Surface energy of free clusters of bubbles: an estimation

We propose an approximate equation for the surface energy of two-dimensional free bubble clusters which we compare with exact calculations of the surface energy of symmetrical clusters consisting of a central bubble surrounded by one or two shells of bubbles of two di erent areas. The accuracy of the equation is good for relatively narrow distributions of the areas and of the number of sides of the bubbles but underestimates the energy for large widths of those distributions. We propose a similar approximate equation for the surface energy of three-dimensional clusters. The energy of foam clusters with an incompressible gas reduces to its surface energy E ˆ P®, where P is the total length (in two-dimensional (2D) clusters) or area (in three-dimensional (3D) clusters) of the ®lms separating adjacent bubbles; ® is the ®lm tension. The energy depends on the strain in the cluster; only free clusters will be considered in this letter. Di erent arrangements (topologies) of a large number of bubbles of given sizes (area or volume) do not seem to a ect the energ