The bubble that grows by counting its sides

The small bubble blows up the big one connected two soap bubbles by a tube and watched the smaller one empty itself into the larger. The reason is Laplace’s: the pressure inside a bubble exceeds the pressure outside by an amount inversely proportional to its radius, so the small bubble is at the higher pressure and its gas flows out. The same rule drives the slow coarsening of every population of bubbles, droplets or crystals in which material can pass from one to another: small ones feed large ones, the average size grows, and the number falls.

A froth is a population of bubbles in which each one touches its neighbours directly, sharing films, and the gas passes through the films rather than through a tube. It coarsens too. Squeeze a soap froth between two sheets of glass so that it is one cell thick, photograph it every few hours, and over a day or two the small cells disappear and the survivors swell. It is natural to assume that the small ones disappear because they are small. Cyril Stanley Smith, who made the first careful photographs of such froths in 1952, noticed that this was not what decided it, and John von Neumann, in a remark printed in the discussion after Smith’s paper, gave the rule that does.

Corners that must be 120° #

The angles a film has no choice about found the constraint that runs everything here. In a dry froth every film has the same tension, so where three films meet they must pull at equal angles — 120° each — or the junction would be dragged along. Plateau found that rule in 1873, and a two-dimensional froth obeys it exactly at every corner.

A polygon whose corners are all 120° and whose sides are straight is a regular hexagon, and nothing else. A cell with fewer than six sides cannot have straight sides and 120° corners at once: a straight-sided pentagon has 108° corners, a square 90°. To make its corners 120° the cell’s sides must bow outward. A cell with more than six sides has corners too wide for straight sides — 135° for an octagon — and its sides must bow inward to bring the corners back to 120°.

A curved film carries a pressure difference across it — the two-dimensional version of Laplace’s law, tension times curvature — with the higher pressure on the concave side. It is the same law that makes a thread of liquid break into drops and a small bubble feed a large one, applied now to each film separately. A cell whose sides all bulge outward is at higher pressure than every one of its neighbours, and gas leaks from it through each of its films. A cell whose sides all curve inward is at lower pressure than every neighbour, and gas leaks into it. The hexagon, with flat sides, is at the same pressure as its neighbours and neither gains nor loses.

The side count has decided the direction of flow already. The size has not been mentioned.

A sum of angles #

How gas gets through a film #

A soap film is a sheet of water a micrometre or less thick, stabilised by soap molecules at both surfaces, and gas does not pass through it as through a sieve. It dissolves into the water on the high-pressure side, diffuses across as dissolved molecules, and comes out of solution on the other side. Henry’s law makes the amount dissolved proportional to the pressure, so the concentration difference across the film is proportional to the pressure difference, and Fick’s law makes the flux proportional to that difference divided by the thickness. The result is a flux per unit length of film equal to a permeability $\kappa$ times the pressure difference, with $\kappa$ set by the gas’s solubility, its diffusion coefficient in water and the film’s thickness.

The solubility is what varies most. Carbon dioxide is about fifty times more soluble in water than nitrogen, and a froth blown with it coarsens fifty-odd times faster. That is the reason some stouts are dispensed with nitrogen rather than carbon dioxide: the bubbles of the head stay small and the head lasts, because the gas that would coarsen it cannot get through the films. And the permeability falls as a film thins until it is black and only a few molecules thick, when the soap layers themselves become the barrier. None of this changes the form of the law. It only sets the constant in front.

The rate follows from adding up the flow through each film. Gas passes through a film at a rate proportional to the pressure difference across it, with a permeability $\kappa$ that depends on the soap and the film’s thickness, and to the film’s length. The pressure difference is the film’s tension $\gamma$ times its curvature. So the rate at which the cell loses area is $\kappa\gamma$ times the sum, over its sides, of each side’s curvature times its length.

Curvature times length is the angle through which a side turns. And the total angle through which the direction of travel turns, going once round any closed boundary, is a full turn, $2\pi$ — whatever the boundary’s shape or size.

That turning comes from two places. At each corner the direction jumps by 60°, a sixth of a turn, because the corner is 120°. The sides supply whatever the corners do not. For a cell with $n$ corners the corners supply $n/6$ of a turn and the sides must supply $1 - n/6$ — positive for $n < 6$, so the sides bulge outward; zero for the hexagon; negative for $n > 6$, so the sides curve back. The total turning of the sides is therefore $2\pi(1 - n/6) = \frac{\pi}{3}(6 - n)$, and the rate of change of area is

$$\frac{dA}{dt} = \frac{\pi\kappa\gamma}{3}\,(n - 6).$$

That is von Neumann’s law. Every step of it is exact. The corners are exactly 120° by Plateau’s rule, the turning round any closed curve is exactly a full turn, and the flux through a film is exactly its permeability times the pressure difference times its length. Nothing in it depends on the lengths of the sides, the area of the cell, or how its sides are curved in detail — only on how many corners there are.

Size does not enter #

The figure computes the rate from drawn cells — each side an arc, its curvature and length measured, their product summed — at three sizes, and the three sizes give the same number for each side count. A larger cell’s sides are longer in proportion to its size and less curved in inverse proportion, and the flux through a side depends on the product.

That is the counter-intuitive result. A pentagon a centimetre across and one a millimetre across lose area at exactly the same rate, in square millimetres per hour. The small pentagon disappears first only because it has less area to lose. A small octagon, meanwhile, grows at twice the rate the pentagon shrinks, even if the pentagon beside it is a hundred times its size. Isolated bubbles coarsen by size; a froth coarsens by topology.

The contrast with droplets is sharp. In a cloud of droplets of different sizes, or a suspension of small crystals in a solution, material passes from small to large through the surrounding medium, and size alone decides: the vapour pressure over a small droplet is higher, by the same curvature that makes a small bubble’s pressure higher, and ice crystals in a cloud grow by stealing from the droplets around them. That is Ostwald ripening, and in it the largest particle always wins. A froth differs because each cell exchanges gas only with the cells it touches, across films whose curvature is set by the corner rule, and the corner rule cares about counting, not size. The surface energy driving the two processes is the same; the geometry that routes it is not.

There is a sense in which size does still matter, and it is worth being exact about it. Size and side count are correlated in a real froth. Larger cells tend to have more neighbours — Lewis found in 1928, in the cells of cucumber skin, that a cell’s area rises roughly in proportion to its number of sides — so the large cells are mostly the many-sided ones and the small cells mostly the few-sided ones, and on average the large do grow and the small shrink. But the correlation is statistical and loose. What decides the fate of a particular cell is its sides, and an exception to the size rule is easy to find in any photograph.

How a cell dies #

The law makes the life of a shrinking cell a sequence of straight lines. A pentagon’s area falls at a constant rate until something changes its number of sides. That happens when a neighbouring film shrinks to nothing and the corners at its ends meet and swap partners — a T1 event, in the jargon — which takes one side from two cells and gives one to two others. Each loss steepens the cell’s decline. A four-sided cell loses area twice as fast as a pentagon, and a triangle three times as fast, so the end comes quickly once a cell is down to three sides, and most cells in a coarsening froth disappear as triangles. A cell that vanishes takes its corners with it, and its former neighbours each lose a side.

The rearrangements are sudden. Between them the froth moves smoothly, each film creeping as gas crosses it; at a T1 event a film shrinks to zero length in a moment, four films meet briefly at a point that Plateau’s rule forbids, and the corners snap apart the other way. A froth sheared slowly between plates shows the same snaps, in bursts, and they are how a froth flows at all — the same intermittent relaxation by local rearrangement that keeps a granular pile from ever quite finishing settling. In coarsening the snaps are driven not by an applied shear but by the steady drift of gas from cell to cell.

This is why a coarsening froth is a cascade. The gas that a cell loses goes into its neighbours. The sides a cell loses are taken by its neighbours’ rearrangements, and when it vanishes it changes their side counts too, which changes their rates. The law fixes each cell’s rate at every moment; the topology of the whole froth decides how the side counts evolve. Following that topology is what makes the full problem hard and why it was solved, for real froths, only by simulation and by photographing thousands of cells over many days, as James Glazier, Stephen Gross and Joel Stavans did in 1987.

Why six is the break-even #

The number six is not an accident of the 120° rule alone. It is also the average number of sides a cell must have in any froth whose corners each join three films.

Euler’s formula for any network drawn on a sphere — vertices minus edges plus faces equals two — together with the condition that every corner joins three edges, fixes the average number of sides at $6 - 12/F$ for a network of $F$ cells. A football, with twelve pentagons and twenty hexagons, averages 5.625. As the network grows the correction shrinks, and for a froth on a flat plane, unbounded, the average is exactly six.

Put that beside the law and the result is that the rates sum to zero. The total rate of change of area over the whole froth is $\frac{\pi\kappa\gamma}{3}\sum(n - 6)$, and the sum vanishes because the average $n$ is six. Every bit of area one cell loses, another gains. That is as it must be, because the gas is conserved and the froth is not being compressed — and it is a check on the law that it respects conservation without being told to. The same arithmetic says that in a froth with no flat boundary, a sphere of cells, there must be cells with fewer than six sides: a football cannot be made of hexagons alone. Twelve pentagons is the least any such network can have, as the panels of a football and the carbon cages of fullerenes both show.

Where the law holds and where it stops #

The law needs the froth to be dry, with films so thin that the liquid sits only in the corners, and two-dimensional, one cell thick between plates or on a surface. In a wet froth the corners swell into curved triangular channels — Plateau borders — the effective corner angle is no longer exactly 120°, and the law acquires corrections that grow with the liquid fraction. It needs the gas to pass through films and not round them, and it needs the films to have one tension.

It also holds for things that are not froths. The grains of a thin metal sheet, annealed, coarsen by moving their boundaries towards their centres of curvature at a speed proportional to the curvature, and the boundaries meet at 120° for the same reason films do. William Mullins showed in 1956 that the same law follows, with a mobility in place of the permeability. Two-dimensional grain growth in metals, in ice sheets and in the domains of magnetic garnet films all obey $n - 6$.

In three dimensions it fails, and the failure is instructive. A three-dimensional cell’s boundary has no single “turning” fixed by counting its corners; the corresponding sum, of curvature over the cell’s faces, depends on the lengths of its edges and on a measure of its size. Robert MacPherson and David Srolovitz found the exact three-dimensional rule in 2007, and it is not a count. Many-faced cells tend to grow and few-faced cells to shrink, but the dividing line is a tendency in the statistics, not a law for each cell.

What the pictures cannot show #

The drawn cells are idealisations: regular, symmetric, every side the same arc. Real cells are irregular, and their sides have different curvatures. The law does not care, which is its point — the sum of the turning is the same — but the figures show only the symmetric case and so cannot show how little the law depends on it. The history figure picks the moments at which a side is lost; in a real froth those are set by the neighbours, and they are what makes the coarsening of a whole froth a hard problem.

None of the figures shows time in seconds. The constant $\pi\kappa\gamma/3$ depends on the permeability of the films, which depends on the gas — carbon dioxide passes through soap films far faster than nitrogen — and on the film’s thickness. A froth of air bubbles between glass plates coarsens over days; one made with a more soluble gas, in hours.

The domain of the law is a dry, two-dimensional froth or grain structure with uniform tension, in which material passes across boundaries in proportion to their curvature. Inside that domain a cell’s fate is set by its side count. Outside it — wet foams, three dimensions, anisotropic grain boundaries — size and shape come back in.

Still open: what the long-time froth looks like #

A coarsening froth is expected to reach a scaling state, in which the distribution of side counts stops changing and the whole pattern looks the same at every time, only larger, with its mean area growing in proportion to time. Experiments and simulations find something like it, and the statistics in that state — the fraction of hexagons, the spread of side counts, the tendency of many-sided cells to be surrounded by few-sided ones — have been measured repeatedly. Whether the scaling state is unique, how long a froth takes to reach it from an ordered start, and how sensitive it is to the small amounts of liquid every real froth contains are questions on which experiments and simulations have not fully agreed, partly because a froth large enough to reach the scaling state has very few cells left by the time it does.

The law itself is settled and short. Every corner of a two-dimensional froth is 120°, so n corners supply n/6 of the full turn round a cell and the sides must supply the rest; the sides’ curvature sets the pressure, the pressure drives gas through the films, and the area changes at (πκγ/3)(n − 6) — the same for a cell of any size, so a small octagon grows while a large pentagon beside it vanishes. Isolated bubbles are ranked by size; a froth ranks its cells by counting.