STUDY #27 · 2026 · IN OBSERVATION

Cellular Froth — Grain Growth

A model-driven visual study of curvature-driven coarsening in a two-dimensional cellular network.

MOVING IMAGE — THE COARSENING ARC cells 9216 → 160 · ⟨n⟩ 5.89 → 6.0000 · kT 0.6 · anodised

WHAT IS THIS

Soap films between two glass plates, metal grains under anneal, dried mud, the cells of a leaf: in two dimensions they all settle into the same polygonal network, and it coarsens — small cells vanish, large ones swell. Two rules are enough. Films meet three at a time at 120°, the only angle at which three equal tensions balance; and every film creeps toward the side it is curved away from, at a speed set by its curvature. From those two rules a startling consequence follows exactly: walk once round a cell with n sides, count the turning, and the area must obey dA/dt = (π/3)·Mγ·(n − 6). An integer decides everything. A cell with five sides shrinks, however large it is. A cell with seven grows, however thin. A cell with six does not age.

The network cannot escape either. On a torus, a cellular network whose vertices are all threefold is forced to ⟨n⟩ = 2E/F = exactly six, so summing the law over every cell gives zero: the froth creates no area and destroys none. It passes area around, and it does so by passing sides around — every time a cell dies, its sides are dealt back out to the neighbours. The froth trades sides, not area.

The engine here is a Q-state Potts model running in real time on the GPU: each lattice site carries a crystallographic orientation, unlike neighbours cost energy, and a Metropolis sweep lets the boundaries wander. Nothing in that rule mentions sides, areas or 120°. All of it comes out.

the working froth — hue is crystallographic orientation, as an anodised section reads under crossed polars anodised · kT 0.6 · nbr 8 · 1100 MCS · 484 cells · ⟨A⟩ 1219 px²

Motif grain growth / soap froth / von Neumann–Mullins law / Plateau's 120° / topology

Method A small simulator was generated and modified with AI assistance, then ported to a real-time GPU (GLSL) renderer. The visual output was selected through parameter exploration.

Observation The model is never told about von Neumann's law, yet it obeys it: the measured area change is 1.09·(n−6) px²/MCS with R² = 0.99922, and six-sided cells drift at 0.7% of one side's worth. The same Mγ read from a single shrinking circle predicts that slope to within 10%, the residual having the sign that finite triple-junction mobility demands. Meanwhile the mean side count is exactly six — an integer identity, 2E = 6F — while the cell count falls 94-fold. The froth trades sides, not area. Two things surprised me: heat turned out to be a look dial rather than a rate dial, and the froth can be frozen outright by taking away the diagonal neighbours at zero temperature, leaving a rectilinear mosaic that never coarsens again.

Reference W. W. Mullins, "Two-Dimensional Motion of Idealized Grain Boundaries," Journal of Applied Physics, vol.27, no.8, 900-904 (1956); M. P. Anderson, D. J. Srolovitz, G. S. Grest & P. S. Sahni, "Computer simulation of grain growth—I. Kinetics," Acta Metallurgica, vol.32, no.5, 783-791 (1984).

Tools Python / NumPy / SciPy / three.js / React / GLSL / ffmpeg / AI coding assistant

Year 2026

This is not a scientific simulation result, but a visual interpretation of the phenomenon.

SAME COARSENING PICTURE, A DIFFERENT MECHANISM

Cahn–Hilliard coarsens by hauling matter. The froth coarsens by counting sides.

Study #08 — Phase Separation Study #27 — Cellular Froth

What is conserved each phase's amount — a concentration field only the total area — no single cell is protected

The interface diffuse — a gradient with thickness a sharp film — ideally zero width

What drives it chemical-potential gradients — matter must diffuse far curvature = tension — the film moves in place

The coarsening law L ~ t^(1/3) L ~ t^(1/2) — measurably faster

A cell's identity shape — labyrinth or droplet topology — the number of its sides

PARAMETERS EXPLORED

param meaning effect on the image

kT / J boundary temperature a look dial, not a rate dial: measured d⟨A⟩/dt only falls 1.12 → 0.92 px²/MCS from kT 0 to 1.3, while the film widens from 13.6% to 16.4% of sites — cold films are crisp, hot films thicken and blur

neighbours 8 (Moore) or 4 (von Neumann) 8 gives a round froth at any temperature; 4 + kT 0 freezes it — the walls fall onto the lattice axes and not one further cell dies

MCS Monte Carlo steps = time the whole arc: 9216 cells → 160, mean area ×94, ⟨A⟩ ∝ t

g initial grains (seeding density) how young the froth starts

seed which froth individual, but the statistics are not

palette imaging anodised (crossed polars) / soap (backlit films) / bone / slag / IPF

colour mode orientation or n − 6 the micrograph, or the law painted onto it

Each image below records its exact parameter set.

THE MATHEMATICS the model behind the images

Two rules — tension balancing at 120°, films moving with their curvature — and one bookkeeping identity. The engine is told none of it; every line below is an output.

\frac{dA}{dt} = -M\gamma \oint \kappa\, ds = \frac{\pi}{3}\, M\gamma\, (n - 6)

The von Neumann–Mullins law. Walking once round a cell turns the heading by 2π; each of the n Plateau corners eats π/3 of it, the smooth film the rest — so the area rate depends on the side count alone. The case n = 0, a lone circle, gives dA/dt = −2πMγ and an independent reading of Mγ: it predicts the polycrystal slope within 10%.

H = J \sum_{\langle i,j \rangle} \bigl( 1 - \delta(s_i, s_j) \bigr)

The Q-state Potts energy — a site pays J for every neighbour whose crystal orientation disagrees. Metropolis proposals only copy a neighbour's orientation, so no grain is ever born, and the boundaries wander with their curvature on their own. Run on the GPU as a 4-sublattice sweep: on the 8-neighbour lattice the two-colour checkerboard of #10 and #17 would let diagonal sites of one colour interact, so parallel and sequential updates only stay identical with four sublattices.

\langle n \rangle = \frac{2E}{F} = 6, \qquad \sum_{\text{cells}} \frac{dA}{dt} = 0

Euler's bookkeeping on a torus of threefold vertices: V − E + F = 0 and 2E = 3V pin the mean side count to exactly six, independent of the dynamics. Summing the law then conserves total area — the froth trades sides, not area. Measured here as integers: 2E = 588 = 6F, with the quadruple-point fraction at zero.

A visual interpretation of the model, not a claim of scientific precision. The measured coarsening exponent is 0.940 against the ideal ⟨A⟩ ∝ t, a known trait of Potts froths; Lewis's law of epithelial cells does not hold here (slope 1.9× Lewis 1928) — grain growth and cell division make different networks.

SELECTED STILLS — 6

the working froth — each grain lit by its own orientation anodised · kT 0.6 · nbr 8 · 1100 MCS · 484 cells · ⟨n⟩ 5.996

late architecture — 128 cells left, the mean side count exactly six soap · kT 0.6 · nbr 8 · 4200 MCS · 128 cells · ⟨n⟩ 6.000 (2E = 6F)

young froth — nearly two thousand cells still paying for their corners soap · kT 0.6 · nbr 8 · 260 MCS · 1974 cells · ⟨A⟩ 299 px²

quenched: the froth stops where it stood — a rectilinear mosaic that never coarsens again bone · nbr 4 · annealed kT 0.5 for 900 MCS, then kT 0 · 1041 cells, unmoved for 800 MCS

ember network — the film brightens at every triple point because more neighbours disagree there slag · kT 0.6 · nbr 8 · 1800 MCS · 294 cells · ⟨n⟩ 6.000

the law, painted — arithmetic made visible, not a micrograph anodised · mode n−6 · 1100 MCS · cool n<6 must shrink · warm n>6 must grow · neutral n=6

PROCESS — PARAMETER SWEEPS

The sweeps behind the stills walked the kT ladder (temperature turned out to move the film, not the rate), the 4- versus 8-neighbour lattice (where the froth can be frozen outright), five imagings of one state, and this sheet — the hero's time axis itself. One run sampled at six times: 6000 cells fall to 64, the mean area climbs 94-fold, and ⟨n⟩ rises 5.843 → 6.000 and then refuses to move.

the coarsening arc — one run, six times L 480 · n0 6000 · kT 0.6 · nbr 8 · t 0 → 3200 MCS · cells 6000 → 64 · ⟨n⟩ 5.843 → 6.000

SIGNATURE — THE VERDICT

Fewer than six sides and it must shrink. More and it must grow. Six, and it does not age.

von Neumann stated the law in 1952 in a few pages of conference discussion; Mullins derived it properly four years later. It is an absurdly strong statement: the area rate of a cell depends on nothing about the cell — not size, not shape, not position — except the integer number of its sides. This engine was never told. It was told only that disagreeing neighbours cost energy. The measured area change is 1.09·(n−6) px²/MCS with R² = 0.99922, six-sided cells drift at 0.7% of one side's worth, and the same Mγ read from a single shrinking circle predicts that slope to within 10% — the residual carrying the sign that finite triple-junction mobility demands.

And the ledger balances exactly. At the final state the mean side count is six as an integer identity — 2E = 588 = 6F, not a quadruple point left — while the cell count has fallen 94-fold. In the Interactive Study below, tap any cell and it states its own fate from that single integer, then watches the arithmetic come true, while ⟨n⟩ beside it stays pinned at 6.00.

the law decomposed — the orientation map, the n−6 verdict painted onto it, the doomed alone, the timeless alone; the side-count histogram peaks at six L 480 · n0 3000 · kT 0.6 · nbr 8 · 900 MCS · n₆ 58 of 208 cells · ⟨n⟩ 6.000

COLOUR = ORIENTATION UNDER CROSSED POLARS

The hue is the grain's crystallographic orientation. Polish and anodise an aluminium section, put it under crossed polars, and each grain lights up in a colour set by which way its lattice happens to point — modern metallography does the same thing digitally with EBSD orientation maps. So mapping each cell to its own hue is not an arbitrary prettification: colour is orientation here, and the colour difference between two neighbouring cells is precisely what makes a boundary exist between them. A boundary is a disagreement about direction; the energy in this model counts exactly those disagreements.

The other imagings are honest alternatives rather than filters: soap is a backlit froth, where the cells are empty and only the films scatter; slag is hot metal, where the network glows. In every one of them the film brightens at the triple points on its own — a vertex has more disagreeing neighbours than a straight wall does, exactly as a Plateau border holds more liquid and scatters more light. One image in the set is not a micrograph and is labelled as such: the n − 6 view, cool for the doomed, warm for the growing, neutral for the timeless — arithmetic made visible.

the network glowing as hot metal — every triple point brighter on its own, as a Plateau border holds more liquid slag · kT 0.6 · nbr 8 · 1800 MCS · 294 cells · film 1.6

The model treats every disagreement as costing the same; real grain-boundary energy depends on the misorientation angle. An artistic approximation of published behaviour, not a measurement.

REFERENCES

John von Neumann. "Discussion: Shape of metal grains." in Metal Interfaces, American Society for Metals, Cleveland, 108-110 (1952).
W. W. Mullins. "Two-Dimensional Motion of Idealized Grain Boundaries." Journal of Applied Physics, vol.27, no.8, 900-904 (1956).
Cyril Stanley Smith. "Grain Shapes and Other Metallurgical Applications of Topology." in Metal Interfaces, American Society for Metals, Cleveland, 65-108 (1952).
M. P. Anderson, D. J. Srolovitz, G. S. Grest, P. S. Sahni. "Computer simulation of grain growth—I. Kinetics." Acta Metallurgica, vol.32, no.5, 783-791 (1984).
D. J. Srolovitz, M. P. Anderson, P. S. Sahni, G. S. Grest. "Computer simulation of grain growth—II. Grain size distribution, topology, and local dynamics." Acta Metallurgica, vol.32, no.5, 793-802 (1984).
J. A. Glazier, S. P. Gross, J. Stavans. "Dynamics of two-dimensional soap froths." Physical Review A, vol.36, no.1, 306-312 (1987).
D. A. Aboav. "The arrangement of grains in a polycrystal." Metallography, vol.3, no.4, 383-390 (1970).
D. Weaire. "Some remarks on the arrangement of grains in a polycrystal." Metallography, vol.7, no.2, 157-160 (1974).
F. T. Lewis. "The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of Cucumis." The Anatomical Record, vol.38, no.3, 341-376 (1928).
J. E. Burke, D. Turnbull. "Recrystallization and grain growth." Progress in Metal Physics, vol.3, 220-292 (1952).
J. A. F. Plateau. "Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires." Gauthier-Villars, Paris (1873).
R. D. MacPherson, D. J. Srolovitz. "The von Neumann relation generalized to coarsening of three-dimensional microstructures." Nature, vol.446, 1053-1055 (2007).

INTERACTIVE STUDY

Soap films, metal grains and dried mud settle into the same polygonal network, and it coarsens: small cells vanish, large ones swell. What decides a cell's fate is not its size — it is the number of sides. Tap any cell to read its verdict — fewer than six and it must shrink, more and it must grow, exactly six and it does not age — then watch the arithmetic come true. Meanwhile the mean side count never leaves 6.00; Euler will not allow it. It is a deliberately simplified instrument, capped at a 192² lattice with a few curated knobs and no export, separate from the full engine used to author the finished works.

SIMPLIFIED INSTRUMENTQ-STATE POTTS · GRAIN GROWTH · kT LIVE

This interactive study is not intended as a scientifically validated reproduction. It is a visual interpretation generated from an implemented model and curated parameter exploration — and it is a deliberately simplified instrument, separate from the full engine used to author the finished works.