Dodecagonal Quasicrystals Research Articles

Particle Dynamics in a Diblock-Copolymer-Based Dodecagonal Quasicrystal and Its Periodic Approximant by X-Ray Photon Correlation Spectroscopy.

Temperature-dependent x-ray photon correlation spectroscopy (XPCS) measurements are reported for a binary diblock-copolymer blend that self-assembles into an aperiodic dodecagonal quasicrystal and a periodic Frank-Kasper σ phase approximant. The measured structural relaxation times are Bragg scattering wavevector independent and are 5 times faster in the dodecagonal quasicrystal than the σ phase, with minimal temperature dependence. The underlying dynamical relaxations are ascribed to differences in particle motion at the grain boundaries within each of these tetrahedrally close-packed assemblies. These results identify unprecedented particle dynamics measurements of tetrahedrally coordinated micellar block polymers, thus expanding the application of XPCS to ordered soft materials.

Hamiltonian system for the inhomogeneous plane elasticity of dodecagonal quasicrystal plates and its analytical solutions

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.

A Mobile Two‐Dimensional Ultrasound Focusing System for Personalized Healthcare Applications through a Dodecagonal Quasicrystal Patterned Planar Lens (Adv. Mater. Technol. 16/2023)

A Mobile Two‐Dimensional Ultrasound Focusing System for Personalized Healthcare Applications through a Dodecagonal Quasicrystal Patterned Planar Lens (Adv. Mater. Technol. 16/2023)

A Mobile Two‐Dimensional Ultrasound Focusing System for Personalized Healthcare Applications through a Dodecagonal Quasicrystal Patterned Planar Lens

Ultrasound technology is widely utilized in applications, including tumor treatments, drug delivery, and skin care. However, improvements are required to prevent unwanted damage to non‐targeted tissues. The ultrasound focusing technology, represented by the highly intensive focused ultrasound (HIFU) technology, is actively researched to handle this problem. However, current technology is primarily limited in the point focusing of the ultrasound. Some applications, such as drug delivery and skin care, require 2D‐focusing of the ultrasound for effective utilization. Based on this necessity, this research proposes the rationally designed dodecagonal quasicrystal patterned (DQP) planar lens, which enables ultrasound focusing in 2D. The custom‐built ultrasound scanning setup confirms the 2D focusing behavior of the DQP lens. Furthermore, the developed DQP lens is integrated into the mobile ultrasound 2D focusing (MU2F) system and solved the bulkiness limitation of the HIFU system. The proposed MU2F system is applied in microneedle‐mediated drug delivery. The dramatic enhancement in the dissolution efficiency of the drug‐containing microneedle (≈2.5×) compared with the case without the MU2F system is confirmed. Through the proposed MU2F system, ultrasound‐based medical devices may widen their approachability from the clinic to the home.

Molecular dynamics simulations for quasicrystals under the Born-Gauss potentials

Molecular dynamics simulations are a tool for the study of the microscopic dynamics in quasicrystals. By choosing suitable potentials, such as the Dzugutov potential and the Lennard-Jones-Gauss potential, two-dimensional quasicrystals with 8-, 10- and 12-fold symmetries have been formed. In this work, we propose a Born-Gauss type potential with more adjusting parameters to do simulations in two dimensions. Decagonal and dodecagonal quasicrystals are successfully obtained. The structural properties are analyzed and the phase transitions between different structures are discussed.

Heptagonal quasicrystals: Construction of 2D lattices and demonstrations using laser pointers – Concluding part

Photographic slides of an aperiodic heptagonal tiling were used as two-dimensional diffraction gratings (as a standard approach) to describe and demonstrate the basic properties of dodecagonal quasicrystals. The paper completes our earlier two publications dealing with (1) Penrose (decagonal) and Ammann (octagonal) quasicrystals and (2) dodecagonal quasicrystals, the diffraction gratings of which were constructed and presented in an analogous manner.

Quasicrystal approximants in the two-dimensional Ba-Ti-O system on Pd(111): A LEED, XPS, and STM study

Long-range aperiodically ordered dodecagonal oxide quasicrystals (OQCs) have been observed on Pt(111), together with a variety of related approximants. All structures contain barium, titanium, and oxygen atoms in ${\mathrm{Ti}}_{n}{\mathrm{O}}_{n}$ ring structures with rings of $n$ = 4, 7, and 10. Barium decorates the larger rings and forms a dodecagonal square-triangle-rhombus tiling. In contrast to these studies on Pt(111), we report here on Ba-Ti-O layers on Pd(111), which has a 1% smaller lattice constant than Pt(111). We find a series of quasicrystal approximants with unit cells consisting of up to 60 triangles, 22 squares, and eight rhombuses. The latter resembles the largest quasicrystal approximant reported to date. The basis for these structures are local patches of the $\ensuremath{\sigma}$-phase square-triangle approximant, into which rhombuses are incorporated due to the formation of ordered antiphase domain boundaries in two orthogonal directions. A structural phase diagram of all quasicrystal approximants on Pd(111) system is presented.

Rapid Phase Transitions of Thermotropic Glycolipid Quasicrystal and Frank-Kasper Mesophases: A Mechanistic Rosetta Stone.

Experimental results are presented that serve to lower the barrier for developing the science and technology of non-classical thermotropic glycolipid mesophases, which now include dodecagonal quasicrystal (DDQC) and Frank-Kasper (FK) A15 and σ mesophases that can be produced under mild conditions from a versatile class of sugar-polyolefin conjugates. By employing "alloys" comprised of mono- and disaccharide-polyolefin conjugates, and optionally with vitamin E as a small molecule phase modulator, we report the spontaneous formation of stable A15 mesophases at ambient temperature. We further document a rich thermotropic phase map that includes DDQC, A15, and σ mesophases of tunable periodicity that are connected through rapid thermotropic phase transitions as a function of increasing temperature in the order: liquid-like packing (LLP)→DDQC → A15→σ→ disorder. This first direct observation of a rapid thermotropic A15→σ phase transition provides support for a diffusionless martensitic process proceeding through strain-induced introduction of planar defects into the A15 lattice.

Rapid Phase Transitions of Thermotropic Glycolipid Quasicrystal and Frank‐Kasper Mesophases: A Mechanistic Rosetta Stone

AbstractExperimental results are presented that serve to lower the barrier for developing the science and technology of non‐classical thermotropic glycolipid mesophases, which now include dodecagonal quasicrystal (DDQC) and Frank–Kasper (FK) A15 and σ mesophases that can be produced under mild conditions from a versatile class of sugar‐polyolefin conjugates. By employing “alloys” comprised of mono‐ and disaccharide‐polyolefin conjugates, and optionally with vitamin E as a small molecule phase modulator, we report the spontaneous formation of stable A15 mesophases at ambient temperature. We further document a rich thermotropic phase map that includes DDQC, A15, and σ mesophases of tunable periodicity that are connected through rapid thermotropic phase transitions as a function of increasing temperature in the order: liquid‐like packing (LLP)→DDQC → A15→σ→ disorder. This first direct observation of a rapid thermotropic A15→σ phase transition provides support for a diffusionless martensitic process proceeding through strain‐induced introduction of planar defects into the A15 lattice.

A columnar liquid quasicrystal with a honeycomb structure that consists of triangular, square and trapezoidal cells.

Quasicrystals are intriguing structures that have long-range positional correlations but no periodicity in real space, and typically with rotational symmetries that are 'forbidden' in conventional periodic crystals. Here, we present a two-dimensional columnar liquid quasicrystal with dodecagonal symmetry. Unlike previous dodecagonal quasicrystals based on random tiling, a honeycomb structure based on a strictly quasiperiodic tessellation of tiles is observed. The structure consists of dodecagonal clusters made up of triangular, square and trapezoidal cells that are optimal for local packing. To maximize the presence of such dodecagonal clusters, the system abandons periodicity but adopts a quasiperiodic structure that follows strict packing rules. The stability of random-tiling dodecagonal quasicrystals is often attributed to the entropy of disordering when strict tiling rules are broken, at the sacrifice of the long-range positional order. However, our results demonstrate that quasicrystal stability may rest on energy minimization alone, or with only minimal entropic intervention.

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.

Electrical discharge triggers quasicrystal formation in an eolian dune

We report the discovery of a dodecagonal quasicrystal Mn72.3Si15.6Cr9.7Al1.8Ni0.6-composed of a periodic stacking of atomic planes with quasiperiodic translational order and 12-fold symmetry along the two directions perpendicular to the planes-accidentally formed by an electrical discharge event in an eolian dune in the Sand Hills near Hyannis, Nebraska, United States. The quasicrystal, coexisting with a cubic crystalline phase with composition Mn68.9Si19.9Ni7.6Cr2.2Al1.4, was found in a fulgurite consisting predominantly of fused and melted sand along with traces of melted conductor metal from a nearby downed power line. The fulgurite may have been created by a lightning strike that combined sand with material from downed power line or from electrical discharges from the downed power line alone. Extreme temperatures of at least 1,710 °C were reached, as indicated by the presence of SiO2 glass in the sample. The dodecagonal quasicrystal is an example of a quasicrystal of any kind formed by electrical discharge, suggesting other places to search for quasicrystals on Earth or in space and for synthesizing them in the laboratory.

Local Chain Feature Mandated Self-Assembly of Block Copolymers

This work demonstrates an effective and robust approach to regulate phase behaviors of a block copolymer by programming local features into otherwise homogeneous linear chains. A library of sequence-defined, isomeric block copolymers with globally the same composition but locally different side chain patterns were elaborately designed and prepared through an iterative convergent growth method. The precise chemical structure and uniform chain length rule out all inherent molecular defects associated with statistical distribution. The local features are found to exert surprisingly pronounced impacts on the self-assembly process, which have yet to be well recognized. While other molecular parameters remain essentially the same, simply rearranging a few methylene units among the alkyl side chains leads to strikingly different phase behaviors, bringing about (i) a rich diversity of nanostructures across hexagonally packed cylinders, Frank-Kasper A15 phase, Frank-Kasper σ phase, dodecagonal quasicrystals, and disordered state; (ii) a significant change of lattice dimension; and (iii) a substantial shift of order-to-disorder transition temperature (up to 40 °C). Different from the commonly observed enthalpy-dominated cases, the frustration due to the divergence between the native molecular geometry originating from side chain distribution and the local packing environment mandated by lattice symmetry is believed to play a pivotal role. Engineering the local chain feature introduces another level of structural complexity, opening up a new and effective pathway for modulating phase transition without changing the chemistry or composition.

2D honeycomb transformation into dodecagonal quasicrystals driven by electrostatic forces

Dodecagonal oxide quasicrystals are well established as examples of long-range aperiodic order in two dimensions. However, despite investigations by scanning tunneling microscopy (STM), low-energy electron diffraction (LEED), low-energy electron microscopy (LEEM), photoemission spectroscopy as well as density functional theory (DFT), their structure is still controversial. Furthermore, the principles that guide the formation of quasicrystals (QCs) in oxides are elusive since the principles that are known to drive metallic QCs are expected to fail for oxides. Here we demonstrate the solution of the oxide QC structure by synchrotron-radiation based surface x-ray diffraction (SXRD) refinement of its largest-known approximant. The oxide QC formation is forced by large alkaline earth metal atoms and the reduction of their mutual electrostatic repulsion. It drives the n = 6 structure of the 2D Ti2O3 honeycomb arrangement via Stone–Wales transformations into an ordered structure with empty n = 4, singly occupied n = 7 and doubly occupied n = 10 rings, as supported by DFT.

Chemical Pressure-Derived Assembly Principles for Dodecagonal Quasicrystal Approximants and Other Complex Frank-Kasper Phases.

The structures of complex intermetallic compounds can often be interpreted in terms of assemblies of units from simpler parent phases. For example, dodecagonal quasicrystals appear, when viewed down their high-symmetry axes, as plane-filling arrangements of square and triangular tiles corresponding to the Cr3Si and Al3Zr4 structure types, respectively. The atomic arrangements and cell-dimensions at the (100) faces of the cells of these structures provide a close geometrical match, which underlies not only dodecagonal quasicrystals and their approximants but also the much more common σ-phase structure. In this article, we show that such intergrowth of parent structures can arise from more than just geometrical coincidences but can be driven by a complementary matching of atomic packing forces. DFT-chemical pressure (CP) analysis on elemental versions of the Cr3Si and Al3Zr4 types reveal that in both cases arrays of positive interatomic pressures inhibit the formation of optimal contacts elsewhere in the structures. When they are lined up at the potential Cr3Si/Al3Zr4 interfaces, however, positive pressures from the two structures interdigitate rather than coincide, providing the opportunity for the relaxation of strained interatomic contacts. That such relief is afforded by the interfaces is confirmed by CP analysis of the σ-phase (FeCr-type) structure. Building on this scheme, we introduce the CPinterface function to represent how the CP features of atoms within a structure impact planes or other surfaces that could serve as interfaces between different structures. Using this function, we then explore how the favorability of interfaces between Cr3Si- and Al3Zr4-type units is tuned by partial elemental substitution with Si, as well as their potential matches with Laves phase units. The emerging picture provides an account for features of the quasicrystal approximants Mn7VSi2 and Mn81.5Si18.5, as well as a framework for approaching intermetallic intergrowth structures more broadly.

Rectangle-triangle soft-matter quasicrystals with hexagonal symmetry.

Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built up from, e.g., kites and darts, squares and equilateral triangles, rhombi- or shield-shaped tiles, and can have a variety of different symmetries. However, almost all quasicrystals occurring in soft matter are of the dodecagonal type. Here we investigate a class of aperiodic tilings with hexagonal symmetry that are based on rectangles and two types of equilateral triangles. We show how to design soft-matter systems of particles interacting via pair potentials containing two length scales that form aperiodic stable states with two different examples of rectangle-triangle tilings. One of these is the bronze-mean tiling, while the other is a generalization. Our work points to how more general (beyond dodecagonal) quasicrystals can be designed in soft matter.

Effect of impurities on tiling in a two-dimensional dodecagonal quasicrystal

Langevin dynamics simulations are performed to examine how impurities affect two-dimensional dodecagonal quasicrystals. We assumed that the interaction potential between two particles is the Lennard–Jones–Gauss potential if at least one of these particles is a matrix particle and that the interaction potential between two impurities is the Lennard–Jones potential. Matrix particles and impurities impinge with constant rates on the substrate created by a part of a dodecagonal quasicrystal consisting of square and triangular tiles. The dependences of the twelve-fold rotational order and the number of shield-like tiles on the impurity density are examined after sufficient solid layers are grown. While the change in the twelve-fold rotational symmetry is small, the number of shield-like tiles in the solid increases greatly with increasing impurity density.

Formation and fluctuation of two-dimensional dodecagonal quasicrystals.

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

Dodecagonal quasicrystals of oil-swollen ionic surfactant micelles

A delicate balance of noncovalent interactions directs the hierarchical self-assembly of molecular amphiphiles into spherical micelles that pack into three-dimensional periodic arrays, which mimic intermetallic crystals. Herein, we report the discovery that adding water to a mixture of an ionic surfactant and n-decane induces aperiodic ordering of oil-swollen spherical micelles into previously unrecognized, aqueous lyotropic dodecagonal quasicrystals (DDQCs), which exhibit local 12-fold rotational symmetry and no long-range translational order. The emergence of these DDQCs at the nexus of dynamically arrested micellar glasses and a periodic Frank-Kasper (FK) σ phase approximant sensitively depends on the mixing order of molecular constituents in the assembly process and on sample thermal history. Addition of n-decane to mixtures of surfactant and water instead leads only to periodic FK A15 and σ approximants with no evidence for aperiodic order, while extended ambient temperature annealing of the DDQC also reveals its transformation into a σ phase. Thus, these lyotropic DDQCs are long-lived metastable morphologies, which nucleate and grow from a stochastic distribution of micelle sizes formed by abrupt segregation of varied amounts of oil into surfactant micelles on hydration. These findings indicate that molecular building block complexity is not a prerequisite for the formation of aperiodic supramolecular order, while also establishing the generic nature of quasicrystalline states across metal alloys and self-assembled micellar materials.

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.