Advertisement

Calculation of phase diagrams for models of metallic alloys

Chapter
Part of the Advances in Solid State Physics book series (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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    For reviews of earlier work see e.g. various articles in J. de Physique 38—C7 (1977), and [2–5]Google Scholar
  2. [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. [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. [4]
    A. G. Khachaturyan, Phys. stat. sol. (b) 60, 9 (1973)CrossRefADSGoogle Scholar
  5. [5]
    W. Warlimont (ed.), Order-Disorder Transformations in Alloys (Springer, Berlin-Heidelberg-New York 1974)Google Scholar
  6. [6]
    H. E. Stanley, An Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford 1971)Google Scholar
  7. [7]
    M. Hansen, Constitution of Binary Alloys (McGraw-Hill, New York 1958)Google Scholar
  8. [8]
    S. V. Semenovskaya, phys. stat. sol. (b) 64, 291 (1974)CrossRefGoogle Scholar
  9. [9]
    P. S. Swann, W. R. Duff, and R. M. Fisher, metallurg. Trans. 3, 409 (1972)ADSGoogle Scholar
  10. [10]
    A. Bieber and F. Gautier, J. Phys. Soc. Japan 53, 2061 (1984), Z. Physik B57, 335 (1984)CrossRefADSGoogle Scholar
  11. [11]
    U. Köbler and K. Binder, J. Magn. Magn. Mat. 15–18, 313 (1980)CrossRefGoogle Scholar
  12. [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., submittedGoogle Scholar
  13. [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. [14]
    W. Shockley, J. Chem. Phys. 6, 130 (1938)CrossRefADSGoogle Scholar
  15. [15]
    Y. Y. Li, J. Chem. Phys. 17, 449 (1949)ADSGoogle Scholar
  16. [16]
    R. Kikuchi, J. Chem. Phys. 60, 1071 (1974).CrossRefADSGoogle Scholar
  17. [17]
    C. M. van Baal, Physica 64, 571 (1973)CrossRefADSGoogle Scholar
  18. [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. [19]
    J. M. Sanchez and D. de Fontaine, Phys. Rev. B21, 216 (1980); B25, 1759 (1982)ADSCrossRefGoogle Scholar
  20. [20]
    J. M. Sanchez, D. de Fontaine and W. Teitler, Phys. Rev. B26, 1465 (1982)ADSCrossRefGoogle Scholar
  21. [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. [22]
    N. S. Golosov, L. E. Popov and L. Y. Pudan, J. Phys. Chem. Solids 34, 1149, 1159 (1973)CrossRefADSGoogle Scholar
  23. [23]
    R. Kikuchi and H. Sato, Acta metall. 22, 1099 (1974)CrossRefGoogle Scholar
  24. [24]
    C. E. Dahmani, M. C. Cadeville, J. M. Sanchez and J. L. Morán-López, Phys. Rev. Lett. 55, 1208 (1985)CrossRefADSGoogle Scholar
  25. [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)CrossRefGoogle Scholar
  26. [26]
    M. Ueda and H. Nakano, J. Phys. Soc. Japan 48, 1829 (1980)ADSCrossRefGoogle Scholar
  27. [27]
    G. D. Mahan and F. H. Claro, Phys. Rev. B16, 1168 (1977)ADSCrossRefGoogle Scholar
  28. [28]
    U. Gahn, phys. stat. sol. (a) 40, 463 (1977)CrossRefADSGoogle Scholar
  29. [29]
    G. Inden, Acta metall. 22, 945 (1974); Z. Metallk. 66, 577, 648 (1975)CrossRefGoogle Scholar
  30. [30]
    R. Kikuchi and C. M. can Baal, Scripta metall. 8, 425 (1974)CrossRefGoogle Scholar
  31. [31]
    N. S. Golosov and A. M. Tolstik, J. Phys. Chem. Solids 36, 899, 903 (1975); 37, 273 (1976)CrossRefADSGoogle Scholar
  32. [32]
    D.-H. Wu and R. A. Tahir-Kheli, J. Phys. Soc. Japan 31, 641 (1971)ADSCrossRefGoogle Scholar
  33. [33]
    K. Binder, Phys. Rev. Lett. 45, 811 (1980)CrossRefADSGoogle Scholar
  34. [34]
    M. K. Phani, J. L. Lebowitz and M. H. Kalos, Phys. Rev. B21, 4027 (1980)ADSCrossRefGoogle Scholar
  35. [35]
    K. Binder, J. L. Lebowitz, M. K. Phani, and M. H. Kalos, Acta metall. 29, 1655 (1981)CrossRefGoogle Scholar
  36. [36]
    K. Binder, Z. Physik B45, 61 (1981)ADSGoogle Scholar
  37. [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 publishedCrossRefADSGoogle Scholar
  38. [38]
    K. Binder, W. Kinzel, and W. Selke, J. Magn. Mag. Mat. 31–34, 1145 (1983)Google Scholar
  39. [39]
    T. L. Polgreen, Phys. Rev. B29, 1468 (1984)ADSCrossRefGoogle Scholar
  40. [40]
    H. Meirovitch, Phys. Rev. B30, 2866, (1984)ADSCrossRefGoogle Scholar
  41. [41]
    R. A. Bond and D. K. Ross, J. Phys. F12, 597 (1982)CrossRefADSGoogle Scholar
  42. [42]
    J. L. Lebovitz, M. K. Phani and D. Styer, J. Stat. Phys. 38, 413 (1985)CrossRefADSGoogle Scholar
  43. [43]
    D. F. Styer, M. K. Phani and J. L. Lebovitz, Acta metall. (1985)Google Scholar
  44. [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. [45]
    H. T. Diep, A. Ghazali, B. Berge and P. Lallemand, Europhys Lett. (1986, in print)Google Scholar
  46. [46]
    B. Dünweg and K. Binder, to be published; see also B. Dünweg, Diplomarbeit (Johannes Gutenberg-Universität Mainz, 1986, unpublished)Google Scholar
  47. [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. [48]
    R. Kutner, K. Binder and K. W. Kehr, Phys. Rev. B26, 2967 (1982)ADSCrossRefGoogle Scholar
  49. [49]
    R. Kutner, K. Binder and K. W. Kehr, Phys. Rev. B28, 1846 (1983)ADSCrossRefGoogle Scholar
  50. [50]
    G. Alefeld, Phys. Stat. Sol. 32, 67 (1969)CrossRefADSGoogle Scholar
  51. [51]
    H. Wagner and H. Horner, Adv. Phys. 23, 587 (1974)CrossRefADSGoogle Scholar
  52. [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)CrossRefADSGoogle Scholar
  53. [53]
    R. Kikuchi, Phys. Rev. 81, 998 (1951)ADSMathSciNetGoogle Scholar
  54. [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. [55]
    K. Binder, J. Comp. Phys. 59, 1 (1985)zbMATHCrossRefMathSciNetADSGoogle Scholar
  56. [56]
    J. M. Cowley, Phys. Rev. 77, 669 (1950)zbMATHCrossRefADSGoogle Scholar
  57. [57]
    J. Kanamori, Progr. Theor. Phys. 35, 66 (1966); M. Kaburagi and J. Kanamori, Progr. Theor. Phys. 54, 30 (1975)CrossRefGoogle Scholar
  58. [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)CrossRefGoogle Scholar
  59. [59]
    T. Kudo and S. Katsura, Progr. Theor. Phys. 56, 435 (1976)CrossRefADSGoogle Scholar
  60. [69]
    A. Danielian, Phys. Rev. Lett. 6, 670 (1961); Phys. Rev. A133, 1344 (1964)CrossRefADSGoogle Scholar
  61. [61]
    S. Alexander and P. Pincus, J. Phys. A13, 263 (1980).CrossRefADSGoogle Scholar
  62. [62]
    O. J. Heilmann, J. Phys. A13, 1803 (1980)CrossRefMathSciNetADSGoogle Scholar
  63. [63]
    G. Toulouse, Commun. Phys. 2, 115 (1977)Google Scholar
  64. [64]
    D. Hadjuković and S. Milosević, J. Phys. A15, 3561 (1982)CrossRefADSGoogle Scholar
  65. [65]
    J. Slawny, J. Stat. Phys. 20, 711 (1979)CrossRefMathSciNetADSGoogle Scholar
  66. [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)CrossRefGoogle Scholar
  67. [67]
    W. L. Bragg and E. J. Williams, Proc. Roy. Soc. A145, 699 (1934); A152, 231 (1935)CrossRefADSGoogle Scholar
  68. [68]
    K. Binder and D. P. Landau, Phys. Rev. B21, 1941 (1980)ADSCrossRefGoogle Scholar
  69. [69]
    R. Brout, Phase Transitions (Benjamin), New York 1965Google Scholar
  70. [70]
    P. C. Clapp and S. C. Moss, Phys. Rev. 142, 418 (1966); 171, 754 (1968)CrossRefADSGoogle Scholar
  71. [71]
    M. A. Krivoglaz, Theory of X-Ray and Thermal Neutron Scattering by Real Crystals (Plenum Press, New York 1969).Google Scholar
  72. [72]
    S. C. Moss and P. C. Clapp, Phys. Rev. 171, 764 (1968)CrossRefADSGoogle Scholar
  73. [73]
    For details, see e.g. S. M. Smart, Effective Field Theories of Magnetism (Saunders, London 1966), or [68].Google Scholar
  74. [74]
    T. Morita, J. Phys. Soc. (Japan) 12, 753, 1060 (1957); J. Math. Phys. 13, 115 (1972)ADSCrossRefGoogle Scholar
  75. [75]
    N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller J. Chem. Phys. 21, 108 (1953)CrossRefGoogle Scholar
  76. [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. [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)CrossRefADSMathSciNetGoogle Scholar
  78. [78]
    L. D. Fosdick, Phys. Rev. 116, 565 (1959)CrossRefADSMathSciNetGoogle Scholar
  79. [79]
    P. C. Gehlen and J. B. Cohen, Phys. Rev. 139A, 844 (1965)CrossRefADSGoogle Scholar
  80. [80]
    R. C. Kittler and L. M. Falicov, Phys. Rev. B18, 2506 (1978); B19, 291 (1979)ADSGoogle Scholar
  81. [81]
    H. Furukawa and K. Binder, Phys. Rev. A26, 556 (1982)CrossRefADSGoogle Scholar
  82. [82]
    K. Binder, Z. Physik B48, 319 (1982).ADSGoogle Scholar
  83. [83]
    E. Domany, Y. Shnidman and D. Mukamel, J. Phys. C15, L495 (1982)ADSGoogle Scholar
  84. [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)zbMATHMathSciNetCrossRefGoogle Scholar
  85. [85]
    H. W. Blöte and R. H. Swendsen, Phys. Rev. Lett. 43, 799 (1979)CrossRefADSGoogle Scholar
  86. [86]
    K. Binder, Solid State Comm. 42, 377 (1982)CrossRefADSGoogle Scholar
  87. [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)ADSCrossRefGoogle Scholar
  88. [88]
    D. T. Keating and B. E. Warren, J. Appl. Phys. 22, 286 (1951)CrossRefADSGoogle Scholar
  89. [89]
    S. C. Moss, J. Appl. Phys. 35, 3547 (1964)CrossRefADSGoogle Scholar
  90. [90]
    J. M. Cowley, J. Appl. Phys. 21, 24 (1950)CrossRefADSGoogle Scholar
  91. [91]
    L. H. Schwartz and J. B. Cohen, J. Appl. Phys. 36, 598 (1965)CrossRefADSGoogle Scholar
  92. [92]
    P. Bardhan and J. B. Cohen, Acta Cryst. A32, 597 (1976)Google Scholar
  93. [93]
    F. A. Shunk, Constitution of Binary Alloys (McGraw-Hill, New York 1969)Google Scholar
  94. [94]
    D. P. Landau, Phys. Rev. B16, 4164 (1976)Google Scholar
  95. [95]
    S. C. Moss and P. C. Clapp, Phys. Rev. 171, 764 (1968)CrossRefADSGoogle Scholar

Copyright information

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH 1986

Authors and Affiliations

  1. 1.Institut für PhysikJohannes-Gutenberg-Universität MainzMainzFederal Republic of Germany

Personalised recommendations