Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-19T08:36:28.828Z Has data issue: false hasContentIssue false
  • English
  • Français

A Flux-Corrected Phase-Field Method for Surface Diffusion

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  21 June 2017

Yujie Zhang  and
Wenjing Ye
Yujie Zhang*
Affiliation:
Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Hong Kong
Wenjing Ye*
Affiliation:
Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Hong Kong Division of Biomedical Engineering, Hong Kong University of Science and Technology, Hong Kong
*
*Corresponding author. Email addresses:[email protected] (W. Ye), [email protected] (Y. Zhang)
*Corresponding author. Email addresses:[email protected] (W. Ye), [email protected] (Y. Zhang)
Get access

Abstract

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 flux-corrected 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.

Keywords

Phase-field methodsurface diffusionbulk diffusionprofile correctionflux correction

MSC classification

Secondary: 82C24: Interface problems; diffusion-limited aggregation
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 2 , August 2017 , pp. 422 - 440
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Kun Xu

References

[1] Wheeler, A., Murray, B., Schaefer, R.: Computation of dendrites using a phase field model. Physica D: Nonlinear Phenomena 66(1), 243–262 (1993).CrossRefGoogle Scholar
[2] Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system. I. Interfacial free energy. The Journal of chemical physics 28(2), 258–267 (1958).Google Scholar
[3] Warren, J.A., Boettinger, W.J.: Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method. Acta Metallurgica et Materialia 43(2), 689–703 (1995).Google Scholar
[4] Beckermann, C., Diepers, H.-J., Steinbach, I., Karma, A., Tong, X.: Modeling melt convection in phase-field simulations of solidification. Journal of Computational Physics 154(2), 468–496 (1999).Google Scholar
[5] Karma, A., Rappel, W.-J.: Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Physical Review E 53(4), R3017 (1996).Google Scholar
[6] Boettinger, W., Warren, J., Beckermann, C., Karma, A.: Phase-field simulation of solidification 1. Annual review of materials research 32(1), 163–194 (2002).Google Scholar
[7] Koyama, T.: Phase-field modeling of microstructure evolutions in magnetic materials. Science and Technology of Advanced Materials 9(1) (2008).Google Scholar
[8] Chen, L.-Q.: Phase-field models for microstructure evolution. Annual review of materials research 32(1), 113–140 (2002).Google Scholar
[9] Jacqmin, D.: Calculation of two-phase Navier-Stokes flows using phase-field modeling. Journal of Computational Physics 155(1), 96–127 (1999).Google Scholar
[10] Steinbach, I., Pezzolla, F., Nestler, B., Seeelberg, M., Prieler, R., Schmitz, G., Rezende, J.: A phase field concept for multiphase systems. Physica D: Nonlinear Phenomena 94(3), 135–147 (1996).CrossRefGoogle Scholar
[11] Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D: Nonlinear Phenomena 179(3), 211–228 (2003).Google Scholar
[12] Bhate, D.N., Kumar, A., Bower, A.F.: Diffuse interface model for electromigration and stress voiding. Journal of Applied Physics 87(4), 1712–1721 (2000).Google Scholar
[13] Torabi, S., Lowengrub, J., Voigt, A., Wise, S.: A new phase-field model for strongly anisotropic systems. In: Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 2009, pp. 1337–1359.Google Scholar
[14] Jiang, W., Bao, W., Thompson, C.V., Srolovitz, D.J.: Phase field approach for simulating solid-state dewetting problems. Acta materialia 60(15), 5578–5592 (2012).Google Scholar
[15] Yeon, D.-H., Cha, P.-R., Grant, M.: Phase field model of stress-induced surface instabilities: Surface diffusion. Acta materialia 54(6), 1623–1630 (2006).Google Scholar
[16] Mahadevan, M., Bradley, R.M.: Phase field model of surface electromigration in single crystal metal thin films. Physica D: Nonlinear Phenomena 126(3), 201–213 (1999).Google Scholar
[17] Wise, S., Kim, J., Lowengrub, J.: Solving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method. Journal of Computational Physics 226(1), 414–446 (2007).Google Scholar
[18] Rätz, A., Ribalta, A., Voigt, A.: Surface evolution of elastically stressed films under deposition by a diffuse interface model. Journal of Computational Physics 214(1), 187–208 (2006).Google Scholar
[19] Cahn, J.W., Elliott, C.M., Novick-Cohen, A.: The Cahn-Hilliard equation with a concentration dependent mobility: motion by minus the Laplacian of the mean curvature. European journal of applied mathematics 7(03), 287–301 (1996).Google Scholar
[20] Cahn, J.W., Taylor, J.: Surface motion by surface diffusion. Acta Metallurgica et Materialia 42(4), 1045–1063 (1994).Google Scholar
[21] Lee, A.A., Müunch, A., Süli, E.: Sharp-Interface Limits of the Cahn-Hilliard Equation with Degenerate Mobility. SIAM Journal on Applied Mathematics 76(2), 433–456 (2016).Google Scholar
[22] Lee, A.A., Müunch, A., Süli, E.: Degenerate mobilities in phase field models are insufficient to capture surface diffusion. Applied Physics Letters 107(8), 081603 (2015).Google Scholar
[23] Dai, S., Du, Q.: Coarsening Mechanism for Systems Governed by the Cahn-Hilliard Equation with Degenerate Diffusion Mobility. Multiscale Modeling & Simulation 12(4), 1870–1889 (2014).Google Scholar
[24] Yue, P., Zhou, C., Feng, J.J.: Spontaneous shrinkage of drops andmass conservation in phase-field simulations. Journal of Computational Physics 223(1), 1–9 (2007).Google Scholar
[25] Li, Y., Choi, J.-I., Kim, J.: A phase-field fluid modeling and computation with interfacial profile correction term. Communications in Nonlinear Science and Numerical Simulation 30(1), 84–100 (2016).Google Scholar
[26] Macklin, P., Lowengrub, J.: Evolving interfaces via gradients of geometry-dependent interior Poisson problems: application to tumor growth. Journal of Computational Physics 203(1), 191–220 (2005). doi:10.1016/j.jcp.2004.08.010.CrossRefGoogle Scholar
[27] Macklin, P., Lowengrub, J.: An improved geometry-aware curvature discretization for level set methods: application to tumor growth. Journal of Computational Physics 215(2), 392–401 (2006).CrossRefGoogle Scholar
[28] Salac, D., Lu, W.: A Local Semi-Implicit Level-Set Method for Interface Motion. Journal of Scientific Computing 35(2-3), 330–349 (2008). doi:10.1007/s10915-008-9188-6.Google Scholar
[29] Ervik, Å., Lervåg, K.Y., Munkejord, S.T.: A robust method for calculating interface curvature and normal vectors using an extracted local level set. Journal of Computational Physics 257, 259–277 (2014). doi:10.1016/j.jcp.2013.09.053.Google Scholar
[30] Lervåg, K.Y.: Calculation of interface curvature with the level-set method. in: Sixth National Conference on Computational Mechanics MekIT11, Trondheim, Norway (2011).Google Scholar
[31] Gugenberger, C., Spatschek, R., Kassner, K.: Comparison of phase-field models for surface diffusion. Physical Review E 78(1), 016703 (2008).Google Scholar
[32] Yue, P., Feng, J.J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. Journal of Fluid Mechanics 515, 293–317 (2004).Google Scholar
[33] FFTW Homepage Website. http://www.fftw.org.Google Scholar
[34] Sudoh, K., Iwasaki, H., Kuribayashi, H., Hiruta, R., Shimizu, R.: Numerical Study on Shape Transformation of Silicon Trenches by High-Temperature Hydrogen Annealing. Japanese Journal of Applied Physics 43(9A), 5937–5941 (2004). doi:10.1143/jjap.43.5937.Google Scholar
[35] Zhang, Y., Ye, W.: A High-Order Level-Set Method with Enhanced Stability for Curvature Driven Flows and Surface Diffusion Motion. Journal of Scientific Computing, 1–30 (2016).Google Scholar
[36] Kolahdouz, E.M., Salac, D.: A Semi-implicit Gradient Augmented Level Set Method. SIAM Journal on Scientific Computing 35(1), A231–A254 (2013). doi:10.1137/120871237.Google Scholar