Science China Technological Sciences

, Volume 62, Issue 9, pp 1565–1576 | Cite as

Temperature dependent Grüneisen parameter

  • Xun Zhang
  • Sheng Sun
  • Tao XuEmail author
  • TongYi ZhangEmail author


Grüneisen parameter is one of the most valuable quantities in thermodynamics, which links the material properties of bulk modulus, heat capacity at constant volume, thermal expansion coefficient, and volume together. A new thermodynamic model of temperature-dependent potential energy is proposed here to investigate the temperature dependent Grüneisen parameter of bulk material. The newly developed thermodynamic model leads to temperature dependent analytical solutions of Grüneisen parameter and other thermo-mechanical properties including the Grüneisen equation of state. Molecular dynamics simulations are conducted on single crystalline Ni, Cu, and Au bulk crystals and the simulation results verify the newly developed thermodynamic model and quantitively evaluate the theoretically derived physical quantities. In addition, the Debye model is also employed in the study of temperature dependent Grüneisen parameter and the results also verify the theoretical approach.


thermo-mechanical properties Grüneisen parameter thermodynamic model temperature-dependent potential energy molecular dynamics simulations 



  1. 1.
    Debye P. Zur theorie der spezifischen wärmen. Ann Phys, 1912, 344: 789–839CrossRefzbMATHGoogle Scholar
  2. 2.
    Moruzzi V L, Janak J F, Schwarz K. Calculated thermal properties of metals. Phys Rev B, 1988, 37: 790–799CrossRefGoogle Scholar
  3. 3.
    Jankovský O, Sofer Z, Vítek J, et al. Structure, oxygen non-stoichiometry and thermal properties of (Bi0.4Sr0.6)Sr2CoO5-δ. Thermo-Chim Acta, 2015, 600: 89–94CrossRefGoogle Scholar
  4. 4.
    Fultz B. Vibrational thermodynamics of materials. Prog Mater Sci, 2010, 55: 247–352CrossRefGoogle Scholar
  5. 5.
    Gillet P, Richet P, Guyot F, et al. High-temperature thermodynamic properties of forsterite. J Geophys Res, 1991, 96: 11805–11816CrossRefGoogle Scholar
  6. 6.
    Narasimhan S, De Gironcoli S. Ab initio calculation of the thermal properties of Cu: Performance of the LDA and GGA. Phys Rev B, 2002, 65: 064302CrossRefGoogle Scholar
  7. 7.
    Skelton J M, Parker S C, Togo A, et al. Thermal physics of the lead chalcogenides PbS, PbSe, and PbTe from first principles. Phys Rev B, 2014, 89: 205203CrossRefGoogle Scholar
  8. 8.
    Huang L F, Zeng Z. Lattice dynamics and disorder-induced contraction in functionalized graphene. J Appl Phys, 2013, 113: 083524CrossRefGoogle Scholar
  9. 9.
    Souadkia M, Bennecer B, Kalarasse F. Elastic, vibrational and thermodynamic properties of based group IV semiconductors and GeC under pressure. J Phys Chem Solids, 2013, 74: 1615–1625CrossRefGoogle Scholar
  10. 10.
    Hellman O, Steneteg P, Abrikosov I A, et al. Temperature dependent effective potential method for accurate free energy calculations of solids. Phys Rev B, 2013, 87: 104111CrossRefGoogle Scholar
  11. 11.
    Errea I, Calandra M, Mauri F. First-principles theory of anharmonicity and the inverse isotope effect in superconducting palladium-hydride compounds. Phys Rev Lett, 2013, 111: 177002CrossRefGoogle Scholar
  12. 12.
    Hooton D J. A new treatment of anharmonicity in lattice thermodynamics: I. London Edinburgh Dublin Philos Mag J Sci, 1955, 46: 422–432MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lavrentiev M Y, Drautz R, Nguyen-Manh D, et al. Monte Carlo study of thermodynamic properties and clustering in the bcc Fe-Cr system. Phys Rev B, 2007, 75: 014208CrossRefGoogle Scholar
  14. 14.
    Karimi M, Stapay G, Kaplan T, et al. Temperature dependence of the elastic constants of Ni: Reliability of EAM in predicting thermal properties. Modell Simul Mater Sci Eng, 1997, 5: 337CrossRefGoogle Scholar
  15. 15.
    Zhou X Y, Huang B L, Zhang T Y. Size- and temperature-dependent Young’s modulus and size-dependent thermal expansion coefficient of thin films. Phys Chem Chem Phys, 2016, 18: 21508–21517CrossRefGoogle Scholar
  16. 16.
    Yang X, Zhai P, Liu L, et al. Thermodynamic and mechanical properties of crystalline CoSb3: A molecular dynamics simulation study. J Appl Phys, 2011, 109: 123517CrossRefGoogle Scholar
  17. 17.
    Cai C L, Chen Q F, Cui S X, et al. The Grüneisen parameter of NaCl at high pressures and temperatures: A molecular dynamics study. Chin Phys Lett, 2005, 22: 514–516CrossRefGoogle Scholar
  18. 18.
    Lagache M, Ungerer P, Boutin A, et al. Prediction of thermodynamic derivative properties of fluids by Monte Carlo simulation. Phys Chem Chem Phys, 2001, 3: 4333–4339CrossRefGoogle Scholar
  19. 19.
    van Duin A C T, Dasgupta S, Lorant F, et al. ReaxFF: Areactive force field for hydrocarbons. J Phys Chem A, 2001, 105: 9396–9409CrossRefGoogle Scholar
  20. 20.
    Stillinger F H, Weber T A. Computer simulation of local order in condensed phases of silicon. Phys Rev B, 1985, 31: 5262–5271CrossRefGoogle Scholar
  21. 21.
    Patterson J D, Bailey B C. Solid-State Physics: Introduction to the Theory. Berlin: Springer, 2007Google Scholar
  22. 22.
    Kittel C. Introduction to Solid State Physics. New York: Wiley, 1976zbMATHGoogle Scholar
  23. 23.
    Cohen R E, Gülseren O. Thermal equation of state of tantalum. Phys Rev B, 2001, 63: 224101CrossRefGoogle Scholar
  24. 24.
    Born M, Huang K. Dynamical Theory of Crystal Lattices. Oxford: Clarendon Press, 1954zbMATHGoogle Scholar
  25. 25.
    Barron T H K. Grüneisen parameters for the equation of state of solids. Ann Phys, 1957, 1: 77–90CrossRefGoogle Scholar
  26. 26.
    Eliezer S, Ghatak A K, Hora H. Fundamentals of Equations of State. Singapore: World Scientific, 2002CrossRefzbMATHGoogle Scholar
  27. 27.
    Decker D L. High-pressure equation of state for NaCl, KCl, and CsCl. J Appl Phys, 1971, 42: 3239–3244CrossRefGoogle Scholar
  28. 28.
    Hanfland M, Beister H, Syassen K. Graphite under pressure: Equation of state and first-order Raman modes. Phys Rev B, 1989, 39: 12598–12603CrossRefGoogle Scholar
  29. 29.
    Birch F. Equation of state and thermodynamic parameters of NaCl to 300 kbar in the high-temperature domain. J Geophys Res-Solid Earth, 1986, 91: 4949–4954CrossRefGoogle Scholar
  30. 30.
    Bukowinski M S T. A theoretical equation of state for the inner core. Phys Earth Planet Inter, 1977, 14: 333–344CrossRefGoogle Scholar
  31. 31.
    Wachtman J B, Tefft W E, Lam D G, et al. Exponential temperature dependence of Young’s modulus for several oxides. Phys Rev, 1961, 122: 1754–1759CrossRefGoogle Scholar
  32. 32.
    Varshni Y P. Temperature dependence of the elastic constants. Phys Rev B, 1970, 2: 3952–3958CrossRefGoogle Scholar
  33. 33.
    Plimpton S. Fast parallel algorithms for short-range molecular dynamics. J Comput Phys, 1995, 117: 1–19CrossRefzbMATHGoogle Scholar
  34. 34.
    Foiles S M, Baskes M I, Daw M S. Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Phys Rev B, 1986, 33: 7983–7991CrossRefGoogle Scholar
  35. 35.
    Nosé S. A molecular dynamics method for simulations in the canonical ensemble. Mol Phys, 1984, 52: 255–268CrossRefGoogle Scholar
  36. 36.
    Martyna G J, Tobias D J, Klein M L. Constant pressure molecular dynamics algorithms. J Chem Phys, 1994, 101: 4177–4189CrossRefGoogle Scholar
  37. 37.
    Parrinello M, Rahman A. Polymorphic transitions in single crystals: A new molecular dynamics method. J Appl Phys, 1981, 52: 7182–7190CrossRefGoogle Scholar
  38. 38.
    Shinoda W, Shiga M, Mikami M. Rapid estimation of elastic constants by molecular dynamics simulation under constant stress. Phys Rev B, 2004, 69: 134103CrossRefGoogle Scholar
  39. 39.
    Thompson A P, Plimpton S J, Mattson W. General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions. J Chem Phys, 2009, 131: 154107CrossRefGoogle Scholar
  40. 40.
    Kresse G, Hafner J. Ab initio molecular dynamics for liquid metals. Phys Rev B, 1993, 47: 558–561CrossRefGoogle Scholar
  41. 41.
    Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B, 1996, 54: 11169–11186CrossRefGoogle Scholar
  42. 42.
    Blöchl P E. Projector augmented-wave method. Phys Rev B, 1994, 50: 17953–17979CrossRefGoogle Scholar
  43. 43.
    Perdew J P, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys Rev Lett, 1996, 77: 3865–3868CrossRefGoogle Scholar
  44. 44.
    Monkhorst H J, Pack J D. Special points for Brillouin-zone integrations. Phys Rev B, 1976, 13: 5188–5192MathSciNetCrossRefGoogle Scholar
  45. 45.
    Callister Jr W D, Rethwisch D G. Fundamentals of Materials Science and Engineering: An Integrated Approach. Hoboken: John Wiley & Sons, 2012Google Scholar
  46. 46.
    Kang M H, Tatar R C, Mele E J, et al. Real-space formulation of the mixed-basis pseudopotential method: Bulk structural properties of elemental copper. Phys Rev B, 1987, 35: 5457–5472CrossRefGoogle Scholar
  47. 47.
    Kong L T. Phonon dispersion measured directly from molecular dynamics simulations. Comput Phys Commun, 2011, 182: 2201–2207CrossRefGoogle Scholar
  48. 48.
    Çağin T, Dereli G, Uludoğan M, et al. Thermal and mechanical properties of some fcc transition metals. Phys Rev B, 1999, 59: 3468–3473CrossRefGoogle Scholar
  49. 49.
    Hasegawa M, Young W H. Gruneisen parameters for simple metals. J Phys F-Met Phys, 1980, 10: 225–234CrossRefGoogle Scholar
  50. 50.
    Gauster W B. Low-temperature Grüneisen parameters for silicon and aluminum. Phys Rev B, 1971, 4: 1288–1296CrossRefGoogle Scholar
  51. 51.
    Singh R N, Arafin S, George A K. Temperature-dependent thermoelastic properties of s-, p- and d-block liquid metals. Phys B-Condensed Matter, 2007, 387: 344–351CrossRefGoogle Scholar
  52. 52.
    Boehler R, Ramakrishnan J. Experimental results on the pressure dependence of the Grüneisen parameter: A review. J Geophys Res, 1980, 85: 6996–7002CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Materials Genome InstituteShanghai UniversityShanghaiChina

Personalised recommendations