Temperature-dependent structural, mechanical, and thermodynamic properties of B2-phase Ti2AlNb for aerospace applications
Source abstract
Abstract
Ti2AlNb intermetallic is a potential substitute for Ni-based superalloys in next-generation aerospace materials. In the current work, the structural, mechanical, and thermodynamic properties of the Ti2AlNb in B2 phase are studied at various temperatures by implementing first principle calculations under quasi-harmonic approximations. The lattice parameter, elastic moduli, and linear coefficient of thermal expansion (CTE) are well reproduced according to the experimental reports. The intermetallic remains mechanically and dynamically stable at all temperatures. The elastic moduli and microhardness decrease slightly from 0 to 1300 K. Furthermore, the calculations reveal that Ti2AlNb remains ductile at all temperatures. Phonon calculations show that Nb atoms are dominant contributor to the vibrational modes. The temperature-dependent heat capacities, entropy, CTE, and isothermal bulk modulus are further investigated. Present calculations predict that the B2 phase Ti2AlNb is a suitable candidate for the manufacturing of high-temperature application parts such as turbine blades of aerospace engines.
Source: expertdata · Quality: candidate unverified · Source record ↗
Research context
OpenAlex topics: MXene and MAX Phase Materials · Intermetallics and Advanced Alloy Properties · Boron and Carbon Nanomaterials Research
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (88)
Show source-provided references
1. Boyer RR (1996) An overview on the use of titanium in the aerospace industry. Mater Sci Eng A 213:103–114. https://doi.org/10.1016/0921-5093(96)10233-110.1016/0921-5093(96)10233-1 ↗
2. Veiga C, Loureiro AJR, Davim JP (2012) Properties and applications of titanium alloys. Rev Adv Mater Sci 32:133–148
3. Huda Z, Edi P (2013) Materials selection in design of structures and engines of supersonic aircrafts: a review. Mater Des 46:552–560. https://doi.org/10.1016/j.matdes.2012.10.00110.1016/j.matdes.2012.10.001 ↗
4. Moiseyev VN (2005) Titanium alloys: Russian aircraft and aerospace applications. CRC Press United States of America, ISBN: 0–8493–3273–710.1201/9781420037678 ↗
5. O’Neal J, Animagraffs. Inside a Jet Engine. http://animagraffs.com/inside-a-jet-engine/
6. Takehiro O (2015) Materials for Aircraft Engines, ASEN 5063 Aircraft Propulsion Final Report. www.colorado.edu
7. Peters M, Kumpfert J, Ward CH, Leyens C (2003) Titanium alloys for aerospace applications. Adv Eng Mater 5:419–427. https://doi.org/10.1002/adem.20031009510.1002/adem.200310095 ↗
8. Meetham GW (1991) High-temperature materials—a general review. J Mater Sci 26:853–860. https://doi.org/10.1007/BF0057675910.1007/BF00576759 ↗
9. Banerjee D, Gogia AK, Nandi TK, Joshi VA (1988) A new ordered orthorhombic phase in a Ti3Al-Nb alloy. Acta Metall 36:871–882. https://doi.org/10.1016/0001-6160(88)90141-110.1016/0001-6160(88)90141-1 ↗
10. Lipsitt HA (1985) Titanium Aluminides—an Overview. Mater Res Soc Symp Proc 39:351–364. https://doi.org/10.1557/proc-39-35110.1557/proc-39-351 ↗
11. Valkov S, Bezdushnyi R, Petrov P (2018) Synthesis, structure and mechanical properties of Ti-Al-Nb coatings formed by electron beam additive technique. Vacuum 156:140–145. https://doi.org/10.1016/j.vacuum.2018.07.02110.1016/j.vacuum.2018.07.021 ↗
12. Hao YL, Xu DS, Cui YY, Yang R, Li D (1999) The site occupancies of alloying elements in TiAl and Ti3Al alloys. Acta Mater 47:1129–1139. https://doi.org/10.1016/S1359-6454(99)00006-310.1016/S1359-6454(99)00006-3 ↗
13. Tan BT, Zhang J, Sopiha KV, Wu P (2019) Multi-doping effect on ductility of TiAl3: a DFT study. Met Mater Int 25:869–879. https://doi.org/10.1007/s12540-018-00213-y10.1007/s12540-018-00213-y ↗
14. Goyal K, Sardana N (2021) Phase stability and microstructural evolution of Ti2AlNb alloys-a review. Mater Today Proc 41:951–968. https://doi.org/10.1016/j.matpr.2020.10.92510.1016/j.matpr.2020.10.925 ↗
15. Zhang H, Yan N, Liang H, Liu Y (2021) Phase transformation and microstructure control of Ti2AlNb-based alloys: a review. J Mater Sci Technol 80:203–216. https://doi.org/10.1016/j.jmst.2020.11.02210.1016/j.jmst.2020.11.022 ↗
16. Gogia AK, Banerjee D, Nandy TK (1990) Structure, tensile deformation, and fracture of a Ti3Al-Nb alloy. Metall Trans A 21:609–625. https://doi.org/10.1007/BF0267193310.1007/BF02671933 ↗
17. Boehlert CJ, Majumdar BS, Seetharaman V, Miracle DB (1999) Part I. The microstructural evolution in Ti-AI-Nb O + Bcc orthorhombic alloys. Metall. Mater Trans A Phys Metall Mater Sci 30:2305–2323. https://doi.org/10.1007/s11661-999-0240-410.1007/s11661-999-0240-4 ↗
18. Goyal K, Sardana N (2021) Mechanical Properties of the Ti2AlNb Intermetallic: a review. Trans Indian Inst Met 74:1839–1853. https://doi.org/10.1007/s12666-021-02307-510.1007/s12666-021-02307-5 ↗
19. Shao B, Shan D, Guo B, Zong Y (2019) Plastic deformation mechanism and interaction of B2, α2, and O phases in Ti-22Al-25Nb alloy at room temperature. Int J Plast 113:18–34. https://doi.org/10.1016/j.ijplas.2018.09.00410.1016/j.ijplas.2018.09.004 ↗
20. Popille F, Douin J (1996) Comparison of the deformation microstructures at room temperature in O and B2 phases of a Ti2AlNb alloy. J Phys IV JP 6:211–216. https://doi.org/10.1051/jp4:199622910.1051/jp4:1996229 ↗
21. Cai XQ, Wang Y, Yang ZW, Wang DP, Liu YC (2016) Transient liquid phase (TLP) bonding of Ti2AlNb alloy using Ti/Ni interlayer: Microstructure characterization and mechanical properties. J Alloys Compd 679:9–17. https://doi.org/10.1016/j.jallcom.2016.03.28610.1016/j.jallcom.2016.03.286 ↗
22. Chen X, Xie FQ, Ma TJ, Li WY, Wu XQ (2015) Microstructure evolution and mechanical properties of linear friction welded Ti2AlNb alloy. J Alloys Compd 646:490–496. https://doi.org/10.1016/j.jallcom.2015.05.19810.1016/j.jallcom.2015.05.198 ↗
23. Feng G, Wei Y, Hu B, Wang Y, Deng D, Yang X (2021) Vacuum diffusion bonding of Ti2AlNb alloy and TC4 alloy. Trans Nonferrous Met Soc China 31:2677–2686. https://doi.org/10.1016/s1003-6326(21)65684-410.1016/s1003-6326(21)65684-4 ↗
24. Niu JB, Wang Y, Yang ZW, Wang DP (2019) Microstructure and mechanical properties of titanium–zirconium–molybdenum and Ti2AlNb joint diffusion bonded with and without a Ni interlayer. Adv Eng Mater 21:1–10. https://doi.org/10.1002/adem.20190071310.1002/adem.201900713 ↗
25. Wang Y, Hu B, Wei Y, Feng G, Deng D (2021) Microstructure and mechanical properties of vacuum diffusion bonded Ti2AlNb/ti/TC4 joint. Crystals 11:770-781 https://doi.org/10.3390/cryst1107077010.3390/cryst11070770 ↗
26. Ravi C, Vajeeston P, Mathijaya S, Asokamani R (1999) Electronic structure, phase stability, and cohesive properties of Ti2XAl (X = Nb, V, Zr). Phys Rev B 60:15683–15689. https://doi.org/10.1103/PhysRevB.60.1568310.1103/PhysRevB.60.15683 ↗
27. Cui X, Jinlong Y, Qunxiang L, Shangda X (1999) Electronic structure of Ti2AlNb (O phase). J Phys Condens Matter 11:6179–6186. https://doi.org/10.1088/0953-8984/11/32/30910.1088/0953-8984/11/32/309 ↗
28. Hu M, Yang R, Xu S, Hao L, Li D, Wu T (2003) Geometric and electronic structure of Ti2AlX (X=V, Nb, or Ta). Phys Rev B Condens Matter Mater Phys 68:054102-1-054102-7. https://doi.org/10.1103/PhysRevB.68.05410210.1103/PhysRevB.68.054102 ↗
29. Pathak A, Singh AK (2015) A first principles study of Ti2AlNb intermetallic. Solid State Commun 204:9–15. https://doi.org/10.1016/j.ssc.2014.12.00210.1016/j.ssc.2014.12.002 ↗
30. Goyal K, Bera C, Sardana N (2021) Phase stability, mechanical, thermal, electronic properties, anisotropy, lattice dynamics and APB -energies of Ti2AlX intermetallics in α2, B2, and O phases: A First Principle Study. Mater Today Commun 29:102864-1-102864-20. https://doi.org/10.1016/j.mtcomm.2021.10286410.1016/j.mtcomm.2021.102864 ↗
31. Wang A, Liu P, Xie J, Ma D, Mao Z (2020) First-principles investigation on the atomic structure, stability and electronic property of O(001)/B2(110) interface in Ti2AlNb alloys. J Alloys Compd 817:152734-152742. https://doi.org/10.1016/j.jallcom.2019.15273410.1016/j.jallcom.2019.152734 ↗
32. Hu K, Huang J, Wei Z, Peng Q, Xie Z, Sa B, Wu B (2017) Elastic and thermodynamic properties of the Ti2AlNb orthorhombic phase from first-principles calculations. Phys Status Solidi 254:1600634-1600642. https://doi.org/10.1002/pssb.20160063410.1002/pssb.201600634 ↗
33. Wei ZY, Hu KM, Sa BS, Wu B (2017) Pressure-induced structure, electronic, thermodynamic, and mechanical properties of Ti2AlNb orthorhombic phase by first-principles calculations. Rare Met 40:1–11. https://doi.org/10.1007/s12598-017-0915-810.1007/s12598-017-0915-8 ↗
34. Esin VA, Mallick R, Dadé M, Denand B, Delfosse J, Sallot P (2020) Combined synchrotron X-ray diffraction, dilatometry and electrical resistivity in situ study of phase transformations in a Ti2AlNb alloy. Mater Charact 169:110654-110663. https://doi.org/10.1016/j.matchar.2020.11065410.1016/j.matchar.2020.110654 ↗
35. Rugut E, Joubert D, Jones G (2021) First principle studies on lattice thermal conductivity and thermoelectric properties of ScCu(S, Se, Te)2. Mater Today Commun 26:101905-101914. https://doi.org/10.1016/j.mtcomm.2020.10190510.1016/j.mtcomm.2020.101905 ↗
36. Duong TC, Singh N, Arróyave R (2013) First-principles calculations of finite-temperature elastic properties of Ti2AlX (X = C or N). Comput Mater Sci 79:296–302. https://doi.org/10.1016/j.commatsci.2013.06.03310.1016/j.commatsci.2013.06.033 ↗
37. Pan R-K, Wang H-C, Shi T-T, Tian X, Tang B-Y (2016) Thermal properties and thermoelasticity of L12 ordered Al3RE (RE=Er, Tm, Yb, Lu) phases: a first-principles study. Mater Des 102:100–105. https://doi.org/10.1016/j.matdes.2016.03.16510.1016/j.matdes.2016.03.165 ↗
38. Xu WW, Han JJ, Wang Y, Wang CP, Liu XJ, Liu Z-K (2013) First-principles investigation of electronic, mechanical, and thermodynamic properties of L12 ordered Co3(M, W) (M=Al, Ge, Ga) phases. Acta Mater 61:5437–5448. https://doi.org/10.1016/j.actamat.2013.05.03210.1016/j.actamat.2013.05.032 ↗
39. Luo D-M, Ma L, Yang J, Ding N, Liu S-Y, Tang B-Y (2021) Temperature-dependent elastic properties of high entropy ceramic (ZrTaNbTi)C from self-consistent quasi-harmonic approximation. Solid State Commun 336:114432-114440. https://doi.org/10.1016/j.ssc.2021.11443210.1016/j.ssc.2021.114432 ↗
40. Shin Y, Jung WS, Lee YS (2016) First-principles study on the thermal expansion of Ni-X binary alloys based on the quasi-harmonic Debye model. Met Mater Int 22:1065–1072. https://doi.org/10.1007/s12540-016-6345-610.1007/s12540-016-6345-6 ↗
41. Kwon H, Moon A, Kim J (2019) Phase stability diagrams of Ti–M–O–C (M = Zr, Hf, Nb, and Ta) systems at 1800 K. Met Mater Int 25:396–407. https://doi.org/10.1007/s12540-018-0185-510.1007/s12540-018-0185-5 ↗
42. Kresse G, Hafner J (1993) Ab initio molecular dynamics for open-shell transition metals. Phys Rev B 48:13115–13118. https://doi.org/10.1103/PhysRevB.48.1311510.1103/PhysRevB.48.13115 ↗
43. Kresse G, Furthmüller J (1996) Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput Mater Sci 6:15–50. https://doi.org/10.1016/0927-0256(96)00008-010.1016/0927-0256(96)00008-0 ↗
44. Kresse G, Furthmüller J (1996) Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys Rev B Condens Matter Mater Phys 54:11169–11186. https://doi.org/10.1103/PhysRevB.54.1116910.1103/PhysRevB.54.11169 ↗
45. Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made simple. Phys Rev Lett 77:3865–3868. https://doi.org/10.1103/PhysRevLett.77.386510.1103/PhysRevLett.77.3865 ↗
46. Kresse D, Joubert G (1999) From ultrasoft pseudopotentials to the projector augmented-wave method. Phys Rev B Condens Matter Mater Phys 59:1758–1775. https://doi.org/10.1103/PhysRevB.59.175810.1103/PhysRevB.59.1758 ↗
47. Payne MC, Teter MP, Allan DC, Arias TA, Joannopoulos JD (1992) Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients. Rev Mod Phys 64:1045–1097. https://doi.org/10.1103/RevModPhys.64.104510.1103/RevModPhys.64.1045 ↗
48. Gonze X (1996) Towards a potential-based conjugate gradient algorithm for order-N self-consistent total energy calculations. Phys Rev B Condens Matter Mater Phys 54:4383–4386. https://doi.org/10.1103/PhysRevB.54.438310.1103/PhysRevB.54.4383 ↗
49. Monkhorst HJ, Pack JD (1976) Special points for brillouin zone integration. Phys Rev B 13:5188–519210.1103/PhysRevB.13.5188 ↗
50. Pathak A (2019) A first principles calculation of site occupancy of the B2 phase in Ti2AlX (X=V, Cr, Fe, Mo, Ta, Nb, Zr, Hf, and Re) intermetallics. Chinese J Phys 60:339–344. https://doi.org/10.1016/j.cjph.2019.05.02010.1016/j.cjph.2019.05.020 ↗
51. Yu H, Cao S, Youssef SS, Ma YJ, Lei JF, Qi Y, Hu QM, Yang R (2021) Generalized stacking fault energies and critical resolved shear stresses of random α-Ti-Al alloys from first-principles calculations. J Alloys Compd 850:156314. https://doi.org/10.1016/j.jallcom.2020.15631410.1016/j.jallcom.2020.156314 ↗
52. Le Page Y, Saxe P (2002) Symmetry-general least-squares extraction of elastic data for strained materials from ab initio calculations of stress. Phys Rev B Condens Matter Mater Phys 65:1–14. https://doi.org/10.1103/PhysRevB.65.10410410.1103/PhysRevB.65.104104 ↗
53. Dove MT (1993) Introduction to Lattice Dynamics. Cambridge University Press, ISBN 9780–5213–929–3810.1017/CBO9780511619885 ↗
54. Baroni S, Giannozzi P, Testa A (1987) Green’s-function approach to linear response in solids. Phys Rev Lett 58:1861–1864. https://doi.org/10.1103/PhysRevLett.58.186110.1103/PhysRevLett.58.1861 ↗
55. Baroni S, de Gironcoli S, Dal Corso A, Giannozzi P (2001) Phonons and related crystal properties from density-functional perturbation theory. Rev Mod Phys 73:515–562. https://doi.org/10.1103/RevModPhys.73.51510.1103/RevModPhys.73.515 ↗
56. Tanaka I (2015) Scripta materialia first principles phonon calculations in materials science. Scr Mater 108:1–5. https://doi.org/10.1016/j.scriptamat.2015.07.02110.1016/j.scriptamat.2015.07.021 ↗
57. Togo A, Oba F, Tanaka I (2008) First-principles calculations of the ferroelastic transition between rutile-type and CaCl2-type SiO2 at high pressure. Phys Rev B 78:134106-1-134106-9. https://doi.org/10.1103/PhysRevB.78.13410610.1103/PhysRevB.78.134106 ↗
58. Allen PB (2020) Theory of thermal expansion: quasi-harmonic approximation and corrections from quasi-particle renormalization. Mod Phys Lett B 34:2050025-1-2050025-15. https://doi.org/10.1142/S021798492050025610.1142/S0217984920500256 ↗
59. Mounet N, Marzari N (2005) First-principles determination of the structural, vibrational and thermodynamic properties of diamond, graphite, and derivatives. Phys Rev B Condens Matter Mater Phys 71:1–14. https://doi.org/10.1103/PhysRevB.71.20521410.1103/PhysRevB.71.205214 ↗
60. Togo A, Chaput L, Tanaka I, Hug G (2010) First-principles phonon calculations of thermal expansion in Ti3SiC2, Ti3AlC2, and Ti3GeC2. Phys Rev B 81:174301-1-174301-6. https://doi.org/10.1103/PhysRevB.81.17430110.1103/PhysRevB.81.174301 ↗
61. Huang LF, Gong PL, Zeng Z (2014) Correlation between structure, phonon spectra, thermal expansion, and thermomechanics of single-layer MoS2. Phys Rev B 90:045409-1-045409-7. https://doi.org/10.1103/PhysRevB.90.04540910.1103/PhysRevB.90.045409 ↗
62. Zhang X, Qin J, Ning J, Sun X, Li X, Ma M, Liu R (2013) First principle study of elastic and thermodynamic properties of FeB4 under high pressure. J Appl Phys 114:183517. https://doi.org/10.1063/1.482992610.1063/1.4829926 ↗
63. Ruan W, Zeng X, Wan H, Mei W, Wen Y (2019) A comparative first-principles study of orthorhombic and full-Heusler phases in Ti2AlNb intermetallic. Mater Res Express 6:0–12. https://doi.org/10.1088/2053-1591/ab2b2310.1088/2053-1591/ab2b23 ↗
64. Tang F, Awane T, Hagiwara M (2002) Effect of compositional modification on Young’s modulus of Ti2AlNb-based alloy. Scr Mater 46:143–147. https://doi.org/10.1016/S1359-6462(01)01211-810.1016/S1359-6462(01)01211-8 ↗
65. Olsson PAT (2015) First principles investigation of the finite temperature dependence of the elastic constants of zirconium, magnesium and gold. Comput Mater Sci 99:361–372. https://doi.org/10.1016/j.commatsci.2015.01.00510.1016/j.commatsci.2015.01.005 ↗
66. Aliofkhazraei M, ed. (2018) Physical properties of yttrium-rhodium and yttrium-copper B2-type rare earth intermetallic compounds: first principles study, in: Intermet. Compd.—Form. Appl., InTechOpen, London, United Kingdom. https://doi.org/10.5772/intechopen.6825610.5772/intechopen.68256 ↗
67. Daoud S, Bouarissa N, Benmakhlouf A, Allaoui O (2020) High-pressure effect on elastic constants and their related properties of MgCa intermetallic compound. Phys Status Solidi 257:1900537-1-1900537-10. https://doi.org/10.1002/pssb.20190053710.1002/pssb.201900537 ↗
68. Hadi MA, Islam MN, Babu MH (2019) Cubic perovskite Pb(Mg1/3Nb2/3)O3: a damage tolerant, machinable, and Thermal barrier coating material. Zeitschrift Fur Naturforsch Sect A J Phys Sci 74:71–81. https://doi.org/10.1515/zna-2018-033410.1515/zna-2018-0334 ↗
69. Born M (1940) On the stability of crystal lattices. I. Math Proc Cambridge Philos Soc 36:160–172. https://doi.org/10.1017/S030500410001713810.1017/S0305004100017138 ↗
70. F.I. Fedorov (1968) General Equations of the Theory of Elasticity, in: Theory Elastic Waves Cryst. Springer US, Boston, MA, 1–33. https://doi.org/10.1007/978-1-4757-1275-9_110.1007/978-1-4757-1275-9_1 ↗
71. Sin’ko GV, Smirnov NA (2002) Ab initio calculations of elastic constants and thermodynamic properties of bcc, fcc, and hcp Al crystals under pressure. J Phys Condens Matter 14:6989–7005. https://doi.org/10.1088/0953-8984/14/29/30110.1088/0953-8984/14/29/301 ↗
72. Nye JF, Lindsay RB (1957) Physical properties of crystals: their representation by tensors and matrices. Phys Today 10:26. https://doi.org/10.1063/1.306020010.1063/1.3060200 ↗
73. Voigt W (1966) Lehrbuch der Kristallphysik, Vieweg+Teubner Verlag. Wiesbaden. https://doi.org/10.1007/978-3-663-15884-410.1007/978-3-663-15884-4 ↗
74. Reuss A (1929) Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. ZAMM - Zeitschrift Für Angew Math Und Mech 9:49–58. https://doi.org/10.1002/zamm.1929009010410.1002/zamm.19290090104 ↗
75. Hill R (1952) The elastic behaviour of a crystalline aggregate. Proc Phys Soc Sect A 65:349–354. https://doi.org/10.1088/0370-1298/65/5/30710.1088/0370-1298/65/5/307 ↗
76. Li CM, Zeng SM, Chen ZQ, Cheng NP, Chen TX (2014) First-principles calculations of elastic and thermodynamic properties of the four main intermetallic phases in Al-Zn-Mg-Cu alloys. Comput Mater Sci 93:210–220. https://doi.org/10.1016/j.commatsci.2014.06.03110.1016/j.commatsci.2014.06.031 ↗
77. Boehlert CJ (2001) Part III. The tensile behavior of Ti-Al-Nb O + Bcc orthorhombic alloys. Metall Mater Trans A Phys Metall Mater Sci 32:1977–1988. https://doi.org/10.1007/s11661-001-0010-410.1007/s11661-001-0010-4 ↗
78. Pugh SF (1954) XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals. London, Edinburgh, Dublin. Philos Mag J Sci 45:823–843. https://doi.org/10.1080/1478644080852049610.1080/14786440808520496 ↗
79. Olijnyk H, Jephcoat AP (2000) The E 2g phonon and the elastic constant C 44 in hexagonal van der Waals bonded solids. J Phys Condens Matter 12:10423–10428. https://doi.org/10.1088/0953-8984/12/50/30510.1088/0953-8984/12/50/305 ↗
80. Chen XQ, Niu H, Li D, Li Y (2011) Modeling hardness of polycrystalline materials and bulk metallic glasses. Intermetallics 19:1275–1281. https://doi.org/10.1016/j.intermet.2011.03.02610.1016/j.intermet.2011.03.026 ↗
81. Ranganathan SI, Ostoja-Starzewski M (2008) Universal elastic anisotropy index. Phys Rev Lett 101:055504-1-055504-4. https://doi.org/10.1103/PhysRevLett.101.05550410.1103/PhysRevLett.101.055504 ↗
82. Kube CM (2016) Elastic anisotropy of crystals. AIP Adv 6:095209-1-095209-6. https://doi.org/10.1063/1.496299610.1063/1.4962996 ↗
83. Ye YX, Musico BL, Lu ZZ, Xu LB, Lei ZF, Keppens V, Xu HX, Nieh TG (2019) Evaluating elastic properties of a body-centered cubic NbHfZrTi high-entropy alloy—A direct comparison between experiments and ab initio calculations. Intermetallics 109:167–173. https://doi.org/10.1016/j.intermet.2019.04.00310.1016/j.intermet.2019.04.003 ↗
84. Tang W, Sanville E, Henkelman G (2009) A grid-based Bader analysis algorithm without lattice bias. J Phys Condens Matter 21:084204-1084204-7. https://doi.org/10.1088/0953-8984/21/8/08420410.1088/0953-8984/21/8/084204 ↗
85. Lucovsky G, Brodsky MH, Burstein E (1970) Extension of a linear diatomic-chain model for the calculation of local-mode frequencies in real crystals. Phys Rev B 2:3295–3302. https://doi.org/10.1103/PhysRevB.2.329510.1103/PhysRevB.2.3295 ↗
86. Braeuer A (2015) Chapter 2-Interaction of Matter and Electromagnetic Radiation in: Supercritical Fluid Science and Technology ISBN: 9780444634221. 7: 41–192. https://doi.org/10.1016/B978-0-444-63422-1.00002-X10.1016/B978-0-444-63422-1.00002-X ↗
87. Naumann RJ (2009) Thermal properties of solids, In: Introd to Phys Chem Mater, CRC Press Taylor and Francis: 321–353
88. Lee WH, Yao XH (2015) First principle investigation of phase transition and thermodynamic properties of SiC. Comput Mater Sci 106:76–82. https://doi.org/10.1016/j.commatsci.2015.04.04410.1016/j.commatsci.2015.04.044 ↗