The structure of foam cells: Isotropic Plateau polyhedra

Abstract. – A mean-field theory for the geometry and diffusive growth rate of soap bubbles in dry 3D foams is presented. Idealized foam cells called isotropic Plateau polyhedra (IPPs), with F identical spherical-cap faces, are introduced. The geometric properties (e.g., surface area S, curvature R, edge length L, volume V) and growth rate G of the cells are obtained as analytical functions of F, the sole variable. IPPs accurately represent average foam bubble geometry for arbitrary F ≥ 4, even though they are only constructible for F = 4, 6, 12. While R/V 1/3, L/V 1/3 and G exhibit F 1/2 behavior, the specific surface area S/V 2/3 is virtually independent of F. The results are contrasted with those for convex isotropic polyhedra with flat faces. Dry soap foams (with liquid volume fraction 1) are space-filling structures composed of polyhedral cells (bubbles) with curved faces. Over short time scales, the bubbles change shape under the influence of surface tension until the total surface area of the foam achieves a local minimum. Over longer times, further area reduction results from gas diffusion between neigh-boring cells with different internal pressures. Some cells shrink in the process and eventually disappear, increasing the average bubble size. This diffusive coarsening [1] is paradigmatic for other surface-minimizing evolution processes, such as grain growth in polycrystalline metals [2