Dynamic evolution of initial instability during non-steady-state growth

Dynamic evolution of initial instability is investigated by an analytic model obtained by modifying the theory of Warren and Langer [Phys. Rev. E 47, 2702 (1993)PLEEE81063-651X10.1103/PhysRevE.47.2702] and the quantitative phase-field model in directional solidification under transient conditions for realistic parameters of a dilute alloy. The evolutions of tip velocity and concentration in the liquid side of the interface predicted by the analytic model agree very well with that from the phase-field simulation in the linear growth stage of the non-steady-state growth, indicating that the model could be used as a convenient method to study the initial instability during non-steady-state growth. The influences of non-steady-state conditions which include the increasing rate of pulling speed and temperature gradient at the onset of initial instability are investigated, and we find that, the initial instability seems to depend strongly on the non-steady-state conditions and the non-steady-state history, and thus, it should be primarily considered in the study of the transient growth.