Self-consistent modeling of anisotropic interfaces and missing orientations: Derivation from phase-field crystal

Abstract

Highly anisotropic interfaces play an important role in the development of material microstructure. Using the diffusive atomistic phase-field crystal (PFC) formalism, we determine the capability of the model to quantitatively describe these interfaces. Specifically, we coarse grain the PFC model to attain both its complex amplitude formulation and its corresponding phase-field limit.Using this latter formulation, in one-dimensional calculations, we determine the surface energy and the properties of the Wulff shape. We find that the model can yield Wulff shapes with missing orientations, the transition to missing orientations, and facet formation. We show that the corresponding phase-field limit of the complex amplitude model yields a self-consistent description of highly anisotropic surface properties that are a function of the surface orientation with respect to the underlying crystal lattice. The phase-field model is also capable of describing missing orientations on equilibrium shapes of crystals and naturally includes a regularizing contribution. We demonstrate, in two dimensions, how the resultant model can be used to study growth of crystals with varying degrees of anisotropy in the phase-field limit.

An efficient numerical framework for the amplitude expansion of the phase-field crystal model

“…This is a crucial step to address large, mesoscale problems still retaining details of the atomic length scale. Future work will be devoted to explicitly include further details, compatible with the APFC model, in the numerical framework presented here as, for instance, an improved description of interface-energy anisotropy [25,46], binary systems [26], an improved description of the dynamics [23,42], the coupling with magnetic fields [47], as well as improvements on scaling properties of the numerical approach to enable larger systems addressing grain growth in 3D.…”

“…Moreover, it has been recently used to study the anisotropic shrinkage in 3D of small-angle spherical grain boundaries (GBs) regardless of the crystal lattice symmetry [22]. In addition, the APFC framework has been proven suitable to allow for the description of hydrodynamics [23], dislocation dynamics [24] and surface-energy anisotropy [25] within the more general PFC framework.…”

Coarse-grained modeling of crystals by the amplitude expansion of the phase-field crystal model: an overview

“…In Refs. [119,129] similar underlying ideas led to a phase-field model connecting anisotropic surface energy and corresponding Wulff shapes to the lattice symmetry of various crystals through the choice of reciprocal lattice vectors. The model remarkably encodes a regularization term leading to corner rounding of faceted shapes similarly to diffuse interface theories [130][131][132].…”

“…The higher-order gradient contribution [∇ 2 φ m ] 2 enforces the rounding of corners appearing among facets. A coefficient may be also introduced to tune its influence [129].…”

Closing the gap between atomic-scale lattice deformations and continuum elasticity

“…Although some intrinsic limitations for large deformations and tilts exist [25], APFC has proved useful in the advanced modeling of materials as illustrated in studies of elasticity effects [20,25], compositional domains [33], binary alloys [34], dislocation dynamics [35,36], morphology and motion of dislocation networks at grain boundaries [37], and control of material properties [38][39][40]. However, the basic concept of APFC, namely the coarse-graining of an explicit lattice representation by fo-cusing on the complex coefficients of Fourier modes, can be readily applied to any atomistic description as obtained, e.g, from theoretical modeling, atomistic simulations, or experimental imaging.…”