Tuned Mullins-Sekerka instability: Exact results

Mullins-Sekerka's instability at 3D self-similar growth of a spherical seed crystal in an undercooled fluid is discussed. The exact solution of the linearized stability problem is obtained. It is quite different from the conventional results of the quasisteady approximation. The instability occurs much weaker, so that instead of exponential growth in time, unstable modes exhibit just power-law-growth. The relative growth rates of different modes vary in time and depend on their initial amplitudes. It allows control over the growth of each mode individually and tailoring the instability, to obtain a desired shape of the growing crystal at a given time.

Figure 1

A temperature profile in an undercooled melt in the vicinity of a growing crystal (schematically). The solid phase is indicated by filling with a wavy pattern. The solid-liquid interface is shown with a thick line. Thin lines designate the sequential isotherms with fixed temperature step $\Delta T$. The unperturbed spherically symmetric crystal (a) and fourfold deformed one (b). The deformation gives rise to an increase in the modulus of the temperature gradient at convex parts of the interface and in decrease of it at concave ones (note the corresponding changes of the distances between the sequential isotherms). Moreover, perturbed heat flow (1) in addition to radial (2) gets an azimuthal component (3) directed from the convex to concave parts of the interface; see arrows in (b). Both effects enhance the heat transfer from the bulges and suppresses the one from the concave parts.

Figure 2

Example, illustrating the opportunity of shape-control. Temporal evolution of the cross-section of a growing crystal by an equatorial plane passing through $z$ axis. The spherical seed is perturbed by two eigenmodes with the same amplitudes; $\ell$ equals 3 and 12, respectively: $\frac{R}{2\sqrt{\chi t}}=0.1+0.01\sqrt{\frac{t}{\tau }}{Y}_3^0+0.01{\left(\frac{t}{\tau }\right)}^5{Y}_{12}^0$. The solid phase is indicated with a wavy pattern. (a) At $t=0.5\tau$ the shape is determined by the smoother perturbation with $\ell =3$. (b) At $t=\tau$ it is strongly affected by the perturbation with $\ell =12$.