Characterizing Order in Amorphous Systems

We measure and compare three correlation lengths proposed to describe the extent of structural order in amorphous systems. In particular, the recently proposed “patch correlation length” is measured as a function of temperature and fragility and shown to be comparable with other measures. In addition, we demonstrate that the patch method also allows us to characterize the symmetries of the local order without any a priori knowledge of it.

Figure 1

Patch entropy as a function of patch volume: temperatures range from $T/T^{*}=4.35$ (left) to $T/T^{*}=0.59$ (right) for a system of 2138 particles. $V_{\sigma }$ is the volume of a single particle and $T^{*}$ the temperature of the avoided crystallization (see Ref. 16). Note the three regimes of the entropy: constant and near zero for small sizes, then increasing linearly, and finally saturation. The patch correlation length is extracted from the inverse of the slope of the linear (middle) regime, measured by a linear fit indicated by a dot-dash line for the lowest temperature. Insert: ratio of extracted length scales for $T/T^{*}=0.35$ and $T/T^{*}=3.93$ for the less fragile system versus the distance threshold involved in patch comparison. In this article, the optimum value $d^{*}=0.425$ given by the maximum has been used.

Figure 2

The structural correlation lengths vs temperature for different fragilities: ${\xi }_6$ (dark blue open symbols) denotes the hexatic length, ${\xi }_{\mathrm{PTS}}$ (light green open symbols) is the point-to-set length, and ${\xi }_S$ (solid red symbols) is the patch correlation length. Errors bars are roughly of the same size as the symbols. The vertical axis corresponds to absolute values of ${\xi }_6$. ${\xi }_{\mathrm{PTS}}$ and ${\xi }_S$ have been linearly scaled but their raw values are of the same order of magnitude as ${\xi }_6$: ${\xi }_{\mathrm{PTS}}$ is multiplied by a factor between 0.56 and 1.1 (depending on the curvature), ${\xi }_S$ by a factor between 0.52 and 0.65, and both have been vertically shifted. The three different curvatures $\kappa$ correspond to three different fragilities of the system, with the smallest curvature being associated with the largest fragility.

Figure 3

Density profile of the mean patch of the largest class of patches: the diameter is equal to $8.4\sigma$, $T/T^{*}=0.52$, and $\kappa \sigma =0.05$. This class contains 1591 repeating patches out of a total of 4279 patches in the system, so roughly one third of the system is used to construct this density profile. Red (dark) spots indicate the most probable positions of particles inside of patch of this diameter, indicating a pronounced hexagonal order out to this scale.