A practically unconditionally gradient stable scheme for the N-component Cahn-Hilliard system

We present a practically unconditionally gradient stable conservative nonlinear numerical scheme for the N-component Cahn–Hilliard system modeling the phase separation of an N-component mixture. The scheme is based on a nonlinear splitting method and is solved by an efficient and accurate nonlinear multigrid method. The scheme allows us to convert the N-component Cahn–Hilliard system into a system of N−1 binary Cahn–Hilliard equations and significantly reduces the required computer memory and CPU time. We observe that our numerical solutions are consistent with the linear stability analysis results. We also demonstrate the efficiency of the proposed scheme with various numerical experiments.

[1]  H. Frieboes,et al.  Three-dimensional multispecies nonlinear tumor growth--I Model and numerical method. , 2008, Journal of theoretical biology.

[2]  Long-Qing Chen,et al.  Computer simulation of 3-D grain growth using a phase-field model , 2002 .

[3]  Yoshiyuki Saito,et al.  Numerical simulation of phase separation in Fe-Cr binary and Fe-Cr-Mo ternary alloys with use of the Cahn-Hilliard equation , 2000 .

[4]  Jeffrey J. Hoyt,et al.  The continuum theory of nucleation in multicomponent systems , 1990 .

[5]  Robert Nürnberg,et al.  Numerical simulations of immiscible fluid clusters , 2009 .

[6]  H. Harada,et al.  Phase-field simulation with the CALPHAD method for the microstructure evolution of multi-component Ni-base superalloys , 2008 .

[7]  A. McHugh,et al.  Kinetics of thermally induced phase separation in ternary polymer solutions. I. Modeling of phase separation dynamics , 1999 .

[8]  Yan Xu,et al.  Local discontinuous Galerkin methods for the Cahn-Hilliard type equations , 2007, J. Comput. Phys..

[9]  Charles M. Elliott,et al.  `A generalised diffusion equation for phase separation of a multi-component mixture with interfacial free energy' , 1991 .

[10]  David Jasnow,et al.  Coarse‐grained description of thermo‐capillary flow , 1996, patt-sol/9601004.

[11]  Harald Garcke,et al.  A parametric finite element method for fourth order geometric evolution equations , 2007, J. Comput. Phys..

[12]  David Jacqmin,et al.  Contact-line dynamics of a diffuse fluid interface , 2000, Journal of Fluid Mechanics.

[13]  Donald M. Anderson,et al.  A diffuse-interface description of internal waves in a near-critical fluid , 1997 .

[14]  J. Strutt.  Scientific Papers: Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density , 2009 .

[15]  T. Ohta,et al.  The kinetics and morphology of phase-separating copolymer mixtures , 1996 .

[16]  Pierre Seppecher,et al.  Moving contact lines in the Cahn-Hilliard theory , 1996 .

[17]  Junseok Kim,et al.  On the long time simulation of the Rayleigh–Taylor instability , 2011 .

[18]  Kim.  An Unconditionally Gradient Stable Adaptive Mesh Refinement for the Cahn-Hilliard Equation , 2008 .

[19]  Dong-Hee Yeon,et al.  Phase-field model for multicomponent alloy solidification , 2005 .

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

[21]  Balasubramanya T. Nadiga,et al.  Investigations of a Two-Phase Fluid Model , 1995, comp-gas/9511003.

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

[23]  Steven M. Wise,et al.  An adaptive multigrid algorithm for simulating solid tumor growth using mixture models , 2011, Math. Comput. Model..

[24]  Junseok Kim,et al.  Phase field computations for ternary fluid flows , 2007 .

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

[26]  D. J. Eyre.  Unconditionally Gradient Stable Time Marching the Cahn-Hilliard Equation , 1998 .

[27]  W. Carter,et al.  Extending Phase Field Models of Solidification to Polycrystalline Materials , 2003 .

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

[29]  Junseok Kim,et al.  A second-order accurate non-linear difference scheme for the N -component Cahn–Hilliard system , 2008 .

[30]  Bruce T. Murray,et al.  Approximation of Cahn–Hilliard diffuse interface models using parallel adaptive mesh refinement and coarsening with C1 elements , 2008 .

[31]  T. Abinandanan,et al.  A study of phase separation in ternary alloys , 2003 .

[32]  G. Taylor.  The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[33]  Héctor D. Ceniceros,et al.  Computation of multiphase systems with phase field models , 2002 .

[34]  W. Carter,et al.  A continuum model of grain boundaries , 2000 .

[35]  Xiangrong Li,et al.  Nonlinear simulations of solid tumor growth using a mixture model: invasion and branching , 2009, Journal of mathematical biology.

[36]  Junseok Kim,et al.  Phase field modeling and simulation of three-phase flows , 2005 .

[37]  R. Chella,et al.  Mixing of a two-phase fluid by cavity flow. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[38]  Yunzhi Wang,et al.  Sparse data structure and algorithm for the phase field method , 2006 .

[39]  Yunxian Liu,et al.  A class of stable spectral methods for the Cahn-Hilliard equation , 2009, J. Comput. Phys..

[40]  Junseok Kim,et al.  A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows , 2009 .

[41]  Chang Shu,et al.  Diffuse interface model for incompressible two-phase flows with large density ratios , 2007, J. Comput. Phys..

[42]  Harald Garcke,et al.  On anisotropic order parameter models for multi-phase system and their sharp interface limits , 1998 .

[43]  M. I. M. Copetti.  Numerical experiments of phase separation in ternary mixtures , 2000 .

[44]  Héctor D. Ceniceros,et al.  A nonstiff, adaptive mesh refinement-based method for the Cahn-Hilliard equation , 2007, J. Comput. Phys..

[45]  Junseok Kim,et al.  A phase-field approach for minimizing the area of triply periodic surfaces with volume constraint , 2010, Comput. Phys. Commun..

[46]  Srikanth Vedantam,et al.  Efficient numerical algorithm for multiphase field simulations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  T. Hughes,et al.  Isogeometric analysis of the Cahn–Hilliard phase-field model , 2008 .

[48]  Andrea Mazzino,et al.  Phase-field model for the Rayleigh–Taylor instability of immiscible fluids , 2008, Journal of Fluid Mechanics.

[49]  Thomas Wanner,et al.  Spinodal Decomposition for Multicomponent Cahn–Hilliard Systems , 2000 .

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

[51]  van de Fn Frans Vosse,et al.  Diffuse-interface modelling of thermocapillary flow instabilities in a Hele-Shaw cell , 2001, Journal of Fluid Mechanics.

[52]  Junseok Kim,et al.  Accurate contact angle boundary conditions for the Cahn–Hilliard equations , 2011 .

[53]  F. Campelo,et al.  Dynamic model and stationary shapes of fluid vesicles , 2006, The European physical journal. E, Soft matter.

[54]  David J. Eyre,et al.  Systems of Cahn-Hilliard Equations , 1993, SIAM J. Appl. Math..

[55]  Yu-qiang Ma.  Domain patterns in ternary mixtures with different interfacial properties , 2001 .

[56]  Harald Garcke,et al.  Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix , 1997 .

[57]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy and Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 2013 .

[58]  Junseok Kim,et al.  A continuous surface tension force formulation for diffuse-interface models , 2005 .

[59]  Luis de Sobrino,et al.  Note on capillary waves in the gradient theory of interfaces , 1985 .

[60]  Junseok Kim,et al.  A numerical method for the Cahn–Hilliard equation with a variable mobility , 2007 .

[61]  Junseok Kim,et al.  A diffuse-interface model for axisymmetric immiscible two-phase flow , 2005, Appl. Math. Comput..

[62]  P. D. Graham,et al.  Kinetics of Thermally Induced Phase Separation in Ternary Polymer Solutions. II. Comparison of Theory and Experiment , 1999 .

[63]  John W. Cahn,et al.  Spinodal decomposition in ternary systems , 1971 .

[64]  Chang-Hun Kim,et al.  Phase-field model for the pinchoff of liquid-liquid jets , 2009 .

[65]  Junseok Kim,et al.  A numerical method for the ternary Cahn--Hilliard system with a degenerate mobility , 2009 .

[66]  Franck Boyer,et al.  Study of a three component Cahn-Hilliard flow model , 2006 .

[67]  Yinnian He,et al.  On large time-stepping methods for the Cahn--Hilliard equation , 2007 .

[68]  E. Mello,et al.  Numerical study of the Cahn–Hilliard equation in one, two and three dimensions , 2004, cond-mat/0410772.

[69]  Jaime Peraire,et al.  A time-adaptive finite volume method for the Cahn-Hilliard and Kuramoto-Sivashinsky equations , 2008, J. Comput. Phys..

[70]  Shiwei Zhou,et al.  Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition , 2006 .

[71]  Junseok Kim,et al.  CONSERVATIVE MULTIGRID METHODS FOR TERNARY CAHN-HILLIARD SYSTEMS ∗ , 2004 .

[72]  J. Lowengrub,et al.  Conservative multigrid methods for Cahn-Hilliard fluids , 2004 .