Universality of Jamming Criticality in Overdamped Shear-Driven Frictionless Disks

We investigate the criticality of the jamming transition for overdamped shear-driven frictionless disks in two dimensions for two different models of energy dissipation: (i) Durian’s bubble model with dissipation proportional to the velocity difference of particles in contact, and (ii) Durian’s “mean-field” approximation to (i), with dissipation due to the velocity difference between the particle and the average uniform shear flow velocity. By considering the finite-size behavior of pressure, the pressure analog of viscosity, and the macroscopic friction $\sigma /p$, we argue that these two models share the same critical behavior.

Figure 1

(a) Finite-size behavior of pressure $p$ in model ${\mathrm{RD}}_0$ (open symbols) and model ${\mathrm{CD}}_0$ (closed symbols) at different strain rates $\dot{\gamma }$. For $\dot{\gamma }=10^{-8}$ we only have results for model ${\mathrm{RD}}_0$. The crossover from power law behavior at small $L$ to a constant at large $L$ determines the correlation length $\xi$, plotted vs $\dot{\gamma }$ in (b). We see that $\xi$ is essentially identical in both models, growing monotonically as $\dot{\gamma }$ decreases, reaching values as large as $\xi \approx 20$ for our smallest $\dot{\gamma }$. As $L$ varies, the number of particles varies from $N=24$ to 4096.

Figure 2

Pressure analog of viscosity ${\eta }_p\equiv p/\dot{\gamma }$ for model ${\mathrm{CD}}_0$. Panel (a) shows the raw data for $\dot{\gamma }=10^{-7}$ through $10^{-5}$ in a narrow interval of $\varphi$ about ${\varphi }_J$. Solid lines interpolate between the data points. Panel (b) shows our scaling collapse of ${\eta }_p$ vs ${\varphi }_{\text{eff}}$, with ${\varphi }_J$, $\beta$, and the parameters in Eqs. (6) and (7) determined from the analysis of data with $\dot{\gamma }\le 10^{-6}$. The solid line is the fitted power law scaling function.

Figure 3

Comparison of critical parameters in models ${\mathrm{RD}}_0$ and ${\mathrm{CD}}_0$ for similar ranges of strain rate $\dot{\gamma }$. Panel (a) shows the jamming packing fraction ${\varphi }_J$ and panel (b) the viscosity exponent $\beta$ that result from fits of our data to Eqs. (6) and (7), for different ranges of ${\dot{\gamma }}_{\min }\le \dot{\gamma }\le {\dot{\gamma }}_{\max }$ and $0.838\le \varphi \le 0.846$. The error bars in the figure represent only the statistical errors from the fitting procedure; quoted errors in the text include rough estimates of systematic errors, such as arise when varying the window in $\varphi$ of the data utilized in the fit. We plot results vs varying ${\dot{\gamma }}_{\min }$ for two different fixed values of ${\dot{\gamma }}_{\max }=1\times 10^{-6}$ and $2\times 10^{-6}$. The open symbols are results for ${\mathrm{RD}}_0$ and the closed symbols are for ${\mathrm{CD}}_0$. For ${\mathrm{RD}}_0$ we have data down to ${\dot{\gamma }}_{\min }=10^{-9}$; however, for ${\mathrm{CD}}_0$ our data go down to only ${\dot{\gamma }}_{\min }=10^{-7}$.

Figure 4

Macroscopic friction $\sigma /p$ vs $\varphi$ for different $\dot{\gamma }$ for models ${\mathrm{RD}}_0$ (open symbols) and ${\mathrm{CD}}_0$ (closed symbols), for a system with $N=1024$ particles. Also shown are results from quasistatic simulations representing the $\dot{\gamma }\to 0$ limit.