Description of hard-sphere crystals and crystal-fluid interfaces: A comparison between density functional approaches and a phase-field crystal model

Abstract

In materials science the phase-field crystal approach has become popular to model crystallization processes. Phase-field crystal models are in essence Landau-Ginzburg-type models, which should be derivable from the underlying microscopic description of the system in question. We present a study on classical density functional theory in three stages of approximation leading to a specific phase-field crystal model, and we discuss the limits of applicability of the models that result from these approximations. As a test system we have chosen the three-dimensional suspension of monodisperse hard spheres. The levels of density functional theory that we discuss are fundamental measure theory, a second-order Taylor expansion thereof, and a minimal phase-field crystal model. We have computed coexistence densities, vacancy concentrations in the crystalline phase, interfacial tensions, and interfacial order parameter profiles, and we compare these quantities to simulation results. We also suggest a procedure to fit the free parameters of the phase-field crystal model. Thereby it turns out that the order parameter of the phase-field crystal model is more consistent with a smeared density field (shifted and rescaled) than with the shifted and rescaled density itself. In brief, we conclude that fundamental measure theory is very accurate and can serve as a benchmark for the other theories. Taylor expansion strongly affects free energies, surface tensions, and vacancy concentrations. Furthermore it is phenomenologically misleading to interpret the phase-field crystal model as stemming directly from Taylor-expanded density functional theory.

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

“…A possible resolution of this contradiction could be that the MD simulations refer to metallic systems that would correspond to small values in the PFC model (where the interface is diffuse), whereas the PFC data of Ref. [23] for the fcc structure refer to very high values (where the interface is sharp and faceted, as in the case of the nearest-neighbor broken-bond model).…”

“…For comparison, we have plotted γ hkl (hkl = 100, 110, and 111) for the fcc-liquid interface at = 0.5, 0.65, and 0.8 from Ref. [23].…”

“…Next, the solution is inserted into the expression of grand-potential density and integrated (the contributions emerge exclusively from the interfacial regions), dividing then the result by twice the cross-sectional area, delivering the bccliquid interface free energy, γ hkl , for the hkl orientation. [23] are also shown (empty symbols). In contrast to the molecular dynamics results [8,9,10], the data for the fcc and bcc structures fall rather close to each other.…”

“…Theoretical predictions for the anisotropy of the crystal-liquid interface in 3D emerge mostly from the early broken-bond models for the fcc, bcc, hcp, and dc structures [14,15,16] (utilizing former results for the crystal-vapor interfaces [17,18,19,20,21]), from the classical density functional theory [22,23], and recently for the fcc and bcc structures from the Phase-Field Crystal (PFC) approach [23,24] (a simple dynamical density functional theory [25,26,27]). Some analytical predictions based on the approximations of the PFC model are also available: A multi-scale analysis has been used by Wu and Karma [28] to evaluate the anisotropy of the interfacial fee energy near the critical point.…”

Assessment of phase-field-crystal concepts using long-time molecular dynamics

“…As stated in the PFC literature, there is a poor correspondence between PFC and DFT even when the PFC parameters for the functional expansions are fit to the DFT energy functional 26,47 . Improved results can be achieved when a more accurate fit of the first peak of the c (2) L function is used 26 and polynomial functions can be used to fit several peaks of c (2) L and bring results very close to the original DFT 41 .…”