Phase Field Crystal Research Articles

Scale-bridging phase-field simulations of microstructure responses on nucleation in metals and colloids

In the present studies we investigate the connection between atomistic simulation methods, i.e. molecular dynamics (MD) and phase-field crystal (PFC), to the mesoscopic phase-field methods (PFM). While the first describes the evolution of a system on the basis of motion equations of particles the second uses a Cahn–Hilliard type equation to described an atomic density field and the third grounds on the evolution of continuous local order parameter field. The first aim is to point out the ability of the mesoscopic phase-field method to make predictions of growth velocity at the nanoscopic length scale. Therefore the isothermal growth of a spherical crystalline cluster embedded in a melt is considered. We also show simulation techniques that enable to computationally bridge from the atomistic up to the mesoscopic scale. We use a PFM to simulate symmetric thermal dendrites started at an early stage of solidification related to nucleation. These techniques allow to simulate three dimensional dendrites from the state of nuclei (≈50 Å) converted from MD up to a size of some μm where ternary side-arms start to grow.

Bridging the phase-field and phase-field crystal approaches for anisotropic material systems

In this paper the amplitude representation of the anisotropic phase-field crystal (APFC) model recently proposed as a generalized model for isotropic as well as anisotropic crystal lattice systems is developed. The relationship between the amplitude equations and the standard phase-field model for solidification of pure substances with elasticity effects is derived which provide an explicit connection between the phase-field and APFC models. On the one hand we present a computationally more efficient model and highlight its potential as a bridge between the PFC and phase-field models with anisotropic interface energies and kinetics on the other hand.

Phase field crystal simulation of grain boundary movement and dislocation reaction

The phase field crystal (PFC) model is used to simulate the premelting dislocation movement of the symmetric tilt grain boundary (STGB) under strain action when the system temperature is at far from the melting point and close to the melting point, respectively. The results show a local premelting occurs surrounding the dislocations as the premelting temperature is approached to from below temperature. The premelting dislocations of the STGB can glide under strain action, and the premelting region is a companion for dislocation gliding. The process of STGB decay is very similar at the two high temperature conditions. As premelting presents, it diminishes the gliding resistance for the dislocations and leads to a faster movement of dislocations, and also brings about more energy reduction of the system during the decay process of STGB. In spite of applying strain to these premelting samples in whole decay processes of STGB, the premelting dislocation region does not obviously develop and extend. This indicates that the external strain action does not promote the premelting at the high temperature, and cannot induce more premelting dislocation, which can be owed to the premelting phase around the dislocation exhibit fluid-like properties and to the premelting dislocation easily gliding and relaxing the strain energy; this is in agreement with the results of experiments and molecular dynamics.

Phase-Field Crystal Model for Fe Connected to MEAM Molecular Dynamics Simulations

A recently developed phase-field crystal (PFC) model incorporates elasticity and plasticity in the microstructural evolution of materials naturally by representing the density field for the crystalline state by periodic functions and by using a constant density for liquid state. PFC is of great interest in nano- and micro-structural modeling of materials because it is a model with atomistic scale details but is applicable to diffusive time scales. However, determining model parameters for specific materials is one of the less developed aspects of PFC modeling. In this article, molecular dynamics (MD) simulations of solid–liquid structures for Fe were performed using the modified embedded-atom method to determine the melting point, latent heat, expansion in melting, density profile, and liquid structure factor. The influence of simulation cell size on the results of MD simulations was also investigated. The melting temperature, density profile, and liquid structure factor were used as inputs to find model parameters required by the PFC model for Fe. The spatial derivative order of the PFC time-evolution equation was reduced from four to two, and the resultant system of partial differential equations was solved numerically using the finite element method. The required simulation domain and element size for the convergence of the PFC simulations were determined, and the expansion in melting, latent heat and solid–liquid surface free energy were calculated. The PFC results were compared with the results of other computational and experimental works in the literature.

Effect of predeformation on the transition from hexagonal phase to square phase near the melting point using phase field crystal method

The two-mode phase field crystal (PFC) method is used to calculate the phase diagram. And in this paper it is used to simulate the effects of predeformation degree and isothermal temperature on the hexagonal grain boundary evolution and on the hexagonal/square phase transition. Results show that when there is no predeformation in the initial phase, the grain boundary defect causes the pre-melting around the melting point; predeformation increases and the interaction between deformation and defects induces the pre-melting around the melting point; and the predeformation further increases, deformation induces liquid phase and square phase simultaneously at the distortion place. The bigger the predeformation and the closer to melting point the maintained temperature, the more obvious the growth of liquid phase is; on the contrary, the square phase grows obviously. The distortion energy is released with time and the phase of grain finally becomes square phase. It can be concluded that keeping the hexagonal phase isothermal near the melting temperature, the liquid phase appears at the grain boundary or at the other defects because the predeformation leads to the increase of atom activity, thus increasing atom disorder degree. Then with the release of distortion energy, the grain phase finally transforms into an equilibrium square phase. In this way the hexagonal/square transition time is extended.

Multistage microstructural evolution caused by deformation in two-mode phase field crystals

The two-mode phase-field-crystal (PFC) method is used to calculate two-dimensional phase diagram and to simulate the process of multistage microstructural evolution in the transformation from hexagonal phase to square phase, which is induced by deformation. And the effect of misorientation and deformation on dislocation, grain boundary, crystal structure and morphology of the new phase is carefully analyzed. Simulation results show that both the nucleation site and growth direction of the square phase are affected by the direction of deformation. Under a tensile deformation, the nucleation of the square phase occurs preferentially in the deformation zone; while under compression deformation, the nucleation of the square phase may begin at dislocations and grain boundary. Moreover, the new phase grows towards the direction along which the degree of atomic mismatch decreases, i.e. the vertical direction of tensile deformation and the parallel direction of compressive deformation. Besides, the free energy varies with misorientation. In small misorientation, the dislocation climbing, slipping and annihilating will result in an energy peak; while in a big misorientation, the dislocation annihilates in several stages and thus offsetting the energy caused by deformation. Furthermore, the process of phase transformation is complex: It is not a pure phase transformation but a composite change of phase transformation and dynamic recrystallization.

Characterizing solute segregation and grain boundary energy in binary alloy phase field crystal models

This paper studies how solute segregation and its relationship to grain boundary energy in binary alloys are captured in the phase field crystal (PFC) formalism, a continuum method that incorporates atomic scale elasto-plastic effects on diffusional time scales. Grain boundaries are simulated using two binary alloy PFC models – the original binary model by Elder et al. [18] and the XPFC model by Greenwood et al. [25]. In both cases, grain boundary energy versus misorientation data is shown to be well described by Read–Shockley theory. The Gibbs adsorption theorem is then used to derive a semi-analytic function describing solute segregation to grain boundaries. This is used to characterize grain boundary energy versus average alloy concentration and undercooling below the solidus. We also investigate how size mismatch between different species and their interaction strength affects segregation to the grain boundary. Finally, we interpret the implications of our simulations on material properties related to interface segregation.

Complex order parameter phase-field models derived from structural phase-field-crystal models

The phase-field-crystal (PFC) modeling paradigm is rapidly emerging as the model of choice when investigating materials phenomena with atomistic scale effects over diffusive time scales. Recent variants of the PFC model, so-called structural PFC (XPFC) models introduced by Greenwood et al., have further increased the capability of the method by allowing for easy access to various structural transformations in pure materials [Greenwood, Provatas, and Rottler, Phys. Rev. Lett. 105, 045702 (2010)] and binary alloys [Greenwood, Ofori-Opoku, Rottler, and Provatas, Phys. Rev. B 84, 064104 (2011)]. We present an amplitude expansion of these XPFC models, leading to a mesoscale complex order parameter, i.e., phase-field, model for two-dimensional square-triangular structures. Amplitude models retain the salient atomic scale features of the underlying PFC models, while resolving microstructures on mesoscales as traditional phase-field models. The applicability and capability of these complex-order parameter models is demonstrated with simulations of peritectic solidification and grain growth exhibiting the emergence of secondary phase structures.

Micromechanics of emergent patterns in plastic flows

Crystalline solids undergo plastic deformation and subsequently flow when subjected to stresses beyond their elastic limit. In nature most crystalline solids exist in polycrystalline form. Simulating plastic flows in polycrystalline solids has wide ranging applications, from material processing to understanding intermittency of earthquake dynamics. Using phase field crystal (PFC) model we show that in sheared polycrystalline solids the atomic displacement field shows spatio-temporal heterogeneity spanning over several orders of length and time scales, similar to that in amorphous solids. The displacement field also exhibits localized quadrupolar patterns, characteristic of two dislocations of the opposite sign approaching each other. This is a signature of crystallinity at microscopic scale. Polycrystals being halfway between single crystals and amorphous solids, in terms of the degree of structural order, descriptions of solid mechanics at two widely different scales, namely continuum plastic flow and discrete dislocation dynamics turns out to be necessary here.

An adaptive time-stepping strategy for solving the phase field crystal model

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 ste...

Phase Field Crystal Modeling for Nanocrystalline Growth

The two-mode phase field-crystal (PFC) method is used to simulate the nanograin growth, including the grain growth in different sets of crystal planes, the grain boundary structure with mismatch, the grain orientation and also the incoherent grain boundary in two dimensional plane. It is obviously observed that there are dislocation structures in nanograin boundary due to mismatch and misorientation of grains. These simulation results can not only be used in artificial controlling the grain boundary of nanograin, but also is of significant for designing new nanograin with a good grain boundary for structure materials.

Phase-field crystal model with a vapor phase

Phase-field crystal (PFC) models are able to resolve atomic length scale features of materials during temporal evolution over diffusive time scales. Traditional PFC models contain solid and liquid phases, however many important materials processing phenomena involve a vapor phase as well. In this work, we add a vapor phase to an existing PFC model and show realistic interfacial phenomena near the triple point temperature. For example, the PFC model exhibits density oscillations at liquid-vapor interfaces that compare favorably to data available for interfaces in metallic systems from both experiment and molecular-dynamics simulations. We also quantify the anisotropic solid-vapor surface energy for a two-dimensional PFC hexagonal crystal and find well-defined step energies from measurements on the faceted interfaces. Additionally, the strain field beneath a stepped interface is characterized and shown to qualitatively reproduce predictions from continuum models, simulations, and experimental data. Finally, we examine the dynamic case of step-flow growth of a crystal into a supersaturated vapor phase. The ability to model such a wide range of surface and bulk defects makes this PFC model a useful tool to study processing techniques such as chemical vapor deposition or vapor-liquid-solid growth of nanowires.

Atomistic investigation of clustering phenomenon in the Al–Cu system: Three-dimensional phase-field crystal simulation and HRTEM/HRSTEM characterization

Early-stage clustering in quenched/aged supersaturated Al–Cu alloys is simulated and characterized at the atomic scale to study the structural and compositional evolution of clusters. A three-dimensional analysis of the nucleation and growth mechanisms of early clusters in binary alloys is built upon a phase-field crystal (PFC) methodology previously developed in two dimensions by Fallah et al. [Phys. Rev. B. 86, 134112 (2012)]. The dislocation-mediated nucleation of clusters, reproduced in three dimensions, is compared against new results of atomistic characterization using high-resolution transmission electron microscopy (HRTEM) and high-resolution scanning transmission electron microscopy (HRSTEM). Consistent with the simulation results, analysis of HRTEM micrographs reveals the association of quenched-in dislocations with the early clusters. Moreover, the PFC model and HRSTEM observations show remarkable agreement on the non-equilibrium evolution of composition within the early clusters as well as their preferred growth orientation and spherical-to-ellipsoidal morphological transition.

Simulation of Solidification Across the Length Scales

Solidification is an incredibly complex process. At the macroscale, it involves both a liquid phase that may be flowing and a solid phase that may be deforming. At the microscale, it involves pattern selection processes, diffusional effects, grain nucleation, and attachment kinetics. There is also heat and mass transfer to consider at all the length scales of interest. Thus, the process of solidification involves strong coupling between these microscopic and macroscopic views. The use of computational methods to investigate theoretical and practical aspects of solidification has become quite popular over the past 30 years as they allow for the construction of accurate and detailed models that both include and provide insight into many important phenomena. At the macroscale, geometry and processing conditions determine the solidification sequence. Various modeling techniques based on finite elements and computational fluid dynamics exist to investigate die filling and the subsequent solidification process in components ranging from automotive components to aero engine turbine blades. As the length scale is reduced to roughly 1 lm–500 lm, the characteristic geometry for simulation becomes the microstructure and thus the modeling of grains, dendrites, and solute segregation is of interest. Key techniques at this scale include phase-field, cellular automata, and microsegregation methods in order to investigate the effects of solidification on the final properties of a cast component. A further reduction in the length scale brings us to the atomic level in which the properties of the solid/liquid interface and also the attachment kinetics can be investigated through molecular dynamics, density functional theory and phase-field crystal (to name a few). A significant current area of research is the anisotropy of the solid/liquid interface since this will affect to a large extent the morphology of the final microstructure. This JOM topic explores recent advances in the simulation of solidification at various length scales to examine pattern selection, defects, and processing effects. Five papers are presented that span the length scales described above. In each paper, different methodologies for modeling solidification are described along with the computational resources needed for performing such complex simulations. The first two papers focus on phase-fields methods to investigate pattern selection and interface effects. The article by I. Steinbach, ‘‘Why solidification, why phase-field’’ provides a wonderful historical perspective on the suitability and application of phase-field to solidification. The article ‘‘Atomistic Modelling of Solidification Phenomena using the Phase Field Crystal Model’’ by H. Humadi, N. OforiOpoku, N. Provatas and J. Hoyt focuses on the extension of phase-field to atomic-scale effects at diffusive time-scales. As shown in both contributions, phase-field is a powerful tool for answering questions relating to solidification pattern selection where rigorous theoretical physics are needed. At the other end of the spectrum, the paper by C. Reilly, J. Duan, L. Yao, D.M. Maijer and S.L. Cockcroft, ‘‘Process Modeling of Low-Pressure Die Casting of Aluminium Alloy Automotive Wheels,’’ presents a study of the use of computational modeling to improve industrial casting processes. In this work, the wheel is used as a case study to show both the importance of understanding heat flow in order to minimize shrinkage porosity, as well as the use of criteria functions. In process-level simulations, criteria functions predict defect formation with empirical equations based on local physical conditions such as cooling rate and strain-rate. Criteria functions are useful since they provide the industrial foundry with a strong indication of the likelihood of defect formation at low computational cost and are easy to interpret. In the case of the work by Reilly and colleagues, the Niyama criterion has been applied to the wheel-casting as a whole. Meso-scale models of solidification attempt to bridge the gap between pattern formation at the

Atomistic Modeling of Solidification Phenomena Using the Phase-Field-Crystal Model

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.

An adaptive time-stepping strategy for solving the phase field crystal model

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.

Energy stable and efficient finite-difference nonlinear multigrid schemes for the modified phase field crystal equation

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.

Simulation of early-stage clustering in ternary metal alloys using the phase-field crystal method

Phase-field crystal methodology is applied, for the first time, to study the effect of alloy composition on the clustering behavior of a quenched/aged supersaturated ternary Al alloy system. An analysis of the work of formation is adapted from a methodology developed in Fallah et al. to describe the dislocation-mediated nucleation and growth mechanisms of early clusters in binary alloys [Phys Rev B 2012;86:134112]. Consistent with the experiments, we demonstrate that the addition of Mg to an Al–1.1Cu alloy increases the nucleation rate of clusters in the quenched/aged state by increasing the effective driving force for nucleation, enhancing the dislocation stress relaxation and decreasing the surface energy associated with the Cu-rich co-clusters of Cu–Mg. Furthermore, we show that it is thermodynamically favorable for small subcritical clusters to have higher affinity for Mg than larger post-critical Cu-rich clusters, particularly depicting a two-stage clustering phenomenon.

Free energy of the bcc–liquid interface and the Wulff shape as predicted by the phase-field crystal model

The Euler–Lagrange equation of the phase-field crystal (PFC) model has been solved under appropriate boundary conditions to obtain the equilibrium free energy of the body centered cubic crystal–liquid interface for 18 orientations at various reduced temperatures in the range ϵ∈[0,0.5]. While the maximum free energy corresponds to the {100} orientation for all ϵ values, the minimum is realized by the {111} direction for smaller ϵ(<0.13), and by the {211} orientation for higher ϵ. The predicted dependence on the reduced temperature is consistent with the respective mean field critical exponent. The results are fitted with an eight-term Kubic harmonic series, and are used to create stereographic plots displaying the anisotropy of the interface free energy. We have also derived the corresponding Wulff shapes that vary with increasing ϵ from sphere to a polyhedral form that differs from the rhombo-dodecahedron obtained previously by growing a bcc seed until reaching equilibrium with the remaining liquid.

Classical density functional theory and the phase-field crystal method using a rational function to describe the two-body direct correlation function

We introduce a new approach to represent a two-body direct correlation function (DCF) in order to alleviate the computational demand of classical density functional theory (CDFT) and enhance the predictive capability of the phase-field crystal (PFC) method. The approach utilizes a rational function fit (RFF) to approximate the two-body DCF in Fourier space. We use the RFF to show that short-wavelength contributions of the two-body DCF play an important role in determining the thermodynamic properties of materials. We further show that using the RFF to empirically parametrize the two-body DCF allows us to obtain the thermodynamic properties of solids and liquids that agree with the results of CDFT simulations with the full two-body DCF without incurring significant computational costs. In addition, the RFF can also be used to improve the representation of the two-body DCF in the PFC method. Last, the RFF allows for a real-space reformulation of the CDFT and PFC method, which enables descriptions of nonperiodic systems and the use of nonuniform and adaptive grids.