A numerical analysis of dendritic and cellular growth of a pure material investigating the transition from “array” to “isolated” growth

Abstract

A time dependent numerical model has been used to analyse the growth of dendrites into a supercooled liquid bath of a pure material. Both dendritic plates and dendrites having axial symmetry have been treated. Solutions were obtained over nearly twelve orders of magnitude of velocity and two orders of magnitude of undercooling. For a finite surface energy, discrete solutions rather than a continuum of solutions were obtained for steady state dendritic growth at a fixed undercooling. The problem was considered in the limit of a zero specific heat so that the results could be compared with those obtained for the similar Hele-Shaw problem for which analytical solutions are available. It was found that there was an excellent agreement with the earlier work. The steady state results are plotted to show the relationship between the Hele-Shaw and dendrite problem. The transition between constrained and isolated dendrites was examined. It was found that this two parameter problem could be reduced to a single line in the limit of a small thermal diffusion layer and a large dendrite spacing (an isolated dendrite) and also when the thermal layer was larger than the dendrite spacing (a constrained cell or dendrite). Two types of solution, cell-like or dendritic, were found. The steady state dendritic results agreed quantitatively with the prediction of marginal stability but not with physical basis of the model. It has previously been suggested that these steady shapes were not possible unless the surface energy is anisotropic. Possible reasons for the different behaviour found in the present work are discussed. The steady state shapes were tested for stability using time dependent growth.