- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Reviews of Modern Physics

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

The effect of lattice vibrations on substitutional alloy thermodynamics

A. van de Walle and G. Ceder

A. van de Walle and G. Ceder

  • Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

PDF

Rev. Mod. Phys. 74, 11 – Published 28 January, 2002

DOI: https://doi.org/10.1103/RevModPhys.74.11

Abstract

A long-standing limitation of first-principles calculations of substitutional alloy phase diagrams is the difficulty in accounting for lattice vibrations. A survey of the theoretical and experimental literature seeking to quantify the effect of lattice vibrations on phase stability indicates that they can be significant. Typical vibrational entropy differences between phases are of the order of 0.1 to 0.2kB/atom, which is comparable to the typical values of configurational entropy differences in binary alloys (at most 0.693kB/atom). This article presents the basic formalism underlying ab initio phase diagram calculations, along with the generalization required to account for lattice vibrations. The authors review the various techniques allowing the theoretical calculation and the experimental determination of phonon dispersion curves and related thermodynamic quantities, such as vibrational entropy or free energy. A clear picture of the origin of vibrational entropy differences between phases in an alloy system is presented that goes beyond the traditional bond counting and volume change arguments. Vibrational entropy change can be attributed to the changes in chemical bond stiffness associated with the changes in bond length that take place during a phase transformation. This so-called “bond stiffness vs bond length” interpretation both summarizes the key phenomenon driving vibrational entropy changes and provides a practical tool to model them.

View figure in article

References (118)

  1. Abrikosov, I. A., A. V. Ruban, D. Ya Kats, and Y. H. Vekilov, 1993, J. Phys.: Condens. Matter 5, 1271.
  2. Ackland, G. J., 1994, in Alloy Modelling and Design, edited by G. Stocks and P. Turchi (The Minerals, Metals and Materials Society, Pittsburgh, PA), p. 149.
  3. Ackland, G. J., M. C. Warren, and S. J. Clark, 1997, J. Phys.: Condens. Matter 9, 7861.
  4. Allen, M. P., and D. J. Tildesley, 1987, Computer Simulation of Liquids (Clarendon Press, Oxford).
  5. Althoff, J. D., D. Morgan, D. de Fontaine, M. Asta, S. M. Foiles, and D. D. Johnson, 1997, Phys. Rev. B 56, R5705.
  6. Althoff, J. D., D. Morgan, D. de Fontaine, M. D. Asta, S. M. Foiles, and D. D. Johnson, 1998, Comput. Mater. Sci. 10, 411.
  7. Anthony, L., L. J. Nagel, J. K. Okamoto, and B. Fultz, 1994, Phys. Rev. Lett. 73, 3034.
  8. Anthony, L., J. K. Okamoto, and B. Fultz, 1993, Phys. Rev. Lett. 70, 1128.
  9. Ashcroft, N. W., and N. D. Mermin, 1976, Solid State Physics (Harcourt Brace College, New York).
  10. Asta, M., R. McCormack, and D. de Fontaine, 1993, Phys. Rev. B 48, 748.
  11. Bakker, H., 1982a, Philos. Mag. A 45, 213.
  12. Bakker, H., 1982b, Phys. Status Solidi B 109, 211.
  13. Bakker, H., and C. Tuijn, 1986, J. Phys. C 19, 5585.
  14. Baroni, S., P. Giannozzi, and A. Testa, 1987, Phys. Rev. Lett. 58, 1861.
  15. Becker, J. D., and J. M. Sanchez, 1993, Mater. Sci. Eng., A 170, 161.
  16. Binder, K., and D. W. Heermann, 1988, Monte Carlo Simulation in Statistical Physics (Springer-Verlag, New York).
  17. Bogdanoff, P. D., and B. Fultz, 1999, Philos. Mag. B 79, 753.
  18. Bogdanoff, P. D., B. Fultz, and S. Rosenkranz, 1999, Phys. Rev. B 60, 3976.
  19. Booth, C., and J. S. Rowlinson, 1955, Trans. Faraday Soc. 51, 463.
  20. Born, M., and K. Huang, 1956, Dynamical Theory of Crystal Lattices (Oxford University Press, London).
  21. Boyer, L. L., and M. J. Mehl, 1993, Ferroelectrics 150, 13.
  22. Ceder, G., 1991, Ph.D. thesis, University of California, Berkeley.
  23. Ceder, G., 1993, Comput. Mater. Sci. 1, 144.
  24. Ceder, G., M. Asta, W. C. Carter, M. Sluiter, M. E. Mann, M. Kraitchman, and D. de Fontaine, 1990, Phys. Rev. B 41, 8698.
  25. Chadi, D. J., and M. Cohen, 1973, Phys. Rev. B 8, 5747.
  26. Cleri, F., and V. Rosato, 1993, Philos. Mag. Lett. 67, 369.
  27. Colinet, C., et al., 1994, J. Phys.: Condens. Matter 6, L47.
  28. Craievich, P. J., and J. M. Sanchez, 1997, Comput. Mater. Sci. 8, 92.
  29. Craievich, P. J., J. Sanchez, R. E. Watson, and M. Weinert, 1997, Phys. Rev. B 55, 787.
  30. Daw, M. S., and M. I. Baskes, 1984, Phys. Rev. B 29, 6443.
  31. de Fontaine, D., 1994, Solid State Phys. 47, 33.
  32. de Fontaine, D., J. Althoff, D. Morgan, M. Asta, S. Foiles, and D. J. A. Quong, 1998, in Phase Transformations and Systems Driven Far From Equilibrium, edited by E. Ma, M. Atzmon, P. Bellon, and R. Trivedi (Materials Research Society, Warrendale, PA), p. 175.
  33. de Gironcoli, S., 1995, Phys. Rev. B 51, 6773.
  34. Dick, B., and A. W. Overhausser, 1958, Phys. Rev. 112, 90.
  35. Ducastelle, F., 1991, Order and Phase Stability in Alloys (Elsevier Science, New York).
  36. Dyson, F. J., 1953, Phys. Rev. 92, 1331.
  37. Fultz, B., L. Anthony, L. J. Nagel, J. K. Okamoto, R. M. Nicklow, and S. Spooner, 1995, Phys. Rev. B 52, 3315.
  38. Fultz, B., L. Anthony, J. L. Robertson, R. M. Nicklow, S. Spooner, and M. Mostoller, 1995, Phys. Rev. B 52, 3280.
  39. Fultz, B., T. A. Stephens, W. Sturhahn, T. S. Toellner, and E. E. Alp, 1998, Phys. Rev. Lett. 80, 3304.
  40. Garbulsky, G. D., 1996, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA.
  41. Garbulsky, G. D., and G. Ceder, 1994, Phys. Rev. B 49, 6327.
  42. Garbulksy, G. D., and G. Ceder, 1995, Phys. Rev. B 51, 67.
  43. Garbulsky, G. D., and G. Ceder, 1996, Phys. Rev. B 53, 8993.
  44. Giannozzi, P., S. de Gironcoli, P. Pavone, and S. Baroni, 1991, Phys. Rev. B 43, 7231.
  45. Gonze, X., 1997, Phys. Rev. B 55, 10337.
  46. Gonze, X., D. C. Allan, and M. P. Teter, 1992, Phys. Rev. Lett. 68, 3603.
  47. Gonze, X., and C. Lee, 1997, Phys. Rev. B 55, 10355.
  48. Grimvall, G., 1986, Thermophysical Properties of Materials, Selected Topics in Solid State Physics, edited by E. P. Wohlfarth (North-Holland, Amsterdam).
  49. Grimvall, G., and I. Ebbsjo, 1975, Phys. Scr. 12, 168.
  50. Grimvall, G., and J. Rosen, 1983, Int. J. Thermophys. 4, 139.
  51. Harrison, W. A., 1989, Electronic Structure and the Properties of Solids (Dover, New York).
  52. Hass, K. C., L. C. Davis, and A. Zunger, 1990, Phys. Rev. B 42, 3757.
  53. Imada, M., A. Fujimori, and Y. Tokura, 1998, Rev. Mod. Phys. 70, 1039.
  54. Johnson, D. D., A. V. Smirnov, J. B. Staunton, F. J. Pinski, and W. A. Shelton, 2000, Phys. Rev. B 62, R11 917.
  55. Kikuchi, R., 1951, Phys. Rev. 81, 988.
  56. Kohan, A. F., P. D. Tepesch, G. Ceder, and C. Wolverton, 1998, Comput. Mater. Sci. 9, 389.
  57. LeSar, R., R. Najafadabi, and D. J. Srolovitz, 1989, Phys. Rev. Lett. 63, 624.
  58. Maradudin, A. A., E. W. Montroll, and G. H. Weiss, 1971, Theory of Lattice Dynamics in the Harmonic Approximation, Second Edition (Academic Press, New York).
  59. Matthew, J. A. D., R. E. Jones, and V. M. Dwyer, 1983, J. Phys. F: Met. Phys. 13, 581.
  60. McCormack, R., and D. de Fontaine, 1996, Phys. Rev. B 54, 9746.
  61. McQuarrie, D. A., 1973, Statistical Thermodynamics (University Science Books, Mill Valley, CA).
  62. Miller, A., 1989, Phys. Rev. B 39, 3616.
  63. Mohri, T., 1994, Prog. Theor. Phys. Suppl. 115, 147.
  64. Mohri, T., S. Takizawa, and K. Terakura, 1993, J. Phys.: Condens. Matter 5, 1473.
  65. Monkhorst, H. J., and J. Pack, 1976, Phys. Rev. B 13, 5188.
  66. Moraitis, G., and F. Gautier, 1977, J. Phys. F: Met. Phys. 7, 1421.
  67. Morgan, D., 1998, Ph.D. thesis, University of California, Berkeley.
  68. Morgan, D., J. D. Althoff, and D. de Fontaine, 1998, J. Phase Equilib. 19, 559.
  69. Morgan, D., A. van de Walle, G. Ceder, J. D. Althoff, and D. de Fontaine, 2000, Modell. Simul. Mater. Sci. Eng. 8, 1.
  70. Moruzzi, V. L., J. F. Janak, and K. Schwarz, 1988, Phys. Rev. B 37, 790.
  71. Mukherjee, G. D., C. Bansal, and A. Chatterjee, 1996, Phys. Rev. Lett. 76, 1876.
  72. Mukherjee, G. D., C. Bansal, and A. Chatterjee, 1998, Int. J. Mod. Phys. B 12, 2233.
  73. Nagel, L. J., L. Anthony, and B. Fultz, 1995, Philos. Mag. Lett. 72, 421.
  74. Nagel, L. J., L. Anthony, J. K. Okamoto, and B. Fultz, 1997, J. Phase Equilib. 18, 551.
  75. Nagel, L. J., B. Fultz, and J. L. Robertson, 1997, Philos. Mag. B 75, 681.
  76. Nagel, L. J., B. Fultz, J. L. Robertson, and S. Spooner, 1996, Phys. Rev. B 55, 2903.
  77. Najafabadi, R., H. Y. Wang, D. J. Srolovitz, and R. LeSar, 1991, Acta Metall. Mater. 39, 3071.
  78. Nakamura, K., and T. Mohri, 1993, Modell. Simul. Mater. Sci. Eng. 1, 143.
  79. Oh, C.-S., T. Mohri, and D. N. Lee, 1994, Mater. Trans., JIM 35, 445.
  80. Ozoliņš, V., 1996, Ph.D. thesis, Royal Institute of Technology, Stockholm, Sweden.
  81. Ozoliņš, V., and M. Asta, 2001, Phys. Rev. B 86, 448.
  82. Ozoliņš, V., C. Wolverton, and A. Zunger, 1998a, Phys. Rev. B 57, 6427.
  83. Ozoliņš, V., C. Wolverton, and A. Zunger, 1998b, Phys. Rev. B 57, 4816.
  84. Ozoliņš, V., C. Wolverton, and A. Zunger, 1998c, Phys. Rev. B 58, R5897.
  85. Persson, K., M. Ekman, and G. Grimvall, 1999, Phys. Rev. B 60, 9999.
  86. Petry, W., A. Heiming, J. Trampenau, M. Alba, C. Herzig, H. R. Schober, and G. Vogl, 1991, Phys. Rev. B 43, 10933.
  87. Pettifor, D. G., 1992, in Electron Theory in Alloy Design, edited by D. Pettifor and A. H. Cottrell (The Institute of Materials, London), p. 81.
  88. Quong, A. A., and A. Y. Lui, 1997, Phys. Rev. B 56, 7767.
  89. Quong, A. A., and B. Klein, 1992, Phys. Rev. B 46, 10734.
  90. Ravelo, R., J. Aguilar, M. Baskes, J. Angelo, B. Fultz, and B. L. Holian, 1998, Phys. Rev. B 57, 862.
  91. Rittner, J. D., S. M. Foiles, and D. N. Seidman, 1994, Phys. Rev. B 50, 12004.
  92. Robertson, J. L., B. Fultz, and H. N. Frase, 1999, Phys. Rev. B 60, 9329.
  93. Russell, T. P., T. E. Karis, Y. Gallot, and A. M. Mayes, 1994, Nature (London) 368, 729.
  94. Sanchez, J. M., and J. D. Becker, 1994, Prog. Theor. Phys. Suppl. 115, 45.
  95. Sanchez, J. M., F. Ducastelle, and D. Gratias, 1984, Physica A 128A, 334.
  96. Sanchez, J. M., J. P. Stark, and V. L. Moruzzi, 1991, Phys. Rev. B 44, 5411.
  97. Shaojun, L., D. Suqing, and M. Benkun, 1998, Phys. Rev. B 58, 9705.
  98. Silverman, A., A. Zunger, A. Kalish, and J. Adler, 1995, Phys. Rev. B 51, 10 795.
  99. Sluiter, M. H., M. Weinert, and Y. Kawazoe, 1999, Phys. Rev. B 59, 4100.
  100. Sutton, A. P., 1989, Philos. Mag. A 60, 147.
  101. Tepesch, P. D., A. F. Kohan, G. D. Garbulsky, G. Ceder, C. Coley, H. T. Stokes, L. L. Boyer, M. J. Mehl, B. P. Burton, K. J. Cho, and J. Joannopoulos, 1996, J. Am. Ceram. Soc. 49, 2033.
  102. Tseng, W. T., and J. P. Stark, 1994, Philos. Mag. B 70, 919.
  103. Tuijn, C., and H. Bakker, 1989, Phys. Status Solidi B 155, 107.
  104. van de Walle, A., 1995, Master’s thesis, University of Waterloo, Waterloo, Canada.
  105. van der Ven, A., M. K. Aydinol, G. Ceder, G. Kresse, and J. Hafner, 1998, Phys. Rev. B 58, 2975.
  106. van de Walle, A., and G. Ceder, 1999, Phys. Rev. B 59, 14 992.
  107. van de Walle, A., and G. Ceder, 2000, Phys. Rev. B 61, 5972.
  108. van de Walle, A., G. Ceder, and U. V. Waghmare, 1998, Phys. Rev. Lett. 80, 4911.
  109. Voter, A. F., and S. P. Chen, 1988, in Characterization of Defects in Materials, edited by R. W. Siegel, J. R. Weertman, and R. Sinclair (Materials Research Society, Pittsburgh), p. 175.
  110. Waegmaekers, A. A. H. J., and H. Bakker, 1984, in Phase Transformations in Solids, edited by T. Tsakalakos, Mater. Res. Soc. Symp. Proc. No. 21 (Materials Research Society, Pittsburgh, PA), p. 343.
  111. Waghmare, U. V., 1996, Ph.D. thesis, Yale University, New Haven, CT.
  112. Wei, S., and M. Y. Chou, 1992, Phys. Rev. Lett. 69, 2799.
  113. Wojtowicz, P. J., and J. G. Kirkwood, 1960, J. Chem. Phys. 33, 1299.
  114. Wolverton, C., and A. Zunger, 1995, Phys. Rev. B 52, 8813.
  115. Wolverton, C., and V. Ozoliņš, 2001, Phys. Rev. Lett. 86, 5518.
  116. Zhao, L., R. Najafabadi, and D. J. Srolovitz, 1993, Modell. Simul. Mater. Sci. Eng. 1, 539.
  117. Zunger, A., 1994, in NATO Advanced Study Institute on Statics and Dynamics of Alloy Phase Transformation, edited by P. E. Turchi, and A. Gonis (Plenum Press, New York), Vol. 319, p. 361.
  118. Zunger, A., S.-H. Wei, L. Ferreira, and J. E. Bernard, 1990, Phys. Rev. Lett. 65, 353.

Outline

Information

Rev. Mod. Phys. 74, 11– Published 28 January, 2002

    © 2002 American Physical Society

    Sign In to Your Journals Account

    Filter

    Filter

    Article Lookup

    Enter a citation