Prediction of dendritic spacings in a directional-solidification experiment

We present a theoretical analysis of the formation of a dendritic array in a directional-solidification experiment. Our calculation contains three sequential ingredients: acceleration of an initially flat interface and the concomitant buildup of a solutal boundary layer in front of it; onset of a morphological instability, triggered by thermal fluctuations, producing a relatively finely spaced array of dendritic tips; coarsening of this array and final selection of a steady-state primary spacing. For sufficiently large growth speeds, where the resulting dendrites interact with each other weakly, we find--with no adjustable parameters--good agreement with the experiments of Trivedi and Somboonsuk [Acta Metall. 33, 1061 (1985)] and Somboonsuk, Mason, and Trivedi [Metall. Trans A 15A, 967 (1984)]