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 (r) over bar =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 Delta(r) over bar 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.