Rigidity transitions and constraint counting in amorphous networks: Beyond the mean-field approach

We study rigidity transitions in covalent amorphous networks using size-increasing cluster approximations and constraint-counting algorithms. Possible consequences of the presence of self-organization are examined. The analysis reveals two transitions instead of the usual (mean-field) single transition: one from a floppy to an isostatic rigid phase at a mean coordination number (r) over bar (c1) where the number of floppy modes vanishes and a second one from an isostatic to a stressed rigid phase at (r) over bar (c2). The value of the two critical mean coordination numbers as well as the width Delta(r) over bar = (r) over bar (c2) - (r) over bar (c1) of the intermediate phase depend very strongly on the presence of medium-range order elements such as rings.