Monte Carlo Simulations for Spinodal Decomposition
 Evelyn Sander,
 Thomas Wanner
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
This paper addresses the phenomenon of spinodal decomposition for the Cahn–Hilliard equation. Namely, we are interested in why most solutions to the Cahn–Hilliard equation which start near a homogeneous equilibrium u _{0}≡μ in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius R in a Hilbert space centered at u _{0}. There are two mathematical explanations for spinodal decomposition, due to Grant and to MaierPaape and Wanner. In this paper, we numerically compare these two mathematical approaches. In fact, we are able to synthesize the understanding we gain from our numerics with the approach of MaierPaape and Wanner, leading to a better understanding of the underlying mechanism for this behavior. With this new approach, we can explain spinodal decomposition for a longer time and larger radius than either of the previous two approaches. A rigorous mathematical explanation is contained in a separate paper. Our approach is to use Monte Carlo simulations to examine the dependence of R, the radius to which spinodal decomposition occurs, as a function of the parameter ε of the governing equation. We give a description of the dominating regions on the surface of the ball by estimating certain densities of the distributions of the exit points. We observe, and can show rigorously, that the behavior of most solutions originating near the equilibrium is determined completely by the linearization for an unexpectedly long time. We explain the mechanism for this unexpectedly linear behavior, and show that for some exceptional solutions this cannot be observed. We also describe the dynamics of these exceptional solutions.
 F. Bai, C. M. Elliott, A. Gardiner, A. Spence, and A. M. Stuart, The viscous Cahn–Hilliard equation. Part I: Computations, Nonlinearity 8:131–160 (1995).
 F. Bai, A. Spence, and A. M. Stuart, Numerical computations of coarsening in the onedimensional Cahn–Hilliard model of phase separation, Physica D 78:155–165 (1994).
 J. W. Cahn, Free energy of a nonuniform system. II. Thermodynamic basis, J. Chem. Phys. 30:1121–1124 (1959).
 J. W. Cahn, Phase separation by spinodal decomposition in isotropic systems, J. Chem. Phys. 42:93–99 (1965).
 J. W. Cahn, Spinodal decomposition, Transactions of the Metallurgical Society of AIME 242:166–180 (1968).
 J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys. 28:258–267 (1958).
 M. I. M. Copetti, Numerical analysis of nonlinear equations arising in phase transition and thermoelasticity, Ph.D. thesis (University of Sussex, 1991).
 M. I. M. Copetti and C. M. Elliott, Kinetics of phase decomposition processes: Numerical solutions to Cahn–Hilliard equation, Materials Science and Technology 6:273–283 (1990).
 K. R. Elder and R. C. Desai, Role of nonlinearities in offcritical quenches as described by the Cahn–Hilliard model of phase separation, Phys. Rev. B 40:243–254 (1989).
 K. R. Elder, T. M. Rogers, and R. C. Desai, Early stages of spinodal decomposition for the Cahn–Hilliard–Cook model of phase separation, Phys. Rev. B 38:4725–4739 (1988).
 C. M. Elliott, The Cahn–Hilliard model for the kinetics of phase separation, in Mathematical Models for Phase Change Problems, J. F. Rodrigues, ed. (Birkhauser, Basel, 1989), pp. 35–73.
 C. M. Elliott and D. A. French, Numerical studies of the Cahn–Hilliard equation for phase separation, IMA J. Appl. Math. 38:97–128 (1987).
 P. C. Fife, Models for phase separation and their mathematics. Preprint, 1991.
 C. P. Grant, Spinodal decomposition for the Cahn–Hilliard equation, Communications in Partial Differential Equations 18:453–490 (1993).
 W. Hardle, Smoothing Techniques (SpringerVerlag, New York–Berlin–Heidelberg, 1991).
 J. E. Hilliard, Spinodal decomposition, in Phase Transformations, H. I. Aaronson, ed. (American Society for Metals, Metals Park, Ohio, 1970), pp. 497–560.
 B. R. Hunt, T. Sauer, and J. A. Yorke, Prevalence: A translationinvariant “almost every” on infinitedimensional spaces, Bull. Amer. Math. Soc. 27:217–238 (1992).
 J. M. Hyde, M. K. Miller, M. G. Hetherington, A. Cerezo, G. D. W. Smith, and C. M. Elliott, Spinodal decomposition in FeCr alloys: Experimental study at the atomic level and comparison with computer models, Acta metallurgica et materialia 43:3385–3426 (1995).
 J. S. Langer, Theory of spinodal decomposition in alloys, Ann. Phys. 65:53–86 (1971).
 S. MaierPaape and T. Wanner. Spinodal decomposition for the Cahn–Hilliard equation in higher dimensions. Part I: Probability and wavelength estimate, Communications in Mathematical Physics 195(2):435–464 (1998).
 S. MaierPaape and T. Wanner, Spinodal decomposition for the Cahn–Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis, to appear.
 E. Sander and T. Wanner, Unexpectedly linear behavior for the Cahn–Hilliard equation. Submitted for publication, 1999.
 B. W. Silverman. Density Estimation for Statistics and Data Analysis (Chapman and Hall, London–New York, 1986).
 A. Stuart, Perturbation theory for infinite dimensional dynamical systems, in Theory and Numerics of Ordinary and Partial Differential Equations (Leicester, 1994), M. Ainsworth, J. Levesley, W. A. Light, and M. Marletta, eds. (Clarendon Press, 1995), pp. 181–290.
 Title
 Monte Carlo Simulations for Spinodal Decomposition
 Journal

Journal of Statistical Physics
Volume 95, Issue 56 , pp 925948
 Cover Date
 19990601
 DOI
 10.1023/A:1004550416829
 Print ISSN
 00224715
 Online ISSN
 15729613
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Cahn–Hilliard equation
 spinodal decomposition
 phase separation
 exit distributions
 density estimation
 Monte Carlo method
 Galerkin approximation
 Industry Sectors
 Authors

 Evelyn Sander ^{(1)}
 Thomas Wanner ^{(2)}
 Author Affiliations

 1. Department of Mathematical Sciences, George Mason University, Fairfax, Virginia, 22030
 2. Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, Maryland, 21250