Self-similarity and growth kinetics driven by surface free energy reduction

Abstract

We consider a configuration of interfaces (area S) moving under the driving force of interfacial free energy reduction in a large volume V. The time dependence of the linear scale l = V S of the configuration is deduced for systems with a single or multicomponent order parameter, conserved or non-conserved, by use of the hypothesis of statistical self-similarity: specifically, we assume that statistical parameters that are invariant under uniform magnification are also independent of time and that local equilibrium of the intersection angles of interfaces holds along lines of intersections (if any). This hypothesis, together with the scaling characteristics of interface velocities in the system under consideration, is sufficient to yield the result l( t) ~ t n with n > 0; in particular, we show that n = 1 2 for curvature driven growth and n = 1 3 for diffusion limited coarsening. We conclude that these values are independent of many of the approximations and restrictions that characterize specific models (e.g. over-simplifications of geometry, mean field approximations, small volume fraction of precipitates). Free energy functional models that exhibit the above growth laws are also discussed.