Bouncing, rolling, energy flows, and cluster formation in a two-dimensional vibrated granular gas

We study the formation of crystalline clusters for a two-dimensional (2D) sinusoidally vibrated granular gas, with maximum vertical acceleration smaller than gravity, using fully 3D simulations. It is found that this phenomenon arises from the spontaneous segregation of the granulate into two dynamical modes: one of grains that bounce in synchrony with the motion of the sustaining plate (“bouncers”) and another of grains that cease to bounce and simply rolls on the plate, without ever loosing contact with it (“rollers”). These two dynamical categories are quite robust with respect to perturbations. The populations for bouncers and rollers depend on the preparation of the granulate and can be made to take arbitrary values in all the range of accelerations where both dynamical modes are present. It is found that the dynamical mode with the largest population coalesces in clusters under the influence of the other mode, whose grains act as a higher pressure gas that compresses the clusters. In this way it is possible to produce clusters of rollers or clusters of bouncers. A gas made of grains from only one dynamical class shows only weak density fluctuations. When the occupation fractions for both modes are similar, one observes segregation and clusters of both types. The clustering of the gas is monitored using both the average coordination number and the local hexatic order parameter ${\psi }_6$. Energy flows in the plane are monitored, and it is shown that roller-bouncer collisions increase horizontal kinetic energy, while all other types of collisions reduce this energy. We find that friction with the substrate is the main sink of horizontal energy for these granular gases.

Figure 1

Evolution of the bouncer populations for two sets of parameters. The upper line (red) shows some drift and corresponds to $\epsilon =0.75$, $v_{z0}\left(\max \right)=6$ cm s${}^{-1}$. The lower line (blue) does not drift and corresponds to $\epsilon =0.9$, $v_{z0}\left(\max \right)=4$ cm s${}^{-1}$.

Figure 2

(a) Bifurcation diagram showing the stable resonances for the bouncing of a collection of noninteracting grains on a vibrating plate, as described in the text. Here we are using the $\epsilon =0.9$ parameters. (b) Same as before, but in the presence of interchange noise, that is, of random interchanges of vertical momentum.

Figure 3

Fractions of snapshots of the granular gas, covering each around $1/3$ of the actually simulated area. Rollers are shown as yellow (light gray) circles, and bouncers are shown as green (gray) circles. (a) Granulate with $\epsilon =0.9$ parameters and $v_{z0}\left(\max \right)=2$ cm s${}^{-1}$, showing a large crystalline cluster. For this sample $N_r/N_b=6.6$. (b) Granulate with $\epsilon =0.75$ parameters and $v_{z0}\left(\max \right)=8$ cm s${}^{-1}$. Here clusters are less dense, and no large hexagonal crystals have yet appeared. The value of $N_r/N_b$ is $=1.1$.

Figure 4

Evolution of clustering $Z$ and the hexatic parameter ${\psi }_6$ for both rollers and bouncers for the simulation with $\epsilon =0.9$ parameters and $v_{z0}\left(\max \right)=6$ cm s${}^{-1}$. The evolution is slow, since the figure covers only the slow coarsening phase. Thick lines are Bezier smoothings, provided only as a guide to the eye.

Figure 5

Work done by parts of the system on each other. The colored lines, starting from top, correspond to the different works: $W_{\mathrm{br}}$, $W_{\mathrm{rb}}$, $W_{\mathrm{bb}}$, $W_{\mathrm{rr}}$, $W_{\mathrm{fb}}$ and $W_{\mathrm{fr}}$, where $\mathrm{r}$ stands for roller, $\mathrm{b}$ for bouncer, and $\mathrm{f}$ for the floor. The work $W_{ab}$ is the work done by elements of type $a$ on elements of type $b$. Notice how the largest sinks of energy correspond to the interactions with the floor. These works are averaged over eight realizations of the dynamics. Some representative error bars are shown, these correspond to the variance of the quantity displayed.