Quasicrystals Research Articles

Semi-analytical solution for mixed supported and multilayered two-dimensional thermo-elastic quasicrystal plates with interfacial imperfections

The tremendous attention of researchers has been attracted to the unusual properties of quasicrystals (QCs). In this article, the thermo-elastic problems of imperfectly bonded, multilayered, two-dimensional decagonal QC plates with mixed boundary conditions are investigated based on the QC linear elasticity theory. The temperature, heat flux, displacement, and stress components are expressed in terms of differential quadrature regional discrete point expansions in any rectangular plate, from which the state equations with thermomechanical coupling effects in a concise and compact matrix form is obtained. The different imperfect interface conditions are introduced to characterize specific structural and thermal contact properties at the interfaces and are further converted into the interface propagator matrix. Different from the conventional propagator matrix method, the new global propagator relation for the composite laminate with the presence of perfect/imperfect interfaces is recursively obtained by means of the reduced propagator distance and the exponential term between the layer to layer propagator matrix. The numerical examples indicate that numerical results are stability and precision, different boundary conditions have hardly influenced temperature and heat fluxes, and interlaminar thermal stresses strengthen with increasing interface coefficients. The methods and results can also serve as a reference for verifying existing or future QC laminate theories.

Synthesis, processing and phase analysis of quasi crystal particle reinforced aluminium matrix composite

ABSTRACT The main goal of the present study involves synthesis of Al-Cu-Fe based icosahedral quasi-crystal (QC) particles reinforced Al matrix composite and analyze its microstructure, phase and hardness property. The icosahedral quasi-crystal phase nominal composition was taken to Al63Cu25Fe12 (%). Mechanical alloying as a viable powder metallurgy route was used to synthesize the quasi-crystal phase with various milling parameters. The X-Ray diffraction analysis immediately after milling revealed that there was no formation of quasi-crystal phase exclusively by milling. However, after annealing the 100 hours milled powders at 600°C for 4 hours, the quasi-crystal phase was observed to form. The quasi-crystal phase was identified to be the 1/1 cubic approximant (α-Al85.5Cu36.95Fe15.5) of the icosahedral phase. The quasi-crystal particles reinforced aluminum composites were prepared with different concentrations (10–50 weight %) of the quasi-crystal phase by cold compaction. The compacted samples were then sintered for 4 hours at different temperatures such as 400°C, 500°C, and 600°C. The icosahedral Al64.3Cu23.5Fe12.2 phase formation was observed in the 600°C sintered composite, whose composition is approximately similar to the nominal composition used. The results uncovered that, while increasing the concentration of the reinforcement quasi-crystal phase, the hardness of the composite increased.

The short-range order in liquid water and amorphous ice

The short-range order in water and ice was determined from experimentally measured partial radial distribution functions by applying the Quasi Crystalline Model (QCM). Partial radial distribution functions were analyzed for water at several pressures and temperatures, crystalline ice, and for the three known phases of amorphous ice: Low-Density Amorphous (LDA), High-Density Amorphous (HDA), and Very-High-Density Amorphous (VHDA). It was found that at low temperatures and pressures, the short-range order of water is similar to that of the hexagonal ice (Ih) structure. At higher pressures and low temperatures, the short-range order of water becomes similar to that of tetragonal ice III structures with a c/a ratio of 0.8. At higher temperatures of 573 K, the short-range order obtained was similar to that of rhombohedral ice II (α = 113°). As for the amorphous ices, we conclude from the QCM analysis that these three forms are structurally distinct with short-range orders corresponding to ice Ih, ice III, and ice II for LDA, HDA, and VHDA ices, respectively.

The evolution of quasicrystal during additive manufacturing and aging treatment of a Si-modified Al–Mn-Sc alloy

By adding Si to Al–Mn-Sc alloy, an aging strengthened aluminum alloy with stable quasicrystal (QC) phase for additive manufacturing (AM) was developed, in which the QC-Al81Mn19 in as-printed sample transforms to QC-Al74Mn20Si6 by the long-range reaction-diffusion of Si atoms (Al81Mn19+Si → Al74Mn20Si6+Al6Mn) during aging strengthening (holding at 325°C for 4 h), rather than crystallizing into intermetallic compounds. In AM process, the Mn-rich clusters (QC-embryo) nucleate from melt, subsequently, the matrix transforms to QC-phase as follows: matrix → clusters (nuclei) → GP-zones → long-period stacking order → QC-phase during subsequent solid-state phase transformation of AM thermal cycling.

Thermoelectric Properties of Co-Substituted Al-Pd-Re Icosahedral Quasicrystals.

The practical application of quasicrystals (QCs) as thermoelectric materials makes icosahedral (i-) Al–Pd–Re QC attractive because of its moderate electrical conductivity (~280 Ω−1 cm−1), relatively high Seebeck coefficient (~100 μV K−1), and low thermal conductivity (~1.3 W m−1 K−1) at room temperature. To develop a thermoelectric Π-shaped power generation module, we need both p- and n-type thermoelectric materials. In this work, we aimed to develop an n-type i-Al–Pd–Re-based QC and investigated the effect of Co substitution for Re on the thermoelectric properties, i.e., the electron-doping effect. We synthesized dense bulk samples with nominal compositions of Al71Pd20(Re1−xCox)9 (x = 0, 0.1, 0.2, 0.3, 0.4, 0.5) via arc-melting, annealing, and sintering methods. We found that Co can produce n-type carriers in dilute substitution amounts of x = 0.1 and 0.2; however, the Seebeck coefficient at 300 K showed an n- to p-type transition with increasing x. This indicates that a simple rigid-band approximation is not applicable for i-Al–Pd–Re QC, which makes it difficult to synthesize an n-type i-Al–Pd–Re-based QC. Although the thermal conductivity was reduced from 1.28 (x = 0) to 1.08 W m−1 K−1 (x = 0.3) at 373 K by lowering of the electron thermal conductivity (electrical conductivity) and the alloying effect via Co substitution, the dimensionless figure of merit was not enhanced because of lowering of the power factor for all samples. The elastic moduli of i-Al–Pd–Re QC decreased by Co substitution, indicating that i-Al–Pd–Re-Co QC had a more ionic and brittle character.

Effect of icosahedral phase formation on the stress corrosion cracking (SCC) behaviors of the as-cast Mg-8%Li (in wt.%) based alloys

Through exploring the stress corrosion cracking (SCC) behaviors of the as-cast Mg-8%Li and Mg-8%Li-6%Zn-1.2%Y alloys in a 0.1 M NaCl solution, it revealed that the SCC susceptibility index (ISCC) of the Mg-8%Li alloy was 47%, whilst the ISCC of the Mg-8%Li-6%Zn-1.2%Y alloy was 68%. Surface, cross-sectional and fractography observations indicated that for the Mg-8%Li alloy, the α-Mg/β-Li interfaces acted as the preferential crack initiation sites and propagation paths during the SCC process. With regard to the Mg-8%Li-6%Zn-1.2%Y alloy, the crack initiation sites included the I-phase and the interfaces of I-phase/β-Li and α-Mg/β-Li, and the preferential propagation paths were the I-phase/β-Li and α-Mg/β-Li interfaces. Moreover, the SCC of the two alloys was concerned with hydrogen embrittlement (HE) mechanism.

Magnetic dynamics of hedgehog in icosahedral quasicrystal

Quasicrystals (QCs) possess a unique lattice structure without translational invariance, which is characterized by the rotational symmetry forbidden in periodic crystals such as the 5-fold rotation. Recent discovery of the ferromagnetic (FM) long-range order in the terbium-based QC has brought about breakthrough but the magnetic structure and dynamics remain unresolved. Here, we reveal the dynamical as well as static structure of the FM hedgehog state in the icosahedral QC. The FM hedgehog is shown to be characterized by the triple-Q state in the reciprocal-lattice {{varvec{q}}} space. Dynamical structure factor is shown to exhibit highly structured {{varvec{q}}} and energy dependences. We find a unique magnetic excitation mode along the 5-fold direction exhibiting the streak fine structure in the {{varvec{q}}}-energy plane, which is characteristic of the hedgehog in the icosahedral QC. Non-reciprocal magnetic excitations are shown to arise from the FM hedgehog order, which emerge in the vast extent of the {{varvec{q}}}-energy plane.

Quasicrystal stability in complex aluminum alloys: Insights from first principles models on structural motifs in crystalline approximants

Al81Cu7Fe4Cr8 is an alloy that readily forms quasicrystalline phases but understanding the complex atomic arrangements in the structure of such quasicrystals is challenging. For the Al-Cu-Fe-Cr system, the structures of crystalline approximants have been identified. These exhibit atomic motifs similar to those for the quasicrystalline configurations. We employ here a comprehensive methodology to analyze statistically generated ground states of orthorhombic approximants using density functional theory to shed light on the role of elemental variations of such complex alloys. Our findings show that the stability of QC approximants has a strong dependence on the specific locations of Cu and Cr atoms. The geometry of the atomic motifs from structural optimization suggests that the icosahedral phase is more favorable than the decagonal phase. The formation energy difference between these is small which indicates that the two quasicrystalline phases may coexist.

Influence of quasi-crystal particles and processing condition on microstructure and tensile properties of a selective laser melted high entropy alloy

Quasi-crystal (QC) Al65Cu20Fe10Cr5 powder particles were added into CoCrFeMnNi high entropy alloy (HEA) and processed by selective laser melting (SLM) under different conditions with the aim of enhancing their strengths. It is shown that increased energy density leads to increased grain size and texture in the SLMed HEA but does not cause significant influence on tensile properties. With the addition of QC particles, the microstructure and tensile properties highly depend on input energy density. Thus, HEA-10 wt% QC samples made between 55.6 J/mm3 and 92.6 J/mm3 are dominated by dispersive columnar γ grains and arc-shaped B2 phase and nano-sized Al2O3 particles while those made with high energy densities such as 133.3 J/mm3 contain mixed large vertical and horizontal columnar γ grains but no B2 phase. The B2-containing HEA-10 wt% QC samples show remarkably enhanced yield strength and ultimate tensile strength, thanks to B2 and nano-sized Al2O3 particles acting as effective dislocation motion barriers. The B2-free HEA-10 wt% QC samples show reduced strengths which is attributed to their large and complex grain structure and dendritic structure.

The generalized method to calculate the real-space winding number for one-dimensional systems with complex multi-band-gap structure

The topological study of the complicated one-dimensional (1D) systems with multi-band-gap structures, including quasi-crystals (QCs), is very hard since the lack of effective topological invariants to describe the non-triviality of gaps. A generalized method, based on the contracted wave-function, is proposed in this work to calculate the real-space winding number for the complicated 1D systems with multi-band-gap structures. First, the effectiveness of the generalized method is demonstrated to obtain the quantized real-space winding number for the gaps and correctly predict the topological phase transition and the existing fractional charge on the edges for the periodic 4-atoms SSH model (4A-SSH model). Then, we apply the generalized method to more complicated 1D Thue–Morse (TM) systems, which is one kind of QCs. The quantized real-space winding number is obtained for two traditional gaps and two fractal gaps for the TM systems and can also correctly predict the existence of topological edge-states and fractional charge on the ends. Several new phenomena are observed, e.g. the topological phase transition and the edge-states for the gaps in multi-band-gap structures, the fractional charge for the 4A-SSH model, the fluctuation of local charge and the asymmetric (but still with a quantized difference) fractional charge at the ends of TM system. The generalized method could be a powerful tool to study the topology of gaps in the complicated periodic systems or QCs.

Intermetallic phases and mechanical properties of a Mg–8Zn–6Al–1Sm (wt%) casting alloy

Intermetallic phases including precipitate constituents in both as-cast and peak-aged Mg–8Zn–6Al-1Sm (wt%) alloys were investigated using transmission electron microscopy. In the as-cast sample, grain boundary intermetallic phases are mainly Mg44Al15Zn41 icosahedral (i) phase, Mg5Zn2Al2 (φ) phase, Mg32(Al,Zn)49 (τ) phase, and Al2Sm phase. During aging treatment, a possible phase transformation from τ to i might occur. Additionally, precipitate constituents in α-Mg matrix of the peak-aged sample are mainly i phase, MgZnAl cubic phase, MgZn2 phase, Mg4Zn7 phase, and aperiodic structures. Two or more orientation relationships between i (or Mg4Zn7, or MgZnAl cubic phase) and Mg matrix were revealed, and ordering planar faults were observed for some MgZn2 precipitates. Finally, calculations indicate that the strengthening effect of Mg–Zn–Al precipitates is more efficient than that of Mg–Zn precipitates, and the high strength of the studied alloy sample is mainly attributed to the semi-continuous intermetallic skeletons and the Mg–Zn–Al based precipitates.

Study of in situ formed quasicrystals in Al-Mn based alloys fabricated by SLM

Quasicrystal (QC) phases are often observed in Al-based alloys with Al content< 85 at%, and their particle size is typically greater than 0.3 µm. In this study, an in situ formed QC phase was observed in Al-4.1Mn-0.98Mg-0.8Sc-0.5Zr alloy (Al ≈ 96 at%) when processed using Selective Laser Melting (SLM). The particle size of the QC phase processed by SLM is 55–85% smaller than when processed by other methods, which is likely due to the lower Mn concentration and higher cooling rate. Moreover, the shape of the QC phase is spherical near the bottom of the melt pool and becomes a strip in the center of melt pool with the variation of solidification conditions. At higher cooling rates, it can be inferred that the formation of QCs is promoted while the growth of QC phase is effectively inhibited. According to calculations and electron spectroscopy (EDS) results, the possible chemical composition of the QC phase is considered as Al80Cu2.83Mn17.16 (at%) with an e/a of 1.9849 and Rav of ~1.4152 Å. During aging, a portion of the QC phase could transform into Al6Mn through a peritectoid reaction. The homogeneously dispersed QC transforms the Al6Mn into a finer and more dispersed phase. Consequently, the mechanical properties are cooperatively improved both by Al6Mn and the residual QC phase. After aging, the ultimate strength is 548.67 ± 4.66 MPa with an elongation of 12.2 ± 1.09%. These results indicate that SLM expands the formation range of QC and is an ideal processing route to fabricate in situ QC-strengthened Al-based alloys.

An Al-5Fe-6Cr alloy with outstanding high temperature mechanical behavior by laser powder bed fusion

This research aims to study the laser powder bed fusion ( L -PBF) processability, the microstructure and the tensile mechanical properties of an AlFeCr alloy at a wide range of temperatures. With this goal, Al-5Fe-6Cr (wt%) pre-alloyed powder was produced by casting and gas atomization. The microstructure of the as-atomized powders and the as-built specimens was characterized via scanning electron microscopy, x-ray diffraction, and transmission electron microscopy. Following a parameter optimization study, dense as-built specimens with a high relative density of 99.8% and a Vickers hardness at room temperature of 192 HV were fabricated. In addition, the newly developed AlFeCr alloy exhibits yield strength values of 273 ± 5 MPa and 179 ± 3 MPa at 300ºC and 400ºC, respectively. The exceptional strength of this alloy at high temperature is attributed to the homogeneous precipitation during processing of large density of nanoscale icosahedral (i-phase) and intermetallic phases, which are endowed with high thermal stability. These values are significantly higher than those corresponding to the conventional wrought Al7075 aluminum alloy in the T6 condition and to all Al alloys manufactured by L -PBF to date. The current research sheds guidelines for the design of high strength aluminum alloys containing transition alloying elements that are suitable to be processed by L -PBF for high temperature structural applications in a cost effective way.

Reinvestigation of the Thermoelectric Properties of Fe‐Substituted Icosahedral Al–Pd–Re Quasicrystals

Icosahedral quasicrystals (QCs) have a glass‐like, low thermal conductivity because of their complex crystal structures, and a moderate power factor because of deep pseudogap formation in their electronic density of states (DOS) at the Fermi level originating from icosahedral symmetry and covalent bonding nature. These features are beneficial for high‐performance thermoelectric materials. Densified bulk samples of icosahedral Al–Pd–Re–Fe QCs are synthesized by arc‐melting, annealing, and spark plasma sintering methods. It is found that a 2/1‐approximant crystal (AC) can be produced by higher Fe substitution [Al71Pd20(Re1–xFex)9 ()]. The thermoelectric properties of Al71Pd20(Re1–xFex)9 (where x = 0, 0.2, 0.4, 0.65, 0.7, and 0.75) are evaluated between 373 and 973 K. Fe substitution enhances the power factor and reduces the phonon thermal conductivity by a concept of “weakly bonded rigid heavy clusters” (WBRHCs) and an increase in the carrier concentration, enhancing the dimensionless figure of merit (zT). The 2/1‐AC phase has a much higher electrical conductivity and a moderate Seebeck coefficient in the mid‐temperature region due to a considerable increase in carrier concentration; this results in the highest power factor of 900 μW m−1 K−2, and zT value for the 2/1‐Al−Pd−Re−Fe AC is 0.26 at 573 K.

Cooper instability and superconductivity of the Penrose lattice

Bulk superconductivity (SC) has recently been observed in the Al-Zn-Mg quasicrystal (QC). To settle several fundamental issues of the SC on the QC, we use an attractive Hubbard model to perform a systematic study on the Penrose lattice. The first issue is the Cooper instability of the QC, i.e., no Fermi surface under an infinitesimal attractive interaction. Starting from the two-electron problem outside the filled Fermi-sea, we analytically prove that an infinitesimal Hubbard attraction can lead to the Cooper instability as long as the density of state is nonzero at the Fermi level, which provides the basis of the SC on the QC. Our numerical results yield that the Cooper pairing always takes place between the two time-reversal states, satisfying the Anderson's theorem. On this theorem, we perform a mean-field (MF) study at both zero and finite temperatures. The MF study also shows that an arbitrarily weak attraction can lead to the pairing order, with the resulted pairing state being well described by the BCS theory and the thermal dynamic behaviors being well consistent with experimental results. The second issue is about the superfluid density on the QC without translational symmetry. It's clarified that although the normal state of the system locates at the critical point of the metal-insulator transition, the pairing state exhibits real SC, carrying finite superfluid density that can be verified by the Meissner effect, consistent with experiment also. These revealed properties of the SC on the Penrose lattice are universal for all QCs.

Displacement discontinuity method for interfacial cracks in one-dimensional hexagonal quasi-crystal coating under thermal-mechanical loading

This study derived Green’s functions of the concentrated interfacial displacement and temperature discontinuities. Green’s functions and the displacement and temperature discontinuity method were used to analyze interfacial cracks in a one-dimensional (1D) hexagonal quasi-crystal (QC) coating under thermal-mechanical loading. The boundary integral-differential equations were established for an interfacial crack, the fundamental solutions were obtained for uniformly distributed temperature, and the displacement discontinuities were applied over a line element. The near-crack-tip oscillatory singularity of stresses was eliminated by introducing the Gaussian distribution function to approximate the delta function. After this treatment, the expressions of the stress and heat flux intensity factors without oscillatory singularity, and the energy release rate (ERR) at the crack tips were obtained. The displacement and temperature discontinuity boundary element method for numerical simulation is proposed, and the influence of relevant factors is briefly discussed.

Effect of cooling rate on the microstructure and thermal expansion properties of Al–Mn–Fe alloy

Abstract Al–Mn–Fe alloy was prepared by traditional casting, injection casting, and melt spinning. The effect of cooling rate on the microstructure of alloy was studied. Results show that the phases in the alloy are the same for the three methods, including quasicrystal icosahedral phase (I-phase), Al6Mn phase, and Al phase. With the increase in cooling rate, the morphology of the I-phase changed from long strips to short rods and gradually to a daisy shape, and the content of the I-phase increased from 28 to 70%. Meanwhile, the morphology of the Al6Mn phase in the alloy changed from long bars to short rods, and the content decreased from 45 to 20%. Meanwhile, the thermal expansion coefficient of the alloy in the injection casting state is significantly reduced by about 15% at all temperature stages compared with that in as-cast state, and the results show that an appropriate cooling rate can improve the microstructure and properties of the alloy.

Analysis of a mode-I crack in a one-dimensional orthorhombic quasicrystal strip

A mode-I crack in a one-dimensional (1D) orthorhombic quasicrystal (QC) strip under in-plane phonon and phason stress loading is considered. Fourier transforms are applied to reduce the mixed boundary value problem of the mode-I crack to solving a system of simultaneous singular integral equations. Asymptotic expressions of the phonon and phason stresses and displacement fields near the crack tips have been obtained in an explicit form. The crack-tip singularities of the mode-I crack have been investigated and the intensity factors of the stresses in the phonon and phason fields are derived explicitly. The stress intensity factors (SIFs) and the hoop stress intensity factors (HSIFs) have been determined to investigate the effect of the geometric size and the crack kinking phenomenon. The effect of the thickness ratio of the cracked strip on the SIFs and energy release rates has been investigated. When the thickness of the cracked strip becomes infinite large, the results obtained for the crack problem can be reduced to the analytic solution for a mode-I crack in an infinite 1D orthorhombic QC media.

Microstructure and Properties after Friction Stir Processing of Twin-Roll Cast Al–Mn–Cu–Be Alloy

We studied the effect of friction stir processing (FSP) on the microstructure and properties of high-speed twin-roll cast strips made of an experimental Al–Mn–Cu–Be alloy. The samples were examined using light, scanning, and transmission electron microscopy, microchemical analysis, X-ray diffraction, and indentation testing. During FSP, the rotational speed varied, while other parameters remained constant. The uniformity of the microstructure increased with the growing rotational speed. In the stir zone, several processes took place, and the most important were: recrystallisation of the matrix grains, fragmentation of the primary intermetallic particles Al15Mn3Be2 and their more uniform distribution in the stir zone, fracture, and dispersion of the eutectic icosahedral quasicrystalline phase (IQC), transformation of tiny Al15Mn3Be2 and IQC particles into the τ1-Al26Mn6Cu4 phase and precipitation of Al–Mn–Cu precipitates. In the thermomechanically affected zone, new dislocations formed as well as dispersion of the IQC eutectic phase and recrystallisation of the matrix grains. In the heat-affected zone, dissolution of θ’-Al2Cu precipitates occurred. The hardness variation was not severe between the stir and heat-affected zones.

The Largest Quasicrystalline Tiling with Dodecagonal Symmetry from a Single Pentablock Quarterpolymer of the AB1CB2D Type.

A quasicrystalline tiling pattern with tile size of ca. 60 nm has been discovered in the bulk state of a four-component pentablock polymer molecule of the AS1IS2P type, where A, S, I, and P denote poly(4-vinylbenzyldimethylamine), polystyrene, polyisoprene, and poly(2-vinylpyridine), respectively. The polymer samples used were prepared by anionic polymerizations and have narrow molecular weight distribution. The sample films were obtained by an extremely slow solvent-cast process from dilute solutions of tetrahydrofuran for 14 days. It has been found by TEM observation that the quarterpolymer, AS1IS2P-4 (Mn = 149 kg/mol, ϕA/ϕS1/ϕI/ϕS2/ϕP = 0.12/0.27/0.20/0.29/0.13), reveals two final stable structures, i.e., a 3.3.4.3.4 periodic tiling pattern as a minor component and a quasicrystalline (QC) tiling with dodecagonal symmetry as a major component, where the former includes a triangle/square number ratio of 2 and the latter has one of approximately 2.28, which is close enough to the ideal ratio, 4/ 2.31, for the triangle/square random tiling of the dodecagonal QC tiling (DDQC). Two structures were also clearly proved by SAXS diffraction patterns. Here, it should be noted this QC structure having a tile side length of ca. 60 nm was created with a single block polymer molecule.