Journals →  CIS Iron and Steel Review →  2021 →  #2 →  Back

Metal Science and Metallography
ArticleName Variational theory of crystal growth and its application for analysis of forming processes for metastable phases in overcooled metallic melts with eutectic composition
DOI 10.17580/cisisr.2021.02.09
ArticleAuthor M. V. Dudorov, A. D. Drozin, A. V. Stryukov, V. E. Roshchin

South Ural State University, Chelyabinsk, Russia:

M. V. Dudorov, Cand. Phys.-Math., Senior Researcher, Dept. “Pyrometallurgical processes”, e-mail:
A. D. Drozin, Dr. Eng., Prof., Dept. “Pyrometallurgical processes”, e-mail:
V. E. Roshchin, Dr. Eng., Prof., Chief Researcher, Dept. “Pyrometallurgical processes”, e-mail:

Ashinsky Metallurgical Plant, Asha, Russia

A. V. Stryukov, Engineer, Head of the Plant Laboratory, e-mail:


The new crystallization theory for overcooled metastable melt is developed; it is based on variational mechanical principles and takes into account regularities of forming and diffusion growth of equilibrium crystals as well as diffusion-free growth of metastable crystals. Calculations for the melt Fe83B17 were conducted on the model; they displayed that simultaneous nucleation and growth of Fe and Fe2B with metastable phase Fe3B are observed in overcooled melt, and growth speed of near-critical dimensional crystals of Fe3B exceeds crystal growth speed of Fe and Fe2B. The effect of diffusion-free growth is observed for Fe3B crystals, when quickly growing Fe3B crystal surface catches boron atoms. Quasi-equilibrium phase diagram for overcooled Fe-B melt was built on the base of the developed theory; it takes into account both equilibrium crystal growth and metastable phase growth. The obtained diagram allows to predict the values of components concentration near the surface of growing crystals both for Fe and Fe2B crystals meeting the requirements of local equi-librium condition on their surface and for Fe3B metastable crystals which are characterized by diffusion-free growth stipulated by high motion speed of crystal surface.

keywords Variation growth theory, crystal growth, metastable phase, diffusion-free growth, amorphous metals, nanocrystal metals Fe-B

1. Suzuki K., Fuzimori H., Hashimoto K. Amorphous metals. Moscow. Metallurgiya. 1987. 328 p.
2. Herlach D. M., Galenko P. Holland-Moritz D. Metastable Solids from Undercooled Melts. Amsterdam. Elsevier. 2007. 448 p.
3. Roshchin V. E., Roshchin A. V. Electrometallurgy and steel metallurgy. A textbook for high schools. Moscow: Vologda. Infra-Inzheneriya. 2021. 572 p.
4. Dudorov M. V., Drozin A. D, Plastinin B. G. Features of the Use of Equilibrium State Diagrams for Description of Crystal Growth from Metastable Melts. Solid State Phenomena. 2020. Vol. 299. pp. 622–627.
5. Baker M., Cahn J. W. Solute trapping by rapid solidification. Acta Metallurgica. 1969. No.17. pp. 575–578.
6. Aziz M. J. Rapid solidification: Growth kinetics. In The Encyclopedia of Advanced Materials. Edited by David Bloor et al. Oxford: Pergamon Press. 1994. pp. 2186–2194.
7. Galejko P. K., Kherlakh D. M. Diffusion-free growth of crystal structure on high-speed solidification of eutectic binary system. Vestnik Udmurtskogo universiteta. Fizika. 2006. No. 4. pp. 77–92.
8. Boettinger W. J., Warren J. A., Beckermann C., Karma A. Phase-Field Simulation of Solidification. Annual Review of Materials Research. 2002. Vol. 32. pp. 163–194.
9. Sekerka R. F. Fundamentals of phase field theory, Advances in Crystal Growth Research. Edited by K. Sato, Y. Furukawa and K. Nakajima. Amsterdam: Elsevier. 2001, pp. 21–41.
10. Sobolev S. L., Poluyanov L. V., Liu F. An analytical model for solute diffusion in multicomponent alloy solidification. Journal of Crystal Growth. 2014. No. 395. pp. 46–54.
11. Pinomaa T., Provatas N. Quantitative phase field modeling of solute trapping and continuous growth kinetics in quasi-rapid solidification. Acta Materialia. 2019. No. 168. pp. 167–177.
12. Jokisaari A. M., Voorhees P. W., Guyer J. E., Benchmark problems for numerical implementations of phase field models. Computational Materials Science. 2017. Vol. 126. pp. 139–151.
13. Dudorov M. V. Decomposition of crystal-growth equations in multicomponent melts. Journal of Crystal Growth. 2014. No. 396. pp. 45–49.
14. Dudorov M. V., Roshchin V. E. Simulation of Crystal Growth in Multicomponent Metastable Alloys. Steel in Translation. 2019. Vol. 49. No. 12. pp. 836–842.
15. Glensdorf P., Prigozhin I. Thermodynamic theory of structure, stability and fluctuations. Moscow. Mir. 1973. 280 p.
16. Skripov V. P., Koverda V. P. Spontaneous crystallization of overcooled liquids. Moscow: Nauka. 1984. 230 p.
17. Palumboa M., Cacciamanib G., Boscoa E., Baricco M. Driving forces for crystal nucleation in Fe–B liquid and amorphous alloys. Intermetallics. 2003. Vol. 11. pp. 1293–1299.
18. Zhang D., Xu J., Liu F. In Situ Observation of the Competition Between Metastable and Stable Phases in Solidification of Undercooled Fe-17at. pctB Alloy Melt. Metallurgical and Materials Transactions A. 2015. Vol. 46, pp. 5232–5239.
19. Palumbo M., Baricco M. Modelling of primary bcc-Fe crystal growth in a Fe85B15 amorphous alloy. Acta Materialia. 2005. Vol. 53. pp. 2231–2239.
20. Prigozhin I., Defay R. Chemical thermodynamics. Moscow: Binom. 2009. 533 p.
21. Mikhailovskiy B. V., Kutsenok I. B., Geiderikh V. A. Assessment of thermodynamic functions in crystallization of amorphous alloys of the system Fe-Si-B. Zhurnal fizicheskoy khimii. 1997. Vol. 71. No. 3. p. 409.
22. Gasik M. I., Lyakishev N. P. Theory and technology of ferroalloys metallurgy. A textbook for high schools. Moscow: Intermet Inzhiniring. 1999. 764 p.
23. Battezzati L., Antonione C., Baricco M. Undercooling of Ni-B and Fe-B alloys and their metastable phase diagrams. Journal of Alloys and Compounds. 1997. Vol. 247. No. 1–2, 30, pp. 164–171.

Full content Variational theory of crystal growth and its application for analysis of forming processes for metastable phases in overcooled metallic melts with eutectic composition