  In living cells, protein-rich condensates can wet the cell membrane and
surfaces of membrane-bound organelles. Interestingly, many phase-separating
proteins also bind to membranes leading to a molecular layer of bound
molecules. Here we investigate how binding to membranes affects wetting,
prewetting and surface phase transitions. We derive a thermodynamic theory for
a three-dimensional bulk in the presence of a two-dimensional, flat membrane.
At phase coexistence, we find that membrane binding facilitates complete
wetting and thus lowers the wetting angle. Moreover, below the saturation
concentration, binding facilitates the formation of a thick layer at the
membrane and thereby shifts the prewetting phase transition far below the
saturation concentration. The distinction between bound and unbound molecules
near the surface leads to a large variety of surface states {and complex
surface phase diagrams with a rich topology of phase transitions. Our work
suggests that surface phase transitions combined with molecular binding
represent a versatile mechanism to control the formation of protein-rich
domains at intra-cellular surfaces.
