Ginzburg-Landau theory of the bcc-liquid interface kinetic coefficient

We extend the Ginzburg-Landau (GL) theory of atomically rough bcc-liquid interfaces [Wu {\it et al.}, Phys. Rev. B \textbf{73}, 094101 (2006)] outside of equilibrium. We use this extension to derive an analytical expression for the kinetic coefficient, which is the proportionality constant $\mu(\hat n)$ between the interface velocity along a direction $\hat n$ normal to the interface and the interface undercooling. The kinetic coefficient is expressed as a spatial integral along the normal direction of a sum of gradient square terms corresponding to different nonlinear density wave profiles. Anisotropy arises naturally from the dependence of those profiles on the angles between the principal reciprocal lattice vectors $\vec K_i$ and $\hat n$. Values of the kinetic coefficient for the$(100)$, $(110)$ and $(111)$ interfaces are compared quantitatively to the prediction of linear Mikheev-Chernov (MC) theory [J. Cryst. Growth \textbf{112}, 591 (1991)] and previous molecular dynamics (MD) simulation studies of crystallization kinetics for a classical model of Fe. Additional MD simulations are carried out here to compute the relaxation time of density waves in the liquid in order to make this comparison free of fit parameter. The GL theory predicts a similar expression for $\mu$ as the MC theory but yields a better agreement with MD simulations for both its magnitude and anisotropy due to a fully nonlinear description of density wave profiles across the solid-liquid interface. GL theory is also used to derive an inverse relation between $\mu$ and the solid-liquid interfacial free-energy. The general methodology used here to derive an expression for $\mu(\hat n)$ also applies to amplitude equations derived from the phase-field-crystal model, which only differ from GL theory by the choice of cubic and higher order nonlinearities in the free-energy density.