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Dynamic evolution of the initial instability of planar interface predicted by the present model with the pulling speed of during directional solidification. (a) Evolution of the amplitude spectrum obtained from Eq. (8). (b) The corresponding interface structures.
Dynamic evolution of the initial instability of planar interface simulated by the quantitative phase field model with the same conditions as those in Fig. 1.
Comparisons of the present model and previous investigations for the initial instability. (a) The crossover time from linear regime to nonlinear regime. (b) The average wavelength at crossover point. (Two data sets of results from the present model and phase field simulations with different random numbers are given. It also should be noted that phase field simulations are only carried out at the relatively high pulling speed due to the time-consuming.)
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