Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 726–734 | Cite as

Frequency Estimate for Multicomponent Crystalline Compounds

Article

Abstract

Among crystal structures of N-component metal alloys, far fewer examples are known with \(N\ge 4\) than with \(N=2\) or 3, in apparent contradiction to the exponentially growing number of possible combinations of elements. Two effects contribute to this shortfall. Since the N-component composition space resides within a d-dimensional simplex with \(d=N-1\), the vanishing volume in high dimensions reduces the distinct N-component compositions. Additionally, the increasing surface area makes it more probable that stable structures reside on the surface of the simplex (containing fewer than N components) as opposed to its interior. Specific estimates are developed through application of the empirical Miedema enthalpy model. Despite their rarity, we propose that the actual number of \(N=4\)- and 5-component alloys greatly exceeds the number that are currently known.

Keywords

Multicomponent alloy Enthalpy Crystal structures Convex hull High entropy alloy 

References

  1. 1.
    Bak, P.: Commensurate phases, incommensurate phases and the devil’s staircase. Rep. Prog. Phys. 45, 587–629 (1982)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.T.: The quickhull algorithm for convex hulls. ACM Trans. Math. Software 22, 469–483 (1996)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    de Boissieu, M., Boudard, M., Hennion, B., Bellissent, R., Kycia, S., Goldman, A., Janot, C., Audier, M.: Diffuse scattering and phason elasticity in the alpdmn icosahedral phase. Phys. Rev. Lett. 75, 89–92 (1995)ADSCrossRefGoogle Scholar
  4. 4.
    Cantor, B., Chang, I.T., Knight, P., Vincent, A.J.: Microstructural development in equiatomic multicomponent alloys. Mater. Sci. Eng. 375, 213–218 (2004)CrossRefGoogle Scholar
  5. 5.
    DiSalvo, F.J.: Challenges and opportunities in solid-state chemistry. Pure Appl. Chem. 72, 1799–1807 (2000)CrossRefGoogle Scholar
  6. 6.
    Ercolessi, F., Adams, J.B.: The force-matching method. Europhys. Lett. 26, 583 (1994)ADSCrossRefGoogle Scholar
  7. 7.
    Fredeman, D.J., Tobash, P.H., Torrez, M.A., Thompson, J.D., Bauer, E.D., Ronning, F., Tipton, W., Rudin, S.P., Hennig, R.G.: Computationally-driven experimental discovery of the CeIr 4 In compound. Phys. Rev. B 83(22), 224102 (2011)ADSCrossRefGoogle Scholar
  8. 8.
    Glass, C.W., Oganov, A.R., Hansen, N.: Uspex—evolutionary crystal structure prediction. Comp. Phys. Comm. 175, 713–720 (2006)ADSMATHCrossRefGoogle Scholar
  9. 9.
    Goncalves, A.P., Almeida, M.: Extended Miedema model: predicting the formation enthalpies of intermetallic phases with more than two elements. Physica B 228, 289–294 (1996)ADSCrossRefGoogle Scholar
  10. 10.
    Levine, D., Steinhardt, P.J.: Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984)ADSCrossRefGoogle Scholar
  11. 11.
    Miedema, A.R., de Chatel, P.F., de Boer, F.R.: Cohesion in alloys—fundamentals of a semi-empirical model. Physica B 100, 1–28 (1980)CrossRefGoogle Scholar
  12. 12.
    Miedema, A.R., Niessen, A.K.: The enthalpy of solution for solid binary alloys of two 4d-transition metals. CALPHAD 7, 27–36 (1983)CrossRefGoogle Scholar
  13. 13.
    Mihalkovic, M., Henley, C.L.: Empirical oscillating potentials for alloys from ab initio fits and the prediction of quasicrystal-related structures in the Al-Cu-Sc system. Phys. Rev. B 85, 092102 (2012)ADSCrossRefGoogle Scholar
  14. 14.
    Mihalkovič, M., Widom, M.: Ab-initio cohesive energies of Fe-based glass-forming alloys. Phys. Rev. B 70, 144107 (2004)ADSCrossRefGoogle Scholar
  15. 15.
    Mousavi, M.S., Abbasi, R., Kashani-Bozorg, S.F.: A thermodynamic approach to predict formation enthalpies of ternary systems based on Miedema’s model. Met. Mat. Trans. A 47, 3761–3770 (2016)CrossRefGoogle Scholar
  16. 16.
    Niessen, A.K., Miedema, A.R.: The enthalpy effect on forming diluted solid solutions of two 4d and 5d transition metals. Ber. Bunsenges. Phys. Chem. 87, 717–725 (1983)CrossRefGoogle Scholar
  17. 17.
    Sadeghi, E., Karimzadeh, F., Abbasi, M.H.: Thermodynamic analysis of TiAlC intermetallics formation by mechanical alloying. J. Alloy Compd. 576, 317–323 (2013)CrossRefGoogle Scholar
  18. 18.
    Senkov, O.N., Miller, J.D., Miracle, D.B., Woodward, C.: Accelerated exploration of multi-principal element alloys with solid solution phases. Nat. Comm. 6, 6529 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    Senkova, O., Wilks, G., Miracle, D., Chuang, C., Liaw, P.: Refractory high-entropy alloys. Intermetallics 18, 1758–1765 (2010)CrossRefGoogle Scholar
  20. 20.
    Shechtman, D., Blech, I., Gratias, D., Cahn, J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984)ADSCrossRefGoogle Scholar
  21. 21.
    Steurer, W., Dshemuchadse, J.: Intermetallics: structures, properties, and statistics. Oxford University Press, Oxford (2016)CrossRefGoogle Scholar
  22. 22.
    ThermoCalc: TCHEA1 - TCS High entropy alloy database, Version 1.0 (2015). URL http://www.thermocalc.com/media/35873/tchea10_extended_info_bh.pdf
  23. 23.
    van de Walle, A., Ceder, G.: Automating first-principles phase diagram calculations. J. Phase Equilib. 23(4), 348–359 (2002)CrossRefGoogle Scholar
  24. 24.
    Wang, T.L., Liu, B.X.: Glass forming ability of the FeZrCu system studied by thermodynamic calculation and ion beam mixing. J. Alloy Compd. 481, 156–160 (2009)CrossRefGoogle Scholar
  25. 25.
    Widom, M.: Elastic stability and diffuse scattering in icosahedral quasicrystals. Philos. Mag. Lett. 64, 297–305 (1991)ADSCrossRefGoogle Scholar
  26. 26.
    Widom, M.: Prediction of structure and phase transformations. In: Gao, M.C., Yeh, J.-W., Liaw, P.K., Zhong, Y. (eds.) High entropy alloys: fundamentals and applications, chap. 8, pp. 267–298. Springer, New York (2016)Google Scholar
  27. 27.
    Woodley, S.M., Catlow, R.: Crystal structure prediction from first principles. Nat. Mat. 7, 937–946 (2008)CrossRefGoogle Scholar
  28. 28.
    Yeh, J.W., Chen, S.K., Lin, S.J., Gan, J.Y., Chin, T.S., Shun, T.T., Tsau, C.H., Chang, S.Y.: Nanostructured high-entropy alloys with multiple principal elements: novel alloy design concepts and outcomes. Adv. Eng. Mater. 6, 299–303 (2004)CrossRefGoogle Scholar
  29. 29.
    Zhang, C., Zhang, F., Chen, S., Cao, W.: Computational thermodynamics aided high-entropy alloy design. JOM 64, 839–845 (2012)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of PhysicsCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations