Ni3Al interface doped with B or P

Materials Science and Engineering A 416 (2006) 169–175

First-principles study of the properties of Ni/Ni3Al interface doped with B or P P. Peng a,b,∗ , D.W. Zhou a , J.S. Liu a , R. Yang b , Z.Q. Hu b a

School of Materials Science and Engineering, Hunan University, Changsha 410082, China b Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, China Received in revised form 21 September 2005; accepted 5 October 2005

Abstract By using a first-principles pseudopotential plane-wave method and the CASTEP program, the energetics and electronic structures as well as bonding characteristics of the Ni/Ni3 Al interface doped with B or P have been investigated. For Ni/Ni3 Al interface with a B or P atom placed at its octahedral center, fracture occurs by cleavage along the (0 0 1) atomic layer in the ␥ -Ni3 Al block in a brittle manner, similar to the clean Ni/Ni3 Al interface, but its rupture strength W increases by 0.736 J/m2 and 0.537 J/m2 , respectively. B-doping can obviously improve the local toughness of the interfacial region between the (0 0 1) atomic layer in the ␥-Ni block and the coherent (0 0 2) atomic layer, i.e., region-2, whereas P-doping is deleterious for the local toughness of the interfacial regions, especially in region-1 bound by the coherent (0 0 2) atomic layer and the (0 0 1) atomic layer in the ␥ -Ni3 Al block. The increase of W in the B- or P-doped systems can be attributed to the enrichment of covalent electron density between the first nearest neighbor (FNN) Ni–Al in region-1. The change of electron interactions between first nearest neighbor atoms in the interfacial regions caused by the displacement of atoms at the interfacial center octahedron is responsible for the toughening effect of B-doping and the brittle influence of P-doping on the Ni/Ni3 Al interface. © 2005 Elsevier B.V. All rights reserved. Keywords: Ni/Ni3 Al interface; Electronic structure; Bond overlap population; Work of separation; Plane-wave pseudopotential method

1. Introduction Ni-based single crystal (SC) superalloys have been widely used for turbine blades and vanes in most advanced gas turbine engines, which mainly consist of a high volume fraction of ␥ phase precipitates coherently dispersed on a ␥ matrix. Improvement of creep resistance and thermal fatigue resistance of SC over polycrystalline superalloys mainly derives from the effective removal of grain boundaries except the interfaces between the ␥-Ni solid–solution phase (a disordered fcc (A1) structure) and the L12 -ordered ␥ precipitates (an intermetallic phase of stoichiometry based upon Ni3 Al). To some extent, their unique high temperature properties mostly depend on the cohesive strength and the bonding characteristics of the ␥-Ni/␥ -Ni3 Al interface. Many experiments have demonstrated that trace elements and minor alloying additions, e.g., B [1], C [2,5], P [3], S [4], Si, H, etc. have a great influence on the strength and ductility

∗

Corresponding author. Tel.: +86 731 8821610; fax: +86 731 8821483. E-mail address: [email protected] (P. Peng).

0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.10.019

of Ni-based SC superalloys. Usually, B is regarded as a typical enhancer to increase the cohesive strength of the grain boundaries [6] and reduce the embrittlement effect of H [7] because of its strong tendency to segregate to the grain boundaries of ␥ Ni3 Al phase, which leads to local disordering in grain boundary region and a strong interaction with adjacent ␥-Ni. The trace element P has an inherent embrittlement effect on the grain boundary of Ni or Ni3 Al as its content becomes excessive [9,10], and is classified as a common impurity and detrimental element [3]. However, several recent investigations have revealed that a moderate content of P is beneficial to prolonging the stress rupture life of Ni-based SC superalloys [3,8]. Up to now, unlike pure Ni [9] and Ni3 Al [10], few studies have been made to investigate the effects of the doping of B or P on the ␥-Ni/␥ -Ni3 Al interface. Hence an in-depth and thorough understanding of their strengthening and toughening mechanism of the Ni/Ni3 Al interface is necessary in order to guide the control of the trace elements and minor alloying additions in Ni-based SC superalloys. In regard to the ␥-Ni/␥ -Ni3 Al interfacial cohesive property, Liu et al. [11] made a first calculation on the doping effect of the Ni/Ni3 Al interfaces with a discrete variational X␣ (DV-X␣ )

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P. Peng et al. / Materials Science and Engineering A 416 (2006) 169–175

method based on the density functional theory (DFT). They found that the binding strength of the doped interfaces gradually reduce in the following order: C, B, N, O, H, clean, P, S. A further investigation [12] of the influence of the lattice misfit (δ) on the binding strength of the Ni/Ni3 Al interface was made using the same method. It was found that the effective range of negative δ for improvement of the binding strength of the Ni/Ni3 Al interface is less than −0.6%. Recently, Chen et al. [13] studied the bonding characteristics of the Ni/Ni3 Al interface with S-doping by means of a DMol program. Their results indicated that a strong bonding between S and Ni atoms lying in the interface is responsible for sulfur embrittlement effect of the Ni/Ni3 Al interface, and revealed that the substitution of Re for Al at the Ni/Ni3 Al interface may relieve the tendency of interface embitterment of sulfur [13]. It is worth noting that the above cluster model calculations seldom consider the influence of the atoms far away from the Ni/Ni3 Al interface on the interface bonding. However, in our recent investigation on the Re alloying effect [14], we found that a Ni/Ni3 Al supercell model is more suitable and reasonable to simulate and calculate the electron interaction and cohesive property of the Ni/Ni3 Al interface than the cluster model [13]. In this paper, we adopt a supercell model to further investigate the electronic and energetic structures of the Ni/Ni3 Al interface doped with B or P. Particular attention will be paid to considering the variation of geometrical structures of the Ni/Ni3 Al interface caused by doping atoms. 2. Method and models of computation Cambridge serial total energy package (CASTEP) [15,16], a first-principles pseudopotential plane-wave method, based on density functional theory, is used in this work. Ultrasoft pseudopotentials [17] in reciprocal space with the exchangecorrelation energy represented by a local density approximation (LDA) improved by Cepeley–Alder [18] were adopted for all elements in our models. In our calculation, the cut-off energy of atomic wave functions (PWs), Ecut , is set at 330 eV. A finite basis set correction [19] and the Pulay scheme of density mixing [20] are applied for evaluation of energy and stress. A supercell model of Ni/Ni3 Al interface, which consists of 64 atoms and two Ni/Ni3 Al interfaces, is devised for the present study. Atomic arrangements in the supercell are shown in Fig. 1. The (0 0 2) atomic layer is taken as a coherent interface of the ␥Ni phase and ␥ -Ni3 Al phase based on the experimental results reported by Harada et al. [21]. The lattice constant of the supercell is taken to be equal for Ni-␥ and Ni3 Al-␥ blocks owing to the assumption of perfect coherency. The interaction between two adjacent interfaces is neglected. Since the (0 0 2) atomic layer can be regarded as either the surface of the ␥-Ni block or that of the ␥ -Ni3 Al block, there exist two orientation relationships of (0 0 2)␥ //(0 0 1)␥ and (0 0 1)␥ //(0 0 2)␥ in the proposed interfacial model. Correspondingly, there are four constructional surface models, i.e., (0 0 2) and (0 0 1) surface models of ␥-Ni phase, and (0 0 1) and (0 0 2) surface models of ␥ -Ni3 Al phase. In order to investigate the effect of the doping of B or P on the strength and toughness of Ni/Ni3 Al interface, we construct Bor P-doping interfacial models in which B or P atoms are placed

Fig. 1. The supercell model of the Ni/Ni3 Al interface for CASTEP calculation. The black balls denote the doping atoms of X (B or P), big white balls, and small grey balls denote Al and Ni atoms, respectively. (0 0 1)␥ , (0 0 1)␥ , and (0 0 2) represent the (0 0 1) atomic layer in ␥-Ni block, the (0 0 1) atomic layer in ␥ Ni3 Al block and the coherent Ni/Ni3 Al interfacial layer, respectively. Region-1 is the region bounded by coherent (0 0 2) and (0 0 1)␥ layers, and region-2 is that bounded by (0 0 1)␥ and coherent (0 0 2) layers.

at coherent (0 0 2) atomic layer and at octahedral interstitial centers of the Ni/Ni3 Al interface. For consideration of symmetry, the interfacial supercell is doped with two B or P atoms, which are partitioned to the two interfaces. All atomic positions in the supercell with and without B- or P-doping have been relaxed according to the total energy and force using the BFGS scheme [22], based on the cell optimization criterion (RMS force of ˚ stress of 0.2 GPa, and displacement of 0.005 A). ˚ The 0.12 eV/A, calculation of total energy and electronic structure is followed by cell optimization with SCF tolerance of 1 × 10−5 eV under LDA Cepeley–Alder potential [18]. 3. Results and discussion 3.1. Geometrical structures To assess the accuracy of our computation method, we perform a series of calculations on the bulk properties of Ni3 Al and Ni correlative to the Ni/Ni3 Al interface models. Tables 1 and 2 list the results of calculated equilibrium lattice constant (a), heat of formation (H), cohesive energy (E), and bulk modulus (B) of the bulk Ni3 Al and Ni. A good agreement between present calculation and experimental [23,24] as well as other first-principles calculation values [25,26] can be seen. Thus, the same culculation condition is adopted to calculate the geometrical and electronic structures of the Ni/Ni3 Al interfaces with and without B- or P-doping. Table 3 tabulates the equilibrium lattice constant (a) and volume (V) of the Ni/Ni3 Al interfacial models with and without B- or P-doping after cell optimization. One can see a distinct decrease in volume and a variation in lattice constant c after B- or P-doping. B-doping causes shrinkage of the interface separation; by contrast, P-doping leads to an extension

P. Peng et al. / Materials Science and Engineering A 416 (2006) 169–175

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Table 1 The equilibrium lattice constant (a), heat of formation (H), and bulk modulus (B) of the bulk Ni3 Al crystal Parameter

Present

Expt. [23]

TB-LMTO [25]

AE-LMTO [23]

PPW-LDA [10]

˚ a (A) H (eV/atom) B (MPa)

3.58 2.12 1.75

3.57 1.59–1.62 2.40

3.51 2.20 1.46

3.55 1.94 2.10

3.49 2.27 2.20

Table 2 The equilibrium lattice constant (a), cohesive energy (E), and bulk modulus (B) of the bulk Ni crystal Parameter

Present

Expt. [24]

LSDA[26]

GGA [26]

˚ a (A) E (eV/atom) B (MPa)

3.55 3.90 2.13

3.52 4.44 1.86

– 4.18 2.50

– 4.44 2.08

˚ outthe initial positions. For example, Ni1 atoms move 0.23 A ˚ inward. Undoubtedly, ward; by contrast, Al4 atoms move 0.18 A the displacements of the doped atoms and their adjacent atoms as well as the variation of interfacial separation will result in changes of energetics and electronic structures of the Ni/Ni3 Al interfaces. 3.2. Work of separation

Table 3 Equilibrium lattice constant (a, b, c) and volume (V) of the Ni/Ni3 Al interfacial supercells after cell optimization Model

˚ a (A)

˚ b (A)

˚ c (A)

˚ 3) V (A

Clean B-doping P-doping

7.195533 6.978190 6.984244

7.195533 6.978190 6.984244

14.290432 13.986472 14.331695

739.837 681.073 699.095

of the separation. Table 4 lists z-coordinates of selected atoms, i.e., atomic positions in the Ni/Ni3 Al interface, at (0 0 1)␥ , coherent (0 0 2), and (0 0 1)␥ atomic layers in Fig. 1. From Table 4, it is easily deduced that for clean interface the interfacial separation (d1 ) in region-1 between the (0 0 1) atomic layer in ␥ -Ni3 Al ˚ and block and the coherent (0 0 2) atomic layer is about 1.80 A, ˚ in region-2 between the (0 0 1) atomic layer in d2 about 1.77 A ␥-Ni block and the coherent (0 0 2) atomic layer. For B-doped ˚ and 1.75 A, ˚ respectively, while for system, d1 and d2 are 1.77 A ˚ and 1.82 A, ˚ respectively. P-doped system, d1 and d2 are 1.77 A The former means that the interfacial separation of region-1 and region-2 slightly drops relative to the clean interface. The latter indicates that the increment of the separation in region-2 in the case of P-doping is much larger than in the case of B-doping. It is worth noting that the B and P atoms situated initially at the octahedral interstitial centers of the Ni/Ni3 Al interface are far away from the coherent (0 0 2) interfacial layer and have a displace˚ and 0.51 A ˚ toward the ␥-Ni block, respectively, ment of 0.15 A after cell optimization. The Ni and Al atoms adjacent to the doped atoms simultaneously take a slight movement away from Table 4 ˚ of atoms at (0 0 1)␥ , coherent (0 0 2), and (0 0 1)␥ atomic z-Coordinates (unit A) layers in the Ni/Ni3 Al interfacial supercells Layer

Species

Clean

B-doping

P-doping

(0 0 1)␥

Ni1 Ni2 (Ni7)

12.52 12.52

12.49 12.26

13.20 12.57

(0 0 2)

X Ni3

– 10.75

10.66 10.51

11.26 10.75

(0 0 1)␥

Ni5 (Al6) Al4

8.95 8.95

8.74 8.56

8.98 8.78

In this section, we will take into account the effect of the doping of B or P on the strength of the Ni/Ni3 Al interface. As is well known, ␥-Ni/␥ -Ni3 Al interface is the weakest region in Ni-based single crystal superalloys. To a certain degree, the binding strength of ␥-Ni/␥ -Ni3 Al interface can be regarded as representative of the rupture strength of SC superalloys. Here, a work of separation, i.e., the Griffith rupture work [10] W, which is defined as the reversible work needed to separate a crystal along the interface into two free surfaces [27], is employed to evaluate the binding strength of the Ni/Ni3 Al interface. W can be calculated by means of a difference in total energy between the interfacial model and corresponding surface models [10]: 
−1  [Ei (n, m, l) − Es␥ (n␥ , m␥ , l␥ ) − Es␥ (n␥ , m␥ , l␥ )] W= 2Si (1) where Si = ai × bi is the area of the coherent interfacial layer, ai and bi are the supercell lattice constants along x and y directions, ␥ ␥ respectively. Es (n␥ , m␥ , l␥ ) and Es (n␥ , m␥ , l␥ ) are the total energies of surface models of ␥-Ni block and ␥ -Ni3 Al block, respectively, corresponding to the Ni/Ni3 Al interface model. n, m, l denote the number of Ni, Al, B (or P) atoms, respectively, where n = n␥ + n␥ , m = m␥ + m␥ , l = l␥ + l␥ . Table 5 lists the total energies of Ni/Ni3 Al interfacial and correlated surface models as well as the work of separation calculated with Eq. (1). An examination of the two orientation relationships of the present Ni/Ni3 Al interfacial model, i.e., (0 0 2)␥ //(0 0 1)␥ and (0 0 1)␥ //(0 0 2)␥ , suggests two potential inter-phase fracture sites: one is adjacent to the coherent (0 0 2) atomic layer, by cleavage along (0 0 1) atomic layer in the ␥ Ni3 Al block, i.e., region-1, as showed in Fig. 1; the other is splitting along (0 0 1) atomic layer in the ␥-Ni block, i.e., region2, also indicated in Fig. 1. From Table 4, we can see that the W of the clean Ni/Ni3 Al interface in region-1 is smaller than that in region-2, which means that the binding strength in region1 is smaller than that in region-2. Generally, fracture emerges at the weakest part of materials; therefore, the work of separation in region-1, 4.359 J/m2 , is the rupture strength of the clean Ni/Ni3 Al interface.

P. Peng et al. / Materials Science and Engineering A 416 (2006) 169–175

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Table 5 The total energies of the Ni/Ni3 Al interfacial models (Ei ) and corresponding surface models (Es ) as well as the work of separation (W) in region-1 and region-2 γ

γ

Model

Site

Ei (eV)

Es (eV)

Es (eV)

˚ 2) Si (A

W (J/m2 )

Clean

Region-1 Region-2

−59532.15 −59532.15

−42173.69 −25297.88

−17330.28 −34204.62

51.78 51.78

4.359 4.588

B-doping

Region-1 Region-2

−76174.82 −76174.82

−54100.09 −32358.15

−22043.60 −43782.33

48.70 48.70

5.122 5.649

P-doping

Region-1 Region-2

−76379.81 −76379.81

−54306.81 −32356.96

−22043.19 −43990.82

48.78 48.78

4.896 5.260

A similar evaluation is made for the Ni/Ni3 Al interface with B- or P-doping. As B or P atoms are placed at the octahedral interstitial centers of the Ni/Ni3 Al interfaces, the weaker regions in the Ni/Ni3 Al interface are still region-1, but the rupture strength of the Ni/Ni3 Al interface does change. This indicates that either the slight shrinkage of separation of region-2 caused by B-doping or the obvious expansion caused by P-doping does not change the fracture site in the Ni/Ni3 Al interface. Compared with the rupture strength of 4.359 J/m2 for the clean Ni/Ni3 Al interface, that of the B- and P-doped systems increases by 0.736 J/m2 and 0.537 J/m2 , respectively. This indicates that although the decrease in separation of region-1 caused by both Bdoping and P-doping leads to the strengthening of the Ni/Ni3 Al interface, B-doping is more beneficial than P-doping to improve the rupture strength of Ni-based SC superalloys [3,11]. 3.3. Population analysis In order to qualitatively evaluate the effect of doping of B or P on the brittleness or toughness of the Ni/Ni3 Al interface, we further calculate the ionic and covalent bonding between first nearest neighbor (FNN) atoms in the Ni/Ni3 Al interfaces by using Mulliken’s population analysis method [28]. Mulliken’s charge Q(A) of A atom and a bond overlap population QA–B between A and B atoms are defined as follows: Q(A) =



QA−B =

wk

A A  

Pµν (k)Sνµ (k)

k

µ



B A  

k

wk

µ

(2)

ν

2Pµν (k)Sµν (k)

(3)

ν

where Pµν (k) and Sνµ (k) are the density matrix and the overlap matrix, respectively. wk is the weight associated with the calculated k-points in Brillouin zone. Usually, the magnitude and sign of Q(A) characterize the ionicity of A atom in the supercell, and QA–B can be used to approximately measure the covalent bonding strength. Table 6 lists the Mulliken charge Q(A) of the atom A located at the interfacial octahedron in the Ni/Ni3 Al supercell. Here, a small magnitude of Q(Ni1) in the clean interfacial model indicates that the ionic bonding between FNN Ni–Ni in the ␥-Ni block is weak. In addition, a significant charge transfer from Al4 atom at (0 0 1)␥ layer to Ni3 atom at the coherent (0 0 2) layer means that there is a strong ionic bonding between FNN Ni–Al

Table 6 Mulliken charge Q(A) (unit: e) of the atom A located at the interfacial octahedron in the Ni/Ni3 Al supercell Q(A)

Clean

B-doping

P-doping

Ni1 X Ni3 Al4

+0.03 – −0.10 +0.22

+0.09 −0.71 +0.06 +0.37

−0.12 +0.15 −0.15 +0.32

atoms at the Ni/Ni3 Al interface as reported in refs. [13,29]. With the doping of B or P atoms, several new ionic bonds emerge in the interfacial regions. For example, partial electrons of P atoms transfer to its FNN Ni1 and Ni3 atoms; in contrast, B atoms obtain partial electrons from its FNN Ni1 and Ni3 atoms. It should be noted that the brittleness or toughness of materials mainly correlates with the directionality of covalent bonding. Hence, in this section, we will focus on the analysis of the bond overlap populations between FNN atoms in the Ni/Ni3 Al interfacial region. Table 7 lists the QA–B between FNN atoms at the interfacial center octahedron. One can see six new bonds with a large bonding strength in the octahedron after B- or P-doping. Four of the new bonds are located at the coherent (0 0 2) layer, while the other two bonds are located in region-1 and region-2. However, a weakening effect of doping of B or P on the bonding strength of Ni5–Ni3 and Al4–Ni3 in region-1 and Ni2–Ni3 in region-2 is also noted. The bonding between Ni1–Ni3 in region-2 even vanishes as after P-doping. Obviously, these variations in bonding strength should be attributed to the influence of the change in bonding length caused by the displacements of atoms at the octahedron, i.e., the movements of Ni1 and B or P atoms toward Table 7 Bond overlap population (QA–B ) between FNN atoms in the Ni/Ni3 Al interfacial supercells QA–B

Clean

B-doping

P-doping

QNi1–Ni2 QNi2–Ni3 QNi1–Ni3 QAl4–Ni3 QNi5–Ni3 QX-Ni3 QX-Ni1 QX-Al4 QX-Ni2

0.12 0.12 0.14 0.19 0.06 – – – –

0.12 0.12 −0.01 0.04 0.06 0.36 0.38 0.38 −0.03

0.08 0.07 – 0.02 0.02 0.37 0.38 0.17 0.03

P. Peng et al. / Materials Science and Engineering A 416 (2006) 169–175

the ␥-Ni block and of Al4 atoms toward the ␥ -Ni3 Al block (refer to Table 4). Similar to a previous investigation on Re alloying effect of the Ni/Ni3 Al interface [14], a local bond overlap population (LBOP) [13] is calculated and adopted to characterize the intralayer and inter-layer bonding strength. LBOP of intra-layer and inter-layer are defined as the sum of bond overlap population along and across the Ni/Ni3 Al interface, respectively. Although, the LBOPs in intra-layer and in inter-layer are not the real shear and cohesive strengths of the Ni/Ni3 Al interface [12,13], respectively, their ratio (RLBOP = LBOPintra-layer /LBOPinter-layer ) has been shown to be an effective parameter for judging the competition of the ductile and brittle fracture modes of the Ni/Ni3 Al interfaces [13,14]. Usually a material will 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 [13]. Hence, if RLBOP > 1, a material will fail in brittle fracture mode. However, if RLBOP < 1, it will fail in ductile fracture mode. In addition, RLBOP is also an excellent indicator of the directionality of local covalent bonding in the Ni/Ni3 Al interfacial region. If the ratio of the number of covalent bonds in two intra-layers to that in the inter-layer is R0 , then the closer the value of RLBOP is to R0 , the weaker the embrittlement of a material, the better the toughness [14]. Table 8 tabulates the RLBOP values in different Ni/Ni3 Al interfacial regions. For region-1 between coherent (0 0 2) and (0 0 1)␥ layers, the RLBOP values are close to 1 but larger than 1 in the clean Ni/Ni3 Al interfacial model, which means that region-1

173

Table 8 The local bond overlap populations in intra-layer and inter-layer as well as their ratios (RLBOP ) in region-1 and region-2 Region

Parameter

Clean

B-doping

P-doping

Intra-layer

LBOP(0 0 1)␥ LBOP(0 0 2)␥ LBOP(0 0 1)␥

1.92 1.12 3.20

1.72 1.76 3.40

1.48 1.20 3.56

Inter-layer

LBOP(0 0 2)–(0 0 1)␥ LBOP(0 0 1)␥–(0 0 2)

4.00 4.16

3.70 3.38

3.13 3.03

Region-1

RLBOP

1.08

1.39

1.52

Region-2

RLBOP

0.73

1.03

0.88

has good toughness but the failure mode would be brittle fracture, while in the region-2 between (0 0 1)␥ and coherent (0 0 2) layers, the RLBOP values are significantly smaller than 1, which indicates that the toughness of region-2 is low but the failure would be via a shear mode, i.e., ductile fracture. After B- or P-doping, an evident increase in RLBOP can be seen. For region1, the increments of RLBOP are 0.31 and 0.44 in B-doped and P-doped systems, respectively, which means that the doping of B or P does not change the brittle fracture mode but degrades its toughness. Relative to the clean interface, P-doping is not as effective as B-doping in improving the local toughness of region-1. In region-2, the RLBOP in B-doped system reaches 1.03 compared with 0.73 in the clean interface. This value is very close to the ratio of the LBOP in intra-layer to that in interlayer R0 = (32 + 4)/(32 + 1) = 1.09 in the doped system. However,

Fig. 2. The total and partial DOS of different atoms at the center of the interfacial region in (a) clean, (b) B-doping, and (c) P-doping supercells.

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P. Peng et al. / Materials Science and Engineering A 416 (2006) 169–175

Fig. 3. The total valence charge density contour plots on (1 0 0) plane in (a) clean, (b) B-doping, and (c) P-doping Ni/Ni3 Al supercells. The region indicated by dashed arrows is region-1, and that by solid arrows is region-2.

for P-doped system, the RLBOP is smaller than and far from 1.09 although its magnitude is increased to 0.884. This indicates that B-doping, in contrast to P-doping, cannot only improve the local toughness of region-2 but also does not change its ductile mode of failure. Hence, on the whole, the improvement of the local toughness of region-2 adjacent to the ␥-Ni block caused by the doping of B is responsible for its toughening effect on the Ni/Ni3 Al interface. In contrast, the brittle influence of P-doping on the Ni/Ni3 Al interface should be attributed to the decrease of its local toughness of region-1 adjacent to the ␥ -Ni3 Al block. 3.4. Electronic structure To reveal the nature of the interfacial bonding, the electronic structures of the B- or P-doped Ni/Ni3 Al interfaces are analyzed and compared. Fig. 2 illustrates the partial and total electronic density of states (DOS) of Ni1, Ni2, Ni3, B or P, Al4, Ni5 atoms showed in Fig. 1. For the clean interface, many Ni 3d valence electrons at the Fermi energy (EF ) level means that the bonding between FNN Ni–Ni atoms in the ␥-Ni block are mainly metallic in nature. A few Al 3p valence electrons at EF level and a broad overlap between the Al4 3p and Ni5 3d valence electrons indicate weak metallic bonding and strong covalent bonding between FNN Ni–Al atoms in the ␥ -Ni3 Al block. With reference to the interfacial region, a similar result can be deduced from the DOS of Al4, Ni5, and Ni3 atoms. Namely, the electronic interaction between FNN Ni–Ni atoms in region-2 is mainly metallic bonding, whereas the electronic interaction between FNN Ni–Ni and Ni–Al atoms in region-1 are mixed, metallic, and covalent bonding. In the case of B- or P-doping, data presented in Table 6 show that there exists ionic bonding between Ni, Al4 atoms,

and B or P atom because of their electron transfer. Furthermore, from Fig. 2(b and c), one can see that the main components of the electronic interaction between X-Al4 and X-Ni1 atoms are covalent bonding caused by their p–p and p–d hybridization effects, respectively. A small portion of the contribution to QAB listed in Table 7 originates from the new DOS peaks of Ni1 and Al4 atoms at −10 eV induced by B-doping or at −13 eV induced by P-doping. Fig. 3 presents the charge density contour plots on the crosssection of the two coherent interfacial layers of the Ni/Ni3 Al interfacial supercell with and without B- or P-doping. This cross-section is (1 0 0) plane for all interfacial models. For the clean Ni/Ni3 Al interface, the contour plots show similar bonding characteristics to that presented in Fig. 2 and Table 7, i.e., the electronic interaction between the FNN Ni–Al atoms (including ionic and covalent bonding) in the ␥ -Ni3 Al block is stronger than that between the FNN Ni–Ni atoms in the ␥-Ni block [12]. In addition, a significant anisotropic build-up of the directional d bonding charge of Ni atoms in the ␥ -Ni3 Al block, which is caused mainly by the polarization of p electrons of Al atoms as a result of the p–d hybridization effect, along the FNN Ni–Al and FNN Ni–Ni directions [30] can also be observed. However, it is worth noting that for Ni atoms in the ␥-Ni block, which is farther from Al atoms in the ␥ -Ni3 Al block, the electronic interaction between FNN Ni–Ni atoms (mostly in a metallic bonding mode or a fractional covalent bonding mode, refer to Fig. 2 and Tables 6 and 7) is obviously isotropic. On the whole Fig. 3(a) clearly illustrates that the valence charge density in region-2 is rich but poor in region-1. In other words, the local electronic interaction in region-2 is stronger than that in region-1. Thus, the total valence charge density contour plots of the Ni/Ni3 Al

P. Peng et al. / Materials Science and Engineering A 416 (2006) 169–175

interface provide a direct visual pattern for understanding the inter-phase rupture site indicated in Table 5. For the B- or Pdoped system, the contour plots of valence charge density are similar to those of the clean interfacial model. In comparison with the clean Ni/Ni3 Al interface, the doping of B or P makes the covalent electron density to accumulate further in the interfacial regions and enhances their electronic interactions between FNN atoms. For example, the bonding between Ni7 and Ni3 in region-2 and between Al6 and Ni3 in region-1 is obviously reinforced. One can even note very strong bonding interactions between X-Ni1 in region-2 and between X-Al4 in region-1, as showed in Table 7. Thus, the increase of W in the B- or P-doped system, listed in Table 5, can be attributed to the enrichment of covalent electron density between the FNN Ni3 and Al6 caused by the decrease in the separation d1 of region-1 (refer to Table 4), especially the enrichment of covalent electron density within region-1 induced by B- or P-doping. Obviously, the less enrichment of covalent electron density within region-1 induced by P-doping (compared to that induced by B-doping) is responsible for its weak strengthening effects on binding the (0 0 1)␥ atomic layer in the ␥ -Ni3 Al block and the coherent (0 0 2)␥/␥ atomic layer. 4. Conclusions A systematic investigation of the strengthening and toughening effects of B- or P-doping on the ␥-Ni/␥ -Ni3 Al interface of a Ni-based superalloy SC has been performed using a firstprinciples plane-wave pseudopotential method. As B or P is doped at the octahedral center in the Ni/Ni3 Al interface, the fracture site does not change compared with the clean Ni/Ni3 Al interface and remains the interface between the (0 0 1) atomic layer in the ␥ -Ni3 Al block and the coherent (0 0 2) atomic layer, i.e., region-1, but their rupture strength increases by 0.763 J/m2 and 0.537 J/m2 , respectively. Either B-doping or P-doping does not change the inter-phase fracture mode in the clean interface, namely, in region-1 the inter-phase fracture occurs along the (0 0 1) atomic layer in the ␥ -Ni3 Al block in a brittle manner, while in region-2 the failure would be along the (0 0 1) atomic layer in the ␥-Ni block via shear, i.e., ductile fracture. In comparison with the clean Ni/Ni3 Al interface, B-doping obviously improves the local toughness of region-2, whereas P-doping is very deleterious to the local toughness of the interfacial regions, especially, in region-1. The increase of W in the B- or P-doped systems can be attributed to the enrichment of covalent electron density between the FNN Ni–Al in region-1, especially the enrichment of covalent electron density within region-1 induced by the doping of B or P. The toughening effect of B-doping and embrittling effect of P-doping on the Ni/Ni3 Al interface originates from the variation of the interfacial separation and displacements of atoms in the interfacial center octahedron. Lattice shrinkage with small displacement of B atoms toward the

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