Front instability and pattern dynamics in the phase-field model for crystal growth

We study front instability and the pattern dynamics in the phase-field model with four-fold rotational symmetry. When the undercooling Δ is 1<Δ<Δc, the flat interface is linearly unstable, and the deformation of the interface evolves to spatio-temporal chaos or nearly stationary cellular structures appear, depending on the growth direction. When Δ<1, the flat interface grows with a power law x∼t1/2 and the growth rates of linear perturbations with finite wave number q decay to negative values. It implies that the flat interface is linearly stable as t→∞, if the width of the interface is finite. However, the perturbations around the flat interface actually grow since the linear growth rates take positive values for a long time, and the flat interface changes into an array of doublons or dendrites. The competitive dynamics among many dendrites is studied more in detail.