Selection criterion of a stable dendrite growth in rapid solidification

We present an analysis of a free dendrite growing in a binary mixture under non-isothermal conditions. The stable growth mode is analyzed through the solvability condition giving the stability criterion for the dendrite tip as a function of the thermal Péclet number, PT, and ratio, W=V/VD, of the dendrite velocity V and solute diffusion speed VD in bulk liquid. We extend previous studies limited to small values of the Péclet numbers, by considering the effect of the anisotropy of surface energy for the needle-like dendrite growing at arbitrary Péclet numbers and under local non-equilibrium solute diffusion described by a hyperbolic type of transport equation. Transitions in growth regimes, namely, from solute diffusion-limited to thermo-solutal regime and, finally, to pure thermally controlled regime of the anisotropic dendrite are derived and revealed. Limiting cases of known criteria for anisotropic dendrite growing at small and high growth Péclet numbers are provided. A comparison with the previously obtained criterion of marginal stability of rapidly growing dendrite is made.