Theory of phase-separation dynamics in quenched binary mixtures

A systematic theory of the dynamics of phase separation of quenched binary systems into metastable states is presented. Not only the kinetic equation for the single-droplet-size distribution function f(R,t) but also the linear equation for the structure function S(k,t), which are valid over the entire time region after the nucleation stage, are derived from a unified point of view. The functions f(R,t) and S(k,t) are shown to satisfy the dynamical scaling relations f(R,t)=[n(t)/〈R〉(t)]F(R/〈R〉,t) and S(k,t)=[${\mathit{k}}_{\mathit{M}}$(t)${]}^{\mathrm{-}\mathit{d}}$[Φ(t)${]}^{\mathrm{\delta }}$Ψ(k/${\mathit{k}}_{\mathit{M}}$,t) with ${\mathit{k}}_{\mathit{M}}^{\mathrm{-}1}$(t)∝〈R〉/${\mathrm{\Phi }}^{1/\mathit{d}}$, where n is the number density of the minority phase, 〈R〉 the average droplet radius, Φ the volume fraction of the minority phase, ${\mathit{k}}_{\mathit{M}}$ the peak position of S(k,t), and d=3 here. Three characteristic stages are then shown to exist after the nucleation stage: the growth stage with δ=2, where the power laws are given by 〈R〉∝${\mathit{t}}^{1/2}$, n∝${\mathit{t}}^0$, ${\mathit{k}}_{\mathit{M}}$∝${\mathit{t}}^0$, and Φ∝${\mathit{t}}^{3/2}$; the intermediate stage with δ=1/d, where 〈R〉∝${\mathit{t}}^{1/4}$, n∝${\mathit{t}}^{\mathrm{-}2/3}$, ${\mathit{k}}_{\mathit{M}}$∝${\mathit{t}}^{\mathrm{-}2/9}$, and Φ∝${\mathit{t}}^{1/12}$; and the coarsening stage with δ=1/d, where 〈R〉∝${\mathit{t}}^{1/3}$, n∝${\mathit{t}}^{\mathrm{-}1}$, ${\mathit{k}}_{\mathit{M}}$∝${\mathit{t}}^{\mathrm{-}1/3}$, and Φ∝${\mathit{t}}^0$. Two crossovers are then observed in the time exponents. The time evolution of the scaling functions F(ρ,t) and Ψ(x,t) are thus studied explicitly, including their asymptotic behavior and their dependence on Φ.