Marangoni Retardation of the Terminal Velocity of a Settling Droplet: The Role of Surfactant Physico-Chemistry

Abstract

Nonlinear adsorption models accounting formonolayer saturationandnonideal surfactant interactionsare used to find the terminal velocityU′ of a droplet settling through a surfactant solution. Most prior research uses a linear adsorption model which cannot capture these effects. The solution concentration[formula]is assumed to be large enough for the surfactant mass transfer to be adsorption-controlled. The Langmuir model accounts for monolayer saturation by incorporating an upper bound for the surface concentration,[formula]Two competing effects result which alterU′ from that predicted by the linear model. For slow adsorption–desorption kinetics, strong Marangoni stresses develop when the surface concentration at the rear pole approaches[formula]. These stresses favorstrong retardationinU′. The adsorption flux is proportional to the unoccupied space on the interface, so depleted regions are supplied more rapidly. This diminishes Marangoni stresses and favorsweak retardation.This effect dominates for rapid sorption kinetics. The Frumkin framework incorporates monolayer saturation and nonideal surfactant interactions which alter the amount of adsorbed surfactant, the sensitivity of the surface tension, and the dynamics of adsorptive–desorptive exchange. For a fixed mass of adsorbed surfactant,U′ retarded for no interactions is increased by intersurfactant repulsion and decreased for cohesion. At elevated[formula]U′ asymptotes to a value less than the Hadamard–Rybczynski velocity[formula]for the Langmuir case and for cohesive interactions. For repulsive interactions,U′ approaches[formula]in this limit. These asymptotes indicate the degree of surface remobilization attainable for finite adsorption–desorption kinetics and nonideal interactions.