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Supersaturation and nucleus seed formation are universal processes that precede all phase transitions. Despite extensive research on nucleation, thermodynamic equations governing supersaturation, formation of nucleus seeds and their crystallization have long remained missing. Here, we present exact statistical thermodynamic formula for the saturation degree, the most-probable size distribution of mesoscopic nuclei, and their crystallization, introducing mesoscopic state defined by temperature, the total monomer concentration, and the largest cluster size (LCS). Our results show how supersaturation emerges for mesoscopic nuclei systems and the supersaturation degree decreases with the LCS. The size-distribution of nucleus seeds is either a unimodal or a monotonically decreasing function of size, depending on the system and temperature. For nucleus seeds with a unimodal size distribution, their most probable size increases with temperature. We discover a critical supersaturation condition under which nucleus seeds undergo a phase transition, characterized by an abrupt increase in the most probable size. We also investigated nonequilibrium dynamics of supersaturation, nucleus seed formation, and phase transition, accounting for the monomer-supply-rate effects. Nucleus seeds attain either nonequilibrium-steady-state (NESS) or transient-oscillatory-state (TOS) depending on the monomer-supply-rate, before undergoing phase transition. In both NESS and TOS, nuclei assume a quasi-equilibrium size distribution. During crystallization, the survival probability of nuclei in solution phase exhibits a power-law relaxation regardless of the monomer-supply-rate. This work can be extended to investigate diverse nucleation and phase transition phenomena prevalent across nature and industry.
Song et al. (Mon,) studied this question.