Domain of oscillatory growth in directional solidification of dilute binary alloys

The oscillatory growth of a dilute binary alloy has recently been described by a nonlinear oscillator equation that applies to small temperature gradients and large growth velocities in the setup of directional solidification. Based on a one-dimensional stability analysis of stationary solutions of this equation, we explore in the present paper the complete region where the solidification front propagates in an oscillatory way. The boundary of this region is calculated exactly, and the nature of the oscillations is evaluated numerically in several segments of the region.

Figure 1

Neutral stability lines in the $m^2,v_P$ plane for $p=100$, $\gamma =0.01$ (solid line), and $\gamma =0.007$ (dotted line).

Figure 2

Neutral stability lines in the $\gamma ,v_P,$ plane for $p=100$, and for $m^2=0$ (solid line), $m^2=1\times {10}^{-6}$ (dashed line), and $m^2=9\times {10}^{-6}$ (dotted line).

Figure 3

Limit cycles in the $h,\dot{h}$ phase space for $p=100$, $m^2=9\times {10}^{-6}$, $\gamma ={10}^{-2}$, and, from left to right, decreasing pulling velocities $v_P=0.3,v_P=0.1,v_P=0.05,v_P=0.03$.

Figure 4

Oscillatory approach of the fixed point $h=\dot{h}=0$ for the same parameter values as in Fig. 3, but $v_P=0.025$.

Figure 5

Trajectories $h\left(t\right),\dot{h}\left(t\right)$, and $c(0,t)$ for $p=100$, $\gamma ={10}^{-2}$, $m^2=9\times {10}^{-6}$, and $v_P=0.3$.

Figure 6

Trajectories $h\left(t\right)$, $\dot{h}\left(t\right)$, and $c(0,t)$ for the same parameter set as in Fig. 5, except for the smaller value $v_P=0.03$ of the pulling velocity.