Anchored boundary conditions for locally isostatic networks

TL;DR

This paper introduces anchored boundary conditions to effectively make finite, locally isostatic networks rigid by fixing surface constraints, with applications to atomic-scale imaging and complex geometries.

Contribution

It proposes a new boundary condition method for rendering finite isostatic networks effectively rigid, applicable to complex geometries and atomic-scale structures.

Findings

Anchored boundary conditions successfully eliminate floppy modes.

Application to 2D networks of corner sharing triangles.

Relevance to atomic-level imaging of vitreous silica.

Abstract

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 needs 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.