Growth of mushy layers with temperature modulations

Directional solidification of aqueous solutions produces a solid-melt coexisting zone whose growth rate can be predicted by the mushy-layer theory. We present measurements of mushy-layer growth when solidifying aqueous ammonium chloride with the cooling temperature modulated periodically $T_B(t)=T_0+Acos({\omega}t)$. The mush-liquid interface $h(t)$ evolves as the square root of time for a constant $T_B$, but exhibits periodical humps in the present of modulations. The growth rate $\dot{h}(t)$ is best approximate to $\dot{h}(t)={\dot{h}_0}e^{-{\gamma}{\omega}t/2{\pi}}cos({\omega}t+\pi+{\phi}(t))$, with a decay rate $\gamma=0.82{\pm}0.05$ independent on the modulation amplitude $A$ and frequency $\omega$, and a phase-shift ${\phi}(t)$ increasingly lag behind $T_B$ as a function of time. We discuss a mushy-layer growth model based on the Neumann solution of the Stefan problem with periodical boundary conditions, and show that the numerical results are in agreement with the experimental observations.