Spatiotemporal intermittency and localized dynamic fluctuations upon approaching the glass transition

We introduce a robust approach for characterizing spatially and temporally heterogeneous behavior within a system based on the evolution of dynamic fluctuations averaged over different space lengths and timescales. We apply it to investigate the dynamics in two canonical systems as the glass transition is approached: simulated Lennard-Jones liquids and experimental dense colloidal suspensions. In both cases the onset of glassiness is marked by spatially localized dynamic fluctuations originating in regions of correlated mobile particles. By removing the trivial system size dependence we show that the spatial heterogeneity of the dynamics extends to large length scales containing tens to hundreds of particles, corresponding to the timescale of maximally non-Gaussian dynamics.

Figure 1

(a) Contour plot of the distance matrix ${\mathrm{\Delta }}^2(t^{\prime },t^{\prime \prime })$ for a binary Lennard-Jones system at $T=0.50$ within a cubical subsystem containing 125 particles. (b) Contour plot of the same system for the full 8000 particle simulation. Data taken from Ref. [29]. (c) $\mathrm{\Omega }(t^{\prime },t^{\prime \prime })$. The data correspond to the same subsystem as in (a). In the image shown, darker points indicate smaller values. (d) The values of ${\mathrm{\Delta }}^2$ as a function of $\mathrm{\Delta }t=|t^{\prime }-t^{\prime \prime }|$ for the data shown in (a). For small $\mathrm{\Delta }t$ this tends to 0, and for intermediate $\mathrm{\Delta }t$ the scatter indicates the temporal heterogeneity in this subsystem. The overall increase mirrors the mean-square displacement $M^2\left(\mathrm{\Delta }t\right)$. (e) ${\mathrm{\Omega }}^2\left(\mathrm{\Delta }t\right)$ for the data shown in (c). The scatter at small $\mathrm{\Delta }t$ reflects the non-Gaussian statistics of the displacements at those time scales.

Figure 2

$\mathrm{\Omega }\left(N\right)$ and ${\mathrm{\Omega }}_R\left(N\right)$ as a function of subsystem size $N$ for the binary Lennard-Jones system [(a) and (c)] for different temperatures as indicated and for the experimental colloidal suspension [(b) and (d)] at different volume fractions $\varphi$ as indicated. In the case of $\mathrm{\Omega }\left(N\right)$ the $N$ particles are part of the same subsystem. In the case of ${\mathrm{\Omega }}_R\left(N\right)$ the $N$ particles are selected randomly among all particles in the system. The dashed lines in (a) indicate power-law decay with the exponents shown, and the inset to (a) shows the decay exponent as a function of $T$ for the data. The decay exponent is found by fitting the data for $N\le 800$, and for all $T$ we find a high-quality fit ($R^2\ge 0.99$).

Figure 3

Subtracting $\mathrm{\Omega }$ calculated for randomly chosen particles from $\mathrm{\Omega }$ calculated for compact subsystems. (a) For the LJ system. The inset shows the maximum of the curves as a function of temperature. (b) The same quantity for the colloidal suspensions. The black curve in (a) shows the same quantity evaluated for a system of ${10}^6$ particles at $T=0.466$, showing that the position of the maximum is not affected by the finite size of the investigated system.

Figure 4

$\mathrm{\Omega }\left(\mathrm{\Delta }t\right)$ averaged over all subsystem sizes $N$, for the LJ data with different temperatures as indicated by the color scale. Red circles indicate the peak of each curve, and yellow circles indicate the timescale $\mathrm{\Delta }{t}^{*}$.