Why are quasicrystals quasiperiodic?

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.

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.

From Laves Phases to Quasicrystal Approximants in the Na-Au-Cd System

The exploratory synthesis of gold-based polar intermetallic phases has revealed many new compounds with unprecedented crystal structures, unique bonding arrangements, and interesting electronic features. Here, we further understand the complexity of gold's crystal chemistry by studying the Na-Au-Cd ternary composition space. A nearly continuous structure transformation is observed between the seemingly simple binary NaAu₂-NaCd₂ phases, yielding three new intermetallic compounds with the compositions Na(Au_0.89(5)Cd_0.11(5))₂, Na(Au_0.51(4)Cd_0.49(4))₂, and Na₈Au_3.53(1)Cd_13.47(1). Two compounds adopt different Laves phases, while the third crystallizes in a complex decagonal quasicrystal approximant. All three compounds are related through Friauf-Laves polyhedral building units with the gold/cadmium ratio found to control the transition among the unique crystal structures. Electronic structure calculations subsequently revealed the metallic nature of all three compounds with a combination of polar covalent Na-(Au/Cd) interactions and covalent (Au/Cd)-(Au/Cd) bonding interactions stabilizing each structure. These results highlight the crystal and electronic structure relationship among Laves phases and quasicrystal approximants enabled by the unique chemistry of gold.

Thermoelectric Properties of Co-Substituted Al-Pd-Re Icosahedral Quasicrystals

The practical application of quasicrystals (QCs) as thermoelectric materials makes icosahedral (i-) Al-Pd-Re QC attractive because of its moderate electrical conductivity (~280 Ω^-1 cm^-1), relatively high Seebeck coefficient (~100 μV K^-1), and low thermal conductivity (~1.3 W m^-1 K^-1) at room temperature. To develop a thermoelectric Π-shaped power generation module, we need both p- and n-type thermoelectric materials. In this work, we aimed to develop an n-type i-Al-Pd-Re-based QC and investigated the effect of Co substitution for Re on the thermoelectric properties, i.e., the electron-doping effect. We synthesized dense bulk samples with nominal compositions of Al₇₁Pd₂₀(Re_1-xCo_x)₉ (x = 0, 0.1, 0.2, 0.3, 0.4, 0.5) via arc-melting, annealing, and sintering methods. We found that Co can produce n-type carriers in dilute substitution amounts of x = 0.1 and 0.2; however, the Seebeck coefficient at 300 K showed an n- to p-type transition with increasing x. This indicates that a simple rigid-band approximation is not applicable for i-Al-Pd-Re QC, which makes it difficult to synthesize an n-type i-Al-Pd-Re-based QC. Although the thermal conductivity was reduced from 1.28 (x = 0) to 1.08 W m^-1 K^-1 (x = 0.3) at 373 K by lowering of the electron thermal conductivity (electrical conductivity) and the alloying effect via Co substitution, the dimensionless figure of merit was not enhanced because of lowering of the power factor for all samples. The elastic moduli of i-Al-Pd-Re QC decreased by Co substitution, indicating that i-Al-Pd-Re-Co QC had a more ionic and brittle character.

Frustrated Packing in Simple Structures: Chemical Pressure Hindrance to Isolobal Bonds in the TiAl3 type and ZrAl2.6Sn0.4

While simple close-packed arrangements convey a sense of optimization, they can, in fact, host competition between different types of interactions. The TiAl₃ structure type, for example, represents one of a series of ordered TE₃ variants (T = transition metal, E = main group element) of the face-centered cubic structure, alongside the AuCu₃ and ZrAl₃ types. These structures differ in their T-T connectivity corresponding to the 18-n rule: electronic pseudogaps occur at electron concentrations of 18-n/T atom, where n is the number of electron pairs each T atom shares with other T atoms in T-T isolobal bonds. Facile stacking variations interrelate these structures, presumably setting the stage for an electronically precise series. However, the prototype of the TiAl₃ type itself violates the 18-n rule, with its count of 13 electrons/Ti atom calling for n = 5 rather than the 4 isolobal T-T bonds/T atom available in this type. Here, we investigate the factors underlying this deviation from the 18-n rule and their relation to the new TiAl₃-type compound ZrAl_3-xSn_x (x ∼ 0.4). First, the relative stabilities of the TiAl₃ and ZrAl₃ types are compared for TAl₃ compounds (T = Zr and Ti). While for T = Zr, the structure adhering to the 18-n rule is highly preferred, for T = Ti, the energy difference essentially vanishes. This trend is connected through DFT-Chemical Pressure (CP) analysis to a tension that emerges in TiAl₃ between the optimization of the T-T isolobal bonds and the space requirements of Al-Al contacts elsewhere. This picture elucidates the transition of ZrAl₃ from its own type to the TiAl₃-type upon partial Sn substitution in ZrAl_2.6Sn_0.4: the incorporation of Sn brings the electron count closer to that predicted for the TiAl₃ type, while electronegativity and CP direct the larger Sn atoms to the site that resists isolobal bond formation in TiAl₃.

Gummelt versus Lück decagon covering and beyond. Implications for decagonal quasicrystals

Specific structural repeat units can be used as quasi-unit cells of decagonal quasicrystals. So far, the most famous and almost exclusively employed one has been the Gummelt decagon. However, in an increasing number of cases Lück decagons have been found to be more appropriate without going into depth. The diversities and commonalities of these two basic decagonal clusters and of some more general ones are discussed. The importance of the type of underlying tiling for the correct classification of a quasi-unit cell is demonstrated.

A fivefold UO 2 2 + node is a path to dodecagonal quasicrystal approximants in coordination polymers

Aperiodic formations continue to focus interest in areas ranging from advanced scientific theories to practical everyday applications. Starting from diverse and tightly bonded intermetallic compounds, this world showed an important breakthrough toward the so-called soft systems of meso/macroscale: liquid crystals, thin films, polymers, proteins, etc. This work opens a route for making bulk quasicrystals (QC) in an unprecedented but very common area, with molecular ligands. Since these systems are, to a large extent, transparent, they extend the possible areas of QC application to previously unreachable corners, e.g., photonics. We combined efficient bridging ligands with uranyl pentagonal bonding centers and, unexpectedly, brought the unique attributes of f-element coordination chemistry to an interdisciplinary area of aperiodic formations. Taking advantage of the planar coordination of uranyl ions, we were able to direct the structure expansion solely in two directions with a characteristic snub square tiling, a predicted but previously unobtainable dodecagonal approximant.

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.

Full real-space analysis of a dodecagonal quasicrystal

The atomically resolved real-space structure of a long-range-ordered dodecagonal quasicrystal is determined based on scanning tunnelling microscopy. For the BaTiO3-derived oxide quasicrystal which spontaneously forms on a Pt(111) surface, 8100 atomic positions have been determined and are compared with an ideal Niizeki-Gähler tiling. Although the Niizeki-Gähler tiling has a complex three-element structure, the abundance of the triangle, square and rhomb tiling elements in the experimental data closely resembles the ideal frequencies. Similarly, the frequencies of all possible next-neighbour tiling combinations are, within the experimental uncertainty, identical to the ideal tiling. The angular and orientational distributions of all individual tiling elements show the characteristics of the dodecagonal quasicrystal. In contrast, the analysis of the orientation of characteristic and more complex tiling combinations indicates the partial decomposition of the quasicrystal into small patches with locally reduced symmetry. These, however, preserve the long-range quasicrystal coherence. The symmetry reduction from dodecagonal to sixfold is assigned to local interaction with the threefold substrate. It leads to atomic flips which preserve the number of quasicrystal tiling elements.

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.

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.

From Quasicrystals to Crystals with Interpenetrating Icosahedra in Ca-Au-Al: In Situ Variable-Temperature Transformation

The irreversible transformation from an icosahedral quasicrystal (i-QC) CaAu_4.39Al_1.61 to its cubic 2/1 crystalline approximant (CA) Ca₁₃Au_56.31(3)Al_21.69 (CaAu_4.33(1)Al_1.67, Pa3̅ (No. 205); Pearson symbol: cP728; a = 23.8934(4)), starting at ∼570 °C and complete by ∼650 °C, is discovered from in situ, high-energy, variable-temperature powder X-ray diffraction (PXRD), thereby providing direct experimental evidence for the relationship between QCs and their associated CAs. The new cubic phase crystallizes in a Tsai-type approximant structure under the broader classification of polar intermetallic compounds, in which atoms of different electronegativities, viz., electronegative Au + Al vs electropositive Ca, are arranged in concentric shells. From a structural chemical perspective, the outermost shell of this cubic approximant may be described as interpenetrating and edge-sharing icosahedra, a perspective that is obtained by splitting the traditional structural description of this shell as a 92-atom rhombic triacontahedron into an 80-vertex cage of primarily Au [Au_59.86(2)Al_17.14□_3.00] and an icosahedral shell of only Al [Al_10.5□_1.5]. Following the proposal that the cubic 2/1 CA approximates the structure of the i-QC and on the basis of the observed transformation, an atomic site analysis of the 2/1 CA, which shows a preference to maximize the number of heteroatomic Au-Al nearest neighbor contacts over homoatomic Al-Al contacts, implies a similar outcome for the i-QC structure. Analysis of the most intense reflections in the diffraction pattern of the cubic 2/1 CA that changed during the phase transformation shows correlations with icosahedral symmetry, and the stability of this cubic phase is assessed using valence electron counts. According to electronic structure calculations, a cubic 1/1 CA, "Ca₂₄Au₈₈Al₆₄" (CaAu_3.67Al_2.67) is proposed.

Quasicrystals: What do we know? What do we want to know? What can we know?

More than 35 years and 11 000 publications after the discovery of quasicrystals by Dan Shechtman, quite a bit is known about their occurrence, formation, stability, structures and physical properties. It has also been discovered that quasiperiodic self-assembly is not restricted to intermetallics, but can take place in systems on the meso- and macroscales. However, there are some blank areas, even in the centre of the big picture. For instance, it has still not been fully clarified whether quasicrystals are just entropy-stabilized high-temperature phases or whether they can be thermodynamically stable at 0 K as well. More studies are needed for developing a generally accepted model of quasicrystal growth. The state of the art of quasicrystal research is briefly reviewed and the main as-yet unanswered questions are addressed, as well as the experimental limitations to finding answers to them. The focus of this discussion is on quasicrystal structure analysis as well as on quasicrystal stability and growth mechanisms.

Three-Dimensional Modeling of Quasicrystal Structures from X-ray Diffraction: An Icosahedral Al-Cu-Fe Alloy

Quasicrystals (QCs) are well-ordered but aperiodic crystals with classically forbidden symmetries (such as 5-fold). High-dimensional (HD) crystallography is a standard method to locate atom positions explicitly. However, in practice, it is still challenging because of its complexity. Here, we report a new simple approach to three-dimensional (3D) atomic modeling derived from X-ray diffraction data, and apply it to the icosahedral QC Al_0.63Cu_0.25Fe_0.12. Electron density maps were calculated directly from 3D diffraction data indexed with noninteger (fractional) numbers as measured, with proper phases; each of 2⁵ = 32 possible phase assignments for the five strongest reflections was used for Fourier synthesis. This resulted in an initial phasing model based on chemically sensible electron density maps. The following procedure was exactly the same as that used to determine ordinary crystal structures, except that fractional indices were assigned to the reciprocal vectors relative to the three orthogonal 2-fold axes in icosahedral (I_h) symmetry to which the observed diffraction data conformed. Finally, ∼30 000 atoms were located within a sphere of a ∼48 Å radius. Structural motifs or basic repeating units with a hierarchical nature can be found. Isolated icosahedral clusters are surrounded by a concentric dodecahedron, beyond which there is a concentric truncated icosahedron. These are strikingly similar to those obtained via HD crystallography, but show very clear real-space relationships between the clusters.

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.

Interface-driven formation of a two-dimensional dodecagonal fullerene quasicrystal

Since their discovery, quasicrystals have attracted continuous research interest due to their unique structural and physical properties. Recently, it was demonstrated that dodecagonal quasicrystals could be used as bandgap materials in next-generation photonic devices. However, a full understanding of the formation mechanism of quasicrystals is necessary to control their physical properties. Here we report the formation of a two-dimensional dodecagonal fullerene quasicrystal on a Pt₃Ti(111) surface, which can be described in terms of a square–triangle tiling. Employing density functional theory calculations, we identify the complex adsorption energy landscape of the Pt-terminated Pt₃Ti surface that is responsible for the quasicrystal formation. We demonstrate the presence of quasicrystal-specific phason strain, which provides the degree of freedom required to accommodate the quasicrystalline structure on the periodic substrate. Our results reveal detailed insight into an interface-driven formation mechanism and open the way to the creation of tailored fullerene quasicrystals with specific physical properties.

Quasicrystals promise exciting technological advances in optical devices, but their formation mechanism is yet not fully understood. Here, the authors describe a two-dimensional dodecagonal fullerene quasicrystal, forming on a Pt₃Ti(111)-surface due to the complex adsorption-energy landscape.

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.

An Icosahedral Quasicrystal and Its 1/0 Crystalline Approximant in the Ca-Au-Al System

A new icosahedral quasicrystalline phase, CaAu_4.5-xAl_1.5+x [0.11 ≤ x ≤ 0.40(6); CaAu_4.4Al_1.6, a_QC = 5.383(4) Å, and Pm3̅ 5̅], and its lowest-order 1/0 cubic crystalline approximant phase, CaAu_3+xAl_1-x [0 ≤ x ≤ 0.31(1); a = 9.0766(5)-9.1261(8) Å, Pa3̅ (No. 205), and Pearson symbol cP40], have been discovered in the Ca-poor region of the Ca-Au-Al system. In the crystalline approximant, eight [Au_3-xAl_1+x] tetrahedra fill the unit cell, and each tetrahedron is surrounded by four Ca atoms, thus forming a three-dimensional network of {Ca_4/4[Au_3-xAl_1+x]} tetrahedral stars. A computational study of Au and Al site preferences concurs with the experimental results, which indicate a preference for near-neighbor Au-Al interactions over Au-Au and Al-Al interactions. Analysis of the electronic density of states and the associated crystal orbital Hamilton population curves was used to rationalize the descriptions of CaAu_4.5-xAl_1.5+x [0.11 ≤ x ≤ 0.46(6)] and CaAu_3+xAl_1-x [0 ≤ x ≤ 0.31(1)] as polar intermetallic species, whereby Ca atoms engage in polar covalent bonding with the electronegative, electron-deficient [Au_3-xAl_1+x] tetrahedral clusters and the observed phase width of the crystalline approximant.

Critical properties of a two-dimensional Ising magnet with quasiperiodic interactions

We address the study of quasiperiodic interactions on a square lattice by using an Ising model with ferromagnetic and antiferromagnetic exchange interactions following a quasiperiodic Fibonacci sequence in both directions of a square lattice. We applied the Monte Carlo method, together with the Metropolis algorithm, to calculate the thermodynamic quantities of the system. We obtained the Edwards-Anderson order parameter q_{EA}, the magnetic susceptibility χ, and the specific heat c in order to characterize the universality class of the phase transition. We also use the finite size scaling method to obtain the critical temperature of the system and the critical exponents β,γ, and ν. In the low-temperature limit we obtained a spin-glass phase with critical temperature around T_{c}≈2.274, and the critical exponents β,γ, and ν, indicating that the quasiperiodic order induces a change in the universality class of the system. Also, we discovered a spin-glass ordering in a two-dimensional system which is rare and, as far as we know, the unique example is an under-frustrated Ising model.

Direct observation of solid-state reversed transformation from crystals to quasicrystals in a Mg alloy

Phase transformation of quasicrystals is of interest in various fields of science and technology. Interestingly, we directly observed unexpected solid-state epitaxial nucleation and growth of Zn₆Mg₃Y icosahedral quasicrystals in a Mg alloy at about 573 K which is about 300 K below the melting point of Zn₆Mg₃Y, in contrast to formation of quasicrystals through solidification that was usually found in many alloys. Maximizing local packing density of atoms associated with segregation of Y and Zn in Mg adjacent to Mg/Zn₃MgY interfaces triggered atomic rearrangement in Mg to form icosahedra coupled epitaxially with surface distorted icosahedra of Zn₃MgY, which plays a critical role in the nucleation of icosahedral clusters. A local Zn:Mg:Y ratio close to 6:3:1, corresponding to a valence electron concentration of about 2.15, should have been reached to trigger the formation of quasicrystals at Mg/Zn₃MgY interfaces. The solid-state icosahedral ordering in crystals opens a new window for growing quasicrystals and understanding their atomic origin mechanisms. Epitaxial growth of quasicrystals onto crystals can modify the surface/interface structures and properties of crystalline materials.

Atomic structure of icosahedral quasicrystals: stacking multiple quasi-unit cells

An effective tiling approach is proposed for the structural description of icosahedral quasicrystals based on the original substitution algorithm.

Substitution rules for icosahedral quasicrystals

An original well-defined substitution algorithm on how to construct the icosahedral packings is proposed, simple inflation/deflation rules for icosahedral quasicrystals are described, and natural local matching rules are derived.

Structural DNA nanotechnology: state of the art and future perspective

Over the past three decades DNA has emerged as an exceptional molecular building block for nanoconstruction due to its predictable conformation and programmable intra- and intermolecular Watson-Crick base-pairing interactions. A variety of convenient design rules and reliable assembly methods have been developed to engineer DNA nanostructures of increasing complexity. The ability to create designer DNA architectures with accurate spatial control has allowed researchers to explore novel applications in many directions, such as directed material assembly, structural biology, biocatalysis, DNA computing, nanorobotics, disease diagnosis, and drug delivery. This Perspective discusses the state of the art in the field of structural DNA nanotechnology and presents some of the challenges and opportunities that exist in DNA-based molecular design and programming.

Growth modes of quasicrystals

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

Computing phase diagrams for a quasicrystal-forming patchy-particle system

We introduce an approach to computing the free energy of quasicrystals, which we use to calculate phase diagrams for systems of two-dimensional patchy particles with five regularly arranged patches that have previously been shown to form dodecagonal quasicrystals. We find that the quasicrystal is a thermodynamically stable phase for a wide range of conditions and remains a robust feature of the system as the potential's parameters are varied. We also demonstrate that the quasicrystal is entropically stabilized over its crystalline approximants.

Fibonacci, quasicrystals and the beauty of flowers

The appearance of Fibonacci sequences and the golden ratio in plant structures is one of the great outstanding puzzles of biology. Here I suggest that quasicrystals, which naturally pack in the golden ratio, may be ubiquitous in biological systems and introduce the golden ratio into plant phyllotaxy. The appearance of golden ratio-based structures as beautiful indicates that the golden ratio may play a role in the development of consciousness and lead to the aesthetic natural selection of flowering plants.