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

Robert D. MacPherson () and David J. Srolovitz ()
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Robert D. MacPherson: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540, USA
David J. Srolovitz: Yeshiva University, New York, New York 10033, USA

Nature, 2007, vol. 446, issue 7139, 1053-1055

Abstract: Abstract 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.

Date: 2007
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DOI: 10.1038/nature05745

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