Three-dimensional dendritic growth

The entire solution for three-dimensional (3D) dendritic growth of needle crystals is described analytically. Its construction involves the existing 3D selection theory for the tip of the dendrite [Ben Amar and Brener, Phys. Rev. Lett. 71 (1993) 589] 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 an Ivantsov paraboloid, is in qualitative agreement with experimental observations. We consider the time-dependent behavior of side-branching deformations, taking into account the actual non-axisymmetric 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 can resolve the puzzle that experimentally observed side branches have much larger amplitudes than can be explained by thermal noise in the framework of the axisymmetric approach. The coarsening behavior of side branches in the non-linear regime is discussed briefly.