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| Fig. 1:(a) 8 mm glass beads on top of 15 mm polypropylene, which show the classical Brazil nut effect (b) 10 mm bronze spheres on 4 mm glass beads showing the reverse Brazil nut effect. Reprinted with permission from Breu et al. [7]. |
The phenomenon of granular separation by size is one observed by every child. A package of granola always has the big clumps on top, a bag of tortilla chips has the crumbs on bottom, and a box of mixed nuts always has the Brazil nuts on top. What does each of these situations have in common? Large particles on top; small particles on bottom.
Granular segregation was first studied by Brown in 1939 [1], and was subsequently studied by engineers until 1987 when it was introduced to the physics community as the "Brazil nut problem" [2]. While the problem may seem simple, it is in fact, quite complex and besides being a current active field of research, has recently generated a great deal of controversy with the advent of what is being called the "reverse Brazil nut effect".
No less than ten different mechanisms have been proposed to explain the Brazil nut effect. Many explanations fall under the category of "void filling". That is, as large particles ascend upwards in the mixture, they leave behind voids that are filled by incoming particles. Other methods include convection, arching and inertia. Yet Huerta and Ruiz-Suarez find that only three mechanisms are necessary to explain granular segregation: inertia, convection, and buoyancy, by which they mean void filling [3]. Their experiment, like most others of the Brazil nut effect, consists of a glass cylinder (in other cases square cylinder) that is fixed to a vibrating table. The vibrating table oscillates the cylinder, which is filled with beads of two different sizes, vertically, at a set frequency for a set period of time. Bead distribution is observed before, during and after the oscillations. They find that one effect or the other dominates at different sphere densities; if ρ represents density, when ρ > 1, inertia dominates, and when ρ < 1, convection does.
A paper by Hong et al. in 2001 predicted the "reverse Brazil nut effect", in which under certain conditions, their numerical observation showed the opposite effect - large beads falling to the bottom of a container, and small beads rising to the top of the container. [4,5] A previous finding of theirs showed that a system of hard sphere condenses in the presence of gravity below a critical temperature Tc [6].
The first known explanation for booming was provided by Poynting and Thomson in 1909: their proposed model was
where m and d are the mass and diameter of the elastic hard spheres, respectively and M and M0 are quantites related to the filling height of the column and packing structures. For simplicity, they consider only a binary mixture of spheres of type A and spheres of type B with masses mA and mB and diameters dA and dB. They then obtain
Setting ρA and ρB equal to the densities of A and B, respectively the following relation is obtained:
This means that if the diameter ratio is smaller than the inverse of the density ratio, the particle mixture should show Brazil-nut effect, but if the diameter ratio is larger than the inverse of the density ratio, the particle mixtures should show the reverse Brazil nut effect.
Breu et al. tested this prediction using beads made of glass, aluminum, bronze, steel, polypropylene, polyurethane, synthetic resin, and wood of diameters ranging from 2 mm to 20 mm [7]. In their two-particle system, they made a prediction of which particle would rise to the top and placed it on the bottom for the pre-shaken mixture. If their prediction was correct, the particles would rise to the top, and if not, the system would be stable. This experiment was successful in finding evidence of both the Brazil-nut effect (Fig. 1a), and the Reverse Brazil nut effect (Fig. 1b). Yet they admitted that their predictions held only in 82% of the cases, and discovered a surprising dependence on filling height within the vessel.
Even more curiously, around the same time that Breu et al. published their results, two other results were published in Nature and Physical Review Letters that claimed not to have found the reverse Brazil nut effect [8,9]. Canul-Chay et al. explained that while Hong et al.'s prediction may work for ideal conditions, it assumes that the two types of particles in the bidisperse granular system do not feel one another. This, they claim, is a crude assumption that does not actually describe the way an actual system of particles behaves. They say that their experiment was carried out under conditions precisely identical to those theorized in the paper.
A paper by Mobius et al. agrees, saying that background air pressure changes the system, and that rather than being monotonic, as Hong et al. proposed, the density differences were sensitive to background air pressure. Breu et al. respond that air pressure should have nothing to do with the system, and about Canul-Chay et al.'s experiment, Breu et al. write, "We cannot explain why the experiment did not work in the case discussed there."
While their remains a lack of agreement on many fronts, all experiments agree that the larger the bead, which is termed 'intruder' in the literature, the faster it will rise through the vibrating bed. Studies have found that merely two minutes of mixing is sufficient to reach a steady state. Moreover, when the Brazil nut effect is observed, it is distinguishable almost immediately after the vibrations begin.
Differences in scientific findings usually point to the necessity for a more nuanced theory, and this case is no exception. More recent papers have found more and more characteristics on which granular segregation hinges. For example, a recent study by Ulrich et al. found that a transition from reverse Brazil nut effect to Brazil nut effect could be induced simply by waiting long enough. That is, they found that after 25 hours of continuous shaking, or aging, a sharp transition occurred for unchanged driving conditions. Perhaps more telling was their find that after an aggressive cleaning procedure of the glass tube in which the beads were, the system returned to its original position.
Ulrich et al. explained this phenomenon by measuring the friction on the sides of the container. They found that aging the system increased the friction, while cleaning the system decreased the system to its original value. They suggest a two-mechanism system. First the dense particles sink to the bottom because their buoyancy is smaller than their weight, then, the sidewall-driven convection begins to play a part. The way this convection works is that as the container vibrates up and down, a convection roll going downwards at the walls of the container and upwards in the center of the container begins. When the larger particles move upwards in the center, they have a lower probability to enter the downstream layer at the walls of the container. Since an increase in the friction of the walls would strengthen the convective motion relative to the buoyancy, it would strengthen that segregation method.
Can a new mechanism be found that will combine all ten of the currently proposed mechanisms into simple equations? In another paper, Schroter et al. seem to think so. Doing so, however, will take many more years of research efforts. The wonder of children at the segregation of the nuts in the mixed nuts box is not so misplaced nor so easily explained after all.
© 2007 Limor S. Spector. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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