Correlation between Dynamic Heterogeneity and Medium-Range Order in Two-Dimensional Glass-Forming Liquids

A glassy state of matter results if crystallization is avoided upon cooling or increasing density. However, the physical factors controlling the ease of vitrification and nature of the glass transition remain elusive. Using numerical simulations of polydisperse hard disks, we find a direct relation between medium-range crystalline ordering and the slow dynamics which characterizes the glass transition. This suggests an intriguing scenario that the strength of frustration controls both the ease of vitrification and nature of the glass transition. Vitrification may be a process of hidden crystalline ordering under frustration, at least in our system.

Figure 1

(a) The $\varphi$ dependence of $F(q_p,t)$ for $\Delta =11\%$. The solid lines represent the fitted functions (see text). (b) Dependence of ${\tau }_{\alpha }$ on $\varphi /{\varphi }_g$ for systems of various $\Delta$’s. The solid lines are the results of the Vogel-Fulcher fittings for ${\tau }_{\alpha }$. The inset shows the $\varphi$ dependence of $\beta$. The lines are guides to the eye. It is worth noting that bifurcation occurs around ${\varphi }_I(\Delta =0)$ (dashed line) and the dynamics become heterogeneous above ${\varphi }_I(\Delta =0)$ (blue region) [Fig. 4a].

Figure 2

(a) Snapshot of a polydisperse colloidal system of $\varphi =0.631$ for $\Delta =9\%$. The color of particles represents ${\overline{\Psi }}_6$ (see the color bar). (b) Distribution function of ${\overline{\Psi }}_6$, $P({\overline{\Psi }}_6)$ for $\Delta =0$ and 9%. (c) $\varphi$ dependence of $g_6(r)/g(r)$ for $\Delta =13\%$. Solid lines are the OZ functions (see text). (d) Dependence of $g(r)$ on $\varphi$ for $\Delta =9\%$ and $g(r)$ of the crystalline structure at $\Delta =0\%$.

Figure 3

Dynamic heterogeneity and MRCO at $\varphi =0.631$ and $\Delta =9\%$. (a) Correlation between ${\overline{\Psi }}_6^i$ and the DW factor. The outer region of each particle is painted red for ${\overline{\Psi }}_6^i>0.75$, otherwise blue. The inner region is, on the other hand, painted white for $⟨\Delta r^2({\tau }_{\beta })⟩<0.023$, otherwise black. Note that the DW factor is negatively correlated with $⟨\Delta r^2({\tau }_{\beta })⟩$ (${\tau }_{\beta }=300$), which characterizes the cage size. (b) Correlation between ${\overline{\Psi }}_6^i$ and the degree of translational motion $⟨\Delta r^2({\tau }_{\alpha })⟩$. The color of the outer region of a particle has the same meaning as in (a). The inner region is, on the other hand, painted yellow for $⟨\Delta r^2({\tau }_{\alpha })⟩<0.063$, otherwise black.

Figure 4

(a) $\varphi$ dependence of $\xi$, ${\xi }_6$, and ${\xi }_4$ for $\Delta =9$, 11, 13, and 16%. (b) Relationship of ${\tau }_{\alpha }$ to $\xi /{\xi }_0$ and ${\xi }_4/{\xi }_{40}$. The solid lines represent ${\tau }_{\alpha }={\tau }_0\exp (D\xi /{\xi }_0)$. (c) $\Delta$ dependence of $D$ and ${\varphi }_0$. The dashed lines are guides to the eye. (d) Correlation between $s_2$ and $\xi /{\xi }_0$. Straight lines are fitted.