Coarsening versus pattern formation

Abstract

It is known that similar physical systems can reveal two quite different ways of behavior, either coarsening, which creates a uniform state or a large-scale structure, or formation of ordered or disordered patterns, which are never homogenized. We present a description of coarsening using simple basic models, the Allen-Cahn equation and the Cahn-Hilliard equation, and discuss the factors that may slow down and arrest the process of coarsening. Among them are pinning of domain walls on inhomogeneities, oscillatory tails of domain walls, nonlocal interactions, and others. Coarsening of pattern domains is also discussed.

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

“…In Sections 5 and 6 we report the numerical results for the cases with two-sided and one-sided degenerate diffusion mobility, respectively, with emphasis on the coarsening mechanisms and rates. Even though there have been numerous computational simulations for CH equations in the literature [3,4,5,7,8,14,16,17,18,19,20,21,23,24,25,27,32,39,40,42,43,44,46,50,51,54,55] and the references cited therein, the results given in this work represent the most comprehensive so far: we take small time steps in our simulations not only to ensure stability but most importantly to ascertain time accuracy; we run the simulations for exceedingly long time intervals so that the solution behaviors reach well into appropriate asymptotic regimes; we conduct numerical studies of a test case involving only an elliptical domain and another test case involving only two spherical domains to demonstrate the consistency of numerical solutions with the physical Gibbs-Thomson effect and to illustrate the communication between disjoint domains and the mechanism behind the coarsening; we also carefully perform large scale simulations on systems involving convoluted spatial patterns to demonstrate the dependence of coarsening rates on the diffusion mobilities and volume fractions. For each case of two-sided and one-sided degenerate mobilities, we analyze three typical scenarios of 75%, 50% and 25% volume fractions corresponding to having large, equal and small positive phases and compare their coarsening rates.…”

Kink dynamics with oscillating forces

“…If surface tension combines with a double well potential Ũ (h), which accounts for the existence of two macroscopic stable states, F = F GL is called Ginzburg-Landau free energy and it plays a relevant role in the theory of phase transitions and phase ordering. In one dimension, a simple description of energetics and dynamics can be given in terms of kinks [4]. A kink h k (x) is the simplest non-homogeneous state which interpolates between the two minima of the potential, ±h m , and it has two main features: it is a monotonous function, and it is localized, i.e.…”

Domain coarsening in a subdiffusive Allen–Cahn equation

“…However, if the system contains several domain walls, they can move because of the mutual interaction, which leads to a coarsening of the domains. That process has been a subject of detailed investigation in the past (see [3][4][5]). A number of physical processes, e.g., in the case of a glass transition, are characterized by memory effects [6][7][8] and described by a reaction-diffusion equation with memory,…”