Mastering the Mixer

The frustrating physics of cake mix and concrete

Part of the fun of experimenting with granular materials, says Stephen W. Morris, is the showmanship. In one stunt that he has demonstrated in settings ranging from high school classrooms to television studios, the University of Toronto physicist loads clear plastic tubes with white table salt and black sand and starts them rotating. What transpires in the tubes usually knocks the socks off of any unsuspecting bystander. Instead of mixing into a drab gray sameness, the sand particles slowly separate into crisp black bands cutting across a long, narrow field of salt. As the spinning continues, some bands disappear and new ones arise.

YIKES, STRIPES! Small changes to either the fill level or rotation speed of this industrial-style double-cone blender (top) produce strikingly different segregation patterns (bottom). Bead diameters are 4 millimeters (red) and 1.4 mm (blue). Shinbrot & Muzzio/Physics Today
ON A ROLL. After quickly forming a spiral (top), sand-grain-size particles that are identical except in color will mix fully after dozens of cycles. If powder-size grains are used instead, the spiral becomes jagged (bottom) as chaotic grain motions hasten mixing. Shinbrot & Muzzio/Physics Today
JUST ADD WATER. In this tumbler experiment, all beads are the same size but coated with a water-attracting film (orange) or a water-repellent one (green). The beads mix well when dry (above). In an odd twist, adding moisture makes them segregate (below). Li et al./Physical Review Letters

Li et al./Physical Review Letters

STRIKE UP THE BAND. When different-size grains rotate in a tumbler, the smaller grains clump into mobile bands that scientists still can’t explain after decades of study. Here, fine sand forms a band amid coarser salt inside a glass tube. Morris

“It’s a parlor trick,” Morris says.

Not to deny its entertainment value, this demonstration of how strangely granular materials can behave is also an authentic experiment in a field both rich in fundamental physics and major practical consequences.

Yet granular mixing today remains more of an art than a science, says chemical engineer Fernando J. Muzzio of Rutgers University in New Brunswick, N.J. Scientists are often at a loss to explain why grains assume surprising patterns instead of simply mixing uniformly (SN: 11/17/01, p. 309: Available to subscribers at The Brazil nut effect gets more jumbled). Most industrial-mixing processes are devised by trial and error.

In the past decade, by using laboratory mixers and mathematical models trimmed of many of the complexities of actual industrial-mixing operations, investigators have made progress toward predicting why mixing does or doesn’t occur. Some scientists are now taking further steps by teasing out on a microscopic level how adhesive forces, collisions, and other interactions among particles affect the outcome of mixing. At the same time, researchers are turning up yet more unexpected phenomena.

A better understanding of mixing would benefit many industries, Muzzio says. “Without good powder mixing,” he says, “you can’t build a road, you can’t make a cake, and you can’t even kill crabgrass, let alone make high potency pharmaceuticals.”

Spill the beans

There’s nothing exotic about granular mixtures. They’re as mundane as the flour, cocoa, and sugar in brownie mix or the sand, gravel, and cement used to make concrete. Regardless of their ingredients, such mixtures have perplexing properties. Depending on how the grain combinations are piled, poured, or shaken, they may behave collectively like solids, liquids, or even gases. Indeed, many physicists consider granular materials to be a distinct state of matter.

Studying this quixotic state of matter entails a whole lot of shaking, rolling, and agitating–and researchers in this arena have devised many ways of doing that (SN: 8/31/96, p. 135). One of the most common and industrially relevant techniques is to rotate loads of grains or powders in tumblers, essentially rotating drums, such as those used widely in the pharmaceutical industry.

Inside a tumbler, contact with the drum’s inner wall drags the pile of compacted grains upward until the surface becomes steeply sloped. Then, a layer of grains on and near the top of that incline breaks loose and cascades down (SN: 3/11/95, p. 159).

Mixing takes place in the cascading surface layer as falling particles collide and are buried by later arrivals. Those particles get reincorporated into the main mass and eventually are carried back up to the leading edge, and the cycle begins again.

“Every grain comes to the surface, and every grain slides down the surface and then gets subducted again,” says Troy Shinbrot of Rutgers University.

Experiments over the past decade have shown that a bewildering number of factors influence whether a given combination of particles in the tumbler will thoroughly mix or one of countless segregation patterns will appear. For instance, particles of two different sizes will tend to segregate into different regions of the drum. The weights of the particles, the rotation speed of the tumbler, and even how the particles are placed in the drum are among the other factors that make a difference.

Although the grains in a tumbler are subjected to just one type of agitation–the drum’s rotations–scientists describe the subsequent patterns of mixing and segregation as the consequence of two nearly independent processes.

To study one of them, called radial mixing, researchers examine the effects of particle motions as seen in circular cross-sections of drums. One way they do this is by stopping the rotating tumbler and then hardening a drum’s contents with glue–freezing the particles in place. Next, they cut through the drum perpendicular to its axis of rotation. The distribution of ingredients reveals how particles travel in that plane.

The other blending process acts along the drum’s length. To study this axial mixing, researchers use tools ranging from video cameras and lasers to magnetic resonance imaging (MRI).

Turn styles

The smallest changes in radial-mixing studies can lead to wildly different and often visually striking patterns.

Some experiments begin by charging a tumbler with two side-by-side portions of sand-size grains that are identical except for color. As soon as the tumbler begins turning, a spiral pattern appears as grains of one color wrap around grains of the other color. Using a so-called continuum model that assumes that individual grains are vanishingly small, theorists have simulated the formation of spiral patterns and their quick onset.

Researchers have also found, both in theory and experiment, that getting from that initial spiral arrangement to a completely mixed state requires dozens of rotations. In food processing, pharmaceutical and chemical production, and other industrial processes, a more thorough knowledge of these dynamics could tell process managers what types of blenders to use and how to fill them to rapidly achieve complete mixing.

Currently, drug makers develop mixing methods by testing many different sets of conditions, says Ajaz Hussain, deputy director of the Food and Drug Administration office that oversees pharmaceutical research and regulations. “That’s not very efficient,” he says. “If we can improve the scientific understanding . . . things can be much, much better.”

Experiments have shown that the size of the particles matters a lot, says Julio M. Ottino of Northwestern University in Evanston, Ill. A downward shift in size transforms circulation patterns of particles in a tumbler. He and his colleagues have found that if the two types of same-size particles are each on the scale of a powder rather than of sand, a single turn of a tumbler produces a splintered and stretched pattern instead of one with clean outlines. Moreover, it takes only a few additional turns to progress from the spiral formation to a homogeneous condition of complete mixing.

Why does particle size make such a large difference? Ottino explains that poppy-seed-size grains, for instance, tend to move in orderly loops, like runners on a track. However, grains the size of powder, such as flour particles, typically move erratically, like players in a soccer game.

Size reduction leads to the famous type of irregularity known as chaos (SN: 8/1/98, p. 70). Chaos refers to situations in which the outcome depends strongly on the initial conditions. Slightly change the initial positions of the particles in the tumbler experiment and those particles’ trajectories can be dramatically different.

“Whenever something mixes well, it is because chaos is present,” Ottino says. Although commercial mixing operations have always unwittingly exploited chaos, only now are they moving toward a scientific footing in the design of blending equipment and processes.

Researchers attribute the chaotic trajectories of very small, tumbled particles to the influence of weak interparticle attractions called van der Waals bonds. For particles smaller than salt grains, or less than about 300 micrometers, those bonds can affect particle flow by competing against both the downward pull of gravity and the propulsive kicks of collisions.

Another type of bonding that can arise between granular particles results from the influence of tiny bridges of water (SN: 1/2/99, p. 6). In production plants, mixing specialists have found that adding some moisture to a combination of dry granules lessens the likelihood of segregation. However, a new study finds that moisture’s effects can have a flip side.

Hongming Li and Joseph J. McCarthy, both chemical engineers at the University of Pittsburgh, applied polymer coatings to glass beads of varying submillimeter diameters. One coating made water cling to the beads’ surfaces; the other repelled water. To examine radial mixing, the researchers used a flattened tumbler, a thin, transparent disk that turns like a wheel, in which they could observe particles.

According to calculations by the team, their bead-surface treatments set the stage for a competition of forces. On one side are the natural tendencies of the beads to mix or to segregate because of size differences. On the other side is a force between beads that can be attractive, repulsive, or absent, depending on how much water is present and how the beads are coated.

In the May 9 Physical Review Letters, Li and McCarthy report that their mixtures of two beads behave normally under dry conditions, regardless of the coatings. That is to say, different-size beads segregate, and same-size beads mix. However, if beads of different sizes are given water-attracting coatings and a little water is added, they mix thoroughly. In an even more atypical behavior, beads of equal size but opposite coatings segregate when dampened.

To look in on interparticle forces at work in such behavior, Muzzio, Shinbrot, and their colleagues have recently used an atomic-force microscope to measure the attractive tug created by films of water on glass beads. These unpublished data reveal that the strongest tugs correlate with the least segregation in tests in an industrial blender, says Muzzio.

“Learning to characterize and manage interparticle forces is the key to understanding and controlling [mixing] processes,” he says.

One potential benefit of doing so would be to find ways of simultaneously using several catalysts in complex industrial-chemical processes and then, by controlling how the catalysts segregate, retrieving each one separately. Says Muzzio: “We might be able to make chemical products faster in smaller factories that are cheaper and have better yields.”

Missing the bandwagon

No less intriguing than the radial component of mixing is the axial component. This segregation of particles into bands along a tumbler’s length is the basis of Morris’ parlor tricks. A topic of study since the 1930s, this type of banding results primarily from interparticle collisions in the tumbler’s flowing layer. But scientists don’t know how those collisions generate bands and can’t predict the bands’ appearance and disappearance.

This impasse is “a stick in the eye of theorists,” Shinbrot says. Forming the banding patterns is “the silliest little thing [the tumbler system] could possibly do, but we cannot come to grips with it and that’s a frustration,” he adds.

In the past decade, researchers have measured many features of axial mixing. For instance, scientists probing beneath the surface of the drum’s contents with MRI have made the surprising discovery that fine particles form a roughly circular core that extends along the axis of the drum. What’s more, that shaft develops bulges along its length that grow and shrink, sometimes reaching the surface to appear as bands of small particles surrounded by larger particles.

The more they look, the more researchers find. For example, some have observed that the height of the particle mass in the drum is different in the band areas than in the intervening spaces. Morris and his coworkers have detected moving, wavelike variations in the distribution of particle sizes.

In new work, Muzzio, Shinbrot, and their Rutgers colleague Albert W. Alexander have found that little particles in big drums readily form bands, whereas big particles in small drums don’t. Between those extremes, bands come and go as the drum is made to rotate at a faster or slower rate.

Meanwhile, Morris and his colleagues are testing predictions of a leading theory of axial mixing that attempts to explain how differences in surface height and particle distribution underlie band behavior. The timing of the changes observed in the tests doesn’t support the model. “It’s really mysterious,” Morris says.

Even as researchers continue scratching their heads about the dynamics of granular mixing in idealized settings such as laboratory tumblers, they’re getting a dose of reality in investigations of commercial-style mixing equipment.

In experiments on one such device called a double-cone blender and used in the pharmaceutical industry, investigators put particles of two sizes into the device until it’s more than a quarter full. The result is rapid and nearly total segregation: The larger particles accumulate in the middle of the blender while the finer particles crowd to the sides. However, even a 1 percent decrease in the volume of particles dumped into the blender–or an equally tiny increase in the tumbler’s rotation rate–makes the pattern flip so that the smaller particles hog the center and the larger ones fill the periphery.

For a field that has barely begun to tackle real-world conditions, such findings suggest that many tough challenges remain in the mix.


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