The thickening effect of interfacial surfactant in the drag-out coating problem

Abstract

In experiments involving dip coating flows on an infinite flat substrate which iswithdrawn from an infinite liquid bath, the thin film deposited far up on theplate usually thickens in the presence of insoluble interfacial surfactant. Usingperturbation analysis within the lubrication approximation we prove that the filmthickens in the presence of interfacial surfactant for low capillary numbers if surfacetension away from the transition and meniscus regions increases in the direction ofwithdrawal of the plate, a condition that should truly emerge from the solutionof the full problem. Thus, we essentially show that fine scale properties of theinterfacial dynamics and the dynamics in the bulk of the fluid near the transitionand meniscus regions are, in fact, not important. We show that it is only thesurface tension gradient far away from the transition and meniscus regions thatmatters. This result is arrived at by first deriving upper and lower bounds onthe film thickness in terms of Marangoni and capillary numbers. An estimatebased on these results and interfacial surfactant dynamics also yields a qualitativeprofile of the interfacial surfactant concentration that results in an increase in filmthickness.