Effect of interfacial kinetics on the settling of a drop in a viscous medium

Multiphase emulsions, such as drops in a continuous medium, tend to have surfactant-like impurities present at the interfaces, either naturally or introduced artificially for stability, which may influence the flow field and, hence, alter the motion of the drops through a host of different mechanisms. Here, we carry out a robust analysis to characterize multiple aspects of such interfacial phenomena by studying the settling of a drop in a quiescent viscous medium. The surface active agents are assumed to be bulk-insoluble and non-ideal, while the interface itself is assumed to have its own rheology, described by the Boussinesq-Scriven model. The diffusive fluxes of the surfactants are expressed in a thermodynamically consistent manner as proportional to the chemical potential gradient, which results in concentration dependent diffusivity. We subsequently derive semi-analytical solutions for approximately spherical drops without any other restrictions on the transport processes. Our results reveal that stresses originating from interfacial rheology tend to decrease the settling velocity and at the same time make the surfactant concentration uniform across the surface. Remarkably, this settling velocity is revealed to be independent of the choice of the free-energy isotherms and the extent of packing of the surfactants when a variable diffusivity is correctly accounted for. These insights will be helpful in better understanding of the underlying dynamics of surfactant-laden drops, having potential applications in microfluidic devices, food and pharmaceutical industries, and separation processes.