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Modeling the structure and thermodynamics of high-entropy alloys

Published online by Cambridge University Press:  24 July 2018

Michael Widom Open the ORCID record for Michael Widom [Opens in a new window]
Michael Widom*
Affiliation:
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
*
a)Address all correspondence to this author. e-mail: [email protected]
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Abstract

High-entropy and multiprincipal element alloys present exciting opportunities and challenges for computational modeling of their structure and phase stability. Recent interest has catalyzed rapid development of techniques and equally rapid growth of new results. This review surveys the essential concepts of thermodynamics and total energy calculation, and the bridge between them provided by statistical mechanics. Specifically, we review the electronic density functional theory of alloy total energy as applied to supercells and special quasirandom structures. We contrast these with the coherent potential approximation and semi-empirical approximations. Statistical mechanical approaches include cluster expansions, hybrid Monte Carlo/molecular dynamics simulations, and extraction of entropy from correlation functions. We also compare first-principles approaches with Calculation of Phase Diagrams (CALPHAD) and highlight the need to augment experimental databases with first-principles derived data. Numerous example applications are given highlighting recent progress utilizing the concepts and methods that are introduced.

Keywords

electronic structurealloyphase equilibria
Type
Invited Review
Information
Journal of Materials Research , Volume 33 , Issue 19: Focus Issue: Fundamental Understanding and Applications of High-Entropy Alloys , 14 October 2018 , pp. 2881 - 2898
Copyright
Copyright © Materials Research Society 2018 

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Footnotes

This section of Journal of Materials Research is reserved for papers that are reviews of literature in a given area.

References

REFERENCES

Cantor, B., Chang, I.T.H., Knight, P., and Vincent, A.J.B.: Microstructural development in equiatomic multicomponent alloys. Mater. Sci. Eng., A 375–377, 213 (2004).CrossRefGoogle Scholar
Yeh, J-W., Chen, S-K., Lin, S-J., Gan, J-Y., Chin, T-S., Shun, T-T., Tsau, C-H., and Chang, S-Y.: Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes. Adv. Eng. Mater. 6, 299 (2004).CrossRefGoogle Scholar
Miracle, D.B. and Senkov, O.N.: A critical review of high entropy alloys and related concepts. Acta Mater. 122, 448 (2017).CrossRefGoogle Scholar
Gao, M.C., Zhang, C., Gao, P., Zhang, F., Ouyang, L.Z., Widom, M., and Hawk, J.A.: Thermodynamics of concentrated solid solution alloys. Curr. Opin. Solid State Mater. Sci. 21, 238 (2017).CrossRefGoogle Scholar
Senkov, O.N., Miller, J.D., Miracle, D.B., and Woodward, C.: Accelerated exploration of multi-principal element alloys with solid solution phases. Nat. Commun. 6, 6529 (2015).CrossRefGoogle ScholarPubMed
Widom, M.: Frequency estimate for multicomponent crystalline compounds. J. Stat. Phys. 167, 726 (2017).CrossRefGoogle Scholar
Zunger, A., Wei, S-H., Ferreira, L.G., and Bernard, J.E.: Special quasirandom structures. Phys. Rev. Lett. 65, 353 (1990).CrossRefGoogle ScholarPubMed
Kaufman, H. and Bernstein, L.: Computer Calculation of Phase Diagrams (Academic Press, New York, 1970).Google Scholar
Murty, S., Yeh, B.S., and Ranganathan, J-W.: High-Entropy Alloys (Butterworth-Heinemann, London, 2014).CrossRefGoogle Scholar
Gao, M.C., Yeh, J-W., Liaw, P.K., and Zhang, Y.: High-Entropy Alloys (Springer International Publishing, Cham, 2016).CrossRefGoogle Scholar
Miracle, D.B.: Critical assessment 14: High entropy alloys and their development as structural materials. Mater. Sci. Technol. 31, 1142 (2015).CrossRefGoogle Scholar
Tsai, M-H. and Yeh, J-W.: High-entropy alloys: A critical review. Mater. Res. Lett. 2, 107 (2014).CrossRefGoogle Scholar
Ye, Y.F., Wang, Q., Lu, J., Liu, C.T., and Yang, Y.: High-entropy alloy: Challenges and prospects. Mater. Today 19, 349 (2016).CrossRefGoogle Scholar
Tian, F.: A review of solid-solution models of high-entropy alloys based on ab initio calculations. Front. Mater. 4, 36 (2017).CrossRefGoogle Scholar
Pickering, E.J. and Jones, N.G.: High-entropy alloys: A critical assessment of their founding principles and future prospects. Int. Mater. Rev. 61, 183 (2016).CrossRefGoogle Scholar
Ikeda, Y., Grabowski, B., and Koermann, F.: Ab initio phase stabilities and mechanical properties of multicomponent alloys: A comprehensive review for high entropy alloys and compositionally complex alloys. Mater. Charact. (2018). (to appear). Available at: https://doi.org/10.1016/j.matchar.2018.06.019.CrossRefGoogle Scholar
Senkov, O.N., Wilks, G.B., Scott, J.M., and Miracle, D.B.: Mechanical properties of Nb25Mo25Ta25W25 and V20Nb20Mo20Ta20W20 refractory high entropy alloys. Intermetallics 19, 698 (2011).CrossRefGoogle Scholar
Senkov, O.N., Senkova, S.V., Miracle, D.B., and Woodward, C.: Mechanical properties of low-density, refractory multi-principal element alloys of the Cr–Nb–Ti–V–Zr system. Mater. Sci. Eng., A 565, 51 (2013).CrossRefGoogle Scholar
Gludovatz, B., Hohenwarter, A., Catoor, D., Chang, E.H., George, E.P., and Ritchie, R.O.: A fracture-resistant high-entropy alloy for cryogenic applications. Science 345, 1153 (2014).CrossRefGoogle ScholarPubMed
Zhang, Y., Stocks, G.M., Jin, K., Lu, C., Bei, H., Sales, B.C., Wang, L., Béland, L.K., Stoller, R.E., Samolyuk, G.D., Caro, M., Caro, A., and Weber, W.J.: Influence of chemical disorder on energy dissipation and defect evolution in concentrated solid solution alloys. Nat. Commun. 6, 8736 (2015).CrossRefGoogle ScholarPubMed
Kurniawan, M., Perrin, A., Xu, P., Keylin, V., and McHenry, M.: Curie temperature engineering in high entropy alloys for magnetocaloric applications. IEEE Magn. Lett. 7, 1 (2016).CrossRefGoogle Scholar
Berardan, D., Meena, A.K., Franger, S., Herrero, C., and Dragoe, N.: Controlled Jahn-Teller distortion in (MgCoNiCuZn)O-based high entropy oxides. J. Alloys Compd. 704, 693 (2017).CrossRefGoogle Scholar
Bérardan, D., Franger, S., Meena, A.K., and Dragoe, N.: Room temperature lithium superionic conductivity in high entropy oxides. J. Mater. Chem. A 4, 9536 (2016).CrossRefGoogle Scholar
Bérardan, D., Franger, S., Dragoe, D., Meena, A.K., and Dragoe, N.: Colossal dielectric constant in high entropy oxides. Phys. Status Solidi RRL 10, 328 (2016).CrossRefGoogle Scholar
Meisenheimer, P.B., Kratofil, T.J., and Heron, J.T.: Giant enhancement of exchange coupling in entropy-stabilized oxide heterostructures. Sci. Rep. 7, 13344 (2017).CrossRefGoogle ScholarPubMed
Shafeie, S., Guo, S., Hu, Q., Fahlquist, H., Erhart, P., and Palmqvist, A.: High-entropy alloys as high-temperature thermoelectric materials. J. Appl. Phys. 118, 184905 (2015).CrossRefGoogle Scholar
Fan, Z., Wang, H., Wu, Y., Liu, X.J., and Lu, Z.P.: Thermoelectric high-entropy alloys with low lattice thermal conductivity. RSC Adv. 6, 52164 (2016).CrossRefGoogle Scholar
Hohenberg, P. and Kohn, W.: Inhomogeneous electron gas. Phys. Rev. 136, B864 (1964).CrossRefGoogle Scholar
Kohn, W. and Sham, L.J.: Self-consistent equations including exchange and correlation effects. Phys. Rev. 140, A1133 (1965).CrossRefGoogle Scholar
Perdew, J.P. and Zunger, A.: Self-interaction correction to density-functional approximations for many-electron systems. Phys. Rev. B 23, 5048 (1981).CrossRefGoogle Scholar
Perdew, J.P., Burke, K., and Ernzerhof, M.: Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).CrossRefGoogle ScholarPubMed
Blöchl, P.E.: Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994).CrossRefGoogle ScholarPubMed
Kresse, G. and Joubert, D.: From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758 (1999).CrossRefGoogle Scholar
Mihalkovič, M. and Widom, M.: Ab initio calculations of cohesive energies of Fe-based glass-forming alloys. Phys. Rev. B: Condens. Matter Mater. Phys. 70, 1 (2004).CrossRefGoogle Scholar
Widom, M. and Mihalkovic, M.: Stability of Fe-based alloys with structure type C6Cr23. J. Mater. Res. 20, 237 (2005).CrossRefGoogle Scholar
Kresse, G. and Hafner, J.: Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558 (1993).CrossRefGoogle ScholarPubMed
Kohn, W.: Density functional and density matrix method scaling linearly with the number of atoms. Phys. Rev. Lett. 76, 3168 (1996).CrossRefGoogle ScholarPubMed
Gao, M.C., Niu, C., Jiang, C., and Irving, D.L.: High-Entropy Alloy (Springer International Publishing, Cham, 2016); pp. 333–368.CrossRefGoogle Scholar
van de Walle, A., Tiwary, P., de Jong, M., Olmsted, D.L., Asta, M., Dick, A., Shin, D., Wang, Y., Chen, L-Q., and Liu, Z-K.: Efficient stochastic generation of special quasirandom structures. Calphad 42, 13 (2013).CrossRefGoogle Scholar
Jiang, C. and Uberuaga, B.P.: Efficient ab initio modeling of random multicomponent alloys. Phys. Rev. Lett. 116, 105501 (2016).CrossRefGoogle ScholarPubMed
Yonezawa, F. and Morigaki, K.: Coherent potential approximation. Prog. Theor. Phys. Suppl. 53, 1 (1973).CrossRefGoogle Scholar
Moriarty, J.A.: Density-functional formulation of the generalized pseudopotential theory. Phys. Rev. B 16, 2537 (1977).CrossRefGoogle Scholar
Moriarty, J.A.: Analytic representation of multi-ion interatomic potentials in transition metals. Phys. Rev. B 42, 1609 (1990).CrossRefGoogle ScholarPubMed
Moriarty, J.A. and Widom, M.: First-principles interatomic potentials for transition-metal aluminides: Theory and trends across the 3d series. Phys. Rev. B 56, 7905 (1997).CrossRefGoogle Scholar
Mihalkovič, M. and 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).CrossRefGoogle Scholar
Daw, M.S. and Baskes, M.I.: Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 29, 6443 (1984).CrossRefGoogle Scholar
Foiles, S.M., Baskes, M.I., and Daw, M.S.: Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Phys. Rev. B 33, 7983 (1986).CrossRefGoogle ScholarPubMed
Finnis, M.W. and Sinclair, J.E.: A simple empirical N-body potential for transition metals. Philos. Mag. A 50, 45 (1984).CrossRefGoogle Scholar
Johnson, R.A. and Oh, D.J.: Analytic embedded atom method model for bcc metals. J. Mater. Res. 4, 1195 (1989).CrossRefGoogle Scholar
Ercolessi, F. and Adams, J.B.: Interatomic potentials from first-principles calculations: The force-matching method. Europhys. Lett. 26, 583 (1994).CrossRefGoogle Scholar
Mendelev, M.I., Han, S., Srolovitz, D.J., Ackland, G.J., Sun, D.Y., and Asta, M.: Development of new interatomic potentials appropriate for crystalline and liquid iron. Philos. Mag. 83, 3977 (2003).CrossRefGoogle Scholar
Miedema, A.R., de Châtel, P.F., and de Boer, F.R.: Cohesion in alloys—Fundamentals of a semi-empirical model. Phys. B 100, 1 (1980).CrossRefGoogle Scholar
Miedema, A.R. and Niessen, A.K.: The enthalpy of solution for solid binary alloys of two 4d-transition metals. Calphad 7, 27 (1983).CrossRefGoogle Scholar
Gonçalves, A.P. and Almeida, M.: Extended Miedema model: Predicting the formation enthalpies of intermetallic phases with more than two elements. Phys. B 228, 289 (1996).CrossRefGoogle Scholar
Mousavi, M.S., Abbasi, R., and Kashani-Bozorg, S.F.: A thermodynamic approach to predict formation enthalpies of ternary systems based on Miedema’s model. Metall. Mater. Trans. A 47, 3761 (2016).CrossRefGoogle Scholar
Takeuchi, A. and Inoue, A.: Mixing enthalpy of liquid phase calculated by Miedema’s scheme and approximated with sub-regular solution model for assessing forming ability of amorphous and glassy alloys. Intermetallics 18, 1779 (2010).CrossRefGoogle Scholar
Takeuchi, A., Amiya, K., Wada, T., Yubuta, K., Zhang, W., and Makino, A.: Entropies in alloy design for high-entropy and bulk glassy alloys. Entropy 15, 3810 (2013).CrossRefGoogle Scholar
Takeuchi, A., Gao, M.C., Qiao, J., and Widom, M.: Applications of special quasi-random structures to high-entropy alloys. In High-Entropy Alloys: Fundamentals and Applications, Gao, Y., Yeh, M.C., Liaw, J-W., and Zhang, P.K., eds. (Springer International Publishing, Cham, 2016); pp. 267–298.Google Scholar
Gao, M.C., Carney, C.S., Doğan, Ö.N., Jablonksi, P.D., Hawk, J.A., and Alman, D.E.: Design of refractory high-entropy alloys. JOM 67, 2653 (2015).CrossRefGoogle Scholar
Zhang, Y., Zhou, Y.J., Lin, J.P., Chen, G.L., and Liaw, P.K.: Solid-solution phase formation rules for multi-component alloys. Adv. Eng. Mater. 10, 534 (2008).CrossRefGoogle Scholar
Yang, X. and Zhang, Y.: Prediction of high-entropy stabilized solid-solution in multi-component alloys. Mater. Chem. Phys. 132, 233 (2012).CrossRefGoogle Scholar
Pomrehn, G.S., Toberer, E.S., Snyder, G.J., and van de Walle, A.: Entropic stabilization and retrograde solubility in Zn4Sb3. Phys. Rev. B 83, 094106 (2011).CrossRefGoogle Scholar
Zhang, H., Yao, S., and Widom, M.: Predicted phase diagram of boron–carbon–nitrogen. Phys. Rev. B 93, 144107 (2016).CrossRefGoogle Scholar
Widom, M. and Huhn, W.P.: Prediction of orientational phase transition in boron carbide. Solid State Sci. 14, 1648 (2012).CrossRefGoogle Scholar
Feng, R., Liaw, P.K., Gao, M.C., and Widom, M.: First-principles prediction of high-entropy-alloy stability. npj Comput. Mater. 3, 50 (2017).CrossRefGoogle Scholar
Rogal, L., Bobrowski, P., Körmann, F., Divinski, S., Stein, F., and Grabowski, B.: Computationally-driven engineering of sublattice ordering in a hexagonal AlHfScTiZr high entropy alloy. Sci. Rep. 7, 2209 (2017).CrossRefGoogle Scholar
Qiu, Y., Hu, Y.J., Taylor, A., Styles, M.J., Marceau, R.K.W., Ceguerra, A.V., Gibson, M.A., Liu, Z.K., Fraser, H.L., and Birbilis, N.: A lightweight single-phase AlTiVCr compositionally complex alloy. Acta Mater. 123, 115 (2017).CrossRefGoogle Scholar
Santodonato, L.J., Zhang, Y., Feygenson, M., Parish, C.M., Gao, M.C., Weber, R.J.K., Neuefeind, J.C., Tang, Z., and Liaw, P.K.: Deviation from high-entropy configurations in the atomic distributions of a multi-principal-element alloy. Nat. Commun. 6, 5964 (2015).CrossRefGoogle ScholarPubMed
Huhn, W.P. and Widom, M.: Prediction of A2 to B2 phase transition in the high-entropy alloy Mo–Nb–Ta–W. JOM 65, 1772 (2013).CrossRefGoogle Scholar
Lederer, Y., Toher, C., Vecchio, K.S., and Curtarolo, S.: The search for high entropy alloys: A high-throughput ab initio approach (2017). https://arxiv.org/abs/1711.03426.Google Scholar
Bragg, W.L. and Williams, E.J.: The effect of thermal agitation on atomic arrangement in alloys. Proc. R. Soc. London, Ser. A 145, 699 (1934).CrossRefGoogle Scholar
Fowler, R.H. and Guggenheim, E.A.: Statistical thermodynamics of super-latices. Proc. R. Soc. London, Ser. A 174, 189 (1939).CrossRefGoogle Scholar
Bethe, H.A.: Statistical theory of superlattices. Proc. R. Soc. London, Ser. A 150, 552 (1935).CrossRefGoogle Scholar
Kikuchi, R.: A theory of cooperative phenomena. Phys. Rev. 81, 988 (1951).CrossRefGoogle Scholar
Widom, M.: Entropy and diffuse scattering: Comparison of NbTiVZr and CrMoNbV. Metall. Mater. Trans. A 47, 3306 (2016).CrossRefGoogle Scholar
Lucas, M.S., Belyea, D., Bauer, C., Bryant, N., Michel, E., Turgut, Z., Leontsev, S.O., Horwath, J., Semiatin, S.L., McHenry, M.E., and Miller, C.W.: Thermomagnetic analysis of FeCoCrxNi alloys: Magnetic entropy of high-entropy alloys. J. Appl. Phys. 113, 17A923 (2013).CrossRefGoogle Scholar
Li, P., Wang, A., and Liu, C.T.: A ductile high entropy alloy with attractive magnetic properties. J. Alloys Compd. 694, 55 (2017).CrossRefGoogle Scholar
Schneeweiss, O., Friák, M., Dudová, M., Holec, D., Šob, M., Kriegner, D., Holý, V., Beran, P., George, E.P., Neugebauer, J., and Dlouhý, A.: Magnetic properties of the CrMnFeCoNi high-entropy alloy. Phys. Rev. B 96, 014437 (2017).CrossRefGoogle Scholar
Van de Walle, A., Asta, M., and Ceder, G.: The alloy theoretic automated toolkit: A user guide. CALPHAD: Comput. Coupling Phase Diagrams Thermochem. 26, 539 (2002).CrossRefGoogle Scholar
Ravi, C., Panigrahi, B.K., Valsakumar, M.C., and van de Walle, A.: First-principles calculation of phase equilibrium of V–Nb, V–Ta, and Nb–Ta alloys. Phys. Rev. B 85, 054202 (2012).CrossRefGoogle Scholar
van de Walle, A.: Multicomponent multisublattice alloys, nonconfigurational entropy and other additions to the alloy theoretic automated toolkit. CALPHAD: Comput. Coupling Phase Diagrams Thermochem. 33, 266 (2009).CrossRefGoogle Scholar
Widom, M., Huhn, W.P., Maiti, S., and Steurer, W.: Hybrid Monte Carlo/molecular dynamics simulation of a refractory metal high entropy alloy. Metall. Mater. Trans. A 45, 196 (2014).CrossRefGoogle Scholar
Yao, S., Gao, Q., and Widom, M.: Phase diagram of boron carbide with variable carbon composition. Phys. Rev. B 95 (2017).CrossRefGoogle Scholar
Niu, C., Windl, W., and Ghazisaeidi, M.: Multi-cell Monte Carlo relaxation method for predicting phase stability of alloys. Scr. Mater. 132, 9 (2017).CrossRefGoogle Scholar
Widom, M.: Prediction of structure and phase transformations. In High-Entropy Alloys: Fundamentals and Applications, Gao, Y., Yeh, M.C., , J, -Liaw, W., and Zhang, P.K., eds. (Springer International Publishing, Cham, 2016); pp. 267–298.CrossRefGoogle Scholar
Feng, B. and Widom, M.: Elastic stability and lattice distortion of refractory high entropy alloys. Mater. Chem. Phys. 210, 309 (2018).CrossRefGoogle Scholar
Oh, H., Ma, D., Leyson, G., Grabowski, B., Park, E., Körmann, F., and Raabe, D.: Lattice distortions in the FeCoNiCrMn high entropy alloy studied by theory and experiment. Entropy 18, 321 (2016).CrossRefGoogle Scholar
Fernández-Caballero, A., Wróbel, J.S., Mummery, P.M., and Nguyen-Manh, D.: Short-range order in high entropy alloys: Theoretical formulation and application to Mo–Nb–Ta–V–W system. J. Phase Equilib. Diffus. 38, 391 (2017).CrossRefGoogle Scholar
Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620 (1957).CrossRefGoogle Scholar
Gao, M.C. and Widom, M.: Information entropy of liquid metals. J. Phys. Chem. B 122, 3550 (2018).CrossRefGoogle ScholarPubMed
Kikuchi, R.: Second Hessian determinant as the criterion for order (first or second) of phase transition. Phys. A 142, 321 (1987).CrossRefGoogle Scholar
Gao, M.C., Gao, P., Hawk, J.A., Ouyang, L., Alman, D.E., and Widom, M.: Computational modeling of high-entropy alloys: Structures, thermodynamics and elasticity. J. Mater. Res. 32, 3627 (2017).CrossRefGoogle Scholar
Mao, H., Chen, H-L., and Chen, Q.: TCHEA1: A thermodynamic database not limited for “high entropy” alloys. J. Phase Equilib. Diffus. 38, 353 (2017).CrossRefGoogle Scholar
Open CALPHAD: http://www.opencalphad.com/.Google Scholar
Wang, W-R., Wang, W-L., Wang, S-C., Tsai, Y-C., Lai, C-H., and Yeh, J-W.: Effects of Al addition on the microstructure and mechanical property of AlxCoCrFeNi high-entropy alloys. Intermetallics 26, 44 (2012).CrossRefGoogle Scholar
Tian, F., Delczeg, L., Chen, N., Varga, L.K., Shen, J., and Vitos, L.: Structural stability of NiCoFeCrAlx high-entropy alloy from ab initio theory. Phys. Rev. B 88, 085128 (2013).CrossRefGoogle Scholar
Kao, Y-F., Chen, T-J., Chen, S-K., and Yeh, J-W.: Microstructure and mechanical property of as-cast, -homogenized, and -deformed AlxCoCrFeNi (0 ≤ x ≤ 2) high-entropy alloys. J. Alloys Compd. 488, 57 (2009).CrossRefGoogle Scholar
Lee, S. and Hoffmann, R.: Bcc and fcc transition metals and alloys: A central role for the Jahn-Teller effect in explaining their ideal and distorted structures. J. Am. Chem. Soc. 124, 4811 (2002).CrossRefGoogle ScholarPubMed
Guo, S., Ng, C., Lu, J., and Liu, C.T.: Effect of valence electron concentration on stability of fcc or bcc phase in high entropy alloys. J. Appl. Phys. 109, 103505 (2011).CrossRefGoogle Scholar
Singh, P., Smirnov, A.V., and Johnson, D.D.: Atomic short-range order and incipient long-range order in high-entropy alloys. Phys. Rev. B: Condens. Matter Mater. Phys. 91, 1 (2015).CrossRefGoogle Scholar
Niu, C., Zaddach, A.J., Oni, A.A., Sang, X., Hurt, J.W., LeBeau, J.M., Koch, C.C., and Irving, D.L.: Spin-driven ordering of Cr in the equiatomic high entropy alloy NiFeCrCo. Appl. Phys. Lett. 106, 161906 (2015).CrossRefGoogle Scholar
Niu, C., LaRosa, C.R., Miao, J., Mills, M.J., and Ghazisaeidi, M.: Magnetically-driven phase transformation strengthening in high entropy alloys. Nat. Commun. 9, 1363 (2018).CrossRefGoogle ScholarPubMed
Wang, Y., Yan, M., Zhu, Q., Wang, W.Y., Wu, Y., Hui, X., Otis, R., Shang, S-L., Liu, Z-K., and Chen, L-Q.: Computation of entropies and phase equilibria in refractory V–Nb–Mo–Ta–W high-entropy alloys. Acta Mater. 143, 88 (2018).CrossRefGoogle Scholar
Troparevsky, M.C., Morris, J.R., Kent, P.R.C., Lupini, A.R., and Stocks, G.M.: Criteria for predicting the formation of single-phase high-entropy alloys. Phys. Rev. X 5, 1 (2015).Google Scholar
Automatic FLOW for Materials Discovery: http://aflowlib.org (2018).Google Scholar
The Materials Project: http://materialsproject.org (2018).Google Scholar
Widom, M. and Mihalkovic, M.: Alloy database: http://alloy.phys.cmu.edu (2018).Google Scholar
The Open Quantum Materials Database: http://oqmd.org (2018).Google Scholar
Körmann, F., Ruban, A.V., and Sluiter, M.H.F.: Long-ranged interactions in bcc NbMoTaW high-entropy alloys. Mater. Res. Lett. 5, 35 (2017).CrossRefGoogle Scholar
Nelson, L.J., Hart, G.L.W., Zhou, F., and Ozoliņš, V.: Compressive sensing as a paradigm for building physics models. Phys. Rev. B: Condens. Matter Mater. Phys. 87, 1 (2013).CrossRefGoogle Scholar
Nelson, L.J., Ozoliņš, V., Reese, C.S., Zhou, F., and Hart, G.L.W.: Cluster expansion made easy with Bayesian compressive sensing. Phys. Rev. B: Condens. Matter Mater. Phys. 88, 1 (2013).CrossRefGoogle Scholar
Taylor, C.D., Lu, P., Saal, J., Frankel, G.S., and Scully, J.R.: Integrated computational materials engineering of corrosion resistant alloys. npj Mater. Degrad. 2, 6 (2018).CrossRefGoogle Scholar
Youssef, K.M., Zaddach, A.J., Niu, C., Irving, D.L., and Koch, C.C.: A novel low-density, high-hardness, high-entropy alloy with close-packed single-phase nanocrystalline structures. Mater. Res. Lett. 3, 95 (2015).CrossRefGoogle Scholar
Niu, C., Zaddach, A.J., Koch, C.C., and Irving, D.L.: First principles exploration of near-equiatomic NiFeCrCo high entropy alloys. J. Alloys Compd. 672, 510 (2016).CrossRefGoogle Scholar
Cao, P., Ni, X., Tian, F., Varga, L.K., and Vitos, L.: Ab initio study of AlxMoNbTiV high-entropy alloys. J. Phys.: Condens. Matter 27, 075401 (2015).Google ScholarPubMed
Tian, L-Y., Hu, Q-M., Yang, R., Zhao, J., Johansson, B., and Vitos, L.: Elastic constants of random solid solutions by SQS and CPA approaches: The case of fcc Ti–Al. J. Phys.: Condens. Matter 27, 315702 (2015).Google ScholarPubMed
Tian, F., Wang, Y., Irving, D.L., and Vitos, L.: High-Entropy Alloy (Springer International Publishing, Cham, 2016); pp. 299–332.CrossRefGoogle Scholar
Tian, F., Varga, L.K., Chen, N., Delczeg, L., and Vitos, L.: Ab initio investigation of high-entropy alloys of 3d elements. Phys. Rev. B 87, 075144 (2013).CrossRefGoogle Scholar
Tian, F., Varga, L.K., Shen, J., and Vitos, L.: Calculating elastic constants in high-entropy alloys using the coherent potential approximation: Current issues and errors. Comput. Mater. Sci. 111, 350 (2016).CrossRefGoogle Scholar
Körmann, F. and Sluiter, M.: Interplay between lattice distortions, vibrations and phase stability in NbMoTaW high entropy alloys. Entropy 18, 403 (2016).CrossRefGoogle Scholar
Körmann, F., Ikeda, Y., Grabowski, B., and Sluiter, M.H.F.: Phonon broadening in high entropy alloys. npj Comput. Mater. 3, 36 (2017).CrossRefGoogle Scholar
Leong, Z., Wróbel, J.S., Dudarev, S.L., Goodall, R., Todd, I., and Nguyen-Manh, D.: The effect of electronic structure on the phases present in high entropy alloys. Sci. Rep. 7, 39803 (2017).CrossRefGoogle ScholarPubMed
Rak, Z., Rost, C.M., Lim, M., Sarker, P., Toher, C., Curtarolo, S., Maria, J-P., and Brenner, D.W.: Charge compensation and electrostatic transferability in three entropy-stabilized oxides: Results from density functional theory calculations. J. Appl. Phys. 120, 095105 (2016).CrossRefGoogle Scholar
Maiti, S. and Steurer, W.: Structural-disorder and its effect on mechanical properties in single-phase TaNbHfZr high-entropy alloy. Acta Mater. 106, 87 (2016).CrossRefGoogle Scholar
Meraj, M. and Pal, S.: Deformation of Ni20W20Cu20Fe20Mo20 high entropy alloy for tensile followed by compressive and compressive followed by tensile loading: A molecular dynamics simulation based study. IOP Conf. Ser.: Mater. Sci. Eng. 115, 012019 (2016).CrossRefGoogle Scholar
Li, J., Fang, Q., Liu, B., Liu, Y., and Liu, Y.: Mechanical behaviors of AlCrFeCuNi high-entropy alloys under uniaxial tension via molecular dynamics simulation. RSC Adv. 6, 76409 (2016).CrossRefGoogle Scholar
Xie, L., Brault, P., Thomann, A-L., Yang, X., Zhang, Y., and Shang, G.: Molecular dynamics simulation of Al–Co–Cr–Cu–Fe–Ni high entropy alloy thin film growth. Intermetallics 68, 78 (2016).CrossRefGoogle Scholar
Liu, Z., Lei, Y., Gray, C., and Wang, G.: Examination of solid-solution phase formation rules for high entropy alloys from atomistic Monte Carlo simulations. JOM 67, 2364 (2015).CrossRefGoogle Scholar
Varvenne, C., Luque, A., and Curtin, W.A.: Theory of strengthening in fcc high entropy alloys. Acta Mater. 118, 164 (2016).CrossRefGoogle Scholar
Choi, W-M., Jo, Y.H., Sohn, S.S., Lee, S., and Lee, B-J.: Understanding the physical metallurgy of the CoCrFeMnNi high-entropy alloy: An atomistic simulation study. npj Comput. Mater. 4, 1 (2018).CrossRefGoogle Scholar
Choi, W-M., Kim, Y., Seol, D., and Lee, B-J.: Modified embedded-atom method interatomic potentials for the Co–Cr, Co–Fe, Co–Mn, Cr–Mn, and Mn–Ni binary systems. Comput. Mater. Sci. 130, 121 (2017).CrossRefGoogle Scholar
Yadav, T.P., Mukhopadhyay, S., Mishra, S.S., Mukhopadhyay, N.K., and Srivastava, O.N.: Synthesis of a single phase of high-entropy Laves intermetallics in the Ti–Zr–V–Cr–Ni equiatomic alloy. Philos. Mag. Lett. 97, 494 (2017).CrossRefGoogle Scholar
Poletti, M.G., Fiore, G., Szost, B.A., and Battezzati, L.: Search for high entropy alloys in the X–NbTaTiZr systems (X = Al, Cr, V, Sn). J. Alloys Compd. 620, 283 (2015).CrossRefGoogle Scholar
King, D.J.M., Middleburgh, S.C., McGregor, A.G., and Cortie, M.B.: Predicting the formation and stability of single phase high-entropy alloys. Acta Mater. 104, 172 (2016).CrossRefGoogle Scholar
Fernandez, J.F., Widom, M., Cuevas, F., Ares, J.R., Bodega, J., Leardini, F., Mihalkovič, M., and Sánchez, C.: First-principles phase stability calculations and estimation of finite temperature effects on pseudo-binary Mg6(PdxNi1−x) compounds. Intermetallics 19, 502 (2010).CrossRefGoogle Scholar
Gao, M.C., Suzuki, Y., Schweiger, H., Doǧan, Ö.N., Hawk, J., and Widom, M.: Phase stability and elastic properties of Cr–V alloys. J. Phys.: Condens. Matter 25 (2013).Google ScholarPubMed
Wang, Z., Li, J., Fang, Q., Liu, B., and Zhang, L.: Investigation into nanoscratching mechanical response of AlCrCuFeNi high-entropy alloys using atomic simulations. Appl. Surf. Sci. 416, 470 (2017).CrossRefGoogle Scholar
Sharma, A., Singh, P., Johnson, D.D., Liaw, P.K., and Balasubramanian, G.: Atomistic clustering-ordering and high-strain deformation of an Al0.1CrCoFeNi high-entropy alloy. Sci. Rep. 6, 31028 (2016).CrossRefGoogle ScholarPubMed
Zhang, F., Zhang, C., Chen, S.L., Zhu, J., Cao, W.S., and Kattner, U.R.: An understanding of high entropy alloys from phase diagram calculations. Calphad 45, 1 (2014).CrossRefGoogle Scholar
Guruvidyathri, K., Hari Kumar, K.C., Yeh, J.W., and Murty, B.S.: Topologically close-packed phase formation in high entropy alloys: A review of calphad and experimental results. JOM 69, 2113 (2017).CrossRefGoogle Scholar
Zhang, C., Zhang, F., Chen, S., and Cao, W.: Computational thermodynamics aided high-entropy alloy design. JOM 64, 839 (2012).CrossRefGoogle Scholar
Gao, M. and Alman, D.: Searching for next single-phase high-entropy alloy compositions. Entropy 15, 4504 (2013).CrossRefGoogle Scholar
Zhang, C. and Gao, M.C.: High-Entropy Alloy (Springer International Publishing, Cham, 2016); pp. 399–444.CrossRefGoogle Scholar
Durga, A., Hari Kumar, K.C., and Murty, B.S.: Phase formation in equiatomic high entropy alloys: CALPHAD approach and experimental studies. Trans. Indian Inst. Met. 65, 375 (2012).CrossRefGoogle Scholar
Saal, J.E., Berglund, I.S., Sebastian, J.T., Liaw, P.K., and Olson, G.B.: Equilibrium high entropy alloy phase stability from experiments and thermodynamic modeling. Scr. Mater. 146, 5 (2018).CrossRefGoogle Scholar
Zhang, B., Gao, M.C., Zhang, Y., and Guo, S.M.: Senary refractory high-entropy alloy CrxMoNbTaVW. Calphad 51, 193 (2015).CrossRefGoogle Scholar
Ma, D., Yao, M., Pradeep, K.G., Tasan, C.C., Springer, H., and Raabe, D.: Phase stability of non-equiatomic CoCrFeMnNi high entropy alloys. Acta Mater. 98, 288 (2015).CrossRefGoogle Scholar
Choi, W-M., Jung, S., Jo, Y.H., Lee, S., and Lee, B-J.: Design of new face-centered cubic high entropy alloys by thermodynamic calculation. Met. Mater. Int. 23, 839 (2017).CrossRefGoogle Scholar