润湿带上液滴的高度
Height of a liquid drop on a wetting stripe
摘要：
考虑液体在由宽度为$L$的亲水性条纹装饰的平面壁上的吸附。在壁仅部分湿润（或干燥）而条纹趋于完全湿润的条件下，在条纹上方形成液滴。液滴的最大高度$\ell_m(δμ)$取决于条纹宽度$L$和与饱和度$δμ$的化学势偏离，其中它采用值$\ell_0=\ell_m(0)$。假设由条纹施加的范德华型长程势，界面哈密顿模型被用来表明$\ell_0$与$δμ$线性地接近，其斜率在满足$L\lesssim ξ_\parallel$的区域上缩放为$L^2$，其中$ξ_\parallel$是与条纹相关的平行相关函数。这表明在接近饱和时，存在一条通用曲线$\ell_m(δμ)$，当适当地重新缩放时，对应于不同$L$值的吸附等温线全部塌陷。虽然基于界面哈密顿模型的级数展开可以通过考虑高阶项来形成，但提出了基于标度参数的有理函数形式的更合适的近似。近似是基于精确的渐近结果，即$\ell_m\simδμ^{-1/3}$为$L\to\infty$和$\ell_m$服从正确的$δμ\to0$行为符合界面哈密顿模型的结果。所有的预测进行了验证，通过与微观密度泛函理论（DFT）的比较，特别是，有理函数近似-即使在其最简单的形式-被证明是在一个非常合理的协议与DFT广泛的$δμ$和$L$。
英文摘要：
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 $\ell_m(δμ)$ of the drop depends on the stripe width $L$ and the chemical potential departure from saturation $δμ$ 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 $δμ$ with a slope which scales as $L^2$ over the region satisfying $L\lesssim ξ_\parallel$, where $ξ_\parallel$ is the parallel correlation function pertinent to the stripe. This suggests that near the saturation there exists a universal curve $\ell_m(δμ)$ 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 $\ell_m\simδμ^{-1/3}$ for $L\to\infty$ and that $\ell_m$ obeys the correct $δμ\to0$ 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 $δμ$ and $L$.
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