Modeling the corrugation of the three-phase contact line perpendicular to a chemically striped substrate

We model an infinitely long liquid bridge confined between two plates chemically patterned by stripes of the same width and different contact angle, where the three-phase contact line runs, on average, perpendicular to the stripes. This allows us to study the corrugation of a contact line in the absence of pinning. We find that, if the spacing between the plates is large compared to the length scale of the surface patterning, the cosine of the macroscopic contact angle corresponds to an average of cosines of the intrinsic angles of the stripes, as predicted by the Cassie equation. If, however, the spacing becomes on the order of the length scale of the pattern, there is a sharp crossover to a regime where the macroscopic contact angle varies between the intrinsic contact angle of each stripe, as predicted by the local Young equation. The results are obtained using two numerical methods, lattice Boltzmann (a diffuse interface approach) and Surface Evolver (a sharp interface approach), thus giving a direct comparison of two popular numerical approaches to calculating drop shapes when applied to a nontrivial contact line problem. We find that the two methods give consistent results if we take into account a line tension in the free energy. In the lattice Boltzmann approach, the line tension arises from discretization effects at the diffuse three phase contact line.