Phase Field Crystal Research Articles

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.

A second-order linear unconditionally energy-stable scheme for the phase field crystal equation

This paper presents a linear implicit scheme for numerically solving the phase field crystal (PFC) equation. The scheme is constructed by the scalar auxiliary variable (SAV) approach and the leapfrog scheme in temporal direction. The proposed scheme is decoupled, and it is easy to be implemented. In contrast, the previous SAV-type schemes for the PFC equation are usually coupled. The scheme satisfies the energy stability at the discrete level. Moreover, it is proved that the scheme is convergent with second-order accuracy in the temporal direction. Several numerical examples are carried out to confirm the theoretical results.

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.

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 linear second-order convex splitting scheme for the modified phase-field crystal equation with a strong nonlinear vacancy potential

The modified phase-field crystal (MPFC) equation was extended to the MPFC equation with a strong nonlinear vacancy potential (VMPFC) to include vacancies. In this paper, we develop a linear, second-order, and unconditionally energy stable scheme for solving the VMPFC equation by proposing a new linear convex splitting, using the Crank–Nicolson formula to the contractive part and the second-order Adams–Bashforth formula to the expansive part, and adding a second-order Douglas–Dupont regularization. The unconditional energy stability of the developed scheme is proved. By solving the VMPFC equation, vacancies which exist in real materials can be observed in numerical simulations.

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 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.

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.

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.

Multiplicity of grain boundary structures and related energy variations

Ideal minimum-energy grain boundary (GB) structures are usually the only ones considered when characterizing GBs. This limited perspective provides an incomplete view of the possible structural variability of GBs. In the present study, phase field crystal (PFC) simulations are employed in a systematic search of GB states based on γ-surface sampling, demonstrating PFC to be a capable tool for this purpose. It is also shown that a set of microscopic degrees of freedom (DOF) must be considered in addition to the GB’s five macroscopic DOF when identifying variants in GB structure. This set of microscopic DOF comprises three components of relative crystal translation, plus one DOF describing the atomic density in the GB region. GB energy is taken as an example of a key property which is shown to exhibit a considerable spread due to variations in these microscopic DOF, at constant macroscopic DOF. The findings underline the need to consider a spectrum of GB energies for each macroscopic GB configuration, rather than a single value corresponding to an idealized minimum-energy structure. As part of the study, some computationally efficient strategies for setting up the PFC simulation model are also proposed.

Strain-induced grain boundary migration and grain rotation in polycrystalline metals: Atomic-and meso-scale phase field simulations

The study reveals that in phase field (PF) simulations with over 300 grains, grain boundary (GB) migration exhibits anisotropic behavior and higher velocities under applied external loading. However, these simulations fail to provide detailed atomic-scale information about the evolution of grain orientation. Therefore, the researchers utilize phase field crystal (PFC) simulations to track the evolution of dislocation configurations and the annihilation of GB dislocations. The PFC simulations confirm a newly proposed plasticity process in nanocrystalline metals, where grain rotation is primarily dominated by dislocation climb and dislocation absorption at triple-junction points on GB, rather than GB sliding or diffusional creep as previously suggested. Furthermore, the PFC simulations demonstrate that the analysis of misorientation angles and GB dislocation evolution between neighboring grains can be quantitatively described using the Frank-Bilby equation. Thus, a hybrid approach of PF and PFC successfully describes in-situ transmission electron microscopy observations of GB migration and grain rotation for the first time. This multi-scale simulation method provides a deeper understanding of plasticity formation and development in nanostructured materials.

Benchmark Computations of the Phase Field Crystal and Functionalized Cahn-Hilliard Equations via Fully Implicit, Nesterov Accelerated Schemes

We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov accelerated gradient descent (PAGD) method. We discretize these problems with a Fourier collocation method in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent (PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account. Benchmark computations of five different schemes for the PFC and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver) perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision. For the PFC, the results are not as conclusive as in the FCH experiments, which, we believe, is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation. We also discuss some practical matters in applying the PAGD. We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters. The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.

Phase-Field Crystal Studies on Grain Boundary Migration, Dislocation Behaviors, and Topological Transition under Tension of Square Polycrystals

In this paper, the tensile deformation behaviors of polycrystals after relaxation were studied using the phase-field-crystal (PFC) method. Here, the free energy density map characterized the 2D energy distribution of atomic configuration effectively. The application of the Read–Shockley equation distinguished high-energy grain boundary (HEGB) and low-energy grain boundary (LEGB) in large-angle grain boundary (LAGB), and they demonstrated different migration behaviors at the early and later stages. The behaviors of small-angle grain boundary (SAGB), including its migration and grains’ rotation, were also studied. Two different mechanisms of dislocation emission and absorption were explored, which demonstrates the possibility of dislocation elevating interfacial energy. The simulated results on the topological transition of grain boundaries prompted us to propose the thinking about the applications of the Neumann–Mullins law and Euler formula.

Improved time integration for phase‐field crystal models of solidification

AbstractWe optimize a numerical time‐stabilization routine for a class of phase‐field crystal (PFC) models of solidification. By numerical experiments, we demonstrate that our simple approach can improve the accuracy of underlying time integration schemes by a few orders of magnitude. We investigate different time integration schemes. Moreover, as a prototypical example for applications, we extend our numerical approach to a PFC model of solidification with an explicit temperature coupling.

Pattern Formation under Deep Supercooling by Classical Density Functional-Based Approach.

Solidification patterns during nonequilibrium crystallization are among the most important microstructures in the natural and technical realms. In this work, we investigate the crystal growth in deeply supercooled liquid using the classical density functional-based approaches. Our result shows that the complex amplitude expanded phase-field crystal (APFC) model containing the vacancy nonequilibrium effects proposed by us could naturally reproduce the growth front nucleation (GFN) and various nonequilibrium patterns, including the faceted growth, spherulite, symmetric and nonsymmetric dendrites among others, at the atom level. Moreover, an extraordinary microscopic columnar-to-equiaxed transition is uncovered, which is found to depend on the seed spacing and distribution. Such a phenomenon could be attributed to the combined effects of the long-wave and short-wave elastic interactions. Particularly, the columnar growth could also be predicted by an APFC model containing inertia effects, but the lattice defect type in the growing crystal is different due to the different types of short-wave interactions. Two stages are identified during the crystal growth under different undercooling, corresponding to diffusion-controlled growth and GFN-dominated growth, respectively. However, compared with the second stage, the first stage becomes too short to be noticed under the high undercooling. The distinct feature of the second stage is the dramatic increments of lattice defects, which explains the amorphous nucleation precursor in the supercooled liquid. The transition time between the two stages at different undercooling is investigated. Crystal growth of BCC structure further confirms our conclusions.

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.

Consistent representation of vapor phases in phase field crystal dynamics

Accurate exploration of processes involving interactions among defects, voids, and density shrinkage in rapid solidification requires the ability to simulate phase transformations over large density ranges. We begin this work by presenting a number of numerical artifacts that arise in previous attempts to model the dynamics of solid-liquid-vapor interactions using phase field crystal (PFC) models based on a single density field coupled to its mean field. We then propose a new PFC formalism for modeling solid-liquid-vapor systems that self-consistently couples two components of the density field, one varying on the usual atomic length-scales, the other on scales much greater than the atomic lattice constant. It is shown that the new formalism is free of the aforementioned artifacts exhibited by previous PFC models. We generalize this new solid-liquid-vapor PFC model to alloys and demonstrate its utility through the nucleation of voids in both a fully solid material and during solidification into a liquid.