Frozen-temperature solutal interface dispersion ($s{C}_I=(Dq-V)[V{C}_I/D-(G+\Gamma k^2)/m]$)

For an interface perturbation $\eta e^{ikx+st}$ with instantaneous interface kinetics, take the liquid concentration perturbation proportional to $e^{-qz}$. Here $\Gamma >0$ is the coefficient of curvature-induced melting-temperature depression. The advection-diffusion equation and interfacial mass conservation giveFor a growing normal mode the unique decaying liquid branch has $q=(V+\sqrt{V^2+4D(s+D{k}^2)})/(2D)$. Writing $b=(G+\Gamma k^2)/(m{C}_I)$, its rate is $s=(D/2)(V/D-b)(\sqrt{b^2+4k^2}-b)$. Instability requires $G+\Gamma k^2<mV{C}_I/D$. The Gibbs-Thomson effect suppresses sufficiently short waves; a quasistatic solute diffusion to solute diffusion changes the rate and must be stated separately.

A quasistatic approximation to the solute field neglects its perturbation time derivative while retaining the interface motion in the solute balance. For steady pulling, a lateral wavenumber $k$ then has decaying exponent $q=(V+\sqrt{V^2+4D^2{k}^2})/(2D)$. It is justified when the interface growth time is long compared with the relevant solute diffusion time. Fixing the thermal field does not by itself impose this solutal timescale ordering.