Decagonal Quasicrystals Research Articles

Extraction of Two-Dimensional Aluminum Alloys from Decagonal Quasicrystals

Atomically thin metallic alloys are receiving increased attention due to their prospective applications as interconnects/contacts in two-dimensional (2D) circuits, sensors, and catalysts, among others. In this work, we demonstrate an easily scalable technique for the synthesis of 2D metallic alloys from their 3D quasicrystalline precursors. We have used aluminum (Al)-based single-phase decagonal quasicrystal Al66Co17Cu17 alloy to extract the corresponding 2D alloy structure. The 2D layered Al alloy possesses 2-fold decagonal quasicrystalline symmetry and consists of two- or three-layer-thick sheets with a lateral dimension of microns. These 2D metallic layers were combined with the atomic layers of tungsten disulfide to form the stacked heterostructures, which is demonstrated to be a stable and efficient catalyst for hydrogen evolution reaction.

A decagonal quasicrystal with rhombic and hexagonal tiles decorated with icosahedral structural units.

The structure of a decagonal quasicrystal in the Zn58Mg40Y2 (at.%) alloy was studied using electron diffraction and atomic resolution Z-contrast imaging techniques. This stable Frank-Kasper Zn-Mg-Y decagonal quasicrystal has an atomic structure which can be modeled with a rhombic/hexagonal tiling decorated with icosahedral units at each vertex. No perfect decagonal clusters were observed in the Zn-Mg-Y decagonal quasicrystal, which differs from the Zn-Mg-Dy decagonal crystal with the same space group P10/mmm. Y atoms occupy the center of 'dented decagon' motifs consisting of three fat rhombic and two flattened hexagonal tiles. About 75% of fat rhombic tiles are arranged in groups of five forming star motifs, while the others connect with each other in a 'zigzag' configuration. This decagonal quasicrystal has a composition of Zn68.3Mg29.1Y2.6 (at.%) with a valence electron concentration (e/a) of about 2.03, which is in accord with the Hume-Rothery criterion for the formation of the Zn-based quasicrystal phase (e/a = 2.0-2.15).

Nonwetting Behavior of Al-Co Quasicrystalline Approximants Owing to Their Unique Electronic Structures.

Good wetting is generally observed for liquid metals on metallic substrates, while poor wetting usually occurs for metals on insulating oxides. In this work, we report unexpected large contact angles for lead on two metallic approximants to decagonal quasicrystals, namely, Al5Co2 and Al13Co4. Intrinsic surface wettability is predicted from first principles, using a thermodynamic model based on the Young equation, and validated by the good agreement with experimental measurements performed under ultra-high vacuum by scanning electron microscopy. The atomistic details of the atomic and electronic structures at the Pb-substrate interface, and the comparison with Pb(111)/Al(111), underline the influence of the specific electronic structures of quasicrystalline approximants on wetting. Our work suggests a possible correlation of the contact angles with the density of states at the Fermi energy and paves the way for a better fundamental understanding of wettability on intermetallic substrates, which has potential consequences in several applications such as supported catalysts, protective coatings, or crystal growth.

EBSD orientation analysis based on experimental Kikuchi reference patterns

Orientation determination does not necessarily require complete knowledge of the local atomic arrangement in a crystalline phase. We present a method for microstructural phase discrimination and orientation analysis of phases for which there is only limited crystallographic information available. In this method, experimental Kikuchi diffraction patterns are utilized to generate a self-consistent master reference for use in the technique of Electron Backscatter Diffraction (EBSD). The experimentally derived master data serves as an application-specific reference in EBSD pattern matching approaches. As application examples, we map the locally varying orientations in samples of icosahedral quasicrystals observed in a Ti40Zr40Ni20 alloy, and we analyse AlNiCo decagonal quasicrystals.

Quasicrystalline phase and crystalline approximant in Ni–Mn–In Heusler alloy system

Intermetallic compounds and micro-structures in Ni–Mn–In alloy system were investigated with transmission electron microscopy, selected-area and nano-beam electron diffraction, and energy-dispersive x-ray spectroscopy. Together with two types of Heusler phases, a decagonal quasicrystalline phase and a structurally related crystallite were found in the Ni–Mn–In system. As is well-known, most of the decagonal quasicrystals are aluminum-based intermetallic phases. It is interesting to investigate the aluminum-free decagonal quasicrystalline phase as a new member of the quasicrystal family. Structural characterization was carried out on the decagonal quasicrystalline phase and the crystalline approximant in the Ni–Mn–In alloy system.

Phasonic Diffusion and Self-confinement of Decagonal Quasicrystals in Hyperspace

We introduce a novel simulation method that is designed to explore fluctuations of the phasonic degrees of freedom in decagonal colloidal quasicrystals. Specifically, we attain and characterise thermal equilibrium of the phason ensemble via Monte Carlo simulations with particle motions restricted to elementary phasonic flips. We find that, at any temperature, the random tiling ensemble is strongly preferred over the minimum phason-strain quasicrystal. Phasonic flips are the dominant carriers of diffusive mass transport in physical space. Sub- diffusive transients suggest cooperative flip behaviour on short time scales. In complementary space, particle mobility is geometrically restricted to a thin ring around the acceptance domain, resulting in self-confinement and persistent phasonic order.

Path-independent J-integral for cracks in decagonal quasicrystals

The path-independent J-integral is derived for fracture mechanics analysis of decagonal quasicrystals (QCs). The gradient theory of quasicrystals is developed here to consider large strain gradients at the crack tip vicinity. The constitutive equations contain phonon and phason stresses, and the higher-order stress tensor. The higher-order elastic material parameters are proportional to the internal length material parameter and the conventional elastic coefficients. The FEM equations are derived to solve general boundary value problems for the strain gradient theory of the QCs.

From a single slit to periodic, modulated, and quasiperiodic crystals – a physical space diffraction analysis of aperiodic systems

Using a statistical approach to diffraction analysis, it was shown that the scale of difficulty in the analysis of the diffraction image of a one-dimensional quasicrystal is not significantly different from the analysis of the diffraction image for a single slit. In both cases we get a rectangular probability distribution, for which the Fourier transform leads to analytical expressions on the envelopes of diffraction peaks expressed by means of simple trigonometric functions (sinc). The generalization to 2d (decagonal quasicrystals) and 3d (icosahedral quasicrystals) requires more complex calculations, but conceptually, it does not significantly differ from the calculations for model one-dimensional quasicrystals.

Natural versus synthetic quasicrystals: analogies and differences in the optical behavior of icosahedral and decagonal quasicrystals

The optical behavior of both synthetic and natural icosahedral and decagonal quasicrystals in the range 400–700 nm has been explored by repeated measurements of the reflectance values on crystallographically oriented fragments. We found differences in the optical behavior between natural and synthetic quasicrystals, and discuss the possible reasons to justify this feature. Natural quasicrystals, both icosahedral and decagonal, exhibit an almost identical chemical composition to their synthetic analogs, but seem to show a higher degree of structural perfection, as indicated from their electron and X-ray diffraction patterns. Such perfection is mainly due to the very low phason strain shown by the minerals with respect to the synthetic compounds and substantiates the differences in the measured reflectance values. The new finding constitutes significant new support for the original proposal that icosahedral and decagonal quasicrystals can be energetically stable states of matter, on the same footing as crystals. Electron diffraction pattern of icosahedrite (left) and decagonite (right). A high-resolution transmission electron image of icosahedrite is on the background.

Analysis of elastic−plastic fracture in two-dimensional decagonal quasicrystals using the displacement discontinuity method

The generalized Dugdale model of cracks in two-dimensional decagonal quasicrystals is studied by developing the extended displacement discontinuity boundary element method. Expressions for the phonon and phason stress intensity factors and energy release rate are given in terms of the phonon and phason displacement discontinuities near the crack front. The extended displacement discontinuity boundary element method, with an iterative approach of determining the size of the yielding zone, is adopted to numerically simulate the nonlinear fracture problem. The effects of phonon and phason loadings, as well as the yield stress on the size of yielding zone and energy release rate, are discussed.

Event-chain Monte Carlo simulations of the liquid to solid transition of two-dimensional decagonal colloidal quasicrystals

In event-chain Monte Carlo simulations, we model colloidal particles in two dimensions that interact according to an isotropic short-ranged pair potential which supports the two typical length scales present in decagonal quasicrystals. We investigate the assembled structures as we vary the density and temperature. Our special interest is related to the transition from quasicrystal to liquid. In contrast to the KTHNY melting theory for quasicrystals which predicts an intermediate pentahedratic phase, we find a one-step first-order melting transition. However, we discover that the slow relaxation of phasonic flips, i.e. rearrangements of the particles due to additional degrees of freedom in quasicrystals, changes the positional correlation functions, to the extent that structures with long-range orientational correlations, but exponentially decaying positional correlations, are observed.

Gradient theory for crack problems in quasicrystals

Constitutive equations in gradient theory of quasicrystals are written for phonon and phason stresses, and the higher-order stress tensor. They are expressed by the phonon and phason strains and the gradient of phonon strains. The higher-order elastic parameters are proportional to the conventional elastic stiffness coefficients by the internal length material parameter. Material parameters in constitutive equations correspond to the decagonal quasicrystals. The principle of virtual work is applied to derive governing equations and corresponding boundary conditions. The finite element method (FEM) is developed to solve general 2D boundary value problems in problems described by governing equations for strain-gradient theory of quasicrystals. The path-independent J-integral is also derived for fracture mechanics analysis of such solids. Numerical examples are presented to demonstrate the veracity of the formulations.

Stabilizing quasicrystals composed of patchy colloids by narrowing the patch width

We explore the behavior of two-dimensional patchy colloidal particles with 8 or 10 symmetrically arranged patches by employing Monte Carlo simulations. The particles interact according to an isotropic pair potential that possesses only one typical length. The patches lead to additional attractions that are anisotropic and depend on the relative orientation of two neighboring particles. We investigate the assembled structures with a special interest in quasicrystals. We found that the patch width is of great importance: Only in the case of narrow patch widths we are able to observe metastable octagonal and decagonal quasicrystals, while dodecagonal quasicrystals can also occur for broad patches. These results are important to understand the role of interactions with preferred binding angles in order to obtain quasicrystals. Our findings suggest that in the case of sharp binding angles, as they occur in metallic alloys, octagonal and decagonal symmetries might be observed more often than in systems with less sharp binding angles as it is the case in soft-matter systems where dodecagonal quasicrystals dominate.

Analytic Solutions to Two-Dimensional Decagonal Quasicrystals with Defects Using Complex Potential Theory

An analytical treatment for two-dimensional point group 10 mm decagonal quasicrystals with defects was suggested based on the complex potential method. On the basis of the assumption of linear elasticity, two new conformal maps were applied to two examples: the first was an arc with an elliptic notch inner surface in a decagonal quasicrystal, where the complex potentials could be exactly obtained; and the second was concerned with a decagonal point group 10 mm quasicrystalline strip weakened by a Griffith crack, which was subjected to a pair of uniform static pressures. Using the basic idea underlying crack theory, the extent of the stress intensity factors was analytically estimated. If the height was allowed to approach infinity, these results can be turned into the known results of an “ordinary” crystal with only phonon elastic parameters when the phason and phonon-phason elastic constants are eliminated.

Decagonal and Dodecagonal Quasicrystals Obtained by Molecular Dynamics Simulations**Supported by the National Natural Science Foundation of China under Grant No 11674102.

Double-well potentials are used for molecular dynamics simulation in monatomic systems. The potentials change as their parameters are adjusted, resulting in different structures. Of particular interest, we obtain decagonal and dodecagonal quasicrystals by simulations. We also verify the results and explain the formation of quasicrystals from the perspective of potential energy.

Isothermal section of Ga-Co-Cu phase diagram at 830 °C and its peculiarities

Recently, the Ga-based ternary alloys have gained interest for the possibility of finding new quasicrystals. Inspired by the previously reported decagonal and icosahedral quasicrystals identified in the Ga50Co25Cu25 alloy at 830 °C, the phase equilibria of 27 alloys of various composition were investigated in this work. Based on the obtained results, an isothermal section of the Ga-Co-Cu phase diagram at 830 °C was proposed. However, neither decagonal nor icosahedral quasicrystals were found to be stable at 830 °C, but instead of that, a superalloy-like structure was found in the Ga35Co25Cu40 alloy consisting of β-GaCo and β-GaCu/ζ-GaCu.

带裂纹十次对称二维准晶平面弹性的无摩擦接触问题

With the classical complex function method, a frictionless contact problem of 2D decagonal quasicrystal semiplane elasticity with arbitraryform cracks was addressed under the action of a rigid convex basal punch. Based on complex expressions of stresses and displacements of 2D decagonal quasicrystals, the problem was converted into solvable boundary value problems with analytic functions, and then reduced to a class of Riemann boundary problems. Solutions to the Riemann boundary problems give the stress functions in closed form, the explicit expressions of the stress intensity factors at crack tips and the contact stress distribution under the punch. The expression of the contact stress shows that, it has singularity at the edge of the contact zone and the crack tips. Without the effect of the phason field, the obtained results match well with those classical conclusions for elastic materials. Numerical examples illustrated the solutions to the frictionless contact problem in 2D decagonal quasicrystal semiplane elasticity with a vertical crack and a horizontal straight crack under a rigid punch. The work provides a theoretical reference for the application of quasicrystalline materials.

Temperature dependence of decagonal quasicrystals heat capacity

The heat capacity of decagonal quasicrystals of the Al–Co–Cu or Al–Co–Ni alloys was calculated at the temperatures of 600, 700, and 900 K in this work. The expression for the heat capacity of the quasicrystals was obtained based on the Debye model. For the quasicrystals, the linear “excessive” heat capacity is observed in the range of temperatures between 400 to 600 К which means the deviation from the 3R Dulong-Petit value. The heat capacity at a temperature of 900 К is about 28.4 J/mol К which is higher than the Dulong-Petit value (~ 25 J/mol К). The “excessive” heat capacity relates to the peculiarities in the decagonal quasicrystal anisotropy. These crystals are quasiperiodic in the x and y directions, and periodic in the z direction. As a result, there is a difference in the dispersive laws in the different directions. The Debye temperature values have essential influence on the temperature dependencies of the heat capacity of the decagonal quasicrystals. Thus, the higher the Debye temperature and the larger “excessive” heat capacity, the more stable are considered the quasicrystals exposed to the temperature effects.

Anisotropic transient thermoelasticity analysis in a two-dimensional decagonal quasicrystal using meshless local Petrov–Galerkin (MLPG) method

The meshless local Petrov–Galerkin (MLPG) method is employed for anisotropic transient thermoelasticity analysis of 2D decagonal quasicrystals (QCs) subjected to transient thermal and mechanical shock loadings. The wave type model and the elasto-hydrodynamic model are applied to derive the phonon and phason governing equations, respectively. The temperature affects only the phonon field. To find the temperature distributions on the assumed 2D domain, the anisotropic heat conduction problem is solved using the MLPG method. Also, the MLPG method is successfully employed to obtain the transient behaviors of both phonon and phason displacements by solving the governing equations in local integral equations (LIEs) forms. Making use a unit step function as the test functions in the local weak-form of governing equations, we derived the local integral equations (LIEs) considered on small subdomains identical with support domains of test functions around each node. The radial basis functions are used for approximation of the spatial variation of field variables. The Laplace-transform technique is utilized to discretize the time variations.

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.