Finite pieces of locally isostatic networks have a large number of floppy modes because of missing constraints at the surface. Here we show that by imposing suitable boundary conditions at the surface the network can be rendered effectively isostatic. We refer to these as anchored boundary conditions. An important example is formed by a two-dimensional network of corner sharing triangles, which is the focus of this paper. Another way of rendering such networks isostatic is by adding an external wire along which all unpinned vertices can slide (sliding boundary conditions). This approach also allows for the incorporation of boundaries associated with internal holes and complex sample geometries, which are illustrated with examples. The recent synthesis of bilayers of vitreous silica has provided impetus for this work. Experimental results from the imaging of finite pieces at the atomic level need such boundary conditions, if the observed structure is to be computer refined so that the interior atoms have the perception of being in an infinite isostatic environment.

Recently determined atomistic scale structures of near-two dimensional bilayers of vitreous silica (using scanning probe and electron microscopy) allow us to refine the experimentally determined coordinates to incorporate the known local chemistry more precisely. Further refinement is achieved by using classical potentials of varying complexity; one using harmonic potentials and the second employing an electrostatic description incorporating polarization effects. These are benchmarked against density functional calculations. Our main findings are that (a) there is a symmetry plane between the two disordered layers; a nice example of an emergent phenomenon, (b) the layers are slightly tilted so that the Si-O-Si angle between the two layers is not $180^{\circ}$ as originally thought but rather $175 \pm 2 ^{\circ}$ and (c) while interior areas that are not completely imagined can be reliably reconstructed, surface areas are more problematical. It is shown that small crystallites that appear are just as expected statistically in a continuous random network. This provides a good example of the value that can be added to disordered structures imaged at the atomic level by implementing computer refinement.Comment: 5 pages, 4 figure

We examine the correlations between rings in random network glasses in two dimensions as a function of their separation. Initially, we use the topological separation (measured by the number of intervening rings), but this leads to pseudo-long-range correlations due to a lack of topological charge neutrality in the shells surrounding a central ring. This effect is associated with the noncircular nature of the shells. It is, therefore, necessary to use the geometrical distance between ring centers. Hence we find a generalization of the Aboav-Weaire law out to larger distances, with the correlations between rings decaying away when two rings are more than about three rings apart.

This article studies the set of equivalent realizations of isostatic frameworks and algorithms for finding all such realizations. It is shown that an isostatic framework has an even number of equivalent realizations that preserve edge lengths and connectivity. The complete set of equivalent realizations for a toy framework with pinned boundary in two dimensions is enumerated and the impact of boundary length on the emergence of these realizations is studied. To ameliorate the computational complexity of finding a solution to a large multivariate quadratic system corresponding to the constraints, alternative methods—based on constraint reduction and distance‐based covering map or Cayley parameterization of the search space—are presented. The application of these methods is studied on atomic clusters, a model of 2D glasses and jamming.

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