Dynamics of phase separations of a dissipative system and a fluid mixture

Abstract

The temporal evolutions of structure functions ${S}_{k}(t)$ of quenched binary mixtures are studied theoretically. With the aid of a Langevin-type equation, basic nonlinear kinetic equations for the composition fluctuation are derived for a purely dissipative system and for a fluid mixture. Predicting that the free energy is expanded on the basis of a cluster gas picture, the equations of motion for structure functions are derived. The inverse nonhydrodynamic susceptibility ${{\ensuremath{\chi}}_{k}}^{\ensuremath{-}1}$, which is the first-derivative coefficient of the free energy, and ${S}_{k}(t)$ are assumed to have the form ${R}^{\ensuremath{-}d}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\chi}}{(\mathrm{kR})}^{\ensuremath{-}1}$ and ${R}^{d}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{S}(\mathrm{kR})$ in $d$ dimensions. Here $R$ is the average cluster diameter, which behaves as ${t}^{\ensuremath{-}{a}^{\ensuremath{'}}}$ [${a}^{\ensuremath{'}}={(d+2)}^{\ensuremath{-}1}or{(d+3)}^{\ensuremath{-}1}$ for a purely dissipative system and ${a}^{\ensuremath{'}}=\frac{l}{d}$ for a fluid mixture]. If ${{\ensuremath{\chi}}_{k}}^{\ensuremath{-}1}$ has a gap of the order ${R}^{\ensuremath{-}d}$, then our calculation of ${S}_{k}(t)$ yields good agreements with experiments (for $d=3$). The renormalizations both the mobility and of susceptibility due to long-range hydrodynamic interactions are treated with the use of the mode-coupling technique.