A Chiral Granular Gas

Inspired by rattleback toys, we created small chiral wires that rotate in a preferred direction on a vertically oscillating platform and quantified their motion with experiment and simulation. We demonstrate experimentally that angular momentum of rotation about particle centers of mass is converted to collective angular momentum of center-of-mass motion in a granular gas of these wires, and we introduce a continuum model that explains our observations.

Figure 1

Schematic representations of bent-wire chiral objects made from a central stem and two arms. (a) Chiral wire with angle $\alpha >0$, defined as the angle by which the second arm rises out of the plane formed by the stem and first arm (marked with a vertical line). (b) Mirror image of (a), formed by reflection through a plane perpendicular to the central stem, for which the angle $\alpha$ is defined to be negative. In both cases, the curved arrows indicate the direction of spin under vibration. (c) Configuration used in the gas experiments with $\alpha =-3\pi /4$. In general, the spin direction is such that the raised arm moves in the direction of its projection onto the plane of the vibrating surface so that the bent wire in (c) rotates in the counterclockwise direction as seen from above.

Figure 2

(a) Experimental and (b) simulated spin angular velocity $\Omega$ as a function of $\alpha$ for $f=60\mathrm{Hz}$ and $A=3g/(2\pi f{)}^2$. These curves are repeated periodically for $\alpha <0$ and $\alpha >\pi$. (c) Plot of experimental scaled spin frequency ($\Omega /f$) for $\alpha =-3\pi /4$ vs vibration amplitude $A$ for different vibration frequencies $f$. Inset: $\Omega /f$ vs $A$ from simulation. In the simulations, $K_n=500K_0=1.25K_t$ and ${\gamma }_n=75{\gamma }_0=2.5{\gamma }_t$, where $K_0=mg/R_p$ and ${\gamma }_0=m\sqrt{g/(10R_p)}$. The simulation error bars reflect both statistical errors and accumulated small numerical errors from each collision.

Figure 3

(a) Still image of $N=200$ wires in the cell illuminated by visible light. (b) Vectors showing the total displacements of individual particles accumulated for 26.6 sec, with their initial positions marked as circles, for $N=350$, $f=60\mathrm{Hz}$, and rms vibrational amplitude $A=2g/(2\pi f{)}^2$. In both (a) and (b), arm angle $\alpha =-3\pi /4$, stem length $=10.0\pm 0.2\mathrm{mm}$, arm length $=4.0\pm 0.2\mathrm{mm}$, and wire diameter $=0.91\pm 0.05\mathrm{mm}$.

Figure 4

Measured center-of-mass angular velocity $v(r)/r$ [for $f=60\mathrm{Hz}$ and $A=2g/(2\pi f{)}^2$] averaged (to reduce noise) over the three annuli described in the text for different numbers of particles. This angular velocity is greatest in the outer region.

Figure 5

Plots of $v(r)/r$ for ${\lambda }_L/R=0.133$, ${\lambda }_S/R=0.025$, and $l/R=10$ (full line) and for ${\lambda }_L/R=0.111$, ${\lambda }_S/R=0.00183$, and $l/R=5$ (dotted curve). Its mean in three bins is shown as discussed in the text. The curves are normalized so that the mean angular velocity in the third bin is equal to the experimentally determined value of $20\times {10}^{-3}\mathrm{rad}/\mathrm{s}$.