The pile that is never finished settling

Pour dry grains into a jar and they fill about fifty-five per cent of it. Tap the jar and they settle: more grains fit, the surface drops, the packing rises. Tap it again and it settles a little more.

The experiment that made this interesting was to keep going. A column of monodisperse beads, tapped a hundred thousand times, with the height measured throughout — and the result was not that the packing approached its final value quickly, or slowly, but that it approached it logarithmically, which is a different kind of statement.

What logarithmic means here #

An exponential relaxation has a time constant. After three of them it is ninety-five per cent finished, after ten it is done for any practical purpose, and the phrase “it settles” is meaningful.

A logarithmic relaxation has no such number. Going from a hundred taps to a thousand improves the packing by as much as going from a thousand to ten thousand, and as much again from ten thousand to a hundred thousand. The improvement per tap keeps falling, and the total never converges within any experiment.

That has a consequence for what the published numbers mean. “Random close packing is 0.64” is a statement about the asymptote of a fitted curve. No experiment has been there. The hundred-thousand-tap runs end measurably short of it, and the shortfall is not noise — it is the tail of a process that has not finished and will not.

Three quantities are given the same name in the literature and they are different: the asymptote of a fit, the packing reached after a stated protocol, and a geometric limit computed for an idealised arrangement. Quoting one and meaning another is the commonest error in this subject.

Where the slowness comes from #

The mechanism is one sentence and it is worth deriving rather than accepting.

A grain can only move if there is somewhere for it to go — a void large enough to accept it. The free volume per grain shrinks as the packing rises, and it is spread thinly over the whole column, so the chance of enough of it being gathered in one place at one time falls exponentially as the packing rises.

Write the rate of rearrangement as $\exp(-a/V_f)$ with the free volume $V_f$ vanishing as the packing approaches its maximum, and integrate. What comes out is a curve that is slower than any power law and very close to the empirical inverse logarithm.

The structure is that each rearrangement makes the next one rarer, because it has consumed some of the space that made it possible. A process whose rate depends exponentially on its own progress relaxes logarithmically in time, and it does so whether the system is a jar of sand, a glass below its transition temperature, or a sheet of paper being crumpled tighter.

That is the same free-volume argument used for the viscosity of a supercooled liquid, and the correspondence is close enough that granular compaction is studied as a model of the glass transition — a glass with no thermal motion at all, where the taps supply the agitation that temperature usually would.

The experiment, and why it was hard #

The measurement sounds trivial and is not, which is worth a paragraph because the difficulty explains why the result waited until the 1990s.

The column has to be tall and narrow so that the height is a sensitive measure of the packing, and narrow columns are dominated by their walls — so the diameter has to be many grain diameters even while the aspect ratio stays large. The taps have to be identical to a few per cent over a hundred thousand repetitions, which rules out anything hand-operated and most shakers. The grains have to be monodisperse, because a spread of sizes packs differently and segregates as it is tapped, which is the big one comes to the top contaminating the measurement. And the humidity has to be controlled, because a monolayer of water between grains adds a capillary bridge that changes the friction entirely.

A hundred thousand taps at one a second is a day and a half of uninterrupted running, and the interesting part of the curve is the last decade of it. An experiment that has to run for days without drift, to measure a quantity changing in the fourth decimal place, is the reason this is a modern result about a phenomenon anybody could have noticed.

Two branches #

The lower branch is a memory of how the column was made. Gentle taps do almost nothing to a freshly poured column, because nothing can rearrange; harder taps compact it; and once it has been shaken hard enough, that memory is gone and it never returns to the lower branch by any sweep of the intensity.

The upper branch has a maximum. Very gentle taps do not agitate enough to rearrange anything, and very hard ones fluidise the column and let it settle loose again, so there is an intensity in between that packs best.

The route to the densest state is therefore to shake hard and then reduce the intensity slowly, which is annealing with taps instead of temperature. Sweeping the intensity down along the reversible branch reaches a denser state than any single intensity does, and the analogy with cooling a glass slowly is exact enough to be useful rather than decorative.

What a tap actually does #

It is worth being concrete about the event the curves are averaging over, because it is not what the word suggests.

A tap of an intensity above about one gravity throws the whole column upwards, briefly, and it comes apart: the grains separate, fall back, and land. The landing is where the packing changes, and only a small fraction of the grains land anywhere new. Below one gravity the column never leaves the floor and almost nothing happens, which is the flat part at the left of the branch figure.

So the useful range of intensities is narrow. Too little and the grains cannot move at all; too much and the column is fluidised on every tap and lands as loosely as it started. The optimum is where the assembly comes apart just enough for a few grains to find better positions and not enough for the whole arrangement to be forgotten.

That is the same trade every annealing schedule makes, and it is why the maximum in the branch figure exists rather than the packing rising forever with the shaking. Agitation is what lets the system search, and enough agitation destroys what it has found.

The packing tapping cannot reach #

Spheres stacked in a crystal fill about 0.7405 of space, and no amount of tapping produces that from a random start. The obstacle is not energy but the route: reaching a crystal from a jammed random state means passing through arrangements that are looser than where the column already is, and tapping only ever makes it denser. The process cannot go downhill even briefly, so it cannot leave the basin it is in.

This is exactly why a glass is a glass. There is a crystalline state of lower energy, the system is not in it, and the barrier is not a height but a path: getting there requires a coordinated rearrangement that no local process supplies.

A tapped column will sometimes crystallise anyway, from the walls inwards, because a flat wall is a template that organises the first layer and each layer then organises the next. That is a different mechanism using the container rather than the physics of the grains, and it is why a careful measurement of random close packing has to be made before the walls have had time to do it.

Compactivity, and what it would buy #

If a tapped column behaves like a glass, the obvious question is whether it has a temperature — and there is a proposal for one.

Edwards’s suggestion was to build a statistical mechanics with volume in place of energy. Count the mechanically stable arrangements of a given volume, take the logarithm as an entropy, and its derivative with respect to volume defines a quantity playing the part of a temperature: the compactivity. A loosely packed column has a high compactivity, a dense one a low one, and two columns brought into contact should equilibrate to a common value.

The programme is attractive because it would turn a history-dependent mess into a state described by two or three numbers. Whether it works is still argued about. The strongest evidence for it is that quantities measured on columns prepared by different protocols do collapse onto single curves when plotted against a fitted compactivity; the strongest evidence against is that other quantities do not, and that the equilibration between two columns in contact — the test the whole idea rests on — is hard to arrange and harder to interpret.

What is not in doubt is the shape of the question. A system with many stable configurations and no thermal motion, agitated from outside, is being asked whether the configurations it visits are the ones a counting argument would predict. That is the same question statistical mechanics answers for a gas, asked where the usual justification — that the dynamics explores everything — is exactly what is missing.

What the arrangement is like at the end #

The state a column ends in is worth a paragraph because it is not simply “dense”.

Every grain is touching several others, and the contacts carry force in a very uneven way: most of the load goes through a small fraction of the grains, in the chains that the heap that becomes a solid is about, while the rest are essentially along for the ride. Compaction adds contacts and makes the network more redundant, so a denser column distributes load more evenly and is stiffer — which is why compacted fill is what a foundation sits on.

It also makes the column harder to shear, and the effect is not small: the difference between a loose and a well-tapped column of the same grains is the difference between something that pours and something a spoon will not enter. The transition between those two behaviours as the packing rises is the heap that becomes a solid approached from the density side rather than from the load side. A dense packing has to expand before it can flow, because the grains are interlocked and one cannot pass another without the whole assembly loosening. That is dilatancy, and it is why a well-tapped column of sand goes rigid and why footprints on wet beach sand go pale: the sand under the foot has to dilate to deform, and the water that filled it is drawn down into the extra space.

The same logarithm, in other places #

A relaxation that slows itself down is not rare, and recognising the signature is worth more than the granular case alone.

The creep of a metal under a constant load follows a logarithm in time for the same reason: each dislocation that moves leaves the remaining ones in a slightly harder configuration. A crumpled sheet of paper held under a weight goes on compacting for weeks, logarithmically, and the same free-volume argument describes it. The electrical properties of a glass drift logarithmically after a temperature change, which is called physical ageing and is a nuisance in precision resistors and capacitors. And the settling of a soil under a building continues logarithmically for decades after the primary consolidation has finished, which is why old buildings keep sinking slowly.

In every case a fitted asymptote is being quoted for a process that has not reached it, and the same care is needed about what the number means.

The general test is easy to apply. Plot the quantity against the logarithm of the time. An exponential relaxation curves over and flattens; a logarithmic one is a straight line, and a straight line on that plot is a warning that the endpoint in the fit is an extrapolation rather than an observation. The last curve to go makes a related point about a different kind of system: what looks like a system settling is often a system that has stopped being able to explore.

Where the model stops #

The grains are identical spheres, and are not. A distribution of sizes packs much better, because small grains fill the gaps between large ones, and a well-graded sand reaches packing fractions well above 0.64 for reasons that have nothing to do with this argument. Concrete aggregate is designed around exactly that.

Friction is absent from the packing numbers. Random close packing at 0.64 is the frictionless value; real grains with friction jam looser, and the packing at which a column jams depends on the friction coefficient — which means “the” random close packing is not a property of the geometry alone.

The free-volume model has one adjustable number in it, and it produces the shape rather than the numbers. It gives a logarithm, which is the point, and it does not give the coefficient without being told.

And the tap is idealised as a single number. A real tap has a waveform, a duration and a direction, and columns tapped with the same peak acceleration and different pulse shapes compact differently. The intensity axis in the branch figure is a summary of something with more in it.

Why anyone cares #

The measurement is not only a model system, and three places it decides something are worth naming.

Pharmaceutical tabletting. A tablet is powder compacted in a die, and the mass in each tablet is set by the volume of the die and the density of the powder that filled it. A powder whose packing depends on how the hopper was vibrated on the way in gives tablets whose dose depends on the machine’s history, which is a regulatory problem before it is a physics one. The industry measures “tapped density” by a standardised protocol precisely because the quantity has no protocol-free value.

Ground improvement. Compacting fill before building on it is the same process at the scale of a site, done with a vibrating roller instead of a tapper, and the same logarithm applies: the first few passes do most of the work and the specification says how many passes rather than what density to reach.

Anything shipped as a powder. A container filled to the brim arrives settled with headspace, and the amount of settling is a logarithmic function of how far it travelled. Packaging that states a weight rather than a volume, and the note about contents settling in transit, are both consequences of this figure.

In each the useful statement is the same one: there is no natural stopping point, so the protocol has to define one.

What the pictures cannot show #

Every figure here is a single number for the whole column, and a tapped column is not uniform: it compacts from the bottom upwards, with a front that moves slowly through it, and the top can still be loose long after the base has jammed. Measuring the height of the free surface gives the average and hides that entirely.

Nor do the figures show a single tap. Between one tap and the next, almost nothing happens — a few grains somewhere in the column find room and take it, and the packing changes by a part in $10^5$. The smooth curves are averages over enormous numbers of individually discrete events, and the fluctuations about them are themselves a measurement that the figures throw away.

Where the ladder goes next #

The granular ladder began with the silo that does not weigh what it holds, where the walls carry most of the load, went on to the angle that does not know the size of the heap and the heap that becomes a solid, where the force chains appear, then to segregation and to the hourglass that keeps time. This rung asks what the assembly does when it is agitated for a very long time, and finds that the answer is a relaxation with no end to it.

The rung after it is the fluctuations around the average: a tapped column at steady state is not still, and the statistics of its density fluctuations are what a thermodynamic description of a granular system would have to reproduce. The habit worth carrying is the one this rung is built on: when a system approaches a limit, ask how — because a logarithm and an exponential are both called “settling”, and only one of them ever gets there.