Complete Wetting and Drying at Sinusoidal Walls

TL;DR

This work analyzes complete wetting and drying on sinusoidally corrugated walls, contrasting SR drying on a hard wall with LR wetting on a wall with dispersion forces. It combines a nonlocal interfacial Hamiltonian theory (for SR) and a sharp-kink approximation (for LR) with fully microscopic DFT to derive and test scaling laws for the mean interfacial height $\ell$ and corrugation $\epsilon$ as $\delta\mu\to 0$. Key findings include a quadratic-to-linear crossover in $\delta\ell$ with wall amplitude for SR, a logarithmic correction at larger corrugation, $\delta\ell\sim\delta\mu^{1/3}$ with $\epsilon\sim\delta\mu^{4/3}$ for LR, and linear dependencies on wall amplitude $A$; all predictions are in strong agreement with DFT. The results reveal how wall geometry and interaction range jointly shape interfacial unbinding, with LR systems showing universal leading-order scaling and SR systems displaying geometry-driven crossovers, validated here by microscopic simulations and offering a basis for exploring more complex surfaces and fluctuations.

Abstract

We investigate complete wetting and drying at sinusoidally corrugated solid walls, focusing on the effects of wall geometry and interaction range. Two distinct interaction models are considered: one incorporating only short-ranged (SR) forces (applied to drying), and another including long-ranged (LR) van der Waals interactions (applied to wetting). The SR model is analyzed within the framework of nonlocal Hamiltonian theory by Parry et al., while the LR model is treated using a sharp-kink approximation. In both cases, we derive scaling relations that describe the dependence of the adsorbed layer's width and morphology on the wall's geometric parameters as the system approaches two-phase coexistence. We identify distinct scaling regimes determined by the degree of wall corrugation and highlight the contrasting effects of SR and LR interactions. Theoretical predictions are corroborated by numerical results from classical density functional theory.

Figures (15)

• Figure 1: A schematic of our model showing a wall with a local height profile $\psi(x)$ (relative to the horizontal, indicated by the thick solid line), which varies smoothly with period $L$ and has amplitude $A$. An adsorbed wetting layer of local height $\ell(x)$ is also depicted. The mean height of the layer above the wall is $\ell$, as defined in Eq. (\ref{['ell_dft']}), and its corrugation is characterized by the parameter $\epsilon$, defined in Eq. (\ref{['eps_dft']}).
• Figure 2: Semi-log plot comparing the growth of the interface height at sinusoidal walls, $\ell$ (red), to that at a planar wall, $\ell_\pi$ (green), near saturation as obtained from DFT. The fitted lines have a slope of $-\xi_b$, confirming the expected logarithmic divergence of both $\ell$ and $\ell_\pi$ in the limit $\delta\mu\to0$. The respective asymptotes are vertically shifted by $\delta\ell=0.7\,\sigma$ for the wall with $L=20\,\sigma$, $A=2\,\sigma$, and $\delta\ell=3\,\sigma$ for the wall with $L=100\,\sigma$, $A=5\,\sigma$, showing excellent agreement with the prediction of Eq. (\ref{['ell']}).
• Figure 3: (a) The deviation of the interface height, $\delta\ell=\ell-\ell_\pi$, from its planar-wall value as a function of the wall amplitude $A$ for hard sinusoidal walls with the fixed period of $L=100\,\sigma$. The solid line represents the solution of the nonlocal Hamiltonian obtained by solving numerically Eq.(\ref{['Hflat']}), while the symbols denote DFT results. The dashed line with slope 1 confirms the linear regime of $\delta\ell(A)$. The inset shows a log-log plot, where the linear fit to the nonlocal Hamiltonian numerics with slope 2 supports the expected quadratic regime of $\delta\ell(A)$ for small $A$. (b) Verification of the negative logarithmic correction to $\delta\ell$ [cf. Eq.(\ref{['ell_A_weak']})] through the dependence of $\delta\tilde{\ell}=\delta\ell-A$ on $A$. The solid line corresponds to the numerical solution of Eq. (\ref{['Hflat']}), while symbols represent DFT results. The dashed line shows a linear fit to the DFT data.
• Figure 4: The deviation of the interface height, $\delta\ell=\ell-\ell_\pi$, from its planar-wall value as a function of the wall wave number $k$ for hard sinusoidal walls with the fixed amplitude of $A=2\,\sigma$. The solid line represents the solution of the interface Hamiltonian theory obtained by numerically solving Eq.(\ref{['Hflat']}), while symbols denote DFT results. The dashed line is a quadratic fit to the DFT data confirming the expected dependence of $\delta\ell$ on $k$ [cf. Eq.(\ref{['ell_k_weak']})] for weakly corrugated walls, with the additive constant $0.61\,\sigma$, which is slightly below the theoretical value $c_0\approx0.74\,\sigma$.
• Figure 5: Same as in Fig. \ref{['fixed-k-small-roughness-ell']}, but for strongly corrugated walls with period $L=20\,\sigma$ and large values of $A$.