Unconstrained motions, dynamic heterogeneities, and relaxation in disordered solids

A disordered network of bonds with a fixed configuration can relax via a variety of unconstrained motions. These motions can be directly inferred from the topological arrangement of constraints without any geometrical information. We use the pebble game algorithm of Jacobs and Thorpe [D. J. Jacobs and M. F. Thorpe, Phys. Rev. Lett. 75, 4051 (1995)] to decompose the system into separate rigid clusters and identify the remaining degrees of freedom. Unconstrained motions can then be resolved in the form of translations and rotations of isolated groups of bonds and the internal motion within bond groups. We show that each motion can be assigned a characteristic thermal velocity and hence a relaxation time scale. We use this information to construct a relaxation function and also examine the spatial distribution of relaxation time scales. We investigate the sensitivity of the relaxation time scales and their spatial distribution when making individual bond changes in the system, and we consider the dependence of these time scales on the underlying structure.

Figure 1

(Color online) Two example systems based on a triangular lattice arrangement of 100 sites in periodic boundary conditions are shown. The left panel shows a system at a bond density of 0.633 (before the percolation transition) and the right panel shows a bond density of 0.667 (after the percolation transition). Lattice sites or vertices are shown either as open circles, for a pivot point, or as black dots. The lines show bonds between the vertices. The different shades indicate the presence of separate rigid clusters. Overconstrained regions, indicated by heavy lines, are present in the system on the right.

Figure 2

The two panels show a system of five sites and five bonds forming a pentagon. Two possible arrangements of pebbles describing the degrees of freedom within the system are shown. The left panel shows a symmetrical pebble arrangement and the right panel shows one possible final arrangement resulting from a pebble game analysis of the system. In the symmetrical arrangement of pebbles, moving any of the free pebbles onto a bond and subsequent shuffling of pebbles to different bonds (ignoring any other free pebbles) allows a pebble to be freed at any of the other vertices of the pentagon; every degree of freedom is delocalized across the whole system. In the right panel, three of the free pebbles remain localized and cannot be moved around the whole system.

Figure 3

(Color online) Distributions of participation ratios $n_p$, for all motions in the system, are shown for the interconnected lattice structure in the left panel of Fig. 1. The results are derived from an analysis of unconstrained motions (left panel) and an analysis of floppy modes (right panel). The graphs compare results for four different site labeling schemes and the histograms have been shifted vertically for clarity. The results are identical in the unconstrained motion analysis, confirming that the representation of unconstrained motions is unique. The same cannot be said for the representation of modes from the floppy mode analysis as we see that different site labels result in different distributions of modes.

Figure 4

(Color online) For a system of five sites joined together to form a pentagon, a pebble game analysis shows that each bond is a separate rigid cluster. Therefore, pivot points connecting different rigid clusters are present at each vertex. The left panel shows a test bond, represented by a dashed line, placed across the leftmost pivot. The right panel shows the resulting rigid cluster of three bonds formed by this test bond. None of the other bonds are affected by this new bond and there are four remaining pivots.

Figure 5

The average number of unconstrained motions per floppy mode, $n_{\text{motions}}/n_{\text{floppy}}$, is shown as a function of bond density $\rho$. The unconstrained motions include three modes—two translations and one rotation—for each free cluster, two modes (translation) for each free particle, and one mode for each group of interdependent pivots. The dashed line shows the ideal ratio of one unconstrained motion per floppy mode.

Figure 6

(Color online) The variation in the moment of inertia $I$ of rigid clusters with their size $s$. Cluster size is given by the number of bonded sites in the cluster. The data are taken from systems with bond densities between 0.433 and 0.67. The solid line shows a linear regression fit to the data, $y=-2.13+2.05x$. The dashed line is the relationship expected for solid disks with a uniform mass.

Figure 7

(Color online) The two panels show a pivot cluster formed from one large rigid cluster and one small rigid cluster, which is a single bond. Motion of the small cluster is independent of the size of the large cluster as indicated in the left panel. By the same token, we can describe the motion of the larger rigid cluster with the small cluster frozen in position (right panel).

Figure 8

(Color online) The total moment of inertia of a group of rigid clusters joined together by interdependent pivots is calculated by adding together the moments for the individual clusters, excluding the largest cluster. This is calculated by summing the exact moments of inertia of each cluster $\left(I_{\text{calc}}\right)$ and by summing moments of inertia given by $I=0.119s^{2.05}$, where $s$ is the size of the rigid cluster $\left(I_{\text{est}}\right)$. The line shows $I_{\text{est}}=I_{\text{calc}}$.

Figure 9

(Color online) Normalized histograms showing average distributions of relaxation frequencies resulting from pivot and free clusters for five different bond densities—shown from bottom to top: 0.433, 0.597, 0.65, 0.66, and 0.67—on a double-logarithmic plot. Frequencies for both translational and rotational modes of free clusters are shown, as well as the two highest relaxational frequencies for each pivot cluster.

Figure 10

(Color online) The overall relaxation $\Phi \left(t\right)$ averaged over 100 different random bond configurations for each of five different bond densities, shown from bottom to top: 0.433, 0.597, 0.65, 0.66, and 0.67. For the high bond densities, complete relaxation of the system does not occur for a fixed bond configuration; full relaxation would require changes in bond configuration.

Figure 11

(Color online) Spatial maps of the shortest relaxation time per particle for a number of different bond densities from left to right: 0.597, 0.65, 0.66, and 0.67. We consider each unconstrained motion in the system involving the translation or rotation of free clusters or the collective motion of groups of rigid clusters connected by interdependent pivots. Each site on the lattice may be involved in any number of unconstrained motions but is assigned a time scale according to the mode with the maximum thermal velocity. The logarithmic scale gives the relaxation time for each site.

Figure 12

(Color online) Spatial maps of the variance in shortest relaxation time for each particle at a number of different bond densities from left to right: 0.597, 0.65, 0.66, and 0.67. The variance is given by comparing time scales for 100 systems formed by one bond change from the same initial configuration as shown in Fig.11. The logarithmic scale gives the variance in relaxation time for each site.

Figure 13

(Color online) The two panels show changes recorded for 100 single bond moves from initial configurations at four different densities, from bottom to top: 0.597, 0.65, 0.66, and 0.67. The left panel shows the distribution of changes in the shortest relaxation time scale for each site $\left(\Delta \tau \right)$ and the right panel shows the changes in rigid cluster size $\left(\Delta s\right)$ for the same bond moves. The change in rigid cluster size has been scaled by 0.345 to allow a direct comparison between the two panels.