Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model

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.

Enabling simulations of grains within a full rotation range in amplitude expansion of the phase-field crystal model

This paper introduces improvements to the amplitude expansion of the phase-field crystal model that enable the simulation of grains within a full range of orientations. The unphysical grain boundary between grains, rotated by a crystal's symmetry rotation, is removed using a combination of the auxiliary rotation field described in our previous work and an algorithm that correctly matches the complex amplitudes according to the differences in local rotation.

Atomic structures and migration mechanisms of interphase boundaries during body- to face-centered cubic phase transformations

Atomic structures and migration mechanisms of interphase boundaries have been of scientific interest for many years owing to their significance in the field of phase transformations. Though the interphase boundary structures can be deduced from crystallographic investigations, the detailed atomic structures and migration mechanisms of interphase boundaries during phase transformations are still poorly understood. In this study, a systematic study on atomic structures and migration mechanisms of interphase boundaries in a body-centered cubic (b.c.c.) to face-centered cubic (f.c.c.) massive transformation was carried out using the phase-field crystal model. Simulation results show that the f.c.c./b.c.c. interphase boundaries can be classified into faceted interphase boundaries and side surfaces. The faceted interphase boundaries are semi-coherent with a group of dislocations, leading to a ledge migration mechanism, while the side surfaces are incoherent and thus migrate in a continuous way. After a careful analysis of the simulated migration process of interphase boundaries at atomic scales, a detailed description of the ledge mechanism based on the motion and nucleation of interphase boundary dislocations is presented. The ledge-forming process is accompanied by the nucleation of new heterogeneous dislocations and motions of original dislocations, and thus the barrier of ledge formation comes from the hindrance of these two dislocation behaviors. Once the ledge is formed, the original dislocations continue to advance until the ledge height reaches 1/|Δg|, where Δg represents the difference in reciprocal lattice vectors between two phases. The new heterogeneous dislocation moves along the radial direction of the interphase boundary, resulting in ledge extension. The interface dislocation behaviors greatly affect the migration of the interphase boundary, leading to different migration kinetics of faceted interphase boundaries under the Kurdjumov–Sachs and the Nishiyama–Wasserman orientation relationships. This study revealed the mechanisms and kinetics of complex structure transition during a b.c.c.–f.c.c. massive phase transformation and can shed some light on the process of solid phase transformations.

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.

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.

Analysis of thermodiffusive cellular instabilities in continuum combustion fronts

We explore numerically the morphological patterns of thermodiffusive instabilities in combustion fronts with a continuum fuel source, within a range of Lewis numbers and ignition temperatures, focusing on the cellular regime. For this purpose, we generalize the recent model of Brailovsky et al. to include distinct process kinetics and reactant heterogeneity. The generalized model is derived analytically and validated with other established models in the limit of infinite Lewis number for zero-order and first-order kinetics. Cellular and dendritic instabilities are found at low Lewis numbers. These are studied using a dynamic adaptive mesh refinement technique that allows very large computational domains, thus allowing us to reduce finite-size effects that can affect or even preclude the emergence of these patterns. Our numerical linear stability analysis is consistent with the analytical results of Brailovsky et al. The distinct types of dynamics found in the vicinity of the critical Lewis number, ranging from steady-state cells to continued tip splitting and cell merging, are well described within the framework of thermodiffusive instabilities and are consistent with previous numerical studies. These types of dynamics are classified as "quasilinear" and characterized by low-amplitude cells that may be strongly affected by the mode selection mechanism and growth prescribed by the linear theory. Below this range of Lewis number, highly nonlinear effects become prominent and large-amplitude, complex cellular and seaweed dendritic morphologies emerge.

Honeycomb and triangular domain wall networks in heteroepitaxial systems

A comprehensive study is presented for the influence of misfit strain, adhesion strength, and lattice symmetry on the complex Moiré patterns that form in ultrathin films of honeycomb symmetry adsorbed on compact triangular or honeycomb substrates. The method used is based on a complex Ginzburg-Landau model of the film that incorporates elastic strain energy and dislocations. The results indicate that different symmetries of the heteroepitaxial systems lead to distinct types of domain wall networks and phase transitions among various surface Moiré patterns and superstructures. More specifically, the results show a dramatic difference between the phase diagrams that emerge when a honeycomb film is adsorbed on substrates of honeycomb versus triangular symmetry. It is also shown that in the small deformation limit, the complex Ginzburg-Landau model reduces to a two-dimensional sine-Gordon free energy form. This free energy can be solved exactly for one dimensional patterns and reveals the role of domains walls and their crossings in determining the nature of the phase diagrams.

Microscopic treatment of solute trapping and drag

The long wavelength limit of a recent microscopic phase-field crystal (PFC) theory of a binary alloy mixture is used to derive an analytical approximation for the segregation coefficient as a function of the interface velocity, and relate it to the two-point correlation function of the liquid and the thermodynamic properties of solid and liquid phases. Our results offer the first analytical derivation of solute segregation from a microscopic model, and support recent molecular dynamics and numerical PFC simulations. Our results also provide an independent framework, motivated from classical density functional theory, from which to elucidate the fundamental nature of solute drag, which is still highly contested in the literature.

Self-assembly via branching morphologies in nematic liquid-crystal nanocomposites

We demonstrate that the morphological diversity in liquid-crystal hybrid systems is much richer than previously anticipated. More importantly, we reveal the existence of a dual mechanism for self-assembly of nanoparticles via morphological instabilities at phase boundaries. Using numerical simulations, we study the growth of isolated nematic droplets in an isotropic liquid crystal (LC) doped with nanoparticles (NPs) and provide insight into the nature of microstructure evolution in LC hybrids. Our work expands the numerically accessible time and length scales in these systems, capturing morphologies which develop under the competition of nonequilibrium elastic interactions, diffusive instabilities mediated by NP transport, and the anisotropy of the nematic field. By mapping nematic morphologies, we also propose a methodology for estimating various important LC material parameters that are difficult to obtain experimentally.

Phase-field-crystal models and mechanical equilibrium

Phase-field-crystal (PFC) models constitute a field theoretical approach to solidification, melting, and related phenomena at atomic length and diffusive time scales. One of the advantages of these models is that they naturally contain elastic excitations associated with strain in crystalline bodies. However, instabilities that are diffusively driven towards equilibrium are often orders of magnitude slower than the dynamics of the elastic excitations, and are thus not included in the standard PFC model dynamics. We derive a method to isolate the time evolution of the elastic excitations from the diffusive dynamics in the PFC approach and set up a two-stage process, in which elastic excitations are equilibrated separately. This ensures mechanical equilibrium at all times. We show concrete examples demonstrating the necessity of the separation of the elastic and diffusive time scales. In the small-deformation limit this approach is shown to agree with the theory of linear elasticity.

Crystallization induced by multiple seeds: dynamical density functional approach

Using microscopic dynamical density functional theory, we calculate the dynamical formation of polycrystals by following the crystal growth around multiple crystalline seeds imposed to an undercooled fluid. Depending on the undercooling and the size ratio as well as the relative crystal orientation of two neighboring seeds, three possibilities of the final state emerge, namely no crystallization at all, formation of a monocrystal, or two crystallites separated by a curved grain boundary. Our results, which are obtained for two-dimensional hard disk systems using a fundamental-measure density functional, shed new light on the particle-resolved structure and growth of polycrystalline material in general.

Density and concentration field description of nonperiodic structures

We propose a simple nonlocal energy functional that is suitable for the continuum field characterization of nonperiodic and localized textures. The phenomenological functional is based on the pairwise direction-dependent interaction of field gradients that are separated by a fixed distance. In an appendix, we describe the numerical minimization of our functional. On that basis, we investigate the kinetic evolution of threadlike stripe patterns that are created by the functional when we start from an initially disordered state. At later stages, we find a coarse graining that shows the same scaling behavior as was obtained for the Cahn-Hilliard equation. In fact, the Cahn-Hilliard model is contained in our characterization as a limiting case. A slight modification of our model omits this coarse graining and leads to nonperiodic stripe phases. For the latter case, we investigate the temporal evolution of the defects (end points) of the threadlike stripes. In view of actual applications of this functional, we discuss the characterization of processes observed for polymeric systems and vesicles. The statistics of the growth of the threadlike structures is compared to the case of step-growth polymerization reactions. Furthermore, we demonstrate that the functional may be applied for the study of vesicles in a continuum field description. Basic features, such as the tendency of tank treading in simple shear flows and parachute folding in pipe flows, are reproduced.

Phase-field study of spacing evolution during transient growth

The primary spacing of a dendritic array grown under transient growth conditions displays a distribution of wavelengths. The average primary spacing is shown, both experimentally and numerically, to evolve between characteristic incubation periods during which the distribution of wavelengths remains essentially stable. Our primary spacing results display a gradual transition period from one spacing range to another, consistent with the fact that the abrupt doubling of spacing predicted by Warren and Langer for an idealized periodic array affects different wavelengths of the distribution at different times. This transition is shown to depend on the rate of change in growth speed using phase-field simulations of directional solidification where the pulling speed is ramped at different rates. In particular, for high rates of change of the pulling speed we observe temporary marginally stable array configurations separated by relatively short lived transitions, while for lower rates of change of the pulling speed the distinction between incubation and transition periods disappears.

Phase-field-crystal calculation of crystal-melt surface tension in binary alloys

A phase field crystal (PFC) density functional for binary mixtures is coarse grained and a formalism for calculating the simultaneous concentration, temperature, and density dependence of the surface energy anisotropy of a solid-liquid interface is developed. The methodology systematically relates bulk free energy coefficients arising from coarse graining to thermodynamic data, while gradient energy coefficients are related to molecular properties. Our coarse-grained formalism is applied to the determination of surface energy anisotropy in two-dimensional Zn-Al films, a situation relevant for quantitative phase field simulations of dendritic solidification in zinc coatings.

A phase-field-crystal approach to critical nuclei

We investigate a phase-field-crystal model for homogeneous nucleation. Instead of obtaining the time evolution of a density field towards equilibrium, we use a string method to identify saddle points in phase space. The saddle points allow us to obtain the nucleation barrier and the critical nucleus. The advantage of using the phase-field-crystal model for this task is that it can be used to resolve atomistic effects. The results obtained indicate different properties of the critical nucleus as compared with those for bulk crystals and provide a detailed description of the nucleation process.

A phase field crystal study of epitaxial island formation on nanomembranes

In this paper the liquid phase heteroepitaxial growth of two-dimensional strained islands on nanomembranes is examined via an amplitude expansion of a binary phase field crystal model. The maximum size that the islands can grow to coherently is shown to be strongly dependent on the nanomembrane thickness and to a lesser extent on the flux rate. For a large membrane-island misfit of about 10%, islands were found to be able to grow coherently with the membrane to almost twice as large as those grown on thick membranes (or infinite substrates). It was also found that when islands are growing on both sides of the membrane, strain relaxation in the membrane leads to more and less favorable growth regions. For thinner membranes this effect increases the degree of ordering of the islands, a result consistent with recent experimental findings.

Phase-field-crystal dynamics for binary systems: Derivation from dynamical density functional theory, amplitude equation formalism, and applications to alloy heterostructures

The dynamics of phase field crystal (PFC) modeling is derived from dynamical density functional theory (DDFT), for both single-component and binary systems. The derivation is based on a truncation up to the three-point direct correlation functions in DDFT, and the lowest order approximation using scale analysis. The complete amplitude equation formalism for binary PFC is developed to describe the coupled dynamics of slowly varying complex amplitudes of structural profile, zeroth-mode average atomic density, and system concentration field. Effects of noise (corresponding to stochastic amplitude equations) and species-dependent atomic mobilities are also incorporated in this formalism. Results of a sample application to the study of surface segregation and interface intermixing in alloy heterostructures and strained layer growth are presented, showing the effects of different atomic sizes and mobilities of alloy components. A phenomenon of composition overshooting at the interface is found, which can be connected to the surface segregation and enrichment of one of the atomic components observed in recent experiments of alloying heterostructures.

Amplitude expansion of the binary phase-field-crystal model

Amplitude representations of a binary phase-field-crystal model are developed for a two-dimensional triangular lattice and three-dimensional bcc and fcc crystal structures. The relationship between these amplitude equations and the standard phase-field models for binary-alloy solidification with elasticity are derived, providing an explicit connection between phase-field-crystal and phase-field models. Sample simulations of solute migration at grain boundaries, eutectic solidification, and quantum dot formation on nanomembranes are also presented.

Nonlinear elasticity of the phase-field crystal model from the renormalization group

The rotationally covariant renormalization group equations of motion for the density wave amplitudes in the phase field crystal model are shown to follow from a dynamical equation driven by an effective free energy density that we derive. We show that this free energy can be written purely as a function of the strain tensor and thence derive the corresponding equations governing the nonlinear elastic response.

Solid-liquid interfacial energies and equilibrium shapes of nanocrystals

We extract the anisotropy of the solid-liquid interfacial energy of small crystals using phase field crystal simulations. The results indicate a strong dependence of the interfacial energy on the parameters in the phase field crystal model determining the position in the solid-liquid coexistence region in the phase diagram. Furthermore a size dependence of the anisotropy is shown if the crystal shape is reduced to the size of a nucleus.

Classical density functional theory: an ideal tool to study heterogeneous crystal nucleation

Density functional theory provides an ideal microscopic theory to address freezing and crystallization problems. We review the application of static density functional theory for the calculation of equilibrium phase diagrams. We also describe the dynamical extension of density functional theory for systems governed by overdamped Brownian dynamics. Applications of density functional theory to crystallization problems, in particular to heterogeneous crystal nucleation and subsequent crystal growth, are summarized. Heterogeneous nucleation at an externally imposed nucleation cluster is discussed in detail, in particular for a simple two-dimensional dipolar system. Finally the relation of dynamical density functional theory and the phase field crystal approach are outlined.

Phase field crystal study of deformation and plasticity in nanocrystalline materials

We introduce a modified phase field crystal (MPFC) technique that self-consistently incorporates rapid strain relaxation alongside the usual plastic deformation and multiple crystal orientations featured by the traditional phase field crystal (PFC) technique. Our MPFC formalism can be used to study a host of important phase transformation phenomena in material processing that require rapid strain relaxation. We apply the MPFC model to study elastic and plastic deformations in nanocrystalline materials, focusing on the "reverse" Hall-Petch effect. Finally, we introduce a multigrid algorithm for efficient numerical simulations of the MPFC model.

Derivation of the phase-field-crystal model for colloidal solidification

The phase-field-crystal model is by now widely used in order to predict crystal nucleation and growth. For colloidal solidification with completely overdamped individual particle motion, we show that the phase-field-crystal dynamics can be derived from the microscopic Smoluchowski equation via dynamical density-functional theory. The different underlying approximations are discussed. In particular, a variant of the phase-field-crystal model is proposed which involves less approximations than the standard phase-field-crystal model. We finally test the validity of these phase-field-crystal models against dynamical density-functional theory. In particular, the velocities of a linear crystal front from the undercooled melt are compared as a function of the undercooling for a two-dimensional colloidal suspension of parallel dipoles. Good agreement is only obtained by a drastic scaling of the free energies in the phase-field-crystal model in order to match the bulk freezing transition point.

Molecular dynamics on diffusive time scales from the phase-field-crystal equation

We extend the phase-field-crystal model to accommodate exact atomic configurations and vacancies by requiring the order parameter to be non-negative. The resulting theory dictates the number of atoms and describes the motion of each of them. By solving the dynamical equation of the model, which is a partial differential equation, we are essentially performing molecular dynamics simulations on diffusive time scales. To illustrate this approach, we calculate the two-point correlation function of a fluid.

Deriving surface-energy anisotropy for phenomenological phase-field models of solidification

The free energy of classical density functional theory of an inhomogeneous fluid at coexistence with its solid is used to describe solidification in two-dimensional hexagonal crystals. A coarse-graining formalism from the microscopic density functional level to the macroscopic single order parameter level is provided. An analytic expression for the surface energy and the angular dependence of its anisotropy is derived and its coefficients related to the two-point direct correlation function of the liquid phase at coexistence.