Ginzburg-Landau theory of crystalline anisotropy for bcc-liquid interfaces

The weak anisotropy of the interfacial free-energy $\gamma$ is a crucial parameter influencing dendritic crystal growth morphologies in systems with atomically rough solid-liquid interfaces. The physical origin and quantitative prediction of this anisotropy are investigated for body-centered-cubic (bcc) forming systems using a Ginzburg-Landau theory where the order parameters are the amplitudes of density waves corresponding to principal reciprocal lattice vectors. We find that this theory predicts the correct sign, $\gamma_{100}>\gamma_{110}$, and magnitude, $(\gamma_{100}-\gamma_{110}) / (\gamma_{100}+\gamma_{110})\approx 1%$, of this anisotropy in good agreement with the results of MD simulations for Fe. The results show that the directional dependence of the rate of spatial decay of solid density waves into the liquid, imposed by the crystal structure, is a main determinant of anisotropy. This directional dependence is validated by MD computations of density wave profiles for different reciprocal lattice vectors for $\{110\}$ crystal faces. Our results are contrasted with the prediction of the reverse ordering $\gamma_{100}<\gamma_{110}$ from an earlier formulation of Ginzburg-Landau theory [Shih \emph{et al.}, Phys. Rev. A {\bf 35}, 2611 (1987)].