The Geometry of Bubbles and Foams

by John M. Sullivan

Published in Foams, Emulsions, and Cellular Materials, (J.-F. Sadoc, ed.), NATO Adv. Sci. Inst. Ser. C, Kluwer, 1998, pp 379-402. Plain-English Abstract

Soap films, bubble clusters, and foams and froths can be modeled mathematically as collections of surfaces which minimize their surface area subject to volume constraints. Each surface in such a solution has constant mean curvature, and is thus called a CMC surface. We first examine a more general class of variational problems for surfaces, and then concentrate on CMC surfaces, where one important result is the balancing of certain forces. Next we look at existence results for bubble clusters, and at the singularities in such clusters and in foams; these are described by Plateau's rules.

These rules for singularities imply that a foam is combinatorially dual to some triangulation of space. Triangulations of a surface are related through their Euler number to the topology of the surface. In three dimensions, not as much can be said along these lines, but we describe certain results. Among the most interesting triangulations, from the point of view of equal-volume foams, are those of the tetrahedrally close-packed (TCP) structures from chemistry. The TCP structures observed in nature fit into an infinite mathematical class; we describe constructions for some infinite families. Finally, we discuss the application of these ideas to Kelvin's problem of partitioning space. The counterexample to Kelvin's conjecture found by Weaire and Phelan was one TCP foam, and the others all seem useful for generating efficient equal-volume foams.