Ni-base superalloys are unique high temperature materials, which display excellent high-temperature properties, such as strength, ductility, fracture toughness, fatigue resistance, creep resistance, and oxidation resistance.^[1,2] Increasing demands for applications of such materials, mostly used in turbine blades of aircraft jet engines and land-based power generators, drive continued development of these alloys to perform in extreme conditions. The superalloys are characterized by cubical γ′ precipitates (L12, ordered face-centered cubic (fcc), intermetallic compounds Ni₃Al) embedded in the γ matrix (disordered fcc, solid solution Ni).^[1] In order to strengthen the single-crystal superalloys, nearly 10 elements are usually doped to improve the high temperature mechanical properties and their lifetime. Transition alloying elements Ti, Co, Cr, Nb, Mo, Ru, Ta, Hf, W, and Re are added to enhance the high-temperature properties. The elements Al and Ta are enriched in the γ′ phase, while the elements Co, Cr, Mo, W, Re, and Ru are enriched in the γ matrix phase. Solid solution strengthening of the γ matrix is one key factor for improving the creep strength of Ni-base superalloys at high temperatures. Creep processes are largely controlled by the thermally activated motion of dislocations in the matrix phase, which requires dislocation climb at the γ/γ′ interfaces. Elemental diffusion strongly influences the dislocation motion, thus it is important to consider the interaction of the solute elements. Among these alloying elements, Re remarkably improves the strength and creep resistance. The content of Re is considered as one of the characteristics in different-generation single-crystal superalloys, and in the newly developed fourth- and fifth-generation single-crystal superalloys.

However, no widely convinced mechanism has been conferred for the strengthening effect of Re until now. It was proposed that Re is present in the form of clusters in γ and acts as efficient obstacles for dislocation motion. Based on the results of one-dimensional atom probe (1DAP), Re clustering postulate was first suggested by Blavette et al.^[3–5] Wanderka et al.^[6] further used atom probe tomography (APT) to study the superalloy CMSX-4 and also concluded the formation of Re cluster. Rüsing et al.^[7] used the more advanced three-dimensional atom probe (3DAP) and concluded that Re forms clusters with sizes of about 1 nm and mutual average distances of 20 nm. Mottura et al.^[8] drew a contrary conclusion by using APT method, which took into account the detection efficiency and positional scatter of the atoms, and a purpose-built algorithm was used to analyze the atom probe datasets. They showed that the fluctuation in the ladder diagrams may be due to the random variation in the solute atoms and no solute clusters were detected in the Ni–Re binary alloys or the CMSX-4 single-crystal superalloys.^[8,9] The experimental x-ray absorption fine structure (EXAFS) spectra were also compared to the simulated spectra, and it was shown that the experimental Ni–Re alloy does not display the double peaks in the spectra and Re will not form clusters.^[10]

In addition to these experimental studies, simulation methods were also used to check whether Re forms clusters or not in superalloys. The ab initio density functional theory simulations showed strong negative binding energies between Re–Re nearest neighbor pairs, and the repulsion energy for Re–Re pairs is rapidly reduced as the atoms are placed further apart.^[10,11] Zhu et al.^[12] used molecular dynamics (MD) simulations to calculate the binding energies of Re clusters in the Ni–Al–Re system and the results showed strong binding energies between Re atoms and indicated that Re tends to form clusters in the γ phase.

From a physical point of view, the co-segregation of Re with other alloying elements may be responsible for the strengthening effect of Re. The clusters built up by a couple of Re-solute atoms may play a major role in improving the creep strength of Re-containing superalloys.^[13] Though not much information is available with the synergistic effects of Re with other alloying elements, some studies do give some indications of the synergy. Smith^[14] studied a series of simple model alloys and showed that Cr strongly affects the partitioning of Re. And Re could not strengthen the superalloys without the addition of Cr. The following work of Chen et al.^[15] showed similar behavior of the effect of Cr on Re and some bright contrast regions in the γ phase near the γ/γ′ interface were found. Recently, Ge et al.^[16] studied the distribution of Re in a second-generation single-crystal superalloy DD6 using high-angle annular dark-field scanning transmission electron microscope (HAADF–STEM) with the energy-dispersive spectroscopy (EDS) mapping. They suggested that Re and W may form clusters with sizes of about 1–2 nm in the γ phase close to the γ/γ′ interface in crept superalloys, while no such clusters could be found near the γ/γ′ interface before creep test. Co-segregation of Re with Co and Cr was observed at the tips of protrusions at the γ/γ′ interface. As a result of coupling between the interfacial dislocation motion and the co-segregation of Re with Co and Cr, the creep rate in the steady creep regime was slowed down.^[17] Yu et al.^[18] used transmission electron microscopy and first-principles calculations to show that Re and Ru have a strong interaction, which results in the repartitioning of Re to the γ phase. Huang et al.^[19] demonstrated that correlated element–element binding energies play a key role in the observed Ni retention-excesses at the γ-Ni/γ′-Ni₃Al interfaces during aging and evolution of the precipitates. In order to obtain deeper insight into the electronic mechanism of the element distribution, further theoretical calculations are needed to understand the interaction of Re element with other alloying elements at atomic-scale.

In the present work, via studying the binding energy of two doping elements, the site preferences of two alloying elements X–Y in γ-Ni of Ni-based superalloy are systematically studied using first-principles calculations with and without spin-polarization. The doping elements X and Y are chosen from the 27 kinds of 3d, 4d, 5d group transition metals (Sc, Ti, V, Cr, Mn, Fe, Co, Cu, Zn, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, Hf, Ta, W, Re, Os, Ir, Pt, Au) and Al. We find that the spin-polarized calculations for Re–Re, Re–Ru, Re–Cr, Ru–Cr show a strong chemical binding affinity between the solute elements and more consistent with the experimental results. The binding energies of pairs between the 28 elements have an obvious periodicity and are closely related the electronic configurations of the elements. When d-electrons of an element is close to the half full-shell state, the doping element possesses an attractive binding energy, reflecting the effect of the Hund’s rule. Meanwhile, the combination of early transition metals (Sc, Ti, V, Y, Zr, Nb, Hf, Ta) has a repulsive interaction in γ-Ni.

In the previous work, Mottura et al.^[11] concluded that magnetism is expected to have little effect on the binding energies of Re–Re, Ta–Ta, W–W pairs in the fcc-Ni lattice. To verify this conclusion, the binding energies between Re–Re, W–W, and Ta–Ta pairs are firstly calculated and the data are given in Table 2 and plotted in Fig. 2. The result based on non-spin-polarized calculations is consistent with that given by Mottura et al.^[11] However, our results show that spin-polarized calculations and non-spin-polarized calculations will result in different conclusions for the Re–Re pairs. Therefore, the effect of magnetism on the binding energies of a X–Y nearest-neighbor pair is studied by comparing the spin-polarized and non-spin-polarized calculations.

Fig. 2.

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	Fig. 2. (color online) The binding energies between Re–Re, W–W, and Ta–Ta pairs in a 3 × 3 × 3 fcc Ni supercell containing 108 atoms as a function of X–X separate distance: (a) spin-polarized calculations, (b) non-spin-polarized calculations.

Table 2.

Table 2.

Table 2. The binding energies (in eV) between Re–Re, W–W, and Ta–Ta pairs at different distances in single-crystal nickel-based superalloys. . Re–Re W–W Ta–Ta ith NN non-spin spin non-spin[11] non-spin spin non-spin[11] non-spin spin non-spin[11] 1st NN 0.434 –0.190 0.425 0.615 0.529 0.598 0.685 0.323 0.679 2nd NN 0.067 –0.260 0.095 0.007 –0.197 0.013 –0.069 –0.191 –0.064 3rd NN 0.034 –0.130 0.046 0.040 –0.066 0.029 0.009 –0.076 0.010 4th NN 0.088 –0.052 0.109 0.075 –0.013 0.067 0.023 –0.004 0.030 5th NN –0.011 –0.155 0.012 –0.015 –0.089 0.002 –0.047 –0.101 –0.042 6th NN –0.005 –0.133 0.015 0.005 –0.014 0.011 –0.004 –0.036 0.003 7th NN 0.002 –0.041 – 0.005 –0.017 – –0.008 –0.035 – 9th NN –0.033 –0.129 – –0.087 –0.114 – –0.105 –0.142 – 11th NN –0.001 –0.077 – 0.006 –0.020 – –0.005 –0.037 – Note: the formula for calculating the binding energy in Ref. [11] has a minus sign compared with the formula used in this paper.

Table 2. The binding energies (in eV) between Re–Re, W–W, and Ta–Ta pairs at different distances in single-crystal nickel-based superalloys. .

As shown in Table 1, the results show positive binding energies for Re–Re 1^st NN and 2^nd NN pairs by non-spin-polarized calculations, but negative binding energies by spin-polarized calculations, and the 2^nd NN pair is energetically more stable. The 2^nd NN rhenium pair is energetically stable at 0 K, suggesting that rhenium clusters would not form in the γ phase of single-crystal nickel-based superalloys. Therefore, it is expected that Re will approach a random distribution when alloyed in the γ phase with the concentration of 1.0–2.0 at.% typically used in these materials. Meanwhile, the results of W–W pairs and Ta–Ta pairs are almost the same in both spin-polarized and no-spin-polarized calculations. The repulsion energies observed for the Ta–Ta and W–W nearest neighbor pairs are rapidly reduced as the atoms are placed further apart, suggesting that the interactions between these solute atoms are strongly localized. It is possible to form short-range ordered structures and a Re–Re periodic chain along the ⟨100⟩ direction. Luo et al.^[28] reported that heavy-atom columns align in the ⟨100⟩ and ⟨110⟩ directions and form heavy-atom stripes of about 1–2 nm in length. Heavy-atom ordering resembles the results reported by Maisel et al.,^[29] which suggested that Re prefers to align in the ⟨420⟩ direction in the form of Ni₄Re. The Ni₄Re compound was shown to be stable by quantum mechanical high-throughput calculations at 0 K. Owing to the high diffusion activation energy of the heavy atoms and the stronger bonding strength of the refractory element with Ni, this short-range ordering will help to improve the mechanical performance.

The binding energies between X(Re, Ru)–Y(Re, W, Ta, Ru, Mo, Co, Cr, Al) pairs in Ni₁₀₆XY system vs. X–Y separate distance are shown in Fig. 3. The binding energies essentially decay to zero when the distances between X–Y pairs increase. The interactions are only significant through the 3^rd NN, which is not surprising as in a metallic alloy, the charges on the ion cores are screened strongly by the conduction free electrons.

Fig. 3.

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	Fig. 3. The binding energies between X(Re, Ru)–Y(Re, W, Ta, Ru, Mo, Co, Cr, Al) pairs in Ni106XY system vs. X–Y separate distance: (a), (b) spin-polarized calculation; (c), (d) non-spin-polarized calculation. Ranges from the 1st NN to 3rd NN distances of pure Ni are indicated by the dashed vertical lines.

We next discuss the effect of magnetism on the binding energies of an X–Y 1^st NN pair by comparing the spin-polarized and non-spin-polarized calculations. The results are plotted in Fig. 4. It is shown that the binding energy strongly depends on the spin polarization. For example, under the spin-polarized calculations, the pairs of Re–Re, Re–Cr, Ru–Cr, Re–Ru, Ru–Mo, Ru–W, Re–Mo, and Mo–Cr at 1^st NN have big attractive binding energies; while the pairs of Co–Cr, Co–Ru, Co–Mo, Co–W, Co–Re at 1^st NN have a weak chemical repulsive tendency. The results of the non-spin polarization calculation are opposite to those of the spin polarization calculation. Under non-spin polarization conditions, most of pairs have big repulsive binding energies and are strongly repulsive in the first neighbor. Short-range order does not form, and it is randomly distributed over long-range. It is worth noted the results of Re–Cr, Ru–Cr, W–Cr, Ta–Cr, Re–Al, W–Al, Ta–Al, and Ru–Al with spin polarization are consistent with the experimental and calculation results reported by Huang et al.^[19] The result of Re–Ru with spin polarization is consistent with the previous calculation result and experimental results reported by our research group.^[18] Based on the existing experimental results, we believe that the results of spin polarized calcuation are more convincing. Therefore the following discussion are mainly based on the results of spin polarized calcuation, meanwhile the data of nonspin polarized calcuation are given to understand the magnetic effect.

Fig. 4.

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	Fig. 4. The binding energies between X(Re, W, Ta, Ru, Mo, Co, Cr, Al)–Y(Re, W, Ta, Ru, Mo, Co, Cr, Al) pairs in Ni106XY system at 1st NN distance. The blue histogram represents the result of spin-polarized calculation, and the red histogram represents the result of non-spin-polarized calculation.

The detailed binding energies between X (Re, W, Ta, Ru, Mo, Co, Cr, Al) and Y (Re, W, Ta, Ru, Mo, Co, Cr, Al) pairs with 1^st NN, 2^nd NN, and 11^th NN distances are listed in Table 3. In the spin-polarized calculations, for the binding energies of pairs of Al–X, Ta–X, and W–X, almost all the solute elements exhibit alternating positive and negative binding energies from the 1^st to 2^nd NN sites, which indicates that all refractory elements are substituting on the Al-sublattice and form an L12 ordered structure with Ni, except for Co. The pairs of Re–Ru, Re–Cr, Ru–Cr, Re–Re, Mo–Cr, Re–Mo, and Ru–Mo have attractive binding energies at 1^st and 2^nd NN distances. The results show that Re–Re, Re–Cr, Mo–Cr, and Re–Mo at 2^nd NN are more stable, Re–Ru and Ru–Cr at 1^st NN are more stable, consistent with the previous experimental and calculation results.^[18,19] The binding energies of Co-X pairs are close to 0, indicating that cobalt and other elements interact relatively weakly. We believe that cobalt and other elements do not have preference of occupancy and are randomly distributed. From the above analysis of the binding energies, the metals can be divided into three classes: (i) Al, Ta, and W exhibit repulsive binding energies with second elements at the 1^st NN distance, but they have attractive binding energies with the second element at the 2^nd NN distance; (ii) Re, Mo, Cr, and Ru have an affinity with each other and attractive binding energies at 1^st NN and 2^nd NN distances; and (iii) Co interacts with other elements relatively weakly.

Table 3.

Table 3.

Table 3. The binding energies (in eV) between X(Re, W, Ta, Ru, Mo, Co, Cr, Al)–Y(Re, W, Ta, Ru, Mo, Co, Cr, Al) pairs in Ni106XY system at 1st NN, 2nd NN, and 11th NN distances. . X–Y system Spin polarized calculation Non-spin polarized calculation 1st NN 2nd NN 11th NN site preference 1st NN 2nd NN 11th NN site preference Re–Ru –0.119, –0.15[18] –0.056, –0.10[18] –0.027 1st NN 0.164 0.039 –0.007 repulsive random Ru–Cr –0.171, –0.211[19] 0.023, –0.172[19] –0.001 1st NN 0.070 0.020 0.001 repulsive random Re–Re –0.190 –0.260 –0.077 2nd NN, 1st NN 0.434 0.067 –0.001 repulsive Re–Cr –0.140, –0.242[19] –0.216, –0.154[19] –0.056 2nd NN, 1st NN 0.311 0.027 0.009 repulsive random Re–Mo –0.028 –0.255 –0.046 2nd NN 0.484 0.024 0.002 repulsive random Mo–Cr –0.013 –0.203 –0.047 2nd NN 0.365 0.018 0.007 repulsive Ru–Ta 0.035 –0.115 –0.017 2nd NN 0.265 –0.020 –0.013 repulsive random Ru–Mo –0.085 –0.090 –0.022 random 0.191 0.009 –0.006 repulsive Ru–W –0.053 –0.091 –0.028 random 0.224 0.017 –0.004 repulsive Co–Cr 0.071 –0.024 –0.011 random –0.044 –0.003 –0.001 random Co–Ru 0.015 –0.010 –0.010 random –0.046 0.013 0.001 random Co–Re 0.055 0.000 –0.008 random –0.032 0.013 0.005 random Co–Mo 0.076 –0.002 –0.005 random –0.003 0.010 0.003 random Co–Co –0.011 –0.015 –0.008 random –0.028 0.002 0.003 random Co–W 0.080 –0.003 –0.014 random 0.006 0.011 0.002 random Co–Al 0.080 –0.010 –0.008 random 0.047 0.001 0.001 random Co–Ta 0.109 –0.005 –0.009 random 0.044 0.003 –0.006 random Ru–Ru 0.054 0.050 0.006 random 0.042 0.023 –0.010 random Al–Al 0.226 –0.027 –0.006 random 0.294 0.023 –0.001 repulsive Cr–Cr –0.003 –0.091 –0.024 2nd NN 0.241 0.002 0.011 repulsive Ru–Al 0.061, 0.372[19] –0.056, –0.134[19] –0.018 2nd NN 0.203 –0.005 –0.007 repulsive Cr–Al 0.091 –0.069 –0.019 2nd NN 0.266 0.013 –0.001 repulsive Re–Al 0.200, 0.281[19] –0.099, –0.110[19] –0.017 2nd NN 0.389 0.003 0.006 repulsive Mo–Al 0.242 –0.080 –0.010 2nd NN 0.369 0.008 0.004 repulsive W–Al 0.284, 0.194[19] –0.086, –0.152[19] –0.013 2nd NN 0.420 0.008 0.006 repulsive Ta–Al 0.323, 0.156[19] –0.074, –0.271[19] –0.013 2nd NN 0.425 0.003 0.002 repulsive W–Cr 0.039, 0.184[19] –0.203, –0.186[19] –0.043 2nd NN 0.399 0.023 0.005 repulsive Re–W 0.055 –0.237 –0.040 2nd NN 0.528 0.034 0.005 repulsive Ta–Cr 0.150, 0.292[19] –0.184, –0.114[19] –0.035 2nd NN 0.428 0.013 0.001 repulsive Mo–Mo 0.148 –0.226 –0.026 2nd NN 0.521 –0.004 0.003 repulsive Re–Ta 0.234 –0.226 –0.029 2nd NN 0.565 –0.014 –0.005 Repulsive W–Mo 0.242 –0.208 –0.031 2nd NN 0.571 0.000 0.007 repulsive W–W 0.323 –0.191 –0.037 2nd NN 0.615 0.007 0.006 repulsive Ta–Mo 0.388 –0.192 –0.020 2nd NN 0.593 –0.037 –0.004 2nd NN W–Ta 0.446 –0.192 –0.026 2nd NN 0.647 –0.033 0.000 2nd NN Ta–Ta 0.529 –0.197 –0.020 2nd NN 0.685 –0.069 –0.005 2nd NN

Table 3. The binding energies (in eV) between X(Re, W, Ta, Ru, Mo, Co, Cr, Al)–Y(Re, W, Ta, Ru, Mo, Co, Cr, Al) pairs in Ni106XY system at 1st NN, 2nd NN, and 11th NN distances. .

Figure 5 shows the binding energies between Re and the other 28 elements. It is not difficult to find that the binding energy has an obvious periodicity, and different groups show different rules. In the spin-polarized calculations (see Fig. 5(a)), the 1^st NN binding energy of Re–X exhibits a V-shaped curve, the lowest binding energy is located at half-full d = 5 and the value at the valley is less than 0. We believe that Re and other elements almost all have the 1^st NN repulsive and 2^nd NN attracting interaction energies, except that the pairs of Re–Re, Re–Cr, Re–Tc, Re–Ru, and Re–Os are attractive both at the 1^st NN and 2^nd NN. The binding energise of Re–Ru and Re–Os at 1^st NN are lower than those at 2^nd NN. It is possible to have short-range ordered structures in the ⟨110⟩ and ⟨100⟩ directions. Forming a Re-X periodic chain along the ⟨110⟩ direction may reflect the strengthening effect of Re. On the other hand, in the non-spin polarized calculation shown in Fig. 5(b), the 1^st NN binding energy also exhibits a V-shaped curve, the lowest binding energy is located at d = 7 and the value at the valley is greater than 0, except for element Co. The binding energies at the second nearest neighbor are almost greater than 0, except that some early transiton metal elements (Sc, Ti, Y, Zr, Hf ) at the 2^nd NN have a weak attraction. The binding energies of Re–Co at the 1^st NN and 2^nd NN distances all approach 0, indicating that there is a random distribution between Re and Co.

Fig. 5.

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	Fig. 5. The binding energies calculated with spinpolarization (a) and without spinpolarization (b) between X(Re)–Y(3d, 4d, 5d transition metal and Al) pairs in Ni106XY system at 1st and 2nd distances.

In order to systematically study the interactions between elements in the periodic table and to explore the rules from the data, we calculate the binding energies between combinations of the 28 elements. The binding energies of X–Y pairs in Ni₁₀₆XY system at 1^st NN by spin polarized calculation and non-spin polarized calculation are shown in Figs. 6 and 7, respectively.

Fig. 6.

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	Fig. 6. The binding energies between X(3d, 4d, 5d transition metals and Al)–Y(3d, 4d, 5d transition metals and Al) pairs in Ni106XY system at 1st NN distance by spin polarized calculation.

Fig. 7.

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	Fig. 7. The binding energies between X(3d, 4d, 5d transition metals and Al)–Y(3d, 4d, 5d transition metals and Al) pairs in Ni106XY system at 1st NN distance by non-spin polarized calculation.

In Figs. 6 and 7, the circle represents the positive binding energy, the triangle represents the negative binding energy, the box represents the binding energy close to zero. The deeper red color represents the greater binding energy, the darker blue color represents the more negative binding energy, the lighter color indicates that the binding energy is closer to zero. From Fig. 6, it can be found that the binding energies between the two elements have an obvious periodicity. The binding energy is closely related the electronic configuration of the elements. The behaviors of different elemental periods are not the same, the properties of 4d and 5d group elements are similar. For the early transition elements (Sc, Ti, V, Y, Zr, Nb, Hf, Ta), the sum of the d-electrons of the two element combinations is less than 6 and it exhibits repulsive effects. From the map, it is found that the triangular area forms a band, especially when the d electronic shell is fully filled (X_d + Y_d = 9, 10, 11), such as Re+Cr, Re+Re, Re+Ru, Ru+Cr, etc. Its magnitude will decrease rapidly with decreasing overlap of the d shells of the adjacent atoms. Its magnitude will therefore be smallest for those elements with X_d + Y_d nearly fully filled: (i) X_d + Y_d = 2, 3, 4, 5, 6, 7, 8, repulsive binding energies for combinations of early transition metals; (ii) X_d + Y_d = 9, 10, 11, attractive binding energies when the d-electrons of the element close to half full-shell state possess attractive binding energies. These results reflect the effect of the Hund’s rule.

This law may be related to the spin coupling between the incomplete d shells and the conduction electrons which leads to a tendency for a ferromagnetic alignment of the d electrons. Specific deep-level physics mechanism will be in-depth studied in future research. Additionally, further work studying the effects of alloy elements from electronic structure will always be of great importance.

The site preferences of two alloying elements X–Y in γ-Ni of Ni-based superalloy are systematically studied using first-principles calculations with and without spin-polarization. The doping elements X and Y are chosen from the 27 kinds of 3d, 4d, 5d group transition metals (Sc, Ti, V, Cr, Mn, Fe, Co, Cu, Zn, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, Hf, Ta, W, Re, Os, Ir, Pt, Au) and Al. We find that the spin-polarized calculations for Re–Re, Re–Ru, Re–Cr, Ru–Cr show a strong chemical binding affinity between the solute elements and more consistent with the experimental results. The binding energies of pairs between the 28 elements have an obvious periodicity and are closely related the electronic configuration of the elements. When the d-electrons of the element is close to the half full-shell state, two alloying elements possess attractive binding energies, this result reflects the effect of the Hund’s rule. The combination of early transition metals (Sc, Ti, V, Y, Zr, Nb, Hf, Ta) have a repulsive interaction in γ-Ni. These results offer insights into the role of alloying elements in strengthening mechanism of superalloy.