Dynamics of Granular Material

Heinrich M Jaegar & Sidney R Nagel. American Scientist. Volume 85, Issue 6. Nov/Dec 1997.

Sand provides children with hours of entertainment; it is the raw stuff of fantasy sandbox kingdoms, and as it trickles through an hourglass provides a mesmerizing vision of the passage of time. Sometimes scientists make sand toys, too. One of our favorites, a tall, clear-plastic cylinder nearly filled with fine sand encasing a single large ball near its bottom, reveals some rather perculiar properties of this material. For example, can a child bring the steel ball to the top of the sand? Perhaps surprisingly, a few vertical shakes of the column will move the ball upward until it reaches the top.

In manufacturing, the same phenomenon demonstrated by our sand toy arises in all sorts of mixes of dry aggregates-from pharmaceutical pills and powders to cereals and nut mixes-in which shaking brings up the larger particles, independent of their density. Many people call this the “Brazil-nut effect,” because of the way those nuts work their way to the top of a mixture of nuts. Although this effect fascinates us, it creates trouble in industrial processes, where vibrations during handling and transport of an initially well-mixed batch may lead to unwanted size separation. Unraveling the cause of this size separation-and many other characteristics of dry mixes-requires a knowledge of the physics of granular materials, which are essentially any large assembly of solid particles, or grains, held together only by gravity.

Here, we shall focus on recent developments in the emerging field of granular dynamics and the phenomena caused by vibrating granular materials. These phenomena help us understand processes that range from shaking’s effects on sand to the relaxation of magnetic flux in superconductors.

Architecture of Assemblies

At first glance, granular assemblies seem simple. We understand most of the macroscopic and microscopic material properties of individual grains. What is more, as long as the grains are dry, we can even assume that no cohesive forces exist between individual grains or between them and the walls of the assembly’s confining container. So the only interactions between grains come from direct contacts. Nevertheless, cooperative phenomena throughout an entire system of grains produce a host of interesting effects.

For example, assemblies of granular materials form arches-very small versions of those found in the design of bridges and entryways. In architectural structures, designers can create large open areas by placing stones in an arch, one on top of another, until at the uppermost level they are kept in position by the clever placement of a keystone, which distributes the weight horizontally. In granular materials, disorder in the packing leads to a large proportion of voids, which are held up by “keystones.” In the absence of any external forces, the arches stay put, keeping the individual grains in a metastable state that persists until it is disturbed by, say, vibration.

Beyond holding up individual arches, the keystones in a granular assembly also serve as nodes in a web of forces throughout the material. Such a force web holds up the weight of the material and any additional loads placed on top of it by deflecting the forces toward the outside-to the material’s container. To rearrange the web, arches must be destroyed and keystones replaced, which requires externally generated vibrations that let the pile escape from one metastable configuration to another. In such changes, an assembly tends to approach a more compact state.

Consequently, vibration also affects the packing properties of a granular assembly. In the early 18th century, English chemist and physiologist Stephen Hales made one of the first examinations of granular packing when he studied the arrangement of randomly packed peas. Today, we know that the densest possible packing of identical spheres-an orderly close-packed hexagonal structure-takes up 74 percent of the total volume, and the remaining 26 percent accounts for voids between the spheres. Of course, a disordered arrangement is much less dense. In most disordered packings of granular materials-even after as many arches as possible have been broken down, perhaps through vibration-the grains only approach the socalled “random close-packed limit” in which they take up 64 percent of the total volume.

Once an assembly is well packed, grains may not be able to move, unless the assembly expands. In 1885, the English engineer and physicist Osborne Reynolds first discussed this phenomenon, called dilatancy. Even in a well-packed assembly, external vibrations can cause temporary expansion so that the density decreases sufficiently for movement among grains. Thus, vibrating a material can rearrange its arches, increase packing and allow grains to flow.

Stimulating Sounds

Other unusual characteristics of granular assemblies can be revealed by subjecting them to sound waves. If the sound has a long wavelength-larger than either the beads themselves or the arches made up of strong bead contacts-it may only propagate vertically in a granular assembly. This happens because a sound’s propagation velocity depends on the pressure holding the grains together: Loosely packed grains lead to slow sound velocities because there is only a very small restoring force to any movement of a grain, and more tightly packed grains lead to larger restoring forces so that the velocity of sound increases. Far away from an assembly’s container walls, the average pressure between grains increases linearly with the depth. Consequently, the velocity of sound increases with depth, too. So if a wave begins in a horizontal direction at any depth, the lower region of the wavefront moves faster than the upper region. In a short distance, the asymmetrical velocities cause the wave to turn upward-moving only in the vertical direction. As the wave travels upward, it moves through grains that are less and less densely packed, and its velocity slows accordingly. At the surface of the assembly, the wave’s velocity falls to zero.

In addition to changing direction, a sound wave’s attenuation in a granular assembly may vary dramatically with time. To demonstrate this, one can fill an insulated box with a granular material, say glass beads that are 5 millimeters across, start a fixed-frequency sound wave at one point and then measure the transmitted sound wave at another point, say just a few centimeters away Surprisingly, the fluctuations in the amplitude of the transmitted wave can equal the average value of the signal itself, and those fluctuations, or noise, include a variety of frequencies, some that are as low as 0.00001 hertz. The sound causes the noise by making grains slowly shift their positions over time. As a result, arches break and reform in different configurations, and that slight motion changes the pattern of sound propagation in the material, making the detected signal very noisy and seemingly unpredictable.

Sound propagation in a granular material also depends delicately on the temperature of the system. We found that extremely small changes in the ambient temperature of our laboratory affected our measurements. For instance, a temperature change of only 0.03 degree Celsius could change the amplitude of the detected signal by a factor of three.

After recognizing that relation, we wondered what would happen to the transmission of a sound signal after heating a small area in an assembly. So we placed a small heater between the sound source and the detector. A short pulse (0.2 second) of current through the heater raised its temperature by less than 1 degree. The increased temperature produced a tiny-only 100 nanometers-thermal expansion of the heater, but an enormous 25 percent decrease in the amplitude of transmitted sound. After the heater’s current pulse, it took tens of seconds for the transmitted sound’s amplitude to return to its previous level.

Waiting on Relaxation

In the preceding examples, the low-intensity vibrations caused only minute, local rearrangements in assemblies of grains. That can happen only in sufficiently stable materials. Conversely, steep slopes on the sides of a pile of granular material may lead to farfrom-stable grain configurations. For instance, a sand pile will keep its shape as long as the slope on its sides-the so-called free-surface angle-does not exceed the maximum angle of stability, Oe, which is measured relative to the horizontal. If the free-surface angle surpasses (Rms then a pile becomes unstable and surface avalanches develop, in which only the most-exterior grains begin tumbling down the sides of the pile. The surface avalanches continue, reducing the free-surface angle to the so-called angle of repose, Or, which provides a new metastable state.

For a pile with a free-surface angle less than Or, only external vibrations can change the pile’s shape by further decreasing the angles through so-called relaxation. Moreover, such changes can continue for a long time. To illustrate this characteristic, we began with a plexiglass cylinder and filled it halfway with a granular material composed of spherical glass beads with an average diameter of 0.54 millimeter. Then, we slowly rotated the cylinder until the free-surface angle of the beads reached 25 degrees-just 1 degree short of this material’s 26-degree angle of repose, or Or-and vibrated it at a fixed frequency and amplitude. Those vibrations caused the pile’s free-surface angle to decrease slowly over time. In fact, the piles of beads relaxed at a logarithmic rate, such that a day might pass before the free-surface angle approached values close to zero degrees. Similar relaxation characteristics exist in many other materials, including type-II superconductors, where a material’s magnetization relaxes along a time course that resembles the relaxation of the free-surface angle of a vibrating pile of beads.

Beyond changing the free-surface angle of an assembly of grains, vibration can also trigger relaxation of an assembly’s bulk structure. You see such bulk relaxation every time you open a box of cereal and find that the contents fill only a portion of the container. Likewise, if you fill a canister with sugar, you can reduce its volume by tapping the sides. In most cases, vibrating a disorderly configuration of a granular material-from sugar to cereal-can reduce its volume by 10 to 20 percent.

Like relaxation of a granular material’s free-surface angle, bulk relaxation proceeds very slowly-also logarithmically in some cases. When we fill a portion of an upright tube with granular material and then vibrate it, even 100,000 cycles of vibration might not completely relax the material-depending on the intensity of the vibrations. What makes bulk settling take so long? Currently, a number of investigators seem to favor what could be called the “parking lot” explanation. Imagine that you want to park in a lot that is nearly filled with equal-size vehicles. Some spaces will be just slightly too small for another car-perhaps because one driver strayed across a space-dividing line just far enough that another car cannot “squeeze” into an open space. If the filling of the lot is to come from a random series of parking and unparking events, then several vehicles may need to move to make room for an additional vehicle. A similar scenario might arise in bulk settling of a granular material: Many grains may need to move to make room for one additional grain. In the end, such rearrangements could stretch bulk relaxation along a logarithmic time scale.

Convection Rolls

In some cases, vibrations cause organized patterns of flow in granular assemblies. To demonstrate such flow, we could begin with a circular container-one with a radius that is larger than its thickness, which allows for a quasi-two-dimensional geometry in which we can observe the movement of the grains. Next, we fill the container with alternating layers of different colors of sand, so that layers can be tracked. Finally, we simply vibrate the container vertically.

At first, the vibrations make the layers of sand “smile”-curving upward along the container’s side walls and downward in the middle. Soon after that, the layers mix so much that the stripes disappear, but the grains continue to move upward along the curving sides of the container. Just before the flow of rising sand reaches the halfway point in climbing the container’s walls, where the sand would need to flow straight up, the flow curves inward, leaving the boundary. This loop of flowing grains-up along the walls of the container and then down through the center of the material-is called a convection roll. In this demonstration, two large convection rolls-one on each side-form two heaps on the top surface of the material. The peak of each heap lies a few centimeters away from the container’s boundary. This situation also produces two other smaller convection loops, just beneath the top of the heaps. The main loops actually turn downward before reaching the peak of the heaps, and a small loop exists above the larger loop on each side. These little loops rotate in a clockwise direction on the right side and in a counterclockwise direction on the left side-that is, reverse of the large loops beneath them.

The pattern generated by convection rolls depends crucially on the shape of the container. The circular side walls in the above example induced an upward motion of particles along the walls. However, in a container with vertical side walls, the grains move downward along the walls. Such a downward movement also exists in the little convection rolls, which move downward along the vertical portion of the walls of the circular container described above.

Scientists have known for many years that vibrations can generate convection rolls in granular materials. The English chemist and physicist Michael Faraday discovered that phenomenon in 1831. Faraday concluded that the global motion of the grains depends on the air surrounding them. Modern experiments point to a different conclusion, because convection rolls develop in granular materials even in the absence of air. Moreover, computer simulations of vibrating granular materials reproduce convection rolls without any sort of interstitial fluid separating the grains. (However, in some cases, when the grains are sufficiently small, interstitial air can help produce a convection roll.)

To better understand the nature of convection rolls, we would like to watch them in a three-dimensional container, such as a cylinder. However, such a container allows us to watch the movement of the grains only near the container’s boundaries. Fortunately, we can now look inside the material by using magnetic-resonance imaging (MRI).

Imagine a cylinder filled with small clear beads, a thin layer of blue beads and one large blue bead near the bottom. With MRI techniques, we can follow the movement of the blue beads, even though most of them lie inside the boundaries of the cylinder. Shaking the cylinder vertically-just like the children’s toy described at the beginning of this article-causes the blue beads near the cylinder’s side to move downward rapidly and those in the middle, including the large one, to move upward, although at a somewhat slower rate. When the upwardmoving middle beads reach the top of the pile, they move sideways, following the slope on the top of the pile, until they reach the cylinder’s side and start moving downward. As the little blue beads move down along the cylinder wall, they experience random kicks from friction with the wall. This drives them inward (often even before they reach the container’s bottom) and lets them join the central, upward flow. What happens to the large bead when it reaches the top of the pile and moves to one side? It stays there. The bead’s size makes it too large to fit in the thin downward sheet of convection along the cylinder’s wall.

What does this demonstration tell us about the mechanism behind size separation in granular materials? According to conventional explanations, size segregation comes from little particles filling the gaps beneath larger particles. However, our MRI studies did not show any evidence of such a phenomenon. Instead, we think that convection rolls cause size separation, at least in some circumstances. For example, the convection rolls in the cylinder could separate the beads according to sizeas shown by the fact that they strand the larger bead on top and allow smaller beads to continue their flow toward the bottom. Moreover, the convectionroll hypothesis can be tested: Differentshaped containers should produce different patterns of size separation, because of the different directions of convection rolls in them. And, indeed, we have tested some that do. For example, a cone-shaped container causes convection rolls that move upward along the side and downward across a wide middle region; in such a container, large particles end up at the bottom after shaking.

Although we know that convection rolls can play a crucial role in size separation, we do not know exactly what causes them. As mentioned above, Faraday believed that air surrounding the grains caused the rolls. Today, some investigators believe that frictional properties of the grains themselves may cause the convection rolls. The effects of friction can be shown rather easily We took a cylinder, coated part of the side with fine sand to make it rough and left the rest of the side smooth (and nearly frictionless), filled the cylinder with clear beads and put a layer of blue beads in the middle, and vibrated the cylinder. Along the sand-covered strip of the cylinder’s wall, the blue beads quickly moved downward before any of the other beads moved at all. These results suggest that friction does play a crucial part in granular convection.

As you might suspect, other factors also affect the characteristics of these rolls. For instance, a roll’s velocity depends on the vibrational acceleration. Moreover, the velocity of beads rising in the middle of a container varies with depth: Deeper beads move slower than ones near the surface. By combining all of these features ascertained from MRI techniques, investigators hope to develop a comprehensive theory for granular flow, similar in stature to that existing for ordinary fluids.

Conclusion

Although this article concentrated largely on describing the physical phenomena of granular materials being vibrated, the findings can also be extended to a variety of applications. In some cases, mechanisms in granular physics may help us understand properties of other materials, such as the magnetization of superconductors mentioned above. In addition, findings in granular dynamics may be used directly in some industrial processes. Transporting and storing granular materials must be done in many industries, including the pharmaceutical industry, which must process a wide variety of powders and pills, and the agricultural and food industries, which must transport and work with seeds and grains. As an example of the importance of understanding the physics of granular assemblies, some industrial analysts think that we could increase the throughput of powder-processing plants by as much as 40 percent if we could solve problems related to the handling and transport of granular materials. So any small improvement in our knowledge of the characteristics of granular materials could be extremely valuable. That knowledge may come from studying the fundamental physical characteristics of a wide range of materials-including the sand in a child’s toy.