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

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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Figure 1: A two-dimensional grain and its extension into three dimensions.
Figure 2: Notation used in description of the three-dimensional von Neumann–Mullins relation.

References

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

    Google Scholar 

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

    Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  5. Burke, J. E. & Turnbull, D. Recrystallization and grain growth. Prog. Metal Phys. 3, 220–292 (1952)

    Article  ADS  CAS  Google Scholar 

  6. Durian, D. J., Weitz, D. A. & Pine, D. J. Scaling behavior in shaving cream. Phys. Rev. A 44, R7902–R7905 (1991)

    Article  ADS  CAS  Google Scholar 

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

    Google Scholar 

  8. Rhines, F. N. & Craig, K. R. Mechanism of steady-state grain growth in aluminum. Metall. Trans. 5, 413–425 (1974)

    Article  CAS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  10. Hilgenfeldt, S., Kraynik, A. M., Koehler, S. A. & Stone, H. A. An accurate von Neumann’s law for three-dimensional foams. Phys. Rev. Lett. 86, 2685–2688 (2001)

    Article  ADS  CAS  Google Scholar 

  11. Hilgenfeldt, S., Kraynik, A. M., Reinelt, D. A. & Sullivan, J. M. The structure of foam cells: Isotropic plateau polyhedra. Europhys. Lett. 67, 484–490 (2004)

    Article  ADS  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  14. Hadwiger, H. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie (Springer, Berlin, 1957)

    Book  Google Scholar 

  15. Klain, D. A. & Rota, G.-C. Introduction to Geometric Probability (Cambridge Univ. Press, Cambridge, UK, 1997)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Bröcker, L. & Kuppe, M. Integral geometry of tame sets. Geom. Dedicata 82, 285–323 (2000)

    Article  MathSciNet  Google Scholar 

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

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

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

    Article  CAS  Google Scholar 

  22. Abbruzzese, G., Heckelmann, I. & Lucke, K. Statistical theory of two-dimensional grain growth. Acta Metall. Mater. 40, 519–532 (1992)

    Article  CAS  Google Scholar 

Download references

Acknowledgements

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

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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. (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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