Slipping sticks: Microscopic description of vortex formation

In a recent series of experiments [13] Neicu, Kudrolli et al. observed that a system of vibrated rods spontaneously formed the long-range orientational order. Above the critical packing fraction the cooperative motion of the rods resulted in a formation of vortex structures. We developed a detailed microscopic model [24] complemented by 3D molecular dynamics simulations which extends the description of the phenomenon proposed by Aranson et al. [14].

First, we considered the mechanics of in-plane, eccentric, oblique collision between the rod and vibrating plate. We constructed a 2D map which related post-collisional velocities of a rod for two successive impacts on the plate. The statistical average of the map possesses a fixed point which describes the average velocity of a ring of tilted rods as a function of the mean tilt, motion of the plate, and micromechanical characteristics of the collision. Surprisingly, this simple solution quantitatively agrees with the quasi-2D experiment with ``dancing pens'' and qualitatively explains the properties of the vortices observed experimentally [13] (movies showing the phenomenon can be found at http://inls.ucsd.edu/~volfson/rods.

**Figure:** (Left) Image of rods contained in a vertically vibrated 1-D annulus from [13]; (Right) similar MD experiment [24]. The rods are always observed to move in the direction of inclination.
$\includegraphics[width=2.5in]{pdfs/rodsexp}$ $\includegraphics[width=2.5in]{pdfs/rodsmd}$

Next we designed and employed a new (to our knowledge) and effective 3D MD algorithm which tracks the interactions between the rods, taking into account inelasticity of the collisions and friction. The idea is to reduce the interactions between rods to interaction between spheres. However, rods are not just made out of a number of spheres. Instead, when two rods approach each other a virtual sphere is placed on the axis each cylinder, so that the centers of the spheres are located at the closest points between the axes.

MD simulations with thousands of rods placed into a vibrating cylindrical cavity are underway. We anticipate to observe the formation and the whole range of dynamics of vortices and plan to match the proposed theories with 3D experimental data quantitatively. This study should provide more insight on the role of the shape anisotropy in granular materials.