Skip to main content
SpringerLink
Log in

Calculation of phase diagrams for models of metallic alloys

  • Chapter
  • First Online:
Festkörperprobleme 26

Part of the book series: Advances in Solid State Physics ((ASSP,volume 26))

Abstract

We briefly review a longstanding problem of metallurgy and statistical physics, namely, the prediction of phase diagrams of binary alloys from simple model assumptions on the atomic interactions, such as Ising-type models. Various methods of statistical mechanics which have been applied to this problem are introduced and compared to each other, such as the cluster-variation method and Monte-Carlo simulation. The merits as well as the limitations of these methods are discussed, emphasizing examples of fcc and bcc lattices which are potentially relevant for the problem of short-range order and long-range order in metallic alloys such as Cu−Au, Ni−Cr, and Fe−Al.A brief comparison with corresponding experimental data is given. It is shown that the models studied here also exhibit interesting critical and multicritical phenomena.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. For reviews of earlier work see e.g. various articles in J. de Physique 38—C7 (1977), and [2–5]

    Google Scholar 

  2. D. de Fontaine, in: Solid State Physics (ed. by H. Ehrenreich, F. Seitz, and D. Turnbull) Vol. 34, p. 73 (Academic Press, New York 1979).

    Google Scholar 

  3. J. Als-Nielsen, in Phase Transitions and Critical Phenomena. vol. 5a (ed. by C. Domb and M. S. Green), p. 88 (Academic Press, New York 1976)

    Google Scholar 

  4. A. G. Khachaturyan, Phys. stat. sol. (b) 60, 9 (1973)

    Article  ADS  Google Scholar 

  5. W. Warlimont (ed.), Order-Disorder Transformations in Alloys (Springer, Berlin-Heidelberg-New York 1974)

    Google Scholar 

  6. H. E. Stanley, An Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford 1971)

    Google Scholar 

  7. M. Hansen, Constitution of Binary Alloys (McGraw-Hill, New York 1958)

    Google Scholar 

  8. S. V. Semenovskaya, phys. stat. sol. (b) 64, 291 (1974)

    Article  Google Scholar 

  9. P. S. Swann, W. R. Duff, and R. M. Fisher, metallurg. Trans. 3, 409 (1972)

    ADS  Google Scholar 

  10. A. Bieber and F. Gautier, J. Phys. Soc. Japan 53, 2061 (1984), Z. Physik B57, 335 (1984)

    Article  ADS  Google Scholar 

  11. U. Köbler and K. Binder, J. Magn. Magn. Mat. 15–18, 313 (1980)

    Article  Google Scholar 

  12. V. Gerold and J. Kern, in: Atomic Transport and Defects in Metals by Neutron Scatterint (ed. by T. Petry and D. Richter), in print (Springer, Berlin-Heidelberg-New York 1986); see also J. Kern, Dissertation (Universität Stuttgart 1983, unpublished), and V. Gerold and J. Kern. Acta Met., submitted

    Google Scholar 

  13. W. Schweika and H.-G. Haubold, in: Atomic Transport and Defects in Metals by Neutron Scattering (ed. by T. Petry and D. Richter), in print (Springer, Berlin-Heidelberg-New York 1986); see also W. Schweika, Dissertation (RWTH Aachen 1985 unpublished)

    Google Scholar 

  14. W. Shockley, J. Chem. Phys. 6, 130 (1938)

    Article  ADS  Google Scholar 

  15. Y. Y. Li, J. Chem. Phys. 17, 449 (1949)

    ADS  Google Scholar 

  16. R. Kikuchi, J. Chem. Phys. 60, 1071 (1974).

    Article  ADS  Google Scholar 

  17. C. M. van Baal, Physica 64, 571 (1973)

    Article  ADS  Google Scholar 

  18. D. de fontaine and R. Kikuchi, in: Applications of Phase Diagrams in Metallurgy and Ceramics, Vol. 2, p. 976 (NBS Special Publication 496, 1978)

    Google Scholar 

  19. J. M. Sanchez and D. de Fontaine, Phys. Rev. B21, 216 (1980); B25, 1759 (1982)

    Article  ADS  Google Scholar 

  20. J. M. Sanchez, D. de Fontaine and W. Teitler, Phys. Rev. B26, 1465 (1982)

    Article  ADS  Google Scholar 

  21. A. Finel and F. Ducastelle, Europhys. Lett. (1986, in print); see also T. Mohri, J. M. Sanchez, and D. de. Fontaine, Acta Metall. 33, 1171 (1985)

    Google Scholar 

  22. N. S. Golosov, L. E. Popov and L. Y. Pudan, J. Phys. Chem. Solids 34, 1149, 1159 (1973)

    Article  ADS  Google Scholar 

  23. R. Kikuchi and H. Sato, Acta metall. 22, 1099 (1974)

    Article  Google Scholar 

  24. C. E. Dahmani, M. C. Cadeville, J. M. Sanchez and J. L. Morán-López, Phys. Rev. Lett. 55, 1208 (1985)

    Article  ADS  Google Scholar 

  25. J. Büth and G. Inden, Acta metall. 30, 213 (1982); U. Gahn, Z. Metallkde 64, 268 (1973); 65, 418, 735 (1974); phys. stat. sol. (a) 29, 529 (1975)

    Article  Google Scholar 

  26. M. Ueda and H. Nakano, J. Phys. Soc. Japan 48, 1829 (1980)

    Article  ADS  Google Scholar 

  27. G. D. Mahan and F. H. Claro, Phys. Rev. B16, 1168 (1977)

    Article  ADS  Google Scholar 

  28. U. Gahn, phys. stat. sol. (a) 40, 463 (1977)

    Article  ADS  Google Scholar 

  29. G. Inden, Acta metall. 22, 945 (1974); Z. Metallk. 66, 577, 648 (1975)

    Article  Google Scholar 

  30. R. Kikuchi and C. M. can Baal, Scripta metall. 8, 425 (1974)

    Article  Google Scholar 

  31. N. S. Golosov and A. M. Tolstik, J. Phys. Chem. Solids 36, 899, 903 (1975); 37, 273 (1976)

    Article  ADS  Google Scholar 

  32. D.-H. Wu and R. A. Tahir-Kheli, J. Phys. Soc. Japan 31, 641 (1971)

    Article  ADS  Google Scholar 

  33. K. Binder, Phys. Rev. Lett. 45, 811 (1980)

    Article  ADS  Google Scholar 

  34. M. K. Phani, J. L. Lebowitz and M. H. Kalos, Phys. Rev. B21, 4027 (1980)

    Article  ADS  Google Scholar 

  35. K. Binder, J. L. Lebowitz, M. K. Phani, and M. H. Kalos, Acta metall. 29, 1655 (1981)

    Article  Google Scholar 

  36. K. Binder, Z. Physik B45, 61 (1981)

    ADS  Google Scholar 

  37. U. Gahn, J. Phys. Chem. Solids 43, 977 (1982); and to be published; see also H. Ackermann, S. Crusius, and G. Inden, to be published

    Article  ADS  Google Scholar 

  38. K. Binder, W. Kinzel, and W. Selke, J. Magn. Mag. Mat. 31–34, 1145 (1983)

    Google Scholar 

  39. T. L. Polgreen, Phys. Rev. B29, 1468 (1984)

    Article  ADS  Google Scholar 

  40. H. Meirovitch, Phys. Rev. B30, 2866, (1984)

    Article  ADS  Google Scholar 

  41. R. A. Bond and D. K. Ross, J. Phys. F12, 597 (1982)

    Article  ADS  Google Scholar 

  42. J. L. Lebovitz, M. K. Phani and D. Styer, J. Stat. Phys. 38, 413 (1985)

    Article  ADS  Google Scholar 

  43. D. F. Styer, M. K. Phani and J. L. Lebovitz, Acta metall. (1985)

    Google Scholar 

  44. K. Binder, in: Atomic Transport and Defects in Metals by Neutron Scattering (ed. by T. Petry and D. Richter), in print (Springer, Berlin-Heidelberg-New York 1986)

    Google Scholar 

  45. H. T. Diep, A. Ghazali, B. Berge and P. Lallemand, Europhys Lett. (1986, in print)

    Google Scholar 

  46. B. Dünweg and K. Binder, to be published; see also B. Dünweg, Diplomarbeit (Johannes Gutenberg-Universität Mainz, 1986, unpublished)

    Google Scholar 

  47. For a review, see K. W. Kehr and K. Binder, in: Applications of the Monte Carlo Method in Statistical Physics, Chap. 6 (ed. by K. Binder), (Springer, Berlin-Heidelberg-New York 1984)

    Google Scholar 

  48. R. Kutner, K. Binder and K. W. Kehr, Phys. Rev. B26, 2967 (1982)

    Article  ADS  Google Scholar 

  49. R. Kutner, K. Binder and K. W. Kehr, Phys. Rev. B28, 1846 (1983)

    Article  ADS  Google Scholar 

  50. G. Alefeld, Phys. Stat. Sol. 32, 67 (1969)

    Article  ADS  Google Scholar 

  51. H. Wagner and H. Horner, Adv. Phys. 23, 587 (1974)

    Article  ADS  Google Scholar 

  52. W. Selke, K. Binder, and W. Kinzel, Surface Sci. 125, 74 (1983); K. Binder, W. Kinzel, and D. P. Landau, Surface Sci. 117, 232 (1982)

    Article  ADS  Google Scholar 

  53. R. Kikuchi, Phys. Rev. 81, 998 (1951)

    ADS  MathSciNet  Google Scholar 

  54. K. Binder (ed.), Monte Carlo Methods in Statistical Physics, (Springer, Berlin-Heidelberg-New York 1979); Applications of the Monte Carlo Method in Statistical Physics, (Springer, Berlin-Heidelberg-New York 1984)

    Google Scholar 

  55. K. Binder, J. Comp. Phys. 59, 1 (1985)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  56. J. M. Cowley, Phys. Rev. 77, 669 (1950)

    Article  MATH  ADS  Google Scholar 

  57. J. Kanamori, Progr. Theor. Phys. 35, 66 (1966); M. Kaburagi and J. Kanamori, Progr. Theor. Phys. 54, 30 (1975)

    Article  Google Scholar 

  58. M. J. Richards and J. W. Cahn, Acta Met. 19, 1263 (1971); S. M. Allen and J. W. Cahn, Acta Met. 20, 423 (1972); Scripta Met. 7, 1261 (1973)

    Article  Google Scholar 

  59. T. Kudo and S. Katsura, Progr. Theor. Phys. 56, 435 (1976)

    Article  ADS  Google Scholar 

  60. A. Danielian, Phys. Rev. Lett. 6, 670 (1961); Phys. Rev. A133, 1344 (1964)

    Article  ADS  Google Scholar 

  61. S. Alexander and P. Pincus, J. Phys. A13, 263 (1980).

    Article  ADS  Google Scholar 

  62. O. J. Heilmann, J. Phys. A13, 1803 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  63. G. Toulouse, Commun. Phys. 2, 115 (1977)

    Google Scholar 

  64. D. Hadjuković and S. Milosević, J. Phys. A15, 3561 (1982)

    Article  ADS  Google Scholar 

  65. J. Slawny, J. Stat. Phys. 20, 711 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  66. W. F. Wolff and J. Zittartz, in: Heidelberg Colloquium on Spin Glasses (ed. by J. L. van Hemmen and I. Morgenstern), p. 252 (Springer, Berlin-Heidelberg-New York 1983); J. Villain, R. Bideaux, J.-P. Carton, and R. Conte, J. Phys. (Paris) 41, 1263 (1980)

    Chapter  Google Scholar 

  67. W. L. Bragg and E. J. Williams, Proc. Roy. Soc. A145, 699 (1934); A152, 231 (1935)

    Article  ADS  Google Scholar 

  68. K. Binder and D. P. Landau, Phys. Rev. B21, 1941 (1980)

    Article  ADS  Google Scholar 

  69. R. Brout, Phase Transitions (Benjamin), New York 1965

    Google Scholar 

  70. P. C. Clapp and S. C. Moss, Phys. Rev. 142, 418 (1966); 171, 754 (1968)

    Article  ADS  Google Scholar 

  71. M. A. Krivoglaz, Theory of X-Ray and Thermal Neutron Scattering by Real Crystals (Plenum Press, New York 1969).

    Google Scholar 

  72. S. C. Moss and P. C. Clapp, Phys. Rev. 171, 764 (1968)

    Article  ADS  Google Scholar 

  73. For details, see e.g. S. M. Smart, Effective Field Theories of Magnetism (Saunders, London 1966), or [68].

    Google Scholar 

  74. T. Morita, J. Phys. Soc. (Japan) 12, 753, 1060 (1957); J. Math. Phys. 13, 115 (1972)

    Article  ADS  Google Scholar 

  75. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller J. Chem. Phys. 21, 108 (1953)

    Article  Google Scholar 

  76. For a review on finite size scaling, see M. N. Barber, in: Phase Transitions and Critical Phenomena, Vol. 8 (ed. by C. Domb and S. L. Lebowitz) Ch. 2 (Academic Press, New York 1983)

    Google Scholar 

  77. K. Binder and D. P. Landau, Phys. Rev. B30, 1477 (1984); M. S. S. Challa, D. P. Landau, and K. Binder, Phys. Rev. B34, (1986)

    Article  ADS  MathSciNet  Google Scholar 

  78. L. D. Fosdick, Phys. Rev. 116, 565 (1959)

    Article  ADS  MathSciNet  Google Scholar 

  79. P. C. Gehlen and J. B. Cohen, Phys. Rev. 139A, 844 (1965)

    Article  ADS  Google Scholar 

  80. R. C. Kittler and L. M. Falicov, Phys. Rev. B18, 2506 (1978); B19, 291 (1979)

    ADS  Google Scholar 

  81. H. Furukawa and K. Binder, Phys. Rev. A26, 556 (1982)

    Article  ADS  Google Scholar 

  82. K. Binder, Z. Physik B48, 319 (1982).

    ADS  Google Scholar 

  83. E. Domany, Y. Shnidman and D. Mukamel, J. Phys. C15, L495 (1982)

    ADS  Google Scholar 

  84. R. B. Potts, Proc. Cambridge Philos. Soc. 48, 106 (1952); for a recent review of the Potts model see F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  85. H. W. Blöte and R. H. Swendsen, Phys. Rev. Lett. 43, 799 (1979)

    Article  ADS  Google Scholar 

  86. K. Binder, Solid State Comm. 42, 377 (1982)

    Article  ADS  Google Scholar 

  87. C. Sykes and F. W. Jones, Proc. Roy. Soc. London, Sec. A157, 213 (1938); R. L. Orr, Acta Metall. 8, 489 (1960); R. L. Orr, J. Luciat-Labry and R. Hultgren, Acta Metall. 8, 431 (1960)

    Article  ADS  Google Scholar 

  88. D. T. Keating and B. E. Warren, J. Appl. Phys. 22, 286 (1951)

    Article  ADS  Google Scholar 

  89. S. C. Moss, J. Appl. Phys. 35, 3547 (1964)

    Article  ADS  Google Scholar 

  90. J. M. Cowley, J. Appl. Phys. 21, 24 (1950)

    Article  ADS  Google Scholar 

  91. L. H. Schwartz and J. B. Cohen, J. Appl. Phys. 36, 598 (1965)

    Article  ADS  Google Scholar 

  92. P. Bardhan and J. B. Cohen, Acta Cryst. A32, 597 (1976)

    Google Scholar 

  93. F. A. Shunk, Constitution of Binary Alloys (McGraw-Hill, New York 1969)

    Google Scholar 

  94. D. P. Landau, Phys. Rev. B16, 4164 (1976)

    Google Scholar 

  95. S. C. Moss and P. C. Clapp, Phys. Rev. 171, 764 (1968)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Physik, Johannes-Gutenberg-Universität Mainz, D-6500, Mainz, Federal Republic of Germany

    Kurt Binder

Authors
  1. Kurt Binder
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

P. Grosse

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH

About this chapter

Cite this chapter

Binder, K. (1986). Calculation of phase diagrams for models of metallic alloys. In: Grosse, P. (eds) Festkörperprobleme 26. Advances in Solid State Physics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0107795

Download citation

  • DOI: https://doi.org/10.1007/BFb0107795

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-528-08032-7

  • Online ISBN: 978-3-540-75359-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation