Relaxation times and rheology in dense athermal suspensions

We study the jamming transition in a model of elastic particles under shear at zero temperature. The key quantity is the relaxation time $\tau$ which is obtained by stopping the shearing and letting energy and pressure decay to zero. At many different densities and initial shear rates we do several such relaxations to determine the average $\tau$. We establish that $\tau$ diverges with the same exponent as the viscosity and determine another exponent from the relation between $\tau$ and the coordination number. Though most of the simulations are done for the model with dissipation due to the motion of particles relative to an affinely shearing substrate, we also examine a model, where the dissipation is instead due to velocity differences of disks in contact, and confirm that the above-mentioned exponent is the same for these two models. We also consider finite size effects on both $\tau$ and the coordination number.

Figure 1

Examples of the pressure relaxation at different $\varphi$. The figure shows the pressure relaxation after the shearing has been switched off. The preceding shearings were performed at very low shear rates in order to stay close to the linear region; the densities and the initial shear rates were $(\varphi ,\dot{\gamma })=(0.8340,{10}^{-8})$, $(0.8380,{10}^{-8})$, $(0.8400,5\times {10}^{-9})$, $(0.8408,2\times {10}^{-9})$, $(0.8416,{10}^{-9})$. To determine the relaxation times $\tau$, we fit pressure to an exponential decay, only using data with $p<{10}^{-7}$.

Figure 2

Relaxation time and dissipation time vs density. Panel (a) shows $\tau$ vs $\varphi$ at several different shear rates. The data increase rapidly with increasing $\varphi$ suggestive of a divergence at ${\varphi }_J$. There is also a clear shear rate dependence, $\tau$ decreases when $\dot{\gamma }$ is decreased towards the hard disk limit, $\dot{\gamma }\to 0$. Panel (b) which shows ${\tau }_{\mathrm{diss}}$ vs $\varphi$ also increases rapidly with ${\varphi }_J$. The shear rate dependence is however the opposite; ${\tau }_{\mathrm{diss}}$ increases with decreasing $\dot{\gamma }$. Panel (c) shows a comparison of $\tau$ and ${\tau }_{\mathrm{diss}}$ which only includes the data with the lowest $\dot{\gamma }$ (i.e., closest to the hard disk limit). $\tau$ and ${\tau }_{\mathrm{diss}}$ behave essentially the same across this density interval; they are very close at the highest density close to ${\varphi }_J$, but the (relative) difference increases with decreasing $\varphi$.

Figure 3

Divergence of $\tau$ and ${\tau }_{\mathrm{diss}}$. We here fix ${\varphi }_J=0.8433$ and determine $\beta$ by fitting the few points of $\tau$ and ${\tau }_{\mathrm{diss}}$, respectively, with ${\varphi }_J-\varphi <0.006$ and sufficiently small $\dot{\gamma }$ to be close to the hard disk limit. (The points for $\varphi =0.8420$ and $\dot{\gamma }={10}^{-9}$ appear to be too far from the hard disk limit and are not included in the fits.)

Figure 4

Comparison of $\tau$ and ${\eta }_p$ which, up to a constant prefactor, are expected to behave the same as $\dot{\gamma }\to 0$. Panel (a) shows the raw data, $\tau$, and $A_p{\eta }_p$, with the constant $A_p=36$. The data for large shear rates (solid triangles) are clearly different, but the respective points approach one another as $\dot{\gamma }\to 0$. The dashed line is $f_{\tau }\left(\varphi \right)\sim {({\varphi }_J-\varphi )}^{-\beta }$ with $\beta =2.60$. Panel (b) show the same data, but now divided by $f_{\tau }\left(\varphi \right)$. The figure clearly suggests that the data should agree in the $\dot{\gamma }\to 0$ limit.

Figure 5

Corresponding values of ${\tau }_1$ and $\delta z_1$. Panel (a) shows 2719 corresponding values of ${\tau }_1$ and $\delta z_1$. Each point is from a relaxation that gives both a relaxation time ${\tau }_1$ and a final configuration from which the contact number $z_1$ is determined. The relaxation time clearly depends algebraically on $\delta z_1$—the distance to isostaticity. A fit of all data with $\delta z_1<0.08$ (1625 points) gives the exponent $\beta /u_z=2.69$. Panel (b) is a zoom-in with a more restricted set of data: $\varphi =0.8412$ and four different shear rates. This shows that the points for different initial shear rates fall on a single curve.

Figure 6

Change in contact number in the relaxation process. The figure shows contact numbers before and after the relaxation. The solid line is $z_1=z_1^{\mathrm{start}}$. The configurations are at density $\varphi =0.8412$; the starting configurations are generated with four different initial shear rates. Both the initial $z_1^{\mathrm{start}}$ and $z_1$, obtained after the relaxation, are calculated after repeatedly removing all particles with less than three contacts.

Figure 7

Determination of $\beta /u_z$ for the ${\mathrm{CD}}_0$ model. By fitting data for $\delta z_1<0.08$ to Eq. (18) we determine $\beta /u_z=2.63$. We note that this is very close to $\beta /u_z=2.69$ of the ${\mathrm{RD}}_0$ model.

Figure 8

Mean values of ${\tau }_1$ and $\delta z_1$ determined in two different ways. Panel (a) shows the ordinary arithmetic mean values. For small $\delta z$ these points deviate clearly from the expected algebraic behavior. This phenomenon is due to the large spread of the data which appears close to jamming as is also described in conjunction with Fig. 9. Panel (b) which shows the geometric means, ${\tau }_g$, and ${\left(\delta z\right)}_g$ of the points $({\tau }_1,\delta z_1)$ in Fig. 5 for the same $\varphi$ and $\dot{\gamma }$. These points obey an algebraic behavior with the exponent $\beta /u_z=2.68$ in very good agreement with the analysis of the individual data points in Fig. 5.

Figure 9

Illustration of the arithmetic mean and the geometric means for some points on the curve $y=x^{-3}$. From the figure with linear scale in panel (b) it is clear that one cannot expect the arithmetic mean to lie on top of the curve. As discussed in the text this effect only becomes important in cases where the relative variance is sizable.

Figure 10

Finite size and the spread of the points $({\tau }_1,\delta z_1)$ for $\varphi =0.838$ and initial shear rate $\dot{\gamma }={10}^{-7}$. Panel (a) which is $({\tau }_1,\delta z_1)$ for three different system sizes shows that the points spread considerably more for smaller $N$. Panel (b) shows different quantitative measures of the spread of these data. The open circles are $s\left(z_1\right)$—the standard deviation of $z_1$. Open squares are $s\left({\tau }_1\right)/{\tau }_a$. (The normalization by ${\tau }_a$ is to get quantities of the same order of magnitude). Solid dots are the standard deviation of ${\tau }_1/f_{\tau }\left(z_1\right)$ which is the relative deviation of ${\tau }_1$ from the solid line in panel (a). Note that both the spread of $z_1$ and the spread around the solid line vanish as $1/\sqrt{N}$, as if the data were averages of $N$ independent samples.