Illustrated Physics
physics, drawn
Concept

Packing — where it appears

How much of a volume is filled by the objects in it, expressed as a fraction. For identical spheres it runs from about 0.55 for a poured heap to 0.64 where random arrangements jam and 0.7405 for a crystal, and which of those a real assembly reaches depends on its history rather than on geometry alone.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The two holes a grain can fall through, and their exact sizes. Three equal spheres in contact, and four, drawn with the largest sphere that passes between them. The numbers are geometry and nothing else. Three mutually touching spheres put their centres on an equilateral triangle of side 2R, whose circumradius is 2R/√3, so the gap admits a sphere of radius 0.154701R — about a seventh. Four in a square admit 0.414214R, nearly half. A real packing contains both arrangements and everything between, so a grain smaller than the first threshold gets through everywhere, one larger than the second gets through nowhere, and one in between percolates slowly through the loosest routes. That is the whole size-dependence of segregation by percolation, and it is why the effect is reliable below about a seventh and erratic between a seventh and a half.

The big one comes to the top

Shake a jar of mixed grains and it sorts itself, which is the opposite of what shaking a mixture of gases does. There is no thermodynamic paradox in it because there is no temperature to speak of — and the mechanism is a piece of geometry with an exact number in it: a sphere fits through the gap between three touching spheres only below a radius ratio of 0.1547.

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 matterAnnealingConvectionEntropyFree volumeGlassHistory dependenceHysteresisIntruderIrreversibilityJammingLogarithmic relaxation

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