Geometric frustration induces the transition between rotation and counterrotation in swirled granular media

Granular material in a swirled container exhibits a curious transition as the number of particles is increased: At low densities, the particle cluster rotates in the same direction as the swirling motion of the container, while at high densities it rotates in the opposite direction. We investigate this phenomenon experimentally and numerically using a corotating reference frame in which the system reaches a statistical steady state. In this steady state, the particles form a cluster whose translational degrees of freedom are stationary, while the individual particles constantly circulate around the cluster's center of mass, similar to a ball rolling along the wall within a rotating drum. We show that the transition to counterrotation is friction dependent. At high particle densities, frictional effects result in geometric frustration, which prevents particles from cooperatively rolling and spinning. Consequently, the particle cluster rolls like a rigid body with no-slip conditions on the container wall, which necessarily counterrotates around its own axis. Numerical simulations verify that both wall-disk friction and disk-disk friction are critical for inducing counterrotation.

Figure 1

(a) Raw image of experimental particles in a swirling container, and the same container at a later time after it has translated through roughly half its circular trajectory. The white dotted line represents the outermost points of the container during its trajectory. The container does not rotate in the laboratory frame. (b) The mean angular velocity ($\varpi$) of the particle cluster about its own center of mass for different particle counts $N$. $\varpi$ is reported as how many radians the cluster gains on each swirling cycle of the dish. As $N$ increases, the cluster transitions from rotation to stalling to counterrotation. Black dashed line indicates the angular velocity of a theoretical single particle or pancake, with perimeter matching that of the cluster, if it were perfectly rolling along the wall. (c) The M frame rotates with the container such that the point (left, labeled “M”) on the container's boundary furthest from the center of swirling is always positioned at the bottom in the M frame (middle and right). As a result, the walls of the container in the M frame appear to be rotating clockwise, as depicted by the black arrows. Also shown are the point $S$, center of swirling, and the point $C$, center of the container. The rightmost M-frame diagram depicts the centrifugal force field ($F_{\text{cent}}$, brown) as well as the Coriolis force ($F_{\text{Cor}}$, red) associated with a particle (green) with indicated velocity ($v$, blue).

Figure 2

(a) Sample experimental image in the M frame, with pinned (red) and loose (blue) region particle centers labeled. A sample angular slice interrogation area is highlighted in green. The angular slice slides around the entire container during analysis; the major angular positions $\theta$ are denoted on the edge of the container. (b) Density histogram of the particles for a rotation case at low $N=28$. The white arrows denote the average local particle velocity deviation from the velocity of the underlying container. (c) Same as panel (b), but for counterrotation at high $N=48$. The coherent pinned region (top left on the container) grows in size with increasing $N$, while the loose region shrinks in size. The particles perform clockwise trajectories around the container while passing between the pinned and loose regions in a cyclic manner.

Figure 3

(a) Experimental average particle angular velocity ${\langle {\varpi }_i^M\left(\theta \right)\rangle }_i$ about the cluster's center of mass, as a function of $\theta$ and $N$. In the loose region, ${\varpi }_i\left(\theta \right)$ increases and decreases in a quantitatively similar manner for all values of $N$, with the only difference being the location on the dish at which this peak occurs. However, in the pinned region, the average value of |${\varpi }_i\left(\theta \right)$| consistently increases with increasing $N$. Plotted in black are the average locations at which particles enter the pinned and loose regions. Note that the apparent decrease of $|{\langle {\varpi }_i^M\left(\theta \right)\rangle }_i|$ in the transition regions is an artifact of the oblong shape of the particle cluster. The transition regions occur at higher distances from the center, but the linear velocity of the particles during the transition regions does not change much. (b) Average experimental angular velocity of particles circling the dish, normalized with respect to the angular velocity of the wall about the cluster's center of mass. This is done by dividing ${\langle {\varpi }_i\left(\theta \right)\rangle }_i$ by the angular velocity of the boundary around the cluster's center of mass, a function of $\theta$ since the center of mass is not at $C$. At low $N$, a normalized velocity <1 in the pinned region indicates that those particles are traveling slower than the dish wall. As $N$ is increased, the normalized velocity of particles in the pinned region approaches 1, meaning the particles are moving at the velocity of the wall.

Figure 4

Experimental data. (a) The slip parameter $s$ is defined as the slip conditions between the particle cluster and the container wall. The value of $s$ of the particles in the pinned region approaches 1 as $N$ is increased, corresponding to the pinned region particles moving more coherently with the moving wall. Additionally, the slip parameter $s$ of roughened particles along the wall of the pinned region is closer to 1 (nonslip conditions) than for smoother particles, meaning that roughened particles also move more coherently with the moving wall in the pinned region. (b) The local rotation, as well as variability in rotation behavior, of individual particles along the wall in the pinned area decreases with increasing $N$, a result of increasing frustration between the particles. Inset shows snapshot of a particle with overlaid particle image velocimetry (PIV) vectors. We use PIV on the surface features of the particles to determine overall feature velocity (blue) and particle spinning about a vertical axis through its own center (orange). (c) Particles with roughened surfaces, and therefore increased friction, transition to counterrotation at a lower $N$ than for smoother particles.

Figure 5

Simulations. (a) Snapshot of the simulated disks. (b) The particle cluster transitions from rotation to counterrotation as $N$ is increased when friction is present. This rotation-counterrotation transition point occurs at higher $N$ when friction is decreased. When either disk-disk or disk-wall friction is completely eliminated, the system never transitions to counterrotation. Here “high friction” is ${\mu }_d={\mu }_w=1.0$, “medium friction” is ${\mu }_d={\mu }_w=0.5$, and “low friction” is ${\mu }_d={\mu }_w=0.1$. When one friction is completely turned off, the other friction is set to 1.0. (c) Density histogram of the disks for a rotating ($N=28$, left) and counterrotating ($N=48$, right) case. (d) Quiver plots showing the average local disk velocity deviation from the container for a rotating ($N=28$, left) and counterrotating ($N=48$, right) case.