Phase diagram and commensurate-incommensurate transitions in the phase field crystal model with an external pinning potential

Nonequilibrium dynamics of a two-defect system under severe load

In the framework of a two-defect model, based on a variation of the Landau technique, the kinetics of structural defect generation in solids under severe external load is investigated. The approach is based on a special form of kinetic equation in terms of internal energy, which is applied here to the description of an important practical problem of fine-grained structure formation in metals under severe plastic deformation. It unifies strengthening curves over the entire range of deformation, including the Hall-Petch and linear strengthening sections.

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.

Rotation-induced grain growth and stagnation in phase-field crystal models

We consider grain growth and stagnation in polycrystalline microstructures. From the phase-field crystal modeling of the coarsening dynamics, we identify a transition from a grain-growth stagnation upon deep quenching below the melting temperature T(m) to a continuous coarsening at shallower quenching near T(m). The grain evolution is mediated by local grain rotations. In the deep quenching regime, the grain assembly typically reaches a metastable state where the kinetic barrier for recrystallization across boundaries is too large and grain rotation with subsequent coalescence or boundary motion is infeasible. For quenching near T(m), we find that the grain growth depends on the average rate of grain rotation, and follows a power-law behavior with time, with a scaling exponent that depends on the quenching depth.

Vacancy diffusion in colloidal crystals as determined by dynamical density-functional theory and the phase-field-crystal model

A two-dimensional crystal of repulsive dipolar particles is studied in the vicinity of its melting transition by using Brownian dynamics computer simulation, dynamical density-functional theory, and phase-field-crystal modeling. A vacancy is created by taking out a particle from an equilibrated crystal, and the relaxation dynamics of the vacancy is followed by monitoring the time-dependent one-particle density. We find that the vacancy is quickly filled up by diffusive hopping of neighboring particles towards the vacancy center. We examine the temperature dependence of the diffusion constant and find that it decreases with decreasing temperature in the simulations. This trend is reproduced by the dynamical density-functional theory. Conversely, the phase-field-crystal calculations predict the opposite trend. Therefore, the phase-field model needs a temperature-dependent expression for the mobility to predict trends correctly.

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.

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.

Glassy phases and driven response of the phase-field-crystal model with random pinning

We study the structural correlations and the nonlinear response to a driving force of a two-dimensional phase-field-crystal model with random pinning. The model provides an effective continuous description of lattice systems in the presence of disordered external pinning centers, allowing for both elastic and plastic deformations. We find that the phase-field crystal with disorder assumes an amorphous glassy ground state, with only short-ranged positional and orientational correlations, even in the limit of weak disorder. Under increasing driving force, the pinned amorphous-glass phase evolves into a moving plastic-flow phase and then, finally, a moving smectic phase. The transverse response of the moving smectic phase shows a vanishing transverse critical force for increasing system sizes.

Nonequilibrium evolution thermodynamics of vacancies

An alternative approach, nonequilibrium evolution thermodynamics, is compared with the classical Landau approach. A statistical justification of the approach is done with the help of a probability distribution function on an example of a solid with vacancies. Two kinds of kinetic equations are derived in terms of the internal energy and the modified free energy.

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.

Phase-field crystal modeling of compositional domain formation in ultrathin films

Bulk-immiscible binary systems often form stress-induced miscible alloy phases when deposited on a substrate. Both alloying and surface dislocation formation lead to the decrease of the elastic strain energy, and the competition between these two strain-relaxation mechanisms gives rise to the emergence of pseudomorphic compositional nanoscale domains, often coexisting with a partially coherent single phase. In this work, we develop a phase-field crystal model for compositional patterning in monolayer aggregates of binary metallic systems. We first demonstrate that the model naturally incorporates the competition between alloying and misfit dislocations, and quantify the effects of misfit and line tension on equilibrium domain size. Then, we quantitatively relate the parameters of the phase-field crystal model to a specific system, CoAg/Ru(0001), and demonstrate that the simulations capture experimentally observed morphologies.

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.

Polymorphism, crystal nucleation and growth in the phase-field crystal model in 2D and 3D

We apply a simple dynamical density functional theory, the phase-field crystal (PFC) model of overdamped conservative dynamics, to address polymorphism, crystal nucleation, and crystal growth in the diffusion-controlled limit. We refine the phase diagram for 3D, and determine the line free energy in 2D and the height of the nucleation barrier in 2D and 3D for homogeneous and heterogeneous nucleation by solving the respective Euler-Lagrange (EL) equations. We demonstrate that, in the PFC model, the body-centered cubic (bcc), the face-centered cubic (fcc), and the hexagonal close-packed structures (hcp) compete, while the simple cubic structure is unstable, and that phase preference can be tuned by changing the model parameters: close to the critical point the bcc structure is stable, while far from the critical point the fcc prevails, with an hcp stability domain in between. We note that with increasing distance from the critical point the equilibrium shapes vary from the sphere to specific faceted shapes: rhombic dodecahedron (bcc), truncated octahedron (fcc), and hexagonal prism (hcp). Solving the equation of motion of the PFC model supplied with conserved noise, solidification starts with the nucleation of an amorphous precursor phase, into which the stable crystalline phase nucleates. The growth rate is found to be time dependent and anisotropic; this anisotropy depends on the driving force. We show that due to the diffusion-controlled growth mechanism, which is especially relevant for crystal aggregation in colloidal systems, dendritic growth structures evolve in large-scale isothermal single-component PFC simulations. An oscillatory effective pair potential resembling those for model glass formers has been evaluated from structural data of the amorphous phase obtained by instantaneous quenching. Finally, we present results for eutectic solidification in a binary PFC model.

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.

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.

Formation of the internal structure of solids under severe load

An alternative form of kinetic equations, involving the internal and free energies symmetrically, has been derived in the framework of the theory of vacancies. Dynamical nature of irreversible phenomena during formation and motion of defects (dislocations) has been analyzed by a computer experiment. Results of this simulation are then extended into a thermodynamic identity, involving the law of conservation of energy at interaction with an environment (the first law of thermodynamics) and the law of energy transformation in the internal degrees of freedom (relaxation). This identity is compared to the analogous Jarzynski identity. The approach is illustrated by simulation of processes during severe plastic deformation; the Rybin kinetic equation for this case has been derived.

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.

Dynamical transitions and sliding friction of the phase-field-crystal model with pinning

We study the nonlinear driven response and sliding friction behavior of the phase-field-crystal (PFC) model with pinning including both thermal fluctuations and inertial effects. The model provides a continuous description of adsorbed layers on a substrate under the action of an external driving force at finite temperatures, allowing for both elastic and plastic deformations. We derive general stochastic dynamical equations for the particle and momentum densities including both thermal fluctuations and inertial effects. The resulting coupled equations for the PFC model are studied numerically. At sufficiently low temperatures, we find that the velocity response of an initially pinned commensurate layer shows hysteresis with dynamical melting and freezing transitions for increasing and decreasing applied forces at different critical values. The main features of the nonlinear response in the PFC model are similar to the results obtained previously with molecular dynamics simulations of particle models for adsorbed layers.

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.

Dynamics of colloids in a narrow channel driven by a nonuniform force

Using Brownian dynamics simulations, we investigate the dynamics of colloids confined in two-dimensional narrow channels driven by a nonuniform force Fdr(y) . We considered linear-gradient, parabolic, and deltalike driving-force profiles. This driving force induces melting of the colloidal solid (i.e., shear-induced melting), and the colloidal motion experiences a transition from elastic to plastic regime with increasing Fdr. For intermediate Fdr (i.e., in the transition region) the response of the system, i.e., the distribution of the velocities of the colloidal chains upsiloni(y) , in general does not coincide with the profile of the driving force Fdr(y), and depends on the magnitude of Fdr, the width of the channel, and the density of colloids. For example, we show that the onset of plasticity is first observed near the boundaries while the motion in the central region is elastic. This is explained by: (i) (in)commensurability between the chains due to the larger density of colloids near the boundaries, and (ii) the gradient in Fdr. Our study provides a deeper understanding of the dynamics of colloids in channels and could be accessed in experiments on colloids (or in dusty plasma) with, e.g., asymmetric channels or in the presence of a gradient potential field.

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.

Nonlinear driven response of a phase-field crystal in a periodic pinning potential

We study numerically the phase diagram and the response under a driving force of the phase field crystal model for pinned lattice systems introduced recently for both one- and two-dimensional systems. The model describes the lattice system as a continuous density field in the presence of a periodic pinning potential, allowing for both elastic and plastic deformations of the lattice. We first present results for phase diagrams of the model in the absence of a driving force. The nonlinear response to a driving force on an initially pinned commensurate phase is then studied via overdamped dynamic equations of motion for different values of mismatch and pinning strengths. For large pinning strength the driven depinning transitions are continuous, and the sliding velocity varies with the force from the threshold with power-law exponents in agreement with analytical predictions. Transverse depinning transitions in the moving state are also found in two dimensions. Surprisingly, for sufficiently weak pinning potential we find a discontinuous depinning transition with hysteresis even in one dimension under overdamped dynamics. We also characterize structural changes of the system in some detail close to the depinning transition.

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.

Thermal fluctuations and phase diagrams of the phase-field crystal model with pinning

We study the influence of thermal fluctuations in the phase diagram of a recently introduced two-dimensional phase field crystal model with an external pinning potential. The model provides a continuum description of pinned lattice systems allowing for both elastic deformations and topological defects. We introduce a nonconserved version of the model and determine the ground-state phase diagram as a function of lattice mismatch and strength of the pinning potential. Monte Carlo simulations are used to determine the phase diagram as a function of temperature near commensurate phases. The results show a rich phase diagram with commensurate, incommensurate, and liquidlike phases with a topology strongly dependent on the type of ordered structure. A finite-size scaling analysis of the melting transition for the c(2x2) commensurate phase shows that the thermal correlation length exponent nu and specific heat behavior are consistent with the Ising universality class as expected from analytical arguments.

Superparamagnetic colloids confined in narrow corrugated substrates

We report a Brownian dynamics simulation study of the structure and dynamics of superparamagnetic colloids subject to external substrate potentials and confined in narrow channels. Our study is motivated by the importance of phenomena like commensurable-incommensurable phase transitions, anomalous diffusion, and stochastic activation processes that are closely related to the system under investigation. We focus mainly on the role of the substrate in the order-disorder mechanisms that lead to a rich variety of commensurate and incommensurate phases, as well as its effect on the single-file diffusion in interacting systems and the depinning transition in one dimension.