Gel Electrophoresis of an Oil Drop

We present a theoretical model for the electrophoresis of a weakly charged oil drop migrating through an uncharged polymer gel medium saturated with an aqueous electrolyte solution. The surface charge of the drop arises from the specific adsorption of ions onto its interface. Unlike solid particles, liquid drops exhibit internal fluidity and interfacial dynamics, leading to distinct electrokinetic behavior. In this study, the drop motion is driven by long-range hydrodynamic effects from the surrounding gel, which are treated using the Debye-Bueche-Brinkman continuum framework. A simplified version of the Baygents-Saville theory is adopted, assuming that no ions are present inside the drop and that the surface charge distribution results from linear ion adsorption. An approximate analytical expression is derived for the electrophoretic mobility of the drop under the condition of low zeta potential. Importantly, the derived expression explicitly includes the Marangoni effect, which arises from spatial variations in interfacial tension due to non-uniform ion adsorption. This model provides a physically consistent and mathematically tractable basis for understanding the electrophoretic transport of oil drops in soft porous media such as hydrogels, with potential applications in microfluidics, separation processes, and biomimetic systems. These results also show that the theory could be applied to more complicated or biologically important soft materials.