Height of a liquid drop on a wetting stripe

Abstract

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 $\delta \mu \to 0$ 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 $L$.

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