Skip to main content
SpringerLink
Log in

Equation of state model for metals with ionization effectively taken into account. Equation of state of tantalum, tungsten, aluminum, and beryllium

  • Published:
Combustion, Explosion, and Shock Waves Aims and scope

Abstract

A model of a wide-range semi-empirical equation of state for metals is presented. The specific heat and Grüneisen coefficients of ions and electrons are functions of temperature and density. At low temperatures, the heat capacity varies according to Debye theory. The removal of the degeneration of the electron gas with increasing temperature is taken into account. The effect of ionization on the thermodynamic functions is effectively taken into account. The equation of state allows the calculation of states in a two-phase liquid-vapor region. This model was used to develop the equations of state for Ta, W, Al, and Be. For its range of applicability, the equation of state contains a relatively small number of free parameters, most of which have a physical meaning. Comparison of calculations of various isolines using equations of state with experimental data and calculations based on other models show that the equations of state for Ta, W, Al, and Be, describe most experimental data for these substances. At ultrahigh pressures and temperatures, calculations using the equations of state are in good agreement with calculations using the Thomas-Fermi model with corrections.

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

Similar content being viewed by others

References

  1. L. V. Al’tshuler, A. V. Buchman, M. V. Zhernokletov, V. N. Zubarev, A. A. Leont’ev, and V. E. Fortov, “Unloading Isentropes and the Equation of State of Metals at High Energy Densities,” Zh. Eksp. Teor. Fiz. 78, 741–760 (1980).

    Google Scholar 

  2. A. V. Bushman, I. V. Lomonosov, and V. E. Fortov, Equation of State of Metals at High Energy Densities (Chernogolovka, 1992) [in Russian].

  3. I. V. Lomonosov, “Phase Diagrams and Thermodynamic Properties of Metals at High Pressures and Temperatures,” Doct. Dissertation in Phys. and Math. (Chernogolovka, 1999).

  4. M. M. Basko, “Equation of State of Metals in the Mean-Ion Approximation,” Teplofiz. Vysok. Temp. 23(3), 483–491 (1985).

    Google Scholar 

  5. R. M. More et al., “A New Quotidian Equation of State (QEOS) for Hot Dense Matter,” Phys. Fluids 31(10), 3059–3078 (1988).

    Article  ADS  MATH  Google Scholar 

  6. L. V. Al’tshuler and S. E. Brusnikin, “Simulation of High-Energy Processes and Wide-Range Equations of State,” in Problems of Atomic Science and Technology. Mathematical Modeling of Physical Processes, No. 1 (1992), pp. 34–42.

  7. V. P. Kopyshev and A. B. Medvedev, Thermodynamic Model of Compressible Covolume (Sarov, VNIIEF, 1995) [in Russian].

  8. A. B. Medvedev, “Modification of the Van der Waals Model for Dense States,” in Shock Waves and Extreme States of Matter, Ed. by E. V. Fortov et al. (Nauka, Moscow, 2000), pp. 315–341 [in Russian].

    Google Scholar 

  9. V. V. Prut, “A Semi-Empirical Equation of State of Condensed Matter,” Teplofiz. Vysok. Temp. 43(5), 713–726 (2005).

    Google Scholar 

  10. K. V. Khishchenko and O. P. Shemyakin, “Semi-Empirical Equation of State of Aluminum Based on the Thomas-Fermi Model,” in Physics of Extreme States of Matter-2006, Ed. by E. V. Fortov et al. (IPCP RAS, Chernogolovka, 2006).

    Google Scholar 

  11. O. P. Shemyakin, P. R. Levashov, and K. V. Khishchenko, “Equation of State of Al Based on the Thomas Fermi Model,” in Physics of Extreme States of Matter 2011, Ed. by Fortov et al. (IPCP RAS, Chernogolovka, 2011).

    Google Scholar 

  12. R. Feynman, N. Metropolis, and E. Teller, “Equations of State of Elements Based on the Generalized Fermi-Thomas Theory,” Phys. Rev. 75(10), 1561–1573 (1949).

    Article  ADS  MATH  Google Scholar 

  13. S. B. Kormer, A. I. Funtikov, V. D. Urlin, and A. N. Kolesnikova, “Dynamic Compression of Porous Metals and the Equation of State with a Variable Specific Heat at High Temperatures,” Zh. Eksp. Teor. Fiz. 42(3), 686–702 (1962).

    Google Scholar 

  14. D. G. Gordeev, L. F. Gudarenko, M. V. Zhernokletov, V. G. Kudel’kin, and M. A. Mochalov, “Semi-Empirical Equation of State of Metals. Equation of State of Aluminum,” Fiz. Goreniya Vzryva 44(2), 61–75 (2008) [Combust., Expl., Shock Waves 44 (2), 177–189 (2008)].

    Google Scholar 

  15. Yu. S. Zav’yalov, B. I. Kvasov, and B. K. Miroshnichenko, Methods of Spline Functions (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  16. G. I. Kerley, “User’s Manual for PANDA: A Computer Code for Calculating Equations of State,” Los Alamos Nat. Lab. Report No. LA-8833-M (November, 1981).

  17. N. N. Kalitkin and L. V. Kuz’mina, “Tables of Thermodynamic Functions of Matter at High Energy Density,” Preprint No. 35 (Inst. Appl. Mech., USSR Acad. of Sci., Moscow, 1975).

    Google Scholar 

  18. D. G. Gordeev and A. I. Lomaikin, “Approximations of the Debye Function D 3(x) in the Range 0 ⩽ x < ∞ by a Function of Class C 2,” in Problems of Atomic Science and Technology. Mathematical Modeling of Physical Processes (2008), Vol. 1, pp 42–50.

    Google Scholar 

  19. Hitose Nagara and Tuto Nakamura, “Theory of Lattice-Dynamical Properties of Compressed Solids,” Phys. Rev. B 31(4), 1844–1855 (1985).

    Article  ADS  Google Scholar 

  20. J. P. Hansen, “Statistical Mechanics of Dense Ionized Matter. I. Equilibrium Properties of the Classical One-Component Plasma,” Phys. Rev. A 8(6), 3096–3109 (1973).

    Article  ADS  Google Scholar 

  21. E. L. Pollock and J. P. Hansen, “Statistical Mechanics of Dense Ionized Matter. II. Equilibrium Properties and Melting Transition of the Crystallized One-Component Plasma,” Phys. Rev. A 8(6), 3110–3122 (1973).

    Article  ADS  Google Scholar 

  22. V. P. Kopyshev, “On the Thermodynamics of Nuclei of Monatomic Substance,” Preprint No. 59 (Inst. Appl. Mech., USSR Acad. of Sci., Moscow, 1978).

    Google Scholar 

  23. R. Grover, “Liquid Metal Equation of State Based on Scaling,” Chem. Phys. 55(7), 3435–3441 (1971).

    ADS  Google Scholar 

  24. B. Zel’dovich and Yu. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Nauka, Moscow, 1966) [in Russian].

    Google Scholar 

  25. H. Cynn and C. S. Yoo, “Equation of State of Tantalum to 174 GPa,” Phys. Rev. B 59(13), 8526–8529 (1999).

    Article  ADS  Google Scholar 

  26. J. H. Eggert, M. Bastea, D. Braun, D. Fujino, R. Rygg, R. Smith, J. Hawreliak, D. G. Hicks, and G. Collins, “Laser-Induced Ramp Compression of Tantalum and Iron to over 300 GPa: EOS and X-ray Diffraction,” Lawrence Livermore Nat. Lab., LLNL-CONF-425256 (March 9, 2010).

  27. J. Xu, H.-K. Mao, and P. M. Bell, “Position-Sensitive X-ray Diffraction: Hydrostatic Compressibility of Argon, Tantalum, and Cooper to 769 kbar,” High Temp. High Pressures 16, 495–499 (1984).

    Google Scholar 

  28. A. Dewaele, P. Loubeyre, and M. Mezouar, “Equations of State of Six Metals above 94 GPa,” Phys. Rev. B 70,094112 (1–8) (2004).

  29. Y. Akahama, M. Nishimura, K. Kinoshita, H. Kawamura, and Y. Ohishi, “Evidence of a FCC-HCP Transition in Aluminum at Multimegabar Pressure,” Phys. Rev. Lett. 96, 045505 (2006).

    Article  ADS  Google Scholar 

  30. W. J. Evans, M. J. Lipp, H. Cynn, C. S. Yoo, M. Somayazulu, D. Häusermann, G. Shen, and V. Prakapenka,, “X-ray Diffraction and Raman Studies of Beryllium: Static and Elastic Properties at High Pressures,” Phys. Rev. B 72,094113 (1–6) (2005).

  31. Velisavljevic, G. N. Chestnut, Y. K. Vohra, S. T. Weir, V. Malba, and J. Akella, “Structural and Electrical Properties of Beryllium Metal to 66 GPa Studied Using Designer Diamond Anvils,” Phys. Rev. B 65,172107 (1–4) (2002).

  32. A. L. Ruoff, H. Xia, H. Luo, Y. K. Vohra, “Miniaturization Techniques for Obtaining Static Pressures Comparable to the Pressure at the Center of the Earth: X-ray Diffraction at 416 GPa,” Rev. Sci. Instrum. 61(12), 3830–3833 (1990).

    Article  ADS  Google Scholar 

  33. Yi. Wang, D. Chen, and X. Zhang, “Calculated Equation of State of Al, Cu, Ta, Mo, and W to 1000 GPa,” Phys. Rev. Lett. 84(15), 3220–3223 (2000).

    Article  ADS  Google Scholar 

  34. M. Foata-Prestavoine, G. Robert, and M. H. Nadal, “First-Principles Study of the Relations between the Elastic Constants, Phonon Dispersion Curves, and Melting Temperatures of BCC Ta at Pressures up to 1000 GPa,” Phys. Rev. B 76,104104 (1–10) (2007).

  35. LASL Shock Hugoniot Data, Ed. by S. P. Marsh (Univ. of California Press, Berkley, 1980).

    Google Scholar 

  36. R. S. Hixson and J. N. Fritz, “Shock Compression of Tungsten and Molybdenum,” J. Appl. Phys. 71(4), 1721–1728 (1992).

    Article  ADS  Google Scholar 

  37. A. C. Mitchell and W. J. Nellis, “Shock Compression of Aluminum, Copper, and Tantalum,” J. Appl. Phys. 52(5), 3363–3374 (1981).

    Article  ADS  Google Scholar 

  38. W. J. Nellis, A. C. Mitchell, and D. A. Young, “Equation-of-State Measurements for Aluminum, Copper, and Tantalum in the Pressure Range 80–440 GPa (0.8–4.4 Mbar),” J. Appl. Phys. 93(1), 304–310 (2003).

    Article  ADS  Google Scholar 

  39. N. C. Holmes, J. A. Moriarty, G. R. Gathers, and W. J. Nellis, “The Equation of State of Platinum to 660 GPa (6.6 Mbar),” J. Appl. Phys. 66(7), 2962–2967 (1989).

    Article  ADS  Google Scholar 

  40. R. F. Trunin, L. F. Gudarenko, M. V. Zhernokletov, and G. V. Simakov, Experimental Data on Shock Compression and Adiabatic Expansion of Condensed Matter, Ed. by R. F. Trunin (Sarov, VNIIEF, 2006) [in Russian].

    Google Scholar 

  41. R. R. Boade, “Dynamic Compression of Porous Tungsten,” J. Appl. Phys. 40(9), 3781–3785 (1969).

    Article  ADS  Google Scholar 

  42. W. J. Nellis, J. A. Moriarty, A. C. Michell, and N. C. Holmes, “Equation of State of Beryllium at Shock Pressure of 0.4-1.1 TPa (4-11 Mbar),” J. Appl. Phys. 82, 2225–2227 (1997).

    Article  ADS  Google Scholar 

  43. R. Cauble, T. S. Perry, D. R. Bach, K. S. Budil, B. A. Hammel, G. W. Collins, D. M. Gold, J. Dunn, P. Celiers, L. B. Da Silva, M. E. Foord, R. J. Wallace, R. E. Stewart, and N. C. Woosley, “Absolute Equationof-State Data in the 10–40 Mbar (1–4 TPa) Regime,” Phys. Rev. Lett. 80(6), 1248–1251 (1998).

    Article  ADS  Google Scholar 

  44. M. D. Knudson, J. R. Asay, and C. Deeney, “Adiabatic Release Measurements in Aluminum from 240 to 500 GPa States on the Principal Hugoniot,” J. Appl. Phys. 97,073514 (1–14) (2005).

  45. C. E. Ragan III, “Shock-Wave Experiments at Threefold Compression,” Phys. Rev. A 29(3), 1391–1402 (1984).

    Article  ADS  Google Scholar 

  46. M. D. Knudson, R.W. Lemke, D. B. Hayes, C. A. Hall, C. Deeney, and J. R. Asay, “Near-Absolute Hugoniot Measurements in Aluminum to 500 GPa Using a Magnetically Accelerated Flyer Technique,” J. Appl. Phys. 94(7), 4420–4431 (2003).

    Article  ADS  Google Scholar 

  47. A. C. Mitchell, W. J. Nellis, J. A. Moriatry, R. A. Heinle, N.C. Holmes, R. E. Tipton, and G. W. Repp, “Equation of State of Al, Cu, Mo, and Pb at Shock Pressures up to 2.4. TPa (24 Mbar),” J. Appl. Phys. 69(5), 2981–2986 (1991).

    Article  ADS  Google Scholar 

  48. C. E. Ragan III, “Shock Compression Measurements at 1 to 7 TPa,” Phys. Rev. A 25(6), 3360–3375 (1982).

    Article  ADS  Google Scholar 

  49. J. R. Wise, L. C. Chhabildas, and J. L. Asay, “Shock Compression of Beryllium,” in Shock Waves in Condensed Matter 1981, Ed. by W. J. Nellis, L. Seaman, and R. A. Graham (Amer. Inst. of Physics, 1982), pp. 417–421.

  50. A. I. Voropinov, L. A. Il’kaeva, M. A. Podurets, G. V. Simakov, and R. F. Trunin, “Hugoniots of Porous Aluminum, Titanium, Copper and Tungsten, and Poisson Adiabats of Porous Copper and Tungsten in the Region of Incomplete Closing of Pores: Thermodynamic Model and Experiment,” in Problems of Atomic Science and Technology. Mathematical Modeling of Physical Processes, Issue Nos. 1–2 (2005), pp. 45–50.

  51. J. R. Asay and D. B. Hayes, “Shock-Compression and Release Behavior near Melt States in Aluminum,” J. Appl. Phys. 46(11), 4789–4799 (1975).

    Article  ADS  Google Scholar 

  52. Qiang Wu and Fuqian Jing. “Thermodynamic Equation of State and Application to Hugoniot Predictions for Porous Substances,” J. Appl. Phys. 80(8), 4343–4349 (1996).

    Article  ADS  Google Scholar 

  53. S. Y. Savrasov, “Linear-Response Theory and Lattice Dynamics: A Muffin-Tin-Orbital Approach,” Phys. Rev. B 54, 16470 (1996).

    Article  ADS  Google Scholar 

  54. R. W. Ohse and H. Tippelskirch, “The Critical Constants of the Elements and of Some Refractory Substances with High Critical Temperatures (A Review),” High Temp. High Pressures 9, 367–385 (1977).

    Google Scholar 

  55. V. E. Fortov, A. N. Dremin, A. A. Leont’ev, “Estimation of the Parameters of the Critical Point,” Teplofiz. Vysok. Temp. 13(5), 1072–1080 (1975).

    Google Scholar 

  56. G. Faussurier, C. Blancard, and P. L. Silvestrelli, “Evaluation of Aluminum Critical Point Using an ab initio Variational Approach,” Phys. Rev. B 79, 134202 (2009).

    Article  ADS  Google Scholar 

  57. G. R. Gathers, “Dynamic Methods for Investigating Thermophysical Properties of Matter at Very High Temperatures and Pressures,” Rep. Progr. Phys. 49, 341–396 (1986).

    Article  ADS  Google Scholar 

  58. A. A. Likalter, “Equation of State of Metallic Fluids near the Critical Point of Phase Transition,” Phys. Rev. Lett. 53, 4386 (1996).

    ADS  Google Scholar 

  59. U. Fucke and W. Seydel, “Improved Experimental Determination of Critical Point Data for Tungsten,” High Temp. High Pressures 12(4), 419–432 (1980).

    Google Scholar 

  60. V. Ternovoi et al., “Liquid-Vapor Phase Boundaries Determination by Dynamic Experimental Method,” Bull. Amer. Phys. Soc. 44(2), 95 (1999).

    Google Scholar 

  61. D. A. Young and B. J. Alder, “Critical Point of Metals from the van der Waals Model,” Phys. Rev., Ser. A 3(1), 364–371 (1971).

    Article  ADS  Google Scholar 

  62. A. D. Rakhel, A. Kloss, and H. Hess, “On the Critical Point of Tungsten,” Int. J. Thermophys 23(5), 1369 (2002).

    Article  Google Scholar 

  63. J. M. Brown and J. W. Shaner, “Rarefaction Velocities in Shocked Tantalum and the High-Pressure Melting Point,” in Shock Waves in Condensed Matter 1983, Ed. by J. R. Asay et al. (North-Holland, Amsterdam, 1984), p. 91.

    Google Scholar 

  64. L.-K. Cai, Z.-Y. Zeng, X.-L. Zhang, and J.-B. Hu, “Experimental Research on High Pressure Phase Transitions of Mo and Ta,” in 8th Int. Conf. New Models and Hydrocodes for Shock Wave Processes in Condensed Matter, Paris, May 24–28, 2010.

  65. L. Burakovsky, S. P. Chen, D. L. Preston, A. B. Belonoshko, A. Rosengren, A. S. Mikhaylushkin, S. I. Simak, and J. A. Moriarty, “High-Pressure-High-Temperature Polymorphism in Ta: Resolving an Ongoing Experimental Controversy,” Phys. Rev. Lett. 104,255702 (1–4) (2010).

  66. T. S. Duffy and T. J. Ahrens, “Sound Velocities at High Pressure and Temperature and their Geophysical Implications,” J. Geophys. Res. 97(84), 4503–4520 (1992).

    Article  ADS  Google Scholar 

  67. Xianwen Ran, Yuying Yu, Hua Tan, and Wenhui Tang. “Behavior of Aluminum Shear Modulus in Solid-Liquid Mixed Phase: Estimation with Percolation Theory,” J. Appl. Phys. 103,103539 (1–5) (2008).

  68. V. V. Dremov, A. V. Karavaev, F. A. Sapozhnikov, M. A. Vorobyova, and L. Soulard, “Molecular Dynamics Simulation of Thermodynamic and Mechanical Properties of Be (Pt II),” in Shock Compression of Condensed Matter-2009, Ed. by M. L. Elert et al. (Amer. Inst. of Physics, 2009) pp. 837–840.

  69. L. C. Chhabildas, J. L. Wise, and J. R. Asay, “Reshock and Release Behavior of Beryllium,” in Shock Waves in Condensed Matter 1981, Ed. by W. J. Nellis et al. (Amer. Inst. of Physics, 1982), pp. 422–426.

  70. D. Errandonea, B. Schwager, R. Ditz, C. Gessmann, R. Boehler, and M. Ross, “Systematics of Transition Metal Melting,” Phys. Rev. B 63,132104 (1–4) (2001).

  71. A. Dewaele et al., “High Melting Points of Tantalum in a Laser-Heated Diamond Anvil Cell,” Phys. Rev. Lett. 104,255701 (1–4) (2010).

  72. E. Yu. Tonkov, Phase Diagrams of Elements at High Pressure (Nauka, Moscow, 1979) [in Russian].

    Google Scholar 

  73. G. V. Sin’ko and N. A. Smirnov, “Existence of a Structural Transition in Aluminum at Pressures of ∼1.5 Mbar and a Temperature of ⩾1000 K,” in IX Kharitonov Readings (Sarov, VNIIEF, 2007), pp. 287–291.

    Google Scholar 

  74. E. D. Chisolm, D. C. Scott, and C. W. Duane, “Test of Theoretical Equation of State for Elemental Solids and Liquids,” Phys. Rev. B 68,104103 (1–12) (2003).

  75. A. Kloss, H. Hess, H. Schneidenbach, and R. Grossjohann, “Scanning the Melting Curve of Tungsten by a Submicrosecond Wire-Explosion Experiment,” Int. J. Thermophys. 20(4), 1199–1209 (1999).

    Article  Google Scholar 

  76. Feng Xi and Lingcang Cai, “Theoretical Study of Melting Curves on Ta, Mo, and W at High Pressures,” Physica. B 403, 2065–2070 (2008).

    Article  ADS  Google Scholar 

  77. A. G. Morachevskii and I. B. Sladkov, Thermodynamic Calculations in Metallurgy: Handbook (Moscow, Metallurgiya, 1993) [in Russian].

    Google Scholar 

  78. The Elements: Handbook, Ed. by J. Emsley (Clarendon Press, Oxford-New York, 1998).

    Google Scholar 

  79. V. E. Zinov’ev, Thermophysical Properties of Metals at High Temperatures (Metallurgiya, Moscow, 1989) [in Russian].

    Google Scholar 

  80. R. S. Hixson and M. A. Winkler, “Thermophysical Properties of Solid and Liquid Tungsten,” Int. J. Thermophys. 11(4), 709 (1990).

    Article  ADS  Google Scholar 

  81. Properties of Elements: Handbook, Ed. by M. E. Drits (Metallurgiya, Moscow, 1985) [in Russian].

    Google Scholar 

  82. G. I. Kerley, Equations of State for Be, Ni, W, and Au, Sandia Report No. SAND 2003-3784 (October, 2003).

  83. Shock Wave DataBase: A Collection of Numerous Shock-Wave Experimental Points; http://teos.ficp.ac.ru/rusbank/catsearch.php.

Download references

Author information

Authors and Affiliations

  1. Institute of Experimental Physics, Russian Federal Nuclear Center (VNIIEF), Sarov, 607188, Russia

    D. G. Gordeev, L. F. Gudarenko, A. A. Kayakin & V. G. Kudel’kin

Authors
  1. D. G. Gordeev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. F. Gudarenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Kayakin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. V. G. Kudel’kin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. G. Gordeev.

Additional information

Original Russian Text © D.G. Gordeev, L.F. Gudarenko, A.A. Kayakin, V.G. Kudel’kin.

__________

Translated from Fizika Goreniya i Vzryva, Vol. 49, No. 1, pp. 106–120, January–February, 2013.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gordeev, D.G., Gudarenko, L.F., Kayakin, A.A. et al. Equation of state model for metals with ionization effectively taken into account. Equation of state of tantalum, tungsten, aluminum, and beryllium. Combust Explos Shock Waves 49, 92–104 (2013). https://doi.org/10.1134/S0010508213010103

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0010508213010103

Keywords

Search

Navigation