Skip to main content
Log in

The shape of dendritic tips: the role of external impacts

  • Regular Article
  • Published:
The European Physical Journal Special Topics Aims and scope Submit manuscript

Abstract

This study is concerned with the question of what is the shape of a dendritic tip grown from an undercooled melt in the presence of external impacts? To answer this question we extend the recent theory (Alexandrov and Galenko in Philos Trans R Soc A 378:20190243, 2020) to the case of external processes influencing the crystal growth phenomenon. The tip shape function is derived and tested against experimental data and numerical simulations when forced convection and dissolved impurities play a decisive role. It is shown that the tip shape function taking external impacts into account is in good agreement with the theory, experiments and computations. Using our well tested formula for the dendrite tip shape we show that the mechanisms of heat and mass transfer in inclined fluid currents can be essentially different. Namely, heat and mass fluxes at the crystal surface can be described by Fick’s or Newton’s laws or even by a more general mixed-type heat and mass transfer formula.

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

Access this article

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 availibility

All data generated or analysed during this study are included in this published article.

References

  1. W. Kurz, D.J. Fisher, Fundamentals of solidification (Trans Tech Publications Ltd, Aedermannsdorf, 1989)

    Google Scholar 

  2. D. Herlach, P. Galenko, D. Holland-Moritz, Metastable solids from undercooled melts (Elsevier, Amsterdam, 2007)

    Google Scholar 

  3. D.V. Alexandrov, A.Y. Zubarev, Philos. Trans. R. Soc. A 377, 20180353 (2019)

    Article  ADS  Google Scholar 

  4. D.V. Alexandrov, P.K. Galenko, Philos. Trans. R. Soc. A 379, 20200325 (2021)

    Article  ADS  Google Scholar 

  5. H.E. Huppert, J. Fluid Mech. 212, 209 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  6. P.K. Galenko, V.A. Zhuravlev, Physics of dendrites (World Scientific, Singapore, 1994)

    Google Scholar 

  7. G.P. Ivantsov, Dokl. Akad. Nauk SSSR 58, 567 (1947)

    Google Scholar 

  8. G.P. Ivantsov, Dokl. Akad. Nauk SSSR 83, 573 (1952)

    Google Scholar 

  9. G. Horvay, J.W. Cahn, Acta Metall. 9, 695 (1961)

    Article  Google Scholar 

  10. R. Ananth, W.N. Gill, J. Fluid Mech. 208, 575 (1989)

    Article  ADS  Google Scholar 

  11. G.B. McFadden, S.R. Coriell, R.F. Sekerka, J. Cryst. Growth 208, 726 (2000)

    Article  ADS  Google Scholar 

  12. D.V. Alexandrov, E.A. Titova, P.K. Galenko, M. Rettenmayr, L.V. Toropova, J. Phys. Condens. Matter 33, 443002 (2021)

    Article  ADS  Google Scholar 

  13. L.V. Toropova, E.A. Titova, D.V. Alexandrov, P.K. Galenko, M. Rettenmayr, A. Kao, G. Demange, J. Phys. Condens. Matter 33, 365402 (2021)

    Article  Google Scholar 

  14. E.A. Titova, D.V. Alexandrov, P.K. Galenko, J. Cryst. Growth 531, 125319 (2020)

    Article  Google Scholar 

  15. D.V. Alexandrov, P.K. Galenko, Philos. Trans. R. Soc. A 378, 20190243 (2020)

    Article  ADS  Google Scholar 

  16. M. Plapp, A. Karma, Phys. Rev. Lett. 84, 1740 (2000)

    Article  ADS  Google Scholar 

  17. R. Almgren, W.-S. Dai, V. Hakim, Phys. Rev. Lett. 71, 3461 (1993)

    Article  ADS  Google Scholar 

  18. E. Brener, Phys. Rev. Lett. 71, 3653 (1993)

    Article  ADS  Google Scholar 

  19. E. Brener, Physica A 263, 338 (1999)

    Article  ADS  Google Scholar 

  20. D.V. Alexandrov, L.V. Toropova, E.A. Titova, A. Kao, G. Demange, P.K. Galenko, M. Rettenmayr, Philos. Trans. R. Soc. A 379, 20200326 (2021)

    Article  ADS  Google Scholar 

  21. L.V. Toropova, Eur. Phys. J. Spec. Top. 231, 1129 (2022)

    Article  Google Scholar 

  22. L.V. Toropova, D.V. Alexandrov, M. Rettenmayr, D. Liu, J. Phys. Condens. Matter 34, 094002 (2022)

    Article  ADS  Google Scholar 

  23. E. A. Titova, D. V. Alexandrov, J. Phys. A: Math. Theor. 55, 485701 (2022)

    Google Scholar 

  24. Q. Guo, P. Cheng, Int. J. Heat Mass Transf. 145, 118658 (2019)

    Article  Google Scholar 

  25. A. Kao, L.V. Toropova, I. Krastins, G. Demange, D.V. Alexandrov, P.K. Galenko, JOM 72, 3123 (2020)

    Article  ADS  Google Scholar 

  26. A. Kao, L.V. Toropova, D.V. Alexandrov, G. Demange, P.K. Galenko, J. Phys. Condens. Matter 32, 194002 (2020)

    Article  ADS  Google Scholar 

  27. J. LaCombe, M. Koss, V. Fradkov, M. Glicksman, Phys. Rev. E 52, 2778 (1995)

    Article  ADS  Google Scholar 

  28. A.J. Melendez Ramirez, PhD Thesis. (University of Iowa, Iowa, 2009)

  29. L. V. Toropova, P. K. Galenko, D. V. Alexandrov, Crystals 12, 965 (2022)

    Article  Google Scholar 

  30. D. Notz, M.G. McPhee, M.G. Worster, G.A. Maykut, K.H. Schl\({\ddot{\rm u}}\)nzen, H. Eicken, J. Geophys. Res. 108, 3223 (2003)

  31. M.G. McPhee, G.A. Maykut, J.H. Morison, J. Geophys. Res. 92, 7017 (1987)

    Article  ADS  Google Scholar 

  32. D.V. Alexandrov, I.G. Nizovtseva, Int. J. Heat Mass Transf. 51, 5204 (2008)

    Article  Google Scholar 

  33. J. Gao, M. Han, A. Kao, K. Pericleous, D.V. Alexandrov, P.K. Galenko, Acta Mater. 103, 184 (2016)

    Article  ADS  Google Scholar 

  34. Wikipedia. https://en.wikipedia.org/wiki/Reynolds_number

  35. P.K. Galenko, L.V. Toropova, D.V. Alexandrov, G. Phanikumar, H. Assadi, M. Reinartz, P. Paul, Y. Fang, S. Lippmann, Acta Mater. 241, 118384 (2022)

    Article  Google Scholar 

  36. D.V. Alexandrov, L.V. Toropova, Sci. Rep. 12, 17857 (2022)

    Article  ADS  Google Scholar 

  37. L.V. Toropova, D.V. Alexandrov, Sci. Rep. 12, 10997 (2022)

    Article  ADS  Google Scholar 

  38. D.V. Alexandrov, A.A. Ivanov, I.G. Nizovtseva, S. Lippmann, I.V. Alexandrova, E.V. Makoveeva, Crystals 12, 949 (2022)

    Article  Google Scholar 

  39. I.V. Alexandrova, A.A. Ivanov, S.V. Bulycheva, D.V. Alexandrov, Eur. Phys. J. Spec. Top. 231, 1123 (2022)

    Article  Google Scholar 

  40. I.V. Alexandrova, A.A. Ivanov, A.P. Malygin, D.V. Alexandrov, M.A. Nikishina, Eur. Phys. J. Spec. Top. 231, 1089 (2022)

    Article  Google Scholar 

Download references

Acknowledgements

The present work is dedicated to the blessed memory of Professor Markus Rettenmayr who provided thorough thermodynamic study and direct practical applications of novel mainly metallic and alloying materials. L.V.T. acknowledges the financial support from the Russian Science Foundation (Project No. 21-79-10012).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dmitri V. Alexandrov.

Additional information

S.I.: Structural Transformations and Non-Equilibrium Phenomena in Multicomponent Disordered Systems. Guest editors: Liubov Toropova, Irina Nizovtseva.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alexandrov, D.V., Kao, A., Galenko, P.K. et al. The shape of dendritic tips: the role of external impacts. Eur. Phys. J. Spec. Top. 232, 1273–1279 (2023). https://doi.org/10.1140/epjs/s11734-023-00853-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjs/s11734-023-00853-1

Navigation