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.

Droplet behavior in the presence of insoluble surfactants

“…This is in correspondence with the results obtained in ͑sub-͒critical flow by Hu and Lips, 8 Janssen et al, 10,13 Eggleton et al, 12 and Velankar et al 14 It remains to be seen, however, whether this should be attributed to the change in ‫*⌫ץ/*ץ‬ around the reference concentration, or due to the proximity of ⌫ ϱ * . Therefore, these cases will be considered separately.…”

“…In general, the influence of the Péclet number on the droplet deformation in supercritical flows ͑Caϭ0.1͒ was found to be similar to its influence on the droplet deformation in subcritical flows. Following Eggleton et al, 12 we investigated the influence of the surfactant coverage on the deformation of a droplet covered with a surfactant described by the Langmuir equation of state with ␤ϭ0.2. Similar to the results obtained by Eggleton et al 12 in subcritical flow, the droplet deformation was found to increase with increasing surfactant coverage ⌫ r /⌫ ϱ for low and moderate ⌫ r /⌫ ϱ .…”

“…4,[9][10][11] Whether the upper bound ⌫ ϱ is reached, depends on the surfactant coverage ⌫/⌫ ϱ and the Péclet number Pe, which is the ratio between convection and diffusion along the droplet interface. 12 For low surfactant coverage, the upper bound is not reached by far, and the droplet deformation will be higher than in the case of a clean droplet. For high surfactant coverage, the relation between the interfacial tension coefficient and the surfactant concentration becomes very strong.…”

“…For simplicity reasons, only equal viscosity ratio systems are considered. The surfactant is assumed to be insoluble, which is a good approximation if the bulk concentration is dilute 12 or if the migration to and from the bulk is slow. 18 In fact, the study by Milliken and Leal 9 shows that in almost all cases, the droplet deformation and breakup in the presence of soluble surfactants fall between the results for insoluble surfactant and those for a clean interface.…”

Microconfined equiviscous droplet deformation: Comparison of experimental and numerical results

“…In this limiting case, the interface velocity can be calculated with a boundary integral method. 27,28 Boundary integral methods have been used successfully to study drop dynamics in bulk flow conditions [29][30][31][32][33] or the movement of solid spheres and more recently drops between parallel walls. [34][35][36] In the boundary integral formulation, the velocity u at a point x 0 on the interface is given by…”