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 $\ell_m(\delta\mu)$ of the drop depends on the stripe width $L$ and the chemical potential departure from saturation $\delta\mu$ where it adopts the value $\ell_0=\ell_m(0)$. Assuming a long-range potential of van der Waals type exerted by the stripe, the interfacial Hamiltonian model is used to show that $\ell_0$ is approached linearly with $\delta\mu$ with a slope which scales as $L^2$ over the region satisfying $L\lesssim \xi_\parallel$, where $\xi_\parallel$ is the parallel correlation function pertinent to the stripe. This suggests that near the saturation there exists a universal curve $\ell_m(\delta\mu)$ 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 $\ell_m\sim\delta\mu^{-1/3}$ for $L\to\infty$ and that $\ell_m$ obeys the correct $\delta\mu\to0$ behaviour 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 $\delta\mu$ and $L$.