Topological Origin of the Surface Singularities in Glasses

Abstract

ABSTRACTA model of CRN in two dimensions is presented, composed of equilateral but not equiangular polygons with a constant coordination number = 3; in a simplified version only pentagons, hexagons and heptagons are present in the network. We introduce the mean potential energy per atom averaged over a typical cell containing three adjacent polygons; we assume that although the thermal equilibrium is not attained for single atoms, it is attained on the level of these cells, so that we can apply the virial theorem for the cells. Then we minimize the free energy which contains the configuration entropy contribution. In terms of two variables P and δ (hexagon frequency and mean bond angle deviation) we get the surfaces of constant energy. Under stress the energy configurations cease to be one-connected, and the 0-th homotopy group is no more trivial. This can give rise to surface singularities (cracks).