Competitive clustering in a bidisperse granular gas: Experiment, molecular dynamics, and flux model

A compartmentalized bidisperse granular gas clusters competitively [R. Mikkelsen, D. van der Meer, K. van der Weele, and D. Lohse, Phys. Rev. Lett. 89, 214301 (2002)]: By tuning the shaking strength, the clustering can be directed either towards the compartment initially containing mainly small particles or to the compartment containing mainly large particles. Here, the conditions under which this competitive clustering occurs are studied experimentally, numerically (by means of molecular dynamics simulations), and analytically. A minimal model is derived that quantitatively accounts for the observed phenomena.

Figure 1

Images from two experiments with a bidisperse mixture of steel beads, starting from the same initial condition. For relatively strong shaking (left column) the clustering is directed towards the left compartment, whereas for mild shaking (right column) it goes into the right compartment. The shaking frequency in the left column is $f=60.0\mathrm{Hz}$, and in the right column 37.5 Hz, while the peak-to-peak amplitude in both cases is $2a=2\mathrm{mm}$. The initial condition (topmost picture) is {180 large, 200 small} in the left compartment, and {120 large, 400 small} in the right one. With the radius of the large beads $(r_1=2.50\mathrm{mm})$ being twice that of the small ones $(r_2=1.25\mathrm{mm})$, this means that initially 55% of the total particle mass is in the left compartment.

Figure 2

Molecular dynamics (MD) simulation results for a monodisperse granular gas (size ratio $\psi =1$) shaken at frequency $f=70\mathrm{Hz}$ and amplitude $a=1\mathrm{mm}$. Shown are the particle number density (left) and the granular temperature (right) as function of the height $z$. The ground area of the container is $\Omega =100{\mathrm{cm}}^2$ (equivalent to the experimental setup of Fig. 1 without the central wall), the number of particles is $P=900$, their radius is 1.25 mm, and the plots are based on ${10}^6$ numerical snapshots of the gas in its steady state, sampled at a rate of 1000 per second. The scale used in these plots is the same as in Figs. 3, 4, to allow for a comparison with the bidisperse case.

Figure 3

Same as Fig. 2, but now for a bidisperse granular gas consisting of 300 large and 600 small particles with size ratio $\psi =2$. The small particles have the same size as those in Fig. 2, i.e., $r_2=1.25\mathrm{mm}$. The density profiles of both the large and the small particles follow straight lines, indicating an exponential decay with $z$ (barometric height distribution). The right plot shows that the granular temperature of the large beads is larger than that of the small beads. The slanting lines at the top of the temperature profiles correspond to free parabolic flights of single particles; see also Fig. 5.

Figure 4

Same as Fig. 3, but now for size ratio $\psi =3$. The densities of the large and small particles still follow an approximately exponential decay. The temperature shows considerable deviations from a constant value, especially for the large particles (see also the inset, in which the same profiles are shown on a different scale); however, the upper region of the temperature profile is made up by only a few particles and has hardly any statistical weight. It is apparent that the temperature difference between the large and small particles has increased with $\psi$; cf. Fig. 6.

Figure 5

The $x,y$ and $z$ components of the granular temperature for the bidisperse mixture of Fig. 3, $\psi =2$. These temperature components are directly proportional to the kinetic energies of the particles: $T_{i,x}=\frac{1}{3}{m}_i{v}_{i,x}^2$, etc. ($i=1$, 2 denoting the large and small particles, respectively).

Figure 6

Temperature ratio $T_1∕T_2$, determined from MD simulations, as function of the size ratio $\psi =r_1∕r_2$. The values are taken at $z=0.075\mathrm{m}$, which lies in the “constant” part of the large-particle temperature profile for each $\psi$.

Figure 7

The small-particle flux $F_2(N_1,N_2)$ as a function of $N_2$, for various numbers of large particles in the compartment: $N_1=0$ (dashed), $N_1=120$ (thin), and $N_1=180$ (thick). For $D=60$ (relatively strong shaking; left plot) the flux from a compartment with {180 large, 200 small} particles (indicated by the left dot) is smaller than from a compartment with {120 large, 400 small} particles (right dot). Hence the clustering is towards the former compartment, i.e., type I clustering, in agreement with the experimental observation of Fig. 1. For $D=200$ (weak shaking; right plot) it is the other way around, leading to type II clustering, again in agreement with experiment. The “tennis ball on basketball effect” is most pronounced on the left flank of the flux function, where the small-particle flux from the compartment actually rises (with respect to the dashed curve) upon adding large particles. Note the different scales of the vertical axis in the two plots: the stronger the shaking, the higher the particle flux.

Figure 8

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 curves represent the steady state according to the theoretical flux model and the squares and diamonds are experimental data. For the experiments we used the setup shown in Fig. 1, filled with a mixture of stainless steel beads: $P_1=300$ large ones with radius $r_1=2.5\mathrm{mm}$, and $P_2=600$ small ones with radius $r_2=1.25\mathrm{mm}$. Every new run was started from the same initial condition: {180 large, 200 small} in compartment A, and {120 large, 400 small} in B. The squares correspond to the large beads, and the diamonds to the small ones. Solid symbols refer to compartment A, and open symbols to B; note that every measurement is thus represented by two points, which accounts for the mirror-symmetry of the plot in the vertical direction.

Figure 9

Evolution of the system calculated from the flux model, starting from the initial condition Eq. (14), for (a) $D=100$ (type-I clustering) and (b) $D=200$ (type-II clustering). The solid curves represent the number of particles in compartment A; the dashed ones compartment B. It is seen that the small particles (lower row) cluster first, followed by the large ones (top row). Note the different time scales between type-I and type-II clustering.

Figure 10

Flow diagrams calculated from the flux model (for the same system as in Figs. 8, 9) showing how the contents of compartment B evolve at five successive values of the shaking parameter: (a) $D=1$, (b) $D=20$, (c) $D=80$, (d) $D=200$, and (e) $D=250$. The cross indicates the initial condition used in the experiments: $\{N_{1B}\left(0\right),N_{2B}\left(0\right)\}=\{120,400\}$. At $D=1$ there is no clustering and all initial conditions lead to the uniform distribution (the central point in the flow diagram). For $D=20$, …, 250 all initial conditions in the light (yellow) basin of attraction lead to a comparatively empty compartment B (type-I clustering) and those in the dark (blue) basin of attraction lead to a well-filled compartment B (type-II clustering). Note that the slope of the boundary between the two regimes shows nonmonotonic behavior as a function of $D$. Part (f) shows the region of competitive clustering: The boundary between the two basins of attraction in (a)–(e) sweeps through this region, and therefore the initial conditions here lead to either type-I or type-II clustering depending on the value of $D$.

Figure 11

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 drawn curves are calculated from the flux model, and the symbols correspond to experiments: open circles$=\text{no}\text{clustering},\text{crosses}=$type‐I clustering, and $\text{red}\text{triangles}=$type-II clustering. The experimental results on the vertical dashed line $\psi =2$ also feature in Fig.8. The initial condition is always taken to be as in Eq. (14).

Figure 12

The same phase diagram as in Fig. 11, for the same initial situation (14), but this time the symbols represent MD simulations. Between the regimes of type-I and type-II clustering, there is also a zone where the clustering in the MD simulations can go either way, depending on statistical fluctuations. This undecided state of affairs is indicated by the open squares: Each of them is based on 10 repetitions of the MD simulation, of which typically half ended in type-I clustering and the other half in type-II. As in all previous figures, the ratio of large to small particles is $\sigma =P_1∕P_2=300∕600=1∕2$.

Figure 13

The same as Fig. 12, but now for (a) $\sigma =P_1∕P_2=200∕1200=1∕6$ and (b) $\sigma =600∕600=1$. For growing $\sigma$ (relative abundance of the large particles) type-I clustering clearly gains ground. The initial condition used here is $\left\{\frac{3}{5}{P}_1,\frac{1}{3}{P}_2\right\}$ in compartment A and (hence) $\left\{\frac{2}{5}{P}_1,\frac{2}{3}{P}_2\right\}$ in compartment B, in analogy with the condition (14) which was taken in all previous figures (where $\sigma =300∕600=1∕2$).