Skip to main content
SpringerLink
Log in

Peculiarities of recrystallization activated by a diffusion flow of an impurity from a thin-film coating

  • Regular Article - Statistical and Nonlinear Physics
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

We propose new model of recrystallization activated by a diffusion flux of impurity atoms from a thin coating of the surface of a metal sample. The model is based on diffusion equation with a stepwise diffusion coefficient and moving boundary corresponding to the front of recrystallization. We obtain the explicit exact solution of the problem formulated, which describes the distribution of impurity concentration. We derive the analytical dependences of the depth of the recrystallized layer and recrystallization rate on time within the framework of the formulated model. The results obtained agree with the fact that the process of migration of grain boundaries, which causes recrystallization, develops under conditions of constant feeding of the boundaries by impurities diffusing from the surface.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theoretical study and no experimental data has been listed.]

References

  1. Y.R. Kolobov, R.Z. Valiev, G.P. Grabovetskaya, Grain Boundary Diffusion and Properties of Nanostructured Materials (Cambridge International Science Publishing, Cambridge, 2007), p. 250

    Google Scholar 

  2. Y.R. Kolobov, G.P. Grabovetskaya, M.B. Ivanov, A.P. Zhilyaev, R.Z. Valiev, Grain boundary diffusion characteristics of nanostructured nickel. Scripta Mater. 44(6), 873–878 (2001). https://doi.org/10.1016/S1359-6462(00)00699-0

    Article  Google Scholar 

  3. Yu.R. Kolobov, G.P. Grabovetskaya, K.V. Ivanov, M.B. Ivanov, Grain boundary diffusion and mechanisms of creep of nanostructured metals. Interface Sci. 10, 3136 (2002). https://doi.org/10.1023/A:1015128928158

    Article  Google Scholar 

  4. K. Marquardt, E. Petrishcheva, E. Gardés et al., Grain boundary and volume diffusion experiments in yttrium aluminium garnet bicrystals at 1,723 K: a miniaturized study. Contrib Mineral Petrol. 162, 739–749 (2011). https://doi.org/10.1007/s00410-011-0622-7

    Article  ADS  Google Scholar 

  5. I. Apyhtina, K. Kovaleva, A. Novikov, D. Orelkina, A. Petelin, E. Yakushko, Diffusion controlled grain boundary and triple junction wetting in polycrystalline solid metal. Defect Diffus. Forum 363, 127–129 (2015). https://doi.org/10.4028/www.scientific.net/ddf.363.127

  6. R.E. Mistler, R.L. Coble, Grain-boundary diffusion and boundary widths in metals and ceramics. J. Appl. Phys. 45, 1507–1509 (1974). https://doi.org/10.1063/1.1663451

    Article  ADS  Google Scholar 

  7. S. Herth, T. Michel, H. Tanimoto, M. Egersmann, R. Dittmar, H.-E. Schaefer, W. Frank, R. Würschum, Self-diffusion in nanocrystalline Fe and Fe-rich alloys. Defect Diffus. Forum 194–199, 1199–1204 (2001). https://doi.org/10.4028/www.scientific.net/ddf.194-199.1199

  8. I. Kaur, Y. Mishin, W. Gust, Fundamentals of Grain and Interphase Boundary Diffusion (Wiley, Chichester, 1995)

    Google Scholar 

  9. S.E. Savotchenko, M.B. Ivanov, O.V. Yurova, Single-phase model of recrystallization of molybdenum activated by diffusion of nickel impurities. Russ. Phys. J. 50, 1118–1123 (2007). https://doi.org/10.1007/s11182-007-0164-7

    Article  Google Scholar 

  10. A. Gupta, V. Kulitcki, B.T. Kavakbasi, Y. Buranova, J. Neugebauer, G. Wilde, T. Hickel, S.V. Divinski, Precipitate-induced nonlinearities of diffusion along grain boundaries in Al-based alloy. Phys. Rev. Mater. 2, 073801 (2018). https://doi.org/10.1103/PhysRevMaterials.2.073801

    Article  Google Scholar 

  11. S.V. Divinski, F. Hisker, Y.-S. Kang, J.-S. Lee, C. Herzig, Fe-59 grain boundary diffusion in nanostructured gamma-Fe–Ni—Part I: radiotracer experiments and Monte-Carlo simulation in the type-A and B kinetic regimes. Z. Met. 93, 256–264 (2002). https://doi.org/10.3139/146.020265

    Article  Google Scholar 

  12. G.P. Grabovetskaya, I.P. Mishin, I.V. Ratochka, S.G. Psakhie, Yu.R. Kolobov, Grain boundary diffusion of nickel in submicrocrystalline molybdenum processed by severe plastic deformation. Tech. Phys. Lett. 34(2), 136–138 (2008). https://doi.org/10.1134/S1063785008020156

    Article  ADS  Google Scholar 

  13. A.O. Rodin, A. Khairullin, Ni grain boundary diffusion in Cu–Co alloys. Defect Diffus. Forum 363, 130–132 (2015). https://doi.org/10.4028/www.scientific.net/ddf.363.130

  14. D. Prokoshkina, V. Esin, G. Wilde, S.V. Divinski, Grain boundary width, energy and self-diffusion in nickel: effect of material purity. Acta Mater. 61, 51885197 (2013). https://doi.org/10.1016/j.actamat.2013.05.010

    Article  Google Scholar 

  15. N. Anento, A. Serra, Y. Osetsky, Effect of nickel on point defects diffusion in Fe–Ni alloys. Acta Mater. 132, 367–373 (2017). https://doi.org/10.1016/j.actamat.2017.05.010

    Article  ADS  Google Scholar 

  16. J.C. Fisher, Calculation of diffusion penetration curves for surface and grain boundary diffusion. J. Appl. Phys. 22, 74–77 (1951). https://doi.org/10.1063/1.1699825

    Article  ADS  Google Scholar 

  17. R. Smoluchowski, Theory of grain boundary diffusion. Phys. Rev. 87, 482 (1952). https://doi.org/10.1103/PhysRev.87.482

    Article  ADS  Google Scholar 

  18. R.T.P. Whipple, Concentration contours in grain boundary diffusion. Philos. Mag. 45, 1225–1236 (1954). https://doi.org/10.1080/14786441208561131

    Article  MATH  Google Scholar 

  19. T. Suzuoka, Lattice and grain boundary diffusion in polycrystals. Trans. Jpn. Inst. Met. 2, 25–32 (1961). https://doi.org/10.2320/matertrans1960.2.25

    Article  Google Scholar 

  20. L.G. Harrison, Influence of dislocations on kinetics in solids with particular reference to the alkali halides. Trans. Faraday Soc. 57, 1191–1199 (1961). https://doi.org/10.1039/tf9615701191

    Article  Google Scholar 

  21. V. E. Wood, A. E. Austin, F. J. Milford, Theoretical solutions of grain boundary diffusion problem. Approximations and interpretation of experiments. J. Appl. Phys. 33, 3574 (1962). https://doi.org/10.1063/1.1702449

    Article  ADS  Google Scholar 

  22. A.D. Le Claire, A. Rabinovitch, A mathematical analysis of diffusion in dislocations. I. Application to concentration ‘tails’. J. Phys. C Solid State Phys. 14, 3863 (1981). https://doi.org/10.1088/0022-3719/14/27/011

  23. A. D. Le Claire, A. Rabinovitch, A mathematical analysis of diffusion in dislocations: II. Influence at low densities on measured diffusion coefficients. J. Phys. C Solid State Phys. 15, 3455 (1982). https://doi.org/10.1088/0022-3719/15/16/007

    Article  ADS  Google Scholar 

  24. A.G. Kesarev, V.V. Kondrat’ev, I.L. Lomaev, Description of grain-boundary diffusion in nanostructured materials for thin-film diffusion source. Phys. Met. Metallogr. 116, 225234 (2015). https://doi.org/10.1134/S0031918X15030072

    Article  Google Scholar 

  25. M.V. Chepak-Gizbrekht, Modeling of grain-boundary diffusion taking into account the grain shape. AIP Conf. Proc. 2167, 020050 (2019). https://doi.org/10.1063/1.5131917

    Article  Google Scholar 

  26. V.V. Krasil’nikov, S.E. Savotchenko, Grain boundary diffusion patterns under nonequilibrium and migration of grain boundaries in nanoctructure materials. Bull. Russ. Acad. Sci. Phys. 73, 12771283 (2009). https://doi.org/10.3103/S1062873809090214

    Article  MATH  Google Scholar 

  27. H. Mehrer, Diffusion in Solids. Fundamentals, Methods, Materials, Diffusion-Controlled Processes (Springer, Berlin, 2007), p. 645

    Google Scholar 

  28. S.E. Savotchenko, The nonlinear wave and diffusion processes in media with a jump change in characteristics depending on the amplitude of the field distribution. Commun. Nonlinear Sci. Numer. Simul. 99, 105785 (2021). https://doi.org/10.1016/j.cnsns.2021.105785

    Article  MathSciNet  MATH  Google Scholar 

  29. A.M. Meirmanov, The Stefan Problem (Walter de Gruyter, Berlin, 1992)

    Book  Google Scholar 

  30. V. Alexiades, A.D. Solomon, Mathematical Modelling of Melting and Freezing Processes (Hemisphere-Taylor & Francis, Washington DC, 1993)

    Google Scholar 

  31. A. Kar, J. Mazumder, Analytic solution of the Stefan problem in finite mediums. Quart. Appl. Math. 52, 49–58 (1994). Available at: https://www.ams.org/journals/qam/1994-52-01/S0033-569X-1994-1262318-3/S0033-569X-1994-1262318-3.pdf

  32. A.C. Briozzo, M.F. Natale, On a twophase Stefan problem with convective boundary condition including a density jump at the free boundary. Math. Methods Appl. Sci. 43(6), 3744–3753 (2020). https://doi.org/10.1002/mma.6152

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. V.F. Khorunov, S.V. Maksymova, Brazing of superalloys and the intermetallic alloy (\(\gamma \)-TiAl). In: D.P. Sekulić (Ed.) Advances in Brazing Woodhead Publishing Series in Welding and Other Joining Technologies (2013), pp. 85–120. https://doi.org/10.1533/9780857096500.2.85

Download references

Author information

Authors and Affiliations

  1. Belgorod State Technological University Named After V.G. Shukhov, Kostukova St., 46, 308012, Belgorod, Russia

    S. E. Savotchenko

Authors
  1. S. E. Savotchenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Author wrote this paper and contributed with analytical calculations supplemented by graph data.

Corresponding author

Correspondence to S. E. Savotchenko.

Ethics declarations

Conflict of interest

The author declares that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Savotchenko, S.E. Peculiarities of recrystallization activated by a diffusion flow of an impurity from a thin-film coating. Eur. Phys. J. B 94, 190 (2021). https://doi.org/10.1140/epjb/s10051-021-00203-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/s10051-021-00203-x

Search

Navigation