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A pattern emerges

From an animal's markings to the ripples on a sandy beach, nature's patterns may all arise from the same simple building block, says Marcus Chown



PAUL Umbanhowar, Francesco Melo and Harry Swinney spend a lot of time playing with sand. It seems to be a popular pastime for physicists these days. It's not that they are immature or nostalgic for the lost days of childhood or anything like that. No, in their lab at the University of Texas at Austin, these three have been playing with sand in the ambitious hope that it might help them understand how and why patterns form. Not just patterns in sand, mind you, but patterns everywhere, from the immense cellular structures of flowing air that form in thunderstorms to the very stripes on a zebra's back.

Is that possible? The idea that a single theory could explain pattern formation in such diverse settings might seem ludicrous. But in the past ten years, scientists have found striking similarities in the patterns forming in everything from tanks of stirred chemicals and dishes of growing amoebae to fields of freezing and thawing tundra. The nature of the underlying stuff is seemingly irrelevant, as if the true roots of pattern formation lay in some deep and as yet unrevealed mathematical similarity between all of these systems.

So to study patterns, Umbanhowar, Melo and Swinney have been looking at the simplest pattern forming system they can think of-a vibrating box of sand. And they haven't been disappointed. In a thin layer of jiggling grains, they've witnessed the spontaneous formation of regular arrays of stripes, squares and hexagons. And they've watched as these have dissolved into irregular, spaghetti-like tangles of ridges and valleys.

Last year, however, the trio stumbled onto a truly remarkable pattern that no one had ever seen before something they call an "oscillon". Much like tiny dimples or bumps on the vibrating surface of the sand, oscillons drift about like tiny particles, and have the remarkable ability to combine to make complex structures which look for all the world like molecules and crystal lattices.

Theorists are scrambling to determine the significance of these strange point-like patterns, but impressed by their ability to gather together to make larger patterns, some have already suggested that the Texas team may have come upon the very "atoms" of a fundamental theory of pattern formation.

The search for such a theory began long ago, and in their efforts to unravel it, even giants such as Galileo, Newton and Faraday were attracted to vibrating sand. If a square metallic tray is made to vibrate- by subjecting it to loud sounds, for example-then sand in the tray will spontaneously arrange itself into patterns known as Chiadni figures. The sand naturally gathers in some regions, and shuns others. Newton, in explaining why the sand moves as it does, highlighted one of the key features of the behaviour of vibrating grains. The higher they bounce in the air, the more violently they collide with other grains when they come down. Since these impacts are apt to impart some sideways motion to grains, sand "diffuses" from places where a plate is vibrating to those where it is stationary-the so-called "nodal" lines-where it collects in piles.

But Chiadni figures ultimately tell you more about the vibrational patterns of the platter than of the sand itself. To get around this, and to really get at the fundamental aspects of pattern formation, the Texas team uses a rigid container. This ensures that the patterns aren't simply enforced by its flexible vibrations, but reflect instead the behaviour of the grains, and their tendency to "self organise".

Umbanhowar, Melo and Swinney began their studies five years ago, trying to improve on other researchers' earlier experiments which were done at atmospheric pressure and with layers of sand almost as thick as they were wide. In these early experiments, not only did grains collide with one another, but their motion was also slowed by the air and the walls. "We've eliminated these complications by using a very thin layer of grains in a vacuum," says Umbanhowar.

The team's "thin layer" is typically 800 particles wide and only a few particles deep-it sits in a shallow metal pan which vibrates between 10 and 200 times per second. "In the beginning, we used a loud speaker to do the vibrating," says Swinney. "But, it had a tendency to tilt the layer Now we use an electromechanical shaker which vibrates the layer evenly."

The final ingredient in this simple experimental setup is illumination, achieved by surrounding the metal pan by a ring of strobe lights. This low-angle illumination is such that when the vibrating layer is observed with a camera from above, those grains which bounced higher than the rest shine brightly while the rest appear dark. The strobe flashes at half the rate at which the tray vibrates, so that the team catches a "snapshot" of the layer's profile after every other up-down cycle of the tray's motion.

The idea is simply to shake the box at different frequencies and amplitudes and see what happens. The "acceleration amplitude" is the maximum acceleration of the tray as it rises-a convenient measure of the violence of the shaking. At low frequencies and amplitudes, the sand just sits there in a smooth layer But as soon as Umbanhowar and his colleagues shake the sand more vigorously, they see patterns. Stripes, squares and hexagons emerge, apparently for no reason, once the amplitude reaches some two and a half times the acceleration of normal gravity.

At this acceleration, light and dark stripes appear for frequencies above 40 hertz, and arrays of small squares appear for frequencies below 40 hertz. If the acceleration amplitude is increased further to a threshold of about four times normal gravity yet another pattern forms, this time an array of tiny hexagons. And as the acceleration amplitude and frequency increase still further, the sand layer enters various regimes characterised by stripes, squares, hexagons, spirals and triangles. The picture that results from this exploration of parameter space is complicated. But it can be neatly encapsulated in a phase diagram (see Diagram), as the team first reported 18 months ago (Physical Review Letters, vol 75, p 3838). Choose a certain frequency and amplitude of vibration, and their diagram tells you roughly which pattern will emerge.

But is there any way to explain why one pattern arises rather than another? Some answers come from a model put forward by Lev Tsimring of the University of California in San Diego and Igor Aranson of the Argonne National Laboratory in Illinois. Their basic idea is that a vibrated layer of sand undergoes what physicists refer to as a "parametric instability". This happens when the periodic kick of a vibrating plate produces oscillating patterns in the sand layer in much the same way that a child makes a swing go by bending her knees twice every oscillation period. As with the swing, the period of the patterns is twice that of the plate's vibration.

In their model, Tsimring and Aranson follow the developing pattern using Newton's laws of mass and momentum conservation applied to the flow of grains. The model doesn't treat the grains individually-which would be computationally very demanding-but "smears" the grains out into a sandy fluid with similar properties. Studies of the model show that grains that go high are involved in violent collisions which push them sideways when they come down. So grains that in one half-cycle rise high to form a peak in the sand layer, have moved into neighbouring valleys by the next half cycle. "What was a valley becomes a peak, and what was dark becomes light," says Tsimring.

Tsimring's simple model reproduces the patterns seen in the experiments, and it doesn't depend on any details of the shapes or sizes of the grains involved. And indeed, the patterns that Umbanhowar and his colleagues observe turn out to be remarkably robust. They've changed the shape of the metal pan, and used dozens of different types of grains, from salt and sugar granules to brass balls with diameters ranging from 0.05 millimetres to 3.0 millimetres. "It really doesn't matter what the particles are," says Swinney. "The patterns are always the same."

And it turns out that similar patterns can also be spotted in vibrating layers of liquid, as Jay Fineberg and his colleagues at the University of Jerusalem have discovered. This may seem odd since liquids would appear to have very little in common with granular media. After all, sand can support the weight of a person, whereas nobody-at least in the past two millennia-has succeeded in walking on water .And sand can accumulate in piles, whereas a liquid spreads out and there is no such a thing as a pile of water ("Sand castles and cocktail nuts", New Scientist, 24 May, p 24)

But Tsimring points out that there are some very basic similarities, too. For instance, in sand or liquid, the interactions between the component parts conserve mass and momentum but not energy. Granular systems and liquids are both strongly dissipative-that is, any internal motion is quickly damped down by friction. And when it is being shaken, sand can flow like a liquid. These fundamental similarities seem to be enough to guarantee that similar patterns will occur in both grains and liquids. It also lets Tsimring model sand grains with continuous, fluid-like equations.

But however impressive these robust patterns of hexagons and squares may be, they don't have the appeal and fascination of the Texas team's most recent discovery-the oscillon.

Oscillons occupy a narrow range of the phase diagram between completely flat sand and sand vibrating at the right frequency to produce squares. But making them takes rather more than simply tuning the vibration frequency and the acceleration amplitude into the right ball park. The Texas trio found them by accident when they were vibrating their sand layer at around 30 hertz and reduced the acceleration amplitude down to the correct range-just below two and a half times gravity.

"Suddenly, we saw a highly localised structure no more than 25 grains across," says Umbanhowar "During one cycle it was a peak and during the next it was a crater" The team announced their remarkable discovery in a paper in Nature last August (vol 382, p 793).

What was so astonishing about their finding was not only that a completely uniform vibration had created a localised structure-qualitatively different from the other patterns-but that the structure was so stable. "It persisted for more than a million vibration cycles," says Swinney

Tsimring thinks he has a simple explanation for oscillons as well. They owe their existence, he believes, to the fact that less energy is dissipated by the grains in a region that is thinner than average. So, if by chance a slight depression appears in the sand layer, the sand grains in the depression will bounce slightly more vigorously than the grains around them. And since the collisions they will be involved in when they come down will be more violent than the collisions of their neighbours, the grains in the dip will stand a greater chance of being knocked out of the dip than grains in the surrounding region will of being knocked in.

"However, the diffusion of grains out of the depression will serve only to make the dip thinner and less dissipative and the surrounding region thicker and more dissipative," says Tsimring. "Consequently, there will be a positive feedback which inevitably makes a big oscillation in the depression."

Choosing a pattern: the pattern that emerges in a tray of vibrating sand depends on both the frequency and vigour of shaking

The closest analogy to an oscillon is a soliton, a localised wave of water that maintains its integrity as it travels along a narrow channel such as a canal.

A famous example is the Severn "bore" in England. Solitons only occur in a relatively friction-free medium and eventually decay. But "oscillons can live for ever since they derive their energy from the vibrator," says Tsimring.

And oscillons are more remarkable than solitons in another way, too. In a water channel, for example, there is a whole family of possible solitons corresponding to waves with slightly different speeds and heights. But there is only one unique oscillon in a vibrating bed of sand at a given set of parameters. Tsimring puts it more precisely: "Oscillons are quantised solutions of a nonequilibrium wave equation-they're the classical analogues of quantum particles like electrons."

Whether there is any deep significance in this comparison, Tsimring is as yet unwilling to say. But there are striking similarities between the behaviour of oscillons and other particle-like entities: atoms. The similarities became apparent to Umbanhowar and his colleagues as soon as they learnt how to create more than one oscillon in their sand layer This could be done, they discovered, simply by varying the rate at which the acceleration amplitude was lowered into the oscillon range. For the first time, it was possible to observe the behaviour of a pair of oscillons.

Although there was no long-range interaction between the two oscillons, imperfections in the metal pan caused them to drift and eventually come together When they did, what happened was weird. If the oscillons were in phase- that is, they formed peaks and craters in perfect step-they repelled each other .If the oscillons were totally out of phase- that is, the crater of one occurred at the same time as the peak of the other and vice versa-they attracted each other.It was reminiscent of seeing like electrical particles repelling each other and unlike particles attracting each other," says Swinney.

But the most amazing thing of all was that when two oscillons attracted each other, they became permanently bound together, just like two atoms in a molecule. When there were more than two oscillons, their behaviour was even more peculiar "They can join together to form chains, triangular associations and even extended lattices-and all are stable," says Swinney.

"In fact," adds Tsimring, "if enough oscillons join together, they will eventually make a square lattice. So, in oscillons we're seeing the basic building blocks of all the other patterns-the 'atoms' of pattern formation." This would make the square lattice simply the lowest-energy configuration of an even mix of attracting and repelling particles.

Such an arrangement is well known in solid state physics. In what is known as an "antiferromagnetic" lattice for instance, each positively charged particle is surrounded by four negatively charged particles in a square lattice. By contrast, the lowest energy configuration of like- charged objects is a hexagonal lattice. So, once the symmetry of the square lattice is broken-that is, one kind of oscillon out numbers the other-the pattern begins to change. If the imbalance in the kinds of oscillons becomes large, then the pattern naturally tends towards one with overall hexagonal symmetry.

It is tempting to believe that with oscillons physicists have finally seen to the very heart of the phenomenon of pattern formation. However, Tsimring warns that we may not even yet have even discovered all the possible patterns that can exist in a vibrating sand layer, let alone explained them.

So what all this means for patterns in more complex systems is still unclear "The vibrating sand pile results are amazingly suggestive," says Tsimring. "But I get a bit nervous when people ask me about its implications for the whole Universe." Swinney agrees. He is not sure whether the discovery of oscillons has brought physicists interested in pattern formation any closer to a Universal Theory of Pattern Formation. "It would be nice to think so," he says. "But at this point, it's still too early to tell."


Further reading: Pattern formation outside of equilibrium, M C Cross and P C Hohenberg, Reviews of Modern Physics, Vol 65, Part II, p 851(1993)
[New Scientist 12/7/1997]



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