Rings and rigidity transitions in network glasses

Three elastic phases of covalent networks, (I) floppy, (II) isostatically rigid, and (III) stressed-rigid, have now been identified in glasses at specific degrees of cross linking (or chemical composition) both in theory and experiments. Here we use size-increasing cluster combinatorics and constraint counting algorithms to study analytically possible consequences of self-organization. In the presence of small rings that can be locally I, II, or III, we obtain two transitions instead of the previously reported single percolative transition at the mean coordination number $\overline{r}=2.4,$ one from a floppy to an isostatic rigid phase, and a second one from an isostatic to a stressed rigid phase. The width of the intermediate phase $\ensuremath{\Delta}\overline{r}$ and the order of the phase transitions depend on the nature of medium-range order (relative ring fractions). We compare the results to the group-IV chalcogenides, such as Ge-Se and Si-Se, for which evidence of an intermediate phase has been obtained, and for which estimates of ring fractions can be made from structures of high-$T$ crystalline phases.