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 approximately (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 4straight phi where |z| is the distance from the tip and all lengths are scaled by rho(Iv). This fit yields A4 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 52, 2778 (1995)], which are reanalyzed here and found to be consistent with a single cos 4straight phi mode nonaxisymmetric 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 approximately 0.68. Finally, the characterization of the operating state of the dendrite tip is revisited in the light of these results.