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Peierls stress (τp) of face-centered-cubic (FCC) structures from the Peierls-Nabarro (PN) model calculation is extremely sensitive to the input parameters including the shear modulus (μ), Poisson's ratio (ν) and generalized stacking fault energy (GSFE), which makes the model not a very reliable tool for the prediction of τp because uncertainties occur inevitably in the theoretical calculation or experimental measurements of these input parameters. In the present work, we scrutinized systematically the sensitivity of τps of the FCC metals to the input parameters within the framework of semi-discrete variational PN (SVPN) model. We showed that τp oscillates drastically with varying input parameters. The period of the oscillation is associated with the dependence of the distance between the two partials of the extended dislocation (D) on μ, ν and GSFE while the amplitude of the oscillation is modulated by the dependence of the half-width of the partial dislocation (ζ) on these input parameters. Based on the explored origin of the oscillation, a modified SVPN model is proposed to relieve the sensitivity of τp on the input parameters. The Peierls stresses evaluated with the modified SVPN model are in general agreement with the available experimental values for all the FCC metals.

Abstract: Stacking fault energy (SFE) plays an important role in the plastic deformation of metals. As compared to those of face-centered cubic metals, the SFEs of hexagonal close-packed (hcp) metals are less reported in literature. In this paper, we derive the expressions of four types (I 1, I 2, E and T 2) of basal plane SFEs of hcp metals in terms of the interlayer interaction energies within the framework of the Ising model. The SFEs of 14 kinds of hcp metals are then evaluated with the interlayer interaction energies extracted from the total energies of four prototypes calculated by using the first-principles full-potential augmented plane-wave method. We show that the hcp metals can be divided into three types according to their interlayer interaction energies. For all the hcp metals involved in this study, I 1 has the lowest SFE, whereas E has the highest. The metals (Mg, Co, Zn and Cd) with principal slip system (0001) generally have low basal plane SFEs. The I 1 and T 2 SFEs increase linearly with the energy difference between double hexagonal close-packed and hcp structures, whereas the I 2 and E SFEs increase linearly with the energy difference between the short-period twin and hcp structures, indicating a trivial contribution of the interaction energy between atomic layers over third nearest neighbors to the SFEs. The SFEs also correlate with the cohesive energy density (cohesive energy of unit volume) with the exception of Be, Co, Tc and Re. [Copyright &y& Elsevier]

The influences of doping elements (Ti, Zr, Hf, V, Ta, Cr, Mo, W and Al) on the heats of formation of Nb, Nb5Si3 and Nb3Si have been studied using first-principles pseudopotential plane-wave method based on density functional theory. Site preferences of dopants in Nb5Si3 and Nb3Si intermetallics are first determined by comparing the heats of formation of the systems with different site occupations of the doping atoms. The partitionings of dopants between the equilibrium phases Nb and Nb5Si3 of Nb-Si in situ composites are then discussed and compared with experimental results. Phase partitioning behaviours of the alloying elements are found to depend strongly on the number of their valence electrons. By calculating the enthalpy of reaction regarding the eutectoid decomposition of Nb3Si phase, we conclude that Ta is a Nb3Si stabiliser while other dopants destabilise Nb3Si at the investigated concentration. [ABSTRACT FROM AUTHOR]

Abstract: The interaction energies between substitutional rare earth (RE) atoms, between RE and interstitial C, N, O, H atoms, as well as between RE and vacancies in α-Ti are calculated via first-principles density-functional theory with projector augmented-wave (PAW) pseudopotentials. The results show that the RE–vacancy and RE–RE interactions are attractive due to the weaker RE–Ti bond than the host Ti–Ti bond. All of the RE atoms investigated in this paper are repulsive to C and N, but attractive to H. RE–O interactions are repulsive for the light RE atoms, though the interactions are very weak for the heavy RE atoms. The mechanism underlying the interactions and their possible influence on the properties of Ti alloys are discussed. [Copyright &y& Elsevier]

Solute atoms such as Fe, Co, and Ni diffuse abnormally fast in α -Ti, which influences significantly the mechanical properties of the titanium alloys. Various mechanisms (e.g., the interstitial diffusion mechanism and solute-vacancy complex mechanism) have been proposed to account for the fast diffusion of these solutes in α -Ti. To elucidate such diffusion mechanism, a first-principles method is employed to calculate the formation energies, migration energy barriers, and solute-vacancy binding energies of the substitutional and interstitial solute atoms including Al, Si, Sn, V, Mn, Fe, Co, Ni, and Cu in α -Ti. Based on the calculated parameters, the diffusion mechanisms are discussed. Comparing the formation energies of the substitutional and interstitial solutes, we find that all the solute atoms prefer the substitutional configuration to the interstitial one. The interstitial migration energy barriers are quite low for all the solutes. Al and Sn diffuse mainly through normal vacancy mediated mechanism due to the high substitutional to interstitial preferential energy that leads to very low fraction of interstitial solutes (about 10 − 15 ∼ 10 − 16 at 1000 K) at thermal equilibrium state and high interstitial diffusion activation energy. The 3d metal solute atoms, especially Mn, Fe, and Co, are fast diffusers and the diffusion coefficients are dominated by the interstitial mechanism because of their sizable thermal equilibrium interstitial fractions (several percent at 1000 K). The solute-vacancy complex mechanism is not likely to account for the fast diffusions in α -Ti. We show that the direct chemical interaction between the solute and matrix atoms determines the site-occupancy of the solute atoms in α -Ti besides the atomic size effect that was commonly believed to be responsible for the fast diffusions. [ABSTRACT FROM AUTHOR]

Hydride induced embrittlement is one of the main causes for the stress corrosion crack of titanium alloys. The formation process of the hydride at atomic level remains controversial as it is not accessible directly by experiments. In the present work, the formation of titanium hydride is investigated by using a first principles method. By calculating the formation energies of TiH m with various types of H occupation state in HCP and FCC Ti matrix, we show that there exists phase equilibrium between α-Ti-H solid solution and δ/ε hydrides with χ and γ hydrides appearing as metastable phases. The energy barrier for the HCP → FCC structure transformation for the formation of the hydrides decreases with increasing H:Ti ratio m. The structure transformation is accompanied with spontaneous shift of half amounts of the H atoms from the octahedral to the tetrahedral interstices. The energy barrier for the migration of the remaining half of the H atoms from the octahedral to the tetrahedral interstices reduces with increasing m and approaches to a constant value. The hydrostatic compressive stress acting on TiH m , resulted from the H-induced lattice dilation tendency and the constraint of the surrounding matrix, does not facilitate the HCP → FCC transformation and the migration of the H atoms. The present work sheds light on the hydride formation process at atomic level. [Display omitted] [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the local asymmetrical distortion (measured as the dipole shape factor) induced by interstitial atoms (IAs: C, N, and O) in bcc-metals (V, Nb, and Ta) and the diffusion barriers of the IAs in the matrix are calculated by the use of a first-principles pseudopotential method. The obtained dipole shape factor and diffusion barrier are in reasonable agreement with the experimental values. The trend of the dipole shape factor and diffusion barrier with respect to the IAs and host metals are discussed in terms of the size difference between the IA and the interstice, the chemical and elastic interaction between the IA and the host metals. It is shown that both the size difference and the chemical interaction affect significantly the dipole shape factor, whereas the diffusion barrier is dominated by the chemical interaction. [Copyright &y& Elsevier]

The curve of the hardness of ternary nitride Ti 1- x Al x N against the Al content x is convex shaped with a maximum at x ∼0.6, of which the mechanism remains to be further explored. The present work scrutinizes the phase stability and elastic modulus of Ti 1- x Al x N against the Al content x by using a first principles method as they are highly pertinent to the hardness of the system. It is shown that the cubic B1 structure is more stable than the hexagonal structure (Bk) for x < ∼0.75 whereas the hexagonal structure (B4) is more stable than the cubic B1 structure for x > ∼0.75. The hardness of both B1- and B4–Ti 1- x Al x N evaluated with the calculated elastic modulus become higher with increasing x. B1–Ti 1- x Al x N is harder than B4–Ti 1- x Al x N. Besides the composition dependent hardness of the individual B1- and B4–Ti 1- x Al x N, the phase decomposition contributes to the convex shaped hardness of Ti 1- x Al x N against x. The work presents a clear picture on how the chemical composition and phase transition influence jointly on the hardness of Ti 1- x Al x N. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • The interface energies and GSFE of β / ω interface in Ti alloy are calculated. • The strong anisotropy of the interface energy contributes to the ellipsoid-like morphology of β particles, with its major axis parallel to 〈1 1 1〉 β direction. • The β / ω interface is prone to crack initiation and propagation as compared to the bulk and contributes to the ω -embrittlement of β -Ti alloys. Abstract ω phase precipitating in a β -Ti alloy is generally detrimental to the ductility and plasticity of the alloy.To understand the influence of ω phase on the mechanical properties of β -Ti alloys, the property of the β / ω interface is demanded.In this regard, we calculate the interface energies ( γ int ), cleavage energies ( γ cl ), and generalized stacking fault energies (γ) of the β / ω interfaces with different orientations using a first-principles method.Our calculations predict interface energies of 0.32 J m−2 and 0.03 J m−2 for the (1 1 1 ¯) β / (0 0 0 1) ω and (1 1 2) β / (1 0 1 ¯ 0) ω interfaces, respectively.The strong anisotropy of the interface energy contributes to the ellipsoid-like morphology of ω phase with its major axis parallel to the 〈 1 1 1 〉 β direction.The influence of the interface on the ductility of the alloy is discussed with the calculated cleavage energies and the unstable stacking fault energies (γ us) extracted from the generalized stacking fault energy (GSFE). The γ cl / γ us ratio of the (1 1 2) β / (1 0 1 ¯ 0) ω interface is lower than those of the main slip planes of the bulk phases, indicating that this interface is prone to crack initiation and propagation as compared to the bulk and contributes to the ω -embrittlement of β -Ti alloys. [ABSTRACT FROM AUTHOR]

A plane wave pseudo-potential method based on density functional theory is employed to calculate the migration energy barrier for the atomic self-diffusion in HCP metals including Mg, Zn, Ti, Zr, and Hf. The influences of some key factors (plane-wave cutoff energy, k -mesh, supercell size, and geometric optimization scheme) on the calculated migration energy barrier and its anisotropy are systematically investigated. We show that the supercell size affects heavily the migration energy barrier and its anisotropy for the metals with valence d electrons (Ti, Zr, and Hf) but not for the ones with only valence s metals (Mg). In general, the anisotropy of the migration energy barrier reduces with increasing size of the supercell especially for Ti, Zr, and Hf. The optimization of the shape and volume of the supercell matters for the migration energy barrier calculated with the small size supercell but not for that calculated with the large supercell. With the calculated migration energy barrier, the self-diffusion coefficients are evaluated based on the transition state theory and compared with other first-principles calculations and the experimental measurements. [ABSTRACT FROM AUTHOR]

The phase stability and elastic property of binary Ti- x TM (TM = V, Cr, Nb, Mo) and ternary Ti-15TM- y Al alloys are investigated systematically by using a first-principles method. The coherent potential approximation is employed to describe the random distribution of the alloying atoms in the alloys. We show that the transition metal (TM) elements V, Cr, Nb, Mo increase the elastic stability of the β phase but decreases that of the α phase. The polycrystalline bulk modulus of the α phase increases with the concentration of the TM alloying elements whereas the Young's modulus and shear modulus are weakened. For the β phase, all the polycrystalline elastic moduli hardens with the addition of the TM alloying elements. The influence of the simple metal (SM) element Al on the elastic properties is much weaker than the TM ones. The TM alloying elements increase the stability of the β relative to the α and ω phases. Al stabilizes significantly the β phase over the ω phase. Our calculations suggest that the d electron density dominates the elastic modulus but not the relative phase stability. The interaction between the TM and SM alloying elements in titanium alloys matters to the relative stability. [ABSTRACT FROM AUTHOR]

Phase decomposition influences significantly the mechanical properties of high entropy alloys (HEAs). Prediction of the phase decomposition of HEA is greatly hindered by the hyper-dimensional composition space of the alloys. In the present work, we propose to represent the HEAs as various pseudo-binary alloys of which the temperature dependent free energies as functions of compositions may be readily calculated by using first-principles methods in combination with thermodynamic models. With the calculated free energies, the phase diagrams of the pseudo-binary alloys may be constructed and the phase decomposition can be predicted. This procedure is applied to Hf-Nb-Ta-Ti-Zr alloy with body-centered cubic (BCC) structure. We predict that the equiatomic HfNbTaTiZr HEA suffers from phase decomposition below critical temperature of 1298 K. The HEA decomposes most favorably to BCC NbTa-rich and HfZr-rich phases. The BCC HfZr-rich phase transfers to a hexagonal close-packed structure (HCP) phase at low temperature. The predicted compositions of the decomposed phases are in good agreement with experiment and Thermal-Calc modeling. Furthermore, the effect of the phase decomposition on the strength of the HEA is evaluated by considering the solid-solution and precipitation strengthening mechanisms. The precipitation strengthening effect is stronger than the solid-solution strengthening at the low annealing temperature but becomes weaker at high annealing temperature. [Display omitted]. [ABSTRACT FROM AUTHOR]

• Al-Zr bond dominates chemical ordering in ordered AlNbVTiZr alloy. • Ordered configurations with lower total energies significantly increase elastic constant of C 11. • (AlV)-(NbZrTi) configuration with lower total energy has a larger tensile strength than disordered configuration. • Chemical ordering mediates thermodynamic properties of AlNbVTiZr, but has little effects at high pressure. Local chemical ordering influences significantly the mechanical properties of refractory high entropy alloys. In this work, the effects of local chemical ordering on the structural stability and mechanical properties of AlNbVTiZr are investigated by using first-principles simulations. It is shown that the Al Zr bond dominates the chemical ordering for ordered configurations. The local ordering enhances the structural stability, elastic property and ideal strength. By taking into account the effects of temperature and pressure, we found that chemical ordering mediates the thermodynamic properties of the AlNbVTiZr alloy, but it has little effect at high pressure. [ABSTRACT FROM AUTHOR]

Using density-functional theory, we determine the generalized stacking fault energy (GSFE) for the { 1 1 ¯ 2 } 〈 1 ¯ 11 〉 twinning system in ferromagnetic (FM) body-centered cubic Fe and Fe–Cr alloys with molar fraction of Cr ≤ 0.5. We adopt both reflection and isosceles twinning pathways and reveal the magnetic ordering effects on the GSFE by contrasting the FM results to those obtained for the magnetically disordered paramagnetic (PM) state. The results show that the isosceles twin boundary configuration is energetically preferred in this binary. The loss of long-range magnetic order lowers the GSFE amplitude but increases the twin boundary migration (TBM) energy regardless of the Cr content. The twin boundary formation (TBF) energy and the TBM energy show non-linear dependences on Cr content in the FM and PM states, and the effect of Cr on these properties critically depends on the magnetic state. We discuss our results in regard to the stable twin boundary structure and deformation twinning experimentally observed in homogeneous Fe-50 wt.% Cr alloy up to temperatures slightly above the magnetic ordering temperature. [Display omitted] [ABSTRACT FROM AUTHOR]

The critical resolved shear stress (CRSS) and its related plastic deformation of titanium with hexagonal close packed structure are highly anisotropic, leading to low ductility of the material. Understanding the alloying effect on the CRSS is crucial for the improvement of the mechanical properties through rational composition design. Accurate prediction of the CRSS is not straightforward due to the atomic randomness of the alloy. The generalized stacking fault energies (GSFEs) for the basal and prismatic plane 〈 a 〉 slips of random α-Ti 1− x Al x alloys (0 ≤ x ≤ 0.1875) are calculated by using first-principles methods including exact muffin-tin orbitals (EMTO) and plane-wave psuedopotential (VASP) methods in this work. The random distribution of Al in the alloy is treated by using both coherent potential approximation (CPA) for EMTO and special quasirandom structure (SQS) techniques for VASP. The CRSSs are then evaluated within the frame work of semi-discrete variational Peierls-Nabarro model. The VASP-SQS calculations with atomic relaxation generate reasonably good GSFE and CRSS compared with the EMTO-CPA and VASP-SQS calculations without atomic relaxation. For pure Ti, the unstable stacking fault energy (γ u s f ) and CRSS (τ a) for the basal 〈 a 〉 slip are higher than those for the prismatic 〈 a 〉 slip. With increasing Al concentration x , γ u s f b for the basal 〈 a 〉 slip decreases whereas γ u s f p for the prismatic 〈 a 〉 increases. The CRSSs for the basal 〈 a 〉 slip (τ a b) and the prismatic 〈 a 〉 slip (τ a p) both increase with x while τ a p increases faster than τ a b such that τ a b approaches to τ a p. The calculated CRSSs explain successfully our recently measured mechanical properties (yield strength, fracture toughness, ultimate strength, and elongation) of the α-Ti 1− x Al x alloy against x. • The article calculated GSFEs and CRSSs for basal and prismatic plane (a) slips of random α-Ti 1-x Al x alloys (0 ≤ x ≤ 0.1875). • The local lattice distortion influences significantly on the GSFEs and CRSSs. • Strength and fracture toughness of α-Ti 1-x Al x should increase whereas the plasticity reduces with increasing x. • Basal and prismatic plane slips may be activated during plastic deformation with increasing x. • The unstable stacking fault energy alone does not always work for the description of the mobility of some slip system. [ABSTRACT FROM AUTHOR]

Introducing high symmetric cubic disordered β or ordered β 0 phases by alloying β -stabilizers improves the strength and high-temperature deformability of intermetallic γ -TiAl. However, the β phase is prone to decomposition of lower symmetric hexagonal phases such as ω 0 and ω ' ' , which are extremely brittle and harmful to the ductility of the alloy. Therefore, understanding the effect of the β -stabilizers on the phase stability of TiAl is crucial for the rational design of TiAl based alloys. In the present work, the equilibrium lattice structures, bulk moduli, and heats of formation of β , β 0 , ω ' ' , and ω 0 of TiAl with some typical β stabilizers V, Nb, Ta, Cr, Mo, W, and Mn were calculated by using a first-principles method, with which the relative phase stability were discussed. We showed that the phase stability sequence is β < β 0 < ω 0 < ω ' ' for Ti 4 Al 3 Nb and Ti 4 Al 3 Ta, β < ω 0 < β 0 < ω ' ' for Ti 4 Al 3 V, Ti 4 Al 3 Mo and Ti 4 Al 3 W, and β < ω 0 < ω ' ' < β 0 for Ti 4 Al 3 Cr and Ti 4 Al 3 Mn. The β -stabilizing effect of the alloying element X increases from Nb/Ta to V/Mo/W to Cr/Mn. The relative stability of the various phases is roughly in accordance with the trends of the unit volume and bulk modulus against the phases with a few exceptions. The phase diagrams of the alloys associated with these phases were estimated by comparing the free energies against the temperature of various phases, evaluated with the calculated unit volumes and bulk moduli. • Phase stability of TiAl alloying with transition metal elements is determined. • Phase stability is roughly in accordance with the unit volume and bulk modulus. • Strategy for the composition design of TiAl based alloys is presented. [ABSTRACT FROM AUTHOR]

Ab initio calculations on the single-crystal elastic constants (C ij), Zener anisotropy ratio (C 44 / C ′), and single-crystal Young's moduli (E [hkl]) for bcc Ti-Mo- M (M = Mg, Mn, Ni, Zr, Nb, and W) ternary solid solutions as a function of Mo and M contents. • The theoretical elastic moduli of Ti-Mo agree with the available calculated and experimental data. • Complex alloying effects are found in the lattice constants and elastic properties. • Mn, Ni, Nb, Zr, and W enhance both the Young's modulus and the stability of the β phase. • The single crystal Young's moduli of the Ti-Mo - based alloys show a strong elastic anisotropy. Using ab initio alloy theory, we systemically investigate the effect of alloying elements on the elastic properties of body-centered cubic (bcc) Ti 1− x − y Mo x M y (0.05 ≤ x ≤ 0.2; 0 ≤ y ≤ 0.4; M = Mg, Mn, Ni, Zr, Nb, and W) alloys. The theoretical single-crystal and polycrystalline elastic moduli of Ti 1− x Mo x (0.05 ≤ x ≤ 0.2) agree well with the available experimental values and previous theoretical data. The lattice parameters of Ti-Mo- M ternary alloys significantly increase (decrease) with increasing Mg and Zr (Mn and Ni) contents, while remain almost constant for Nb and W additions. It is found that Mg is a promising alloying element that could decrease the Young's modulus of bcc Ti-Mo alloys, but its content should be as small as possible since the stability of the β phase decreases with increasing Mg concentration. On the other hand, Mn, Ni, Nb, Zr, and W enhance the Young's modulus and the stability of the β phase. [ABSTRACT FROM AUTHOR]