Equilibrium electro-deformation of a surfactant-laden viscous drop

We theoretically investigate the deformation of a viscous drop covered with non-diffusing insoluble surfactant under a uniform DC electric field. At equilibrium, surfactant immobilizes the spheroidal drop surface and completely suppresses the fluid flow. In this work we focus on the equilibrium electro-deformation of a surfactant-laden drop in the leaky dielectric framework by developing (1) a second-order small-deformation analysis and (2) a spheroidal model for a highly deformed (prolate or oblate) drop. Both models are compared against experimental data and numerical simulation results in the literature. Our analysis shows how the existence of equilibrium spheroidal drop depends on the permittivity ratio, conductivity ratio, surfactant coverage, and the elasticity number. Furthermore, the spheroidal model highlights that differences between surfactant effects, such as tip stretching and surface dilution effects, are greatly amplified at large surfactant coverage and high electric capillary number. These surfactant effects are well captured in the spheroidal model, but cannot be described in the second-order small-deformation theory.