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Quasicrystalline alloys and their composites have been extensively studied due to their complex atomic structures, mechanical properties, and their unique tribological and thermal behaviors. However, technological applications of these materials have not yet come of age and still require additional developments. In this review, we discuss the recent advances that have been made in the last years toward optimizing fabrication processes and properties of Al-matrix composites reinforced with quasicrystals. We discuss in detail the high-strength rapid-solidified nanoquasicrystalline composites, the challenges involved in their manufacturing processes and their properties. We also bring the latest findings on the fabrication of Al-matrix composites reinforced with quasicrystals by powder metallurgy and by conventional metallurgical processes. We show that substantial developments were made over the last decade and discuss possible future studies that may result from these recent findings.

Quasicrystals are non-periodic solids that were discovered in 1982 by Dan Shechtman, Nobel Prize Laureate in Chemistry 2011. The underlying mathematics, known as the theory of aperiodic order, is the subject of this comprehensive multi-volume series. This first volume provides a graduate-level introduction to the many facets of this relatively new area of mathematics. Special attention is given to methods from algebra, discrete geometry and harmonic analysis, while the main focus is on topics motivated by physics and crystallography. In particular, the authors provide a systematic exposition of the mathematical theory of kinematic diffraction. Numerous illustrations and worked-out examples help the reader to bridge the gap between theory and application. The authors also point to more advanced topics to show how the theory interacts with other areas of pure and applied mathematics.

Quasicrystalline alloys and their composites have been extensively studied due to their complex atomic structures, mechanical properties, and their unique tribological and thermal behaviors. However, technological applications of these materials have not yet come of age and still require additional developments. In this review, we discuss the recent advances that have been made in the last years toward optimizing fabrication processes and properties of Al‐matrix composites reinforced with quasicrystals. We discuss in detail the high‐strength rapid‐solidified nanoquasicrystalline composites, the challenges involved in their manufacturing processes and their properties. We also bring the latest findings on the fabrication of Al‐matrix composites reinforced with quasicrystals by powder metallurgy and by conventional metallurgical processes. We show that substantial developments were made over the last decade and discuss possible future studies that may result from these recent findings.

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.

Frank–Kasper (F-K) and quasicrystal phases were originally identified in metal alloys and only sporadically reported in soft materials. These unconventional sphere-packing schemes open up possibilities to design materials with different properties. The challenge in soft materials is how to correlate complex phases built from spheres with the tunable parameters of chemical composition and molecular architecture. Here, we report a complete sequence of various highly ordered mesophases by the self-assembly of specifically designed and synthesized giant surfactants, which are conjugates of hydrophilic polyhedral oligomeric silsesquioxane cages tethered with hydrophobic polystyrene tails. We show that the occurrence of these mesophases results from nanophase separation between the heads and tails and thus is critically dependent on molecular geometry. Variations in molecular geometry achieved by changing the number of tails from one to four not only shift compositional phase boundaries but also stabilize F-K and quasicrystal phases in regions where simple phases of spheroidal micelles are typically observed. These complex self-assembled nanostructures have been identified by combining X-ray scattering techniques and real-space electron microscopy images. Brownian dynamics simulations based on a simplified molecular model confirm the architecture-induced sequence of phases. Our results demonstrate the critical role of molecular architecture in dictating the formation of supramolecular crystals with “soft” spheroidal motifs and provide guidelines to the design of unconventional self-assembled nanostructures.

Integrable models form pillars of theoretical physics because they allow for full analytical understanding. Despite being rare, many realistic systems can be described by models that are close to integrable. Therefore, an important question is how small perturbations influence the behavior of solvable models. This is particularly true for many-body interacting quantum systems where no general theorems about their stability are known. Here, we show that no such theorem can exist by providing an explicit example of a one-dimensional many-body system in a quasiperiodic potential whose transport properties discontinuously change from localization to diffusion upon switching on interaction. This demonstrates an inherent instability of a possible many-body localization in a quasiperiodic potential at small interactions. We also show how the transport properties can be strongly modified by engineering potential at only a few lattice sites.

The quasicrystal (QC) possesses a unique lattice structure with rotational symmetry forbidden in periodic crystals. The electric property is far from complete understanding. It has been a long-standing issue whether magnetic long-range order is realized in the QC. Here, we report our theoretical discovery of the ferromagnetic long-range order in the Tb-based QC. The difficulty in past theoretical studies on the QC was lack of the microscopic theory of the crystalline electric field (CEF), which is crucially important in the rare earth systems. By analyzing the CEF in the Tb-based QC, we clarify that magnetic anisotropy plays a key role in realizing unique magnetic textures in the Tb-based QC and approximant crystal (AC). By constructing the minimal model, we show that various magnetic textures on the icosahedron, at whose vertices Tb atoms are located, are realized. We find that the hedgehog state is characterized by the topological charge of one and the whirling-moment state is characterized by an unusually large topological charge of three. The hedgehog and whirling-moment states are shown to be realized as antiferromagnetic orders transcribed as the emergent monopole and antimonopole in the 1/1 AC. We find that these states exhibit the topological Hall effect under applied magnetic field accompanied by the topological as well as metamagnetic transition. Our model and the determined phase diagram are expected to be relevant to the broad range of the rare earth–based QCs and ACs with strong magnetic anisotropy, which are useful not only to understand magnetism but also, to explore topological properties.

This study presents a unique Mg-based alloy composition in the Mg–Zn–Yb system which exhibits bulk metallic glass, metastable icosahedral quasicrystals (iQCs), and crystalline approximant phases in the as-cast condition. Microscopy revealed a smooth gradual transition from glass to QC. We also report the complete melting of a metastable eutectic phase mixture (including a QC phase), generated via suppression of the metastable-to-stable phase transition at high heating rates using fast differential scanning calorimetry (FDSC). The melting temperature and enthalpy of fusion of this phase mixture could be measured directly, which unambiguously proves its metastability in any temperature range. The kinetic pathway from liquid state to stable solid state (an approximant phase) minimizes the free-energy barrier for nucleation through an intermediate state (metastable QC phase) because of its low solid–liquid interfacial energy. At high undercooling of the liquid, where diffusion is limited, another approximant phase with near-liquid composition forms just above the glass-transition temperature. These experimental results shed light on the competition between metastable and stable crystals, and on glass formation via system frustration associated with the presence of several free-energy minima.

Elemental powder mixtures of Al, Cu, and Fe with nominal composition of Al 65 Cu 20 Cu 15 were proceeded in a planetary ball mill for up to 5 h. Phase analysis of as-milled powders shows that only diffraction peaks of metallic elements were detected after milling for 60 min. The long milling times resulted in formation of an Al (Cu, Fe) solid solution (β-phase), but no icosahedral quasicrystalline phase (i-phase). To induce solid-state transformation, milled powders were annealed at 600–700 °C for 4 h. i-phase, β, and ω-Al 7 Cu 2 Fe 1 phases were detected in samples milled 5–45 min. The maximum amount of i-phase was obtained after milling for 15 min, annealing at 700 °C and after milling for 30 min, and annealing at 650–700 °C. The annealed powders present soft ferromagnetic behavior at room temperature, but the magnetic loss was reduced with the increase of the i-phase and its size and reduced amounts of β and ω phases. Graphical abstract SEM images of mechanically alloyed powders milled for 30 min and annealed at (a-b) 600 °C, (c) 650 °C, and (d) 700 °C. Insets 1-4 in (a) show the coexistence of different QC morphologies (scale bars in the insets represent 10 μm. White arrowheads in (b) indicate the concurrency of two distinct morphologies.

Moiré lattices consist of two superimposed identical periodic structures with a relative rotation angle. Moiré lattices have several applications in everyday life, including artistic design, the textile industry, architecture, image processing, metrology and interferometry. For scientific studies, they have been produced using coupled graphene–hexagonal boron nitride monolayers 1 , 2 , graphene–graphene layers 3 , 4 and graphene quasicrystals on a silicon carbide surface 5 . The recent surge of interest in moiré lattices arises from the possibility of exploring many salient physical phenomena in such systems; examples include commensurable–incommensurable transitions and topological defects 2 , the emergence of insulating states owing to band flattening 3 , 6 , unconventional superconductivity 4 controlled by the rotation angle 7 , 8 , the quantum Hall effect 9 , the realization of non-Abelian gauge potentials 10 and the appearance of quasicrystals at special rotation angles 11 . A fundamental question that remains unexplored concerns the evolution of waves in the potentials defined by moiré lattices. Here we experimentally create two-dimensional photonic moiré lattices, which—unlike their material counterparts—have readily controllable parameters and symmetry, allowing us to explore transitions between structures with fundamentally different geometries (periodic, general aperiodic and quasicrystal). We observe localization of light in deterministic linear lattices that is based on flat-band physics 6 , in contrast to previous schemes based on light diffusion in optical quasicrystals 12 , where disorder is required 13 for the onset of Anderson localization 14 (that is, wave localization in random media). Using commensurable and incommensurable moiré patterns, we experimentally demonstrate the two-dimensional localization–delocalization transition of light. Moiré lattices may feature an almost arbitrary geometry that is consistent with the crystallographic symmetry groups of the sublattices, and therefore afford a powerful tool for controlling the properties of light patterns and exploring the physics of periodic–aperiodic phase transitions and two-dimensional wavepacket phenomena relevant to several areas of science, including optics, acoustics, condensed matter and atomic physics. A superposition of tunable photonic lattices is used to create optical moiré patterns and demonstrate the resulting localization of light waves through a mechanism based on flat-band physics.