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We consider a model of a free-growing dendrite in a binary dilute system solidifying under nonequilibrium conditions. The numerical solution of the model equations was obtained by finite-difference technique on a two-dimensional square lattice. A special case in which the liquid-solid surface tension is zero and a stabilizing action on the dendritic form is produced by both the surface kinetics and the anisotropic influence of the computational lattice was studied. We find that, depending on the initial undercooling and computational lattice scale, an interesting behavior in the dendrite sidebranch surface is expected. Except for the evolution of the sidebranch surface realized by regularly repeated doubling of the distances between the secondary branches by the Feigenbaum scenario, there is a clear tendency for the formation of a needlelike dendrite, structured after a Hopf-type bifurcation, chaotic structure with random period of branching, packet structure with the branching period that is not defined by the Feigenbaum scenario. Simulation data are correlated with known conclusions of the thermodynamical approach to phase transformations, marginal stability theory, and analytical treatments of the local model of the boundary layer. Satisfactory qualitative agreement with the results given by the continuum diffusion-limited aggregation model and the modeling of three-dimensional heat flow dendrites has been found.

Using the model of local non-equilibrium solidification, which takes into account deviations from local thermodynamical equilibrium at the interface and bulk phases, a model for pattern formation of crystals in isothermal undercooled binary alloys is presented. The model equations describe the kinetics of the liquid-solid interface motion and local non-equilibrium solute diffusion in the liquid phase. For the self-consistency of the model, we have also taken into account the solute trapping at the interface and deviation of the liquidus line from its equilibrium value using local non-equilibrium diffusion. A main feature of the model equations is the inclusion of crystal growth velocities which are of the order of the diffusion speed in the bulk liquid. This allows one to describe rapid solidification in deeply undercooled alloys. To solve numerically the problem of the growth of crystals and obtain their morphology, an approximation for the model differential equations is formulated using the finite-difference technique.

A detailed finite-difference approximation of equations of the model for crystal pattern formation in a solidifying binary alloy is given. The stability of the numerical scheme obtained is investigated and the criterion of von Neumann for computing stability is defined. A solution of the finite-difference equations simulates crystal patterns of binary alloys in a wide range of undercooling. The results of numerical solutions testify that the model of local non-equilibrium solidification used adequately describes the pattern formation for natural experimental data both at slow and high velocities of solidification.

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