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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Robert D. MacPherson & David J. Srolovitz, 2007. "The von Neumann relation generalized to coarsening of three-dimensional microstructures," Nature, Nature, vol. 446(7139), pages 1053-1055, April.
    • Handle: RePEc:nat:nature:v:446:y:2007:i:7139:d:10.1038_nature05745
    DOI: 10.1038/nature05745
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    Cited by:

    1. Bo-Yu Liu & Zhen Zhang & Fei Liu & Nan Yang & Bin Li & Peng Chen & Yu Wang & Jin-Hua Peng & Ju Li & En Ma & Zhi-Wei Shan, 2022. "Rejuvenation of plasticity via deformation graining in magnesium," Nature Communications, Nature, vol. 13(1), pages 1-10, December.

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