Electrophoresis of a weakly charged oil drop in an electrolyte solution: ion adsorption and Marangoni effects

We present a simple analytic expression for the electrophoretic mobility of a weakly charged oil drop in an aqueous electrolyte solution, where the drop acquires its surface charge through ion adsorption. The derivation is based on a simplified Baygents-Saville model, in which no ions are present inside the drop. This model incorporates the Marangoni effect arising from interfacial tension gradients. The resulting analytic expression shows excellent agreement with the numerical results of Baygents and Saville for low zeta potential values, validating the accuracy of the approximation. It is found that, under the assumption of no internal ions, the electrophoretic mobility of the drop is independent of its internal dielectric permittivity. This behavior stands in contrast to the case of an oil drop with a uniform, constant surface charge density, where the mobility depends on the drop's internal dielectric permittivity. However, it is similar to the case of rigid particles, as shown by O’Brien and White. Furthermore, it is found that in the analytic expression for the electrophoretic mobility of a mercury drop, the leading term proportional to κa (where κ is the Debye–Hückel parameter and a is the drop radius) completely cancels out for an oil drop due to the tangential Maxwell stress and the Marangoni effect—both of which are absent in the case of a mercury drop. As a result, the electrophoretic mobility of an oil drop does not increase linearly with increasing κa when plotted at a fixed zeta potential.

Graphical abstract