Numerical simulation of dip-coating in the evaporative regime

A hydrodynamic model is used for numerical simulations of a polymer solution in a dip-coating-like experiment. We focus on the regime of small capillary numbers where the liquid flow is driven by evaporation, in contrast to the well-known Landau-Levich regime dominated by viscous forces. Lubrication approximation is used to describe the flow in the liquid phase. Evaporation in stagnant air is considered (diffusion-limited evaporation), which results in a coupling between liquid and gas phases. Self-patterning due to the solutal Marangoni effect is observed for some ranges of the control parameters. We first investigate the effect of evaporation rate on the deposit morphology. Then the role of the spatial variations in the evaporative flux on the wavelength and mean thickness of the dried deposit is ascertained, by comparing the 2D and 1D diffusion models for the gas phase. Finally, for the very low substrate velocities, we discuss the relative importance of diffusive and advective components of the polymer flux, and consequences on the choice of the boundary conditions.