Journal of Materials Science

, Volume 50, Issue 2, pp 678–687 | Cite as

Approximated equations for molar volumes of pure solid fcc metals and their liquids from zero Kelvin to above their melting points at standard pressure

Original Paper


Approximated equations have been constructed to describe the temperature (T) dependence of the molar volumes (V) of fcc solid metals (Ag, Al, Au, Cu, Ir, Ni, Pb, Pd, Pt, and Rh) and their liquids below and above their melting points at standard pressure of 1 bar from zero Kelvin. Below the melting point, the following new approximated equation is suggested for both the solid and liquid metals: V = a + b * T (to the power of n), where a, b, and n are semi-empirical parameters (at n larger than 1; this equation obeys the boundary condition that the thermal expansion coefficient becomes zero at T = 0 K). This approximated equation reproduces the measured molar volume of solids from zero Kelvin to melting point with an accuracy of 0.2 % or better. As a compromise, the derivative of this equation reproduces the measured thermal expansion coefficient of solids only with an accuracy 10 % or better and only above 100 K. Above the melting point, the following well-known equation is used for both liquid and solid phases: V = c + d * T, where c and d are semi-empirical parameters. This equation implies that the thermal expansion coefficient above the melting point has an approximately constant value. It is found that the volume change upon melting extrapolated to zero K is about 58 % of that at the melting point for all the 10 fcc metals. The tabulated 4 equations (below and above the melting point/for fcc and liquid states) are provided for each of the 10 fcc metals. These equations will be useful for estimating phase equilibria of nano-materials.


Melting Point Molar Volume Liquid Metal Thermal Expansion Coefficient Solid Metal 



The authors acknowledge the financial support from the Hungarian Academy of Science, under the grant number K101781 and the “Science and Industry: the road from scientific achievements to social benefits” TÁMOP-4.2.3.-12/1/KONV-2012-0029 project supported by the European Union and co-financed by the European Social Fund. This work was partially carried out in the framework of the Center of Applied Materials Science and Nano-Technology at the University of Miskolc.


  1. 1.
    Paradis PF, Ishikawa T, Lee GW, Holland-Moritz D, Brillo J, Rhim WK, Okada JT (2014) Materials properties measurements and particle beam interactions studies using electrostatic levitation. Mater Sci Eng R 76:1–53CrossRefGoogle Scholar
  2. 2.
    Kobatake H, Schmitz J, Brillo J (2014) Density and viscosity of ternary Al–Cu–Si liquid alloys. J Mater Sci 49:3541–3549. doi: 10.1007/s10853-014-8072-z CrossRefGoogle Scholar
  3. 3.
    Poirier DR (2014) Density, viscosity, and diffusion coefficients in hypoeutectic Al–Si liquid alloys: an assessment of available data. Metal Mater Trans B. doi: 10.1007/s11663-014-0037-8 Google Scholar
  4. 4.
    Gancarz T, Moser Z, Gasior W, Pstrus J, Henein H (2011) A comparison of surface tension, viscosity, and density of Sn and Sn–Ag alloys using different measurement techniques. Int J Thermophys 32:1210–1233CrossRefGoogle Scholar
  5. 5.
    Terzieff P (2010) Some physico-chemical properties of liquid Ag–Sn–Zn. Phys. B 405:2668–2672CrossRefGoogle Scholar
  6. 6.
    Hallstedt B, Dupin N, Hillert M, Höglund L, Lukas HL, Schuster JC, Solak N (2007) Thermodynamic models for crystalline phases. Composition dependent model for volume, bulk modulus and thermal expansion. CALPHAD 31:28–37CrossRefGoogle Scholar
  7. 7.
    Kucharski M, Fima P (2004) The surface tension and density of Cu–Pb–Fe alloys. Arch Metall Mater 49:565–573Google Scholar
  8. 8.
    Lu XG, Selleby M, Sundman B (2005) Implementation of a new model for pressure dependence of condensed phases in Thermo-Calc. CALPHAD 29:49–55CrossRefGoogle Scholar
  9. 9.
    Guenther G, Guillon O (2014) Models of size-dependent nanoparticle melting tested on gold. J Mater Sci 49:7915–7932. doi: 10.1007/s10853-014-8544-1 CrossRefGoogle Scholar
  10. 10.
    Lee J, Sim KJ (2014) General equations of CALPHAD-type thermodynamic description for metallic nanoparticle systems. CALPHAD 44:129–132CrossRefGoogle Scholar
  11. 11.
    Kaptay G, Janczak-Rusch J, Pigozzi G, Jeurgens LPH (2014) Theoretical analysis of melting point depression of pure metals in different initial configurations. J Mater Eng Perform 23:1600–1607CrossRefGoogle Scholar
  12. 12.
    Junkaew A, Ham B, Zhang X, Arróyave R (2014) Tailoring the formation of metastable Mg through interfacial engineering: a phase stability analysis. CALPHAD 45:145–150CrossRefGoogle Scholar
  13. 13.
    Sopousek J, Vrestal J, Pinkas J, Broz P, Bursik J, Styskalik A, Skoda D, Zobac O, Lee J (2014) Cu–Ni nanoalloy phase diagram: prediction and experiment. CALPHAD 45:33–39CrossRefGoogle Scholar
  14. 14.
    Garzel G, Janczak-Rusch J, Zabdyr L (2012) Reassessment of the Ag–Cu phase diagram for nanosystems including particle size and shape effect. CALPHAD 36:52–56CrossRefGoogle Scholar
  15. 15.
    Kaptay G (2012) Nano-Calphad: extension of the Calphad method to systems with nano-phases and complexions. J Mater Sci 47:8320–8335. doi: 10.1007/s10853-012-6772-9 CrossRefGoogle Scholar
  16. 16.
    Tang C, Sung YM, Lee J (2012) Nonlinear size-dependent melting of the silica-encapsulated silver nanoparticles. Appl Phys Lett 100:201903CrossRefGoogle Scholar
  17. 17.
    Koukkari P, Pajarre R, Hack K (2007) Constrained Gibbs energy minimization. Int J Mater Res 98:926–934CrossRefGoogle Scholar
  18. 18.
    Touloukian YS, Kirby RK, Taylor RE, Lee TYR (1977) Thermal expansion. IFI/Plenum, New YorkCrossRefGoogle Scholar
  19. 19.
    Emsley J (1989) The elements. Clarendon Press, OxfordGoogle Scholar
  20. 20.
    Lide DR (ed) (1993–1994) CRC Handbook of chemistry and physics. CRC Press, Boca RatonGoogle Scholar
  21. 21.
    Iida I, Guthrie RIL (1993) The physical properties of liquid metals. Clarendon Press, OxfordGoogle Scholar
  22. 22.
    Lu XG, Selleby M, Sundman B (2005) Assessments of molar volume and thermal expansion for selected bcc, fcc and hcp metallic elements. CALPHAD 29:68–89CrossRefGoogle Scholar
  23. 23.
    Lu XG, Selleby M, Sundman B (2005) Theoretical modeling of molar volume and thermal expansion. Acta Mater 53:2259–2272CrossRefGoogle Scholar
  24. 24.
    Arblaster JW (1997) Crystallographic properties of platinum. Plat Met Rev 41:12–21Google Scholar
  25. 25.
    Arblaster JW (1997) Crystallographic properties of rhodium. Plat Met Rev 41:184–189Google Scholar
  26. 26.
    Arblaster JW (2006) Crystallographic properties of platinum. New methodology and erratum. Plat Met Rev 50:118–119CrossRefGoogle Scholar
  27. 27.
    Arblaster JW (2010) Crystallographic properties of iridium. Plat Met Rev 54:93–102CrossRefGoogle Scholar
  28. 28.
    Arblaster JW (2012) Crystallographic properties of palladium. Plat Met Rev 56:181–189CrossRefGoogle Scholar
  29. 29.
    Ishikawa T, Paradis PF, Fujii R, Saita Y, Yoda S (2005) Thermophysical property measurements of liquid and supercooled iridium by containerless methods. Int J Thermophys 26:893–904CrossRefGoogle Scholar
  30. 30.
    Paradis PF, Ishikawa T, Saita Y, Yoda S (2004) Containerless property measurements of liquid palladium. Int J Thermophys 25:1905–1921CrossRefGoogle Scholar
  31. 31.
    Ishikawa T, Paradis PF, Koike N (2006) Non-contact thermophysical property measurements of liquid and supercooled platinum. Jpn J Appl Phys 45:1719–1728CrossRefGoogle Scholar
  32. 32.
    Paradis PF, Ishikawa T, Yoda S (2003) Thermophysical property measurements of supercooled and liquid rhodium. Int J Thermophys 24:1121–1136CrossRefGoogle Scholar
  33. 33.
    Dinsdale AT (1991) SGTE data for pure elements. CALPHAD 15:317–425CrossRefGoogle Scholar
  34. 34.
    Paradis PF, Ishikawa T, Koike N (2008) Density of liquid gold measured by a non-contact method. Gold Bull 41:242–245CrossRefGoogle Scholar
  35. 35.
    Chung SK, Thiessen DB, Rhim WK (1996) A noncontact measurement technique for the density and thermal expansion coefficient of solid and liquid materials. Rev Sci Instrum 67:3175–3181CrossRefGoogle Scholar
  36. 36.
    Ishikawa T, Paradis PF, Saita Y (2004) Thermophysical property measurement of molten nickel using an electrostatic levitation furnace. J Jpn Inst Met 68:781–786CrossRefGoogle Scholar
  37. 37.
    Kaptay G (2008) A unified model for the cohesive enthalpy, critical temperature, surface tension and volume thermal expansion coefficient of liquid metals of bcc, fcc and hcp crystals. Mater Sci Eng A 495:19–26CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of NanotechnologyUniversity of MiskolcMiskolcHungary
  2. 2.Department of Nanomaterials, BAY-LOGIBay Zoltan Nonprofit LtdMiskolcHungary

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