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.