Crystalline Granular Arrays

When you stand on a sand dune, the grains holding up your foot are supported by their neighboring grains. These, in turn are supported by their neighbors, and so forth, to the bottom of the dune. What paths do the forces take from the force applied by the foot to the supporting bottom? This simple question has befuddled researchers in granular materials for the past decade. The pathways are much different in a pack of grains with no cohesive forces than they are in a solid piece of material. But in accounting for these differences, competing theories give mutually contradictory conclusions. The theories arrive at different answers because they make different assumptions about the effect of randomness in the granular arrangements.

In order to get to the heart of this puzzle, Materials Research Center scientists decided to study the forces in granular packs with no randomness: crystal lattices. Solid state physicists know much about these lattices, since atomic crystals follow these same regular arrangements. They know that there are two lattices that pack into the smallest possible volume: face centered cubic (fcc) and hexagonal close packed (hcp). Though these two arrangements have identical densities, they differ dramatically in their other properties. But do they differ in the pathways followed by an applied force on its way to the bottom support?

To answer this question, Ph.D. student Nathan Mueggenburg, working with Profs. Heinrich Jaeger and Sidney Nagel, arranged thousands of 3mm glass beads into large, perfectly-ordered, three-dimensional “crystals” in a specially designed apparatus (figure 1). Applying an external force to a small area at the top of the pack and placing carbon paper at both the top and bottom of their granular array, they could determine which areas of the array “experienced” the most force, and what relative fraction of the original force was transmitted through the ordered array. The darkest marks left by the carbon paper corresponds to the region through which the most force was transmitted. Repeating this experiment a number of times, and averaging over all experiments, they could reconstruct profiles for the intensity of the force experienced by the ordered granular array relative to the applied force.

At the bottom of the face-centered cubically packed granular crystal, they observed three primary regions of large force (figure 2a), with lines of smaller force connecting these regions, forming a triangular pattern. Increasing the depth of the packing, resulted in a similar triangular pattern, although broader and with less intensity (figure 2b).

In contrast, at the bottom of their hexagonally-close packed array, they observed a ring of maximum force (figure 3a). Again, after increasing the depth of the packing, they observed a larger circular pattern (figure 3b).

The differences in these experimental results reflect the differences in structure of the two different granular arrays. In the fcc packed arrangement, each bead is supported by the three beads directly underneath it and these contacts are arranged along straight lines downward through the packing (figure 4) . 

Thus, applied forces are transmitted via a straight line, from bead to bead, forming the triangular pattern at the bottom surface. In the hcp packing, the possibility for a direct line of force transmission no longer exists. In fact, the original force must be split, across every other layer, due to its staggered packing arrangement (figure 5). This results in a ring of maximum force.

These experiments have been important in elucidating what types of order and disorder control specific aspects of force propagation within granular materials. The patterns observed at the bottom of a granular pack, shown in the figures above, might be thought of a "fingerprints" of the arrangement of the grains. However, there is a key difference with other, more familiar experiments often used to map out the structure of solids (e.g., x-rays): while different crystalline arrangements can be identified by strikingly different patterns, the ring-shape for the hcp packing demonstrates that crystallinity does not always result in a pattern composed of well-isolated peals (as it would for x-rays). The reason is that the experiments described here give information that goes beyond details about the structural arrangement of the beads and deliver something that cannot be done by other methods: they show fingerprints of the force propagation.