Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model

Pattern Formation under Deep Supercooling by Classical Density Functional-Based Approach

Solidification patterns during nonequilibrium crystallization are among the most important microstructures in the natural and technical realms. In this work, we investigate the crystal growth in deeply supercooled liquid using the classical density functional-based approaches. Our result shows that the complex amplitude expanded phase-field crystal (APFC) model containing the vacancy nonequilibrium effects proposed by us could naturally reproduce the growth front nucleation (GFN) and various nonequilibrium patterns, including the faceted growth, spherulite, symmetric and nonsymmetric dendrites among others, at the atom level. Moreover, an extraordinary microscopic columnar-to-equiaxed transition is uncovered, which is found to depend on the seed spacing and distribution. Such a phenomenon could be attributed to the combined effects of the long-wave and short-wave elastic interactions. Particularly, the columnar growth could also be predicted by an APFC model containing inertia effects, but the lattice defect type in the growing crystal is different due to the different types of short-wave interactions. Two stages are identified during the crystal growth under different undercooling, corresponding to diffusion-controlled growth and GFN-dominated growth, respectively. However, compared with the second stage, the first stage becomes too short to be noticed under the high undercooling. The distinct feature of the second stage is the dramatic increments of lattice defects, which explains the amorphous nucleation precursor in the supercooled liquid. The transition time between the two stages at different undercooling is investigated. Crystal growth of BCC structure further confirms our conclusions.

Simulation of TSV Protrusion in 3DIC Integration by Directly Loading on Coarse-Grained Phase-Field Crystal Model

As thermal management in 3DIC integration becomes increasingly important in advanced semiconductor node processes, novel experimental and modeling approaches are in great demand to reveal the critical material issues involving multiscale microstructures that govern the behavior of through-silicon-via (TSV) protrusion. Here, a coarse-grained phase-field crystal model properly coupled with mechanics through the atomic density field is used to simulate the formation of polycrystalline structures and protrusion of nano-TSVs from the atomic scale. TSVs with different grain structures are directly loaded, and protrusion/intrusion profiles are obtained along with displacement, stress, and strain fields. Thermodynamic driving forces from external loadings and the mismatch of Young’s modulus between adjoining grains as well as detailed displacement and strain distributions are ascribed to control the complex deformation in TSVs. TSVs with sizes up to around 30 nm and an aspect ratio of 4 are successfully investigated, and a further increase in the size and aspect ratio to cover the micrometer range is feasible, which lays down a solid basis toward a multiscale material database for simulation inputs to the design of TSV-based 3DIC integration and relevant electronic design automation (EDA) tools.

Mesoscale Defect Motion in Binary Systems: Effects of Compositional Strain and Cottrell Atmospheres

The velocity of dislocations is derived analytically to incorporate and predict the intriguing effects induced by the preferential solute segregation and Cottrell atmospheres in both two-dimensional and three-dimensional binary systems of various crystalline symmetries. The corresponding mesoscopic description of defect dynamics is constructed through the amplitude formulation of the phase-field crystal model, which has been shown to accurately capture elasticity and plasticity in a wide variety of systems. Modifications of the Peach-Koehler force as a result of solute concentration variations and compositional stresses are presented, leading to interesting new predictions of defect motion due to effects of Cottrell atmospheres. These include the deflection of dislocation glide paths, the variation of climb speed and direction, and the change or prevention of defect annihilation, all of which play an important role in determining the fundamental behaviors of complex defect network and dynamics. The analytic results are verified by numerical simulations.

Traveling waves of the solidification and melting of cubic crystal lattices

Using the phase field crystal model (PFC model), an analysis of slow and fast dynamics of solid-liquid interfaces in solidification and melting processes is presented. Dynamical regimes for cubic lattices invading metastable liquids (solidification) and liquids propagating into metastable crystals (melting) are described in terms of the evolving amplitudes of the density field. Dynamical equations are obtained for body-centered cubic (bcc) and face-centered cubic (fcc) crystal lattices in one- and two-mode approximations. A universal form of the amplitude equations is obtained for the three-dimensional dynamics for different crystal lattices and crystallographic directions. Dynamics of the amplitude's propagation for different lattices and PFC mode's approximations is qualitatively compared. The traveling-wave velocity is quantitatively compared with data of molecular dynamics simulation previously obtained by Mendelev et al. [Modell. Simul. Mater. Sci. Eng. 18, 074002 (2010)MSMEEU0965-039310.1088/0965-0393/18/7/074002] for solidification and melting of the aluminum fcc lattice.

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.

Tunable mosaic structures in van der Waals layered materials

Intrinsic mosaic structures composed of distinctive stacking domains separated by domain walls (DWs) show the potential to regulate many outstanding properties of van der Waals layered materials. A comprehensive simulation at the atomic scale is performed to explore how the lattice/twist mismatch and the interlayer interaction influence the mosaic configuration from the incommensurate Moiré pattern to commensurate mosaic structures by adapting a complex amplitude version of the phase field crystal method. It is found that after an incommensurate-commensurate transition occurs, the topology of the mosaic structure indicated by different domain wall (DW) patterns can be drastically changed. An experimentally observed intriguing spiral domain wall (SDW) network is revealed as result of the emergent mixed dislocation driven by minimizing the elastic and interlayer energies in the presence of both lattice and twist mismatches. The transition process from a herringbone domain wall (HBDW) network to a SDW network is also simulated, elucidated by a dislocation reaction and in good agreement with the experimental observations.

Travelling-wave amplitudes as solutions of the phase-field crystal equation

The dynamics of the diffuse interface between liquid and solid states is analysed. The diffuse interface is considered as an envelope of atomic density amplitudes as predicted by the phase-field crystal model (Elder et al. 2004 Phys. Rev. E70, 051605 (doi:10.1103/PhysRevE.70.051605); Elder et al. 2007 Phys. Rev. B75, 064107 (doi:10.1103/PhysRevB.75.064107)). The propagation of crystalline amplitudes into metastable liquid is described by the hyperbolic equation of an extended Allen-Cahn type (Galenko & Jou 2005 Phys. Rev. E71, 046125 (doi:10.1103/PhysRevE.71.046125)) for which the complete set of analytical travelling-wave solutions is obtained by the [Formula: see text] method (Malfliet & Hereman 1996 Phys. Scr.15, 563-568 (doi:10.1088/0031-8949/54/6/003); Wazwaz 2004 Appl. Math. Comput.154, 713-723 (doi:10.1016/S0096-3003(03)00745-8)). The general [Formula: see text] solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general [Formula: see text] solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko et al. 2015 Phys. D308, 1-10 (doi:10.1016/j.physd.2015.06.002)). The influence of the driving force on the phase-field profile, amplitude velocity and correlation length is investigated for various relaxation times of the gradient flow.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Bimodal Grain-Size Scaling of Thermal Transport in Polycrystalline Graphene from Large-Scale Molecular Dynamics Simulations

Grain boundaries in graphene are inherent in wafer-scale samples prepared by chemical vapor deposition. They can strongly influence the mechanical properties and electronic and heat transport in graphene. In this work, we employ extensive molecular dynamics simulations to study thermal transport in large suspended polycrystalline graphene samples. Samples of different controlled grain sizes are prepared by a recently developed efficient multiscale approach based on the phase field crystal model. In contrast to previous works, our results show that the scaling of the thermal conductivity with the grain size implies bimodal behavior with two effective Kapitza lengths. The scaling is dominated by the out-of-plane (flexural) phonons with a Kapitza length that is an order of magnitude larger than that of the in-plane phonons. We also show that, to get quantitative agreement with the most recent experiments, quantum corrections need to be applied to both the Kapitza conductance of grain boundaries and the thermal conductivity of pristine graphene, and the corresponding Kapitza lengths must be renormalized accordingly.

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.

Energetics and structure of grain boundary triple junctions in graphene

Grain boundary triple junctions are a key structural element in polycrystalline materials. They are involved in the formation of microstructures and can influence the mechanical and electronic properties of materials. In this work we study the structure and energetics of triple junctions in graphene using a multiscale modelling approach based on combining the phase field crystal approach with classical molecular dynamics simulations and quantum-mechanical density functional theory calculations. We focus on the atomic structure and formation energy of the triple junctions as a function of the misorientation between the adjacent grains. We find that the triple junctions in graphene consist mostly of five-fold and seven-fold carbon rings. Most importantly, in addition to positive triple junction formation energies we also find a significant number of orientations for which the formation energy is negative.

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.

Long-wavelength properties of phase-field-crystal models with second-order dynamics

The phase-field-crystal (PFC) approach extends the notion of phase-field models by describing the topology of the microscopic structure of a crystalline material. One of the consequences is that local variation of the interatomic distance creates an elastic excitation. The dynamics of these excitations poses a challenge: pure diffusive dynamics cannot describe relaxation of elastic stresses that happen through phonon emission. To this end, several different models with fast dynamics have been proposed. In this article we use the amplitude expansion of the PFC model to compare the recently proposed hydrodynamic PFC amplitude model with two simpler models with fast dynamics. We compare these different models analytically and numerically. The results suggest that in order to have proper relaxation of elastic excitations, the full hydrodynamical description of the PFC amplitudes is required.

Scaling of alloy interfacial properties under compositional strain

Complex morphologies and microstructures that emerge during materials growth and solidification are often determined by both equilibrium and kinetic properties of the interface and their crystalline anisotropies. However, limited knowledge is available on alloying and, particularly, compositionally generated elastic effects on these interface characteristics. Here we systematically investigate such compositional effects on the interfacial properties of an alloy model system based on a phase-field-crystal analysis, including the solid-liquid interfacial free energy, kinetic coefficient, and lattice pinning strength. Scaling relations for these interfacial quantities over various ranges of material parameters are identified and predicted. Our results indicate the important effects of couplings among mesoscopic and microscopic length scales of alloy structure and concentration, and the influence of compressive and tensile interface stresses induced by composition variations. The approach developed here provides an efficient way to systematically identify these key material properties beyond the traditional atomistic and continuum methods.

Consistent Hydrodynamics for Phase Field Crystals

We use the amplitude expansion in the phase field crystal framework to formulate an approach where the fields describing the microscopic structure of the material are coupled to a hydrodynamic velocity field. The model is shown to reduce to the well-known macroscopic theories in appropriate limits, including compressible Navier-Stokes and wave equations. Moreover, we show that the dynamics proposed allows for long wavelength phonon modes and demonstrate the theory numerically showing that the elastic excitations in the system are relaxed through phonon emission.

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.

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.

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

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

Exploring the complex world of two-dimensional ordering with three modes

The world of two-dimensional crystals is of great significance for the design and study of structural and functional materials with novel properties. Here we examine the mechanisms governing the formation and dynamics of these crystalline or polycrystalline states and their elastic and plastic properties by constructing a generic multimode phase field crystal model. Our results demonstrate that a system with three competing length scales can order into all five Bravais lattices, and other more complex structures including honeycomb, kagome, and other hybrid phases. In addition, nonequilibrium phase transitions are examined to illustrate the complex phase behavior described by the model. This model provides a systematic path to predict the influence of lattice symmetry on both the structure and dynamics of crystalline and defected systems.

Scale-coupling and interface-pinning effects in the phase-field-crystal model

Effects of scale coupling between mesoscopic slowly varying envelopes of liquid-solid profile and the underlying microscopic crystalline structure are studied in the phase-field-crystal (PFC) model. Such scale coupling leads to nonadiabatic corrections to the PFC amplitude equations, the effect of which increases strongly with decreasing system temperature below the melting point. This nonadiabatic amplitude representation is further coarse-grained for the derivation of effective sharp-interface equations of motion in the limit of small but finite interface thickness. We identify a generalized form of the Gibbs-Thomson relation with the incorporation of coupling and pinning effects of the crystalline lattice structure. This generalized interface equation can be reduced to the form of a driven sine-Gordon equation with Kardar-Parisi-Zhang (KPZ) nonlinearity, and can be combined with two other dynamic equations in the sharp interface limit obeying the conservation condition of atomic number density in a liquid-solid system. A sample application to the study of crystal layer growth is given, and the corresponding analytic solutions showing lattice pinning and depinning effects and two distinct modes of continuous vs nucleated growth are presented. We also identify the universal scaling behaviors governing the properties of pinning strength, surface tension, interface kinetic coefficient, and activation energy of atomic layer growth, which accommodate all range of liquid-solid interface thicknesses and different material elastic moduli.

Description of hard-sphere crystals and crystal-fluid interfaces: a comparison between density functional approaches and a phase-field crystal model

In materials science the phase-field crystal approach has become popular to model crystallization processes. Phase-field crystal models are in essence Landau-Ginzburg-type models, which should be derivable from the underlying microscopic description of the system in question. We present a study on classical density functional theory in three stages of approximation leading to a specific phase-field crystal model, and we discuss the limits of applicability of the models that result from these approximations. As a test system we have chosen the three-dimensional suspension of monodisperse hard spheres. The levels of density functional theory that we discuss are fundamental measure theory, a second-order Taylor expansion thereof, and a minimal phase-field crystal model. We have computed coexistence densities, vacancy concentrations in the crystalline phase, interfacial tensions, and interfacial order parameter profiles, and we compare these quantities to simulation results. We also suggest a procedure to fit the free parameters of the phase-field crystal model. Thereby it turns out that the order parameter of the phase-field crystal model is more consistent with a smeared density field (shifted and rescaled) than with the shifted and rescaled density itself. In brief, we conclude that fundamental measure theory is very accurate and can serve as a benchmark for the other theories. Taylor expansion strongly affects free energies, surface tensions, and vacancy concentrations. Furthermore it is phenomenologically misleading to interpret the phase-field crystal model as stemming directly from Taylor-expanded density functional theory.

Patterning of heteroepitaxial overlayers from nano to micron scales

Thin heteroepitaxial overlayers have been proposed as templates to generate stable, self-organized nanostructures at large length scales, with a variety of important technological applications. However, modeling strain-driven self-organization is a formidable challenge due to different length scales involved. In this Letter, we present a method for predicting the patterning of ultrathin films on micron length scales with atomic resolution. We make quantitative predictions for the type of superstructures (stripes, honeycomb, triangular) and length scale of pattern formation of two metal-metal systems, Cu on Ru(0001) and Cu on Pd(111). Our findings are in excellent agreement with previous experiments and call for future experimental investigations of such systems.

Dynamic density functional theory of solid tumor growth: Preliminary models

Cancer is a disease that can be seen as a complex system whose dynamics and growth result from nonlinear processes coupled across wide ranges of spatio-temporal scales. The current mathematical modeling literature addresses issues at various scales but the development of theoretical methodologies capable of bridging gaps across scales needs further study. We present a new theoretical framework based on Dynamic Density Functional Theory (DDFT) extended, for the first time, to the dynamics of living tissues by accounting for cell density correlations, different cell types, phenotypes and cell birth/death processes, in order to provide a biophysically consistent description of processes across the scales. We present an application of this approach to tumor growth.

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.

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.

Emergence of foams from the breakdown of the phase field crystal model

The phase field crystal (PFC) model captures the elastic and topological properties of crystals with a single scalar field at small undercooling. At large undercooling, new foamlike behavior emerges. We characterize this foam phase of the PFC equation and propose a modified PFC equation that may be used for the simulation of foam dynamics. This minimal model reproduces von Neumann's rule for two-dimensional dry foams and Lifshitz-Slyozov coarsening for wet foams. We also measure the coordination number distribution and find that its second moment is larger than previously reported experimental and theoretical studies of soap froths, a finding that we attribute to the wetness of the foam increasing with time.

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.

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.

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.

Comment on "Renormalization-group theory for the phase-field crystal equation"

Athreya, Goldenfeld, and Dantzig [Phys. Rev. E 74, 011601 (2006)] claim that the current implementation of the renormalization-group method neglects the proper ordering of renormalization and differentiation. Their analysis is, however, based on the wrong multiple-scales method results.

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.

Mesoscopic and microscopic modeling of island formation in strained film epitaxy

The instability of strained films for island formation is examined through an approach incorporating both discrete microscopic details and continuum mechanics. A linear relationship between the island wave number and misfit strain is found for large strains, while only in the small strain limit is a crossover to the continuum elasticity result obtained. A universal scaling relation accommodating all range of misfit strains is identified. Our results indicate that continuum mechanics may break down even at relatively small misfit stress due to the discrete nature of crystalline surfaces.

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.

Analysis and visualization of multiply oriented lattice structures by a two-dimensional continuous wavelet transform

The phase-field-crystal model [K. R. Elder and M. Grant, Phys. Rev. E 70, 051605 (2004)] produces multigrain structures on atomistic length scale but on diffusive time scales. Since individual atoms are resolved but are treated identically it is difficult to distinguish the exact position of grain boundaries and defects within grains. In order to analyze and visualize the whole grains a two-dimensional wavelet transform has been developed, which is capable of extracting grain boundaries and the lattice orientation of a grain relative to a laboratory frame of reference. This transformation makes it possible not only to easily visualize the multigrain structure, but also to perform exact measurements on low- and high-angle boundaries, grain size distributions and boundary-angle distributions, which can then be compared to experimental data. The presented wavelet transform can also be applied to results of other atomistic simulations, e.g., molecular dynamics or granular materials.

Phase-field simulations of velocity selection in rapidly solidified binary alloys

Time-dependent simulations of two-dimensional isothermal Ni-Cu dendrites are simulated using a phase-field model solved with a finite-difference adaptive mesh refinement technique. Dendrite tip velocity selection is examined and found to exhibit a transition between two markedly different regimes as undercooling is increased. At low undercooling, the dendrite tip growth rate is consistent with the kinetics of the classical Stefan problem, where the interface is assume to be in local equilibrium. At high undercooling, the growth velocity selected approaches a linear dependence on melt undercooling, consistent with the continuous growth kinetics of Aziz and with a one-dimensional steady-state phase-field asymptotic analysis of Ahmad [Phys. Rev. E 58, 3436 (1998)]. Our simulations are also consistent with other previously observed behaviors of dendritic growth as undercooling is increased. These include the transition of dendritic morphology to absolute stability and nonequilibrium solute partitioning. Our results show that phase-field models of solidification, which inherently contain a nonzero interface width, can be used to study the dynamics of complex solidification phenomena involving both equilibrium and nonequilibrium interface growth kinetics.

Renormalization-group theory for the phase-field crystal equation

We derive a set of rotationally covariant amplitude equations for use in multiscale simulation of the two-dimensional phase-field crystal model by a variety of renormalization-group (RG) methods. We show that the presence of a conservation law introduces an ambiguity in operator ordering in the RG procedure, which we show how to resolve. We compare our analysis with standard multiple-scale techniques, where identical results can be obtained with greater labor, by going to sixth order in perturbation theory, and by assuming the correct scaling of space and time.

Diffusive atomistic dynamics of edge dislocations in two dimensions

The fundamental dislocation processes of glide, climb, and annihilation are studied on diffusive time scales within the framework of a continuum field theory, the phase field crystal model. Glide and climb are examined for single edge dislocations subjected to shear and compressive strain, respectively, in a two-dimensional hexagonal lattice. It is shown that the natural features of these processes are reproduced without any explicit consideration of elasticity theory or ad hoc construction of microscopic Peierls potentials. Particular attention is paid to the Peierls barrier for dislocation glide or climb and the ensuing dynamic behavior as functions of strain rate, temperature, and dislocation density. It is shown that the dynamics are accurately described by simple viscous motion equations for an overdamped point mass, where the dislocation mobility is the only adjustable parameter. The critical distance for the annihilation of two edge dislocations as a function of separation angle is also presented.

Coarse-graining of cellular automata, emergence, and the predictability of complex systems

We study the predictability of emergent phenomena in complex systems. Using nearest-neighbor, one-dimensional cellular automata (CA) as an example, we show how to construct local coarse-grained descriptions of CA in all classes of Wolfram's classification. The resulting coarse-grained CA that we construct are capable of emulating the large-scale behavior of the original systems without accounting for small-scale details. Several CA that can be coarse-grained by this construction are known to be universal Turing machines; they can emulate any CA or other computing devices and are therefore undecidable. We thus show that because in practice one only seeks coarse-grained information, complex physical systems can be predictable and even decidable at some level of description. The renormalization group flows that we construct induce a hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity and is therefore a good candidate for a complexity measure and a classification method. Finally we argue that the large-scale dynamics of CA can be very simple, at least when measured by the Kolmogorov complexity of the large-scale update rule, and moreover exhibits a novel scaling law. We show that because of this large-scale simplicity, the probability of finding a coarse-grained description of CA approaches unity as one goes to increasingly coarser scales. We interpret this large-scale simplicity as a pattern formation mechanism in which large-scale patterns are forced upon the system by the simplicity of the rules that govern the large-scale dynamics.