Self-organized criticality in the intermediate phase of rigidity percolation

Experimental results for covalent glasses have highlighted the existence of a self-organized phase due to the tendency of glass networks to minimize internal stress. Recently, we have shown that an equilibrated self-organized two-dimensional lattice-based model also possesses an intermediate phase in which a percolating rigid cluster exists with a probability between zero and one, depending on the average coordination of the network. In this paper, we study the properties of this intermediate phase in more detail. We find that microscopic perturbations, such as the addition or removal of a single bond, can affect the rigidity of macroscopic regions of the network, in particular, creating or destroying percolation. This, together with a power-law distribution of rigid cluster sizes, suggests that the system is maintained in a critical state on the rigid-floppy boundary throughout the intermediate phase, a behavior similar to self-organized criticality, but, remarkably, in a thermodynamically equilibrated state. The distinction between percolating and nonpercolating networks appears physically meaningless, even though the percolating cluster, when it exists, takes up a finite fraction of the network. We point out both similarities and differences between the intermediate phase and the critical point of ordinary percolation models without self-organization. Our results are consistent with an interpretation of recent experiments on the pressure dependence of Raman frequencies in chalcogenide glasses in terms of network homogeneity.

Figure 1

(Color online) The fraction of bonds belonging to the percolating rigid cluster among all bonds in the network, averaged over all percolating self-organized networks, for different network sizes indicated in the figure. All network sizes here and in other figures in the paper are given in terms of sites.

Figure 2

(Color online) The standard deviation of the fraction of bonds in the percolating cluster for different network sizes, for self-organized networks.

Figure 3

(Color online) The distribution of sizes (given in terms of bonds) of nonpercolating rigid clusters in nonpercolating self-organized networks at $⟨r⟩=3.92$ [panel (a)] and in both percolating and nonpercolating self-organized networks at $⟨r⟩=3.97$ [panel (b)]. The details of the simulation and the fits (lines) are given in the text.

Figure 4

(Color online) The probability that a bond added to a nonpercolating self-organized network at a random allowed place makes this network percolating.

Figure 5

(Color online) The fraction of nonpercolating self-organized networks such that there exists a bond whose insertion causes percolation.

Figure 6

(Color online) The comparison between the average size of the percolating cluster in originally percolating self-organized networks (“without insertion”) and in originally nonpercolating networks that become percolating after bond insertion (“after insertion”). The latter quantity is calculated as the average over all networks that can become percolating, by using “judicious placement” of a bond, as described in the text. Networks of 90 000 sites are used.

Figure 7

(Color online) The probability that removal of a randomly chosen bond from a percolating self-organized network destroys percolation.

Figure 8

(Color online) The probability that removal of a random bond destroys percolation (the quantity in Fig. 7) divided by the average fraction of bonds in the percolating cluster (the quantity in Fig. 1). This serves as an estimate of the probability that a bond chosen at random among those belonging to the percolating cluster destroys percolation.

Figure 9

(Color online) The average fraction of rigid bonds in the network in the random and self-organized cases (in the latter case, also separately for percolating and nonpercolating networks). All simulations are for networks of $90000$ sites, the overall averages are over 200 networks, the averages restricted to percolating and nonpercolating networks are over those of the 200 self-organized networks that are, respectively, percolating and nonpercolating.

Figure 10

(Color online) The standard deviation of the distribution of fractions of rigid bonds in percolating (a) and nonpercolating (b) self-organized networks for different network sizes.

Figure 11

(Color online) The average change in the number of rigid bonds upon addition of a single bond, for random and self-organized networks of $90000$ sites.

Figure 12

(Color online) The average fraction of floppy bonds converting to rigid upon addition of a single bond to a self-organized network. (a) The overall average, as well as partial averages for the cases when the network converts from nonpercolating to percolating, remains nonpercolating, and remains percolating, for networks of $90000$ sites. (b)–(d) Partial averages in the nonpercolating $\to$ percolating, percolating $\to$ percolating, and nonpercolating $\to$ nonpercolating cases, respectively, for three different network sizes. The inset in (d) shows the average number of converted bonds in the nonpercolating $\to$ nonpercolating case.

Figure 13

(Color online) Two examples of rigidification of self-organized networks upon addition of a bond. In the top panel, the network switches from nonpercolating to percolating. In the bottom panel, it remains nonpercolating. In both cases, the added bond is pointed with an arrow, thick green bonds (gray in print) are those that are originally rigid, thin bonds (blue online) remain floppy, thick black bonds switch from floppy to rigid. Both networks contain $10000$ sites.

Figure 14

(Color online) The average fraction of rigid bonds converting to floppy upon removal of a single bond from a self-organized network. (a) The overall average, as well as partial averages for the cases when the network converts from percolating to nonpercolating, remains nonpercolating and remains percolating, for networks of $90000$ sites. (b)–(d) The partial averages in the percolating $\to$ nonpercolating, percolating $\to$ percolating, and nonpercolating $\to$ nonpercolating cases, respectively, for three different network sizes. The inset in (d) shows the average number of converted bonds in the nonpercolating $\to$ nonpercolating case.

Figure 15

(Color online) The average fractions of bonds: converting from rigid to floppy when a bond is removed and the network switches from percolating to nonpercolating (marked “removal”); converting from floppy to rigid when a bond is inserted and the network switches from nonpercolating to percolating (“addition”). These are compared to the difference between the average fractions of rigid bonds in the percolating and nonpercolating cases (marked “difference”). All quantities are for networks of $90000$ sites.

Figure 16

(Color online) The probability that a percolating stressed region forms when a bond is inserted at a random “disallowed” place into a network with a percolating rigid cluster.

Figure 17

(Color online) The average fraction of stressed bonds in the self-organized network after a single “disallowed” bond is inserted. (a) The overall average, as well as the partial averages for cases when stress does and does not percolate, for networks of $90000$ sites. (b) The partial average for the case when stress percolates, for three different network sizes. (c) The partial average when stress does not percolate, again, for three different network sizes; in the inset the average number of stressed bonds is shown for the same case.

Figure 18

Variations in the vibrational frequency of CS tetrahedral units as a function of pressure for different ${\mathrm{Ge}}_x{\mathrm{Se}}_{1-x}$ glasses. Triangles are results taken from the work of Murase and Fukunaga [33] and filled circles are results of the work of Wang et al. [18]. The figure is taken from Ref. 18.