Surface phase separation in a confined space

Biomolecular condensates frequently form near surfaces, including lipid membranes. However, the effect of a surface on phase separation within a cell, a confined system, remains elusive. In this work, we study surface phase separation under the constraint of a finite number of molecules in a confined space and identify three wetting states: thin, intermediate, and thick. Notably, the intermediate state, which is unstable in a fixed-chemical-potential system, can be stabilized in confinement below a critical width. Intriguingly, in wider systems, the intermediate solution becomes unstable due to a negative relationship between the chemical potential and the particle number, resulting in partial-wetting or prewetting droplets. Importantly, starting from a homogeneous surface, droplets can form only when the average volume fraction reaches the threshold of the intermediate state. Our theory elucidates the kinetic pathways of droplet formation in a confined system, providing crucial insights into how biomolecular condensates near surfaces respond to cellular events, such as the synthesis of biomolecules.