https://doi.org/10.1140/epjs/s11734-024-01394-x
Regular Article
Towards the theory of crystal growth ahead of the moving solid–liquid phase interface: a U-shaped anomalous behaviour of the crystallization velocity of melt undercooling
1
Laboratory of Mathematical Modeling of Physical and Chemical Processes in Multiphase Media, Department of Theoretical and Mathematical Physics, Ural Federal University, Lenin ave., 51, 620000, Ekaterinburg, Russian Federation
2
Otto-Schott-Institut für Materialforschung, Friedrich-Schiller-Universität-Jena, 07743, Jena, Germany
3
Laboratory of Multi-Scale Mathematical Modeling, Department of Theoretical and Mathematical Physics, Ural Federal University, Lenin ave., 51, 620000, Ekaterinburg, Russian Federation
Received:
7
September
2024
Accepted:
29
October
2024
Published online:
18
November
2024
Motivated by the experimental data on a U-shaped anomalous behaviour of the crystallization velocity of melt undercooling, we develop a new mathematical model of combined directional and bulk crystallization of the undercooled binary melt. The model is based on the crystallization along a spatial direction caused by temperature difference between the solid and liquid phases as well as nucleation and growth of crystals in an undercooled two-phase region placed between these phases. Appearance, evolution, and attachment of particles to the phase interface accelerate its motion and lead to a U-shaped anomalous behaviour. A high non-linear mathematical model describing the heat and mass transfer, crystal growth, and motion of phase transition boundaries in a self-similar manner is analytically solved by means of the saddle-point technique for evaluating the Laplace-type integral and the method of small parameter expansion of the temperature, solute concentration, and melt undercooling in Taylor series. As a result, temperature and solute concentration profiles, solid-phase fraction, crystal-size distribution, and two-phase region boundaries are analytically found. This solution enables to get the crystallization velocity as a function of melt undercooling leading to a U-shaped anomalous behaviour. The theory is in agreement with the experimental data on crystallization of Al-rich Al–Ni alloys.
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