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

Abstract

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.

arXiv comment57 pages, 15 figures

References in corpus (61)

• Modeling Elasticity in Crystal Growth
• Modeling Elastic and Plastic Deformations in Non-Equilibrium Processing Using Phase Field Crystals
• Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview
• Phase-field Crystals with Elastic Interactions
• Derivation of the phase field crystal model for colloidal solidification
• Free energy functionals for efficient phase field crystal modeling of structural phase transformations
• Sharp interface limits of phase-field models
• Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model
• Phase-field modeling of crystal nucleation in undercooled liquids -- A review
• Traveling and resting crystals in active systems
• Amplitude expansion of the binary phase field crystal model
• Geometry of logarithmic strain measures in solid mechanics
• Renormalization group theory for the phase field crystal equation
• Model of surface instabilities induced by stress
• Phase field crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures
• Exploring the complex world of two-dimensional ordering with three modes
• Phase-field-crystal study of grain boundary premelting and shearing in bcc iron
• Multiscale modeling of polycrystalline graphene: A comparison of structure and defect energies of realistic samples from phase field crystal models
• Active crystals and their stability
• Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model
• An energy-stable convex splitting for the phase-field crystal equation
• Phase Field Crystal Modeling as a Unified Atomistic Approach to Defect Dynamics
• Defect stability in phase-field crystal models: Stacking faults and partial dislocations
• Consistent Hydrodynamics for Phase Field Crystals
• Grain growth beyond the Mullins model, capturing the complex physics behind universal grain size distributions
• Amplitude equations for polycrystalline materials with interaction between composition and stress
• New density functional approach for solid-liquid-vapor transitions in pure materials
• Dislocation dynamics and crystal plasticity in the phase field crystal model
• Dynamical Crystallites of Active Chiral Particles
• Mesoscale model of dislocation motion and crystal plasticity
• Phase-field-crystal models and mechanical equilibrium
• Segregation--Assisted Spinodal and Transient Spinodal Phase Separation at Grain Boundaries
• A coarse-grained phase-field crystal model of plastic motion
• Direct atomistic modeling of solute drag by moving grain boundaries
• Phase Field Crystals as a Coarse-Graining in Time of Molecular Dynamics
• Non-singular dislocation continuum theories: Strain gradient elasticity versus Peierls-Nabarro model
• Phase-field crystal model for ordered crystals
• Wetting layer thickness and early evolution of epitaxially strained thin films
• Orientation gradients in rapidly solidified pure aluminum thin films: comparison of experiments and phase-field crystal simulations
• A mesoscopic field theoretical approach for active systems
• Closing the gap between atomic-scale lattice deformations and continuum elasticity
• Complex order-parameter phase-field models derived from structural phase-field-crystal models
• A Navier Stokes Phase Field Crystal Model for Colloidal Suspensions
• Controlling the energy of defects and interfaces in the amplitude expansion of the phase-field crystal model
• Morphological instability, evolution, and scaling in strained epitaxial films: An amplitude equation analysis of the phase field crystal model
• A microscopic field theoretical approach for binary mixtures of active and passive particles
• Development and Analysis of a Block-Preconditioner for the Phase-Field Crystal Equation
• Non-linear elastic effects in phase field crystal and amplitude equations: Comparison to ab initio simulations of bcc metals and graphene
• Defects at grain boundaries: A coarse-grained, three-dimensional description by the amplitude expansion of the phase-field crystal model
• Mesoscale Defect Motion in Binary Systems: Effects of Compositional Strain and Cottrell Atmospheres
• An efficient numerical framework for the amplitude expansion of the phase-field crystal model
• Scale coupling and interface pinning effects in the phase-field-crystal model
• Stress in ordered systems: Ginzburg-Landau type density field theory
• Multiscale analysis of crystal defect formation in rapid solidification of pure aluminium and aluminium-copper alloys
• Anisotropic velocity statistics of topological defects under shear flow
• Self-consistent modeling of anisotropic interfaces and missing orientations: Derivation from phase-field crystal
• Modeling of grain boundary dynamics using amplitude equations
• Defect Statistics in the Two Dimensional Complex Ginsburg-Landau Model
• Elastically-mediated interactions between grain boundaries and precipitates in two-phase coherent solids
• The elastic inclusion problem in the (amplitude) phase field crystal model
• Renormalization-group for amplitude equations in cellular pattern formation with and without conservation law

Cited by in corpus (5)

• A phase field crystal theory of the kinematics of dislocation lines
• The elastic inclusion problem in the (amplitude) phase field crystal model
• Amplitude expansion of the phase-field crystal model for complex crystal structures
• Magnetic APFC modeling and the influence of magneto-structural interactions on grain shrinkage
• Explicit temperature coupling in phase-field crystal models of solidification