New density functional approach for solid-liquid-vapor transitions in pure materials

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.

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.

Multiscale analysis of crystalline defect formation in rapid solidification of pure aluminium and aluminium-copper alloys

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 potential-based 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. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Orientation Gradients in Rapidly Solidified Pure Aluminum Thin Films: Comparison of Experiments and Phase-Field Crystal Simulations

Rapid solidification experiments on thin film aluminum samples reveal the presence of lattice orientation gradients within crystallizing grains. To study this phenomenon, a single-component phase-field crystal (PFC) model that captures the properties of solid, liquid, and vapor phases is proposed to model pure aluminium quantitatively. A coarse-grained amplitude representation of this model is used to simulate solidification in samples approaching micrometer scales. The simulations reproduce the experimentally observed orientation gradients within crystallizing grains when grown at experimentally relevant rapid quenches. We propose a causal connection between defect formation and orientation gradients.

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.

A phase-field-crystal alloy model for late-stage solidification studies involving the interaction of solid, liquid and gas phases

We present a multiphase binary alloy phase-field-crystal model. By introducing density difference between solid and liquid into a previous alloy model, this new fusion leads to a practical tool that can be used to investigate formation of defects in late-stage alloy solidification. It is shown that this model can qualitatively capture the liquid pressure drop due to solidification shrinkage in confined geometry. With an inherited gas phase from a previous multiphase model, cavitation of liquid from shrinkage-induced pressure is also included in this framework. As a unique model that has both solute concentration and pressure-induced liquid cavitation, it also captures a modified Scheil-Gulliver-type segregation behaviour due to cavitation. Simulation of inter-dendritic channel solidification using this model demonstrates a strong cooling rate dependence of the resulting microstructure.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Two-component structural phase-field crystal models for graphene symmetries

We extend the three-point XPFC model of Seymour & Provatas (Seymour & Provatas 2016 Phys. Rev. B93, 035447 (doi:10.1103/PhysRevB.93.035447)) to two components to capture chemical vapour deposition-grown graphene, and adapt a previous two-point XPFC model of Greenwood et al. (Greenwood et al. 2011 Phys. Rev. B84, 064104 (doi:10.1103/PhysRevB.84.064104)) into a simple model of two-component graphene. The equilibrium properties of these models are examined and the two models are compared and contrasted. The first model is used to study the possible roles of hydrogen in graphene grain boundaries. The second model is used to study the role of hydrogen in the dendritic growth morphologies of graphene. The latter results are compared with new experiments.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

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.

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.

Phase-Field-Crystal Model for Electromigration in Metal Interconnects

We propose an atomistic model of electromigration (EM) in metals based on a recently developed phase-field-crystal (PFC) technique. By coupling the PFC model's atomic density field with an applied electric field through the EM effective charge parameter, EM is successfully captured on diffusive time scales. Our framework reproduces the well-established EM phenomena known as Black's equation and the Blech effect, and also naturally captures commonly observed phenomena such as void nucleation and migration in bulk crystals. A resistivity dipole field arising from electron scattering on void surfaces is shown to contribute significantly to void migration velocity. With an intrinsic time scale set by atomic diffusion rather than atomic oscillations or hopping events, as in conventional atomistic methods, our theoretical approach makes it possible to investigate EM-induced circuit failure at atomic spatial resolution and experimentally relevant time scales.

Emergence of Chirality from Isotropic Interactions of Three Length Scales

Chirality is known to play a pivotal role in determining material properties and functionalities. However, it remains a great challenge to understand and control the emergence of chirality and the related enantioselective process particularly when the building components of the system are achiral. Here we explore the generic mechanisms driving the formation of two-dimensional chiral structures in systems characterized by isotropic interactions and three competing length scales. We demonstrate that starting from isotropic and rotationally invariant interactions, a variety of chiral ordered patterns and superlattices with anisotropic but achiral units can self-assemble. The mechanisms for selecting specific states are related to the length-scale coupling and the selection of resonant density wave vectors. Sample phase diagrams and chiral elastic properties are identified. These findings provide a viable route for predicting chiral phases and selecting the desired handedness.

Interfacial free energy adjustable phase field crystal model for homogeneous nucleation

To describe the homogeneous nucleation process, an interfacial free energy adjustable phase-field crystal model (IPFC) was proposed by reconstructing the energy functional of the original phase field crystal (PFC) methodology. Compared with the original PFC model, the additional interface term in the IPFC model effectively can adjust the magnitude of the interfacial free energy, but does not affect the equilibrium phase diagram and the interfacial energy anisotropy. The IPFC model overcame the limitation that the interfacial free energy of the original PFC model is much less than the theoretical results. Using the IPFC model, we investigated some basic issues in homogeneous nucleation. From the viewpoint of simulation, we proceeded with an in situ observation of the process of cluster fluctuation and obtained quite similar snapshots to colloidal crystallization experiments. We also counted the size distribution of crystal-like clusters and the nucleation rate. Our simulations show that the size distribution is independent of the evolution time, and the nucleation rate remains constant after a period of relaxation, which are consistent with experimental observations. The linear relation between logarithmic nucleation rate and reciprocal driving force also conforms to the steady state nucleation theory.

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.