Electrophoresis of droplets laden with insoluble ionic surfactants incorporating correlations among finite-sized ions

The present study deals with the electrophoresis of a non-polarizable droplet with irreversibly adsorbed ionic surfactants suspended in monovalent or multivalent electrolyte solutions. The impact of the non-uniform surface charge density, governed by the interfacial surfactant concentration, along with Marangoni, hydrodynamic and Maxwell stresses on droplet electrophoresis is analysed. At a large ionic concentration, the hydrodynamic steric interactions and correlations among finite-sized ions manifest. In this case the viscosity of the medium rises as the local volume fraction of the finite-sized ions is increased. The governing equations, incorporating these short-range effects, are solved numerically based on the regular linear perturbation analysis under a weak applied electric field consideration. We find that the electrophoretic velocity consistently decreases with an increase in the droplet-to-electrolyte viscosity ratio due to the Marangoni stress caused by inhomogeneous surfactant distribution. This monotonic relationship with droplet viscosity is absent for the case of constant surface charge density, where a low-viscosity droplet may exhibit a lower mobility than a high-viscosity droplet. In the presence of ionic surfactant, a continuous variation of mobility with surfactant concentration is found. For a monovalent electrolyte, mobility decreases significantly at an elevated ionic concentration due to the short-range effects described above. Reversal in mobility is observed in multivalent electrolytes due to the correlations among finite-sized ions, attributed to overscreening of surface charge and formation of a coion-rich layer within the electric double layer. In this case a toroidal vortex develops adjacent to the droplet and the reversed mobility enhances as the Marangoni number is increased. This mobility reversal is delayed for low-viscosity droplets.