Skip to main content
SpringerLink
Log in

Non-equilibrium Nucleation: Application to Solidification and Molar-Specific Heat Capacity of Pure Metals and Phases

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

Abstract

A recently derived equation for nucleation is applied for pure aluminum and Al–6 wt% Cu–3 wt% Si alloy under upward solidification conditions to calculate the surface stress, surface energy, nucleus radius, and Gibbs–Thomson Coefficient as a function of the distance from the chill. The microscopic and macroscopic fields are coupled through a Representative Elementary Volume—REV approach. As expected, higher surface energy and lower critical radius values are observed in positions in which high cooling rates occurred. Then, a nucleation model is carried out to simulate the effect of cooling rates for the molar specific heat capacity of pure Al, Fe, and Nb, and for Al2Cu and Al3Ni2 phases. In the case of Nb, a set of experimental data deviates from high temperatures from theoretical predictions, probably due to high O2 activity as previously observed for the case of pure Fe. A nucleation model for the alloy is proposed as a function of pressure, concentration, and temperature gradients in view to permit the calculation of nucleation radius, surface energy, surface stress, and Gibbs–Thomson Coefficient under non-equilibrium any given cooling rate.

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
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. M. Aniolek, T. Smith, F. Czerwinski, Combining differential scanning calorimetry and cooling-heating curve thermal analysis to study the melting and solidification behavior of Al–Ce binary alloys. Metals 11–372, 1–17 (2021)

    Google Scholar 

  2. G.V. Poel, V.B.F. Mathot, High-speed/high performance differential scanning calorimetry (HPer DSC): temperature calibration in heating and cooling mode and minimization of thermal lag. Thermochim. Acta 446, 41–54 (2006)

    Google Scholar 

  3. J. Piatkowski, R. Przeligorz, A. Gontarczyk, The study of phase transformation of AlSi9Cu3 alloy by DSC method. Arch. Foundry Eng. 16, 109–112 (2016)

    Google Scholar 

  4. W. Zhai, W.L. Wang, D.L. Geng, B. Wei, A DSC analysis of thermodynamic properties and solidification characteristics for binary Cu-Sn alloys. Acta Mater. 60, 6518–6527 (2012)

    ADS  Google Scholar 

  5. I.L. Ferreira, A Non-equilibrium nucleation model to calculate the density of state and its application to the heat capacity of stoichiometric UO2. Int. J. Thermophys. 42, 148 (2021)

    ADS  Google Scholar 

  6. R. Shuttleworth, The surface tension in solids. Proc. Phys. Soc. 63A, 444–457 (1950)

    ADS  Google Scholar 

  7. J.S. Vermaak, C.W. Mays, D. Juhlmann-Wilsdorf, On the surface stress and surface tensor: I. Theoretical considerations. Surf. Sci. 12, 128–133 (1968)

    ADS  Google Scholar 

  8. M.E. Gurtin, A.I. Murdoch, Surface stress in solids. Int. J. Solids Struct. 14, 431–440 (1978)

    MATH  Google Scholar 

  9. P. Müller, A. Saul, Elastic effects on surface physics. Surf. Sci. Rep. 29, 157–258 (2004)

    ADS  Google Scholar 

  10. P. Müller, A. Saul, F. Leroy, Simple views on surface stress and surface energy concepts. Nanosci. Nanotechnol. 5, 013002 (2014)

    Google Scholar 

  11. M. Bobadilla, J. Lacaze, G. Lesoult, Influence des conditions de solidification sur le déroulement de la solidification de aciers inoxydable austénitique. J. Cryst. Growth 89, 531–544 (1988)

    ADS  Google Scholar 

  12. M. Rappaz, S.A. David, J.M. Vitek, L.A. Boatner, Analysis of solidification microstructures in Fe-Ni-Cr single-crystal welds. Metall. Trans. A 21A, 1767–1782 (1990)

    ADS  Google Scholar 

  13. M. Rappaz, W.J. Boettinger, On dendritic solidification of multicomponent alloys with unequal liquid diffusion coefficients. Acta Mater. 47, 3205–3219 (1999)

    ADS  Google Scholar 

  14. J. Ni, C. Beckermann, A volume-averaged two-phase model for transport phenomena during solidification. Metall. Trans. B 22B, 349–361 (1991)

    ADS  Google Scholar 

  15. C.R. Swaminathan, V.R. Voller, Towards a general numerical scheme for solidification systems. Int. J. Mass Transf. 40, 2859–2868 (1997)

    MATH  Google Scholar 

  16. I.L. Ferreira, C.A. Santos, A. Garcia, V.R. Voller, Analytical, numerical and experimental analysis of inverse macrosegregation during upward unidirectional solidification of Al-Cu alloys. Metall. Trans. B 35B, 285–297 (2004)

    Google Scholar 

  17. I.L. Ferreira, V.R. Voller, B. Nestler, A. Garcia, Two-dimensional numerical model for the analysis of macrosegregation during solidification. Comp. Mater. Sci. 46, 358–366 (2009)

    Google Scholar 

  18. I.L. Ferreira, J.F.C. Lins, D.J. Moutinho, L.G. Gomes, A. Garcia, Numerical and experimental investigation of microporosity formation in a ternary Al-Cu-Si alloy. J. Alloy. Compds. 503, 31–39 (2010)

    Google Scholar 

  19. I.L. Ferreira, J.A. de Castro, A. Garcia, Determination of heat capacity of pure metals, compounds and alloys by analytical and numerical methods. Thermochim. Acta 682, 178418 (2019)

    Google Scholar 

  20. I.L. Ferreira, On the heat capacity of pure elements and phases. Mater. Res. 24, e20200529 (2021)

    Google Scholar 

  21. I.L. Ferreira, J.A. Castro, A. Garcia, On the Determination of Molar Heat Capacity of Transition Elements: From the Absolute to the Melting Point in Book: Recent Advances on Numerical Simulation (INTECHOPEN, London, 2021)

    Google Scholar 

  22. E.H. Kim, B.J. Lee, Size dependency of melting point of crystalline nano particles and nano wires: a thermodynamic modeling. Met. Mater. Int. 15, 531–537 (2009)

    Google Scholar 

  23. N. Wu, X. Lu, R. An, X. Ji, Thermodynamic analysis and modification of Gibbs–Thomson equation for melting point depression of metal nanoparticles. Chin. J. Chem. Eng. 31, 198–205 (2021)

    Google Scholar 

  24. I.L. Ferreira, A.L.S. Moreira, J. Aviz, T.A. Costa, O.F.L. Rocha, A.S. Barros, A. Garcia, On an expression for the growth of secondary dendrite arm spacing during non-equilibrium solidification of multicomponent alloys: validation against ternary aluminum-based alloys. J. Manuf. Proc. 3, 634–650 (2018)

    Google Scholar 

  25. V.R. Voller, On a general back-diffusion parameter. J. Crystal Growth 226, 562–568 (2001)

    ADS  Google Scholar 

  26. J.C. Eriksson, Thermodynamics of surface phase systems: V. Contribution to the thermodynamics of the solid-gas interface. Surf. Sci. 14, 221–246 (1969)

    ADS  Google Scholar 

  27. M. Moser, L. Völgyesi, The inner structure of the Earth. Period Polytech. Chem. Eng. 26, 155–204 (1982)

    Google Scholar 

  28. S.H. Simon, The Oxford Solid State Basics, 1st edn. (Oxford University Press, Oxford, 2013)

    MATH  Google Scholar 

  29. Q. Chen, B. Sundman, Calculation of Debye temperature for crystalline structures—a case study on Ti, Zr, and Hf. Acta Mater. 49, 947–961 (2001)

    ADS  Google Scholar 

  30. F.C. Nascimento, M.C.C. Paresque, J.A. de Castro, P.A.D. Jácome, A. Garcia, I.L. Ferreira, Application of computational thermodynamics to the determination of thermophysical properties as a function of temperature for multicomponent Al-based alloys. Thermochim. Acta 619, 1–7 (2015)

    Google Scholar 

  31. W. Giauque, J. Meads, The heat capacities and entropies of aluminum and copper from 15 to 300°K. J. Chem. Phys. 63, 423–432 (1941)

    Google Scholar 

  32. C.R. Brooks, R.E. Bingham, The specific heat of aluminum from 330 to 890 K and contributions from the formation of vacancies and anharmonic effects. J. Phys. Chem. Solids 29, 1553–1560 (1968)

    ADS  Google Scholar 

  33. J.H. Awbery, E. Griffiths, The thermal capacity of pure iron. Proc. R. Soc. Lond. A 174, 1–15 (1940)

    ADS  Google Scholar 

  34. K.K. Kelley, The specific heat of pure iron at low temperatures. J. Chem. Phys. 11, 16–18 (1943)

    ADS  Google Scholar 

  35. A. Eucken, H. Werth, Die spezifische Wärme einiger Metalle und Metallegierungen bei tiefen Temperaturen. Z. Anorg. Allgem. Chem. 188, 152–172 (1930)

    Google Scholar 

  36. F. Simon, R.L. Swain, Untersuchungen über die spezifische Wärme bei tiefen temperaturen. Z. Phys. Chem. B 28, 189–198 (1935)

    Google Scholar 

  37. J.J. Valencia, P. Quested, Thermophysical properties. Casting. ASM Handb. ASM Int. 15, 468–481 (2008)

    Google Scholar 

  38. V.A. Kirillin, A.E. Scheindlin, V.Y. Chekhovskoi, I.A. Zhukova, Themodynamic properties of niobium from 0K to the melting point, 2740 K. in Advances in Thermophysical Properties at Extreme Temperature and Pressures Proceedings of the Third Symposium on Thermophysical Properties. ASME, 1965, p. 152.

  39. A.E. Scheindlin, B.Y. Berezin, V.Y. Chekhovskoi, Enthalpy of niobium in the solid and liquid state. High Temp. High Press. 4, 611–619 (1972)

    Google Scholar 

  40. I.I. Novikov, V.V. Roshchupkin, A.G. Mozgovoi, N.A. Semashko, Specific heat of nickel and niobium in the temperature interval 300–1300 K. High Temp. 19, 694 (1981)

    Google Scholar 

  41. F. Righini, R.B. Roberts, A. Rosso, Measurements of thermophysical properties by a pulse-heating method: niobium in the range 1000–2500 K. Int. J. Thermophys. 6, 681 (1985)

    ADS  Google Scholar 

  42. K. Morohoshi, M. Uchikoshi, M. Isshki, H. Fukuyama, Surface tension of liquid iron as functions of oxygen activity and temperature. ISIJ Int. 51, 1580–1586 (2011)

    Google Scholar 

  43. M.L.M. Sistiaga, R. Mertens, B. Vrancken, X. Wang, B. Van Hooreweder, J.P. Kruth, J. Van Humbeeck, Changing the alloy composition of Al7075 for better processability by selective laser melting. J. Mater. Process. Technol. 238, 437–445 (2016)

    Google Scholar 

  44. P.A. Rometsch, H. Zhong, K.M. Nairn, T. Jarvis, X. Wu, Characterization of a laser fabricated hypereutectic Al-Sc alloy bar. Scr. Mater. 87, 13–16 (2014)

    Google Scholar 

  45. S. Ghosh, Predictive modeling of solidification during laser additive manufacturing of nickel superalloys: recent developments, future directions. Mater. Res. Express 5, 012001 (2018)

    ADS  Google Scholar 

  46. Y. Qin, P. Wen, M. Voshage, Y. Chen, P.G. Schückler, L. Jauer, D. Xia, H. Guo, Y. Zheng, J.H. Schleifenbaum, Additive manufacturing of biodegradable Zn-xWE43 porous scaffolds: formation quality, microstructure and mechanical properties. Mater. Des. (2019). https://doi.org/10.1016/j.matdes.2019.107937

    Article  Google Scholar 

  47. C. Wei, Z. Zhang, D. Cheng, Z. Sun, M. Zhu, L. Li, An overview of laser-based multiple metallic material additive manufacturing: from macro- to micro-scales. Int. J. Extrem. Manuf. 3, 012003 (2021)

    Google Scholar 

  48. H. Dobbelstein, E.P. George, E.L. Gurevich, A. Kostka, A. Ostendorf, G. Laplanche, Laser metal deposition of refractory high-entropy alloys for high throughput synthesis and structure-property characterization. Int. J. Extrem. Manuf. 3, 015201 (2021)

    Google Scholar 

Download references

Acknowledgments

The authors acknowledge the financial support provided by FAPERJ (The Scientific Research Foundation of the State of Rio de Janeiro), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil—Finance Code 001) and CNPq (National Council for Scientific and Technological Development). A.L.S. Moreira, from Federal University of Pará, is also acknowledged for reviewing of the manuscript.

Funding

The funding was provided by CNPq (302381/2019-8).

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Federal University of Pará-UFPA, Belém, PA, Augusto Correa Avenue 1, 66075-110, Brazil

    Ivaldo Leão Ferreira

Authors
  1. Ivaldo Leão Ferreira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivaldo Leão Ferreira.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferreira, I.L. Non-equilibrium Nucleation: Application to Solidification and Molar-Specific Heat Capacity of Pure Metals and Phases. Int J Thermophys 43, 33 (2022). https://doi.org/10.1007/s10765-021-02956-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10765-021-02956-0

Keywords

Search

Navigation