  James A. Warren,et al.  PHASE-FIELD SIMULATION OF SOLIDIFICATION 1 , 2002 .

[18]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[19]  James J. Feng,et al.  A diffuse-interface method for simulating two-phase flows of complex fluids , 2004, Journal of Fluid Mechanics.

[20]  Ashish Kumar,et al.  Diffuse interface model for electromigration and stress voiding , 2000 .

[21]  Karl Yngve Lervaag.  Calculation of interface curvature with the level-set method , 2014, 1407.7340.

[22]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[23]  A. Karma,et al.  Regular Article: Modeling Melt Convection in Phase-Field Simulations of Solidification , 1999 .

[24]  Endre Süli,et al.  Sharp Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility , 2015 .

[25]  Steven M. Wise,et al.  Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method , 2007, J. Comput. Phys..

[26]  Qiang Du,et al.  Coarsening Mechanism for Systems Governed by the Cahn-Hilliard Equation with Degenerate Diffusion Mobility , 2014, Multiscale Model. Simul..

[27]  J. Warren,et al.  Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method , 1995 .

[28]  Yujie Zhang,et al.  A High-Order Level-Set Method with Enhanced Stability for Curvature Driven Flows and Surface Diffusion Motion , 2016, J. Sci. Comput..

[29]  Charles M. Elliott,et al.  The Cahn–Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature , 1996, European Journal of Applied Mathematics.

[30]  R. Shimizu,et al.  Numerical Study on Shape Transformation of Silicon Trenches by High-Temperature Hydrogen Annealing , 2004 .

[31]  Ebrahim M. Kolahdouz,et al.  A Semi-implicit Gradient Augmented Level Set Method , 2012, SIAM J. Sci. Comput..

[32]  Toshiyuki Koyama,et al.  Phase-field modeling of microstructure evolutions in magnetic materials , 2008, Science and technology of advanced materials.

[33]  J. Lowengrub,et al.  Evolving interfaces via gradients of geometry-dependent interior Poisson problems: application to tumor growth , 2005 .

[34]  Axel Voigt,et al.  Surface evolution of elastically stressed films under deposition by a diffuse interface model , 2006, J. Comput. Phys..

[35]  Alpha A Lee,et al.  Degenerate mobilities in phase field models are insufficient to capture surface diffusion , 2015, 1505.06381.

[36]  Robert Spatschek,et al.  Comparison of phase-field models for surface diffusion. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.