Shapes, wavelength selection, and the cellular-dendritic ‘‘transition’’ in directional solidification

We have studied cellular shapes in directionally solidified ${\mathrm{CBr}}_4$-${\mathrm{Br}}_2$ and have shown that they follow a scaling law from the planar-cellular threshold ${\mathit{V}}_{\mathit{c}}$ up to about (8–9)${\mathit{V}}_{\mathit{c}}$: the shapes depend on the relative velocity ε=(V-${\mathit{V}}_{\mathit{c}}$)/${\mathit{V}}_{\mathit{c}}$, and not separately on the absolute velocity V or the temperature gradient G. A comparison with Saffman-Taylor profiles is found to be unsatisfactory. We then examined cellular widths and their variation along a given solidification front from threshold up to the appearance of dendrites. Again up to about (8–9)${\mathit{V}}_{\mathit{c}}$, the dispersion is comparable to experimental uncertainties but beyond that point the width of some cells begins to increase spectacularly. These abnormally wide cells change shape, and the widest ones become dendrites once the Péclet number reaches 2–3.