Surfactant-induced retardation of the thermocapillary migration of a droplet

Abstract

A neutrally buoyant droplet in a fluid possessing a temperature gradient migrates under the action of thermocapillarity. The drop pole in the high-temperature region has a reduced surface tension. The surface pulls away from this low-tension region, establishing a Marangoni stress which propels the droplet into the warmer fluid. Thermocapillary migration is retarded by the adsorption of surfactant: surfactant is swept to the trailing pole by surface convection, establishing a surfactant-induced Marangoni stress resisting the flow (Barton & Subramanian 1990).The impact of surfactant adsorption on drop thermocapillary motion is studied for two nonlinear adsorption frameworks in the sorption-controlled limit. The Langmuir adsorption framework accounts for the maximum surface concentration Γ′∞ that can be attained for monolayer adsorption; the Frumkin adsorption framework accounts for Γ′∞ and for non-ideal surfactant interactions. The compositional dependence of the surface tension alters both the thermocapillary stress which drives the flow and the surfactant-induced Marangoni stress which retards it. The competition between these stresses determines the terminal velocity U′, which is given by Young's velocity U′0 in the absence of surfactant adsorption. In the regime where adsorption–desorption and surface convection are of the same order, U′ initially decreases with surfactant concentration for the Langmuir model. A minimum is then attained, and U′ subsequently increases slightly with bulk concentration, but remains significantly less than U′0. For cohesive interactions in the Frumkin model, U′ decreases monotonically with surfactant concentration, asymptoting to a value less than the Langmuir velocity. For repulsive interactions, U′ is non-monotonic, initially decreasing with concentration, subsequently increasing for elevated concentrations. The implications of these results for using surfactants to control surface mobilities in thermocapillary migration are discussed.