Short-Time Beta Relaxation in Glass-Forming Liquids Is Cooperative in Nature

Temporal relaxation of density fluctuations in supercooled liquids near the glass transition occurs in multiple steps. Using molecular dynamics simulations for three model glass-forming liquids, we show that the short-time $\beta$ relaxation is cooperative in nature. Using finite-size scaling analysis, we extract a growing length scale associated with beta relaxation from the observed dependence of the beta relaxation time on the system size. We find, in qualitative agreement with the prediction of the inhomogeneous mode coupling theory, that the temperature dependence of this length scale is the same as that of the length scale that describes the spatial heterogeneity of local dynamics in the long-time $\alpha$-relaxation regime.

Figure 1

Four-point susceptibility ${\chi }_4(t)$ of the 3dKALJ model for different system sizes at $T=0.450$, plotted vs time $t$. Note the appearance of a peak at short times. The height of the peak increases with increasing system size. The filled and open symbols represent data from molecular dynamics and Monte Carlo simulations, respectively (Monte Carlo results are not shown for the largest system with $N=28160$). The peak of ${\chi }_4$ at short time scales disappears in Monte Carlo simulations. Inset: The height of the short-time peak of ${\chi }_4$ is plotted as a function of the length $L$ of the sample, given by $L=(N/\rho {)}^{1/3}$. The height increases linearly with $L$.

Figure 2

Mean square displacement (MSD) and the derivative of the logarithm of MSD with respect to the logarithm of time $t$, shown as a function of $t$, for the 3dKALJ model with $N=28160$. The minimum of the derivative defines ${\tau }_{\beta }$ (see the text for details).

Figure 3

Top panel: ${\tau }_{\beta }$ for the 3dKALJ model, shown as a function of the system size for different temperatures in the interval $T\in [0.45,0.90]$. ${\tau }_{\beta }$ increases with increasing system size, saturating at a value that increases with decreasing temperature. Bottom panel: Scaling collapse of ${\tau }_{\beta }$. For each temperature, the linear size $L$ is scaled by a length $\xi (T)$ such that the data for all temperatures collapse to a master curve. Inset: Comparison of the correlation length extracted from the system-size dependence of ${\tau }_{\beta }$ with the length scale of dynamic heterogeneity in the $\alpha$ regime [41]. The two length scales exhibit the same temperature dependence to a good accuracy.

Figure 4

Top left panel: System-size dependence of ${\tau }_{\beta }$ for the 3dIPL model for all the temperatures studied, including high temperatures. Top middle panel: Data collapse of ${\tau }_{\beta }(N,T)$ to obtain the length scale associated with the cooperativity in $\beta$ relaxation. Top right panel: Comparison of the correlation length extracted from the system-size dependence of ${\tau }_{\beta }$ with that obtained from the system-size dependence of ${\chi }_4^P$. The two length scales exhibit the same temperature dependence. Bottom panels: Results of a similar analysis performed for the 3dR10 model system. One can see that the length scales obtained from ${\tau }_{\beta }$ scaling and ${\chi }_4^P$ scaling exhibit the same temperature dependence.

Figure 5

The relaxation time ${\tau }_{\beta }$ obtained for large system sizes, plotted as a function of the correlation length $\xi$, exhibits a power-law dependence, albeit over a small range of values. The line drawn through the data points represents a power law with exponent $z=0.80$.