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The von Neumann relation generalized to coarsening of three-dimensional microstructures

Nature volume 446, pages 10531055 (26 April 2007) | Download Citation



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.

Author information


  1. School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, USA

    • Robert D. MacPherson
  2. Department of Physics, Yeshiva University, New York, New York 10033, USA

    • David J. Srolovitz


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Reprints and permissions information is available at The authors declare no competing financial interests.

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Correspondence to Robert D. MacPherson or David J. Srolovitz.

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    Supplementary Information

    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.

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