Skip to main content
SpringerLink
Log in

Thermodynamic Properties and Equation of State for Solid and Liquid Copper

  • Published:
International Journal of Thermophysics Aims and scope Submit manuscript

Abstract

High-temperature equations of state for solid and liquid copper were constructed herein using experimental data on thermodynamic properties, thermal expansion, compressibility, bulk modulus and sound velocity measurements, supplemented with phase diagram data (melting curve). The totality of experimental data were optimized using the temperature-dependent Tait equation over a pressure range of up to 1000 kbar and over a temperature range from 20 K to the melting point for solid copper and to 3900 K for liquid copper. The temperature-dependence of thermodynamic and thermophysical parameters was described by an expanded Einstein model. The resultant equations of state describe well the entire set of experimental data within measurement errors of individual parameters.

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

Access this article

Log in via an institution

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

Similar content being viewed by others

Data Availability

The data that supports the findings of this study are available within the article.

References

  1. H.K. Mao, J. Appl. Phys. 49, 3276 (1978)

    Article  ADS  Google Scholar 

  2. M.I. McMahon, R.J. Nelmes, Chem. Soc. Rev. 35, 943 (2006)

    Article  Google Scholar 

  3. L. Dubrovinsky, N. Dubrovinskaia, W.A. Crichton, A.S. Mikhaylushkin, S.I. Simak, I.A. Abrikosov, J.S. de Almeida, R. Ahuja, W. Luo, B. Johansson, Phys. Rev. Lett. 98, 045503 (2007)

    Article  ADS  Google Scholar 

  4. P. Yue, X. Long, X. Jiang, Z. Zhang, Propell. Explos. Pyrot. 45, 600 (2020)

    Article  Google Scholar 

  5. R.D. Cowan, W. Fickett, J. Chem. Phys. 24, 932 (1956)

    Article  ADS  Google Scholar 

  6. Y. Wang, J. Zhang, H. Xu, Z. Lin, L.L. Daemen, Y. Zhao, L. Wang, Appl. Phys. Lett. 94, 071904 (2009)

    Article  ADS  Google Scholar 

  7. N.V. Kozyrev, V.V. Gordeev, Metals 12, 1 (2022)

    Article  Google Scholar 

  8. M.W. Chase, I. Ansara, A. Dinsdale, G. Eriksson, G. Grimvall, L. Hoglund, H. Yokokawa, CALPHAD 19, 437 (1995)

    Article  Google Scholar 

  9. G.F. Voronin, I.B. Kutsenok, J. Chem. Eng. Data. 58, 2083 (2013)

    Article  Google Scholar 

  10. P. Jacobson, S. Stoupin, Diamond Relat. Mat. 97, 107469 (2019)

    Article  ADS  Google Scholar 

  11. J.R. MacDonald, Rev. Mod. Phys. 41, 316 (1969)

    Article  ADS  Google Scholar 

  12. J.H. Dymond, R. Malhotra, Int. J. Thermophys. 9, 941 (1988)

    Article  ADS  Google Scholar 

  13. A.R. Hansen, J. Chem. Eng. Data. 36, 252 (1991)

    Article  Google Scholar 

  14. H. Preston-Thomas, Metrologia 27, 3 (1990)

    Article  ADS  Google Scholar 

  15. G. Deffrennes, B. Oudot, CALPHAD 74, 102291 (2021)

    Article  Google Scholar 

  16. A.T. Dinsdale, CALPHAD 15, 317 (1991)

    Article  Google Scholar 

  17. Y. Chang, W. Oates, Materials Thermodynamics (Wiley, New York, 2010)

    Google Scholar 

  18. R.A. Robie, B.S. Hemingway, J.R. Fisher, Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (105 pascals) pressure and at higher temperatures (U.S. Geological Survey Bulletin 1452. United States Government Printing Office, Washington, 1978)

  19. L.B. Pankratz, J.M. Stuve, N.A. Gokcen, Thermodynamic data for mineral technology (Bulletin No. 677. US Bureau of Mines. Washington, 1984)

  20. I. Barin, Thermochemical Data of Pure Substances (VCH Verlagsgesellschaft mbH, Weinheim, 1995)

    Book  Google Scholar 

  21. R.A. Robie, B.S. Hemingway, Thermodynamic properties of minerals and related substances at 298.15 K and 1 bar (105 pascals) pressure and at higher temperatures (U.S. Geological Survey Bulletin 2131. United States Government Printing Office, Washington, 1995).

  22. M.W. Chase, J. Phys. Chem. Ref. Data Monogr. 9, 1 (1998)

    Google Scholar 

  23. K.C. Mills, Recommended Values of Thermophysical Properties for Selected Commercial Alloys (Woodhead Publishing Ltd, Cambridge, 2002)

    Book  Google Scholar 

  24. D.R. Lide, The CRC Handbook of Chemistry and Physics, 90th edn (Internet Version, CRC Press/Taylor and Francis, Boca Raton, Florida, 2010)

  25. J.W. Arblaster, J. Phase Equilib. Diff. 36, 422 (2015)

    Article  Google Scholar 

  26. I.S. Grigoriev, E.Z. Meilikhov, Handbook of Physical Quantities (CRC Press, Boca Raton, 1996)

    Google Scholar 

  27. J.W. Ablaster, Selected Values of the Crystallographic Properties of the Elements (ASM International, Materials Park, 2018)

    Google Scholar 

  28. Y.S. Touloukian, R.K. Kirby, R.E. Taylor, P.D. Desai, Thermophysical Properties of Matter, vol. 12 (IFI/Plenum, New York, 1975)

  29. G.K. White, Int. J. Thermophys. 18, 1269 (1997)

    Article  ADS  Google Scholar 

  30. K. Wang, R.R. Reeber, High Temp. Mater. Sci. 35, 181 (1996)

    Google Scholar 

  31. S.N. Vaidya, G.C. Kennedy, J. Phys. Chem. Solids 31, 2329 (1970)

    Article  ADS  Google Scholar 

  32. L. Liu, M. Liu, H. Verbeek, C. Hoffner, G. Will, J. Phys. Chem. Solids 51, 435 (1990)

    Article  ADS  Google Scholar 

  33. J. Xu, H. Mao, P.M. Bell, High Temp. High Press. 16, 495 (1984)

    Google Scholar 

  34. A. Dewaele, P. Loubeyre, M. Mezouar, Phys. Rev. B 70, 094112 (2004)

    Article  ADS  Google Scholar 

  35. P.M. Bell, J. Xu, H.K. Mao, Static compression of gold and copper and calibration of the ruby pressure scale to pressures to 1.8 megabars (static.rno), in Shock Waves in Condensed Matter, ed. by Y.M. Gupta (Plenum Press, New York, 1985), p. 125

  36. D.E. Fratanduono, R.F. Smith, S.J. Ali, D.G. Braun, A. Fernandez-Pañella, S. Zhang, R.G. Kraus, F. Coppari, J.M. McNaney, M.C. Marshall, L.E. Kirch, D.C. Swift, M. Millot, J.K. Wicks, J.H. Eggert, Phys. Rev. Lett. 124, 015701 (2020)

    Article  ADS  Google Scholar 

  37. W.J. Carter, S.P. Marsh, J.N. Fritz, R.G. McQueen, The equation of state of selected materials for high-pressure references, in Accurate Characterization of the High Pressure Environment, ed. by E.C. Lloyd (N.B.S. Spec. Pub. 326, Washington, 1971), p. 147.

  38. L.V. Al’tshuler, S.E. Brusnikin, E.A. Kuz’menkov, J. Appl. Mech. Tech. Phys. 28, 129 (1987)

    Article  ADS  Google Scholar 

  39. D.E. Gray, American Institute of Physics Handbook (McGraw-Hill, New York, 1972)

    MATH  Google Scholar 

  40. D. Lazarus, Phys. Rev. 76, 545 (1949)

    Article  ADS  Google Scholar 

  41. L.S. Cain, J.F. Thomas, Phys. Rev. B. 4, 4245 (1971)

    Article  ADS  Google Scholar 

  42. K. Salama, G.A. Alers, Phys. Status Solidi A 41, 241 (1977)

    Article  ADS  Google Scholar 

  43. H. Ledbetter, S. Kim, C. Fortunko, P. Heyliger, Int. J. Thermophys. 17, 263 (1996)

    Article  ADS  Google Scholar 

  44. W.C. Overton, J. Gaffney, Phys. Rev. 98, 969 (1955)

    Article  ADS  Google Scholar 

  45. P. Van’t Klooster, N.J. Trappeniers, S.N. Biswas, Physica B&C 97, 65 (1979)

    Article  ADS  Google Scholar 

  46. H.M. Ledbetter, Phys. Status Solidi A 66, 477 (1981)

    Article  ADS  Google Scholar 

  47. Y.A. Chang, R. Hultgren, J. Phys. Chem. 69, 4162 (1965)

    Article  Google Scholar 

  48. Y.A. Chang, L. Himmel, J. Appl. Phys. 37, 3567 (1966)

    Article  ADS  Google Scholar 

  49. L.E. Kinsler, A.R. Fray, A.B. Coppens, J.V. Sanders, Fundamentals of Acoustics (Wiley, New York, 2000)

    Google Scholar 

  50. M.J. Assael, A.E. Kalyva, K.D. Antoniadis, R.M. Banish, I. Egry, J. Wu, E. Kaschnitz, W.A. Wakeham, J. Phys. Chem. Ref. Data 39, 033105 (2010)

    Article  ADS  Google Scholar 

  51. T. Iida, R.I.L. Guthrie, The Thermophysical Properties of Metallic Liquids, vols. 1, 2 (Oxford University Press, New York, 2015).

  52. J.A. Cahill, A.D. Kirshenbaum, J. Phys. Chem. 66, 1080 (1962)

    Article  Google Scholar 

  53. M.G. Frohberg, R. Weber, Arch. Eisenhuttenwes. 35, 877 (1964)

    Google Scholar 

  54. L.D. Lucas, Mem. Sci. Rev. Metall. 61, 1 (1964)

    Article  Google Scholar 

  55. S. Watanabe, T. Saito, J. Jpn. Inst. Met. 35, 554 (1971)

    Article  Google Scholar 

  56. Y.N. Ivashchenko, G.P. Khil, Prib. Tekh. Eksp. 6, 208 (1972). [in Russian]

    Google Scholar 

  57. N.A. Bykova, V.G. Shevchenko, Physicochemical Studies on Liquid Metals and Alloys (Ural Branch of the AS USSR, Sverdlovsk City, 1974), p. 42 [in Russian

  58. B.I. Yavoisky, A.A. Ezhov, V.F. Kravchenko, Izv. Acad. Sci. USSR Met. 4, 61 (1974). [in Russian]

    Google Scholar 

  59. C.O. Ruud, M.T. Hepworth, J.M. Fernandez, Metall Trans. B 6, 351 (1975)

    Article  Google Scholar 

  60. M. Bian, J. Wang, Acta Metall. Sin. 22, 134 (1986)

    Google Scholar 

  61. V.V. Makeev, E.L. Demina, P.S. Popel, E.L. Arkhangelskiy, High Temp. 27, 701 (1989)

    Google Scholar 

  62. K. Nogi, K. Oishi, K. Ogino, Mater. Trans. 30, 137 (1989)

    Article  Google Scholar 

  63. S.V. Stankus, P.V. Tyagelsky, Russ. J. Eng. Thermophys. 2, 93 (1992)

    Google Scholar 

  64. J. Brillo, I. Egry, Int. J. Mater. Res. 95, 691 (2004)

    Article  Google Scholar 

  65. P.S. Popel, V.E. Sidorov, D.A. Yagodin, G.M. Sivkov, A.G. Mozgovoj, in 17th European Conference on Thermophysical Properties, Bratislava, 2005, ed. by L. Vozár, I. Medved’, L’. Kubičár (University of Pau, Slovakia. 2005), p. 242

  66. J. Brillo, I. Egry, J. Westphal, Int. J. Mater. Res. 99, 162 (2008)

    Article  Google Scholar 

  67. S.I. Bakhtiyarov, D.A. Siginer, Fluid Dyn. Mater. Proc. 4, 163 (2008)

    Google Scholar 

  68. A.R. Kurochkin, P.S. Popel’, D.A. Yagodin, A.V. Borisenko, A.V. Okhapkin, High Temp. 51, 197 (2013)

    Article  Google Scholar 

  69. Y.C. Cho, B. Kim, H. Yoo, J.Y. Kim, S. Lee, Y.H. Lee, G.W. Lee, S.Y. Jeong, CrystEngComm. 16, 7575 (2014)

    Article  Google Scholar 

  70. I.I. Korobeinikov, D. Chebykin, X. Yu, S. Seetharaman, O. Volkova, Arch. Mater. Sci. Eng. 91, 28 (2018)

    Article  Google Scholar 

  71. M. Watanabe, M. Adachi, H. Fukuyama, J. Chem. Thermodyn. 130, 9 (2019)

    Article  Google Scholar 

  72. S. Blairs, Int. Mater. Rev. 52, 321 (2007)

    Article  Google Scholar 

  73. M.B. Gitis, I.G. Mikhailov, Sov. Phys. Acoust. 11, 372 (1966)

    Google Scholar 

  74. S.I. Filippov, N.B. Kazakov, L.A. Pronin, Izv. Vyssh. Uchebn. Zaved. Chern. Met. 9, 8 (1966)

    Google Scholar 

  75. R.T. Smith, G.M.B. Webber, F.R. Young, R.W.B. Stephens, Adv. Phys. 16, 515 (1967)

    Article  ADS  Google Scholar 

  76. R. Turner, D. Crozier, J.F. Cochran, Can. J. Phys. 50, 2735 (1972)

    Article  ADS  Google Scholar 

  77. O.N. Kallistratov, C.I. Filippov, Izv. Vyssh. Uchebn. Zaved. Chern. Met. 1, 5 (1972)

    Google Scholar 

  78. R. Turner, E.D. Crozier, J.F. Cochran, J. Phys. C Solid State Physics. 6, 3359 (1973)

    Article  ADS  Google Scholar 

  79. V.I. Stremousov, V.V. Tekuchev, Russ. J. Phys. Chem. 51, 206 (1977)

    Google Scholar 

  80. Y. Tsu, H. Suenaga, K. Takano, Y. Shiraishi, Trans. Jpn. Inst. Met. 23, 1 (1982)

    Article  Google Scholar 

  81. P.W. Mirwald, G.C. Kennedy, J. Geophys. Res. 84, 6750 (1979)

    Article  ADS  Google Scholar 

  82. J. Akella, G.C. Kennedy, J. Geophys. Res. 76, 4969 (1971)

    Article  ADS  Google Scholar 

  83. N.R. Mitra, D.L. Decker, H.B. Vanfleet, Phys. Rev. 161, 613 (1967)

    Article  ADS  Google Scholar 

  84. H. Brand, D.P. Dobson, L. Vocadlo, I.G. Wood, High Press. Res. 26, 185 (2006)

    Article  ADS  Google Scholar 

  85. D. Errandonea, Phys. Rev. B. 87, 054108 (2013)

    Article  ADS  Google Scholar 

  86. S. Japel, B. Schwager, R. Boehler, M. Ross, Phys. Rev. Lett. 95, 167801 (2005)

    Article  ADS  Google Scholar 

  87. F. Simon, G. Glatzel, Z. Anorg, Allg. Chem. 178, 309 (1929)

    Article  Google Scholar 

  88. Q. An, S. Luo, L. Han, L. Zheng, O. Tschauner, J. Phys. Condens. Matter 20, 095220 (2008)

    Article  ADS  Google Scholar 

  89. Z. Liu, X. Zhang, L. Cai, J. Chem. Phys. 143, 114101 (2015)

    Article  ADS  Google Scholar 

  90. S.R. Baty, L. Burakovsky, D. Errandonea, Crystals 11, 537 (2021)

    Article  Google Scholar 

  91. C. Pu, X. Yang, D. Xiao, J. Cheng, Mater. Today Commun. 26, 101990 (2021)

    Article  Google Scholar 

  92. J.A. Nelder, R. Mead, Comput. J. 7, 308 (1965)

    Article  Google Scholar 

  93. D.A. Young, H. Cynn, P. Söderlind, A. Landa, J. Phys. Chem. Ref. Data. 45, 043101 (2016)

    Article  ADS  Google Scholar 

  94. W.B. Holzapfel, M.F. Nicol, High Press. Res. 27, 377 (2007)

    Article  ADS  Google Scholar 

  95. P.I. Dorogokupets, T.S. Sokolova, B.S. Danilov, K.D. Litasov, Geodyn. Tectonophys. 3, 129 (2012)

    Article  Google Scholar 

Download references

Acknowledgements

This study was performed under the auspices of the Ministry of Science and Higher Education of the Russian Federation (contract No. FUFE-2021-0005 (0308-2021-0005)).

Author information

Authors and Affiliations

  1. Laboratory for Physicochemical Fundamentals of Energetic Condensed Systems, Institute for Problems of Chemical and Energetic Technologies, Siberian Branch of the Russian Academy of Sciences (IPCET SB RAS), Biysk, Altai Krai, Russia, 659322

    Nikolay V. Kozyrev

Authors
  1. Nikolay V. Kozyrev
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The author has contributed solely to the manuscript.

Corresponding author

Correspondence to Nikolay V. Kozyrev.

Ethics declarations

Conflict of interest

The author declares no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kozyrev, N.V. Thermodynamic Properties and Equation of State for Solid and Liquid Copper. Int J Thermophys 44, 31 (2023). https://doi.org/10.1007/s10765-022-03136-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10765-022-03136-4

Keywords

Search

Navigation