An Intrinsic Equation of Interfacial Motion for the Solidification of a Pure Hypercooled Melt

Solidification of a pure hypercooled melt is studied in the case where heat diffusion is localized within a narrow boundary layer along the solid-liquid interface. An intrinsic coordinate system, based on the distance to the interface, and a perturbation expansion in the boundary layer thickness are used to derive an expression for the normal velocity of the interface as a function of the local surface geometry. This intrinsic equation of motion represents an extension of the ideas of long-wave theories. We identify and examine a co-dimension two limit in this theory in which two terms may vanish corresponding to small front speed and marginal long-wave stability. Two special geometries are considered: near-planar and near-spherical interfaces. For near-planar interfaces, long-wave theory yields a modified Kuramoto-Sivashinsky equation. We analytically examine the bifurcation of steady planform solutions (i.e. rolls, squares and hexagons) from a planar interface, and numerically extend the roll solutions into the fully non-linear regime. In certain cases, an analytical solution suggests that some periodic disturbances will grow without bound. For spherical particles, we identify a critical growth radius (below which particles shrink) and an instability radius, above which non-spherical deformations are amplified. The intrinsic coordinate methodology developed here should be applicable to many other problems in interface motion including directional solidification, solidification of a dilute binary alloy and reaction-diffusion and phase-field models that mimic solidification dynamics