Thin liquid films on rough or heterogeneous solids

We study the conformation of thin liquid films on rough or heterogeneous solid substrates. The liquid-substrate interaction dominates for sufficiently thin films, and heterogeneity roughens the liquid interface. As the film thickens, surface tension becomes increasingly important, and the liquid interface flattens. A general equation for the equilibrium interface shape is derived. Analytic results are obtained in the limit of weak disorder for rough or self-affine surfaces as well as chemically heterogeneous solids. The effect of disorder depends strongly on the wave vector. Fluctuations at scales smaller than the film thickness or a healing length produce little roughness. At larger wavelengths, the film conforms to the local fluctuations. Exact numerical solutions of the general equation are presented for surfaces with square grooves. These confirm the qualitative predictions of the analytic theory, and are in quantitative agreement when the depth of the grooves is small. The variation of roughness with film thickness, as well as the calculated adsorption isotherms, are compared to recent experimental results. We show that previously measured isotherms can be reproduced by corrugated surfaces with a single characteristic length scale, and do not necessarily imply that the surfaces studied were self-similar.