Dodecagonal Quasicrystals Research Articles

Modeling of dodecagonal quasicrystal lattices

We consider a method for constructing dodecagonal quasi-lattices, which involves the recurrent multiplication of the initial group of sites that are vertices of a regular dodecagon. The algorithm of modeling of reciprocal lattices of dodecagonal quasicrystals has been analyzed. The proposed modeling method is compared with the traditional projection method. We have performed indexing of the diffraction pattern from dodecagonal quasicrystals, which is analogous to indexing proposed by Cahn for icosahedral quasicrystals. The ambiguity of the choice of the basis vectors of the quasicrystal reciprocal lattice has been considered.

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.

Phase behaviour of quasicrystal forming systems of core-corona particles

Using Monte Carlo simulations and free-energy calculations, we study the phase behaviour of a two-dimensional system of particles interacting with a hard core of diameter σHD and a repulsive square shoulder potential. The interest in this system lies in the formation of quasicrystals of different symmetries at specific square-shoulder widths δ as previously reported by Dotera et al. [Nature 506, 208 (2014)]. However, an insight into other possible periodic phases formed in these systems and the thermodynamic stability of both the periodic and quasicrystal phases is yet to be addressed. Here, we study the phase behaviour and map out the phase diagrams for three different shoulder widths δ=1.27σHD,1.40σHD, and 1.60σHD, where octadecagonal, dodecagonal, and decagonal quasicrystals were previously reported. In addition, we verify the thermodynamic stability of these quasicrystals with respect to their periodic approximants. In general, we find that the system at all three shoulder widths forms hexagonal phases in two distinct density ranges due to the two characteristic length scales in the interaction potential. Further, we find that the dodecagonal and octadecagonal quasicrystals are stable in between two crystal phase regimes. In contrast, the decagonal quasicrystal is not bounded by a low-density crystal phase regime due to the lower density of this quasicrystal. From the free-energy calculations, we find indications that the decagonal and dodecagonal quasicrystals are thermodynamically stable with respect to their approximants, and the octadecagonal quasicrystal is stabilised by a configurational entropy contribution.

Defect structures in Frank–Kasper type square–triangle tiling of multimodal cage-type mesoporous silicas

Multimodal cage-type mesoporous silicas (MCMSs) with Frank–Kasper type square–triangle tiling show a unique defect structure, so-called three-fold symmetric hexagons, or shields, which are caused by phason fluctuations in dodecagonal quasicrystals. We observed and characterized three types of configurations inside shields in both quasiperiodic and periodic 32.4.3.4 tiling of MCMSs by transmission electron microscopy (TEM). The high-resolution TEM images of the shields were well explained by polyhedral models, which are the constituents of the Frank–Kasper type tetrahedrally close-packed structures of MCMSs. Shield defects invariably formed because of mismatch in periodic and/or aperiodic square–triangle tiling, and they were also catalyzed by other defects. Multiple shields overlapped with sharing of 30° rhombus units and showed characteristic motifs in the tiling, such as defect-mediated 12-fold wheel and stripe bundle arrangements. Hence, MCMSs with square–triangle tiling would be governed by a random-tiling-like structure stabilized by entropy rather than energy, which results in defect-free tiling.

Aging effects of dodecagonal quasicrystal formed in Mn-Cr-Ni-Si alloys

The formation of Mn-Cr-Ni-Si dodecagonal quasicrystal has been studied at 600 and 700 °C. The growth process of the quasicrystal from as-cast β-Mn phase was explained by Johnson-Mehl-Avrami equation. In order to evaluate structural quality of the quasicrystal, distortion in the diffraction pattern observed along the 12-fold axis was analyzed. This analysis indicated that the aging at 600 °C for 100 h is the best condition to synthesize the quasicrystal.

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 . 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 and . 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.

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 σHD surrounded by a purely repulsive soft corona of diameter δ = 1.95σHD. 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.

Universal self-assembly of one-component three-dimensional dodecagonal quasicrystals

Using molecular dynamics simulations, we study computational self-assembly of one-component three-dimensional dodecagonal (12-fold) quasicrystals in systems with two-length-scale potentials. Existing criteria for three-dimensional quasicrystal formation are quite complicated and rather inconvenient for particle simulations. So to localize numerically the quasicrystal phase, one should usually simulate over a wide range of system parameters. We show how to universally localize the parameter values at which dodecagonal quasicrystal order may appear for a given particle system. For that purpose, we use a criterion recently proposed for predicting decagonal quasicrystal formation in one-component two-length-scale systems. The criterion is based on two dimensionless effective parameters describing the fluid structure which are extracted from the radial distribution function. The proposed method allows reduction of the time spent for searching the parameters favoring a certain solid structure for a given system. We show that the method works well for dodecagonal quasicrystals; this result is verified on four systems with different potentials: the Dzugutov potential, the oscillating potential which mimics metal interactions, the repulsive shoulder potential describing effective interactions for the core/shell model of colloids and the embedded-atom model potential for aluminum. Our results suggest that the mechanism of dodecagonal quasicrystal formation is universal for both metallic and soft-matter systems and it is based on competition between interparticle scales.

Phonon–phason dynamics and hydrodynamics of fivefold and tenfold symmetry quasicrystals

In view of the same model of fivefold and tenfold symmetry quasicrystals, with the same boundary and the same initial conditions, we have obtained a lot of results by phonon–phason dynamics and hydrodynamics and have performed a detailed comparative analysis. The quantitative results on mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, and viscosity stresses and their time–space variations help us to understand the motion of solid quasicrystals. The analysis for octagonal and dodecagonal quasicrystals can be easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. These results reveal that the phonon field is dominated; moreover, the coupling between phonon and phason fields is important for the studied dynamic process, and these two elementary excitations are the main figures in the dynamic process. In contrast, the effects of the compressibility and viscosity are very weak, and they make almost no contribution to the dynamic process. For this dynamic process, the hydrodynamics and phonon–phason dynamics are equivalent, and the hydrodynamics can be simplified by phonon–phason dynamics for solid quasicrystals.

Solutions for hydrodynamics of 5- and 10-fold symmetry quasicrystals

This study carries out a complete analysis of time-space solution of hydrodynamics of pentagonal/decagonal quasicrystals. The behaviors of wave propagation for phonons and diffusion for phasons and coupling between phonon-phason fields are explored explicitly. Comprehensive discussion on physical time-space variations of all hydrodynamic field variables of the alloy quasicrystals is given. The computational specimen is simple, convenient in testing computational results, and provides a possibility that is easy to test experimentally. The quantitative results of mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, viscosity stresses, and their time-space variations help us understand the motion of solid quasicrystals in a hydrodynamic condition (long-wavelength and low-frequency). The analysis presented in this paper can be used for octagonal and dodecagonal quasicrystals and is easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. Some problems explored by the computational results are also discussed.

Two Kinds of Contact Problems in Dodecagonal Quasicrystals of Point Group 12 mm

In this paper, two kinds of contact problems in 2-D dodecagonal quasicrystals were discussed using the complex variable function method: one is the finite frictional contact problem, the other is the adhesive contact problem. The analytic expressions of contact stresses in the phonon and phason fields were obtained for a flat rigid punch, which showed that: (1) for the finite frictional contact problem, the contact stress exhibited power-type singularities at the edge of the contact zone; (2) for the adhesive contact problem, the contact stress exhibited oscillatory singularities at the edge of the contact zone. The distribution regulation of contact stress under punch was illustrated; and the low friction property of quasicrystals was verified graphically.

Self-assembly of a space-tessellating structure in the binary system of hard tetrahedra and octahedra.

We report the formation of a binary crystal of hard polyhedra due solely to entropic forces. Although the alternating arrangement of octahedra and tetrahedra is a known space-tessellation, it had not previously been observed in self-assembly simulations. Both known one-component phases - the dodecagonal quasicrystal of tetrahedra and the densest-packing of octahedra in the Minkowski lattice - are found to coexist with the binary phase. Apart from an alternative, monoclinic packing of octahedra, no additional crystalline phases were observed.

Stability of two-dimensional soft quasicrystals in systems with two length scales.

The relative stability of two-dimensional soft quasicrystals in systems with two length scales is examined using a recently developed projection method, which provides a unified numerical framework to compute the free energy of periodic crystal and quasicrystals. Accurate free energies of numerous ordered phases, including dodecagonal, decagonal, and octagonal quasicrystals, are obtained for a simple model, i.e., the Lifshitz-Petrich free-energy functional, of soft quasicrystals with two length scales. The availability of the free energy allows us to construct phase diagrams of the system, demonstrating that, for the Lifshitz-Petrich model, the dodecagonal and decagonal quasicrystals can become stable phases, whereas the octagonal quasicrystal stays as a metastable phase.

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

Using computer simulations, we study the phase behavior of a model system of colloidal hard disks with a diameter σ and a soft corona of width 1.4σ. The particles interact with a hard core and a repulsive square-shoulder potential. We calculate the free energy of the random-tiling quasicrystal and its crystalline approximants using the Frenkel-Ladd method. We explicitly account for the configurational entropy associated with the number of distinct configurations of the random-tiling quasicrystal. We map out the phase diagram and find that the random tiling dodecagonal quasicrystal is stabilised by entropy at finite temperatures with respect to the crystalline approximants that we considered, and its stability region seems to extend to zero temperature as the energies of the defect-free quasicrystal and the crystalline approximants are equal within our statistical accuracy.

Controlled self-assembly of periodic and aperiodic cluster crystals.

Soft particles are known to overlap and form stable clusters that self-assemble into periodic crystalline phases with density-independent lattice constants. We use molecular dynamics simulations in two dimensions to demonstrate that, through a judicious design of an isotropic pair potential, one can control the ordering of the clusters and generate a variety of phases, including decagonal and dodecagonal quasicrystals. Our results confirm analytical predictions based on a mean-field approximation, providing insight into the stabilization of quasicrystals in soft macromolecular systems, and suggesting a practical approach for their controlled self-assembly in laboratory realizations using synthesized soft-matter particles.

Controlled Self-Assembly of Soft-Matter Quasicrystals

A large number of soft-matter systems, whose building blocks range in size from several nanometers to almost a micron, have been shown in recent years to form ordered phases with dodecagonal (12-fold) symmetry (for recent reviews see [1]). Contrary to metallurgic quasicrystals, whose source of stability remains a question of great debate to this day, we show that the stability of certain soft-matter quasicrystals–interacting via pair potentials with repulsive cores, which are either bounded or only slowly diverging–can directly be explained. Their stability is attributed to the existence of two natural length scales in their isotropic pair potentials, along with an effective three-body interaction arising from entropy. We establish the validity of this mechanism at the level of a mean-field theory [2], and then use molecular dynamics simulations in two dimensions to confirm it beyond mean field, and to show that it leads to the formation of cluster crystals [3]. We demonstrate that our understanding of the stability mechanism allows us to generate a variety of desired structures, including decagonal and dodecagonal quasicrystals [3], suggesting a practical approach for their controlled self-assembly in laboratory realizations using synthesized soft-matter particles.

Five-Vertex Lanthanide Coordination on Surfaces: A Route to Sophisticated Nanoarchitectures and Tessellations

We report a combined scanning tunneling microscopy and density functional theory study on the formation of intricate networks involving a flexible 5-fold carbonitrile–lanthanide (cerium or gadolinium) coordination. By employing linear linkers equipped with terminal carbonitrile functional groups, and by tuning the local rare-earth to molecule stoichiometry, architectures evidencing high spatial complexity become manifested, including disordered islands of pentameric and nonameric supramolecules, a 2D hierarchical short-range orientational disordered network, and a 2D Archimedean snub square tessellation of the surface, which coexists with minority 2D Archimedean elongated triangular tiling motifs. The combination of both the intricate structural features and the unique properties of lanthanide elements prospects great potential in a variety of fields such as magnetism and catalysis. Furthermore, the expression of Archimedean tiling motifs based on 5-fold vertexes suggests a route to the design of self-assembled dodecagonal quasicrystals on surfaces.

Bragg Extraction and Diffraction Properties of Active Organic/Inorganic 2D Dodecagonal Photonic Quasi-Crystals

Bragg Extraction and Diffraction Properties of Active Organic/Inorganic 2D Dodecagonal Photonic Quasi-Crystals

The symmetry of the thermoelastic properties of quasicrystals

The thermoelastic properties of quasicrystals, possessing 5-th, 8-th, 10-th and 12-th order rotational axes, not characteristic of crystalline bodies, are considered. Using group analysis, canonical representations of the second and fourth rank tensors, characterizing the properties of quasicrystals, are constructed. The fact that orthogonal, decagonal and dodecagonal quasicrystals belong to the class of transversely isotropic materials and the isotropy of icosahedric quasicrystals and fullerenes are proved.

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.