Skip to main content

Caloric and isothermal equations of state of solids: empirical modeling with multiply broken power-law densities

Abstract

Empirical equations of state (EoSs) are developed for solids, applicable over extended temperature and pressure ranges. The EoSs are modeled as multiply broken power laws, in closed form without the use of ascending series expansions; their general analytic structure is explained and specific examples are studied. The caloric EoS is put to test with two carbon allotropes, diamond and graphite, as well as vitreous silica. To this end, least-squares fits of broken power-law densities are performed to heat capacity data covering several logarithmic decades in temperature, the high- and low-temperature regimes and especially the intermediate temperature range where the Debye theory is of limited accuracy. The analytic fits of the heat capacities are then temperature integrated to obtain the entropy and caloric EoS, i.e. the internal energy. Multiply broken power laws are also employed to model the isothermal EoSs of metals (Al, Cu, Mo, Ta, Au, W, Pt) at ambient temperature, over a pressure range up to several hundred GPa. In the case of copper, the empirical pressure range is extended into the TPa interval with data points from DFT calculations. For each metal, the parameters defining the isothermal EoS (i.e. the density–pressure relation) are inferred by nonlinear regression. The analytic pressure dependence of the compression modulus of each metal is obtained as well, over the full data range.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

References

  1. W.B. Holzapfel, High Press. Res. 16, 81 (1998)

    ADS  Google Scholar 

  2. W.B. Holzapfel, Z. Kristallogr. 216, 473 (2001)

    Google Scholar 

  3. J.S. Tse, W.B. Holzapfel, J. Appl. Phys. 104, 043525 (2008)

    ADS  Google Scholar 

  4. J. Hama, K. Suito, J. Phys. Condens. Matter 8, 67 (1996)

    ADS  Google Scholar 

  5. F.D. Stacey, Rep. Prog. Phys. 68, 341 (2005)

    ADS  Google Scholar 

  6. R. Tomaschitz, Physica A (2020). https://doi.org/10.1016/j.physa.2019.123188

    Article  Google Scholar 

  7. R. Tomaschitz, Physica A 483, 438 (2017)

    ADS  MathSciNet  Google Scholar 

  8. R. Tomaschitz, Fluid Phase Equilib. 496, 80 (2019)

    Google Scholar 

  9. K.V. Khishchenko, J. Phys: Conf. Ser. 946, 012082 (2018)

    Google Scholar 

  10. K.V. Khishchenko, J. Phys. Conf. Ser. 1147, 012001 (2019)

    Google Scholar 

  11. K.V. Khishchenko, Tech. Phys. Lett. 30, 829 (2004)

    ADS  Google Scholar 

  12. D.V. Minakov, P.R. Levashov, K.V. Khishchenko, AIP Conf. Proc. 1426, 836 (2012)

    ADS  Google Scholar 

  13. D.V. Minakov, P.R. Levashov, K.V. Khishchenko, V.E. Fortov, J. Appl. Phys. 115, 223512 (2014)

    ADS  Google Scholar 

  14. M.A. Kadatskiy, K.V. Khishchenko, J. Phys: Conf. Ser. 653, 012079 (2015)

    Google Scholar 

  15. M.A. Kadatskiy, K.V. Khishchenko, J. Phys. Conf. Ser. 774, 012005 (2016)

    Google Scholar 

  16. M.A. Kadatskiy, K.V. Khishchenko, Phys. Plasmas 25, 112701 (2018)

    ADS  Google Scholar 

  17. K.V. Khishchenko, J. Phys. Conf. Ser. 121, 022025 (2008)

    Google Scholar 

  18. K.V. Khishchenko, J. Phys. Conf. Ser. 653, 012081 (2015)

    Google Scholar 

  19. J.R. Macdonald, Rev. Mod. Phys. 38, 669 (1966)

    ADS  Google Scholar 

  20. B.G. Yalcin, Appl. Phys. A 122, 456 (2016)

    ADS  Google Scholar 

  21. S. Khatta, S.K. Tripathi, S. Prakash, Appl. Phys. A 123, 582 (2017)

    ADS  Google Scholar 

  22. M. Kaddes, K. Omri, N. Kouaydi, M. Zemzemi, Appl. Phys. A 124, 518 (2018)

    ADS  Google Scholar 

  23. W. Ouerghui, M.S. Alkhalifah, Appl. Phys. A 125, 374 (2019)

    ADS  Google Scholar 

  24. A. Laroussi, M. Berber, B. Doumi, A. Mokaddem, H. Abid, A. Boudali, H. Bahloul, H. Moujri, Appl. Phys. A 125, 676 (2019)

    ADS  Google Scholar 

  25. A.D. Chijioke, W.J. Nellis, I.F. Silvera, J. Appl. Phys. 98, 073526 (2005)

    ADS  Google Scholar 

  26. R.G. Kraus, J.-P. Davis, C.T. Seagle, D.E. Fratanduono, D.C. Swift, J.L. Brown, J.H. Eggert, Phys. Rev. B 93, 134105 (2016)

    ADS  Google Scholar 

  27. Y. Wang, R. Ahuja, B. Johansson, J. Appl. Phys. 92, 6616 (2002)

    ADS  Google Scholar 

  28. C.W. Greeff, J.C. Boettger, M.J. Graf, J.D. Johnson, J. Phys. Chem. Solids 67, 2033 (2006)

    ADS  Google Scholar 

  29. L.E. Fried, W.M. Howard, Phys. Rev. B 61, 8734 (2000)

    ADS  Google Scholar 

  30. K.V. Khishchenko, V.E. Fortov, I.V. Lomonosov, M.N. Pavlovskii, G.V. Simakov, M.V. Zhernokletov, AIP Conf. Proc. 620, 759 (2002)

    ADS  Google Scholar 

  31. K.V. Khishchenko, V.E. Fortov, I.V. Lomonosov, Int. J. Thermophys. 26, 479 (2005)

    ADS  Google Scholar 

  32. S.Sh. Rekhviashvili, Kh.L. Kunizhev, High Temp. 55, 312 (2017)

    Google Scholar 

  33. J.E. Desnoyers, J.A. Morrison, Philos. Mag. 3, 42 (1958)

    ADS  Google Scholar 

  34. W. DeSorbo, J. Chem. Phys. 21, 876 (1953)

    ADS  Google Scholar 

  35. A.C. Victor, J. Chem. Phys. 36, 1903 (1962)

    ADS  Google Scholar 

  36. B.J.C. van der Hoeven, P.H. Keesom, Phys. Rev. 130, 1318 (1963)

    ADS  Google Scholar 

  37. W. DeSorbo, G.E. Nichols, J. Phys. Chem. Solids 6, 352 (1958)

    ADS  Google Scholar 

  38. W. DeSorbo, W.W. Tyler, J. Chem. Phys. 21, 1660 (1953)

    ADS  Google Scholar 

  39. M.W. Chase, NIST-JANAF Thermochemical Tables, 4th ed. (AIP, Woodbury, 1998), https://janaf.nist.gov

  40. A.T.D. Butland, R.J. Maddison, J. Nucl. Mater. 49, 45 (1973)

    ADS  Google Scholar 

  41. T. Nihira, T. Iwata, Phys. Rev. B 68, 134305 (2003)

    ADS  Google Scholar 

  42. V.N. Senchenko, R.S. Belikov, J. Phys: Conf. Ser. 891, 012338 (2017)

    Google Scholar 

  43. J.C. Lasjaunias, A. Ravex, M. Vandorpe, S. Hunklinger, Solid State Commun. 17, 1045 (1975)

    ADS  Google Scholar 

  44. R.O. Pohl, in: Amorphous Solids, W.A. Phillips, ed. (Springer, Berlin, 1981)

  45. R.B. Stephens, Phys. Rev. B 8, 2896 (1973)

    ADS  Google Scholar 

  46. P. Flubacher, A.J. Leadbetter, J.A. Morrison, B.P. Stoicheff, J. Phys. Chem. Solids 12, 53 (1959)

    ADS  Google Scholar 

  47. R.C. Lord, J.C. Morrow, J. Chem. Phys. 26, 230 (1957)

    ADS  Google Scholar 

  48. P.W. Anderson, B.I. Halperin, C.M. Varma, Philos. Mag. 25, 1 (1972)

    ADS  Google Scholar 

  49. W.A. Phillips, Rep. Prog. Phys. 50, 1657 (1987)

    ADS  Google Scholar 

  50. I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, 8th edn. (Academic Press, Waltham, 2015)

    MATH  Google Scholar 

  51. W.B. Holzapfel, Rep. Prog. Phys. 59, 29 (1996)

    ADS  Google Scholar 

  52. W.B. Holzapfel, High Press. Res. 22, 209 (2002)

    ADS  Google Scholar 

  53. G.M. Amulele, M.H. Manghnani, S. Marriappan, X. Hong, F. Li, X. Qin, H.P. Liermann, J. Appl. Phys. 103, 113522 (2008)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  55. W.B. Holzapfel, High Press. Res. 30, 372 (2010)

    ADS  Google Scholar 

  56. K. Katahara, M. Manghnani, E. Fisher, J. Appl. Phys. 47, 434 (1976)

    ADS  Google Scholar 

  57. K.W. Katahara, M.H. Manghnani, E.S. Fisher, J. Phys. F: Met. Phys. 9, 773 (1979)

    ADS  Google Scholar 

  58. P. van’t-Klooster, N.J. Trappeniers, S.N. Biswas, Physica B + C 97, 65 (1979)

  59. S.N. Biswas, P. van’t-Klooster, N.J. Trappeniers, Physica B + C 103, 235 (1981)

  60. J.L. Tallon, A. Wolfenden, J. Phys. Chem. Solids 40, 831 (1979)

    ADS  Google Scholar 

  61. D. Steinberg, J. Phys. Chem. Solids 43, 1173 (1982)

    ADS  Google Scholar 

  62. W. Holzapfel, M. Hartwig, W. Sievers, J. Phys. Chem. Ref. Data 30, 515 (2001)

    ADS  Google Scholar 

  63. K. Syassen, W.B. Holzapfel, J. Appl. Phys. 49, 4427 (1978)

    ADS  Google Scholar 

  64. K. Takemura, A. Dewaele, Phys. Rev. B 78, 104119 (2008)

    ADS  Google Scholar 

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

    ADS  Google Scholar 

  66. E.E. Salpeter, Astrophys. J. 134, 669 (1961)

    ADS  MathSciNet  Google Scholar 

  67. F.D. Stacey, Geophys. J. Int. 143, 621 (2000)

    ADS  Google Scholar 

  68. F.D. Stacey, P.M. Davis, Phys. Earth Planet. Inter. 142, 137 (2004)

    ADS  Google Scholar 

  69. F.D. Stacey, J.H. Hodgkinson, Phys. Earth Planet. Inter. 286, 42 (2019)

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Roman Tomaschitz.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tomaschitz, R. Caloric and isothermal equations of state of solids: empirical modeling with multiply broken power-law densities. Appl. Phys. A 126, 102 (2020). https://doi.org/10.1007/s00339-019-3256-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00339-019-3256-7

Keywords

  • Multi-parameter equation of state (EoS)
  • Caloric EoS of carbon allotropes
  • Specific heat of vitreous silica
  • Thermal EoS and compression modulus of metals
  • High-pressure regime
  • Multiply broken power laws