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Solute Profile during Diffusion Annealing of Thinly Coated Semi-Infinite Metallic Substrate

  • Advanced Functional and Structural Thin Films and Coatings
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Abstract

A mathematical model of interdiffusion with a layer of finite coating thickness has been formulated. The method has adopted the Rayleigh–Ritz technique along with the mass balance integral method on the interface to predict the solute profile in the substrate, as well as the effective interdiffusion coefficient over a range of compositions in the matrix. The theoretical results are compared with the experimental data of Zn/CuZn and Pt/β-NiAl systems. The model precisely estimates the solute penetration depth in the substrate. The activation energy and pre-exponent factor of the predicted effective interdiffusion coefficient has been determined.

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References

  1. Z. Wang, L. Fang, I. Cotton, and R. Freer, Mater. Sci. Engg. B 198, 86 (2015).

    Article  Google Scholar 

  2. S. Hayashi, W. Wang, D.J. Sordelet, and B. Gleeson, Metall. Mater. Trans. A 36A, 1769 (2005).

    Article  Google Scholar 

  3. A.D. Smigelskas, and E.O. Kirkendall, Trans. AIME 171, 130 (1947).

    Google Scholar 

  4. L.S. Darken, Trans. AIME 75, 184 (1948).

    Google Scholar 

  5. C. Wagner, Acta Metall. 17, 99 (1969).

    Article  Google Scholar 

  6. F.J.A. De Broeder, Scr. Metall. 3, 321 (1969).

    Article  Google Scholar 

  7. A.R. Allnat, and A.B. Lidiard, Atomic Transport in Solids (Cambridge University Press, UK, 1993).

    Book  Google Scholar 

  8. H. Mehrer, Diffusion in solids (Fundamentals, Methods, Materials Diffusion–Controlled Processes) (Springer, Verlag Berlin Heidelburg, 2007).

    Google Scholar 

  9. L.V. Kantarovich, and V.I. Kryolov, Approximate Methods of Higher Analysis (Interscience Publishing, New York, NY, 1964).

    Google Scholar 

  10. B.A. Finlayson, The Method of Weighted Residuals and Variational Principles (Academic Press, New York, NY, 1972).

    MATH  Google Scholar 

  11. A. S. Gupta, Calculus of Variations with Applications, (PHI Learning, Delhi, 2022).

  12. H. G. Ahmad: Studies and Modelling of High Temperature Diffusion Process in Selected High Performance Structural Coating Systems, PhD Thesis, (Northumbria University, UK, 2010). http://nrl.northumbria.ac.uk/1587.

  13. R. Filipek, P.K. Datta, M. Danielewski, L. Bednerz, R. Best, and A. Rakowska, Defect. Diffus. Forum 194–199, 571 https://doi.org/10.4028/www.scientific.net/DDF.194-199.571 (2001).

    Article  Google Scholar 

  14. J. Crank, The Mathematics of Diffusion (Oxford University Press, London, 1975).

    MATH  Google Scholar 

  15. S.L. Mitchell, and T.G. Myers, Soc. Ind. Appl. Math. 52(1), 57 https://doi.org/10.1137/0807733036 (2010).

    Article  Google Scholar 

  16. P.S. Basak, S N Appl. Sci. 3, 432 https://doi.org/10.1007/s42452-021-04435-5 (2021).

    Article  Google Scholar 

  17. M.E. Christopher Jr., Q. Zhang, and J. Zhao, J. Phase Equilib. Diffus. 41, 642 https://doi.org/10.1007/s11669-020-00831-3 (2020).

    Article  Google Scholar 

  18. A. Hoxha, and J. Jani, Int. J. Sci. Technol. Res. 4(6), 173 (2015).

    Google Scholar 

  19. P. Audigie, A.R.V. Put, A. Malie, S. Hamadi, and D. Monceau, Surf. Coat. Technol. 260, 9 https://doi.org/10.1016/j.surfcoat.2014.08.83 (2014).

    Article  Google Scholar 

  20. V.D. Divya, U. Ramammurthy, and A. Paul, J. Mater. Res. 26(18), 2384 https://doi.org/10.1557/jmr.2011.203 (2011).

    Article  Google Scholar 

  21. E. Copland, Partial thermodynamic properties of γ’-(NiPt)3Al in the Ni–Al–Pt system, Paper presented in 2006 TMS Annual Meeting & Exhibition, San Antonio, 12–16 (2006).

Download references

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Basak, P.S. Solute Profile during Diffusion Annealing of Thinly Coated Semi-Infinite Metallic Substrate. JOM 75, 3345–3352 (2023). https://doi.org/10.1007/s11837-023-05915-2

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  • DOI: https://doi.org/10.1007/s11837-023-05915-2

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