The Science of Sticky Spheres
On the strange attraction of spheres that like to stick together
Sticky Spheres in Action
Both the Harvard and the Yale groups were inspired to undertake this exercise by an interest in aggregations of material spheres rather than mathematical ones. Hoy, Harwayne-Gidansky and O’Hern discuss the mysteries of crystallization. Bulk materials tend to favor configurations that maximize density, such as the Kepler packing. But crystal growth must start from clusters of just a few atoms, where the configurations that minimize energy are not the same as those in the bulk. Some high-contact-number clusters exhibit motifs seen in the Kepler packing, but other clusters are incompatible with space-filling structures. For example, the n=7 pentagonal dipyramid is not a pattern that can tile three-dimensional space.
Arkus is particularly interested in the self-assembly of nanostructures, both natural ones (such as viral capsids) and engineered materials. Guangnan Meng of Harvard, working with Arkus, Brenner and Manoharan, has developed an experimental system that offers another way to explore small clusters of sticky spheres. The spheres are polystyrene beads one micrometer in diameter, suspended in water along with a large population of much smaller plastic nanoparticles. When two spheres come into contact, the nanoparticles are excluded from the space between them; this phenomenon creates a short-range attractive force between the spheres. Hence the system is a good model of idealized sticky spheres.
In Meng’s experiments the microspheres were spread over glass plates with thousands of cylindrical microwells, where they formed clusters with an average of about 10 spheres per well. The wells were scanned with a microscope to tabulate the relative abundance of various configurations. If potential energy were the sole criterion, then clusters with more contacts would be more common, but entropy also enters into this calculus: Structures that can be formed in many different ways are more probable. For the most part, results were broadly in accord with theoretical expectations. Entropy favored clusters with lower symmetry, and also enhanced the representation of nonrigid structures. But because extra contacts lower the potential energy, structures with more than 3n–6 bonds were also overrepresented.
Apart from these physical applications of sticky spheres, the contact-counting model also evokes a celebrated open problem in pure mathematics. The question was raised by Paul Erdös in 1946: Given n points in d-dimensional space, how many pairs of points can be separated by the same distance? By scaling all distances appropriately, the repeated distance can always be set equal to 1, and so the problem is sometimes called the unit-distance problem. In three dimensions, the maximum-contact problem for unit spheres is equivalent to the Erdös unit-distance problem with the additional constraint that no distance is allowed to be less than 1. Thus the recent results on sticky spheres solve this restricted version of the problem for all n≤11.
What lies beyond n=11? Arkus suggests that the main roadblock to enumerating maximum-contact clusters for higher n is not the geometric problem of solving for coordinates and distances but the combinatorial one of generating all appropriate adjacency matrices. Because so few of the matrices correspond to valid packings, the process becomes hideously wasteful. Arkus suggests a possible alternative approach, although it has not yet been successfully implemented. Through n=10 she has shown that every cluster with 3n–6 contacts can be converted into any other 3n–6 cluster by some chain of simple transformations, in which a single bond is broken and another bond is formed. She conjectures that this property holds true for all n. If it does, the maximum-contact problem might be solved by generating any one structure with 3n–6 contacts and then systematically traversing the tree of all single-bond-exchange transformations.
At some large enough value of n, the diversity of these curious geometric structures will necessarily begin to diminish, as all larger clusters come to look more and more like pieces of the Kepler packing. But we’re not there yet, and there may still be oddities to discover.
- Arkus, N., V. N. Manoharan and M. P. Brenner. 2009. Minimal energy clusters of hard spheres with short range attractions. Physical Review Letters 103:118303.
- Arkus, N., V. N. Manoharan and M. P. Brenner. 2011. Deriving finite sphere packings. SIAM Journal on Discrete Mathematics 25(4):1860–1901.
- Aste, T., and D. Weaire. 2008. The Pursuit of Perfect Packing, 2nd ed. New York: Taylor & Francis.
- Biedl, T. E., et al. 2001. Locked and unlocked polygonal chains in three dimensions. Discrete and Computational Geometry 26:269–281.
- Conway, J. H., and N. J. A. Sloane. 1999. Sphere Packings, Lattices, and Groups, 3rd ed. New York: Springer.
- Erdös, P. 1946. On sets of distances of n points. American Mathematical Monthly 53:248–250.
- Hales, T. C. 2005. A proof of the Kepler conjecture. Annals of Mathematics 162:1065–1185.
- Hoare, M. R., and J. McInnes. 1976. Statistical mechanics and morphology of very small atomic clusters. Faraday Discussions of the Chemical Society 61:12–24.
- Hoy, R. S., J. Harwayne-Gidansky and C. S. O’Hern. 2012. Structure of finite sphere packings via exact enumeration: Implications for colloidal crystal nucleation. Physical Review E 85:051403.
- Hoy, R. S., and C. S. O’Hern. 2010. Minimal energy packings and collapse of sticky tangent hard-sphere polymers. Physical Review Letters 105:068001.
- Kepler, J. 2010. The Six-Cornered Snowflake: A New Year’s Gift. Philadelphia: Paul Dry Books.
- Meng, G., N. Arkus, M. P. Brenner and V. N. Manoharan. 2010. The free-energy landscape of clusters of attractive hard spheres. Science 327:560–563.
- Schütte, K., and B. L. van der Waerden. 1953. Das Problem der dreizehn Kugeln. Mathematische Annalen 125:325–334.
- Sloane, N. J. A., R. H. Hardin, T. D. S. Duff and J. H. Conway. 1995. Minimal-energy clusters of hard spheres. Discrete and Computational Geometry 14:237–259.