Lower bounds for the surface energy of two-dimensional foams

Abstract:

Amongst the two-dimensional cellular patterns that fill a plane, dry foams at stable equilibrium typify a particular subset for which the total perimeter P of cell boundaries (i.e., films between bubbles) has a local minimum. For a given set of bubble areas A_i ( i = 1,..., N), P can be written in the form P = R( )/2, where R is topology dependent. We seek the set of areas A_i and the cluster topology that minimise R, and propose lower bounds for R that set lower bounds for the surface energy of i) individual bubbles, with circular edges meeting at 2π/3 angles at vertices (Plateau cells), and ii) infinite periodic bubble clusters.