The coordination number of granular cylinders

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Soft Matter Keywords

Coordination Number, Jamming, Volume Fraction, Packing

Summary

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

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


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, <math>\langle z\rangle</math> could be determined by counting the number of paintless spots on each sphere. Bernal and Mason determined <math>\langle z\rangle = 6.4</math> for their system. A later experiment (Donev et al., 2004) used the same procedure to determine that for M&Ms, <math>\langle z\rangle=9.8</math>. What is the problem then? Previous studies have found that using the same arguments used to obtain the values given above, <math>\langle z\rangle</math> for ellipsoids has a lower limit of 10. However, this means that <math>\langle z\rangle</math> 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 <math>\langle z\rangle</math> of rods (cylinders) also has a similar problem. The conclusion is that for rods, <math>\langle z\rangle</math> 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 <math>\langle z\rangle</math> much at all, in contrary to the values given above. Similarly, no <math>\langle z\rangle</math> 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 <math>\langle z\rangle=6</math>).

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 difference between boundary rods and bulk rods, the authors determined <math>\langle z\rangle</math> 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.

Another factor to take into consideration is excitation. Each time the rods are dumped into the bucket, they will have different random arrangements with different volume fractions. Shaking the bucket will most likely decrease the volume they take up (increase volume fraction) while increasing average coordination number. To provide consistency across experiments, the authors would shake the container until a quasi-steady-state value was reached (about 500-1000 vertical excitations at 50Hz). This is called the compacted form. The figure below shows the dependence of volume fraction on number of excitations and aspect ratio. We also expect that higher aspect ratios lead to lower volume fraction (the way I think of it is that with a longer rod, there are less degrees of freedom than two rods of half the length).

L/D indicates aspect ratio. a)After 1000 excitations, the sample is presumed to have reached a quasi-steady-state volume fraction. b)<math>\phi_{comp}</math> refers to the volume fraction at the quasi-steady state. Solid line indicates the 5.4 rule.

Interestingly, the product of compacted volume fraction and aspect ratio is a constant, <math>\phi_{comp}*\frac{L}{D} = 5.4</math>. Finally, the most important data is the plot of <math>\langle z\rangle</math> versus aspect ratio.

average coordination length versus aspect ratio.

The figure about contains several comment-worthy features. Firstly, note that as expected, the compacted (1000 vertical excitations) rods have higher coordination numbers than uncompacted rods. The range of <math>\langle z\rangle</math> is within the theoretically calculated values of 4 and 10. The bowtie mark is the sphere data taken by the early experiment from Bernal and Mason. The M&Ms data is also indicated. We can conclude pretty decisively from the graph that for cylinders <math>\langle z\rangle</math> increases with aspect ratio, though the M&M data see to be an outlier, and the simulation (by another paper) seems very different (though remember the M&Ms are ellipsoids).

Reactions and Relevance to Soft Matter

The conclusion that <math>\langle z\rangle</math> increases with aspect ratio seems like an obvious one. After all, if I increase the length of a rod by huge amount, then I would expect it to touch more of the other rods. In fact, the fact that <math>\langle z\rangle</math> is bounded by 10 in the first place surprises me; however, upon thinking more, it makes some sense because the rods are stiff objects and every point of contact between two rods restricts the possible placements of other rods. Moreover, the compacted state is not the most densely packed state. The most densely packed state would be when all of the rods are lying parallel to each other, and in this case, the number of touching rods would theoretically be infinite (I believe the theoretical prediction assumes non-parallel contact).

How does this study of hard rods relate at all to soft matter? The idea of a coordination number is very useful concept, because it determines how many interactions can take place. In soft matter, it is important to know when interactions between particles need to be taken into account. For example, in the ideal gas model (gas is not condensed matter, but many things can be modeled as ideal gases), we essentially assume that the coordination number is 0 because none of the gases interact; however, taking into account the interaction between particles, we get corrections revealing new phenomena, such as the Van der Waals gas. If we can assign coordination numbers to system of particles in soft matter, it will help us determine when to factor in different interactions, and how strong these interactions are, and how they depend on particle aspect ratio. The rods in this paper are analogous to a hard sphere model, because none of the rods can overlap in space. Even such a simple interaction model can lead to profound results, as indicated by difference between fractal dimension of 2 versus 5/3 in a random walk versus self-avoiding polymer. Coordination number also appears to be directly correlated with jamming in systems, since the compacted system is essentially a jammed state. Therefore, we can correlate the aspect ratio of a particle with its coordination number and how it jams or unjams. (Unjamming a Polymer Glass)

It seems to me that in many models we often take ideal limits of cases, such as spherical particles or 1-dimensional links. Taking aspect ratio into account would be a good correction to these models, especially since coordination number (number of interactions) and volume fraction depend on aspect ratio. In the future, it would interesting to investigate how putting the rods in a medium would affect these results (coordination number, effect of excitations, effect of friction, etc). Perhaps it would only increase the number of excitations necessary to reach a quasi-stable state, or perhaps it would not affect anything at all, since the paper states that friction was not a large factor in their results. Finally, these results would be even more applicable to soft matter if they were conducted for flexible rods, though this would considerably increase the complexity of the theoretical models by adding many degrees of freedom. Perhaps we could start going about it by defining a persistence length as in polymer analysis.


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.