  Using molecular simulation, we study the nucleation of liquid droplets from
binary mixtures and determine the free energy of nucleation along entropic
pathways. To this aim, we develop the $\mu_1 \mu_2 VT-S$ method, based on the
grand-canonical ensemble modeling the binary mixture, and use the entropy of
the system $S$ as the reaction coordinate to drive the formation of the liquid
droplet. This approach builds on the advantages of the grand-canonical
ensemble, which allows for the direct calculation of the entropy of the system
and lets the composition of the system free to vary throughout the nucleation
process. Starting from a metastable supersaturated vapor, we are able to form a
liquid droplet by gradually decreasing the value of $S$, through a series of
umbrella sampling simulations, until a liquid droplet of a critical size has
formed. The $\mu_1 \mu_2 VT-S$ method also allows us to calculate the free
energy barrier associated with the nucleation process, to shed light on the
relation between supersaturation and free energy of nucleation, and to analyze
the interplay between the size of the droplet and its composition during the
nucleation process.
