Diffusion-limited osmotic swelling of droplets

Abstract

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.

Particle capture by expanding droplets: effects of inner diffusion

“…For high Péclet numbers, the absolute value of can be very large. In fact, the second derivative of the short-time expansion in our earlier work (Roure & Davis 2021 a ) is singular at , indicating that although the initial value of the collision efficiency is not affected by the Péclet number, there is a sharp decay of for at very short times (physically, this decay occurs because of the sharp reduction in salt concentration and, hence, osmotic expansion rate when diffusion of salt from the drop interior to its inner edge is slow). For , we can use this short-time expansion to determine the critical value of engulfment, Eg, for which the particle capture at short times will transition from flow dominated to engulfment dominated.…”

“…Davis & Zinchenko (2018) estimated values in the ranges of to for small drops and particles with reduced radii of 1–100 m in water, whereas even smaller values may be estimated from experiments with larger (millimetre-sized) agglomerates (DeIuliis et al. 2021; Roure & Davis 2021 a ). These same experiments yielded , but smaller values are expected for smaller drops (Roure & Davis 2021 a ).…”

“…2021; Roure & Davis 2021 a ). These same experiments yielded , but smaller values are expected for smaller drops (Roure & Davis 2021 a ). The engulfment parameter can be recast as and so can take on a wide range of values of or less.…”

“…The initial and boundary conditions presented in respectively correspond to the initial drop radius, initially uniform salt concentration inside the droplet, axisymmetry of the problem, and the osmotic water permeation through the interface (Roure & Davis 2021 a ). Note that the salt concentration is made dimensionless by the initial uniform concentration .…”

“…However, in most cases, the influx of fresh water and the non-instantaneous diffusion of salt from the drop centre to its interface diminish the concentration of salt at the inner drop interface, slowing down subsequent osmosis and drop expansion considerably. In recent works, this interplay between diffusion and expansion was investigated theoretically and experimentally (DeIuliis et al 2021(DeIuliis et al , 2022Roure & Davis 2021a). Roure & Davis (2021a) modelled the drop expansion as a diffusion problem with nonlinear, moving boundary conditions similar to those found in Stefan problems with kinetic undercooling (Evans & King 2000;McCue et al 2011;Back et al 2014).…”