Dendritic solidification morphology viewed from the perspective of constructal theory

Abstract

In this paper we focus on the application of the constructal theory in predicting the dendritic solid structure. First we analyse the marginal stability criterion from the perspective of the constructal principle. Having as a guiding principle the constructal law we have shown that among the whole range of possible dendrite tip radiuses predicted by the stability analysis the dendrite tip will choose the smallest one, that is a radius equal with the smallest perturbation wavelength leading to instabilities. We identify as well the existence of a competition between the diffusion controlled growth and the dendritic growth. Second, we develop a model for the secondary arm spacing. We identify a competition between the lateral diffusion controlled growth of a needle and the dendritic growth of lateral secondary arms. By analysing this competition we are able to characterize the sidebranching mechanism and to finally compute the secondary arm spacing. The result is in good agreement with the experimental results. Finally, the primary arm spacing is analysed from the perspective of the constructal law. The constructal law predicts that the only way the columnar tips can optimize the solidification process is to minimize the spacing between two adjacent tips, namely λ1. By quantifying the two mechanisms responsible for the selection of λ1, the dendrite division and the dendrite overgrown mechanisms, we were finally able to obtain a model for the primary arm spacing. This model is also validated against various experimental data.