Film drainage between two surfactant-coated drops colliding at constant approach velocity

Abstract

The drainage of the intervening continuous phase film between two drops approaching each other at constant velocity under the influence of insoluble surfactant is investigated. The mathematical model to be solved is a coupled pair of fourth-order nonlinear partial differential equations which arise from the relationships governing the evolution of the film thickness and the surfactant interfacial concentration in the lubrication approximation. We adopt a simplified approach which uses lubrication theory to describe the flow within the drop, marking a departure from the conventional framework in which Stokes flow is assumed. When the model is solved numerically together with the relevant initial and boundary conditions, the results obtained are compared with those found in the literature using the “boundary integral” method to solve for the flow in the drop phase. The close agreement between the results inspires confidence in the predictions of the simplified approach adopted. The analysis on the effect of insoluble surfactant indicates that its presence retards the drainage of the film: The fully immobile interface limit is recovered even in the presence of a small amount of surfactant above a critical concentration; film rupture is either prolonged or prevented. The retardation of the film was attributed to gradients of interfacial tension which gave rise to the Marangoni effect. A study of the influence of various system parameters on the drainage dynamics was conducted and three regimes of drainage and possible rupture were identified depending on the relative magnitudes of the drop approach velocity and the van der Waals interaction force: Nose rupture, rim rupture, and film immobilization and flattening. Finally, the possibility of forming secondary droplets by encapsulating the continuous phase film into the coalesced drop at rupture was examined and quantified in light of these regimes.