A Regular Solution Model for a Single-Phase High Entropy and Enthalpy Alloy


A regular solution, 3-component model suggested by J.L. Meijering in which binary interaction parameters were equal and positive has been extended to 5 and 6-component high entropy alloys (HEAs). On cooling, Meijering’s model develops miscibility gaps containing a low temperature eutectoid at the equiatomic composition. Similar behavior is found in this work on HEAs with the eutectoid temperature decreasing, while both the entropy and enthalpy are increasing, as additional components are added to the system. An equation for the chemical spinodal at the equiatomic composition is derived from the same thermodynamic model that was used to predict miscibility gaps. The spinodal temperature is at a cone point where multiple spinodal surfaces meet and is dominated by entropy. A proposal is made to categorize HEAs as having low, medium or high enthalpy. Low enthalpy HEAs are defined as having mixing enthalpies less than 1.25 kJ/mol, high enthalpy HEAs having mixing enthalpies greater than 2.9 kJ/mol, and medium HEA as between the extremes. A possible approach for designing high enthalpy HEAs is suggested to incorporate Meijering’s method of analyzing potential HEAs according to their individual binary interaction parameters instead of their total mixing enthalpy.

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  1. 1.

    J.L. Meijering, Segregation in Regular Ternary Solutions, Part I, Philips Res. Rep., 1950, 5, p 333-356, Part II, Philips Res. Rep., 1951, 6, p 183-210

  2. 2.

    D.B. Miracle and O.N. Senkova, A Critical Review of High Entropy Alloys and Related Concepts, Acta Mater., 2017, 122, p 448-511

    Article  Google Scholar 

  3. 3.

    A. Takeuchi, K. Amiya, T. Wada, K. Yubuta, W. Zhang, and A. Makino, Designs of High-Entropy Crystalline and Bulk Glassy Alloys by Evaluating Mixing Enthalpy and Delta Parameter for Quinary to Decimal Equi-atomic Alloys, Mater. Trans., 2014, 55, p 165-170

    Article  Google Scholar 

  4. 4.

    S. Guo and C.T. Liu, Phase Stability in High Entropy Alloys: Formation of Solid-Solution Phase or Amorphous Phase, Prog. Nat. Sci. Mater. Int., 2011, 21, p 433-467

    Article  Google Scholar 

  5. 5.

    J.E. Morral, On Characterizing Stability Limits for Ternary Systems, Acta Metall., 1972, 20, p 1061-1067

    Article  Google Scholar 

  6. 6.

    J.E. Morral, Stability Limits for Ternary Regular Systems, Acta Metall., 1972, 20, p 1069-1076

    Article  Google Scholar 

  7. 7.

    R. Kikuchi, D. deFontaine, M. Murakami, and T. Nakamura, Ternary Phase Diagram Calculations, Acta Metall., 1977, 25, p 207

    Article  Google Scholar 

  8. 8.

    D. deFontaine, Configurational Thermodynamics of Solid Solutions, in Solid State Physics, vol. 34 (Academic Press, New York, 1979) p 74-371

  9. 9.

    J.E. Morral and S.L. Chen, High Entropy Alloys, Miscibility Gaps, and the Rose Geometry, J. Phase Equilib. Diff., 2017, 38(3), p 319-331

  10. 10.

    J.E. Morral and R.H. Davies, Thermodynamics of Isolated Miscibility Gaps, J. Chim. Phys., 1997, 94, p 861-868

    Article  Google Scholar 

  11. 11.

    S.L. Chen, J.-Y. Zhang, X.-G. Lu, K.C. Chou, W.A. Oates, R. Schmid-Fetzer, and Y.A. Chang, Calculation of Rose Geometry, Acta Mater., 2007, 55, p 243-250

    Article  Google Scholar 

  12. 12.

    L.S. Palatnik and A.I. Landau, Phase Equilibria in Multicomponent Systems, Holt Rinehart and Winston, New York, 1964, p 238

    Google Scholar 

  13. 13.

    E.M. Slyusarenko, E.Y. Kerimov, and M.V. Sofin, Analysis of the Phase Equilibria in Multicomponent Systems Using Graphs, Mendeleev Commun., 1999, 9, p 56-59

    Article  Google Scholar 

  14. 14.

    E.Y. Kerimov, S. Nikolaev, and E.M. Slyusarenko, Phase Equilibria in the Quaternary Ni-Re-Nb-Cr System at 1375 K Determined Using the Graph Method, J. Phase Equilib. Diff., 2016, 37, p 135-148

    Article  Google Scholar 

  15. 15.

    J.L. Meijering and H.K. Hardy, Acta Metall., 1956, 4, p 249-256

    Article  Google Scholar 

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The authors are grateful to the Ohio State University and CompuTherm LLC for providing facilities to perform this research.

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Correspondence to J. E. Morral.

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Morral, J.E., Chen, S. A Regular Solution Model for a Single-Phase High Entropy and Enthalpy Alloy. J. Phase Equilib. Diffus. 38, 382–390 (2017). https://doi.org/10.1007/s11669-017-0578-z

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  • binary interaction parameters
  • high enthalpy and entropy
  • high entropy alloys
  • multicomponent regular solutions
  • multicomponent spinodal