Predicting the patterns in lamellar growth

The removal of the degeneracy in the steady-state diffusion-capillarity relation for the undercooling of lamellar eutectics as a function of forced growth velocity and spacing requires the statement of a global principle. The consequences of Langer's conjecture that "each lamella must grow in a direction which is perpendicular to the solidification front" have been explored. In the first instance it is demonstrated that the principle is equivalent on the isotherm to a conditional minimum in the frontal-surface free energy. Secondly, it is deduced for forced-velocity eutectics that the stable spacing is $\sqrt{2}{\lambda }_m$, where ${\lambda }_m$ is given by a minimum in the undercooling. Langer and co-workers have been led to favor the value ${\lambda }_m$ on the basis of an unjustified approximation. In contradistinction to Langer's identification of a "diffusive mode" for relaxation of lamellar spacing, we find that the mechanism of stabilization is best described as a damped oscillation in the spacing. The present stable coordinate is identical with that obtained for isothermal structures via Langer's conjecture and by a number of earlier related perturbation arguments. It corresponds to an isothermal state of the spacing which coincides with a maximum in the entropy-production rate. The thermodynamic validation of this principle is briefly discussed.