Three-dimensional dendrite tip morphology at low undercooling

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 $\rho$ (which is crudely approximated by $\rho\approx (1-\alpha)\rho_{Iv}$ where $\rho_{Iv}$ is the Ivantsov tip radius of an isothermal paraboloid with the same tip velocity and $\alpha$ is the stiffness anisotropy). The universal tip shape, which excludes this distortion, is well fitted by the form $z=-r^2/2+A_4 r^4\cos 4\phi$ where $|z|$ is the distance from the tip and all lengths are scaled by $\rho_{Iv}$. 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 $\cos 4\phi$ 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.