Abstract
Supersaturation, nucleation, and phase separation are ubiquitous phenomena of great interest in both science and industry. However, a unified, quantitative understanding of these phenomena has yet to be achieved for mesoscopic systems. Here, we present a set of general equations that determine the monomer saturation degree, the size distribution, and the free energy of mesoscopic systems, as well as their phase-transition conditions. These equations reveal that, under supersaturation, the largest cluster size (LCS) is an important state variable; the supersaturation degree decreases with the LCS, approaching unity in the macroscopic limit. We identify the critical supersaturation, at which the nuclei undergo the phase transition to form large crystals. Below this critical supersaturation, the nucleus size distribution is either a unimodal function or a monotonically decreasing function of size, depending on the system and temperature. We also predict the most probable nucleus size and the direction of spontaneous changes of the LCS. Our theory provides a unified, quantitative explanation of the nucleus-size-distribution across six different systems, including nanoparticles and biological condensates. This work serves as a general theoretical framework useful for understanding and designing nucleation and phase transitions of mesoscopic systems.