The big one comes to the top

Put muesli in a jar, shake it, and the nuts end up on top. Put two gases in a box, shake it, and they mix. Both are collections of particles being agitated, and they do opposite things.

The gas is obeying the second law, and the temptation is to look for a reason why the muesli is not. There is no need. The two systems differ in something more basic than their statistics: the muesli has no temperature.

No temperature, no thermodynamics #

The energy scale that decides whether thermal motion can rearrange something is $k_BT$, which at room temperature is $4\times10^{-21}$ joules.

The energy required to lift a one-millimetre sand grain by its own diameter is about $2\times10^{-8}$ joules. The ratio is $5\times10^{12}$.

That number ends the discussion. Thermal fluctuations cannot lift a grain, cannot rotate one, and cannot move one past another, so a granular material never explores the configurations that a thermodynamic argument averages over. It sits in whatever arrangement it was put into until something with a great deal more energy than $k_BT$ rearranges it — a hand, a motor, a passing lorry.

There is no temperature here, and that is the first thing to be clear about. Grains are far too heavy for thermal energy to move them — a sand grain’s thermal velocity is unmeasurably small — so a granular assembly does not explore its configurations the way a gas does. It sits in whatever arrangement it was left in until something shakes it, and every statistical statement in this essay is about a driven system rather than an equilibrium one.

That is why the usual machinery does not apply. Entropy in the ordinary sense counts arrangements a system will actually visit, and a granular packing visits none of them unless it is being driven. So there is no free energy to minimise, no equilibrium distribution to compute, and no guarantee that anything settles at all — which is why the sorting below is possible. A system that had a temperature would mix.

So a shaken granular material is not approaching equilibrium. It is being driven, and a driven system may go wherever the driving takes it — including to states of lower entropy, at the cost of the energy the shaking supplies. There is no more paradox in sand sorting itself than in a refrigerator getting cold.

The question is therefore not why the second law is violated but what the mechanism is, and there turn out to be several.

What replaces temperature #

Since the ordinary statistical apparatus does not apply, it is worth asking what does — and the honest answer is that the substitute is much weaker.

The parameter that plays the role of a temperature is the dimensionless acceleration of the shaking, usually written $\Gamma$: the peak acceleration divided by $g$. Below $\Gamma \approx 1$ the bed never leaves the container’s floor and nothing happens at all. Above it the bed lifts each cycle and rearranges, and the character of the behaviour changes as $\Gamma$ rises — sorting, then convection, then a fluidised state, then a granular gas.

It is not a temperature. It does not appear in a Boltzmann factor, two systems at the same $\Gamma$ in contact do not equilibrate, and there is no zeroth law. What it shares with a temperature is only that it measures how vigorously the system is being agitated, and the failure of the analogy at every other point is why granular physics has no equation of state and why its literature is a catalogue of mechanisms rather than a theory.

There have been serious attempts to build a genuine statistical mechanics for these systems, counting jammed configurations rather than microstates and defining a quantity conjugate to the volume — Edwards’ compactivity. It works in some restricted settings and has not become a general tool, and the reason is the one this section started with: an ensemble is a statement about what a system explores, and a granular material explores nothing unless something shakes it.

The geometric threshold #

The oldest and simplest mechanism is percolation, and it has an exact number in it.

Ask when a small sphere can pass between large ones. Three equal spheres of radius $R$ in mutual contact put their centres on an equilateral triangle of side $2R$, whose circumradius is $2R/\sqrt3$; a sphere fitting in the middle therefore has radius

$$\left(\frac{2}{\sqrt3} - 1\right)R = 0.154700\,R.$$

Four in a square arrangement give a larger opening, with circumradius $R\sqrt2$ and a fitting radius of $(\sqrt2 - 1)R = 0.414214\,R$.

A real packing contains both arrangements and everything in between, so a grain smaller than a seventh of the large radius passes everywhere, one larger than four-tenths passes nowhere, and one in between percolates slowly by finding the loose routes. That is the entire size dependence of the mechanism, and it explains why the effect is reliable for a large size ratio and erratic for a modest one.

The threshold is geometric rather than statistical, and it is bounded rather than sharp. The openings in a real random packing are not all the same size, so the two exact thresholds for a regular arrangement bound a distribution instead of defining a single number — and a real mixture sorts over a range of size ratios rather than at one.

The ratchet #

Percolation on its own would not sort anything. What sorts is that it happens in one direction only, which is the shape of every irreversible mechanism whatever it is made of.

Each shake momentarily unloads the packing, so voids open. A void beneath a large grain can be filled by whatever can reach it: a small grain slips down through the gaps between its neighbours, and the large grain cannot, because it does not fit. The small grain goes down and the large one is left one layer higher.

Running the same process backwards would require a small grain to climb up through a gap it fitted through going down — which it could do, if something lifted it, and nothing does. Gravity supplies the direction, and the geometry supplies the selection.

That is a ratchet: a mechanism with a preferred direction driven by an unbiased agitation. The computed curves show it directly — every trajectory is monotone, with no downward step at any size ratio, which is the property that distinguishes a ratchet from a random walk that happens to drift.

The rate depends on the size ratio in the way the thresholds predict: fast when the small grains fit through every gap, slower when they fit through only the loose ones, and not at all when they fit through none.

Two shakes, and what a cycle has to contain #

The ratchet needs more than agitation; it needs the agitation to have a particular structure, and spelling that out shows why the effect has a threshold.

Each cycle must contain a moment when the packing is unloaded enough for voids to open. In a bed thrown clear of the floor that is the flight time; in one merely vibrated it is the part of the cycle when the acceleration exceeds gravity downward and the grains are momentarily in free fall relative to the container.

It must also contain a moment when the packing settles under gravity, so that whatever fell into a void stays there. A container shaken so violently that the material never settles is a granular gas, and the sorting mechanism of this essay stops working — a different one takes over.

So the useful range is bounded on both sides, and the sorting is fastest somewhere in the middle. That is a general feature of ratchets: they need both a stroke that unlocks and a stroke that locks, and driving harder eventually removes the second. The same shape appears in every molecular ratchet, in vibratory feeders, and in the mechanism that walks a nail out of a vibrating board.

The other mechanisms, which are not optional #

If percolation were the whole story the effect would vanish for grains of similar size. It does not, and the reason is that a shaken bed does something else as well.

Convection. A shaken container develops a circulation: material rises in a broad column up the middle and returns in a thin layer down the walls, driven by friction at the walls acting asymmetrically on the up and down strokes. A large intruder is carried up with the bulk — along force chains that carry the stress — and then cannot return, because the descending stream at the wall is narrower than it is. It parks at the top, and it will stay there while the convection runs, which is a steady state maintained by driving rather than a minimum of anything.

This mechanism does not care about the ratio of grain sizes at all — only about whether the intruder fits in the downward stream — so it works where percolation cannot, and it is the dominant one in tall containers with rough walls.

Interstitial air. The gaps between grains contain air, which has to move when the packing rearranges. For fine powders the air’s viscosity dominates the grain inertia and changes the answer completely; the same experiment run in a vacuum can sort the opposite way. Anybody comparing two published results on this subject has to check whether the air was pumped out.

And condensation of the shaking. At high accelerations the bed becomes a granular gas, and in a gas of particles that lose energy on collision the denser regions cool faster and collapse — which sorts by density rather than by size and can put the large particles anywhere.

It can run backwards #

The honest summary of the literature is that the effect has a name, several mechanisms, and no criterion.

Under some conditions a large intruder sinks. The reverse Brazil nut effect appears when the intruder is considerably denser than the bed, when the container is wide, and when interstitial air is present — and the boundary in the parameter space of size ratio and density ratio has been mapped experimentally without being derived.

That is not a failure of the physics so much as an accurate reflection of what granular materials are. There is no small number of state variables that determines the behaviour, because the material has no equation of state: its history matters, the preparation matters, the container matters, and the same jar shaken two ways behaves two ways.

What all the mechanisms share is the requirement that the packing be unlocked. A dense granular material cannot deform without first expanding — dilatancy, which is why wet sand goes dry underfoot — so each shake has to lift the bed enough to open the voids before anything can move through them. Below that acceleration nothing sorts at all, and the threshold is close to one gravity for the obvious reason.

Where it matters, and it matters expensively #

Segregation is the standing problem of every industry that handles powders.

A pharmaceutical tablet must contain a stated mass of active ingredient, typically a few per cent of the tablet, blended with excipients. If the blend segregates between the mixer and the press — in the hopper, on a conveyor, during a single pour — the tablets at the start of a run and the end of it have different doses. The regulatory apparatus around blend uniformity exists because of the mechanisms above.

The same problem afflicts breakfast cereal, cement, coal, ore, catalyst pellets and every powder metallurgy process. Pouring a mixture into a heap segregates it radially, because larger grains roll further down the slope; discharging a silo segregates it in time, because the flow draws from different regions in sequence; and vibrating anything during transport segregates it vertically.

The engineering response is mostly to avoid the situation: blend as late as possible, keep drops short, avoid free surfaces, and use mass-flow hoppers that discharge in the order material entered. None of that removes the mechanism; it removes the opportunity.

The name, and what was found when #

The phenomenon is old and its study is not. Anybody who has carried a sack of mixed grain has seen it, and the first quantitative attention it received was industrial: the segregation of ores and of pharmaceutical blends was a recognised nuisance well before anybody wrote down the geometry.

The percolation argument goes back to work on the mechanics of mixtures in the middle of the twentieth century, and the phrase “Brazil nut effect” attached itself after a 1987 simulation paper that put the vertical-shaking case on a computer and demonstrated the rise of a large intruder in a two-dimensional bed of small ones. That paper is why the effect has a name, and the name has since caused some trouble: it suggests a single phenomenon with a single cause, and the subsequent thirty years established that it is at least three phenomena that happen to point the same way in the usual laboratory conditions.

The reverse effect was reported in 2001 and was initially disbelieved for the reason such results usually are — it contradicted a well-known effect, and the well-known effect had a name. It survived, and its parameter range was mapped, and the mapping is empirical.

That trajectory is worth recognising because it is the ordinary one for a subject with no governing equation. A phenomenon is noticed, a mechanism is proposed and demonstrated, the mechanism is found not to be unique, and the field settles into cataloguing regimes. Granular physics is nearly all like this, and it is one of the few remaining parts of classical mechanics where that is true.

Where it stops #

The simulation here is a model of the ratchet, not of a granular bed. Its rule is the geometry of the first figure: a void is filled by a grain that fits, and the intruder rises when one does. What it demonstrates is the character of the mechanism — monotone, irreversible, size-selective — and not the rate, which depends on the packing’s details, the acceleration, the air and the container.

The two thresholds are for equal spheres. Real grains are neither equal nor spherical, and an angular grain jams in openings a sphere would pass. The thresholds are therefore an upper bound on how easily percolation happens, and a real mixture of irregular grains segregates less readily than the geometry suggests.

The simulation’s rule is imposed, and that is the honest description. The passage criterion in it comes from the geometry of the first figure rather than from a mechanical calculation of grains in contact, so the model demonstrates the consequence of the geometry rather than deriving the geometry from mechanics. A discrete-element simulation that integrates the contact forces between several thousand grains does derive it, at a cost of several processor-days per experiment, and reproduces both the thresholds and the reversal. The figure here is a statement about what follows from the geometry once it is granted.

And a shaken bed is not the only way to drive one. Shearing a granular mixture sorts it too, by a mechanism involving the shear gradient rather than gravity; so does flowing it down a chute, where the large grains rise to the surface and then run ahead. Those have their own thresholds and their own reversals, and the vertical-shaking case treated here is one member of a family.

Where it stops is where thermal motion takes over. Particles small enough to be moved by thermal energy explore their configurations continuously, and exploration is mixing: a colloidal suspension of two sizes comes to equilibrium rather than sorting. The size at which that happens is where granular physics ends and ordinary statistical mechanics begins, and it is why the two subjects share so little machinery despite describing the same shapes.

What the pictures cannot show #

Every figure here is two-dimensional, and the thresholds are three-dimensional numbers. That is not a small discrepancy: in two dimensions a small disc cannot pass between two touching large ones at all, whatever its size, because the gap between two circles in contact has zero width. Percolation in two dimensions requires the packing to be loose to begin with, and the exact thresholds quoted have no two-dimensional counterpart.

So the drawn geometry is a section through a three-dimensional arrangement, and the circles in it are the equators of spheres whose centres lie in the plane. That is the standard convention and it is worth stating, because a reader who takes the drawing literally will conclude that nothing can percolate anywhere.

The second thing no figure shows is time. Segregation is slow: the intruder in a laboratory demonstration rises over tens or hundreds of shakes, and an industrial hopper segregates over a discharge lasting minutes. A picture of a sorted state and a picture of a mixed one look like two states of a system, and what separates them is a process with a rate that depends on everything.

The ladder from here #

Later rungs on this anchor: convection in a vibrated bed derived from the wall friction, with the roll’s direction predicted rather than observed; the phase diagram of the reverse effect in the plane of size ratio and density ratio; segregation in chute flow, where the mechanism is shear-driven and the large grains end up at the free surface; and the statistical mechanics of jammed states, where an attempt is made to build a temperature-like quantity for a system that has no temperature.

The neighbouring ladders are the angle that does not know the size of the heap, which is the same material at rest; the heap that becomes a solid, which is the unlocking every mechanism here requires; and mixing what is already mixed, where the entropy argument that fails here is doing its proper work.