Height of a liquid drop on a wetting stripe

Adsorption of liquid on a planar wall decorated by a hydrophilic stripe of width L is considered. Under the condition that the wall is only partially wet (or dry) while the stripe tends to be wet completely, a liquid drop is formed above the stripe. The maximum height ₌ () of the drop depends on the stripe width L and the chemical potential departure from saturation where it adopts the value ₀=₌ (0). Assuming a long-range potential of van der Waals type exerted by the stripe, the interfacial Hamiltonian model is used to show that ₀ is approached linearly with with a slope which scales as L^2 over the region satisfying L_, where _ is the parallel correlation function pertinent to the stripe. This suggests that near the saturation there exists a universal curve ₌ () to which the adsorption isotherms corresponding to different values of L all collapse when appropriately rescaled. Although the series expansion based on the interfacial Hamiltonian model can be formed by considering higher order terms, a more appropriate approximation in the form of a rational function based on scaling arguments is proposed. The approximation is based on exact asymptotic results, namely, that ₌^-1/3 for L and that ₌ obeys the correct 0 behavior in line with the results of the interfacial Hamiltonian model. All the predictions are verified by the comparison with a microscopic density functional theory (DFT) and, in particular, the rational function approximation---even in its simplest form---is shown to be in a very reasonable agreement with DFT for a broad range of both and L.