Excitations Are Localized and Relaxation Is Hierarchical in Glass-Forming Liquids

For several atomistic models of glass formers, at conditions below their glassy-dynamics–onset temperatures, $T_{\mathrm{o}}$, we use importance sampling of trajectory space to study the structure, statistics, and dynamics of excitations responsible for structural relaxation. Excitations are detected in terms of persistent particle displacements of length $a$. At supercooled conditions, for $a$ of the order of or smaller than a particle diameter, we find that excitations are associated with correlated particle motions that are sparse and localized, occupying a volume with an average radius that is temperature-independent and no larger than a few particle diameters. We show that the statistics and dynamics of these excitations are facilitated and hierarchical. Excitation-energy scales grow logarithmically with $a$. Excitations at one point in space facilitate the birth and death of excitations at neighboring locations, and space-time excitation structures are microcosms of heterogeneous dynamics at larger scales. This nature of dynamics becomes increasingly dominant as temperature $T$ is lowered. We show that slowing of dynamics upon decreasing temperature below $T_{\mathrm{o}}$ is the result of a decreasing concentration of excitations and concomitantly growing length scales for dynamical correlations that develop in a hierarchical manner, and further that the structural-relaxation time $\tau$ follows the parabolic law, $\log (\tau /{\tau }_{\mathrm{o}})=J^2(1/T-1/T_{\mathrm{o}}{)}^2$, for $T<T_{\mathrm{o}}$, where $J$, ${\tau }_{\mathrm{o}}$ and $T_{\mathrm{o}}$ can be predicted quantitatively from dynamics at short time scales. Particle motion is facilitated and directional, and we show that this becomes more apparent with decreasing $T$. We show that stringlike motion is a natural consequence of facilitated, hierarchical dynamics.

Popular Summary

Supercooling a viscous liquid fast enough leads to the formation of a glass, a material that is mechanically rigid, like crystalline solids, but structurally amorphous, similar to liquids. Our understanding of the molecular processes underlying the glass formation has remained uncertain. Manifestations of the complexity and cooperativity of these processes include stretched exponential decay of time-correlation functions and precipitous slowing of dynamics upon cooling, with relaxation times growing faster than exponentials of reciprocal temperature. Theoretical treatments of the glass structure and dynamics fall into two classes. In one, originating from Adam and Gibbs’s work in 1965, a supercooled liquid is pictured as a mosaic of nearly rigid domains that cannot move or change shape or size without moving all the atoms within the same domain; relaxation follows from collective reorganization of atoms within these so-called “cooperative rearranging regions.” In the other, dating back to Glarum’s work in 1960, the material contains a gas of point-like excitations or local soft spots comprising only a few atoms or molecules; relaxation follows from hierarchical dynamics in which these defects facilitate the creation and destruction of neighboring excitations. In the former case, dynamics slows because mosaic domains grow, while in the latter, dynamics slows because excitation separations grow. There is support for both perspectives, and possibly the two can be related, but the most basic underlying features—the cooperative regions in the former and localized excitations in the latter—have yet to be demonstrated. Here, we show that the latter emerges from Newtonian dynamics.

We do so by considering the molecular dynamics of several different atomistic models for glass-forming liquids in both two and three dimensions. For each, we ask the question: How does an atom move between distinct neighboring positions separated by a length $a$? We are able to answer this question by augmenting standard but extensive numerical simulation with methods of importance sampling of trajectory space. Our findings show that, in glass-forming liquids, microscopic dynamics on one microscopic scale is a microcosm of dynamics on larger scales, much as imagined in the 1980s by P. W. Anderson and co-workers. The specific nature of the hierarchy we elucidate allows us to establish a universal class of models that accurately predict the behaviors of glass-forming liquids.

Figure 1

Excitations can be identified from dynamical processes localized in space and time. Here, identifications are illustrated with rendered representative trajectories for the 2D-$68:32$ system (models in Table ). The unit of length is $\sigma$ and the unit of time is $(ϵ/m{\sigma }^2{)}^{-1/2}$; i.e., see text. (a) The upper panels shows the displacement for the unprocessed coordinate of a tagged particle at time $t$ together with that displacement in the inherent-structure coordinate and in the time-coarse-grained structure, i.e., $|{\overline{\mathbf{r}}}_1(t)-{\overline{\mathbf{r}}}_1(-\Delta t/2)|$. Here $\Delta t$ is the instanton time and $t_a$ is the plateau time. The averaging window used is $\delta t=0.6$. The snapshots are drawn to show the space-filling sizes of particles and are colored to indicate the distance of a particle from its position at $t=-\Delta t/2$. The arrows indicate the direction of that motion. (b) The lower panels show similarly rendered snapshots for the full 10 000-particle system over one instanton time for a tagged particle at the center of the box. Enlargements show detailed structures of the excitation dynamics, where arrows indicate the direction of the color-labeled displacements. Blue indicates relative immobility while red indicates relative mobility. The depictions are for particle density $\rho =0.75$ at four different temperatures (as indicated). The glassy-dynamics–onset temperature for this system at this density is $T_{\mathrm{o}}\approx 2.1$.

Figure 2

Excitation dynamics exhibits separation of time scales with instanton times and plateau times. (a) The probability distribution of the instanton time, $P_a(\Delta t)$, for the W system (see Table ) at the lowest equilibrated temperature for three different displacement lengths, $a=\sigma /4$, $\sigma /2$ and $\sigma$. (b) The mean value of the rate function $R(t_a)$, demonstrating that a plateau value exists for $T$ smaller than the onset temperature, $T_{\mathrm{o}}$. The particular system considered here is the W model at the density $\rho =1.296$, where $T_{\mathrm{o}}\simeq 0.88$. Lines in panel (b) are drawn through data points as guides to the eye.

Figure 3

Excitations are localized and follow Boltzmann statistics of a dilute gas. (a) Distribution of instanton times for excitations with displacement length $a=\sigma$ for the W model at several temperatures below its onset temperature, $T_{\mathrm{o}}\simeq 0.88$. (b) Excess mobility density a distance $r$ from an excitation, relative to its value at $r=\sigma$ for a displacement length $a=\sigma$. The system considered here is the KA model at density $\rho =1.2$ at equilibrium conditions in the supercooled regime. (c) Same as (b) but for very cold nonequilibrium conditions, sampled using transition-path sampling. The insets of (b) and (c) show the corresponding temperature dependence of $\mu (t_a)=⟨|{\overline{\mathbf{r}}}_i(t_a)-{\overline{\mathbf{r}}}_i(0)|⟩$. (d) Temperature dependence of excitation concentration $c_a$ for the 12 systems considered. The inset shows plots of $-\ln c_a$ versus $1/T-1/T_{\mathrm{o}}$ for excitations on a length scale $a=\sigma$. The data collapse to the single curve when scaled by the energy parameter $J_a$. Values of $T_{\mathrm{o}}$ and $J_a$ are given in Table .

Figure 4

Hierarchical excitation-energy scales predict observed structural-relaxation times. (a) Demonstration of logarithmic growth of $J_a$ with respect to $a$ [Eq. (2)]. Fit parameters are given in Table . The inset shows the uncollapsed data. The data set 2D-$68:32$, $\rho =0.75$ is omitted for clarity in the inset, as its scale obscures the other data sets. (b) Structural-relaxation times predicted from results shown in Figs. 3d, 4a, which are compared to measured relaxation times. Parameters from statistics are listed in Table . Data symbols are the same as those in Fig. 3d.

Figure 5

Dynamics is facilitated, directional, and stringlike, and more apparently so as temperature is lowered. (a) Dynamical facilitation volume, $v_{\mathrm{F}}(t)$, for a representative W system at several different temperatures. (b) The peak of the facilitation volume as a function of temperature for the 12 atomistic models studied. (c) $F(r,\cos \theta ,t,t^{\prime };a)$ for a single temperature with parameters $a=\sigma$, $t=t_a/2$, and $t^{\prime }=3t_a/2$ as a function of both the facilitation direction $\theta$ and the distance $r$ from the tagged particle at the origin at time $t_a/2$. (d) $F(r,\cos \theta ,t,t^{\prime };a)$ as a function of the temperature with the same parameters as in (c) for a single facilitation angle $\theta \sim \pi$. (e) $F(r,\cos \theta ,t,t^{\prime };a)$ for a single temperature with parameters $a=\sigma$, $t=t_a/2$, and $t^{\prime }=\tau$, where $\tau$ is the structural-relaxation time, as a function of both the facilitation direction $\theta$ and the distance $r$ from the tagged particle at the origin at time $t_a/2$. (f) $F(r,\cos \theta ,t,t^{\prime };a)$ as a function of temperature with the same parameters as in (e) for a single facilitation angle $\theta \sim \pi$. (g) Mean string length $l_s$ versus the concentration of excitations of displacement length $\sigma$, $c_{\sigma }$ at several different temperatures $T$ for the 10 three-dimensional systems studied. Data symbols are the same as those in Fig. 3d.