Minimal network partitions using average Ν-hedra

Network structures, such as polycrystals and dry foams, consist of space-filing, contiguous, irregular polyhedral grains or bubbles. The irregularity of such polyhedra derives from the fact that their curved faces are of mixed form, their edges are of varying length, and the positions of the vertices are non-symmetrical. The geometric complexity and variability of the component polyhedra in irregular networks make their energetic and kinetic analyses daunting. To help circumvent this difficulty, average N-hedra (ANHs) were proposed as a substitute set of highly symmetric polyhedra consisting of N identical faces. ANHs act only as topological 'proxies' for each corresponding class of irregular network polyhedra that contains the same number of faces. This study provides a comparison of the areas and volumes of ANHs with data estimated numerically using Surface Evolver simulations for a wide range of polyhedral network cells. Surface Evolver data show that for every topological class tested, ANHs consistently provide an upper bound for the isoperimetric quotient, Q(N), which is a measure of the inverse energy cost per unit volume of the cell. Of special interest here is Kusner's proof for the bounds on the average number of faces in minimally partitioned networks in compact 3-manifolds. We show that in Euclidean 3-space the requisite number of faces per cell for the minimal partition is satisfied exactly by the critical ANH. The critical ANH, therefore, statistically represents the abstract 'unit cell' for a network with minimum average free energy per unit volume. Analysis shows that the isoperimetric quotient for the critical ANH, Q(star) approximate to 0: 779 635..., is the highest value attainable for this quotient for any polyhedral cell exhibiting the requisite number of faces stipulated by Kusner's network face formula. The critical ANH thus provides the theoretical minimum partition area cost for its random network. This limit remains of particular interest in the case of annealed polycrystals and constant-pressure dry foams, as it establishes the limiting least bound of the free energy cost of such random structures.