Onsager relation for electrokinetics at surfactant-covered bubble film

We analytically describe the slip length of the surfactant-covered bubble film under the joint actions of pressure gradient and electric field. Considering the Marangoni effect, the slip length and consequent zeta potential of the liquid‒vapor interface significantly reduced compared to the Marangoni-free interface at low surfactant concentrations, due to the surfactant accumulation at downstream of the bubble liquid film. In addition, we discovered that the friction coefficient of the liquid‒vapor interface becomes field dependent in a regime of strong coupling among volume flow, surfactant transport, and ionic current at the liquid‒vapor interface. We use the Onsager reciprocal relationship to describe the electrokinetic effects within a bubble film, including flow velocity, ionic current, and surfactant transport, which can describe the Marangoni effects while considering multi-physical effects.