Computational studies of coarsening rates for the Cahn-Hilliard equation with phase-dependent diffusion mobility

We study computationally coarsening rates of the Cahn-Hilliard equation with a smooth double-well potential, and with phase-dependent diffusion mobilities. The latter is a feature of many materials systems and makes accurate numerical simulations challenging. Our numerical simulations confirm earlier theoretical predictions on the coarsening dynamics based on asymptotic analysis. We demonstrate that the numerical solutions are consistent with the physical Gibbs-Thomson effect, even if the mobility is degenerate in one or both phases. For the two-sided degenerate mobility, we report computational results showing that the coarsening rate is on the order of l ~ c t 1 / 4 , independent of the volume fraction of each phase. For the one-sided degenerate mobility, that is non-degenerate in the positive phase but degenerate in the negative phase, we illustrate that the coarsening rate depends on the volume fraction of the positive phase. For large positive volume fractions, the coarsening rate is on the order of l ~ c t 1 / 3 and for small positive volume fractions, the coarsening rate becomes l ~ c t 1 / 4 .

[1]  Carlos J. García-Cervera,et al.  A new approach for the numerical solution of diffusion equations with variable and degenerate mobility , 2013, J. Comput. Phys..

[2]  Harald Garcke,et al.  Finite Element Approximation of the Cahn-Hilliard Equation with Degenerate Mobility , 1999, SIAM J. Numer. Anal..

[3]  Francisco Guillén-González,et al.  On linear schemes for a Cahn-Hilliard diffuse interface model , 2013, J. Comput. Phys..

[4]  Qiang Du,et al.  A Fourier Spectral Moving Mesh Method for the Cahn-Hilliard Equation with Elasticity , 2009 .

[5]  Amy Novick-Cohen,et al.  Upper bounds for coarsening for the degenerate Cahn-Hilliard equation , 2009 .

[6]  Xinfu Chen,et al.  The Hele-Shaw problem and area-preserving curve-shortening motions , 1993 .

[7]  John W. Cahn,et al.  Linking anisotropic sharp and diffuse surface motion laws via gradient flows , 1994 .

[8]  Barbara Niethammer,et al.  Non-Self-Similar Behavior in the LSW Theory of Ostwald Ripening , 1999 .

[9]  F. Otto,et al.  Upper Bounds on Coarsening Rates , 2002 .

[10]  David Andrs,et al.  A quantitative comparison between C0 and C1 elements for solving the Cahn-Hilliard equation , 2013, J. Comput. Phys..

[11]  Robert Nürnberg,et al.  The degenerate and non-degenerate deep quench obstacle problem: A numerical comparison , 2013, Networks Heterog. Media.

[12]  Andreas Prohl,et al.  Numerical analysis of the Cahn-Hilliard equation and approximation for the Hele-Shaw problem, Part II: Error analysis and convergence of the interface , 2001 .

[13]  Desai,et al.  Numerical study of late-stage coarsening for off-critical quenches in the Cahn-Hilliard equation of phase separation. , 1989, Physical review. B, Condensed matter.

[14]  D. J. Eyre,et al.  An Unconditionally Stable One-Step Scheme for Gradient Systems , 1997 .

[15]  Peter W Voorhees,et al.  Phase-field simulation of 2-D Ostwald ripening in the high volume fraction regime , 2002 .

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

[17]  J. Langer,et al.  New computational method in the theory of spinodal decomposition , 1975 .

[18]  Jaemin Shin,et al.  A conservative numerical method for the Cahn-Hilliard equation in complex domains , 2011, J. Comput. Phys..

[19]  Emmott,et al.  Lifshitz-Slyozov scaling for late-stage coarsening with an order-parameter-dependent mobility. , 1995, Physical review. B, Condensed matter.

[20]  Harald Garcke,et al.  Transient coarsening behaviour in the Cahn–Hilliard model , 2003 .

[21]  V. Ozoliņš,et al.  Trans-interface diffusion-controlled coarsening , 2005, Nature materials.

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

[23]  Jie Shen,et al.  Coarsening kinetics from a variable-mobility Cahn-Hilliard equation: application of a semi-implicit Fourier spectral method. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  A. Nepomnyashchy.  Coarsening versus pattern formation , 2015, 1501.05514.

[25]  Zi-kui Liu,et al.  Coarsening kinetics of γ′ precipitates in the Ni–Al–Mo system , 2008 .

[26]  Chert,et al.  Applications of semi-implicit Fourier-spectral method to phase field equations , 2004 .

[27]  Amy Novick-Cohen,et al.  Upper Bounds for Coarsening for the Deep Quench Obstacle Problem , 2010 .

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

[29]  C. M. Elliott,et al.  On the Cahn-Hilliard equation with degenerate mobility , 1996 .

[30]  Barbara Niethammer,et al.  Derivation of the LSW‐Theory for Ostwald Ripening by Homogenization Methods , 1999 .

[31]  Sancho,et al.  Domain growth in binary mixtures at low temperatures. , 1992, Physical review. B, Condensed matter.

[32]  I. Lifshitz,et al.  The kinetics of precipitation from supersaturated solid solutions , 1961 .

[33]  Thomas P. Witelski,et al.  Transient and self-similar dynamics in thin film coarsening , 2009 .

[34]  Yin Jingxue,et al.  On the existence of nonnegative continuous solutions of the Cahn-Hilliard equation , 1992 .

[35]  M. Fine,et al.  Effect of lattice disregistry variation on the late stage phase transformation behavior of precipitates in NiAlMo alloys , 1989 .

[36]  D. Seidman,et al.  The use of 3-D atom-probe tomography to study nickel-based superalloys , 2006 .

[37]  R. Nicolaides,et al.  Numerical analysis of a continuum model of phase transition , 1991 .

[38]  Qiang Du,et al.  Coarsening Kinetics of a Two Phase Mixture with Highly Disparate Diffusion Mobility , 2010 .

[39]  Jian Zhang,et al.  Fast and accurate algorithms for simulating coarsening dynamics of Cahn–Hilliard equations , 2015 .

[40]  J. Langer.  Models of Pattern Formation in First-Order Phase Transitions , 1986 .

[41]  Carl Wagner,et al.  Theorie der Alterung von Niederschlägen durch Umlösen (Ostwald‐Reifung) , 1961, Zeitschrift für Elektrochemie, Berichte der Bunsengesellschaft für physikalische Chemie.

[42]  Robert L. Pego,et al.  Front migration in the nonlinear Cahn-Hilliard equation , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[43]  Cheng Wang,et al.  Second order convex splitting schemes for periodic nonlocal Cahn-Hilliard and Allen-Cahn equations , 2014, J. Comput. Phys..

[44]  Peter W Voorhees,et al.  The theory of Ostwald ripening , 1985 .

[45]  Qiang Du,et al.  Motion of Interfaces Governed by the Cahn-Hilliard Equation with Highly Disparate Diffusion Mobility , 2012, SIAM J. Appl. Math..

[46]  R. Pego.  Lectures on Dynamics in Models of Coarsening and Coagulation , 2006 .

[47]  T. Küpper,et al.  Simulation of particle growth and Ostwald ripening via the Cahn-Hilliard equation , 1994 .

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

[49]  Jian Zhang,et al.  Numerical Studies of Discrete Approximations to the Allen--Cahn Equation in the Sharp Interface Limit , 2009, SIAM J. Sci. Comput..

[50]  Katsuyo Thornton,et al.  Modelling the evolution of phase boundaries in solids at the meso- and nano-scales , 2003 .

[51]  Peter W. Bates,et al.  Convergence of the Cahn-Hilliard equation to the Hele-Shaw model , 1994 .

[52]  Shibin Dai,et al.  Weak Solutions for the Cahn–Hilliard Equation with Degenerate Mobility , 2016 .

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

[54]  Xiaofeng Yang,et al.  Numerical approximations of Allen-Cahn and Cahn-Hilliard equations , 2010 .