Carbon diffusion in α-Ti and α₂-Ti₃Al

The interstitial defect formation energies of carbon in various interstices, crystal orbital Hamilton populations, electron localization function, Voronoi volumes, and other characteristics in α-Ti and α₂-Ti₃Al were calculated by the projector augmented wave method. The climbing image nudged elastic band method was used to estimate migration barriers along paths between stable interstitials. The temperature-dependent diffusion coefficients of carbon were evaluated along two nonequivalent crystallographic directions (a and c) in α-Ti  and, for the first time, in α₂-Ti₃Al. The energetically preferred site in both α-Ti and α₂-Ti₃Al is an octahedral one, whose local environment contains only Ti atoms. This preference depends on material and is primarily due to low mechanical contribution to the defect formation energy in α-Ti and chemical one in the case of the alloy. The presence of Al leads to an increase in the defect formation energy in the alloy. In the case of α-Ti, the obtained diffusion coefficients are in good agreement with experiment. Aluminum in the alloy leads to a slowdown in carbon diffusion due to higher both defect formation and migration energies.

1. Berdovsky Y.N. Intermetallics research progress. Nova Science Publishers, New York, 2008, 300 p.

2. Leyens C., Peters M. Titanium and titanium alloys: fundamentals and applications. Wiley-VCH, Weinheim, 2003, 532 p.

3. Sha W., Malinov S. Titanium alloys: modelling of microstructure, properties and applications. CRC Press, Boca Raton, 2009, 586 p.

4. Heumann T. Diffusion in Metallen: Grundlagen, Theorie, Vorgange in Reinmetallen und Legierungen. Springer-Verlag, Berlin, 1992, 323 p.

5. Mehrer H. Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes. Springer, Berlin, 2007, 671 p.

6. Breuer J., Wilger T., Friesel M., Herzig Chr. Interstitial and substitutional diffusion of metallic solutes in Ti3Al. Intermetallics, 1999, 7, P. 381–388.

7. Ru¨sing J., Herzig Chr. Concentration and temperature dependence of titanium self-diffusion and interdiffusion in the intermetallic phase Ti3Al. Intermetallics, 1996, 4, P. 647–657.

8. Herzig Chr., Friesel M., Derdau D., Divinski S.V. Tracer diffusion behavior of Ga as an Al-substituting element in Ti3Al and TiAl intermetallic compounds. Intermetallics, 1999, 7, P. 1141–1151.

9. Koizumi Y., Sakakibara Y., Minamino Y., Tsuji N. Diffusion of Si in Ti3Al intermetallic compound. Defect Diffus. Forum, 2005, 237–240, P. 334– 339.

10. Koizumi Y., Kishimoto M., Minamino Y., Nakajima H. Oxygen diffusion in Ti3Al single crystals. Philos. Mag., 2008, 88, P. 2991–3010.

11. Mishin Y., Herzig Chr. Diffusion in the Ti–Al system. Acta Mater., 2000, 48, P. 589–623.

12. Zhang L.J., Chen Z.Y., Hu Q.M., Yang R. On the abnormal fast diffusion of solute atoms in α-Ti: A first-principles investigation. J. Alloys Compd., 2018, 740, P. 156–166.

13. Gorev N.D., Bakulin A.V., Kulkova S.E. Mechanisms of iron diffusion in α-Ti. J. Exp. Theor. Phys., 2024, 165 (6), P. 818–827.

14. Gorev N.D., Bakulin A.V., Kulkova S.E. Diffusion of Co and Ga in α-Ti. Comput. Mater. Sci., 2024, 244, P. 113231.

15. Gorev N.D., Bakulin A.V., Kasparyan S.O., Kulkova S.E. Peculiarities of the formation of impurity defects in α-Ti and α2-Ti3Al. J. Exp. Theor. Phys., 2025, 168 (1), P. 110–124. (in Russian)

16. Bakulin A.V., Gorev N.D., Kulkova S.E. Diffusion of Fe, Co, Al and Ga in α2-Ti3Al. J. Alloys Compd., 2025, 1039, P. 183180.

17. Wagner F.C., Bucur E.J., Steinberg M.A. The rate of diffusion of carbon in alpha and beta titanium. Trans. Am. Soc. Met., 1956, 48, P. 742–761.

18. Nakonetchnikov A.I., Pavlinov L.V. Diffusion of carbon in α-phases of zirconium and titanium. Izvestija AN SSSR. Metalli, 1972, 2, P. 213. (in Russian)

19. Minkwitz Ch., Herzig Ch. Diffusion of 14C in single and polycrystalline α-titanium. Defect Diffus. Forum, 1997, 143–147, P. 61–66.

20. De Barros M.I., Rats D., Vandenbulcke L., Farges G. Influence of internal diffusion barriers on carbon diffusion in pure titanium and Ti–6Al–4V during diamond deposition. Diamond Relat. Mater., 1999, 8, P. 1022–1032.

21. Scotti L., Mottura A. Interstitial diffusion of O, N, and C in α-Ti from first-principles: Analytical model and kinetic Monte Carlo simulations. J. Chem. Phys., 2016, 144, P. 084701.

22. Blo¨chl P.E. Projector augmented-wave method. Phys. Rev. B, 1994, 50, P. 17953–17979.

23. Kresse G., Joubert D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B, 1999, 59, P. 1758–1775.

24. Perdew J.P., Burke K., Ernzerhof M. Generalized gradient approximation made simple. Phys. Rev. Lett., 1996, 77, P. 3865–3868.

25. Lozovoi A.Y., Paxton A.T., Finnis M.W. Structural and chemical embrittlement of grain boundaries by impurities: A general theory and firstprinciples calculations for copper. Phys. Rev. B, 2006, 74, P. 155416.

26. Dronskowski R., Blo¨chl P.E. Crystal orbital Hamilton populations (COHP). Energy-resolved visualization of chemical bonding in solids based on density-functional calculations. J. Phys. Chem., 1993, 97, P. 8617–8624.

27. Maintz S., Deringer V.L., Tchougre´eff A.L., Dronskowski R. Analytic projection from plane-wave and PAW wavefunctions and application to chemical-bonding analysis in solids. J. Comput. Chem., 2013, 34, P. 2557.

28. Maintz S., Deringer V.L., Tchougre´eff A.L., Dronskowski R. LOBSTER: A tool to extract chemical bonding from plane-wave based DFT. J. Comput. Chem., 2016, 37, P. 1030–1035.

29. Silvi B., Savin A. Classification of chemical bonds based on topological analysis of electron localization functions. Nature, 1994, 371, P. 683–686.

30. Manz T.A., Limas N.G. Introducing DDEC6 atomic population analysis: part 1. Charge partitioning theory and methodology. RSC Adv., 2016, 6, P. 47771–47801.

31. Limas N.G., Manz T.A. Introducing DDEC6 atomic population analysis: part 2. Computed results for a wide range of periodic and nonperiodic materials. RSC Adv., 2016, 6, P. 45727–45747.

32. Henkelman G., Uberuaga B.P., Jo´nsson H. A climbing image nudged elastic band method for finding saddle points and minimum energy paths. J. Chem. Phys., 2000, 113, P. 9901–9904.

33. Vineyard G.H. Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids, 1957, 3, P. 121–127.

34. Landman U., Shlesinger M.F. Stochastic theory of multistate diffusion in perfect and defective systems. I. Mathematical formalism. Phys. Rev. B, 1979, 19, P. 6207–6219.

35. Landman U., Shlesinger M.F. Stochastic theory of multistate diffusion in perfect and defective systems. II. Case studies. Phys. Rev. B, 1979, 19, P. 6220–6237.

36. Bakulin A.V., Latyshev A.M., Kulkova S.E. Absorption and diffusion of oxygen in the Ti3Al alloy. J. Exp. Theor. Phys., 2017, 125 (1), P. 138–147.

37. Zhang L.J., Spiridonova T.I., Kulkova S.E., Yang R., Hu Q.M. Atomic self-diffusion anisotropy of HCP metals from first-principles calculations. Comput. Mater. Sci., 2017, 128, P. 236–242.

38. Tanaka K., Okamoto K., Inui H., Minonishi Y., Yamaguchi M., Koiwa M. Elastic constants and their temperature dependence for the intermetallic compound Ti3Al. Philos. Mag. A, 1996, 73 (5), P. 1475–1488.

39. Flowers J.W., Jr., O’Brien K.C., McEleney P.C. Elastic constants of alpha-titanium single crystals at 25°C. J. Less-Common Metals, 1964, 7, P. 393–395.

40. Kulkov S.S., Bakulin A.V., Kulkova S.E. Effect of boron on the hydrogen-induced grain boundary embrittlement in α-Fe. Int. J. Hydrogen Energy, 2018, 43, P. 1909–1925.

41. Bakulin A.V., Chumakova L.S., Kulkova S.E. Influence of impurities on the oxygen adsorption on the Ti5Si3(0001) surface. J. Exp. Theor. Phys., 2023, 136 (6), P. 711–719.