Critical Scaling of Shear Viscosity at the Jamming Transition

We carry out numerical simulations to study transport behavior about the jamming transition of a model granular material in two dimensions at zero temperature. Shear viscosity $\eta$ is computed as a function of particle volume density $\rho$ and applied shear stress $\sigma$, for diffusively moving particles with a soft core interaction. We find an excellent scaling collapse of our data as a function of the scaling variable $\sigma /|{\rho }_c-\rho {|}^{\Delta }$, where ${\rho }_c$ is the critical density at $\sigma =0$ (“point $J$”), and $\Delta$ is the crossover scaling critical exponent. We define a correlation length $\xi$ from velocity correlations in the driven steady state and show that it diverges at point $J$. Our results support the assertion that jamming is a true second-order critical phenomenon.

Figure 1

(a) Plot of inverse shear viscosity ${\eta }^{-1}$ vs volume density $\rho$ for several different numbers of particles $N$, at constant small applied shear stress $\sigma ={10}^{-5}$. As $N$ increases, one see jamming at a limiting value of the density ${\rho }_c\sim 0.84$. (b) Log-log replot of the data of (a) as ${\eta }^{-1}$ vs ${\rho }_c-\rho$, with ${\rho }_c=0.8415$. The dashed line has slope $\beta =1.65$ indicating the continuous algebraic vanishing of ${\eta }^{-1}$ at ${\rho }_c$ with a critical exponent $\beta$.

Figure 2

Plot of inverse shear viscosity ${\eta }^{-1}$ vs applied shear stress $\sigma$ for several different values of the volume density $\rho$. The dashed line represents the power law dependence expected exactly at $\rho ={\rho }_c$ and has a slope $\beta /\Delta =1.375$. Solid lines are guides to the eye. Points labeled $\sigma =0.0012$ correspond to densities $\rho =0.870$, $0.872$, 0.874, 0.876, and 0.878.

Figure 3

Plot of scaled inverse viscosity ${\eta }^{-1}/|\rho -{\rho }_c{|}^{\beta }$ vs scaled shear stress $z\equiv \sigma /|\rho -{\rho }_c{|}^{\Delta }$ for the data of Fig. 2. We find an excellent collapse to the scaling form of Eq. (6) using values ${\rho }_c=0.8415$, $\beta =1.65$, and $\Delta =1.2$. The dashed line represents the large $z$ asymptotic dependence, $\sim z^{\beta /\Delta }$. Data point symbols correspond to those used in Fig. 2.

Figure 4

Inset: normalized transverse velocity correlation function $g(x)/g(0)$ vs longitudinal position $x$ for $N=1024$ particles, applied shear stress $\sigma ={10}^{-4}$, and volume densities $\rho =0.830$, 0.834, and 0.838. The position of the minimum determines the correlation length $\xi$. Main figure: plot of scaled inverse correlation length ${\xi }^{-1}/|\rho -{\rho }_c{|}^{\nu }$ vs scaled shear stress $z\equiv \sigma /|\rho -{\rho }_c{|}^{\Delta }$ for the data of Fig. 2. We find a good scaling collapse using values ${\rho }_c=0.8415$, $\Delta =1.2$ (the same as in Fig. 3), and $\nu =0.6$. Data point symbols correspond to those used in Fig. 2.