Long-period Superstructures Research Articles

Revaling the mechanism of microscopic kinking phenomena in Hf6Ta2O17 superstructures

As a promising candidate for a new-generation thermal barrier coating material, Hf6Ta2O17 is characterized by its unique long-period superstructures and excellent mechanical properties. In this study, the unique "kinking" phenomenon of the atomic arrangement in Hf6Ta2O17 crystals is directly visualized via high-resolution transmission electron microscopy for the first time. The evolution of the microstructure, and particularly the micro-mechanism of the "kinking" phenomenon, have been further explored by first-principles calculations. The study suggests that this phenomenon is primarily triggered by the rotation of the polyhedral structure, resulting from the partial elongation or shortening of the Ta-O bond length. These structural alterations are believed to directly influence the macroscopic mechanical properties and can improve the tensile properties of the Hf6Ta2O17 crystals. The findings of this study provide an important microscopic perspective for understanding the strain-induced crystal structure transformation. The quantitative analysis of the strengths of Ta-O and Hf-O bonds in different coordination polyhedra reveals the electronic structure changes behind the crystal structure transition, thus providing a new theoretical basis for an in-depth understanding of the behavior of crystals under the external stress.

QH-POCC: Taming tiling entropy in thermal expansion calculations of disordered materials

Disordered materials are attracting considerable attention because of their enhanced properties compared to their ordered analogs, making them particularly suitable for high-temperature applications. The feasibility of incorporating these materials into new devices depends on a variety of thermophysical properties. Among them, thermal expansion is critical to device stability, especially in multi-component systems. Its calculation, however, is quite challenging for materials with substitutional disorder, hindering computational screenings. In this work, we introduce QH-POCC to leverage the local tile-expansion of disorder. This method provides an effective partial partition function to calculate thermomechanical properties of substitutionally disordered compounds in the quasi-harmonic approximation. Two systems, AuCu3 and CdMg3, the latter a candidate for long-period superstructures at low temperature, are used to validate the methodology by comparing the calculated values of the coefficient of thermal expansion and isobaric heat capacity with experiment, demonstrating that QH-POCC is a promising approach to study thermomechanical properties of disordered systems.

Formation of lamellar microstructure in Ti-48Al-7Nb-2.5V-1Cr alloy

The reasonable design and application of phase transformation is an efficient approach to coordinate the microstructure and properties of alloys. The sequence of compositional and structural transformation determines the type and mechanism of phase transformation. Because of their low density and excellent high-temperature properties, TiAl alloys that contain fully lamellar microstructure have been applied to low-pressure turbine engine blades. However, there is no consensus on the mechanism of lamellar formation. Here, we propose a new mechanism that is dominated by spinodal decomposition, in which compositional transformation occurs prior to structural transformation during lamellar formation. Transitional lamellae with fcc-based long-period superstructures are introduced to prove the mechanism of sequentially spinodal decomposition, shear transformation, compositional modulation and ordering.

Theoretical Study on the Structural, Elastic, Electronic and Thermodynamic Properties of Long-Period Superstructures h- and r-Al2Ti under High Pressure.

The formations of long-period superstructures strongly influence the properties of Al-rich L1-TiAl intermetallic alloys. To soundly understand the role of the superstructures in the alloys, fundamentals about them have to be known. In the present work, the structural, elastic, electronic and thermodynamic properties of h- and r-AlTi long-period superstructures under pressure up to 30 GPa were systematically investigated using first-principles calculations based on density functional theory. The pressure dependence of structural parameters, single-crystal elastic constants, polycrystalline elastic modulus, Cauchy pressures and elastic anisotropy were successfully calculated and discussed. The total and partial densities of states at different pressures were also successfully calculated and discussed. Furthermore, combining with quasi-harmonic approximation, the effects of the pressure on the temperature dependent volume, isothermal bulk modulus, thermal expansion coefficient, heat capacity and Gibbs free energy difference were successfully obtained and discussed. Our results were consistent with the available experimental and theoretical values.

Orientation-dependent transport properties of Cu3Sn

Cu3Sn, a well-known intermetallic compound with a high melting temperature and thermal stability, has found numerous applications in microelectronics, 3D printing, and catalysis. However, the relationship between the material's thermal conductivity anisotropy and its complex anti-phase boundary superstructure is not well understood. Here, frequency domain thermoreflectance was used to map the thermal conductivity variation across the surface of arc-melted polycrystalline Cu3Sn. Complementary electron backscatter diffraction and transmission electron microscopy revealed the thermal conductivity in the principal a, b, and c orientations to be 57.6, 58.9, and 67.2 W/m-K, respectively. Density functional theory calculations for several Cu3Sn superstructures helped examine thermodynamic stability factors and evaluate the direction-resolved electron transport properties in the relaxation time approximation. The analysis of computed temperature- and composition-dependent free energies suggests metastability of the known long-period Cu3Sn superstructures while the transport calculations indicate a small directional variation in the thermal conductivity. The ∼15% anisotropy measured and computed in this study is well below previously reported experimental values for samples grown by liquid-phase electroepitaxy.

Photon-mediated electronic correlation effects in irradiated two-dimensional Dirac systems

Periodically driven systems can host many interesting phenomena. Two-dimensional Dirac systems irradiated by circularly polarized light are especially attractive thanks to the special absorption and emission of photons near Dirac cones. Here, letting the light travel in the two-dimensional plane, we treat the light-driven Dirac systems by using a unitary transformation, instead of usual Floquet theory, to capture the photon-mediated electronic correlation effects. In this approach, the direct electron–photon interaction terms can be removed and the resulting effective electron–electron interactions can produce important effects. The effective interactions can produce topological band structure in the case of irradiated 2D Dirac fermion system, and can lift the energy degeneracy of the Dirac cones for irradiated graphene. This method can be applied to other light-driven Dirac systems to investigate their photon-mediated electronic effects. These phenomena would be observed with ultraviolet light in some effective two-dimensional Dirac systems of honeycomb long-period superstructures.

Recursive alloy Hamiltonian construction and its application to the Ni-Al-Cr system

Navigating the high dimensional composition spaces of multi-principle element alloys, also referred to as high entropy alloys, will require new methods to construct first-principles alloy Hamiltonians. We introduce a recursive approach to parameterizing multi-component alloy Hamiltonians using interaction parameters from simpler subsystems as Bayesian informative priors. We applied this approach to perform a first-principles statistical mechanics study of the Ni-Al-Cr system. Ternary cluster expansions for the Ni-Al-Cr alloy were constructed by building on optimized Ni-Al and Ni-Cr binary cluster expansions. Monte Carlo simulations predict a sizable Cr solubility in the Ni-rich FCC based γ and γ′ phases. The L12 -ordered γ′ phase is predicted to dissolve Cr primarily on its Al-sublattice. We also identify a family of hierarchical long-period super structure orderings as groundstates in the Ni-Cr and Al-Cr binaries. The recursive approach to parameterizing alloy Hamiltonians opens the door to rigorous first-principles treatments of the elevated temperature thermodynamics of alloys in high dimensional composition spaces.

Critical stresses estimation by crystal viscoplasticity modeling of rate-dependent anisotropy of Al-rich TiAl alloys at high temperature

Determining critical stresses for different slip systems is one of the most important parts in crystal plasticity modeling of anisotropy. However, the task of finding individual critical resolved shear stress (CRSS) for every single slip system, if not impossible, is formidable and a delicate one especially if the microstructure is very complex. Slip family-based, mechanism-based and morphology-based (e.g., phase interface) slip systems classification and hence determining CRSS consistent with experimental measurements are often used in crystal plasticity. In this work, a novel approach to determining CRSS at high homologous temperature has been proposed by crystal plasticity modeling of rate-dependent anisotropy. Two-internal-variable-based phenomenological crystal viscoplasticity model is adopted for simulating isothermal, two-phase, single-crystal-like Al-rich lamellar Ti–61.8at.%Al binary alloy at high-temperature compression state ( $$1050\,^\circ \hbox {C}$$ ) by employing finite strain and finite rotation framework. To the best of authors’ knowledge, this is the first micromechanical modeling attempt with long-period superstructures. Conventional approaches related to CRSS estimation are also compared with the proposed one. Our material parameters are based on calibrating three different sets of compressive stain rate-controlled plasticity data taken from the loading of two different lamellar directions. It is revealed that the proposed approach works fine for rate-dependent anisotropy modeling, while other conventional approaches highly under- or overestimate available anisotropic experimental behavior of this alloy.

Elastic Properties, Phonon Focusing and Electronic Structures of Typical Long-Period Superstructures Al<sub>5</sub>Ti<sub>2</sub> and Al<sub>11</sub>Ti<sub>5</sub>

The first-principles calculations were performed to systematically study the elastic properties, phonon focusing and electronic structures of typical long-period superstructures Al5Ti2 and Al11Ti5. The obtained lattice parameters are in good agreement with the experimental data. The elastic constants were calculated, and bulk modulus B, shear modulus G, Young’s modulus E and Poison’s ration ν were further obtained. Both of the numerical indicators and the three dimensional images show that Al5Ti2 and Al11Ti5 are anisotropic, and the anisotropy is slightly high for Al5Ti2. Three slowness surfaces were constructed for Al5Ti2 and Al11Ti5 to investigate the phonon focusing patterns, and the group velocity surfaces were also obtained to gain more insight into the elastic anisotropy. Due to elastic anisotropy, both of the slowness surfaces and the group velocity surfaces are nonspherical, and the anisotropy is highest for slow transverse mode, and then followed by fast transverse mode and longitudinal mode. The electronic density of states and charge density distribution of Al5Ti2 and Al11Ti5 indicate that due to strong hybridization between Al-2p and Ti-3d, there is a strong directional bonding between the Al and Ti atoms. Electronic structures reveal the underling mechanism of elastic properties and phonon focusing for Al5Ti2 and Al11Ti5.

Ordering effect on the mechanical, electronic and magnetic properties of the β-based non-canonical approximant phases: β-Al50Cu33Fe17, η-Al50Cu44Fe6 and φ-Al47.5Cu49.5Fe3

Using transmission electron microscopy and X-ray diffraction, we established that the ordered η1-Al50Cu44Fe6 and φ-Al47.5Cu49.5Fe3 (Fmm2) alloys with nano-sized domain structure are formed by slowly cooling, whereas β-solid solutions with a short-range order were found in quenched states. The φ′-modification which exhibits the additional long-period superstructure was also observed in Al47.5Cu49.5Fe3. The studies of low temperature magnetic susceptibility and heat capacity did not reveal any another phase transitions in these alloys. The indentation test showed that hardness and Young’s modulus consistently grow as β-Al50Cu33Fe17 → η1-Al50Cu44Fe6 → (φ+φ′)-Al47.5Cu49.5Fe3 and approach to those in icosahedral phase. The same trend in the Young’s modulus was obtained for alloys containing β-solid solution with a short-range order. Ab initio calculations, however, predicted the opposite tendency in cubic β-Al50Cu50−xFex with a decrease in x, which was explained by the weakening of the covalent Fe 3d – Al sp bonding. This discrepancy between the results for β- and ordered phases, we related to a crucial effect of ordering which is accompanied by a progressive distortion of cubic local structure in the series β-Al50Cu33Fe17 → η1-Al50Cu44Fe6 → φ-Al47.5Cu49.5Fe3. As we demonstrated for η-Al(Cu, Fe), these distortions lead to the strengthening of the both covalent Fe–Al and Cu–Al bonds and the higher modules.

Testing predictions from density functional theory at finite temperatures:β2-like ground states in Co-Pt

We perform a critical assessment of the accuracy of DFT-based methods in predicting stable phases within the Co-Pt binary alloy. Statistical mechanical analysis applied to zero kelvin DFT predictions yields finite-temperature results that can be directly compared with experimental measurements. The predicted temperature-composition phase diagram is qualitatively incompatible with experimental observations, indicating that the predicted stability of long-period superstructures as ground states in the Co-Pt binary is incorrect. We also show that recently suggested methods to better align DFT and experiment via the hybrid functional HSE06 are unable to resolve the discrepancies in this system. Our results indicate a need for better verification of DFT based phase stability predictions, and highlight fundamental flaws in the ability of DFT to treat late 3$d$-5$d$ binary alloys.

Cu3Sn – understanding the systematic absences

The intermetallic compound Cu3Sn has previously been described as a long-period antiphase boundary superstructure of the Cu3Ti structure type. While the compound itself has been reported as a tenfold and an eightfold superstructure, ternary doped alloys show shorter repetitions. Interestingly, the diffraction patterns of these compounds show non-crystallographic absences that cannot be explained using the superstructure models. Since the compound exhibits phase broadening, these models are not satisfactory because the paucity of observed data does not allow for a refinement of the composition. Here, an alternative, superspace model in the orthorhombic space group Xmcm(0β0)000 is proposed, with the centering vectors (0,0,0,0) and (½,0,0,½). The presence of the non-crystallographic absences is explained as a result of a dominating occupational modulation that is accompanied by a weaker displacive modulation. In consistency with the EDXS results, the composition has been refined to Cu3 + xSn from single-crystal X-ray diffraction data. It is further demonstrated that by varying the length and the direction of the modulation wavevector in the superspace model, the ternary Cu3Sn compounds and other colored hexagonal close packing (h.c.p.) structures can be produced.

Ab-initio study of long-period superstructures and anti-phase boundaries in Al-rich γ-TiAl (L10)-based alloys

In this work, we report first-principles investigation of structural stability of all experimentally observed ordered long-period superstructures (LPSs), viz., r-Al2Ti, h-Al2Ti, Al5Ti3 along with Al5Ti3′, Al11Ti7 and Al3Ti2 LPSs, which are observed only as short-range ordered clusters at nanoscale level in Al-rich TiAl-based alloys. We adopt a procedure based on space-filling tiling arrangement of ordered Ti2Al, Ti3Al, Ti4Al motifs and their combination along with a symmetry analysis programme to determine the unit cell and the crystallographic information of Al5Ti3′, Al11Ti7 and Al3Ti2 LPSs in terms of L10 fcc unit cell. First-principles calculations are performed to further refine these crystallographic parameters (Wyckoff positions and lattice parameters) obtained from the above procedure. Moreover, it is found that the family of five LPSs have subgroup–supergroup relationships with γ-TiAl (Sp. gr. P4/mmm) and among themselves. Further, we find the inherent stability of r-Al2Ti + γ-TiAl and 2Al5Ti3 + γ-TiAl phase mixtures at 0 K compared to isomolecular Al3Ti2 and Al11Ti7 LPSs at their respective concentrations. The calculations of single-crystal elastic constants of Al5Ti3, Al11Ti7, Al3Ti2 and Al5Ti3′ LPSs show all these four structures are mechanically stable. We also calculate antiphase boundary (APB) formation energies for two types of APBs, viz., type-A and type-C in ordered Al5Ti3 LPS using the supercell approach. The relaxed APB energies for type-A and type-C APBs are 15.44 and 124.16 mJ/m2, respectively.

First-Principles Elucidation of Atomic Size Effects Using DFT-Chemical Pressure Analysis: Origins of Ca36Sn23's Long-Period Superstructure.

The space requirements of atoms are empirically known to play key roles in determining structure and reactivity across compounds ranging from simple molecules to extended solid state phases. Despite the importance of this concept, the effects of atomic size on stability remain difficult to extract from quantum mechanical calculations. Recently, we outlined a quantitative yet visual and intuitive approach to the theoretical analysis of atomic size in periodic structures: the DFT-Chemical Pressure (DFT-CP) analysis. In this Article, we describe the methodological details of this DFT-CP procedure, with a particular emphasis on refinements of the method to make it useful for a wider variety of systems. A central improvement is a new integration scheme with broader applicability than our earlier Voronoi cell method: contact volume space-partitioning. In this approach, we make explicit our assumption that the pressure at each voxel is most strongly influenced by its two closest atoms. The unit cell is divided into regions corresponding to individual interatomic contacts, with each region containing all points that share the same two closest atoms. The voxel pressures within each contact region are then averaged, resulting in effective interatomic pressures. The method is illustrated through the verification of the role of Ca-Ca repulsion (deduced earlier from empirical considerations by Corbett and co-workers) in the long-period superstructure of the W5Si3 type exhibited by Ca36Sn23.

Special features of the long-range order – short-range order phase transition in FCC-based alloys

Results of x-ray analysis of the long-range order in FCC-based alloys with L12, L12(M), L12(MM), and D1a superstructures are generalized. Regularities of the behavior of the long-range order parameter and average size of the thermal and periodic antiphase boundaries (APB) are traced, main mechanisms of temperature order-disorder phase transition are elucidated, and influence of the thermal and periodic APB on the special features of the order-disorder phase transition is revealed. The character of change of the periodic antiphase boundary density determines the special features of the order-disorder phase transition in alloys with longperiod superstructures.

Atomic ordering in Au-(42 to 50) at.%Pd: A diffuse scattering and first-principles investigation

Atomic ordering in Au-Pd alloys was studied by diffuse x-ray scattering and first-principles methods. Diffuse scattering was done of a single crystal of Au-48 at.% Pd that was aged at 703 K for 24 days. The weakly modulated short-range-order scattering exhibits diffuse maxima with an incommensurate wave vector, which can be related to a Fermi-surface nesting mechanism. From effective pair interaction parameters determined by the inverse Monte Carlo method, a one-dimensional long-period superstructure of the CH structure, LPS1, was found for AuPd. Concurrent electronic-structure calculations of the effective cluster interaction (ECI) parameters indicated the presence of another closely related superstructure, LPS2, at 0 K. At the same time, direct first-principles calculations of the total energies of the CH structure and further one-dimensional long-period superstructures predicted the stabilization of LPS4. Although the energy differences between these structures are small and a complex behavior of the effective interactions is expected due to the Fermi-surface nesting, experimental data and theoretical results both support the stabilization of a long-period superstructure of the CH structure for AuPd at 0 K. The ECI parameters determined by the screened generalized perturbation method also predicted a ground-state structure different from Au${}_{7}$Pd${}_{5}$, previously obtained from cluster expansion calculations. Its energetic preference was confirmed by direct total-energy calculations.

First-principles studies of typical long-period superstructures Al5Ti3, h-Al2Ti and r-Al2Ti in Al-rich TiAl alloys

First-principles calculations are performed to study the structural stabilities, electronic and elastic properties of typical long-period superstructures Al5Ti3, h-Al2Ti and r-Al2Ti in Al-rich TiAl alloys together with γ-TiAl. The obtained lattice parameters by relaxation of crystalline cells are in good agreement with the experimental data. The calculated formation enthalpies show that r-Al2Ti has the highest structure stability from energetic point of view, and then followed by h-Al2Ti, Al5Ti3 and γ-TiAl. The electronic density of states and charge density distribution indicate that due to strong hybridization between Al-2p and Ti-3d, there is a strong directional bonding between Ti and Al atoms. The elastic constants are calculated, suggesting that these structures are mechanically stable. Bulk modulus B, shear modulus G, Young’s modulus E and Poison’s ratio ν of polycrystalline materials are derived from the elastic constants. By several criteria, elastic anisotropies are analyzed, showing that these structures possess different degree of anisotropies.

Stability and instability of long-period superstructures in binary Cu–Pd alloys: A first principles study

Whereas binary intermetallic compounds often appear as ordered structures with fewer than 10 atoms per cell, large-supercell structures consisting of one- and two-dimensional superstructures have long been observed in CuPd. However, whereas the stability of the ordinary one-dimensional long-period superstructures (1-D LPS) has been previously investigated by first-principles total-energy methods, two-dimensional superstructures were not amenable to such calculations because of their large number of atoms ( O (10 3) atoms/cell). Using a cluster expansion extracted from a set of first-principles total energy calculations, we show that 2-D LPSs are likely kinetically-stabilized structures which transform into the 1-D LPS ground-state structures at thermodynamic equilibrium.

Origins of Superstructure Ordering and Incommensurability in Stuffed CoSn-Type Phases

The CoSn structure type contains large interstitial void spaces that frequently host electropositive guest atoms, such as rare earth elements. In this stuffing process, an intriguing ordering occurs between the neighboring void spaces leading to a family of long-period superstructures comprising intergrowths of the ScFe6Ge6 and ScFe6Ga6 structure types. This superstucture ordering culminates in incommensurability in the REFe6Ge6-deltaGa delta systems with RE = Sc, Tb, and Lu. In this work, we derive a 3 + 1D superspace model encompassing this series of structures and investigate the origins of the structural trends in this family with electronic structure calculations, at both the LDA-DFT and extended Hückel levels. Using our 3 + 1D model, we refine the structures of four new ErFe6Ge6-deltaGa delta (0 < or = delta < or = 6) phases, two commensurate and two incommensurate, from powder X-ray diffraction data. The refinement results confirm trends observed in the Sc-, Tb-, and Lu-based series: a gradual lengthening and, eventually, turning of the q-vector as Ge is progressively exchanged for Ga. These trends, and the incommensurate ordering as a whole, are traced to a tension between two modes by which the host lattice responds to stuffing atom insertion: (1) an atomic charge modulation enhancing the anionic character of the cavity walls around the guest atoms, and (2) a positional modulation expanding the cavities occupied by guest atoms. These two modes direct the stuffing atom ordering pattern toward opposite ends of the ScFe6Ge6-ScFe6Ga6 intergrowth series. The full series of structures, complex and incommensurate, reflects various degrees of balance between these two factors.

Low-energy antiphase boundaries, degenerate superstructures, and phase stability in frustrated fcc Ising model and Ag-Au alloys

An Ising model exhibits zero-energy antiphase boundaries (APBs) and frustration on close-packed face-centered cubic (fcc) and triangular lattices. The frustration results in degenerate structures and chains of long-period superstructures forming a quasicontinuous ground-state ``hull'' in the formation energy versus composition $(c)$ diagram. In alloys, a nonzero but small APB energy yields a $c$-dependent reduction in this degeneracy that affects the phase diagram topology and range of the two-phase coexistence. Using density functional theory combined with cluster expansions (CEs), we study Ag-Au alloys as a prototype and find the effective cluster interactions (dominated by nearest-neighbor pairs), predict energetics of millions of structures, and construct the temperature versus $c$ phase diagrams. We then compare the CE interactions for Ag-Au with those calculated directly from a supercell approach, and visualize the electronic origins of pair and multibody interactions, highlighting the physical nature of the chemical interactions implicit in the CE methods. We discuss generality of the results for close-packed alloys.