Pattern formation in three-dimensional dendritic growth

The entire solution for three-dimensional (3D) nonaxisymmetric dendrites is described analytically. Its construction involves the 3D selection theory for the tip of the dendrite plus matching of the tail to this tip. Both intermediate and final asymptotics of the tail shape are given. This shape, which deviates strongly from the Ivantsov paraboloid, is in qualitative agreement with experimental observations. We consider the time-dependent behavior of sidebranching deformations taking into account the actual nonaxisymmetric shape of the needle crystal. It is found that the amplitude of the deformation grows faster than for the axisymmetric paraboloid shape of the needle. We argue that this effect resolves the puzzle that experimentally observed side-branches have much larger amplitudes that can be explained by thermal noise in the framework of the axisymmetric approach. The coarsening behavior of sidebranches in the nonlinear regime is shortly discussed.