Distribution of local relaxation events in an aging three-dimensional glass: Spatiotemporal correlation and dynamical heterogeneity

We investigate the spatiotemporal distribution of microscopic relaxation events, defined through particle hops, in a model polymer glass using molecular dynamics simulations. We introduce an efficient algorithm to directly identify hops during the simulation, which allows the creation of a map of relaxation events for the whole system. Based on this map, we present density-density correlations between hops and directly extract correlation scales. These scales define collaboratively rearranging groups of particles and their size distributions are presented as a function of temperature and age. Dynamical heterogeneity is spatially resolved as the aggregation of hops into clusters, and we analyze their volume distribution and growth during aging. A direct comparison with the four-point dynamical susceptibility ${\chi }_4$ reveals the formation of a single dominating cluster prior to the ${\chi }_4$ peak, which indicates maximally correlated dynamics. An analysis of the fractal dimension of the hop clusters finds slightly noncompact shapes in excellent agreement with independent estimates from four-point correlations.

Figure 1

Sample trajectory of a particle and hop identifier function $p_{\text{hop}}$. (a) shows the localization into two cages and the “hop” is marked by rapid changes in the trajectory (e). Corresponding to this, $p_{\text{hop}}$ in (d) is sharply peaked at the transition and the maximum defines the hop time $t_{\text{hop}}$. Plots (b) and (c) are overlays of $p_{\text{hop}}$ and the trajectory just before and after the hop ($z$ comp. only for better visibility). The colored fields (A and B) in both plots indicate the evaluation window for $p_{\text{hop}}$ [see Eq. (1)]. Initial and final position of the particle is calculated from time averages of the trajectory sections that are highlighted as zoom, i.e., (b) initial position (orange) and (c) final position (cyan).

Figure 2

Histograms of hop identifier $p_{\text{hop}}$ peak heights calculated at glass temperature $T=0.3$ and time window parameters (order of legend):$(N_{\text{hist}},N_{\text{obs}})=(20,100),(40,50),(10,200),(10,100),(40,100)$. The data for the three parameter sets with $t_{\text{eval}}=15{\tau }_{\text{LJ}}$ are nearly identical (blue $\text{}$ overlay other markers). The solid line is an exponential fit of the distribution tail and the dashed line indicates the resulting parameter choice $p_{\mathrm{th}}$. The inset shows the mean square displacement for temperatures $T=0.2$(blue $◯$), $T=0.25$(green $▽$), and $T=0.3$(red $\diamond$). The horizontal lines indicate $p_{\text{th}}$ at the corresponding temperature.

Figure 3

Snapshots of a single configuration showing (a) all particles and (b) only those particles that are in the middle of a hop. There are only four hops at that time step and their position is highlighted by arrows. (c) shows all hops that are detected in a time window of $3000{\tau }_{\text{LJ}}$. The configurations are taken from a glass at $T=0.2$ and age $20000{\tau }_{\text{LJ}}$.

Figure 4

Top: Distribution of persistence time $\tau$ of cages at three temperatures. The solid lines indicate power laws. Bottom: Distribution of hop distances $\left|d\right|$, i.e., the distance between old and new cage. Exponential fits are indicated by the solid lines and the vertical dashed lines indicate $\sqrt{p_{\text{th}}}$ and $2\sqrt{p_{\text{th}}}$ for the respective temperatures; see legend in top panel. Error bars in both plots are smaller than the markers and are omitted for visibility.

Figure 5

The hop frequency $f_{\text{hop}}$ is the number of hops observed in the whole system per time ${\tau }_{\text{LJ}}$. Shown is the frequency over the duration of the simulation run in a log-log plot, with solid lines indicating power-law fits.

Figure 6

Hop displacement autocorrelation for three glass temperatures (top) at age $t_{\text{age}}={10}^5{\tau }_{\text{LJ}}$. The histogram (bottom) is calculated using normalized displacement vectors ($T=0.25$ and age as above) and reveals an anisotropy in the direction of consecutive hops. The lines are guides to the eye.

Figure 7

(a) Probability density surface for spatiotemporal separation of two different hopping particles based on Eq. (4) for a glass at $T=0.25$ and $t_{\text{age}}={10}^5{\tau }_{\text{LJ}}$. The color scale is logarithmic (scale at the top-right corner) and the dashed lines indicate integration limits used to calculate the one-dimensional probability functions. Center plots show the probability function of (b) separation and (c) time delay between hops for $T=0.2$ (blue $◯$), $T=0.25$ (green $▽$), and $T=0.3$ (red $\diamond$) at the same age. The gray vertical dashed lines in (b,c) illustrate the correlation ranges and the black dashed curve in (b) indicates the radial distribution function. (d) Probability density surface following Eq. (4) for the same glass as the left panel, but with $r^{*}$ calculated from initial position (at the origin) to the final position of the second particle after the hop. The color scale is again logarithmic.

Figure 8

Size distribution of cooperatively rearranging particles. The main panel shows distributions at six ages for a glass at temperature $T=0.3$; see legend in Fig. 9. The inset shows the size distribution of three glasses at age $t_{\text{age}}={10}^5{\tau }_{\text{LJ}}$ and temperatures $T=0.2$ (blue $◯$), $0.25$ (green $▽$), $0.3$ (red $\diamond$). Both plots have the same axes ranges, and the solid black lines indicate $P\left(s\right)\propto \exp \left(-s\right)$.

Figure 9

The top panel shows the number of caged, i.e., not yet hopped, particles $N_{\text{caged}}$ averaged over independent simulations as a function of time for six glass ages. The four-point susceptibility ${\chi }_4$ shown in the bottom panel is calculated from the variance of $N_{\text{caged}}$, Eq. (5).

Figure 10

Snapshots of the growth of a single cluster over time. The particles are visualized at their initial positions (before the hop) and the coloring indicates depth. The plot on the right shows the cluster volume of 15 example clusters as a function of time. Examples were recorded at glass age $t_{\text{age}}={10}^5{\tau }_{\text{LJ}}$ and an animation that further illustrates the growth shown in the snapshots can be found online [37].

Figure 11

Mean cluster volume (top) given as a fraction of the total simulation box volume and the number of simultaneous clusters (bottom) as a function of time. Results for six ages are shown, see legend in Fig. 9. The insets show data collapse when time is rescaled by the time of the ${\chi }_4$ peak.

Figure 12

Mean hole volume (top) given as a fraction of the total simulation box volume and the number of simultaneous holes (bottom) as a function of time. Results for six ages are shown, see legend in Fig. 9. The insets show data collapse when time is rescaled by the time of the ${\chi }_4$ peak.

Figure 13

Collection of cluster volume distributions of a glass at age $t_{\text{age}}={10}^5{\tau }_{\text{LJ}}$ measured at various times in double-log scale. Each distribution is plotted in the $V_{\text{cl}}$-$p\left(V_{\text{cl}}\right)$ plane and placed along the $t$ axis according to the size of the time window used for the cluster analysis. The back wall shows an overlay of distributions at small times in a single plane. We include data at times $\lesssim {10}^3{\tau }_{\text{LJ}}$ (up to and including the second blue distribution, see arrow) and the same colors as the separate distributions are used to indicate the origin of the data points. The solid line on the back wall indicates a power law with exponent $-2$. The black solid curve on the floor wall indicates ${\chi }_4$ as a function of time.

Figure 14

Main panel shows the mean fractal dimension of the hop clusters over time for six ages, see legend in Fig. 9. The inset shows data collapse when time is rescaled by the time of the ${\chi }_4$ peak.