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In this article, we present the behavior of two‐mode phase field crystal (2MPFC) method under a concentration dependent deformation. A mixed finite element formulation is proposed for the 2MPFC method that solves a 10th‐order parabolic equation. Lithium concentration diffusion in the electrode particle is captured by the Cahn–Hilliard (CH) equation and the host electrode material, Li$_{x}$Mn$_{2}$O$_{4}$ (LMO), which has a face‐centered cubic (fcc) lattice structure, is modeled using 2MPFC. The coupling between 2MPFC and CH models brings about the concentration dependent deformation in the polycrystalline LMO electrode particle. The atomistic dynamics is assumed to operate on a faster time‐scale compared to the diffusion of lithium, thereby both the 2MPFC and CH models evolve on two different time‐scales. The coupled 2MPFC–CH system models the diffusion induced grain boundary migration in LMO capturing the charging and discharging state of the battery.

Since the edge-to-edge matching relationship of close-packed planes on an incoherent interphase boundary was found, the one-dimensional ledge migration mechanism has been put forward. However, owing to the lack of direct experimental evidence, the existence of the one-dimensional ledge is still questioned and it is thus usually treated as just an assumption. In this study, focusing on the existence of one-dimensional ledges and the migration mechanism of incoherent interphase boundaries, an atomic scale investigation on the migration of incoherent interphase boundaries in a body- to face-centered cubic transformation has been carried out using the phase-field crystal model. Simulation results demonstrated the presence of one-dimensional ledges on incoherent interphase boundaries, but only on those boundaries with high atomic densities. The simulation results further showed that the interphase boundaries with one-dimensional ledges migrate as a result of the nucleation and extension of the one-dimensional ledge, similar to the mechanism for two-dimensional ledges; meanwhile the interphase boundaries without one-dimensional ledges migrate according to a continuous mechanism by random atomic jumping. Because it is difficult for one-dimensional ledges to nucleate under low driving forces, interphase boundary migration based on the one-dimensional ledge mechanism is slower than that based on the continuous mechanism. This study reveals the structures and mechanisms of complex transitions of incoherent interphase boundaries and can aid a deeper understanding of solid phase transformations.

In this paper, the second-order scalar auxiliary variable approach in time and linear finite element method in space are employed for solving the Cahn-Hilliard type equation of the phase field crystal model. The energy stability of the fully discrete scheme and the boundedness of numerical solution are studied. The rigorous error estimates of order \begin{document}$ O(\tau^2+h^2) $\end{document} in the sense of \begin{document}$ L^2 $\end{document} -norm is derived. Finally, some numerical results are given to demonstrate the theoretical analysis.

In this paper, we consider time marching numerical schemes to solve a two-mode phase field crystal model with face-centered-cubic ordering. We propose two type, linear, and unconditionally energy stable schemes by combining the recently developed IEQ and SAV approaches with the stabilization technique, where several extra stabilizing terms are added to enhance the stability and keep the required accuracy while using large time steps. We further prove the unconditional energy stabilities of the developed schemes rigorously. Through the comparisons with some other prevalent schemes like the fully-implicit, semi-implicit, and convex-splitting schemes for some benchmark numerical examples in 2D and 3D, we demonstrate the stability and the accuracy of the schemes numerically.

Electromigration (EM) is the motion of lattice atoms under high-density electric current. It is an important mechanism for structural changes in nanoelectronic devices and a major contributor to the electroplastic effect in structural metals. Recently, a phase-field-crystal (PFC) model for studying EM in metals was developed and shown to successfully capture many important EM-driven structural evolutions at experimentally relevant timescales. In this work, connections between the previous PFC EM model and existing EM theories are established. It is shown that the PFC model can be linked to both the electron-density-based quantum EM theory and the drift-diffusion-based continuum EM theory, therefore filling an important gap in the theoretical and computational study of EM-related phenomena. The numerical method for implementing the PFC EM model is discussed in detail. The well-established Blech effect is quantitatively reproduced using the model.

Most field theories for active matter neglect effects of memory and inertia. However, recent experiments have found inertial delay to be important for the motion of self-propelled particles. A major challenge in the theoretical description of these effects, which makes the application of standard methods very difficult, is the fact that orientable particles have both translational and orientational degrees of freedom which do not necessarily relax on the same time scale. In this work, we derive the general mathematical form of a field theory for soft matter systems with two different time scales. This allows to obtain a phase field crystal model for polar (i.e., nonspherical or active) particles with translational and orientational memory. Notably, this theory is of third order in temporal derivatives and can thus be seen as a spatiotemporal jerky dynamics. We obtain the phase diagram of this model, which shows that, unlike in the passive case, the linear stability of the liquid state depends on the damping coefficients. Moreover, we investigate sound waves in active matter. It is found that, in active fluids, there are two different mechanisms for sound propagation. For certain parameter values and sufficiently high frequencies, sound mediated by polarization waves experiences less damping than usual passive sound mediated by pressure waves of the same frequency. By combining the different modes, acoustic frequency filters based on active fluids could be realized., Comment: 15 pages, 4 figures

In this paper, we propose a linear, unconditionally energy stable and second-order (in time) numerical scheme based on a convex splitting scheme and the semi-implicit spectral deferred correction (SISDC) method for the phase field crystal equation. The convex splitting scheme we use is linear, uniquely solvable and unconditionally energy stable but is of first-order, so we take the SISDC method to improve the rate of convergence. The resulted scheme inherits the advantages of the convex splitting scheme and thus leads to linear equations at each time step, which is easy to implement. We also prove that the scheme is unconditionally weak energy stable and of second-order accuracy in time. Numerical experiments are presented to validate the accuracy, efficiency and energy stability of the proposed numerical strategy.

An adaptive BDF2 implicit time-stepping method is analyzed for the phase field crystal model. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels if the time-step ratios $r_k:=\tau_k/\tau_{k-1}, Comment: 29 pages, 18 figures, 2 tables

Interface energy and kinetic coefficient of crystal growth strongly depend on the face of the crystalline lattice. To investigate the kinetic anisotropy and velocity of different crystallographic faces we use the hyperbolic (modified) phase field crystal model which takes into account atomic density (as a slow thermodynamic variable) and atomic flux (as a fast thermodynamic variable). Such model covers slow and rapid regimes of interfaces propagation at small and large driving forces during solidification. In example of BCC crystal lattice invading the homogeneous liquid, dynamical regimes of advancing front propagating along the selected crystallographic directions are studied. The obtained velocity and the velocity sequences for different faces are compared with known results., Comment: Accepted manuscript to Journal of Crystal Growth

We study the behavior of the commensurate-incommensurate (CI) transitions by using a phase field crystal model. The model is capable of modeling both elastic and plastic deformation and can simulate the evolution of the microstructure of the material at the atomic scale and the diffusive time scale, such as for adsorbed monolayer. Specifically, we study the behavior of the CI transitions as a function of lattice mismatch and the amplitude of substrate pinning potential. The behavior of CI phase transitions is revealed with the increase of the amplitude of pinning potential in some certain lattice mismatches. We find that for the negative lattice mismatch absorbed monolayer undergoes division, reorganization and displacement as increasing the amplitude of substrate pinning potential. In addition, for the positive mismatch absorbed monolayer undergoes a progress of phase transformation after a complete grain is split. Our results accord with simulations for atomic models of absorbed monolayer on a substrate surface.

The present work focuses on the development of a relatively simple phase field crystal model for materials with nanoscale porous inclusions. We found that the pore’s main effect is to act as a nucleation agent, promoting crystallization of material at the pore’s interface, followed by micro-structural evolution of the solid in the supercooled liquid. Details of the crystal around the pore are investigated in terms of the pore radius and density of material outside the pore. Moreover, details of the pore-material interface is investigated through the interfacial tension and pressure. Finally, the model is extended to investigate the effect of multiple pores on the kinetics of crystallization.

Migration of phase boundaries in crystalline solids eliminates one set of lattice sites and establishes another. Using a combination of phase field crystal modeling and crystallographic analysis, we present here a complete atomistic description of the migration mechanism of a high-index planar interface during a diffusional hexagon to square phase transformation. In particular we show that a terrace-step interface advances macroscopically in the form of growth ledges, while microscopically its migration occurs by opposite shearing on the terraces and a one-to-two splitting of lattice sites, giving a new class of lattice site correspondence and superabundant vacancies. In addition, a new approach capable of finding a critical nucleus with atomic resolution is developed by combining the phase field crystal energetics with the free-end nudged elastic band algorithm.

In this paper, we develop a series of linear, unconditionally energy stable numerical schemes for solving the classical phase field crystal model. The temporal discretizations are based on the first order Euler method, the second order backward differentiation formulas (BDF2) and the second order Crank-Nicolson method, respectively. The schemes lead to linear elliptic equations to be solved at each time step, and the induced linear systems are symmetric positive definite. We prove that all three schemes are unconditionally energy stable rigorously. Various classical numerical experiments in 2D and 3D are performed to validate the accuracy and efficiency of the proposed schemes.

Dissertation, RWTH Aachen University, 2019; Aachen 1 Online-Ressource (167 Seiten) : Illustrationen, Diagramme (2019). = Dissertation, RWTH Aachen University, 2019, Steel has a wide range of excellent but flexible properties, which makes this material so versatile and popular. The origin of these features can be understood by analysing the microstructure and crystal structure, which can consequently be used to develop new materials for specific purposes. A precise knowledge of properties and processes on these small scales is therefore necessary, which explains the ongoing research of steel despite its deployment for over thousand years. In this work, elastic effects and their influences on the microstructure are presented by utilising a variety of scale bridging modelling approaches. Nonlinear elasticity appears at large deformations. A full description of this material behaviour is then challenging as a large number of parameters is needed. The phase field crystal method (PFC method) is a modelling technique that still resolves atomistic structures on diffusive timescales and is able to describe elastic behaviour intrinsically. By expressing the PFC model with amplitude equations and combining the result with continuum mechanics, an elegant and compact way of describing nonlinear elasticity in crystal structures is found. Important is the understanding, that the phase field crystal model uses a Eulerian description. Thus, only Eulerian strain tensors are suited to describe a nonlinear elastic deformation in the PFC model. Further investigations of an uniaxial strained BCC system shows, that only the Lagrange-Karni-Reiner strain tensor provides a compact description of nonlinear elasticity. Additional comparisons with the Birch-Murnaghan equation of state and ab initio simulations indicate, that the PFC results are indeed correct and quite accurate, even when only one-mode approximations of the amplitude equations are used. Elastic effects play also a role during microstructure evolution and are in this work considered in a transformation from austenite to bainite. The phase evolution is described by a multi phase field model and a distinction between lower and upper bainite is made. The lower bainite formation is modelled via a displacive transformation, which produces a bainitic ferrite sub-unit that is supersaturated with carbon. Cahn-Hilliard modelling of the carbon diffusion leads to phase separation and uphill diffusion and further to precipitation of cementite inside the sub-unit. Mechanical contributions are gathered in the eigenstrains of the different phases and in the boundary conditions of the simulation, which allows an imitation of manufacturing conditions. By compression and tension of the system during isothermal transformation, the phase distribution in the final product changes, especially the amount of cementite. Macroscopic properties of a steel with bainite microstructure are therefore influenced during the manufacturing process by elasticity. Additionally, the onset of transformation plasticity in the elastic regime is addressed and nonlinear behaviour is not found. The upper bainite transformation uses paraequilibrium conditions at the phase boundary between austenite and ferrite. Thus, the cementite precipitates correctly in the austenite near the ferrite phase, due to a large difference in diffusion coefficients. The growth rate of bainite is furthermore strongly influenced by the cementite formation. The precipitation of cementite results in an increase of growth rate, because the carbide acts as a carbon sink. Another interface in microstructures is a grain boundary. The interaction of precipitates with grain boundaries is investigated on the mesoscale and on the microscale. The mesoscale perspective treats the grain boundary as an elastically softer/harder layer compared to the surrounding matrix, which leads to elastic energy changes near the interface when a precipitate is close. This elastic energy alteration leads to solubility limit adjustments. On the microscale, the shear-coupled motion of a grain boundary and its interaction with precipitates, that possess an eigenstrain, is considered. The elastic interaction energy density is analytically derived and numerically evaluated. The result is an elastic energy reduction due to the interaction. A maximisation of the energy-gain leads to grain boundary shape changes in order to accommodate the inclusions. The precipitates, e.g. cementite, are "attracted" to the interface and stay close to it; this can even lead to break-ups of the boundary. A penetration of the grain boundary is energetically unfavourable. Similar to the mesoscopic case, the decrease of the elastic energy leads to a significant solubility limit reduction, examined by an exemplary data set of cementite. A comparison of meso- and microscale results leads to the conclusion, that the solubility limit decrease, due to the interaction between shear-coupled motion of the grain boundary and precipitates, can be described by an effective elastic parameter on larger scales., Published by Aachen

We consider simple mean field continuum models for first order liquid–liquid demixing and solid–liquid phase transitions and show how the Maxwell construction at phase coexistence emerges on going from finite-size closed systems to the thermodynamic limit. The theories considered are the Cahn–Hilliard model of phase separation, which is also a model for the liquid-gas transition, and the phase field crystal model of the solid–liquid transition. Our results show that states comprising the Maxwell line depend strongly on the mean density with spatially localized structures playing a key role in the approach to the thermodynamic limit.

Phase field crystal model is used for studying the grain boundary premelting system under an applied strain at high temperature. The evolution of the dislocation configuration and proliferation and annihilation process of the dislocation and also the characteristic of the morphology of the premelting regions are observed. The results show that the extension premelting domain of the dislocation induced by the applied strain can be regarded as a new phase region (the softening crystal phase domain, SCPD). Due to the lattice softening of the SCPD, two alternant slipping systems exist in it. The cycle processes of the dislocation proliferation by blocking and annihilation by gliding in the SCPD can be regarded as two processes of dynamic recoveries. The evolution of the dislocation configuration and the microscopic mechanism of the cycle dynamic recovery-controlled dislocation at high temperature under plastic deformation in SCPD can well be revealed by the improved KM model.

In this paper, we extend the phase field crystal model to the modified phase field crystal model which includes diffusive dynamics, elastic interaction, and stochastic noises effect. We present a second-order accurate semi-implicit finite difference scheme for the modified phase field crystal model. The resulting scheme is based on the stabilized splitting method and Crank–Nicolson method. The nonlinear term is linearized by the Taylor series. The resulting scheme is linear at each time step, which makes it easy to be implemented and efficient to be solved by using the linear multigrid solver. We prove that the resulting scheme is unconditionally energy stable. Various numerical experiments are conducted to verify the accuracy and efficiency of our proposed algorithm.

This paper introduces improvements to an amplitude expansion of the phase-field-crystal model. An auxiliary field describing local grain rotation is introduced and used to enable the adaptive mesh to be coarsened in all grains, regardless of their orientation. Only a Cartesian representation of the amplitude equations is used.

The phase-field crystal (PFC) method is a promising computational model with atomistic resolution and diffusive time scale. In this work, the Fourier-spectral-method (FSM) scheme was developed for evaluating the PFC free energy of a system subjected homogeneous deformation. This scheme addresses the complication where, in numerical implementation of FSM using discrete Fourier transform (DFT), the discretized data may no longer lie along the directions of the Cartesian basis due to deformation. In this scheme, the real-space coordinate transformation is employed so that the (continuous) Fourier transform is performed on the function of the undeformed coordinates. This transformation allows straightforward DFT implementation because the sampling at the undeformed configuration is unaffected by the deformation. This scheme is also shown to be applicable to both the original PFC model and a “CDFT”-type PFC model containing a two-body correlation function.

International audience; We study a Phase-Field-Crystal model described by a free energy functional involving second order derivatives of the order parameter in a periodic setting and under a fixed mass constraint. We prove a Γ-convergence result in an asymptotic thin-film regime leading to a reduced 2-dimensional model. For the reduced model, we prove necessary and sufficient conditions for the global minimality of the uniform state. We also prove similar results for the Ohta-Kawasaki model.

A general procedure to investigate the elastic response and calculate the elastic constants of stressed and unstressed materials through continuum field modeling, particularly the phase field crystal (PFC) models, is presented. It is found that for a complete description of system response to elastic deformation, the variations of all the quantities of lattice wave vectors, their density amplitudes (including the corresponding anisotropic variation and degeneracy breaking), the average atomic density, and system volume should be incorporated. The quantitative and qualitative results of elastic constant calculations highly depend on the physical interpretation of the density field used in the model, and also importantly, on the intrinsic pressure that usually pre-exists in the model system. A formulation based on thermodynamics is constructed to account for the effects caused by constant pre-existing stress during the homogeneous elastic deformation, through the introducing of a generalized Gibbs free energy and an effective finite strain tensor used for determining the elastic constants. The elastic properties of both solid and liquid states can be well produced by this unified approach, as demonstrated by an analysis for the liquid state and numerical evaluations for the bcc solid phase. The numerical calculations of bcc elastic constants and Poisson's ratio through this method generate results that are consistent with experimental conditions, and better match the data of bcc Fe given by molecular dynamics simulations as compared to previous work. The general theory developed here is applicable to the study of different types of stressed or unstressed material systems under elastic deformation., 14 pages, 4 figures. To be published in Phys. Rev. B

Our aim in this paper is to study the well-posedness for the phase-field-crystal model with logarithmic nonlinear terms. More precisely, we prove the existence and uniqueness of variational solutions, based on a variational inequality.

Order-disorder transition is an attractive topic in the research field of phase transformation. However, how to describe order-disorder transitions on atomic length scales and diffusional time scales is still challenging. Inspired from high-resolution transmission electron microscopy, we proposed an approach to describe ordered structures by introducing an order parameter into the original phase-field-crystal model to reflect the atomic potential distribution. This new order parameter contains information about kinds of atoms, showing that different kinds of sublattices in ordered structures can be distinguished by the amplitude of the order parameter. Two case studies, growth of ordered precipitations and evolution of antiphase domains, are also presented to demonstrate the capabilities of this approach.

Fronts dynamics of periodic crystalline state, which invades the homogeneous state (liquid phase), are analysed. These fronts are considered as traveling waves of atomic density amplitudes. The propagation of amplitudes is described by the hyperbolic equation of an extended Allen-Cahn type for which the complete set of analytical traveling-wave solutions are obtained by tanh-method. The set of solutions includes previously known traveling waves for the parabolic Allen-Cahn equation of both extended and standard form.

Understanding and controlling the properties and dynamics of topological defects is a lasting challenge in the study of two-dimensional materials, and is crucial to achieve high-quality films required for technological applications. Here grain boundary structures, energies, and dynamics of binary two-dimensional materials are investigated through the development of a phase field crystal model that is parameterized to match the ordering, symmetry, energy and length scales of hexagonal boron nitride. Our studies reveal some new dislocation core structures for various symmetrically and asymmetrically tilted grain boundaries, in addition to those obtained in previous experiments and first-principles calculations. We also identify a defect-mediated growth dynamics for inversion domains governed by the collective atomic migration and defect core transformation at grain boundaries and junctions, a process that is related to inversion symmetry breaking in binary lattice., Comment: 5 pages, 4 figures, 1 supplemental material

We use the simplified string method in order to examine two dimensional heterogeneous nucleation at a wall and on a substrate. The material is described by a phase field crystal model and the influence of the wall or substrate is included by an external potential. Tuning the external potential we show that we can control the contact angle in equilibrium and misfit to a substrate. The nucleation barrier is reduced by a wall, but cannot be explained by classical nucleation theory due to non-classical nucleation paths. For small misfits a substrate also decreases the nucleation barrier, while large misfits increases the nucleation barrier.

In this work, we will propose an adaptive time step method for simulating the dynamics of the phase field crystal (PFC) model. The numerical simulation of the PFC model needs long time to reach steady state, and then large time-stepping method is necessary. Unconditionally energy stable schemes are used to solve the PFC model. The time steps are adaptively determined based on the time derivative of the corresponding energy. It is found that the use of the proposed time step adaptivity cannot only resolve the steady state solution, but also the dynamical development of the solution efficiently and accurately. The numerical experiments demonstrate that the CPU time is significantly saved for long time simulations.

In recent years, a fundamental understanding of solidification and its behavior has been gained through molecular dynamics simulations and the phase-field method, the first of which is limited to short time scales and the latter of which does not represent interface and elastoplastic properties accurately. Recently, the phase-field-crystal (PFC) method, a continuum method operating on atomistic length scales and diffusive time scales, has helped bridge the multiple scale gap between molecular dynamics and phase field. This review surveys the advances of PFC models in the context of various solidification phenomena.