Heterogeneities in the glassy state

We study heterogeneities in a binary Lennard-Jones system below the glass transition using molecular dynamics simulations. We identify mobile and immobile particles and measure their distribution of vibrational amplitudes. For temperatures near the glass transition the distribution of vibrational amplitudes obeys scaling and compares reasonably well with a mean-field theory for the amorphous solid state. To investigate correlations among the immobile and mobile particles we identify clusters and analyze their size and shape. For a fixed number of immobile particles we observe that the immobile particles cluster more strongly together as the temperature is increased which allows the particles to block each other more effectively and to therefore stay immobile. For the mobile particles, on the other hand, the clustering is most pronounced at low temperatures, indicating that mobility at low temperatures can only be sustained in cooperative motion.

Figure 1

(Color online) Vibrational amplitude $⟨d_i^2⟩$ as a function of temperature for the 5% fastest and 5% slowest particles. See Fig. 5 for an enlargement of $⟨d_i^2⟩\left(T\right)$ for the 5% slowest $A$ and $B$ particles.

Figure 2

Sketch of a particle trajectory to illustrate the definition of irreversible and reversible jumps.

Figure 3

The fraction of immobile particles $Q$ as a function of temperature $T$ using definition I for immobility. $Q\left(T\right)$ is given both for $A$ and $B$ particles separately and for $d_{\mathrm{cut}}^2=0.2$ and $d_{\mathrm{cut}}^2=0.01$. The simulations are with initial temperature $T_{\mathrm{init}}=0.5$.

Figure 4

Same as Fig. 3, but for definitions III and IV, for $A$ and $B$ particles and for different initial temperatures.

Figure 5

(Color online) Same as Fig. 1 for immobile particles using definitions I–IV and for $A$ and $B$ particles. The inset shows an enlargement for low temperatures with linear fits through the origin and a line at $⟨d_i^2⟩=0.01$ for the Lindemann criterion.

Figure 6

Distribution of the normalized vibrational amplitude compared to a mean-field theory [3]. For the simulation data the definition III for immobility has been used and the distribution is shown for $A$ particles.

Figure 7

Average number of cluster members $s$ as a function of temperature. Shown are an average of $s$ over only the biggest cluster of each independent configuration $⟨s_{\mathrm{bcl}}⟩$ and an average over all clusters $⟨s⟩$ are shown for definitions I, III, and IV.

Figure 8

Same as Fig. 7 but for definition II and only for $⟨s_{\mathrm{bcl}}⟩$.

Figure 9

Average number of clusters $⟨K⟩$ as a function of temperature using definition II.

Figure 10

The average cluster size $⟨S⟩$ as a function of percentage $p$ using definition II for immobility. The average is over all nonpercolating clusters and over independent initial configurations. The inset shows the fraction of percolating configurations as a function of $p$.

Figure 11

Log-log plot of the radius of gyration, $R_{\mathrm{G}}$, as a function of the number of cluster members $s$ using definition I for immobility. The line is a linear fit with slope 0.55.

Figure 12

Average coordination number of members of the biggest cluster $⟨z_{\mathrm{bcl}}⟩$ as a function of temperature using definition II.

Figure 13

(Color online) Distribution of coordination numbers $P\left(z_{i,\mathrm{bcl}}\right)$ of the 5% fastest particles in the biggest cluster for simulations with $T_{\mathrm{init}}=0.5$. The inset shows $P\left(z_{i,\mathrm{bcl}}\right)$ for the 5% slowest particles for simulations with $T_{\mathrm{init}}=0.446$.

Figure 14

Distribution of ring sizes $P\left(n_r\right)$ within the largest cluster of each independent configuration using definition II.

Figure 15

Average ring size $⟨n_{\mathrm{bcl}}⟩$ as a function of temperature using definitions III and IV and the comparison of $⟨n⟩$ for the complete system (100%).