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

Abstract

A drop in an axisymmetric extensional ow is studied using boundary integral methods to understand the effects of a monolayer-forming surfactant on a strongly deforming interface. Surfactants occupy area, so there is an upper bound to the surface concentration that can be adsorbed in a monolayer, Γ∞. The surface tension is a highly nonlinear function of the surface concentration Γ because of this upper bound. As a result, the mechanical response of the system varies strongly with Γ for realistic material parameters. In this work, an insoluble surfactant is considered in the limit where the drop and external fluid viscosities are equal.For Γ<Γ∞, surface convection sweeps surfactant toward the drop poles. When surface diffusion is negligible, once the stable drop shapes are attained, the interface can be divided into stagnant caps near the drop poles, where Γ is non-zero, and tangentially mobile regions near the drop equator, where the surface concentration is zero. This result is general for any axisymmetric fluid particle. For Γ near Γ∞, the stresses resisting accumulation are large in order to prevent the local concentration from reaching the upper bound. As a result, the surface is highly stressed tangentially while Γ departs only slightly from a uniform distribution. For this case, Γ is never zero, so the tangential surface velocity is zero for the steady drop shape.This observation that Γ dilutes nearly uniformly for high surface concentrations is used to derive a simplified form for the surface mass balance that applies in the limit of high surface concentration. The balance requires that the tangential flux should balance the local dilatation in order that the surface concentration profile will remain spatially uniform. Throughout the drop evolution, this equation yields results in agreement with the full solution for moderate deformations, and underscores the dominant mechanism at high deformation. The simplified balance reduces to the stagnant interface condition at steady state.Drop deformations vary non-monotonically with concentration; for Γ<Γ∞, the reduction of the surface tension near the poles leads to higher deformations than the clean interface case. For Γ near Γ∞, however, Γ dilutes nearly uniformly, resulting in higher mean surface tensions and smaller deformations. The drop contribution to the volume averaged stress tensor is also calculated and shown to vary non-monotonically with surface concentration.