Fingering phenomena created by a soluble surfactant deposition on a thin liquid film

A striking fingering instability accompanies surfactant droplet deposition upon thin films. The fingers apparently emerge from the droplet and are preceded by a circular rim of thickened fluid. In this work, we present a theory, based on the lubrication approximation, capable of explaining the physical mechanism responsible for these phenomena and the trends observed experimentally. The model comprises evolution equations for the film thickness and surfactant interfacial and bulk concentrations, which are parametrized by appropriately defined solubility and sorption kinetics parameters, and surface and bulk Peclet numbers; closure is achieved via use of a nonlinear equation of state. The results of our transient growth analysis and two-dimensional numerical simulations illustrate the delicate interplay, which exists between surfactant solubility and the nonlinearity in the equation of state. In particular, we show that solubility acts to destabilize the spreading process for a given value of the nonlinearity except for very highly soluble surfactants. In this case, the magnitude of the Marangoni stresses available to drive spreading and instability diminishes leading to flow stability.