Definition and properties of ideal amorphous structures

Full text: Amorphous structure is usually defined by what it is not (ie, no crystalline peaks in XRS, no bond correlation in NMR), rather than by what it is. The interest in defining the structure of non-crystalline materials is long standing; packing geometry of spheres, molecular structure of glassy SiO2, or the structure of atactic polymers are prime examples. The earliest definitions of amorphous structure were in terms of a microcrystallite model of Valenkov, or continuous random network by Zachariasen. The random close packing of spheres of equal size, and an amorphous structure, composed of freely jointed linear chains of hard spheres, has been described mathematically in terms of a linear homogeneous Poisson process. This paper aims to describe some geometrical, kinematic, and topological properties of these two ideal amorphous structures, which belong to the same amorphous class. The geometry of packing is elucidated, and the use of Voronoi tessellation method for measuring the structures is described. The ideal amorphous solid has no symmetry elements; its volume can not be divided into identical unit cells. However, there is a volume element small enough to allow the distinction of its nanoscopic inhomogeneities, and sufficiently large enough to represent, accurately the overall behaviour. We define this volume element, the representative volume element. Suitable boundary conditions must be prescribed for a choice of RVE, and satisfy certain requirements. Topologically, a catchment region on the Born-Oppenheimer potential energy surface over nuclear configuration space, is defined by Mezey and Bader as an energetically stable geometry of the open region of R3 traversed by all the trajectories which terminate at a local maximum. Two topological properties will be described: (i) the boundaries of the catchment region as a direct geometrical correspondence to the Voronoi polyhedron for a given atom in a given structure, and (ii) the constriction points, described previously, as an inherent characteristic of IAS type II (and possibly others)