Effect of surface free energy anisotropy on dendrite tip shape

Abstract In previous work, approximate solutions were found for paraboloids having perturbations with four-fold axial symmetry in order to model dendritic growth in cubic materials. These solutions provide self-consistent corrections through second order in a shape parameter e to the Peclet number vs supercooling relation of the Ivantsov solution. The parameter e is proportional to the amplitude of the four-fold correction to the dendrite shape, as measured from the Ivantsov paraboloid of revolution. The equilibrium shape for anisotropic surface free energy to second order in the anisotropy is calculated. The value of e is determined by comparing the dendrite tip shape with the portion of the equilibrium shape near the growth direction, [001], for anisotropic surface free energy of the form γ = γ 0 [1+4 e 4 ( n 1 4 + n 2 4 + n 3 4 )], where the n i are components of the unit normal of the crystal surface. This comparison results in e =−2 e 4 −24 e 4 2 + O ( e 4 3 ), independent of the Peclet number. From the experimental value of e 4 , it is found that e ≈−0.012±0.004, in good agreement with the measured value e ≈−0.008 of LaCombe et al . ( Phys. Rev. E , 1995, 52 , 2778)