Linear morphological stability analysis of the solid-liquid interface in rapid solidification of a binary system

The interface stability against small perturbations of the planar solid-liquid interface is considered analytically in linear approximation. Following the analytical procedure of Trivedi and Kurz [R. Trivedi and W. Kurz, Acta Metall. 34, 1663 (1986)], which is advancing the original treatment of morphological stability by Mullins and Sekerka [W. W. Mullins and R. F. Sekerka, J. Appl. Phys. 35, 444 (1964)] to the case of rapid solidification, we extend the model by introducing the local nonequilibrium in the solute diffusion field around the interface. A solution to the heat- and mass-transport problem around the perturbed interface is given in the presence of the local nonequilibrium solute diffusion. Using the developing local nonequilibrium model of solidification, the self-consistent analysis of linear morphological stability is presented with the attribution to the marginal (neutral) and absolute morphological stability of a rapidly moving interface. Special consideration of the interface stability for the cases of solidification in negative and positive thermal gradients is given. A quantitative comparison of the model predictions for the absolute morphological stability is presented with regard to experimental results of Hoglund and Aziz [D. E. Hoglund and M. J. Aziz, in Kinetics of Phase Transformations, edited by M.O. Thompson, M. J. Aziz, and G. B. Stephenson, MRS Symposia Proceedings No. 205 (Materials Research Society, Pittsburgh, 1991), p. 325] on critical solute concentration for the interface breakdown during rapid solidification of $\text{Si-Sn}$ alloys.

Figure 1

Morphological diagram for solidification of binary systems, which is illustrating the microstructural transitions “planar front” $\to$ “cellular structure” $\to$ “dendrites” $\to$ “cellular structure” $\to$ “planar front,” with the increasing of the solidification velocity $V$. Here $V_C$ is the velocity given by the criterion of constitutional undercooling, and $V_A$ is the velocity for absolute morphological stability of the interface.

Figure 2

Optical micrograph of longitudinal through-thickness section of a melt-spun ribbon of Ni-18 at.% B alloy [22]. The ribbon was produced with the spinning velocity of $120\mathrm{m}∕\mathrm{s}$ and with the thickness of $35\mu \mathrm{m}$. The crystal microstructure exhibits a transition from planar interface with solute segregation-free to cellular-dendritic patterns. The transition proceeds due to decreasing of the interface velocity from $V>V_A$ up to $V<V_A$. Wheel surface at bottom of micrograph.

Figure 3

Velocity $V_A$ of absolute chemical stability vs solute concentration $C_{\infty }$ for Al-Fe alloy. Dashed curve corresponds to solution of Eqs. (68, 69, 71) with solute-drag effect. Solid curve corresponds to solution of Eqs. (68, 70, 72). Dashed-dotted line $V=V_D$ represents the limiting velocity for the absolute interface stability.

Figure 4

Critical concentration $C_{\infty }$ above which planar interface is unstable. Experimental points correspond to solidification of the Si-Sn alloy [42]. Circles are taken from measurements performed on bulk single crystal Si(100), and squares are taken from measurements using Sn-implanted Si-on-sapphire (SOS) samples. Curves are given by the models for interfacial absolute stability: dashed, with local equilibrium diffusion and solute-drag effect, Eqs. (68, 69, 71); solid, with the local nonequilibrium diffusion, Eqs. (68, 70, 72). Dashed-dotted line $V=V_D$ represents the limiting velocity for the absolute interface stability.