  We study the spontaneous charging and the crystallization of spherical
micron-sized water-droplets dispersed in oil by numerically solving, within a
Poisson-Boltzmann theory in the geometry of a spherical cell, for the density
profiles of the cations and anions in the system. We take into account
screening, ionic Born self-energy differences between oil and water, and
partitioning of ions over the two media. We find that the surface charge
density of the droplet as induced by the ion partitioning is significantly
affected by the droplet curvature and by the finite density of the droplets. We
also find that the salt concentration and the dielectric constant regime in
which crystallization of the water droplets is predicted is enhanced
substantially compared to results based on the planar oil-water interface,
thereby improving quantitative agreement with recent experiments.
