Enhanced dip-coating on a soft substrate

A solid, withdrawn from a liquid bath, entrains a thin liquid film. This simple process, initially discussed by Landau and Levich, can be observed in everyday's life and is widely used in industry for surface coating purposes. Nevertheless, the height of fluid entrained can be altered by various properties of the withdrawn solid. Here, we develop a novel model that includes a thin coating on a plate. For simplicity, a Winkler's foundation is assumed for the elastic response. A new power-law regime is found for the height of fluid entrained $h \sim \sqrt{l_{cap} l_{ec}}Ca^{1/2}$ (where $Ca$ denotes the capillary number, $l_{cap}$ the capillary length and $l_{ec}$ an elastocapillary length) at small capillary numbers. At large $Ca$, the Landau-Levich scaling laws $h \sim l_{cap} Ca^{2/3}$ is recovered. The transition occurs when the substrate's deformation becomes comparable to the thickness of the entrained film. Finally, using this model, we discuss recent experimental results.