Diffusion-limited osmotic swelling of droplets

Swelling phenomena due to permeation appear in problems, such as the swelling of hydrogels and water-in-oil-in-water (W/O/W) emulsions. In the osmotic swelling of W/O/W emulsions driven by an inner salt concentration, diffusive effects inside the drop can decrease its expansion rate considerably. Although these inner-diffusion effects can play a large role on hindering drop swelling, they have not usually been taken into account in most works concerning the swelling kinetics of W/O/W emulsions. We perform numerical simulations of the expansion-diffusion problem governing the diffusion inside an expanding spherical droplet containing salt and with a semi-permeable interface. We also present asymptotic solutions for the limiting cases of slow and fast diffusion, which we compare with our numerical results. The results indicate that diffusive resistance significantly reduces the swelling kinetics of droplets. Moreover, in the regime of large Péclet numbers, diffusive effects are localized near the drop's interface in a concentration boundary layer, as predicted by our theory. The numerical results presented in this paper are in agreement with the behavior observed in recent experiments on W/O/W emulsion swelling.