Phason Strain Research Articles

Nearly Ideal Random Tiling with Dodecagonal Symmetry from Pentablock Quarterpolymers of the AB1CB2D Type.

Two-dimensional tiling manners as cross-sectional views of cylindrical domain assembly formed by pentablock quarterpolymers of the AB1CB2D type in bulk were investigated. Several binary and ternary blends from three mother polymers having different ϕB1/ϕB2 ratios (ϕB1 and ϕB2 are the volume fractions of the B1 and B2 blocks, respectively) represent nonperiodic but ordered triangle/square tilings, where the N3/N4 ratios (N3 and N4 are the numbers of triangles and squares in the observed area, respectively) are all close enough to the theoretical value of 4/√3 ≑ 2.31 for the dodecagonal quasicrystalline (DDQC) state, irrespective of the total number of polygons. The TEM images, having almost the same N3/N4 ratios, were proved to show 4- and 6-fold symmetries in terms of the angular appearance of equilateral polygon sides via image analyses. Among them, a ternary blend showed a nearly ideal random tiling pattern that is almost equivalent to the theoretically predicted tiling by SCFT. Moreover, the magnitude of phason strain estimated for a TEM image from the ternary blend was proved to be quite small when the observing area is narrow, while it deviates from the ideal quasicrystalline tiling with an increasing number of vertices in the observing area.

Unraveling the effects of phason strain on the thermoelectric properties in Ag-In-Yb icosahedral quasicrystals

This study represents the experimental comparative analysis of the thermoelectric properties in an Ag-In-Yb phason-free icosahedral quasicrystal (iQC) and four phason-strained iQCs. The aim is to understand how phason strain affects these properties. The four phason-strained iQCs are 1/1 and 2/1 approximant crystals (1/1 AC and 2/1 AC), a sample grown by the Bridgman method with a high growth rate (phason QC-1), and a sample obtained by phonon–phason coupling (phason QC-2), the magnitude of phason strain decreasing in this order. Their compositions are similar to that of iQC, and in particular, the compositions of phason QC-1 and 2 are exactly the same as that of iQC, making them suitable for clarifying the correlation between physical properties and the magnitude of phason strain. Systematic changes were observed in the electrical resistivity, Seebeck coefficient, and electronic specific heat coefficient with increasing phason-strain magnitude, which can be interpreted as due to the effect of the phason-strain blunting the pseudogap at the Fermi level in the electronic density of states. In contrast, no appreciable dependence of phonon thermal conductivity on the phason-strain magnitude was observed; very low value and a plateau feature observed consistently for all the samples do not arise directly from the quasiperiodicity or high rotational symmetry but from complex structures with a large unit cell.

Phonon–Phason Coupling Strength in a Tsai-Type Ag–In–Yb Icosahedral Quasicrystal

Owing to the quasiperiodic order, one independent term in the elasticity of quasicrystals (phonon–phason coupling) is characteristic of quasicrystals. It is not seen in conventional crystals and has continued to be a significant subject in the research field of quasicrystals. Recently, a novel method was applied to a Mackay-type Al–Pd–Mn icosahedral quasicrystal to prove the existence of phonon–phason coupling and to evaluate its strength based on the elasticity of quasicrystals. This study applied the method to a Tsai-type Ag–In–Yb icosahedral system to evaluate its phonon–phason coupling strength. We applied phonon strain to the quasicrystal at a temperature with active phason, and the induction of phason strain was successfully detected and evaluated using powder X-ray diffraction. We evaluated the phonon–phason coupling elastic constant of the Ag–In–Yb icosahedral quasicrystal to be 0.17 ± 0.04 GPa by quantitatively comparing the measured induced phason strain with our calculation results.

Evaluating the phonon-phason coupling strength of an Al-Ni-Co decagonal quasicrystal

Recently, a novel method was proposed to probe the phonon-phason coupling effect in quasicrystals, and it was applied to an Al-Pd-Mn icosahedral quasicrystal. In this study, the method was applied to an Al-Ni-Co decagonal quasicrystal with a periodic layered structure of a two-dimensional quasicrystal. We applied elastic phonon strain along a 2-fold (aperiodic) axis in the quasicrystal plane at high temperatures with active phasons. Then, the phason strain was induced to the quasicrystal, which was demonstrated by the phason momentum (G ⊥) dependences of the powder X-ray diffraction peak widths. In contrast, when we applied phonon strain along the 10-fold periodic axis, no appreciable phason strain was observed to be induced. These results present clear evidence on the presence of phonon-phason coupling in a decagonal quasicrystal. The shape and width of the powder X-ray diffraction peaks were calculated based on the generalized elasticity of quasicrystals. The absolute value of the phonon-phason coupling constant was evaluated by quantitatively comparing the calculated results with the measured diffraction peaks.

Three-dimensional analysis of elliptical inclusions in two-dimensional decagonal quasicrystal

An infinite quasicrystal composites with elliptical inclusions is considered by the Green's function method. The general analytical expressions for the phonon and the phason field strain inside the inclusions are derived, respectively. Then the effective constants of the quasicrystal composites are calculated by utilizing these results. In particular, the elliptical cylindrical inclusions is analyzed in detail, the analytical expressions for the elastic constants of quasicrystal composites are given. Finally, the effects of inclusions volume fraction and shape on the effective elastic constants of quasicrystal composites are discussed.

Fingerprinting phason strain

Fingerprinting phason strain

Direct experimental evidence of phonon–phason coupling in an Al-Pd-Mn icosahedral quasicrystal

ABSTRACT Due to their unique structural order, quasicrystals have special elastic degrees of freedom called the phason degrees of freedom, in addition to the conventional phonon degrees of freedom. In this study, we demonstrate the presence of phonon–phason coupling, expected from the generalized elastic theory of quasicrystals, in an Al-Pd-Mn icosahedral quasicrystal by a new experimental method that probes directly the elastic free energy function of the QC. Conventional phonon strain was applied to the quasicrystal at a high temperature where phasons are active. In response to the applied phonon strain, phason strain was observed to be induced by X-ray diffraction measurements, which demonstrated clearly the presence of phonon–phason coupling. Then, the coupling strength was quantitatively evaluated by comparing the magnitude of the induced phason strain with the theoretical values. In view of the evaluated coupling strength, the origin of an internal friction peak observed previously for an Al-Pd-Mn icosahedral quasicystal and the possible effects of the coupling on the onset of phasonic elastic instability are discussed.

A four-dimensional model for the Ba–Ti–O dodecagonal quasicrystal

A four-dimensional (4D) model is presented of the first oxide dodecagonal quasicrystal found in a Ba-Ti-O ultra-thin film on a Pt(111) single-crystal substrate. The 4D model, with a 4D dodecagonal lattice constant ad= 8.39 Å, was derived by considering a tile decoration model of dodecagonal Niizeki-Gähler tiling composed of squares, triangles and 30° rhombuses. The model consists of four kinds of occupation domain, and 4D positional vectors defining the shape of each occupation domain are given. Moreover, the atomic arrangement of two Ba-Ti-O periodic approximants, the sigma-phase approximant and a 25.6 Å approximant were derived from the 4D model by the introduction of linear phason strains.

Entropic formation of a thermodynamically stable colloidal quasicrystal with negligible phason strain

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.

Insight into the structure of decagonite - the extraterrestrial decagonal quasicrystal.

A set of X-ray data collected on a fragment of decagonite, Al71Ni24Fe5, the only known natural decagonal quasicrystal found in a meteorite formed at the beginning of the Solar System, allowed us to determine the first structural model for a natural quasicrystal. It is a two-layer structure with decagonal columnar clusters arranged according to the pentagonal Penrose tiling. The structural model showed peculiarities and slight differences with respect to those obtained for other synthetic decagonal quasicrystals. Interestingly, decagonite is found to exhibit low linear phason strain and a high degree of perfection despite the fact it was formed under conditions very far from those used in the laboratory.

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.

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.

Direct observation of growth and stability of Al-Cu-Fe quasicrystal thin films

Al-Cu-Fe based quasicrystal thin films exhibit unique surface and mechanical properties. To better understand the formation of the quasicrystal thin films, we observe direct growth of quasicrystals, prepared in a multilayer Al-Cu-Fe thin films with subsequent heat treatment, by in-situ synchrotron x-ray diffraction and in-situ transmission electron microscopy during heating and cooling. Using these two methods, we show that the ternary phase is more thermodynamically stable compared to the binary phases at temperature higher than 470 °C during the heating process, and quasicrystal formation occurs during the cooling process, specifically at 660 °C, after the sample has reached a liquid state. To distinguish quasicrystal from approximant crystals in the obtained thin film samples, we use high resolution x-ray diffraction to analyze the sample at room temperature. We reveal that the peak broadening increases monotonically along the twofold, threefold, and fivefold high-symmetry directions with the physical scattering vector but does not have systematic dependence on the phason momentum, which suggests that the thin film sample is indeed a quasicrystal instead of approximant crystals and it is almost free of phason strain. Our study provides a complete understanding of the growth mechanism for thin film Al-Cu-Fe quasicrystals, which is of particular importance for developing versatile applications of quasicrystal thin films.

Direct Observation of Growth and Stability of Al-Cu-Fe Quasicrystal Thin Films

Al-Cu-Fe based quasicrystal thin films exhibit unique surface and mechanical properties. To better understand the formation of the quasicrystal thin films, we observe direct growth of quasicrystals, prepared in a multilayer Al-Cu-Fe thin films with subsequent heat treatment, by in-situ synchrotron x-ray diffraction and in-situ transmission electron microscopy during heating and cooling. Using these two methods, we show that the ternary phase is more thermodynamically stable compared to the binary phases at temperature higher than 470 °C during the heating process, and quasicrystal formation occurs during the cooling process, specifically at 660 °C, after the sample has reached a liquid state. To distinguish quasicrystal from approximant crystals in the obtained thin film samples, we use high resolution x-ray diffraction to analyze the sample at room temperature. We reveal that the peak broadening increases monotonically along the twofold, threefold, and fivefold high-symmetry directions with the physical scattering vector but does not have systematic dependence on the phason momentum, which suggests that the thin film sample is indeed a quasicrystal instead of approximant crystals and it is almost free of phason strain. Our study provides a complete understanding of the growth mechanism for thin film Al-Cu-Fe quasicrystals, which is of particular importance for developing versatile applications of quasicrystal thin films.

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.

X-ray Study of Phason Strains in an AlCuFeMn Decagonal Phase

X-ray Study of Phason Strains in an AlCuFeMn Decagonal Phase

Shock synthesis of quasicrystals with implications for their origin in asteroid collisions

We designed a plate impact shock recovery experiment to simulate the starting materials and shock conditions associated with the only known natural quasicrystals, in the Khatyrka meteorite. At the boundaries among CuAl5, (Mg0.75Fe(2+) 0.25)2SiO4 olivine, and the stainless steel chamber walls, the recovered specimen contains numerous micron-scale grains of a quasicrystalline phase displaying face-centered icosahedral symmetry and low phason strain. The compositional range of the icosahedral phase is Al68-73Fe11-16Cu10-12Cr1-4Ni1-2 and extends toward higher Al/(Cu+Fe) and Fe/Cu ratios than those reported for natural icosahedrite or for any previously known synthetic quasicrystal in the Al-Cu-Fe system. The shock-induced synthesis demonstrated in this experiment reinforces the evidence that natural quasicrystals formed during a shock event but leaves open the question of whether this synthesis pathway is attributable to the expanded thermodynamic stability range of the quasicrystalline phase at high pressure, to a favorable kinetic pathway that exists under shock conditions, or to both thermodynamic and kinetic factors.

2D quasicrystals from perovskites

Perovskite oxides represent a versatile class of materials with a simple cubic or pseudo-cubic crystal structure. The family of perovskite oxides contains insulators, metals, semiconductors, and superconductors with nearly identical lattice parameters. This structural equivalence additionally allows to combine perovskites with different properties in multilayer systems to produce functional materials with unique properties. We report here on the formation of a quasicrystal (QC) thin film on a threefold Pt(111) surface. This QC film is derived from the classical perovskite oxide BaTiO3 which is the most intensely studied ferroelectric perovskite oxide. An easily accessible ferroelectric to paraelectric phase transition at 400 K makes the material so interesting for basic and applied research. Due to matching lattice conditions BaTiO3 can be grown epitaxially on selected metal substrates. Periodic thin films of either BaTiO3(100) or BaTiO3(111) have been grown depending on substrate orientation and preparation conditions on Pt(001) and on Pt(111) [1, 2]. As we demonstrate here, astonishingly also a two-dimensional dodecagonal quasicrystalline structure can be formed by annealing an initially 1.4 nm thick BaTiO3 film on Pt(111) [3]. It develops at a temperature of 1250 K from a wetting layer spreading between a few thicker BaTiO3(111) islands. Surface sensitive electron diffraction (LEED) shows a bright and sharp pattern with dodecagonal symmetry. High-resolution scanning tunneling microscopy (STM) images reveal an arrangement of quadratic, triangular, and rhombic elements which compares well to a Gähler tiling. The development of higher-order self-similar structures is widely suppressed by a linear phason strain. This is supported by the fine structure of the diffraction data.

Magnetic and transport properties of i- R -Cd icosahedral quasicrystals (R=Y,Gd-Tm)

We present a detailed characterization of the recently discovered i-$R$-Cd ($R$ = Y, Gd-Tm) binary quasicrystals by means of x-ray diffraction, temperature-dependent dc and ac magnetization, temperature-dependent resistance and temperature-dependent specific heat measurements. Structurally, the broadening of x-ray diffraction peaks found for i-$R$-Cd is dominated by frozen-in phason strain, which is essentially independent of $R$. i-Y-Cd is weakly diamagnetic and manifests a temperature-independent susceptibility. i-Gd-Cd can be characterized as a spin-glass below 4.6 K via dc magnetization cusp, a third order non-linear magnetic susceptibility peak, a frequency-dependent freezing temperature and a broad maximum in the specific heat. i-$R$-Cd ($R$ = Ho-Tm) is similar to i-Gd-Cd in terms of features observed in thermodynamic measurements. i-Tb-Cd and i-Dy-Cd do not show a clear cusp in their zero-field-cooled dc magnetization data, but instead show a more rounded, broad local maximum. The resistivity for i-$R$-Cd is of order 300 $\mu \Omega$ cm and weakly temperature-dependent. The characteristic freezing temperatures for i-$R$-Cd ($R$ = Gd-Tm) deviate from the de Gennes scaling, in a manner consistent with crystal electric field splitting induced local moment anisotropy.

Non-centro-symmetric electron diffraction pattern of icosahedral quasicrystal induced by combination of linear phason strain and curvature of Ewald sphere

Non-centro-symmetric characteristics are observed in the experimental electron diffraction patterns (EDPs) from the icosahedral quasicrystalline precipitates in ZrAlNiCuNb alloys. Different from the well-known breaking of the Friedel's law, where a strong dynamical effect will reveal in EDPs the concealed non-centro-symmetry originated from the crystal structures themselves, the current results can be interpreted in terms of changes in deviation parameters due to a delicate combination of the linear phason strain characteristic of quasicrystals and the curvature of Ewald sphere. After taking this effect into consideration, the corresponding simulated EDPs fit quite well to the experimental data.