A cumulant expansion for the size distribution of liquid droplets during phase separation

We consider a collection of droplets growing from a supersaturated vapor or (in general fluid) solution for the time well after the nucleation. Its coarsening is driven by surface energy and leads asymptotically to a linear growth of the mean droplet volume with time (Ostwald ripening). The droplets grow either from the supersaturated uncondensed phase (coalescence) or by collisions with subsequent fusion (coagulation). We derive the evolution equation for a scaled size distribution of the droplets, which includes both mechanisms, and obtain the temporal behavior of the average size, variance and skewness by a cumulant expansion method. Since the cumulant expansion contains the boundary value at zero droplet radius, an assumption about this value must be made. In this context we use a piecewise linear approximation of the distribution. A comparison with earlier results and with the calculated asymptotic distribution shows that such an expansion in low order reproduces the main features of the system for the whole time evolution in most cases.