Free energy of the bcc–liquid interface and the Wulff shape as predicted by the phase-field crystal model

Abstract

The Euler-Lagrange equation of the phase-field crystal (PFC) model has been solved under appropriate boundary conditions to obtain the equilibrium free energy of the body centered cubic crystal-liquid interface for 18 orientations at various reduced temperatures in the range ∈ [0, 0.5]. While the maximum free energy corresponds to the {100} orientation for all values, the minimum is realized by the {111} direction for small (< 0.13), and by the {211} orientation for higher . The predicted dependence on the reduced temperature is consistent with the respective mean field critical exponent. The results are fitted with an eight-term Kubic harmonic series, and are used to create stereographic plots displaying the anisotropy of the interface free energy. We have also derived the corresponding Wulff shapes that vary with increasing from sphere to a polyhedral form that differs from the rhombo-dodecahedron obtained previously by growing a bcc seed until reaching equilibrium with the remaining liquid.

Phase-field modeling of crystal nucleation in undercooled liquids – A review

“…Remarkably, in the single-mode PFC model the fcc-liquid and bcc-liquid interfacial free energies fall quite close to each other ( Fig. 41) [193]. The same model was employed to address homogeneous and heterogeneous crystal nucleation in 2D and 3D.…”

Phase-field modeling of eutectic structures on the nanoscale: the effect of anisotropy

“…While a very recent work [35] indicates that the handling of anisotropies can be far more complex owing to mechanical effects than the "usual" treatment adopted here, the latter proved to be a practical way to address anisotropy for various complex problems [36][37][38][39]. Anisotropy functions of the types employed here were used to represent the results from molecular dynamics studies [36,40,41] and molecular scale theories [42][43][44]. Future work is planned to consider the influence of mechanical effects on the eutectic patterns.…”

Phase-Field Modeling of Polycrystalline Solidification: From Needle Crystals to Spherulites—A Review

“…Other anisotropy functions can be taken from atomistic simulations. [50,51] This approach was used to address a broad range of polycrystalline solidification morphologies in three dimensions, including multidendritic solidification, [15,30] disordered dendrites, [30] spherulites, [15,30] and shish-kebab structure, [40] and was adopted for modeling grain boundary dynamics. [52] It also served as a basis for developing the quantitative OFPF model for solidification.…”