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γ′ interface of Ni-base single crystal superalloys
Acta Materialia 51 (2003) 1079–1086 www.actamat-journals.com
Sulfur embrittlement on γ/γ⬘ interface of Ni-base single crystal superalloys K. Chen a,∗, L.R. Zhao a, J.S. Tse b a
Structures, Materials and Propulsion Laboratory, Institute for Aerospace Research, National Research Council Canada, Ottawa, Canada K1A 0R6 b Steacie Institute for Molecular Sciences, National Research Council Canada, Ottawa, Canada Received 16 May 2002; accepted 16 October 2002
Abstract The impurity sulfur embrittlement on γ / γ⬘ interface of Ni-base single crystal superalloys has been investigated by first principles quantum mechanics DMol calculations. Using a local sum of vertical (
冘 冘 冘
冘
BOv) and horizontal
( BOh) Mayer bond orders, we proposed a new method to evaluate the competition between the shear and cohesive strengths of the interface. Coupled with the phenomenological theory of fracture, we define the ratio of RBO ⫽ BOh / BOv to assess the embrittlement trend of the interface related to sulfur doping or sulfur doping combined with Re substitution. It is shown that sulfur increases RBO by 121% relative to the sulfur-free γ / γ⬘ interface, which could induce interface embrittlement from the electron bonding point of view. Calculations of both BO and charge density distribution reveal that it is the strong bonds between sulfur and Ni atoms lying within the interface that contribute to the interface embrittlement. The substitution of Re for Al at the γ / γ⬘ interface results in reduced RBO, thus relieving the tendency of interface embrittlement. Furthermore, our model on sulfur embrittlement is compared with previous models. 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: First principles calculations; Sulfur embrittlement; γ/γ⬘ interface; Ni-base superalloys
1. Introduction It was widely reported that sulfur reduces malleability of nickel, causes cold and hot brittleness, and reduces tensile strength, ductility and creep rupture strength of superalloys [1–6]. Earlier
∗
Corresponding author. E-mail address: [email protected] (K. Chen).
attempts to deduce the atomic mechanisms responsible for sulfur embrittlement indicate that there is a depletion of intermetallic electron density accompanying the sulfur segregation to interface boundaries [7,8]. This depletion is suggested to cause a weakening of intermetallic bonds and thus reduces the cohesive strength across the interface boundary. Due to the lack of specifics of crack propagation in earlier models, Eberhart et al. [9] argued that any of the models which explain
1359-6454/03/$30.00 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(02)00512-8
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K. Chen et al. / Acta Materialia 51 (2003) 1079–1086
embrittlement caused by either the ideal shear strength or the ideal cohesive strength alone, should be considered incomplete. They [9] suggested that the brittle fracture is the result of a sensitive interplay of shear and cohesive strengths at a crack tip. Their calculations show that it is the sulfur–sulfur network bonds lying within the interface boundary that increase the shear strength, while the cohesive strength continues to diminish, giving rise to the essential condition for the brittle fracture. However, it should be pointed out that Eberhart et al.’s model is unable to account for the sulfur embrittlement in INCONEL Ni and pure Ni when sulfur content is below 30 ppm [10]. In these cases, the sulfur content is so low that sulfur–sulfur network bonds are unlikely to form and make appreciable contributions to the shear strength within the interface boundary. In this study, a new first principles approach, coupled with the phenomenological theory of fracture [11–13], is proposed to study the sulfur embrittlement of γ/γ⬘ interface of Ni-base single crystal (SC) superalloys. By comparison with previous models, our model does not require the formation of sulfur– sulfur network bonds, and thus is applicable to any sulfur content in the materials. Furthermore, the effect of Re, which is typically added in SC superalloys as a refractory strengthening element [14– 16], on sulfur embrittlement is calculated. These results show that Re can reduce the embrittlement of the interface.
which contains double set valence functions plus a single d or p polarization function, as this technique has proven to be very efficient in studying molecular systems. Furthermore, the frozen-core approximation for Ni, Al, Re and S atoms is used in the calculations. The degree of convergence of the self-consistent iterations, measured by (rms) changes in the charge density, is set to be 10⫺5, which allows the energy to converge to 10⫺5 Ryd. All calculations in this paper were performed with the generalized-gradient approximation (GGA) proposed by Perdew and Wang [19]. The charge density and charge density difference are calculated and used to evaluate electronic properties. In addition, the Hirschfeld charge [20] and Mulliken orbital population techniques [21] are adopted to analyze the valence electron occupation on atomic orbitals. Moreover, the Mayer bond order (BO) [22] is employed to evaluate the atomic bond strength. Fig. 1 shows the 64 atoms cluster model used in this study. It consists of an upper Ni region and a lower Ni3Al region, representing the γ matrix and γ⬘ precipitate in Ni-base SC alloys, respectively. We define the γ/γ⬘ interface without both sulfur residing at the interstitial site and Re substitution for Al site as the pure γ/γ⬘ interface. Sulfur atoms usually occupy the octahedral interstitial sites
2. Methodology The DMol molecular cluster orbital approach is used to analyze the orbital electron occupations. As a molecular cluster approach, DMol3 is based on the density functional theory with a local-density approximation [17]. In DMol3, the local atomic wavefunctions, which are used as the basis functions for the cluster model, are generated numerically from the same local-density functional theory solutions for the free atoms. In general, the molecular cluster approach is suitable for studying electronic properties that are primarily a function of local environment [18]. The model uses the double numerical basis set for Ni, Al, Re and S atoms,
Fig. 1. Cluster model for DMol calculations. Open circles, Ni atoms; black circles, Al atoms; grey circle M1, Re atom; grey circle M4, sulfur atom. The 0–0 layer represents the highest plane of the Ni3Al region, while the 1–1 layer represents the lowest plane of the upper Ni region.
K. Chen et al. / Acta Materialia 51 (2003) 1079–1086
[23,24], and the Re atom, as reported experimentally to substitute for Al in γ⬘ precipitates [25–27], is placed at the Al(1) site. As part of the γ⬘ phase, the 0–0 layer (x–y plane), as labeled in Fig. 1, contains mixed Ni and Al atoms, in which Ni atoms occupy the face center sites, while Al atoms occupy the corner sites. The next layer adjacent to the 0–0 layer in the upper γ region is defined as the 1–1 layer. Inequivalent atoms in the cluster as shown in Fig. 1 are denoted by the numerical labels depending on the C4v point group symmetry of the cluster. The lattice constant of the cluster is chosen to be equal for both Ni (γ) and Ni3Al (γ⬘) due to the assumption of complete coherence. The cluster is then relaxed with the variation in the lattice constant when energetic calculations are performed. 3. Results and discussions Pugh [11] proposed that the resistance to plastic deformation is related to the product Gb, where G and b are the shear modulus and Burgers vector, respectively. The fracture strength was assumed to be proportional to the product Ba, where B and a are Bulk modulus and lattice parameter, respectively. If Gb/Ba is high, a material will behave in a brittle behavior. The Gb/Ba reflects the competition between the shear and cohesive strengths at the crack tip of fracture. Rice and Thompson [12] speculated that brittle vs. ductile behavior may be governed at the tip of an atomically sharp crack. A material will, in principle, fail in a brittle manner if the ideal cohesive strength is reached along the extension of the crack before the ideal shear strength is reached. Any complete theory of embrittlement must deal not only with the way the cohesive strength changes as a result of segregation of impurities, but also the effect the segregation has on the shear strength. In order to qualitatively evaluate the competition between the shear and cohesive strengths of the γ/γ⬘ interface, we need to consider the BO values between atoms. The BO in nature is the overlap of electron wavefunctions between atoms, which can be used to qualitatively measure the strength of atomic bonds. In this study, we use the Mayer BO form and calculation approach [22]. The definition of Mayer BO between atoms A and B is given by:
BOAB ⫽ 2
冘冘
1081
[(PaS)mn(PaS)nm
(1)
m苸Am苸B
⫹ (PbS)mn(PbS)nm] where Pa, Pb are the density matrices, S is the overlap matrix of the wavefunctions [22]. We calculate BOAB values within the first nearest neighbor (FNN) B atoms around A site. Our calculations indicate that the BOAB value, if the separation between atoms A and B is beyond FNN range, is small and can be neglected without altering the conclusions. In order to examine the interplay of shear and cohesive strengths at the interface in detail, we divide the BOs into these vertical and horizontal components. The vertical BO (BOv) is defined and calculated for atoms along the [001] z direction to qualitatively measure the cohesive strength across the γ/γ⬘ interface. While, the horizontal BO (BOh) is defined and calculated for inplane atoms perpendicular to the [001] z direction to qualitatively represent shear strength. Further-
冘 冘
more, we define a ratio of RBO ⫽ BOh / BOv to assess the embrittlement trend of the interface related to sulfur doping or sulfur doping combined
冘
with Re substitution, where the BOh is a local sum of BOs within both the 0–0 and 1–1, while the
冘 冘
BOv is a local sum of BOs across 0–1 layer.
It is necessary to point out that although
冘
BOh
and BOv are not the real shear and cohesive strengths of the γ/γ⬘ interface in terms of the continuum mechanics viewpoint, it is suggested that
冘 冘
the ratio RBO ⫽ BOh / BOv can be used to represent the trend of shear strength vs. cohesive strength of the interface. In this paper, we use RBO at the γ/γ⬘ interface to examine the embrittlement trend induced by impurity sulfur doping. Fig. 2(a) shows the BOs for pure, sulfur-doped and sulfur-doped plus Re added γ/γ⬘ interface. For
冘
冘 冘 冘
BOhp and BOvp to repcomparison, we use resent the shear and cohesive strengths by per pair BO, respectively, and use RBO ⫽ BOhp / BOvp to evaluate the embrittlement trend of the interface as a function of sulfur doping or sulfur doping plus
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冘BO and 冘BO for pure (Ni/Ni Al), sulfur-doped (Ni/Ni Al–S) and sulfur-doped plus Re substitution (Ni/Ni Al–Re–S) γ/γ⬘ interface. (b) The ratio R ⫽ 冘BO / 冘BO for pure (Ni/Ni Al), sulfur-doped (Ni/Ni Al–S) and sulfur-
Fig. 2.
(a) A local sum of BOs 3
h p
v p
3
BO
h p
3
v p
3
3
doped plus Re substitution (Ni/Ni3Al–Re–S) γ/γ⬘ interfaces.
Re substitution. As shown in Fig. 2(b), the calculated RBO ⫽ 0.7877 for a pure γ/γ⬘ interface is set as a baseline. In the case of the sulfur-doped interface, the BOv between Ni atoms normal to the γ/γ⬘ interface is reduced from 0.2914 of pure interface to 0.2597. This indicates that the bonding between metal Ni atoms responsible for the interface cohesion is reduced due to the sulfur doping. This result is consistent with the conclusions of Briant and Messmer [7], suggesting that sulfur causes
embrittlement by weakening the metal bonding across the interface. Our calculations, however, demonstrate that the cohesive strength across the 0–1 layer alone cannot completely determine the interface embrittlement. The bonding of sulfur and metal atoms within the 1–1 plane and bonding between metal atoms in the 0–0 plane will contribute to the shear strength of the interface. It is the interplay between the shear and cohesive strengths that determines the brittle or ductile trend of the
K. Chen et al. / Acta Materialia 51 (2003) 1079–1086
冘
冘
interface. Therefore, we need to examine the h
v
BO p and BO p which include both sulfur– metal bonds and metal–metal bonds. Calculations reveal that the BOh between Ni atoms within the 1–1 layer is reduced from 0.2636 for the pure interface to 0.2259 for the sulfur-doped interface. The BOh between sulfur and Ni atoms within the 1–1 layer reaches 0.8552, 3.8 times larger than Ni–Ni bonds. This part of BOh obviously makes large contributions to the shear strength, thus gives rise to a large ratio RBO. From Fig. 2(a), we have
冘
冘
BOhp ⫽ 0.4157, and BOvp ⫽ 0.2382. Therefore, for sulfur-doped interface, the ratio RBO ⫽ 1.745 is obtained, representing 121% increase compared to the pure γ/γ⬘ interface. From an electronic structure point of view, it can be concluded that the strong sulfur–metal bonds (BOhS⫺Ni ⫽ 0.8552) lying within the interface boundary cause the interface embrittlement. It should be noted that our model is independent of the sulfur concentration as no sulfur–sulfur network bonds [9] are necessary to explain the sulfur embrittlement. Thus, it is reasonably believed that if the sulfur content at the interface increases, each individual sulfur atom would form a strong bond between sulfur and Ni atoms lying within the interface boundary, and would contribute to the embrittlement behavior as a single sulfur atom does. As a refractory element, Re was proven to be a potent strengthener and key element for the improvement of mechanical properties of SC superalloys [28]. Recently, we have performed first principles calculations on the Re strengthening effect on the γ/γ⬘ interface of Ni-base SC superalloys [15]. Our results show that refractory Re can significantly strengthen γ/γ⬘ interface. In order to examine Re effects on sulfur embrittlement of the interface, we substitute Re for Al atom as suggested by experiments [25–27] in the model of a sulfur-doped interface. In this case, the calculated BO between sulfur and Ni atoms within the 1–1 layer is significantly reduced from 0.8552 of Refree to 0.6808 of Re substitution for Al. This change partially reduces the contribution of sulfur– metal bonds to the interface shear strength. However, due to the substitution of Re for Al, the metal
1083
冘
Re–Ni bonds within the 0–0 plane increase significantly. As a combined result, the BOhp is 0.4176, almost the same as the purely sulfur-doped interface. Calculations of the BOv between sulfur and Re atoms across the 0–1 plane shows that BOvRe⫺S ⫽ 1.1246, 1.65 times larger than sulfur– Ni bonds on the 0–0 plane. Similar to BOvRe⫺S, the BOvRe⫺Ni increases from 0.0959 of sulfur-doped interface to 0.3619 of Re substitution, 3.8 times larger than the purely sulfur-doped interface. As
冘
BOvp ⫽ 0.34414. shown in Fig. 2(a), we have The increased BOv lowers the ratio RBO, and thus reduces the tendency of interface embrittlement. For the case of sulfur doping plus Re substitution, we have RBO ⫽ 1.235, a reduction of 30% from 1.745 of purely sulfur-doped interface. One of the important issues in our study is the bond redistributions in the interface region (shear and cohesion) induced by either the substitution element, such as Re, or by the interstitial atom, such as sulfur. The sensitive interplay of the shear and cohesive strengths as a function of additional elements may control the tendency of interface embrittlement. To further study sulfur embrittlement on the interface, we calculated the spatial distribution of electrons associated with the interface structure. Of the electronic properties, the total valence charge density and the charge density difference ⌬rind(r) are important parameters. The latter is defined as: ⌬rb(r) ⫽ rcluster(r,X)⫺rcluster(r)
(2)
where rcluster(r,X) is the charge density with Re(S) addition at the interface. Fig. 3(a) presents the total valence charge density of sulfur-doped γ/γ⬘ interface on the 0–1 (x–z) plane in units of 10⫺3 e/(au)3. Strong bonds between FNN sulfur and metal (Ni and Al) atoms are observed. These strong bonds include sulfur–Ni bonds lying on the 1–1 plane and a sulfur–Al bond normal to the γ/γ⬘ interface, with the latter linking the interface together. These bonds contribute to both shear and cohesive strengths of the interface. Fig. 3(b) shows the charge density difference between sulfur-doped and sulfur-free interface on the 0–1 (x–z) plane. Electron accumulation around sulfur and depletion around Ni and Al sites are observed, which could weaken the Ni–Ni and Ni–Al bonds linking the
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interface. This is consistent with the BOvNi⫺Ni and BOvAl⫺Ni calculations. The weakened metal bonds normal to the interface make less contributions to the cohesive strength. Fig. 3(c) shows the charge density difference between sulfur-doped and sulfur-free on the 1–1 plane. Strong bonds between FNN sulfur and Ni atoms are formed, which significantly contribute to the shear strength of γ/γ⬘ interface [9]. This bonding character is consistent with the BOhS⫺Ni calculations. As analyzed in the proceeding section, it is this part of bonding that makes major contributions to the shear strength, thus leading to the sulfur embrittlement of the interface. Fig. 4(a) presents the charge density difference between sulfur doping plus Re substitution and the pure interface on the 0–1 (x–z) plane. It is clear to see that a strong bond between Re and sulfur is formed, which is consistent with the BOvS⫺Re ( ⫽ 1.1246) calculation. This strong bond between Re and sulfur could enhance the interface cohesion. To further examine the sulfur effect on Re substitution for Al at the interface, we present the charge density difference between sulfur doping plus Re substitution (denoted as ReS) and Re substitution in Fig. 4(b). Results indicate that sulfur induces charge depletion around Re atoms. By comparing Figs. 3(b) and 4(b), we find electron accumulation (solid lines) between sulfur and Re atoms shown in Fig. 4(b), which prompts the formation of a strong covalent bond between Re and sulfur. In an attempt to examine the effect of Re on sulfur charge distribution, we calculated the charge density difference between ReS and sulfur doping at the interface. The result in Fig. 4(c) shows that Re induces the electron accumulation around its FNN atoms including sulfur and Ni atoms, thus Fig. 3. (a) Total valence charge density for sulfur-doped γ/γ⬘ interface on the (010) (x–z) plane. Contours start from 0.001 e/(au)3, and increased successively by 0.025 e/(au)3. (b) The charge density difference between sulfur-doped and sulfur-free interface on the (010) (x–z) plane. Contours start from ± 1.0 × 10⫺3 e / (au)3, and increased successively by ± 1.0 × 10⫺3 e / (au)3. (c) The charge density difference between sulfur-doped and sulfur-free γ/γ⬘ interface on the 1–1 (x–y) plane. Contours start from ± 1.0 × 10⫺3 e / (au)3, and increased successively by 3.0 × 10⫺2 e / (au)3.fore, Re substitution promotes the ductile behavior of the interface.
K. Chen et al. / Acta Materialia 51 (2003) 1079–1086
1085
increasing the bonds across the interface, i.e. increases the interface cohesion strength. There We calculated the Hischfeld charge and Mulliken orbital occupation to study the charge transfer of electrons associated with the bond formation. For the pure γ/γ⬘ interface, it is found that the Ni atoms lying within the γ/γ⬘ interface (0–0 and 1–1 layers) gain electrons, while Al atoms donate electrons. In the case of sulfur doping at the octahedral interstitial site at the interface, Ni atoms gain less electrons than the sulfur-free interface. This gives rise to the weakening of the metal bonding due to less electrons taking part in the formation of bonds. Orbital charge transfer indicates that for Al (1) at the sulfur-doped interface, 3s and 3p electrons transfer to their 3d orbitals, while 3s electrons of sulfur atom transfer to their 3p orbitals. They form strong bond between Al and sulfur atoms, which agrees with the calculated BOvAl⫺S result (0.6360). For Ni atoms in the case of the addition of sulfur and Re, the 4s electrons transfer to their 3d and 4p orbitals, while for Re atoms, the 6s electrons transfer to their 5d and 6p orbital. This yields strong bonds between Re–sulfur and Re–Ni atoms, thus benefiting the cohesive strength. In the past decades, sulfur embrittlement has been extensively studied from both experimental and theoretical viewpoints. The major conclusion from the earlier first principles calculations done by Briant et al. [8] was that as a impurity element, sulfur causes weakening of metal bonding across the grain boundary, thus reduces the cohesive strength. Latter work by Eberhart et al., based on the phenomenological theory of fracture, emphasizes [9] that the sulfur–sulfur network bonds within the boundary increase the shear strength, and thus induce the embrittlement of the grain boundary. In this paper, we use the vertical and horizontal BOs to qualitatively represent the shear and cohesive strengths by a
冘
冘
local sum of BOv and BOh, respectively. The ratio RBO of shear strength over cohesive strength is then employed to judge the sulfur embrittlement of the interFig. 4. (a) The charge density difference between sulfurdoped plus Re substitution and pure γ/γ⬘ interface on the (010) (x–z) plane. Contours start from ± 1.0 × 10⫺3 e /(au)3, and increased successively by 2.0 × 10⫺2. (b) The charge density difference between sulfur-doped plus Re substitution and Re substitution γ/γ⬘ interface on the (010) (x–z) plane. Contours start from ± 1.0 × 10⫺3 e / (au)3, and increased successively by 2.0 × 10⫺2. (c) The charge density difference between sulfurdoped plus Re substitution and sulfur-doped γ/γ⬘ interface on the (010) (x–z) plane. Contours start from ± 1.0 × 10⫺3 e / (au)3, and increased successively by 5.0 × 10⫺3.
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face. We conclude that it is the sulfur and metal (S–Ni) bonds lying within the shear plane that cause interface embrittlement. As our model is independent of the sulfur, it could be used as a new approach to study the impurity embrittlement of the interface boundary.
[3] [4] [5] [6] [7] [8] [9]
4. Conclusions Fully first principles quantum mechanics DMol calculations have been performed to investigate impurity sulfur embrittlement on the γ/γ⬘ interface of Ni-base SC superalloys. Strong bonds between sulfur and Ni atoms lying within the interface are formed, which significantly increase the shear strength. The large shear strength gives rise to a high ratio RBO of shear strength over cohesive strength, and thus leads to interface embrittlement. Refractory Re substitution at γ/γ⬘ interface increases the cohesive strength and hence reduces RBO of the interface. This in turn relieves the sulfur embrittlement of the interface. Acknowledgements This project is funded by the Department of National Defense (Canada) through the Technology Investment Fund Program.
References [1] Merica PD, Waltenberg RT. Trans AIME 1925;71:709. [2] Lozinsky MG. Russ Metall 1967;5:65.
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Kraai DA, Floreen S. Trans AIME 1964;230:833. Bay AS. Phys Met Metallogr 1968;25:179. Floreen S, Wrestbrook JH. Acta Metall 1974;17:1175. Holt RT, Wallace W. Int Metals Rev 1976;203:1. Briant CL, Messmer RP. Philos Mag B 1980;42:569. Briant CL, Messmer RP. Acta Metall 1982;30:457. Eberhart ME, Johnson KH, Latanision RM. Acta Metall 1984;32:955. Loier C, Boos J. Metall Trans A 1981;12A:1223. Pugh SF. Philos Mag 1954;45:823. Rice J, Thomson R. Philos Mag 1974;29:73. Rice J. J Mech Phys Solids 1992;40:239. Caron P, Khan T. Aerosp Sci Technol 1999;3:513. Chen K, Zhao LR, Tse JS. Philos Mag, in press. Chen K, Zhao LR, Tse JS. Philos Mag Lett, in press. Dixon DA, Andzelm J, Fitzerald G, Wimmer E, Jasien P. In: Labanowski JK, Andzelm JW, editors. Density functional methods in chemistry. Berlin: Springer; 1991. p. 33. Tang SP, Freeman AJ, Olson GB. Phys Rev B 1993;47:2441. Perdew JP, Wang Y. Phys Rev B 1986;33:8800. Hirschfeld FL. Theor Chim Acta 1977;44:129. Mulliken RS. J Chem Phys 1955;23:1833. Mayer I. Int J Quantum Chem 1986;29:477. Sun S, Kioussis N. Phys Rev B 1996;54:3074. Liu Y, Chen K, Lu G, Zhang J, Hu ZQ. Acta mater 1997;45:1837. Blavette D, Caron P, Khan T. Scr Metall 1986;20:1395. Wanderka N, Glatzel U. Mater Sci Eng A 1995;203:69. Warren PJ, Cerezo A, Smith GDW. Mater Sci Eng A 1998;250:88. Cetel AD, Duhl DN. Superalloys 1988. In: Warrendale, PA: The Metallurgical Society, Inc.; 1988. p. 235.
Sulfur embrittlement on γ/γ⬘ interface of Ni-base single crystal superalloys K. Chen a,∗, L.R. Zhao a, J.S. Tse b a
Structures, Materials and Propulsion Laboratory, Institute for Aerospace Research, National Research Council Canada, Ottawa, Canada K1A 0R6 b Steacie Institute for Molecular Sciences, National Research Council Canada, Ottawa, Canada Received 16 May 2002; accepted 16 October 2002
Abstract The impurity sulfur embrittlement on γ / γ⬘ interface of Ni-base single crystal superalloys has been investigated by first principles quantum mechanics DMol calculations. Using a local sum of vertical (
冘 冘 冘
冘
BOv) and horizontal
( BOh) Mayer bond orders, we proposed a new method to evaluate the competition between the shear and cohesive strengths of the interface. Coupled with the phenomenological theory of fracture, we define the ratio of RBO ⫽ BOh / BOv to assess the embrittlement trend of the interface related to sulfur doping or sulfur doping combined with Re substitution. It is shown that sulfur increases RBO by 121% relative to the sulfur-free γ / γ⬘ interface, which could induce interface embrittlement from the electron bonding point of view. Calculations of both BO and charge density distribution reveal that it is the strong bonds between sulfur and Ni atoms lying within the interface that contribute to the interface embrittlement. The substitution of Re for Al at the γ / γ⬘ interface results in reduced RBO, thus relieving the tendency of interface embrittlement. Furthermore, our model on sulfur embrittlement is compared with previous models. 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: First principles calculations; Sulfur embrittlement; γ/γ⬘ interface; Ni-base superalloys
1. Introduction It was widely reported that sulfur reduces malleability of nickel, causes cold and hot brittleness, and reduces tensile strength, ductility and creep rupture strength of superalloys [1–6]. Earlier
∗
Corresponding author. E-mail address: [email protected] (K. Chen).
attempts to deduce the atomic mechanisms responsible for sulfur embrittlement indicate that there is a depletion of intermetallic electron density accompanying the sulfur segregation to interface boundaries [7,8]. This depletion is suggested to cause a weakening of intermetallic bonds and thus reduces the cohesive strength across the interface boundary. Due to the lack of specifics of crack propagation in earlier models, Eberhart et al. [9] argued that any of the models which explain
1359-6454/03/$30.00 2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. doi:10.1016/S1359-6454(02)00512-8
1080
K. Chen et al. / Acta Materialia 51 (2003) 1079–1086
embrittlement caused by either the ideal shear strength or the ideal cohesive strength alone, should be considered incomplete. They [9] suggested that the brittle fracture is the result of a sensitive interplay of shear and cohesive strengths at a crack tip. Their calculations show that it is the sulfur–sulfur network bonds lying within the interface boundary that increase the shear strength, while the cohesive strength continues to diminish, giving rise to the essential condition for the brittle fracture. However, it should be pointed out that Eberhart et al.’s model is unable to account for the sulfur embrittlement in INCONEL Ni and pure Ni when sulfur content is below 30 ppm [10]. In these cases, the sulfur content is so low that sulfur–sulfur network bonds are unlikely to form and make appreciable contributions to the shear strength within the interface boundary. In this study, a new first principles approach, coupled with the phenomenological theory of fracture [11–13], is proposed to study the sulfur embrittlement of γ/γ⬘ interface of Ni-base single crystal (SC) superalloys. By comparison with previous models, our model does not require the formation of sulfur– sulfur network bonds, and thus is applicable to any sulfur content in the materials. Furthermore, the effect of Re, which is typically added in SC superalloys as a refractory strengthening element [14– 16], on sulfur embrittlement is calculated. These results show that Re can reduce the embrittlement of the interface.
which contains double set valence functions plus a single d or p polarization function, as this technique has proven to be very efficient in studying molecular systems. Furthermore, the frozen-core approximation for Ni, Al, Re and S atoms is used in the calculations. The degree of convergence of the self-consistent iterations, measured by (rms) changes in the charge density, is set to be 10⫺5, which allows the energy to converge to 10⫺5 Ryd. All calculations in this paper were performed with the generalized-gradient approximation (GGA) proposed by Perdew and Wang [19]. The charge density and charge density difference are calculated and used to evaluate electronic properties. In addition, the Hirschfeld charge [20] and Mulliken orbital population techniques [21] are adopted to analyze the valence electron occupation on atomic orbitals. Moreover, the Mayer bond order (BO) [22] is employed to evaluate the atomic bond strength. Fig. 1 shows the 64 atoms cluster model used in this study. It consists of an upper Ni region and a lower Ni3Al region, representing the γ matrix and γ⬘ precipitate in Ni-base SC alloys, respectively. We define the γ/γ⬘ interface without both sulfur residing at the interstitial site and Re substitution for Al site as the pure γ/γ⬘ interface. Sulfur atoms usually occupy the octahedral interstitial sites
2. Methodology The DMol molecular cluster orbital approach is used to analyze the orbital electron occupations. As a molecular cluster approach, DMol3 is based on the density functional theory with a local-density approximation [17]. In DMol3, the local atomic wavefunctions, which are used as the basis functions for the cluster model, are generated numerically from the same local-density functional theory solutions for the free atoms. In general, the molecular cluster approach is suitable for studying electronic properties that are primarily a function of local environment [18]. The model uses the double numerical basis set for Ni, Al, Re and S atoms,
Fig. 1. Cluster model for DMol calculations. Open circles, Ni atoms; black circles, Al atoms; grey circle M1, Re atom; grey circle M4, sulfur atom. The 0–0 layer represents the highest plane of the Ni3Al region, while the 1–1 layer represents the lowest plane of the upper Ni region.
K. Chen et al. / Acta Materialia 51 (2003) 1079–1086
[23,24], and the Re atom, as reported experimentally to substitute for Al in γ⬘ precipitates [25–27], is placed at the Al(1) site. As part of the γ⬘ phase, the 0–0 layer (x–y plane), as labeled in Fig. 1, contains mixed Ni and Al atoms, in which Ni atoms occupy the face center sites, while Al atoms occupy the corner sites. The next layer adjacent to the 0–0 layer in the upper γ region is defined as the 1–1 layer. Inequivalent atoms in the cluster as shown in Fig. 1 are denoted by the numerical labels depending on the C4v point group symmetry of the cluster. The lattice constant of the cluster is chosen to be equal for both Ni (γ) and Ni3Al (γ⬘) due to the assumption of complete coherence. The cluster is then relaxed with the variation in the lattice constant when energetic calculations are performed. 3. Results and discussions Pugh [11] proposed that the resistance to plastic deformation is related to the product Gb, where G and b are the shear modulus and Burgers vector, respectively. The fracture strength was assumed to be proportional to the product Ba, where B and a are Bulk modulus and lattice parameter, respectively. If Gb/Ba is high, a material will behave in a brittle behavior. The Gb/Ba reflects the competition between the shear and cohesive strengths at the crack tip of fracture. Rice and Thompson [12] speculated that brittle vs. ductile behavior may be governed at the tip of an atomically sharp crack. A material will, in principle, fail in a brittle manner if the ideal cohesive strength is reached along the extension of the crack before the ideal shear strength is reached. Any complete theory of embrittlement must deal not only with the way the cohesive strength changes as a result of segregation of impurities, but also the effect the segregation has on the shear strength. In order to qualitatively evaluate the competition between the shear and cohesive strengths of the γ/γ⬘ interface, we need to consider the BO values between atoms. The BO in nature is the overlap of electron wavefunctions between atoms, which can be used to qualitatively measure the strength of atomic bonds. In this study, we use the Mayer BO form and calculation approach [22]. The definition of Mayer BO between atoms A and B is given by:
BOAB ⫽ 2
冘冘
1081
[(PaS)mn(PaS)nm
(1)
m苸Am苸B
⫹ (PbS)mn(PbS)nm] where Pa, Pb are the density matrices, S is the overlap matrix of the wavefunctions [22]. We calculate BOAB values within the first nearest neighbor (FNN) B atoms around A site. Our calculations indicate that the BOAB value, if the separation between atoms A and B is beyond FNN range, is small and can be neglected without altering the conclusions. In order to examine the interplay of shear and cohesive strengths at the interface in detail, we divide the BOs into these vertical and horizontal components. The vertical BO (BOv) is defined and calculated for atoms along the [001] z direction to qualitatively measure the cohesive strength across the γ/γ⬘ interface. While, the horizontal BO (BOh) is defined and calculated for inplane atoms perpendicular to the [001] z direction to qualitatively represent shear strength. Further-
冘 冘
more, we define a ratio of RBO ⫽ BOh / BOv to assess the embrittlement trend of the interface related to sulfur doping or sulfur doping combined
冘
with Re substitution, where the BOh is a local sum of BOs within both the 0–0 and 1–1, while the
冘 冘
BOv is a local sum of BOs across 0–1 layer.
It is necessary to point out that although
冘
BOh
and BOv are not the real shear and cohesive strengths of the γ/γ⬘ interface in terms of the continuum mechanics viewpoint, it is suggested that
冘 冘
the ratio RBO ⫽ BOh / BOv can be used to represent the trend of shear strength vs. cohesive strength of the interface. In this paper, we use RBO at the γ/γ⬘ interface to examine the embrittlement trend induced by impurity sulfur doping. Fig. 2(a) shows the BOs for pure, sulfur-doped and sulfur-doped plus Re added γ/γ⬘ interface. For
冘
冘 冘 冘
BOhp and BOvp to repcomparison, we use resent the shear and cohesive strengths by per pair BO, respectively, and use RBO ⫽ BOhp / BOvp to evaluate the embrittlement trend of the interface as a function of sulfur doping or sulfur doping plus
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冘BO and 冘BO for pure (Ni/Ni Al), sulfur-doped (Ni/Ni Al–S) and sulfur-doped plus Re substitution (Ni/Ni Al–Re–S) γ/γ⬘ interface. (b) The ratio R ⫽ 冘BO / 冘BO for pure (Ni/Ni Al), sulfur-doped (Ni/Ni Al–S) and sulfur-
Fig. 2.
(a) A local sum of BOs 3
h p
v p
3
BO
h p
3
v p
3
3
doped plus Re substitution (Ni/Ni3Al–Re–S) γ/γ⬘ interfaces.
Re substitution. As shown in Fig. 2(b), the calculated RBO ⫽ 0.7877 for a pure γ/γ⬘ interface is set as a baseline. In the case of the sulfur-doped interface, the BOv between Ni atoms normal to the γ/γ⬘ interface is reduced from 0.2914 of pure interface to 0.2597. This indicates that the bonding between metal Ni atoms responsible for the interface cohesion is reduced due to the sulfur doping. This result is consistent with the conclusions of Briant and Messmer [7], suggesting that sulfur causes
embrittlement by weakening the metal bonding across the interface. Our calculations, however, demonstrate that the cohesive strength across the 0–1 layer alone cannot completely determine the interface embrittlement. The bonding of sulfur and metal atoms within the 1–1 plane and bonding between metal atoms in the 0–0 plane will contribute to the shear strength of the interface. It is the interplay between the shear and cohesive strengths that determines the brittle or ductile trend of the
K. Chen et al. / Acta Materialia 51 (2003) 1079–1086
冘
冘
interface. Therefore, we need to examine the h
v
BO p and BO p which include both sulfur– metal bonds and metal–metal bonds. Calculations reveal that the BOh between Ni atoms within the 1–1 layer is reduced from 0.2636 for the pure interface to 0.2259 for the sulfur-doped interface. The BOh between sulfur and Ni atoms within the 1–1 layer reaches 0.8552, 3.8 times larger than Ni–Ni bonds. This part of BOh obviously makes large contributions to the shear strength, thus gives rise to a large ratio RBO. From Fig. 2(a), we have
冘
冘
BOhp ⫽ 0.4157, and BOvp ⫽ 0.2382. Therefore, for sulfur-doped interface, the ratio RBO ⫽ 1.745 is obtained, representing 121% increase compared to the pure γ/γ⬘ interface. From an electronic structure point of view, it can be concluded that the strong sulfur–metal bonds (BOhS⫺Ni ⫽ 0.8552) lying within the interface boundary cause the interface embrittlement. It should be noted that our model is independent of the sulfur concentration as no sulfur–sulfur network bonds [9] are necessary to explain the sulfur embrittlement. Thus, it is reasonably believed that if the sulfur content at the interface increases, each individual sulfur atom would form a strong bond between sulfur and Ni atoms lying within the interface boundary, and would contribute to the embrittlement behavior as a single sulfur atom does. As a refractory element, Re was proven to be a potent strengthener and key element for the improvement of mechanical properties of SC superalloys [28]. Recently, we have performed first principles calculations on the Re strengthening effect on the γ/γ⬘ interface of Ni-base SC superalloys [15]. Our results show that refractory Re can significantly strengthen γ/γ⬘ interface. In order to examine Re effects on sulfur embrittlement of the interface, we substitute Re for Al atom as suggested by experiments [25–27] in the model of a sulfur-doped interface. In this case, the calculated BO between sulfur and Ni atoms within the 1–1 layer is significantly reduced from 0.8552 of Refree to 0.6808 of Re substitution for Al. This change partially reduces the contribution of sulfur– metal bonds to the interface shear strength. However, due to the substitution of Re for Al, the metal
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冘
Re–Ni bonds within the 0–0 plane increase significantly. As a combined result, the BOhp is 0.4176, almost the same as the purely sulfur-doped interface. Calculations of the BOv between sulfur and Re atoms across the 0–1 plane shows that BOvRe⫺S ⫽ 1.1246, 1.65 times larger than sulfur– Ni bonds on the 0–0 plane. Similar to BOvRe⫺S, the BOvRe⫺Ni increases from 0.0959 of sulfur-doped interface to 0.3619 of Re substitution, 3.8 times larger than the purely sulfur-doped interface. As
冘
BOvp ⫽ 0.34414. shown in Fig. 2(a), we have The increased BOv lowers the ratio RBO, and thus reduces the tendency of interface embrittlement. For the case of sulfur doping plus Re substitution, we have RBO ⫽ 1.235, a reduction of 30% from 1.745 of purely sulfur-doped interface. One of the important issues in our study is the bond redistributions in the interface region (shear and cohesion) induced by either the substitution element, such as Re, or by the interstitial atom, such as sulfur. The sensitive interplay of the shear and cohesive strengths as a function of additional elements may control the tendency of interface embrittlement. To further study sulfur embrittlement on the interface, we calculated the spatial distribution of electrons associated with the interface structure. Of the electronic properties, the total valence charge density and the charge density difference ⌬rind(r) are important parameters. The latter is defined as: ⌬rb(r) ⫽ rcluster(r,X)⫺rcluster(r)
(2)
where rcluster(r,X) is the charge density with Re(S) addition at the interface. Fig. 3(a) presents the total valence charge density of sulfur-doped γ/γ⬘ interface on the 0–1 (x–z) plane in units of 10⫺3 e/(au)3. Strong bonds between FNN sulfur and metal (Ni and Al) atoms are observed. These strong bonds include sulfur–Ni bonds lying on the 1–1 plane and a sulfur–Al bond normal to the γ/γ⬘ interface, with the latter linking the interface together. These bonds contribute to both shear and cohesive strengths of the interface. Fig. 3(b) shows the charge density difference between sulfur-doped and sulfur-free interface on the 0–1 (x–z) plane. Electron accumulation around sulfur and depletion around Ni and Al sites are observed, which could weaken the Ni–Ni and Ni–Al bonds linking the
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interface. This is consistent with the BOvNi⫺Ni and BOvAl⫺Ni calculations. The weakened metal bonds normal to the interface make less contributions to the cohesive strength. Fig. 3(c) shows the charge density difference between sulfur-doped and sulfur-free on the 1–1 plane. Strong bonds between FNN sulfur and Ni atoms are formed, which significantly contribute to the shear strength of γ/γ⬘ interface [9]. This bonding character is consistent with the BOhS⫺Ni calculations. As analyzed in the proceeding section, it is this part of bonding that makes major contributions to the shear strength, thus leading to the sulfur embrittlement of the interface. Fig. 4(a) presents the charge density difference between sulfur doping plus Re substitution and the pure interface on the 0–1 (x–z) plane. It is clear to see that a strong bond between Re and sulfur is formed, which is consistent with the BOvS⫺Re ( ⫽ 1.1246) calculation. This strong bond between Re and sulfur could enhance the interface cohesion. To further examine the sulfur effect on Re substitution for Al at the interface, we present the charge density difference between sulfur doping plus Re substitution (denoted as ReS) and Re substitution in Fig. 4(b). Results indicate that sulfur induces charge depletion around Re atoms. By comparing Figs. 3(b) and 4(b), we find electron accumulation (solid lines) between sulfur and Re atoms shown in Fig. 4(b), which prompts the formation of a strong covalent bond between Re and sulfur. In an attempt to examine the effect of Re on sulfur charge distribution, we calculated the charge density difference between ReS and sulfur doping at the interface. The result in Fig. 4(c) shows that Re induces the electron accumulation around its FNN atoms including sulfur and Ni atoms, thus Fig. 3. (a) Total valence charge density for sulfur-doped γ/γ⬘ interface on the (010) (x–z) plane. Contours start from 0.001 e/(au)3, and increased successively by 0.025 e/(au)3. (b) The charge density difference between sulfur-doped and sulfur-free interface on the (010) (x–z) plane. Contours start from ± 1.0 × 10⫺3 e / (au)3, and increased successively by ± 1.0 × 10⫺3 e / (au)3. (c) The charge density difference between sulfur-doped and sulfur-free γ/γ⬘ interface on the 1–1 (x–y) plane. Contours start from ± 1.0 × 10⫺3 e / (au)3, and increased successively by 3.0 × 10⫺2 e / (au)3.fore, Re substitution promotes the ductile behavior of the interface.
K. Chen et al. / Acta Materialia 51 (2003) 1079–1086
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increasing the bonds across the interface, i.e. increases the interface cohesion strength. There We calculated the Hischfeld charge and Mulliken orbital occupation to study the charge transfer of electrons associated with the bond formation. For the pure γ/γ⬘ interface, it is found that the Ni atoms lying within the γ/γ⬘ interface (0–0 and 1–1 layers) gain electrons, while Al atoms donate electrons. In the case of sulfur doping at the octahedral interstitial site at the interface, Ni atoms gain less electrons than the sulfur-free interface. This gives rise to the weakening of the metal bonding due to less electrons taking part in the formation of bonds. Orbital charge transfer indicates that for Al (1) at the sulfur-doped interface, 3s and 3p electrons transfer to their 3d orbitals, while 3s electrons of sulfur atom transfer to their 3p orbitals. They form strong bond between Al and sulfur atoms, which agrees with the calculated BOvAl⫺S result (0.6360). For Ni atoms in the case of the addition of sulfur and Re, the 4s electrons transfer to their 3d and 4p orbitals, while for Re atoms, the 6s electrons transfer to their 5d and 6p orbital. This yields strong bonds between Re–sulfur and Re–Ni atoms, thus benefiting the cohesive strength. In the past decades, sulfur embrittlement has been extensively studied from both experimental and theoretical viewpoints. The major conclusion from the earlier first principles calculations done by Briant et al. [8] was that as a impurity element, sulfur causes weakening of metal bonding across the grain boundary, thus reduces the cohesive strength. Latter work by Eberhart et al., based on the phenomenological theory of fracture, emphasizes [9] that the sulfur–sulfur network bonds within the boundary increase the shear strength, and thus induce the embrittlement of the grain boundary. In this paper, we use the vertical and horizontal BOs to qualitatively represent the shear and cohesive strengths by a
冘
冘
local sum of BOv and BOh, respectively. The ratio RBO of shear strength over cohesive strength is then employed to judge the sulfur embrittlement of the interFig. 4. (a) The charge density difference between sulfurdoped plus Re substitution and pure γ/γ⬘ interface on the (010) (x–z) plane. Contours start from ± 1.0 × 10⫺3 e /(au)3, and increased successively by 2.0 × 10⫺2. (b) The charge density difference between sulfur-doped plus Re substitution and Re substitution γ/γ⬘ interface on the (010) (x–z) plane. Contours start from ± 1.0 × 10⫺3 e / (au)3, and increased successively by 2.0 × 10⫺2. (c) The charge density difference between sulfurdoped plus Re substitution and sulfur-doped γ/γ⬘ interface on the (010) (x–z) plane. Contours start from ± 1.0 × 10⫺3 e / (au)3, and increased successively by 5.0 × 10⫺3.
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face. We conclude that it is the sulfur and metal (S–Ni) bonds lying within the shear plane that cause interface embrittlement. As our model is independent of the sulfur, it could be used as a new approach to study the impurity embrittlement of the interface boundary.
[3] [4] [5] [6] [7] [8] [9]
4. Conclusions Fully first principles quantum mechanics DMol calculations have been performed to investigate impurity sulfur embrittlement on the γ/γ⬘ interface of Ni-base SC superalloys. Strong bonds between sulfur and Ni atoms lying within the interface are formed, which significantly increase the shear strength. The large shear strength gives rise to a high ratio RBO of shear strength over cohesive strength, and thus leads to interface embrittlement. Refractory Re substitution at γ/γ⬘ interface increases the cohesive strength and hence reduces RBO of the interface. This in turn relieves the sulfur embrittlement of the interface. Acknowledgements This project is funded by the Department of National Defense (Canada) through the Technology Investment Fund Program.
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