In this paper, based on the steady solution of deep cellular crystal growth, the matching asymptotic method and multiple variable method are used to obtain the dispersion relation and the quantization condition of the interfacial morphology in directional solidification process when the interfacial dynamics is anisotropic. The stability of interfacial morphology of deep cell growth during directional solidification under the influence of anisotropic interfacial dynamics is studied.

The mathematical model of the oriented solidification system is established, and the overall ground state solution of the constant cellular growth is taken as the ground state, and the unsteady state solution of the deep cellular growth is expressed as the superposition of the ground state solution and the perturbation dynamics solution when the stability analysis is carried out. The thermodynamic conditions in the mathematical model of the problem constitute a regenerative problem together with the boundary conditions. The asymptotic solution of the cellular crystal growth when \varepsilon \to 0 can be found by dividing the cellular crystal growth region into an outer region far from the root and a region near the root, with an asymptotic solution found in the external region and the root region, respectively, and then matching them to obtain a consistent and effective asymptotic solution in the whole region. The asymptotic solution of the model in the external region is derived to obtain a first-order approximation of the eigenvalues. The inner solutions are matched with the outer solutions based on the vicinity of the singularity to obtain the global solutions and quantization conditions of the system, and finally the stability analysis is conducted. The results show that the directional solidification system of deep cellular crystal growth considering anisotropic interfacial kinetics contains two global instability mechanisms: global oscillation instability and low-frequency instability. The stability analysis shows that the anisotropic interfacial kinetics has a significant effect on the global oscillatory instability mechanism in the low order approximation. With the increase of the anisotropic interfacial kinetics parameter \beta _4, the global oscillatory instability region \left( \textOs\text.U \right)of the dendrite structure with strong oscillation in neutral mode decreases. At the same time, the influence of interfacial dynamic anisotropy parameters on the overall oscillation instability of the system increases with interfacial dynamic parameters increasing.