Collision dynamics of particle clusters in a two-dimensional granular gas

In a granular gas, inelastic collisions produce an instability in which the constituent particles cluster heterogeneously. These clusters then interact with each other, further decreasing their kinetic energy. We report experiments of the free collisions of dense clusters of particles in a two-dimensional geometry. The particles are composed of solid CO${}_2$, which float nearly frictionlessly on a hot surface due to sublimated vapor. After two dense clusters of $\approx$100 particles collide, there are two distinct stages of evolution. First, the translational kinetic energy rapidly decreases by over 90% as a “jamming front” sweeps across each cluster. Subsequently, the kinetic energy decreases more slowly as the particles approach the container boundaries. In this regime, the measured velocity distributions are non-Gaussian with long tails. Finally, we compare our experiments to computer simulations of colliding, two-dimensional, granular clusters composed of circular, viscoelastic particles with friction.

Figure 1

(a) Schematic of the experimental apparatus. An anodized, aluminum plate with tilted boundaries is heated to $\approx$100${}^{\circ }$C. Two clusters of solid CO${}_2$ (dry ice) particles collide in the middle of the plate. Rectangular, silicone rubber strips prevent the particles from escaping from the plate edges. The dynamics are filmed from above with a high-speed camera. (b) Sublimated gas from beneath the dry-ice particles creates a high-pressure region that supports the particle's weight so that the particles float with nearly zero friction above the hot substrate. (Figure adapted from Ref. [29]).

Figure 2

(a) Images showing the collision of two isolated CO${}_2$ particles. The collision induces changes in each particle's velocity and rotation. The time between the first and last image is 0.85 s. The arrows indicate the direction and relative magnitude of the velocity. (b) Final momentum versus initial momentum, and (c) final angular momentum versus initial angular momentum of both particles for 15 separate collisions. The errors in the data are less than the point size. The dotted lines indicate perfect conservation of translational and angular momentum.

Figure 3

(a) Images from a typical movie showing the collision in the $x$ direction of two clusters after sliding down the sloped edges of the apparatus. The clusters collide at $t=0.0$ s. The particles then spread out in the $y$ direction and eventually come to rest in clusters near opposite edges of the plate. (b) Simulation of the same experiment using viscoelastic particles with friction. The inner color corresponds to the magnitude of angular velocity $\omega$ in the clockwise (red) and counterclockwise (blue) direction, where white indicates $\omega =0$. Clustering also occurs near the boundaries in the simulation.

Figure 4

(a) Translational kinetic energy $E_T\left(t\right)$ for four different experiments. The first particles collide at $t=0$. Each data set is normalized by the initial kinetic energy prior to collision, $E(t=0)$. The initial drop in kinetic energy (regime I) is discussed in Sec. 4d, and the late-time decay (regime II) is discussed in Sec. 4e. (b) $E_T\left(t\right)$ (solid) and $E_R\left(t\right)$ (dashed) from the direct simulations of the experiments, both normalized by $E(t=0)$. Colors refer to the corresponding experimental data set in (a).

Figure 5

Images of the particles from an experiment during the initial moments of the collision. The color indicates the magnitude of the velocity, as denoted by the scale on the left ($v_0=52$ cm/s). The front of reduced velocity (jammed particles) travels faster than the initial speed of the particles and eventually encompasses all of the particles. The cluster then proceeds to elongate in the $y$ direction until it touches the boundary of the system, inducing further collisions among the particles.

Figure 6

(a) $\Sigma$ versus time averaged over four different experiments from Fig. 4. The error bars (experiment) represent one standard deviation from the mean, and the solid red (gray) area shows the maximum and minimum values from the simulations. At early times the data is flat because only two particles have collided. Both the data and simulations are consistent with $\Sigma \propto t^{3/2}$. (b) $\Sigma$ versus time from simulations of elliptical-cluster collisions with 5000 particles per cluster. The cluster shapes are shown on the right with the corresponding colors. The dashed lines are predictions from Eq. (4) with parameters ${\varphi }_J=0.907$, ${\varphi }_0=0.71$, $v_0=1.6\times {10}^{-4}$, and $a=$ {52.7, 83.7, 131.8} {black, red (dark gray), green (light gray)}.

Figure 7

(a) Translational kinetic energy $E_T\left(t\right)$ for four different experiments. Each data set is normalized by the initial kinetic energy prior to collision, $E(t=0)$. Just before $t$ $\approx$ 1.0 s, the particles hit the boundaries of the plate and begin a slow decay of the kinetic energy due to multiple collisions. The error bars (experiment) represent one standard deviation from the mean, and the solid red (gray) area shows the maximum and minimum values from the simulations. At late times, both experiments and simulations are consistent with $E_T\left(t\right)/E(t=0)\propto {(t-t_{\mathrm{jam}})}^{1.8\pm 0.2}$.

Figure 8

Single component velocity probability distributions [$P\left(v_x\right)$ and $P\left(v_y\right)$] for both the experiment (a) and corresponding simulation (b). The distributions are averaged over four experiments. The clusters impact at $t=0$ s. All velocities are normalized by $v_0=52$ cm/s, which corresponds to 1.0 on the horizontal axis. (c, d) Log-linear plot of $P\left(v_x\right)$ and $P\left(v_y\right)$ at $t=0.5$ s after the initial impact. Data in the light-gray region (below density $=6\times {10}^{-2}$) is not reliable and represents the velocity of a single particle in one bin size. Since particles are not directly tracked in the experiment, this limit shows up as an increase in noise.