Ring correlations in random networks

We examine the correlations between rings in random network glasses in two dimensions as a function of their separation. Initially, we use the topological separation (measured by the number of intervening rings), but this leads to pseudo-long-range correlations due to a lack of topological charge neutrality in the shells surrounding a central ring. This effect is associated with the noncircular nature of the shells. It is, therefore, necessary to use the geometrical distance between ring centers. Hence we find a generalization of the Aboav-Weaire law out to larger distances, with the correlations between rings decaying away when two rings are more than about three rings apart.

Figure 1

A piece of a two-dimensional cellular network generated by a bond-switching algorithm from a honeycomb lattice. Rings are colored based on their size, pentagons (black), hexagons (yellow), and heptagons (green). On the bottom left corner, a group of six-fold rings can be seen, which also happens in experimental samples and is a feature of amorphous materials, due to statistical correlations. A central six-fold ring has been left uncolored and shells of rings will be found around this. Any ring can be used as a central location.

Figure 2

(a) Partitioning of the random network in Fig. 1 into topological shells. The shells grow roughly in circular shapes. This piece also has a triplet inclusion in the fourth (blue) shell where a five-fold ring is isolated from the fifth (purple) shell. (b) Although shells are roughly circular, no circle can sweep all rings within a single shell; hence, ring distributions with topological and geometrical definitions are different.

Figure 3

Dependence of the number of rings, $K_t\left(n\right)$, on topological distance $t$ and size of the central ring $n$. $K_t\left(n\right)$ grows linearly for $t\ge 4$. Solid lines are fitted lines to the last three points. Points are offset for clarity with an additional $6n$ added.

Figure 4

Shell charge $q_t\left(n\right)$ versus topological distance $t$. The shell charge settles to a constant nonzero number for $t\ge 4$. The dashed line shows the asymptotic offset $-0.4$.

Figure 5

The topological charge $q_r\left(n\right)$ per shell is plotted against the geometrical distance $r$. The shell charge approaches zero for distances about three rings away. This figure should be compared to Fig. 4. The two dashed lines represent the geometrical distance corresponding to the minimum and maximum values for the first shell with $t=1$. Curves are offset for clarity where horizontal solid lines show the expected asymptotic values of zero.

Figure 6

Plot of two coefficients in the generalized Aboav-Weaire law, ${\alpha }_r$ and ${\mu }_r$, with their topological counterparts, ${\alpha }_t$ and ${\mu }_t$. Geometrical definitions show that correlation quickly decays to zero while pseudocorrelations in the topological case last over a long range for ${\mu }_t$. The geometrical distances are chosen so the geometrical and topological distances agree for the first shell.

Figure 7

Plot of weighted mean ring size $nm\left(n\right)$ versus ring size $n$ for the third neighbors using both geometrical and topological distance. This plot shows that the mean ring size follows the generalized Aboav-Weaire law [Eq. (9)]. Note that the topological definition leads to a slightly larger mean ring size. A plot of the coefficients $\alpha$ and $\mu$ is shown in Fig. 6.

Figure 8

Ring-ring correlation $C_t(n,n^{\prime })$ versus topological distance $t$. The correlations are short or medium range depending on the size of the interacting rings. Although hexagons are weakly correlated with their neighbor rings, other rings show a high degree of correlations up to three rings away. Very similar results are obtained using geometrical distances. Note that correlations are symmetric so that a 5-6 correlation is the same as a 6-5 correlation, etc. (a) Five-fold rings, (b) six-fold rings, (c) seven-fold rings, and (d) eight-fold rings.