Effect of surface tension anisotropy on the interface morphological stability of deep cellular crystal

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In this paper, we study the effect of anisotropic surface tension on the interface morphological stability of deep cellular crystal during directional solidification. We assume that the process of solidification is viewed as a two-dimensional problem, the anisotropic surface tension is a four-fold symmetry function, the solute diffusion in the solid phase is negligible, the thermodynamic properties are the same for both solid and liquid phases, and there is no convection in the system. On the basis of the basic state solution for the deep cellular crystal in directional solidification, by the matched asymptotic expansion method and the multiple variable expansion method, we obtain the asymptotic solution, and then the quantization condition of interfacial morphology for deep cellular crystal is obtained. The results show that by comparison with the directional solidification system of surface tension isotropy, the interface morphological stability of surface tension anisotropy also possesses two types of global instability mechanisms: the global oscillatory instability (GTW-mode), whose neutral modes yield strong oscillatory dendritic structures, and the low-frequency instability (IF-mode), whose neutral modes yield weakly oscillatory cellular structures. Both of the two global instability mechanisms have the symmetrical mode (S-mode) and the anti-symmetrical mode (A-mode), and the growth rate of the S-mode with the same index n is greater than that of the A-mode. In this sense we say that the S-mode is more dangerous than the A-mode. All the neutral curves of the GTW-S-modes and LF-S-modes divide the parameter plane into two subdomains: the stable domain and the unstable domain. In the paper we show the neural curves of the GTW-S-modes and LS-S-modes for various n, respectively. It is seen that among all the GTW-S-modes (n=0, 1, 2), the GTW-S-mode with n=0 is the most dangerous oscillatory mode, while among all the LF-S-modes (n=0, 1, 2), the LF-S-mode with n=0 is the most dangerous weakly oscillatory mode. We also show the neural curves of the GTW-S-mode (n=0) and LS-S-mode (n=0) for various anisotropic surface tension parameters, respectively. It is seen that as the anisotropic surface tension increases, the unstable domain of global oscillatory instability decreases, and the unstable domain of the low-frequency instability increases.