When is a surface foam-phobic or foam-philic?

Abstract

It is commonly assumed that the liquid making up a sessile bubble completely wets the surface upon which the bubble lies. However, this need not be so, and the degree of wetting will determine how well a collection of bubbles - a foam - sticks to a surface. As a preliminary to this difficult problem, we study the shape of a single vertical soap film spanning the gap between two flat, horizontal solid substrates of given wettabilities. For this simple geometry, the Young-Laplace equation can be solved (quasi-)analytically to yield the equilibrium shapes, under gravity, of the two-dimensional Plateau borders along which the film contacts the substrates. We thus show that these Plateau borders, where most of a foam's liquid resides, can only exist if the values of the Bond number Bo and of the liquid contact angle θc lie within certain domains in (θc,Bo) space: under these conditions the substrate is foam-philic. For values outside these domains, the substrate cannot support a soap film and it is foam-phobic. In other words, on a substrate of a given wettability, only Plateau borders of a certain range of sizes can form. For given (θc,Bo), the top Plateau border can never have greater width or cross-sectional area than the bottom one. Moreover, the top Plateau border cannot exist in a steady state for contact angles above 90°. Our conclusions are validated by comparison with both experimental and numerical (Surface Evolver) data. We conjecture that these results will hold, with slight modifications, for non-planar soap films and bubbles. Our results are also relevant to the motion of bubbles and foams in channels, where the friction force of the substrate on the Plateau borders plays an important role.

The shape of two-dimensional liquid bridges

“…This generalises to a physically more realistic situation our earlier work in which it was assumed that the menisci at the two substrates were connected by a liquid film of zero thickness. As shown in [1], the solution method we use yields results that are virtually indistinguishable from those obtained using the Surface Evolver, which gives us confidence that they are correct, in spite of the absence of experimental data to corroborate them.…”

“…where z is height measured from the bottom substrate, x is the distance measured horizontally from the plane of symmetry (the midplane of the 2D bridge), ∆p(z) is the pressure difference across the bridge surface at each height, and γ is the surface tension of the liquid. Our aim is to solve equation (1) for one of the surfaces bounding the bridge. Naturally, the other surface is mirrorsymmetric with respect to x = 0.…”

“…where θ is the inclination of the bridge surface (see figure 1), defined as the angle between the tangent to the bridge surface at point (x, z) and the horizontal axis (0 θ π). Using equations (2) and (3), equation (1) becomes…”

“…In a recent paper [1] we investigated the equilibrium shapes, under gravity, of the two-dimensional (2D) Plateau borders along which a single vertical soap film contacts two flat, horizontal solid substrates of given wettabilities. For this simple geometry, the Young-Laplace equation can be solved (quasi-) analytically, and we showed that these Plateau borders, where most of a foam's liquid resides, can only exist if the values of the Bond number Bo and of the liquid contact angle θ c lie within certain domains in (θ c , Bo) space: under these conditions the substrate is foam-philic.…”

“…It should be noted that the presence of gravity complicates matters substantially, as it precludes an analytical solution for the bridge shape. Here we shall follow a different route and generalise our work on 2D Plateau borders [1] to consider a slab of liquid between two flat, unbounded horizontal substrates at which the contact angles are fixed. In the terminology introduced by Fortes [23] these are θ-bridges, and perhaps the nearest experimental realisation of the slab geometry is a slit pore (see, e.g.…”