On the stability of a quasicrystal and its crystalline approximant in a system of hard disks with a soft corona

Understanding the Role of Trapezoids in Honeycomb Self-Assembly-Pathways between a Columnar Liquid Quasicrystal and its Liquid-Crystalline Approximants

Quasiperiodic patterns and crystals-having long range order without translational symmetry-have fascinated researchers since their discovery. In this study, we report on new p-terphenyl-based T-shaped facial polyphiles with two alkyl end chains and a glycerol-based hydrogen-bonded side group that self-assemble into an aperiodic columnar liquid quasicrystal with 12-fold symmetry and its periodic liquid-crystalline approximants with complex superstructures. All represent honeycombs formed by the self-assembly of the p-terphenyls, dividing space into prismatic cells with polygonal cross-sections. In the perspective of tiling patterns, the presence of unique trapezoidal tiles, consisting of three rigid sides formed by the p-terphenyls and one shorter, incommensurate, and adjustable side by the alkyl end chains, plays a crucial role for these phases. A delicate temperature-dependent balance between conformational, entropic and space-filling effects determines the role of the alkyl chains, either as network nodes or trapezoid walls, thus resulting in the order-disorder transitions associated with emergence of quasiperiodicity. In-depth analysis suggests a change from a quasiperiodic tiling involving trapezoids to a modified one with a contribution of trapezoid pair fusion. This work paves the way for understanding quasiperiodicity emergence and develops fundamental concepts for its generation by chemical design of non-spherical molecules, aggregates, and frameworks based on dynamic reticular chemistry.

Quasicrystal of Binary Hard Spheres on a Plane Stabilized by Configurational Entropy

Because of their aperiodic nature, quasicrystals are one of the least understood phases in statistical physics. One significant complication they present in comparison to their periodic counterparts is the fact that any quasicrystal can be realized as an exponentially large number of different tilings, resulting in a significant contribution to the quasicrystal entropy. Here, we use free-energy calculations to demonstrate that it is this configurational entropy which stabilizes a dodecagonal quasicrystal in a binary mixture of hard spheres on a plane. Our calculations also allow us to quantitatively confirm that in this system all tiling realizations are essentially equally likely, with free-energy differences less than 0.0001k_{B}T per particle-an observation that could be related to the observation of only random tilings in soft-matter quasicrystals. Owing to the simplicity of the model and its available counterparts in colloidal experiments, we believe that this system is an excellent candidate to achieve the long-awaited quasicrystal self-assembly on the micron scale.

Structures Formed by Particles with Shoulderlike Repulsive Interaction in Thin Systems

When particles are constructed in thin systems between two parallel flat walls, structures that are not observed in bulk systems are created and the created structures change, depending on the width between the walls. In this study, the structures formed by particles constructed in thin systems were investigated through performing isothermal-isobaric Monte Carlo simulations, where the interaction between the particles is given by the hard-core square shoulder potential. By controlling the width of the shoulder-like repulsive interaction and the system width, several novel structures such as the connection of rhombuses and the square lattice of the (100) face of the body-centered cubic lattice were created.

Self-assembly of dodecagonal and octagonal quasicrystals in hard spheres on a plane

Hard spheres are one of the most fundamental model systems in soft matter physics, and have been instrumental in shedding light on nearly every aspect of classical condensed matter. Here, we add one more important phase to the list that hard spheres form: quasicrystals. Specifically, we use simulations to show that an extremely simple, purely entropic model system, consisting of two sizes of hard spheres resting on a flat plane, can spontaneously self-assemble into two distinct random-tiling quasicrystal phases. The first quasicrystal is a dodecagonal square-triangle tiling, commonly observed in a large variety of colloidal systems. The second quasicrystal has, to our knowledge, never been observed in either experiments or simulations. It exhibits octagonal symmetry, and consists of three types of tiles: triangles, small squares, and large squares, whose relative concentration can be continuously varied by tuning the number of smaller spheres present in the system. The observed tile composition of the self-assembled quasicrystals agrees very well with the theoretical prediction we obtain by considering the four-dimensional (lifted) representation of the quasicrystal. Both quasicrystal phases form reliably and rapidly over a significant part of parameter space. Our results demonstrate that entropy combined with a set of geometrically compatible, densely packed tiles can be sufficient ingredients for the self-assembly of colloidal quasicrystals.

Phase diagrams-Why they matter and how to predict them

Understanding the thermodynamic stability and metastability of materials can help us to, for example, gauge whether crystalline polymorphs in pharmaceutical formulations are likely to be durable. It can also help us to design experimental routes to novel phases with potentially interesting properties. In this Perspective, we provide an overview of how thermodynamic phase behavior can be quantified both in computer simulations and machine-learning approaches to determine phase diagrams, as well as combinations of the two. We review the basic workflow of free-energy computations for condensed phases, including some practical implementation advice, ranging from the Frenkel-Ladd approach to thermodynamic integration and to direct-coexistence simulations. We illustrate the applications of such methods on a range of systems from materials chemistry to biological phase separation. Finally, we outline some challenges, questions, and practical applications of phase-diagram determination which we believe are likely to be possible to address in the near future using such state-of-the-art free-energy calculations, which may provide fundamental insight into separation processes using multicomponent solvents.

Density-tunable pathway complexity in a minimalistic self-assembly model

An open challenge in self-assembly is learning how to design systems that can be conditionally guided towards different target structures depending on externally-controlled conditions. Using a theoretical and numerical approach, here we discuss a minimalistic self-assembly model that can be steered towards different types of ordered constructs at the equilibrium by solely tuning a facile selection parameter, namely the density of building blocks. Metadynamics and Langevin dynamics simulations allow us to explore the behavior of the system in and out of equilibrium conditions. We show that the density-driven tunability is encoded in the pathway complexity of the system, and specifically in the competition between two different main self-assembly routes. A comprehensive set of simulations provides insight into key factors allowing to make one self-assembling pathway prevailing on the other (or vice versa), determining the selection of the final self-assembled products. We formulate and validate a practical criterion for checking whether a specific molecular design is predisposed for such density-driven tunability of the products, thus offering a new, broader perspective to realize and harness this facile extrinsic control of conditional self-assembly.

Stability of the high-density Jagla liquid in 2D: sensitivity to parameterisation

We computed the pressure-temperature phase diagram of the hard-core two-scale ramp potential in two-dimensions, with the parameterisation originally suggested by Jagla [E. A. Jagla, Phys. Rev. E, 63, 061501 (2001)], as well as with a series of systematically modified variants of the model to reveal the sensitivity of the stability of phases. The nested sampling method was used to explore the potential energy landscape, allowing the identification of thermodynamically relevant phases, such as low- and high-density liquids and various crystalline forms, some of which have not been reported before. We also proposed a smooth version of the potential, which is differentiable beyond the hard-core. This potential reproduces the density anomaly, but forms a dodecahedral quasi-crystal structure at high pressure. Our results allow to hypothesise on the necessary modifications of the original model in order to improve the stability of the metastable high-density liquid phase in 3D.

Inverse design of soft materials via a deep learning-based evolutionary strategy

Colloidal self-assembly—the spontaneous organization of colloids into ordered structures—has been considered key to produce next-generation materials. However, the present-day staggering variety of colloidal building blocks and the limitless number of thermodynamic conditions make a systematic exploration intractable. The true challenge in this field is to turn this logic around and to develop a robust, versatile algorithm to inverse design colloids that self-assemble into a target structure. Here, we introduce a generic inverse design method to efficiently reverse-engineer crystals, quasicrystals, and liquid crystals by targeting their diffraction patterns. Our algorithm relies on the synergetic use of an evolutionary strategy for parameter optimization, and a convolutional neural network as an order parameter, and provides a way forward for the inverse design of experimentally feasible colloidal interactions, specifically optimized to stabilize the desired structure.

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.

Programming patchy particles to form three-dimensional dodecagonal quasicrystals

Model patchy particles have been shown to be able to form a wide variety of structures, including symmetric clusters, complex crystals, and even two-dimensional quasicrystals. Here, we investigate whether we can design patchy particles that form three-dimensional quasicrystals, in particular targeting a quasicrystal with dodecagonal symmetry that is made up of stacks of two-dimensional quasicrystalline layers. We obtain two designs that are able to form such a dodecagonal quasicrystal in annealing simulations. The first is a one-component system of seven-patch particles but with wide patches that allow them to adopt both seven- and eight-coordinated environments. The second is a ternary system that contains a mixture of seven- and eight-patch particles and is likely to be more realizable in experiments, for example, using DNA origami. One interesting feature of the first system is that the resulting quasicrystals very often contain a screw dislocation.

Semi-supervised learning for the study of structural formation in colloidal systems via image recognition

The analysis of the structural formation of colloidal systems using machine learning techniques has recently attracted much attention. In many of these studies, local bond-order parameters (LBOPs) were employed as descriptors, where such LBOPs are suitable mainly for the detection of crystal structures. On the other hand, image-based convolutional neural networks (CNNs) are quite effective in detecting not only crystals but also random structures, and the author demonstrated their efficiency in a previous paper. However, in supervised learning, it is difficult to obtain a correct result when there is an unexpected new phase that was unknown when training the CNN. In this paper, we propose a hybrid scheme that consists of supervised and unsupervised learning techniques, employing two different approaches: image-based CNN and generalized LBOP. The proposed method was applied to two-dimensional colloidal systems, and its efficiency was demonstrated.

Nonconventional Phases of Colloidal Nanorods with a Soft Corona

Using computer simulations, we investigate the phase behavior of hard-core spherocylinders with a length-to-diameter ratio L/σ=5 and coated by a soft deformable corona of length λ/σ=1.35. When quasi-two-dimensional layers are formed in smectic and solid phases at low temperatures, the competition between the two intrinsic length scales of the parallel aligned particles leads to the stabilization of different in-plane lattices of nonconventional symmetry, including low-density hexagonal, square, and high-density hexagonal crystals, as well as an intriguing dodecagonal quasicrystal. Our Letter opens up the opportunity to control the assembly of anisotropic nanoparticles into structures with preengineered symmetry-dependent physical properties.

Phase behavior of a two-dimensional core-softened system: new physical insights

In this work, we report results of extensive computer simulations regarding the phase behavior of a core-softened system. By using structural and thermodynamic descriptors, as well as self-diffusion coefficients, we provide a comprehensive view of the rich phase behavior displayed by the particular instance of the model studied in here. Our calculations agree with previously published results focused on a smaller region in the temperature-density parameter space (Dudalov et al 2014 Soft Matter 10 4966). In this work, we explore a broader region in this parameter space, and uncover interesting fluid phases with low-symmetry local order, that were not reported by previous works. Solid phases were also found, and have been previously characterized in detail by (Kryuchkov et al 2018 Soft Matter 14 2152). Our results support previously reported findings, and provide new physical insights regarding the emergence of order as disordered phases transform into solids by providing radial distribution function maps and specific heat data. Our results are summarized in terms of a phase diagram.

Pattern formation in two-dimensional hard-core/soft-shell systems with variable soft shell profiles

Hard-core/soft shell (HCSS) particles have been shown to self-assemble into a remarkably rich variety of structures under compression due to the simple interplay between the hard-core and soft-shoulder length scales in their interactions. Most studies in this area model the soft shell interaction as a square shoulder potential. Although appealing from a theoretical point of view, the potential is physically unrealistic because there is no repulsive force in the soft shell regime, unlike in experimental HCSS systems. To make the model more realistic, here we consider HCSS particles with a range of soft shell potential profiles beyond the standard square shoulder form and study the model using both minimum energy calculations and Monte Carlo simulations. We find that by tuning density and the soft shell profile, HCSS particles in the thin shell regime (i.e., shell to core ratio ) can form a large range of structures, including hexagons, chains, squares, rhomboids and two distinct zig-zag structures. Furthermore, by tuning the density and r1/r0, we find that HCSS particles with experimentally realistic linear ramp soft shoulder repulsions can form honeycombs and quasicrystals with 10-fold and 12-fold symmetry. Our study therefore suggests the exciting possibility of fabricating these exotic 2D structures experimentally through colloidal self-assembly.

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.

Moiré and honeycomb lattices through self-assembly of hard-core/soft-shell microgels: experiment and simulation

Moiré and honeycomb lattices result from the sequential double deposition of monolayers of core/shell microgels in dependence of the drying conditions.

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.

Two-dimensional mixture of amphiphilic dimers and spheres: Self-assembly behaviour

The emergence of supramolecular aggregates from simple microscopic interaction rules is a fascinating feature of complex fluids which, besides its fundamental interest, has potential applications in many areas, from biological self-assembly to smart material design. We here investigate by Monte Carlo simulation the equilibrium structure of a two-dimensional mixture of asymmetric dimers and spheres (disks). Dimers and disks are hard particles, with an additional short-range attraction between a disk and the smaller monomer of a dimer. The model parameters and thermodynamic conditions probed are typical of colloidal fluid mixtures. In spite of the minimalistic character of the interaction, we observe-upon varying the relative concentration and size of the two colloidal species-a rich inventory of mesoscale structures at low temperature, such as clusters, lamellæ (i.e., polymer-like chains), and gel-like networks. For colloidal species of similar size and near equimolar concentrations, a dilute fluid of clusters gives way to floating lamellæ upon cooling; at higher densities, the lamellæ percolate through the simulation box, giving rise to an extended network. A crystal-vapour phase-separation may occur for a mixture of dimers and much larger disks. Finally, when the fluid is brought in contact with a planar wall, further structures are obtained at the interface, from layers to branched patterns, depending on the nature of wall-particle interactions.

Waterlike anomalies in a two-dimensional core-softened potential

We investigate the structural, thermodynamic, and dynamic behavior of a two-dimensional (2D) core-corona system using Langevin dynamics simulations. The particles are modeled by employing a core-softened potential which exhibits waterlike anomalies in three dimensions. In previous studies in a quasi-2D system a new region in the pressure versus temperature phase diagram of structural anomalies was observed. Here we show that for the two-dimensional case two regions in the pressure versus temperature phase diagram with structural, density, and diffusion anomalies are observed. Our findings indicate that, while the anomalous region at lower densities is due the competition between the two length scales in the potential at higher densities, the anomalous region is related to the reentrance of the melting line.

Low-Temperature Crystal Structures of the Hard Core Square Shoulder Model

In many cases, the stability of complex structures in colloidal systems is enhanced by a competition between different length scales. Inspired by recent experiments on nanoparticles coated with polymers, we use Monte Carlo simulations to explore the types of crystal structures that can form in a simple hard-core square shoulder model that explicitly incorporates two favored distances between the particles. To this end, we combine Monte Carlo-based crystal structure finding algorithms with free energies obtained using a mean-field cell theory approach, and draw phase diagrams for two different values of the square shoulder width as a function of the density and temperature. Moreover, we map out the zero-temperature phase diagram for a broad range of shoulder widths. Our results show the stability of a rich variety of crystal phases, such as body-centered orthogonal (BCO) lattices not previously considered for the square shoulder model.

Periodic layers of a dodecagonal quasicrystal and a floating hexagonal crystal in sedimentation-diffusion equilibria of colloids

We investigate the behaviour of a system of colloidal particles interacting with a hard-core and a repulsive square shoulder potential under the influence of a gravitational field using event-driven Brownian dynamics simulations. We use a fixed square shoulder diameter equal to 1.4 times the hard-core diameter of the colloids, for which we have previously calculated the equilibrium phase diagram considering two-dimensional disks [H. Pattabhiraman et al., J. Chem. Phys. 143, 164905 (2015) and H. Pattabhiraman and M. Dijkstra, J. Phys.: Condens. Matter 20, 094003 (2017)]. The parameters in the simulations are chosen such that the pressure at the bottom of the sediment facilitates the formation of phases in accordance with the calculated phase diagram of the two-dimensional system. It is surprising that we observe the formation of layers with dodecagonal, square, and hexagonal symmetries at the relevant pressures in the three-dimensional sedimentation column. In addition, we also observe a re-entrant behaviour exhibited by the colloidal fluid phase, engulfing a hexagonal crystal phase, in the sedimentation column. In other words, a floating crystal is formed between the colloidal fluid regions.

Assembly of multi-flavored two-dimensional colloidal crystals

We systematically investigate the assembly of binary multi-flavored colloidal mixtures in two dimensions. In these mixtures all pairwise interactions between species may be tuned independently. This introduces an additional degree of freedom over more traditional binary mixtures with fixed mixing rules, which is anticipated to open new avenues for directed self-assembly. At present, colloidal self-assembly into non-trivial lattices tends to require either high pressures for isotropically interacting particles, or the introduction of directionally anisotropic interactions. Here we demonstrate tunable assembly into a plethora of structures which requires neither of these conditions. We develop a minimal model that defines a three-dimensional phase space containing one dimension for each pairwise interaction, then employ various computational techniques to map out regions of this phase space in which the system self-assembles into these different morphologies. We then present a mean-field model that is capable of reproducing these results for size-symmetric mixtures, which reveals how to target different structures by tuning pairwise interactions, solution stoichiometry, or both. Concerning particle size asymmetry, we find that domains in this model's phase space, corresponding to different morphologies, tend to undergo a continuous "rotation" whose magnitude is proportional to the size asymmetry. Such continuity enables one to estimate the relative stability of different lattices for arbitrary size asymmetries. Owing to its simplicity and accuracy, we expect this model to serve as a valuable design tool for engineering binary colloidal crystals from multi-flavored components.

On the formation of stripe, sigma, and honeycomb phases in a core-corona system

Using Monte Carlo simulations and free-energy calculations, we investigate the phase behaviour of a two-dimensional core-corona system. We model this system as particles consisting of an impenetrable hard core of diameter σ_HD surrounded by a purely repulsive soft corona of diameter δ = 1.95σ_HD. At low densities, we observe the spontaneous formation of a phase with a stripe texture as well as a honeycomb-like phase driven by both energy and entropy considerations. At high densities, we find that a two-dimensional analogue of the periodic sigma phase, considered as an approximant of dodecagonal quasicrystals, is energetically stabilised with respect to two distinct dodecagonal quasicrystals, namely, a square-triangle tiling and a square-triangle-shield tiling. We also find the formation of stable hexagonal phases at three distinct density ranges, which are energetically driven, i.e. by minimising the overlap of coronas. Furthermore, our calculations show that the low-density dodecagonal quasicrystal that was previously reported by Dotera et al., [Nature, 2014, 506, 208] is kinetically formed in the coexistence region between the honeycomb and the medium-density hexagonal phase.

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.

The effect of temperature, interaction range, and pair potential on the formation of dodecagonal quasicrystals in core-corona systems

A two-dimensional dodecagonal quasicrystal was previously reported by Dotera et al (2014 Nature 506 208) in a system of particles interacting with a hard core of diameter σ and a repulsive square shoulder of diameter [Formula: see text]. In the current work, we examine the formation of this quasicrystal using bond orientational order parameters, correlation functions and tiling distributions. We find that this dodecagonal quasicrystal forms from a fluid phase. We further study the effect of the width of the repulsive shoulder by simulating the system over a range of values of δ. For the range of densities and temperatures considered, we observe the formation of the dodecagonal quasicrystal between [Formula: see text] and [Formula: see text]. We also study the effect of shape of the interaction potential by simulating the system using three other interaction potentials with two length scales, namely hard-core plus a linear ramp, modified exponential, or Buckingham (exp-6) potential. We observe the presence of the quasicrystal in all three systems. However, depending on the shape of the potential, the formation of the quasicrystal takes place at lower temperatures (or higher interaction strengths). Using free-energy calculations, we demonstrate that the quasicrystal is thermodynamically stable in the square-shoulder and linear-ramp system.

Self-assembly of two-dimensional binary quasicrystals: a possible route to a DNA quasicrystal

We use Monte Carlo simulations and free-energy techniques to show that binary solutions of penta- and hexavalent two-dimensional patchy particles can form thermodynamically stable quasicrystals even at very narrow patch widths, provided their patch interactions are chosen in an appropriate way. Such patchy particles can be thought of as a coarse-grained representation of DNA multi-arm 'star' motifs, which can be chosen to bond with one another very specifically by tuning the DNA sequences of the protruding arms. We explore several possible design strategies and conclude that DNA star tiles that are designed to interact with one another in a specific but not overly constrained way could potentially be used to construct soft quasicrystals in experiment. We verify that such star tiles can form stable dodecagonal motifs using oxDNA, a realistic coarse-grained model of DNA.

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.

Observation and Structure Determination of an Oxide Quasicrystal Approximant

We report on the first observation of an approximant structure to the recently discovered two-dimensional oxide quasicrystal. Using scanning tunneling microscopy, low-energy electron diffraction, and surface x-ray diffraction in combination with ab initio calculations, the atomic structure and the bonding scheme are determined. The oxide approximant follows a 3^{2}.4.3.4 Archimedean tiling. Ti atoms reside at the corners of each tiling element and are threefold coordinated to oxygen atoms. Ba atoms separate the TiO_{3} clusters, leading to a fundamental edge length of the tiling 6.7 Å.