Competitive Clustering in a Bidisperse Granular Gas

A bidisperse granular gas in a compartmentalized system is experimentally found to cluster competitively: Depending on the shaking strength, the clustering can be directed either towards the compartment initially containing mainly small particles or to the one containing mainly large particles. The experimental observations are quantitatively explained within a flux model.

Figure 1

Images from two experiments with a mixture of $P_1=300$ large steel beads ($r_1=2.50\mathrm{m}\mathrm{m}$, $m_1=0.500\mathrm{g}$) and $P_2=600$ smaller ones ($r_2=1.25\mathrm{m}\mathrm{m}$, $m_2=0.063\mathrm{g}$). The initial distribution of the beads is the same in both experiments: $\{\frac{3}{5}{P}_1,\frac{1}{3}{P}_2\}$ in compartment $A$ and $\{\frac{2}{5}{P}_1,\frac{2}{3}{P}_2\}$ in $B$. For moderate shaking strength ($f=60.0\mathrm{H}\mathrm{z}$, $a=2.0\mathrm{m}\mathrm{m}$, top row) a cluster is formed in the compartment initially dominated by large particles ($A$). For milder shaking ($f=37.5\mathrm{H}\mathrm{z}$, $a=2.0\mathrm{m}\mathrm{m}$, bottom row) the clustering takes place in the compartment initially containing most of the small particles ($B$).

Figure 2

Bifurcation diagram showing the three different clustering regimes O, I, and II. The particle numbers $N_i$ ($i=1,2$) are given relative to the symmetric solution: $N_i-\frac{1}{2}{P}_i$. The squares and diamonds are experimental data, corresponding to the numbers of large and small particles, respectively, in compartment $A$ (solid symbols) and $B$ (open symbols). The curves represent the equilibrium situation according to the theoretical flux model. The initial condition is always the same as in Fig. 1.

Figure 3

Evolution of the bidiperse system, calculated from the flux model, for (a) $D=1$, (b) $D=100$, and (c) $D=200$. The top rows show the number of large and small particles as a function of time (note the different time scales): the solid curves represent compartment $A$, and the dashed ones compartment $B$. The flow diagrams (bottom row) show how the contents of $B$ evolve: $N_{1B}$ (large) and $N_{2B}$ (small). The cross indicates the initial condition $\{N_{1B},N_{2B}\}=\{120,400\}$. The transition from (a) to (b) comes about through a bifurcation, whereas the transition from (b) to (c) is seen to be the result of a shifting basin boundary.

Figure 4

Phase diagram, showing the three clustering regimes as a function of the inverse shaking strength $D$ and the size ratio $\psi =r_1/r_2$. The shaded areas are calculated from the flux model, and the open circles (no clustering), stars (type I clustering), and triangles (type II clustering) correspond to experiments. The experimental results on the vertical dashed line $\psi =2$ also feature in Fig. 2. The initial distribution of particles is always taken to be the same as in Figs. 1, 2, 3.