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Oxygen Diffusion Modeling in Titanium Alloys: New Elements on the Analysis of Microhardness Profiles

Abstract

This study focuses on the diffusion of oxygen in titanium alloys during high-temperature oxidation. In particular, the model used to obtain thermokinetic coefficients from microhardness profiles was investigated. A literature review shows that microhardness profiles are modeled by a simple error function in the same way as oxygen concentration profiles obtained by microprobe analysis (EPMA). The analysis of literature shows that the hypothesis of a linear relationship between microhardness and oxygen content is not true over the entire oxygen concentration range and that a parabolic relationship is empirically more accurate. A new modeling equation taking into account this parabolic law is proposed as well as a simplified and easier to use form. The relative error of the diffusion coefficients obtained using the simplified equation was then determined. This new model was applied to the experimental microhardness profile of a Ti-6242s sample oxidized at 625 °C. The resulting oxygen diffusion coefficient is in excellent agreement with the one determined from EPMA profile using the classic error function model. Finally, other data from literature were analyzed with the new model to obtain an Arrhenius diagram of oxygen diffusivity in Ti-64 alloy between 550 and 850 °C. This diagram gives thermokinetic coefficients \(D_{0} = 1.1 \times 10^{ - 5} { \exp }\left( {\frac{{ - 191\,{\text{kJ/mol}}}}{RT}} \right)\) that are close to those reported for pure α-Ti in the temperature range 550–850 °C.

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References

  1. 1.

    M. Yan, W. Xu, M. S. Dargusch, H. P. Tang, M. Brandt and M. Qian, Powder Metallurgy57, 251 (2014).

  2. 2.

    C. E. Shamblen and T. K. Redden, in The Science, Technology and Application of Titanium, eds. R. I. Jaffe and N. E. Promisel (1970), p. 199.

  3. 3.

    K. S. Chan, M. Koike, B. W. Johnson and T. Okabe, Metallurgical and Materials Transactions A39, 171 (2008).

  4. 4.

    H. Mehrer, H. Bakker, K.-H. Hellwege, H. Landolt, R. Börnstein and O. Madelung (eds.), Numerical Data and Functional Relationships in Science and Technology, (Springer, Berlin, 1990).

  5. 5.

    D. David, G. Beranger and E. A. Garcia, Journal of the Electrochemical Society130, 4 (1983).

  6. 6.

    J. Unnam, R. N. Shenoy and R. K. Clark, Oxidation of Metals26, 231 (1986).

  7. 7.

    Z. Liu and G. Welsch, Metallurgical Transactions A19, 1121 (1988).

  8. 8.

    W. L. Finlay and J. A. Snyder, JOM2, 277 (1950).

  9. 9.

    L. Bendersky and A. Rosen, Engineering Fracture Mechanics20, 303 (1984).

  10. 10.

    R. Gaddam, M.-L. Antti and R. Pederson, Materials Science and Engineering: A599, 51 (2014).

  11. 11.

    D. P. Satko, et al., Acta Materialia107, 377 (2016).

  12. 12.

    R. H. Olsen, Metallography3, 183 (1970).

  13. 13.

    R. Gaddam, B. Sefer, R. Pederson and M.-L. Antti, Materials Characterization99, 166–174 (2015).

  14. 14.

    Z. Liu and G. Welsch, Metallurgical Transactions A19, 527 (1998).

  15. 15.

    K. E. Wiedemann, R. N. Shenoy and J. Unnam, Metallurgical Transactions A18, 1503 (1987).

  16. 16.

    B. Sefer, J. J. Roa, A. Mateo, R. Pederson, and M.-L. Antti, in Proceedings of the 13th World Conference on Titanium, eds. V. Venkatesh, A. L. Pilchak, J. E. Allison, S. Ankem, R. Boyer, J. Christodoulou, H. L. Fraser, M. A. Imam, Y. Kosaka, H. J. Rack, A. Chatterjee, and A. Woodfield (Wiley, Hoboken, 2016), p. 1619.

  17. 17.

    C. Dupressoire, A. Rouaix-Vande Put, P. Emile, C. Archambeau-Mirguet, R. Peraldi and D. Monceau, Oxidation of Metals87, 343 (2017 ).

  18. 18.

    A. Vande Put, C. Thouron, P. Emile, R. Peraldi, B. Dod, and D. Monceau, in Proceedings of the 14th World Conference on Titanium (2019).

  19. 19.

    J. Crank, The Mathematics of Diffusion, 2nd edn. (Clarendon Press, Oxford, 1975).

  20. 20.

    K. N. Strafford and J. M. Towell, Oxidation of Metals10, 41 (1976).

  21. 21.

    C. J. Rosa, Metallurgical Transactions1, 2517 (1970).

  22. 22.

    M. Göbel, V. A. C. Haanappel and M. F. Stroosnijder, Oxidation of Metals55, 137 (2001).

  23. 23.

    S. Zabler, Materials Characterization62, 1205 (2011).

  24. 24.

    J. Baillieux, D. Poquillon and B. Malard, Journal of Applied Crystallography49, 175 (2016).

  25. 25.

    H. Conrad, Progress in Materials Science26, 123 (1981).

  26. 26.

    C. Leyens and M. Peters (eds.), Titanium and Titanium Alloys: Fundamentals and Applications, 1st edn. (Wiley, New York, 2003).

  27. 27.

    R. N. Shenoy, J. Unnam and R. K. Clark, Oxidation of Metals26, 105 (1986).

  28. 28.

    J. Qu, P. J. Blau, J. Y. Howe and H. M. Meyer III, Scripta Materialia60, 886 (2009).

  29. 29.

    K. Maeda, et al., Journal of Alloys and Compounds776, 519 (2019).

  30. 30.

    H. Guleryuz and H. Cimenoglu, Journal of Alloys and Compounds472, 241 (2009).

  31. 31.

    C. Ciszak, I. Popa, J.-M. Brossard, D. Monceau and S. Chevalier, Corrosion Science110, 91 (2016).

  32. 32.

    H. Okamoto, Journal of Phase Equilibria and Diffusion32, 473 (2011).

  33. 33.

    S. Winitzki, A Handy Approximation for the Error Function and Its Inverse, A Lecture Note Obtained Through Private Communication (2008)

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Acknowledgements

The authors gratefully acknowledge the support of Airbus Operations SAS and the fruitful discussions with Benjamin Dod and Yannick Cadoret.

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Correspondence to Nicolas Vaché.

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Appendix: Mathematical Development Associated with Fig. 2

Appendix: Mathematical Development Associated with Fig. 2

The ratio between the two oxygen diffusion coefficients can be expressed from Eqs. 10 and 12 [20].

$$\frac{{D_{{{\text{simplified}}\,{\text{model}}}} }}{{D_{{{\text{complete}}\,{\text{model}}}} }} = \frac{{\frac{1}{4t}*\left[ {\frac{x}{{{\text{erf}}^{ - 1} \left[ {1 - \frac{{\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)^{2} }}{{\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)^{2} }}} \right]}}} \right]^{2} }}{{\frac{1}{4t}*\left[ {\frac{x}{{{\text{erf}}^{ - 1} \left[ {1 - \frac{{\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)^{2} + 2b\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)\sqrt {C_{0} } }}{{\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)^{2} + 2b\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)\sqrt {C_{0} } }}} \right]}}} \right]^{2} }}$$

Terms \(\frac{{x^{2} }}{4t}\) can be simplified,

$$\frac{{D_{{{\text{simplified}}\,{\text{model}}}} }}{{D_{{{\text{complete}}\,{\text{model}}}} }} = \frac{{\left[ {\frac{1}{{{\text{erf}}^{ - 1} \left[ {1 - \frac{{\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)^{2} }}{{\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)^{2} }}} \right]}}} \right]^{2} }}{{\left[ {\frac{1}{{{\text{erf}}^{ - 1} \left[ {1 - \frac{{\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)^{2} + 2b\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)\sqrt {C_{0} } }}{{\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)^{2} + 2b\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)\sqrt {C_{0} } }}} \right]}}} \right]^{2} }}$$

Then,

$$\frac{{D_{{{\text{simplified}}\,{\text{model}}}} }}{{D_{{{\text{complete}}\,{\text{model}}}} }} = \frac{{\left( {{\text{erf}}^{ - 1} \left[ {1 - \frac{{\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)^{2} + 2b\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)\sqrt {C_{0} } }}{{\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)^{2} + 2b\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)\sqrt {C_{0} } }}} \right]} \right)^{2} }}{{\left( {{\text{erf}}^{ - 1} \left[ {1 - \frac{{\left( {{\text{HV}}_{x} - {\text{HV}}_{0} } \right)^{2} }}{{\left( {{\text{HV}}_{\text{s}} - {\text{HV}}_{0} } \right)^{2} }}} \right]} \right)^{2} }}$$

As there is no explicit form of the reciprocal error function erf−1(x), the approximation of Winitzki was used [33]:

$${\text{erf}}^{ - 1} \left( x \right) \approx \left[ { - \frac{2}{\pi a} - \frac{{\ln \left( {1 - x^{2} } \right)}}{2} + \sqrt {\left( {\frac{2}{\pi a} + \frac{{\ln \left( {1 - x^{2} } \right)}}{2}} \right)^{2} - \frac{1}{a}{ \ln }\left( {1 - x^{2} } \right) } } \right]^{{\frac{1}{2}}}$$

With a \(\approx\) 0.147, the largest relative error of this equation is about 2 × 10−3.

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Vaché, N., Monceau, D. Oxygen Diffusion Modeling in Titanium Alloys: New Elements on the Analysis of Microhardness Profiles. Oxid Met 93, 215–227 (2020). https://doi.org/10.1007/s11085-020-09956-9

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Keywords

  • Titanium
  • Oxidation
  • Oxygen diffusion coefficient
  • Microhardness profile
  • Modeling