Unique Non-Planar Solutions to the Thin Film Cellular Solidification Problem

A theoretical procedure is demonstrated for calculating two-dimensional binary cellular single-valued interfaces and extended to triple-valued (remelt) shapes. The numerical calculations for the single-valued local equilibrium problem which yield unique shapes as functions of the supersaturation parameter a and the partition coefficient k are reviewed. The cell amplitude L and wavelength A scale identically as the inverse velocity to the 112 power and the aspect ratio r is a constant equal to 312, independently of a and k for a < 0.5.. For 0.5 < a < m, L, A and r decrease asymptotically to zero indicating the finite amplitude nature of the instability at a = 1 and the existence of sub-marginal states which converge to a flat interface at a- m (Glv- m). Scaling comparisons between equilibrium single-valued and triple-valued and deep-rooted non- equilibrium cells are made.