Dimensionality and Viscosity Exponent in Shear-driven Jamming

Collections of bidisperse frictionless particles at zero temperature in three dimensions are simulated with a shear-driven dynamics with the aim to compare with the behavior in two dimensions. Contrary to the prevailing picture, and in contrast to results from isotropic jamming from compression or quench, we find that the critical exponents in three dimensions are different from those in two dimensions and conclude that shear-driven jamming in two and three dimensions belong to different universality classes.

Figure 1

Relaxation time vs distance to the transition as measured by $\delta z\equiv z_c-z$. The figure shows results for both 2D and 3D and gives strong evidence that the exponents $\beta /u_z$, given by the slopes at small $\delta z$ in 3D and 2D, are different.

Figure 2

Results from scaling analyses of the pressure. Shown here are (a) the quality of the fits in terms of ${\chi }^2/\mathrm{DOF}$, (b) the exponent $\beta$, (c) ${\varphi }_J$, and, (d) $1/z\nu$. All quantities are plotted against ${\dot{\gamma }}_{\max }$ to examine whether the fittings are stable against a changing range of shear rates, which would be a requirement for a good fit. The solid dots are from fitting to the simple Eq. (6) without corrections to scaling, whereas open circles are from fitting to the full Eq. (5). The simple fits (solid dots) are clearly unsatisfactory as they give bad quality fits and fitting parameters that vary strongly with ${\dot{\gamma }}_{\max }$.

Figure 3

Attempts to refine the determination of $\beta /u_z$ for 3D in Fig. 1. Panel (a) shows $\beta /u_z$ from fits with different $(\delta z{)}_{\max }$. The open squares are from linear fits to Eq. (3), whereas the open circles are from fitting $\tau$ to Eq. (7), which includes corrections to scaling. The dashed line is $\beta /u_z=3.82$ with $\beta$ from the scaling analysis shown in Fig. 2, assuming $u_z=1$. Panel (b) shows the correction-to-scaling exponent.