Kinetics of droplet growth in systems with arbitrary degeneracy

The dynamics of the first-order phase transition in systems with $p$-fold degeneracy ($p\ge 2$) is discussed. A scaling analysis of a phenomenological kinetic equation is discussed to obtain various prototypes of droplet growth rates from a unified viewpoint. A process of repetitious switching of the growth mechanisms is discussed relying on the dynamic-scaling assumption. The effect of this process gives the growth rate of droplet radius $R \propto t^{a^{*}}$ with $a^{*}$ being a weighted average of the exponents of elementary processes associated. The crossover from a slow growth rate to large one need not occur, nor does the growth rate need to approach a universal law. The final growth rate may depend on system parameters such as concentration, temperature, degeneracy, and so on. Using a simplified cell model, we calculate the droplet growth rate in the case where $p$ phases have the same average density. Droplet radius $R$ grows by means of a curvature-driven force for $p\le p_c-1\mathrm{}\approx 3^d-1\mathrm{}$, and mainly by means of an entropy (or thermal) force for $p\ge p_c^{\prime }\approx 5^d$, where $d$ is the dimensionality. Here $p_c=p_c^{\prime }=2\mathrm{}$ for $d=1\mathrm{}$ and $p_c=7\mathrm{}$ and $p_c^{\prime }=19\mathrm{}$ for $d=2\mathrm{}$, however. For such a temperature as $T<\mathrm{}{T}_c{e}^{-{\xi }^{-1\mathrm{}}R}$ the entropy force does not apply and an exponentially weak attractive force drives droplet growth instead. Here $\xi$ is the thermal correlation length and $T_c$ the transition temperature. The exponent $a^{*}$ takes constant values both for $p<\mathrm{}{p}_c$ and $p>\mathrm{}{p}_c^{\prime }$. For $p_c\le p\le p_c^{\prime }$, $a^{*}$ monotonically decreases as $p$ increases. These predictions are consistent with recent numerical simulations of Sahni et al. and of Sadiq et al.