Surfactant-induced fingering phenomena beyond the critical micelle concentration

Abstract

We consider the spreading of a droplet of soluble surfactant, at concentrations beyond its critical micelle concentration (CMC), on a pre-existing thin liquid layer. Lubrication theory is used to derive a coupled system of four two-dimensional nonlinear evolution equations for the film thickness and surfactant concentration at the interface and in the bulk as both monomers and micelles. These equations are parameterized by a number of dimensionless groups that reflect the relative importance of Marangoni stresses, surface and bulk diffusion, capillarity, surfactant solubility, sorption kinetics and the nonlinearity of the equation of state. Our results for the base state indicate that two parameters in particular exert a significant influence on the flow profiles: the dimensionless mass of surfactant deposited, $M$, and a parameter reflecting the preference of the surfactant to form micelles, $R$. For a fixed value of $R$, increasing $M$ leads to the development of a protuberance that appears at the edge of the drop; upon increasing $M$ further, this protuberance separates from the drop to form a distinct secondary front that lies behind a leading front that usually accompanies the spreading process. Our examination of the linear and nonlinear stability of the system through a transient growth analysis and transient numerical simulations, respectively, indicates that these features are vulnerable to transverse perturbations, leading to the formation of fingers. The results obtained in the present work are in qualitative agreement with recently available experimental data.