Direct Numerical Simulation of Wetting and Spreading Behavior on Heterogeneous and Roughened Substrates

A method of calculation is presented that allows the simulation of the time-dependent three-dimensional motion of thin liquid layers on solid substrates for systems with finite equilibrium contact angles. The contact angle is a prescribed function of position on the substrate. Similar mathematical models are constructed for substrates with a pattern of roughness. Evolution equations are given, using the lubrication approximation, that include viscous, capillary and disjoining forces. Motion to and from dry substrate regions is made possible by use of a thin energetically-stable wetting layer. We simulate motion on heterogeneous substrates with periodic arrays of high contact-angle patches. Two different problems are treated for heterogenous substrates. The first is spontaneous motion driven only by wetting forces. If the contact-angle difference is sufficiently high, the droplet can find several different stable positions, depending on the previous history of the motion. A second simulation treats a forced cyclical motion. Energy dissipation per cycle for a heterogeneous substrate is found to be larger than for a uniform substrate with the same total energy. The Landau-Levich solution for plate removal from a liquid bath is extended to account for a pattern of roughness on the plate.