Diffusive atomistic dynamics of edge dislocations in two dimensions

Grain boundaries are Brownian ratchets

We demonstrate that grain boundaries (GBs) behave as Brownian ratchets, exhibiting direction-dependent mobilities and unidirectional motion under oscillatory driving forces or cyclic thermal annealing. We observed these phenomena for nearly all nonsymmetric GBs but not for symmetric ones. Our observations build on molecular dynamics and phase-field crystal simulations for a wide range of GB types and driving forces in both bicrystal and polycrystalline microstructures. We corroborate these simulation results through in situ experimental observations. We analyze these results with a Markov chain model and explore the implications of GB ratchet behavior for materials processing and microstructure tailoring.

Classification and Simulation of Structural Phase Transformation-Induced Interfacial Defects in Group VI Transition-Metal Dichalcogenide Monolayers

Polymorphic 2D materials have recently emerged as promising candidates for use in nanoelectronic devices by way of their ability to undergo structural phase transformations induced by external fields. Under cyclic transformations, however, induced interfacial defects may proliferate and compromise the system properties. Herein, we first employ geometric analysis to classify such defects generated during the 2H ↔ 1T and 2H ↔ 1T' transformations in group VI transition-metal dichalcogenide monolayers. Then, simulations of a mesoscale model with atomistic spatial resolution are conducted to assess the proliferation of such defects during cyclic 2H ↔ 1T transformations. It is shown that defect densities reach a steady state, with the 2H phase remaining more pristine than the 1T and 1T' states. We expect that the effects of these defects on the device performance are application-dependent and will require further inquiry.

Atomistic simulation model on a diffusive timescale based on the extension of the cluster-activation method to continuous space

Recently the phase-field crystal method has attracted considerable attention because it can simulate the atomic behavior of a system on a diffusive timescale. In this study an atomistic simulation model is proposed, which is an extension of the cluster-activation method (CAM) from discrete to continuous space. This approach, called the continuous CAM, can simulate various physical phenomena of atomistic systems on diffusive timescales and employs well-defined atomistic properties, such as interatomic interaction energies, as the main input parameters. The versatility of the continuous CAM was investigated by performing simulations of crystal growth in an undercooled melt, homogeneous nucleation during solidification, and formation of grain boundaries in pure metal.

A Second-Order Crank-Nicolson Leap-Frog Scheme for the Modified Phase Field Crystal Model with Long-Range Interaction

In this paper, we construct a fully discrete and decoupled Crank-Nicolson Leap-Frog (CNLF) scheme for solving the modified phase field crystal model (MPFC) with long-range interaction. The idea of CNLF is to treat stiff terms implicity with Crank-Nicolson and to treat non-stiff terms explicitly with Leap-Frog. In addition, the scalar auxiliary variable (SAV) method is used to allow explicit treatment of the nonlinear potential, then, these technique combines with CNLF can lead to the highly efficient, fully decoupled and linear numerical scheme with constant coefficients at each time step. Furthermore, the Fourier spectral method is used for the spatial discretization. Finally, we show that the CNLF scheme is fully discrete, second-order decoupled and unconditionally stable. Ample numerical experiments in 2D and 3D are provided to demonstrate the accuracy, efficiency, and stability of the proposed method.

Radiation-Induced Patterning at the Nanometric Scale: A Phase Field Approach

The phase field approach was developed in the last 20 years to handle radiation damage in materials. This approach bridges the gap between atomistic simulations extensively used to model first step of radiation damage at short time and continuum approach at large time. The main advantage of such an approach lies in its ability to compute not only the microstructure at the nanometric scale but also to calculate generalized susceptibilities such as elastic constants under irradiation. After a brief description of the rate theory, used to model the microstructure induced by irradiation, we briefly discuss the foundation of the phase field method, highlighting not only its advantages, but also its limitations in comparison with the rate theory. We conclude this presentation by proposing future orientations for computing the microstructure in irradiated materials.

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.

Structure diagram and dynamics of formation of hexagonal boron nitride in phase-field crystal model

The phase-field crystal (PFC-model) is a powerful tool for modelling of the crystallization in colloidal and metallic systems. In the present work, the modified hyperbolic phase-field crystal model for binary systems is presented. This model takes into account slow and fast dynamics of moving interfaces for both concentration and relative atomic number density (which were taken as order parameters). The model also includes specific mobilities for each dynamical field and correlated noise terms. The dynamics of chemical segregation with origination of mixed pseudo-hexagonal binary phase (the so-called 'triangle phase') is used as a benchmark in two spatial dimensions for the developing model. Using the free energy functional and specific lattice vectors for hexagonal crystal, the structure diagram of co-existence of liquid and three-dimensional hexagonal phase for the binary PFC-model was carried out. Parameters of the crystal lattice correspond to the hexagonal boron nitride (BN) crystal, the values of which have been taken from the literature. The paper shows the qualitative agreement between the developed structure diagram of the PFC model and the previously known equilibrium diagram for BN constructed using thermodynamic functions. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

On the Well-Posedness of A High Order Convective Cahn-Hilliard Type Equations

High order convective Cahn-Hilliard type equations describe the faceting of a growing surface, or the dynamics of phase transitions in ternary oil-water-surfactant systems. In this paper, we prove the well-posedness of the classical solutions for the Cauchy problem, associated with this equation.

Variable-order particle dynamics: formulation and application to the simulation of edge dislocations

This study presents the application of variable-order (VO) fractional operators to modelling the dynamics of edge dislocations under the effect of a static state of shear stress. More specifically, a particle dynamic approach is used to simulate the microscopic structure of a material where the constitutive atoms or molecules are modelled via discrete masses and their interaction via inter-particle forces. VO operators are introduced in the formulation in order to capture the complex linear-to-nonlinear dynamic transitions following the translation of dislocations as well as the creation and annihilation of bonds between particles. Remarkably, the motion of the dislocation does not require any a priori assumption in terms of either possible trajectory or sections of the model that could undergo the nonlinear transition associated with the creation and annihilation of bonds. The model only requires the definition of the initial location of the dislocations. Results will show that the VO formulation is fully evolutionary and capable of capturing both the sliding and the coalescence of edge dislocations by simply exploiting the instantaneous response of the system and the state of stress. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Mechanical relaxation and fracture of phase field crystals

A computational method is developed for the study of mechanical response and fracture behavior of phase field crystals (PFC), to overcome a limitation of the PFC dynamics which lacks an effective mechanism for describing fast mechanical relaxation of the material system. The method is based on a simple interpolation scheme for PFC (IPFC) making use of a condition of the displacement field to satisfy local elastic equilibration, while preserving key characteristics of the original PFC model. We conduct a systematic study on the mechanical properties of a sample nanoribbon system with honeycomb lattice symmetry subjected to uniaxial tension, for numerical validation of the IPFC scheme and the comparison with the original PFC and modified PFC methods. Results of mechanical response, in both elasticity and fracture regimes, show the advantage and efficiency of the IPFC method across different system sizes and applied strain rates, due to its effective process of mechanical equilibration. A brittle fracture behavior is obtained in IPFC calculations, where effects of system temperature and chirality on the fracture strength and Young's modulus are also identified, with results agreeing with those found in previous atomistic simulations of graphene. The IPFC scheme developed here is generic and applicable to the mechanical studies using different types of PFC free-energy functionals designed for various material systems.

Separation of Elastic and Plastic Timescales in a Phase Field Crystal Model

A consistent small-scale description of plasticity and dislocation motion in a crystalline solid is presented based on the phase field crystal description. By allowing for independent mass motion and lattice distortion, the crystal can maintain elastic equilibrium on the timescale of plastic motion. We show that the singular (incompatible) strains are determined by the phase field crystal density, while the smooth distortions are constrained to satisfy elastic equilibrium. A numerical implementation of the model is presented and used to study a benchmark problem: the motion of an edge dislocation dipole in a triangular lattice. The time dependence of the dipole separation agrees with continuum elasticity with no adjustable parameters.

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.

Hydrodynamic theory of freezing: Nucleation and polycrystalline growth

Structural aspects of crystal nucleation in undercooled liquids are explored using a nonlinear hydrodynamic theory of crystallization proposed recently [G. I. Tóth et al., J. Phys.: Condens. Matter 26, 055001 (2014)JCOMEL0953-898410.1088/0953-8984/26/5/055001], which is based on combining fluctuating hydrodynamics with the phase-field crystal theory. We show that in this hydrodynamic approach not only homogeneous and heterogeneous nucleation processes are accessible, but also growth front nucleation, which leads to the formation of new (differently oriented) grains at the solid-liquid front in highly undercooled systems. Formation of dislocations at the solid-liquid interface and interference of density waves ahead of the crystallization front are responsible for the appearance of the new orientations at the growth front that lead to spherulite-like nanostructures.

Numerical determination of the interfacial energy and nucleation barrier of curved solid-liquid interfaces in binary systems

The phase-field crystal (PFC) technique is a widely used approach for modeling crystal growth phenomena with atomistic resolution on mesoscopic time scales. We use a two-dimensional PFC model for a binary system based on the work of Elder et al. [Phys. Rev. B 75, 064107 (2007)PRBMDO1098-012110.1103/PhysRevB.75.064107] to study the effect of the curved, diffuse solid-liquid interface on the interfacial energy as well as the nucleation barrier. The calculation of the interfacial energy and the nucleation barrier certainly depends on the proper definition of the solid-liquid dividing surface and the corresponding nucleus size. We define the position of the sharp interface at which the interfacial energy is to be evaluated by using the concept of equimolar dividing surface (r^{e}) and the minimization of the interfacial energy (r^{s}). The comparison of the results based on both radii shows that the difference r^{e}-r^{s} is always positive and has a limit for large cluster sizes which is comparable to the Tolman length. Furthermore, we found the real nucleation barrier for small cluster sizes, which is defined as a function of the radius r^{s}, and compared it with the classical nucleation theory. The simulation results also show that the extracted interfacial energy as function of both radii is independent of system size, and this dependence can be reasonably described by the nonclassical Tolman formula with a positive Tolman length.

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.

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.

Morphological instability of grain boundaries in two-phase coherent solids

We show both computationally and analytically that grain boundaries that exhibit shear-coupled motion become morphologically unstable in solid alloys that phase separate into coherent domains of distinct chemical compositions. We carry out simulations of continuum models demonstrating that this instability is mediated by long-range elastic interaction between compositional domains and grain boundaries. In addition, we perform a linear stability analysis that predicts the range of unstable wavelengths in good quantitative agreement with simulations. In nonlinear stages, this pattern-forming instability leads to the breakup of low-angle grain boundaries, thereby strongly impacting microstructural evolution in a wide range of phase-separating materials.

Solid-liquid surface tensions of critical nuclei and nucleation barriers from a phase-field-crystal study of a model binary alloy using finite system sizes

Phase-field-crystal (PFC) modeling has emerged as a computationally efficient tool to address crystal growth phenomena on atomistic length and diffusive time scales. We use a two-dimensional phase-field-crystal model for a binary system based on Elder et al. [Phys. Rev. B 75, 064107 (2007)] to study critical nuclei and their liquid-solid phase boundaries, in particular the nucleus size dependence of the liquid-solid interface tension as well as of the nucleation barrier. Critical nuclei are stabilized in finite systems of various sizes, however, the extracted interface tension as function of the nucleus radius r is independent of system size. We suggest a phenomenological expression to describe the dependence of the extracted interface tension on the nucleus radius r for the liquid-solid system. Moreover, the numerical PFC results show that this dependency can not be fully described by the nonclassical Tolman formula.

Phase-field-crystal modeling of glass-forming liquids: spanning time scales during vitrification, aging, and deformation

Two essential elements required to generate a glass transition within phase-field-crystal (PFC) models are outlined based on observed freezing behaviors in various models of this class. The central dynamic features of glass formation in simple binary liquids are qualitatively reproduced across 12 orders of magnitude in time by applying a physically motivated time scaling to previous PFC simulation results. New aspects of the equilibrium phase behavior of the same binary model system are also outlined, aging behavior is explored in the moderate and deeply supercooled regimes, and aging exponents are extracted. General features of the elastic and plastic responses of amorphous and crystalline PFC solids under deformation are also compared and contrasted.

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.

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.

Unified theoretical framework for polycrystalline pattern evolution

The rate of curvature-driven grain growth in polycrystalline materials is well known to be limited by interface dissipation. We show analytically and by simulations that, for systems forming modulated phases or nonequilibrium patterns with crystal ordering, growth is limited by bulk dissipation associated with lattice translation, which dramatically slows down grain coarsening. We also show that bulk dissipation is reduced by thermal noise and that this reduction leads to faster coarsening behavior dominated by interface dissipation for a high Peierls-Nabarro barrier to dislocation motion and high noise. Those results provide a unified theoretical framework for understanding and modeling polycrystalline pattern evolution in diverse systems over a broad range of length and time scales.

Solidification fronts in supercooled liquids: how rapid fronts can lead to disordered glassy solids

We determine the speed of a crystallization (or, more generally, a solidification) front as it advances into the uniform liquid phase after the system has been quenched into the crystalline region of the phase diagram. We calculate the front speed by assuming a dynamical density functional theory (DDFT) model for the system and applying a marginal stability criterion. Our results also apply to phase field crystal (PFC) models of solidification. As the solidification front advances into the unstable liquid phase, the density profile behind the advancing front develops density modulations and the wavelength of these modulations is a dynamically chosen quantity. For shallow quenches, the selected wavelength is precisely that of the crystalline phase and so well-ordered crystalline states are formed. However, when the system is deeply quenched, we find that this wavelength can be quite different from that of the crystal, so the solidification front naturally generates disorder in the system. Significant rearrangement and aging must subsequently occur for the system to form the regular well-ordered crystal that corresponds to the free energy minimum. Additional disorder is introduced whenever a front develops from random initial conditions. We illustrate these findings with simulation results obtained using the PFC model.

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.

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.

DDFT calibration and investigation of an anisotropic phase-field crystal model

The anisotropic phase-field crystal model recently proposed and used by Prieler et al (2009 J. Phys.: Condens. Matter 21 464110) is derived from microscopic density functional theory for anisotropic particles with fixed orientation. Its morphology diagram is also explored. In particular we have investigated the influence of anisotropy and undercooling on the process of nucleation and microstructure formation from the atomic to the microscale. To that end numerical simulations were performed varying those dimensionless parameters which represent anisotropy and undercooling in our anisotropic phase-field crystal model. The results from these numerical simulations are summarized in terms of a morphology diagram of the stable state phases. These stable phases are also investigated with respect to their kinetics and characteristic morphological features.

Nucleation, growth, and coarsening of crystalline domains in order-order transitions between lamellar and hexagonal phases

Using the numerical solution of the time-dependent Ginzburg-Landau equation, we study the entire process of transformation between the lamellar and the hexagonal phases from the early-stage nucleation and growth to the late-stage coarsening regime. The metastable crystalline structure that nucleates first is identified in terms of the mean-field theory under the single-wave-number approximation. This has been borne out by the numerically efficient preparation of single-crystal structure developed via the noise-induced self-organization. We also present results for the scaling of the late-time domain growth, which is quantified by two measures: the structure factor and the orientational correlation function. In particular, the growth exponent is shown to be robust and indifferent to conservation of the order parameter.

Phase-field-crystal methodology for modeling of structural transformations

We introduce and characterize free-energy functionals for modeling of solids with different crystallographic symmetries within the phase-field-crystal methodology. The excess free energy responsible for the emergence of periodic phases is inspired by classical density-functional theory, but uses only a minimal description for the modes of the direct correlation function to preserve computational efficiency. We provide a detailed prescription for controlling the crystal structure and introduce parameters for changing temperature and surface energies, so that phase transformations between body-centered-cubic (bcc), face-centered-cubic (fcc), hexagonal-close-packed (hcp), and simple-cubic (sc) lattices can be studied. To illustrate the versatility of our free-energy functional, we compute the phase diagram for fcc-bcc-liquid coexistence in the temperature-density plane. We also demonstrate that our model can be extended to include hcp symmetry by dynamically simulating hcp-liquid coexistence from a seeded crystal nucleus. We further quantify the dependence of the elastic constants on the model control parameters in two and three dimensions, showing how the degree of elastic anisotropy can be tuned from the shape of the direct correlation functions.

Unconditionally gradient-stable computational schemes in problems of fast phase transitions

Equations of fast phase transitions, in which the phase boundaries move with velocities comparable with the atomic diffusion speed or with the speed of local structural relaxation, are analyzed. These equations have a singular perturbation due to the second derivative of the order parameter with respect to time, which appears due to phenomenologically introduced local nonequilibrium. To develop unconditionally stable computational schemes, the Eyre theorem [D. J. Eyre, unpublished] proved for the classical equations, based on hypotheses of local equilibrium, is used. An extension of the Eyre theorem for the case of equations for fast phase transitions is given. It is shown that the expansion of the free energy on contractive and expansive parts, suggested by Eyre for the classical equations of Cahn-Hilliard and Allen-Cahn, is also true for the equations of fast phase transitions. Grid approximations of these equations lead to gradient-stable algorithms with an arbitrary time step for numerical modeling, ensuring monotonic nonincrease of the free energy. Special examples demonstrating the extended Eyre theorem for fast phase transitions are considered.

Dynamics of dislocations in a two-dimensional block copolymer system with hexagonal symmetry

Block copolymer thin films have attracted considerable attention for their ability to self-assemble into nanometre-scale architectures. Recent advances in the use of block copolymer thin films as nano-lithographic masks have driven research efforts in order to have better control of long-range ordering in the plane of the film. Irrespective of the method of sample preparation, different quasi-two-dimensional systems with hexagonal symmetry unavoidably contain translational defects, called dislocations. Dislocations control the process of coarsening in the nano/meso-scales and provide one of the most important mechanisms of length-scale selection in hexagonal patterns. Although in the last decade the nonlinear dynamics of topological defects in quasi-two-dimensional systems has witnessed significant progress, still little is known about the role of external fields on the creation and annihilation mechanisms involved in the relaxation process towards equilibrium states. In this paper, the dynamics of dislocations in non-optimal hexagonal patterns is studied in the framework of the Ohta-Kawasaki model for a diblock copolymer. Measurements of the climb and glide velocities as a function of the wave vector deformation reveal the main mechanisms of relaxation associated with the motion of dislocations.

Eighth-order phase-field-crystal model for two-dimensional crystallization

We present a derivation of the recently proposed eighth-order phase-field crystal model [A. Jaatinen, Phys. Rev. E 80, 031602 (2009)] for the crystallization of a solid from an undercooled melt. The model is used to study the planar growth of a two-dimensional hexagonal crystal, and the results are compared against similar results from dynamical density functional theory of Marconi and Tarazona, as well as other phase-field crystal models. We find that among the phase-field crystal models studied, the eighth-order fitting scheme gives results in good agreement with the density functional theory for both static and dynamic properties, suggesting it is an accurate and computationally efficient approximation to the density functional theory.

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.

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.

Plasticity and dislocation dynamics in a phase field crystal model

The critical dynamics of dislocation avalanches in plastic flow is examined using a phase field crystal model. In the model, dislocations are naturally created, without any ad hoc creation rules, by applying a shearing force to the perfectly periodic ground state. These dislocations diffuse, interact and annihilate with one another, forming avalanche events. By data collapsing the event energy probability density function for different shearing rates, a connection to interface depinning dynamics is confirmed. The relevant critical exponents agree with mean field theory predictions.

Phase-field-crystal model for fcc ordering

We develop and analyze a two-mode phase-field-crystal model to describe fcc ordering. The model is formulated by coupling two different sets of crystal density waves corresponding to <111> and <200> reciprocal lattice vectors, which are chosen to form triads so as to produce a simple free-energy landscape with coexistence of crystal and liquid phases. The feasibility of the approach is demonstrated with numerical examples of polycrystalline and (111) twin growth. We use a two-mode amplitude expansion to characterize analytically the free-energy landscape of the model, identifying parameter ranges where fcc is stable or metastable with respect to bcc. In addition, we derive analytical expressions for the elastic constants for both fcc and bcc. Those expressions show that a nonvanishing amplitude of [200] density waves is essential to obtain mechanically stable fcc crystals with a nonvanishing tetragonal shear modulus (C11-C12)/2. We determine the model parameters for specific materials by fitting the peak liquid structure factor properties and solid-density wave amplitudes following the approach developed for bcc [K.-A. Wu and A. Karma, Phys. Rev. B 76, 184107 (2007)]. This procedure yields reasonable predictions of elastic constants for both bcc Fe and fcc Ni using input parameters from molecular dynamics simulations. The application of the model to two-dimensional square lattices is also briefly examined.

Extended phase diagram of the three-dimensional phase field crystal model

We determine the phase diagram of the phase field crystal model in three dimensions by using numerical free energy minimization methods. Previously published results, based on single mode approximations, have indicated that in addition to the uniform (liquid) phase, there would be regions of stability of body-centered cubic, hexagonal and stripe phases. We find that in addition to these, there are also regions of stability of face-centered cubic and hexagonal close packed structures in this model.

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.

Effect of noise-induced nucleation on grain size distribution studied via the phase-field crystal method

We contribute to the more detailed understanding of the phase-field crystal model recently developed by Elder et al (2002 Phys. Rev. Lett. 88 245701), by focusing on its noise term and examining its impact on the nucleation rate in a homogeneously solidifying system as well as on successively developing grain size distributions. In this context we show that principally the grain size decreases with increasing noise amplitude, resulting in both a smaller average grain size and a decreased maximum grain size. Despite this general tendency, which we interpret based on Panfilis and Filiponi (2000 J. Appl. Phys. 88 562), we can identify two different regimes in which nucleation and successive initial growth are governed by quite different mechanisms.

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.

An anisotropic phase-field crystal model for heterogeneous nucleation of ellipsoidal colloids

We derive a generalized model for isotropic as well as anisotropic crystal lattice systems of arbitrary Poisson ratio within the framework of the continuum phase-field crystal (PFC) approach (Elder and Grant 2004 Phys. Rev. E 70 051606). To this end we extend the simplest PFC model defined by a free energy functional, which is based upon the Swift-Hohenberg model of pattern formation (Swift and Hohenberg 1993 Phys. Rev. A 15 851) to a conservative, anisotropic Langevin equation. By studying heterogeneous nucleation of ellipsoidal colloids at a wall, we demonstrate the capacity of our approach to contribute to the more precise understanding of condensed matter systems built up from non-spherical units at the atomic scale. In particular we address the question of how (a) the orientation of the ellipsoids as well as (b) the interaction potential with the wall determine the resulting contact angle.

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.

Phase-field-crystal and Swift-Hohenberg equations with fast dynamics

A phenomenological description of transition from an unstable to a (meta)stable phase state, including microscopic and mesoscopic scales, is presented. It is based on the introduction of specific memory functions which take into account contributions to the driving force of transformation from the past. A region of applicability for phase-field crystals and Swift-Hohenberg-type models is extended by inclusion of inertia effects into the equations of motion through a memory function of an exponential form. The inertia allows us to predict fast degrees of freedom in the form of damping perturbations with finite relaxation time in the instability of homogeneous and periodic model solutions.

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.

Atomistic studies of strain relaxation in heteroepitaxial systems

We present a review of recent theoretical studies of different atomistic mechanisms of strain relaxation in heteroepitaxial systems. We explore these systems in two and three dimensions using different semi-empirical interatomic potentials of Lennard-Jones and many-body embedded atom model type. In all cases we use a universal molecular static method for generating minimum energy paths for transitions from the coherent epitaxial (defect free) state to the state containing an isolated defect (localized or extended). This is followed by a systematic search for the minimum energy configuration as well as self-organization in the case of a periodic array of islands. In this way we are able to understand many general features of the atomic mechanisms and energetics of strain relaxation in these systems. Finally, for the special case of Pd/Cu(100) and Cu/Pd(100) heteroepitaxy we also use conventional molecular dynamics simulation techniques to compare the compressively and tensilely strained cases. The results for this case are in good agreement with the existing experimental data.

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.

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

We implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes. Our numerical approach uses the systematic renormalization-group formulation of a phase-field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field--a quantity that is spatially uniform except near topological defects, grain boundaries, and other lattice imperfections. Our algorithm employs a hybrid formulation of the amplitude equations, combining Cartesian and polar decompositions of the complex amplitude. We show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions.

Phase-field crystals with elastic interactions

We report on a novel extension of the recently introduced phase-field crystal (PFC) method [Elder, Phys. Rev. Lett. 88, 245701 (2002)10.1103/PhysRevLett.88.245701], which incorporates elastic interactions as well as crystal plasticity and diffusive dynamics. In our model, elastic interactions are mediated through wave modes that propagate on time scales many orders of magnitude slower than atomic vibrations but still much faster than diffusive time scales. This allows us to preserve the quintessential advantage of the PFC model: the ability to simulate atomic-scale interactions and dynamics on time scales many orders of magnitude longer than characteristic vibrational time scales. We demonstrate the two different modes of propagation in our model and show that simulations of grain growth and elastoplastic deformation are consistent with the microstructural properties of nanocrystals.