Topological modelling of disordered cellular structures

Abstract

The authors model the structure of space-filling disordered cellular systems. These systems are cellular networks with minimum incidence numbers (D+1 edges incident on a vertex in D-dimension). In the literature such systems are known as froths since the soap froth is the archetype of these structures. They present a method where the structure of froths is analyzed as organized in concentric layers of cells around a given, arbitrary, central cell. A simple map gives, by recursion, the number of cells in each layer. The map has one parameter, given as a function of the average topological properties of the cells in the neighbouring layers. From the behaviour of the number of cells per layer with the topological distance, one obtains the curvature of the space tiled by the froth. By using the map it is therefore possible to characterize the shape of the manifold tiled by the froth in term of the topological arrangements of its tiles. In two dimensions, they propose a method to deduce the Gaussian curvature of surfaces from a set of sampled points. In three dimensions, they use the map to investigate the freedom in constructing disordered Euclidean cellular structures. Among the closed packed structures, they find the average shape of the cells that maximize this freedom in filling space.