Glass Transition for Driven Granular Fluids

We investigate the dynamics of a driven system of dissipative hard spheres within mode-coupling theory. The dissipation is modeled by normal restitution, and driving is applied to individual particles in the bulk. In such a system, a glass transition is predicted for a finite transition density. With increasing dissipation, the transition shifts to higher densities. Despite the strong driving at high dissipation, the transition persists up to the limit of totally inelastic normal restitution.

Figure 1

(a) The jamming diagram for sheared systems as a function of inverse packing fraction ${\phi }^{-1}$, temperature $T$, and shear stress $\sigma$. (b) Jamming diagram for driven inelastic hard spheres where the driving power $v_{\mathrm{dr}}^2$ replaces the shear stress. In this latter case, the temperature dependence is trivial for $T>0$, and the origin of the graph lies at random-close packing.

Figure 2

Transition density ${\phi }^c$ as a function of the coefficient of restitution $\epsilon$. The short horizontal bar indicates the result for the elastic case for $\epsilon =1$. The inset shows the evolution of the critical exponent $a$ with $\epsilon$.

Figure 3

Dynamics of the coherent density correlator ${\varphi }_q(t)$ for the wave vector $qd=4.2$ at the respective critical densities ${\phi }^c(\epsilon )$ and at $\phi =0.999{\phi }^c(\epsilon )$ for $\epsilon =1.0$ (dashed lines, elastic case), 0.5 (full lines), and 0.0 (dotted lines).

Figure 4

Critical glass form factors $f_q^c$ for coefficient of restitution $\epsilon =1.0$ (empty squares, elastic case), 0.5 (filled circles), and 0.0 (filled diamonds) as a function of wave number $qd$.

Figure 5

Dynamic scattering function ${\varphi }_q(t)$ for $qd=4.2$ at the glass transition, $\phi ={\phi }_c$, and for slightly lower volume fraction, $\phi =0.999{\phi }_c$ for Newtonian dynamics ($N$, dotted curves) with ${\nu }_q=0$, Brownian dynamics ($B$, full curves), and granular dynamics ($G$, dashed curves) with coefficient of restitution $\epsilon =0.5$. The Newtonian dynamics is scaled along the time axis to match the Brownian dynamics at long times. No such rescaling is possible for the granular dynamics. The inset shows three lines of equal diffusivity (from top to bottom: $D=0.7$, 0.8, and 0.9 in relative units) taken from the data of a computer simulation [12].