Transition-metal-element dependence of ideal shear strength and elastic behaviors of γ′-Ni3Al: ab initio study to guide rational alloy design

Journal of Alloys and Compounds 806 (2019) 1260e1266

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Transition-metal-element dependence of ideal shear strength and elastic behaviors of g0 -Ni3Al: ab initio study to guide rational alloy design Minru Wen a, Xing Xie a, Yifan Gao a, Huafeng Dong a, *, Zhongfei Mu b, Fugen Wu c, Chong-Yu Wang d, ** a

School of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou, 510006, China Experimental Teaching Department, Guangdong University of Technology, Guangzhou, 510006, China School of Materials and Energy, Guangdong University of Technology, Guangzhou, 510006, China d Department of Physics, Tsinghua University, Beijing, 100084, China b c

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 May 2019 Received in revised form 16 July 2019 Accepted 23 July 2019 Available online 24 July 2019

When doped with transition-metal (TM) elements, modern commercial Ni-based superalloys achieve an extraordinary mechanical performance, including elastic behaviors and shear strength. By using density functional theory, the TM-element (3d:SceZn, 4d:YeCd, 5d:HfeAu) dependence of the elastic properties and the ideal shear strength of g0 -Ni3Al were determined. According to the positive definiteness requirement of the elastic stiffness coefficients matrix, we present the necessary and sufficient mechanical stability in terms of Born criteria for a monoclinic crystal system. By examining the mechanical stability and calculating the stressestrain curve at every single strain, the shear behavior of pure Ni3Al in the weakest shear-slip system was investigated in detail. Our results show that the d-orbital occupancy of TM elements can affect the mechanical properties of the g0 phase significantly when they occupy the Al site in Ni3Al, where the elements toward the center part of each series (e.g. V, Cr, Mn, Nb, Mo, Ru, Ta, W, Re, and Os) can enhance the elastic constants, elastic moduli, and shear strength. Elements at the beginnings and ends of the series (such as Cu, Zn, Y, Ag, Cd, and Au) will decrease the elastic properties and shear strength of g0 -Ni3Al. © 2019 Elsevier B.V. All rights reserved.

Keywords: Transition-metal elements Shear strength Elastic behaviors g0 -Ni3Al Ni-Based superalloys ab initio calculations

1. Introduction As important elevated-temperature structural materials, Nibased single-crystal superalloys are used extensively in advanced turbine engine blades because of their superior high-temperature mechanical properties [1,2]. The ideal strength of solid materials is the minimum stress that is required to yield a defect-free crystal material and sets the upper limit of the strength that a real material can achieve under deformation (e.g., hydrostatic tension/ compression, uniaxial tension/compression, and shear) without defects [3,4]. The ideal strength is an intrinsic mechanical parameter of solid materials and is related to the breaking and exchange

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (H. Dong), [email protected] (C.-Y. Wang). https://doi.org/10.1016/j.jallcom.2019.07.284 0925-8388/© 2019 Elsevier B.V. All rights reserved.

of chemical bonds and the generation and failure of cracks and dislocations [4]. Studies of the ideal strength are fundamental to understand the mechanical behaviors of engineering materials and to develop practical applications. When doped with transitionmetal elements (e.g., Re, Mo, Ta, Co, W, Cr, and Ru), modern commercial Ni-based superalloys achieve an extraordinary elevatedtemperature mechanical performance [1,2,5e11]. In general, the ordered g0 -Ni3Al phase, which is one of the vital strengthening phases with a volume fraction as high as nearly 70% in commercial nickel-based alloys, is largely responsible for the improvement of creep resistance and the yield strength of alloys at a high temperature [2,12]. Consequently, an investigation of the TM-element dependence of the ideal shear strength of g0 -Ni3Al is essential. Advances in computational methods and modern computers allow for the ideal material strength to be predicted reliably by ab initio calculations [13e16], whereas it is unfeasible to determine the strength directly in experiments except by nanoindentation experiments [17,18]. Wang and Wang [19] studied the ideal tensile

M. Wen et al. / Journal of Alloys and Compounds 806 (2019) 1260e1266

and shear strength of Ni3Al with and without Re by using firstprinciples density functional theory (DFT) calculations. By combining the first-principles and quasi-harmonic approximation, Wu and Wang [20] investigated the effect of the alloying elements (Re, Ru, Mo, Cr, Co, W, and Ta) on the temperature-dependent ideal shear strength of g0 -Ni3Al. Recently, Chen et al. [21] studied the impact of single Re and double Re additions on the ideal shear strength of g0 -Ni3Al by first-principles calculations. Similar to the face-centered cubic (fcc) crystals, the weakest shear-slip system of Ni3Al is f111g < 112 > , and the calculated ideal shear strengths are 5.8 GPa, 5.768 GPa, and 4.23 GPa according to Refs. [19,20], and [21], respectively. Besides these reported studies, to the best of our knowledge, no additional theoretical studies have investigated the effect of alloying elements on the ideal shear strength of the g0 phase. No knowledge exists on the influence of Ti, V, Nb, and Hf on the ideal shear strength of Ni3Al, although these alloying elements are added to commercial Ni-based single superalloys. Similar to the ideal strength, the elastic properties of materials (such as the elastic constant Cij, bulk modulus B, Young's modulus E, shear modulus G, and Poisson's ratio y) reflect the ability of the material to resist elastic deformation, which are basic parameters for understanding the mechanical properties of structural materials. Thus far, extensive studies have focused on the influences of alloying elements (e.g., Re [22e27], W [22e24,26], Ta [22e24,26,27], Cr [22e25,27e29], Mo [22e27], Ru [22e24,26], Co [22e24], Y [24,28], Pt [28], Zr [28,29], and Hf [28]) on the elastic properties of g0 -Ni3Al by first-principles calculations. Moreover, by constructing ternary embedded-atom-method potential, the effects of Re [30,31], Co [31], Ta [31], Ru [31], and W [32] on the elastic properties of g0 phase were investigated by molecular-dynamics simulations. However, limited detailed knowledge exists on the effects of all TM elements on the elastic behaviors for g0 -Ni3Al phase. In fact, a systematic investigation of the TM-element (3d:SceZn, 4d:YeCd, 5d:HfeAu) dependence of the elastic properties of g0 -Ni3Al is necessary for the practical application of Nibased alloys, like the scenario for the Ni matrix according to Ref. [33]. We investigate the TM-element (3d:SceZn, 4d:YeCd, 5d:HfeAu) dependence of the ideal shear strength with the elastic properties of g0 -Ni3Al using ab initio calculations. The relationships between the d-electron number of the TM element, the shear modulus of the alloying system, and the ideal shear strength of the alloying system are discussed in detail.

t¼

1261

1 vEðgÞ ; VðgÞ vg

(2)

where E(g) is the total energy and V(g) is the volume of the calculated system. The ideal strength refers to the first maximum or minimum point of the stressestrain curve that is achieved during uniaxial loading, hydrostatic (isotropic) loading or shear deformation if the crystal lattice does not fail via other mechanisms. Herein, we defined the corresponding stress and strain at this critical point during the shear deformation as the first maximum (minimum) stress tFM and the first maximum (minimum) strain gFM . As verified in previous studies [34e39], the crystal may become unstable prior to reaching the maxima (minima) during the deformation, such as the mechanical instability (elastic instability) and/or phonon instability (inelastic instability). The mechanical-stability criteria, which were introduced by Born and Furth [40,41], were derived from the general positive definiteness requirement of the elastic stiffness coefficients matrix. A soft phonon can lower the crystal energy and a verification of the phonon stability of the deformed crystal is necessary in principle [36e38]. However, ab initio phonon calculations are time consuming. We verified the elastic stability in terms of the Born criteria and ignored the phonon instability. Thus, our predictions of an ideal shear strength for the g0 phase may be overestimated. In the shear deformation of g0 -Ni3Al, the method reported in Refs. [13,42] and a 48-atom ½112 2½110  ½111 single-impurity supercell (shown in Fig. 1(a)) were used to simulate the curve of the stress t vs. strain g for all studied systems. The initial orthorhombic lattice becomes monoclinic for a f111g < 112 > shear deformation and the elastic constant matrix for the deformed monoclinic g0 -Ni3Al is:

2 6 C11 6 C12 6   6 C13 Cij ¼ 6 6 0 6 6 C15 4 0

C12 C22 C23 0 C25 0

C13 C23 C33 0 C35 0

0 0 0 C44 0 C46

C15 C25 C35 0 C55 0

3 0 7 0 7 7 0 7 7 C46 7 7 0 7 5 C66

(3)

Therefore, the mathematically necessary and sufficient conditions of the positive definiteness requirement of the elastic stiffness coefficients matrix for this monoclinic lattice are given by:

2. Computational methods

C11 > 0;

(4a)

The shear deformation of the close-packed {111} plane along the < 112 > direction, which is the weakest shear-slip system of g0 Ni3Al, was studied in this work. For the f111g < 112 > slip system, the deformed lattice vector R was obtained as R ¼ R0D, where R0 is the un-deformed vector, and the deformation matrix D is

C44 > 0

(4b)

C11 C22  C 212 > 0;

(4c)

C44 C66  C 246 > 0;

(4d)

3

2 61 D¼6 40

g

0 1 0

07 07 5; 1

(1)

T ¼ C11 C22 C33 þ 2C12 C13 C23  C11 C 223  C22 C 213  C33 C 212 > 0 (4e)

where g is the shear strain. The stress t is given by:

      C 215 C 223  C22 C33 þ C 225 C 213  C11 C33 þ C 235 C 212  C11 C22 þ 2C15 C35 ðC13 C22  C12 C23 Þþ 2C15 C25 ðC12 C33  C13 C23 Þ þ 2C25 C35 ðC23 C11  C12 C13 Þ þ C55 T > 0:

(4f)

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modulus B were obtained by fitting the energyevolume curve with the Murnaghan equation of state [46]. The shear modulus G was determined from empirical Hill's values GH ¼ (GV þ GR)/2, where GV ¼ (C11eC12 þ 3C44)/5 and GR ¼ {4/[5(C11  C12)]þ3/(5C44)}1 are the upper Voigt [47] and lower Reuss [48] bounds, respectively. The Young's modulus E, Poisson ratio y, and anisotropic factor A [49] are obtained as E ¼ 9GHB/(GH þ 3B), y ¼ (3B e 2GH)/[2(3B þ GH)], and A ¼ 2C44/(C11  C12), respectively. We used total-energy methods based on the DFT that was implemented in the VASP code [50] for all calculations. The ioneelectron interaction was described by the projector augmented-wave (PAW) method [51] with the plane wave cutoff of 400 eV, whereas the exchangeecorrelation function was described by the generalized gradient approximation of Perdew et al. [52]. A MonkhorstePack [53] k-mesh of 7  5  9 and 9  9  9 was used for the 48- and 32-atom supercell, respectively. The energy convergence of the electronic self-consistency was 105 eV and the force criterion for ionic relaxation was 0.01 eV/Å. 3. Results and discussion

Fig. 1. (a) 48-Atom model used to test the ideal shear process in the f111g < 112 > slip direction. Blue and purple balls represent Ni and Al atoms, respectively. (b) Energy and stress and (c) mechanical stability criteria as a function of shear strain in the f111g < 112 > slip system for Ni3Al. The vertical dotted line in (c) marks the strain at which the mechanical stability criterion is violated first. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

The mechanical stability conditions of the monoclinic crystals are not all linear, but they are polynomial functions of elastic constants (up to four degree), which are different from those of [43]. Wu et al. [43] derived simple linear mechanical stability criteria for a monoclinic lattice. In fact, as mentioned by Mouhat [44], the reported conditions derived by Wu et al. [43] are necessary but not sufficient. In this work, we calculated the elastic constants and examined the mechanical stability criteria (Eq. (4)) at every single strain during the shear process. The maximum mechanical stability stress (tMMS ) and maximum mechanical stability strain (gMMS ), respectively, represent the corresponding stress and strain at which the mechanical stability criteria were violated first during the shear deformation. Here, the thirteen independent elastic constants of the monoclinic lattice were calculated by using the strainestress methods [45]. To investigate the elastic properties for all alloying systems, we used a 32-atom 2 [100]  2 [010]  2 [001] supercell with a singlepoint defect at its center. The equilibrium volume V and bulk

We conducted ab initio computational shear tests in pure Ni3Al for the weakest shear-slip system. The calculated energyestrain and stressestrain relationships of the non-doped Ni3Al are shown in Fig. 1(b). As shown in Fig. 1(b), the strain energy increases continuously with an increasing strain and exhibits an inflection at g ¼ 0.18. Consequently, the stress increases with an increasing strain until it reaches a maximum of 5.848 GPa at this inflection point. Our calculated tFM (the first maximum stress of the stressestrain curve) of the non-doped Ni3Al agrees well with the results of Wu et al. [20] and Wang et al. [19]. Fig. 1(c) plots the mechanical stability criteria of Ni3Al at every shear strain during the deformation. As shown in Fig. 1(c), the mechanical stability conditions are almost related linearly to the shear strain, and the lattice first becomes mechanically unstable in the mode of Eq. (4f) at a strain of ~17%. Consequently, the maximum mechanical stress (tMMS ) and maximum mechanical strain gMMS were determined to be 5.822 GPa and 17%, respectively, according to Fig. 1(b) and (c). The difference between the first maximum (minimum) strain (gFM ¼ 0:17) and the maximum mechanical stability strain (gMMS ¼ 0:18) of the f111g < 112 > slip system is negligible for g0 Ni3Al (this discrepancy may derive from computational error, such as the fact that the strain step may be not small). Although the crystal lattice here maintains its mechanical stability until the peak point of the stressestrain curve during the deformation tests, an examination of the elastic stability of the deformed crystal in terms of Born criteria is necessary to ascertain the stability of the solid materials, as verified in previous works [34,35,38,39,44,54e56]. Prior to predicting the effect of TM elements on the ideal shear strength of g0 -Ni3Al, we studied their effect on the elastic properties of g0 -Ni3Al. Table 1 summarizes the d-electron number of the TM elements and calculated equilibrium volume and elastic properties of the doped Ni3Al together with the corresponding values from experiments and other ab initio DFT calculations. The ordered g0 Ni3Al (L12 structure with space group of Pm3m) possesses two sublattices comprising the a-sublattice (face center) that was occupied by the Ni and the b-sublattice (cube corner) that was occupied by Al. According to our previous study (Ref. [39]), Pd, Pt, and Au exhibit a strong Ni site preference in g0 -Ni3Al. In this work, we predicted the elastic behaviors of Ni3Al-X (X ¼ 3d:SceZn, 4d:YeCd, 5d:HfeAu) with TM elements occupying the Al site (normal font in Table 1) and Ni3Al-X (X ¼ Ni, Pd, Pt and Au) with X replacing the Ni atom (italic font in Table 1). Here, the values for stoichiometric L12-Ni3Al are shown in bold italic font in Table 1. Note that the tFM =G ratio is 0.076 for L12-Ni3Al, which is close to

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Table 1 d-Electron number (d) of TM elements and calculated equilibrium volume (V, Å3/atom), elastic constants (Cij, GPa), bulk modulus (B, GPa), shear modulus (GH, GPa), Young's modulus (E, GPa), Poisson ratio (y), anisotropic factor (A), and GH/B of Ni3Al alloys doped with TM elements, along with experiments and other DFT-calculated values. X

d

V

C11

C12

C44

B

GH

E

y

A

GH/B

Sc Ti Tia V Cr Cra Crb Crc Mn Fe Co Ni Ni Nid Nie Nia Nib Nic Nif Nig Nih Cu Zn Y Ya Yc Zr Zrc Nb Nba Mo Moa Mob Mog Tc Ru Rua Rug Rh Pd Pd Ag Cd Hf Hfc Ta Taa Tag W Wa Wg Re Rea Reb Reg Os Ir Pt Ptc Pt Ptc Au Au

1 2 2 3 5 5 5 5 5 6 7 8 8 8 8 8 8 8 8 8 8 10 10 1 1 1 2 2 4 4 5 5 5 5 5 7 7 7 8 10 10 10 10 2 2 3 3 3 4 4 4 5 5 5 5 6 7 9 9 9 9 10 10

11.552 11.418 11.507 11.338 11.304 11.388 11.270 11.293 11.293 11.295 11.304 11.322 11.346 11.375 11.375 11.464 11.384 11.365 11.413 11.232 11.365 11.354 11.392 11.759 11.977 11.748 11.613 11.596 11.512 11.570 11.434 11.513 11.490 11.300 11.414 11.417 11.496 11.280 11.442 11.486 11.539 11.551 11.601 11.573 11.560 11.492 11.655 11.365 11.436 11.520 11.302 11.4162 11.493 11.384 11.279 11.419 11.449 11.497 11.496 11.554 11.556 11.573 11.622

223.29 232.99 246.80 237.36 238.36 244.30 246.30 249.00 237.16 233.77 230.18 226.10 227.25 227.00 225.00 243.80 232.10 242.00 225.30 233.50 228.42 224.10 223.95 215.73 233.00 232.00 228.79 237.00 235.62 249.80 239.09 252.00 240.10 246.40 239.48 236.46 246.40 245.40 230.11 223.45 224.60 219.18 218.23 230.54 244.00 237.31 248.60 242.40 241.14 251.20 249.30 242.71 257.60 249.70 251.60 242.29 235.69 228.28 243.00 229.22 235.00 221.57 224.82

144.12 149.52 148.80 150.99 151.48 151.90 157.80 155.00 151.79 152.66 153.16 153.11 150.01 148.00 149.00 148.70 151.10 152.00 157.60 157.20 151.76 150.98 149.38 138.76 146.40 151.00 146.84 149.00 150.66 152.40 152.25 152.70 153.10 158.20 152.30 152.62 150.20 159.80 153.53 152.61 149.71 148.45 146.30 148.14 158.00 151.57 150.50 156.70 153.00 155.10 160.10 153.01 158.30 157.35 159.30 152.57 153.93 154.29 155.00 151.36 150.00 151.42 147.94

116.32 123.29 128.00 126.60 127.61 126.80 129.80 127.00 126.57 122.88 119.90 117.04 122.60 120.00 124.00 123.40 124.00 125.00 121.10 128.90 116.89 118.01 119.51 105.89 112.30 108.00 116.83 119.00 123.72 125.50 127.15 128.90 125.60 132.50 127.89 124.70 124.80 129.30 118.48 112.37 118.68 112.06 113.28 119.47 122.00 125.19 125.80 130.70 128.48 130.10 133.90 129.59 131.90 131.50 134.80 128.68 122.22 115.35 117.00 120.24 122.00 110.51 116.66

170.51 177.34 181.50 179.78 180.44 182.70 187.30 186.00 180.25 179.70 178.83 177.44 175.76 174.30 174.30 180.40 178.10 182.00 180.20 182.60 177.31 175.35 174.24 164.42 175.30 178.00 174.16 178.00 178.98 184.90 181.20 185.80 182.10 187.60 181.36 180.57 182.30 188.30 179.05 176.23 174.67 172.02 170.28 175.61 187.00 180.15 183.20 185.30 182.38 187.20 189.80 182.91 191.40 188.10 190.10 182.48 181.18 178.96 184.00 177.31 178.00 174.80 173.57

75.57 79.93 87.10 82.33 82.92 84.60 84.39 85.00 81.94 78.88 76.16 73.49 77.29 76.95 77.30 84.20 79.28 83.00 72.90 79.30 74.85 73.92 75.06 70.59 76.60 73.00 76.81 80.00 80.66 86.00 82.72 87.90 82.16 85.30 83.14 80.63 85.10 83.10 75.44 70.85 74.87 70.69 71.63 78.02 80.00 81.54 86.20 83.70 83.73 87.30 86.30 84.75 89.20 86.49 87.80 84.39 78.87 73.23 79.00 76.61 80.00 69.87 74.84

197.54 208.48 225.30 214.27 215.71 219.90 220.11 222.00 213.47 206.44 200.07 193.73 202.22 201.25 202.10 218.50 207.11 217.00 192.74 207.90 196.85 194.43 196.90 185.27 200.70 192.00 200.89 209.00 210.37 223.30 215.39 227.80 214.27 222.00 216.36 210.55 221.00 217.30 198.45 187.44 196.52 186.53 188.46 203.87 210.00 212.54 223.60 220.10 217.85 226.70 224.80 220.24 231.60 224.99 228.20 219.36 206.63 193.31 208.00 200.90 209.00 184.97 196.31

0.307 0.304 0.293 0.301 0.301 0.299 0.304 0.301 0.303 0.309 0.314 0.318 0.308 0.308 0.307 0.298 0.306 0.301 0.322 0.310 0.315 0.315 0.312 0.312 0.309 0.320 0.308 0.304 0.304 0.299 0.302 0.296 0.304 0.302 0.301 0.306 0.298 0.308 0.315 0.323 0.312 0.319 0.316 0.307 0.313 0.303 0.297 0.304 0.301 0.298 0.303 0.299 0.298 0.301 0.300 0.300 0.310 0.320 0.312 0.311 0.304 0.324 0.311

2.94 2.95 2.61 2.93 2.94 2.74 2.93 2.73 2.97 3.03 3.11 3.21 3.17 3.04 3.26 2.60 3.06 2.78 3.58 3.38 3.05 3.23 3.21 2.75 2.59 2.67 2.85 2.70 2.91 2.57 2.93 2.59 2.89 3.01 2.93 2.98 2.60 3.02 3.09 3.17 3.17 3.17 3.15 2.90 2.84 2.92 2.56 3.05 2.92 2.71 3.00 2.89 2.65 2.85 2.92 2.87 2.99 3.12 2.67 3.09 2.87 3.15 3.04

0.443 0.451 0.480 0.458 0.460 0.463 0.451 0.451 0.455 0.439 0.426 0.414 0.440 0.442 0.443 0.467 0.445 0.445 0.405 0.434 0.422 0.422 0.431 0.429 0.437 0.410 0.441 0.449 0.451 0.465 0.457 0.473 0.451 0.455 0.458 0.447 0.467 0.441 0.421 0.402 0.429 0.411 0.421 0.444 0.428 0.453 0.471 0.452 0.459 0.466 0.455 0.463 0.466 0.460 0.462 0.462 0.435 0.409 0.429 0.432 0.449 0.400 0.431

Values with roman and italic letters represent Ni3Al alloys doped with TM elements at Al and Ni sites, respectively. a [24]. b [25]. c [28]. d [58]. e [59]. f [62]. g [26]. h [61].

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occupied Ni-site exhibit better elastic constants and elastic moduli (except for C12 and B) than the corresponding Al-site doped systems. As shown in Table 1, our calculated elastic constants and elastic moduli are consistent with previous calculated results [24e26,28,60,61]. For instance, Li et al. [25] studied the effect of Re, Cr, and Mo on the elastic properties of Ni3Al and found that improvements in elastic moduli (B, G, and E) occur in the following order: Re > Cr > Mo. Moreover, the ability of W, Re, Mo, Ta, and Ru to increase the elastic moduli, studied by Gong et al. [26], was in the order: Re > W > Mo > Ta > Ru, which agrees well with our predicted results. Fig. 3(a) plots the variation of tFM as a function of d-electron number of the TM elements for all studied Ni3Al alloys. Similar to the elastic properties, we performed shear deformation tests for Ni3Al-X (X ¼ 3d:SceZn, 4d:YeCd, 5d:HfeAu) with the X occupied Al-site (solid symbol in Fig. 3(a)) and Ni3Al-X (X ¼ Pd, Pt and Au) with the X occupied Ni-site (open symbol in Fig. 3(a)). Similar to the shear modulus, tFM displays a concave and a non-strict parabolic dependence on the number of d electrons for 3d, 4d, and 5d series TM elements when they occupied the Al-site. Here, alloying elements with d ¼ 4, 5, and 6 (such as Re, Tc, Os, W, Mo, V, and Ta) can increase the shear strength of the g0 phase, and Re manifests the

Fig. 2. Elastic constants (a) and elastic moduli (b) for Ni3Al alloys with doped 3d, 4d, and 5d TM elements. The solid and open symbols represent the TM elements that occupied Al and Ni sites in g0 -Ni3Al, respectively. The dashed lines mark the corresponding value of the elastic properties for undoped Ni3Al.

the Frenkel's ratio (0.1) [55,57]. As shown in Table 1, our calculated equilibrium volume, elastic constants, and modulus of the pure L12Ni3Al agree well with the experimental values [58,59] and other calculations [24e26,28,60,61], which confirms the reliability of our computational methods. To analyze the influence of the TM elements on the elastic properties of the Ni3Al alloys, the calculated elastic constants and elastic moduli for all calculated systems in Table 1 are plotted in Fig. 2(a) and Fig. 2(b), respectively. C11 and C44 measure the ability of the crystal to resist normal strain and shear deformation. The bulk modulus, shear modulus, and Young's modulus represent the resistance of the crystal to volume, shear shape change, and stiffness, respectively. According to Table 1 and Fig. 2(a), besides Sc, Ni, Cu, Zn, Y, Pd, Ag, Cd, and Au, the other alloying elements can increase the C11 of the system to varying degrees when they replace the Al atom in Ni3Al; in particular, alloying elements with half-filled d bonds in each series (Cr, Tc, and Re for 3d, 4d, and 5d, respectively) manifest the greatest efficacy. Similar to C11, the TM elements at the center of each series (i.e., V, Cr, Mn, Fe, Nb, Mo, Tc, Ru, Ta, W, Re, and Os) can increase C44, B, G, and E of g0 -Ni3Al when they occupy the Al site, which indicates that these elements can improve the resistance to deformation and mechanical strength of alloys in the practical application significantly. Unlike these elements, the TM elements at the beginning and end of the series (such as Sc, Cu, Zn, Y, Ag, Cd, and Au) will decrease the elastic constants and elastic moduli when they occupy the Al site in Ni3Al, and Y exhibits the greatest reduction. Note that the doped g0 -Ni3Al with Pd, Pt, and Au

Fig. 3. (a) First maximum stress (tFM ) of alloyed g0 phase doped with 3d, 4d, and 5d Ni3 AlX 3 Al TM elements. (b) Correlation between DtFM (DtFM ¼ tFM  tNi FM ) expressed in percent (%) and DG (DG ¼ GNi3 AlX  GNi3 Al ) expressed in percent (%) of stoichiometric Ni3Al for all Ni3Al alloys. Data points mark where TM elements substitute the Al-atom (solid symbols) and the Ni-atom (open symbols) in Ni3Al.

M. Wen et al. / Journal of Alloys and Compounds 806 (2019) 1260e1266

greatest efficacy for improving the tFM of Ni3Al. However, Ni3Al alloys that were doped with elements at the beginning and end of the series (i.e., Y, Pd, Pt, Cu, Zn, Ag, Cd, and Au) exhibit a smaller tFM than pure Ni3Al, where the Ni3AleAg displays the smallest shear strength of all researched systems. A 48-atom ½112 2½110  ½111 supercell with a single impurity was used to research the shear strength in the f111g < 112 > slip direction. Unlike the Al-site defect models, two non-equivalent sites of Ni sublattices exist for the Ni-site defect Ni3Al during the f111g < 112 > shear process. As shown in Fig. 1(a), one Ni sublattice is the Ni-site of the ½110 crystal column that is composed of Ni atoms (i.e., Ni1 in Fig. 1(a)), whereas the other Ni sublattice is the Ni-site of the ½110 crystal column that is composed of Ni atoms that alternate with Al atoms (i.e., Ni2 in Fig. 1(a)). Hence, it was essential to test the first-principles shear calculations for these two nonequivalent Ni-sites for Ni3Al-X (X ¼ Pd, Pt, and Au). The smallest tFM for the Ni-site defect g0 phase are presented in Fig. 3(a) by open symbols. Compared with the Al-site defected model, the Ni-site defected Ni3Al alloy doping with Pd and Au exhibits a larger tFM , whereas doping with Pt shows a smaller tFM . Clearly, either the Nior the Al-site is occupied, and the doping of Pd, Pt, and Au reduces the ideal strength, which is similar to the situation of the shear modulus (see Fig. 2(b)). Our predicted values of tFM of Ni3Al-X (X ¼ Re, W, Mo, Cr, Ta, Ru, and Co) are consistent with the values of [20]. We plot the relationship between the shear modulus and the ideal shear strength of alloying Ni3Al in Fig. 3(b). The G has a nearly linear dependence with tFM . The elements in the first quadrant (such as Re, Os, W, Mo, Cr, Ru, V, Ta, Nb, Ir, and Hf) can increase G and tFM . It is expected that the doping of Re, Mo, W, Cr, Ru, Ta, Nb, and V can improve the mechanical strength of Ni-based alloys, which agrees with previous experiments [6,7,63e65]. As a key alloying element, the concentration of Re is an important criterion for distinguishing the first, second, and third generations of Nibased superalloys [2,66]. Here, our results show that the Redoped system exhibits the most outstanding elastic properties and shear strength. From our previous research [39,60], we can predict the strengthening effect of the alloying elements on the ideal strength and shear modulus of Ni3Al mainly because of the interaction between impurities and host atoms. Pd, Pt, and Au, along with other elements in the third quadrant (such as Cu, Zn, and Y) will decrease G and tFM . It is well known that the addition of Y, along with Hf, Cr, and Zr, can improve the anti-oxidation behavior of Ni-based alloys [67e71], rather than the mechanical strength. Finally, the addition of elements in the second quadrant (i.e., Zr, Rh, Sc, and Co) increases the tFM of the g0 phase slightly, whereas they decrease the shear modulus of the alloys slightly. 4. Conclusions Ab initio calculations based on DFT have been performed to investigate the 3d, 4d, and 5d TM-element dependence of the elastic properties and ideal shear strength of g0 -Ni3Al. By combining the calculated stressestrain curve and the mechanical stability conditions, the values of tFM and tMMS of stoichiometric L12-Ni3Al in the f111g < 112 > slip direction are 5.848 GPa and 5.822 GPa, respectively. The TM elements at the center of each period (i.e., V, Cr, Mn, Fe, Nb, Mo, Tc, Ru, Ta, W, Re, and Os) can increase the C11, C44, B, G, E, and tFM of g0 -Ni3Al when they occupy the Al site. Alloying elements with half-filled d-orbitals in each series (Cr, Tc, and Re for 3d, 4d, and 5d, respectively) exhibit the greatest enhancement. However, the TM elements at the beginnings and ends of the series (such as Cu, Zn, Y, Ag, Cd, and Au) will decrease the elastic constants, elastic moduli and ideal shear strength when they occupy the Al site in Ni3Al, where Y-doped

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Ni3Al manifests the smallest elastic constants and elastic moduli and Au-doped Ni3Al exhibits the smallest tFM . Our results show that Ni-site-defected Ni3Al alloy doping with Pd and Au exhibits a larger C11, C44, G, E and tFM than the Al-site defected Ni3Al. However, either a Ni or an Al site is occupied, and the doping of Pd, Pt, and Au could reduce the C44, G, E, and tFM of the g0 phase. Our results agree well with previous experimental and computational data and will help understand and guide the design of rational Nibased superalloys. Acknowledgments We are grateful to Y. Hu, B. Lu, and S. Liu for beneficial discussions. This work was supported by the National Natural Science Foundation of China (Grant No. 11804057), Natural Science Foundation of Guangdong, China (Grants No. 2017B030306003) and National Key R&D Program of China (Grant No. 2017YFB0701500). The simulations were carried out on the “Lvliang” cluster system of the National University of Defense Technology, Hunan, China. References [1] C.T. Sims, N.S. Stoloff, W.C. Hagel, Superalloys II, John Wiley & Sons, New York, 1987. [2] R.C. Reed, The Superalloys: Fundamentals and Applications, Cambridge university press, New York, 2006. [3] A. Kelly, N.H. Macmillan, Strong Solids, third ed. ed., Clarendon Press, Oxford, 1986.

[4] J. Pokluda, M. Cerný, M. Sob, Y. Umeno, Ab initio calculations of mechanical properties: methods and applications, Prog. Mater. Sci. 73 (2015) 127e158. [5] R. Reed, T. Tao, N. Warnken, Alloys-by-design: application to nickel-based single crystal superalloys, Acta Mater. 57 (2009) 5898e5913. [6] A.C. Yeh, S. Tin, Effects of Ru and Re additions on the high temperature flow stresses of Ni-base single crystal superalloys, Scr. Mater. 52 (2005) 519e524. [7] E. Fleischmann, M.K. Miller, E. Affeldt, U. Glatzel, Quantitative experimental determination of the solid solution hardening potential of rhenium, tungsten and molybdenum in single-crystal nickel-based superalloys, Acta Mater. 87 (2015) 350e356. [8] A. Sato, H. Harada, T. Yokokawa, T. Murakumo, Y. Koizumi, T. Kobayashi, H. Imai, The effects of ruthenium on the phase stability of fourth generation Ni-base single crystal superalloys, Scr. Mater. 54 (2006) 1679e1684. [9] M. Xie, R. Helmink, S. Tin, The influence of Ta on the solidification microstructure and segregation behavior of g(Ni)/g0 (Ni3Al)ed(Ni3Nb) eutectic Nibase superalloys, J. Alloy. Comp. 562 (2013) 11e18. [10] C. Ai, L. Liu, J. Zhang, M. Guo, Z. Li, T. Huang, J. Zhou, S. Li, S. Gong, G. Liu, Influence of substituting Mo for W on solidification characteristics of Recontaining Ni based single crystal superalloy, J. Alloy. Comp. 754 (2018) 85e92. [11] S.V. Raju, B.K. Godwal, A.K. Singh, R. Jeanloz, S.K. Saxena, High-pressure strengths of Ni3Al and Ni-Al-Cr, J. Alloy. Comp. 741 (2018) 642e647. [12] T. Murakumo, T. Kobayashi, Y. Koizumi, H. Harada, Creep behaviour of Ni-base single-crystal superalloys with various g0 volume fraction, Acta Mater. 52 (2004) 3737e3744. [13] D. Roundy, C.R. Krenn, M.L. Cohen, J.W. Morris Jr., Ideal shear strengths of fcc aluminum and copper, Phys. Rev. Lett. 82 (1999) 2713.
k, D. Legut, J. Fiala, V. Vitek, The role of ab initio electronic [14] M. Sob, M. Fria structure calculations in studies of the strength of materials, Mater. Sci. Eng., A 387 (2004) 148e157. [15] C.R. Krenn, D. Roundy, J.W. Morris, M.L. Cohen, Ideal strengths of bcc metals, Mater. Sci. Eng., A 319 (2001) 111e114.
[16] M. Cerný, J. Pokluda, Ideal tensile strength of cubic crystals under superimposed transverse biaxial stresses from first principles, Phys. Rev. B 82 (2010) 174106. [17] A. Gouldstone, H.J. Koh, K.Y. Zeng, A.E. Giannakopoulos, S. Suresh, Discrete and continuous deformation during nanoindentation of thin films, Acta Mater. 48 (2000) 2277e2295. [18] C.R. Krenn, D. Roundy, M.L. Cohen, D.C. Chrzan, J.W. Morris Jr., Connecting atomistic and experimental estimates of ideal strength, Phys. Rev. B 65 (2002) 134111. [19] Y.J. Wang, C.Y. Wang, Influence of the alloying element Re on the ideal tensile and shear strength of g0 -Ni3Al, Scr. Mater. 61 (2009) 197e200. [20] X. Wu, C. Wang, Effect of the alloying element on the temperature-dependent ideal shear strength of g0 -Ni3Al, RSC Adv. 6 (2016) 20551e20558. [21] Y. Chen, S. He, Z. Yi, P. Peng, Impact of correlative defects induced by double Re-addition on the ideal shear strength of g0 -Ni3Al phases, Comput. Mater. Sci. 152 (2018) 408e416. [22] Y. Wang, C. Wang, Effect of alloying elements on the elastic properties of g-Ni and g'-Ni3Al from first-principles calculations, in: MRS Proceedings,

1266

M. Wen et al. / Journal of Alloys and Compounds 806 (2019) 1260e1266

Cambridge Univ Press, 2009. 1224-FF1205-1231. [23] X. Wu, C. Wang, Density functional theory study of the thermodynamic and elastic properties of Ni-based superalloys, J. Phys. Condens. Matter 27 (2015) 295401. [24] Q. Wu, S. Li, Alloying element additions to Ni3Al: site preferences and effects on elastic properties from first-principles calculations, Comput. Mater. Sci. 53 (2012) 436e443. [25] C. Li, J. Shang, L. Kou, Z. Yue, Synergistic effect of alloying elements doping and external pressure on the elastic property of Ni3Al: a first-principles study, AIP Adv. 5 (2015), 077136. [26] W. Gong, W. Zhao, N. Miao, J. Zhou, Z. Sun, S. Li, S. Gong, Strengthening effects of alloying elements W and Re on Ni3Al: a first-principles study, Comput. Mater. Sci. 144 (2018) 23e31. [27] W. Zhao, Z. Sun, S. Gong, Synergistic effect of co-alloying elements on site preferences and elastic properties of Ni3Al: a first-principles study, Intermetallics 65 (2015) 75e80. [28] D.E. Kim, S.L. Shang, Z.K. Liu, Effects of alloying elements on elastic properties of Ni3Al by first-principles calculations, Intermetallics 18 (2010) 1163e1171. [29] A. Kumar, A. Chernatynskiy, M. Hong, S.R. Phillpot, S.B. Sinnott, An ab initio investigation of the effect of alloying elements on the elastic properties and magnetic behavior of Ni3Al, Comput. Mater. Sci. 101 (2015) 39e46. [30] J.P. Du, C.Y. Wang, T. Yu, Construction and application of multi-element EAM potential (NieAleRe) in g/g0 Ni-based single crystal superalloys, Model. Simul. Mater. Sci. Eng. 21 (2013), 015007. [31] X. Yang, W. Hu, The alloying element dependence of the local lattice deformation and the elastic properties of Ni3Al: a molecular dynamics simulation, J. Appl. Phys. 115 (2014) 153507. [32] Q.N. Fan, C.Y. Wang, T. Yu, J.P. Du, A ternary NieAleW EAM potential for Nibased single crystal superalloys, Phys. B Condens. Matter 456 (2015) 283e292. [33] S.L. Shang, D.E. Kim, C.L. Zacherl, Y. Wang, Y. Du, Z.K. Liu, Effects of alloying elements and temperature on the elastic properties of dilute Ni-base superalloys from first-principles calculations, J. Appl. Phys. 112 (2012), 053515. [34] F. Milstein, Theoretical strength of a perfect crystal, Phys. Rev. B 3 (1971) 1130. [35] R. Hill, F. Milstein, Principles of stability analysis of ideal crystals, Phys. Rev. B 15 (1977) 3087. [36] D.M. Clatterbuck, C.R. Krenn, M.L. Cohen, J.W. Morris Jr., Phonon instabilities and the ideal strength of aluminum, Phys. Rev. Lett. 91 (2003) 135501. [37] S.M.M. Dubois, G.M. Rignanese, T. Pardoen, J.C. Charlier, Ideal strength of silicon: an ab initio study, Phys. Rev. B 74 (2006) 235203.


a k, M. Cerný, [38] P. Reh M. Sob, Mechanical stability of Ni and Ir under hydrostatic and uniaxial loading, Model. Simul. Mater. Sci. Eng. 23 (2015), 055010. [39] M. Wen, C.-Y. Wang, Transition-metal alloying of g’-Ni3Al: effects on the ideal uniaxial compressive strength from first-principles calculations, Phys. Rev. B 97 (2018), 024101. [40] M. Born, On the stability of crystal lattices. I, in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 1940, pp. 160e172. [41] M. Born, R. Fürth, The stability of crystal lattices. III: an attempt to calculate the tensile strength of a cubic lattice by purely static considerations, in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 1940, pp. 454e465. [42] D. Roundy, C.R. Krenn, M.L. Cohen, J.W. Morris Jr., The ideal strength of tungsten, Philos. Mag. A 81 (2001) 1725e1747. [43] Z.J. Wu, E.J. Zhao, H.P. Xiang, X.F. Hao, X.J. Liu, J. Meng, Crystal structures and elastic properties of superhard IrN2 and IrN3 from first principles, Phys. Rev. B 76 (2007), 054115. [44] F. Mouhat, F.-X. Coudert, Necessary and sufficient elastic stability conditions in various crystal systems, Phys. Rev. B 90 (2014) 224104. [45] S. Shang, Y. Wang, Z.-K. Liu, First-principles elastic constants of a- and q-Al2O3, Appl. Phys. Lett. 90 (2007) 101909. [46] F.D. Murnaghan, The compressibility of media under extreme pressures, Proc. Natl. Acad. Sci. 30 (1944) 244e247.

[47] W. Voigt, Lehrbuch der Kristallphysik, 1928. Ann Arbor, Mich, Leipzig Berlin. [48] A. Reuss, Berechnung der fließgrenze von mischkristallen auf grund der plastizit€ atsbedingung für einkristalle, J. Appl. Math. Mech. 9 (1929) 49e58. [49] D.H. Chung, W.R. Buessem, The elastic anisotropy of crystals, J. Appl. Phys. 38 (1967) 2010e2012. [50] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54 (1996) 11169e11186. [51] G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59 (1999) 1758e1775. [52] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865e3868. [53] H.J. Monkhorst, J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188e5192. [54] A. Breidi, S.G. Fries, A.V. Ruban, Ideal compressive strength of fcc Co, Ni, and Ni-rich alloys along the <001> direction: a first-principles study, Phys. Rev. B 93 (2016) 144106.


[55] J. Pokluda, M. Cerný, P. Sandera, M. Sob, Calculations of theoretical strength: state of the art and history, J. Comput. Aided Mater. Des. 11 (2004) 1e28. [56] J. Wang, S. Yip, S.R. Phillpot, D. Wolf, Crystal instabilities at finite strain, Phys. Rev. Lett. 71 (1993) 4182e4185. €tsgrenze und der festigkeit kristallinischer [57] J. Frenkel, Zur theorie der elastizita €rper, Z. Phys. 37 (1926) 572e609. ko [58] M.J. Mehl, B.M. Klein, D.A. Papaconstantopoulos, First principles calculations of elastic properties, in: Intermetallic Compounds Principles and Practice, John Wiley & Sons, New York, 1995, pp. 195e210. [59] S.V. Prikhodko, H. Yang, A.J. Ardell, J.D. Carnes, D.G. Isaak, Temperature and composition dependence of the elastic constants of Ni3Al, Metall. Mater. Trans. A 30 (1999) 2403e2408. [60] M. Wen, C.-Y. Wang, Effect of alloying elements on the ideal strength and charge redistribution of g0 -Ni3Al: a first-principles study of tensile deformation, RSC Adv. 6 (2016) 77489e77498. [61] X. Gao, J. Wang, X. Wu, R. Wang, Z. Jia, Effects of alloying atoms on antiphase boundary energy and yield stress anomaly of L12 intermetallics: firstprinciples study, Crystals 8 (2018) 96. [62] H. Hou, Z. Wen, Y. Zhao, L. Fu, N. Wang, P. Han, First-principles investigations on structural, elastic, thermodynamic and electronic properties of Ni3X (X ¼ Al, Ga and Ge) under pressure, Intermetallics 44 (2014) 110e115. [63] M. Fahrmann, W. Hermann, E. Fahrmann, A. Boegli, T. Pollock, H. Sockel, Determination of matrix and precipitate elastic constants in (geg0 ) Ni-base model alloys, and their relevance to rafting, Mater. Sci. Eng., A 260 (1999) 212e221. [64] M.V. Nathal, R.A. MacKay, R.G. Garlick, Temperature dependence of g-g'lattice mismatch in nickel-base superalloys, Mater. Sci. Eng. 75 (1985) 195e205. € ken, Micromechanical characterisation of the influence of [65] K. Durst, M. Go rhenium on the mechanical properties in nickel-base superalloys, Mater. Sci. Eng., A 387 (2004) 312e316. [66] P. Caron, T. Khan, Evolution of Ni-based superalloys for single crystal gas turbine blade applications, Aero. Sci. Technol. 3 (1999) 513e523. [67] D.P. Whittle, J. Stringer, Improvements in high temperature oxidation resistance by additions of reactive elements or oxide dispersions, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 295 (1980) 309e329. [68] S.-J. Park, S.-M. Seo, Y.-S. Yoo, H.-W. Jeong, H. Jang, Statistical study of the effects of the composition on the oxidation resistance of Ni-based superalloys, J. Nanomater. (2015) 11, 2015. [69] H. Guo, L. Sun, H. Li, S. Gong, High temperature oxidation behavior of hafnium modified NiAl bond coat in EB-PVD thermal barrier coating system, Thin Solid Films 516 (2008) 5732e5735. [70] D. Prajitno, B. Gleeson, D.J. Young, The cyclic oxidation behaviour of a-Cr þ bNiAl alloys with and without trace Zr addition, Corros. Sci. 39 (1997) 639e654. [71] C.A. Barrett, Effect of 0.1 at.% zirconium on the cyclic oxidation resistance ofbNiAl, Oxid. Metals 30 (1988) 361e390.