Modeling the capture of fine particles by swelling emulsions

Fast agglomeration by emulsion binders to capture fine, hydrophobic particles was recently developed as an alternative to froth flotation. It involves mixing a suspension of particles in water with a saltwater-in-oil emulsion. The droplets in the emulsion phase swell due to an osmotic flux caused by the presence of salt. In this work, we developed a framework to model this particle-capture method, consisting of (i) using mobility dynamics to investigate the binary interactions between swelling droplets and particles, (ii) an extension of classical collision theory to moving interfaces, which we use to calculate the capture rate of fine particles, (iii) a continuous diffusion-limited swelling model for a single droplet, and (iv) a permeation network model for the swelling of a droplet agglomerate, for which the mass transfer is limited by the permeation of water within the internal oil layers. Our results indicate that drop swelling substantially influences particle capture, even for low swelling rates and high Péclet numbers, where the swelling of a single droplet is governed by a concentration boundary layer and displays a universal behavior at short times. For droplet agglomerates, a similar trend for swelling is observed, and structure-dependent effective diffusivities and Péclet numbers can be determined by matching the results with the effective-diffusion model.