Theory for the dynamics of "clusters." II. Critical diffusion in binary systems and the kinetics of phase separation

Abstract

A semiphenomenological cluster theory is developed for the dynamics of systems with a conserved (one component) order parameter, which is not limited to small deviations from equilibrium. Concentration fluctuations of the binary system are parametrized in terms of clusters of various "sizes" $l$; these fluctuations decay by cluster reactions and cluster diffusion. The cluster diffusivity ${D}_{l}$ is estimated using the master equation for atomic exchange processes, and is confirmed by recent computer simulations of Rao et al. Close to equilibrium the nonlinear set of kinetic equations is reduced to a Fokker-Planck equation for the concentration of large clusters, which contains an effective chemical potential produced by the small clusters. Due to the conservation law this potential slowly varies with time. From this equation, we obtain as special cases the critical behavior of the diffusion constant both in solid and liquid binary systems close to ${T}_{c}$, and the Lifshitz-Slyozov theory of grain growth (below ${T}_{c}$). Additional terms describing the coagulation of large clusters have to be included in the latter case, however. At intermediate times the Lifshitz-Slyozov mechanism may even be neglected. A dynamic scaling solution of the coagulation equation predicts that the typical linear dimension should increase $\ensuremath{\propto}{t}^{\frac{1}{(3+d)}}$ in $d$ dimensions, in agreement with our previous heuristic arguments. The results are compared to computer simulations and to experiments on real systems. For the nonlinear relaxation above ${T}_{c}$ both scaling analysis and cluster dynamics give identical predictions. We also compare our approach to other theories of spinodal decomposition, deriving them in a unified way by factorization approximations of a rigorous kinetic equation, and thus elucidate their validity.