https://doi.org/10.1140/epjs/s11734-026-02385-w
Review
Unsteady stage of dendritic growth and transition time to steady state
1
Laboratory of Mathematical Modeling of Physical and Chemical Processes in Multiphase Media, Ural Federal University, Lenin ave., 51, 620000, Ekaterinburg, Russian Federation
2
Laboratory of Multi-Scale Mathematical Modeling, Department of Theoretical and Mathematical Physics, Ural Federal University, Lenin ave., 51, 620000, Ekaterinburg, Russian Federation
a
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Received:
6
January
2026
Accepted:
13
May
2026
Published online:
29
May
2026
Abstract
This paper presents an analytical study of the non-stationary stage of dendritic crystal growth. A modified Gibbs–Thomson equation, derived from a hyperbolic phase-field model with velocity- and acceleration-dependent boundary conditions, is solved to describe transient growth dynamics in a pure undercooled melt. The model reveals a sharp decrease in the duration of the unsteady growth period as melt undercooling increases. Analytical solutions for both constant and arbitrary interface curvature demonstrate that dendrite tips rapidly approach steady-state motion. The analysis is extended to secondary branches and to parabolic or circular interface shapes, showing that non-stationary behavior is primarily governed by heat and mass transport integrals. The results indicate that dendritic growth tends to become stationary across a wide range of undercoolings, with the transient stage becoming negligible at high driving forces.
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© The Author(s), under exclusive licence to EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2026
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.
