Leveling of a model paint film with a yield stress

The coating dynamics of a drying paint film with a yield stress is studied. The liquid is modeled as a binary mixture with one volatile component, solvent, and one nonvolatile component, resin. When the solvent has a different surface tension than the resin, solvent evaporation can lead to the creation of surface tension gradients which can potentially overcome the yield stress and dramatically affect the flow history. Using the lubrication approximations to derive the flux of the liquid film parallel to the substrate, we find that the presence of the yield stress causes several distinct flow regimes. The total flux of each of these regimes is summed, and using the continuity equation we derive an evolution equation giving the height of the free surface as a function of the distance along the substrate and time. The resulting equations are discretized and solved numerically using finite differences. High order derivative is treated implicitly, allowing for large time steps and reducing the computational requirements. We find that the presence of a yield stress greatly affects the leveling behavior of the coating. Critical yield stresses exist that can cause maximal leveling of the coating film.