Phase-field crystal modeling and classical density functional theory of freezing

Abstract

In this paper the relationship between the classical density functional theory of freezing and phase-field modeling is examined. More specifically a connection is made between the correlation functions that enter density functional theory and the free energy functionals used in phase-field crystal modeling and standard models of binary alloys ͑i.e., regular solution model͒. To demonstrate the properties of the phase-field crystal formalism a simple model of binary alloy crystallization is derived and shown to simultaneously model solidification, phase segregation, grain growth, elastic and plastic deformations in anisotropic systems with multiple crystal orientations on diffusive time scales.

Phase-field-crystal study of grain boundary premelting and shearing in bcc iron

“…We use the simplest PFC model 40,41 with the same freeenergy function as the Swift-Hohenberg model of pattern formation 47 , which favors hexagonal and bcc ordering in two and three dimensions (2D and 3D), respectively. This model can be interpreted 43 as a considerably simplified version of classical density function theory (DFT) [48][49][50][51][52][53] where the crystal density field is dominated by the set of primary reciprocal lattice vectors. With suitable choices of parameters for Fe (see Ref.…”

“…In this paper, we investigate GB premelting and shearing and their relationship using the PFC method [40][41][42][43][44][45][46] . We use the simplest PFC model 40,41 with the same freeenergy function as the Swift-Hohenberg model of pattern formation 47 , which favors hexagonal and bcc ordering in two and three dimensions (2D and 3D), respectively.…”

Marginal stability analysis of the phase field crystal model in one spatial dimension

“…The PFC model describes a field that is related to the local atomic number density, such that it is spatially periodic in the solid and constant in the liquid. It can be related to other continuum fields theories such as classical density-functional theory 8,9 and the atomic density function theory 10 . The PFC-model may also be considered as a conserved version of the Swift-Hohenberg equation and provides an efficient method for simulating liquid-solid transitions 11,12 , colloidal solidification 13 , dislocation motion and plasticity 14,15 , glass formation 16 , epitaxial growth 6,17 , grain boundary premelting 18 , surface reconstructions 19 , and grain boundary energies 20 .…”

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

“…Eq. (1) can be deduced [26,27] from the perturbative density functional theory of Ramakrishnan and Yussouff [36]. The solutions that extremize the free energy functional can be obtained by solving the respective Euler-Lagrange equation (ELE) [31]:…”

“…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.…”