Three-dimensional dendrite tip morphology at low undercooling

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 $ρ\approx (1-α)ρ_{Iv}$ where $ρ_{Iv}$ 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 $z=-r^2/2+A_4 r^4\cos 4ϕ$ where $|z|$ is the distance from the tip and all lengths are scaled by $ρ_{Iv}$. 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 $\cos 4ϕ$ 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.