Scaling functions, self-similarity, and the morphology of phase-separating systems

In the late stages of phase separation in liquids or solids with negligible coherency stresses, the structure function S(k,t) is known to follow a scaling behavior in the form S(k,t)km-3(t)F(k/km(t)), where km(t) is the value of k that maximizes S at a given t. Previous work has shown that, for many real systems and for three-dimensional computer models, the scaling function F(x) depends only on the volume fraction of the minority phase but not on the temperature T for a given. Results from a Monte Carlo simulation of the two-dimensional Ising model, and also from a recently published numerical solution of the two-dimensional Cahn-Hilliard equation, are shown here to give a scaling function that can be fitted, as in the three-dimensional case, by an analytical expression containing just one adjustable parameter, independent of T but dependent on. We analyze and interpret some universal features of these scaling functions, including their behavior at small x and at large x, and their dependence on. Our discussion is based on a two-phase model, i.e., a mixture of two types of domains separated by thin interfaces, with kinetics based on the Cahn-Hilliard equation. We introduce an assumption of self-similar evolution (in the sense of self-similar probability ensembles) and show that it leads to the well-known t1/3 growth rate for the average domain size and to the above-mentioned universal properties of the scaling function. Simple geometric considerations also allow the calculation of the parameter, so that the scaling function may be obtained without any adjustment of parameters. The influence of droplet-size distributions on the scaling function, the limit of very dilute alloys, and the temperature dependence of the coarsening rate are also considered. © 1991 The American Physical Society.