Energy and tension of films and Plateau borders in a foam

We discuss a number of problems relating to the energy and geometry of liquid foams in three dimensions. We use Plateau’s concept of a skeletonised or ideal dry foam, consisting of geometrical surfaces, lines and points, which correspond, respectively, to the films, Plateau borders and nodes of the real foam, but including the contributions of all these elements to the total foam energy (and not just the film energy, as is usually done). In order to be able to write down all these contributions for a real foam, we conjecture that the same can be skeletonised by successively replacing the films by their mid-surfaces, the Plateau borders by the edges along which the mid-surfaces meet, and the nodes by the points where four edges are joined. We then relate the film energy γγ and the film tension σσ (energy per unit area), and define a Plateau border line tension ττ (energy per unit length) and a node energy νν. The Plateau border line tension is negative and causes deviations from 120°° of the angles between films, and from cos⁡−1(−1/3)cos⁡−1(−1/3) of the angles between lines at nodes. We further define the excess energy ϵϵ of a Plateau border relative to the dry line. We illustrate our findings with calculations performed for a few simple systems – the double bubble, the lens bubble, and a bubble at a plate – for which skeletonisation is straightforward, and show that τ=ϵ/2τ=ϵ/2 when ϵ∝A1/2ϵ∝A1/2, with A the Plateau border cross-sectional area, i.e., for not too high liquid fractions.