{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T03:48:01Z","timestamp":1777607281979,"version":"3.51.4"},"reference-count":34,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2016,8,20]],"date-time":"2016-08-20T00:00:00Z","timestamp":1471651200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["KO 5080\/1-1"],"award-info":[{"award-number":["KO 5080\/1-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Refractory high entropy alloys (HEA), such as BCC NbMoTaW, represent a promising materials class for next-generation high-temperature applications, due to their extraordinary mechanical properties. A characteristic feature of HEAs is the formation of single-phase solid solutions. For BCC NbMoTaW, recent computational studies revealed, however, a B2(Mo,W;Nb,Ta)-ordering at ambient temperature. This ordering could impact many materials properties, such as thermodynamic, mechanical, or diffusion properties, and hence be of relevance for practical applications. In this work, we theoretically address how the B2-ordering impacts thermodynamic properties of BCC NbMoTaW and how the predicted ordering temperature itself is affected by vibrations, electronic excitations, lattice distortions, and relaxation energies.<\/jats:p>","DOI":"10.3390\/e18080403","type":"journal-article","created":{"date-parts":[[2016,8,22]],"date-time":"2016-08-22T10:40:33Z","timestamp":1471862433000},"page":"403","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":73,"title":["Interplay between Lattice Distortions, Vibrations and Phase Stability in NbMoTaW High Entropy Alloys"],"prefix":"10.3390","volume":"18","author":[{"given":"Fritz","family":"K\u00f6rmann","sequence":"first","affiliation":[{"name":"Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands"}]},{"given":"Marcel","family":"Sluiter","sequence":"additional","affiliation":[{"name":"Department of Materials Science and Engineering, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands"}]}],"member":"1968","published-online":{"date-parts":[[2016,8,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"189","DOI":"10.3390\/e18050189","article-title":"MoNbTaV medium-entropy alloy","volume":"18","author":"Yao","year":"2016","journal-title":"Entropy"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"7748","DOI":"10.1038\/ncomms8748","article-title":"Ultrastrong ductile and stable high-entropy alloys at small scales","volume":"6","author":"Zou","year":"2015","journal-title":"Nat. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1758","DOI":"10.1016\/j.intermet.2010.05.014","article-title":"Refractory high-entropy alloys","volume":"18","author":"Senkov","year":"2010","journal-title":"Intermetallics"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"698","DOI":"10.1016\/j.intermet.2011.01.004","article-title":"Mechanical properties of Nb25Mo25Ta25W25 and V20Nb20Mo 20Ta20W20 refractory high entropy alloys","volume":"19","author":"Senkov","year":"2011","journal-title":"Intermetallics"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.actamat.2016.01.018","article-title":"Structural-disorder and its effect on mechanical properties in single-phase TaNbHfZr high-entropy alloy","volume":"106","author":"Maiti","year":"2016","journal-title":"Acta Mater."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1016\/j.actamat.2013.11.049","article-title":"Size-dependent plasticity in an Nb25Mo25Ta25W25 refractory high-entropy alloy","volume":"65","author":"Zou","year":"2014","journal-title":"Acta Mater."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Murty, B.S., Yeh, J.W., and Ranganathan, S. (2014). High-Entropy Alloys, Butterworth-Heinemann.","DOI":"10.1016\/B978-0-12-800251-3.00002-X"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.pmatsci.2013.10.001","article-title":"Microstructures and properties of high-entropy alloys","volume":"61","author":"Zhang","year":"2014","journal-title":"Prog. Mater. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Gao, M.C., Yeh, J.W., Liaw, P.K., and Zhang, Y. (2016). High-Entropy Alloys: Fundamentals and Applications, Springer. [1st ed.].","DOI":"10.1007\/978-3-319-27013-5"},{"key":"ref_10","unstructured":"Huhn, W.P. (2014). Thermodynamics from First Principles: Prediction of Phase Diagrams and Materials Properties Using Density Functional Theory. [Ph.D. Thesis, Carnegie Mellon University]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1772","DOI":"10.1007\/s11837-013-0772-3","article-title":"Prediction of A2 to B2 phase transition in the high-entropy alloy Mo-Nb-Ta-W","volume":"65","author":"Huhn","year":"2013","journal-title":"JOM"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1007\/s11661-013-2000-8","article-title":"Hybrid monte carlo\/molecular dynamics simulation of a refractory metal high entropy alloy","volume":"45","author":"Widom","year":"2014","journal-title":"Metall. Mater. Trans. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.jallcom.2012.04.053","article-title":"Determination of the transition to the high entropy regime for alloys of refractory elements","volume":"534","author":"Bozzolo","year":"2012","journal-title":"J. Alloy. Compd."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3285","DOI":"10.1016\/j.physb.2011.12.088","article-title":"Modeling of high entropy alloys of refractory elements","volume":"407","author":"Bozzolo","year":"2012","journal-title":"Physica B"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"K\u00f6rmann, F., Ruban, A.V., and Sluiter, M.H. (2016). Long-ranged interactions in bcc NbMoTaW high-entropy alloys. Mater. Res. Lett., 1\u20136.","DOI":"10.1080\/21663831.2016.1198837"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"11169","DOI":"10.1103\/PhysRevB.54.11169","article-title":"Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set","volume":"54","author":"Kresse","year":"1996","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0927-0256(96)00008-0","article-title":"Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set","volume":"6","author":"Kresse","year":"1996","journal-title":"Comput. Mater. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"17953","DOI":"10.1103\/PhysRevB.50.17953","article-title":"Projector augmented-wave method","volume":"50","year":"1994","journal-title":"Phys. Rev. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3865","DOI":"10.1103\/PhysRevLett.77.3865","article-title":"Generalized gradient approximation made simple","volume":"77","author":"Perdew","year":"1996","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1103\/PhysRevLett.65.353","article-title":"Special quasirandom structures","volume":"65","author":"Zunger","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Gao, M.C., Niu, C., Jiang, C., and Irving, D.L. (2016). High-Entropy Alloys, Springer.","DOI":"10.1007\/978-3-319-27013-5_11"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"9319","DOI":"10.1029\/JB092iB09p09319","article-title":"Compressibility of solids","volume":"92","author":"Vinet","year":"1987","journal-title":"J. Geophys. Res. Solid Earth"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1103\/PhysRevB.37.790","article-title":"Calculated thermal properties of metals","volume":"37","author":"Moruzzi","year":"1988","journal-title":"Phys. Rev. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1016\/j.actamat.2015.08.050","article-title":"Ab initio thermodynamics of the CoCrFeMnNi high entropy alloy: Importance of entropy contributions beyond the configurational one","volume":"100","author":"Ma","year":"2015","journal-title":"Acta Mater."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Oh, H.S., Ma, D., Leyson, G.P., Grabowski, B., Park, E.S., K\u00f6rmann, F., and Raabe, D. (2016). Lattice distortions in the FeCoCrNiMn high entropy alloy studied by theory and experiment. Entropy, submitted.","DOI":"10.3390\/e18090321"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"5792","DOI":"10.1103\/PhysRevB.39.5792","article-title":"Electronic theory for phase stability of nine AB binary alloys, with A = Ni, Pd, or Pt and B = Cu, Ag, or Au","volume":"39","author":"Takizawa","year":"1989","journal-title":"Phys. Rev. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"12587","DOI":"10.1103\/PhysRevB.46.12587","article-title":"Efficient cluster expansion for substitutional systems","volume":"46","author":"Laks","year":"1992","journal-title":"Phys. Rev. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"512","DOI":"10.1103\/PhysRevB.44.512","article-title":"First-principles statistical mechanics of structural stability of intermetallic compounds","volume":"44","author":"Lu","year":"1991","journal-title":"Phys. Rev. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"015012","DOI":"10.1088\/0965-0393\/18\/1\/015012","article-title":"Visualization and analysis of atomistic simulation data with OVITO\u2013the Open Visualization Tool","volume":"18","author":"Stukowski","year":"2009","journal-title":"Modell. Simul. Mater. Sci. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1016\/j.actamat.2015.07.010","article-title":"Interatomic spacing distribution in multicomponent alloys","volume":"97","author":"Dudarev","year":"2015","journal-title":"Acta Mater."},{"key":"ref_31","unstructured":"Widom, M. (2016). High-Entropy Alloys, Springer."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"773","DOI":"10.1016\/j.jallcom.2016.04.320","article-title":"Thermodynamic properties of refractory high entropy alloys","volume":"682","author":"Song","year":"2016","journal-title":"J. Alloy. Compd."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"395401","DOI":"10.1088\/0953-8984\/21\/39\/395401","article-title":"Phonon spectra of PdxFe1\u2212x alloys with transferable force constants","volume":"21","author":"Dutta","year":"2009","journal-title":"J. Phys. Condens. Matter"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"485403","DOI":"10.1088\/0953-8984\/23\/48\/485403","article-title":"Phonon dispersions in random alloys: A method based on special quasi-random structure force constants","volume":"23","author":"Wang","year":"2011","journal-title":"J. Phys. Condens. 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