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

The temporal evolutions of structure functions $S_k\mathrm{}\left(t\right)\mathrm{}$ 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 ${{\chi }_k}^{-1\mathrm{}}$, which is the first-derivative coefficient of the free energy, and $S_k\mathrm{}\left(t\right)\mathrm{}$ are assumed to have the form $R^{-d}\tilde{\chi }{\mathrm{}\left(\mathrm{kR}\right)\mathrm{}}^{-1\mathrm{}}$ and $R^d\tilde{S}\mathrm{}\left(\mathrm{kR}\right)\mathrm{}$ in $d$ dimensions. Here $R$ is the average cluster diameter, which behaves as $t^{-a^{\prime }}$ [$a^{\prime }={\mathrm{}(d+2\mathrm{})\mathrm{}}^{-1\mathrm{}}or{\mathrm{}(d+3\mathrm{})\mathrm{}}^{-1\mathrm{}}$ for a purely dissipative system and $a^{\prime }=\frac{l}{d}$ for a fluid mixture]. If ${{\chi }_k}^{-1\mathrm{}}$ has a gap of the order $R^{-d}$, then our calculation of $S_k\mathrm{}\left(t\right)\mathrm{}$ yields good agreements with experiments (for $d=3\mathrm{}$). The renormalizations both the mobility and of susceptibility due to long-range hydrodynamic interactions are treated with the use of the mode-coupling technique.