Packing in the Spheres

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Original entry: William Bonificio, AP 225, Fall 2009


Packing in the Spheres David A. Weitz, Science , vol 303 968 (2004)


close, packing, spheres, ellipsoid


The paper discusses the different types of spherical packing, and how ellipsoidal objects pack compared to spherical ones. Surprisingly, ellipsoidal objects take very little energy to pack tightly - resulting in a very high volume fraction for random close packing. A reason for this unintuitive result is discussed.

The left picture shows spheres which can only be traslated in space, not rotated. The middle picture demonstrates how an ellipsoid could be rotated to fit into an unoccupied space. The picture on the right shows that more ellipsoids are needed to balance the forces on one another, and thus closer packing occurs.

Soft Matter Example

Colloidal physics often deals with the problem of packing. For the most part, scientists will approximate all object that are being packed as spheres and then calculate the packing efficiency with that basis. One such type of packing that can occur is when the spheres still have a completely random orientation, but are packed as tightly as they can be, this is called random close packing, or RCP. For spheres, the volume fraction for RCP is 0.64.

If we allow the spheres to be ordered, and pack them as tightly as possible they take a configuration called hexagonal close packed, or HCP. In this configuration the spheres form hexagonal layers, and each layer above will fall into the crevices formed by the layer below. The volume fraction for spheres in HCP is 0.74.

The final kind of packing discussed is random loose packing, RLP. In this packing the spheres are in a random orientation, but have been packed in a fluid that proved buoyancy. One can imagine dumping marbles in an empty bucket (RCP) versus dumping marbles in a bucket full of water (RLP). The volume fraction for spheres in RLP is 0.56.

What are the packing values for shapes that are not spheres, such as monodisperse ellipsoids? It appears that the volume fraction of RCP for ellipsoids is generally 0.74 and decreases as the ellipsoid approaches the spherical shape. This is a surprising result that suggests that spheres are perhaps an anomaly with a very low RCP Volume fraction.

The reason for the anomaly suggested by the paper is that spheres in RCP have several nearest neighbors that are all jammed into place. The sphere can only be translated by the forces from these neighbors, not rotated. However, ellipsoids can experience a torque along with a translation. The result of this is several more nearest neighbors needed to balance the forces, and much closer packing.