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In this article, we propose a linear, second-order, semi-discrete time stepping scheme for the phase field crystal equation based on generalized positive auxiliary variable (GPAV) approach. This scheme reduces the operation counts by half compared to the GPAV and the scalar auxiliary variable methods in previous works. We prove the unconditionally energy stability and provide bounds and error estimates for the field function. Numerical experiments are carried out to verify our theoretical results and demonstrate the robustness and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

We present a C0 interior penalty finite element method for the sixth‐order phase field crystal equation. We demonstrate that the numerical scheme is uniquely solvable, unconditionally energy stable, and convergent. We remark that the novelty of this paper lies in the fact that this is the first C0 interior penalty finite element method developed for the phase field crystal equation. Additionally, the error analysis presented develops a detailed methodology for analyzing time dependent problems utilizing the C0 interior penalty method. We furthermore benchmark our method against numerical experiments previously established in the literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [Display omitted] [ABSTRACT FROM AUTHOR]

We introduce a phase field crystal (PFC) model for particles with n -fold rotational symmetry in two dimensions. Our approach is based on a free energy functional that depends on the reduced one-particle density, the strength of the orientation, and the direction of the orientation, where all these order parameters depend on the position. The functional is constructed such that for particles with axial symmetry (i.e. n = 2) the PFC model for liquid crystals as introduced by Löwen (2010 J. Phys.: Condens. Matter 22 364105) is recovered. We discuss the stability of the functional and explore phases that occur for 1 ⩽ n ⩽ 6. In addition to isotropic, nematic, stripe, and triangular order, we also observe cluster crystals with square, rhombic, honeycomb, and even quasicrystalline symmetry. The n -fold symmetry of the particles corresponds to the one that can be realized for colloids with symmetrically arranged patches. We explain how both, repulsive as well as attractive patches, are described in our model. [ABSTRACT FROM AUTHOR]

Different methods for evaluation of displacement and strain fields based on phase field crystal (PFC) simulations are shown. Methods originally devised for molecular dynamics (MD) simulations or analysis of high-resolution microscopy images are adapted to a PFC setting, providing access to displacement and strain fields for systems of discrete atoms, such as in MD, as well as to continuous deformation fields. The latter being achieved by geometrical phase analysis. As part of the study, the application of prescribed non-affine deformations in a 3D structural PFC (XPFC) setting is demonstrated as well as an efficient numerical scheme for evaluation of PFC phase diagrams, such as, for example, those required to stabilize solid/liquid coexistence. The present study provides an expanded toolbox for using PFC simulations as a versatile numerical method in the analysis of material behavior at the atomic scale. [ABSTRACT FROM AUTHOR]

Grain boundary interaction with second-phase particles having different degrees of coherency is investigated using the phase field crystal (PFC) method. Both the enveloping and pass-through mechanisms are studied with regards to grain boundary pressure, passage time and interface evolution. It is found that coherent particles exert a stronger retardation effect on grain boundaries compared to incoherent particles, with regards to both pressure and time, but also that this benefit is limited to a small range of misfit values. The simulations also show that the mobility is not a constant during particle passage, as commonly assumed, which means that grain boundary pressure cannot easily be extracted from the grain boundary velocity. Furthermore, the complex evolution of the pass-through mechanism and the transient behavior for intermediate coherencies is also investigated. The highest drag force is found to occur at the switching point between enveloping and pass-through. As part of the study, the advantages of using PFC for this type of analyses are also highlighted.

Rapid solidification leads to unique microstructural features, where a less studied topic is the formation of various crystalline defects, including high dislocation densities, as well as gradients and splitting of the crystalline orientation. As these defects critically affect the material's mechanical properties and performance features, it is important to understand the defect formation mechanisms, and how they depend on the solidification conditions and alloying. To illuminate the formation mechanisms of the rapid solidification induced crystalline defects, we conduct a multiscale modelling analysis consisting of bond-order potentialbased molecular dynamics (MD), phase field crystal-based amplitude expansion simulations, and sequentially coupled phase field-crystal plasticity simulations. The resulting dislocation densities are quantified and compared to past experiments. The atomistic approaches (MD, PFC) can be used to calibrate continuum level crystal plasticity models, and the framework adds mechanistic insights arising from the multiscale analysis. [ABSTRACT FROM AUTHOR]

A two-mode phase field crystal (PFC) model is employed to investigate the equilibrium configurations of a range of grain boundaries in fcc-structured materials. A total of 80 different symmetrical tilt grain boundaries are evaluated by PFC simulations in 3D and the results are shown to agree well with data taken from the literature, both regarding the variation of grain boundary energy and also in terms of the resulting grain boundary structures. This verification complements existing PFC studies which are almost exclusively focused either on grain boundaries found in 2D systems or in bcc lattices in 3D. The present work facilitates application of PFC in the analysis of grain boundary mechanics in an extended range of materials, in particular such mechanics that take place at extended time scales not tractable for molecular dynamics (MD) simulations. In addition to the verification of predicted grain boundary energies and structures, wavelet transforms of the density field are used in the present work to obtain phase fields from which it is possible to identify grain boundary fluctuations that provide the means to evaluate grain boundary stiffness based on the capillarity fluctuation method. It is discussed how PFC provides benefits compared to alternative methods, such as MD simulations, for this type of investigations. [ABSTRACT FROM AUTHOR]

Different methods for evaluation of displacement and strain fields based on phase field crystal (PFC) simulations are shown. Methods originally devised for molecular dynamics (MD) simulations or analysis of high-resolution microscopy images are adapted to a PFC setting, providing access to displacement and strain fields for systems of discrete atoms, such as in MD, as well as to continuous deformation fields. The latter being achieved by geometrical phase analysis. As part of the study, the application of prescribed non-affine deformations in a 3D structural PFC (XPFC) setting is demonstrated as well as an efficient numerical scheme for evaluation of PFC phase diagrams, such as, for example, those required to stabilize solid/liquid coexistence. The present study provides an expanded toolbox for using PFC simulations as a versatile numerical method in the analysis of material behavior at the atomic scale.

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, LixMn2O4 (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. [ABSTRACT FROM AUTHOR]

In this paper we present two unconditionally energy stable finite difference schemes for the modified phase field crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic phase field crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully second-order scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes. © 2013 Elsevier Inc.

In this paper we present two unconditionally energy stable finite difference schemes for the modified phase field crystal (MPFC) equation, a sixth-order nonlinear damped wave equation, of which the purely parabolic phase field crystal (PFC) model can be viewed as a special case. The first is a convex splitting scheme based on an appropriate decomposition of the discrete energy and is first order accurate in time and second order accurate in space. The second is a new, fully second-order scheme that also respects the convex splitting of the energy. Both schemes are nonlinear but may be formulated from the gradients of strictly convex, coercive functionals. Thus, both are uniquely solvable regardless of the time and space step sizes. The schemes are solved by efficient nonlinear multigrid methods. Numerical results are presented demonstrating the accuracy, energy stability, efficiency, and practical utility of the schemes. In particular, we show that our multigrid solvers enjoy optimal, or nearly optimal complexity in the solution of the nonlinear schemes. © 2013 Elsevier Inc.

We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order O(Δt2 + N−m), where Δt, N, and m are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes. [ABSTRACT FROM AUTHOR]

Despites of some efforts in deformation simulations by the Phase field crystal (PFC) method, simulations of phase transitions and plasticity of crystals under continuous deformations are still lack and some related fundamental issues remain open as well, for example the definition of stresses and the non-zero stresses of unstrained system. In the present work, we propose a deformation simulation method which conforms to the well-established framework of the PFC model. In contrast to traditional deformation simulation methods, our method could naturally mimic melting/freezing, solid-solid phase transition and plasticity of materials under continuous deformations without any additional parameters. Within the frameworks of our method, the stress is well-defined and isothermal-isobaric simulation method is developed. The isothermal-isobaric simulation method enables us to overcome the drawback of previous PFC simulations, for example the nonzero stress of unstrained system. Numerical examples given in present work confirm our conclusions. Particularly, the physical natures of the plasticity are uncovered at the temporal and spatial scale accessible to the PFC method. • Deformation simulations are performed by PFC method without additional parameters. • Multiple phase transitions and plasticity under deformations are naturally simulated. • Stress defined well conforms to present PFC formalisms. • Isochoric and isobaric simulation techniques are delicately developed. • Equilibrium conditions and instabilities are discussed within the established framework. [ABSTRACT FROM AUTHOR]

Abstract Using the advantage of phase field crystal model in microstructure evolution, we simulated the details of two-dimensional triangular phase on atomic scale and diffusion time scale. The results show that the annihilation process of grain boundaries can be divided into three stages. The energy changes of the system at different stages are accurately calculated and the process is quantitatively analyzed. By comparing the atomic density distributions at different temperatures and orientation angles, it is shown that the grain boundary is sensitive to temperature. Within the range of low-angle, the increase of orientation angle inhibits the occurrence of annihilation and decreases the grain boundary migration rate. Under the applied stress, the orientation angle has a certain response relationship with the yield strength of the material. This work has some practical significance in the influence of microstructure on the mechanical properties of materials. Graphical abstract Image 107272 Highlights • Analyzing the grain boundary structure of a pure material at low-angle symmetrical tilt grain boundaries. • The annihilation process of the grain boundary during the isovolumic assumption deformation of the crystal was revealed. • Combined with the change of system energy, the influence of parameters on grain boundary annihilation was analyzed. [ABSTRACT FROM AUTHOR]

(a) Phase field and (b) phase field crystal simulation of grain boundary migration and grain rotation, and (c) schematic of grain boundary dislocation near triple junction point O. [Display omitted] • GB migration and grain rotation are treated by a hybrid phase field approach. • A newly proposed plasticity mechanism of nanograin confirmed by PFC simulation. • GB dislocation climb and absorption at triple-junction points are monitored. • Misorientation angle of grains are quantitatively verified by Frank-Bilby theory. • GB evolution observed in situ TEM were reproduced by multi-scale simulation. 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. [ABSTRACT FROM AUTHOR]

We study heat transport across individual grain boundaries in suspended monolayer graphene using extensive classical molecular dynamics (MD) simulations. We construct bicrystalline graphene samples containing grain boundaries with symmetric tilt angles using the two-dimensional phase field crystal method and then relax the samples with MD. The corresponding Kapitza resistances are then computed using nonequilibrium MD simulations. We find that the Kapitza resistance depends strongly on the tilt angle and shows a clear correlation with the average density of defects in a given grain boundary, but is not strongly correlated with the grain boundary line tension. We also show that quantum effects are significant in quantitative determination of the Kapitza resistance by applying the mode-by-mode quantum correction to the classical MD data. The corrected data are in good agreement with quantum mechanical Landauer-Bütticker calculations. [ABSTRACT FROM AUTHOR]

A phase field crystal (PFC) model is employed to study morphology evolution of nanoheteroepitaxy and misfit dislocation generation when applied with enhanced supercooling, lattice mismatch and substrate vicinal angle conditions. Misfit strain that rises due to lattice mismatch causes rough surfaces or misfit dislocations, deteriorates film properties, hence, efforts taken to reveal their microscopic mechanism are significant for film quality improvement. Uniform islands, instead of misfit dislocations, are developed in subcritical thickness film, serving as a way of strain relief by surface mechanism. Misfit dislocations generate when strain relief by surface mechanism is deficient in higher supercooling, multilayers of misfit dislocations dominate, but the number of layers reduces gradually when the supercooling is further enhanced. Rough surfaces like islands or cuspate pits are developed which is ascribed to lattice mismatch, multilayers of misfit dislocations generate to further enhance lattice mismatch. Layers of misfit dislocations generate at a thickening position at enhanced substrate vicinal angle, this further enhancing the angle leading to sporadic generation of misfit dislocations. [ABSTRACT FROM AUTHOR]

The excess interaction term of classical density functional theory (cDFT) represents the effect of the neighboring atoms through the two-body direct correlation function (DCF). The DCF plays a crucial role in cDFT and the first peak of DCF is important in the phase field crystal (PFC) model. Unlike the reciprocal-space formulation, the real-space formulation of cDFT would not be restricted to only periodic systems, and thus, it has considerable potential for exploring non-periodic complex phenomena at the atomic scale. An accurate representation of the DCF is very important for the accurate and efficient implementation of cDFT. In this work, we propose a two-step process for systematically deriving the real-space DCF for any crystal structure. In the first step, we fit a set of rational functions to the experimental or molecular simulation data in reciprocal space, as shown in a previous study (Pisutha-Arnond et al., 2013). Then, in the second step, we obtain the analytical expression for the DCF in real space using an inverse Fourier–Bessel transform. One advantage of this method is that the functional fit to the DCF in reciprocal space is accomplished automatically. The proposed analytical technique is validated by comparing against numerically obtained DCF for bcc iron. This technique can be used to obtain a real-space DCF for any material. Finally, the domain of influence for DCF is investigated by integrating the DCF over a 3D domain. [ABSTRACT FROM AUTHOR]

[Display omitted] • Grain boundary dislocation core extension is simulated by PFC method. • Decomposition of dislocation core reaching maximum critical width occurs. • Extended dislocation core is decomposed into three dislocation cores. • A energy equation of extended dislocation core system is given and analyzed. The evolution of grain boundary dislocation structure under biaxial tension–compression loading is simulated by the phase field crystal (PFC) method for small angle symmetric tilt grain boundary (STGB) of body center cubic (BCC) bi-crystal. Extension of the STGB dislocation core under the strain and the extended width of the core with the strain increasing are studied. In the elastic deformation process under the loading, the extension of the grain boundary dislocation core is accompanied by increasing the strain energy of the system. When the extension of the dislocation core reaches maximum critical width, the extended dislocation core occurs to decompose into three dislocation cores, in which two cores is in no-extended state, the Burgers vector direction of which is in the same direction of the original extended dislocation. The other dislocation core is of the extended type which Burgers vector direction is opposite to the Burgers vector direction of the original extended dislocation core. By establishing the macroscopic energy equation of the extended dislocation core system, it can be well revealed that the elastic strain energy is mainly stored in the extended dislocation defects and the complete lattice of the matrix. At the same time, it also reveals the function relationship between the energy of the extended dislocation core and the width of the extended dislocation during the biaxial loading process of the bi-crystal, as well as the critical condition of energy instability and the change of the energy barrier of the system for the dislocation extension core decomposition. [ABSTRACT FROM AUTHOR]

In this paper we construct an energy stable finite difference scheme for the amplitude expansion equations for the two-dimensional phase field crystal (PFC) model. The equations are formulated in a periodic hexagonal domain with respect to the reciprocal lattice vectors to achieve a provably unconditionally energy stable and solvable scheme. To our knowledge, this is the first such energy stable scheme for the PFC amplitude equations. The convexity of each part in the amplitude equations is analyzed, in both the semi-discrete and fully-discrete cases. Energy stability is based on a careful convexity analysis for the energy (in both the spatially continuous and discrete cases). As a result, unique solvability and unconditional energy stability are available for the resulting scheme. Moreover, we show that the scheme is point-wise stable for any time and space step sizes. An efficient multigrid solver is devised to solve the scheme, and a few numerical experiments are presented, including grain rotation and shrinkage and grain growth studies, as examples of the strength and robustness of the proposed scheme and solver. [ABSTRACT FROM AUTHOR]

Understanding and controlling the selection of crystalline structure are of great significance in crystal growth, in which the preparation of metastable crystal structures is one of the most important issues to be understood. Although plenty of experimental and simulation investigations have been carried out to reveal the formation of metastable structures, the mechanism of metastable phase existence and forming still remains unclear. In the present work, by using the phase field crystal (PFC) model, we further explored the existence and formation mechanism of metastable structure by modeling the crystallization process with prestructured seeds in 2D. We found that the metastable structure can be survived in a certain region of the phase diagram. Our results show that the selection of final structures is determined not only by the thermodynamic driving force but also by competitions between the metastable and stable structures. [ABSTRACT FROM AUTHOR]

In this paper we present a numerical method for hydrodynamic models that arise from time dependent density functional theories of freezing. The models take the form of compressible Navier–Stokes equations whose pressure is determined by the variational derivative of a free energy, which is a functional of the density field. We present unconditionally energy stable and mass conserving implicit finite difference methods for the models. The methods are based on a convex splitting of the free energy and that ensures that a discrete energy is non-increasing for any choice of time and space step. The methods are applicable to a large class of models, including both local and non-local free energy functionals. The theoretical basis for the numerical method is presented in a general context. The method is applied to problems using two specific free energy functionals: one local and one non-local functional. A nonlinear multigrid method is used to solve the numerical method, which is nonlinear at the implicit time step. The non-local functional, which is a convolution operator, is approximated using the Discrete Fourier Transform. Numerical simulations that confirm the stability and accuracy of the numerical method are presented. [ABSTRACT FROM AUTHOR]

Effects of a preexisting disconnection dipole on the migration of a Σ 5 ( 3 ¯ 1 0 ) / [ 0 0 1 ] grain boundary (GB) at low shear rates have been investigated by using the minimized bulk dissipation phase field crystal model. Simulation results show that the two disconnections play different roles during the process of shear-coupled GB migration via different behaviors regarding heterogeneous nucleation and propagation of disconnections. Specifically, the different roles can be attributed to their structural deviation from the equilibrated crystallographic structure caused by shear loading: one disconnection contributes to the onset of GB migration by heterogeneous nucleation of disconnections, while the other one slows down the corresponding GB migration dynamics through the impediment of homogeneous nucleation in the flat GB plane. [ABSTRACT FROM AUTHOR]

In this paper, we present operator splitting methods for solving the phase field crystal equation which is a model for the microstructural evolution of two-phase systems on atomic length and diffusive time scales. A core idea of the methods is to decompose the original equation into linear and nonlinear subequations, in which the linear subequation has a closed-form solution in the Fourier space. We apply a nonlinear Newton-type iterative method to solve the nonlinear subequation at the implicit time level and thus a considerably large time step can be used. By combining these subequations, we achieve the first- and second-order accuracy in time. We present numerical experiments to show the accuracy and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

In recent years, the phase field crystal (PFC) model has been confirmed as a good candidate to describe grain boundary (GB) structures and their nearby atomic arrangement. To further understand the mechanical behaviours of nanocrystalline materials, strain fields near GBs need to be quantitatively characterized. Using the strain mapping technique of geometric phase approach (GPA), we have conducted strain mapping across the GBs in nanocrystalline grains simulated by the PFC model. The results demonstrate that the application of GPA in strain mapping of low and high angles GBs as well as polycrystalline grains simulated by the PFC model is very successful. The results also show that the strain field around the dislocation in a very low angle GB is quantitatively consistent with the anisotropic elastic theory of dislocations. Moreover, the difference between low angle GBs and high angle GBs is revealed by the strain analysis in terms of the strain contour shape and the structural GB width. [ABSTRACT FROM AUTHOR]

• A kinetic amplitude-expanded phase-field-crystal (KAPFC) model is proposed for exploring microstructure evolutions in crystals under vacancy super- or undersaturation states. • A kinetic disturbance is introduced to allow for necessary rare events during system evolutions. • The kinetic disturbance is systematically formulated within the APFC model framework and validated both analytically and numerically. • No adjustable parameters and additional empirical rules are required when applying such method to explore evolutions of lattice defects, like dislocations and grain boundaries. • For the first time, different evolution behaviors of dislocation loops in irradiated BCC crystals known in experiments are naturally predicted from atom scale. Exploring atomic mechanism of microstructure evolutions at long time still remains a great challenge at present. Amplitude-expanded phase field crystal (APFC) model derived from the classical density functional theory is a promising candidate to access this issue. However, it fails to describe dislocation evolutions in systems under super- or undersaturation states because of the lack of necessary rare events, which hampers its applications in the related realms, such as quick quenching, impacting, irradiating and so on. In this work, we find that the necessary rare events in solids are attributed to the kinetic disturbances due to the motion of local lattice elements instead of the traditional Gaussian noise. The kinetic disturbance is evaluated by the long-time-averaged motions of the local lattice elements as well as a general energy variation principle with respect to the virtual variation of the reciprocal lattice vector. The results by these two approaches are mutually verified. It is demonstrated that the APFC model with the kinetic disturbance converges to the mechanical-equilibrium-condition coupled APFC models at the long time limit and reduces to the original one when the high-energy events are forbidden. Further, the kinetic model is rationalized through theoretical analysis combined with numerical testing on the vacancy-mediated dislocation climb. As a practical application of the kinetic model, we explore the long-time annealing behaviors of dislocation loops in irradiated BCC crystals with different vacancy supersaturations. It is the first time that the vacancy-mediated shrinking, 1-D diffusive motion as well as changing habit plane of interstitial dislocation loops, known in experiments, are correctly predicted at atom scale. [ABSTRACT FROM AUTHOR]

Highlights: [•] We simulate the grain rotation process by using the phase field crystal model. [•] The simulation results are well consistent with the coupled motion theory. [•] The grain rotation with small misorientation is driven by the motion of GBDs. [•] The grain rotation can be well interpreted via the strain field analysis. [Copyright &y& Elsevier]

Deformation behaviors of bicrystalline and nano-polycrystalline structures of various tilt angles and inclination angles in two dimensions are investigated in detail using a two-mode phase field crystal model. The interaction between grain boundary (GB) and dislocation is also examined in bicrystals and nano-polycrystals that both contain asymmetric and symmetric tilt GBs, with energy analysis being carried out to analyze these processes. During deformation simulations, we assume the volume of each simulation cell at every time step is coincident with that of the initial state just before deformation. Our simulation results show that the behaviors of symmetric and asymmetric GBs in bicrystals and nano-polycrystals differ from each other depending on tilt angle and inclination angle. A new dislocation emission mechanism of interest is observed in bicrystals which contain low angle symmetric tilt GBs. Low angle GB has a higher mobility relative to high angle GB in both bicrystalline and nano-polycrystalline structures, as does asymmetric GB to symmetric GB. The generation, motion, pileup and annihilation of dislocations, grain rotation and grain coalescence are observed, which is consistent with the simulation results obtained by molecular dynamics. These simulation results can provide strong guidelines for experimentation. [ABSTRACT FROM AUTHOR]

Abstract: Grain growth experiments on thin metallic films have shown the geometric and topological characteristics of the grain structure to be universal and independent of many experimental conditions. The universal size distribution, however, is found to differ both qualitatively and quantitatively from classical curvature driven models of Mullins type, which reduce grain growth to an evolution of a grain boundary network, with the experiments exhibiting an excess of small grains (termed an “ear”) and an excess of very large grains (termed a “tail”) compared with the models. While a plethora of extensions of the original Mullins model have been proposed to explain these characteristics, none have been successful. In this work, large-scale simulations of a model that resolves the atomic scale on diffusive time scales, the phase field crystal model, are used to examine the complex phenomena of grain growth. The results are in remarkable agreement with the prior experimental results, recovering the characteristic “ear” and “tail” features of the experimental grain size distribution. The simulations also indicate that, while the geometric and topological characteristics are universal, the dynamic growth exponent is not. [Copyright &y& Elsevier]