SpringerLink
Log in

Entropy of Alloy Phases: A Macroscopic Perspective

  • Original Research Article
  • Published:
Metallurgical and Materials Transactions A Aims and scope Submit manuscript

Abstract

This study presents a comprehensive analysis of the entropy of condensed phases, its temperature, pressure, and composition dependence on a macroscopic correlative platform. Two principal contributions to total nonconfiguration entropy (ST) are outlined. They are: (i) the pure thermal (Sth) contribution arising from the isochoric temperature dependence of Gibbs energy (GT) and (ii) the elastic contribution (Sel) representing the dilatational volume effects. It is then argued that entropy variation among a group of alloy phases can be exclusively related to molar volume, only when both thermal pressure (pth) and thermal entropy terms assume common values for all members. This argument is extended to establish a linear relationship between transformation entropy (ΔStr) and transformation-induced volumetric strain (ΔVtr/V). The temperature and pressure dependencies of entropy have been discussed in terms of the complementing roles of Sth and Sel and simple approximations to these effects are suggested. A macroscopic power law relation for systematizing the standard entropy variation using a composite scaling parameter (MV2/3/Tm) has been proposed, and its validity is demonstrated for both solid and liquid metals. This power law correlation has been exploited to deduce the following outcome: (i) a simple approximation for the initial slope (dp/dTm) of p–Tm melting curve, (ii) self-consistent correlation of entropy with specific heat and Debye temperature, (iii) estimation of entropy of metastable phases, and (iv) correlating dilute solution entropy with volume effects of alloying.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. H.S. Leff: Found. Phys., 2007, vol. 37, pp. 1744–66.

    Article  Google Scholar 

  2. E.I. Kozliak: J. Chem. Ed., 2004, vol. 81, pp. 1595–8.

    Article  CAS  Google Scholar 

  3. M. Popovich: J. Subj. Didact., 2018, vol. 2, pp. 73–80.

    Google Scholar 

  4. G.M. Anderson: Thermodynamics of Natural Systems: Theory and Applications in Geochemistry and Environmental Science, 3rd ed. Cambridge University Press, Cambridge, 2017, pp. 66–107.

    Google Scholar 

  5. S. Stolen and T. Grande: Chemical Thermodynamics of Materials: Macroscopic and Microscopic Aspects, Wiley, Chichester, 2004, pp. 1–407.

    Google Scholar 

  6. P. Richet: The Physical Basis of Thermodynamics: With Applications to Chemistry, Springer, Heidelberg, 2001, pp. 1–24.

    Book  Google Scholar 

  7. S.-A. Cho: J. Solid State Chem., 1976, vol. 16, pp. 335–61.

    Article  Google Scholar 

  8. M.W. Zemansky and R.H. Dittman: Heat and Thermodynamics: An Intermediate Textbook, 7th ed. McGraw-Hill, New York, 1997, p. 84.

    Google Scholar 

  9. D.L. Anderson: Theory of the Earth, Blackwell Scientific Publications, Boston, 1989, pp. 79–102.

    Google Scholar 

  10. O. Kubaschewski, C.B. Alcock, and P.J. Spencer: Materials Thermochemistry, Sixth Revised Edition, Pergamon Press, Oxford, 1993, pp. 1–361.

  11. O.L. Anderson: Equations of State of Solids for Geophysics and Ceramic Science, Oxford University Press, New York, 1995, pp. 1–405.

    Google Scholar 

  12. P. Richet, P. Gillet, and G. Fiquet: in Thermodynamic Data: Systematics and Estimation. S.K. Saxena, ed., Springer, New York, 1992, pp. 98–132.

    Chapter  Google Scholar 

  13. D. Bhattacharya and J.M. Dawlaty: J. Chem. Educ., 2019, vol. 96, pp. 2208–16.

    Article  Google Scholar 

  14. A.A. Sobko: Dokl. Phys., 2007, vol. 52, pp. 13–8.

    Article  CAS  Google Scholar 

  15. A.A. Sobko: Dokl. Phys., 2007, vol. 52, pp. 577–8.

    Article  CAS  Google Scholar 

  16. V. Ivlev: Res. Dev. Mater Sci., 2017, vol. 2, pp. 121–2.

    Google Scholar 

  17. Z.-K. Liu and Y. Wang: Computational Thermodynamics of Materials, Cambridge University Press, Cambridge, 2016, pp. 104–49.

    Google Scholar 

  18. P.E.A. Turchi, I.A. Abrikosov, B. Burton, S.G. Fries, G. Grimvall, L. Kaufman, P. Korzhavy, V. Rao Manga, M. Ohno, A. Pisch, A. Scott, and W.Q. Zhang: Calphad., 2007, vol. 31, pp. 4–27.

    Article  CAS  Google Scholar 

  19. B. Fultz: Prog. Mater. Sci., 2010, vol. 55, pp. 247–352.

    Article  CAS  Google Scholar 

  20. J.A. Alonso and N.H. March: Electrons in Metals and Alloys, Academic, London, 1989, pp. 1–603.

    Book  Google Scholar 

  21. M. Hillert and M. Jarl: Calphad., 1978, vol. 2, pp. 227–38.

    Article  CAS  Google Scholar 

  22. J. Rogal, S.V. Divinski, M.W. Finnis, A. Glensk, J. Neugebauer, J.H. Perepezko, S. Schuwalow, and M.H.F. Sluiter: Phys. Status Solidi B., 2014, vol. 251, pp. 97–129.

    Article  CAS  Google Scholar 

  23. Z.-K. Liu: J. Phase Equilib. Diffus., 2009, vol. 30, pp. 517–34.

    Article  CAS  Google Scholar 

  24. W.A. Oates: J. Phase Equilib. Diffus.., 2007, vol. 28, pp. 78–89.

    Article  Google Scholar 

  25. C.A. Becker, J. Agren, M. Baricco, Q. Chen, S.A. Decterov, U.R. Kattner, J.H. Perepezko, G.R. Pottlacher, and M. Selleby: Phys. Status Solidi B., 2014, vol. 251, pp. 33–52.

    Article  CAS  Google Scholar 

  26. L. Wang: Metals., 2014, vol. 4, pp. 570–85.

    Article  Google Scholar 

  27. I. Yokoyama, K. Maruyama, and Y. Waseda: High Temp. Mater. Proc., 2003, vol. 22, pp. 94–102.

    Article  Google Scholar 

  28. D.C. Wallace: Int. J. Quantum Chem., 1994, vol. 52, pp. 425–35.

    Article  CAS  Google Scholar 

  29. E.D. Chisholm and D.C. Wallace: J. Phys. Condens. Matter., 2001, vol. 13, pp. R739–69.

    Article  Google Scholar 

  30. D.C. Wallace, S. Rudin, G. De Lorenzi-Venneri, and T. Sjostrom: Phys. Rev. B., 2019, vol. 99, pp. 104204(1-7).

    Article  Google Scholar 

  31. K. Trachenko and V.V. Brazkin: Rep. Prog. Phys., 2016, vol. 79, p. 016502.

    Article  CAS  Google Scholar 

  32. D. Bolmatov, V.V. Brazhkin, and K. Trachenko: Sci. Rep., 2012, vol. 2, pp. 421–7.

    Article  CAS  Google Scholar 

  33. M.P. Desjarlais: Phys. Rev. E., 2013, vol. 88, pp. 062145(1-9).

    Article  Google Scholar 

  34. A.V. Granato: J. Noncryst. Solids., 2002, vol. 307–310, pp. 376–86.

    Article  Google Scholar 

  35. R.H. Henchman: J. Chem. Phys., 2003, vol. 119, pp. 400–6.

    Article  CAS  Google Scholar 

  36. D. Lilley, A. Jain, and R. Prasher: Appl. Phys. Lett., 2021, vol. 118, pp. 083902(1-5).

    Article  Google Scholar 

  37. V.N. Chuvildeev and A.V. Semenycheva: Inorg. Mater., 2017, vol. 53, pp. 770–3.

    Article  CAS  Google Scholar 

  38. A. Pasturel and N. Jakse: J. Phys. Condens. Matter., 2014, vol. 27, pp. 325104(1-9).

    Google Scholar 

  39. A.F. Guillermet and G. Grimvall: J. Phys. Chem. Solids., 1992, vol. 53, pp. 105–25.

    Article  Google Scholar 

  40. A.F. Guillermet and G. Grimvall: J. Less Common Met., 1991, vol. 169, pp. 257–81.

    Article  Google Scholar 

  41. M. Lasocka: Phys. Lett., 1975, vol. 51A, pp. 137–8.

    Article  CAS  Google Scholar 

  42. J.L. Tallon and W.H. Robinson: Phys. Lett. A., 1982, vol. 87, pp. 365–8.

    Article  Google Scholar 

  43. Z. Akdeniz and M.P. Tosi: Proc. R. Soc. Lond. A., 1992, vol. 437, pp. 85–96.

    Article  CAS  Google Scholar 

  44. G. Grimvall: Int. J. Thermophys., 1983, vol. 4, pp. 363–7.

    Article  CAS  Google Scholar 

  45. W.M. Latimer: J. Am. Chem. Soc., 1951, vol. 73, pp. 1480–2.

    Article  CAS  Google Scholar 

  46. F. Chen, Z. Yang, H. Wen, and Z. Xu: Acta Phys. Chim., 1997, vol. 13, pp. 712–6.

    Article  CAS  Google Scholar 

  47. E.T. Turkdogan and J. Pearson: J. Appl. Chem., 1953, vol. 3, pp. 495–501.

    Article  CAS  Google Scholar 

  48. L.I. Ivanova: Russ. J. Phys. Chem., 1984, vol. 58, pp. 80–4.

    CAS  Google Scholar 

  49. D. Ai, R. Zeng, and Z. Zhang: Chin. Sci. Bull., 1999, vol. 44, pp. 270–3.

    Article  Google Scholar 

  50. L.I. Ivanova: Russ. J. Phys. Chem., 1976, vol. 50, pp. 133–6.

    Google Scholar 

  51. B. Chatterjee: Metall. Trans. A., 1977, vol. 8, pp. 1485–6.

    Article  Google Scholar 

  52. G.K. Moiseev and J. Sestak: Prog. Cryst. Growth Charact., 1995, vol. 30, pp. 23–81.

    Article  CAS  Google Scholar 

  53. P.J. Spencer: Thermochim. Acta., 1998, vol. 314, pp. 1–21.

    Article  CAS  Google Scholar 

  54. DSh. Tsagareishvili, G.V. Tsagareishvili, and MCh. Tushishvili: Diam. Relat. Mater., 1996, vol. 5, pp. 475–7.

    Article  CAS  Google Scholar 

  55. S. Raju: Metall. Mater. Trans. A., 2021, vol. 52A, pp. 242–56.

    Article  Google Scholar 

  56. Z.-K. Liu, Y. Wang, and S. Shang: Sci. Rep., 2014, vol. 4, pp. 7043(1-6).

    Google Scholar 

  57. H.D.B. Jenkins and L. Glasser: Inorg. Chem., 2003, vol. 42, pp. 8702–8.

    Article  CAS  Google Scholar 

  58. S. Raju, K. Sivasubramanian, and E. Mohandas: Solid State Commun., 2002, vol. 124, pp. 151–6.

    Article  CAS  Google Scholar 

  59. T.J.B. Holland: Am. Mineral., 1989, vol. 74, pp. 5–13.

    CAS  Google Scholar 

  60. S.K. Saxena: Science., 1976, vol. 193, pp. 1241–2.

    Article  CAS  Google Scholar 

  61. S. Cantor: Science., 1977, vol. 198, pp. 206–7.

    Article  CAS  Google Scholar 

  62. S. Cantor: J. Appl. Phys., 1972, vol. 43, pp. 706–9.

    Article  CAS  Google Scholar 

  63. G. Grimvall and A.F. Guillermet: in Thermochemical Data: Advances in Physical Geochemistry, vol. 10, S.K. Saxena, ed., Springer, New York, 1992, pp. 272–82.

    Chapter  Google Scholar 

  64. S.S. Penner: J. Chem. Phys., 1948, vol. 16, pp. 745–6.

    Article  CAS  Google Scholar 

  65. R.K. Kirby, R.E. Taylor, and P.D. Desai: in TRPC Data Series, vol. 12, Y.S. Touloukian and C.Y. Ho, eds., Plenum Press, New York, 1975, pp. 1–1442.

    Google Scholar 

  66. P. Villars and J.L.C. Daams: J. Alloys Compd., 1993, vol. 197, pp. 177–96.

    Article  CAS  Google Scholar 

  67. J.W. Arblaster: Selected Values of the Crystallographic Properties of the Elements, ASM, Materials Park, OH, 2018, pp. 1–683.

    Google Scholar 

  68. T. Gorecki: Mater. Sci. Eng., 1980, vol. 43, pp. 225–30.

    Article  CAS  Google Scholar 

  69. H. Warlimont and W. Martienssen, eds.: Springer Handbook of Materials Data, 2nd ed. Springer, Heidelberg, 2018, pp. 1–1140.

    Google Scholar 

  70. I. Barin: Thermochemical Data on Pure Substances, 3rd ed. VCH, Weinheim, 1995, pp. 1–1885.

    Google Scholar 

  71. A.T. Dinsdale: Calphad., 1991, vol. 15, pp. 317–425.

    Article  CAS  Google Scholar 

  72. D.G. Pettifor: Solid State Phys., 1987, vol. 40, pp. 43–92.

    Article  CAS  Google Scholar 

  73. Y. Marcus: J. Chem. Thermodyn., 2017, vol. 109, pp. 11–5.

    Article  CAS  Google Scholar 

  74. T. Iida and R.I. Guthrie: Thermophysical Properties of Metallic Liquids, vol. 1 and 2, Oxford University Press, Oxford, 2015, pp. 497–542.

    Google Scholar 

  75. S. Raju and A.K. Rai: J. Nucl. Mater., 2011, vol. 408, pp. 40–4.

    Article  CAS  Google Scholar 

  76. M. Zinkevich: Prog. Mater. Sci., 2007, vol. 52, pp. 517–47.

    Article  Google Scholar 

  77. L.J. Wittenberg and R. De Witt: J. Chem. Phys., 1972, vol. 56, pp. 4526–33.

    Article  CAS  Google Scholar 

  78. D.C. Wallace: Proc. R. Soc. Lond. A., 1991, vol. 433, pp. 631–61.

    Article  CAS  Google Scholar 

  79. A.C. Lawson: Philos. Mag., 2009, vol. 89, pp. 1757–70.

    Article  CAS  Google Scholar 

  80. TYa. Kosolopova: Handbook of High Temperature Compounds: Properties, Production and Application, Hemisphere, New York, 1990, pp. 1–933.

    Google Scholar 

  81. D. Errandonea: J. Appl. Phys., 2010, vol. 108, pp. 033517(1-10).

    Google Scholar 

  82. A. van de Walle, C. Nataraj, and Z.-K. Liu: Calphad., 2018, vol. 61, pp. 173–8.

    Article  Google Scholar 

  83. N. Saunders, A.P. Miodownik, and A.T. Dinsdale: Calphad., 1988, vol. 12, pp. 351–74.

    Article  CAS  Google Scholar 

  84. J.Z. Liu, A. van de Walle, G. Ghosh, and M. Asta: Phys. Rev. B., 2005, vol. 72, pp. 144109(1-16).

    Google Scholar 

  85. O.L. Anderson, L. Dubrovinsky, S.K. Saxena, and T. Le Bihan: Geophys. Res. Lett., 2001, vol. 28, pp. 349–402.

    Google Scholar 

  86. L.S. Dubrovinsky, S.K. Saxena, N.A. Dubrovinskaya, S. Rekhi, and T. LeBihan: Am Mineral., 2001, vol. 85, pp. 386–9.

    Article  Google Scholar 

  87. T. Komabayashi and Y.W. Fei: J. Geophys. Res. B., 2010, vol. 115, pp. 032029(1-12).

    Article  Google Scholar 

  88. S.L. Shang, A. Saengdee**g, Z.G. Mei, D.E. Kim, H. Zhang, S. Ganeshan, Y. Wang, and Z.K. Liu: Comput. Mater. Sci., 2010, vol. 48, pp. 813–26.

    Article  CAS  Google Scholar 

  89. F. Guinea, J.H. Rose, J.R. Smith, and J. Ferrante: Appl. Phys. Lett., 1984, vol. 44, pp. 53–5.

    Article  CAS  Google Scholar 

  90. C. Li and P. Wu: Chem. Mater., 2002, vol. 14, pp. 4833–6.

    Article  CAS  Google Scholar 

  91. Q. Chen and B. Sundman: Acta Mater., 2001, vol. 49, pp. 947–61.

    Article  CAS  Google Scholar 

  92. A. Dewaele, P. Loubeyre, F. Occelli, M. Mezouar, P.I. Dorogokupets, and M. Torrent: Phys. Rev. Lett., 2006, vol. 97, pp. 4–8.

    Article  Google Scholar 

  93. R. Lubbers, H.F. Grunsteudel, A.I. Chumakov, and G. Wortmann: Science., 2000, vol. 287, pp. 1250–3.

    Article  CAS  Google Scholar 

  94. G. Shen, W. Sturhahn, E.E. Alp, J. Zhao, T.S. Toellner, V.B. Prakapenka, Y. Meng, and H.-R. Mao: Phys. Chem. Miner., 2004, vol. 31, pp. 353–9.

    Article  CAS  Google Scholar 

  95. J. Zaretski and C. Stassis: Phys. Rev. B., 1987, vol. 35, pp. 4500–2.

    Article  Google Scholar 

  96. Q. Chen and B. Sundman: J. Phase Equilb., 2001, vol. 22, pp. 631–44.

    Article  CAS  Google Scholar 

  97. S. Curtarolo, W. Setyawan, S. Wang, J. Xue, K. Yang, R.H. Taylor, L.J. Nelson, G.L.W. Hart, S. Sanvito, and M. Buongiorno-Nardelli: No Mingo, and O Levy: Comput. Mater. Sci., 2012, vol. 58, pp. 227–35.

    Article  CAS  Google Scholar 

  98. J. Hafner, C. Wolverton, and G. Ceder: MRS Bull., 2006, vol. 31, pp. 659–65.

    Article  Google Scholar 

  99. G. Bozzolo, R.D. Noebe, and P.B. Abel, eds.: Applied Computational Materials Modelling, Springer, New York, 2007, pp. 1–483.

    Google Scholar 

  100. W. **ong and G.B. Olson: MRS Bull., 2015, vol. 40, pp. 1035–42.

    Article  Google Scholar 

  101. S.R. Kalidindi: J. Appl. Phys., 2020, vol. 128, pp. 041103(1-15).

    Article  Google Scholar 

  102. A.R. Miedema and A.K. Niessen: Physica B + C., 1982, vol. 114, pp. 367–74.

    Article  CAS  Google Scholar 

  103. V.A. Lubarda: Mech. Mater., 2003, vol. 35, pp. 53–68.

    Article  Google Scholar 

  104. J. Hafner: J. Phys. F., 2000, vol. 15, pp. L43–5.

    Article  Google Scholar 

  105. D.E. Passoja, K.A. Chao, and G.S. Ansell: J. Appl. Phys., 1967, vol. 40, pp. 1967–71.

    Article  Google Scholar 

  106. H.J. Schaller: Ber. Bunsenges. phys. Chem., 1983, vol. 87, pp. 734–41.

    Article  CAS  Google Scholar 

  107. E. Mayer and J.-P. Bros: J. Alloys Compd., 1995, vol. 220, pp. 193–6.

    Article  Google Scholar 

Download references

Acknowledgments

The author acknowledges Dr. Shaju K Albert, Director, Metallurgy and Materials Group, and Dr. A. K. Bhaduri, Director, IGCAR, for their support and encouragement. The attention of author to the work of Dr. Bivabasu Chatterjee (Ref. [51]) was drawn by late Prof. P. Ramachandra Rao. This work is dedicated in fond memories of Dr. Srikumar Banerjee, formerly Chairman, DAE, and Secretary to the Government of India.

Conflict of interest

The author declares no conflict of interest.

Author information

Authors and Affiliations

  1. Physical Metallurgy Division, Materials Characterization Group, Metallurgy & Materials Group, Indira Gandhi Centre for Atomic Research (IGCAR), Kalpakkam, 603102, India

    Subramanian Raju

Authors
  1. Subramanian Raju
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Subramanian Raju.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Manuscript submitted June 16, 2021; accepted Sep 19, 2021.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raju, S. Entropy of Alloy Phases: A Macroscopic Perspective. Metall Mater Trans A 52, 5274–5292 (2021). https://doi.org/10.1007/s11661-021-06466-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11661-021-06466-4

Search

Navigation