Noise-induced sidebranching in the three-dimensional nonaxisymmetric dendritic growth.

Abstract

We consider the time-dependent behavior of sidebranching deformations taking into account the actual nonaxisymmetric shape of the needle crystal. The Green's function of the linearized problem is presented by a functional integral with the help of the Mullins-Sekerka local spectrum. For the short-wavelength perturbations the functional integral can be calculated by the steepest descent method, where the Green's function behavior is determined by the extremal trajectories governed by Hamilton equations. The local spectrum plays the role of the Hamilton's function. As in the axisymmetric approach [J.S. Langer, Phys. Rev. A 36, 3350 (1987)], noise-induced wave packets generated in the tip region grow in amplitude, and spread and stretch as they move down the sides of the dendrite producing a train of sidebranches. The amplitude grows exponentially as a function of (\ensuremath{\Vert}Z${\mathrm{\ensuremath{\Vert}}}^{2/5}$/${\mathrm{\ensuremath{\sigma}}}^{1/2}$), where \ensuremath{\Vert}Z\ensuremath{\Vert} is the distance from the dendritic tip and \ensuremath{\sigma} is the stability parameter. The important result is that the amplitude of the sidebranches for the anisotropic needle crystal grows faster than for the axisymmetric paraboloid shape [in the latter case it grows exponentially but only as a function of (\ensuremath{\Vert}Z${\mathrm{\ensuremath{\Vert}}}^{1/4}$/${\mathrm{\ensuremath{\sigma}}}^{1/2}$)]. We argue that this effect can resolve the puzzle that experimentally observed sidebranches have much larger amplitudes than can be explained by thermal noise in the framework of the axisymmetric approach. The coarsening behavior of sidebranches in the nonlinear regime is briefly discussed.