The Dynamics of Nucleation in Stochastic Cahn-Morral Systems

Cahn-Morral systems serve as models for several phase separation phenomena in multicomponent alloys. In this paper we study the dynamical aspects of nucleation in a stochastic version of these models using numerical simulations, concentrating on ternary, i.e., three-component, alloys on two- dimensional square domains. We perform numerical studies and give a statistical classification for the distribution of droplet types as the component structure of the alloy is varied. We relate these statistics to the low-energy equilibria of the deterministic equation.

[1]  E. J. Doedel,et al.  AUTO: a program for the automatic bifurcation analysis of autonomous systems , 1980 .

[3]  Giuseppe Da Prato,et al.  Stochastic Cahn-Hilliard equation , 1996 .

[4]  J. Hoyt.  Linear spinodal decomposition in a regular ternary alloy , 1990 .

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

[6]  H. E. Cook,et al.  Brownian motion in spinodal decomposition , 1970 .

[7]  Jonathan P. Desi,et al.  Complex transient patterns on the disk , 2006 .

[8]  John E. Hilliard,et al.  Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 1959 .

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

[10]  Jeffrey J. Hoyt,et al.  Spinodal decomposition in ternary alloys , 1989 .

[11]  Thomas Wanner,et al.  A semi-implicit spectral method for stochastic nonlocal phase-field models , 2009 .

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

[13]  Christopher P. Grant.  SLOW MOTION IN ONE-DIMENSIONAL CAHN-MORRAL SYSTEMS , 1995 .

[14]  Evelyn Sander,et al.  Unexpectedly Linear Behavior for the Cahn-Hilliard Equation , 2000, SIAM J. Appl. Math..

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

[16]  Evelyn Sander,et al.  Monte Carlo Simulations for Spinodal Decomposition , 1999 .

[17]  P. Hohenberg,et al.  Theory of Dynamic Critical Phenomena , 1977 .

[18]  J. Waals.  The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density , 1979 .

[19]  Dirk Blömker,et al.  Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation , 2001 .

[20]  D. J. Eyre.  Cascades of spinodal decompositions in theternary , 2007 .

[21]  Anthony C. Davison,et al.  Bootstrap Methods and Their Application , 1998 .

[22]  C. Cardon-Weber.  Cahn-Hilliard stochastic equation: existence of the solution and of its density , 2001 .

[23]  Dirk Blömker,et al.  Spinodal Decomposition¶for the Cahn–Hilliard–Cook Equation , 2001 .

[24]  Dirk Blömker,et al.  Nucleation in the one-dimensional stochastic Cahn-Hilliard model , 2010 .

[25]  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 .

[26]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[27]  Peter W. Bates,et al.  The Dynamics of Nucleation for the Cahn-Hilliard Equation , 1993, SIAM J. Appl. Math..

[28]  S. Maier-Paape,et al.  SURFACE ROUGHNESS IN MOLECULAR BEAM EPITAXY , 2001 .

[29]  Thomas Wanner,et al.  Maximum norms of random sums and transient pattern formation , 2003 .

[30]  Thomas Wanner,et al.  Spinodal Decomposition for the Cahn–Hilliard Equation in Higher Dimensions.¶Part I: Probability and Wavelength Estimate , 1998 .

[31]  G. Burton.  Sobolev Spaces , 2013 .

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

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

[34]  Thomas Wanner,et al.  Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics , 2000 .

[35]  S. Maier-Paape,et al.  Path-following the equilibria of the Cahn–Hilliard equation on the square , 2002 .

[36]  Konstantin Mischaikow,et al.  Evolution of pattern complexity in the Cahn–Hilliard theory of phase separation , 2005 .

[37]  B. Gawron,et al.  Nucleation in the one-dimensional Cahn-Hilliard Model , 2006 .

[38]  J. Zabczyk,et al.  Stochastic Equations in Infinite Dimensions , 2008 .

[39]  Eric Vanden-Eijnden,et al.  Rare Events in Stochastic Partial Differential Equations on Large Spatial Domains , 2008 .

[40]  Paul C. Fife,et al.  Models for phase separation and their mathematics. , 2000 .

[41]  John W. Cahn,et al.  Free Energy of a Nonuniform System. II. Thermodynamic Basis , 1959 .