Translational and rotational velocities in shear-driven jamming of ellipsoidal particles

We study shear-driven jamming of ellipsoidal particles at zero temperature with a focus on the microscopic dynamics. We find that a change from spherical particles to ellipsoids with aspect ratio $\alpha =1.02$ gives dramatic changes of the microscopic dynamics with much lower translational velocities and a new role for the rotations. Whereas the velocity difference at contacts—and thereby the dissipation—in collections of spheres is dominated by the translational velocities and reduced by the rotations, the same quantity is in collections of ellipsoids instead totally dominated by the rotational velocities. By also examining the effect of different aspect ratios we find that the examined quantities show either a peak or a change in slope at $\alpha \approx 1.2$, which thus gives evidence for a crossover between different regions of low and high aspect ratio.

Figure 1

Discussion of ellipses and the quartic modes. The thick solid lines are the ellipses, whereas the dashed line is a circle with radius $r_0$.

Figure 2

Shear viscosity for spheres, $\alpha =1.00$, and ellipsoids, $\alpha =1.02$. Panel (a) shows direct comparisons for shear strain rates $\dot{\gamma }={10}^{-8}$, ${10}^{-6}$, and ${10}^{-4}$, which show that a small asphericity only has significant effects at low shear strain rates. Panel (b), which is the same data plotted vs $\varphi /{\varphi }_J\left(\alpha \right)$, where ${\varphi }_J\left(1.00\right)=0.648$ and ${\varphi }_J\left(1.02\right)=0.652$, shows that the change in $\sigma /\dot{\gamma }$ at small $\dot{\gamma }$ can be understood as an effect of a shift in jamming density. This has also been found to be the case for spherocylinders in 2D [19].

Figure 3

Measures of translational and rotational particle velocities for spheres, $\alpha =1.00$, and ellipsoids with $\alpha =1.02$. Panels (a) and (b) show $v/\omega$ vs $\varphi$ for spheres and ellipsoids. [Legends for the different $\dot{\gamma }$ are in panel (d).] For spheres this quantity only depends slowly on $\varphi$ but for ellipsoids it decreases in a dramatic way, and even more so for the lower shear rates. The vertical dashed lines are the respective ${\varphi }_J$. Panel (c) shows an extrapolation of $v/\omega$ at ${\varphi }_J\left(1.02\right)$ to the $\dot{\gamma }\to 0$ limit, giving ${[v/\omega ]}_0\approx 0.11$. Panel (d) shows $W$, which is the ratio of the root-mean-square rotational velocities along the minor axes and the major axis, as described in the main text. The figure shows that the rotations are predominantly around the minor axes, which are the ones that can help the particles fit in with their neighbors.

Figure 4

Contributions of the different terms of Eq. (5) to ${\left(v_{ij}^C\right)}^2$—and thus to the dissipation—for both spheres (open circles, $\dot{\gamma }={10}^{-5}$, no significant $\dot{\gamma }$ dependence) and ellipsoids for many different shear rates, $\dot{\gamma }=1\times {10}^{-8}$, $2\times {10}^{-8}$, $5\times {10}^{-8}$, ..., ${10}^{-4}$. The figure shows the difference between spheres and ellipsoids in two ways. (i) For spheres the velocity difference is mainly due to the translational velocity difference and the effect of the rotations—panels (b) and (c)—is to reduce the velocity difference at contacts. (ii) For ellipsoids at densities around ${\varphi }_J\left(1.02\right)=0.652$ and in the limit of small shear rates (crosses for $\dot{\gamma }={10}^{-8}$) the velocity difference is mainly due to the rotational velocity. The translational velocity—panels (a) and (b)—gives a very small contribution.

Figure 5

Determination of our approximate ${\varphi }_J\left(\alpha \right)$. Panel (a) shows $p\left(\dot{\gamma }\right)$ for spheres (big open circles) and 10 different $\alpha >1$ at the respective ${\varphi }_J\left(\alpha \right)$ chosen such that each data set is algebraic in $\dot{\gamma }$ to a decent approximation. Panel (b) is ${\varphi }_J\left(\alpha \right)$.

Figure 6

Dependency on the aspect ratio, $\alpha$. All quantities are obtained after extrapolating to the $\dot{\gamma }\to 0$ limit at the respective ${\varphi }_J\left(\alpha \right)$. Panels (a) and (b) are $W_0$ and ${[v/\omega ]}_0$ from Fig. 3. Panel (c) relates to Fig. 4 and shows the relative contribution to the dissipation of the three terms in Eq. (5) vs $\alpha$. (The points for $\alpha =1.00$ are not included since they are far off and would only clutter the figure.) The data in panel (d) are particle rotations ${[{\omega }_z/\dot{\gamma }]}_0$ due to the shearing. These figures suggest a crossover to different behaviors at $\alpha \approx 1.2$.

Figure 7

Comparison of different kinds of dissipative forces. In dry friction the tangential dissipative force is limited through $F_t^{\mathrm{dis}}\le \mu F^{\mathrm{el}}$. We here compare our data from the main simulations—which may be described as being produced with $\mu =\infty$—with simulations with $\mu =0.1$. At and close to the transition at $\varphi \approx 0.652$ there are very small changes, though some differences start to be visible below $\varphi =0.62$.

Figure 8

Translational and rotational velocities for ellipsoidal and spherical particles. The data are shown at their respective ${\varphi }_J$, ${\varphi }_J\left(1.00\right)=0.648$ and ${\varphi }_J\left(1.02\right)=0.652$. Panel (a) shows that the nonaffine translational velocity at low shear strain rates is much lower for ellipsoids than for spheres. The rotational velocity, panel (b), is in contrast somewhat higher for ellipsoids than for spheres. These behaviors of $v$ and $\omega$ together give a dramatic decrease in $v/\omega$, shown in Figs. 3 and 3.

Figure 9

Velocity difference at the point of contact. This is a quantity which is related to the dissipation and thereby to the shear viscosity. Panel (a) shows the behavior of spheres and ellipsoids at their respective ${\varphi }_J$. Panel (b) which is data at the same $\varphi$ for both cases is included to show that the ellipsoids, at a given density, have lower contact velocities than the spheres, and thereby also a lower shear viscosity.

Figure 10

Correlations between different contributions to the contact velocity difference for both spheres (one set of open circles) and ellipsoids for several different shear rates. The figure shows the (anti)correlations between the translational contribution ${\mathbf{v}}_{ij}$ and the rotational contribution ${\mathbf{v}}_{ij}^R$ to the velocity difference at contacts. It is found that ${\mathbf{v}}_{ij}$ and ${\mathbf{v}}_{ij}^R$ are almost perfectly anticorrelated at low densities, $\varphi <0.58$, which means that the rotations almost entirely compensate for the translational velocity differences. For the spherical particles ${\mathbf{v}}_{ij}$ and ${\mathbf{v}}_{ij}^R$ remain strongly anticorrelated but, for the aspherical particles at the lowest shear rates, these correlations almost vanish and the two contributions are almost independent.

Figure 11

Extrapolations behind the data in Figs. 6, 6, and 6.