Filling and wetting transitions on sinusoidal substrates: a mean-field study of the Landau-Ginzburg model

We study the interfacial phenomenology of a fluid in contact with a microstructured substrate within the mean-field approximation. The sculpted substrate is a one-dimensional array of infinitely long grooves of sinusoidal section of periodicity length L and amplitude A. The system is modelled using the Landau-Ginzburg functional, with fluid-substrate couplings which correspond to either first-order or critical wetting for a flat substrate. We investigate the effect of the roughness of the substrate in the interfacial phenomenology, paying special attention to filling and wetting phenomena, and compare the results with the predictions of the macroscopic and interfacial Hamiltonian theories. At bulk coexistence, for values of L much larger than the bulk correlation, we observe first-order filling transitions between dry and partially filled interfacial states, which extend off-coexistence, ending at a critical point; and wetting transitions between partially filled and completely wet interfacial states with the same order as for the flat substrate (if first-order, wetting extends off-coexistence in a prewetting line). On the other hand, if the groove height is of order of the correlation length, only wetting transitions between dry and complete wet states are observed. However, their characteristics depend on the order of the wetting transition for the flat substrate. So, if it is first-order, the wetting transition temperature for the rough substrate is reduced with respect to the wetting transition temperature for a flat substrate, and coincides with the Wenzel law prediction for very shallow substrates. On the contrary, if the flat substrate wetting transition is continuous, the roughness does not change the wetting temperature.