First-principles study of the influence of lattice misfit on the segregation behaviors of hydrogen and boron in the Ni–Ni3Al system

Acta Materialia 55 (2007) 4845–4852 www.elsevier.com/locate/actamat

First-principles study of the influence of lattice misfit on the segregation behaviors of hydrogen and boron in the Ni–Ni3Al system Y.X. Wu a, X.Y. Li b, Y.M. Wang a

a,*

Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Science, 72 Wenhua Road, Shenyang 110016, China b Materials for Special Environment Department, Institute of Metal Research, Chinese Academy of Science, 72 Wenhua Road, Shenyang 110016, People’s Republic of China Received 1 September 2006; received in revised form 5 May 2007; accepted 5 May 2007 Available online 27 June 2007

Abstract A first-principles method is employed to investigate the segregation behaviors of hydrogen and boron in Ni-based and Ni3Al-based alloys using two models. Chemical binding energy analysis shows that both boron and hydrogen are able to segregate to the interstices in the Ni phase, Ni3Al phase and Ni/Ni3Al interface. Boron, however, is bound to its neighbor atoms more tightly than hydrogen in both models and its stable state exists over a broader lattice misfit range compared with hydrogen. The bond order analysis we have proposed reveals the origin of the boron-induced ductility and hydrogen-induced embrittlment at the Ni/Ni3Al interface with different lattice misfit. The calculations indicate that hydrogen causes more severe embrittlement at the Ni/Ni3Al interface in Ni3Al-based than in Ni-based alloys. Furthermore, it is found that the boron-induced ductility and hydrogen-induced embrittlement are changed, and thus controllable, by the lattice misfit. Our results provide a quantitative explanation of many experimental phenomena caused by the addition of boron and hydrogen to Ni-based and Ni3Al-based alloys.  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Interface; Embrittlement; Ductility; Lattice misfit; First-principles

1. Introduction Ni-based superalloys have been successfully utilized in both land-based and aeroplane gas turbine industries for their excellent oxidation resistance and high-temperature strength – generally resulting from the incorporation of coherent, ordered Ni3Al c 0 phase into the disordered, solid solution Ni c matrix. However, Ni3Al c 0 phase incorporation leads to an increased propensity for hydrogen-induced embrittlment in Ni-based superalloys [1–6] and thus limits the application of Ni3Al-incorporated Ni-based alloys at high temperatures. Liu et al. [7] found that the addition of a small amount of boron could suppress hydrogeninduced embrittlment in Ni-rich Ni3Al alloys. It is believed that it is the segregatation of boron to grain boundaries which increases the ductility of such alloys [7–13]. How*

Corresponding author. Tel.: +86 24 2397 1840; fax: +86 24 2389 1320. E-mail address: [email protected] (Y.M. Wang).

ever, Sun et al. [14] suggested that the ‘‘bulk effect’’ of boron should also be considered since some experiments [15] have shown that the addition of boron could also improve the ductility of single Ni3Al crystals. Later on, Al was proposed to be the active element in the extraction of hydrogen from H2O [16,17]. In the Ni3Al-incorporated Ni-based superalloys and Ni-rich Ni3Al intermetallics, the phase interface containing Al may not only extract hydrogen but also be a possible channel for hydrogen uptake in these alloys. To date, much of research attention has been paid to the effects of impurities on different kinds of interfaces due to the important role of impurities in mechanical properties of materials, e.g. sulfur on the Ni/Ni3Al interface [18], and carbon, oxygen and sulfur on the NiAl/Mo interface [19]. Although many experiments and calculations [14– 18,20–28] studied the strengthening mechanism of c 0 (Ni3Al) precipitates in Ni-based alloys and the mechanism of boron-induced ductility in Ni3Al, these studies mainly

1359-6454/$30.00  2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2007.05.006

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Y.X. Wu et al. / Acta Materialia 55 (2007) 4845–4852

focused on the effects of impurities on the electronic structure of a single Ni phase, a single Ni3Al phase or a Ni/ Ni3Al interface. Few reports in the literature are concerned with the occupation behavior of impurities when three phases exist simultaneously, namely the Ni-Ni3Al alloy with a Ni/Ni3Al interface. Studying the segregation behaviors of impurities in such an alloy is of interest because the tritium-filled experiments of Cheˆne and Brass [29] found that hydrogen segregates in the c 0 phase rather than c/c 0 interface of CMSX-2 Ni-based superalloys. Takasugi et al. [30], in an experimental study on environmental embrittlement in Ni3(Si, Ti) and Co3Ti alloys also emphasized the need for further study of the competition between the segregation behaviors of hydrogen and boron to better understand the environmental embrittlement of L12 polycrystals. In addition, it has been reported that lattice misfit can influence the mechanical properties of different materials [31–37]. In their first-principles study of the Ni/Ni3Al interface with a lattice misfit, Peng et al. [31] reported that when the lattice misfit is carefully controlled, a desired mechanical property of the alloy can be obtained. Thompson et al. [36,37] proposed that the c/c 0 interface acts like a kind of trap for hydrogen, closely related to the lattice misfit d on the interface between c 0 and Fe c phases, and thus the hydrogen-induced ductility loss can be modified by the lattice misfit. However, the results reported by West et al. [38] raise the question of why a lattice parameter difference of less 0.1% for Fe c phases in A-286 superalloys, aged for 0.25 and 450 h respectively, resulted in approximately a 10 times difference in hydrogen-induced ductility loss. Accordingly, it is worth investigating quantitatively the relationship between the lattice misfit and the mechanical properties of alloys. Experimental study of alloys containing impurities such as hydrogen is complicated by the fact that low-atomicnumber species scatter X-photons, electrons and neutrons poorly. Thus, in such a case theoretical calculations can be useful for investigating the effect of impurities on the crystalline and electronic structure determining the mechanical properties of alloys. All these factors motivated us to study the segregation behaviors of hydrogen and boron and their effects on the mechanical properties of a Ni–Ni3Al alloy including the interface with different lattice misfit to gain an insight into the hydrogen-induced embrittlement, the boron-induced ductility and the relationship between these characteristics. 2. Models and computational details Since one of our objectives was to study the segregation behaviors of dilute hydrogen and boron in Ni–Ni3Al alloys, the model needed be large enough to: (i) include at least a Ni phase, a Ni3Al phase and a coherent phase interface between them; and (ii) guarantee the impurity to be dilute. For this purpose a cluster model may be appropriate. First, as early as in 1971, Keller [39] pointed out

that a cluster model can obtain the correct density of state (DOS) if the cluster is large enough to include the nearest-neighbor atoms. Second, Keller’s conclusion was confirmed by Lindgren et al. [40] in their study on the hyperfine fields and electronic structure of hydrogen impurities in transition metals. Furthermore, Eberhart et al. [10] have found that the DOS profile and band width obtained from only a 19-atom cluster are close to those from a linear muffin-tin band calculation for Ni3Al. Recently a cluster model was used to investigate the effects of impurities on the Ni/Ni3Al interface cohesion [23] and the hydrogen embrittlement in polycrystalline Ni3Al [26,27]. All these studies [10,23,26,27,39,40] suggest that the cluster model is an appropriate method to represent the basic characteristics of materials. First-principle computational results about impurities in Ni3Al [24,25,41] indicated that hydrogen and boron preferentially occupy the Ni-rich interstices. A nuclear magnetic resonance (NMR) experiment conducted by Shinohara and Takasugi [42] also suggested that boron prefers to occupy an interstice surrounded by Ni. Based on these results, our calculations started by placing a hydrogen or boron atom at an interstice surrounded by 6 Ni atoms in an 11layer cluster model as shown in Fig. 1, where the large and small grey spheres denote Al atoms and impurity X (X = H, B) atoms, respectively, and the black spheres are Ni atoms. The upper five layers of the model were of Ni phase and the lower six layers were of Ni3Al phase. The interstice possibly occupied by hydrogen or boron was labeled by Xn (n = I–III). The locations of Xn were chosen at those sites as far as possible from the cluster surface to reduce the influence of the cluster surface on the calculated results. The lattice constant for the upper five layers of Ni phase was denoted aU and that for the lower six layers of Ni3Al phase aD. The coherent Ni/Ni3Al interface with different lattice misfit was constructed by changing aD ˚ in Model 1 to simulate (aNi3 Al) but fixing aU (aNi) to 3.56 A the case of Ni phases in a Ni3Al matrix, namely a Ni3Albased alloy. To simulate a Ni3Al phase in a Ni matrix, ˚ and aU was varied in Model 2. aD was fixed at 3.60 A The lattice misfit d was defined as d1 = 2(aD  aU)/(aD + aU) · 100% for Model 1 but d2 = 2(aU  aD)/(aD + aU) · 100% for Model 2. The discrete variational Xa (DV-Xa) method [43–47] within the framework of the linear combination of atomic orbital (LCAO) approximation was used to calculate the binding energy and electronic structure. The exchange-correlation potential of von Barth and Hedin was adopted [48]. The atomic orbitals used in the calculations were: (1s2s2p3s3p)3d4s4p for nickel, (1s2s2p)3s3p3d for aluminum, 1s2s for hydrogen and (1s)2s2p3s for boron. The atomic orbitals in the brackets were frozen. To guarantee the accuracy of calculated results the number of spatial integration points for each model was 75,000, i.e. about 1000 points per atom. This number of points was larger than the 800 points per atom adopted by Monma et al. [49] in their study of the electronic structure of LaNi5

Y.X. Wu et al. / Acta Materialia 55 (2007) 4845–4852

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maintaining the C4v symmetry to obtain the lowest value of Eb(Xn, d). To compare the effect of hydrogen and boronm the impurity formation energy was calculated as follows: DEimp ðXn; dÞ ¼ Eb ðXn; dÞ  Eb ðclean;dÞ:

ð2Þ

By using Eq. (2), the surface effect of the clusters on the calculated results can be minimized to give a good description of the change in Eb due to the presence of impurity at Xn. In the calculation the atomic bond order BOA–B between atoms A and B, which can be evaluated by the Mulliken population analysis [50], was defined as: X XX BOA–B ¼ nl alm alm0 S mm0 ; ð3Þ l

m0 2B m2A

where alm and alm0 are the coefficients of the atomic orbital m and m 0 in the molecular orbital l. S mm0 is an overlap ma0 trix element between the two atomic orbitals m and m , and nl is the occupied charge of the molecular orbital l. BOA–B was used to evaluate the strength of the covalent bond between atoms A and B. 3. Results and discussions 3.1. Energy analyses

Fig. 1. Original cluster model used in the calculation. The large and small grey spheres denote Al atoms and impurity Xn (X = H, B; n = I–III) atoms, respectively, and the black spheres Ni atoms. The nearest- and next-nearest-neighbor atoms surrounding Xn are numbered.

and LaNi5H0.2 using the DV-Xa method. In calculating the binding energies, a geometric relaxation to each configuration of both Model 1 and 2 was done with the cluster point symmetry of C4v kept unchanged. The binding energy Eb was obtained directly as a result of the DV-Xa method and Eb was defined as: X Eb ¼ Etotal  Eref ¼ Etotal  Eatom ðiÞ; ð1Þ i

where Etotal is the total energy of cluster after the geometric relaxation and Eref the sum of the energies of the free atoms, Eatom(i), included in this cluster. To evaluate the effect of the impurity, the binding energies, Eb(Xn, d) and Eb (clean, d), of the clusters with and without Xn were calculated respectively at the same lattice misfit value d in the range from 25% to 30%. In the calculation of Eb(Xn, d) with an impurity at Xn, besides the geometric relaxation mentioned above, the position Xn was also adjusted whilst

The effect of an impurity on the stability of Model 1 or 2 at various values of lattice misfit d can be approximated by the curve of the impurity formation energy DEimp for the impurity as a function of d. These curves were shown in Fig. 2a and b. Physically DEimp(Xn, d) consists of the deformation energy ED(Xn, d) and the chemical (ionic plus covalent) binding energy EC(Xn, d). ED(Xn, d) can be considered as the deformation energy of the host cluster to accommodate impurity Xn at a given lattice misfit d. EC(Xn, d) is the chemical binding energy for an impurity Xn to remain in the pre-strained host clusters. The relationship between ED(BII, 4) and EC(BII, 4) was shown as an example in Fig. 3 where the two curves show the binding energies calculated with BII and without BII (clean) as a function of d. By means of this relationship the curves of EC(Xn, d) for Xn = HI, HII, HIII, BI, BII and BIII as a function of d were obtained and are shown in Fig. 4a for Model 1 (a Ni3Al-based alloy) and Fig. 4b for Model 2 (a Ni-based alloy). Note that EC(Xn, d) in Fig. 4 instead of DEimp(Xn, d) in Fig. 2 was used to estimate the effect of impurity on the stability of those models. This was because the component ED(Xn, d) out of DEimp(Xn, d) corresponds to the work needed to put an impurity (hydrogen or boron) at Xn in the stressed host cluster, which makes no contribution to the stability of the impurity in the cluster. Fig. 4 illustrates that EC (Hn, d) is only negative in the lattice misfit range of 4.7% < d1 < 8.0% for Model 1 and 7.2% < d2 < 9.3% for Model 2, indicating that hydrogen is able to segregate and to stay at HI, HII and HIII only in these d ranges. However, the corresponding ranges

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Impurity formation energy (eV)

18

12

6

18

b

HI HII HIII BI BII BIII

HI HII HIII BI BII BIII

12

6

c

c 1

0

0

-6

-6

-12

-12

-18 -25 -20 -15 -10 -5 0

Misfit

-25 -20 -15 -10 -5 0

5 10 15 20 25 30 1

Misfit

(%)

-18 5 10 15 20 25 30 2

(%)

Fig. 2. Impurity formation energy DEimp(Xn, d) as a function of the lattice misfit d: (a) for Model 1; (b) for Model 2.

-246

-248 D

Binding energy E b (eV)

E (BII, 4) -250

-252

Eimp(BII, 4)

C

E (BII, 4) -254

-256

clean BII -258 -10

-5

0

5

Misfit

10

15

(%)

Fig. 3. Illustration of the relationship between the deformation energy ED(Xn, d) and the chemical binding energy EC(Xn, d) of which DEimp(Xn, d) consists.

for BI, BII and BIII are 5.4% < d1 < 11.4% and 8.7% < d2 < 13.4%, more than 1.3 times broader than those for hydrogen. Additionally EC (Bn, d) is about 6 eV lower than EC (Hn, d) over a broad lattice misfit range. These results indicate that boron has a greater tendency than hydrogen to occupy sites Xn (n = I–III) and boron is stable at these sites over a broader lattice misfit range than hydrogen. Furthermore, the preference for occupying

Xn can be controlled by changing the lattice misfit. For example, when d2 is changed to 10.5%, boron is stable due to EC = 4.91 eV (BI), 3.91 eV (BII) and 2.74 eV (BIII), while hydrogen is unstable due to EC = 2.94 eV (HI), 2.50 eV (HII) and 2.73 eV (HIII). The boron addition at Xn strengthens the cohesion of BI, BII or BIII with its surrounding atoms and thus may decrease the hydrogen diffusivity but increase its activation energy. This conclu-

Y.X. Wu et al. / Acta Materialia 55 (2007) 4845–4852

HI BI

180

Chemical binding energy EC (eV)

160 140

HII BII

HIII BIII

HI BI

0

0

-2

-2

40

120 100

-8

-8

-10

-10

80

-12

-12

-14 -10

180 160

-6

-6

60

HIII BIII

140

120

80

HII BII

-4

-4

100

4849

60 -5

0

5

-14 -10

10

-5

0

5

10

40

20

20

0

0

-20 -25 -20 -15 -10 -5 0

5 10 15 20 25 30

-25 -20 -15 -10 -5 0

Misfit δ1 (%)

-20 5 10 15 20 25 30

Misfit δ2 (%)

Fig. 4. Chemical binding energy EC(Xn, d) as a function of lattice misfit d: (a) for Model 1; (b) for Model 2.

sion is confirmed by the experimental results of Wan et al. [51]. They studied the effect of boron on hydrogen diffusion in boron-doped Ni3Al and obtained diffusivity D = 0.02 lm2 s1 and D = 0.001 lm2 s1 for Ni3Al with 120 ppm boron and Ni3Al with 1000 ppm boron, respectively, which resulted in reduced dissolved hydrogen penetration depth from 28 lm for the former to 7 lm for the latter. A careful examination of the insets in Fig. 4a and b indicates that EC(BI, d) is the smallest among EC(BI, d), EC(BII, d) and EC(BIII, d) over a broad lattice misfit range. This result may explain to some extent the finding of Liu et al. [7] that boron effectively improves only the ductility of the Ni-rich Ni3Al. In addition, EC (HI, d) is smaller than EC (HII, d) and EC (HIII, d) in the insets of Fig. 4a and b though there is almost no difference among the EC for these three locations. This suggests that the Ni/Ni3Al cohesion interface in our models is not a trap for hydrogen. The issue of the improvement of the ductility of Ni-based and Ni3Al-based alloys by boron will be discussed in next section. To illustrate the behavior of boron when both hydrogen and lattice misfit exist, boron and hydrogen atoms were put at site i, j (i, j = I–III, i 6¼ j) in Model 1, respectively, and the binding energies with all the possible configurations at d1 = 0 and d1 = 6.5% were calculated. Model 1 was chosen due to the importance of the suppressing effect of boron on hydrogen-induced embrittlement in Ni3Al-based alloys [7]. The results were plotted in Fig. 5. When d1 = 0, the cluster with BIHII is the most stable, but when d1 = 6.5%, the cluster with BIIHI rather than BIHII is the most stable. This implies that boron is able to reduce the hydrogen-induced boundary embrittlement by adjusting the lattice misfit.

3.2. Electronic structure analysis Here the bond orders (BOs) of hydrogen and boron with their nearest- and next-nearest-neighbor atoms at the coherent Ni/Ni3Al interface (XII) were calculated to elucidate the boron-induced ductility and hydrogen-induced embrittlment in the hosts. According to their relative direction to XII, these BOs were divided into vertical and parallel parts for each model: BOVXII ðdÞ ¼ BOX–Nið2Þ ðdÞþ 4BOX–Nið7Þ ðdÞ þ BOX–Nið3Þ ðdÞ þ 4BOX–Alð9Þ ðdÞ and BOPXII ðdÞ ¼ 4BOX–Nið8Þ ðdÞ. In the present work the Rice–Wang model [52] was used to evaluate the boron-induced ductility and hydrogeninduced embrittlement with BOs. It is widely accepted that the Rice–Wang thermodynamic model successfully predicts the embrittlement potential of segregations in many systems [53–60]. The fracture will be brittle if the work of dislocation emission in the crack tip, Gdis, is larger than the Griffith work of interfacial cleavage 2cint. If Gdis is less than 2cint, the fracture will be ductile. As 2cint represents the work needed to break an interface, it can be defined as Ni=Ni3 Al

2cint ¼ fsNi þ fsNi3 Al  fb fsNi

fsNi3 Al

;

ð4Þ

and are the surface energies of pure Ni and where Ni=Ni Al Ni3Al per unit area respectively, and fb 3 is the binding energy for joining the Ni and Ni3Al surface together. In our study, fsNi þ fsNi3 Al can be physically and approximately understood as BOPXII  E, where E is the average energy per electron of which the clusters are composed. Ni=Ni3 Al fb can be similarly represented by BOVXII  E. Table 1 lists the 2cint =E ¼ BOPXII  BOVXII for hydrogen and boron at various degrees of lattice misfit d in both models. It can be seen that the Ni/Ni3Al interface is embrittled by

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-264.5 1 1

=0 =6.5%

Binding energy E b (eV)

-265.0

-265.5

-266.0

- 266.5

-267.0 BIHII

BIHIII

BIIHI

BIIHIII

BIIIHI

BIIIHII

Configuration of BiHj Fig. 5. Dependence of binding energy Eb on the configuration of BiHj (i, j = I–III, i 6¼ j) at the two lattice misfit d1 = 0 and 0.65% in Model 1.

the segregation of hydrogen in Model 1 (a Ni3Al-based alloy) because most of the 2cint/E values are negative, except for those at d1 = 1.12% and 1.67%. The condition is a little better in Model 2 (a Ni-based alloy) than in Model 1. For example, at the same lattice misfit value of 0.56%, 2cint/E is 0.0440 in Model 1 but 0.0135 in Model 2. These results reflect that hydrogen causes more severe embrittlement at the Ni/Ni3Al interface in Ni3Al-based than in Ni-based alloys. Comparing with the values obtained for hydrogen, the values of 2cint/E calculated for boron are all positive and as large as 0.65 in the entire lattice misfit range, while the maximum 2cint/E value for hydrogen is only 0.026. The main cause for the large difference in 2cint/E is that boron enhances BOP to almost 2-fold more than BOV, strengthening greatly the cohesion of the Ni/ Ni3Al interface, while values of BOP and BOV for hydrogen are similar. This result is consistent with the conclusion drawn by Liu et al. [7] and Aoki and Izumi [61] that a small amount of boron could greatly improve the ductility of Ni3Al. Experimental data reported by George et al.

[62,63] also illustrated that the ductility of boron-doped Ni3Al and boron-doped FeAl in oxygen are much higher than that of boron-free alloys, and attributed this to the enhancement of grain boundary cohesion by the segregation of boron [7]. Lee and White [64] noted that the intrinsic component of the beneficial effect of boron on the ductility of Ni3Al is greater than the environmental (extrinsic) component. Furthermore, it can be seen that for both hydrogen- and boron-doping in our models, 2cint/E increases with lattice misfit. At d1  1.12% for Model 1 and d2  1.68% for Model 2, both corresponding to aU < aD, 2cint/E changes from negative to positive. Therefore, a transition from the hydrogen-induced embrittlement to ductility appears, indicating that the ductility of the hosts can be controlled and adjusted by the lattice misfit. Generally highly ductile materials have small maximum theoretical shear stress smax, e.g silver (0.77 GPa), gold (0.74 GPa) and copper (1.2 GPa), while brittle crystals have much larger smax, e.g. tungsten (16.5 GPa). Therefore, smax can also be used to estimate the ductility or brittle

Table 1 BOVXII , BOPXII and 2cint/E for XII = HII and BII at the Ni/Ni3Al interface with the lattice misfit d in both models BOV XII

BOPXII

2cint =E ¼ BOPXII  BOV XII

Model

Misfit d (%)

HII

BII

HII

BII

HII

BII

1

1.7 0.56 0.00 0.56 1.12 1.67

0.4499 0.4396 0.4346 0.4291 0.4235 0.4178

0.6288 0.6482 0.6569 0.6647 0.6709 0.6760

0.3707 0.3956 0.4074 0.4191 0.4305 0.4418

1.1406 1.2070 1.2379 1.2672 1.2948 1.3207

0.0792 0.0440 0.0272 0.0100 0.0070 0.0240

0.5118 0.5588 0.5810 0.6025 0.6239 0.6447

2

2.82 1.68 1.12 0.56 0.00 0.55

0.4256 0.4242 0.4235 0.4227 0.4219 0.4210

0.6743 0.6722 0.6709 0.6692 0.6670 0.6647

0.4127 0.4247 0.4305 0.4362 0.4417 0.4470

1.2765 1.2890 1.2948 1.3001 1.3050 1.3096

0.0129 5E-4 0.0070 0.0135 0.0198 0.0260

0.6022 0.6168 0.6239 0.6309 0.6380 0.6449

Y.X. Wu et al. / Acta Materialia 55 (2007) 4845–4852

b

τ a

τ Fig. 6. Definition of a and b in Eq. (5).

potential of materials. Frenkel [65,66] performed a simple calculation of the theoretical shear strength of crystal by considering two adjacent and parallel lines of atoms subjected to a shear stress. The maximum theoretical shear stress smax is expressed as: Gb ; ð5Þ 2pa where G is the shear modulus, a is the separation between the adjacent planes and b is the interatomic distance in one line as shown in Fig. 6. Since the distance between two atoms reduces as bond strength increases, smax can be writBO0V V P ten here as smax / BOXII P . The values of BOXII =BOXII reflect smax ¼

XII

the influence of the impurity at site XII on the maximum shear stress. The resulting values of BOVXII =BOPXII for hydrogen and boron in the two models are plotted in Fig. 7. It can be seen that the value of BOVHII =BOPHII is almost twice that of BOVBII =BOPBII , i.e. the maximum shear stress of the hydrogen-doped hosts is almost twice that of the boron-doped hosts. This means that boron can lower

1.25

4851

the smax and increase the possibility of shear slip along the interface. Schulson et al. [67] have established experimentally that the addition of 750 ppm boron to stoichiometric Ni3Al appeared to improve the mobility of grain boundary dislocation and, thereby, to increase the ease with which grain boundaries accommodate slip. The linearly decreasing dependence of BOVXII =BOPXII on d in Fig. 7 indicates that smax, and thus the ductility of the interface, can be controlled and improved by manipulation of d. It has been reported by Fredholm et al. [68] that the creep-deformed microstructure of a superalloy at creep temperature depends on the value and sign of the lattice misfit. Although our model is simple, our calculations show clearly that the hydrogen-induced embrittlement and boron-induced ductility for both Ni-based and Ni3Al-based alloys are closely related to the lattice misfit, supporting the conclusion from Fredholm et al. [68]. The different influences of hydrogen and boron on the mechanical characteristics of the Ni/Ni3Al interface are mainly due to the difference in their electronic structures: 1s2s2p3s for boron and 1s2s for hydrogen. The directionality of the p orbital of boron is responsible for the strong covalent bonds formed between boron and the nearestneighbor atoms, Ni(2), Ni(3) and four Ni(8). Since four Ni(8) atoms are on the horizontal plane (contributing to BOPBII ) and Ni(2) and Ni(3) are in the vertical direction (contributing to BOVBII ), the ratio of BOPBII to BOVBII is about twice as large for boron as for hydrogen, consequently leading to the higher 2cint/E and lower smax.

1.25

HII BII

HII BII

1.00

0.75

0.75

V

P

BO XII /BO XII

1.00

0.50 -2.0 -1.5 -1.0 -0.5 0.0

0.5

Misfit δ1 (%)

1.0

1.5

0.50 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0

0.5

Misfit δ2 (%)

Fig. 7. BOVXII =BOPXII as a function of the lattice misfit d for XII = HII and BII respectively: (a) in Model 1; (b) in Model 2.

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