Self-organization with equilibration: A model for the intermediate phase in rigidity percolation

Recent experimental results for covalent glasses suggest the existence of an intermediate phase attributed to the self-organization of the glass network resulting from the tendency to minimize its internal stress. However, the exact nature of this experimentally measured phase remains unclear. We modified a previously proposed model of self-organization by generating a uniform sampling of stress-free networks. In our model, studied on a diluted triangular lattice, an unusual intermediate phase appears, in which both rigid and floppy networks have a chance to occur, a result also observed in a related model on a Bethe lattice by Barré et al. [Phys. Rev. Lett. 94, 208701 (2005)]. Our results for the bond-configurational entropy of self-organized networks, which turns out to be only about 2% lower than that of random networks, suggest that a self-organized intermediate phase could be common in systems near the rigidity percolation threshold.

Figure 1

The fraction of networks in which the percolating rigid cluster is present as a function of $⟨r⟩$ for the model of self-organization without equilibration. Each curve is obtained from 100 separate runs on a bond-diluted triangular lattice; the lattice sizes are indicated in the legend.

Figure 2

Same as in Fig. 1, but now for the model with equilibration, for triangular networks of $10000$ sites, and for several different equilibration times indicated in the legend as the number of equilibration steps per each inserted bond above $⟨r⟩=3.5$; below $⟨r⟩=3.5$, there are 10 equilibration steps per bond in all cases. The data are likewise from 100 separate runs, but in addition, from each run all networks obtained during the equilibration procedure at the given mean coordination are taken into account.

Figure 3

Same as in Fig. 2, but for different network sizes indicated in the legend, always using 10 equilibration steps per inserted bond below $⟨r⟩=3.5$ and 100 equilibration steps above.

Figure 4

The fraction $\nu$ of “allowed bonds” (places where a bond can be inserted without creating stress) as a function of the mean coordination $⟨r⟩$, for a triangular network of $50176$ sites, with an equilibration time of 1000 steps per inserted bond above $⟨r⟩=3.5$ and 10 steps below. The result is obtained by a Monte Carlo procedure described in the text; at each mean coordination, it is averaged over the networks obtained during equilibration.

Figure 5

The difference in the bond-configurational entropy between random and self-organized triangular networks, $\mathrm{\Delta }s$, as a function of the mean coordination $⟨r⟩$, shown for several network sizes and equilibration times, as well as the asymptotic value (i.e., the extrapolation to the infinite size and equilibration time, as described in the text). Quantity $\nu$ used to calculate $\mathrm{\Delta }s$ is obtained by the Monte Carlo procedure described in the text and then averaged over networks obtained during equilibration at the given $⟨r⟩$. In the main plot, all thin lines are results for $50176$ sites; the equilibration times are, from top to bottom: 10 steps, 30 steps, 100 steps, and 300 steps per added bond. The thick line is the asymptotic. In the inset, all thin lines are results for 1000 equilibration steps per added bond. The sizes are, from top to bottom: 5476 sites, 15 376 sites, and 50 176 sites. The thick line is again the asymptotic. The asymptotics are themselves extrapolated to $⟨r⟩=4$, as described in the text. All equilibration times listed are for $⟨r⟩>3.5$; for $⟨r⟩<3.5$, 10 equilibration steps per added bond are always used.

Figure 6

A summary of results for different models of rigidity percolation on the bond-diluted triangular lattice. In the case of random bond dilution without self-organization [panel (a)], there is a single rigidity and stress transition, at which the rigidity and stress percolation probabilities jump simultaneously from 0 to 1 in the thermodynamic limit (this is shown by the solid line); there is no intermediate phase. In the case of the model of self-organization without equilibration by Thorpe et al. [panel (b)], the probabilities of rigidity percolation (solid line) and stress percolation (dashed line) also jump from 0 to 1, but at different $⟨r⟩$, and there is an intermediate phase in between, which is rigid but stress-free. Finally, in our model of self-organization with equilibration [panel (c)], there is also an intermediate phase, but the probability of rigidity percolation changes continuously between 0 and 1 (solid line) within this phase, with the dependence on the distance from the lower boundary of the intermediate phase being either exactly linear or very close to linear in the thermodynamic limit. The network is still stress-free in the intermediate phase; the stress will have to appear inevitably above $⟨r⟩=4$, but we do not specify how to handle stress in our model, so it cannot be extended beyond $⟨r⟩=4$, and the stress percolation probability is not shown in panel (c).