Capillary-Wave Description of Rapid Directional Solidification

TL;DR

This paper extends the capillary-wave model to directional solidification, deriving a universal dispersion relation and a differential equation for interface oscillations, providing insights into solute band formation and stability.

Contribution

It generalizes the capillary-wave description to include directional solidification and derives a universal dispersion relation for unstable modes.

Findings

Derived a universal dispersion relation for unstable eigenmodes.

Established a differential equation for oscillatory interface motions.

Provided a limit-cycle scenario for solute band formation.

Abstract

A recently introduced capillary-wave description of binary-alloy solidification is generalized to include the procedure of directional solidification. For a class of model systems a universal dispersion relation of the unstable eigenmodes of a planar steady-state solidification front is derived, which readjusts previously known stability considerations. We, moreover, establish a differential equation for oscillatory motions of a planar interface that offers a limit-cycle scenario for the formation of solute bands, and, taking into account the Mullins-Sekerka instability, of banded structures.