Journal of Materials Science

, Volume 40, Issue 9–10, pp 2149–2154

Energetics of polycrystals

  • M. E. Glicksman
Proceedings of the IV International Conference High Temperature Capillarity

Abstract

The energetics of polycrystalline solids at high temperatures is treated using topological methods. The theory developed represents individual irregular polyhedral grains as a set of symmetrical abstract geometric objects called average N-hedra (ANH’s), where N, the topological class, equals the number of contacting neighbor grains in the polycrystal. ANH’s satisfy network topological averages in three-dimensions for the dihedral angles and quadrajunction vertex angles, and, most importantly, can act as “proxies” for irregular grains of equivalent topology. The present analysis describes the energetics of grains represented as ANH’s as a function of their topological class. This approach provides a quantitative basis for constructing more accurate models of three-dimensional well-annealed polycrystals governed by capillarity. Rigorous mathematical relations, derived elsewhere, for the curvatures, areas, and volumes of ANH’s yields quantitative predictions for the excess free energy. Agreement is found between the analytic results and recently published computer simulations.

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References

  1. 1.
    C. S. SMITH, “Grain Shapes and other Metallurgical Applications of Topology,” Chapter in Metal Interfaces (American Society for Metals, Cleveland, OH, 1952) p. 65.Google Scholar
  2. 2.
    J. VON NEUMANN, in “Metal Interfaces, written discussion” (American Society for Metals, Cleveland, OH, 1952) p. 108.Google Scholar
  3. 3.
    W. W. MULLINS, J. Appl. Phys 27 (1956) 900.CrossRefMathSciNetADSGoogle Scholar
  4. 4.
    V. FRADKOV, M. PALMER, J. NORDBERG, M. E. GLICKSMAN and K. RAJAN, Physica D 66 (1993) 50.MATHCrossRefADSGoogle Scholar
  5. 5.
    V. E. FRADKOV, M. PALMER, M. E. GLICKSMAN and K. RAJAN, Acta Metall. Mater 42(8) (1994) 2719.CrossRefADSGoogle Scholar
  6. 6.
    M. A. PALMER, V. E. FRADKOV, M. E. GLICKSMAN and K. RAJAN, Scripta Met. et Mat 30 (1994) 633.CrossRefGoogle Scholar
  7. 7.
    M. A. PALMER, M. E. GLICKSMAN, K. RAJAN, V. FRADKOV and J. NORDBERG, Metall. Mat. Trans. A 26A (1995) 1061.CrossRefGoogle Scholar
  8. 8.
    E. B. MATZKE, Am. J. Botany 33 (1946) 58.CrossRefGoogle Scholar
  9. 9.
    Idem., in “Metal Interfaces, written discussion” (American Society for Metals, Cleveland, OH, 1952) p. 110.Google Scholar
  10. 10.
    F. GRANER, Y. JIANG, E. JANIAUD and C. FLAMENT, Phys. Rev. E 63 (2001) 402.Google Scholar
  11. 11.
    M. F. VAZ, M. A. FORTES and F. GRANER, Phil. Mag. Lett 82 (2002) 575.CrossRefADSGoogle Scholar
  12. 12.
    W. W. MULLINS, J. Appl. Phys 59 (1986) 1341.CrossRefADSGoogle Scholar
  13. 13.
    Idem., Acta Metall37 (1989) 2979.CrossRefGoogle Scholar
  14. 14.
    D. WEAIRE and J. A. GLAZIER, Phil. Mag. Lett 68 (1998) 363.CrossRefADSGoogle Scholar
  15. 15.
    J. A. GLAZIER, Phys. Rev. Lett 70 (1993) 2170.CrossRefPubMedADSGoogle Scholar
  16. 16.
    C. MONNEREAU and M. VIGNES-ADLER, ibid 80 (1998) 5228.CrossRefADSGoogle Scholar
  17. 17.
    D. WU, Private Communication, 2003.Google Scholar
  18. 18.
    K. BRAKKE, Exper. Math 1 (1992) 141.MATHMathSciNetGoogle Scholar
  19. 19.
  20. 20.
    C. MONNEREAU, N. PITTET and D. WEAIRE, Europhys. Lett 52 (2000) 361.CrossRefADSGoogle Scholar
  21. 21.
    S. HILGENFELDT, A. M. KRAYNIK, S. A. KOEHLER and H. A. STONE, Phys. Rev. Lettr 86 (2001) 2685.CrossRefADSGoogle Scholar
  22. 22.
    S. J. COX and M. A. FORTES, Phil. Mag. Lettr 83 (2003) 28.Google Scholar
  23. 23.
    S. J. COX, Private Communication, 2003.Google Scholar
  24. 24.
    C. P. GONATUS, J. S. LEIGH, A. G. YODH, J. A. GLAZIER and B. PRAUSE, Phys. Rev. Lettr 75 (1995) 573.CrossRefADSGoogle Scholar
  25. 25.
    J. A. GLAZIER and B. PRAUSE, in “Foams, Emulsions and Their Applications” edited by P. Zitha et al. (MIT-Verlag, Bremen, 2000) p. 120.Google Scholar
  26. 26.
    D. STRUIK, “Lectures on Classical Differential Geometry” (Addison-Wesley, Reading, MA, 1950).MATHGoogle Scholar
  27. 27.
    M. M. LIPSCHUTZ, “Differential Geometry, Schaums Outline Series” (McGraw-Hill, New York, 1981).Google Scholar
  28. 28.
    D. A. DREW, SIAM J. Appl. Math 50 (1990) 649.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    R. T. DeHOFF, Acta Metall. Mater 42(8) (1994) 2633.CrossRefADSGoogle Scholar
  30. 30.
    M. E. GLICKSMAN, Phil. Mag 85 (2005) 3.CrossRefADSGoogle Scholar
  31. 31.
    E. W. WEISSTEIN, “Reuleaux Tetrahedron,” MathWorld, 2004, http://mathworld.wolfram.com/ReuleauxTetrahedron.html..
  32. 32.
    H. S. M. COXETER, M. S. LONGUET-HIGGINS and J. C. P. MILLER, Phil. Trans. Roy. Soc. London, Series A 236 (1954) 401.CrossRefMathSciNetADSGoogle Scholar
  33. 33.
    E. E. UNDERWOOD, “Quantitative Metallography, in Metals Handbook,” 9th ed. (American Society for Metals, Metals Park, OH, 1985) p. 123.Google Scholar
  34. 34.
    J. C. RUSS, “The Image Processing Handbook,” 2nd ed. (CRC Press, Boca Raton, 1995).Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • M. E. Glicksman
    • 1
  1. 1.Materials Science & Engineering DepartmentRensselaer Polytechnic InstituteTroyUSA

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