#
The Impact of Vibrational Entropy on the Segregation of Cu to Antiphase Boundaries in Fe_{3}Al

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Sauthoff, G. Intermetallics; VCH Verlagsgesellschaft: Weinheim, Germany, 1995. [Google Scholar]
- Liu, C.T.; Stringer, J.; Mundy, J.N.; Horton, L.L.; Angelini, P. Ordered intermetallic alloys: An assessment. Intermetallics
**1997**, 5, 579–596. [Google Scholar] [CrossRef] - Stoloff, N.S. Iron aluminides: Present status and future prospects. Mater. Sci. Eng. A
**1998**, 258, 1–14. [Google Scholar] [CrossRef] - Liu, C.T.; Lee, E.H.; McKamey, C.G. An environmental-effect as the major cause for room-temperature embrittlement in FeAl. Scr. Metall. Mater.
**1989**, 23, 875–880. [Google Scholar] [CrossRef] - Lynch, R.J.; Heldt, L.A.; Milligan, W.W. Effects of alloy composition on environmental embrittlement of B2 ordered iron aluminides. Scr. Metall. Mater.
**1991**, 25, 2147–2151. [Google Scholar] [CrossRef] - Liu, C.T.; McKamey, C.G.; Lee, E.H. Environmental-effects on room-temperature ductility and fracture in Fe
_{3}Al. Scr. Metall. Mater.**1990**, 24, 385–389. [Google Scholar] [CrossRef] - Lynch, R.J.; Gee, K.A.; Heldt, L.A. Environmental embrittlement of single-crystal and thermomechanically processed B2-ordered iron aluminides. Scr. Metall. Mater.
**1994**, 30, 945–950. [Google Scholar] [CrossRef] - Zamanzade, M.; Barnoush, A.; Motz, C. A Review on the Properties of Iron Aluminide Intermetallics. Crystals
**2016**, 6, 10. [Google Scholar] [CrossRef] [Green Version] - Kattner, U.; Burton, B. Al-Fe (Aluminium-Iron). In Phase Diagrams of Binary Iron Alloys; Okamoto, H., Ed.; ASM International: Materials Park, OH, USA, 1993; pp. 12–28. [Google Scholar]
- Palm, M.; Inden, G.; Thomas, N. The Fe-Al-Ti system. J. Phase Equilibria
**1995**, 16, 209–222. [Google Scholar] [CrossRef] - Vernieres, J.; Benelmekki, M.; Kim, J.H.; Grammatikopoulos, P.; Bobo, J.F.; Diaz, R.E.; Sowwan, M. Single-step gas phase synthesis of stable iron aluminide nanoparticles with soft magnetic properties. APL Mater.
**2014**, 2, 116105. [Google Scholar] [CrossRef] - Jirásková, Y.; Pizúrová, N.; Titov, A.; Janičkovič, D.; Friák, M. Phase separation in Fe-Ti-Al alloy–Structural, magnetic, and Mössbauer study. J. Magn. Magn. Mater.
**2018**, 468, 91–99. [Google Scholar] [CrossRef] - Palm, M.; Lacaze, J. Assessment of the Al-Fe-Ti system. Intermetallics
**2006**, 14, 1291–1303. [Google Scholar] [CrossRef] [Green Version] - Dobeš, F.; Dymáček, P.; Friák, M. Force-to-Stress Conversion Methods in Small Punch Testing Exemplified by Creep Results of Fe-Al Alloy with Chromium and Cerium Additions. IOP Conf. Ser. Mater. Sci. Eng.
**2018**, 461, 012017. [Google Scholar] [CrossRef] - Dobeš, F.; Dymáček, P.; Friák, M. Small punch creep of Fe-Al-Cr alloy with Ce addition and its relation to uniaxial creep tests. Kovové Materiály Met. Mater.
**2018**, 56, 205. [Google Scholar] [CrossRef] [Green Version] - Palm, M.; Sauthoff, G. Deformation behaviour and oxidation resistance of single-phase and two-phase L2
_{1}-ordered Fe-Al-Ti alloys. Intermetallics**2004**, 12, 1345–1359. [Google Scholar] [CrossRef] - Sundman, B.; Ohnuma, I.; Dupin, N.; Kattner, U.R.; Fries, S.G. An assessment of the entire Al-Fe system including D0(3) ordering. Acta Mater.
**2009**, 57, 2896–2908. [Google Scholar] [CrossRef] - Dymáček, P.; Dobeš, F.; Jirásková, Y.; Pizúrová, N.; Friák, M. Tensile, creep and fracture testing of prospective Fe-Al-based alloys using miniature specimens. Theor. Appl. Fract. Mech.
**2019**, 99, 18–26. [Google Scholar] [CrossRef] - Dobeš, F.; Dymáček, P.; Friák, M. The Influence of Niobium Additions on Creep Resistance of Fe-27 at.% Al Alloys. Metals
**2019**, 9, 739. [Google Scholar] [CrossRef] [Green Version] - Grigorchik, A.N.; Astrashab, V.E.; Kukareko, V.A.; Belotserkovsky, M.A.; Sosnovsky, V.A. High-temperature heat treatment of hypersonic metallization coatings from pseudoalloy “Fe-Al”. Lett. Mater.
**2021**, 11, 198–203. [Google Scholar] [CrossRef] - Deevi, S.C. Advanced intermetallic iron aluminide coatings for high temperature applications. Prog. Mater. Sci.
**2021**, 118. [Google Scholar] [CrossRef] - Tolochyn, O.I.; Baglyuk, G.A.; Tolochyna, O.V.; Evych, Y.I.; Podrezov, Y.M.; Molchanovska, H.M. Structure and Physicomechanical Properties of the Fe
_{3}Al Intermetallic Compound Obtained by Impact Hot Compaction. Mater. Sci.**2021**, 56, 499–508. [Google Scholar] [CrossRef] - Komarov, O.N.; Zhilin, S.G.; Predein, V.V.; Popov, A.V. Mechanisms for Forming Iron-Containing Intermetallics Prepared by Aluminothermy and the Effect of Special Treatment Methods on their Properties. Metallurgist
**2020**, 64, 810–821. [Google Scholar] [CrossRef] - Vodickova, V.; Svec, M.; Hanus, P.; Novak, P.; Zadera, A.; Keller, V.; Prokopcakova, P.P. The Effect of Simultaneous Si and Ti/Mo Alloying on High-Temperature Strength of Fe
_{3}Al-Based Iron Aluminides. Molecules**2020**, 25, 4268. [Google Scholar] [CrossRef] - Luo, X.; Cao, J.; Meng, G.; Chuan, Y.; Yao, Z.; Xie, H. Systematical investigation on the microstructures and tribological properties of Fe-Al laser cladding coatings. Appl. Surf. Sci.
**2020**, 516. [Google Scholar] [CrossRef] - Luo, X.; Cao, J.; Meng, G.; Yu, F.; Jiang, Q.; Zhang, P.; Xie, H. Double Glow Plasma Surface Metallurgy Technology Fabricated Fe-Al-Cr Coatings with Excellent Corrosion Resistance. Coatings
**2020**, 10, 575. [Google Scholar] [CrossRef] - Teker, T.; Yilmaz, S.O. Synthesis and structural characterization of Fe based Ti+Ni
_{3}Al+Al_{2}O_{3}reinforcement composite produced by mechanical alloying. Rev. Metal.**2020**, 56. [Google Scholar] [CrossRef] - Zhang, X.; Sun, Y.; Niu, M.; Shao, M.; Geng, X. Microstructure and mechanical behavior of in situ TiC reinforced Fe
_{3}Al (Fe-23Al-3Cr) matrix composites by mechanical alloying and vacuum hot-pressing sintering technology. Vacuum**2020**, 180. [Google Scholar] [CrossRef] - Ghazanfari, H.; Blais, C.; Gariepy, M.; Savoie, S.; Schulz, R.; Alamdari, H. Improving wear resistance of metal matrix composites using reinforcing particles in two length-scales: Fe
_{3}Al/TiC composites. Surf. Coat. Technol.**2020**, 386. [Google Scholar] [CrossRef] - Khodaei, M. Characterization of Al
_{2}O_{3}in Fe_{3}Al-30 vol.% Al_{2}O_{3}Nanocomposite Powder Synthesized by Mechanochemical Process. J. Nanostruct.**2020**, 10, 456–462. [Google Scholar] [CrossRef] - Altunin, R.R.; Moiseenko, E.T.; Zharkov, S.M. Structural Phase Transformations during a Solid-State Reaction in a Bilayer Al/Fe Thin-Film Nanosystem. Phys. Solid State
**2020**, 62, 200–205. [Google Scholar] [CrossRef] - Tolochyn, O.I.; Tolochyna, O.V.; Bagliuk, H.A.; Yevych, Y.I.; Podrezov, Y.M.; Mamonova, A.A. Influence of Sintering Temperature on the Structure and Properties of Powder Iron Aluminide Fe
_{3}Al. Powder Metall. Met. Ceram.**2020**, 59, 150–159. [Google Scholar] [CrossRef] - Adler, L.; Fu, Z.; Koerner, C. Electron beam based additive manufacturing of Fe
_{3}Al based iron aluminides - Processing window, microstructure and properties. Mater. Sci. Eng. A Struct. Mater. Prop. Microstruct. Process.**2020**, 785. [Google Scholar] [CrossRef] - Michalcova, A.; Ozkan, M.; Mikula, P.; Marek, I.; Knaislova, A.; Kopecek, J.; Vojtech, D. The Influence of Powder Milling on Properties of SPS Compacted FeAl. Molecules
**2020**, 25, 2263. [Google Scholar] [CrossRef] [PubMed] - Peska, M.; Karczewski, K.; Rzeszotarska, M.; Polanski, M. Direct Synthesis of Fe-Al Alloys from Elemental Powders Using Laser Engineered Net Shaping. Materials
**2020**, 13, 531. [Google Scholar] [CrossRef] [Green Version] - Luo, X.; Cao, J.; Meng, G.; Zhou, Y.; Xie, H. Long-range-ordered Fe
_{3}Al with excellent electromagnetic wave absorption. J. Mater. Sci. Mater. Electron.**2020**, 31, 15608–15615. [Google Scholar] [CrossRef] - Ismail, A.; Bahanan, W.; Bin Hussain, P.; Saat, A.M.; Shaik, N.B. Diffusion Bonding of Al-Fe Enhanced by Gallium. Processes
**2020**, 8, 824. [Google Scholar] [CrossRef] - Watson, R.E.; Weinert, M. Transition-metal aluminide formation: Ti, V, Fe, and Ni aluminides. Phys. Rev. B
**1998**, 58, 5981–5988. [Google Scholar] [CrossRef] - Gonzales-Ormeno, P.; Petrilli, H.; Schon, C. Ab-initio calculations of the formation energies of BCC-based superlattices in the Fe-Al system. Calphad-Comput. Coupling Phase Diagrams Thermochem.
**2002**, 26, 573. [Google Scholar] [CrossRef] - Connetable, D.; Maugis, P. First principle calculations of the kappa-Fe
_{3}AlC perovskite and iron-aluminium intermetallics. Intermetallics**2008**, 16, 345–352. [Google Scholar] [CrossRef] [Green Version] - Kellou, A.; Grosdidier, T.; Raulot, J.M.; Aourag, H. Atomistic study of magnetism effect on structural stability in Fe
_{3}Al and Fe_{3}AlX (X = H, B, C, N, O) alloys. Phys. Status Solidi B-Basic Solid State Phys.**2008**, 245, 750–755. [Google Scholar] [CrossRef] - Šesták, P.; Friák, M.; Holec, D.; Všianská, M.; Šob, M. Strength and brittleness of interfaces in Fe-Al superalloy nanocomposites under multiaxial loading: An ab initio and atomistic study. Nanomaterials
**2018**, 8, 873. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lechermann, F.; Fähnle, M.; Meyer, B.; Elsässer, C. Electronic correlations, magnetism, and structure of Fe-Al subsystems: An LDA+U study. Phys. Rev. B
**2004**, 69, 165116. [Google Scholar] [CrossRef] - Airiskallio, E.; Nurmi, E.; Heinonen, M.H.; Vayrynen, I.J.; Kokko, K.; Ropo, M.; Punkkinen, M.P.J.; Pitkanen, H.; Alatalo, M.; Kollar, J.; et al. High temperature oxidation of Fe-Al and Fe-Cr-Al alloys: The role of Cr as a chemically active element. Corros. Sci.
**2010**, 52, 3394–3404. [Google Scholar] [CrossRef] - Lechermann, F.; Welsch, F.; Elsässer, C.; Ederer, C.; Fähnle, M.; Sanchez, J.; Meyer, B. Density-functional study of Fe
_{3}Al: LSDA versus GGA. Phys. Rev. B**2002**, 65, 132104. [Google Scholar] [CrossRef] [Green Version] - Friák, M.; Slávik, A.; Miháliková, I.; Holec, D.; Všianská, M.; Šob, M.; Palm, M.; Neugebauer, J. Origin of the low magnetic moment in Fe
_{2}AlTi: An Ab initio study. Materials**2018**, 11, 1732. [Google Scholar] [CrossRef] [Green Version] - Ju, J.; Kang, M.; Zhou, Y.; Yang, C.; Wang, K.; Li, J.; Wang, R.; Fu, H.; Wang, J. First-principles investigations of the stability, electronic structures, mechanical properties and thermodynamic properties of Fe
_{x}Al_{y}C_{z}compounds in Fe-Cr-B-Al-C alloy. J. Phys. Chem. Solids**2020**, 143. [Google Scholar] [CrossRef] - Miháliková, I.; Friák, M.; Jirásková, Y.; Holec, D.; Koutná, N.; Šob, M. Impact of Nano-Scale Distribution of Atoms on Electronic and Magnetic Properties of Phases in Fe-Al Nanocomposites: An Ab Initio Study. Nanomaterials
**2018**, 8, 1059. [Google Scholar] [CrossRef] [Green Version] - Friák, M.; Holec, D.; Šob, M. Quantum-Mechanical Study of Nanocomposites with Low and Ultra-Low Interface Energies. Nanomaterials
**2018**, 8, 1057. [Google Scholar] [CrossRef] [Green Version] - Kulikov, N.I.; Postnikov, A.V.; Borstel, G.; Braun, J. Onset of magnetism in B2 transition-metal aluminides. Phys. Rev. B
**1999**, 59, 6824–6833. [Google Scholar] [CrossRef] [Green Version] - Friák, M.; Neugebauer, J. Ab initio study of the anomalous volume-composition dependence in Fe-Al alloys. Intermetallics
**2010**, 18, 1316–1321. [Google Scholar] [CrossRef] - Ipser, H.; Semenova, O.; Krachler, R. Intermetallic phases with D0(3)-structure: A statistical-thermodynamic model. J. Alloys Compd.
**2002**, 338, 20–25. [Google Scholar] [CrossRef] - Fähnle, M.; Drautz, R.; Lechermann, F.; Singer, R.; Diaz-Ortiz, A.; Dosch, H. Thermodynamic properties from ab-initio calculations: New theoretical developments, and applications to various materials systems. Phys. Status Solidi B-Basic Solid State Phys.
**2005**, 242, 1159–1173. [Google Scholar] [CrossRef] - Kirklin, S.; Saal, J.E.; Hegde, V.I.; Wolverton, C. High-throughput computational search for strengthening precipitates in alloys. Acta Mater.
**2016**, 102, 125–135. [Google Scholar] [CrossRef] [Green Version] - Liu, S.; Duan, S.; Ma, B. First-principles calculation of vibrational entropy for Fe-Al compounds. Phys. Rev. B
**1998**, 58, 9705–9709. [Google Scholar] - Čížek, J.; Lukáč, F.; Procházka, I.; Kužel, R.; Jirásková, Y.; Janičkovič, D.; Anwand, W.; Brauer, G. Characterization of quenched-in vacancies in Fe-Al alloys. Phys. B
**2012**, 407, 2659–2664. [Google Scholar] [CrossRef] - Miháliková, I.; Friák, M.; Koutná, N.; Holec, D.; Šob, M. An Ab Initio Study of Vacancies in Disordered Magnetic Systems: A Case Study of Fe-Rich Fe-Al Phases. Materials
**2019**, 12, 1430. [Google Scholar] [CrossRef] [Green Version] - Amara, H.; Fu, C.C.; Soisson, F.; Maugis, P. Aluminum and vacancies in α-iron: Dissolution, diffusion, and clustering. Phys. Rev. B
**2010**, 81, 174101. [Google Scholar] [CrossRef] - Friák, M.; Černý, M.; Všianská, M.; Šob, M. Impact of Antiphase Boundaries on Structural, Magnetic and Vibrational Properties of Fe
_{3}Al. Materials**2020**, 13, 4884. [Google Scholar] [CrossRef] - Li, Y.; Liu, Y.; Yang, J. First principle calculations and mechanical properties of the intermetallic compounds in a laser welded steel/aluminum joint. Opt. Laser Technol.
**2020**, 122. [Google Scholar] [CrossRef] - Wang, K.; Wang, Y. The partitioning behavior of dual solutes at the antiphase domain boundary in the B2 intermetallic: A microscopic phase-field study. J. Alloys Compd.
**2020**, 824, 153923. [Google Scholar] [CrossRef] - Koizumi, Y.; Allen, S.M.; Ouchi, M.; Minamino, Y. Effects of solute and vacancy segregation on antiphase boundary migration in stoichiometric and Al-rich Fe
_{3}Al: A phase-field simulation study. Intermetallics**2010**, 18, 1297–1302. [Google Scholar] [CrossRef] [Green Version] - Koizumi, Y.; Allen, S.M.; Minamino, Y. Effects of solute and vacancy segregation on migration of a/4〈111〉 and a/2〈100〉 antiphase boundaries in Fe
_{3}Al. Acta Mater.**2009**, 57, 3039–3051. [Google Scholar] [CrossRef] [Green Version] - Koizumi, Y.; Allen, S.M.; Ouchi, M.; Minamino, Y.; Chiba, A. Phase-Field Simulation of D0
_{3}-Type Antiphase Boundary Migration in Fe_{3}Al with Vacancy and Solute Segregation. Solid State Phenom.**2011**, 172–174, 1313–1319. [Google Scholar] [CrossRef] - Marcinkowski, M.; Brown, N. Theory and direct observation of dislocations in the Fe3Al superlattices. Acta Metall.
**1961**, 9, 764–786. [Google Scholar] [CrossRef] - Marcinkowski, M.J.; Brown, N. Direct Observation of Antiphase Boundaries in the Fe
_{3}Al Superlattice. J. Appl. Phys.**1962**, 33, 537–552. [Google Scholar] [CrossRef] - Friák, M.; Všianská, M.; Šob, M. A Quantum-Mechanical Study of Clean and Cr-Segregated Antiphase Boundaries in Fe
_{3}Al. Materials**2019**, 12, 3954. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Friák, M.; Buršíková, V.; Pizúrová, N.; Pavlů, J.; Jirásková, Y.; Homola, V.; Miháliková, I.; Slávik, A.; Holec, D.; Všianská, M.; et al. Elasticity of Phases in Fe-Al-Ti Superalloys: Impact of Atomic Order and Anti-Phase Boundaries. Crystals
**2019**, 9, 299. [Google Scholar] [CrossRef] [Green Version] - Friák, M.; Golian, M.; Holec, D.; Koutná, N.; Šob, M. An Ab Initio Study of Magnetism in Disordered Fe-Al Alloys with Thermal Antiphase Boundaries. Nanomaterials
**2020**, 10, 44. [Google Scholar] [CrossRef] [PubMed] [Green Version] - McKamey, C.G.; Horton, J.A.; Liu, C.T. Effect of chromium on properties of Fe
_{3}Al. J. Mater. Res.**1989**, 4, 1156–1163. [Google Scholar] [CrossRef] - Morris, D.; Dadras, M.; Morris, M. The influence of Cr addition on the ordered microstructure and deformation and fracture-behavior of a Fe-28-%-Al intermetallic. Acta Metall. Mater.
**1993**, 41, 97–111. [Google Scholar] [CrossRef] - Kral, F.; Schwander, P.; Kostorz, G. Superdislocations and antiphase boundary energies in deformed Fe
_{3}Al single crystals with chromium. Acta Mater.**1997**, 45, 675–682. [Google Scholar] [CrossRef] - Allen, S.; Cahn, J. Microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall.
**1979**, 27, 1085–1095. [Google Scholar] [CrossRef] - Wang, K.; Wang, Y.; Cheng, Y. The Formation and Dynamic Evolution of Antiphase Domain Boundary in FeAl Alloy: Computational Simulation in Atomic Scale. Mater. Res. Ibero-Am. J. Mater.
**2018**, 21. [Google Scholar] [CrossRef] [Green Version] - Balagurov, A.M.; Bobrikov, I.A.; Sumnikov, V.S.; Golovin, I.S. Antiphase domains or dispersed clusters? Neutron diffraction study of coherent atomic ordering in Fe
_{3}Al-type alloys. Acta Mater.**2018**, 153, 45–52. [Google Scholar] [CrossRef] - Murakami, Y.; Niitsu, K.; Tanigaki, T.; Kainuma, R.; Park, H.S.; Shindo, D. Magnetization amplified by structural disorder within nanometre-scale interface region. Nat. Commun.
**2014**, 5, 4133. [Google Scholar] [CrossRef] [Green Version] - Oguma, R.; Matsumura, S.; Eguchi, T. Kinetics of B2-and D0
_{3}type ordering and formation of domain structures in Fe-Al alloys. J. Phys. Condens. Matter**2008**, 20, 275225. [Google Scholar] [CrossRef] - Nishino, Y.; Kumada, C.; Asano, S. Phase stability of Fe
_{3}Al with addition of 3d transition elements. Scr. Mater.**1997**, 36, 461–466. [Google Scholar] [CrossRef] - Friák, M.; Deges, J.; Stein, F.; Palm, M.; Frommeyer, G.; Neugebauer, J. Ab Initio Study of Elastic Properties in Fe
_{3}Al-based Alloys. MRS Proc.**2008**, 1128, 1128-U02-04. [Google Scholar] [CrossRef] - Nishino, Y.; Asano, S.; Ogawa, T. Phase stability and mechanical properties of Fe
_{3}Al with addition of transition elements. Mater. Sci. Eng. A**1997**, 234–236, 271–274. [Google Scholar] [CrossRef] - Friák, M.; Deges, J.; Krein, R.; Frommeyer, G.; Neugebauer, J. Combined ab initio and experimental study of structural and elastic properties of Fe
_{3}Al-based ternaries. Intermetallics**2010**, 18, 1310. [Google Scholar] [CrossRef] - Rosalbino, F.; Carlini, R.; Zanicchi, G.; Scavino, G. Effect of copper alloying addition on the electrochemical corrosion behaviour of Fe
_{3}Al intermetallic in sulphuric acid solution. Mater. Corros.**2016**, 67, 1042–1048. [Google Scholar] [CrossRef] - Park, N.; Lee, S.C.; Cha, P.R. Effects of alloying elements on the stability and mechanical properties of Fe
_{3}Al from first-principles calculations. Comput. Mater. Sci.**2018**, 146, 303–309. [Google Scholar] [CrossRef] - Liu, Y.; Zhang, L.; Cui, S.; Li, W. Effects of transition metal (Cr, Mn, Mo, Ni, Ti, and V) doping on the mechanical, electronic and thermal properties of Fe
_{3}Al. Vacuum**2021**, 185. [Google Scholar] [CrossRef] - Gomell, L.; Katnagallu, S.; Diack-Rasselio, A.; Maier, S.; Perrière, L.; Scheu, C.; Alleno, E.; Gault, B. Chemical segregation and precipitation at anti-phase boundaries in thermoelectric Heusler-Fe
_{2}VAl. Scr. Mater.**2020**, 186, 370–374. [Google Scholar] [CrossRef] - Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B
**1993**, 47, 558–561. [Google Scholar] [CrossRef] - Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B
**1996**, 54, 11169–11186. [Google Scholar] [CrossRef] - Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. B
**1964**, 136, B864–B871. [Google Scholar] [CrossRef] [Green Version] - Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev. A
**1965**, 140, A1133–A1138. [Google Scholar] [CrossRef] [Green Version] - Blöchl, P.E. Projector augmented-wave method. Phys. Rev. B
**1994**, 50, 17953–17979. [Google Scholar] [CrossRef] [Green Version] - Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B
**1999**, 59, 1758–1775. [Google Scholar] [CrossRef] - Perdew, J.P.; Wang, Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B
**1992**, 45, 13244–13249. [Google Scholar] [CrossRef] - Vosko, S.H.; Wilk, L.; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis. Can. J. Phys.
**1980**, 58, 1200. [Google Scholar] [CrossRef] [Green Version] - Všianská, M.; Friák, M.; Šob, M. An ab initio study of Fe3Al: A critical review of generalized gradient approximation. to be published.
- Togo, A.; Tanaka, I. First principles phonon calculations in materials science. Scr. Mater.
**2015**, 108, 1–5. [Google Scholar] [CrossRef] [Green Version] - Momma, K.; Izumi, F. An integrated three-dimensional visualization system VESTA using wxWidgets. Comm. Crystallogr. Comput. IUCr Newslett.
**2006**, 7, 106. [Google Scholar] [CrossRef] - Momma, K.; Izumi, F. VESTA: A three-dimensional visualization system for electronic and structural analysis. J. Appl. Crystallogr.
**2008**, 41, 653–658. [Google Scholar] [CrossRef] - Momma, K.; Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr.
**2011**, 44, 1272–1276. [Google Scholar] [CrossRef]

**Figure 1.**Supercells used in our study. Part (

**a**) shows a 64-atom supercell as an 1 × 1 × 4 multiple of the 16-atom elementary cell of Fe${}_{3}$Al. Parts (

**b**,

**c**) visualize the studied types of antiphase boundaries (APBs) with Cu atoms located either directly at the APB interfaces (

**b**) or away from them (

**c**), respectively. The subfigures (

**b**,

**c**) include red vectors defining a 〈100〉 shift that is characteristic for the studied APBs and applied to the upper halves of the 64-atom supercells.

**Figure 2.**Calculated local magnetic moments of iron atoms in Fe${}_{3}$Al with Cu substituents without antiphase boundaries (

**a**), with Cu atoms located at the interface of the studied antiphase boundaries (

**b**), and with the Cu substituents away from the APBs (

**c**). The magnetic moments are compared with values for the Cu-free and defect-free bulk Fe${}_{3}$Al (horizontal dashed lines).

**Figure 3.**Computed phonon frequencies along selected directions in the reciprocal space and the density of phonon states in the case of Cu-containing crystals of Fe${}_{3}$Al (without APBs).

**Figure 4.**Calculated Helmholtz free energy F (

**a**) and the harmonic phonon energy E (

**b**) of defect-free Fe${}_{3}$Al containing the Cu atoms together with derived vibrational entropy S (

**c**) and constant-volume heat capacity ${C}_{v}$ (

**d**), respectively. The elementary entity for defining one mol is the 64-atom supercell, i.e., we talk about one mole of 64-atom supercells.

**Figure 5.**Computed phonon frequencies along selected directions in the reciprocal space and the density of phonon states in Fe${}_{3}$Al containing antiphase boundaries (APBs) with the Cu atoms located either within the APB interface plane (

**a**) or as far from the APB interface as possible within our computational supercell (

**b**).

**Figure 6.**Calculated segregation-related energy difference (per 64-atom supercell) that includes both the static lattice energy ${U}^{\mathrm{static}}$ and the phonon free energy F. The difference is between the states with (i) two Cu atoms at the APB interfaces and (ii) far away from them as a function of temperature. It is compared with the segregation-related energy difference of static lattices $\Delta {U}^{\mathrm{static}}$ (see horizontal dashed line). As one mol, we mean one mol of 64-atom supercells.

**Figure 7.**Computed entropy difference $\Delta S$ (

**a**) and phonon energy difference $\Delta E$ (

**b**) (both per 64-atom supercell) between the states with Cu at the APB interfaces and far away from them.

**Figure 8.**Calculated APB-related energy difference $\Delta ({U}^{\mathrm{static}}+F)$ (per 64-atom supercell) between (i) the state with two Cu atoms at the APB interfaces and (ii) the state without APBs as a function of temperature in comparison with the static lattice difference (see horizontal dashed line). The elementary entity for defining one mole is the 64-atom supercell.

**Figure 9.**Computed entropy difference $\Delta S$ (

**a**) and phonon energy difference $\Delta E$ (

**b**) (both per 64-atom supercell) between (i) the states of Fe${}_{3}$Al with Cu but without APBs and (ii) with the Cu atoms at the APB interfaces.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Friák, M.; Černý, M.; Šob, M.
The Impact of Vibrational Entropy on the Segregation of Cu to Antiphase Boundaries in Fe_{3}Al. *Magnetochemistry* **2021**, *7*, 108.
https://doi.org/10.3390/magnetochemistry7080108

**AMA Style**

Friák M, Černý M, Šob M.
The Impact of Vibrational Entropy on the Segregation of Cu to Antiphase Boundaries in Fe_{3}Al. *Magnetochemistry*. 2021; 7(8):108.
https://doi.org/10.3390/magnetochemistry7080108

**Chicago/Turabian Style**

Friák, Martin, Miroslav Černý, and Mojmír Šob.
2021. "The Impact of Vibrational Entropy on the Segregation of Cu to Antiphase Boundaries in Fe_{3}Al" *Magnetochemistry* 7, no. 8: 108.
https://doi.org/10.3390/magnetochemistry7080108