Random thoughts on densest packing

Chaikin replies: The readers’ comments remind me of occasions when I’ve run over the time limit while giving a talk, and the first question is, “What would you say if you had another 10 minutes?”

In response to David Rosen, I offer the following. Several infinite structures have the same 0.74 packing fraction as the face-centered cubic structure. In referring to FCC, I was following the spirit of Thomas Hales in his 2005 paper on the proof of the densest packing in three dimensions. ¹ According to his theorem 1.1, the Kepler conjecture, no packing of congruent balls in Euclidean three space has density greater than that of the FCC packing.

Equivalently high packings are found by stacking planes of hexagonal close-packed spheres. Each sheet of spheres is placed so that it sits in the interstices of the previous hexagonal sheet. There are two sets of interstitial sites: B and C; the first layer is called A. The following are some common stackings: ABCABCABC, the FCC structure; ABABABABAB, hexagonal close packed or HCP; and ABACABAC, double HCP, a structure found for some lanthanide and actinide elements. And ABCBACBCBACABCA is random HCP, the structure found in many colloidal crystals. However, the number of such configurations grows exponentially with the length of a sample rather than its volume and thus does not contribute to the sample’s entropy.

I also found Terry Goldman’s comments interesting. Strangely enough, several research groups, including my own, have performed crystallization experiments on colloids in micro-gravity during orbit. The physics of the liquid-to-crystal transition is the same in space as on the ground. That is because the transition has its basis in the purely geometric problem of particle packing. Using the crystal and random packing limits that researchers have found in experiments on granular systems, we can evaluate the entropy of a system at lower densities, and the ordered state has the higher entropy. For molecules on Earth and for colloids in space, thermal energy dominates gravitational potential energy, and entropic effects produce the crystallization with which we are familiar.