We have recently shown that a wire, and more generally a solid, drawn out of a solution containing surfactants entrains a thicker film than one drawn out of a pure liquid (Marangoni coating). The thickening factor, generally of order 2, was found to be independent of the capillary number (for surfactant concentration of the order of the cmc) but depends on the wire radius : the thinner the wire, the larger the thickening. After summarizing these results, we propose a model for understanding this effect, which is shown to result from a balance between convection and adsorption. Finally, a scaling form is proposed for the thickening factor.