Density of topological constraints as a metric for predicting glass hardness

Topological constraint theory has previously been applied to predict the composition dependence of glass hardness for a variety of different composition families. Some recent findings have cast doubt on the correlation between glass hardness and the number of rigid constraints per atom in silicate glasses. In this letter, we revisit the prediction of hardness for borosilicate and phosphosilicate glasses using four different types of constraint counting approaches: total number of constraints per atom, angular constraints per atom, total constraint density, and angular constraint density. We find that the counting approaches using total constraint density or angular constraint density give an improved prediction of glass hardness. We therefore conclude that glass hardness is governed by the density of rigid constraints under an indenter, rather than by the number of rigid constraints per atom.