# The coordination number of granular cylinders

(in progress)

## Summary

In a granular material, the average coordination number $\langle z\rangle$ is defined as the average number of touching neighbors per particle. For hard spheres, $\langle z\rangle$ has been well-studied and is widely-agreed upon. This is paper discusses $\langle z\rangle$ for cylindrical grains. Using constraint equations, the authors prove the following equations:

Coordination Numbers
Grain Type Average Coordination Number
Frictionless Cylinders $10\,$
Friction Cylinders $4\leq\langle z\rangle\leq 10$
$D\,$ dimensional Frictionless Spheres $2 D\,$
$D\,$ dimensional Friction Spheres $D + 1\leq\langle z\rangle\leq 2 D$

Does this match with experimental values? The authors cite an experiment (Bernal and Mason, 1960) where frictional spheres were placed in a container then paint was poured inside. After the paint was drained and dried, $\langle z\rangle$ could be determined by counting the number of paintless spots on each sphere. Bernal and Mason determined $\langle z\rangle = 6.4$ for their system. A later experiment (Donev et al., 2004) used the same procedure to determine that for M&Ms, $\langle z\rangle=9.8$. What is the problem then? Previous studies have found that using the same arguments used to obtain the values given above, $\langle z\rangle$ for ellipsoids has a lower limit of 10. However, this means that $\langle z\rangle$ jumps from 6 to 10 as soon as a sphere is even infinitesimally deformed into an ellipsoid, which contradicts experiments. This paper investigates experimentally the similar problem of whether the values above for $\langle z\rangle$ of rods (cylinders) also has a similar problem. The conclusion is that for rods, $\langle z\rangle$ increases from about 6 at an aspect ratio of unity to about 10 for very high aspect ratios (about 30). Moreover, friction does not seem to affect $\langle z\rangle$ much at all, in contrary to the values given above. Similarly, no $\langle z\rangle$ values were observed below 6 even though the theoretical limit is 4, which leads to questions about how this value might be related to that of the 3D frictionless sphere (also $\langle z\rangle=6$).

## Data

(all images from paper) The data from this paper is very interesting, so I have included it below. The authors followed a similar approach as Bernal and Mason, placing rods in a bucket and pouring paint in. Firstly, an important consideration that must be taken into account when doing the experiments is the effect of boundary conditions. One would expect that rods against the side of the bucket have fewer touching neighbors than rods in the center. The authors separated the rods into two categories: boundary rods and bulk rods. The figure below is a picture of the rods before and after removing the boundary rods.

c) contains all rods, d) contains only bulk rods. The aspect ratio is 32, and the scale line indicates 1cm.

To test whether there is actually a significant different between boundary rods and bulk rods, the authors determined $\langle z\rangle$ for both types of rods. The plot below shows the distribution of coordination numbers for each type of rod. As expected, the boundary rods show on average fewer touching neighbors than the bulk rods.

y-axis is the fraction of rods with a given coordination number. Circles represent bulk rods (454 total), triangles represent boundary rods (904 total). The aspect ratio is 32.

## References

J. Blouwolff and S. Fraden. "The coordination number of granular cylinders." Europhys. Letters, 76 (6), pp. 1095-1101 (2006).

J. D. Bernal and J. Mason. "Co-ordination of Randomly Packed Spheres." Nature, 188 (1960) 910.

A. Donev, I. Cisse, D. Sachs, et al., "Improving the density of jammed disordered packings using ellipsoids",Science, 303 (2004) 990.