Phase Field Crystal Model Research Articles

A Relaxed Decoupled Second-Order Energy-Stable Scheme for the Binary Phase-Field Crystal Model

A Relaxed Decoupled Second-Order Energy-Stable Scheme for the Binary Phase-Field Crystal Model

A phase field crystal model for real-time grain boundary formation and motion in complex concentration alloy

The complex concentration alloys exhibit the excellent mechanical property due to the dual roles of their heterogeneous compositions and microstructures. However, the formation and motion of grain boundaries to significantly regulate the mechanical properties remain unknown at several microsecond and nanoscale in the complex concentration alloys. Here, we utilize a phase field crystal model to investigate the real-time grain boundary formation and motion in complex concentration alloys, specifically, in comparison to traditional alloys, using the example of AlCu alloys. Our findings reveal that the low concentration alloys exhibit segregation primarily at grain boundaries with minimal compositional fluctuations over a long period of grain growth, and the complex concentration alloys display a high concentration gradient within the grains to cause rapid grain rotation and merging in a short time frame. In the complex concentration alloys, the large component fluctuations not only cause the local lattice distortion to hinder dislocation movement during the deformation process, but also lead to the occurrence from dislocation locking to self-unlocking, which is beneficial to improve the strength and ductility. The present work provides a real-time atomic-scale understanding of grain boundary evolution using in situ atomic-resolution simulations.

Computational and analytical studies of a new nonlocal phase-field crystal model in two dimensions

A nonlocal phase-field crystal (NPFC) model is presented as a nonlocal counterpart of the local phase-field crystal (LPFC) model and a special case of the structural PFC (XPFC) derived from classical field theory for crystal growth and phase transition. The NPFC incorporates a finite range of spatial nonlocal interactions that can account for both repulsive and attractive effects. The specific form is data-driven and determined by a fitting to the materials structure factor, which can be much more accurate than the LPFC and previously proposed fractional variant. In particular, it is able to match the experimental data of the structure factor up to the second peak, an achievement not possible with other PFC variants studied in the literature. Both LPFC and fractional PFC (FPFC) are also shown to be distinct scaling limits of the NPFC, which reflects the generality. The advantage of NPFC in retaining material properties suggests that it may be more suitable for characterizing liquid–solid transition systems. Moreover, we study numerical discretizations using Fourier spectral methods, which are shown to be convergent and asymptotically compatible, making them robust numerical discretizations across different parameter ranges. Numerical experiments are given in the two-dimensional case to demonstrate the effectiveness of the NPFC in simulating crystal structures and grain boundaries.

Coupling Phase Field Crystal and Field Dislocation Mechanics for a consistent description of dislocation structure and elasticity

This work addresses differences in predicted elastic fields created by dislocations either by the Phase Field Crystal (PFC) model, or by static Field Dislocation Mechanics (FDM). The PFC order parameter describes the topological content of the lattice, but it fails to correctly capture the elastic distortion. In contrast, static FDM correctly captures the latter but requires input about defect cores. The case of a dislocation dipole in two dimensional, isotropic, elastic medium is studied, and a weak coupling is introduced between the two models. The PFC model produces compact and stable dislocation cores, free of any singularity, i.e., diffuse. The PFC predicted dislocation density field (a measure of the topological defect content) is used as the source (input) for the static FDM problem. This coupling allows a critical analysis of the relative role played by configurational (from PFC) and elastic (from static FDM) fields in the theory, and of the consequences of the lack of elastic relaxation in the diffusive evolution of the PFC order parameter.

Mesoscale modeling of deformations and defects in thin crystalline sheets

We present a mesoscale description of deformations and defects in thin, flexible sheets with crystalline order, tackling the interplay between in-plane elasticity, out-of-plane deformation, as well as dislocation nucleation and motion. Our approach is based on the Phase-Field Crystal (PFC) model, which describes the microscopic atomic density in crystals at diffusive timescales, naturally encoding elasticity and plasticity effects. In its amplitude expansion (APFC), a coarse-grained description of the mechanical properties of crystals is achieved. We introduce surface PFC and surface APFC models in a convenient height-function formulation encoding deformation in the normal direction. This framework is proven consistent with classical aspects of strain-induced buckling, defect nucleation on deformed surfaces, and out-of-plane relaxation near dislocations. In particular, we benchmark and discuss the results of numerical simulations by looking at the continuum limit for buckling under uniaxial compression and at evidence from microscopic models for deformation at defects and defect arrangements, demonstrating the scale-bridging capabilities of the proposed framework. Results concerning the interplay between lattice distortion at dislocations and out-of-plane deformation are also illustrated by looking at the annihilation of dislocation dipoles and systems hosting many dislocations. With the novel formulation proposed here, and its assessment with established approaches, we envision applications to multiscale investigations of crystalline order on deformable surfaces.

A class of efficient high-order time-stepping methods for the anisotropic phase-field dendritic crystal growth model

In this paper, we develop and analyze a novel class of arbitrarily high-order and unconditionally energy stable schemes for the anisotropic phase-field dendritic crystal growth model, which is a highly nonlinear system that combines the anisotropic Allen–Cahn equation with the thermal equation. The proposed schemes are based on an extrapolated and linearized Runge–Kutta method for an auxiliary variable reformulation of the crystal growth model. A delicate implementation demonstrates that the proposed method can be realized in a very efficient way, requiring only the solution of a coupled linear elliptic system at each time step. We prove that the constructed schemes satisfy the energy dissipation property and demonstrate the convergence order through a consistency error analysis. To the best of our knowledge, this is the first unconditionally stable scheme of arbitrarily high-order for the anisotropic phase-field dendritic crystal growth model. Numerical experiments for two and three dimensional dendritic crystal growth are simulated to verify our theoretical accuracy as well as the efficiency of the proposed method. Furthermore, to emphasize the superiority of the high-order schemes, we present a numerical comparison that directly contrasts its performance with the lower-order schemes.

Isogeometric collocation method to simulate phase-field crystal model

PurposeThe purpose of this study is to develop a new numerical algorithm to simulate the phase-field model.Design/methodology/approachFirst, the derivative of the temporal direction is discretized by a second-order linearized finite difference scheme where it conserves the energy stability of the mathematical model. Then, the isogeometric collocation (IGC) method is used to approximate the derivative of spacial direction. The IGC procedure can be applied on irregular physical domains. The IGC method is constructed based upon the nonuniform rational B-splines (NURBS). Each curve and surface can be approximated by the NURBS. Also, a map will be defined to project the physical domain to a simple computational domain. In this procedure, the partial derivatives will be transformed to the new domain by the Jacobian and Hessian matrices. According to the mentioned procedure, the first- and second-order differential matrices are built. Furthermore, the pseudo-spectral algorithm is used to derive the first- and second-order nodal differential matrices. In the end, the Greville Abscissae points are used to the collocation method.FindingsIn the numerical experiments, the efficiency and accuracy of the proposed method are assessed through two examples, demonstrating its performance on both rectangular and nonrectangular domains.Originality/valueThis research work introduces the IGC method as a simulation technique for the phase-field crystal model.

Passive and active field theories for disease spreading

The worldwide COVID-19 pandemic has led to a significant growth of interest in the development of mathematical models that allow to describe effects such as social distancing measures, the development of vaccines, and mutations. Several of these models are based on concepts from soft matter theory. Considerably less well investigated is the reverse direction, i.e. how results from epidemiological research can be of interest for the physics of colloids and polymers. In this work, we consider the susceptible-infected-recovered (SIR)-dynamical density functional theory (DDFT) model, a combination of the SIR model from epidemiology with DDFT from nonequilibrium soft matter physics, which allows for an explicit modeling of social distancing. We extend the SIR-DDFT model both from an epidemiological perspective by incorporating vaccines, asymptomaticity, reinfections, and mutations, and from a soft matter perspective by incorporating noise and self-propulsion and by deriving a phase field crystal (PFC) model that allows for a simplified description. On this basis, we investigate via computer simulations how epidemiological models are affected by the presence of non-reciprocal interactions. This is done in a numerical study of a zombie outbreak.

Linear relaxation method with regularized energy reformulation for phase field models

In this paper, we establish a novel linear relaxation method with regularized energy reformulation for phase field models, which we name the RRER method. We employ the molecular beam epitaxy (MBE) model and the phase-field crystal (PFC) model as test beds, along with several coupled phase field models, to illustrate the concept. Our proposed RRER strategy is applicable to a broad class of phase field models that can be derived through energy variation. The RRER method consists of two major steps. First, we introduce regularized auxiliary variables to reformulate the original phase field models into equivalent forms where the free energy is transformed under the auxiliary variables. Then, we discretize the reformulated PDE model under these variables on a staggered time mesh for the phase field models. We incorporate the energy reformulation idea in the first step and the linear relaxation concept in the second step to derive a general numerical algorithm for phase field models that is linear and second-order accurate. Our approach differs from the classical invariant energy quadratization (IEQ) and scalar auxiliary variable (SAV) approaches, as we don't need to take time derivatives for the auxiliary variables. The resulting schemes are linear, i.e., only linear algebraic systems need to be solved at each time step. Rigorous theoretical analysis demonstrates that these resulting schemes satisfy the modified discrete energy dissipation laws and preserve the discrete mass conservation for the PFC and MBE models. Furthermore, we present numerical results to demonstrate the effectiveness of our method in solving phase field models.

A second-order Strang splitting scheme for the generalized Allen–Cahn type phase-field crystal model with FCC ordering structure

In this paper, we consider the generalized Allen–Cahn-type phase-field crystal model with face-centered-cubic ordering structure (PFC-FCC). Due to the combined complexity of the eighth-order spatial derivative and inherent nonlinearity, it poses a significant challenge to design a numerical scheme of high accuracy, stability, and efficiency to solve the PFC-FCC model. Endeavoring towards this objective, we adopt operator splitting method to address the PFC-FCC model. Based on Fourier spectral method for spatial discretization and SSP-RK method for temporal discretization, we propose an effective and easy-to-implement second-order scheme. We are also able to proffer an analytical proof for discrete mass conservation, and engage a discussion on an optimal error estimate for the second-order scheme. In addition, the energy stability of the numerical scheme is derived. Ultimately, a series of numerical experiments specifically designed to substantiate its accuracy, efficiency, and capability for phase transition are performed.

Temporal error analysis of an unconditionally energy stable second-order BDF scheme for the square phase-field crystal model

In this paper, we first propose and study the second-order time-discrete numerical scheme for the sixth-order nonlinear parabolic problem of the square phase-field crystal model. Then, we demonstrate the two-step backward differentiation formula (BDF-2) scheme with mass conservation and energy dissipation, where the higher order nonlinear term is treated implicitly. Moreover, a rigorous error analysis is presented and we prove the optimal second-order convergence rate O(τ2) in H1- norm, where τ is the time step. Finally, some numerical results are provided to confirm our theoretical analysis.

Gradient elasticity in Swift–Hohenberg and phase-field crystal models

The Swift–Hohenberg (SH) and phase-field crystal (PFC) models are minimal yet powerful approaches for studying phenomena such as pattern formation, collective order, and defects via smooth order parameters. They are based on a free-energy functional that inherently includes elasticity effects. This study addresses how gradient elasticity (GE), a theory that accounts for elasticity effects at microscopic scales by introducing additional characteristic lengths, is incorporated into SH and PFC models. After presenting the fundamentals of these theories and models, we first calculate the characteristic lengths for various lattice symmetries in an approximated setting. We then discuss numerical simulations of stress fields at dislocations and comparisons with analytic solutions within first and second strain-gradient elasticity. Effective GE characteristic lengths for the elastic fields induced by dislocations are found to depend on the free-energy parameters in the same manner as the phase correlation length, thus unveiling how they change with the quenching depth. The findings presented in this study enable a thorough discussion and analysis of small-scale elasticity effects in pattern formation and crystalline systems using SH and PFC models and, importantly, complete the elasticity analysis therein. Additionally, we provide a microscopic foundation for GE in the context of order-disorder phase transitions.

OpenPFC: an open-source framework for high performance 3D phase field crystal simulations

We present OpenPFC (https://github.com/VTT-ProperTune/OpenPFC), a state-of-the-art phase field crystal (PFC) simulation platform designed to be scalable for massive high-performance computation environments. OpenPFC can efficiently handle large-scale simulations, as demonstrated by our strong and weak scaling analyses up to an 81923 grid on 65 536 cores. Our results indicate that meaningful PFC simulations can be conducted on grids of size 20483 or even 40963, provided there is a sufficient number of cores and ample disk storage available. In addition, we introduce an efficient implementation of moving boundary conditions that eliminates the need for copying field values between MPI processes or adding an advection term to the evolution equations. This scheme enhances the computational efficiency in simulating large scale processes such as long directional solidification. To showcase the robustness of OpenPFC, we apply it to simulations of rapid solidification in the regime of metal additive manufacturing using a recently developed quantitative solid-liquid-vapor PFC model, parametrized for pure tungsten (body-centered cubic) and aluminum (face-centered cubic).

Phase field crystal modeling of grain boundary structures in diamond cubic systems

Phase field crystal (PFC) modeling has proved to be a versatile numerical tool in the analysis of crystalline microstructures. Most often, however, the focus is put on bulk crystal behavior, while crystal defects such as grain boundaries (GBs) are less explored. This is, in particular, the case for crystal structures beyond fcc and bcc. In this work, the possibilities and challenges in adopting PFC to diamond cubic (DC) crystal structures is investigated. Three different PFC models are considered for this purpose. One of them was published previously, and two are modifications proposed in the present work. The models are compared in terms of both DC phase stabilization and their ability to provide relevant GB structures. The models employ combinations of two- and three-point correlations, and the addition of a three-point correlation is found to be required for stabilization of the expected DC GB structures. It is concluded that although each of the models has limitations in terms of the GB structures which can be stabilized and performance in terms of phase stability, key PFC components for successful modeling of DC structures can be identified. Published by the American Physical Society 2024

Phase field crystal study on the size and strain rate dependent evolution of Kirkendall voids in binary alloy

The evolution of Kirkendall voids in binary alloy under a uniaxial strain rate loading is investigated by the phase field crystal (PFC) model. The simulation results show that model size and the applied strain rate greatly affect the formation and growth of Kirkendall voids. It is found that the grains are prone to rotate when the model size is smaller than 34.7a×34.7a (a is the lattice constant) and the initially set grain boundaries will disappear after relaxation, no Kirkendall voids form in this case. As the model size increases, Kirkendall voids arise and interconnect with each other, and large voids will dissociate to produce small voids. Moreover, the applied strain rate also affects the formation and growth path of Kirkendall voids. Voids only form at the grain boundaries in the absence of strain rate loading, whereas voids form both inside the grains and at the grain boundaries under strain rate loading. The growth of voids can easily form oblique structures caused by dislocation slip, and they tend to grow perpendicular to the direction of applied strain rate.

Active smectics on a sphere

The dynamics of active smectic liquid crystals confined on a spherical surface is explored through an active phase field crystal model. Starting from an initially randomly perturbed isotropic phase, several types of topological defects are spontaneously formed, and then annihilate during a coarsening process until a steady state is achieved. The coarsening process is highly complex involving several scaling laws of defect densities as a function of time where different dynamical exponents can be identified. In general the exponent for the final stage towards the steady state is significantly larger than that in the passive and in the planar case, i.e. the coarsening is getting accelerated both by activity and by the topological and geometrical properties of the sphere. A defect type characteristic for this active system is a rotating spiral of evolving smectic layering lines. On a sphere this defect type also determines the steady state. Our results can in principle be confirmed by dense systems of synthetic or biological active particles.

Phase field crystal models with applications to laser deposition: A review.

In this article, we address the application of phase field crystal (PFC) theory, a hybrid atomistic-continuum approach, for modeling nanostructure kinetics encountered in laser deposition. We first provide an overview of the PFC methodology, highlighting recent advances to incorporate phononic and heat transport mechanisms. To simulate laser heating, energy is deposited onto a number of polycrystalline, two-dimensional samples through the application of initial stochastic fluctuations. We first demonstrate the ability of the model to simulate plasticity and recrystallization events that follow laser heating in the isothermal limit. Importantly, we also show that sufficient kinetic energy can cause voiding, which serves to suppress shock propagation. We subsequently employ a newly developed thermo-density PFC theory, coined thermal field crystal (TFC), to investigate laser heating of polycrystalline samples under non-isothermal conditions. We observe that the latent heat of transition associated with ordering can lead to long lasting metastable structures and defects, with a healing rate linked to the thermal diffusion. Finally, we illustrate that the lattice temperature simulated by the TFC model is in qualitative agreement with predictions of conventional electron-phonon two-temperature models. We expect that our new TFC formalism can be useful for predicting transient structures that result from rapid laser heating and re-solidification processes.

Phase-field crystal modeling of graphene/hexagonal boron nitride interfaces.

Two-dimensional (2D) materials such as graphene and hexagonal boron nitride (h-BN) are an essential class of materials with enhanced structural and electronic properties compared to their bulk counterparts. The phase-field crystal (PFC) model can reach diffusive time scales to study nucleation, growth of crystallites, and relaxation of strain-driven 2D monolayers that are much larger in comparison to molecular dynamics (MD) and quantum mechanical density functional theory (QMDFT) methods while retaining atomic resolution. The model also naturally incorporates an atomic length scale and elastic and plastic deformations. We simulate the morphological transition of the crystal growth of various equilibrium crystal shapes. In this work, we generalize the one-mode PFC model to study the graphene/h-BN heterostructure interface by using conserved dynamics to describe the dynamics of the model. The model was used to find the equilibrium shape of the crystal of the h-BN crystal embedded in a graphene monolayer.

Noise-Induced Defects in Honeycomb Lattice Structure: A Phase-Field Crystal Study

One of the classes of the kinetic phase-field model in the form of the two-mode hyperbolic phase-field crystal model (modified PFC model) is used for the study of the noise effect of the crystalline structure. Special attention is paid to the origin of the defect’s microstructure in the crystalline honeycomb lattice due to induced colored noise. It shows that the noise–time correlation coefficient τζ, comparable to the diffusion time, enhances the grain boundary mobilities. Instead, a small spatial correlation coefficient, λζ, close to the first lattice parameter of the honeycomb crystal, stabilizes the structure. The finite non-zero value of the relaxation time τ for the atomic flux significantly slows the local relaxation of the fluctuated field and leads to the grains’ fragmentation and formation of the disordered phases. The obtained results are applicable to the hexagonal atomic structures and, in particular, to honeycomb crystals, such as boron nitride, in which the lattice defects might be simulated through the induced colored noise.

Phase-field crystal simulation of the evolution of voids in polycrystalline Au–Al bonds with elastic interaction and the Kirkendall effect

In this paper, the modified phase-field crystal (MPFC) model is employed to study the evolution of Kirkendall voids in polycrystalline Au–Al bonds under mechanical loading. The MPFC model can simulate the structural evolution of deformed pure nanocrystalline materials considering instantaneous elastic interactions. Therefore, coupling a concentration field evolved by the Cahn-Hilliard equation with a density field evolved by the existing MPFC model makes it possible to study the phase transformation of a binary system in the presence of mechanical deformation. Using this modified model, we focus on the morphology evolution of the voids in Au–Al bonds under different mechanical loadings. The semi-implicit Fourier spectral algorithm is applied to solve the six-order coupling dynamics equations. The Kirkendall porosity of the diffusion couple is computed and analyzed. The effects of quasi-static and cyclic axial tensile loads on the morphology of voids, and porosity of the system are investigated. It is found that the morphology of Kirkendall voids that result from applying a quasi-static load is different from those that result from applying a cyclic load.