On the solidification of dendritic arrays: An asymptotic theory for the directional solidification of slender needle crystals

We consider a mathematical model for directional solidification of a binary alloy as a periodic array of three-dimensional needle crystals. Arguing from an analysis of published experimental data for dendritic growth, we identify a natural separation of characteristic length scales for dendrites. We use these observed disparities in length scales to define a small parameter for dendritic growth and identify scalings for all the process parameters in terms of this small parameter. We then solve the resulting free boundary problem using matched asymptotic expansions. Our analysis results in an integral equation for the shape of the needle crystal. We suggest that the integral equation contains a mechanism for the unique selection of the tip radius of the needle crystal independent of surface energy. This is in sharp contrast to previous studies regarding determination of the tip radius of an isolated, single-component, isothermal dendrite. Our results suggest that selection of the tip radius is linked to the spacing of the array.