Compression-driven jamming of athermal frictionless spherocylinders in two dimensions

We simulate numerically the compression-driven jamming of athermal, frictionless, soft-core spherocylinders in two dimensions, for a range of particle aspect ratios $\alpha$. We find the critical packing fraction ${\varphi }_J\left(\alpha \right)$ for the jamming transition and the average number of contacts per particle $z_J\left(\alpha \right)$ at jamming. We find that both are nonmonotonic, with a peak at $\alpha \approx 1$. We find that configurations at the compression-driven jamming point are always hypostatic for all $\alpha$, with $z_J<z_{\mathrm{iso}}=2d_f=6$ the isostatic value. We show that, for moderately elongated spherocylinders, there is no orientational ordering upon athermal compression through jamming. We analyze in detail the eigenmodes of the dynamical matrix close to the jamming point for a few different values of the aspect ratio, from nearly circular to moderately elongated. We find that there are low frequency bands containing $N(z_{\mathrm{iso}}-z_J)/2$ modes, such that the frequencies of these modes vanish as $\varphi \to {\varphi }_J$. We consider the extended versus localized nature of these low frequency modes, and the extent to which they involve translational or rotational motion, and find many low frequency sliding modes where particles can move with little rotation. We highlight the importance of treating side-to-side contacts, along flat sides of the spherocylinder, properly for the correct determination of $z_J$. We note the singular nature of taking the $\alpha \to 0$ limit. We discuss the similarities and differences with previous work on jammed ellipses and ellipsoids, to illustrate the effects that different particle shapes have on configurations at jamming.

Figure 1

Geometry of the spherocylinders: (a) an isolated spherocylinder indicating the spine half-length $A_i$, end cap radius $R_i$, center of mass position ${\mathbf{r}}_i$, and angle of orientation ${\theta }_i$; (b) two spherocylinders in tip-to-side contact, indicating the minimal spine separation $r_{ij}$ and the moment arms ${\mathbf{s}}_{ij}$ and ${\mathbf{s}}_{ji}$; (c) two spherocyliners in tip-to-tip contact. (d) Two spherocylinders in side-to-side contact with the contact point and separation distance $r_{ij}$ taken midway between the spine end points (indicated by the vertical dashed lines).

Figure 2

For systems with $N=1024$ (circles), 2048 (squares), and 4096 (diamonds) spherocylinders of aspect ratio of $\alpha =4.0$, scaled nematic order parameter $\sqrt{N}\langle S_2\rangle$ vs $\varphi$ for (a) bidisperse size distribution with $R_b/R_s=1.4$ and (b) monodisperse distribution; and scaled tetratic order parameter $\sqrt{N}\langle S_4\rangle$ vs $\varphi$ for (c) bidispserse size distribution and (d) monodisperse distribution. Averages are calculated over $M_s=20$ to 40 independent samples compressed at a rate $\kappa ={10}^{-7}$, starting from random zero-energy configurations at ${\varphi }_{\mathrm{init}}=0.2$.

Figure 3

Average pressure $\langle p\rangle$ vs packing fraction $\varphi$ during compression of bidisperse systems of $N=1024$ spherocylinders with aspect ratio (a) $\alpha =0.01$ and (b) $\alpha =4.0$, at compression rates from $\kappa ={10}^{-6}$ (circles) to $\kappa ={10}^{-10}$ (diamonds). Each rate is averaged over 6 to 10 independent samples starting from random zero-energy configurations at ${\varphi }_{\mathrm{init}}=0.4$. The straight line to the right serves to guide the eye.

Figure 4

Pressure $p$ versus packing fraction $\varphi$ for two different samples at compression rate $\kappa ={10}^{-10}$ in bidisperse systems of $N=1024$ spherocylinders with aspect ratio $\alpha =4.0$. Linear fits overlaying the plots of pressure show the extrapolation to the configuration specific jamming density ${\varphi }_{Js}$ for each sample. There are $5\times {10}^6$ compression steps between the data points shown in the curves.

Figure 5

The compression-driven jamming density $\langle {\varphi }_J\rangle$ vs spherocylinder aspect ratios $\alpha$, for a bidispserse system of $N=1024$ spherocylinders.

Figure 6

Configuration of two spherocylinders that are in approximate side-to-side contact. For energy minimization and for computing the dynamical matrix, we consider this case as if there were two contacts at the points indicated by the separations $r_{ij}^{\left(a\right)}$ and $r_{ij}^{\left(b\right)}$.

Figure 7

Snapshots of energy minimized configurations of $N=1024$ bidisperse spherocylinders, at $U/L^2={10}^{-15}$, very close to the jamming transition. Results are shown for aspect ratios (a) $\alpha =0.01$ and (b) $\alpha =4.0$. Big and small spherocylinders are colored blue and green, respectively, while rattlers are gray. Solid black lines on each spherocylinder denote the direction of the spine. Red dots indicate contacts between adjacent spherocylinders; for side-to-side contacts, we show the two contact bonds as illustrated in Fig. 6.

Figure 8

(a) Average number of contacts per particle at jamming $\langle z_J\rangle$ vs spherocylinder aspect ratio $\alpha$. Rattler particles do not enter the average. Circular data points give the values of $\langle z_J\rangle$ when counting each side-to-side contact as two bonds (see discussion in the main text). For comparison, diamond shaped data points give the values of $\langle {\tilde{z}}_J\rangle$ when counting each side-to-side contact as only a single bond. (b) $\langle z_J\rangle -4$ and $\langle \tilde{z}\rangle -4$ vs $\alpha$ on a log-log scale, so as to highlight the behavior as $\alpha \to 0$; for perfectly circular particles, with $\alpha =0,\langle z_J\rangle =4$. Solid lines are fits to $a+b{\alpha }^x$. For $\langle z_J\rangle$ we find $x=0.68\pm 0.13$ and $a=0.31\pm 0.08$, demonstrating that the $\alpha \to 0$ limit of $\langle z_J\rangle$ is discontinuous. For $\langle {\tilde{z}}_J\rangle$ we find $x=0.67\pm 0.11$ and $a=-0.048\pm 0.065$, consistent with $a=0$, and demonstrating that the discontinuity in $\langle z_J\rangle$ at $\alpha =0$ is due to the side-to-side contacts. The system has $N=1024$ spherocylinders.

Figure 9

Fraction of side-to-side, tip-to-side, and tip-to-tip contact bonds in configurations of $N=1024$ bidisperse spherocylinders at the jamming transition, as a function of spherocylinder aspect ratio $\alpha$. Each side-to-side contact is counted as two contact bonds, as explained in the main text.

Figure 10

Probability distribution $P\left(\phi \right)$ for there to be a contact on the surface of a spherocylinder at an angle $\phi$ relative to the direction of the spherocylinder spine, as illustrated in the inset, for mechanically stable configurations at $U/L^2={10}^{-15}$ very close to jamming. Results are shown for aspect ratios $\alpha =0.01$, 0.12, and 1.0, and represent averages over both big and small spherocylinders.

Figure 11

For a bidisperse system of $N=1024$ nearly circular spherocylinders with aspect ratio $\alpha =0.01$, at energy $U/L^2={10}^{-15}$ close to the jamming transition, (a) the density of vibrational modes $D\left(\omega \right)$ vs $\omega$, and also for comparison $D\left(\omega \right)$ for circular disks with $\alpha =0$; (b) the participation ratio $P\left(\omega \right)$ vs $\omega$; (c) the average rotational motion $u_{\theta }^2\left(\omega \right)$ (circles), parallel translational motion $u_{\parallel }^2\left(\omega \right)$ (squares), and perpendicular translational motion $u_{\perp }^2\left(\omega \right)$ (diamonds) vs $\omega$. “Parallel” and “transverse” refer to directions relative to the direction of the spherocylinder spine. Each point is averaged over all of the modes in the same frequency bin for six different independent samples.

Figure 12

For $N=1024$ nearly circular spherocylinders with aspect ratio $\alpha =0.01$, at energy $U/L^2={10}^{-15}$ very close to the jamming transition, (a) eigenmode at ${\omega }_m=3\times {10}^{-3}$ near the middle of the low frequency band, and (b) eignemode at ${\omega }_m=3\times {10}^{-2}$ near the upper edge of the low frequency band. Arrows on the spherocylinders indicate the relative translational motion, and color the relative rotational motion, according to the legend on the right hand side.

Figure 13

Change in energy $\mathrm{\Delta }U$ vs displacement $\delta$ for perturbations of spherocylinder positions $\delta {\widehat{\mathbf{u}}}_m$ along different eigenmode directions ${\widehat{\mathbf{u}}}_m$ at frequencies ${\omega }_m$ as indicated. The system of $N=1024$ bidisperse spherocylinders is at energy $U/L^2={10}^{-15}$, very close to jamming, and the aspect ratio is $\alpha =0.01$. Solid black lines indicate the behaviors $\mathrm{\Delta }U\sim {\delta }^2$ and $\mathrm{\Delta }U\sim {\delta }^4$.

Figure 14

(a) The density of vibrational states $D\left(\omega \right)$ for $N=1024$ bidisperse spherocylinders with aspect ratio $\alpha =0.01$ at configuration energies $U/L^2={10}^{-15},{10}^{-14}$, and ${10}^{-12}$, corresponding to packing fractions of $\varphi -{\varphi }_J\approx 7.4\times {10}^{-8},2.4\times {10}^{-7}$, and $2.4\times {10}^{-6}$, respectively. (b) Average frequency of modes in the lower frequency band ${\overline{\omega }}_0$ vs energy $U/L^2$. We find ${\overline{\omega }}_0\sim {(U/L^2)}^{1/4}$.

Figure 15

For a bidisperse system of $N=1024$ moderately elongated spherocylinders with aspect ratio $\alpha =4.0$, at energy $U/L^2={10}^{-15}$ close to the jamming transition, (a) the density of vibrational modes $D\left(\omega \right)$ vs $\omega$; (b) the participation ratio $P\left(\omega \right)$ vs $\omega$; (c) the average rotational motion $u_{\theta }^2\left(\omega \right)$ (circles), parallel translational motion $u_{\parallel }^2\left(\omega \right)$ (squares), and perpendicular translational motion $u_{\perp }^2\left(\omega \right)$ (diamonds) vs $\omega$. “Parallel” and “transverse” refer to directions relative to the direction of the spherocylinder spine. Each point is averaged over all of the modes in the same frequency bin for six different independent samples.

Figure 16

For $N=1024$ moderately elongated spherocylinders with aspect ratio $\alpha =4.0$, at energy $U/L^2={10}^{-15}$ very close to the jamming transition: (a) eigenmode at ${\omega }_m={10}^{-8}$ in the low frequency band; the red arrow denotes the spherocylinder with the largest motion. And (b) eigenmode at ${\omega }_m={10}^{-4}$ in the middle band; the dark blue spherocylinder denotes the one with the largest motion. Arrows on the spherocylinders indicate the relative translational motion, and color the relative rotational motion, according to the legend on the right hand side. In each case we show only a subregion of the entire system.

Figure 17

Change in energy $\mathrm{\Delta }U$ vs displacement $\delta$ for perturbations of spherocylinder positions $\delta {\widehat{\mathbf{u}}}_m$ along different eigenmode directions ${\widehat{\mathbf{u}}}_m$ at frequencies ${\omega }_m$ as indicated. The system of $N=1024$ bidisperse spherocylinders is at energy $U/L^2={10}^{-15}$, very close to jamming, and the aspect ratio is $\alpha =4.0$. Solid black lines indicate the behaviors $\mathrm{\Delta }U\sim {\delta }^2$ and $\mathrm{\Delta }U\sim {\delta }^4$.

Figure 18

(a) The density of vibrational states $D\left(\omega \right)$ for $N=1024$ bidisperse spherocylinders with aspect ratio $\alpha =4.0$ at configuration energies $U/L^2={10}^{-15},{10}^{-11}$, and ${10}^{-9}$, corresponding to packing fractions of $\varphi -{\varphi }_J\approx 9.1\times {10}^{-8},9.3\times {10}^{-6}$, and $4.3\times {10}^{-4}$, respectively. (b) Average frequency ${\overline{\omega }}_0^t$ of modes in the low frequency band, and average frequency ${\overline{\omega }}_0^r$ of the modes in the middle frequency band, vs energy $U/L^2$. We find ${\overline{\omega }}_0^r\sim {(U/L^2)}^{1/4}$, while ${\overline{\omega }}_0^t\sim {(U/L^2)}^{1/2}$.

Figure 19

For a bidisperse system of $N=1024$ spherocylinders with aspect ratio $\alpha =1.0$, at energy $U/L^2={10}^{-15}$ close to the jamming transition, (a) the density of vibrational modes $D\left(\omega \right)$ vs $\omega$; (b) the participation ratio $P\left(\omega \right)$ vs $\omega$; (c) the average rotational motion $u_{\theta }^2\left(\omega \right)$ (circles), parallel translational motion $u_{\parallel }^2\left(\omega \right)$ (squares), and perpendicular translational motion $u_{\perp }^2\left(\omega \right)$ (diamonds) vs $\omega$. “Parallel” and “transverse” refer to directions relative to the direction of the spherocylinder spine. Each point is averaged over all of the modes in the same frequency bin for six different independent samples.

Figure 20

For a bidisperse system of $N=1024$ spherocylinders with aspect ratio $\alpha =1.0$, the average frequency ${\overline{\omega }}_0^t$ of modes in the low frequency band, and average frequency ${\overline{\omega }}_0^r$ of the modes in the middle frequency band, vs energy $U/L^2$. We find ${\overline{\omega }}_0^r\sim {(U/L^2)}^{1/4}$, while ${\overline{\omega }}_0^t\sim {(U/L^2)}^{1/2}$.

Figure 21

Distribution of (a) net residual forces on spherocylinders $|{\mathbf{F}}_i^{\mathrm{el}}|$ relative to the average contact force ${\overline{F}}_{ij}^{\mathrm{el}}$, and (b) net residual torques on spherocylinders $|{\tau }_i^{\mathrm{el}}|$ relative to the average contact torque ${\overline{\tau }}_{ij}^{\mathrm{el}}$, for configurations of $N=1024$ bidisperse spherocylinders at energy $U/L^2={10}^{-15}$, very close to jamming, as obtained by our conjugate gradient energy minimization. In each panel, circles are for $\alpha =0.01$ and diamonds are for $\alpha =4.0$. Results are averaged over six independent samples and the contact averages ${\overline{F}}_{ij}^{\mathrm{el}}$ and ${\overline{\tau }}_{ij}^{\mathrm{el}}$ are computed separately for each sample.