Laplacian and diffusional growth: A unified theoretical description for symmetrical and parity-broken patterns

Abstract

Nowadays, Laplacian growth patterns such as the Saffman-Taylor instability are well understood, mainly because of exact results derived by conformal mapping techniques. This is not the case for crystal growth processes except for the free needle-crystal. Normally, it is thought that the crystal growth process can be understood by an analogy with the equivalent steady Laplacian process. The results that follow from this analogy, however, are in contradiction with numerical and experimental results even at very low velocities. We present here a way to mimic the diffusion instabilities in a simple way which allow to describe the formation and the dynamics of new patterns. We show that the problem of crystal growth in a channel is closely related to the Saffman-Taylor problem in a sector. By using this analogy we obtain the family of zero-surface-tension solutions for small Peclet numbers including an asymmetric degree of freedom. We study analytically the selection problem and confirm that parity-broken solutions originate from the symmetrical solutions through a bifurcation in agreement with recent numerical investigations. We obtain scaling laws for the parity-broken solutions for a wide range of parameters. We extend these results to the infinite geometry and show that parity-broken double fingers can exist even in the range of small undercooling with fully isotropic surface tension. The scaling law for the selected velocity of double-fingering structures is very different from the scaling law of a free dendrite with anisotropy.