Von Neumann's law generalized to 3D foam coarsening

Settled in 2007 by Robert D. MacPherson and David J. Srolovitz.

In 1952, John von Neumann derived an exact, remarkably simple law relating the growth rate of a 2D soap-froth cell purely to its number of sides, a cornerstone result in the study of coarsening cellular structures. For over 50 years, no analogous exact law was known for real three-dimensional foams, grain microstructures in metals, or biological tissues, despite their practical importance in materials science. In 2007, mathematician Robert MacPherson and materials scientist David Srolovitz derived the exact 3D (and higher-dimensional) generalization, showing the growth rate depends on a cell's mean width and total triple-line length rather than simply its number of faces, published in Nature. The result reframed decades of empirical grain-growth models and is now used to interpret 3D microstructural data from X-ray and electron tomography.

A law about soap bubbles that took over fifty years to generalize from two to three dimensions turned out to hinge not on counting a bubble's neighbors, but on subtler geometric quantities nobody had thought to track.

Explanations that circulated

• Naive extensions assuming the 3D growth rate depends only on a cell's number of faces (as in the 2D case, where it depends only on number of sides) turn out to be wrong; the exact 3D law instead depends on a cell's mean width and the total length of its triple-line edges, not simply its topological face count

References

• The von Neumann relation generalized to coarsening of three-dimensional microstructures, Robert D. MacPherson, David J. Srolovitz, Nature, 2007