MATERIALS SCIENCE:
Shape Matters

Imagine the following: Take a pen, close your eyes, and uniformly and randomly fill a sheet of paper with dots. Now open your eyes, draw a little box on the sheet, and determine the local density of dots by counting their number in the box. You have just taken a sample--a statistical measurement used in opinion polls and blood samples alike.

Common sense suggests that the overall dot density can be estimated by their density in the sample and that this estimate improves with the size of the sample. On page 105 of this issue, Narayan et al. (1) report a spectacular breakdown of this assumption. In their elegant experiment, sticks made of pieces of copper wire are vibrated and rattle around between two closely spaced horizontal plates. Counting, in snapshots of the system, the number of sticks N in boxes of increasing size, Narayan et al. find that their sample-to-sample fluctuations grow proportionally to N. In other words, the local estimate for the overall density does not improve with sample size, and density is ill defined. The authors refer to this observation as giant number fluctuations.

Mathematics rules out such giant fluctuations for a broad class of systems. The central limit theorem states that when fluctuations are independent, sample-to-sample fluctuations grow only as the square root of N. This theorem applies to simple equilibrium systems (such as a gas at constant temperature) but does not automatically apply to nonequilibrium systems (such as living systems).

Swarms and swirls. In the experiments of Narayan et al., agitated sticks form swarming states that exhibit giant number fluctuations. Similar patterns are observed in fish swarms (top left). Swirls are also observed in systems that are close to jamming, for example, in the motion of bubbles in a sheared foam (right). CREDIT: MAIN IMAGE, NARAYAN/NDIAN INSTITUTE OF SCIENCE; TOP LEFT, JOHN FOXX/GETTYIMAGES; RIGHT, M. MÖBIUS/LEIDEN UNIVERSITY

Swarming and giant number fluctuations are a hallmark of the alignment displayed by driven collections of nonspherical particles. Theoretical models have been developed to describe swarming and alignment observed in schools of fish, flocks of birds, herds of sheep, or bacterial colonies--often borrowing from equilibrium models for magnetization, which consider the alignment of arrowlike objects. A very simple nonequilibrium model that exhibits cooperative motion arises when these arrows are allowed to propagate (2). In similar models, a collective response to predators and decision-making can arise (3). Toner and Tu first pointed out the giant fluctuations in such models (4).

In these systems, the particles have a preferred direction of propagation--just like real fish and birds. In 2003, Ramaswamy et al. (5) wondered what would happen for "active nematics," liquid crystals in which the particles have an orientation but have identical heads and tails (like the sticks in the present experiment). Their theory predicted that nematic systems also should exhibit giant number fluctuations, and these were recently observed in computer simulations (6).

However, when Narayan et al. tried to find such fluctuations in experiments, they encountered a surprising hurdle: cylindrical rods, arguably the simplest nematic particles, do not form nematic states and do not exhibit giant fluctuations (7). The authors achieved their present breakthrough only after etching the rods to obtain sticks with thinner ends (see the figure); for unknown reasons, these sticks exhibit nematic order. To complicate matters further, Aranson et al. recently performed similar experiments and observed that weak coupling between the nematic order and spurious in-plane vibrations of the support plate may strongly influence the swirling motion (8). Clearly, swarming is a subtle problem, and the precise nature of the swarming state and the transition to swarming is not yet fully understood.

The experiments of Narayan et al. are part of a bigger story, where nonequilibrium systems of nonspherical particles exhibit surprising behavior: We do not yet understand the consequence of shape. An earlier striking example of this is the finding that, contrary to expectation, M&M candies can be packed more effectively than spheres (9).

Imagine further increasing the density of the sticks in the experiment of Narayan et al. At some point, the particles will hinder each other so much that their dynamics will slow down dramatically, and collectively the sticks will behave as a rigid system--the system will be jammed (10). Studies of jamming in collections of spherical particles have uncovered many aspects of the rich structure of these disordered systems (11-14). Despite the absence of nematic order, these systems show swirling near the jamming transition (see the figure). Probing the spatial ordering and mechanical properties of nematic particles, such as ellipses, promises to provide fresh insights into swarming and jamming.

Does the transition to swarming precede the jamming transition or coincide with it? Do giant fluctuations persist near jamming? Are swarming and swirling two manifestations of the same underlying mechanism? To answer these questions, physicists will increasingly subject rice grains, rods, needles, disks, and other nonspherical objects to shaking and shearing in the coming years. What is clear already is that shape matters.

References

1. V. Narayan, S. Ramaswamy, N. Menon, Science 317, 105 (2007).
2. T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, O. Shochet, Phys. Rev. Lett. 75, 1226 (1995).
3. I. D. Couzin, J. Krause, N. R. Franks, S. A. Levin, Nature 433, 513 (2005).
4. J. Toner, Y. Tu, Phys. Rev. E 58, 4828 (1998).
5. S. Ramaswamy, R. A. Simha, J. Toner, Europhys. Lett. 62, 196 (2003).
6. H. Chaté, F. Ginelli, R. Montagne, Phys. Rev. Lett. 96, 180602 (2006).
7. V. Narayan, N. Menon, S. Ramaswamy, J. Stat. Mech. 2006, P01005 (2006).
8. I. S. Aranson, D. Volfson, L. S. Tsimring, Phys. Rev. E 75, 051301 (2007).
9. A. Donev et al., Science 303, 990 (2004).
10. A. J. Liu, S. R. Nagel, Nature 396, 21 (1998).
11. O. Dauchot, G. Marty, G. Biroli, Phys. Rev. Lett. 95, 265701 (2005).
12. A. S. Keys, A. R. Abate, S. C. Glotzer, D. J. Durian, Nature Phys. 3, 260 (2007).
13. W. G. Ellenbroek, E. Somfai, M. van Hecke, W. van Saarloos, Phys. Rev. Lett. 97, 258001 (2006).
14. F. Lechenault, O. Dauchot, G. Biroli, J.-P. Bouchaud; available online at http://arXiv.org/abs/0706.1531v1.

10.1126/science.1145113