Contribution of Lattice Distortion to Solid Solution Strengthening in a Series of Refractory High Entropy Alloys


We present an experimental approach for revealing the impact of lattice distortion on solid solution strengthening in a series of body-centered-cubic (bcc) Al-containing, refractory high entropy alloys (HEAs) from the Nb-Mo-Cr-Ti-Al system. By systematically varying the Nb and Cr content, a wide range of atomic size difference as a common measure for the lattice distortion was obtained. Single-phase, bcc solid solutions were achieved by arc melting and homogenization as well as verified by means of scanning electron microscopy and X-ray diffraction. The atomic radii of the alloying elements for determination of atomic size difference were recalculated on the basis of the mean atomic radii in and the chemical compositions of the solid solutions. Microhardness (μH) at room temperature correlates well with the deduced atomic size difference. Nevertheless, the mechanisms of microscopic slip lead to pronounced temperature dependence of mechanical strength. In order to account for this particular feature, we present a combined approach, using μH, nanoindentation, and compression tests. The athermal proportion to the yield stress of the investigated equimolar alloys is revealed. These parameters support the universality of this aforementioned correlation. Hence, the pertinence of lattice distortion for solid solution strengthening in bcc HEAs is proven.

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  1. 1.

    This phase remained unidentified in our previous investigation.[14]

  2. 2.

    \( R_{\text{adj}}^{2} = 1 - \left( {\frac{{(1 - R^{2} ) \cdot (n - 1)}}{n - k - 1}} \right) \) is a modified coefficient of determination R 2 taking the degrees of freedom k and the number of data points for data fitting n into account. The coefficient of determination R 2 corresponds to the ratio of the covariance and the product of the standard deviations of the fitted data.


  1. 1.

    J.-W. Yeh, S.-K. Chen, S.-J. Lin, J.-Y. Gan, T.-S. Chin, T.-T. Shun, C.-H. Tsau, and S.-Y. Chang: Adv. Eng. Mater., 2004, vol. 6, pp. 299–303.

    Article  Google Scholar 

  2. 2.

    D.B. Miracle and O.N. Senkov: Acta Mater., 2017, vol. 122, pp. 448–511.

    Article  Google Scholar 

  3. 3.

    C. Varvenne, G.P.M. Leyson, M. Ghazisaeidi, and W.A. Curtin: Acta Mater., 2017, vol. 124, pp. 660–83

    Article  Google Scholar 

  4. 4.

    R.L. Fleischer: Acta Metall., 1961, vol. 9, pp. 996–1000.

    Article  Google Scholar 

  5. 5.

    R.L. Fleischer: Acta Metall., 1963, vol. 11, pp. 203–09.

    Article  Google Scholar 

  6. 6.

    R. Labusch: Physica Status Solidi (b), 1970, vol. 41, pp. 659–69.

    Article  Google Scholar 

  7. 7.

    I. Toda-Caraballo: Scripta Mater., 2017, vol. 127, pp. 113–17.

    Article  Google Scholar 

  8. 8.

    I. Toda-Caraballo and P.E.J. Rivera-Díaz-del-Castillo: Acta Mater., 2015, vol. 85, pp. 14–23.

    Article  Google Scholar 

  9. 9.

    N.L. Okamoto, K. Yuge, K. Tanaka, H. Inui, and E.P. George: AIP Adv., 2016, vol. 6, p. 125008.

    Article  Google Scholar 

  10. 10.

    C. Varvenne, A. Luque, and W.A. Curtin: Acta Mater., 2016, vol. 118, pp. 164–76.

    Article  Google Scholar 

  11. 11.

    Z. Wu, H. Bei, G.M. Pharr, and E.P. George: Acta Mater., 2014, vol. 81, pp. 428–41.

    Article  Google Scholar 

  12. 12.

    S.I. Rao, C. Varvenne, C. Woodward, T.A. Parthasarathy, D. Miracle, O.N. Senkov, and W.A. Curtin: Acta Mater., 2017, vol. 125, pp. 311–20.

    Article  Google Scholar 

  13. 13.

    A. Fernandez-Caballero, J.S. Wrobel, P.M. Mummery and D. Nguyen-Manh: arXiv preprint http://arXiv:1705.01844.

  14. 14.

    H. Chen, A. Kauffmann, B. Gorr, D. Schliephake, C. Seemüller, J.N. Wagner, H.-J. Christ, and M. Heilmaier: J. Alloys Compds., 2016, vol. 661, pp. 206–15.

    Article  Google Scholar 

  15. 15.

    B. Gorr, F. Mueller, H.-J. Christ, T. Mueller, H. Chen, A. Kauffmann, and M. Heilmaier: J. Alloys Compd., 2016, vol. 688, Part B, pp. 468–77.

    Article  Google Scholar 

  16. 16.

    B. Gorr, F. Müller, M. Azim, H.-J. Christ, T. Müller, H. Chen, A. Kauffmann, and M. Heilmaier: Oxid. Met., 2017, vol. 88, pp. 1–11.

    Article  Google Scholar 

  17. 17.

    B. Gorr, M. Azim, H.-J. Christ, T. Mueller, D. Schliephake, and M. Heilmaier: J. Alloys Compds., 2015, vol. 624, pp. 270–78.

    Article  Google Scholar 

  18. 18.

    B. Gorr, M. Azim, H.-J. Christ, H. Chen, D.V. Szabo, A. Kauffmann, and M. Heilmaier: Metall. Mater. Trans. A, 2016, vol. 47A, pp. 961–70.

    Article  Google Scholar 

  19. 19.

    J.B. Nelson and D.P. Riley: Proc. Phys. Soc., 1945, vol. 57, p. 160.

    Article  Google Scholar 

  20. 20.

    M.J. Mayo and W.D. Nix: Acta Metall., 1988, vol. 36, pp. 2183–92.

    Article  Google Scholar 

  21. 21.

    I.-C. Choi, C. Brandl and R. Schwaiger: Acta Mater., 2017, Vol. 140, pp. 107-15.

    Article  Google Scholar 

  22. 22.

    K.L. Johnson: Contact Mechanics, Cambridge University Press, Cambridge, United Kingdom, 1987.

    Google Scholar 

  23. 23.

    K. Lichtenberg, E. Orsolani-Uhlig, R. Rössler and K.A. Weidenmann: J. Compos. Mater., 2017,

  24. 24.

    O.N. Senkov, J.M. Scott, S.V. Senkova, D.B. Miracle, and C.F. Woodward: J. Alloys Compds., 2011, vol. 509, pp. 6043–48.

    Article  Google Scholar 

  25. 25.

    G. Chiarotti: in Structure, G. Chiarotti, ed., Springer-Verlag, Berlin/Heidelberg, 1993, vol. 24a, pp. 21–26.

    Google Scholar 

  26. 26.

    N. Zárubová: Scripta Metall., 1977, vol. 11, pp. 441–44.

    Article  Google Scholar 

  27. 27.

    N. Zárubová and B. Šesták: Physica Status Solidi (a), 1975, vol. 30, pp. 365–74.

    Article  Google Scholar 

  28. 28.

    A. Seeger: Z. Metallkd., 2002, vol. 93, pp. 760–77.

    Article  Google Scholar 

  29. 29.

    V. Maier, A. Hohenwarter, R. Pippan, and D. Kiener: Scripta Mater., 2015, vol. 106, pp. 42–45.

    Article  Google Scholar 

  30. 30.

    B. Šesták and A. Seeger: Z. Metallkd., 1978, vol. 69, pp. 195–202.

    Google Scholar 

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The financial support by the Deutsche Forschungsgemeinschaft (DFG), Grant No. HE 1872/31-1, is gratefully acknowledged. One of the authors (AK) thanks the Carl Zeiss Foundation for financial support via a Postdoctoral Grant. IC acknowledges the financial support by the Alexander von Humboldt Foundation. The authors acknowledge the chemical analysis by ICP-OES at the Institute for Applied Materials (IAM-AWP), Karlsruhe Institute of Technology (KIT). We also thank D. Schliephake, U. Hauf, P. Eyer, F. Hinrichs, and M. Swetik for experimental support. We thank Professor W.A. Curtin (EPFL, Lausanne, Switzerland) for fruitful discussions and suggestions. This work was partly carried out with the support of the Karlsruhe Nano Micro Facility (KNMF,, a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT,

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Correspondence to H. Chen.

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Dedicated to Dr. Martin Palm on the occasion of his 60th birthday.

Manuscript submitted June 29, 2017.

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Chen, H., Kauffmann, A., Laube, S. et al. Contribution of Lattice Distortion to Solid Solution Strengthening in a Series of Refractory High Entropy Alloys. Metall and Mat Trans A 49, 772–781 (2018).

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