Structural Relaxation is a Scale-Free Process

We show that in deeply supercooled liquids, structural relaxation proceeds via the accumulation of Eshelby events, i.e. local rearrangements that create long-ranged and anisotropic stresses in the surrounding medium. Such events must be characterized using tensorial observables and we construct an analytical framework to probe their correlations using local stress data. By analyzing numerical simulations, we then demonstrate that events are power-law correlated in space, with a time-dependent amplitude which peaks at the alpha relaxation time ${\tau }_{\alpha }$. This effect, which becomes stronger near the glass transition, results from the increasingly important role of local stress fluctuations in facilitating relaxation events. Our finding precludes the existence of any length scale beyond which the relaxation process decorrelates.

Figure 1

In an IS of an $L=80$ system equilibrated at $T=0.26$. (a) Pressure, ${\sigma }_1$ (b) ${\sigma }_2\equiv \frac{1}{2}({\sigma }_{xx}-{\sigma }_{yy})$, and (c) ${\sigma }_3={\sigma }_{xy}$.

Figure 2

For $L=80$, $T=0.26$, $t=100$. (a) $\delta {\sigma }_3(\underset{\_}{r};t_0,t_0+t)$, (b),(c) the associated displacement and force fields, respectively, in the upper-left quarter of the cell (see $x,y$ coordinates).

Figure 3

For $L=160$, $T=0.26$, $t=100$. 2D plots of (a) ${\mathcal{C}}_{13}(\underset{\_}{r};t)$, (b) ${\mathcal{C}}_{23}(\underset{\_}{r};t)$, and (c) ${\mathcal{C}}_{33}(\underset{\_}{r};t)$.

Figure 4

Temperature $T=0.26$. (a) At $t=100$ cuts of ${\mathcal{C}}_{13}$, ${\mathcal{C}}_{22}$, and ${\mathcal{C}}_{33}$ along appropriate axes; the dashed line has slope $-2$. (b) ${\mathcal{C}}_{11}(0,t)$, ${\mathcal{C}}_{33}(0,t)$, and their ratio vs $t$. (c) At $t=100$, $Y_2^{13}(r;t)$ (red) and $X_4^{33}(r;t)$ (black) vs $r$, for increasing $L\mathrm{s}$. (d) $Y_2^{13}$ (dashed) and $X_4^{33}$ (full lines) vs $r$ at increasing times up to ${\tau }_{\alpha }$.

Figure 5

For different $T\mathrm{s}$: ${\kappa }_{\mathcal{C}}$, the height of the $X_4^{33}$ plateau, and ${\kappa }_{\mathcal{S}}$ as inferred from [36]; arrows points to the $t\to \infty$ values obtained from $⟨{\sigma }_i{\sigma }_j⟩$ data; filled symbols mark the times ${\tau }_{\alpha }(T)$.