Direct Observation of Medium-Range Crystalline Order in Granular Liquids Near the Glass Transition

Collective behavior of driven granular matter is often strikingly analogous to that of thermal systems. Here we use a vibrated quasi-two-dimensional granular matter as a model system and investigate the mechanism of the liquid-glass transition. We demonstrate by direct observation the existence of long-lived medium-range crystalline order, which is found to be closely related to both dynamic heterogeneity and slow dynamics. Our findings are remarkably similar to recent numerical results on model thermal liquids and thus open an intriguing possibility of understanding the dynamic arrest in both thermal and athermal systems in a unified manner.

Figure 1

(a) $\varphi$ dependence of $g(r)$. (b) Low-$q$ region of $S(q)$. (c) ${\tau }_{\alpha }$ as a function of $\varphi$. The solid curve is a fit to the Vogel-Fulcher function ${\tau }_{\alpha }={\tau }_0\exp [D\varphi /({\varphi }_0-\varphi )]$, with ${\tau }_0=14\mathrm{ms}$, $D=0.41$, and ${\varphi }_0=0.838$. The broken curve, on the other hand, is a power-law fit ${\tau }_{\alpha }={\tilde{\tau }}_0({\varphi }_c-\varphi {)}^{-\gamma }$, where ${\tilde{\tau }}_0=3.7\mathrm{ms}$, $\gamma =1.4$, and ${\varphi }_c=0.785$. Note that this power-law function fails to fit ${\tau }_{\alpha }$ for $\varphi =0.789$.

Figure 2

Relationship between dynamic heterogeneity, medium-range crystalline order, and topological defects for $\varphi =0.774$. (a) Particle trajectory during $t_0<t<t_0+{\tau }_{\alpha }$ ($t_0$: arbitrary). (b) Spatial distribution of the time-averaged ($t_0<t<t_0+{\tau }_{\alpha }$) bond-orientational order parameter ${\Psi }_6$. (c) Spatial distribution of the coordination number $Z$ at $t=t_0$. Clusters of particles with high crystalline order in (b) and the corresponding region in (a) and (c) are circled for a guide to the eye. The region depicted here is 83.9 mm ($\simeq 30⟨d⟩$) in diameter. Particle positions in (b) and (c) are those at $t=t_0$. Since we cannot precisely distinguish the particle sizes in the present experiment, the diameter of particles are shown as the average size.

Figure 3

(a) $\varphi$ dependence of $g_6(r)/g(r)$. The solid curves are the OZ fittings for the envelope. The broken curve represents $g_6(r)/g(r)\propto r^{-1/4}$ [18]. (b) Self part of the four-point susceptibility. (c) Structure factor of immobile particles. The solid curves are the fits to the OZ function. (d) $\varphi$ dependence of ${\xi }_6$ and ${\xi }_4$. The solid curves are the power-law fittings $\xi ={\xi }_0[\varphi /({\varphi }_0-\varphi )]$, where ${\varphi }_0$ is determined by Vogel-Fulcher fitting, and the bare correlation lengths are ${\xi }_{60}=0.37⟨d⟩$ and ${\xi }_{40}=0.26⟨d⟩$, respectively. The small difference between ${\xi }_{60}$ and ${\xi }_{40}$ is perhaps due to the difference in the definition of ${\xi }_6$ and ${\xi }_4$.

Figure 4

${\tau }_{\alpha }$ (solid symbols) and $s_2$ (open symbols) as a function of the correlation lengths (${\xi }_6$, ${\xi }_4$). The solid line shows the relation ${\tau }_{\alpha }={\tau }_0\exp (D\xi /{\xi }_0)$ with ${\tau }_0$, $D$, and ${\xi }_0$ determined by other fittings. The broken line is a fit to the linear function $s_2=a\xi /{\xi }_0+b$, where $a=-0.093$ and $b=-0.96$.