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

In experiments involving dip coating flows on an infinite flat substrate which is withdrawn from an infinite liquid bath, the thin film deposited far up on the plate usually thickens in the presence of insoluble interfacial surfactant. Using perturbation analysis within the lubrication approximation we prove that the film thickens in the presence of interfacial surfactant for low capillary numbers if surface tension away from the transition and meniscus regions increases in the direction of withdrawal of the plate, a condition that should truly emerge from the solution of the full problem. Thus, we essentially show that fine scale properties of the interfacial dynamics and the dynamics in the bulk of the fluid near the transition and meniscus regions are, in fact, not important. We show that it is only the surface tension gradient far away from the transition and meniscus regions that matters. This result is arrived at by first deriving upper and lower bounds on the film thickness in terms of Marangoni and capillary numbers. An estimate based on these results and interfacial surfactant dynamics also yields a qualitative profile of the interfacial surfactant concentration that results in an increase in film thickness. (letter)