杏吧原创

Chicago

TOYS clutter the large table in Sidney Nagel鈥檚 office at the University of Chicago. A Plexiglas container of avalanching mustard seeds, an hourglass that works in only one direction, and numerous vessels containing marbles, ball bearings or sand-anything granular goes. The table is an island in a sea of books and physics journals. And for over a decade, Nagel and his colleagues have been trying to bridge the gap between the toys and the equations in the journals.

This may seem a dubious occupation for serious physicists. But the curious tumblings of seeds and marbles hold clues to the fundamental laws of an unwieldy but extremely important category of stuff that includes soil, curry powder, coffee beans and frozen peas. Just about any collection of objects counts-鈥渆ven those鈥�, says Nagel, nodding warily at a towering heap of back issues of Physical Review Letters that threatens to avalanche down from the shelf above his head.

Sugar, cereal, gravel, detergent-granular materials are everywhere. And they offer an unforgettable lesson in humility to physicists, who understand far more about quarks and electrons than they do a simple pile of salt. In fact, things that form heaps and piles seem to laugh at the ordinary laws of physics. We all know too well the world鈥檚 irritating tendency towards disorder. It takes work to separate jumbled silverware in a drawer, or socks in the laundry, or to make sense of the devastation that so routinely engulfs our desktops. But shake up a can of cocktail nuts, and instead of mixing, they sort themselves out by size-Brazil nuts on top, peanuts on the bottom. It鈥檚 order from disorder-all for free. And the same happens in a pile of gravel, a mix of sugar and starch, or a jar of instant coffee.

This intransigence against natural law sticks in the craw of physicists. But it causes more than a few headaches for industrial engineers as well. For nearly every food product or drug, at some stage of production, is a pile of grains or powders. The nuclear industry mixes powders in exact proportions to make fuel rods for reactors. And mining companies have mountains of grains on their hands. But in drum mixers, silos and grain chutes the world over, these things sort when they鈥檙e supposed to mix, clump together when they should flow, and generally make fools of scientists who try to predict their next move. As a result, factories handling granular materials waste as much as 40% of their production capacity just grappling with the unwieldy stuff. Unmixing isn鈥檛 a terrible problem if it just puts your nuts in the wrong order. But in the pharmaceutical industry, unmixing can prove deadly if it disturbs the sensitive balance of filler and active ingredient in an anti-malarial pill.

Seedy science

But a growing number of physicists and engineers hope to change all that. Piles of sand and rice have reached the best research labs in the world. And in the last few years, by playing with toys like Nagel鈥檚 and unleashing some sophisticated mathematics, scientists have begun to wring some secrets from the unpredictable grains, and to develop some unusual techniques to tame them.

The trouble with heaps and piles is that they鈥檙e not really like anything else. 杏吧原创s sort the world into gases, liquids and solids, but granular materials stubbornly defy such categorisation.

鈥淭hey鈥檙e not exactly liquid, and not exactly solid,鈥� says Susan Coppersmith, a physicist whose office shares a wall with Nagel鈥檚. 鈥淵ou can pour sand like a liquid,鈥� she points out, 鈥渂ut at the same time you can stand on the sand at a beach-while most people have a hard time standing on water.鈥�

It鈥檚 tempting to think of each grain as being like an atom in a solid or liquid. But in a solid, strong forces between atoms hold them in place. This is why a chunk of ice, for example, holds its shape. In a liquid, forces between atoms are also strong, but the atoms jiggle about their positions in a way that grows ever more violent with increasing temperature. This jiggling allows atoms in a liquid to squirt past one another and rearrange themselves. So you can鈥檛 keep water in a pile-it flows. But grains of sand neither attract nor repel one another, and they pay no attention whatever to temperature. Heat up a pile of sand, and the grains don鈥檛 jiggle about at all-they just get warmer.

This lack of identity is more than a matter of semantics. For while sandy soils support buildings with no trouble, their solidity is rather fickle. Port Island, near Kobe, Japan, is an island of loosely-packed soil that has been reclaimed from the sea. The earthquake in January 1995 gave it a serious shake, and turned it from solid ground into something much more like a churning liquid. In this case, sand鈥檚 split personality ruptured a tank storing liquefied gas, causing the evacuation of thousands of residents. In other earthquakes, vibrating soil has been known to swallow entire homes.

So if granular materials are neither liquids nor solids, then what are they? Their refusal to adopt a consistent character frustrates theorists no end, who鈥檝e even suggested that sunflower seeds and piles of fresh chillies might be a new state of matter. If so, it鈥檚 a seemingly inexplicable state-as yet, there is no grand theory of granular stuff.

鈥淲e could give up and model sand grain by grain,鈥� says Heinrich Jaeger, also of the University of Chicago. But that鈥檚 neither elegant nor practical. 鈥淚t would take a supercomputer to simulate a spoonful of sugar,鈥� he adds.

Hoping to throw up some clues to a theory of grains, Nagel, Jaeger and their colleagues have looked closely at sand鈥檚 most infamous effrontery: its apparent disregard for the second law of thermodynamics which states that entropy, or disorder, tends naturally to increase. To crack the Brazil nut problem, they have carried out a series of experiments using ever fancier versions of one of Nagel鈥檚 toys. The basic idea is to bury a single large bead (the Brazil nut) in a jar of smaller beads (the peanuts). Upon shaking, the big bead rises to the top.

King of the heap

When the Chicago group first did the experiment in 1993, there was already one legendary explanation for the effect, laid out by Anthony Rosato and colleagues from the New Jersey Institute of Technology. Their 1987 paper 鈥淲hy the Brazil nuts are on top鈥� claimed that vibrations are more likely to open up small gaps than larger ones-so when a Brazil nut becomes slightly airborne, peanuts rush in underneath, gradually nudging the Brazil nut to the top. Computer simulations even backed up the idea.

But by colour coding layers of beads, and tracking their motion, the Chicago team finds something very different-convection currents. Vibrations drive the beads in circles, up the centre and down the sides. But the downward currents are too narrow to accommodate the larger bead, stranding it on top. It is these currents, they conclude, that maroon the big nuts (see Diagram).

鈥淭hat was a painful experiment,鈥� says Jaeger, for at first a student had to excavate the beads one by one between vibrations to see what was going on. Today, they use magnetic resonance imaging which spares the students鈥� sanity.

But beads aren鈥檛 nuts, contends engineer Troy Shinbrot from his office at Northwestern University on the other side of Chicago. He favours the original explanation, which he calls percolation. 鈥淭he diabolical thing is that both explanations are right,鈥� he says. 鈥淭here鈥檚 obviously convection, I just don鈥檛 think it鈥檚 the major effect with the nuts.鈥� Percolation, he contends, is also why rocks rise to the surface in a garden. Over the winter, he says, 鈥渇rost moves the ground and deforms it. When a big thing goes up, smaller things go below.鈥�

Industry is eager to tame these processes, and to see why, just look at a simple bag of self-raising flour-it contains at least five different powders. Powder mixing is one of the biggest manufacturing activities of all. And without theoretical understanding, doing it right is a matter of experience. Drum mixers will mix things together at some speeds, and separate them at others. Even making a batch of aspirin is taxing.

鈥淥nce they鈥檝e mixed the stuff,鈥� says Shinbrot, 鈥渢hey put it in a bin, and load it on a truck. It bounces about. They get to their site and what do they have? A bunch of big stuff on top and a bunch of little stuff on bottom.鈥�

鈥淵ou want to know how they decide when aspirin is properly mixed?鈥� he continues, incredulous. 鈥淭hey have a guy who has worked for the company for a very long time, who puts on a latex glove, reaches into this huge mixing vat-roughly the size of a large office. He feels it with his hand. If it feels right, he says it鈥檚 done.鈥� Quality controls ensure safety, he insists, 鈥渂ut the point is we don鈥檛 have any good predictive laws to tell how granular materials mix. I say this absolutely, there are no accepted equations for granular materials.鈥�

Experiments illustrating the true perversity of the unmixing problem were reported in March in Nature (vol 386, p 379). A research group based at Boston University found that a mixture of sand and sugar could be unmixed simply by pouring it into a box. 鈥淎 miracle occurs,鈥� says Eugene Stanley, one of the collaborators. 鈥淚t鈥檚 like throwing a deck of cards on the table and having all the blacks fall on one side and the reds on the other.鈥� What gives? Well, it seems the pouring causes miniature avalanches during which the smaller sand particles find their way to the bottom. The sand and sugar sort themselves into zebra-like layers in the resulting pile.

Powder persuasion

Fortunately, the food and pharmaceuticals industries aren鈥檛 without weapons to combat unmixing. Their simplest trick is to craft their powders so that the grains have similar sizes and shapes. Get rid of such differences and mixed powders stay mixed. Still, this isn鈥檛 always possible. Different chemicals form grains of different sorts, and grinding or building them up into another shape takes a lot of effort.

But a more sophisticated method may be on its way. Les Woodcock, a chemical engineer at Bradford University in West Yorkshire, has been studying powders for some 20 years. In a fortunate coincidence, Woodcock鈥檚 graduate student Adrian Marland shared a flat last summer with the bass guitarist of a local pop band. Curiosity made him borrow the band鈥檚 bass amplifier and drag it into the lab, where the researchers made a fascinating discovery.

In a simple experiment, they used a glass jar holding an 鈥渦nmixable blend鈥� of two fine powders, one white and one blue. Ordinary shaking or tumbling of the container always left streaks of segregated powders. But leaving the jar in front of a speaker for a time, Woodcock and Marland were amazed to find that the contents separated into three regions. The upper two contained segregated white and blue powders, but the lowest held a near perfect mixture of the two. Somehow, the bass speaker had induced tiny acoustic vibrations that either segregated or mixed the powder, depending on its position in the container.

The details of how this works are not yet clear, but Woodcock has an idea. 鈥淢ost devices for shaking powders tend to have too large an amplitude,鈥� he says. But acoustic vibrations, even with loud sound, are much weaker, and jiggle the sand about more gently and at higher frequencies. These vibrations cause the grains to 鈥渄istribute their energy just as in a system of molecules鈥�, says Woodcock. In effect, the sound makes the mixture of particles become just like a mixture of two liquids.

A classic chemistry experiment shows almost identical behaviour. Mix some ethanol and water in a glass, and it separates into several regions-with the lower region a perfectly well mixed solution. Both SmithKline Beecham, mixers of fillers and drugs, and Nestl茅, who wrestle with starches and sugars, are interested in the discovery, and you can imagine why-the solution to their mixing troubles may be no further away than the nearest hi-fi shop.

Unmixing is one of the more perplexing activities of granular materials. But even the stoic sand pile harbours mystery. Common sense says that it would be more uncomfortable to be buried under 30 metres of sand than 3 metres, but that鈥檚 not necessarily the case. In a silo, for instance, the grains sit like poorly assembled bricks, and the resulting network carries weight from the centre out to the sides, as a keystone does in an arch. That鈥檚 what keeps cathedral ceilings from falling in on the faithful. So in a silo, the walls, rather than the base, support a good deal of the weight. For this reason, building a good silo isn鈥檛 easy. More than 1000 collapse every year in North America alone.

But what do the arches look like. Nagel and his gang have taken pictures of the lines along which weight is directed. Their trick is to shine light through a pile of special beads which let through more or less light depending on their state of mechanical stress-how much they鈥檙e being 鈥減inched鈥� by neighbouring beads. The photos reveal jagged force lines reminiscent of lightning bolts. Nearly all the weight is supported by only a few crooked columns of beads, while most carry very little at all (see Diagram).

Putting some numbers to these patterns will take some ingenuity. So far, researchers have made do by studying the forces at a pile鈥檚 base, essentially by placing it on a set of tiny scales. Such experiments go back to 1981, when two Czech engineers first showed that the force of the pile on the ground is greatest not in the middle, where the pile is tallest, but somewhat off to one side. The force shows a peculiar dip in the centre.

Over the years, the enigmatic dip has inspired at least a dozen papers. In 1996, physicists at the University of Edinburgh and the Service de Physique de l鈥橢tat Condens茅e in France, published a paper in Nature offering a possible explanation. To understand the pile, they reasoned, one has to understand its history. A pile formed by sand falling from a funnel builds up regularly, layer by layer. As a result, they postulated, the weight travels along fixed lines, oriented at a constant angle.

鈥淭he pile is supported by a series of nested arches,鈥� says Michael Cates, one of the authors, 鈥渆ach like a triangle, supported at the base, with a bridge at the top.鈥� Under the bridge, they calculated, was a dip that seemed to match the observations of the Czech engineers.

Soon after, Science picked up the story, saying the weight dip quandary was to granular mechanics what Fermat鈥檚 last theorem was to mathematics, and offering the Nature paper as a solution.

But all is not well in the sandbox. 鈥淭he physicists just haven鈥檛 read the literature,鈥� says Stuart Savage, an engineering professor at McGill University in Montreal, pointing indignantly to a list of studies he says contradict the theory of Cates and colleagues. Savage cites numerous studies of sand pile cross sections which show no pressure dip. 鈥淚n most experiments, the stress dips occur when the base is allowed to sag. When the bed is prevented from sagging, there is no dip,鈥� says Savage.

But Cates says that these studies may have been looking at a different problem. 鈥淚n our model,鈥� he emphasises, 鈥渢he dip occurs if the pile is made by dropping grains from a single point. There aren鈥檛 many engineering studies that even specify how the pile was constructed.鈥� And the grain-by-grain construction history is, Cates believes, crucial. In Nagel鈥檚 bead experiments, for example, changing the position of even a single bead radically alters the distribution of weight within the pile. According to Cates, 鈥淲hen the experiment is done carefully,鈥� that is, by funnelling grains onto a rigid support, 鈥渢he data show a clear dip that is fit by our theory鈥�.

Most likely, this turmoil is an unfortunate consequence of the academic walls separating physicists and engineers. 鈥淲e鈥檝e been working on the same things,鈥� says Savage, 鈥渂ut publishing in separate journals. We鈥檙e oblivious to each other鈥檚 work,鈥� he says, 鈥淚 hope we can get these communities working together.鈥� Savage plans to air his grievances about the force dip at a conference in the US this month.

Whatever the reception that greets Savage, it is clear that forces do not propagate through a pile of grains in the same way as they do in a solid or liquid. It鈥檚 coming to some precise mathematical theory that is so difficult.

Fennel seeds and fermions

For sand in motion, a theory is just as elusive. Most physicists look to familiar equations-like those for fluids-hoping that minor modifications might give the right equations for granular materials. But Daniel Hong of Lehigh University in Pennsylvania has coopted concepts from the ethereal world of quantum mechanics to describe how grains move around in a vibrating bed. In a paper in Physical Review Letters last month (vol 78, p 2764), Hong, along with physicist Hisao Hayakawa of Kyoto University in Japan, recast the grains as quantum particles in a Fermi gas. In a Fermi gas, particles shuffle around like cars looking for parking spots, but no two particles can occupy the same quantum state. This is the famous Pauli exclusion principle which, when applied to electrons in atoms, accounts for the order of the periodic table. 鈥淚t鈥檚 the same thing here,鈥� says Hong, 鈥渁fter all you can鈥檛 have two grains in the same place.鈥�

So far, the theory seems to line up with experimental data, and Hong hopes it will form the scaffolding for a larger theory. 鈥淲e are trying to develop a universal framework,鈥� he says. Chances are that a complete theory will take years. Until then, aspirin engineers will still be testing their batches with their fingers. But that a grand framework exists is more than just an article of faith.

鈥淵ou feel like if there isn鈥檛 anything to explain,鈥� says Susan Coppersmith of the University of Chicago, 鈥渢hen why all this structure?鈥� Coppersmith moves to illustrate the point on the whiteboard in her office. 鈥淭here鈥檚 always an aesthetic quality to devising a theory,鈥� she says, accidentally upsetting a beaker of marbles borrowed from Nagel鈥檚 toychest. With a crash, the floor is covered with granular materials, bouncing away to the rhythm of an equation that no one has yet written down.