Three-dimensional dendrite tip morphology at low undercooling

Alain Karma; Youngyih H. Lee; Mathis Plapp

doi:10.1103/PhysRevE.61.3996

paper

Three-dimensional dendrite tip morphology at low undercooling

arXiv:cond-mat/9909021 · doi:10.1103/PhysRevE.61.3996

Abstract

We investigate the three-dimensional morphology of the dendrite tip using the phase-field method. We find that, for low undercoolings, this morphology is ostensibly independent of anisotropy strength except for a localized shape distortion near the tip that only affects the value of the tip radius $ρ$ (which is crudely approximated by $ρ \approx (1 - α) ρ_{I v}$ where $ρ_{I v}$ is the Ivantsov tip radius of an isothermal paraboloid with the same tip velocity and $α$ is the stiffness anisotropy). The universal tip shape, which excludes this distortion, is well fitted by the form $z = - r^{2} /2 + A_{4} r^{4} cos 4 ϕ$ where $∣ z ∣$ is the distance from the tip and all lengths are scaled by $ρ_{I v}$ . This fit yields $A_{4}$ in the range $0.004 - 0.005$ in good quantitative agreement with the existing tip morphology measurements in succinonitrile [LaCombe et al., Phys. Rev. E {\bf 52}, 2778 (1995)], which are reanalyzed here and found to be consistent with a single $cos 4 ϕ$ mode non-axisymmetric deviation from a paraboloid. Moreover, the fin shape away from the tip is well fitted by the power law $z = - a ∣ x ∣^{5/3}$ with $a \approx 0.68$ . Finally, the characterization of the operating state of the dendrite tip is revisited in the light of these results.

12 pages, 17 eps figures, submitted to Phys. Rev. E

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Numerical Simulation of Three-Dimensional Dendritic Growth

Three-dimensional dendrite tip morphology at low undercooling

Abstract

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