Mean width and caliper characteristics of network polyhedra

Abstract

The kinetic modelling of space-filling grain networks has been approached traditionally by representing the grains as spheres of equivalent volume. A spherical approximation used to describe polyhedral grains, unfortunately, relinquishes most geometrical and all topological details of the grain structure. Techniques developed by Hilgenfeldt et al., and by Glicksman, describe network structures comprised of space-filling irregular polyhedra and their kinetics with regular polyhedra, which act as ‘proxies’ that preserve both local topology and network constraints. Analytical formulas based on regular polyhedra and Surface Evolver simulations are used in this study to calculate the average caliper width and mean width for extended sets of polyhedra that vary systematically from convex to concave objects. Of importance, caliper width and mean width measures allow estimation of the growth rates of grains. Comparison of these calculations and simulations, however, reveal a weak dependence between average caliper width and mean width measures and the detailed shapes of polyhedra, especially their face curvatures. This finding might, in fact, affect the application and use of linear measures as kinetic tools in quantitative microstructure measurements.

Keywords:

• mean width
• caliper width
• network polyhedra
• grain growth
• bubbles

Notes

Note

1. Note that the Platonic solids with N = 8 and N = 20, the octahedron and the icosahedron, respectively, are precluded from the set of GNHs, as they lack trivalent vertices. Although the case N = 3 does produce a pod-shaped regular polyhedron with two trivalent vertices, it does not result in a Platonic solid in the limit of flat faces, but merely collapses to a line.