Phase-field-crystal methodology for modeling of structural transformations

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.

The Effect of Lattice Misfits on the Precipitation at Dislocations: Phase-Field Crystal Simulation

An atomic-scale approach was employed to simulate the formation of precipitates with different lattice misfits in the early stages of the aging of supersaturated aluminum alloys. The simulation results revealed that the increase in lattice misfits could significantly promote the nucleation rate of precipitates, which results in a larger number and smaller size of the precipitates. The morphologies of the precipitates also vary with the degree of a lattice misfit. Moreover, the higher the lattice misfit, the earlier the nucleation of the second phase occurs, which can substantially inhibit the movement of dislocations. The research on the lattice misfit of precipitation can provide theoretical guidance for the design of high-strength aluminum alloys.

Atomic-Scale Insights into the Self-Assembly of Alternating AlN/TiN Lamellar Nanostructures via Spinodal Decomposition in AlTiN Coating

The self-assembly mechanism of alternating AlN/TiN nano-lamellar structures in AlTiN coating is still a mystery, though this coating has been widely used in industry. Here, by using the phase-field crystal method, we studied the atomic-scale mechanisms of the formation of nano-lamellar structures during spinodal decomposition transformation of an AlTiN coating. The results show that the formation of a lamella is characterized by four distinct stages including the generation of dislocations (stage I), formation of islands (stage II), merging of islands (stage III), and flattening of lamellae (stage IV). The locally periodic fluctuation of the concentration along the lamella leads to the generation of periodically distributed misfit dislocations and then AlN/TiN islands, while the fluctuation of the composition in the direction normal to the lamella is responsible for the merging of islands and flattening of a lamella and more importantly the cooperative growth between neighboring lamellae. Moreover, we found that misfit dislocations play a crucial role in all the four stages, promoting the cooperative growth of TiN and AlN lamellae. Our results demonstrate that the TiN and AlN lamellae could be produced through the cooperative growth of AlN/TiN lamellae in spinodal decomposition of the AlTiN phase.

Atomic-Scale Insights into the Deformation Mechanism of the Microstructures in Precipitation-Strengthening Alloys

Clarifying the deformation behaviors of microstructures could greatly help us understand the precipitation-strengthening mechanism in alloys. However, it is still a formidable challenge to study the slow plastic deformation of alloys at the atomic scale. In this work, the phase-field crystal method was used to investigate the interactions between precipitates, grain boundary, and dislocation during the deformation processes at different degrees of lattice misfits and strain rates. The results demonstrate that the pinning effect of precipitates becomes increasingly strong with the increase of lattice misfit at relatively slow deformation with a strain rate of 10^-4. The cut regimen prevails under the interaction between coherent precipitates and dislocations. In the case of a large lattice misfit of 19.3%, the dislocations tend to move toward the incoherent phase interface and are absorbed. The deformation behavior of the precipitate-matrix phase interface was also investigated. Collaborative deformation is observed in coherent and semi-coherent interfaces, while incoherent precipitate deforms independently of the matrix grains. The faster deformations (strain rate is 10^-2) with different lattice misfits all are characterized by the generation of a large number of dislocations and vacancies. The results contribute to important insights into the fundamental issue about how the microstructures of precipitation-strengthening alloys deform collaboratively or independently under different lattice misfits and deformation rates.

Control of phase ordering and elastic properties in phase field crystals through three-point direct correlation

Effects of three-point direct correlation on properties of the phase field crystal (PFC) modeling are examined for the control of various ordered and disordered phases and their coexistence in both three-dimensional and two-dimensional systems. Such effects are manifested via the corresponding gradient nonlinearity in the PFC free-energy functional that is derived from classical density functional theory. Their significant impacts on the stability regimes of ordered phases, phase diagrams, and elastic properties of the system, as compared to those of the original PFC model, are revealed through systematic analyses and simulations. The nontrivial contribution from three-point direct correlation leads to the variation of the critical point of order-disorder transition to which all the phase boundaries in the temperature-density phase diagram converge. It also enables the variation and control of system elastic constants over a substantial range as needed in modeling different types of materials with the same crystalline structure but different elastic properties. The capability of this PFC approach in modeling both solid and soft matter systems is further demonstrated through the effect of three-point correlation on controlling the vapor-liquid-solid coexistence and transitions for body-centered cubic phase and on achieving the liquid-stripe or liquid-lamellar phase coexistence. All these provide a valuable and efficient method for the study of structural ordering and evolution in various types of material systems.

An atomic scale study of two-dimensional quasicrystal nucleation controlled by multiple length scale interactions

Formation of quasicrystal structures has always been mysterious since the discovery of these magic structures. In this work, the nucleation of decagonal, dodecagonal, heptagonal, and octagonal quasicrystal structures controlled by the coupling among multiple length scales is investigated using a dynamic phase-field crystal model. We observe that the nucleation of quasicrystals proceeds through local rearrangement of length scales, i.e., the generation, merging and stacking of 3-atom building blocks constructed by the length scales, and accordingly, propose a geometric model to describe the cooperation of length scales during structural transformation in quasicrystal nucleation. Essentially, such cooperation is crucial to quasicrystal formation, and controlled by the match and balance between length scales. These findings clarify the scenario and microscopic mechanism of the structural evolution during quasicrystal nucleation, and help us to understand the common rule for the formation of periodic crystal and quasicrystal structures with various symmetries.

Phase-field crystal for an antiferromagnet with elastic interactions

We introduce a model which contains the essential elements to formulate and study antiferromagnetism, using the phase-field crystal framework. We focus on the question of how magneto-elastic coupling could lift the frustration in the two-dimensional hexagonal antiferromagnetic phase. Using simulations we observe a rich variety of different phases stable in this model. To characterize different phases we calculate the chiral order parameter and identify the scaling behavior of this order parameter. Furthermore, we observe that vortices appear and are stable close to the nonmagnetic defects. Finally, we studied the ferrimagnetic and spin-flop phase transition in the presence of an external magnetic field.

A Study of Strain-Driven Nucleation and Extension of Deformed Grain: Phase Field Crystal and Continuum Modeling

The phase-field-crystal (PFC) method is used to investigate migration of grain boundary dislocation and dynamic of strain-driven nucleation and growth of deformed grain in two dimensions. The simulated results show that the deformed grain nucleates through forming a gap with higher strain energy between the two sub-grain boundaries (SGB) which is split from grain boundary (GB) under applied biaxial strain, and results in the formation of high-density ensembles of cooperative dislocation movement (CDM) that is capable of plastic flow localization (deformed band), which is related to the change of the crystal lattice orientation due to instability of the orientation. The deformed grain stores the strain energy through collective climbing of the dislocation, as well as changing the orientation of the original grain. The deformed grain growth (DGG) is such that the higher strain energy region extends to the lower strain energy region, and its area increase is proportional to the time square. The rule of the time square of the DGG can also be deduced by establishing the dynamic equation of the dislocation of the strain-driven SGB. The copper metal is taken as an example of the calculation, and the obtained result is a good agreement with that of the experiment.

Controlling the energy of defects and interfaces in the amplitude expansion of the phase-field crystal model

One of the major difficulties in employing phase-field crystal (PFC) modeling and the associated amplitude (APFC) formulation is the ability to tune model parameters to match experimental quantities. In this work, we address the problem of tuning the defect core and interface energies in the APFC formulation. We show that the addition of a single term to the free-energy functional can be used to increase the solid-liquid interface and defect energies in a well-controlled fashion, without any major change to other features. The influence of the newly added term is explored in two-dimensional triangular and honeycomb structures as well as bcc and fcc lattices in three dimensions. In addition, a finite-element method (FEM) is developed for the model that incorporates a mesh refinement scheme. The combination of the FEM and mesh refinement to simulate amplitude expansion with a new energy term provides a method of controlling microscopic features such as defect and interface energies while simultaneously delivering a coarse-grained examination of the system.

Phase-field-crystal investigation of the morphology of a steady-state dendrite tip on the atomic scale

Through phase-field-crystal (PFC) simulations, we investigated, on the atomic scale, the crucial role played by interface energy anisotropy and growth driving force during the morphological evolution of a dendrite tip at low growth driving force. In the layer-by-layer growth manner, the interface energy anisotropy drives the forefront of the dendrite tip to evolve to be highly similar to the corner of the corresponding equilibrium crystal from the aspects of atom configuration and morphology, and thus affects greatly the formation and growth of a steady-state dendrite tip. Meanwhile, the driving force substantially influences the part behind the forefront of the dendrite tip, rather than the forefront itself. However, as the driving force increases enough to change the layer-by-layer growth to the multilayer growth, the morphology of the dendrite tip's forefront is completely altered. Parabolic fitting of the dendrite tip reveals that an increase in the influence of interface energy anisotropy makes dendrite tips deviate increasingly from a parabolic shape. By quantifying the deviations under various interface energy anisotropies and growth driving forces, it is suggested that a perfect parabola is an asymptotic limit for the shape of the dendrite tips. Furthermore, the atomic scale description of the dendrite tip obtained in the PFC simulation is compatible with the mesoscopic results obtained in the phase-field simulation in terms of the dendrite tip's morphology and the stability criterion constant.

Description of order-disorder transitions based on the phase-field-crystal model

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.

Phase-field-crystal model for ordered crystals

We describe a general method to model multicomponent ordered crystals using the phase-field-crystal (PFC) formalism. As a test case, a generic B2 compound is investigated. We are able to produce a line of either first-order or second-order order-disorder phase transitions, features that have not been incorporated in existing PFC approaches. Further, it is found that the only elastic constant for B2 that depends on ordering is C_{11}. This B2 model is then used to study antiphase boundaries (APBs). The APBs are shown to reproduce classical mean-field results. Dynamical simulations of ordering across small-angle grain boundaries predict that dislocation cores pin the evolution of APBs.

Calculating the role of composition in the anisotropy of solid-liquid interface energy using phase-field-crystal theory

This work uses Ginzburg-Landau theory derived from a recent structural phase-field-crystal model of binary alloys developed by the authors to study the roles of concentration, temperature, and pressure on the interfacial energy anisotropy of a solid-liquid front. It is found that the main contribution to the change in anisotropy with concentration arises from a change in preferred crystallographic orientation controlled by solute-dependent changes in the two-point density correlation function of a binary alloy, a mechanism that leads to such phenomena as solute-induced elastic strain and dislocation-assisted solute clustering. Our results are consistent with experimental observations in recent studies by Rappaz et al. [J. Fife, P. Di Napoli, and M. Rappaz, Metall. Mater. Trans. A 44, 5522 (2013)]. This is the first PFC work, to our knowledge, to incorporate temperature, pressure, and density into the thermodynamic description of alloys.

Modified phase-field-crystal model for solid-liquid phase transitions

A modified phase-field-crystal (PFC) model is proposed to describe solid-liquid phase transitions by reconstructing the correlation function. The effects of fitting parameters of our modified PFC model on the bcc-liquid phase diagram, numerical stability, and solid-liquid interface properties during planar interface growth are examined carefully. The results indicate that the increase of the correlation function peak width at k=k(m) will enhance the stability of the ordered phase, while the increase of peak height at k=0 will narrow the two-phase coexistence region. The third-order term in the free-energy function and the short wave-length of the correlation function have significant influences on the numerical stability of the PFC model. During planar interface growth, the increase of peak width at k=k(m) will decrease the interface width and the velocity coefficient C, but increase the anisotropy of C and the interface free energy. Finally, the feasibility of the modified phase-field-crystal model is demonstrated with a numerical example of three-dimensional dendritic growth of a body-centered-cubic structure.

Phase-field crystal model for a diamond-cubic structure

We present a structural phase-field crystal model [M. Greenwood et al., Phys. Rev. Lett. 105, 045702 (2010)] that yields a stable dc structure. The stabilization of a dc structure is accomplished by constructing a two-body direct correlation function (DCF) approximated by a combination of two Gaussian functions in Fourier space. A phase diagram containing a dc-liquid phase coexistence region is calculated for this model. We examine the energies of solid-liquid interfaces with normals along the [100], [110], and [111] directions. The dependence of the interfacial energy on a temperature parameter, which controls the heights of the peaks in the two-body DCF, is described by a Gaussian function. Furthermore, the dependence of the interfacial energy on the peak widths of the two-body DCF, which controls the excess energy associated with interfaces, defects, and strain, is described by an inverse power law. These relationships can be used to parametrize the phase-field crystal model for the dc structure to match solid-liquid interfacial energies to those measured experimentally or calculated from atomistic simulations.

Phase-field-crystal model for magnetocrystalline interactions in isotropic ferromagnetic solids

An isotropic magnetoelastic phase-field-crystal model to study the relation between morphological structure and magnetic properties of pure ferromagnetic solids is introduced. Analytic calculations in two dimensions were used to determine the phase diagram and obtain the relationship between elastic strains and magnetization. Time-dependent numerical simulations in two dimensions were used to demonstrate the effect of grain boundaries on the formation of magnetic domains. It was shown that the grain boundaries act as nucleating sites for domains of reverse magnetization. Finally, we derive a relation for coercivity versus grain misorientation in the isotropic limit.

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.

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.

Phase-field-crystal model of phase and microstructural stability in driven nanocrystalline systems

We present a phase-field-crystal model for driven systems which describes competing effects between thermally activated diffusional processes and those driven by externally imposed ballistic events. The model demonstrates how the mesoscopic Enrique and Bellon [Phys. Rev. Lett. 84, 2885 (2000)] model of externally induced ballistic mixing can be incorporated into the atomistic phase-field-crystal formalism. The combination of the two approaches results in a model capable of describing the microstructural and compositional evolution of a driven system while incorporating elastoplastic effects. The model is applied to the study of grain growth in nanocrystalline materials subjected to an external driving.

Morphology of monolayer films on quasicrystalline surfaces from the phase field crystal model

We present a computational study of the morphology of adsorbed monolayers on quasicrystalline surfaces with five- and seven-fold symmetry. The phase field crystal model is employed to first simulate the growth of the quasicrystal surfaces and then to elastically couple a two-dimensional film to the substrate. We find several distinct pseudomorphic phases that depend on the position of adsorption sites as well as the strength of the monolayer/substrate interaction, and quantify them by computing local order parameters. In qualitative agreement with recent experiments using colloids in quasiperiodic light fields, we find that the formation of quasicrystalline order is greatly inhibited on seven-fold surfaces.