Trans. Nonferrous Met. Soc. China 31(2021) 2013-2023

Impact of replacement of Re by W on dislocation slip mediated creeps of γ′-Ni₃Al phases

Zhou YI, Yun-lei XU, Ping PENG, Jiang-hua CHEN

School of Materials Science and Engineering, Hunan University, Changsha 410082, China

Received 6 August 2020; accepted 18 February 2021

Abstract: The anomalous flow behavior of γ′-Ni₃Al phases at high temperature is closely related to the cross-slip of 1/2＜110＞{111} super-partial dislocations. Generalized stacking fault energy curves (i.e., Γ-surfaces) along the lowest energy path can provide a great deal of information on the nucleation and movement of dislocations. With the first-principles calculation, the interplay between Re and W, Mo, Ta, Ti doped at preferential sites and their synergetic influence on Γ-surfaces and ideal shear strength (τ_max) in γ′-Ni₃Al phases are investigated. Similar to single Re-addition, the Suzuki segregation of W at stacking faults is demonstrated to enable to impede the movement of 1/6＜112＞{111} Shockley partial dislocations and promote the cross-slip of 1/2＜110＞{111} super-partial dislocations. With the replacement of a part of Re by W, a decreased  indicates that the anomalous flow behavior of γ′ phases at high temperature is not as excellent as the double Re-addition, but an increased τ_max means that the creep rupture strength of Ni-based single crystal superalloys can be benefited from this replacement to some extent, especially in the co-segregation of Re and W at Al-Al sites. As the interaction between X1_Al and X2_Al point defects is characterized by an correlation energy function , it is found that both strong attraction and strong repulsion are unfavarable for the improvement of yield strengths of γ′ phase.

Key words: Ni-based single crystal superalloy; γ′-Ni₃Al; generalized stacking fault energy; ideal shear strength; dislocation; cross-slip

1 Introduction

Ni-based single crystal (SC) superalloys are widely applied in most aero-engine combustors, because of outstanding ability to retain strength and withstand creep at high temperature [1,2]. Their excellent comprehensive performances rely primarily on the microstructure of γ′ precipitates with L1₂ structure embedded coherently in fcc γ matrixes [3], as well as many kinds of refractory elements such as Re, W, Mo, Ta, Ti, Cr, and Ru. Re is proven to be a crucial element in the creep of Ni-based SC superalloys, and the indispensable strengthening is usually called as Re-effect [4]. However, an excessive addition of Re leads to some detrimental topologically close-packed (TCP) phases to be deposited [5]. Also, a limited reserve of this expensive Re element motivates the development of low Re-addition superalloys. Recently, several investigations [6,7] have demonstrated that Re can be partially replaced by other refractory metals. For example, as a half amount of Re in CMSX-4 superalloys is replaced by W, almost the same creep properties, e.g., minimum creep rate, time to reach 2% strain and time for failure at 1123 and 1373 K, can be obtained [6]. Moreover, calculated thermal expansion coefficients and bulk moduli of γ′-Ni₃Al phases at finite temperature and certain pressure suggest that W is of potential possibility to replace Re in Ni-based SC superalloys [7]. However, for the flow behavior of γ′-Ni₃Al phases, these results don’t provide how to be affected by partial replacements of W for Re. Therefore, a cooperative effect of duplex additions of Re and W on the dislocation slip mediated creep in γ′-Ni₃Al phases is desired.

Previous investigations showed that the configuration and movement of dislocations play an important role in the plastic deformation of γ′ phases [8,9]. At the early stage of creep, the flow behavior of Ni-based SC superalloys mainly depends on the activation of 1/2＜110＞{111} dislocations in γ channels [10]. With the increase of dislocation densities in γ channels, some 1/2＜110＞{111} matrix pairs can combine to form ＜110＞{111} super dislocation at the γ/γ′ interface. Once inside the γ′ phase, these ＜110＞{111} super dislocations are easily dissociated [11]. At high temperature and low stress, the most possible deformation mechanism is to be sheared by 1/2＜110＞{111} pairs connected by an anti-phase boundary (APB) fault, i.e., ＜110＞{111}→1/2＜110＞{111}+APB+1/2＜110＞{111} [12], and the 1/2＜110＞{111} super-partial dislocation can be further dissociated into two 1/6＜112＞{111} Shockley partial dislocations separated by a complex stacking fault (CSF), i.e., 1/2＜110＞{111}→1/6＜112＞{111}+CSF+1/6＜112＞{111} [12]. In this case, one ＜110＞{111} super dislocation can be described by four Shockley partial dislocations (i.e., P₁=P₃ and P₂=P₄) bounded by one APB and two CSFs (Fig. 1(a)), and a mutual elastic interaction among 1/2＜110＞{111} super-partial dislocations can generate a tangential force perpendicular to {111} slip plane [13]. In general, a high APB energy will make APB ribbons among 1/2＜110＞{111} super-partial dislocations narrow, leading to stronger tangential force and increased driving force of cross-slips of 1/2＜110＞{111} super-partial dislocations. It is widely accepted that this cross-slip of screw dislocations should be responsible for the anomalous flow behavior of γ′ phases at high temperature [14]. For the cross-slip of 1/2＜110＞{111} dislocations, a temporary recombination of 1/6＜112＞{111} Shockley partial dislocations [13] is required, and a lifted CSF energy can lessen the width of CSF regions and lower the constriction energy of 1/6＜112＞{111} Shockley partial dislocations [14]. Owing to low APB energy in cubic {001} planes relative to octahedral {111} planes, Shockley partial dislocations P₃ and P₄ as a leading pair may cross- slip to {001} planes from {111} planes. As a result, Kear–Wilsdorf (K-W) lock [15] can be formed. This K-W lock will restrict the movement of P₃ and P₄, which makes Shockley partial dislocations P₁ and P₂ as a trailing pair stay still on {111} planes (Fig. 1(b)). Thus, the tendency of cross-slips of leading 1/2＜110＞{111} super-partial dislocations can be utilized to evaluate the ability of plastic deformations of γ′ phases to some extent, and the easier the cross-slip of 1/2＜110＞{111} super-partial dislocations is, the larger the yield strength of γ′ phases at high temperature is [16].

To obtain the information of dislocation slip mediated creeps, lots of researches on cross-slips of 1/2＜110＞{111} super-partial dislocations in γ' phases have been reported [13-15,17]. For example, the influence of refractory metals on generalized stacking fault energy curves (Γ-surfaces) [14,18] and ideal shear strengths (τ_max) [19,20] in  and slip systems were systematically investigated. It is pointed out that W-addition not only remarkably raises the APB energy γ_APB and CSF energy γ_CSF on (111) planes, but also hoists unstable stacking fault energies γ_usf and ideal shear strengths τ_max, and the magnitude is preceded only by Re in the sequence of Re>W> Mo>Ta>Ti>Ru, indicating that W-addition is beneficial to impeding the emission, propagation and movement of 1/2＜110＞{111} super-partial dislocations and 1/6＜112＞{111} Shockley partial dislocations. Moreover, it is demonstrated that the cross-slip activation enthalpy of 1/2＜110＞{111} super-partial dislocations can be lowered by the addition of Re, W and Mo [14], which means that W-addition may also accelerate the cross-slips and promote sessile dislocation locks to be generated, similarly to Re-addition. Recently, in the view of multi-component characteristics of Ni-based SC superalloys, a density functional theory (DFT) based cluster expansion calculation of composition- dependent γ_APB was also performed by DODARAN et al [13]. Their results showed that the impact of individual solute elements X on γ_APB obtained in a ternary L1₂-Ni₃Al(X) crystal cannot be directly translated to a multi-solute γ' phases due to a correlation and synergistic effect among solute elements. For this reason, the influence of multi-additions of Re and W on τ_max in  and slip directions was further investigated [8,9]. It is found that a double Re-addition at preferential Al-Al sites may further elevate τ_max in the single Re-addition system, and τ_max in a multi-addition of Re and W is even larger than that of the double Re-addition, but an extra addition of Re or W at a Ni site is not conducive  to the improvement of τ_max in and  slip directions. This result once again indicates that the correlation and synergistic effect among solute elements cannot be ignored, and the interplay among refractory metals plays an important role in the dislocation slip mediated creeps of γ′ phases.

Fig. 1 Schematic diagrams of cross-slip of screw dislocations in L1₂-Ni₃Al crystals

In this work, several stacking fault models of L1₂-Ni₃Al(X) crystals with X=Re, W, Mo, Ta and Ti are constructed in order to investigate the interplay among refractory metals in APB and CSF in γ′ phases. And then, exact calculations of Γ-surfaces and τ_max in  and  slip directions are carried out. Finally, the influence of various multiple additions on the cross-slip of 1/2＜110＞{111} super-partial dislocations and the nucleation and movement of 1/6＜112＞{111} Shockley partial dislocations is discussed, and a special attention is paid to the impact of a partial replacement of Re by W.

2 Calculation

The first-principles calculation is performed by Vienna ab initio simulation package (VASP) [21] based on the density functional theory (DFT), in which a plane-wave basis set with the projector augmented wave (PAW) [22] is used to characterize the ion-electron interaction, and the exchange correlation term is described within the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) functional [23]. In our self-consistent field (SCF) calculation, the cutoff energy of plane wave functions is set to be 350 eV. A conjugate gradient method [24] has been utilized to optimize the geometry and all atomic positions in the supercell are relaxed until the  force on each atom is less than 0.009 eV/. The calculation of total energies and electronic structures is followed by cell optimization with SCF tolerance of 1×10^-5 eV under GGA-PBE potential.

A series of 24-atom supercells have been constructed to calculate the ideal shear strength  τ_max in  and  slip systems (Fig. 2(a)). Here, the shear direction  or  and the slip plane (111) have been rotated to be parallel to the lattice vector a and normal to the lattice vector c as proposed by ROUNDY et al [25] and OGATA et al [26], respectively. In the shear deformation, a particle swarm optimization algorithm [27] has been adopted to yield the curve of stress τ versus strain ε. After being sheared, the deformed lattice vectors R can be obtained by R=R₀D, where the matrix R₀ represents the un-deformed lattice vector. The deformation matrix D [19] is

                           (1)

where ε is the shear strain, i.e., a ratio of displacement △a with respect to height c in the rotated supercell. In the first-principles calculation, atom positions are represented by fractional coordinates. At each step of shear deformations, the configuration of previous steps is used as a starting point [28]. The atom position and cell shape are relaxed under the constraint of fixed shear strain ε.

To calculate generalized stacking fault energy curves (Γ-surfaces) of L1₂-Ni₃Al crystals along  and  slip directions, several (111) surface models are also built by a similar redefined lattice method for the construction of aforementioned shear models. In these surface models, a slab model of 96 atoms is composed of 12 (111) atomic layers with the stacking sequence of ABCABCABCABC, and a 12  vacuum region is added in the direction normal to the slip plane to avoid the interaction among faults in two neighbor slabs (Fig. 2(b)). In the calculation of Γ-surfaces, the atoms over a designated (111) slip plane will be moved along the  or  direction. As a result of shearing b=a₀/2 in the  direction and b=a₀/6 in the  direction, an anti-phase boundary (APB) fault and complex stacking fault (CSF) will be generated, respectively. As for the calculation of Γ-surfaces of APBs on a (001) plane, the surface model is shown in Fig. 2(c). Its APB can be obtained by slipping a part of atoms over a designated (001) slip plane along [110] direction until a distance b=a₀/2, in which a₀ is an equilibrium lattice parameter of L1₂-Ni₃Al crystals. In the movement of slabs, all atoms in the supercell will be relaxed, but only the direction perpendicular to (111) or (001) plane is allowed.

3 Results and discussion

3.1 Effect of alloying elements on dislocation slip mediated creep of γ′ phases

Previous investigations [29-31] revealed most of Ta and Ti partition to the γ′ phase, while W uniformly distributes between the γ matrix and γ′ precipitates. Re and Mo mainly partition to the γ phase, but some of them enable to enter into γ′ phases [18]. In the γ′-Ni₃Al phase, either Ta and  Ti or W, Re and Mo preferentially occupy Al sites [31,32]. As one Al atom on the (111) slip plane is substituted by X=Ta, Ti, W, Re, Mo (marked as X_Al), the Γ-surfaces along  and  slip directions in γ′-Ni₃Al(X) phases are calculated firstly. From Figs. 3(a, b), one can see that the APB energy  and the CSF energy γ_CSF in a perfect L1₂-Ni₃Al crystal are 203 and 173 mJ/m², respectively. These values agree well with previous calculation results [14]. As for single X-addition, Fig. 3(a) shows (=371 mJ/m²) in the Re_Al model is the largest, and then (=362 mJ/m²) in the W_Al model and (=346 mJ/m²) in the Mo_Al model, followed by (=329 mJ/m²) in the Ta_Al model and (=299 mJ/m²) in the Ti_Al model. As well known,  strongly affects the cross-slip of 1/2＜110＞{111} super-partial dislocations. As  is low, the tangential force component generated by the interaction of 1/2＜110＞{111} super-partial dislocations may significantly reduce cross-slip  rate [13]. Obviously, all of Ta-, Ti-, W-, Re- and Mo-addition can improve the cross-slip of 1/2＜110＞{111} super-partial dislocations from {111} to {001} planes, among which the Re-addition is the best. Meanwhile, a similar variation of γ_CSF to  can also be observed from Fig. 3(b). That is: (=332 mJ/m²) in Re_Al model >  (=319 mJ/m²) in W_Al model > (=315 mJ/m²) in Mo_Al model >(=301 mJ/m²) in Ta_Al model > (=260 mJ/m²) in Ti_Al model. Since γ_CSF determines the driving force of cross-slips to some extent and affects the width of two CSF regions related to the energy of constrictions of 1/6＜112＞{111} Shockley partial dislocations [14], an increasing γ_CSF relative to the perfect L1₂-Ni₃Al crystal indicates that Re-, W-, Mo-, Ta- and Ti-addition can also decrease the width of CSF regions. Therefore, they can all lower the constriction energy of 1/6＜112＞{111} Shockley partial dislocations.

Fig. 2 Schematic diagrams of Ni₃Al(X) stacking fault models for calculation of τ_max and Γ-surfaces

Fig. 3 Curves of fault energy (a, b) and stress τ vs strain ε (c, d) in Ni₃Al(X) stacking fault models

The unstable stacking fault energy γ_usf [33-35], as a measure of energy release rate for dislocation nucleation relevant to ductile response of the material, represents an intrinsic energy barrier for activated stacking faults. In the separation of ＜110＞{111} super dislocations, a high γ_usf means a large shear stress to be requested for the nucleation of 1/2＜110＞{111} super-partial dislocations [18]. Figures 3(a, b) show that γ_usf values in  and  slip directions are simultaneously elevated by single X-addition. For example, relative to the perfect L1₂-Ni₃Al crystal, γ_usf in  and slip directions are lifted by 162 and 159 mJ/m² in the Re-addition model, respectively. These results indicate that either emission of 1/2＜110＞{111} super-partial dislocations or nucleation of 1/6＜112＞{111} Shockley partial dislocations is restricted by Re-, W-, Mo-, Ta- and Ti-addition, and the order of Re_Al>W_Al>Mo_Al>Ta_Al> Ti_Al>Ni₃Al reveals that Suzuki segregations [36,37] of Re, W, Mo at APBs and CSFs are more profitable than the partitioning of Ta and Ti in γ′ phases.

Another parameter related to γ_usf is the ideal shear strength τ_max, which is a critical stress required to plastically deform a perfect crystal [26]. τ_max provides an upper bound of creep strength a material can achieve [28], although it can be affected by dislocations, grain boundaries, cracks and other micro-structural defects in a real crystal. To obtain some crucial information of resistances  to creep deformations, τ_max in  and  slip systems are further calculated. Figures 3(c, d) exhibit the variation of stress τ versus strain ε in the γ′-Ni₃Al(X) phase with and without single X-addition. For the perfect L1₂-Ni₃Al crystal, Fig. 3(c) shows that the shear stress τ along  direction reaches its maximum value of 4.79 GPa at ε=0.16, which means that τ_max in the  slip direction is 4.79 GPa. Similarly, τ_max (=4.23 GPa) at ε=0.18 in the  slip direction can also be deduced from Fig. 3(d). In Re_Al, W_Al, Mo_Al, Ta_Al and Ti_Al models, τ_max is found to increase by 3.17, 3.04, 2.90, 2.39 and 2.25 GPa in the  slip direction and 2.39, 2.24, 1.98, 1.58 and 1.49 GPa in the  slip direction relative to the perfect L1₂-Ni₃Al crystal, respectively, which means that all of Re, W, Mo, Ta and Ti can act as an efficient obstacle in the movement of initial 1/2＜110＞{111} super-partial dislocations and subsequent 1/6＜112＞{111} Shockley partial dislocations. Consequently, from the resistance to dislocation movements concerned, Re-, W-, Mo-, Ta- and Ti-additions are also profitable for the improvement of creep strengths of Ni-based SC superalloys.

Since the anomalous flow behavior of L1₂-Ni₃Al crystals at high temperature mainly arises from thermally activated cross-slips of screw dislocations from primary {111} slip planes onto the {001} cubic planes, a variation of configuration energies of ＜110＞{111} super dislocations induced by cross-slips of 1/2＜110＞{111} super-partial dislocations is further calculated on the basis of PPV model (Fig. 1). In this PPV model [38], the difference in configuration energies of super dislocations per unit length before and after cross-slips can be evaluated by

                     (2)

where  is APB energy in {001} stacking faults, and w is the distance of cross-slips from  {111} to {001} planes.

Obviously, the cross-slip of 1/2＜110＞{111} super-partial dislocations can take place only if . A large  means that γ′ phases have a higher driving force of 1/2＜110＞{111} super-partial dislocations from {111} to {001} planes. In this case, some K-W locks [16] are easy to be formed with the help of APBs on {001} planes. To this end, we employ  to evaluate the influence of single X-addition on the yield strength of γ′ phases at high temperature. The calculated results are tabulated in Table 1. From Table 1, one can see that  in the perfect L1₂-Ni₃Al crystal indeed complies with the condition of cross-slips of 1/2＜110＞{111} super- partial dislocations as reported in literatures [17]. As a part of Al atoms in γ′-Ni₃Al(X) phases are substituted by Re, W and Mo,  values are found to increase by 0.18, 0.16 and 0.09, respectively, while only a small rise can be seen in the case of X=Ta and Ti, e.g., 0.05 in the Ta_Al model and 0.03 in the Ti_Al mode. These results clearly indicate that Re- and W-additions can remarkably promote the cross-slip of 1/2＜110＞{111} super- partial dislocations, but Ta- and Ti-additions are almost invalid. As for the advantage of Suzuki segregations, a careful comparison reveals that either  associated with anomalous yield strength of γ′ phases at high temperature or τ_max relevant to resistances to plastic deformations of γ′ phases in the W-addition model is almost equal to that in the Re-addition model. For example, for the substitution of W for Al in γ′-Ni₃Al(X) phases, Table 1 shows that τ_max (=6.47 GPa) in the  slip direction is very close to that (=6.62 GPa) in the single Re-addition model. These results seem to imply that W-addition indeed has a similar influence to Re-addition on the creep of Ni-based SC superalloys at high temperature [9].

Table 1 Formation enthalpies of point defects H, correlation energies between X1_Al and X2_Al defects, ideal shear strengths τ_max, anti-phase boundary energies γ_APB, complex stacking fault energies γ_CSF, unstable stacking fault energies γ_usf and  in γ′-Ni₃Al(X) phases with X=Re, W, Mo, Ta and Ti

3.2 Comparison of strengthening effects caused by W-addition with Re-addition

As it is well-known, Re-addition induced reinforcement of creep resistances in Ni-based SC superalloys is called as Re-effect [4]. Re not only may reduce stacking fault energies of γ phases and raise  of γ′ phases, but also can slow down coarsening kinetics of γ′ precipitates [39]. Since almost same creep properties have been observed in CMSX-4 superalloys with partial substitution of W for Re [6], a multiple addition of Re and W in γ′ phases is further investigated. In these complexes  of X1_Al+X2_Al defects, only two typical models and with preferred substitution of X1 and X2 for Al atoms are calculated. The calculated results are also listed in Table 1. Herein, X1 and X2 represent the species of alloying elements. d=2nd and d=4th are the distances between X1_Al and X2_Al point defects, which correspond to the second and fourth nearest neighbors of assigned sites (Fig. 2(a)), respectively. From Fig. 2(a), one can see that d=2nd denotes that  X2_Al locates at an adjacent plane of the assigned (111) slip plane, while d=4th means that X2_Al situates at the same (111) slip planes as X1_Al. The former indicates that only Suzuki segregation of X1 exists, but the latter means that co-segregation of X1 and X2 appears in these stacking faults. Corresponding to X_Al and X1_Al+X2_Al, the contents of solutes in the shear models are 4.17 and 8.33 at.%, while they are 1.04 and 2.08 at.% in the calculation models for Γ-surfaces, respectively. Relative to the single Re-addition, Table 1 shows that  in  and  models rises by 7% and 20%, respectively, and τ_max in the  slip direction increases by 0.28 and 1.01 GPa, respectively. Obviously, an extra Re-addition is conducive to yield strengths and creep rupture strengths of γ′ phases at high temperature [8], especially in the Suzuki segregation of Re with high concentration. As X2 is replaced by W, it is found that both of γ_usf in  and  slip directions are large relative to their corresponding double Re-addition models, which means that a partial replacement of W for Re makes the emission of 1/2＜110＞{111} super-partial dislocations and the nucleation of 1/6＜112＞{111} Shockley partial dislocations more difficult [18], and the inhibition caused by W-segregation on the (111) slip plane is stronger than W-addition on its adjacent plane, e.g., 57 mJ/m² in the  model larger than  that in the model, corresponding to  the emission of 1/2＜110＞{111} super-partial dislocations. Moreover, an upraised γ_CSF in the complex of Re_Al+W_Al indicates that the substitution of Re by W also facilitates the constriction of 1/2＜110＞{111} super-partial dislocations by two 1/6＜112＞{111} Shockley partial dislocations [14]. Additionally, relative to the double Re-addition with d=2nd and d=4th, τ_max in the  slip direction increases by 0.46 and 0.07 GPa, respectively, which means that both of replacements of W for Re on the (111) slip plane and its adjacent plane can significantly impede the movement of 1/6＜112＞{111} Shockley partial dislocations, and its resistance to dislocation slip mediated creeps of γ′ phases seems to be large relative to double Re-additions. But, it is noticed that  in  and  models drops by 4% and 2% relative to and models, respectively. From the viewpoint of cross-slips of 1/2＜110＞{111} super-partial dislocations, this reduced  undoubtedly indicates that the anomalous flow behavior of γ′ phases at high temperature in Re_Al+W_Al complexes is not as excellent as that of the double Re-addition.

As X1 on the (111) slip plane is also replaced by W, Table 1 shows although most of  and τ_max in the  slip direction are  lower than those in Re_Al+X2_Al complexes, their decrements are found to be very small. For example, relative to the  model, only 0.01 GPa is cut down for τ_max, and the lost  is less than 1% in the  model. Moreover, a similar variation can also be seen as X2 departs from the (111) slip plane. For example, τ_max (=7.37 GPa) in the  model is very close to that (7.36 GPa) in the  model, and their corresponding difference in  is not more than 0.06. These results clearly indicate that W-addition indeed has a similar strengthening effect to Re-addition on the dislocation slip mediated creep of γ′ phases, especially in the co-segregation of W and Mo. Interestingly, for the complex of W_Al+Mo_Al, it is found that either =2.22 at d=2nd or =2.33 at d=4th is higher than that in the  model (1.78) and that in the  model (1.94). This result further indicates that an interplay between X1 and X2 has a more critical impact than their species on the anomalous flow behavior of γ′ phases.

To investigate the interplay between X1 and X2 in dislocation slip mediated creeps of γ′ phases, a correlation energy function  has been adopted to characterize and assess the interaction between X1_Al and X2_Al point defects (Table 1). Herein,  is a difference of formation enthalpies of point defects, which can be calculated by the following expression [40]:

    (3)

where and are the formation enthalpies of L1₂-Ni₃Al(X) crystals per atom, corresponding to X1_Al, X2_Al and  models, respectively. Table 1 shows that almost all  in  models are positive, e.g., 0.085 eV in the  model and eV in the model, which means that most of interactions between X1_Al and X2_Al point defects are repulsive forces. However, it is noticed that a negative-0.058 eV exists in the model, indicating that the interaction between W_Al and Mo_Al point defects is attractive. As compared with 0.095 eV in the  model, this negative interaction seems to imply that an attraction should be responsible  for a large  in the  model relative to the  model. To test and verify this conjecture, in the complexes of Mo_Al+Ta_Al, Mo_Al+Ti_Al and Ta_Al+Ti_Al  as well as their  and τ_max are also calculated and listed in Table 1. From Table 1, one can see that relative to  and  models,  indeed increases in the model with negative . Also,  in  model with -0.080 eV rises by 0.03 and 0.06 eV, respectively, compared with  and  models with positive . As for the influence of magnitudes of  Fig. 4 further illustrates the variation of  and τ_max versus  For d=2nd, one can see that with the increase of  from -0.039 to 0.190 eV, both of  and τ_max in the  slip direction ascend firstly, and then gradually fall down, and the transformation takes place at 0.050 eV. In the case of d=4th, a similar variation of  and τ_max  versus  can be seen with the increase of  from -0.090 to 0.119 eV. Also, the largest  and τ_max emerge at 0.050 eV, although their drop is more abrupt after the maximum. Similar to the influence of Re-P pairs on the Griffith work of γ-Ni/γ′-Ni₃Al interfaces [41], these results distinctly indicate that a weak interaction between X1_Al and X2_Al defects is more profitable for the improvement of creep strengths of γ′ phases at high temperature, and the smaller the magnitude of  is, the larger the creep rupture strength of γ′-Ni₃Al phases is. That is said, the anomalous flow behavior of γ′ phases at high temperature cannot benefit from a strong interaction between X1_Al and X2_Al defects regardless of attraction or repulsion. Thus, relative to the species of X1 and X2 and their defective configurations in γ′ phases,  provides a simple and direct criterion for the estimation of yield strengths and creep rupture strengths of γ′ phases at high temperature to some extent.

Fig. 4 Dependence of  (a) and τ_max in slip direction (b) on  in complex of X1_Al and X2_Al

4 Conclusions

(1) Similar to Re-addition, the Suzuki segregation of W at {111} stacking faults not only can impede the movement of 1/6＜112＞{111} Shockley partial dislocations, but also may promote the cross-slip of 1/2＜110＞{111} super-partial dislocations.

(2) With the partial replacement of W for Re, an increased τ_max along the  slip direction in the duplex addition model of Re and W suggests that the resistance to creep of γ′ phases can be reinforced, although their  associated with the anomalous flow behavior of γ′ phases drops slightly.

(3) The anomalous flow behavior of γ′ phases at high temperature can benefit from the co-segregation of Re and W in {111} stacking  faults, but a strong attraction and repulsion between X1_Al and X2_Al point defects are disadvantageous for the improvement of creep rupture strengths of Ni-based single crystal superalloys.

Acknowledgments

The authors are grateful for the financial supports from the National Natural Science Foundation of China (Nos. 51871096, 52071136 ).

References

[1] REED R C. The superalloys fundamentals and applications [M]. Cambridge: Cambridge University Press, 2006.

[2] LONG Hai-bo, MAO Sheng-cheng, LIU Yi-nong, ZHANG Ze, HAN Xiao-dong. Microstructural and compositional design of Ni-based single crystalline superalloys—A review [J]. Journal of Alloys and Compounds, 2018, 743: 203-220.

[3] HUANG Yan-yan, MAO Zu-gang, NOEBE R D, SEIDMAN D N. The effects of refractory elements on Ni-excesses and Ni-depletions at γ(f.c.c.)/γ'(L1₂) interfaces in model Ni-based superalloys: Atom-probe tomographic experiments and first-principles calculations [J]. Acta Materialia, 2016, 121: 288-298.

[4] ERICKSON G L. The development and application of CMSX-10 [J]. Superalloys, 1996, 1: 35-44.

[5] RAE C M F, REED R C. The precipitation of topologically close-packed phases in rhenium containing superalloys [J]. Acta Materialia, 2001, 49: 4113-4125.

[6] WOLLMER S, MACK T, GLATZELl U. Influence of tungsten and rhenium concentration on creep properties of a second generation superalloy [J]. Materials Science and Engineering A, 2001, 321: 792-795.

[7] GONG Wei, ZHAO Wen-yue, MIAO Nai-hua, SUN Zhi-mei, LI Shu-suo, GONG Sheng-kai. Strengthening effects of alloying elements W and Re on Ni₃Al: A first-principles study [J]. Computational Materials Science, 2018, 144: 23-31.

[8] CHEN Yi, HE Shuang, YI Zhou, PENG Ping. Impact of correlative defects induced by double Re-addition on the ideal shear strength of γ'-Ni₃Al phases [J]. Computational Materials Science, 2018, 152: 408-416.

[9] CHEN Yi, HE Shuang, YI Zhou, PENG Ping. A synergistic reinforcement of Re and W for ideal shear strengths of γ'-Ni₃Al phases [J]. Journal of Physics and Chemistry of Solids, 2019, 131: 34-43.

[10] LV Xian-zi, ZHANG Jian-xin. Core structure of a ＜100＞ interfacial superdislocations in a nickel-base superalloy during high-temperature and low-stress creep [J]. Materials Science and Engineering A, 2017, 683: 9-14.

[11] LIU Li-rong, JIN Tao, ZHAO Nai-ren, WANG Zi-hui, SUN Xiao-feng, GUAN Heng-rong, HU Zhuang-qi. Creep deformation mechanism in a Ni base single crystal superalloy [J]. Acta Metallurgica Sinica, 2005, 41: 1215-1220.

[12] MILLIGAN W W, ANTOLOVICH S D. The mechanisms and temperature dependence of superlattice stacking fault formation in the single-crystal superalloy PWA 1480 [J]. Metallurgical Transactions A, 1990, 22: 2309-2318.

[13] DODARAN M, ETTEFAGH A H, GUO S M, KHONSARI M M, MENG W J, SHAMSAEI N, SHAO S. Effect of alloying elements on the γ' antiphase boundary energy in Ni-base superalloys [J]. Intermetallics, 2020, 117: 106670.

[14] YU Xiao-xiang, WANG Chong-yu. Effect of alloying element on dislocation cross-slip in γ'-Ni₃Al: A first- principles study [J]. Philosophical Magazine, 2012, 92: 4028-4039.

[15] WANG K Y M. Understanding the yield behaviour of L1₂-ordered alloys [J]. Materials Science and Technology, 2016, 33: 934-943.

[16] YOO M H. On the theory of anomalous yield behavior of Ni₃Al-Effect of elastic anisotropy [J]. Scripta Metallurgica, 1986, 20: 915-920.

[17] SHANG Shun-li, SHIMANEK J, QIN Shi-pin, WANG Yi, BEESE A M, LIU Zi-kui. Unveiling dislocation characteristics in Ni₃Al from stacking fault energy and ideal strength: A first-principles study via pure alias shear deformation [J]. Physical Review B, 2020, 101: 024102.

[18] YU Xiao-xiang, WANG Chong-yu. The effects of alloying elements on generalized stacking fault energies, strength and ductility of γ'-Ni₃Al [J]. Materials Science and Engineering A, 2012, 539: 38-41.

[19] WANG Yu-jiang, WANG Chong-yu. Influence of the alloying element Re on the ideal tensile and shear strength of γ'-Ni₃Al [J]. Scripta Materialia, 2009, 61: 197-200.

[20] WU Xiao-xia, WANG Chong-yu. Effect of the alloying element on the temperature-dependent ideal shear strength of γ'-Ni₃Al [J]. RSC Advances, 2016, 6: 20551-20558.

[21] KRESSE G, HAFNER J. Ab initio molecular dynamics for liquid metals [J]. Physical Review B, 1993, 47: 558-561.

[22] KRESSE G, JOUBERT D. From ultrasoft pseudopotentials to the projector augmented-wave method [J]. Physical Review B, 1999, 59: 1758-1775.

[23] PERDEW J P, BURKE K, ERNZERHOF M. Generalized gradient approximation made simple [J]. Physical Review Letters, 1996, 77: 3865-3868.

[24] HESTENES M R, STIEFEL E. Methods of conjugate gradients for solving linear systems [J]. Journal of Research of the National Bureau Standards, 1952, 49: 409-436.

[25] ROUNDY D, KRENN C R, COHEN M L, MORRIS J J W. Ideal shear strengths of fcc aluminum and copper [J]. Physical Review Letters, 1999, 82: 2713-2716.

[26] OGATA S, LI J, YIP S. Ideal pure shear strength of Aluminum and Copper [J]. Science, 2002, 298: 807-811.

[27] LI Quan, LIU Han-yu, ZHOU Dan, ZHENG Wei-tao, WU Zhi-jian, MA Yan-ming. A novel low compressible and superhard carbon nitride: Body-centered tetragonal CN₂ [J]. Physical Chemistry Chemical Physics, 2012, 14: 13081-13087.

[28] JAHNATEK M, HAFNER J, KRAJCI M. Shear deformation, ideal strength, and stacking fault formation of fcc metals: A density-functional study of Al and Cu [J]. Physical Review B, 2009, 79: 224101-224117.

[29] BAGOT P A J, SILK O B W, DOUGLAS J O, PEDRAZZINI S, CRUDDEN D J, MARTIN T L, HARDY M C, MOODY M P, REED R C. An atom probe tomography study of site preference and partitioning in a nickel-based superalloy [J]. Acta Materialia, 2017, 125: 156-165.

[30] REED R C, YEH A C, TIN S, BABU S S, MILLER M K. Identification of the partitioning characteristics of ruthenium in single crystal superalloys using atom probe tomography [J]. Scripta Materialia, 2004, 51: 327-331.

[31] XU Yu-lai, ZHANG Lei, LI Jun, XIAO Xue-shan, CAO Xiu-li, JIA Guo-qing, SHEN Zhi. Relationship between Ti/Al ratio and stress-rupture properties in nickel-based superalloy [J]. Materials Science and Engineering A, 2012, 544: 48-53.

[32] JIANG Chao, GLEESON B. Site preference of transition metal elements in Ni₃Al [J]. Scripta Materialia, 2006, 55: 433-436.

[33] RICE J R. Dislocation nucleation from a crack tip: An analysis based on the peierls concept [J]. Journal of Mechanics and Physics of Solids, 1992, 40: 239-271.

[34] SUN Yue-min, BELTZ G E. RICE J R. Estimates from atomic models of tension-shear coupling in dislocation nucleation from a crack tip [J]. Materials Science and Engineering A, 1993, 170: 67-85.

[35] RICE J R, BELTZ GLENN E. The activation energy for dislocation nucleation at a crack [J]. Journal of Mechanics and Physics of Solids, 1994, 42: 333-360.

[36] HIDEJI S. Segregation of solute atoms to stacking faults [J]. Journal of the Physical Society Japan, 1962, 17: 322-325.

[37] EURICH N C, BRISTOWE P D. Segregation of alloying elements to intrinsic and extrinsic stacking faults in γ'-Ni₃Al via first principles calculations [J]. Scripta Materialia, 2015, 102: 87-90.

[38] PAIDAR V, POPE D P, VITEK V. A theory of the anomalous yield behavior in L1₂ ordered alloys [J]. Acta Materialia, 1984, 32: 435-448.

[39] YI Zhou, PENG Ping. Impact of Re-clustering on resistances to dislocation slip mediated plastic deformations in γ matrix phases [J]. Computational Materials Science, 2020, 172: 109314.

[40] LI Yu-juan, HU Qing-miao, XU Dong-sheng, YANG Rui. Strengthening of γ-TiAl-Nb by short-range ordering of point defects [J]. Intermetallics, 2011, 19: 793-796.

[41] PENG Li, PENG Ping, LIU Yun-guo, HE Shuang, WEI Hua, JIN Tao, HU Zhuang-qi. The correlation between Re and P and their synergetic effect on the rupture strength of the γ-Ni/γ'-Ni₃Al interface [J]. Computational Materials Science, 2012, 63: 292-302.

以W取代Re对γ'-Ni₃Al相位错滑移调制蠕变的影响

易 洲，胥云雷，彭 平，陈江华

湖南大学 材料科学与工程学院，长沙 410082

摘  要：γ′-Ni₃Al相在高温下的反常流变行为与1/2＜110＞{111}超级部分位错的交滑移密切相关。沿最低能量路径获得的广义堆垛层错能曲线(即Γ面)可以提供大量有关位错形核和运动的信息。采用第一性原理计算，研究Re、W、Mo、Ta、Ti单掺杂与双掺杂对γ′-Ni₃Al相Γ面和理想剪切强度τ_max 的影响。与Re掺杂的情形类似，W在堆垛层错处的Suzuki偏聚可阻止1/6＜112＞{111} Shockley部分位错的运动和促进1/2＜110＞{111}超级部分位错的交滑移。当Re被W部分取代时，的降低表明γ′相在高温下的反常流变行为不如双Re掺杂，但τ_max的增加却表明镍基单晶高温合金的蠕变断裂强度在一定程度上可从中受益，尤其是Re和W在Al-Al位的共偏聚。当点缺陷X1_Al和X2_Al之间的相互作用采用关联能函数来表征时，无论是吸引还是排斥，强关联都不利于γ′相屈服强度的提高。

关键词：镍基单晶高温合金；γ′-Ni₃Al；广义堆垛层错能；理想剪切强度；位错；交滑移

(Edited by Bing YANG)

Corresponding author: Ping PENG, Tel: +86-731-88821727, E-mail: ppeng@hnu.edu.cn

DOI: 10.1016/S1003-6326(21)65634-0

1003-6326/ 2021 The Nonferrous Metals Society of China. Published by Elsevier Ltd & Science Press