Approach to jamming in an air-fluidized granular bed

Quasi-two-dimensional bidisperse amorphous systems of steel beads are fluidized by a uniform upflow of air, so that the beads roll on a horizontal plane. The short-time ballistic motion of the beads is stochastic, with non-Gaussian speed distributions and with different average kinetic energies for the two species. The approach to jamming is studied as a function of increasing bead area fraction and also as a function of decreasing air speed. The structure of the system is measured in terms of both the Voronoi tessellation and the pair distribution function. The dynamics of the system is measured in terms of both displacement statistics and the density of vibrational states. These quantities all exhibit tell-tale features as the dynamics become more constrained closer to jamming. In particular the pair distribution function and the Voronoi cell shape distribution function both develop split peaks. And the mean-squared displacement develops a plateau of subdiffusive motion separating ballistic and diffusive regimes. Though the system is driven and athermal, this behavior is remarkably reminiscent of that in dense colloidal suspensions and supercooled liquids. One possible difference is that kurtosis of the displacement distribution peaks at the beginning of the subdiffusive regime.

Figure 1

(a) Example configurations for area fractions $\varphi =48.7\%$ and $\varphi =80.9\%$, with the big beads colored darker than the small beads. (b),(c) Voronoi tessellations for these configurations with cells shaded darker for increasing coordination number and circularity, respectively.

Figure 2

(Color online) (a) Contour plot of the coordination number distribution; red is for large probability density and blue is for small. The dashed white line indicates $\varphi =0.74$. (b) Coordination number distributions for three area fractions, as labeled.

Figure 3

(Color online) (a) Contour plot of the noncircularity shape factor distributions for the Voronoi tessellation polygons. Beyond about 74% (dashed white line) a well formed second peak develops and the distribution does not change much. (b) Shape factor distributions for a sequence of area fractions; the thick green curve is for $\varphi =74.4\%$. The labels 7, 6, 5, and 4 show the minimum shape factors for polygons with that number of sides.

Figure 4

(Color online) Noncircularity shape factor distributions for the Voronoi tessellation polygons for (a) $\varphi =48.7\%$ and (b) $\varphi =80.9\%$. Thick black curves are the actual distributions, and the thin colored curves are the contributions from cells with different numbers of sides, as labeled.

Figure 5

(Color online) The radial distribution function computed between (a),(e) all beads; (b),(f) big beads; (c),(g) small and big beads; and (d),(h) small beads. The top row shows contour plots, where blue is for large $g$ and red is for small; the dashed white line represents $\varphi =0.74$. The bottom row shows data curves for different area fractions; the thick green curves are for $\varphi =0.744$. The grayed region represents the distance excluded by hard-core contact, for $r$ less than the sum $d_{ij}$ of radii.

Figure 6

(Color online) The area fraction dependence of (a) the occurrence probability $P\left(Z\right)$ of Voronoi cells with $Z$ sides, and peak and valley values of (b) the shape factor distribution and (c) the pair distribution function for small beads.

Figure 7

(Color online) (a),(b) Mean-squared displacement, and (c),(d) kurtosis of displacement probability distribution, all as a function of delay time. The left-hand plots are for the big beads and the right-hand plots are for the small beads. Area fraction color codes for the all plots are labeled in (c),(d); the thick green curve is for $\varphi =74.4\%$. Note that the mean-squared displacement saturates at the square of the sample cell radius. The squares of bead diameters are $d_b^2=0.76{\mathrm{cm}}^2$ and $d_s^2=0.40{\mathrm{cm}}^2$. The square of the position resolution is ${\left(0.0016\mathrm{cm}\right)}^2=3\times {10}^{-6}{\mathrm{cm}}^2$.

Figure 8

(Color online) Density of vibrational states of frequency $f$, for various packing fractions; the thick green curve is for $\varphi =74.4\%$.

Figure 9

(Color online) (a),(b) Kinetic energy vs area fraction for big and small beads, respectively. The time average for each bead is shown by a small dot; the ensemble average over all beads is shown by +. (c) The kinetic energy ratio of small-to-big beads. The line ${(d_s∕d_b)}^3$ shows where beads move with the same mean-squared velocity. Insets show the same quantities on a logarithmic scale.

Figure 10

(Color online) (a),(b) Diffusion coefficient vs area fraction for big and small beads, respectively. The time-average for each bead is shown by a small dot; the ensemble average over all beads is shown by +. (c) The diffusion coefficient ratio of small-to-big beads. Insets show the same quantities on a logarithmic scale.

Figure 11

(Color online) (a) Logarithmic derivative $\lambda$ of the mean-squared displacement vs delay time, for big beads at area fraction $\varphi =80\%$. Four special time scales can be defined from such data, as depicted: ${\tau }_b$ where $\lambda =1.5$, ${\tau }_{\min }$, where $\lambda$ is minimum, and ${\tau }_c$ and ${\tau }_r$ where $\lambda$ is halfway between 1 and its minimum. (b) Density of state, for big beads at $\varphi =80\%$, marked with the four special time scales. (c) Contour plot of the logarithmic derivative for the big beads, where color indicates the value of $\lambda$, as a function of both area fraction and delay time. Red is slope two and blue is slope zero; caged dynamics are when the mean-squared displacement has a slope less than one, which is aqua blue. The four special times defined by the behavior of $\lambda$, as well as a fifth time ${\tau }^{*}$ at which the displacement kurtosis is maximal, are superposed on the contour plot; symbol key is given in the legend.

Figure 12

(Color online) The zero-stress plane of the jamming phase diagram showing our trajectories in phase space. The primary trajectory, which corresponds to sequences shown in all previous plots, is given by the diagonal dashed purple curve and open circles that intersects the “prejammed” boundary at 74% and that ends at point $J$ $\{\varphi \approx 0.83,T=0\}$. The grayed region is forbidden for impenetrable beads. The solid circles, triangles, and square denote measurements of the phase boundary for large hollow polypropylene beads, solid steel beads, and tiny solid steel beads, as shown in the images. Each boundary point corresponds to the trajectory indicated by the dashed going through it.