Solidification attachment mobility ($U=\mathcal{M}(T_{\mathrm{eq}}-T_i)$)

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.