Vol. 25 No. 2 2023 3 EDITORIAL COUNCIL EDITORIAL BOARD EDITOR-IN-CHIEF: Anatoliy A. Bataev, D.Sc. (Engineering), Professor, Rector, Novosibirsk State Technical University, Novosibirsk, Russian Federation DEPUTIES EDITOR-IN-CHIEF: Vladimir V. Ivancivsky, D.Sc. (Engineering), Associate Professor, Department of Industrial Machinery Design, Novosibirsk State Technical University, Novosibirsk, Russian Federation Vadim Y. Skeeba, Ph.D. (Engineering), Associate Professor, Department of Industrial Machinery Design, Novosibirsk State Technical University, Novosibirsk, Russian Federation Editor of the English translation: Elena A. Lozhkina, Ph.D. (Engineering), Department of Material Science in Mechanical Engineering, Novosibirsk State Technical University, Novosibirsk, Russian Federation The journal is issued since 1999 Publication frequency – 4 numbers a year Data on the journal are published in «Ulrich's Periodical Directory» Journal “Obrabotka Metallov” (“Metal Working and Material Science”) has been Indexed in Clarivate Analytics Services. Novosibirsk State Technical University, Prospekt K. Marksa, 20, Novosibirsk, 630073, Russia Tel.: +7 (383) 346-17-75 http://journals.nstu.ru/obrabotka_metallov E-mail: metal_working@mail.ru; metal_working@corp.nstu.ru Journal “Obrabotka Metallov – Metal Working and Material Science” is indexed in the world's largest abstracting bibliographic and scientometric databases Web of Science and Scopus. Journal “Obrabotka Metallov” (“Metal Working & Material Science”) has entered into an electronic licensing relationship with EBSCO Publishing, the world's leading aggregator of full text journals, magazines and eBooks. The full text of JOURNAL can be found in the EBSCOhost™ databases.
OBRABOTKAMETALLOV Vol. 25 No. 2 2023 4 EDITORIAL COUNCIL EDITORIAL COUNCIL CHAIRMAN: Nikolai V. Pustovoy, D.Sc. (Engineering), Professor, President, Novosibirsk State Technical University, Novosibirsk, Russian Federation MEMBERS: The Federative Republic of Brazil: Alberto Moreira Jorge Junior, Dr.-Ing., Full Professor; Federal University of São Carlos, São Carlos The Federal Republic of Germany: Moniko Greif, Dr.-Ing., Professor, Hochschule RheinMain University of Applied Sciences, Russelsheim Florian Nürnberger, Dr.-Ing., Chief Engineer and Head of the Department “Technology of Materials”, Leibniz Universität Hannover, Garbsen; Thomas Hassel, Dr.-Ing., Head of Underwater Technology Center Hanover, Leibniz Universität Hannover, Garbsen The Spain: Andrey L. Chuvilin, Ph.D. (Physics and Mathematics), Ikerbasque Research Professor, Head of Electron Microscopy Laboratory “CIC nanoGUNE”, San Sebastian The Republic of Belarus: Fyodor I. Panteleenko, D.Sc. (Engineering), Professor, First Vice-Rector, Corresponding Member of National Academy of Sciences of Belarus, Belarusian National Technical University, Minsk The Ukraine: Sergiy V. Kovalevskyy, D.Sc. (Engineering), Professor, Vice Rector for Research and Academic Aff airs, Donbass State Engineering Academy, Kramatorsk The Russian Federation: Vladimir G. Atapin, D.Sc. (Engineering), Professor, Novosibirsk State Technical University, Novosibirsk; Victor P. Balkov, Deputy general director, Research and Development Tooling Institute “VNIIINSTRUMENT”, Moscow; Vladimir A. Bataev, D.Sc. (Engineering), Professor, Novosibirsk State Technical University, Novosibirsk; Vladimir G. Burov, D.Sc. (Engineering), Professor, Novosibirsk State Technical University, Novosibirsk; Aleksandr N. Korotkov, D.Sc. (Engineering), Professor, Kuzbass State Technical University, Kemerovo; Dmitry V. Lobanov, D.Sc. (Engineering), Associate Professor, I.N. Ulianov Chuvash State University, Cheboksary; Aleksey V. Makarov, D.Sc. (Engineering), Corresponding Member of RAS, Head of division, Head of laboratory (Laboratory of Mechanical Properties) M.N. Miheev Institute of Metal Physics, Russian Academy of Sciences (Ural Branch), Yekaterinburg; Aleksandr G. Ovcharenko, D.Sc. (Engineering), Professor, Biysk Technological Institute, Biysk; Yuriy N. Saraev, D.Sc. (Engineering), Professor, Institute of Strength Physics and Materials Science, Russian Academy of Sciences (Siberian Branch), Tomsk; Alexander S. Yanyushkin, D.Sc. (Engineering), Professor, I.N. Ulianov Chuvash State University, Cheboksary
Vol. 25 No. 2 2023 5 CONTENTS OBRABOTKAMETALLOV TECHNOLOGY Kisel’ A.G., Churankin V.G. Predicting the coolant lubricating properties based on its density and wetting eff ect.................................................................................................................................................................... 6 Berezin I.M., Zalazinsky A.G., Kryuchkov D.I. Analytical model of equal-channel angular pressing of titanium sponge.............................................................................................................................................. 17 EQUIPMENT. INSTRUMENTS Kuts V.V., Chevychelov S.A. Theoretical study of the curvature of the treated surface during oblique milling with prefabricated milling cutters....................................................................................................................... 32 Skeeba V.Yu., Zverev E.A., Skeeba P.Yu., Chernikov A.D., Popkov A.S. Hybrid technological equipment: on the issue of a rational choice of objects of modernization when carrying out work related to retrofi tting a standard machine tool system with an additional concentrated energy source................................................ 45 MATERIAL SCIENCE Vorontsov A.V., Filippov A.V., Shamarin N.N., Moskvichev E.N., Novitskaya O.S., Knyazhev E.O., Denisova Yu.A., Leonov A.A., Denisov V.V. In-situ analysis of ZrN/CrN multilayer coatings under heating................................................................................................................................................................. 68 Kornienko E.E., Gulyaev I.P., Kuzmin V.I., Tambovtsev A.S., Tyryshkin P.A. Structure and properties of WC-10Co4Cr coatings obtained with high velocity atmospheric plasma spraying.................................... 81 Balanovsky A.E., Nguyen V.V., Astafi eva N.A., Gusev R.Yu. Structure and properties of low carbon steel after plasma-jet hard-facing of boron-containing coating............................................................................. 93 Emurlaeva Yu.Yu., Lazurenko D.V., Bataeva Z.B., Petrov I.Yu., Dovzhenko G.D., Makogon L.D., Khomyakov M.N., Emurlaev K.I., Bataev I.A. Evaluation of vacancy formation energy for BCC-, FCC-, and HCP-metals using density functional theory................................................................................................ 104 EDITORIALMATERIALS 117 FOUNDERS MATERIALS 127 CONTENTS
OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 Evaluation of vacancy formation energy for BCC-, FCC-, and HCP-metals using density functional theory Yulia Emurlaeva 1, a,*, Daria Lazurenko 1, b, Zinaida Bataeva 2, c, Ivan Petrov 3, d, Gleb Dovzhenko 4, e, Lubov Makogon 2, f, Maksim Khomyakov 5, g, Kemal Emurlaev 1, h, Ivan Bataev 1, i 1 Novosibirsk State Technical University, 20 Prospekt K. Marksa, Novosibirsk, 630073, Russian Federation 2 Siberian State University of water transport, 33 Schetinkina str., Novosibirsk, 630099, Russian Federation 3 Novosibirsk State University, 1 Pirogova str., Novosibirsk, 630090, Russian Federation 4 Siberian Circular Photon Source “SKlF” Boreskov Institute of Catalysis of Siberian Branch of the Russian Academy of Sciences (SRF “SKIF”), 1 Nikol’skii pr., Kol’tsovo, 630559, Russian Federation 5 Institute of Laser Physics of Siberian Branch of the Russian Academy of Sciences, 15B Prospekt Ak. Lavrentieva, Novosibirsk, 630090, Russian Federation a https://orcid.org/0000-0003-4835-4134, emurlaeva@corp.nstu.ru, b https://orcid.org/0000-0002-2866-5237, pavlyukova_87@mail.ru, c https://orcid.org/0000-0001-5027-6193, bataevazb@ngs.ru, d https://orcid.org/0000-0002-7968-1130, ivan77766600@outlook.com, e https://orcid.org/0000-0003-0615-0643, g.dovjenko@skif.ru, f https://orcid.org/0009-0006-1463-0697, ledimakagon@mail.ru, g https://orcid.org/0000-0001-8095-2092, mnkhomy@gmail.com, h https://orcid.org/0000-0002-1114-6799, emurlaev@corp.nstu.ru, i https://orcid.org/0000-0003-2871-0269, i.bataev@corp.nstu.ru Obrabotka metallov - Metal Working and Material Science Journal homepage: http://journals.nstu.ru/obrabotka_metallov Obrabotka metallov (tekhnologiya, oborudovanie, instrumenty) = Metal Working and Material Science. 2023 vol. 25 no. 2 pp. 104–116 ISSN: 1994-6309 (print) / 2541-819X (online) DOI: 10.17212/1994-6309-2023-25.2-104-116 ART I CLE I NFO Article history: Received: 10 April 2023 Revised: 18 April 2023 Accepted: 27 April 2023 Available online: 15 June 2023 Keywords: Metals Vacancy formation energy Diffusion Simulation Density functional theory Funding This study was funded by the Federal Task of Ministry of Education and Science of the Russian Federation (project FSUN2020- 0014 (2019-0931): “Investigations of Metastable Structures Formed on Material Surfaces and Interfaces under Extreme External Impacts”). Acknowledgements: Researches were conducted at core facility of NSTU “Structure, mechanical and physical properties of materials”. ABSTRACT Introduction. Vacancies are among the crystal lattice defects that have a significant effect on the structural transformations processes during thermal, chemical-thermal, thermomechanical, and other types of alloys treatment. The vacancy formation energy is one of the most important parameters used to describe diffusion processes. An effective approach to its definition is based on the use of the density functional theory (DFT). The main advantage of this method is to carry out computations without any parameters defined empirically. The purpose of the work is to estimate vacancy formation energy of BCC-, FCC- and HCPmetals widely used in mechanical engineering and to compare these findings obtained using various exchangecorrelation functionals (GGA and meta-GGA). Computation procedure. The computations were carried out using the projector-augmented wave method using the GPAW code and the atomic simulation environment (ASE). The Perdew-Burke-Ernzerhof, MGGAC and rMGGAC functionals were used. The wave functions were described by plane waves within simulations. Vacancies formation energy was evaluated using supercells approach with a size 3 × 3 × 3. Computations were carried out for BCC-metals (Li, Na, K, V, Cr, Fe, Rb, Nb, Mo, Cs, Ta, W), FCC-metals (Al, Ni, Cu, Rh, Pd, Ag, Ir, Pt, Au, Pb, Co) and HCP-metals (Be, Ti, Zr, Mg, Sc, Zn, Y, Ru, Cd, Hf, Os, Co, Re). Results and discussion. A comparison of the defined vacancy formation energies indicates the validity of the following ratio of values: PBE MGGAC rMGGAC f f f E E E < ≤ . The values obtained using the open source GPAW code are characterized by the same patterns as for widely spread commercially distributed program VASP. It was revealed that the use of the PBE and MGGAC functionals leads to a slight deviation relative to the experimentally determined vacancies formation energy in contrast to the computations using rMGGAC. For citation: Emurlaeva Yu.Yu., Lazurenko D.V., Bataeva Z.B., Petrov I.Yu., Dovzhenko G.D., Makogon L.D., Khomyakov M.N., Emurlaev K.I., Bataev I.A. Evaluation of vacancy formation energy for BCC-, FCC-, and HCP-metals using density functional theory. Obrabotka metallov (tekhnologiya, oborudovanie, instrumenty) = Metal Working and Material Science, 2023, vol. 25, no. 2, pp. 104–116. DOI: 10.17212/1994-6309-2023-25.2-104-116. (In Russian). ______ * Corresponding author Emurlaeva Yu. Yu., Assistant Novosibirsk State Technical University, 20 Prospekt K. Marksa, 630073, Novosibirsk, Russian Federation Tel.: 8 (383) 346-06-12, e-mail: emurlaeva@corp.nstu.ru
OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 Introduction Point defects, particularly vacancies, determine considerably the nature of various phenomena that occur in metals and alloys. The presence of vacancies is one of the most critical factors that is taken into consideration in the qualitative and quantitative description of diffusion processes accompanying thermal and thermochemical treatment of metals. For instance, recrystallization that develops in plastically deformed materials is based on the phenomenon of self-diffusion, which is closely related to the characteristics of vacancy migration [1, 2]. Polygonization in deformed metals is closely associated with a climb of edge dislocations which is accompanied by the emission or absorption of vacancies [1, 2]. Vacancies have a significant effect on the kinetics of diffusive phase transformations. The coagulation of multiple vacancies is considered as one of the main reasons for the formation of the so-called Kirkendall porosity observed during diffusion welding of some alloys [3–6]. Interstitial defects and vacancies are also essential underway irradiation-induced swelling that is one of the major tasks in nuclear engineering. The key parameter to describe vacancies is the energy of its formation. There are a number of experimental methods to evaluate the vacancy formation energy (VFE) to date. Methods based on precision measurement of heat capacity, electrical resistivity analysis, and positron annihilation spectroscopy (PAS) are among it [7–9]. It should be noted that the experimental determination of the formation energy of point defects is an extremely time-consuming process and is characterized by insufficient accuracy. The appearance and development of effective computational methods, among which the densityfunctional theory (DFT) should especially emphasized, is the result of the intensive development of computational materials science methods used, among other things, for the analysis of defects in the crystal structure. Using DFT, one can easily evaluate the ground state energy for any substance [10] without the introduction of some sort of empirically determined parameters for the calculations. Thus, point defect formation energy can be defined as the difference between the energy values of a supercell containing a vacancy (vacancy supercell, ) vac tot E and a defect-free supercell (bulk supercell, ) bulk tot E . The value obtained by the DFT requires a number of additional corrections to compare with the empirically determined parameters. The features of this approach are described in detail in review publications [11, 12]. One of the stages of DFT computation is associated with the choice of the exchange-correlation (XC) functional. The exact shape of functionals is currently unknown [13] therefore its approximations are used in practice. It should be noted that even if the chosen approximation of XC functional gives the correct result in evaluating some physical property, it may not be appropriate for evaluating another one. There are two widespread approximations among the great number of possible models of XC functionals, namely: the local density approximation (LDA) based on the free electron model [13, 14] and the generalized gradient approximation (GGA) that takes into account not only the electron density, but also its gradient at the considered point in space [15]. Both LDA and GGA functionals are based on a number of simplifications and, for this reason, are characterized by a certain inaccuracy. The choice of a particular XC functional depends on the type of task being solved. For instance, the cohesive energy using GGA-model can be defined more precisely [16] and therefore GGA can be effectively used to calculate the point defects formation energy including vacancies (VFE). However, the inaccuracy of the VFE using the GGA functional turned out to be quite high in practice [17]. In the review paper [11], Freysoldt et al. highlight that using of LDA functional provides a higher accuracy of the VFE evaluating in comparison with computation using GGA. It is associated with the assessment of the inner surface energy contribution arising when one of the atoms is removed from the supercell. The development of new XC functionals and its application for the calculation of various characteristics of materials, including the VFE, make it possible to minimize the deviation of the calculated data from the experimental ones. In particular, the paper [18] reports about efficiency of meta-GGA-functionals. MetaGGA functionals contain the second derivative of the electron density and also take into account the kineticenergy density of electrons, and therefore provide better precision. However, the computations of VFE via the revTPSS functional (one of the most commonly used meta-GGA functionals) did not confirm this
OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 hypothesis [8]. Thus, the search for XC functionals that make it possible to improve the accuracy of VFE calculations for metals remains an urgent task. The aim of this work was to evaluate the VFE of BCC-, FCC- and HCP-metals widely used in mechanical engineering using DFT and to compare the results obtained by application of various types of XC functionals (GGA and meta-GGA). The results obtained are important for analyzing the effectiveness of DFT computations of point defect formation energy. In addition, the obtained data can be used for reference purposes in the simulation of diffusion processes. Theoretical background The calculations were carried out using the projector-augmented wave (PAW) method with the application of the GPAW code [19, 20] and the atomic simulation environment (ASE) [21], implemented in the Python programming language. The widely used Perdew-Burke-Ernzerhof (PBE) functional of GGA family [22], as well as MGGAC [23] and rMGGAC [24] functionals were used to describe XC potential. The MGGAC functional proposed in [23] is developed for quantum chemistry computations and solid state physics. The authors of this model combined the resulting meta-GGA exchange functional with the GGA correlation one. Using this combination, one can determine the structural and energy properties of solids with high accuracy. The rMGGAC functional proposed by Jana et al. [24] accounts for mismatches in the correlation energy of MGGAC for atoms and ions. The wave functions were described with plane waves. The cut-off energy of 500 eV was used for the plane-wave basis set. Total numbers of k-points generated according to Monkhorst-Pack method was 27 (3 × 3 × 3 along the X, Y and Z axes) for the chosen functionals (PBE, MGGAC and rMGGAC). To improve convergence with respect to Brillouin zone sampling, Marzari-Vanderbilt distribution (cold smearing) with the temperature broadening parameter of 0.2 eV was applied [25]. The energy of vacancy formation was evaluated using supercells approach with a size of 3 × 3 × 3. Detailed information about the parameters used in the computations is given in Appendix A. Calculations were carried out for the following metals: 1) BCC-metals: Li, Na, K, V, Cr, Fe, Rb, Nb, Mo, Cs, Ta, W. 2) FCC-metals: Al, Ni, Cu, Rh, Pd, Ag, Ir, Pt, Au, Pb, Co. 3) HCP-metals: Be, Ti, Zr, Mg, Sc, Zn, Y, Ru, Cd, Hf, Os, Co, Re. To calculate the formation energy of point defect X via DFT, one can use the following formula [11]: [ ] [ ] [ ] f q q tot tot i i F corr i E X E X E bulk n qE E = - - m + + ∑ , (1) where [ ] f q E X is the energy of defect X in charge state q; [ ] q tot E X is the total energy of a supercell containing the defect; [ ] tot E bulk is the total energy of the perfect supercell; ni is a number of atoms of type i that have been added to (in this case it is assumed that ni > 0) or removed from (in this case it is assumed that ni < 0) the supercell to form the defect; μi are the corresponding chemical potentials of the added or removed atoms; f E is the Fermi energy and corr E is a correction term that accounts for finite k-point sampling in the case of shallow impurities (a common term used in the physics of semiconductor). In the case of single vacancy (or monovacancy) formed in a pure metal q = 0; Ecorr = 0; i = 1; n1 = n = –1. Thus, the equation (1) is significantly simplified and takes the following form: [ ] [ ] [ ] , f tot tot E vac E vac E bulk = - + m (2) where m is the chemical potential of the metal analyzed (the chemical potential of a single-element compound is typically used in DFT calculations [26]). This means that the VFE can be derived as the energy difference between a supercell that contains a vacancy and a perfect one (defect-free supercell). However, the total energy is an extensive quantity. In
OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 other words, the energy of the system increases proportionally to the number of atoms contained therein. A supercell containing a monovacancy obviously has one atom less than a perfect one. Thus, its energy (excluding the vacancy effect) will be lower compared to the energy of a perfect supercell. Therefore, to distinguish the vacancy contribution one needs to add the chemical potential of the removed atom to the resulting energy difference according to the equation (2). It should be noted that the issue of point defect energy formation definition is more complex for semiconductors and ionic crystals in contrast to metals [11]. Results and Discussion It is known, that the VFE in metals is well described through the following relationship: f m E AkT ≈ , (3) where Tm is a melting temperature (К); k is the Boltzmann constant; A is a proportionality constant close to 10 [27]. Hayashiuchi et al. believed that a relationship between the VFE and the melting point is caused by the similarity between processes of atomic movement during vacancy formation and also its movement at the “solid – liquid” boundary during melting. According to this theory A ≈ 9.7. Fig. 1 shows the research findings in the coordinates “Ef – Tm”. It can be noted that the trend of ViFE growth with the melting temperature of the material is confirmed by data obtained using various methods. The trends defined in this work have a similar character with the DFT computations carried out by Medasani et al. using the VASP computer program [8]. This fact testifies about the appropriateness of using of open source GPAW code as to alternative to widely used commercial software package VASP. The computational results carried out within this work and those obtained by other authors (including the experimental findings) are summarized in Appendix B. The proportionality constant А, evaluated based on the results obtained using PAS, is close to ~12.1. It is slightly above the А = 10 proposed in [27–29]. The proportionality constant was found to be approxiFig. 1. Vacancy formation energy in various metals according to its melting point
OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 mately equal to ~11.6, ~13.9 and ~17.0 within the computation of VFE using PBE, MGGAC and rMGGAC respectively. Thus, the results obtained using the widespread PBE XC functional are considerably closer to the experimental data. The scatter in the computed results relative to the experimental data can be evaluated using mean square error (MSE). In this paper, it was calculated according to the following equation: ( ) exp 2 i i calc f f E E MSE n - = ∑ , (4) where i calc f E and expi f E are the calculated and experimental VFE for element of type i, respectively. It should be noted that only experimental values of the VFE, measured using PAS [7], were used in this study. Since the experimental data are presented only for some metals [7], the deviation of the calculated energies remained unknown for the rest, and, for this reason, was not taken into account to calculate the MSE. The MSEs are close for PBE and MGGAC functionals (0.66 and 0.64 eV2, respectively). When using rMGGAC, the MSE is significantly higher (1.11 eV2). Using fig. 2, one can compare the VFE calculated within this study with the experimental results. The comparison of findings was carried out according to the approach proposed by Medasani et al. [8]. From the calculated data, it is clear that the use of the rMGGAC and MGGAC functionals results in VFE overestimation as compared to the experimental values. The VFE computed using the widespread PBE functional quite uniformly distributed relative to the y = x line. In general, the results obtained are characterized by the following trend: Ef PBE < E f MGGAC ≤ E f rMGGAC that is well correlated with the findings of Medasani et al. [8]. a b c Fig. 2. Comparison of experimental and computed values of the vacancy formation energy for the exchange-correlation functionals PBE (a), rMGGAC (b) and MGGAC (c). The dotted line representing the function y = x is shown on the graphs for the convenience of analyzing the obtained data Analyzing the obtained results, it can be noted that patterns defined by computing correspond to the experimental data. The typical dependence of the VFE from the melting temperature was mentioned above. Nevertheless, it is difficult to use the VFE estimated using DFT in subsequent calculations without introducing additional corrections. In particular, the equilibrium concentration of vacancies and the diffusion coefficient depend exponentially on the VFE. It means that these parameters extremely sensitive to the error in determining the latter one. According to T. Mattsson and A. Mattsson [30], to obtain a reasonable value of the defects’ equilibrium concentration at room temperature one need to know the VFE with an accuracy of 0.025 eV. From the presented data it follows that this accuracy is unreachable without additional corrections. One of the approaches used for a posteriori correction of the VFE is to account the energy of the inner surface inside the crystal created by removing one of the atoms [30].
OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 Conclusions The analysis of VFE in BCC-, FCC- and HCP-metals was carried out using DFT simulation. Based on the conducted study, the following conclusions can be made. 1. The use of DFT is an effective approach to evaluate the formation energy of point defects. The VFE obtained via open source GPAW code are characterized by the same trends as the widely spread commercial software package VASP. It is reasonable to compare the application efficiency of both programs in terms of calculation accuracy and rate in further studies. 2. In most cases, the use of the PBE and MGGAC functionals provides a slighter deviation relative to the experimentally defined VFE in comparison with the calculation via rMGGAC. 3. A comparison of the computed VFE indicates the validity of the following ratio: PBE MGGAC rMGGAC f f f E E E < ≤ . 4. Despite common patterns, the calculated VFE may differ significantly from the experimental data. Thus, the VFE evaluated at T = 0 K can be used only in comparative studies. To increase the accuracy, the calculated VFE should be subject to additional correction. References 1. Gorelik S.S., Dobatkin S.V., Kaputkina L.M. Rekristallizatsiya metallov i splavov [Recrystallization of metals and alloys]. Moscow, MISiS Publ., 2005. 432 p.ISBN: 5-87623-103-7. 2. Humphreys F.J., Hatherly M. Hatherly recrystallization and related annealing phenomena. 2nd ed. Elsevier, 2004. 605 p. DOI: 10.1016/B978-0-08-044164-1.X5000-2. 3. Siegel R.W. Vacancy concentrations in metals. Journal of Nuclear Materials, 1978, vol. 69–70, pp. 117–146. DOI: 10.1016/0022-3115(78)90240-4. 4. Mehrer H. Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes. Springer, 2007. 673 p. DOI: 10.1007/978-3-540-71488-0. 5. Smigelskas A.D., Kirkendall E.O. Zinc diffusion in alpha brass. Transactions of AIME, 1947, vol. 171, pp. 130–142. 6. Paul A., Laurila T., Vuorinen V., Divinski S. Thermodynamics, diffusion and the Kirkendall effect in solids. Springer, 2014. 530 p. DOI: 10.1007/978-3-319-07461-0. 7. Kraftmakher Y. Equilibrium vacancies and thermophysical properties of metals. Physics Reports, 1998, vol. 299, iss. 2–3, pp. 79–188. DOI: 10.1016/s0370-1573(97)00082-3. 8. Medasani B., Haranczyk M., Canning A., Asta M. Vacancy formation energies in metals: A comparison of MetaGGA with LDA and GGA exchange–correlation functionals. Computational Materials Science, 2015, vol. 101, pp. 96–107. DOI: 10.1016/j.commatsci.2015.01.018. 9. Gong Y., Grabowski B., Glensk A., Körmann F., Neugebauer J., Reed R.C. Temperature dependence of the Gibbs energy of vacancy formation of fcc Ni. Physical Review B, 2018, vol. 97, p. 214106. DOI: 10.1103/ physrevb.97.214106. 10. Lazurenko D.V., Dovzhenko G.D., Lozanov V.V., Petrov I.Y., Ogneva T.S., Emurlaev K.I., Bataev I.A. Stabilization of Ti5Al11 at room temperature in ternary Ti-Al-Me (Me = Au, Pd, Mn, Pt) systems. Journal of Alloys and Compounds, 2023, vol. 944, p. 169244. DOI: 10.1016/j.jallcom.2023.169244. 11. Freysoldt C., Grabowski B., Hickel T., Neugebauer J., Kresse G., Janotti A., Van deWalle C.G. First-principles calculations for point defects in solids. Reviews of Modern Physics, 2014, vol. 86, iss. 1, pp. 253–305. DOI: 10.1103/ revmodphys.86.253. 12. Zhang X., Grabowski B., Hickel T., Neugebauer J. Calculating free energies of point defects from ab initio. Computational Materials Science, 2018, vol. 148, pp. 249–259. DOI: 10.1016/j.commatsci.2018. 13. Giustino F. Materials modelling using density functional theory: properties and predictions. Oxford University Press, 2014. 286 p. 14. Kohn W., Sham L.J. Self-consistent equations including exchange and correlation effects. Physical Review, 1965, vol. 140, iss. 4A, pp. A1133–A1138. DOI: 10.1103/PhysRev.140.A1133. 15. Perdew J.P., Chevary J.A., Vosko S.H., Jackson K.A., Pederson M., Singh D.J., Fiolhais C. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Physical Review B, 1992, vol. 46, iss. 11, pp. 6671–6687. DOI: 10.1103/PhysRevB.46.6671.
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OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 Appendix A Detailed information about the parameters used in the computations Ta b l e 1 Details about metals used for DFT computations Metal Lattice type Space group Lattice parameters, Å a b c Al FCC 225 4.0509 Ni 3.5240 Cu 3.6149 Rh 3.8000 Pd 3.8889 Ag 3.8889 Ir 3.8390 Pt 3.9230 Au 4.0773 Pb 4.9500 Co 3.4200 Li BCC 229 3.5100 Na 4.2830 K 5.3100 V 3.0235 Cr 2.8848 Fe 2.8620 Rb 5.6600 Nb 3.3030 Mo 3.1463 Ta 3.3110 W 3.1648 Be HCP 194 2.2860 3.5840 Zr 3.2340 5.1480 Mg 3.2092 5.2099 Sc 3.3130 5.2760 Zn 2.6575 4.9340 Y 3.6435 5.7272 Ru 2.7040 4.4000 Cd 2.9790 5.6140 Hf 3.1930 5.0520 Os 2.7350 4.3200 Ti 2.9400 4.6800 Co 2.5071 4.0686 Re 2.7600 4.4000
OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 Ta b l e 2 Parameters for calculating the energy of bulk and vacancy supercells Functional Lattice type N k EPW MV n nv PBE FCC 3 × 3 × 3 3 × 3 × 3 500 0.2 108 107 BCC 54 53 HCP MGGAC FCC 108 107 BCC 54 53 HCP rMGGAC FCC 108 107 BCC 54 53 HCP Note: EPW – kinetic energy cutoff that determines the number of plane waves, eV; MV – the magnitude of the temperature broadening in the Marzari-Vanderbilt distribution, eV; n и nv – the number of atoms in an ideal supercell and a supercell with a single vacancy. For all computations, periodic boundary conditions were set. Appendix B Values of vacancy formation energies in various elements Ta b l e 3 Values of vacancy formation energies (eV) calculated in this work using the correlation-exchange functionals PBE, MGGAC, MetaGGA, along with the data from [8] (calculated values) and [7] (results of PAS) No. Metal Lattice PBE MGGAC rMGGAC LDA [8] PBE [8] PW91 [8] PAS [7] 1 Be HCP 0.96 1.65 1.75 - - - - 2 Mg 0.85 0.96 1.07 0.8 0.77 0.72 - 3 Sc 2.01 2.4 2.51 1.97 1.86 1.8 - 4 Zn 0.41 0.68 0.76 0.5 0.42 0.49 - 5 Y 1.92 2.28 2.37 1.91 1.87 1.82 - 6 Ru 2.84 3.48 3.62 3.03 2.71 2.62 - 7 Cd 0.28 0.66 0.66 – – – - 8 Hf 2.29 3.18 - 2.17 2.24 2.16 - 9 Os 3.04 3.8 - 3.35 3.08 3.02 - 10 Ti 2.23 2.87 2.99 2.08 2.08 1.99 - 11 Co 2.04 2.39 2.56 2.22 1.96 1.9 - 12 Re 3.24 3.86 - 3.65 3.4 3.26 - 13 Zr 2.19 2.82 2.95 - - - - 14 Li BCC 0.64 0.61 0.67 - - - - 15 Na 0.43 0.48 - 0.34 0.33 0.31 - 16 K 0.37 0.41 0.44 0.33 0.3 0.29 0.34 17 V 2.98 3.49 3.76 - 2.27 2.2 2.07 18 Cr 3.05 3.93 4.1 2.85 2.77 2.65 2.0 19 Fe 1.86 2.58 2.71 2.3 2.2 2.14 - 20 Rb 0.32 0.37 0.4 - - - - 21 Nb 3.0 3.49 3.71 3.01 2.77 2.71 2.65
OBRABOTKAMETALLOV MATERIAL SCIENCE Vol. 25 No. 2 2023 Conflicts of Interest The authors declare no conflict of interest. 2023 The Authors. Published by Novosibirsk State Technical University. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0). E n d o f t h e t a b . 3 No. Metal Lattice PBE MGGAC rMGGAC LDA [8] PBE [8] PW91 [8] PAS [7] 22 Mo 2.81 3.5 3.67 2.87 2.74 2.56 3.0 23 Cs 0.31 0.32 - - - - - 24 Ta 3.43 4.12 - 2.99 2.82 2.74 - 25 W 3.29 3.79 - 3.48 3.31 3.18 4.0 26 Al FCC 0.74 0.7 0.96 0.71 0.65 0.56 0.66 27 Ni 1.51 2.09 2.19 1.68 1.46 1.89 - 28 Cu 1.04 1.77 1.8 1.29 1.09 1.05 1.28 29 Rh 1.64 2.22 2.31 2.02 1.74 1.66 - 30 Pd 1.06 1.74 1.75 1.48 1.21 1.18 1.85 31 Ag 0.03 0.77 0.77 1.05 0.78 0.77 1.11 32 Ir 1.57 2.52 - 1.89 1.62 1.57 1.79 33 Pt 0.67 1.46 - 0.99 0.74 0.72 1.32 34 Au 0.17 1.18 - 0.66 0.41 0.39 0.89 35 Co 1.75 2.58 2.66 2.1 1.8 1.76 1.34 36 Pb - 0.81 - - - - -
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