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Ni3Al interface doped with Re, Ta and W
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First-principles study of Ni/Ni3Al interface doped with Re, Ta and W ⁎
⁎
Chuanxi Zhua, Tao Yua, , Chongyu Wanga,b, , Dianwu Wanga a b
Central Iron and Steel Research Institute, Beijing 100081, China Department of Physics, Tsinghua University, Beijing 100084, China
A R T I C LE I N FO
A B S T R A C T
Keywords: First-principles calculations Ni-based single crystal superalloys Griffith work Generalized stacking fault energy
The strengthening effects of (0 0 1) Ni/Ni3Al interface alloying with Re, Ta and W in Ni-based single crystal superalloys were studied using first-principles density functional theory. By placing alloying elements on (0 0 1) Ni/Ni3Al interface, the substitution formation energy, the Griffith work, the generalized stacking fault energies, the partial density of states and the charge density difference were all calculated. It is figured out that Re, Ta and W atoms prefer to substitute Al positions of γ ' phase. It is consistent with the experimental results. The strengthening effect of Re atom on (0 0 1) Ni/Ni3Al interface is stronger than that of Ta and W atoms addition. The order of strengthening effects of Re, Ta and W atoms from our calculations is consistent with relevant experimental observations. By analyzing the charge density difference and the partial density of states, our results show that the hybridization between Re atom and the nearest neighbor Ni atoms is the strongest, which is conducive to explore the essential reason for the different strengthening effects of alloying atoms at electronic level.
1. Introduction Ni-based single crystal(SC) superalloys have been widely used as aerospace turbine blades and industrial gas turbine blades. Ni-based SC superalloys consists of a large volume fraction of γ ' (L12 structure)precipitates and nickel solid solution γ (FCC structure)-matrix, where the γ ' precipitates are coherently embedded in the γ matrix [1,2]. Nibased SC superalloys contain a large number of alloying elements: Re, Ru, Co, Hf, Ti, Ta, W, Mo, Nb, Cr, etc [3]. The addition of alloying elements makes Ni-based SC superalloys have relatively excellent high temperature creep performance, oxidation resistance and thermal fatigue resistance [4]. It has been observed that the interface is widely present in the Ni-based SC superalloys [5], the structure of the interface plays a significant role in determining the mechanical properties in multiphase systems [6–8], and the properties of the superalloys are found to depend critically on the coherency of the γ / γ ' interface [9]. Therefore, it is very important to analyze the strengthening effects of alloying elements on the γ / γ ' interface, which is helpful for us to understand how alloying elements affect the mechanical properties of the Ni-based SC superalloys. Re, Ta and W are very important creep strengthening elements. The creep strength of nickel-based SC superalloys was reported to be improved by adding Re, Ta and W [10]. Experiment has shown that improvements in creep-strengthening occur in the following order: Re >
⁎
W > Ta > Cr > Co [11]. Zhu et al. [12] chose the six clusters of the interface for the first-principles discrete variational method(DVM) study, in which the interatomic energy, the charge distribution, the partial density of states and the electron density difference were all calculated. It is found that Re atom can strongly enhance the bonding strength with its nearest neighboring atoms. Chen and Peng et al. [13–15] evaluated the strengthening effect of alloying elements on the Ni/Ni3Al interface by introducing the bond order of the overlap population. By placing alloying elements on the Ni/Ni3Al interface, Wang et al. [16] calculated the Griffith energy of the doped system using firstprinciples plane-wave pseudopotential method, and found that the elements of both Re and Ru enhance the coherent strength of the interface. Gong et al. [17] investigated the strengthening effects induced by alloying elements (Re, Ru, Cr, Co, Mo, W, Ta) on Ni/Ni3Al interface in Ni-based SC superalloys in terms of Griffith work using first-principles density functional theory (DFT). The previous research of the γ / γ ' interface is mainly concerned with calculating the Griffith work to measure the strengthening effects of alloying elements on the γ / γ ' interface. Rice [18] pointed out that, when a dislocation is emitted, the atoms slip on the slip plane and the maximum energy(γus ) associated with this progress decides the resistance to the dislocation emission. The value of γus depends on the maximum value of the generalized stacking fault (GSF) energy [19]. Yu et al. [20] introduced the generalized stacking fault energies to investigate the strengthening effects
Corresponding authors at: Central Iron and Steel Research Institute, Beijing 100081, China. E-mail addresses: [email protected] (T. Yu), [email protected] (C. Wang).
https://doi.org/10.1016/j.commatsci.2020.109586 Received 14 November 2019; Received in revised form 16 January 2020; Accepted 2 February 2020 0927-0256/ © 2020 Elsevier B.V. All rights reserved.
Please cite this article as: Chuanxi Zhu, et al., Computational Materials Science, https://doi.org/10.1016/j.commatsci.2020.109586
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of alloying elements in γ ' -Ni3Al. So, in this paper we can use the generalized stacking fault energy to assess the strengthening effects of alloying elements on the γ / γ ' interface. Experiment has shown that improvements in creep-strengthening occur in the following order: Re > W > Ta [11]. The elements Ta, W and Re elements differ only by one electron, however their effects on the creep strength are different. This problem is still under discussion. To solve this problem, we should analyze how the doped atoms interact with neighboring atoms from the aspects of charge density difference and partial density of states. In this paper, the mechanical properties of the γ / γ ' interface doped with Re, Ta and W were investigated by the first-principles plane-wave pseudopotential method, which was implemented by the Vienna ab initio simulation package (VASP) [23,24]. Meanwhile, we calculated the substitution formation energy of the doped system and combined it with the atom probe tomography (APT) experiment to study the site preference of the doped atoms on the γ / γ ' interface. The Griffith work and the GSF energies were introduced to investigate the strengthening effect of the doped atoms on the γ / γ ' interface. Finally, we shall explore from the charge density difference and partial density of states (PDOS) why the Ta, W, and Re atoms differ only by one electron but have different strengthening effects on the γ / γ ' interface.
Table 1 Compositions of the ternary model alloys investigated in this work. Alloy (wt%)
W
Ta
Re
Al
Ni
M1 M2 M3
– 6.0 –
– – 5.6
5.0 – –
8.3 7.7 7.6
Bal. Bal. Bal.
The total-energy calculation presented in this paper was performed using the projector -augmented wave (PAW) method [21,22] as implemented in the Vienna ab initio simulation package (VASP) [23,24]. The Perdew-Burke-Ernzerhof (PBE) electronic exchange-correlation functional within the generalized gradient approximation (GGA) was taken into account [25]. The plane wave basis cutoff energy was 350 eV. An 11 × 11 × 1 Monkhorst-Pack k-mesh was converged for 96-atoms supercell model of the Ni/Ni3Al interface. A first-order Methfessel–Paxton smearing method [26] was used with a smearing width of 0.1 eV. The relaxation of the electronic degrees of freedom will be stopped if the total energy changes between two steps is smaller than 10−5 eV. The relaxation will stop if all the Hellmann–Feynman forces acting on the atoms are less than 0.02 eV/Å. 2.2. Experimental details
2. Computational model and experimental details The nominal compositions of the ternary model alloys investigated in this paper are listed in Table 1. The M1, M2 and M3 ternary model alloys were used to investigate the partitioning behavior of Re, Ta and W atoms in the γ and γ ' phases. Cast ingots of the ternary model alloy were prepared by vacuum induction melting. The master ingots were then directionally solidified to form 〈0 0 1〉-oriented single crystals using the Bridgman technique. The single crystal bars were homogenized at 1330℃for 20 h and subsequently aged at 870 °C for 32 h to produce a homogeneous distribution of γ ' phase. The partitioning behaviour of Re, Ta and W atoms in the ternary model alloys between the γ and theγ ' phases was measured by the atom probe tomography (APT). By cutting 0.5 mm × 0.5 mm × 15 mm rods from heat treated single crystal bars we got APT specimens, and then the standard two-step polishing procedure was performed [27]. The specimen temperature was ~50 K. Proximity histogram (proxigram) methodology [28] was adopted to determine the concentration information.
2.1. Computational model and methods In this paper, the 96-atoms supercell model of the Ni/Ni3Al interface with three-dimensional translational symmetry was studied. The atomic arrangements of our model are shown in Fig. 1. The lattice parameters of γ -Ni and γ ' -Ni3Al in the DFT calculations were 3.526 Å and 3.570 Å, respectively [16]. In order to determine the lattice parameter of the two-phase model, we relaxed the model with different lattice constants (3.526–3.570 Å), and the Hellmann-Feynman forces acting on the atoms are less than 0.02 eV/Å. When the energy of the system is the lowest, the lattice constant (3.545 Å) was selected as our model of the Ni/Ni3Al interface. The model took periodic boundary conditions in the X-axis and Y-axis directions, and added a vacuum layer in the Z-axis direction. After the energy convergence test, the thickness of the vacuum layer was selected as 12 Å. In this paper, the doped elements Ta, W and Re were considered. The substituting positions were: γ -Ni1, γ ' -Ni2, γ ' -Al, as shown in Fig. 1. The GSF energies as a function of the displacement u were calculated by shifting the γ ' part relative to the fixed γ part. For each rigid shift, the atomic positions were relaxed in the [0 0 1] direction and fixed in the other directions.
3. Results and discussion 3.1. Elemental partitioning between γ and γ ' phases The concentration profiles of the alloying atoms in the ternary model alloys are shown in Fig. 2. The APT experiment results show that Re, Ta and W atoms tend to partition to the γ ' phase in the ternary model alloys. From Fig. 2(a), the sum of the concentration for Al and Re in the γ ' phase is approximately 25 at.%, which indicates that Re atoms tend to occupy the Al positions in the γ ' phase. From Fig. 2(b) and (c), we also find that W and Ta atoms tend to occupy the Al positions in the γ ' phase. In order to study the site preference of alloying elements on the Ni/ Ni3Al interface, we calculated the substitution formation energy of the doped system. Alloying elements substitute for γ -Ni, γ '-Ni, γ ' -Al, respectively, as shown in Fig. 1. The substitution formation energy can be defined as follows [29,30]: '
'
'
Minγ Esub,NiorAl = (EMinγ + μ Ni/Al ) − (Eγ + μ M) Minγ Esub,Ni
(1)
= (EMinγ + μ Ni ) − (Eγ + μ M)
(2) Minγ '
where M represents Ta, W, and Re, and E is the total energy of the doped system for alloying elements substitute γ '-Ni and γ ' -Al, and EMinγ
Fig. 1. The supercell model of the Ni/Ni3Al interface with 96 atoms. 2
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Fig. 2. Concentration profile determined from APT experiment. (a) the concentration profile of Ni, Al and Re atoms in the ternary model alloy (M1); (b) the concentration profile of Ni, Al and Ta atoms in the ternary model alloy (M3); (c) the concentration profile of Ni, Al and W atoms in the ternary model alloy (M2).
the Al position, the substitution formation energy is the lowest, indicating that Re atom has a strong tendency to partition to the γ ' phase and preferentially substitutes at the Al positions in ternary Ni-Al-Re alloy. We observe the doping of Ta and W atoms and find that both Ta and W atoms also tend to occupy the Al positions in ternary Ni-Al-Ta alloy and in ternary Ni-Al-W alloy. The analysis results are consistent with the results given by the APT experiment.
Table 2 The substitution formation energies (eV) of alloying elements substituting for Ni1, Ni2 and Al, respectively. Substitution atom
Ni1 Ni2 Al
Alloying atom Re
Ta
W
0.39 1.46 −0.60
−0.73 1.37 −1.82
0.16 1.77 −0.96
3.2. The effect of doping on the cleavage properties Griffith energy is defined as the work required to separate an interface into two surfaces. It is worth noting that this energy is the critical energy required for crack propagation. The Griffith equation gives [31]: σF = (Wad Y/cπ )1/2 , where σF denotes the critical stress required for crack propagation, c denotes the length of a surface crack, and Y denotes Young's modulus. From the above formula, we can see that σF is proportional to Wad, so we can use Wad to express mechanical properties of alloying elements on the Ni/Ni3Al interface in SC superalloys. Wad can be calculated from the following equation [32]:
is the total energy of the doped system for alloying elements substitute γ -Ni. μ Ni/Al is the chemical potential of Ni or Al in the bulk system. μ Ni is the energy of a single Ni atom in a fcc unit cell. The chemical potential of bulk Ni3Al is equal to 3μ Ni + μAl , so μAl = μ Ni3Al − 3μ Ni . μ M is the chemical potential of alloying elements, where μRe is the energy of a single Re atom in an hcp unit cell, μTa is the energy of a single Ta atom in a bcc unit cell, and μW is the energy of a single W atom in a bcc unit cell. The calculation results are listed in Table 2. It can be seen from Table 2, when Re atom replaces the three doped positions (see in Fig. 1), respectively, the order of substitution forma− Ni1 − Ni2 − Al tion energy is ERe < ERe < ERe . When the Re atom replaces sub sub sub
Wad = (EslabA + EslabB − EA/B)/Ai '
(3)
EA / B is the energy of the γ / γ interface model.EslabA and EslabB are the 3
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Substitution atom
'
Re
Ta
W
4.53 4.91 5.26
4.21 4.99 5.01
4.41 5.04 5.23
4.35 Ni1 Ni2 Al
energies of the γ -Ni and γ '-Ni3Al, respectively, after the γ / γ ' interface model splits into two blocks (γ -Ni block and γ ' -Ni3Al block). Ai is the area of the (0 0 1) atomic layer in the Ni/Ni3Al interface model. The calculation results are listed in Table 3. From Table 3, we can see that the Griffith work increases significantly when Re atom replaces any of the three atoms. When Al atom is replaced by Re atom, the Griffith work is the greatest and the interface is the most stable in the Re-doped system. For the case of Ta atom replacing the atoms in the three positions, when Ta atom replaces the Ni1 atom the Griffith work is reduced relative to the Griffith work of the pure system. When the other two atoms are replaced, the Griffith work is increased. When the Ta atom replaces the Al atom, the Griffith work is the greatest and the interface is the most stable in the Ta-doped system. For the W atom, it has the similar trend as the Re atom in affecting the Griffith work when it substitutes for the three atoms, respectively. The doping of W atom at the three positions can increase the Griffith work relative to the pure system. When Al atom is replaced by the W atom, the Griffith work is the greatest and the interface is the most stable in the W-doped system. By comparing the Griffith work when the Re, Ta, and W atoms replace the Al position, respectively, we can get the rule: the Griffith work is the smallest when Ta atom replaces the Al position, the second smallest is given when the W atom replaces the Al position, and the maximum Griffith work is given when Re atom replaces the Al position. That is to say, when Re, Ta and W are doped at the γ / γ ' interface, respectively, the order of improvement of the mechanical properties of the γ / γ ' interface is Ta < W < Re. The effect on creep performance can also be demonstrated experimentally Ta < W < Re [11].
3.4. Electronic structure analysis 3.4.1. The charge density difference analysis The charge density difference is defined as [35]:
Δρ = [ρ (X − doped) − ρfree (X − doped)] − [ρ (clean) − ρfree (clean)] (4) where X is the substitution elements Re, Ta and W; ρ (X − doped) and ρ (clean) are the charge densities of the X-doped system and the pure system, respectively; ρfree (X − doped) and ρfree (clean) are the charge densities of the free atoms of the X-doped system and the pure system, respectively. From the above analysis in Sections 3.1–3.3, we can conclude that Re, Ta and W atoms tend to occupy the Al positions, and the mechanical properties of the Ni/Ni3Al interface increases the most when they occupy the Al positions. Thus, when we analyze the electronic structure of the doped system we only analyze the situation that the doped atoms occupy the Al positions. In Fig. 4, we know that when Re atom replaces the Al atom on the Ni/Ni3Al interface, electrons accumulate between the Re atom and its NN Ni atoms. The interaction between the Re atom and its NN Ni atoms is enhanced, indicating that the interaction between the Re atom and its NN Ni atoms is stronger than the interaction between the Al atom and its NN Ni atoms. We also find this rule when the Ta and the W atoms replace the Al atom on the Ni/Ni3Al interface. It shows that Re, Ta, W can enhance the stability of the interface when replacing the Al positions, and have the function of strengthening the interface. Looking closely at the Figs. 4 and 5, we can find that although there are 12 NN Ni atoms around the Re atom, the aggregation of electrons between the Re atom and different Ni atoms is different. Electrons aggregate less between Re atom and Ni54 and Ni53 atoms, however more between Re atom and Ni43, Ni44, Ni45, Ni46, Ni49, Ni50, Ni51 and Ni52 atoms (Ni43, Ni44, Ni45 and Ni46 are at equivalent positions, Ni49, Ni50, Ni51 and Ni52 are at equivalent positions, so we will take Ni43 and Ni49 as our representative). The agglomeration of electrons is localized, and the bonding between Re atom and the NN Ni atoms are directional. Because electrons aggregate more between the doped atoms and Ni43 and Ni49 compared with between Al atom and Ni43 and
3.3. The effect of doping on the slip properties It has been experimentally confirmed that there are a large number of [1 1 0](0 0 1) dislocations on the γ / γ ' interface [33]. Wang et al. also confirmed that the slip direction on the γ / γ ' interface is [1 1 0] direction [16]. So we chose the [1 1 0] direction as the slip direction on the γ / γ ' interface. In order to nucleate a dislocation with Burgers vector b, it is necessary to overcome the critical energy. Rice [18] called this energyunstable stacking fault energy, γus . γus is defined as the maximum of the GSF energy [19,34], which characterizes the resistance to dislocation nucleation. The energy cost by relative displacement of γ -Ni and γ ' -Ni3Al was taken to achieve the GSF energy curve. The γGSF can be defined as: γGSF = (E u − E0)/s , where E0 and E u are the energies of the supercell before and after slip deformation, and s is the area of the faulted region. The calculation results are shown in Fig. 3. From Fig. 3(a), the values of γus are 1.108, 1.089, 1.133, 1.250 J/m2 '
'
W γ − Ni2 , W γ − Al substitutions, respectively. We also find the same rule in Fig. 3(b) and (c), when Ta atom and W atom replace γ ' - Al the improvement of the mechanical properties is the most obvious. By comparing the γus of replacing Al position with Re, Ta and W atoms, we can draw the conclusion that the γus of replacing Al position with Ta atom is the smallest, followed by W atom, and the γus of replacing Al position with Re atom is the largest. That is to say, when Re, Ta and W are doped at the γ / γ ' interface, respectively, the order of improving the mechanical properties of the γ / γ ' interface is Ta < W < Re, which is consistent with the calculation results of Griffith work and the experimental results [11]. In order to study the effect of doping on local structure, we calculated the average distance between the doped atom and its nearest neighboring (NN) Ni atoms. Because Re, Ta and W atoms tend to occupy the Al positions and the mechanical properties of the Ni/Ni3Al interface increases the most when they occupy the Al positions, so we only analyze the situation that the doped atoms occupy the Al positions. The calculation results are listed in Table 4. From Table 4, we can see that the average distance between the Re atom and its NN Ni atoms is the smallest, and the average distance between the Ta atom and its NN Ni atoms is the biggest. It is indicated that the coherent strength of interface is the strongest when Re atom occupies the Al positions, and the coherent strength of interface is the weakest when Ta atom occupies the Al positions.
Alloying atom Pure
'
1.184 J/m2 for Taγ − Ni1, Taγ − Ni2 , Taγ − Al substitutions, respectively. From Fig. 3(c), the values of γus are 1.101, 1.152, 1.246 J/m2 for W γ − Ni1,
Table 3 Griffith work (J/m2) of alloying elements substituting for Ni1, Ni2, Al (see in Fig. 1), respectively.
'
for the pure, Reγ − Ni1, Re γ − Ni2 , and Re γ − Al substitutions, respectively. We can see that the γus decreases when Re atom substitutes for γ -Ni1 compared to the pure system, and increases when Re atom substitutes for γ ' -Ni2 and γ ' -Al, respectively. This shows that the stability and the mechanical properties of the interface system are improved when Re atom substitutes for γ ' -Ni2 and γ ' -Al. Further observation shows that the improvement of mechanical properties is the most obvious when Re atom replaces γ '-Al. From Fig. 3(b), the values of γus are 1.069, 1.171, 4
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Fig. 3. The generalized stacking fault energy γGSF curve verse the slipping displacement (u/b) of γ / γ ' interface doped with Re, Ta and W, and b is a Burgers vector. (a) 〈1 1 0〉(0 0 1) slip for the pure and Re substitution interface; (b) 〈1 1 0〉(0 0 1) slip for the pure and Ta substitution interface; (c) 〈1 1 0〉(0 0 1) slip for the pure and W substitution.
stacking fault energy of the Ni/Ni3Al interface are significantly increased and the stability of interface is improved. We observe the Fig. 5 and further analyze the difference of interaction between Re, Ta, W atoms and their NN Ni atoms. From the point of electron transfer, Re atom and its NN Ni atoms lose more electrons, more electrons are transferred to the space between the Re atom and its NN Ni atoms. W atom and its NN Ni atoms lose less electrons relative to Re atom and its NN Ni atoms. Ta atoms and its NN Ni atoms lose the least electrons. The order of electron aggregation between the doped
Table 4 The average distance (Å) between the doped atom and its NN Ni atoms. Re
Ta
W
2.513
2.545
2.522
Ni49, the bonding ability and interaction between the doped atoms and Ni43 and Ni49 are improved significantly. Therefore, when Re, Ta and W atoms replace Al positions, respectively, the Griffith energy and
Fig. 4. The charge density difference of the γ / γ ' interface doped with Re, Ta and W. (a) Re-Al model; (b) Ta-Al model; (c) W-Al model. The isosurfaces is 0.0045 e/Å3. The yellow and blue regions represent charge accumulation and charge loss, respectively. 5
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Fig. 5. Charge density contour plots of γ / γ ' interface doped with Re, Ta and W on the (0 0 1) plane and the (0 1 0) plane. (a) Charge density contour plot on the (0 0 1) plane by the addition of Re atom; (b) Charge density contour plot on the (0 1 0) plane by the addition of Re atom; (c) Charge density contour plot on the (0 0 1) plane by the addition of Ta atom; (d) Charge density contour plot on the (0 1 0) plane by the addition of Ta atom; (e) Charge density contour plot on the (0 0 1) plane by the addition of W atom; (f) Charge density contour plot on the (0 1 0) plane by the addition of W atom. The contour spacing is 0.002 e/a.u.3. Solid lines and dashed lines correspond to the gain and the loss of electrons, respectively.
without Re, Ta and W, respectively. In Fig. 6, Ni43, Ni49 and Ni54 are the nearest neighbor nonequivalent atoms of the doped atoms. It is found that the s-orbital PDOS and the p-orbital PDOS of the Ni atoms exhibit extended character, and the d-orbital PDOS of Ni atoms are more localized. Careful observation shows that the s-orbital PDOS and the p-orbital PDOS of Ni atoms don’t change significantly. On the contrary, the d-orbital PDOS of Ni atoms has changed significantly. Because the s-orbital and the p-orbital contribute little to orbital hybridization, we mainly analyze the d-orbitals of the doped atoms and their NN Ni atoms. In the interaction between the doped atoms and their NN Ni atoms, the d-orbital of the doped atoms plays a major role. When the doped
atoms and its NN Ni atoms is as follows: Re > W > Ta. Therefore, the interaction between the Re atom and its NN Ni atoms is the strongest, followed by that between the W atom and its NN Ni atoms, and the interaction between the Ta atom and its NN Ni atoms is the weakest. It is consistent with the conclusions of the Griffith work and the GSF energy. 3.4.2. The partial density of states analysis The electronic states and the interaction of elements in the system can be expressed by the partial density of states (PDOS). The PDOSs of Re, Ta, W and their NN Ni atoms on the interface are shown in Fig. 6, and the solid lines and the dashed lines denote the PDOSs with and 6
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Fig. 6. PDOSs of Re, Ta, W and their NN Ni atoms on the interface. The solid lines and the dashed lines denote the PDOSs with and without Re, Ta and W, respectively. The Fermi level is shifted to zero.
Ni atoms. We can see that the d-orbital of the doped atoms has strong hybridization with the s-orbital, p-orbital and d-orbital of the NN Ni atoms. The d-orbital PDOS of the NN Ni atoms is reduced at the Fermi level and the d-orbital electrons move to deep energy level, which means that the transition probability of the electronic states is confined and the stability of the Ni/Ni3Al interface is enhanced [12]. Because the d-orbital PDOS of Re and the d-orbital PDOS of Ni43 have a peak at the same low energy level, there is a strong hybridization
atoms replace the Al positions, the main peaks of d-orbital PDOS of the NN Ni atoms are higher in the low energy level and lower in the high energy level, which indicates that electrons of the d-orbital move to the deep energy level. Therefore, compared with the pure system, the doped system has lower energy and becomes more stable. Al atom only has s-orbital and p-orbital, and no d-orbital. The interaction between the s-orbital and p-orbital of Al atom and its NN Ni atoms is much weaker than that between the d-orbital of the doped atoms and its NN 7
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while the hybridization between the Ta, W and Re atoms with their NN Ni atoms is different. We found that the hybridization between the Re atom with its NN Ni atoms is the strongest, followed by W, and the hybridization between the Ta atom with its NN Ni atoms is the weakest. The essential reason for the influence of Re, Ta and W on the creep strength of Ni-based SC superalloys is explored at electronic level.
between the d-orbital of Re and the d-orbital of Ni43, and the hybridization peak occurs at −4.84 eV. The intensity of hybridization peak of Ni43 is 0.35 eV−1. Similarly, we can find that there is a hybridization peak at −4.64 eV in the W-doped system, and the intensity of hybridization peak of Ni43 is 0.34 eV−1. Compared with the case of the Re-doped system, the hybridization peak of Ni43 in the W-doped system moves from low energy level to high energy level, and the corresponding peak decreases, indicating that the hybridization between W-Ni43 is weaker compared to Re-Ni43 hybridization. There is no hybridization peak between the d-orbital of Ta and the d-orbital of Ni43 at low energy level, so the hybridization between Ta-Ni43 is the weakest. There is also a strong hybridization between the d-orbital of Re and the d-orbital of Ni49. The hybridization peaks occur at −4.84 eV and −2.03 eV, and the intensities of hybridization peak of Ni49 are 0.33 eV−1 and 2.38 eV−1, respectively. The d-orbital of W and the dorbital of Ni49 are hybridized, and the intensities of hybridization peak of Ni49 are 0.32 eV−1 and 2.35 eV-1at −4.64 eV and −1.85 eV, respectively. The hybridization between Ta-Ni49 is the weakest, and there is no hybridization peak at the low energy level. Compared with the case of the Re-doped system, the hybridization peak of Ni49 in the W-doped system shifts from low energy level to high energy level, and the intensity of hybridization peak decreases, which indicates that the hybridization between Re-Ni49 is stronger than that between W-Ni49. Similarly, we analyzed the interaction between the doped atoms and Ni54. There are the hybridization peaks at −4.84 eV and −2.03 eV in the Re-doped system, respectively. The corresponding intensities of hybridization peak of Ni54 are 0.95 eV−1 and 3.09 eV−1, respectively. The d-orbital of Ni54 has a hybridization peak at −4.64 eV in the Wdoped system, and the intensity of hybridization peak is 0.82 eV−1. The d-orbital of Ni54 has a hybridization peak at −4.52 eV in the Ta-doped system, and the intensity of hybridization peak is 0.62 eV−1. For the hybridization between Re-Ni54, there are the hybridization peaks not only at low energy level but also at −2.03 eV, which makes the hybridization between Re-Ni54 strong. Compared with the W-doped system and the Ta- doped system, the hybridization peak of Ni54 in the Re-doped system moves to the lower energy level, and the intensity of hybridization peak is the largest, which indicating that the hybridization between Re-Ni54 is the strongest. In the case of the Ta-doped system, the hybridization peak of Ni54 is at the highest energy level and the intensity of hybridization peak is the smallest, so the hybridization between Ta-Ni54 is the weakest. There is only one electron difference between the elements Ta, W and Re, which is in the 5d orbital of the doped elements. It leads to different hybridization between the d-orbital of Ta, W and Re and the dorbital of their NN Ni atoms. We analyzed the hybridization of Ta, W and Re with their NN Ni atoms. It is found that the interaction between Re and its NN Ni atoms is the strongest, followed by W, and the interaction between Ta and its NN Ni atoms is the weakest. It is consistent with the results of charge density difference, the Griffith work and the GSF energy.
CRediT authorship contribution statement Chuanxi Zhu: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Writing - review & editing. Tao Yu: Conceptualization, Writing - review & editing, Investigation. Chongyu Wang: Project administration, Funding acquisition. Dianwu Wang: Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We are grateful to Prof. Jun Jiang for helpful discussion. This work was supported by the National Key R&D Program of China (Grant Nos. 2017YFB0701503). Simulations were performed on the “Explorer 100” cluster system of Tsinghua National Laboratory for Information Science and Technology, Beijing, China. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time due to legal or ethical reasons. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.commatsci.2020.109586. References [1] R.C. Reed, The Superalloys: Fundamentals and Applications, Cambridge University Press, 2006, p. 102. [2] C.T. Sims, N.S. Stoloff, W.C. Hagel, Superalloys II, Wiley, New York, NY, 1987. [3] T.M. Pollock, S. Tin, Nickel-based superalloys for advanced turbine engines: chemistry, microstructure and properties, J. Propul. Power. 22 (2006) 361–374. [4] K. Kawagishi, A.C. Yeh, T. Yokokawa, T. Kobayashi, Y. Koizumi, H. Harada, Development of an oxidation-resistant high-strength sixth-generation single-crystal superalloy TMS-238, Superalloys TMS. Warrendale PA, 2012. pp. 189–195. [5] L. Dirand, J. Cormier, A. Jacques, J.-P. Chateau-Cornu, T. Schenk, O. Ferry, P. Bastie, Measurement of the effective γ/γ′ lattice mismatch during high temperature creep of Ni-based single crystal superalloy, Mater. Charact. 77 (2013) 32–46. [6] H. Yamada, Y. Ogawa, Y. Ishii, H. Sato, M. Kawasaki, H. Akoh, Y. Tokura, Engineered interface of magnetic oxides, Science 305 (2004) 646–648. [7] A. Ziegler, J.C. Idrobo, M.K. Cinibulk, C. Kisielowski, N.D. Browning, R.O. Ritchie, Interface structure and atomic bonding characteristics in silicon nitride ceramics, Science 306 (2004) 1768–1770. [8] D.A. Muller, T. Sorsch, S. Moccio, F.H. Baumann, K. Evans-Lutterodt, G. Timp, The electronic structure at the atomic scale of ultrathin gate oxides, Nature 399 (1999) 758. [9] R.C. Reed, The Superalloys: Fundamentals and applications, Cambridge University Press, 2006, p. 46. [10] G.L. Erickson, A new, third generation, single-crystal, casting superalloy, JOM 47 (1995) 36–39. [11] R.C. Reed, The Superalloys: Fundamentals and Applications, Cambridge University Press, 2006, pp. 157–158. [12] T. Zhu, C.Y. Wang, Y. Gan, Effect of Re in phase, phase and interface of Ni-based single-crystal superalloys, Acta Mater. 58 (2010) 2045–2055. [13] Y. Liu, K.Y. Chen, G. Lu, J.H. Zhang, Z.Q. Hu, Impurity effects on the Ni/Ni3Al interface cohesion, Acta Mater. 45 (1997) 1837–1849. [14] K. Chen, L.R. Zhao, J.S. Tse, Sulfur embrittlement on interface of Ni-base single crystal superalloys, Acta Mater. 51 (2003) 1079–1086.
4. Conclusion By doping Re, Ta and W atoms on the Ni/Ni3Al interface, the substitution formation energy of the doped system was studied. It concluded that Re, Ta and W atoms tend to occupy the Al positions, which is consistent with the experimental results. Further, the Griffith work and the GSF energy of the Ni/Ni3Al interface in the doped system were calculated, and it concluded that Re, Ta and W atoms all have the effect of strengthening the interface. When the doped atoms occupied the γ ' -Al position, the strengthening effect is the best. The strengthening order of the doped atoms for the interface is as follows: Re > W > Ta, which is consistent with the experimental results. Finally, we analyzed the charge density difference and density of states of the doped system. It concluded that the Ta, W and Re atoms differ only by one electron, 8
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in metals, Phys. Rev. B 40 (1989) 3616–3621. [27] M.K. Miller, Atom Probe Tomography: Analysis at the Atomic Level, Springer Science & Business Media, 2012. [28] O.C. Hellman, J.A. Vandenbroucke, J. Rüsing, D. Isheim, D.N. Seidman, Analysis of three-dimensional atom-probe data by the proximity histogram, Microsc. Microanal. 6 (2000) 437–444. [29] S.H. Liu, C.P. Liu, W.Q. Liu, X.N. Zhang, P. Yan, C.Y. Wang, Investigation of the elemental partitioning behaviour and site preference in ternary model nickel-based superalloys by atom probe tomography and first-principles calculations, Philos. Mag. 96 (2016) 2204–2218. [30] A.F. Kohan, G. Ceder, D. Morgan, C.G. Van de Walle, First-principles study of native point defects in ZnO, Phys. Rev. B 61 (2000) 15019–15027. [31] J.E. Raynolds, J.R. Smith, G.L. Zhao, D.J. Srolovitz, Adhesion in NiAl-Cr from first principles, Phys. Rev. B 53 (1996) 13883. [32] M.W. Finnis, The theory of metal-ceramic interfaces, J. Phys.: Condens. Matter. 8 (1996) 5811–5836. [33] S.G. Tian, X.F. Yu, J.H. Yang, N.R. Zhao, Y.B. Xu, Z.Q. Hu, Deformation features of a nickel-base single crystal superalloy during compression creep, Mater. Sci. Eng. A 379 (2004) 141–147. [34] J.W. Christian, V. Vitek, Dislocations and stacking faults, Rep. Prog. Phys. 33 (1970) 307–411. [35] H.L. Dang, C.Y. Wang, T. Yu, First-principles investigation of 3d transition elements in L10 TiAl, J. Appl. Phys. 101 (2007) 083702.
[15] P. Peng, Z.H. Jin, R. Yang, Z.Q. Hu, First-principles study of effect of lattice misfit on the bonding strength of Ni/Ni3Al interface, J. Mater. Sci. 39 (2004) 3957–3963. [16] C. Wang, C.Y. Wang, Density functional theory study of Ni/Ni3Al interface alloying with Re and Ru, Surf. Sci. 602 (2008) 2604–2609. [17] X.F. Gong, G.X. Yang, Y.H. Fu, Y.Q. Xie, J. Zhuang, X.J. Ning, First-principles study of Ni/Ni3Al interface strengthening by alloying elements, Comput. Mater. Sci. 47 (2009) 320–325. [18] J.R. Rice, Dislocation nucleation from a crack tip: an analysis based on the Peierls concept, J. Mech. Phys. Solids 40 (1992) 239–271. [19] V. Vitek, Intrinsic stacking faults in body-centred cubic crystals, Philos. Mag. 18 (1968) 773–786. [20] X.X. Yu, C.Y. Wang, The effects of alloying elements on generalized stacking fault energies, strength and ductility of -Ni3Al, Mater. Sci. Eng. A 539 (2012) 38–41. [21] P.E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50 (1994) 17953–17979. [22] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmentedwave method, Phys. Rev. B 59 (1999) 1758–1775. [23] G. Kresse, J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47 (1993) 558–561. [24] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996) 11169–11186. [25] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865–3868. [26] M. Methfessel, A.T. Paxton, High-precision sampling for Brillouin-zone integration
9
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
First-principles study of Ni/Ni3Al interface doped with Re, Ta and W ⁎
⁎
Chuanxi Zhua, Tao Yua, , Chongyu Wanga,b, , Dianwu Wanga a b
Central Iron and Steel Research Institute, Beijing 100081, China Department of Physics, Tsinghua University, Beijing 100084, China
A R T I C LE I N FO
A B S T R A C T
Keywords: First-principles calculations Ni-based single crystal superalloys Griffith work Generalized stacking fault energy
The strengthening effects of (0 0 1) Ni/Ni3Al interface alloying with Re, Ta and W in Ni-based single crystal superalloys were studied using first-principles density functional theory. By placing alloying elements on (0 0 1) Ni/Ni3Al interface, the substitution formation energy, the Griffith work, the generalized stacking fault energies, the partial density of states and the charge density difference were all calculated. It is figured out that Re, Ta and W atoms prefer to substitute Al positions of γ ' phase. It is consistent with the experimental results. The strengthening effect of Re atom on (0 0 1) Ni/Ni3Al interface is stronger than that of Ta and W atoms addition. The order of strengthening effects of Re, Ta and W atoms from our calculations is consistent with relevant experimental observations. By analyzing the charge density difference and the partial density of states, our results show that the hybridization between Re atom and the nearest neighbor Ni atoms is the strongest, which is conducive to explore the essential reason for the different strengthening effects of alloying atoms at electronic level.
1. Introduction Ni-based single crystal(SC) superalloys have been widely used as aerospace turbine blades and industrial gas turbine blades. Ni-based SC superalloys consists of a large volume fraction of γ ' (L12 structure)precipitates and nickel solid solution γ (FCC structure)-matrix, where the γ ' precipitates are coherently embedded in the γ matrix [1,2]. Nibased SC superalloys contain a large number of alloying elements: Re, Ru, Co, Hf, Ti, Ta, W, Mo, Nb, Cr, etc [3]. The addition of alloying elements makes Ni-based SC superalloys have relatively excellent high temperature creep performance, oxidation resistance and thermal fatigue resistance [4]. It has been observed that the interface is widely present in the Ni-based SC superalloys [5], the structure of the interface plays a significant role in determining the mechanical properties in multiphase systems [6–8], and the properties of the superalloys are found to depend critically on the coherency of the γ / γ ' interface [9]. Therefore, it is very important to analyze the strengthening effects of alloying elements on the γ / γ ' interface, which is helpful for us to understand how alloying elements affect the mechanical properties of the Ni-based SC superalloys. Re, Ta and W are very important creep strengthening elements. The creep strength of nickel-based SC superalloys was reported to be improved by adding Re, Ta and W [10]. Experiment has shown that improvements in creep-strengthening occur in the following order: Re >
⁎
W > Ta > Cr > Co [11]. Zhu et al. [12] chose the six clusters of the interface for the first-principles discrete variational method(DVM) study, in which the interatomic energy, the charge distribution, the partial density of states and the electron density difference were all calculated. It is found that Re atom can strongly enhance the bonding strength with its nearest neighboring atoms. Chen and Peng et al. [13–15] evaluated the strengthening effect of alloying elements on the Ni/Ni3Al interface by introducing the bond order of the overlap population. By placing alloying elements on the Ni/Ni3Al interface, Wang et al. [16] calculated the Griffith energy of the doped system using firstprinciples plane-wave pseudopotential method, and found that the elements of both Re and Ru enhance the coherent strength of the interface. Gong et al. [17] investigated the strengthening effects induced by alloying elements (Re, Ru, Cr, Co, Mo, W, Ta) on Ni/Ni3Al interface in Ni-based SC superalloys in terms of Griffith work using first-principles density functional theory (DFT). The previous research of the γ / γ ' interface is mainly concerned with calculating the Griffith work to measure the strengthening effects of alloying elements on the γ / γ ' interface. Rice [18] pointed out that, when a dislocation is emitted, the atoms slip on the slip plane and the maximum energy(γus ) associated with this progress decides the resistance to the dislocation emission. The value of γus depends on the maximum value of the generalized stacking fault (GSF) energy [19]. Yu et al. [20] introduced the generalized stacking fault energies to investigate the strengthening effects
Corresponding authors at: Central Iron and Steel Research Institute, Beijing 100081, China. E-mail addresses: [email protected] (T. Yu), [email protected] (C. Wang).
https://doi.org/10.1016/j.commatsci.2020.109586 Received 14 November 2019; Received in revised form 16 January 2020; Accepted 2 February 2020 0927-0256/ © 2020 Elsevier B.V. All rights reserved.
Please cite this article as: Chuanxi Zhu, et al., Computational Materials Science, https://doi.org/10.1016/j.commatsci.2020.109586
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of alloying elements in γ ' -Ni3Al. So, in this paper we can use the generalized stacking fault energy to assess the strengthening effects of alloying elements on the γ / γ ' interface. Experiment has shown that improvements in creep-strengthening occur in the following order: Re > W > Ta [11]. The elements Ta, W and Re elements differ only by one electron, however their effects on the creep strength are different. This problem is still under discussion. To solve this problem, we should analyze how the doped atoms interact with neighboring atoms from the aspects of charge density difference and partial density of states. In this paper, the mechanical properties of the γ / γ ' interface doped with Re, Ta and W were investigated by the first-principles plane-wave pseudopotential method, which was implemented by the Vienna ab initio simulation package (VASP) [23,24]. Meanwhile, we calculated the substitution formation energy of the doped system and combined it with the atom probe tomography (APT) experiment to study the site preference of the doped atoms on the γ / γ ' interface. The Griffith work and the GSF energies were introduced to investigate the strengthening effect of the doped atoms on the γ / γ ' interface. Finally, we shall explore from the charge density difference and partial density of states (PDOS) why the Ta, W, and Re atoms differ only by one electron but have different strengthening effects on the γ / γ ' interface.
Table 1 Compositions of the ternary model alloys investigated in this work. Alloy (wt%)
W
Ta
Re
Al
Ni
M1 M2 M3
– 6.0 –
– – 5.6
5.0 – –
8.3 7.7 7.6
Bal. Bal. Bal.
The total-energy calculation presented in this paper was performed using the projector -augmented wave (PAW) method [21,22] as implemented in the Vienna ab initio simulation package (VASP) [23,24]. The Perdew-Burke-Ernzerhof (PBE) electronic exchange-correlation functional within the generalized gradient approximation (GGA) was taken into account [25]. The plane wave basis cutoff energy was 350 eV. An 11 × 11 × 1 Monkhorst-Pack k-mesh was converged for 96-atoms supercell model of the Ni/Ni3Al interface. A first-order Methfessel–Paxton smearing method [26] was used with a smearing width of 0.1 eV. The relaxation of the electronic degrees of freedom will be stopped if the total energy changes between two steps is smaller than 10−5 eV. The relaxation will stop if all the Hellmann–Feynman forces acting on the atoms are less than 0.02 eV/Å. 2.2. Experimental details
2. Computational model and experimental details The nominal compositions of the ternary model alloys investigated in this paper are listed in Table 1. The M1, M2 and M3 ternary model alloys were used to investigate the partitioning behavior of Re, Ta and W atoms in the γ and γ ' phases. Cast ingots of the ternary model alloy were prepared by vacuum induction melting. The master ingots were then directionally solidified to form 〈0 0 1〉-oriented single crystals using the Bridgman technique. The single crystal bars were homogenized at 1330℃for 20 h and subsequently aged at 870 °C for 32 h to produce a homogeneous distribution of γ ' phase. The partitioning behaviour of Re, Ta and W atoms in the ternary model alloys between the γ and theγ ' phases was measured by the atom probe tomography (APT). By cutting 0.5 mm × 0.5 mm × 15 mm rods from heat treated single crystal bars we got APT specimens, and then the standard two-step polishing procedure was performed [27]. The specimen temperature was ~50 K. Proximity histogram (proxigram) methodology [28] was adopted to determine the concentration information.
2.1. Computational model and methods In this paper, the 96-atoms supercell model of the Ni/Ni3Al interface with three-dimensional translational symmetry was studied. The atomic arrangements of our model are shown in Fig. 1. The lattice parameters of γ -Ni and γ ' -Ni3Al in the DFT calculations were 3.526 Å and 3.570 Å, respectively [16]. In order to determine the lattice parameter of the two-phase model, we relaxed the model with different lattice constants (3.526–3.570 Å), and the Hellmann-Feynman forces acting on the atoms are less than 0.02 eV/Å. When the energy of the system is the lowest, the lattice constant (3.545 Å) was selected as our model of the Ni/Ni3Al interface. The model took periodic boundary conditions in the X-axis and Y-axis directions, and added a vacuum layer in the Z-axis direction. After the energy convergence test, the thickness of the vacuum layer was selected as 12 Å. In this paper, the doped elements Ta, W and Re were considered. The substituting positions were: γ -Ni1, γ ' -Ni2, γ ' -Al, as shown in Fig. 1. The GSF energies as a function of the displacement u were calculated by shifting the γ ' part relative to the fixed γ part. For each rigid shift, the atomic positions were relaxed in the [0 0 1] direction and fixed in the other directions.
3. Results and discussion 3.1. Elemental partitioning between γ and γ ' phases The concentration profiles of the alloying atoms in the ternary model alloys are shown in Fig. 2. The APT experiment results show that Re, Ta and W atoms tend to partition to the γ ' phase in the ternary model alloys. From Fig. 2(a), the sum of the concentration for Al and Re in the γ ' phase is approximately 25 at.%, which indicates that Re atoms tend to occupy the Al positions in the γ ' phase. From Fig. 2(b) and (c), we also find that W and Ta atoms tend to occupy the Al positions in the γ ' phase. In order to study the site preference of alloying elements on the Ni/ Ni3Al interface, we calculated the substitution formation energy of the doped system. Alloying elements substitute for γ -Ni, γ '-Ni, γ ' -Al, respectively, as shown in Fig. 1. The substitution formation energy can be defined as follows [29,30]: '
'
'
Minγ Esub,NiorAl = (EMinγ + μ Ni/Al ) − (Eγ + μ M) Minγ Esub,Ni
(1)
= (EMinγ + μ Ni ) − (Eγ + μ M)
(2) Minγ '
where M represents Ta, W, and Re, and E is the total energy of the doped system for alloying elements substitute γ '-Ni and γ ' -Al, and EMinγ
Fig. 1. The supercell model of the Ni/Ni3Al interface with 96 atoms. 2
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Fig. 2. Concentration profile determined from APT experiment. (a) the concentration profile of Ni, Al and Re atoms in the ternary model alloy (M1); (b) the concentration profile of Ni, Al and Ta atoms in the ternary model alloy (M3); (c) the concentration profile of Ni, Al and W atoms in the ternary model alloy (M2).
the Al position, the substitution formation energy is the lowest, indicating that Re atom has a strong tendency to partition to the γ ' phase and preferentially substitutes at the Al positions in ternary Ni-Al-Re alloy. We observe the doping of Ta and W atoms and find that both Ta and W atoms also tend to occupy the Al positions in ternary Ni-Al-Ta alloy and in ternary Ni-Al-W alloy. The analysis results are consistent with the results given by the APT experiment.
Table 2 The substitution formation energies (eV) of alloying elements substituting for Ni1, Ni2 and Al, respectively. Substitution atom
Ni1 Ni2 Al
Alloying atom Re
Ta
W
0.39 1.46 −0.60
−0.73 1.37 −1.82
0.16 1.77 −0.96
3.2. The effect of doping on the cleavage properties Griffith energy is defined as the work required to separate an interface into two surfaces. It is worth noting that this energy is the critical energy required for crack propagation. The Griffith equation gives [31]: σF = (Wad Y/cπ )1/2 , where σF denotes the critical stress required for crack propagation, c denotes the length of a surface crack, and Y denotes Young's modulus. From the above formula, we can see that σF is proportional to Wad, so we can use Wad to express mechanical properties of alloying elements on the Ni/Ni3Al interface in SC superalloys. Wad can be calculated from the following equation [32]:
is the total energy of the doped system for alloying elements substitute γ -Ni. μ Ni/Al is the chemical potential of Ni or Al in the bulk system. μ Ni is the energy of a single Ni atom in a fcc unit cell. The chemical potential of bulk Ni3Al is equal to 3μ Ni + μAl , so μAl = μ Ni3Al − 3μ Ni . μ M is the chemical potential of alloying elements, where μRe is the energy of a single Re atom in an hcp unit cell, μTa is the energy of a single Ta atom in a bcc unit cell, and μW is the energy of a single W atom in a bcc unit cell. The calculation results are listed in Table 2. It can be seen from Table 2, when Re atom replaces the three doped positions (see in Fig. 1), respectively, the order of substitution forma− Ni1 − Ni2 − Al tion energy is ERe < ERe < ERe . When the Re atom replaces sub sub sub
Wad = (EslabA + EslabB − EA/B)/Ai '
(3)
EA / B is the energy of the γ / γ interface model.EslabA and EslabB are the 3
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Substitution atom
'
Re
Ta
W
4.53 4.91 5.26
4.21 4.99 5.01
4.41 5.04 5.23
4.35 Ni1 Ni2 Al
energies of the γ -Ni and γ '-Ni3Al, respectively, after the γ / γ ' interface model splits into two blocks (γ -Ni block and γ ' -Ni3Al block). Ai is the area of the (0 0 1) atomic layer in the Ni/Ni3Al interface model. The calculation results are listed in Table 3. From Table 3, we can see that the Griffith work increases significantly when Re atom replaces any of the three atoms. When Al atom is replaced by Re atom, the Griffith work is the greatest and the interface is the most stable in the Re-doped system. For the case of Ta atom replacing the atoms in the three positions, when Ta atom replaces the Ni1 atom the Griffith work is reduced relative to the Griffith work of the pure system. When the other two atoms are replaced, the Griffith work is increased. When the Ta atom replaces the Al atom, the Griffith work is the greatest and the interface is the most stable in the Ta-doped system. For the W atom, it has the similar trend as the Re atom in affecting the Griffith work when it substitutes for the three atoms, respectively. The doping of W atom at the three positions can increase the Griffith work relative to the pure system. When Al atom is replaced by the W atom, the Griffith work is the greatest and the interface is the most stable in the W-doped system. By comparing the Griffith work when the Re, Ta, and W atoms replace the Al position, respectively, we can get the rule: the Griffith work is the smallest when Ta atom replaces the Al position, the second smallest is given when the W atom replaces the Al position, and the maximum Griffith work is given when Re atom replaces the Al position. That is to say, when Re, Ta and W are doped at the γ / γ ' interface, respectively, the order of improvement of the mechanical properties of the γ / γ ' interface is Ta < W < Re. The effect on creep performance can also be demonstrated experimentally Ta < W < Re [11].
3.4. Electronic structure analysis 3.4.1. The charge density difference analysis The charge density difference is defined as [35]:
Δρ = [ρ (X − doped) − ρfree (X − doped)] − [ρ (clean) − ρfree (clean)] (4) where X is the substitution elements Re, Ta and W; ρ (X − doped) and ρ (clean) are the charge densities of the X-doped system and the pure system, respectively; ρfree (X − doped) and ρfree (clean) are the charge densities of the free atoms of the X-doped system and the pure system, respectively. From the above analysis in Sections 3.1–3.3, we can conclude that Re, Ta and W atoms tend to occupy the Al positions, and the mechanical properties of the Ni/Ni3Al interface increases the most when they occupy the Al positions. Thus, when we analyze the electronic structure of the doped system we only analyze the situation that the doped atoms occupy the Al positions. In Fig. 4, we know that when Re atom replaces the Al atom on the Ni/Ni3Al interface, electrons accumulate between the Re atom and its NN Ni atoms. The interaction between the Re atom and its NN Ni atoms is enhanced, indicating that the interaction between the Re atom and its NN Ni atoms is stronger than the interaction between the Al atom and its NN Ni atoms. We also find this rule when the Ta and the W atoms replace the Al atom on the Ni/Ni3Al interface. It shows that Re, Ta, W can enhance the stability of the interface when replacing the Al positions, and have the function of strengthening the interface. Looking closely at the Figs. 4 and 5, we can find that although there are 12 NN Ni atoms around the Re atom, the aggregation of electrons between the Re atom and different Ni atoms is different. Electrons aggregate less between Re atom and Ni54 and Ni53 atoms, however more between Re atom and Ni43, Ni44, Ni45, Ni46, Ni49, Ni50, Ni51 and Ni52 atoms (Ni43, Ni44, Ni45 and Ni46 are at equivalent positions, Ni49, Ni50, Ni51 and Ni52 are at equivalent positions, so we will take Ni43 and Ni49 as our representative). The agglomeration of electrons is localized, and the bonding between Re atom and the NN Ni atoms are directional. Because electrons aggregate more between the doped atoms and Ni43 and Ni49 compared with between Al atom and Ni43 and
3.3. The effect of doping on the slip properties It has been experimentally confirmed that there are a large number of [1 1 0](0 0 1) dislocations on the γ / γ ' interface [33]. Wang et al. also confirmed that the slip direction on the γ / γ ' interface is [1 1 0] direction [16]. So we chose the [1 1 0] direction as the slip direction on the γ / γ ' interface. In order to nucleate a dislocation with Burgers vector b, it is necessary to overcome the critical energy. Rice [18] called this energyunstable stacking fault energy, γus . γus is defined as the maximum of the GSF energy [19,34], which characterizes the resistance to dislocation nucleation. The energy cost by relative displacement of γ -Ni and γ ' -Ni3Al was taken to achieve the GSF energy curve. The γGSF can be defined as: γGSF = (E u − E0)/s , where E0 and E u are the energies of the supercell before and after slip deformation, and s is the area of the faulted region. The calculation results are shown in Fig. 3. From Fig. 3(a), the values of γus are 1.108, 1.089, 1.133, 1.250 J/m2 '
'
W γ − Ni2 , W γ − Al substitutions, respectively. We also find the same rule in Fig. 3(b) and (c), when Ta atom and W atom replace γ ' - Al the improvement of the mechanical properties is the most obvious. By comparing the γus of replacing Al position with Re, Ta and W atoms, we can draw the conclusion that the γus of replacing Al position with Ta atom is the smallest, followed by W atom, and the γus of replacing Al position with Re atom is the largest. That is to say, when Re, Ta and W are doped at the γ / γ ' interface, respectively, the order of improving the mechanical properties of the γ / γ ' interface is Ta < W < Re, which is consistent with the calculation results of Griffith work and the experimental results [11]. In order to study the effect of doping on local structure, we calculated the average distance between the doped atom and its nearest neighboring (NN) Ni atoms. Because Re, Ta and W atoms tend to occupy the Al positions and the mechanical properties of the Ni/Ni3Al interface increases the most when they occupy the Al positions, so we only analyze the situation that the doped atoms occupy the Al positions. The calculation results are listed in Table 4. From Table 4, we can see that the average distance between the Re atom and its NN Ni atoms is the smallest, and the average distance between the Ta atom and its NN Ni atoms is the biggest. It is indicated that the coherent strength of interface is the strongest when Re atom occupies the Al positions, and the coherent strength of interface is the weakest when Ta atom occupies the Al positions.
Alloying atom Pure
'
1.184 J/m2 for Taγ − Ni1, Taγ − Ni2 , Taγ − Al substitutions, respectively. From Fig. 3(c), the values of γus are 1.101, 1.152, 1.246 J/m2 for W γ − Ni1,
Table 3 Griffith work (J/m2) of alloying elements substituting for Ni1, Ni2, Al (see in Fig. 1), respectively.
'
for the pure, Reγ − Ni1, Re γ − Ni2 , and Re γ − Al substitutions, respectively. We can see that the γus decreases when Re atom substitutes for γ -Ni1 compared to the pure system, and increases when Re atom substitutes for γ ' -Ni2 and γ ' -Al, respectively. This shows that the stability and the mechanical properties of the interface system are improved when Re atom substitutes for γ ' -Ni2 and γ ' -Al. Further observation shows that the improvement of mechanical properties is the most obvious when Re atom replaces γ '-Al. From Fig. 3(b), the values of γus are 1.069, 1.171, 4
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Fig. 3. The generalized stacking fault energy γGSF curve verse the slipping displacement (u/b) of γ / γ ' interface doped with Re, Ta and W, and b is a Burgers vector. (a) 〈1 1 0〉(0 0 1) slip for the pure and Re substitution interface; (b) 〈1 1 0〉(0 0 1) slip for the pure and Ta substitution interface; (c) 〈1 1 0〉(0 0 1) slip for the pure and W substitution.
stacking fault energy of the Ni/Ni3Al interface are significantly increased and the stability of interface is improved. We observe the Fig. 5 and further analyze the difference of interaction between Re, Ta, W atoms and their NN Ni atoms. From the point of electron transfer, Re atom and its NN Ni atoms lose more electrons, more electrons are transferred to the space between the Re atom and its NN Ni atoms. W atom and its NN Ni atoms lose less electrons relative to Re atom and its NN Ni atoms. Ta atoms and its NN Ni atoms lose the least electrons. The order of electron aggregation between the doped
Table 4 The average distance (Å) between the doped atom and its NN Ni atoms. Re
Ta
W
2.513
2.545
2.522
Ni49, the bonding ability and interaction between the doped atoms and Ni43 and Ni49 are improved significantly. Therefore, when Re, Ta and W atoms replace Al positions, respectively, the Griffith energy and
Fig. 4. The charge density difference of the γ / γ ' interface doped with Re, Ta and W. (a) Re-Al model; (b) Ta-Al model; (c) W-Al model. The isosurfaces is 0.0045 e/Å3. The yellow and blue regions represent charge accumulation and charge loss, respectively. 5
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Fig. 5. Charge density contour plots of γ / γ ' interface doped with Re, Ta and W on the (0 0 1) plane and the (0 1 0) plane. (a) Charge density contour plot on the (0 0 1) plane by the addition of Re atom; (b) Charge density contour plot on the (0 1 0) plane by the addition of Re atom; (c) Charge density contour plot on the (0 0 1) plane by the addition of Ta atom; (d) Charge density contour plot on the (0 1 0) plane by the addition of Ta atom; (e) Charge density contour plot on the (0 0 1) plane by the addition of W atom; (f) Charge density contour plot on the (0 1 0) plane by the addition of W atom. The contour spacing is 0.002 e/a.u.3. Solid lines and dashed lines correspond to the gain and the loss of electrons, respectively.
without Re, Ta and W, respectively. In Fig. 6, Ni43, Ni49 and Ni54 are the nearest neighbor nonequivalent atoms of the doped atoms. It is found that the s-orbital PDOS and the p-orbital PDOS of the Ni atoms exhibit extended character, and the d-orbital PDOS of Ni atoms are more localized. Careful observation shows that the s-orbital PDOS and the p-orbital PDOS of Ni atoms don’t change significantly. On the contrary, the d-orbital PDOS of Ni atoms has changed significantly. Because the s-orbital and the p-orbital contribute little to orbital hybridization, we mainly analyze the d-orbitals of the doped atoms and their NN Ni atoms. In the interaction between the doped atoms and their NN Ni atoms, the d-orbital of the doped atoms plays a major role. When the doped
atoms and its NN Ni atoms is as follows: Re > W > Ta. Therefore, the interaction between the Re atom and its NN Ni atoms is the strongest, followed by that between the W atom and its NN Ni atoms, and the interaction between the Ta atom and its NN Ni atoms is the weakest. It is consistent with the conclusions of the Griffith work and the GSF energy. 3.4.2. The partial density of states analysis The electronic states and the interaction of elements in the system can be expressed by the partial density of states (PDOS). The PDOSs of Re, Ta, W and their NN Ni atoms on the interface are shown in Fig. 6, and the solid lines and the dashed lines denote the PDOSs with and 6
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Fig. 6. PDOSs of Re, Ta, W and their NN Ni atoms on the interface. The solid lines and the dashed lines denote the PDOSs with and without Re, Ta and W, respectively. The Fermi level is shifted to zero.
Ni atoms. We can see that the d-orbital of the doped atoms has strong hybridization with the s-orbital, p-orbital and d-orbital of the NN Ni atoms. The d-orbital PDOS of the NN Ni atoms is reduced at the Fermi level and the d-orbital electrons move to deep energy level, which means that the transition probability of the electronic states is confined and the stability of the Ni/Ni3Al interface is enhanced [12]. Because the d-orbital PDOS of Re and the d-orbital PDOS of Ni43 have a peak at the same low energy level, there is a strong hybridization
atoms replace the Al positions, the main peaks of d-orbital PDOS of the NN Ni atoms are higher in the low energy level and lower in the high energy level, which indicates that electrons of the d-orbital move to the deep energy level. Therefore, compared with the pure system, the doped system has lower energy and becomes more stable. Al atom only has s-orbital and p-orbital, and no d-orbital. The interaction between the s-orbital and p-orbital of Al atom and its NN Ni atoms is much weaker than that between the d-orbital of the doped atoms and its NN 7
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while the hybridization between the Ta, W and Re atoms with their NN Ni atoms is different. We found that the hybridization between the Re atom with its NN Ni atoms is the strongest, followed by W, and the hybridization between the Ta atom with its NN Ni atoms is the weakest. The essential reason for the influence of Re, Ta and W on the creep strength of Ni-based SC superalloys is explored at electronic level.
between the d-orbital of Re and the d-orbital of Ni43, and the hybridization peak occurs at −4.84 eV. The intensity of hybridization peak of Ni43 is 0.35 eV−1. Similarly, we can find that there is a hybridization peak at −4.64 eV in the W-doped system, and the intensity of hybridization peak of Ni43 is 0.34 eV−1. Compared with the case of the Re-doped system, the hybridization peak of Ni43 in the W-doped system moves from low energy level to high energy level, and the corresponding peak decreases, indicating that the hybridization between W-Ni43 is weaker compared to Re-Ni43 hybridization. There is no hybridization peak between the d-orbital of Ta and the d-orbital of Ni43 at low energy level, so the hybridization between Ta-Ni43 is the weakest. There is also a strong hybridization between the d-orbital of Re and the d-orbital of Ni49. The hybridization peaks occur at −4.84 eV and −2.03 eV, and the intensities of hybridization peak of Ni49 are 0.33 eV−1 and 2.38 eV−1, respectively. The d-orbital of W and the dorbital of Ni49 are hybridized, and the intensities of hybridization peak of Ni49 are 0.32 eV−1 and 2.35 eV-1at −4.64 eV and −1.85 eV, respectively. The hybridization between Ta-Ni49 is the weakest, and there is no hybridization peak at the low energy level. Compared with the case of the Re-doped system, the hybridization peak of Ni49 in the W-doped system shifts from low energy level to high energy level, and the intensity of hybridization peak decreases, which indicates that the hybridization between Re-Ni49 is stronger than that between W-Ni49. Similarly, we analyzed the interaction between the doped atoms and Ni54. There are the hybridization peaks at −4.84 eV and −2.03 eV in the Re-doped system, respectively. The corresponding intensities of hybridization peak of Ni54 are 0.95 eV−1 and 3.09 eV−1, respectively. The d-orbital of Ni54 has a hybridization peak at −4.64 eV in the Wdoped system, and the intensity of hybridization peak is 0.82 eV−1. The d-orbital of Ni54 has a hybridization peak at −4.52 eV in the Ta-doped system, and the intensity of hybridization peak is 0.62 eV−1. For the hybridization between Re-Ni54, there are the hybridization peaks not only at low energy level but also at −2.03 eV, which makes the hybridization between Re-Ni54 strong. Compared with the W-doped system and the Ta- doped system, the hybridization peak of Ni54 in the Re-doped system moves to the lower energy level, and the intensity of hybridization peak is the largest, which indicating that the hybridization between Re-Ni54 is the strongest. In the case of the Ta-doped system, the hybridization peak of Ni54 is at the highest energy level and the intensity of hybridization peak is the smallest, so the hybridization between Ta-Ni54 is the weakest. There is only one electron difference between the elements Ta, W and Re, which is in the 5d orbital of the doped elements. It leads to different hybridization between the d-orbital of Ta, W and Re and the dorbital of their NN Ni atoms. We analyzed the hybridization of Ta, W and Re with their NN Ni atoms. It is found that the interaction between Re and its NN Ni atoms is the strongest, followed by W, and the interaction between Ta and its NN Ni atoms is the weakest. It is consistent with the results of charge density difference, the Griffith work and the GSF energy.
CRediT authorship contribution statement Chuanxi Zhu: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Writing - review & editing. Tao Yu: Conceptualization, Writing - review & editing, Investigation. Chongyu Wang: Project administration, Funding acquisition. Dianwu Wang: Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments We are grateful to Prof. Jun Jiang for helpful discussion. This work was supported by the National Key R&D Program of China (Grant Nos. 2017YFB0701503). Simulations were performed on the “Explorer 100” cluster system of Tsinghua National Laboratory for Information Science and Technology, Beijing, China. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time due to legal or ethical reasons. Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.commatsci.2020.109586. References [1] R.C. Reed, The Superalloys: Fundamentals and Applications, Cambridge University Press, 2006, p. 102. [2] C.T. Sims, N.S. Stoloff, W.C. Hagel, Superalloys II, Wiley, New York, NY, 1987. [3] T.M. Pollock, S. Tin, Nickel-based superalloys for advanced turbine engines: chemistry, microstructure and properties, J. Propul. Power. 22 (2006) 361–374. [4] K. Kawagishi, A.C. Yeh, T. Yokokawa, T. Kobayashi, Y. Koizumi, H. Harada, Development of an oxidation-resistant high-strength sixth-generation single-crystal superalloy TMS-238, Superalloys TMS. Warrendale PA, 2012. pp. 189–195. [5] L. Dirand, J. Cormier, A. Jacques, J.-P. Chateau-Cornu, T. Schenk, O. Ferry, P. Bastie, Measurement of the effective γ/γ′ lattice mismatch during high temperature creep of Ni-based single crystal superalloy, Mater. Charact. 77 (2013) 32–46. [6] H. Yamada, Y. Ogawa, Y. Ishii, H. Sato, M. Kawasaki, H. Akoh, Y. Tokura, Engineered interface of magnetic oxides, Science 305 (2004) 646–648. [7] A. Ziegler, J.C. Idrobo, M.K. Cinibulk, C. Kisielowski, N.D. Browning, R.O. Ritchie, Interface structure and atomic bonding characteristics in silicon nitride ceramics, Science 306 (2004) 1768–1770. [8] D.A. Muller, T. Sorsch, S. Moccio, F.H. Baumann, K. Evans-Lutterodt, G. Timp, The electronic structure at the atomic scale of ultrathin gate oxides, Nature 399 (1999) 758. [9] R.C. Reed, The Superalloys: Fundamentals and applications, Cambridge University Press, 2006, p. 46. [10] G.L. Erickson, A new, third generation, single-crystal, casting superalloy, JOM 47 (1995) 36–39. [11] R.C. Reed, The Superalloys: Fundamentals and Applications, Cambridge University Press, 2006, pp. 157–158. [12] T. Zhu, C.Y. Wang, Y. Gan, Effect of Re in phase, phase and interface of Ni-based single-crystal superalloys, Acta Mater. 58 (2010) 2045–2055. [13] Y. Liu, K.Y. Chen, G. Lu, J.H. Zhang, Z.Q. Hu, Impurity effects on the Ni/Ni3Al interface cohesion, Acta Mater. 45 (1997) 1837–1849. [14] K. Chen, L.R. Zhao, J.S. Tse, Sulfur embrittlement on interface of Ni-base single crystal superalloys, Acta Mater. 51 (2003) 1079–1086.
4. Conclusion By doping Re, Ta and W atoms on the Ni/Ni3Al interface, the substitution formation energy of the doped system was studied. It concluded that Re, Ta and W atoms tend to occupy the Al positions, which is consistent with the experimental results. Further, the Griffith work and the GSF energy of the Ni/Ni3Al interface in the doped system were calculated, and it concluded that Re, Ta and W atoms all have the effect of strengthening the interface. When the doped atoms occupied the γ ' -Al position, the strengthening effect is the best. The strengthening order of the doped atoms for the interface is as follows: Re > W > Ta, which is consistent with the experimental results. Finally, we analyzed the charge density difference and density of states of the doped system. It concluded that the Ta, W and Re atoms differ only by one electron, 8
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