Cellular structures or tessellations are ubiquitous in nature. Metals and ceramics commonly consist of space-filling arrays of single-crystal grains separated by a network of grain boundaries, and foams (froths) are networks of gas-filled bubbles separated by liquid walls. Cellular structures also occur in biological tissue, and in magnetic, ferroelectric and complex fluid contexts. In many situations, the cell/grain/bubble walls move under the influence of their surface tension (capillarity), with a velocity proportional to their mean curvature. As a result, the cells evolve and the structure coarsens. Over 50 years ago, von Neumann derived an exact formula for the growth rate of a cell in a two-dimensional cellular structure (using the relation between wall velocity and mean curvature, the fact that three domain walls meet at 120° and basic topology). This forms the basis of modern grain growth theory. Here we present an exact and much-sought extension of this result into three (and higher) dimensions. The present results may lead to the development of predictive models for capillarity-driven microstructure evolution in a wide range of industrial and commercial processing scenarios—such as the heat treatment of metals, or even controlling the ‘head’ on a pint of beer.
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D.J.S. was supported by the US Department of Energy and the US National Science Foundation.
Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests.
This file contains Supplementary Notes with the derivation of the main result in the Article, Supplementary Equation illustrating the generalization of the von Neumann-Mullins relation to three dimensions and Supplementary Discussion of practical approaches to calculating the mean width of a domain – including exact methods for polyhedra and a few special shapes plus a numerical method for determining the mean width of arbitrary three-dimensional domains. (PDF 167 kb)
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MacPherson, R., Srolovitz, D. The von Neumann relation generalized to coarsening of three-dimensional microstructures. Nature 446, 1053–1055 (2007). https://doi.org/10.1038/nature05745
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