Solute redistribution around crystal shapes growing under hyperbolic mass transport. (October 2015) - Galenko, P.K.
Highlights: A method based on curvilinear coordinates is developed for the rapidly growing dendrite. Analytical treatments of the rapidly growing crystals to obtain the results useful for their further application in analytical and practical computations are carried out. For the growth velocity smaller than the solute diffusion speed, the steady-state growth shapes are: the paraboloid of revolution, the parabolic platelet, and the planar interface. The crystals have arbitrary shape if the growth velocity is equal to or greater than the solute diffusion speed. Special attention is given to mathematical transformations from parabolic (paraboloidal) coordinates to usual Cartesian co-ordinates for analytical solutions extended to the case of rapid dendritic growth in which the solidification velocity is comparable with the solute diffusion speed in bulk liquid. The obtained analytical solutions can be naturally used for the model of crystals having anisotropic properties of their interface. Abstract: Using a model of local non-equilibrium diffusion during rapid solidification of a binary system, the isosolutal shapes of growing crystals in steady-state approximation are obtained. It is found that for crystals growing with constant velocity along a selected coordinate direction, two isosolutal growth shapes can occur. These are: the parabolic platelet in two-dimensional case and the paraboloid of revolution in three-dimensional case. In the isothermal case of diffusionless solidification, when the velocity of solidification is equal to or greater than the solute diffusive speed in the bulk system, these interfaces can have an arbitrary configuration. Special attention is given to mathematical transformations from parabolic (paraboloidal) coordinates to usual Cartesian coordinates for Ivantsov solutions extended to the case of rapid dendritic growth in which the solidification velocity V is comparable with the solute diffusion speed V D in bulk liquid.