What Can Topological Models Tell Us About Glass Structure and Properties?

Atomic arrangements in condensed matter partition three-dimensional space into polyhedra whose edges are interatomic vectors. These polyhedra, formally known as void polytopes, fill (tesselate) space, and their identity and arrangement can provide one description of a given atomic arrangement (Figure 1a) [1]. Other tessellations associated with space-filling of random structures are Voronoi polyhedral cells [2] and their dual the Delauney network [3]. These tessellations are relatively intuitive in two dimensions, but considerably more complex in three-dimensions—for example in tetrahedral networks like SiO2—where a set of as many as 126 void polyhedra may be required to model interstitial space [1]. Because many arrangements favor particular coordination of one atom by others, owing to bond orbital, radius ratio, or local electrostatic neutrality considerations, discrete coordination polyhedra comprise a subset of the possible void polytopes, and the structure may be described by the way in which coordination polyhedra are connected together and fill space by defining the remaining void polytopes. Space filling by connected structural units was a favorite description tool of early crystal chemists [4], and in fact the connectivity of such structural polytopes (number of polytopes sharing vertices, edges and faces) has been shown to correlate with glass-forming ability and extendability of aperiodic networks [5] and to govern the amorphizability of crystalline solids [6].