Supersaturation, Nucleation, and Phase Separation of Mesoscopic Materials

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 materials. Here, we present a set of general equations that determine the monomer saturation degree, the size distribution and free energy of mesoscopic materials, 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 condition, above 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 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 will serve as a general theoretical framework for understanding, predicting, and designing nucleation and phase transitions in mesoscopic material systems.