Growth modes of quasicrystals

Nucleation and phase transition of decagonal quasicrystals

In this work, we study the nucleation of quasicrystals from liquid or periodic crystals by developing an efficient order-order phase transition algorithm, namely, the nullspace-preserving saddle search method. In particular, we focus on nucleation and phase transitions of the decagonal quasicrystal (DQC) based on the Lifshitz-Petrich model. We present the nucleation path of DQC from the liquid and demonstrate one- and two-stage transition paths between DQC and periodic crystals. We provide a perspective of the group-subgroup phase transition and nucleation rates to understand the nucleation and phase transition mechanisms involving DQC. These results reveal the one-step and multi-step modes of symmetry breaking or recovery in the phase transition from DQC, where the multi-step modes are more probable.

Design of Linear Block Copolymers and ABC Star Terpolymers That Produce Two Length Scales at Phase Separation

Quasicrystals (materials with long-range order but without the usual spatial periodicity of crystals) were discovered in several soft matter systems in the last 20 years. The stability of quasicrystals has been attributed to the presence of two prominent length scales in a specific ratio, which is 1.93 for the 12-fold quasicrystals most commonly found in soft matter. We propose design criteria for block copolymers such that quasicrystal-friendly length scales emerge at the point of phase separation from a melt, basing our calculations on the Random Phase Approximation. We consider two block copolymer families: linear chains containing two different monomer types in blocks of different lengths, and ABC star terpolymers. In all examples, we are able to identify parameter windows with the two length scales having a ratio of 1.93. The models that we consider that are simplest for polymer synthesis are, first, a monodisperse ALBASB melt and, second, a model based on random reactions from a mixture of AL, AS, and B chains: both feature the length scale ratio of 1.93 and should be relatively easy to synthesize.

Structural studies of local environments in high-symmetry quasicrystals

The statistics of how the local environment of a particle looks like, e.g., given by the distribution of nearest neighbor distances or the sizes of Voronoi cells, is important as a starting point for the calculation of many material properties like electronic or photonic band structures. Here we study local environments that occur in quasicrystals with large rotational symmetry. Both with analytical considerations based on geometric arguments and with an analysis of a large number of numerically created patches of high-symmetry quasicrystals we find that the Voronoi area's distribution reaches a bimodal curve and that in the limit of large rotational symmetries the distribution of nearest neighbor distance converges against a universal curve, where [Formula: see text] of the vertices have their nearest neighbor at a normalized distance equal to 1, while for the other [Formula: see text] the nearest neighbor is at a distance less than 1. Therefore, the statistics of local environments is non-trivial but independent of the specific rotational symmetry. Thus properties that only depend on local environments are expected to be universal for all high-symmetry quasicrystals.

Bayesian modeling of pattern formation from one snapshot of pattern

Partial differential equations (PDEs) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying pattern formation. Although many PDE models have been proposed, they rely on the pre-request knowledge of physical laws and symmetries, and developing a model to reproduce a given desired pattern remains difficult. We propose a method, referred to as Bayesian modeling of PDEs (BM-PDEs), to estimate the best dynamical PDE for one snapshot of a objective pattern under the stationary state without ground truth. We apply BM-PDEs to nontrivial patterns, such as quasicrystals (QCs), a double gyroid, and Frank-Kasper structures. We also generate three-dimensional dodecagonal QCs from a PDE model. This is done by using the estimated parameters for the Frank-Kasper A15 structure, which closely approximates the local structures of QCs. Our method works for noisy patterns and the pattern synthesized without the ground-truth parameters, which are required for the application toward experimental data.

Formation and fluctuation of two-dimensional dodecagonal quasicrystals

The self-assembly of two-dimensional dodecagonal quasicrystals (DDQCs) from patchy particles is investigated by Brownian dynamics simulations. The patchy particle has a five-fold rotational symmetry pattern described by the spherical harmonics Y₅₅. From the formation of the DDQC obtained by an annealing process, we find the following mechanism. The early stage of the dynamics is dominated by hexagonal structures. Then, nucleation of dodecagonal motifs appears by particle rearrangement, and finally the motifs span the whole system. The transition from the hexagonal structure into the dodecagonal motif is coincident with the collective motion of the particles. The DDQC consists of clusters of dodecagonal motifs, which can be classified into several packing structures. By the analyses of the DDQC under fixed temperature, we find that the fluctuations are characterised by changes in the network of the dodecagonal motifs. Finally we compare the DDQCs assembled from the patchy particle system and isotropic particle system. The two systems share a similar mechanism of the formation and fluctuation of DDQCs.

Control of phase ordering and elastic properties in phase field crystals through three-point direct correlation

Effects of three-point direct correlation on properties of the phase field crystal (PFC) modeling are examined for the control of various ordered and disordered phases and their coexistence in both three-dimensional and two-dimensional systems. Such effects are manifested via the corresponding gradient nonlinearity in the PFC free-energy functional that is derived from classical density functional theory. Their significant impacts on the stability regimes of ordered phases, phase diagrams, and elastic properties of the system, as compared to those of the original PFC model, are revealed through systematic analyses and simulations. The nontrivial contribution from three-point direct correlation leads to the variation of the critical point of order-disorder transition to which all the phase boundaries in the temperature-density phase diagram converge. It also enables the variation and control of system elastic constants over a substantial range as needed in modeling different types of materials with the same crystalline structure but different elastic properties. The capability of this PFC approach in modeling both solid and soft matter systems is further demonstrated through the effect of three-point correlation on controlling the vapor-liquid-solid coexistence and transitions for body-centered cubic phase and on achieving the liquid-stripe or liquid-lamellar phase coexistence. All these provide a valuable and efficient method for the study of structural ordering and evolution in various types of material systems.

Transition pathways connecting crystals and quasicrystals

Due to structural incommensurability, the emergence of a quasicrystal from a crystalline phase represents a challenge to computational physics. Here, the nucleation of quasicrystals is investigated by using an efficient computational method applied to a Landau free-energy functional. Specifically, transition pathways connecting different local minima of the Lifshitz-Petrich model are obtained by using the high-index saddle dynamics. Saddle points on these paths are identified as the critical nuclei of the 6-fold crystals and 12-fold quasicrystals. The results reveal that phase transitions between the crystalline and quasicrystalline phases could follow two possible pathways, corresponding to a one-stage phase transition and a two-stage phase transition involving a metastable lamellar quasicrystalline state, respectively.

Entropic formation of a thermodynamically stable colloidal quasicrystal with negligible phason strain

Quasicrystals have been discovered in a variety of materials ranging from metals to polymers. Yet, why and how they form is incompletely understood. In situ transmission electron microscopy of alloy quasicrystal formation in metals suggests an error-and-repair mechanism, whereby quasiperiodic crystals grow imperfectly with phason strain present, and only perfect themselves later into a high-quality quasicrystal with negligible phason strain. The growth mechanism has not been investigated for other types of quasicrystals, such as dendrimeric, polymeric, or colloidal quasicrystals. Soft-matter quasicrystals typically result from entropic, rather than energetic, interactions, and are not usually grown (either in laboratories or in silico) into large-volume quasicrystals. Consequently, it is unknown whether soft-matter quasicrystals form with the high degree of structural quality found in metal alloy quasicrystals. Here, we investigate the entropically driven growth of colloidal dodecagonal quasicrystals (DQCs) via computer simulation of systems of hard tetrahedra, which are simple models for anisotropic colloidal particles that form a quasicrystal. Using a pattern recognition algorithm applied to particle trajectories during DQC growth, we analyze phason strain to follow the evolution of quasiperiodic order. As in alloys, we observe high structural quality; DQCs with low phason strain crystallize directly from the melt and only require minimal further reduction of phason strain. We also observe transformation from a denser approximant to the DQC via continuous phason strain relaxation. Our results demonstrate that soft-matter quasicrystals dominated by entropy can be thermodynamically stable and grown with high structural quality--just like their alloy quasicrystal counterparts.

Entropy scaling close to criticality: From simple to metallic systems

Entropy has recently drawn considerable interest both as a marker to detect the onset of phase transitions and as a reaction coordinate, or collective variable, to span phase transition pathways. We focus here on the behavior of entropy along the vapor-liquid phase coexistence and identify how the difference in entropy between the two coexisting phases vary in ideal and metallic systems along the coexistence curve. Using flat-histogram simulations, we determine the thermodynamic conditions of coexistence, critical parameters, including the critical entropy, and entropies along the binodal. We then apply our analysis to a series of systems that increasingly depart from ideality and adopt a metal-like character, through the gradual onset of the Friedel oscillation in an effective pair potential, and for a series of transition metals modeled with a many-body embedded-atoms force field. Projections of the phase boundary on the entropy-pressure and entropy-temperature planes exhibit two qualitatively different behaviors. While all systems modeled with an effective pair potential lead to an ideal-like behavior, the onset of many-body effects results in a departure from ideality and a markedly greater exponent for the variation of the entropy of vaporization with temperature away from the critical temperature.

An atomic scale study of two-dimensional quasicrystal nucleation controlled by multiple length scale interactions

Formation of quasicrystal structures has always been mysterious since the discovery of these magic structures. In this work, the nucleation of decagonal, dodecagonal, heptagonal, and octagonal quasicrystal structures controlled by the coupling among multiple length scales is investigated using a dynamic phase-field crystal model. We observe that the nucleation of quasicrystals proceeds through local rearrangement of length scales, i.e., the generation, merging and stacking of 3-atom building blocks constructed by the length scales, and accordingly, propose a geometric model to describe the cooperation of length scales during structural transformation in quasicrystal nucleation. Essentially, such cooperation is crucial to quasicrystal formation, and controlled by the match and balance between length scales. These findings clarify the scenario and microscopic mechanism of the structural evolution during quasicrystal nucleation, and help us to understand the common rule for the formation of periodic crystal and quasicrystal structures with various symmetries.

Event-chain Monte Carlo simulations of the liquid to solid transition of two-dimensional decagonal colloidal quasicrystals

In event-chain Monte Carlo simulations, we model colloidal particles in two dimensions that interact according to an isotropic short-ranged pair potential which supports the two typical length scales present in decagonal quasicrystals. We investigate the assembled structures as we vary the density and temperature. Our special interest is related to the transition from quasicrystal to liquid. In contrast to the KTHNY melting theory for quasicrystals which predicts an intermediate pentahedratic phase, we find a one-step first-order melting transition. However, we discover that the slow relaxation of phasonic flips, i.e. rearrangements of the particles due to additional degrees of freedom in quasicrystals, changes the positional correlation functions, to the extent that structures with long-range orientational correlations, but exponentially decaying positional correlations, are observed.

Hierarchical complex self-assembly in binary nanoparticle mixtures

Hierarchical self-assembly of soft matter provides a powerful route to create complex materials with enhanced physical properties. The understanding of the fundamental processes leading to such organization can provide design rules to create new functional materials. In this work, we use a simple model of polymer-grafted nanoparticles to explore the self-assembly of binary mixtures. By using Monte Carlo simulations we study the interplay of composition, density and particle sizes on the self-organization of such nanoparticle systems. It is found that complex hierarchical organization can take place for conditions where one-component systems form simple lattices. In particular, a mixture where one component forms a structure with 18-fold symmetry in a sea of an apparent disordered phase of the second component is observed to emerge for certain parameter combinations.

Which Wave Numbers Determine the Thermodynamic Stability of Soft Matter Quasicrystals?

For soft matter to form quasicrystals an important ingredient is to have two characteristic length scales in the interparticle interactions. To be more precise, for stable quasicrystals, periodic modulations of the local density distribution with two particular wave numbers should be favored, and the ratio of these wave numbers should be close to certain special values. So, for simple models, the answer to the title question is that only these two ingredients are needed. However, for more realistic models, where in principle all wave numbers can be involved, other wave numbers are also important, specifically those of the second and higher reciprocal lattice vectors. We identify features in the particle pair interaction potentials that can suppress or encourage density modes with wave numbers associated with one of the regular crystalline orderings that compete with quasicrystals, enabling either the enhancement or suppression of quasicrystals in a generic class of systems.

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Phase field crystal (PFC) theory is extensively used for modeling the phase behavior, structure, thermodynamics, and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇·[n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L with a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to -∞ while being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalized exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallization, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artifacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution lnρ(x) rather than for ρ(x) is surprisingly accurate. This approach gives a tantalizing hint that accurate PFC-type theories may instead be derived as theories for the field lnρ(x), rather than for the density profile itself.

Growth of two-dimensional dodecagonal colloidal quasicrystals: Particles with isotropic pair interactions with two length scales vs. patchy colloids with preferred binding angles

We explore the growth of colloidal quasicrystals with dodecagonal symmetry in two dimensions by employing Brownian dynamics simulations. On the one hand, we study the growth behavior of structures obtained in a system of particles that interact according to an isotropic pair potential with two typical length scales. On the other hand, we consider patchy colloids that possess only one typical interaction length scale but prefer given binding angles. In case of the isotropic particles, we show that an imbalance in the competition between the two distances might lead to defects with wrong nearest-neighbor distances in the resulting structure. In contrast, during the growth of quasicrystals with patchy colloids such defects do not occur due to the lack of a second interaction length scale. However, as a downside, the diffusion of patchy particles along a surface typically is slower such that domains occur where the particles possess different phononic and phasonic offsets. Our results are important to understand how soft matter quasicrystals can be grown as perfectly as possible and to obtain a deeper insight into the mechanisms of the growth of quasicrystals in general.

Structural crossover in a model fluid exhibiting two length scales: Repercussions for quasicrystal formation

We investigate the liquid state structure of the two-dimensional model introduced by Barkan et al. [Phys. Rev. Lett. 113, 098304 (2014)10.1103/PhysRevLett.113.098304], which exhibits quasicrystalline and other unusual solid phases, focusing on the radial distribution function g(r) and its asymptotic decay r→∞. For this particular model system, we find that as the density is increased there is a structural crossover from damped oscillatory asymptotic decay with one wavelength to damped oscillatory asymptotic decay with another distinct wavelength. The ratio of these wavelengths is ≈1.932. Following the locus in the phase diagram of this structural crossover leads directly to the region where quasicrystals are found. We argue that identifying and following such a crossover line in the phase diagram towards higher densities where the solid phase(s) occur is a good strategy for finding quasicrystals in a wide variety of systems. We also show how the pole analysis of the asymptotic decay of equilibrium fluid correlations is intimately connected with the nonequilibrium growth or decay of small-amplitude density fluctuations in a bulk fluid.

Growth of two-dimensional decagonal colloidal quasicrystals

The growth of quasicrystals, i.e. structures with long-range positional order but no periodic translational symmetry, is more complex than the growth of periodic crystals. By employing Brownian dynamics simulations in two dimensions for colloidal particles that interact according to an isotropic pair potential with two incommensurate lengths, we study the growth of quasicrystalline structures by sequentially depositing particles at their surface. We quantify the occurrence of quasicrystalline order as a function of the temperature and the rate of added particles. In addition, we explore defects like local triangular order or gaps within the quasicrystalline structure. Furthermore, we analyze the shapes of the surfaces in grown structures which tend to build straight lines along the symmetry axes of the quasicrystal. Finally, we identify phasonic flips which are rearrangements of the particles due to additional degrees of freedom. The number of phasonic flips decreases with the distance to the surface.

Multiple-scale structures: from Faraday waves to soft-matter quasicrystals

For many years, quasicrystals were observed only as solid-state metallic alloys, yet current research is now actively exploring their formation in a variety of soft materials, including systems of macromolecules, nanoparticles and colloids. Much effort is being invested in understanding the thermodynamic properties of these soft-matter quasicrystals in order to predict and possibly control the structures that form, and hopefully to shed light on the broader yet unresolved general questions of quasicrystal formation and stability. Moreover, the ability to control the self-assembly of soft quasicrystals may contribute to the development of novel photonics or other applications based on self-assembled metamaterials. Here a path is followed, leading to quantitative stability predictions, that starts with a model developed two decades ago to treat the formation of multiple-scale quasiperiodic Faraday waves (standing wave patterns in vibrating fluid surfaces) and which was later mapped onto systems of soft particles, interacting via multiple-scale pair potentials. The article reviews, and substantially expands, the quantitative predictions of these models, while correcting a few discrepancies in earlier calculations, and presents new analytical methods for treating the models. In so doing, a number of new stable quasicrystalline structures are found with octagonal, octadecagonal and higher-order symmetries, some of which may, it is hoped, be observed in future experiments.

Precipitation of binary quasicrystals along dislocations

Dislocations in crystals naturally break the symmetry of the bulk, introducing local atomic configurations with symmetries such as fivefold rings. But dislocations do not usually nucleate aperiodic structure along their length. Here we demonstrate the formation of extended binary quasicrystalline precipitates with Penrose-like random-tiling structures, beginning with chemical ordering within the pentagonal structure at cores of prismatic dislocations in Mg-Zn alloys. Atomic resolution observations indicate that icosahedral chains centered along [0001] pillars of Zn interstitial atoms are formed templated by the fivefold rings at dislocation cores. They subsequently form columns of rhombic and elongated hexagonal tiles parallel to the dislocation lines. Quasicrystalline precipitates are formed by random tiling of these rhombic and hexagonal tiles. Such precipitation may impact dislocation glide and alloy strength.

Dislocation-free growth of quasicrystals from two seeds due to additional phasonic degrees of freedom

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.

Non-close-packed three-dimensional quasicrystals

Quasicrystals are frequently encountered in condensed matter. They are important candidates for equilibrium phases from the atomic scale to the nanoscale. Here, we investigate the computational self-assembly of four quasicrystals in a single model system of identical particles interacting with a tunable isotropic pair potential. We reproduce a known icosahedral quasicrystal and report a decagonal quasicrystal, a dodecagonal quasicrystal, and an octagonal quasicrystal. The quasicrystals have low coordination number or occur in systems with mesoscale density variations. We also report a network gel phase.

Properties of quasiperiodic functions

Today, quasiperiodic tilings are well known and have been studied in great detail since they are very useful to describe the properties of metallic and soft matter quasicrystals. A closely related topic are quasiperiodic functions which have also gained large interest recently. Different types of such functions and there interrelation will be presented here. The main topic will be quasiperiodic potentials generated by laser beams and their variability. The distribution of extremal points and local isomorphisms of quasiperiodic functions will also be addressed.

Defect structures in Frank-Kasper type square-triangle tiling of multimodal cage-type mesoporous silicas

Multimodal cage-type mesoporous silicas (MCMSs) with Frank-Kasper type square-triangle tiling show a unique defect structure, so-called three-fold symmetric hexagons, or shields, which are caused by phason fluctuations in dodecagonal quasicrystals. We observed and characterized three types of configurations inside shields in both quasiperiodic and periodic 3².4.3.4 tiling of MCMSs by transmission electron microscopy (TEM). The high-resolution TEM images of the shields were well explained by polyhedral models, which are the constituents of the Frank-Kasper type tetrahedrally close-packed structures of MCMSs. Shield defects invariably formed because of mismatch in periodic and/or aperiodic square-triangle tiling, and they were also catalyzed by other defects. Multiple shields overlapped with sharing of 30° rhombus units and showed characteristic motifs in the tiling, such as defect-mediated 12-fold wheel and stripe bundle arrangements. Hence, MCMSs with square-triangle tiling would be governed by a random-tiling-like structure stabilized by entropy rather than energy, which results in defect-free tiling.

Detection of phonon and phason modes in intrinsic colloidal quasicrystals by reconstructing their structure in hyperspace

Phasons are additional degrees of freedom which occur in quasicrystals alongside the phonons known from conventional periodic crystals. The rearrangements of particles that are associated with a phason mode are hard to interpret in physical space. We reconstruct the quasicrystal structure by an embedding into extended higher-dimensional space, where phasons correspond to displacements perpendicular to the physical space. In dislocation-free decagonal colloidal quasicrystals annealed with Brownian dynamics simulations, we identify thermal phonon and phason modes. Finite phononic strain is pinned by phasonic excitations even after cooling down to zero temperature. For the phasonic displacements underlying the flip pattern, the reconstruction method gives an approximation within the limits of a multi-mode harmonic ansatz, and points to fundamental limitations of a harmonic picture for phasonic excitations in intrinsic colloidal quasicrystals.

Pushing the limits of crystallography

A very serious concern of scientists dealing with crystal structure refinement, including theoretical research, pertains to the characteristic bias in calculated versus measured diffraction intensities, observed particularly in the weak reflection regime. This bias is here attributed to corrective factors for phonons and, even more distinctly, phasons, and credible proof supporting this assumption is given. The lack of a consistent theory of phasons in quasicrystals significantly contributes to this characteristic bias. It is shown that the most commonly used exponential Debye-Waller factor for phasons fails in the case of quasicrystals, and a novel method of calculating the correction factor within a statistical approach is proposed. The results obtained for model quasiperiodic systems show that phasonic perturbations can be successfully described and refinement fits of high quality are achievable. The standard Debye-Waller factor for phonons works equally well for periodic and quasiperiodic crystals, and it is only in the last steps of a refinement that different correction functions need to be applied to improve the fit quality.

Molecular dynamics study of colloidal quasicrystals

Colloidal quasicrystals have received increased interest recently due to new insight in exploring their potential for photonic materials as well as for optical devices [Vardeny et al., Nat. Photonics, 2013, 7, 177]. Colloidal quasicrystals in aqueous solutions have been found in systems of micelles with impenetrable cores [Fischer et al., Proc. Natl. Acad. Sci. U. S. A., 2011, 108, 1810]. A simple model potential for micelle-micelle interaction is the step potential, which is infinite for core overlaps and constant for shell overlaps. Dotera et al. performed Monte Carlo simulations of the step potential model and found quasicrystals for specific values of the packing fraction η and the shell-core ratio λ [Dotera et al., Nature, 2014, 506, 208 ]. However, the overlap of real micelles causes repulsive forces, which increase with decreasing core distance. We consider this by introducing a novel model potential with repulsive forces depending on a third parameter α. In a systematic manner we study this more realistic potential with two-dimensional molecular dynamics simulations. For α = 0 the model is similar to the step potential model. For the first time, we provide a comprehensive overview of crystalline, quasicrystalline, and disordered structures as a function of η and λ. Simulations performed with α > 0 show the impact of the repulsive forces. We find that quasicrystalline structures at high densities vanish while new quasicrystalline structures appear at intermediate densities. Our results help to tailor colloidal systems for today's advanced applications in photonics and optical devices.

Three-Dimensional Icosahedral Phase Field Quasicrystal

We investigate the formation and stability of icosahedral quasicrystalline structures using a dynamic phase field crystal model. Nonlinear interactions between density waves at two length scales stabilize three-dimensional quasicrystals. We determine the phase diagram and parameter values required for the quasicrystal to be the global minimum free energy state. We demonstrate that traits that promote the formation of two-dimensional quasicrystals are extant in three dimensions, and highlight the characteristics required for three-dimensional soft matter quasicrystal formation.

Impact of Friedel oscillations on vapor-liquid equilibria and supercritical properties in two and three dimensions

We determine the impact of the Friedel oscillations on the phase behavior, critical properties, and thermodynamic contours in films [two dimensions (2D)] and bulk phases [three dimensions (3D)]. Using expanded Wang-Landau simulations, we calculate the grand-canonical partition function and, in turn, the thermodynamic properties of systems modeled with a linear combination of the Lennard-Jones and Dzugutov potentials, weighted by a parameter X (0<X<1). Varying X allows us to control the height of the first Friedel oscillation and to provide a complete characterization of the effect of the metal-like character in the potential on the thermodynamic properties over a wide range of conditions. For 3D systems, we are able to show that the critical parameters exhibit a linear dependence on X and that the loci for the thermodynamic state points, for which the system shows the same compressibility factor or enthalpy as an ideal gas, are two straight lines spanning the subcritical and supercritical regions of the phase diagram for all X values. Reducing the dimensionality to 2D results in a loss of impact of the Friedel oscillation on the critical properties, as evidenced by the virtually constant critical density across the range of X values. Furthermore, our results establish that the straightness of the two ideality lines is retained in 2D and is independent from the height of the first Friedel oscillation in the potential.

Flow of colloidal solids and fluids through constrictions: dynamical density functional theory versus simulation

Using both dynamical density functional theory and particle-resolved Brownian dynamics simulations, we explore the flow of two-dimensional colloidal solids and fluids driven through a linear channel with a constriction. The flow is generated by a constant external force acting on all colloids. The initial configuration is equilibrated in the absence of flow and then the external force is switched on instantaneously. Upon starting the flow, we observe four different scenarios: a complete blockade, a monotonic decay to a constant particle flux (typical for a fluid), a damped oscillatory behaviour in the particle flux, and a long-lived stop-and-go behaviour in the flow (typical for a solid). The dynamical density functional theory describes all four situations but predicts infinitely long undamped oscillations in the flow which are always damped in the simulations. We attribute the mechanisms of the underlying stop-and-go flow to symmetry conditions on the flowing solid. Our predictions are verifiable in real-space experiments on magnetic colloidal monolayers which are driven through structured microchannels and can be exploited to steer the flow throughput in microfluidics.

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.

One-Dimensional Quasicrystals from Incommensurate Charge Order

Artificial quasicrystals are nowadays routinely manufactured, yet only two naturally occurring examples are known. We present a class of systems with the potential to be realized both artificially and in nature, in which the lowest energy state is a one-dimensional quasicrystal. These systems are based on incommensurately charge-ordered materials, in which the quasicrystalline phase competes with the formation of a regular array of discommensurations as a way of interpolating between incommensurate charge order at high temperatures and commensurate order at low temperatures. The nonlocal correlations characteristic of the quasicrystalline state emerge from a free-energy contribution localized in reciprocal space. We present a theoretical phase diagram showing that the required material properties for the appearance of such a ground state allow for one-dimensional quasicrystals to form in real materials. The result is a potentially wide class of one-dimensional quasicrystals.

Soft-core particles freezing to form a quasicrystal and a crystal-liquid phase

Systems of soft-core particles interacting via a two-scale potential are studied. The potential is responsible for peaks in the structure factor of the liquid state at two different but comparable length scales and a similar bimodal structure is evident in the dispersion relation. Dynamical density functional theory in two dimensions is used to identify two unusual states of this system: a crystal-liquid state, in which the majority of the particles are located on lattice sites but a minority remains free and so behaves like a liquid, and a 12-fold quasicrystalline state. Both are present even for deeply quenched liquids and are found in a regime in which the liquid is unstable with respect to modulations on the smaller scale only. As a result, the system initially evolves towards a small-scale crystal state; this state is not a minimum of the free energy, however, and so the system subsequently attempts to reorganize to generate the lower-energy larger-scale crystals. This dynamical process generates a disordered state with quasicrystalline domains and takes place even when this large scale is linearly stable, i.e., it is a nonlinear process. With controlled initial conditions, a perfect quasicrystal can form. The results are corroborated using Brownian dynamics simulations.

Effective substrate potentials with quasicrystalline symmetry depend on the size of the adsorbed particles

We explore the effective potential landscapes that extended particles experience when adsorbed on the surface of quasicrystals. Commonly, these are solids with long-ranged order but no translational symmetry. The effective potentials significantly depend on the size of the adsorbed particles. We show how changing the particle radius changes the so-called local isomorphism class of the effective quasicrystalline pattern. This means effective potentials for different particle sizes cannot directly be mapped onto each other. Our theoretical predictions are confirmed by Monte Carlo simulations. The results are important for colloidal particles with different sizes that are subjected to laser fields with quasicrystalline symmetry as well as for systems where extended molecules are deposited onto the surface of metallic quasicrystals.

Computational self-assembly of a one-component icosahedral quasicrystal

Icosahedral quasicrystals (IQCs) are a form of matter that is ordered but not periodic in any direction. All reported IQCs are intermetallic compounds and either of face-centred-icosahedral or primitive-icosahedral type, and the positions of their atoms have been resolved from diffraction data. However, unlike axially symmetric quasicrystals, IQCs have not been observed in non-atomic (that is, micellar or nanoparticle) systems, where real-space information would be directly available. Here, we show that an IQC can be assembled by means of molecular dynamics simulations from a one-component system of particles interacting via a tunable, isotropic pair potential extending only to the third-neighbour shell. The IQC is body-centred, self-assembles from a fluid phase, and in parameter space neighbours clathrates and other tetrahedrally bonded crystals. Our findings elucidate the structure and dynamics of the IQC, and suggest routes to search for it and design it in soft matter and nanoscale systems.

Controlled self-assembly of periodic and aperiodic cluster crystals

Soft particles are known to overlap and form stable clusters that self-assemble into periodic crystalline phases with density-independent lattice constants. We use molecular dynamics simulations in two dimensions to demonstrate that, through a judicious design of an isotropic pair potential, one can control the ordering of the clusters and generate a variety of phases, including decagonal and dodecagonal quasicrystals. Our results confirm analytical predictions based on a mean-field approximation, providing insight into the stabilization of quasicrystals in soft macromolecular systems, and suggesting a practical approach for their controlled self-assembly in laboratory realizations using synthesized soft-matter particles.