67 research outputs found
Closing the gap between atomic-scale lattice deformations and continuum elasticity
Crystal lattice deformations can be described microscopically by explicitly
accounting for the position of atoms or macroscopically by continuum
elasticity. In this work, we report on the description of continuous elastic
fields derived from an atomistic representation of crystalline structures that
also include features typical of the microscopic scale. Analytic expressions
for strain components are obtained from the complex amplitudes of the Fourier
modes representing periodic lattice positions, which can be generally provided
by atomistic modeling or experiments. The magnitude and phase of these
amplitudes, together with the continuous description of strains, are able to
characterize crystal rotations, lattice deformations, and dislocations.
Moreover, combined with the so-called amplitude expansion of the phase-field
crystal model, they provide a suitable tool for bridging microscopic to
macroscopic scales. This study enables the in-depth analysis of elasticity
effects for macro- and mesoscale systems taking microscopic details into
account.Comment: 9 pages, 7 figures, Supporting Information availabl
Controlling the energy of defects and interfaces in the amplitude expansion of the phase-field crystal model
One of the major difficulties in employing phase field crystal (PFC) modeling
and the associated amplitude (APFC) formulation is the ability to tune model
parameters to match experimental quantities. In this work we address the
problem of tuning the defect core and interface energies in the APFC
formulation. We show that the addition of a single term to the free energy
functional can be used to increase the solid-liquid interface and defect
energies in a well-controlled fashion, without any major change to other
features. The influence of the newly added term is explored in two-dimensional
triangular and honeycomb structures as well as bcc and fcc lattices in three
dimensions. In addition, a finite element method (FEM) is developed for the
model that incorporates a mesh refinement scheme. The combination of the FEM
and mesh refinement to simulate amplitude expansion with a new energy term
provides a method of controlling microscopic features such as defect and
interface energies while simultaneously delivering a coarse-grained examination
of the system.Comment: 14 pages, 9 figure
Coarse-grained modeling of crystals by the amplitude expansion of the phase-field crystal model: an overview
Comprehensive investigations of crystalline systems often require methods
bridging atomistic and continuum scales. In this context, coarse-grained
mesoscale approaches are of particular interest as they allow the examination
of large systems and time scales while retaining some microscopic details. The
so-called Phase-Field Crystal (PFC) model conveniently describes crystals at
diffusive time scales through a continuous periodic field which varies on
atomic scales and is related to the atomic number density. To go beyond the
restrictive atomic length scales of the PFC model, a complex amplitude
formulation was first developed by Goldenfeld et al. [Phys. Rev. E 72, 020601
(2005)]. While focusing on length scales larger than the lattice parameter,
this approach can describe crystalline defects, interfaces, and lattice
deformations. It has been used to examine many phenomena including liquid/solid
fronts, grain boundary energies, and strained films. This topical review
focuses on this amplitude expansion of the PFC model and its developments. An
overview of the derivation, connection to the continuum limit, representative
applications, and extensions is presented. A few practical aspects, such as
suitable numerical methods and examples, are illustrated as well. Finally, the
capabilities and bounds of the model, current challenges, and future
perspectives are addressed.Comment: 57 pages, 15 figure
Amplitude phase-field crystal model for the hexagonal close-packed lattice
The phase field crystal model allows the study of materials on atomic length
and diffusive time scales. It accounts for elastic and plastic deformation in
crystal lattices, including several processes such as growth, dislocation
dynamics, and microstructure evolution. The amplitude expansion of the phase
field crystal model (APFC) describes the atomic density by a small set of
Fourier modes with slowly-varying amplitudes characterizing lattice
deformations. This approach allows for tackling large, three-dimensional
systems. However, it has been used mostly for modeling basic lattice
symmetries. In this work, we present a coarse-grained description of the
hexagonal closed-packed (HCP) lattice that supports lattice deformation and
defects. It builds on recent developments of the APFC model and introduces
specific modeling aspects for this crystal structure. After illustrating the
general modeling framework, we show that the proposed approach allows for
simulating relatively large three-dimensional HCP systems hosting complex
defect networks
Overcoming timescale and finite-size limitations to compute nucleation rates from small scale Well Tempered Metadynamics simulations
Condensation of a liquid droplet from a supersaturated vapour phase is
initiated by a prototypical nucleation event. As such it is challenging to
compute its rate from atomistic molecular dynamics simulations. In fact at
realistic supersaturation conditions condensation occurs on time scales that
far exceed what can be reached with conventional molecular dynamics methods.
Another known problem in this context is the distortion of the free energy
profile associated to nucleation due to the small, finite size of typical
simulation boxes. In this work the problem of time scale is addressed with a
recently developed enhanced sampling method while contextually correcting for
finite size effects. We demonstrate our approach by studying the condensation
of argon, and showing that characteristic nucleation times of the order of
magnitude of hours can be reliably calculated, approaching realistic
supersaturation conditions, thus bridging the gap between what standard
molecular dynamics simulations can do and real physical systems.Comment: 9 pages, 7 figures, additional figures and data provided as
supplementary information. Submitted to the Journal of Chemical Physisc
An efficient numerical framework for the amplitude expansion of the phase-field crystal model
The study of polycrystalline materials requires theoretical and computational
techniques enabling multiscale investigations. The amplitude expansion of the
phase field crystal model (APFC) allows for describing crystal lattice
properties on diffusive timescales by focusing on continuous fields varying on
length scales larger than the atomic spacing. Thus, it allows for the
simulation of large systems still retaining details of the crystal lattice.
Fostered by the applications of this approach, we present here an efficient
numerical framework to solve its equations. In particular, we consider a real
space approach exploiting the finite element method. An optimized
preconditioner is developed in order to improve the convergence of the linear
solver. Moreover, a mesh adaptivity criterion based on the local rotation of
the polycrystal is used. This results in an unprecedented capability of
simulating large, three-dimensional systems including the dynamical description
of the microstructures in polycrystalline materials together with their
dislocation networks.Comment: 12 pages, 7 figure
Amplitude expansion of the phase-field crystal model for complex crystal structures
The phase-field crystal (PFC) model describes crystal lattices at diffusive
timescales. Its amplitude expansion (APFC) can be applied to the investigation
of relatively large systems under some approximations. However, crystal
symmetries accessible within the APFC model are limited to basic ones, namely
triangular and square in two dimensions, and body-centered cubic and
face-centered cubic in three dimensions. In this work, we propose a general,
amplitudes-based description of virtually any lattice symmetry. To fully
exploit the advantages of this model, featuring slowly varying quantities in
bulk and localized significant variations at dislocations and interfaces, we
consider formulations suitable for real-space numerical methods supporting
adaptive spatial discretization. We explore approaches originally proposed for
the PFC model which allow for symmetries beyond basic ones through extended
parametrizations. Moreover, we tackle the modeling of non-Bravais lattices by
introducing an amplitude expansion for lattices with a basis and further
generalizations. We study and discuss the stability of selected, prototypical
lattice symmetries. As pivotal examples, we show that the proposed approach
allows for a coarse-grained description of the kagome lattice, exotic square
arrangements, and the diamond lattice, as bulk crystals and, importantly,
hosting dislocations.Comment: 14 pages, 7 figure
The elastic inclusion problem in the (amplitude) phase field crystal model
In many processes for crystalline materials such as precipitation,
heteroepitaxy, alloying, and phase transformation, lattice expansion or
compression of embedded domains occurs. This can significantly alter the
mechanical response of the material. Typically, these phenomena are studied
macroscopically, thus neglecting the underlying microscopic structure. Here we
present the prototypical case of an elastic inclusion described by a mesoscale
model, namely a coarse-grained phase-field crystal model. A spatially-dependent
parameter is introduced into the free energy functional to control the local
spacing of the lattice structure, effectively prescribing an eigenstrain. The
stress field obtained for an elastic inclusion in a 2D triangular lattice is
shown to match well with the analytic solution of the Eshelby inclusion
problem.Comment: 6 pages; 3 figure
Explicit temperature coupling in phase-field crystal models of solidification
We present a phase-field crystal (PFC) model for solidification that accounts
for thermal transport and a temperature-dependent lattice parameter. Elasticity
effects are characterized through the continuous elastic field computed from
the microscopic density field. We showcase the model capabilities via selected
numerical investigations which focus on the prototypical growth of
two-dimensional crystals from the melt, resulting in faceted shapes and
dendrites. This work sets the grounds for a comprehensive mesoscale model of
solidification including thermal expansion
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