Brazil-nut effect

Why does the largest and heaviest nut end up on top when a can of mixed nuts is shaken?

The instinctive answer is buoyancy: big things float. But the effect is not fussy about density — drop a steel ball into a jar of sand, shake it, and the steel comes up through material several times less dense than itself. Buoyancy running backwards is not buoyancy. So we need to ask what else in a shaken pile could care about a particle at all.

Notice a second oddity before we start. Tilt the can, pour it, stir it slowly: nothing separates. Only shaking does it. Whatever the mechanism is, it lives in the shake.

Take the shake seriously first, because it is a threshold rather than a nuisance. A packed bed of grains is jammed: every particle is wedged by its neighbours and nothing can move. For anything to happen the pile must open up — dilate — so that gaps appear.

When does a shake do that? Let the container move as x = A sin(wt), so its peak acceleration is A w^2. A grain on the surface can only be pulled downward by gravity, at g. The moment the container's downward acceleration exceeds g, the grain cannot keep up: it leaves contact and the bed comes apart. The condition is A w^2 > g, and the ratio Gamma = A w^2 / g governs everything.

Put a domestic shake in. A millimetre of travel at 50 Hz gives w = 2 pi x 50 = 314 rad/s, so A w^2 = 0.001 x 314^2, about 99 m/s^2 — ten times g. A shake you would call gentle is ten g at the grain scale. That is why casual handling separates a can of nuts and slow tilting never does.

Given that the bed opens and closes many times a second, why up rather than down? Here is the step that needs no force at all. Each dilation opens gaps; each settling fills them. A small grain can fall into the gap that momentarily opens beneath the Brazil nut. The Brazil nut cannot fall into the small gaps beneath the small grains — it does not fit. So on every cycle material sneaks underneath the big particle and almost never above it. Nothing pushes the nut up. It is never allowed back down, and the pile builds beneath it. That is a ratchet made of geometry and chance, and it is why size rather than weight is the controlling variable.

A second mechanism is a companion rather than a rival. Vibration also drives a bulk circulation: friction against the container walls drags grains down in a thin layer at the sides while the interior rises, so the material turns over like a slow convection roll. A large intruder rides the broad upward flow easily, arrives at the surface, and then cannot enter the narrow downward stream at the wall because it is wider than the stream. So it parks at the top. Knight, Jaeger and Nagel demonstrated this circulation directly in the early 1990s; I quote the attribution from memory rather than from the paper.

The honest position is that this is a family of competing mechanisms, and which dominates depends on shaking amplitude and frequency, the size and density ratios, the container's shape, and even the air between the grains. The reverse Brazil-nut effect is real: under some conditions a large dense intruder sinks. Work around 2001 — Moebius and colleagues, again a recalled attribution — showed that interstitial air and the density ratio can flip the direction outright, which is why the effect differs in a vacuum. Any account claiming one clean cause is overselling.

But that plurality is a gift, because the mechanisms make different predictions, and one of them is beautifully sharp. The convection roll exists only because of friction against the walls, so its direction depends on the walls. Straight or inward-sloping walls drag material down at the sides and carry it up the middle. Flare the walls outward and the roll runs the other way — and then the intruder should be carried down. Void-filling percolation knows nothing about wall slope and predicts a rise either way.

So: shake the same mixture in a straight jar and in a jar that flares outward. If the big particle rises in one and sinks in the other, convection is doing the work in that regime. If it rises in both, the geometric ratchet is. That is a test that separates two live explanations rather than merely confirming that something happens.

Two further refutations are worth stating because they would kill the account outright. First, the mechanism is size-driven: replace the Brazil nut with a same-sized hollow, very light sphere and it must still rise, while a small dense ball bearing must not. If a large light particle sank while a small heavy one rose, size would not be the variable and the whole argument fails. Second, the dilation threshold: below Gamma = 1 the bed never comes apart, so no amount of patient sub-g shaking should segregate anything. Segregation at Gamma well under 1 would demolish the starting step.

A boundary worth marking against a neighbouring piece here: a shear-thickening paste is about how a dense suspension resists flowing when stressed. This is the opposite situation — no flow and no stress worth the name, just repeated jamming and unjamming, and a sorting that emerges from the geometry of who fits where.

• Why the same segregation is a serious industrial problem in transporting powders and pharmaceutical blends, and what mixer designs do about it.

Every term the collection defines is gathered in the glossary.