Phase dynamics in directional solidification

A phase diffusion equation valid at arbitrary distances above the Mullins-Sekerka threshold is derived from the front integral equation. The steady-state solutions and the associated adjoint functions are computed numerically. As a first exploitation of the phase equation we determine the range of the phase-stable band of steady solutions. Even very close to threshold we find a strong deviation from the Landau-Ginzburg theory. Far from threshold we find that the wavelength ${\lambda }_{\mathrm{Eck}}$ on the long-wavelength edge of the Eckhaus band scales with the growth velocity V as ${\lambda }_{\mathrm{Eck}}$≊${\mathit{V}}^{\mathrm{-}1/2}$, as does the experimentally observed wavelength.