Weaire-Phelan structure

The combined results are shown in Fig. 21, where it is seen that the transition from spheres to completely developed polyhedra is accompanied by an increase in surface area of 8.3%. As mentioned above, for the monodisperse case one predicts an increase in surface area of 9.7% on the basis of Kelvin s polyhedron as the ultimate drop shape, or 9.4% for the Weaire-Phelan structure. Polydispersity appears to give rise to an even somewhat smaller overall change in surface area. Recent computer simulations of various monodisperse and polydisperse structures by Kraynik et al. (66) confirm this result almost quantitatively. 

It is interesting to note that the so-called Weaire-Phelan structure of mono-disperse liquid foam, made of connected bubbles with the least area of surface between them, theoretically predicted and experimentally observed, is the same as that of a type 1 clathrate [44, 45]. 

R. Gabrielli, A.J. Meagher, D. Weaire, K.A. Brakke, S. Hutzler, An experimental realization of the Weaire-Phelan structure in monodisperse liquid foam. Phil. Mag. Lett. 92(1), 1-6 (2012)... 

These local structural rules make it impossible to construct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fill space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal structure that can serve as a convenient mathematical idealization of polyhedral foam structure. Lord Kelvin considered this problem, and his tet-rakaidecahedron is still considered the best periodic structure of identical polyhedra that can nearly satisfy the mechanical constraints, while providing the smallest surface energy (or area). A more efficient structure for minimizing the surface energy, has more recently been proposed (known as the Weaire-Phelan structure (10)), but it consists of bubbles of two different types, and whether it is the optimal structure remains an open question. 

FIGURE 15.17 IGFs without grains, (a and b) Models of the Kelvin cell and a Weaire-Phelan foam used to describe soap bubbles (c-e) how these relate to the structure of TJs and QJs. 

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