Thermodynamic Origin of Nuclei Seed Formation and Monomer Supply Rate-Dependent Crystallization Dynamics

Supersaturation and nucleus seed formation are universal processes that precede all phase transitions. Despite extensive research on nucleation, our understanding on supersaturation, nucleus seed formation and phase transition dynamics remains rudimentary. Here, we present exact statistical thermodynamic formula for the saturation degree, the most-probable size distribution of mesoscopic nuclei, and their phase transition, introducing the mesoscopic state defined by temperature, the total monomer concentration, and the largest cluster size (LCS). These results show that supersaturation emerges even at equilibrium for mesoscopic nuclei systems and 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. There exists a critical supersaturation condition under which nucleus seeds undergo a phase transition, during which the most probable size exhibits an abrupt change. 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; their population exhibits a universal power-law relaxation during the phase transition. This work can be extended to investigate diverse nucleation and phase transition phenomena prevalent across nature and industry.