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The solid black lines delineate regions of commensurate, striped, and honeycomb equilibrium states. The dashed blue line corresponds to the 1D sine-Gordon solution for the stripe-commensurate transition. The dashed green line is V0*. The data shown in Fig. 1 correspond to wavelength or state selection for Cu/Ru(0001). The relative strain for Cu on Ru(0001) and Pd(111) are also shown in the plot. The red dots from top to bottom correspond to the parameters used in Figs. 3a, 3b, 3c, 3d, respectively.
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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...
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Context 1
... estimate the value of V 0 for a monolayer of Cu/Pd(111) (7.1% mismatch) as discussed in the Supplemental Material [30]. For a single monolayer, Fig. 2 this corresponds to a striped superstructure, which is consistent with experiments and earlier MD studies [18]. The semiempirical calculations also indicate that the ratio E int =E str does indeed decrease as a function of the number of layers, as speculated earlier. While the patterns shown in Fig. 3 are for a mismatch of 5.5%, a ...
Context 2
... this Letter a method for modeling the structure of stable and metastable heteroepitaxial films on length scales of several hundred nanometers with atomic resolution was Fig. 1 and panels (a)-(d) correspond to the red dots shown in Fig. 2 for Cu/Ru(0001). In the inset, the number density field was reconstructed from the amplitudes for the small region enclosed by the white box. The lines in the inside are a guide to the eye to highlight the point dislocation. presented. The results of simulations of the model are consistent with existent experimental studies of ...
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The direct contact of ultrathin polymer films with a solid substrate may result in thin film rupture caused by dewetting. With crystallisable polymers such as polyethyleneoxide (PEO), molecular self-assembly into partial ordered lamella structures is studied as an additional source of pattern formation. Morphological features in ultrathin PEO films...
Citations
... Ultrathin films: strain induced ordering. When a monolayer (or several layers) of one material are grown on a substrate, the lattice mismatch can lead to interesting strain induced patterns [159,160] and the APFC model is ideally suited to model such patterns [28,[161][162][163][164][165]. Their nature depends on the misfit strain, ε m = (a s − a f )/a s , where a s and a f are the substrate and film lattice constants, the relative crystal symmetry of the layer/substrate system and the film/substrate coupling strength. ...
... Sample patterns for the TT system are shown in figures 12(a), (b) and (c). In the case of the 1 × 1 the junction energy is so high that it can create dislocation pairs and lead to zig-zag type patterns [164,165]. ...
Comprehensive investigations of crystalline systems often require methods bridging atomistic and continuum scales. In this context, coarse-grained mesoscale approaches are of particular interest as they allow the examination of large systems and time scales while retaining some microscopic details. The so-called Phase-Field Crystal (PFC) model conveniently describes crystals at diffusive time scales through a continuous periodic field which varies on atomic scales and is related to the atomic number density. To go beyond the restrictive atomic length scales of the PFC model, a complex amplitude formulation was first developed by Goldenfeld et al. [Phys. Rev. E 72, 020601 (2005)]. While focusing on length scales larger than the lattice parameter, this approach can describe crystalline defects, interfaces, and lattice deformations. It has been used to examine many phenomena including liquid/solid fronts, grain boundary energies, and strained films. This topical review focuses on this amplitude expansion of the PFC model and its developments. An overview of the derivation, connection to the continuum limit, representative applications, and extensions is presented. A few practical aspects, such as suitable numerical methods and examples, are illustrated as well. Finally, the capabilities and bounds of the model, current challenges, and future perspectives are addressed.
... 6.5.1. Ultrathin films: strain induced ordering When a monolayer (or several layers) of one material are grown on a substrate, the lattice mismatch can lead to interesting strain induced patterns [156,157] and the APFC model is ideally suited to model such patterns [26,[158][159][160][161][162]. Their nature depends on the misfit strain, ε m = (a s − a f )/a s , where a s and a f are the substrate and film lattice constants, the relative crystal symmetry of the layer/substrate system and the film/substrate coupling strength. ...
... 12(a), (b) and (c). In the case of the 1 × 1 the junction energy is so high that it can create dislocation pairs and lead to zig-zag type patterns [161,162]. ...
Comprehensive investigations of crystalline systems often require methods bridging atomistic and continuum scales. In this context, coarse-grained mesoscale approaches are of particular interest as they allow the examination of large systems and time scales while retaining some microscopic details. The so-called Phase-Field Crystal (PFC) model conveniently describes crystals at diffusive time scales through a continuous periodic field which varies on atomic scales and is related to the atomic number density. To go beyond the restrictive atomic length scales of the PFC model, a complex amplitude formulation was first developed by Goldenfeld et al. [Phys. Rev. E 72, 020601 (2005)]. While focusing on length scales larger than the lattice parameter, this approach can describe crystalline defects, interfaces, and lattice deformations. It has been used to examine many phenomena including liquid/solid fronts, grain boundary energies, and strained films. This topical review focuses on this amplitude expansion of the PFC model and its developments. An overview of the derivation, connection to the continuum limit, representative applications, and extensions is presented. A few practical aspects, such as suitable numerical methods and examples, are illustrated as well. Finally, the capabilities and bounds of the model, current challenges, and future perspectives are addressed.
... The PFC model is an intensively studied microscopic mean field model for the dynamics of crystallization processes on diffusive timescales [53]. It was introduced by Elder and coworkers [54] and applies to passive colloidal particles as well as to atomic systems [55,56]. Mathematically it corresponds to the conserved Swift-Hohenberg (cSH) equation [57] in the form of a continuity equation. ...
The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations, and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.
... For an isothermal one-component system, the free energy is written in the form of Landau-Brazovskii functional. 7,[21][22][23][24] Let one consider the one-mode approximation of the PFC model, which is specified by the operator 1 given by previous studies: 2,25,26 1 = r 0 + (q 2 0 + ∇ 2 ) 2 , where q 0 and r 0 define the stability of crystalline structure and the parameters of reciprocal lattice. We consider the simplest case of q 0 = 1 and r 0 = 0: ...
Modeling of crystal micro‐structures and their dynamics during fast phase transitions can be performed by the phase‐field crystal (PFC) model in the hyperbolic formulation (Modified Phase Field Crystal [MPFC] model). This method is suitable for a continual modeling of the atomic density field at diffusion time intervals (slow diffusion dynamics) and short intervals of atomic flux relaxation (fast structural relaxation). Since the PFC model describes transitions of the first and second order, we present a description of both transitions in a unified manner. We show how phase transitions of each order can be treated using specific analytical transformations. To justify the unified approach to description of the first‐ and second‐order transformations, we provide results of numerical simulation of phase changes between homogeneous structure and Body Centered Cubic (BCC) crystal lattice. The set of benchmarks for different domains shows coincidence of instantaneous atomic distributions and free energies in both forms.
... In the past few years the PFC has been successfully used to many fields of the research [35][36][37][38][39][40][41][42][43][44][45][46][47][48]. Although there have been several studies [48][49][50][51][52][53][54][55] focused on cooperative dislocation movement (CDM) [46,51,52,55] of the GB and grain growth by the PFC approach, so far, the mechanism of the strain-driven CDM of the SGB split from the original GB [13] and also of the strain-driven nucleation and growth of the deformed grain with localized strain energy are still unclear. ...
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.
... The misfit strain plays a particularly important role in the microstructure evolution of large lattice-mismatched systems. The straininduced self-assembly and patterning of nanostructure arrays with characteristic shapes have been observed in heteroepitaxial systems [22][23][24][25][26], revealing the power of the strain in controlling the shape of the heteroepitaxial nanostructures. However, the strain effect on the size evolution of the epilayer in a heteroepitaxial system and the corresponding mechanism are still elusive. ...
We find that the misfit strain may lead to the oscillatory size distributions of heteroepitaxial nanostructures. In heteroepitaxial FePt thin films grown on single-crystal MgO substrate, ⟨110⟩-oriented mazelike and granular patterns with “quantized” feature sizes are realized in scanning-electron-microscope images. The physical mechanism responsible for the size oscillations is related to the oscillatory nature of the misfit strain energy in the domain-matching epitaxial FePt/MgO system, which is observed by transmission electron microscopy. Based on the experimental observations, a model is built and the results suggest that when the FePt island sizes are an integer times the misfit dislocation period, the misfit strain can be completely canceled by the misfit dislocations. With applying the mechanism, small and uniform grain is obtained on the TiN (200) polycrystalline underlayer, which is suitable for practical application. This finding may offer a way to synthesize nanostructured materials with well-controlled size and size distribution by tuning the lattice mismatch between the epitaxial-grown heterostructure.
... The PFC model is an intensively studied microscopic continuum model for the dynamics of crystallization processes on diffusive time scales [38]. It was introduced by Elder and coworkers [39] and is applied for passive colloidal particles but also used for atomic systems [40,41]. Mathematically, it corresponds to the conserved Swift-Hohenberg equation (cSH) [42], i.e., the counterpart with conserved dynamics (i.e., of the form of a continuity equation) of the Swift-Hohenberg (SH) equation that represents non-conserved dynamics [43]. ...
The conserved Swift-Hohenberg equation (or Phase-Field-Crystal [PFC] model) provides a simple microscopic description of the thermodynamic transition between fluid and crystalline states. Combining it with elements of the Toner-Tu theory for self-propelled particles Menzel and L\"owen [Phys. Rev. Lett. 110, 055702 (2013)] obtained a model for crystallization (swarm formation) in active systems. Here, we study the occurrence of resting and traveling localized states, i.e., crystalline clusters, within the resulting active PFC model. Based on linear stability analyses and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of periodic and localized, resting and traveling states in a one-dimensional active PFC model. This allows us, for instance, to explore how the slanted homoclinic snaking of steady localized states found for the passive PFC model is amended by activity. A particular focus lies on the onset of motion, where we show that it occurs either through a drift-pitchfork or a drift-transcritical bifurcation. A corresponding general analytical criterion is derived.
... The phase-field crystal (PFC) model has been used to examine the dynamics of liquid-solid transformation, grain boundary migration and dislocation motion [1][2][3]. The PFC model is a continuum model that describes processes on atomic length scales and pattern on the nano-and micro-length scales [4]. ...
... where the driving force B 0 describes B 0 = B 0 > 0, the transition from a metastable state with K 0 = +1, B 0 < 0, the transition from an unstable state with K 0 = −1, (1. 3) a and v are the coefficients in the free energy, which has the form of Landau-de Gennes potential: ...
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. E 70 , 051605 ( doi:10.1103/PhysRevE.70.051605 ); Elder et al. 2007 Phys. Rev. B 75 , 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. E 71 , 046125 ( doi:10.1103/PhysRevE.71.046125 )) for which the complete set of analytical travelling-wave solutions is obtained by the 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 solution of travelling waves is based on the function of hyperbolic tangent. Together with its set of particular solutions, the general solution is analysed within an example of specific task about the crystal front invading metastable liquid (Galenko et al. 2015 Phys. D 308 , 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’.
... These systems typically show 2D moiré patterns that are defined by a triangular (honeycomb) array of commensurate regions when grown on a substrate with an adsorption surface potential having a triangular (honeycomb) array of minima. More complex patterns emerge in other adsorption systems, for example, when Cu is grown on Ru(0001), in addition to honeycomb patterns, zig-zag, and one-dimensional (1D) stripes can form [23][24][25]. In this work, additional patterns that resemble twisted honeycombs are predicted to occur in some adsorption systems. ...
... The purpose of this work is to present a general method for studying the type of atomistically thin films described above and to show how the competition between the elastic and adsorption energies, the relative symmetry of the film and substrate, and the degeneracy of the higher-order commensurate states can influence the patterns that form. The method incorporated here is essentially an extension of the one used in the authors previous studies of (1 × 1)systems [11,24,25,46] and the ( [47]. The method is based on the amplitude expansion [48][49][50][51][52][53] of the phase field crystal (PFC) model [54][55][56], which describes a field related to local number density fluctuations. ...
... The phase diagram for a film of triangular symmetry adsorbed on a substrate that has a triangular pattern of maxima (or honeycomb pattern of minima) gives rise to much richer features than the TH system. The case of a (1 × 1)or (j,m) = (0,1) system was explored in prior publications [24,25], but has not been examined for higherorder systems. Detailed phase diagrams were calculated for the ( ...
... These systems typically show 2D moiré patterns that are defined by a triangular (honeycomb) array of commensurate regions when grown on a substrate with an adsorption surface potential having a triangular (honeycomb) array of minima. More complex patterns emerge in other adsorption systems, for example, when Cu is grown on Ru(0001), in addition to honeycomb patterns, zig-zag, and one-dimensional (1D) stripes can form [23][24][25]. In this work, additional patterns that resemble twisted honeycombs are predicted to occur in some adsorption systems. ...
... The purpose of this work is to present a general method for studying the type of atomistically thin films described above and to show how the competition between the elastic and adsorption energies, the relative symmetry of the film and substrate, and the degeneracy of the higher-order commensurate states can influence the patterns that form. The method incorporated here is essentially an extension of the one used in the authors previous studies of (1 × 1)systems [11,24,25,46] and the ( [47]. The method is based on the amplitude expansion [48][49][50][51][52][53] of the phase field crystal (PFC) model [54][55][56], which describes a field related to local number density fluctuations. ...
... The phase diagram for a film of triangular symmetry adsorbed on a substrate that has a triangular pattern of maxima (or honeycomb pattern of minima) gives rise to much richer features than the TH system. The case of a (1 × 1)or (j,m) = (0,1) system was explored in prior publications [24,25], but has not been examined for higherorder systems. Detailed phase diagrams were calculated for the ( ...
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 (1×1), (3×3)R30∘, (2×2), (7×7)R19.1∘, 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.
