Formation Of Quasicrystals Research Articles

Two-Dimensional Crystallization of Hexagonal Bilayer with Moiré Patterns

Direct observation of crystallization dynamics in real space is of special interest to scientists in various disciplines. Although direct observation of transient structural transformation in a nanocrystalline system has been recently achieved using the state-of-the-art aberration-corrected transmission electron microscopy (AC-TEM), the small length scales of individual species in molecular systems still preclude routine observation of crystallization dynamics. Unidirectional packing of microbeads can serve as an experimental model system, as their dynamics can be observed and recorded readily in the laboratory due to their larger size and slower time scale. Herein, we present direct observation of a two-dimensional (2D) crystallization enabled by such a packing process. The direct imaging approach not only allows observation of the dynamics in a bead-by-bead fashion but also reveals intriguing phenomena, such as the formation of grain boundaries, disorder-order transitions, and the Moiré patterns which arise when two periodic monolayers are overlaid at certain angles. In addition, the imaging afforded by confocal microscopy facilitates a structural analysis of height-dependent polygonal tiling of the top monolayer, which has implication to the formation of 2D quasicrystals.

Formation of dodecagonal quasicrystals in two-dimensional systems of patchy particles

The behaviour of two-dimensional patchy particles with five and seven regularly arranged patches is investigated by computer simulation. For higher pressures and wider patch widths, hexagonal crystals have the lowest enthalpy, whereas at lower pressures and for narrower patches, lower density crystals with five nearest neighbours that are based on the (3(2),4,3,4) tiling of squares and triangles become lower in enthalpy. Interestingly, in regions of parameter space near to that where the hexagonal crystals become stable, quasicrystalline structures with dodecagonal symmetry form on cooling from high temperature. These quasicrystals can be considered as tilings of squares and triangles and are probably stabilized by the large configurational entropy associated with all the different possible such tilings. The potential for experimentally realizing such structures using DNA multi-arm motifs is also discussed.

Why are quasicrystals quasiperiodic?

In the past two decades significant progress has been made in the search for stable quasicrystals, the determination of their structures and the understanding of their physical properties. Now, quasiperiodic ordering states are not only known for intermetallic compounds, but also for mesoscopic systems such as ABC-star terpolymers, liquid crystals or different kinds of colloids. However, in spite of all these achievements fundamental questions concerning quasicrystal formation, growth and stability are still not fully answered. This tutorial review introduces the research in these fields and addresses some of the open questions concerning the origin of quasiperiodicity.

Self-assembly of soft-matter quasicrystals and their approximants

The surprising recent discoveries of quasicrystals and their approximants in soft-matter systems poses the intriguing possibility that these structures can be realized in a broad range of nanoscale and microscale assemblies. It has been theorized that soft-matter quasicrystals and approximants are largely entropically stabilized, but the thermodynamic mechanism underlying their formation remains elusive. Here, we use computer simulation and free-energy calculations to demonstrate a simple design heuristic for assembling quasicrystals and approximants in soft-matter systems. Our study builds on previous simulation studies of the self-assembly of dodecagonal quasicrystals and approximants in minimal systems of spherical particles with complex, highly specific interaction potentials. We demonstrate an alternative entropy-based approach for assembling dodecagonal quasicrystals and approximants based solely on particle functionalization and shape, thereby recasting the interaction-potential-based assembly strategy in terms of simpler-to-achieve bonded and excluded-volume interactions. Here, spherical building blocks are functionalized with mobile surface entities to encourage the formation of structures with low surface contact area, including non-close-packed and polytetrahedral structures. The building blocks also possess shape polydispersity, where a subset of the building blocks deviate from the ideal spherical shape, discouraging the formation of close-packed crystals. We show that three different model systems with both of these features-mobile surface entities and shape polydispersity-consistently assemble quasicrystals and/or approximants. We argue that this design strategy can be widely exploited to assemble quasicrystals and approximants on the nanoscale and microscale. In addition, our results further elucidate the formation of soft-matter quasicrystals in experiment.

Electrons in Quasicrystals

AbstractQuasicrystals, discovered by Dan Shechtman in 1982, provide unique materials with long‐range aperiodic order. Their electronic structure and the electronic transport in quasicrystalline alloys has been a focus of research, motivated by trying to understand the formation and stability of quasicrystals, and their pronounced transport anomalies. This brief review on the motion of electrons concentrates on one perspective of this problem. It focuses on results obtained for simple models which capture the essence of the aperiodic order, rather than on results which attempt a realistic description of quasicrystalline alloys by employing empirical interaction potentials.

Toward the discovery of new soft quasicrystals: From a numerical study viewpoint

AbstractThis is a progress review of an emerging research front: soft quasicrystals including liquid crystalline dendrons, nanoparticles, mesoporous silica, colloids, ABC star and linear terpolymers, and even water and silicon. As an aid to readers, we explain the basics of quasicrystals developed in solid‐state physics: orders in quasicrystals, higher dimensional crystallography, approximants, phason randomness, and the origin of quasicrystal formation. Then we review some numerical studies from early to recent ones. Our main purpose is to elucidate how to construct quasicrystalline structures: The introduction of additional components or a new length‐scale is the key to discover new quasicrystals. As a case study, we describe our recent studies on ABC star terpolymer systems and present the results of simulations of dodecagonal polymeric quasicrystals. In the case of dodecagonal quasicrystals, one easily finds that the key is to search square‐triangle tiling structures with changing components. Application to photonic quasicrystals is reviewed as well. Our hope is that this review will contribute furthering quasicrystal chemistry. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012

Reactivity of metal-containing monomers 71. Synthesis of nanosized quasicrystals and related metallopolymer composites

The methods for the preparation of quasicrystalline intermetallic compounds in the protective matrix in the systems Al65Cu22Fe13 and Al54Cu9Mg37 were developed. They are formed both earlier and from elements during the formation of a metallopolymer composite material. Three methods of preparation are optimal: from low-dimensional powders of quasicrystals and film-forming polymers, by the in situ formation of quasicrystals during the thermal decomposition of the corresponding precursors, and via the thermal polymerization of metal-containing monomers (components of quasicrystals) followed by the controlled pyrolysis of the metallopolymers that formed. The metallopolymer composite obtained by the incorporation of Al63Cu22Fe13 into the high-density polyethylene or polyacrylamide matrix affords up to 10 wt.% cubic phase of β-Al50(Cu,Fe)50. The products of thermal decomposition of the low-molecular-weight precursors are a mixture of the quasicrystalline (Al65Cu22Fe13) and crystalline (cubic β-Al50(Cu,Fe)50 and tetragonal ω-Al7Cu2Fe, Al2Cu) phases. The thermolysis of the metallo-polymer composition yields a finely dispersed powder, whose main component is an alloy Al7Cu3Mg6 with an admixture of the Al2CuMg phase.

Phase diagram studies of Ti–Zr–Ni alloys containing less than 40 at.% Ni and estimated critical cooling rate for icosahedral quasicrystal formation from the liquid

A detailed phase diagram study of Ti–Zr–Ni alloys has been conducted using conventional X-ray scattering, microstructural and thermal annealing, and levitation techniques, to improve on the existing phase diagram. Various phases, such as simple crystalline, approximant, icosahedral quasicrystalline, and complex polytetrahedral phases, have been identified. A eutectic composition has been found near 21 at.% Ni for nearly equal Ti and Zr concentrations. From containerless processing studies, using electrostatic levitation, the phase boundary of the icosahedral quasicrystal (i-phase) is placed between 29 and 45 at.% Ti with 21 at.% Ni. A first-time estimate of the critical cooling rate for the formation of the i-phase directly from the liquid in alloys with 21 at.% Ni is less than 100 °C s −1, which is much smaller than for most of the known metastable quasicrystals.

Change in local environment upon quasicrystallization of Zr–Cu glassy alloys by addition ofPd and Pt

The effects of Pd and Pt, which are known quasicrystal (QC)-forming elements, on the local atomic structurein Zr70Cu30 glassy alloys are investigated. A QC phase precipitates from a glassy phase above a certaintemperature by a cooperative-like motion of icosahedral clusters. Quasicrystallization isaccompanied by a significant change in the local environment around the Zr atoms and aslight change around the noble metal. However, the local environment around the Cu atomsremains almost the same during QC formation. It is suggested that two types of icosahedralpolyhedra exist in the glassy state: one has a relatively perfect icosahedral structure formedaround the Zr atoms. The other is in a distorted state around the Cu atoms. We speculatethat the medium-range order (i.e. QC nucleus) has a Zr-centered icosahedral clusteras its core, and the QC grows via aggregation of possible clusters in the initialstage. Pd or Pt atoms stabilize and/or connect individual Zr-centered icosahedralclusters, facilitating the formation of the nucleus and growth of the QC phase.

Formation of Icosahedral Quasicrystals and 1/1 Crystal Approximants in Al-Pd-RE (RE: Rare Earth Metals) Systems

We have investigated the formation of an icosahedral quasicrystal (i-phase) and its 1/1-crystal approximant (1/1-phase) in the (Al, Ga)-Pd-RE (Rare earth metals) systems. Consequently, the Ga-Pd-Sc 1/1-phase, the Al-Pd-RE (RE = Yb, Tm and Er) 1/1-phase and the Al-Pd-Yb i-phase have been newly found by the substitution of Al for Ga, or Sc for other RE in the constituent elements of the Al54Pd30Sc16 i-phase previously reported. For the i- and the 1/1-phases studied in this work, the number of valence electrons per atom (e⁄a) ratio is 2.10 and the atomic radius ratio of the rare earth element to that of the other base elements is in the range 1.15–1.24, which fulfilled the formation conditions previously reported for other Tsai-type i-phases. On the other hand, the stability of the i-phases becomes lower with increasing the atomic radii of the RE elements, which indicates that the atomic radius ratio plays important role in the formation of the Al-Pd-RE i- and 1/1-phases.

Bond orientational ordering in a metastable supercooled liquid: a shadow of crystallizationand liquid–liquid transition

It is widely believed that a liquid state can be characterized by a single order parameter,density, and that a transition from a liquid to solid can be described by density ordering(translational ordering). For example, this type of theory has had great success indescribing the phase behaviour of hard spheres. However, there are some features thatcannot be captured by such theories. For example, hard spheres crystallize into either hcpor fcc structures, without a tendency of bcc ordering which is expected by theAlexander–McTague theory based on the Landau-type free energy of the density orderparameter. We also found hcp-like bond orientational ordering in a metastablesupercooled liquid, which promotes nucleation of hcp crystals. Furthermore, theoriesbased on the single order parameter cannot explain water-like thermodynamic andkinetic anomalies of a liquid and liquid–liquid transition in a single-componentliquid. Based on these facts, we argue that we need an additional order parameterto describe a liquid state. It is bond orientational order, which is induced bydense packing in hard spheres or by directional bonding in molecular and atomicliquids. Bond orientational order is intrinsically of local nature, unlike translationalorder which is of global nature. This feature plays a unique role in crystallizationand quasicrystal formation. We also reveal that bond orientational ordering isa cause of dynamic heterogeneity near a glass transition and is linked to slowdynamics. In relation to this, we note that, for describing the structuring of ahighly disordered liquid, we need a structural signature of low configurationalentropy, which is more general than bond orientational order. Finally, the water-likeanomaly and liquid–liquid transition can be explained by bond orientational orderingdue to hydrogen or covalent bonding and its cooperativity, respectively. So weargue that bond orientational ordering is a key to the physical understanding ofcrystallization, quasicrystallization, glass transition, water-like anomaly and liquid–liquidtransition. A unified description of these phenomena may be possible along thisline.

Hydrogen and Zr-based metallic glasses: Gas/solid absorption process and structure evolution

The absorption of hydrogen by means of gas–solid reaction and its consequence on the structure have been studied for fully amorphous alloys as well as quasicrystals/glassy composite alloys based on the composition Zr 59Ti 3Cu 20Al 10Ni 8. The process of hydrogen absorption has been performed and monitored under 20 bar of H 2 using high pressure-differential scanning calorimetry (HP-DSC). The structure evolution of the samples has been followed by X-ray diffractometry (XRD). Results show that the nature of the surface oxide layer strongly affects the process of hydrogen absorption, especially its starting temperature. The structure evolves nevertheless along the same basic sequence, regardless of the sample: (i) the alloys keep a global amorphous structure up to roughly H/M = 0.8 and T = 350 °C; (ii) then ZrH 2 and at higher temperature Cu 2AlZr are formed. The stability of the glass is weakened and the formation of quasicrystals is inhibited under 20 bar of H 2. An heterogeneous distribution of hydrogen atoms inside the amorphous matrix has been inferred from the results.

Effects of Cooling Rate on Morphology, Microhardness and Volume Fraction of Mg-Zn-Y Quasicrystals

Mg72.5Zn26Y1.5 (at.%) quasicrystal alloys were investigated under different solidification conditions. The cooling curves were gathered by the multi-channel temperature acquiring system and the corresponding microstructures were analyzed. The morphology, microhardness and volume fraction of quasicrystals were detailedly studied. The effects of cooling rate on the above three aspects were also studied and the relation schema among them were exhibited. The results show that the quasicrystal size tend to smaller, its microhardness and volume fraction gradually increased, and the quasicrystal morphology changed from bulk polygon to petal-like and finally changed to plat X-shape with the elevated cooling rate from 2.3K/s to 181.2K/s. However, the quasicrystal nucleation will be restrained and amorphous matter will be created if the cooling rate exceed the critical point which make against the formation of quasicrystals with high performance.

Formation and stability of quasicrystals and other complex intermetallics

Formation and stability of quasicrystals and other complex intermetallics

Basic Co-rich decagonal Al-Co-Ni: Superstructure

The four-layer superstructure of basic Co-rich decagonal ${\text{Al}}_{72.5}{\text{Co}}_{18.5}{\text{Ni}}_{9}$ was determined by single-crystal x-ray diffraction. Based on our previous work [A. Strutz, A. Yamamoto, and W. Steurer, Phys. Rev. B 80, 184102 (2009)], a superstructure model was derived with five-dimensional (5D) noncentrosymmetric space-group symmetry $P\overline{10}2c$ with some additional constraints resulting from normal-mode analysis. The 5D structure model was refined with 250 parameters, resulting in values of $wR=0.039$ and $R=0.186$ for 1222 unique reflections. Its close relationship with the structure of W-Al-Co-Ni, a $⟨3/2,2/1⟩$ approximant, proves the physical validity of our structure model and justifies the use of the W phase for the derivation of structural principles underlying the formation of Al-based decagonal quasicrystals.

The formation of quasicrystals and their hydrides in Ti–Zr–Ni system

In the present work, an essentially new approach to production of quasicrystals in Ti–Zr–Ni system is described. It consists in co-compaction and dehydrogenation-sintering of mixture of powders of Ni with two and more hydrides of transition metals of IV group (Ti, Zr). The interaction of some of received quasicrystals with hydrogen in combustion mode is studied, the hydrides of these quasicrystals are synthesized and investigated.

Relation between glass and quasicrystal formation in the Zr–Nb–Cu–Ni–Al alloys upon solidification

We reported the relationship among the icosahedral short-range order (ISRO), glass formation and quasicrystal formation in the Zr–Nb–Cu–Ni–Al alloys. The alloys with the Nb addition stabilized ISRO, making the alloys form the amorphous, quasicrystal and crystal with decreasing the cooling rate. Not only bulk glass but also bulk quasicrystal were found to form for these alloys. The structural evolution from the amorphous phase, to the quasicrystal and crystals with the variation in the Nb content or the cooling rate is present. These findings were related to the degree of ISRO, which are advantageous for understanding the glass formation phenomenon.

Frustration of the stable Zr-Ti-Ni quasicrystal as the basis of glass formation

The origin of glass formation and the systematics of quasicrystal formation in Zr-Ti-Ni-(Cu-Be)-based metallic glasses is unraveled. Both are found to rely on frustration of the stable ${\text{Zr}}_{41.5}{\text{Ti}}_{41.5}{\text{Ni}}_{17}$ quasicrystal. A systematic change in composition of the stable Zr-Ti-Ni quasicrystal prevents icosahedral order from direct growth in the deeply undercooled liquid and induces the glass transition. Quasicrystals form in the amorphous phase to reduce frustration, i.e., to recover the ideal symmetry and composition of the stable Zr-Ti-Ni quasicrystal. This relation is established especially by the discovery of a new type of sequential phase transformations of icosahedral quasicrystalline phases involving chemical redistribution.

Proliferation of anomalous symmetries in colloidal monolayers subjected to quasiperiodic light fields

Quasicrystals provide a fascinating class of materials with intriguing properties. Despite a strong potential for numerous technical applications, the conditions under which quasicrystals form are still poorly understood. Currently, it is not clear why most quasicrystals hold 5- or 10-fold symmetry but no single example with 7- or 9-fold symmetry has ever been observed. Here we report on geometrical constraints which impede the formation of quasicrystals with certain symmetries in a colloidal model system. Experimentally, colloidal quasicrystals are created by subjecting micron-sized particles to two-dimensional quasiperiodic potential landscapes created by n = 5 or seven laser beams. Our results clearly demonstrate that quasicrystalline order is much easier established for n = 5 compared to n = 7. With increasing laser intensity we observe that the colloids first adopt quasiperiodic order at local areas which then laterally grow until an extended quasicrystalline layer forms. As nucleation sites where quasiperiodicity originates, we identify highly symmetric motifs in the laser pattern. We find that their density strongly varies with n and surprisingly is smallest exactly for those quasicrystalline symmetries which have never been observed in atomic systems. Since such high-symmetry motifs also exist in atomic quasicrystals where they act as preferential adsorption sites, this suggests that it is indeed the deficiency of such motifs which accounts for the absence of materials with e.g., 7-fold symmetry.

Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra

All hard, convex shapes are conjectured by Ulam to pack more densely than spheres, which have a maximum packing fraction of phi = pi/ radical18 approximately 0.7405. Simple lattice packings of many shapes easily surpass this packing fraction. For regular tetrahedra, this conjecture was shown to be true only very recently; an ordered arrangement was obtained via geometric construction with phi = 0.7786 (ref. 4), which was subsequently compressed numerically to phi = 0.7820 (ref. 5), while compressing with different initial conditions led to phi = 0.8230 (ref. 6). Here we show that tetrahedra pack even more densely, and in a completely unexpected way. Following a conceptually different approach, using thermodynamic computer simulations that allow the system to evolve naturally towards high-density states, we observe that a fluid of hard tetrahedra undergoes a first-order phase transition to a dodecagonal quasicrystal, which can be compressed to a packing fraction of phi = 0.8324. By compressing a crystalline approximant of the quasicrystal, the highest packing fraction we obtain is phi = 0.8503. If quasicrystal formation is suppressed, the system remains disordered, jams and compresses to phi = 0.7858. Jamming and crystallization are both preceded by an entropy-driven transition from a simple fluid of independent tetrahedra to a complex fluid characterized by tetrahedra arranged in densely packed local motifs of pentagonal dipyramids that form a percolating network at the transition. The quasicrystal that we report represents the first example of a quasicrystal formed from hard or non-spherical particles. Our results demonstrate that particle shape and entropy can produce highly complex, ordered structures.