On the capillary interaction between solid plates forming menisci on the surface of a liquid

Abstract

A hydrophilic or a hydrophobic long rigid solid plate of finite width, forming a meniscus with a liquid in a uniform gravitational field is considered. The one-dimensional meniscus with prescribed heights of the triple point from the far-field liquid surface is investigated analytically using the Young–Laplace equation. It is found that for a hydrophilic plate, the vertical force necessary to break the meniscus during removal of the plate from the liquid is larger than the force necessary to break the meniscus during submersion of the plate into the liquid. Furthermore, the capillary force on the plate reaches a maximum before the meniscus collapses during removal, but no maximum exists before collapse during submersion. The reverse is true when the plate is hydrophobic. The study is then extended to investigate the interaction force between two plates, each forming a meniscus with the liquid. The elevations of the plates from the far-field liquid surface are prescribed, in contrast to earlier studies where interaction between long cylinders floating under self weight was considered. Here, the menisci are determined exactly using the Young–Laplace equation. It is shown that for prescribed plate elevations, there can be at most two possible pairs of menisci between them. Each pair bifurcates from a meniscus that is determined by the elevations of the plates and the gap between them. Furthermore, as known for solids floating under self-weight, the horizontal component of the interaction force is attractive for similar menisci (e.g. when the two plates are equally displaced in or out of the liquid), and repulsive when they form opposite menisci. It is shown that if the two menisci are of the same type, but not similar (e.g. one plate is pushed more into the liquid than the other), then the force is attractive at long distances, and may be repulsive at shorter distances with a stable equilibrium at a finite distance between the plates, depending on the elevations of the plates. Such interaction can be between two hydrophilic or two hydrophobic plates or between a hydrophilic and a hydrophobic plate.