Vertical ordering of rods under vertical vibration

Granular media composed of elongated particles rearrange and order vertically upon vertical vibration. We perform pseudo-two-dimensional discrete element model simulations and show that this phenomenon also takes place with no help from vertical walls. We quantitatively analyze the sizes of voids forming during vibrations and consider a void-filling mechanism to explain the observed vertical ordering. Void filling can explain why short rods are less prone to align vertically than long ones. We cannot, however, explain, invoking just void filling, the existence of an optimum acceleration to promote vertical ordering and its dependence on particle length. We finally introduce an interpretation of the phenomenon, by considering the energetic barriers that particles have to overcome to exit a horizontal or a vertical lattice. By comparing these energetic thresholds with the peak mean particle fluctuant kinetic energy, we identify three different regimes. In the intermediate regime a vertical lattice is stable, while a horizontal is not. This interpretation succeeds in reconciling both dependencies on vibration acceleration and on particle length.

Figure 1

Experiments from Ref. 16. Initial configuration on the left. Rod configuration after $1650\mathrm{s}$ vibration on the right.

Figure 2

Simulations 4: Rearrangement after $1200\mathrm{s}$ of vibrations. Initially the particles were ordered horizontally. Lateral walls are periodic.

Figure 3

(Color online) Simulations 4, 8, and 11: Influence of particle length on vertical ordering induced by $3\mathrm{g}$ acceleration. Plot against time of the mean angle with the horizontal plane of the bottom-most two layers of the medium.

Figure 4

(Color online) Effect of rod length on the vertical ordering achieved after $1400\mathrm{s}$ vibration at $3\mathrm{g}$: Mean angle with the horizontal plane of the bottom-most two layers of the medium (left axis) and vertical ordering ratio (right axis).

Figure 5

(Color online) Influence of the acceleration on the vertical ordering ratio achieved after $1400\mathrm{s}$ of short (simulation 11), intermediate (simulation 6) and long rods (simulation 4).

Figure 6

Simulation 4: Left: detail of the time evolution of the medium height ($H$, top curve, left axis), and of the probability $P_s$, $P_l$ of occurrence of short and long voids (bottom curves, central axis). Dots on each curve ($H^B$, $P_s^B$, $P_l^B$) identify the rebounds. Right: $P_s^B$, $P_l^B$ along several vibration cycles. The dashed area corresponds to the time window magnified on the left side of the figure.

Figure 7

Influence of particle length on the porosity ${\epsilon }_s^B$ inducing rods to verticalize. From left to right: simulations 4, 8, 10, 11. The porosity due to long holes ${\epsilon }_l^B$ is similar for these four simulations and is omitted.

Figure 8

Effect of acceleration on the porosity ${\epsilon }_s^B$ inducing rods to verticalize. Acceleration changes from 2 to 3 to 4 times gravity moving from left to right. Simulations 11 with short particles are at top; simulations 4 with long particles are at bottom.

Figure 9

(Color online) Phase diagram summarizing the peak values of fluctuant kinetic energy normalized by the rotational potential barrier on x axis and by the vertical potential barrier on $y$ axis. Simulations with same rod geometry and increasing acceleration describe a ray moving away from the origin. Contours represent an interpolation of the VOR: zero corresponds to a horizontal orientation, one to the most vertical orientation that rods can achieve. Colors help grouping simulations based on their acceleration: blue identify acceleration lower than $2\mathrm{g}$, red $3\mathrm{g}$ acceleration, green $4\mathrm{g}$ and black $6\mathrm{g}$.