Modeling polycrystals with regular polyhedra

Polycrystalline structure is of paramount importance to materials science and engineering. It provides an important example of a space-filling irregular network structure that also occurs in foams as well as in certain biological tissues. Therefore, seeking an accurate description of the characteristics of polycrystals is of fundamental importance. Recently, one of the authors (MEG) published a paper in which a method was devised of representation of irregular networks by regular polyhedra with curved faces. In Glicksman's method a whole class of irregular polyhedra with a given number of faces, N, is represented by a single symmetrical polyhedron with N curved faces. This paper briefly describes the topological and metric properties of these special polyhedra. They are then applied to two important problems of irregular networks: the dimensionless energy 'cost' of irregular networks, and the derivation of a 3D analogue of the von Neumann-Mullins equation for the growth rate of grains in a polycrystal