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 which is crudely approximated by (1-) ₈ₕ where ₈ₕ is the Ivantsov tip radius of an isothermal paraboloid with the same tip velocity and is the stiffness anisotropy. The universal tip shape, which excludes this distortion, is well fitted by the form z=-r^2/2+A₄r^4cos4 where |z| is the distance from the tip and all lengths are scaled by ₈ₕ. 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 cos4 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 a0. 68. Finally, the characterization of the operating state of the dendrite tip is revisited in the light of these results.