In this paper, a two-dimensional phase field crystal model of graphene and hexagonal boron nitride (hBN) is extended to include out-of-plane deformations in stacked multilayer systems. As proof of principle, the model is shown analytically to reduce to standard models of flexible sheets in the small deformation limit. Applications to strained sheets, dislocation dipoles, and grain boundaries are used to validate the behavior of a single flexible graphene layer. For multilayer systems, parameters are obtained to match existing theoretical density functional theory calculations for graphene/graphene, hBN/hBN, and graphene/hBN bilayers. More precisely, it is shown that the parameters can be chosen to closely match the stacking energies and layer spacing calculated by Zhou et al. [Phys. Rev. B 92, 155438 (2015)]. Further validation of the model is presented in a study of rotated graphene bilayers and stacking boundaries. The flexibility of the model is illustrated by simulations that highlight the impact of complex microstructures in one layer on the other layer in a graphene/graphene bilayer.

We present an approximate second-order consistent smoothed particle hydrodynamics method which uses the 1D solutions to approximate the 2D second order derivatives. The numerical tests of the analytic functions show that the method is exact for regular arrangements of interpolation points, while in the disordered areas the accuracy is lower than the exact solution of the second-order consistent modified smoothed particle hydrodynamics, but still better that the standard version or the so-called decoupled finite particle method. We applied the new model to the flow of a fluid around a circular solid obstacle and found that the use of a corrected semi-decoupled second-order consistent SPH gives better accuracy for lower resolutions allowing for a more efficient numerical model and also easier to extend to 3D.

While the microscopic structure of defected solid crystalline materials has significant impact on their physical properties, efficient and accurate determination of a given polycrystalline microstructure remains a challenge. In this paper, we present a highly generalizable and reliable variational method to achieve this goal for two-dimensional crystalline and quasicrystalline materials. The method is benchmarked and optimized successfully using a variety of large-scale systems of defected solids, including periodic structures and quasicrystalline symmetries to quantify their microstructural characteristics, e.g., grain size and lattice misorientation distributions. We find that many microstructural properties show universal features independent of the underlying symmetries.

We explore the growth of two-dimensional quasicrystals, i.e., aperiodic structures that possess long-range order, from two seeds at various distances and with different orientations by using dynamical phase-field crystal calculations. We compare the results to the growth of periodic crystals from two seeds. There, a domain border consisting of dislocations is observed in case of large distances between the seed and large angles between their orientation. Furthermore, a domain border is found if the seeds are placed at a distance that does not fit to the periodic lattice. In the case of the growth of quasicrystals, we only observe domain borders for large distances and different orientations. Note that all distances do inherently not match to a perfect domain wall-free quasicrystalline structure. Nevertheless, we find dislocation-free growth for all seeds at a small enough distance and for all seeds that approximately have the same orientation. In periodic structures, the stress that occurs due to incommensurate distances between the seeds results in phononic strain fields or, in the case of too large stresses, in dislocations. In contrast, in quasicrystals an additional phasonic strain field can occur and suppress dislocations. Phasons are additional degrees of freedom that are unique to quasicrystals. As a consequence, the additional phasonic strain field helps to distribute the stress and facilitates the growth of dislocation-free quasicrystals from multiple seeds. In contrast, in the periodic case the growth from multiple seeds most likely leads to a structure with multiple domains. Our work lays the theoretical foundations for growing perfect quasicrystals from different seeds and is therefore relevant for many applications.

Atomistically thin adsorbate layers on surfaces with a lattice mismatch display complex spatial patterns and ordering due to strain-driven self-organization. In this work, a general formalism to model such ultrathin adsorption layers that properly takes into account the competition between strain and adhesion energy of the layers is presented. The model is based on the amplitude expansion of the two-dimensional phase field crystal (PFC) model, which retains atomistic length scales but allows relaxation of the layers at diffusive time scales. The specific systems considered here include cases where both the film and the adsorption potential can have either honeycomb (H) or triangular (T) symmetry. These systems include the so-called $(1\ifmmode\times\else\texttimes\fi{}1)$, $(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})\phantom{\rule{0.16em}{0ex}}\mathrm{R}{30}^{\ensuremath{\circ}}$, $(2\ifmmode\times\else\texttimes\fi{}2)$, $(\sqrt{7}\ifmmode\times\else\texttimes\fi{}\sqrt{7})\phantom{\rule{0.16em}{0ex}}\mathrm{R}19.{1}^{\ensuremath{\circ}}$, and other higher order states that can contain a multitude of degenerate commensurate ground states. The relevant phase diagrams for many combinations of the H and T systems are mapped out as a function of adhesion strength and misfit strain. The coarsening patterns in some of these systems is also examined. The predictions are in good agreement with existing experimental data for selected strained ultrathin adsorption layers.

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.

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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.

We introduce an effective one-mode phase-field crystal model for studying the commensurate-incommensurate transition and domain wall dynamics of the phase found in systems such as Xe/Pt(111), or Xe and Kr on graphite. The model allows us to study large systems where the domain walls can be separated over large macroscopic distances and at the same time incorporate the microscopic details of the domain wall structures. The resulting phase diagram shows that an intermediate stripe incommensurate phase always separates the commensurate phase from the honeycomb incommensurate phases. The energy of the domain wall crossing is investigated. We also find that near a step edge, the domain walls tend to align perpendicularly to the step edge, in agreement with recent experimental observations.

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We study the phase diagram and the commensurate-incommensurate phase transitions of a twodimensional phase field crystal model for adsorbed layers. The model allows for both elastic and plastic deformations on atomic and diffusive time-scales, and provides a continuum description of lattice systems, such as adsorbed atomic layers or two-dimensional vortex lattices. Analytically, mode expansion analysis and numerical minimization of the free energy are used to determine the ground states as a function of the pinning potential and lattice mismatch parameter. The results show a rich phase diagram with several different types of commensurate and incommensurate phases.

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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.

The growth of quasicrystals, i.e., aperiodic structures with long-range order, seeded from the melt is investigated using a dynamical phase field crystal model. Depending on the thermodynamic conditions, two different growth modes are detected, namely defect-free growth of the stable quasicrystal and a mode dominated by phasonic flips which are incorporated as local defects into the grown structure such that random tilinglike ordering emerges. The latter growth mode is unique to quasicrystals and can be verified in experiments on one-component mesoscopic systems.

A computational study of the self-organization of heteroepitaxial ultrathin metal films is presented. By means of a continuum complex field model, the relationship of the equilibrium surface patterns of the film to the adsorbate-substrate adhesion energy, as well as to the mismatch between the adsorbate and the substrate bulk lattice parameters, are obtained in both the tensile and the compressive regimes. Our approach captures pattern periodicities over large length scales, up to several hundreds of nm, retaining atomistic resolution. Thus, the results can be directly compared with experimental data, in particular for systems such as Cu/Ru(0001) and Ag/Cu(111). Three nontrivial, stable superstructures for the overlayer, namely, stripe, honeycomb, and triangular, are identified that closely resemble those observed experimentally. Simulations in nonequilibrium conditions are performed as well to identify metastable structural configurations and the dynamics of ordering of the overlayer.