Solid–Melt Interface Stability during Directional Solidification: A Phenomenological Theory

O. P. Fedorov$^{1,2}$, A. G. Mashkovsky$^1$, and Ye. L. Zhivolub$^2$

$^1$Space Research Institute of the N.A.S. of Ukraine and State Space Agency of Ukraine, 40 Glushkov Prosp., Bldg. 4/1, UA-03187 Kyiv, Ukraine
$^2$G. V. Kurdyumov Institute for Metal Physics of the N.A.S. of Ukraine, 36 Academician Vernadsky Blvd., UA-03142 Kyiv, Ukraine

Received 11.12.2022; final version — 05.06.2023 Download PDF logo PDF

A mathematical model is developed that makes it possible, within the framework of a single phenomenological approach, to investigate the stability of a planar phase boundary during directional solidification of binary alloy, taking into account the effect of a density change and heat transfer in the solid phase. The study reveals a complex picture of alternating areas of stability and instability, which is sensitive to changes in parameters of growth and temperature gradient at the interface. Areas of instability are formed by the development of a set of disturbances of different frequencies and rates of propagations. As shown, the irreducible liquid-phase flow caused by density change plays a major role in the loss of stability of the solidification front and is realized for perturbations with any wave number k > 0.

Keywords: binary alloy, solid–liquid interface, directional solidification, morphological stability, density jump, latent melting heat, dispersion equation.


Citation: O. P. Fedorov, A. G. Mashkovsky, and Ye. L. Zhivolub, Solid–Melt Interface Stability during Directional Solidification: A Phenomenological Theory, Progress in Physics of Metals, 24, No. 2: 366–395 (2023)

  1. M.C. Flemings, Metall. Trans., 5: 2121 (1974);
  2. A.A. Chernov, Modern Crystallography III. Crystal Growth (Berlin–Heidelberg: Springer: 1984);
  3. W. Kurz and D.J. Fisher, Fundamentals of Solidification, 4th Edition (CRC Press: 1998).
  4. Materials Sciences in Space. A Contribution to the Scientific Basis of Space Processing (Eds. B. Feuerbacher, H. Hamacher, and R.J. Naumann) (Berlin–Heidelberg: Springer: 1986);
  5. V.I. Strelov, I.P.Kuranova, B.G. Zakharov, and A.E. Voloshin, Crystallography Reports, 59, No. 6: 781 (2014);
  6. W.W. Mullins and R.F. Sekerka, J. Appl. Phys., 35, No. 2: 444 (1964);
  7. B. Caroli, C. Caroli, C. Misbah, and B. Roulet, J. Physique, 46: 401 (1985);
  8. N. Noel, H. Jamgotchian, and B. Billia, J. Crystal Growth, 181, Nos. 1–2: 117 (1997);
  9. S.H. Davis and T.P. Schulze, Metall. Mater. Trans. A, 27: 583 (1996);
  10. T. Jiang, M. Georgelin, and A. Pocheau, EPL, 102, No. 5: 54002 (2013);
  11. N. Noel, H. Jamgotchian, and B. Billia, J. Crystal Growth, 187, Nos. 3–4: 516 (1998);
  12. H. Jamgotchian, N. Bergeon, D. Benielli, P. Voge, and B. Billia, Microscopy, 203, No. 1: 119 (2001);
  13. O.P. Fedorov and A.G. Mashkovskiy, Crystallogr. Rep., 60, No. 2: 236, (2015);
  14. S. Coriell, G. McFadden, W.F. Mitchell, B. Murray, J. Andrews, and Y. Arikawa, J. Crystal Growth, 224, Nos. 1–2: 145 (2001);
  15. A. Mori, M. Sato, and Y. Suzuki, Jpn. J. Appl. Phys., 58, No. 4: 045506 (2019);
  16. D. Oxtoby, J. Chem. Phys., 96, No. 5: 3834 (1992);
  17. M. Conti, Phys. Rev. E, 64, No. 5: 051601 (2001);
  18. M. Conti and M. Fermani, Phys. Rev. E, 67, No. 2: 026117 (2003);
  19. L. Sedov, Mechanics of Continuous Media (World Scientific: 1997), vol. 2;
  20. L.D. Landau and E.M. Lifshitz, Fluid Mechanics (Pergamon Press: 1987).
  21. T.C. Lee and R.A. Brown, Phys. Rev. B, 47, No. 9: 4937 (1993);
  22. R. Graham, E. Knuth, and O. Patashnik, Concrete Mathematics – A Foundation for Computer Science (2nd Edition) (Addison-Wesley: 1994).