Spring arrives late in Snowbird, Utah. By May the snow was still several feet thick and melting in rivulets. The ski resort was deserted, except for a few hundred pale-skinned mathematicians who had been cooped up for three days watching one another draw equations on blackboards. It was late in the evening, and 20 or so bleary-eyed souls were seated in a small conference room, hoping for something different. It was Bob Behringer’s turn to speak. Wisely eschewing the blackboard, he popped a videotape into the vcr.
On the television screen was a tray that held an inch-thick layer of sand--ordinary sand, beach sand, sandbox sand. Suddenly the sand appeared to come to life, rising for an instant off the tray, then collapsing again, rising and collapsing, like a formless rag doll. (Evidently Behringer had made the video by strapping a camcorder to the same machine that was moving the tray up and down, so that to a viewer only the sand seemed to be moving, while the tray gave the impression of being stationary.) Gradually the tray picked up speed. In those instants when the sand was suspended in air, it began to take shape. Soon it formed the perfectly rounded hills and valleys of a sine wave. Nice trick. Around the room, a few heads nodded in sleepy acknowledgment--the academic equivalent of polite applause.
Then something extraordinary happened. As the tray reached maximum speed, the sine wave began to break down and its smooth contours gave way to what appeared to be square corners. Yes, it was unmistakable; there on the screen was a square wave, looking for all the world like the menacing grin of a jack-o’-lantern. Now, mathematicians have grown almost fond of seeing sine waves occurring in nature, but square waves, with their taint of artifice, came to this audience as a complete surprise. Suddenly the group was animated. Grunts of bewilderment erupted around the room. From the back, a mathematician blurted out, Why does it do that?
Behringer shot his interrogator a look, slightly comical but not unsympathetic, and shrugged. I don’t know, he said.
Behringer is not a mathematician but a physicist. His expertise is granular flow, which is to say that he studies sand. His role at this conference was to give the mathematicians a dose of the real world, show them some physical curiosities, shake them up a bit. He himself is not particularly flashy. He is a boyish 47, he wears metal-framed glasses with thick lenses, and he possesses the natural modesty of the experimentalist. Over the past decade, he has performed an impressive variety of experiments in his laboratory at Duke University, most of them involving sand in one way or another. Yet he insists that he cannot explain much--perhaps most-- of what he has observed. Simple, ordinary, modest sand, neglected by engineers and physicists for decades, adopted by well-meaning but experientially challenged mathematicians, defies explanation.
That not even a physicist can explain why sand behaves the way it does seems astonishing. Sand is neither invisibly small nor impossibly distant; observing it requires neither particle accelerators nor orbiting telescopes. The interactions of grains of sand are entirely governed by the same Newtonian laws that describe the motion of a bouncing ball or the orbit of Earth about the sun. The odd behavior of a layer of sand bounced up and down on a tray should, in principle, be entirely knowable and entirely predictable. Why, then, can’t Behringer simply take a bunch of equations describing the motion of all of the individual grains, put them in a very large computer, and wait--for years, if necessary--until it spits out a prediction?
The problem, Behringer explains, is not one of computation but of knowledge: though sand is acted on by Newtonian forces, we simply don’t know enough about how those forces operate when let loose in a pile of sand. If you have a ball and you drop it from so high, it will lose some energy when it bounces off the floor, and it won’t rise again quite as high, Behringer says. You can describe that with a very simple equation taught in high school physics, and you can use it to predict very accurately how high the ball is going to bounce. But toward the end, when the ball has lost almost all of its energy and it no longer rises very high off the floor, all kinds of other effects enter into it--acoustic vibrations, temperature, tiny irregularities on the ball’s surface. It turns out that we really don’t know much about these effects, and we don’t know which of them is going to predominate.
Taken alone, of course, a single, spherical grain of sand is very much like an ordinary bouncing ball. Drop it on the floor and it bounces-- pretty high, in fact. Like dogs, however, sand grains have a pack mentality. Put a few million of them in a sack or on a tray and suddenly you don’t know them anymore. Drop a sack of sand on the floor and it absorbs the energy of the fall quite well, which means it doesn’t bounce at all. Look at it in Behringeresque slow motion: the bag is falling, falling; the first few grains that come into contact with the floor do indeed bounce, but they don’t make it very far before hitting other grains, which are in turn nestled right next to even more grains, and so on. As the grains topple down in close proximity, they pass along the force of the fall from one to another, through hundreds, thousands, millions of grains, from one end of the bag to the other and back again. In an instant, the energy of the fall is gone, completely dissipated in an unspeakably large number of very tiny collisions among the grains. This ability to absorb so much energy so quickly is what makes walking along the beach such a slog-- the sand absorbs the impact of your foot, taking away the spring in your step. A single grain in this great collective behaves like a ball, all right, but one that is neither quite in motion nor at rest.
If you’re a physicist seeking to explain sand, this is only the beginning of your woes. It’s not enough to know that individual grains act collectively to absorb energy. If you’re going to make useful predictions, you need to know precisely how they will interact. The large size of sand grains--relative, that is, to things like molecules or subatomic particles- -may seem reassuring, but it is actually the root of all the trouble. With sand, as with grass seed, wheat, millet, cornflakes, or any other granular material, the grains are too big to ignore. For this reason, describing the behavior of sand is different from describing the behavior of, say, a liquid or a gas, which technically is also an agglomeration of smaller units. You can, of course, think of the molecules of a liquid or a gas as very tiny grains, but there is absolutely no need to do so. An engineer can completely ignore this particulate quality without losing a shred of precision. It is enough merely to look at the average behavior of the particles. The individual molecules are so far apart that any single one is not likely to collide with more than one other at a time. These so-called pairwise collisions are a cakewalk compared with what goes on among grains of sand.
If I could look at water on the scale of atoms or molecules, I would see that they’re fluctuating madly, Behringer says. But if I look on a human scale, those fluctuations average out. All you have to do is tell me the temperature and pressure or whatever of the water, and I know everything about it. Of course, in granular systems this doesn’t work. The particles are human scale already, so you’re going to see this inherent granularity. You can’t ignore it. It’s a fact of life.
Yet ignore it is precisely what engineers have had to do. Because the language of physics does not contain a vocabulary for granularity, engineers must treat granular material as either a liquid or a solid. These approximations work most of the time, but occasionally they lead to disaster. Grain silos, for instance, are designed under the dubious assumption that the grains distribute their weight uniformly, as though they were water molecules. In fact, when the grains come to rest against one another they form intricate, quasi-self-supporting structures. That is why adding more grains to the top of a silo often does not increase the pressure delivered to the bottom at all, but rather increases pressure outward against the sides of the silo.
More to the point, these grain structures can deliver enormous force to almost any spot in the silo, in a manner impossible to predict. Every so often, pressure builds up on one of the metal rings that encircle the silo and breaks it, rupturing the corrugated steel skin. It happened in the Canadian town of Whitby in 1990; a grain silo suddenly burst, burying 25 cattle under 500 tons of feed grain. Occasionally, interlocking grains form an arch strong enough to support the entire weight of the grain above it, and when the chute at the bottom is opened, grains pour out and leave a cavity. In 1994, in the English county of Cumbria, a farmer and his son were unloading barley from a silo when they noticed that the flow came to a stop even though the silo was nearly full. When the son tried to clear the blockage, the cavity collapsed and grain poured out suddenly with great force, burying him alive. The flow of grain itself can vary unpredictably from a trickle to a gush--to the constant annoyance of engineers in the food, mining, and shipping industries.
Engineers who design buildings and roads, on the other hand, assume that under stress the supporting (and granular) soil will behave like a deforming solid, much the way plastic does. Once again, this convenient approximation occasionally leads to disasters. In the farming town of Rissa, Norway, for example, in 1978, a farmer digging a foundation for an extension on his barn piled the dirt near the edge of a lake. The added weight caused the strip of shoreline to give way and slip into the water. The shock triggered another, bigger landslide, which in turn caused a whole hillside to give way, dragging a farm and a schoolhouse into the water as well. In 1982, a particularly severe storm in the San Francisco Bay area led to thousands of landslides, killing 25 people and causing more than $66 million in damage. If engineers understood the physics of soil better, these disasters might have been avoided.
Despite such problems, few researchers cared to look into the nature of granular material until the late 1980s, when Per Bak, a physicist at Brookhaven National Laboratory in New York, began publishing papers that purported to explain some of sand’s odder characteristics. Bak--a theorist, not an experimentalist--wasn’t interested in sand per se but in a whole range of phenomena that could be grouped under the rubric of complexity. Among them were such seemingly impenetrable mysteries as the evolution and extinction of biological species, the frequency of earthquakes, and the behavior of the stock market. To describe these complicated, seemingly disorderly systems, Bak was developing an overarching mathematical theory-- Behringer calls it a global scenario--called self-organized criticality, and he felt that if this theory applied to anything at all, it should apply to a pile of sand.
Drop sand a grain at a time and the pile it forms will get higher and higher until, at some critical point, the very next grain causes an avalanche. Sometimes the avalanche occurs almost immediately and constitutes only a few grains sliding down the slope. At other times the grains collect for longer than seems possible, until a great many of them come crashing down at once. (Over time, the avalanches serve to balance out the addition of new grains, so that after each avalanche the slope of the pile remains the same. In this sense, the pile is self-organizing.) If you watch such a sand pile long enough, and count the grains as they fall on top, and keep track of when the avalanches occur and how big they are, you can make a plot of this avalanching phenomenon. The question that Bak thought he had an answer to, without ever having touched a single experimental sand pile, was: What shape will the plot be? What pattern will emerge?
That question might seem academic, but it speaks to the very heart of the sand issue. Avalanching is a metaphor for sand’s annoying unpredictability. Water, by contrast, is predictable. Add a gallon to the bathtub and the level rises, add two gallons and the level rises twice as much, and you know precisely when the tub will overflow. If you let a drop of water fall into the tub now and then, the molecules flow freely every time. There is no suspenseful, unpredictable piling--no avalanching. If you measure the amount of time it takes for the momentary swell of water to flatten itself out, you get a classic bell-shaped curve. The precise time will vary only slightly about a certain characteristic time, which is marked by the peak of the curve.
Complex phenomena behave differently. Add sand, a grain at a time, to the top of a sand pile, then plot the resulting avalanches, and you will find, Bak maintained, that they yield no bell curve but instead conform to what is known as a power law, which essentially means that the frequency of avalanches will have an inverse relation to their size--that is, there will be a great number of very small avalanches and just a few very big ones. (Picture your graph with time as the x axis and number of avalanches as the y axis. Your plot will start high on the left and swoop down sharply and low to the right, which means that many events occur after a small number of grains are added to the pile, while very few wait until a huge number of grains have been added.) The same is true for other complex systems, says Bak. Thus there will be few massive extinctions, many tiny earthquakes, few catastrophic stock market crashes, and so on.
Although this seems reassuring in some sense, it unfortunately means that specific prediction for these systems is impossible. Unlike the system characterized by a bell curve, there is no pattern around which the event revolves. The number of sand grains that will form an avalanche varies over a very wide range. Do a hundred such experiments, a thousand, or a million--you will have no better idea when the next avalanche is likely to occur, or how big it will be.
No wonder Bak’s theory triggered so much interest in sand. Bak was arguing that the old approximations of granular material as fluids or solids were not merely rough but hopelessly flawed. If he was correct, physicists and engineers could no longer count on sand as having an average behavior, as fluids and solids do. If you have no average, and no way to predict how widely the sand’s behavior will fluctuate around that average, it becomes exceedingly difficult to judge how big a safety margin should be incorporated into a silo or a highway to avert disaster. Engineers would have to abandon that time-honored practice, and physicists would have to invent a new physics for sand.
Behringer was drawn into the fray in the autumn of 1987, at the urging of a theorist who had more conventional ideas about sand. David Schaeffer is a Duke mathematician who specializes in the kind of mathematics used to describe the behavior of fluids, which also happens to be the same math that forms the basis of the continuum model (that is, the liquid model) of granular flow, precisely the model Bak’s theory purports to refute. Schaeffer had taken a close look at the equations of the continuum model and managed to tease out of them a prediction of how sand ought to behave under certain conditions. Namely, he thought he could predict how, on average, a tiny disturbance in flowing sand--caused, for instance, by the sharp edge of a particular grain as it rubbed against its neighbors--would propagate throughout the grains. As one grain hit another that hit another and so forth, the disturbance would tend to grow until it got so big that it required more energy to keep growing than the overall stampede of grains could provide.
What differentiated Schaeffer’s ideas from Bak’s was his conviction that such a disturbance would take a characteristic, and therefore predictable, amount of time to grow and to die out--that a plot of these disturbances would indeed generate a bell curve. He also believed that this characteristic time scale should depend directly on inherent properties of the sand itself, such as the shape of its grains and their exact composition. Although Schaeffer and Bak talked about different phenomena--disturbances in flowing sand versus avalanches in sand piles-- they both asked essentially the same question: How do you describe in some useful way what sand does? Schaeffer had worked out the mathematics of his hypothesis, and he wanted to test it. What he needed was somebody who could whip up a nifty experiment.
Behringer is, to put it mildly, an experiment-oriented physicist. It is difficult to carry on a conversation with him without being dragged, repeatedly, from his office down to his basement laboratory. There he flits among a menagerie of makeshift machines, each of them dedicated to shaking, compressing, rubbing, grinding, or irradiating sand.
He is convinced that experimentation is the first step toward acquiring a physics of sand. Coming up with global scenarios such as self- organized criticality is exciting, but unless you know something about the phenomenon you’re trying to describe, it’s useless. In many ways people have only just begun to ask the right questions, he says. People have been trying to fit this square peg into a round hole, but that’s not the right way to go about it. You have to step back and say, ‘Look, I’m really dealing with a material in which the granularity is endemic. I’m going to sit down and look at it and ask, What is the right way to depict it?’ If you take that approach, then I think you can make some progress. You have to do the right experiments, and the experiments will drive the theory.
In 1988, at Schaeffer’s prompting, Behringer decided to try to measure the flow of sand in a hopper by means of an elegantly simple experiment. He built a container with a funnel at the bottom, similar to a grain silo, filled it with sand, and let the sand slowly run out the bottom. As grains fell out, the ones that were left behind shifted and jostled one another, but rather than doing so in an orderly manner, they tended to pile up and avalanche, pile up and avalanche. Each avalanche made a small noise, which Behringer picked up with a microphone attached to the hopper’s side.
When he analyzed the noise of the avalanching sand, he found, to his surprise, that the data did not fit Schaeffer’s theory at all. The pattern of avalanches predicted by Bak’s theory was evident in Behringer’s data, lending support to self-organized criticality, but the avalanches appeared garbled together with other patterns, like the ghost images you see in poor television reception. The point was, he says, we needed to do an experiment that was, in some sense, cleaner.
In retrospect, he realized that listening to vibrations was an inexact way of measuring avalanching because the sounds that reached his microphone could be coming from anywhere in the hopper. To gain precision, he decided to measure the stresses that are transmitted from one grain to another. If you push on a bag of sand with your finger, the resulting stress does not get distributed evenly throughout the bag, as it would with a uniform solid. Instead the grains rub against one another willy-nilly and, depending on their shape and orientation, the stress shoots down through chains of grains that form spontaneously, then break up and re-form somewhere else. Behringer wanted to know just how these stress chains fluctuated--whether they came and went in random fashion or according to some kind of order. Of course, stress fluctuations are not, strictly speaking, the same as avalanching, but the two phenomena should be related, he reasoned, in the same way that the collapse of a bridge is related to the amount of stress placed on its beams.
Rather than use sand grains for his experiment, Behringer opted for much larger glass marbles, which would be easier to measure and, because of their spherical shape, would simplify things a bit. He poured the marbles into a ring 16 inches in diameter and 2.5 inches high and covered them with a metal plate. The plate pressed downward on the several layers of marbles and rotated, thus dragging its surface over the topmost layer of marbles and exerting what physicists call a shearing force. As the friction of the moving board pulled the marbles beneath it one way, the marbles in the layer below exerted another frictional force in the opposite direction, in an effort to keep the marbles stationary. The tension between the shearing force of the board and the friction of the marbles sent the marbles squeaking and popping and pushing upward against the plate. By mounting a pressure sensor at an arbitrary spot at the bottom of the box, Behringer kept track of how the stress was being transmitted down from the top.
What happened was, the stress was carried by these chains from the top somewhere down to the bottom, Behringer says. A lot of the time, the stress actually delivered to the detector was relatively small. But from time to time, in semirandom fashion, we got a chain that carried a large portion of the weight of that top plate all the way down to the detector. That was a major avalanche, so to speak.
Despite Behringer’s efforts to simplify the experiment as much as possible, the results were annoyingly ambiguous. Self-organized criticality was clearly playing some role in what was going on, but it was only part of the picture and probably, Behringer was beginning to suspect, a rather minor one. What was becoming clearer, however, was that Schaeffer’s characteristic time scales were nowhere to be seen. Either Schaeffer’s theories were dead wrong or the forces they described were such a small part of the total picture that they were being overwhelmed by other things. What these other things were, Behringer did not know. But whatever they were, he concluded, apparently neither Schaeffer’s nor Bak’s theory came close to capturing them.
Here Behringer decided to change tack. Rather than go for numerical data, he conjured up a way of actually seeing the stress chains with his own eyes. For this, further simplification was needed. Instead of balls, he used flat polyurethane disks, which have the handy property of changing the way they polarize light when squeezed. He arranged the disks on a platter that had a rotating hub at its center. As the hub turned, it rubbed against the disks that came into contact with it, causing them to push against the other disks on the platter. Where the stress was greatest, the disks allowed polarized light from below to pass through, giving them the appearance of glowing. Watching from above, Behringer observed these stress chains flickering through the disks like flashes of lightning; and it occurred to him that he could very well describe this flickering by saying that he was seeing the stress chains vibrating, like a plucked violin string. Was it possible that much of the behavior of sand could be explained as acoustic effects?
It was just a fanciful thought, and one that might prove misguided. Even so, it suggested something fundamental about sand that Behringer sensed only intuitively. Perhaps his data really represented not irreconcilable models but many different modes of behavior, each of them operating in different domains, usually overlapping several at a time, rarely distinct. And perhaps these modes were so different from one another that they could not be captured in any single, overarching theory of sand. In some of these modes, self-organized criticality does indeed hold sway, and nothing matters except the complex mathematical relationships between the grains. In others, sand really does act as a liquid, and you have to think in terms of fluid dynamics. Sometimes sand operates in an acoustic mode, and you have to think in terms of vibrating stress chains. If you really want to describe what sand is doing in any given situation, you have to know which modes are dominant and which sets of equations you’ll need to employ. But how many modes were there? And which ones were accentuated under what conditions?
Once Behringer had opened this Pandora’s box, he was hooked. Everywhere he looked, it seemed, sand revealed some startling new quirk. He returned to the hopper of his earlier experiment and began questioning some basic assumptions about how sand flowed through it. When the bottom spout is opened and sand begins to pour out, the flow actually occurs only in a cone-shaped region just above the opening--like the small tornado spout that appears when a bathtub is draining. People had always assumed that the sand at the center of this cone slid past the stationary sand surrounding it as though they were two solid objects--that, in other words, they exerted a shearing force on each other. Behringer wasn’t so sure. To see what was going on inside this cone, he wheeled a hopper full of sand to Duke’s medical center to have it X-rayed. Under the X-rays, the denser areas of sand looked bright, while the less dense areas were darker.
When the X-ray videos came back, they confirmed his skepticism. As long as he filled his hopper with Ottawa sand, which consisted of almost perfectly spherical grains formed from the moving water of rivers and streams, the conical flow indeed matched the assumption--the sand appeared bright and dense, except along the edges of the cone, where, as the grains rubbed against one another, the sand became looser and darker. But when he used a rougher, coarser sand, whose grains had been chipped and broken by the wind into irregular shapes, the results were dramatically different. When the hopper was opened, dark waves emanated from the spout and moved upward, against the flow.
What I think is happening, Behringer says, is that the materials that are smooth and round always tend to pack densely. On the other hand, something with a lot of sharp points and edges can pack in many different ways. Think of salt coming out of a salt shaker. There you have fairly regular little cubes. So now imagine that the salt can actually pack with its faces touching, a very dense packing, or it can pack with an edge or even a corner of a cube touching a face. When you see regions that are bright, the material has packed in a very high-density way. In the dark regions, the material has expanded, but the grains are still touching and they are relatively stable. In other words, though to all appearances the coarser sand flows in the same way as its smoother counterpart, there are dynamic processes going on that might, under the right conditions, significantly affect its behavior. As to why these density waves spread upward from the hopper, against the flow, Behringer shrugs. I don’t know, he says.
Behringer continues to try to put sand in novel situations, either to isolate the factors that affect its behavior or to test long-held assumptions. The motivation for the tray experiments, in which sand is thrown up into the air and allowed to fall, was to study the effects of the air. At first, Behringer had a hunch that the quirky wave patterns had something to do with air trapped by clumps of sand grains, but repeating the experiments in a vacuum, as some of his colleagues have done, quickly disproved this hypothesis. What causes the sand to form these standing wave patterns, particularly the square wave Behringer displayed at the mathematics conference, remains a mystery. He can reconcile how some grains would bounce up at different times from others, but he has no idea why they act coherently.
The same aura of mystery surrounds a similar standing-wave phenomenon announced last September by a group of researchers at the University of Texas at Austin. By vibrating a thin layer of spherical balls (which are nothing more than platonically ideal sand grains), they repeatedly created some intriguingly quirky patterns--alternating peaks and dimples appeared in latticelike arrangements of stripes, squares, and hexagonal honeycombs, depending on the strength and frequency of the vibration. These oscillons, as the group calls the odd structures, sometimes seemed to attract and repel each other as though they were electrically charged, which they were not.
According to Behringer, oscillons are just another of the many modes of complex behavior already observed in granular materials. These patterns fit in with a whole set of complex dynamics that we’re all struggling to sort out, he says. Now it’s pretty patchy as to the best way of describing these things. Hopefully people will construct models that knit these phenomena together, but right now it’s an experimentalist’s game.
For his part, Behringer plans to put the basic premise of soil mechanics to the test--specifically, the notion that when a piece of ground gives way and begins to slide over another piece of ground, the two pieces act like two chunks of solid matter, even though they are both composed of grains. In his experiment, he will fill a tall, thin plastic sack with sand, forming a column, and then apply pressure to the top. Eventually a break will appear somewhere in the middle, and the top portion of sand will begin to slide over the bottom portion. By sending sound waves through the column and measuring their speed, he expects to be able to determine what is happening in the layer between the two chunks of sand. If the sand behaves the way soil mechanics says it should, a thin layer at the break should soften and slow the sound waves down. In that case, Behringer will have shown that grains do not play much of a role in soil mechanics. My suspicion, he says, is that this will turn out to be a vain hope.
Behringer thinks that the principles of granular flow may yield insight far afield. In particular, it may turn out that grains of sand behave similarly to, say, the way chunks of rock in Earth’s crust jostle one another when placed under stress. At this early stage, however, it’s difficult to tell where the experiment will lead. Behringer believes that the field is where high-energy physics was in the early part of this century. Back then, physicists had discovered that the light we see from a star contains only a few specific wavelengths, but they had no theory to explain why. They had to content themselves with making observations for decades until quantum mechanics provided an explanation in 1925. That’s what Behringer’s trying to do: he’s collecting observations, categorizing the various behaviors, and trying to sort out the ghost images in his data. If physicists succeed in coming up with a theory of sand, it is likely to consist of a patchwork of different physical points of view.
You just have to recognize that not everything you do is going to shake loose major pieces of knowledge, he says. But collectively, and on rare occasions, experiments or ideas will come along and make a significant impact. It’s like looking at a distribution of avalanches--you have a lot of little ones and, every once in a while, a big one.