Physics of the Solid State

, Volume 61, Issue 11, pp 2096–2103 | Cite as

Nonlinear Dynamics of the Formation of a Periodic Superstructure of Impurity Bands during Rapid Directional Solidification of Binary Alloys

  • A. A. Chevrychkina
  • N. M. Bessonov
  • A. L. KorzhenevskiiEmail author


An analytical description is proposed for the nonlinear dynamics of formation of regular impurity superstructures during a rapid solidification of binary alloys supplemented by a numerical model calculation. It is shown that there is a stable limit cycle, and spatial profiles of the impurity concentration in a solid product are calculated. The inclusion of a finiteness of the impurity atom jump velocity is found to lead to very substantial decrease in the cycle sizes and a change in the profiles of impurity superstructures.


phase transitions temperature gradient mobile impurities capillary-wave model formation of periodic impurity superstructures 



This work was supported by the Russian Science Foundation, project no. 19-19-00552.


The authors declare that they have no conflicts of interest.


  1. 1.
    Kh. S. Bagdasarov, E. I. Givargizov, L. N. Dem’yanets, V. A. Kuznetsov, A. N. Labochev, and A. A. Chernov, Modern Crystallography, Ed. by B. K. Vainshtein (Nauka, Moscow, 1980), Vol. 3, p. 401 [in Russian].Google Scholar
  2. 2.
    M. Shore and A. D. Fowler, Can. Mineral. 34, 1111 (1996).Google Scholar
  3. 3.
    I. Sunagawa, Crystals, Growth, Morphology, and Perfection (Cambridge Univ. Press, New York, 2005).CrossRefGoogle Scholar
  4. 4.
    H. J. Scheel and P. Capper, Crystal Growth Technology (Wiley-VCH, Weinheim, 2008).CrossRefGoogle Scholar
  5. 5.
    G. Dhanaraj, K. Byrappa, V. Prasad, and M. Dudley, Springer Handbook of Crystal Growth (Springer, Berlin, Heidelberg 2010).CrossRefGoogle Scholar
  6. 6.
    I. L’Heureux, Phil. Trans. R. Soc. London, Ser. A 371, 20120356 (2013).ADSCrossRefGoogle Scholar
  7. 7.
    A. A. Chevrychkina, N. M. Bessonov, and A. L. Korzhenevskii, Fiz. Tverd. Tela 61, 1904 (2019).Google Scholar
  8. 8.
    Y. Wang and E. Merino, Geochim. Cosmochim. Acta 56, 587 (1992).ADSCrossRefGoogle Scholar
  9. 9.
    P. Ortoleva, Geochemical Self-Organization (Oxford Univ. Press, Clarendon, New York, 1994).Google Scholar
  10. 10.
    M. G. Cadjan and I. A. Lubashevsky, Phys. Lett. A 244, 285 (1998).ADSCrossRefGoogle Scholar
  11. 11.
    F. Kalischevski, I. Lubashevsky, and A. Heuer, Phys. Rev. E 75, 021601 (2007).ADSCrossRefGoogle Scholar
  12. 12.
    I. Lubashevsky, T. Mues, and A. Heuer, Phys. Rev. E 78, 041606 (2008).ADSCrossRefGoogle Scholar
  13. 13.
    T. Mues, A. Heuer, M. Burger, and I. Lubashevsky, Phys. Rev. E 81, 051605 (2010).ADSCrossRefGoogle Scholar
  14. 14.
    W. J. Boettinger, D. Schechtman, R. J. Schaffer, and F. S. Biancaniello, Met. Trans., A 15, 55 (1984).Google Scholar
  15. 15.
    M. J. Aziz and W. J. Boettinger, Acta Met. Mater. 42, 527 (1994).CrossRefGoogle Scholar
  16. 16.
    W. Kurz and R. Trivedi, Met. Mater. Trans. A 27, 625 (1996).CrossRefGoogle Scholar
  17. 17.
    M. Conti, Phys. Rev. E 56, R6267 (1997).ADSCrossRefGoogle Scholar
  18. 18.
    M. Conti, Phys. Rev. E 58, 2071 (1998).ADSCrossRefGoogle Scholar
  19. 19.
    P. K. Galenko and D. M. Herlach, Phys. Rev. Lett. 96, 150602 (2006).ADSCrossRefGoogle Scholar
  20. 20.
    S. Walder and P. L. Ryder, Acta Met. Mater. 43, 4007 (1995).CrossRefGoogle Scholar
  21. 21.
    M. Carrard, M. Gremaud, M. Zimmerman, and W. Kurz, Acta Met. 40, 983 (1992).CrossRefGoogle Scholar
  22. 22.
    K. Brattkus and D. I. Meiron, SIAM J. Appl. Math. 52, 1303 (1992).ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    G. J. Merchant, R. J. Braun, K. Brattkus, and S. H. Davis, SIAM J. Appl. Math. 52, 1279 (1992).MathSciNetCrossRefGoogle Scholar
  24. 24a.
    A. Karma and A. Sarkissian, Phys. Rev. Lett. 27, 2616 (1992);ADSCrossRefGoogle Scholar
  25. 24b.
    Phys. Rev. E 47, 513 (1993).Google Scholar
  26. 25.
    A. L. Korzhenevskii, R. Bausch, and R. Schmitz, Phys. Rev. Lett. 108, 046101 (2012).ADSCrossRefGoogle Scholar
  27. 26.
    A. L. Korzhenevskii, R. Bausch, and R. Schmitz, Phys. Rev. E 85, 021605 (2012).ADSCrossRefGoogle Scholar
  28. 27.
    A. L. Korzhenevskii, R. Bausch, and R. Schmitz, Phys. Rev. E 83, 041609 (2011).ADSCrossRefGoogle Scholar
  29. 28.
    A. L. Korzhenevskii, R. E. Rozas, and J. Horbach, J. Phys. 28, 035001 (2016).Google Scholar
  30. 29.
    D. Herlach, Mater. Sci. Forum 539, 1977 (2007).CrossRefGoogle Scholar
  31. 30.
    S. Li, J. Zhang, and P. Wu, J. Cryst. Growth 312, 982 (2010).ADSCrossRefGoogle Scholar
  32. 31.
    H. Wang, P. K. Galenko, X. Zhang, W. Kuang, F. Liu, and D. M. Herlach, Acta Mater. 90, 282 (2015).CrossRefGoogle Scholar
  33. 32.
    J. Zhang, H. Wang, W. Kuang, Y. Zhang, S. Li, Y. Zhao, and D. M. Herlach, Acta Mater. 148, 86 (2018).CrossRefGoogle Scholar
  34. 33.
    D. Jou, J. Casas-Vazquez, and G. Lebon, Extended Irreversible Thermodynamics, 4th ed. (Springer, Berlin, 2010).zbMATHCrossRefGoogle Scholar
  35. 34.
    S. L. Sobolev, Sov. Phys. Usp. 34, 217 (1991).ADSCrossRefGoogle Scholar
  36. 35.
    P. Galenko, Phys. Lett. A 190, 292 (1994).ADSCrossRefGoogle Scholar
  37. 36.
    N. A. Ahmad, A. A. Wheeler, W. J. Boettinger, and G. B. McFadden, Phys. Rev. E 58, 3436 (1998).ADSCrossRefGoogle Scholar
  38. 37.
    M. J. Aziz and T. Kaplan, Acta Met. 36, 2335 (1988).CrossRefGoogle Scholar
  39. 38.
    W. J. Boettinger and J. H. Perepezko, Rapidly Solidified Crystalline Alloys, Ed. S. K. Das, B. H. Kear, and S. M. Adam (The Metallurg. Soc., Warrendale, PA, 1985).Google Scholar
  40. 39.
    M. J. Aziz, J. Appl. Phys. 53, 1158 (1982).ADSCrossRefGoogle Scholar
  41. 40.
    S. L. Sobolev, Phys. Status Solidi 156, 293 (1996).ADSCrossRefGoogle Scholar
  42. 41.
    J. A. Kittl, P. G. Sanders, M. J. Aziz, D. P. Brunco, and M. O. Thompson, Appl. Phys. Lett. 64, 2359 (1994).ADSCrossRefGoogle Scholar
  43. 42.
    J. A. Kittl, M. J. Aziz, D. P. Brunco, and M. O. Thompson, J. Cryst. Growth 148, 172 (1995).ADSCrossRefGoogle Scholar
  44. 43.
    K. Eckler, D. M. Herlach, and M. J. Aziz, Acta Met. Mater. 42, 975 (1994).CrossRefGoogle Scholar
  45. 44.
    Y. Yang, H. Humadi, D. Buta, B. B. Laird, D. Sun, J. J. Hoyt, and M. A. Asta, Phys. Rev. Lett. 107, 025505 (2011).ADSCrossRefGoogle Scholar
  46. 45.
    C. Henager, Jr. and J. R. Morris, Phys. Rev. B 80, 245309 (2009).ADSCrossRefGoogle Scholar
  47. 46.
    H. Humadi, J. J. Hoyt, and N. Provatas, Phys. Rev. E 93, 010801(R) (2016).Google Scholar
  48. 47.
    C. Qi, J. F. Li, B. Xu, L. T. Kong, and S. Zhao, Comput. Mater. Sci. 125, 172 (2016).CrossRefGoogle Scholar
  49. 48.
    R. Yan, S. Ma, T. Jing, and H. Dong, Metals 8, 602 (2018).CrossRefGoogle Scholar
  50. 49.
    D. Turnbull, J. Appl. Phys. 21, 1022 (1950).ADSCrossRefGoogle Scholar
  51. 50.
    K. N. Kowal, A. L. Altieri, and S. H. Davis, Phys. Rev. E 96, 042801 (2017).ADSCrossRefGoogle Scholar
  52. 51.
    M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, and R. Trivedi, Acta Mater. 57, 941 (2009).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  • A. A. Chevrychkina
    • 1
  • N. M. Bessonov
    • 1
  • A. L. Korzhenevskii
    • 1
    Email author
  1. 1.Institute of Problems of Mechanical Engineering, Russian Academy of SciencesSt. PetersburgRussia

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