Two rigidity-percolation transitions on binary Bethe networks and the intermediate phase in glass

Rigidity percolation is studied analytically on randomly bonded networks with two types of nodes, respectively, with coordination numbers $z_1$ and $z_2$, and with $g_1$ and $g_2$ degrees of freedom each. For certain cases that model chalcogenide glass networks, two transitions, both of first order, are found, with the first transition usually rather weak. The ensuing intermediate pase, although not isostatic in its entirety, has very low self-stress. Our results suggest a possible mechanism for the appearance of intermediate phases in glass that does not depend on a self-organization principle.

Figure 1

(a) The thin continuous line indicates the multivalued solution $T\left(b\right)$ obtained from Eqs. (4) with $0\le x\le 1$, for a Cayley tree with $g=2$ and $z=6$. This solution includes $T=0\forall b$, which actually corresponds to a single point $x=0$ and is always a locally stable solution. This branch is continued, for $x>0$, by an unstable branch with $b$ a decreasing function of $x$, up to the reversal point $b_c^{\mathrm{Cayley}}$, where it becomes locally stable, with $b$ an increasing function of $x$. On a Cayley tree (dashed; red online), a first-order critical RP transition happens at bond density $b_c^{\mathrm{Cayley}}=0.602789$ if the boundary is perfectly rigid. On a Bethe network with the same $g$ and $z$ the transition happens later (thick line), becoming an ordinary first-order RP transition. The exact location $b_c^{\mathrm{Bethe}}=0.656511134$ of the transition on the Bethe network can be determined analytically [see (12)] from the requirement that the density $r$ of redundant bonds (b) be continuous [38].

Figure 2

Example of $b$-driven phase transitions on mixed Bethe networks with $g_1=g_2=2$, $z_1=4$, and $z_2=20$. The “weak” fraction $p_1$ is, from left to right: $p_1=0.12,0.24,0.36,...,0.96$. (a) Average o-rigid probability $T$, defined by (18), as a function of link density $b$. The leftmost case shows two phase transitions. (b) Density or redundant links $r$, as given by (28), as a function of link density $b$. Notice the double Maxwell loop in the rightmost case.

Figure 3

Example of $p_1$-driven phase transitions on mixed Bethe networks with $g_1=g_2=2$, $z_1=4$, and $z_2=20$. The link density is, from left to right, $b=0.2,0.3,0.4,...$ (a) Average o-rigid probability $T$, defined by (18), as a function of chemical composition $p_1$. Double phase transitions, which happen for $b\approx 0.85$ are not seen in this figure, but are displayed in detail in Fig. 5. (b) Density or redundant links $r$, as given by (28), as a function of chemical composition $p_1$.

Figure 4

Zoom of double phase transitions in the $b$-driven case, on mixed Bethe networks with $g_1=g_2=2$, $z_1=4$, and $z_2=20$. The “weak” fraction $p_1$ is, from left to right, $p1=0.940,0.945,0.950,...,0.980.$ (a) Average o-rigid probability $T$, defined by (18). (b) o-rigid probabilities for each of the components (dashed; blue online: strong component, continuous; red online: weak component). (c) Density or redundant links $r$, as given by (28).

Figure 5

Zoom of double phase transitions in the $p_1$-driven case, on mixed Bethe networks with $g_1=g_2=2$, $z_1=4$, and $z_2=20$. The link density $b$ is, from left to right, $b=0.830,0.835,0.840,...,0.865$. (a) Average o-rigid probability $T$, defined by (18). (b) o-rigid probabilities for each of the components (dashed; blue online: strong component, continuous; red online: weak component). (c) Density or redundant links $r$, as given by (28).