Kinetic undercooling

Kinetic undercooling makes the speed of a phase boundary proportional to the difference between its equilibrium and actual interface temperatures. A curvature correction raises the required undercooling for protrusions and therefore stabilizes short wavelengths.

The attachment mobility $\mathcal{M}>0$ is the coefficient converting interfacial undercooling into a solidification speed. In the linear isotropic law, $U=\mathcal{M}(T_{\mathrm{eq}}-T_i)$, and capillarity makes $T_{\mathrm{eq}}=T_m-{\gamma }_T\mathcal{K}$. Thus $T_i=T_m-U/\mathcal{M}-{\gamma }_T\mathcal{K}$. This kinetic coefficient refers to a moving phase boundary, rather than an order-parameter relaxation coefficient. The Gibbs--Thomson relation determines ${\gamma }_T=T_m\gamma /(\rho L)$ for isotropic solid-liquid surface energy $\gamma$.

Instantaneous interface kinetics is the limit of negligible kinetic undercooling at a moving solidification interface. Its temperature is set by local liquidus equilibrium and curvature-induced melting-temperature depression, without a further velocity-dependent attachment term. It does not imply that diffusion or heat transport is instantaneous.

A planar front growing into a supercooled melt can amplify corrugations because protrusions enter colder liquid. Interface curvature opposes this effect. Linearization couples the perturbation growth rate to the decaying thermal exponent ahead of the front.

Use undercooling $\Delta T=T_m-T_{\infty }$, solidification attachment mobility $\mathcal{M}$, velocity scale $U_0=\mathcal{M}\Delta T$, length $\ell =\kappa /U_0$ and time $\kappa /U_0^2$. Let $S=L/(c_p\Delta T)$ be the latent-to-sensible heat ratio, $\theta =(T-T_{\infty })/\Delta T$, and $\Gamma ={\gamma }_T/(\Delta T\ell )$. In the one-sided liquid-diffusion model, a travelling planar front has $\theta =S{e}^{-V\xi }$ and $V=1-S$. For a corrugation $\zeta e^{\sigma t+i\alpha y}$, the decaying thermal perturbation is $A_1{e}^{-p\xi }$, with $p^2-Vp=\sigma +{\alpha }^2$ and the branch $p=[V+\sqrt{V^2+4(\sigma +{\alpha }^2)}]/2$. The displaced kinetic condition gives $A_1=(SV-\sigma -\Gamma {\alpha }^2)\zeta$. The displaced Stefan condition gives $S\sigma \zeta =p{A}_1-S{V}^2\zeta$. Eliminating $A_1$ proves the displayed dispersion relation. Neglect of solid-side perturbation heat flux is part of this model.

The cubic follows by eliminating $\sigma =p^2-Vp-{\alpha }^2$ from one-sided kinetic solidification dispersion, with $V=1-S$. For $\Gamma =0$, the interface branch tends to $\sigma =SV-SV/\alpha +O({\alpha }^{-2})$. For fixed $0<\Gamma <1$, substituting $p=\sqrt{1-\Gamma }\alpha +d$ gives $2d-V=S\Gamma /(1-\Gamma )$, hence $\sigma =-\Gamma {\alpha }^2+[S\Gamma /\sqrt{1-\Gamma }]\alpha +O(1)$. At $\Gamma =1$ the balance is $p^3\sim S{\alpha }^2$, giving $\sigma =-{\alpha }^2+S^{2/3}{\alpha }^{4/3}+o({\alpha }^{4/3})$. For $1<\Gamma <1+2S/V$, $p\to S/(\Gamma -1)>V/2$ and $\sigma =-{\alpha }^2+p^2-Vp+o(1)$. Stronger capillarity can remove the real isolated interface mode from the chosen thermal branch; it does not create growth. These distinctions explain why replacing $p$ by $\alpha$ is not a controlled fixed-capillarity limit.

For $0<S<1$ and $V=1-S$, the translational branch of one-sided kinetic solidification dispersion has $\sigma =(S/V-\Gamma ){\alpha }^2+O({\alpha }^4)$. Expand $p=V+(\sigma +{\alpha }^2)/V+O({\alpha }^4)$ and substitute to prove this limit. At zero growth, $p_0=[V+\sqrt{V^2+4{\alpha }^2}]/2$, and the destabilizing right side has sign ${\alpha }^2(SV/p_0^2-\Gamma )$. For nonnegative real $\sigma$, the residual has derivative at least one, so there is one growing root precisely when $\Gamma <SV/p_0^2$. Thus instability exists exactly when $\Gamma <S/V$, with unstable band ${\alpha }^2<p_c(p_c-V)$, $p_c=\sqrt{SV/\Gamma }$ for $\Gamma >0$. At threshold the next term is $\sigma =-2S{\alpha }^4/V^3+O({\alpha }^6)$. Surface energy removes short-wave growth even when long waves remain unstable.