Insoluble surfactants on a drop in an extensional flow: a generalization of the stagnated surface limit to deforming interfaces

A drop in an axisymmetric extensional flow is studied using boundary integr al methods to understand the effects of a monolayer-forming surfactant on a strongly deforming interface. Surfactants occupy area, so there is an uppe r bound to the surface concentration that can be adsorbed in a monolayer, G amma(infinity). The surface tension is a highly nonlinear function of the s urface concentration Gamma because of this upper bound. As a result, the me chanical response of the system varies strongly with Gamma for realistic ma terial parameters. In this work, an insoluble surfactant is considered in t he limit where the drop and external fluid viscosities are equal. For Gamma much less than Gamma(infinity), surface convection sweeps surfact ant toward the drop poles. When surface diffusion is negligible, once the s table drop shapes are attained, the interface can be divided into stagnant caps near the drop poles, where Gamma is non-zero, and tangentially mobile regions near the drop equator, where the surface concentration is zero. Thi s result is general for any axisymmetric fluid particle. For Gamma near Gam ma(infinity), the stresses resisting accumulation are large in order to pre vent the local concentration from reaching the upper bound, As a result, th e surface is highly stressed tangentially while Gamma departs only slightly from a uniform distribution. For this case, Gamma is never zero, so the ta ngential surface velocity is zero for the steady drop shape. This observation that Gamma dilutes nearly uniformly for high surface conce ntrations is used to derive a simplified form for the surface mass balance that applies in the limit of high surface concentration. The balance requir es that the tangential flux should balance the local dilatation in order th at the surface concentration profile will remain spatially uniform. Through out the drop evolution, this equation yields results in agreement with the full solution for moderate deformations, and underscores the dominant mecha nism at high deformation. The simplified balance reduces to the stagnant in terface condition at steady state. Drop deformations vary non-monotonically with concentration; for Gamma much less than Gamma(infinity), the reduction of the surface tension near the p oles leads to higher deformations than the clean interface case. For Gamma near Gamma(infinity), however, Gamma dilutes nearly uniformly, resulting in higher mean surface tensions and smaller deformations. The drop contributi on to the volume averaged stress tensor is also calculated and shown to var y non-monotonically with surface concentration.