Models of recrystallization activated by a diffusion flow of impurities from a thin-film coating with a convection term at the crystal surface: exact solutions

Sergey Savotchenko¹ ; Aleksei Cherniakov²

¹Belgorod State Technological University named after V. G. Shukhov, 308012, Kostukova St., 46, Belgorod, Russia
²Financial University Under the Government of the Russian Federation 125167, Leningradsky Pr., 49, Moscow, Russia

Mathematical modelling of natural phenomena, Tome 18 (2023), article no. 1

Résumé

Two models of recrystallization are proposed taking into account the convective flux of impurity exchange between the polycrystalline and the thin-film coating. The special boundary modes of recrystallization described by the single-phase and two-phase Stefan problems with the boundary condition at coated surface containing the convective term. The exact solutions of the formulated problems corresponding to the grain-boundary concentration of impurities are obtained. The detail theoretical analysis focused on the third type problem shows that the concentration of impurities and the width of the recrystallized layer increase with an increase in the annealing time. An increase in intensity of impurity exchange between the polycrystalline and the coating promotes an increase in the width of the recrystallized layer. The recrystallization front position increases with an increase in the surface concentration of impurities and it decreases with an increase in the intensity of the impurity flux from the surface. The rate of recrystallization kinetics increases with an increase in the intensity of impurity exchange between the polycrystalline and the coating.

Détail

DOI : 10.1051/mmnp/2022046

Bibliographie

[1] O.M. Alifanov, Inverse Heat Transfer Problems. Springer-Verlag, Berlin (1994).

[2] N. Anento, A. Serra, Y. Osetsky Effect of nickel on point defects diffusion in Fe-Ni alloys Acta Mater 2017 367 373

[3] 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 2015 127 129

[4] T. Beierling, R. Gorny, G. Sadowski Modeling growth rates in static layer melt crystallization Cryst. Growth Des 2013 5229 5240

[5] A.C. Briozzo, M.F. Natale Ona two-phase Stefan problem with convective boundary condition including a density jump at the free boundary Math. Methods Appi. Sci 2020 3744 3753

[6] J. Cheng, S. Lu, M. Yamamoto Reconstruction of the Stefan-Boltzmann coefficients in a heat-transfer process Inverse Prob 2012 045007

[7] J. Cheng, Y. Xing, E. Dong, L. Zhao, H. Liu, T. Chang, M. Chen, J. Wang, J. Lu, J. Wan An overview of laser metal deposition for cladding: defect formation mechanisms, defect suppression methods and performance improvements of laser-cladded layers Materials (Basel, Switzerland) 2022 5522

[8] M.V. Chepak-Gizbrekht Modeling of grain-boundary diffusion taking into account the grain shape AIP Conf. Proc 2019 020050

[9] C. Evans, D. Indjin, Z. Ikonic, P. Harrison, M. Vitiello, V. Spagnolo, G. Scamarcio Thermal modeling of terahertz quantum-cascade lasers: comparison of optical waveguides IEEE J. Quantum Electr 2008 680 685

[10] J. Fuller, E. Marotta Thermal contact conductance of metal/polymer joints: an analytical and experimental investigation J. Thermophys. Heat Transfer 2001 228 238

[11] R. Ghai, K. Chen, N. Baddour Modeling thermal conductivity of porous thermal barrier coatings Coatings 2019 101

[12] M.A. Gonik, F. Baltaretu Problem of attaining constant impurity concentration over ingot height Modern Electr. Mater 2018 41 51

[13] 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 2018 073801

[14] 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 2008 136 138

[15] S. Herth, T. Michel, H. Tanimoto, M. Egersmann, R. Dittmar, H.-E. Schaefer, W. Frank, R. Wurschum Self-diffusion in nanocrystalline Fe and Fe-rich alloys Defect Diffus. Forum 2001 1199 1204

[16] R. Jendrzejewski, I. Kreja, G. Sliwinski Temperature distribution in laser-clad multi-layers Mater. Sci. Eng. A 2004 313 320

[17] E.M. Kartashov, G.S. Krotov Analytical solution of the single-phase Stefan problem Math. Models Comput. Simul 2009 180 188

[18] I. Kaur, Y. Mishin and W. Gust, Fundamentals of grain and interphase boundary diffusion. Wiley, Chichester (1995).

[19] A.G. Kesarev, V.V. Kondrat’Ev, I.L. Lomaev Description of grain-boundary diffusion in nanostructured materials for thin-fìlm diffusion source Phys. Metals Metallogr 2015 225 234

[20] Y.R. Kolobov, G.P. Grabovetskaya, M.B. Ivanov, A.P. Zhilyaev, R.Z. Valiev Grain boundary diffusion characteristics of nanostructured nickel Scr. Mater 2001 873 878

[21] Yu.R. Kolobov, G.P. Grabovetskaya, K.V. Ivanov, M.B. Ivanov Grain boundary diffusion and mechanisms of creep of nanostructured metals Interface Science 2002 31 36

[22] Y.R. Kolobov, R.Z. Valiev, G.P. Grabovetskaya et al., Grain boundary diffusion and properties of nanostructured materials. Cambridge International Science Publishing, Cambridge, UK (2007), 250 p.

[23] 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 2009 1277 1283

[24] Z. Li, G. Yu, X. He, S. Li, H. Li, Q. Li Study of thermal behavior and solidifìcation characteristics during laser welding of dissimilar metals Res. Phys 2019 1062 1072

[25] K. Marquardt, E. Petrishcheva, E. Gardés Grain boundary and volume diffusion experiments in yttrium aluminium garnet bicrystals at 1,723 K: a miniaturized study Contrib. Mineral Petrol 2011 739 749

[26] H. Mehrer, Diffusion in Solids. Fundamentals, Methods, Materials, Diffusion-Controlled Processes, Springer-Verlag, Berlin, Heidelberg (2007), p. 645.

[27] A.M. Meirmanov, The Stefan Problem, Walter de Gruyter, Berlin, Germany (1992).

[28] M.F. Natale, D.A. Tarzia Explicit solutions to the one-phase Stefan problem with temperature-dependent thermal conductivity and a convective term Int. J. Eng. Sci 2003 1685 1689

[29] V. Niziev, F. Mirzade, V. Panchenko, M. Khomenko, R. Grishaev, S. Pityana, C. Rooyen Numerical study to represent non-isothermal melt-crystallization kinetics at laser-powder cladding Model. Numer. Simul. Mater. Sci 2013 61 69

[30] D. Prokoshkina, V. Esin, G. Wilde, S.V. Divinski Grain boundary width, energy and self-diffusion in nickel: effect of material purity Acta Mater 2013 5188 5197

[31] A.O. Rodin, A. Khairullin Ni grain boundary diffusion in Cu-Co alloys Defect Diffus. Forum 2015 130 132

[32] S.E. Savotchenko Peculiarities of recrystallization activated by a diffusion flow of an impurity from a thin-fìlm coating Eur. Phys. J. B 2021 190

[33] S.E. Savotchenko The nonlinear wave and diffusion processes in media with a jump change in characteristics depending on the amplitude of the fìeld distribution Commun. Nonlinear Sci. Numer. Simul 2021 105785

[34] S.E. Savotchenko Models of activated recrystallization on isolated grain boundaries in polycrystals Modern Phys. Lett. B 2022 2150536

[35] S.E. Savotchenko, A.N. Cherniakov Diffusion from a constant source along nonequilibrium dislocation pipes Int. J. Heat Mass Transfer 2022 122655

[36] S.E. Savotchenko, A.N. Cherniakov The nonlinear diffusion model of recrystallization J. Heat Transfer 2022 064501

[37] 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 2007 1118 1123

[38] L. Shumylyak, V. Zhykharevych, S. Ostapov Modeling of impurities segregation phenomenon in the melt crystallization process by continuous cellular automata method Prikladnaya diskretnaya matematika 2016 104 118

[39] L.N. Tao The Stefan problem with an imperfect thermal contact at the interface J. Appl. Mech 1982 715 720

[40] D.A. Tarzia Exact solution for a Stefan problem with convective boundary condition and density jump PAMM Proc. Appl. Math. Mech 2007 1040307 1040308

[41] W.A. Tiller, K. Jackson, J.W. Rutter, B. Chalmers The redistribution of solute atoms during the solidifìcation of metals Acta Metall 1953 428 437

[42] C. Wang, M. He, X. Liua, J.A. Malen Modeling the tunable thermal conductivity of intercalated layered materials with three-directional anisotropic phonon dispersion and relaxation time J. Mater. Chem. C 2022 11686 11696

[43] T. Wei Uniqueness of moving boundary for a heat conduction problem with nonlinear interface conditions Appl. Math. Lett 2010 600 604

[44] Y. Xian, P. Zhang, S. Zhai, P. Yang, Z. Zheng Re-estimation of thermal contact resistance considering near-fìeld thermal radiation effect Appl. Therm. Eng 2019 113601

[45] G. Yang, M. Yamamoto, J. Cheng Heat transfer in composite materials with Stefan-Boltzmann interface conditions Math. Methods Appl. Sci 2008 1297 1314

[46] L. Zhuo, D. Lesnic, S. Meng Reconstruction of the heat transfer coefficient at the interface of a bi-material Inverse Prob. Sci. Eng 2020 374 401