Electrophoresis of an Oil Drop in a Charged Polymer Gel Medium: Coupled Effects of Drop Electrohydrodynamics and Gel Electroosmosis

We develop a theoretical description of the electrophoretic migration of a weakly charged oil drop dispersed in a dilute polymer gel carrying fixed charges and saturated with an aqueous electrolyte solution. In contrast to neutral gels, a charged polymer network generates electroosmotic flow under an applied electric field, which couples with the electrohydrodynamic motion of the drop. The observed electrophoretic velocity therefore reflects the combined effects of drop-induced flow and gel-driven electroosmosis. On the basis of the Baygents-Saville theory, the drop surface charge is assumed to originate from specific ion adsorption at the oil-water interface, while no mobile ions are present inside the drop. Working within the Brinkman-Debye-Bueche porous-medium model for the gel and employing a linearized treatment valid for low zeta potential, we obtain a simple analytical expression for the electrophoretic mobility. The formulation consistently incorporates Marangoni stresses arising from spatial variations in interfacial tension, and hydrodynamic slip at the oil-water interface, which can be significant for hydrophobic drops in aqueous media. The resulting mobility expression explicitly separates the contribution associated with the intrinsic electrohydrodynamic response of the drop from that due to electroosmosis of the charged gel matrix. This analytical form enables experimental mobility data to be used not only to estimate the zeta potential of the drop but also to evaluate the electroosmotic mobility of the polymer gel medium. The present theory thus provides a physically transparent and experimentally useful framework for characterizing electrokinetic transport in charged soft porous media.