Phase-field crystal modeling of equilibrium bcc-liquid interfaces

We investigate the equilibrium properties of bcc-liquid interfaces modeled with a continuum phase-field crystal (PFC) approach [K. R. Elder and M. Grant, Phys. Rev. E 70, 051605 (2004)]. A multiscale analysis of the PFC model is carried out which exploits the fact that the amplitudes of crystal density waves decay slowly into the liquid in the physically relevant limit where the freezing transition is weakly first order. This analysis yields a set of coupled equations for these amplitudes that is similar to the set of equations derived from Ginzburg-Landau (GL) theory [K.-A. Wu et al., Phys. Rev. B 73, 094101 (2006)]. The two sets only differ in the details of higher order nonlinear couplings between different density waves, which is determined by the form of the nonlinearity assumed in the PFC model and by the ansatz that all polygons with the same number of sides have equal weight in GL theory. Despite these differences, for parameters (liquid structure factor and solid density wave amplitude) of Fe determined from molecular dynamics (MD) simulations, the PFC and GL amplitude equations yield very similar predictions for the overall magnitude and anisotropy of the interfacial free-energy and density wave profiles. These predictions are compared with MD simulations as well as numerical solutions of the PFC model.

Figure 1

Phase diagram of the PFC model obtained under the approximation that the crystal density field is as a sum of density waves corresponding to the set of principal reciprocal lattice vectors for a given crystal structure.

Figure 2

(Color online) Comparison of numerically calculated nonlinear order parameter profiles $u$ and $v$ for {110} crystal faces obtained from the PFC amplitude equations (blue dashed line) and the GL theory (Ref. 44) (red solid line) and computed form MD simulations with ${\vec{K}}_{101}$ and ${\vec{K}}_{110}$ for $u$ and $v$, respectively (solid circles).