Probing a Critical Length Scale at the Glass Transition

We give evidence of a clear structural signature of the glass transition, in terms of a static correlation length with the same dependence on the system size, which is typical of critical phenomena. Our approach is to introduce an external, static perturbation to extract the structural information from the system’s response. In particular, we consider the transformation behavior of the local minima of the underlying potential energy landscape (inherent structures), under a static deformation. The finite-size scaling analysis of our numerical results indicate that the correlation length diverges at a temperature $T_c$, below the temperatures where the system can be equilibrated. Our numerical results are consistent with random first order theory, which predicts such a divergence with a critical exponent $\nu =2/3$ at the Kauzmann temperature, where the extrapolated configurational entropy vanishes.

Figure 1

Coarse-grained nonaffine displacement field ${\mathbf{D}}_b(\mathbf{r})$ defined in Eq. (1) for high ($T=1.0$, left panel) and low ($T=0.4$, right panel) temperature. The coarse-graining length was chosen as $b=2.0$. The color code illustrates surfaces of constant $|\mathbf{D}{|}^2$ values. For better visibility, only particles with $|{\mathbf{D}}_j{|}^2\ge 0.5$ are shown.

Figure 2

Histogram of mismatch lengths $h_b({\mathbf{D}}^2)$ at high (a $T=1.0$) and low (b $T=0.4$) temperature for different values of the coarse-graining length $b$. At high temperatures, the typical value of ${\mathbf{D}}^2$ jumps from an ordered (${\mathbf{D}}^2\approx 1$) to a disordered one (${\mathbf{D}}^2\approx 0$) when $b$ becomes slightly larger than the particle diameter. At low temperatures, instead, this transition happens at much larger values of $b$.

Figure 3

Top: Static correlation length ${\xi }_B$ of coarse-grained mismatch vectors as a function of temperature for different system sizes. Bottom: Data collapse of ${\xi }_{B,L}(T)$ according to Eq. (2).

Figure 4

(a) The structural relaxation time $\tau$ is plotted versus static correlation length ${\xi }_B$ for the system with $N=64000$ particles. The solid line is the best fit to the data and shows an exponential dependence of the form $\tau \sim \exp [A^{\prime }({\xi }_B/k_{B}T{)}^{1.14}]$. (b) The static correlation length is shown as a function of the configurational entropy $S_{\mathrm{conf}}$. The solid line is the best fit to the data and shows the power-law relation ${\xi }_B\sim (T{S}_{\mathrm{conf}}{)}^{-1.02}$.