Today I heard Prof. D J Srolovitz talk on *Curvature driven growth in polycrystalline materials and froths: beyond von Neumann–Mullins*.

In 2D,

- using Euler’s polyhedron formula,
- assuming that triple junctions are points at which the grain boundaries make 120 degrees with each other (in other words, achieve equilibrium), and,
- in systems with isotropic interfacial energy,

von Neumann showed that if the curvature of any given domain wall is a constant (as it is in soap froths), then six sided grains are stable; the grains with sides less than 6 will shrink while ones with sides greater than 6 will grow. The contribution of Mullins is to show that this result holds even if the curvature is not a constant (mean curvature is what matters) on any given domain wall (as in the case of grain boundaries). This result can also be derived using Gauss-Bonnet theorem. This is a purely topological result.

Apparently, there had been many attempts to generalise von Neumann-Mullins to 3D; however, none of these attempts have been successful in obtaining an exact result. Prof. Srolovitz and R D MacPherson have not only managed to generalise the results to 3D, but to any higher dimension. Unfortunately, I am not able to get any reference to their work on the net. So, here is my attempt at summarising the idea:

- Generalise 2D von Neumann-Mullins to multiply connected domains;
- Volume integrate the result; this is equivalent to considering all possible sections of a 3D structure, and doing a von Neumann-Mullins analysis on each, and putting the results together.

Apparently doing all this introduces a natural measure of length called mean width (which is a Hadwiger measure); and, using such a measure, von Neumann-Mullins can be generalised to N-dimensional domain wall networks and the 2D and 3D networks are just the special cases of such a general result. And, as it turns out, in 3D (and in higher dimensions), curvature driven growth is not purely topological.

Finally, ‘curvature’ in this post means mean curvature (and not Gaussian); by the way, while you are at it, take a look at this wiki page on curvature; and, this history of curvature page which it links to. Have fun!

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