Illustrated Physics
physics, drawn
Concept

Jamming — where it appears

The transition at which a disordered collection of grains stops flowing and supports a load, with no change in the material or in the temperature. It happens when the density or the confinement passes a threshold, and near it the flow becomes intermittent and stops and restarts.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

Where a heap stops flowing. The number of motions that cost nothing, against the mean number of contacts each grain has, for a patch of 37 grains. Every point is the rank of an actual rigidity matrix — one row per contact, one column per coordinate — subtracted from the number of coordinates, so it is a measurement of the network rather than a formula about it. The dashed line is Maxwell's count, two coordinates a grain minus half a contact each, which is what the counting alone predicts. The measured curve reaches its floor of three free motions — the rigid-body ones — at a coordination of 4.11, and Maxwell's line reaches zero at exactly four, which is twice the dimension of the plane. That is the isostatic point and the difference between the two numbers is the boundary of a finite patch, whose outer grains have fewer neighbours than they would in an infinite one. Above the threshold the two curves separate: the measured floor stays at three while the count keeps falling, and the gap is the redundancy — contacts whose forces no equilibrium equation can determine. Below it the pack has genuine mechanisms and will rearrange under any load at all. The transition is in the count and not in the material, which is why a fluid and a solid here are made of exactly the same grains.

The heap that becomes a solid

Sand poured into a jar flows; the same sand shaken down and pressed does not. Nothing about the grains has changed — not their size, their hardness, their friction or their density by more than a per cent — and the thing that changed is a count of contacts, which crosses a threshold set by the dimension of space and by nothing else.

fluids · Granular matter The hourglass that keeps time. Discharge against how much is left above the opening, for grain and for liquid through the same 40 mm hole, each as a fraction of its own rate at a full hopper. The grain rate is a horizontal line: it does not appear in Beverloo's law at all, because the pressure at the outlet does not depend on the head — the walls carry the weight, which is what Janssen's argument establishes, so the grains at the opening are pushed by their immediate neighbours and by nothing else. The liquid falls as the square root of the head and is down to 32 per cent by the time a tenth is left. That is why an hourglass keeps time and a water clock does not, and why the water clocks that worked were built with a float and an overflow to hold the head constant.

The hourglass that keeps time

Grain leaves a hopper at a rate that does not depend on how much is above it, and that goes as the orifice to the five-halves power rather than the one half a liquid gives. Both facts follow from the same thing: the weight is carried by the walls, not by the grains at the opening.

fluids · Granular matter A hundred thousand taps, and still not finished. The packing fraction of a column of grains against the number of taps it has been given, on a logarithmic horizontal scale, for several tap intensities. The grains start where pouring leaves them, around 0.55, and climb towards something near 0.64. At an intensity of 1.2 the packing reaches 0.6318 after a hundred thousand taps; At an intensity of 2 the packing reaches 0.6327 after a hundred thousand taps; At an intensity of 3 the packing reaches 0.6338 after a hundred thousand taps, which is still 0.0097 short of the asymptote. The shape is what matters. On a logarithmic axis the curve is close to a straight line over four decades, which means the packing improves by about the same amount for each factor of ten in the number of taps — not for each additional thousand. Going from a hundred taps to a thousand buys as much as going from a thousand to ten thousand. An exponential relaxation is over after a few time constants and this is not one. There is no number of taps after which the column is packed; there is only a number after which the next improvement is too small to measure.

The pile that is never finished settling

Tap a jar of grains and it settles. Keep tapping and it goes on settling — logarithmically, so that each factor of ten in the number of taps buys the same small improvement as the last. There is no number of taps after which the column is packed; there is only a number after which the next improvement is too small to measure, and the asymptote everyone quotes is a fitted number rather than a measured one.

fluids · Granular matter

Named alongside it

The objects these essays reach for when they reach for this one.

Granular matterRandom close packingAnnealingArchingConstraint countingCoordination numberDegrees of freedomDimensional analysisDischargeForce-chainForce chainsFree fall

All concepts