As a key strengthening phase of commercial Ni-based single-crystal superalloys, the ordered γ′-Ni₃Al precipitate phase is of great technological importance owing to its excellent elevated-temperature creep and oxidation resistance as well as the anomalous temperature dependence of the yield stress.^[1–5] The stoichiometric γ′-Ni₃Al L1₂ structure with space group consists of two sublattices (a face-centered α sublattice occupied by Ni and a cube-corner β sublattice occupied by Al). The mechanical strength of γ′-Ni₃Al can be improved by adding certain ternary alloying elements (e.g., Re, Ta, Ru, W, Ir, and Co).^[6–11] Various theoretical studies have focused on the site preference of substitutional ternary elements in γ′-Ni₃Al and the effect of these alloying elements on the elastic properties of the γ′ phase at certain Al or Ni sublattice positions. However, the site occupancy behavior of many alloying elements is dependent on the temperature and composition^[12–14] owing to the effect of entropy. Consequently, a better understanding of the effect of ternary alloying elements with different sites occupancy on the mechanical properties of γ′-Ni₃Al is fundamental to the practical application of Ni-based single-crystal superalloys.

Various experimental methods can be used to explore the site occupancy of alloying elements in Ni₃Al, including x-ray diffraction, transmission electron microscopy, and atom probe methods.^[15–21] However, these experimental studies are costly, and only some transition-metal (TM) elements are considered. Theoretical research on the site preference of alloying elements can make up for the deficiencies of current experimental research methods. Early theoretical investigation was mainly based on the phenomenological Ising-type models.^[22–24] For example, Wu et al.^[23] studied the site preference in ordered L1₂-Ni₃Al and observed that the temperature and alloy composition affect the occupancy behavior of alloying elements. Ruban and Skriver^[12,25] studied the site occupation of 3d, 4d, and 5d TM elements in the γ′ phase using the coherent potential approximation in conjunction with the linear muffin-tin orbitals method. The currently popular theoretical method to calculate the site substitution behavior in ordered compounds is first-principles density functional theory (DFT) calculations using the Wagner–Schottky model.^{[13,14,18,26–31]} By adopting this method, our previous calculations^[29] of the site occupancy of TM elements in Ni₃Al indicated that the site behaviors of Fe, Co, Cu, Zn, Ru, Rh, Ag, Cd, and Ir are strongly related to the alloy composition at T = 0 K.

Similar to elastic constants, the ideal strength (the minimum stress required to yield a defect-free crystal) of a solid material is a primary intrinsic parameter for understanding the mechanical behaviors of solid structural materials. However, unlike the extensive theoretical efforts devoted to investigating the effect of substitutional TM elements on the elastic properties of the γ′-Ni₃Al phase, the study of the TM-element dependence of the ideal strength of the γ′ phase is limited. For L1₂-Ni₃Al, the weakest uniaxial tensile direction is the 〈 110〉 direction, and the weakest shear slip system is the slip system.^[32–35] The effects of Re,^[32,33] W,^[33] Ta,^[33] Mo,^[33] and Co^[33] on the ideal uniaxial tensile strength of γ′-Ni₃Al have been studied using first-principles calculations with the alloying elements occupying the Al sublattice. In Ref. [32], the authors also studied the ideal shear strength of Ni₃Al with and without Re. By combining first-principles calculations and the quasi-harmonic approximation, the temperature-dependent ideal shear strength of the γ′ phase doped with alloying elements (Re, Ru, Mo, Cr, Co, W, and Ta) at the Al site was investigated by Wu and Wang.^[34] Chen et al. studied the site preference and ideal shear strength of γ′-Ni₃Al doped with single Re,^[36] single W,^[35] double Re,^[36] and Re–W^[35] using DFT calculations. In our previous study, we investigated the ideal shear strength of Ni₃Al–X (X = 3 d:Sc–Zn, 4d:Y–Cd, 5d:Hf–Au) with TM elements occupying the Al site and Ni₃Al–X (X = Ni, Pd, Pt, and Au) with X replacing the Ni atom for shear deformation. However, limited attention has been paid to the effect of Ru, Ir, and Fe on the ideal uniaxial tensile behavior of Ni-based superalloys even though these alloying elements play a crucial role in strengthening the mechanical properties of commercial Ni-based superalloys.^[9,37–39] Moreover, to the best of our knowledge, there have been no theoretical studies on the dependence of the tensile behavior on the site occupancy of ternary additions (e.g., Co, Ru, Ir, Rh, and Fe) in γ′-Ni₃Al.

In light of this background, the key purpose of the present research is to investigate the effect of Fe, Co, Cu, Zn, Ru, Rh, Ag, Cd, and Ir additions on the ideal shear and uniaxial tensile strengths with different site occupancies in γ′-Ni₃Al using ab initio calculations. In addition, the charge redistribution and the chemical bonding between the alloying elements and host atoms in the tensile and shear processes were analyzed to unveil the underlying electronic mechanism.

In this work, we studied tension in the [110] direction (the weakest uniaxial tensile orientation) and shear in the slip direction (the weakest shear slip system) for all substitutional γ′-Ni₃Al. Figure 1 shows the original model of a 32-atom 2× 2× 2 supercell used in the present study, where the (111) plane is denoted by the yellow shadowed area. To explore the dependence of mechanical properties on the site occupancy of TM elements in γ′-Ni₃Al, we adopted a single-impurity model based on the hypothesis of a dilute alloy.^[40] Herein, three types of non-equivalent sites in γ′-Ni₃Al, including an Al site and two Ni sites marked by Ni1 and Ni2 in γ′-Ni₃Al in Fig. 1, were substituted by alloying elements to investigate the effect on the shear and tensile behaviors.

Fig. 1.

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	Fig. 1. Schematic illustration of Ni3Al 32-atom 2× 2× 2 supercell. The Al and Ni atoms are denoted by purple and blue balls, respectively. The yellow shadow area represents the (111) slip plane.

By adopting Wagner–Schottky model,^[41] our previous study^[29] calculated the transfer energy (the energy of moving an X atom from a Ni sublattice to an Al sublattice) of 3d (Sc–Zn), 4d (Y–Cd), and 5d (Hf–Au) TM elements in γ′-Ni₃Al. The results show that the transfer energy of Fe, Co, Cu, Zn, Ru, Rh, Ag, Cd, and Ir is 0.22 eV, 0.74 eV, 0.90 eV, 0.42 eV, 0.04 eV, 0.93 eV, 0.86 eV, 0.28 eV, and 0.54 eV, respectively (shown in Table 3 of Ref. [29]). Since (the energy of the exchange antisite defect E^ant in γ′-Ni₃Al is determined to be 1.05 eV in our previous study,^[29]) the site behaviors of Fe, Co, Cu, Zn, Ru, Rh, Ag, Cd, and Ir are depending on the alloy composition. Hence, we focused on the dependence of mechanical properties on the site occupancy of Fe, Co, Cu, Zn, Ru, Rh, Ag, Cd, and Ir in Ni₃Al in the present research.

To simulate the shear and tensile processes, we adopted the standard approach described in Refs. [26,27]. For [110] uniaxial tension, we set the loading direction along the x axis. Hence, the deformed lattice vector R is given by R = R₀ D, where R₀ is the primitive non-deformed vector, and the tensile deformation matrix D_tension is

In the present study, the lattice vector was deformed from 0.0 to 0.18 and 0.28 in steps of 0.02 for the tensile and shear process, respectively. The structures of all the calculated systems were completely relaxed until reaching the force and energy criteria at each deformed strain. The stress–strain relationship was obtained from the calculated tensile and shear energies using Eqs. (3) and (4). As verified in our previous studies,^[33,44] the mechanical stability of the crystal lattice of γ′-Ni₃Al will be maintained until the first maximum point of the stress–strain curve during [110] tensile and (111)[] shear deformation. If there is no other instability mode (e.g., soft phonon instability or magnetic spin arrangement^[45–48]), the corresponding stress of this peak point, denoted by σ_FM and τ_FM respectively, for the tensile and shear process, is exactly the ideal strength of γ′-Ni₃Al under the specified loading. Herein, the corresponding strain of the peak point of the stress–strain curve is defined as the first maximum strain (expressed by ε_FM and γ_FM for tensile and shear deformation, respectively). In general, first-principles phonon calculations based on DFT are time consuming and were not performed in this work. Thus, our determined ideal strength may be overestimated as the other stability modes were not verified in the present work.

The DFT calculations were performed using the Vienna Ab Initio simulation package (VASP)^[49] with projector augmented-wave (PAW) pseudopotentials^[50] for all calculations. The exchange–correlation functional was described by the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA).^[51] A Monkhorst–Pack^[52] k-mesh of 11× 11× 11 was adopted in this work, and the plane-wave kinetic energy cutoff was set to 400 eV. The crystal structure was relaxed until the force converged to less than 0.01 eV/Å, and the energy convergence for electronic self-consistency was 10⁻⁵ eV.

Dominated by the behavior of chemical bonding, common slip, and nucleation of dislocations,^[53–55] the ideal strength of solid materials sets the upper bound of the strength achievable by a real material under certain loading conditions. Investigations of the ideal strength are fundamental to understand the behavior of fracture, failure,^[56–58] and creep^[59,60] for advanced materials as well as to determine the gap between the real strength of materials and their ideal strength.^[61] _{Ab initio} computational [110] tensile and shear tests were implemented in stoichiometric Ni₃Al, and the ideal strength and corresponding strains calculated according to Eqs. (3) and (4) are presented in Table 1. The maximum stress for the non-doped Ni₃Al (σ_FM = 7.20 GPa) was achieved at a strain of 8% for [110] tension, and the peak stress (τ_FM = 5.84 GPa) was achieved at a strain of 18% for shear. As observed in Table 1, we almost exactly reproduced the ideal strengths for γ′-Ni₃Al for both tensile and shear processes determined in earlier studies,^[32,34] except for the work of Chen et al.^[36]

Table 1.

Table 1.

Table 1. Calculated ideal strengths (in unit GPa) and corresponding strains of Ni3Al alloys doped with alloying element, along with other DFT predicted values. . System σFM/εFM of [110] tension τFM/γFM of (111)[] shear Non-doped 7.20/0.08 7.40/0.08a 5.84/0.18 5.80/0.18a 4.23/0.18c 5.768/0.18b 5.848/0.18d Substitutional sublattice Substitutional sublattice Al Ni1 Ni2 Al Ni1 Ni2 Fe 8.33/0.10 6.99/0.08 7.47/0.08 6.27/0.18 5.73/0.16 5.85/0.18 Co 7.38/0.08 7.11/0.08 7.55/0.08 5.81/0.18 5.75/0.16 5.89/0.18 Cu 6.40/0.08 7.78/0.10 7.16/0.08 5.37/0.18 6.11/0.18 5.66/0.18 Zn 6.76/0.08 8.49/0.10 6.91/0.08 5.55/0.18 6.49/0.18 5.47/0.18 Ru 8.80/0.10 6.35/0.08 7.63/0.08 6.54/0.18 5.44/0.16 5.89/0.18 Rh 7.36/0.08 6.72/0.08 7.64/0.08 5.76/0.18 5.51/0.16 5.89/0.18 Ag 6.12/0.08 7.41/0.10 6.86/0.08 5.13/0.16 5.89/0.18 5.46/0.18 Cd 6.48/0.08 8.01/0.10 6.63/0.08 5.39/0.18 6.22/0.18 5.30/0.18 Ir 8.27/0.10 7.01/0.08 7.80/0.08 6.28/0.18 5.71/0.16 6.02/0.18 a Ref. [32], b Ref. [34], c Ref. [36], d Ref. [44].

Table 1. Calculated ideal strengths (in unit GPa) and corresponding strains of Ni3Al alloys doped with alloying element, along with other DFT predicted values. .

The tensile and shear behavior of X_s-doped Ni₃Al, where X_s denotes the TM elements X (X = Fe, Co, Cu, Zn, Ru, Rh, Ag, Cd, and Ir) occupying the s site (s = Al, Ni1, and Ni2 sites), were systematically studied in this work. The calculated ideal strengths for all the substitutional systems are summarized in Table 1. It can be seen that the ideal strengths of γ′ phase doped with different TM elements and a certain X-doped Ni₃Al with the alloying element occupying different sublattices all show significant diversity.

When TM elements X occupied the Al site, according to Table 1, Fe, Ru, and Ir clearly improve σ_FM of the γ′ phase, with Ru resulting in the greatest enhancement, followed by Fe and Ir. The doping of Co and Rh will slightly increase the ideal tensile strength, whereas the addition of Cu, Zn, Ag, and Cd will weaken σ_FM of γ′-Ni₃Al. In terms of shear, the TM elements have a similar effect on the ideal shear strength when they occupy the Al site. The effect of the alloying elements on τ_FM is in the order: Ru > Ir > Fe > Co > Rh > Zn > Cd > Cu > Ag. Ru, Ir, and Fe clearly improve the ideal shear strength of γ′-Ni₃Al, whereas alloying elements with fully filled d shells such as Cu (3d¹⁰4s), Zn (3d¹⁰4s²), Ag (4d¹⁰5s), and Cd (4d¹⁰5s²) are expected to decrease τ_FM, with Ag exhibiting the strongest decreasing effect. Unlike the effect of Rh and Co on the ideal tensile strength, Rh and Co will slightly decrease the ideal shear strength of the γ′ phase.

Compared with Al-site defected alloys, Ni-site defected Ni₃Al results in a different scenario. There are two non-equivalent Ni sites in γ′-Ni₃Al: the Ni1 site of the [] crystal column composed of only Ni atoms and the Ni2 site of the [] crystal column composed of Ni and Al atoms (see Fig. 1). Hence, we evaluated the ab initio [110] tensile and shear deformations for both X_Ni1-doped and X_Ni2-doped alloys in the present study. As demonstrated in Table 1, if a certain alloying element X improves τ_FM and σ_FM when occupying the Ni1 site, this element will decrease τ_FM and σ_FM when occupying the Ni2 site, and vice versa. None of the studied elements could simultaneously increased the ideal strength of the γ′ phase for both Ni1-site and Ni2-site substitution. Elements with partially filled d orbitals, i.e., Fe (3d⁶4s²), Co (3d⁷4s²), Ru (4d⁷5s), Rh (4d⁸5s), and Ir (5d⁷6s²), resulted in a decrease of τ_FM and σ_FM when occupying the Ni1 site, whereas they enhanced the ideal tensile and shear strength of γ′ alloys when occupying Al site and Ni2 site. In contrast, the elements with fully filled d-shells (i.e., Cu, Zn, Ag, and Cd) possessed the opposite strengthening efficacy on the ideal strength of γ′-Ni₃Al.

To better illustrate the effect of the ternary alloying additions on the ideal strength of Ni₃Al alloys, the relations between Δσ_FM and Δτ_FM (both expressed in percent of the ideal strength of stoichiometric Ni₃Al) are plotted in Fig. 2, where Δσ_FM and Δτ_FM represent the difference in σ_FM and τ_FM, respectively, between Ni₃Al–X alloys and non-doped Ni₃Al (i.e., and ). Although the effects of different TM elements on the ideal strength of γ′-Ni₃Al varied, as described in Fig. 2, a linear relationship for Δσ_FM versus Δτ_FM can be observed when the TM elements occupy the same s site (i.e., Al-, Ni1-, or Ni2-site). The slope of the linear fit is 1.60, 1.58, and 1.33 for Al-site, Ni1-site, and Ni2-site defected alloys, respectively. Thus, it can be expected that the effect of alloying elements on the ideal tensile strength is overall greater than the effect on the ideal shear strength. Among all the calculated substitutional systems, Ni₃Al with Ru occupying the Al site exhibited the greatest σ_FM and τ_FM, whereas Ni₃Al with Ag occupying the Al site displayed the smallest σ_FM and τ_FM (see Fig. 2 and Table 1).

Fig. 2.

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	Fig. 2. Correlations between expressed in percent (%) and expressed in percent (%) for all calculated Ni3Al alloys. The black squares, red circles, and blue triangles represent systems with TM elements substituting the Al site, Ni1 site, and Ni2 site, respectively.

To obtain deeper insight into the underlying electronic mechanism responsible for the various efficiencies of ternary alloy element additions on the tensile and shear behavior, we investigated the charge redistribution induced by the substitutional impurities in γ′-Ni₃Al during the tensile and shear processes. Figures 3(a)–3(i) presents the charge density differences (Δρ) of the () plane containing a ternary alloying element for Ru_Al-doped and Zn_Al-doped Ni₃Al alloys (i.e., Ni₂₄Al₇Ru and Ni₂₄Al₇Zn) along with non-doped Ni₃Al at different loading strains.

Fig. 3.

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	Fig. 3. Charge density difference of () plane containing the alloying element for Ni24Al7Ru, Ni24Al7Zn, and stoichiometric Ni3Al at [(a)–(c)] a strain of 0%, [(d)–(f)] a tensile strain of 8%, and [(g)–(i)] a shear strain of 18%. Positive (negative) values [e/(a.u.)3] denote charge accumulation (depletion).

As observed in Figs. 3(a)–3(c), compared with Δρ of non-doped Ni₃Al, the electrons of Ni₂₄Al₇Ru are clearly accumulated around the Ru atom flowing from both Ru and its first nearest neighbor (FNN) host atoms (i.e., Ni atoms), and Δρ exhibits robust directionality (mainly appearing along the [110] direction) at zero strain. In contrast, Δρ of the Zn_Al-doped system shows completely different behavior: Zn doping does not enhance the electron accumulation, and the bonding between Zn and the host atoms shows obvious metallic character. Consequently, the substitution of Ru on the Al site enhances the mechanical properties of Ni₃Al alloys, whereas the doping of Zn on the Al site weakens the mechanical behavior, which is consistent with our previous calculations of σ_FM and τ_FM (see Table 1 and Fig. 2) and previous experimental results.^[9,37–39] Herein, it can be predicted that the bonding character of Fe–Ni and Ir–Ni bonds will be similar to that of the Ru–Ni bond, whereas the interactions of the fully filled d-shell elements (i.e., Cu, Ag, and Cd) and Ni atoms will be similar to those of Zn–Ni bonds. Indeed, Ru, Ir, and Fe are important elements in practical commercial Ni-based superalloys.^{[1–3,9,37–39]} As the strains increase under [110] tension, the distances between atoms along the loading orientation increase, whereas the bond length between atoms along the [001] direction (the perpendicular direction to the applied strain) decreases. Hence, the charge density difference of Ni₂₄Al₇Ru, Ni₂₄Al₇Zn, and stoichiometric Ni₃Al along the [110] direction are significantly weakened at a tensile strain of 8% (see Figs. 3(d)–3(f)).

Figures 3(g)–3(i) display Δρ of the () plane at a strain of γ = 18% during shear deformation for Ni₂₄Al₇Ru, Ni₂₄Al₇Zn, and non-doped Ni₃Al. With increasing shear strain, the bonding in the [] direction becomes weaker for all the calculated systems. As observed in Figs. 3(d)–3(i), Δρ in the Ru_Al-doped system exhibited stronger charge accumulation than non-doped Ni₃Al, and Δρ of the Zn_Al-doped system was the weakest. Moreover, the Ru_Al-doped system has the strongest covalent-like bonds (i.e., Ru–Ni bonds), followed by non-doped Ni₃Al, with the weakest bonds observed for the Zn_Al-doped system. Doping of Ru on the Al site in γ′-Ni₃Al can significantly strengthen the shear strength of the alloys, whereas substitution of Zn on the Al site weakens the shear strength of the system.

In terms of Ni-site defect alloys, the ideal tensile and shear strength are dependent on the ternary elements as well as the doping site. To reveal the strengthening mechanism, we also studied the variation of the bond lengths between the TM element X (X = Ru, Zn, and Ni) and its FNN host atoms in Ni₂₃Al₈Ru, Ni₂₃Al₈Zn, and stoichiometric Ni₃Al during the uniaxial tensile processes. At zero strain, the impurity atom X possesses 12 FNN atoms consisting of 4 Al atoms, 4 Ni1 atoms, and 4 Ni2 atoms (see insets of Fig. 4). With increasing loading strain, the bond length along the [110] tensile direction increases, whereas the bond length along the [101] orientation decreases for all the studied alloys. Consequently, the shortest X–Ni bonds are along the [101] direction during the tensile process for both X_Ni1-doped and X_Ni2-doped alloys. The bond lengths of X_s – Ni bonds (X = Ru and Zn, s = Ni1 and Ni2 site), denoted by L(X – Ni), along the [101] direction in X_s-doped Ni₃Al as a function of tensile strain are plotted in Fig. 4. The bond lengths of the Ni–Ni bond along the [101] direction of pure Ni₃Al are also plotted in Fig. 4 for comparison.

Fig. 4.

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Fig. 4. Variation of bond lengths between the TM element X (X = Ru, Zn, and Ni) and its FNN host atoms normalized with respect to their respective equilibrium values during [110] tensile tests for Ni23Al8Ru, Ni23Al8Zn, and stoichiometric Ni3Al. The left (right) inset shows the substitutional impurity atom X and its FNN atoms for XNi1-doped (XNi2-doped) alloys at zero strain. The purple, blue, and green balls represent Al, Ni, and X atoms, respectively. The red dashed lines in the insets mark the shortest X–Ni bond during the tensile loading.

As demonstrated in Fig. 4, with the application of tensile strain step by step in pure Ni₃Al, the bond length of the Ni–Ni bond, L(Ni–Ni), in the [101] direction decreases and contracts to 0.9898 times the original length at the peak strain of 8%. For Ru-doped Ni₃Al, the shrinkage of L(Ru–Ni) at the strain of 8% varies with the Ni site occupancy. Compared with non-doped Ni₃Al, the length of the Ru–Ni bond in X_Ni1-doped Ni₃Al decreases much more, whereas L(Ru–Ni) in X_Ni2-doped Ni₃Al decreases much less. It can be predicted that Ru doping can effectively improve (weaken) the deformation resistance and mechanical properties of Ni₃Al alloys when Ru occupies the Ni2 (Ni1) site, which is consistent with our previous calculations of ideal tensile strength (see Table 1 and Fig. 2). For Zn-doped systems, the opposite is true. The length of the Zn–Ni bond in the [101] orientation contracts to 0.9978 times and 0.9896 times the original at a strain of 8% for the Ni1-site and Ni2-site defect model, respectively. Consequently, Zn_Ni1-doped alloys exhibit a much higher σ_FM than pure Ni₃Al and Zn_Ni2-doped alloys exhibit a slightly smaller σ_FM than non-doped Ni₃Al, which agrees well with our calculated results in Table 1 and Fig. 2.

The effects of Fe, Co, Cu, Zn, Ru, Rh, Ag, Cd, and Ir on the ideal shear and uniaxial tensile behaviors with different site occupancy in γ′-Ni₃Al were studied in detail using ab initio total-energy methods. Our results indicate that the dependence of the ideal strength on the site preference of alloying element X in γ′-Ni₃Al is associated with the d-shell occupancy of X. Elements with partially filled d orbitals (i.e., Fe, Co, Ru, Rh, and Ir) enhance the ideal tensile strength when occupying the Al site in Ni₃Al, whereas elements with fully filled d orbitals (i.e., Cu, Zn, Ag, and Cd) lead to a decrease of τ_FM and σ_FM. Furthermore, the strengthening effect of Ru arises from the covalent-like bonds between impurities and host atoms according to analysis of the charge redistribution of Ni₃Al alloys, whereas the detrimental effect of Zn is caused by the metallic bonds between Zn and Ni. In terms of Ni-site defect Ni₃Al, elements with partially filled d shells are expected to weaken (improve) the ideal tensile and shear strength of the alloys when they occupy the Ni1 (Ni2) site, and the elements with fully filled d shells have the reverse effect. Moreover, the variations of the bond lengths between the TM element X (X = Ru, Zn, and Ni) and its FNN host atoms during uniaxial tensile tests indicate that Ru can effectively enhance (weaken) the deformation resistance and mechanical behaviors of the γ′ phase when occupying the Ni2 (Ni1) site, whereas the opposite is true for Zn. In addition, a linear relationship for Δσ_FM versus Δτ_FM was observed for the alloys with Al-, Ni1-, and Ni2-site defects in this study. Our findings provide insight into the reinforcement effects of ternary alloying elements on the ideal strength with different site preferences in γ′-Ni₃Al, which have not been previously studied using first-principles calculations.