Bifurcations in a sidebranch surface of a free-growing dendrite

Abstract

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.