On the theoretical description of nucleation in confined space

In a recent paper, Kozisek et al. [J. Chem. Phys. 134, 094508 (2011)] have demonstrated for four different cases of phase formation that the work of formation of critical clusters required to form in the system in some given time a first experimentally measurable cluster of the new phase depends in a logarithmic way on the volume of the system. This result was obtained based on the numerical solution of the kinetic equations describing nucleation and growth processes and the obtained in this way steady-state cluster size distributions. Here a straightforward alternative analytical interpretation of this result is proposed by computing directly the mean expectation times of formation of supercritical clusters. It is proven strictly that this result is generally independent of the kind of nucleation (homogeneous or heterogeneous) or specific realization (condensation, cavitation, crystallization, segregation, etc.) considered. It is shown that such behavior is simply a consequence of the linear dependence of the steady-state nucleation rate on the volume of the system, neither time-lag or primary depletion (due to the establishment of steady-state cluster size distributions for subcritical clusters) or secondary depletion (caused by the change of the state of the ambient phase due to the formation and growth of supercritical clusters and connected with finite size effects) are required for the interpretation of such result. In a second step, this analytical result is extended accounting for the growth of the supercritical cluster to directly measurable sizes. Finally, an analytical foundation of the method of determination of the critical supersaturation as employed by Kozisek et al. is developed and the results obtained via the computation and analysis of steady-state cluster size distributions and calculation of mean expectation times for formation of the first supercritical clusters are compared. Some further general problems of nucleation and growth in finite closed systems are discussed, in addition