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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

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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.

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References

  1. 1.

    in Metal Interfaces (ed. Herring, C.) 108–110 (American Society for Metals, Cleveland, 1952)

  2. 2.

    in Metal Surfaces: Structure, Energetics, and Kinetics (eds Robertson, W. D. & Gjostein, N. A.) 17–66 (American Society for Metals, Metals Park, Ohio, 1963)

  3. 3.

    Two-dimensional motion of idealized grain boundaries. J. Appl. Phys. 27, 900–904 (1956)

  4. 4.

    Grain growth in three dimensions depends on grain topology. Phys. Rev. Lett. 70, 2170–2173 (1993)

  5. 5.

    & Recrystallization and grain growth. Prog. Metal Phys. 3, 220–292 (1952)

  6. 6.

    , & Scaling behavior in shaving cream. Phys. Rev. A 44, R7902–R7905 (1991)

  7. 7.

    in Metal Interfaces (ed. Herring, C.) 65–113 (American Society for Metals, Cleveland, 1952)

  8. 8.

    & Mechanism of steady-state grain growth in aluminum. Metall. Trans. 5, 413–425 (1974)

  9. 9.

    On the structure of random tissues or froths, and their evolution. Phil. Mag. B 47, L45–L49 (1983)

  10. 10.

    , , & An accurate von Neumann’s law for three-dimensional foams. Phys. Rev. Lett. 86, 2685–2688 (2001)

  11. 11.

    , , & The structure of foam cells: Isotropic plateau polyhedra. Europhys. Lett. 67, 484–490 (2004)

  12. 12.

    Capillarity-mediated grain growth in 3-D. Mater. Sci. Forum 467–470, 1025–1031 (2004)

  13. 13.

    Analysis of 3-D network structures. Phil. Mag. 85, 3–31 (2005)

  14. 14.

    Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, Berlin, 1957)

  15. 15.

    & Introduction to Geometric Probability (Cambridge Univ. Press, Cambridge, UK, 1997)

  16. 16.

    Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)

  17. 17.

    Curvature measures of subanalytic sets. Am. J. Math. 116, 819–880 (1994)

  18. 18.

    & Integral geometry of tame sets. Geom. Dedicata 82, 285–323 (2000)

  19. 19.

    Theories of normal growth in pure, single phase systems. Acta Metall. 36, 469–491 (1988)

  20. 20.

    Estimation of the geometrical rate constant in idealized three dimensional grain growth. Acta Metall. 37, 2979–2984 (1989)

  21. 21.

    On the theory of normal and abnormal grain growth. Acta Metall. 13, 227–238 (1965)

  22. 22.

    , & Statistical theory of two-dimensional grain growth. Acta Metall. Mater. 40, 519–532 (1992)

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Acknowledgements

D.J.S. was supported by the US Department of Energy and the US National Science Foundation.

Author information

Affiliations

  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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Competing interests

Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests.

Corresponding authors

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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https://doi.org/10.1038/nature05745

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