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Generalized stacking fault energies of Cr23C6 carbide: A first-principles study
Computational Materials Science 158 (2019) 20–25
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Generalized stacking fault energies of Cr23C6 carbide: A first-principles study Fangfang Xia
a,1
, Weiwei Xu
a,1
a,⁎
b,c
, Lijie Chen , Shunqing Wu , Michael D. Sangid
d,⁎
T
a
School of Aerospace Engineering, Xiamen University, Xiamen 361005, China Collaborative Innovation Center for Optoelectronic Semiconductors and Efficient Devices, Department of Physics, Xiamen University, Xiamen 361005, China Key Laboratory of Low Dimensional Condensed Matter Physics (Department of Education of Fujian Province), Xiamen University, Xiamen 361005, China d School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, USA b c
A R T I C LE I N FO
A B S T R A C T
Keywords: Nickel-based superalloys Cr23C6 carbide First-principles calculations Generalized stacking fault energy
In many high temperature-resistant Ni-based superalloys, M23C6 carbides form during processing and remain as non-metallic inclusions during the materials use in application. M23C6 type inclusions are principal sites for potential fatigue crack initiation, thus the accurate quantification of their energy barriers for deformation is necessary for in-depth understanding of the overarching mechanical behavior of the material system. In this study, the generalized stacking fault energies (GSFE) are theoretically determined for chromium-rich types of M23C6 carbides (i.e., Cr23C6) in two prominent slip systems: {1 1 1}〈1 1 2¯ 〉 and {1 1 1}〈1 0 1¯〉. The results present GSFE curves with double saddle points in both slip systems. The double saddle points of the GSFE curves are explained via the electronic structure of atoms near the slip plane. At deformation quantified by half of a Burgers vector (i.e., 0.5 bp), a pentacyclic charge density distribution appears representing a local minimum energy configuration, which results in a valley of the GSFE curve. Furthermore, the effective coordination number (n) and the distortion index (Δd) of the coordination polyhedrons are employed to characterize the structural change. The analysis reveals that the values of the indexes n and Δd at a displacement value of 0.5 bp are close to the values of a perfect Cr23C6 crystal, while comparatively the distortion index is higher at the saddle points. These results provide new insights into the deformation pathways of M23C6 carbides.
1. Introduction Despite incredible advances in material cleanliness through processing improvements, carbides are inherent within many structural materials. In many superalloys, MC carbides (typically rich in Ti, Ta, or Hf) form at high temperatures and are metastable at low temperatures, while M23C6 carbides (e.g. Cr23C6) form at low temperatures and remain as the majority carbide inclusion phase, in equilibrium and volume fraction, during time-dependent deformation at elevated temperatures [1]. M23C6 type carbides, as prototypical inclusions prefer to precipitate at grain boundaries, are known to distinctly influence the fatigue properties of materials [2]. During fatigue loading, strain rapidly accumulates in the form of a dislocation pile-up at M23C6 inclusions [3], which is a precursor for crack initiation [4,5]. In order to explore the fatigue behavior of Ni-based superalloys, the mechanical and thermodynamic properties of M23C6 carbides, especially Cr23C6 (which is as a representation of M23C6 in most Ni-based superalloys [6]), are desired to identify the mechanism of
heterogeneous deformation and crack initiation. Unfortunately, there are limited experimental reports on these mechanical properties of M23C6. Using first-principles calculations, various foundational properties have been theoretically investigated for Cr23C6 carbides based on density functional theory (DFT) analysis [7–10]. For example, Jiang [7] studied the structural, elastic, and electronic properties of Cr23C6, it is found that Cr23C6 exhibits a high elastic moduli and low Poisson’s ratio, which is indicative of a hard material. Yet, further investigation is needed to study the heterogeneous deformation and the failure mechanism of Cr23C6. The generalized stacking fault energy (GSFE) is inherently related to the strength of materials and is used to determine the mechanical properties of material constituents in the form of a comprehensive description of the energy barrier to deformation or dislocation motion. The GSFE represents the interplanar potential energy required to displace one elastic half space of the crystal along a specific crystallographic direction with respect to the remaining lattice [11]. However, the complete GSFE is experimentally not accessible, hence the GSFE
⁎
Corresponding authors at: School of Aerospace Engineering, Xiamen University, Xiamen 361005, China (L.J. Chen). School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-0307, USA (M.D. Sangid). E-mail addresses: [email protected] (L. Chen), [email protected] (M.D. Sangid). 1 Fangfang Xia and Weiwei Xu contributed equally to this work. https://doi.org/10.1016/j.commatsci.2018.11.006 Received 19 June 2018; Received in revised form 1 November 2018; Accepted 2 November 2018 0927-0256/ © 2018 Published by Elsevier B.V.
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studies with first-principles have achieved considerable attentions in the simulations of plasticity of bulk crystalline materials [12–14]. In previous research, Sangid et al. [15,16], modeled GSFEs of Ni and Ni3Al within Ni-based superalloy and used this information to study the stability of a persistent slip band with respect to dislocation motion as failure criterion for fatigue crack initiation. Cr23C6 carbide is an interstitial compound with a complex facecentered cubic (fcc) structure, for which the conventional cell has 116 atoms [17]. Due to this complex structure, the GSFE of Cr23C6 is investigated to a far less extent, compared to fcc pure metals or binary alloys, both experimentally and theoretically. It is of the believe that M23C6 carbides are generally intermetallic particles that are brittle, however, Rong et al. [18] observed dislocations, stacking faults, and microtwins composing the defect structure of M23C6 type carbides in Nickel-based superalloys based on transmission electron microscopy. Furthermore, Xu et al. [19] reported stacking faults along the boundaries and extended stacking faults along 〈1¯ 1 1〉 directions within M23C6 type carbides in high-manganese steel. Although these studies [18,19] reported the existence of stacking faults in M23C6 type carbides along the {1 1 1} planes indicative of crystallographic slip, there is little information about the energy barriers to achieve these structures, and the associated GSFE of the M23C6. The incomplete description pertaining to the mechanical behavior of such material motivates the present study. In this scenario, the GSFEs of Cr23C6 carbide are investigated in two typical {1 1 1}〈1 1 2¯ 〉 and {1 1 1}〈1 0 1¯〉 slip systems by employing firstprinciples calculations. Through the analysis of atomic structures, electron density distributions, and coordination polyhedrons near the slip plane, the present theoretical results are further discussed.
constructing a supercell model with 6 layers, as shown in Fig. 1(a). The GSFEs are calculated by rigidly shearing the upper layers of the cell, numbers 4–6 relatively to the lower layers number 1–3, along typical 〈1 1 2¯ 〉 and 〈1 0 1¯〉 directions in the (1 1 1) plane. 2.2. Computational details The present calculations were performed by using the projector augmented wave (PAW) method within the density functional theory (DFT) [20,21] as implemented in the Vienna Ab initio Simulation Package (VASP) [22,23]. The exchange-correlation functional was treated within the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) [24,25]. Wave functions were expanded in a plane wave, up to a cutoff energy of 550 eV. The non-spin polarized calculations were considered to perform the calculations, since previous studies indicate that Cr23C6 exhibits paramagnetic characteristic with an almost zero magnetic moment [8,26]. Brillouin zone integrations were approximated by using a special kpoint sampling of Monkhorst-Pack scheme [27] with Γ-centered 15 × 15 × 15 and 5 × 5 × 1 grids for lattice constant and GSFE calculation, respectively. The convergence threshold of total energy is 10−4 eV. All the atoms were relaxed in the direction perpendicular to the glide plane until the force on each atom is smaller than 10−2 eV/Å. In order to reduce the effect of periodic arrangements of atoms on the GSFE, as much as possible, a vacuum layer of 10 Å is added on the top of the (1 1 1) supercell. 3. Results and discussion
2. Computational methodology
3.1. Generalized stacking fault energy
2.1. Modeling methodology
The obtained reference lattice constant a for Cr23C6 is 10.54 Å, which agrees well with experimental (10.64 Å [28]) and theoretical values (10.53 Å [7,8], 10.56 Å [26]). To confirm the convergence for the number of (1 1 1) stacking layers, both 6- and 10-layer supercells are used to calculate the GSFE for Cr23C6. It is found that the difference in the γ values based on the 6- and 10-layer model is extremely small (less than 2%). Therefore, a 6-layer supercell is constructed, considering the computational efficiency. Fig. 2 shows the Burgers vectors of possible slip and the γ curve as a function of the Burgers vector for a bulk Cr23C6 carbide in two typical slip systems: {1 1 1}〈1 1 2¯ 〉 and {1 1 1}〈1 0 1¯〉. Some critical γ values are listed in Table 1. Note that bp represents the Burgers vector for a partial dislocation along 〈1 1 2¯ 〉 directions at {1 1 1} crystal planes; while b represents the Burgers vector for a full dislocation along a 〈1 1 0〉 directions. As we know, the GSFE curve of fcc metals and alloys, such as Ni and Ni3Al, exhibits a typical bell-shaped profile with one peak at 0.5 bp (or 0.5b). However, the GSFE curves of Cr23C6 show double saddle points along 〈1 0 1¯〉 and 〈1 1 2¯ 〉 directions (as shown in Fig. 2(b)). Table 1 presents the saddle points γus1 (2201 mJ/m2) and γus2 (2603 mJ/m2) are nine orders of magnitude larger than those of Ni (277 mJ/m2) and Ni3Al (257 mJ/m2) in {1 1 1}〈1 1 2¯ 〉 slip system (the detailed data for the GSFE of Ni and Ni3Al is discussed in the Supplementary material). As a higher γus value represents a greater energy barrier for emission and associated plastic deformation, it implies that the dislocations within Cr23C6 in this slip system are difficult to form. This extremely large value of the γus within Cr23C6 is possibly due to the complex topological structure of carbides. Although such GSFE curve has not been observed for any fcc structure, it was previously reported for some inorganic materials, such as Mg2SiO4-forsterite [29]. The valley of the GSFE curve along 〈1 1 2¯ 〉 implies there is a local stable structural configuration at 0.5 bp (m position shown in Fig. 2(a)). At position m, there are two potential slip paths for the next step: (i) slip continues along the 〈1 1 2¯ 〉 (m → M) or (ii) slip occurs along the 〈2 1¯ 1¯〉 (m → h), as shown in Fig. 2(a). In Fig. 2(b), the energies associated with slip along m → h reach a stable high value at 0.6 bp, while for the slip
As previous investigations [18,19] reported that the stacking faults of M23C6 were observed along the {1 1 1} planes and there is little information along other planes, we can focus on the stacking faults on the (1 1 1) plane. A stacking fault is formed if the regular fcc stacking of {1 1 1} planes is altered, thus it is possible to generate a stacking fault on {1 1 1} planes by shifting the upper half of the crystal relative to the lower half by a displacement vector u. The GSFE (γ) can be computed as a function of displacement u, defined as γ(u), and the local maximum energy of the GSFE curve is termed the unstable stacking fault energy (γus) [11]. The GSFE can be calculated by:
γ (u ) =
E (u ) − E0 A
(1)
where E (u) is the total energy of the supercell for a given fault vector, u, E0 is the total energy of the perfect supercell, and A is the area of the fault plane. In fcc structures, the calculated supercells for the GSFEs are built up from (1 1 1) layers via an ABC stacking to produce a disruption in the stacking sequence and a single layer intrinsic stacking fault (ISF) is formed when the stacking of the {1 1 1} planes is …ABC|BCA…, where | indicates the position of the fault. The complex fcc crystal structure of Cr23C6 contains 92 chromium and 24 carbon atoms. Cr is located at positions of 4a (0 0 0) (Cr1), 8c (0.25 0.25 0.25) (Cr2), 32f (0.385 0.385 0.385) (Cr3), 48 h (0 0.165 0.165) (Cr4) Wyckoff sites, and C is located at 24e (0.275 0 0) Wyckoff site, respectively [17]. Although Cr23C6 possesses a complex structure, the stacking sequences along the {1 1 1} planes can be defined by repeated ABC, where each unit A (B or C) consists of a “superatom” with 29-atom basis (i.e., primitive cell of Cr23C6). Fig. 1 displays the generation of the computational supercell used in the present calculations. The supercell is created from a primitive cell of pure Cr23C6 in [1 0 1¯], [0 1 1¯] and [1 1 1] directions as illustrated in Fig. 1(c). Then, the primitive cell is used as a “superatom” for 21
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C B
A C
c
B
b t a
(c)
c
Cr (4a)
Cr (8c)
Cr (32f)
Cr(48h)
C
t
A (a)
(b)
Fig. 1. Generation of a 174-atom supercell for Cr23C6 along (1 1 1) planes. (a) A two-dimensional crystal consisting of 6-layer stacking form (ABCABC) along the (1 1 1) planes for Cr23C6 with the 29-atom basis as a “superatom”. (b) The Bravais lattice is specified by two repeat vectors t ([1 1 2¯ ]) and c ([1 1 1]). (c) The “superatom” contains 29 atoms. (4a, 8c, 32f and 48 h represent Wyckoff sites. a: [1 0 1¯], b: [0 1 1¯]).
along the path m → M, the energies reach a smaller value of 1248 mJ/ m2 (as seen in Table 1). The energy barrier for the slip along m → M is slightly bigger than slip along m → h, as seen in Fig. 2(b), which implies the slip along possible path m → h is not energetically favorable compared to slip along 〈1 1 2¯ 〉. Therefore, slip along path 〈1 1 2¯ 〉 at 0.5 bp will continue in the same direction, which can be expressed as: O → m → M. Comparing the GSFE curves of the two prominent slip systems: {1 1 1}〈1 1 2¯ 〉 (O → M) and {1 1 1}〈1 1 2¯ 〉 (O → R), except near 0.7 bp (or 0.7b), the energies associated with slip along the 〈1 0 1¯〉 from 0.1b to 0.8b is always greater than those of slip along the 〈1 1 2¯ 〉 direction correspondingly (detailed data listed in Table 1). As shown in Fig. 2(b), there is only a small gap between the energies required for slipping along O → M and O → R near 0.7b (or 0.7 bp), which implies that slip along 〈1 0 1¯〉 is more energetically difficult than 〈1 1 2¯ 〉. Therefore, the most prominent slip system for dislocation activity in Cr23C6 is {1 1 1} 〈1 1 2¯ 〉, thus we focus on this slip system for further analysis.
Table 1 GSFE values (γ) of Cr23C6 at distinct crystallographic displacements (mJ/m2). ¯
γ
¯
¯
¯
¯
¯
< 101 > γ0.5 b
< 101 > γ0.7 b
< 112 > γ0.35 bp
< 112 > γ0.5 bp
< 112 > γ0.7 bp
< 112 > γ1.0 bp
2434
2315
2496
2201
1660
2603
1248
To clarify the influence of atomic redistribution on the GSFE curve of Cr23C6 during the process of crystallographic slip, the atoms were relaxed and held fixed (termed as no-relaxed in Fig. 2(b)) in the direction perpendicular to the glide plane, respectively. Noteworthy that, if the GSFE for Cr23C6 is calculated directly without the relaxation of atoms, then the obtained GSFE for Cr23C6 contains artificially large values and does not possess a local minimum energy, as shown in Fig. 2(b). The existence of an energy valley at 0.5 bp may be attributed to the stabilization of a local cluster of atoms during the slip process. To 10000
Ȗus2
Cr23C6 2500
Top layer Middle layer
m O
h
r
h
h
Ȗus1
8000
2000
R
enlarge
6000
m
1500
Ȗisf M
Ȗ(m-m2)
Ȗ(m-m2)
Bottom layer
M
¯
< 101 > γ0.4 b
4000
1000
2@ [11m
O
M
[2
1 r 1@
R h
RHOax a2 101!(OĺR) RHOax a6 112!(OĺM) RHOax a6 112!(Oĺm)a6 211!(mĺh) 1oRHOax a6 112!(OĺM)
500
0 0
O
0.1
0.2
0.3
0.4
0.5
[101@
ux(b or bp)
(a)
(b)
0.6
0.7
0.8
2000
R 0.9
0 1.0
Fig. 2. (a) Illustration of three successive (1 1 1) planes of Cr23C6 and the Burgers vectors forming three possible slip paths. The circles in the figure present a “superatom” of Cr23C6 on different planes. m, h, and r represent the midpoints of the segment OM, OR, and MR, respectively. (b) GSFE (γ) as a function of the Burgers vector for Cr23C6 in the {1 1 1} crystal plane along different slip directions. Note that the relaxed energies correspond to the y-axis on the left and unrelaxed energies correspond to the energies on the right y-axis. a stands for lattice constant. In x-axis label, bp = a/6{1 1 1}〈1 1 2¯ 〉 and b = a/2{1 1 1}〈1 0 1¯〉. 22
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(a) 7
2500
(d)
Ȗ(m-m2)
2000
8
(b)
1500
(c)
1000
(e)
7
(a) 0
0.1
Relax <11-2>
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
C
Cr
0.026
(c)
7
(d)
5
6
4
2
7 8 3
2
1 7
5
6
8 1
7
4 2
2
4
4
8 3
9
2
1
3
4
1
3
2
7
9
6
8
4
1
5
6
9
5
6
9
3
3
9
(e)
8 1
2
8
9 Cr 5
6
4
11
4
1
5
6
8
5
7
99 99
3
7
9
C
8
-0.074
8
3
2 6
8
ux(bp)
4
11
slip plane
500
0
6
(b)
5
2 7
4
3 6
5 4
4
9 55
8
4 4
1 2
3
9
Fig. 3. Atomic structure near the slip plane and the relevant contour plots of charge density differences (CDD) (the unit in eV/Å3) in the (1 1¯ 0 ) plane under a normalized displacement of (a) 0 bp, (b) 0.35 bp, (c) 0.5 bp, (d) 0.7 bp, and (e) 1.0 bp. (bp = a/6{1 1 1}〈1 1 2¯ 〉).
among the Cr atoms 1, 2, and 6 after a slip distance of 0.35 bp (Fig. 3(b)). The observed charge distributions change the overall structural configuration from stable to metastable, and thereby lead to a relative large energy value of the GSFE at 0.35 bp and an associated saddle point, as shown in Fig. 2(b). Upon consecutively shearing, the stable six-membered charge density among the Cr atoms 1, 2, 3, 4, 5 and 6 would be more significantly distorted and lead to a larger GSFE value. Interestingly, after shearing to a value of 0.5 bp as illustrated in Fig. 3(c), the Cr atom 2 jumps from its original position to an equilibrium position to accommodate the region that was previously chargefree and participate in the electronic interaction with the Cr atoms 1 and 6. In addition, the neighboring Cr atoms 7 and 8 also participate in the electronic interaction with the Cr atoms 1, 2, and 6, re-forming a local stable structural configuration with the geometry of the fivemembered ring from the distorted six-membered ring. The formation of such five-membered ring remarkably decreases the total energy of Cr23C6, and it results in the local minimal of the GSFE curve at a displacement of 0.5 bp. For a shearing displacement of 0.7 bp (Fig. 3(d)), a similar configuration is observed as the case of 0.35 bp. The local stable five-membered ring is distorted again, which implies the local stable structural configuration is changed from stable back to metastable, leading to the second saddle point of the GSFE curve at 0.7 bp. As the crystal structure is sheared to a displacement value of 1.0 bp, the stacking sequences change from ABC|ABC to ABC|BCA, implies there may be a stable structural configuration. As shown in Fig. 3(e), with more neighboring Cr atoms participating in the bonding interaction corresponding to atoms 1, 2, 3, 4, 5, 6, 7, 8 and 9, both six- and fivemembered rings are formed, which stabilizes the structural configuration of Cr23C6 by reducing its corresponding value of the GSFE, as plotted in Fig. 2(b).
gain deeper insight into the double saddle of GSFE curves, the electron density distributions and the change of coordination polyhedrons near the slip plane in the {1 1 1}〈1 1 2¯ 〉 slip system are analyzed. 3.2. Chemical bonding characteristics The GSFEs are related to the atomic bonding characteristics and a fundamental understanding of the characteristics could reveal the reasons for the double saddle configuration of the GSFE curves. Charge density differences (CDD), which denote charge redistributions, represent a means to further study the bonding characteristics of the material. To gain additional insights about the bonding structure during the slip mediate process of Cr23C6, we plot the structural and CDD maps for atoms near the slip plane under a normalized displacement, as shown in Fig. 3. The scale for CDD is displayed on the left of Fig. 3. The blue color represents the maximum delocalization of electrons and the red color indicates the maximum localization of electrons. For a perfect crystal structure, without a normalized displacement (0 bp), Fig. 3(a) shows that the electrons are oriented among the Cr atoms indicated by 1, 2, 3, 4, 5 and 6 numbering, forming a localized charge distribution with the geometry of a six-membered ring. This electronic hybridization indicates the non-polar covalent bonding characteristics, which is believed to be beneficial to the phase stability of Cr23C6. From the CDD field distributions with a normalized displacement of 0.35 bp as shown in Fig. 3(b), the stable six-membered charge density is distorted and the electronic charge effectively localized between the Cr atoms 5 and 6 in the upper part of the plane, as well as the Cr atoms 2 and 3 in the lower part of the plane. The bond length between Cr atoms 1, 6 increases from 2.42 Å to 3.01 Å with a elongation of 24%, and between Cr atoms 3, 4 increases from 2.42 Å to 3.38 Å with a elongation of 40%. Based on the Lindemann criterion [30], this large elongation results in the breaking of chemical bonds. Meanwhile, an almost charge-free region is observed 23
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3.3. Coordination polyhedrons
Table 2 Effective coordination number (n) and distortion indexes (Δd) of the coordination polyhedrons at distinct crystallographic displacements of the {1 1 1} 〈1 1 2¯ 〉.
To quantify the structural changes during shear deformation, we also investigate the coordination polyhedrons of Cr23C6 near the fault plane of the {1 1 1}〈1 1 2¯ 〉 slip system. In the polyhedral model, crystal structures are represented by coordination polyhedrons where the central atoms, bonds, and apex atoms are included [31]. The coordination number denotes the number of atoms coordinated to a central atom in a coordination polyhedron. In general, the coordination number correlates with electronic interaction and thereby is associated with the structural stability. However, it is difficult to state the coordination of the central atom as a simple number in relatively distorted coordination polyhedron. Hence, the effective coordination number (n) [32,33] is determined by adding the surrounding atoms with a weighting scheme, which is adopted in this paper. The distortion in a coordination polyhedron can be quantified with the distortion index (Δd), which based on the bond lengths is defined as
Δd =
1 n
n
∑ i=1
|di −d¯| d¯
n Δd
0 bp
0.35 bp
0.5 bp
0.7 bp
A, B, C
A, B, C
A
B, C
A
B, C
A, B, C
7.9 0.04
6.8 0.12
6.6 0.067
7.7 0.072
6.7 0.076
6.6 0.077
7.8 0.057
indicate the structural configuration at 0.5 bp form a local stable structural configuration as shown in Fig. 3(c). Upon further continuous shearing to a displacement of 0.7 bp, the coordination polyhedrons appear similar to the structure at a displacement of 0.35 bp, as shown in Fig. 4(b) and (d), indicating the local stable structural configuration is destroyed. However, with shearing to the ISF configuration corresponding to a displacement of 1.0 bp, as shown in Fig. 4(e), the coordination polyhedrons look similar to the perfect structure (Fig. 4(a)). In addition, the values of Δd at a displacement of 1.0 bp are close to the values at 0 bp. Therefore, it reveals the ISF configuration (1.0 bp) is more stable than structural configuration at 0.5 bp. In other words, upon consecutively shearing, atoms are redistributed and re-form stable structural configuration at 0.5 bp and 1.0 bp.
(2)
where di is the bond lengths of Cr-Cr atoms in a coordination polyhedron and d¯ is the average di value. Fig. 4 shows the coordination polyhedrons for the structural model of Cr23C6 under a series of normalized displacements. The coordination polyhedrons n and Δd near the slip plane are calculated and summarized in Table 2. The value of n is the reflection of the structural stability, and for a perfect structure, the value of n is about 8, as shown in Fig. 4(a). A smaller Δd exhibits a smaller distortion of the atomic structure; as expected the smallest Δd value corresponds to a perfect crystal structure (see Table 2). Fig. 4(b) shows that the values of n for the coordination polyhedrons A, B, C are about 7, under the shearing displacement of 0.35 bp. Meanwhile, the polyhedrons values of Δd significantly increase; thus implying that the stable structural configuration at 0 bp is significantly destroyed, and the result is consistent with the features of the CDD map shown in Fig. 3(b). Upon consecutively shearing of the crystal structure to a displacement of 0.5 bp, due to the Cr atoms 1 and 2 drastic swing (shown in Fig. 4(c)), the n of coordination polyhedrons B and C are about 8, while their values are about 7 at 0.35 bp. The values of Δd for the polyhedrons A, B, C at a displacement of 0.5 bp are smaller than those at 0.35 bp. While the value of n for coordination polyhedron A is similar between a displacement of 0.35 bp and 0.5 bp, the values of n for coordination polyhedrons B and C represent a more stable structural configuration for a displacement of 0.5 bp. These numerical values
4. Conclusions By constructing a 174-atom supercell, the present work quantified the GSFE of Cr23C6 for two typical slip systems in a fcc–type crystal structure: {1 1 1}〈1 1 2¯ 〉 and {1 1 1}〈1 0 1¯〉 using first-principles calculations. The charge density difference and the coordination polyhedron around the slip plane were analyzed to rationalize the unique features of the GSFE curves for Cr23C6. Based on the present calculations, results of this study can be summarized as follows: (1) Unlike Ni and Ni3Al, Cr23C6 exhibits a double bell-shaped GSFE curve, which represents a series of saddle points. A valley point representing a stable stacking fault energy is found at the shearing displacement value of 0.5 bp. (2) Energetically, the most favorable slip system for dislocation activity in Cr23C6 is {1 1 1}〈1 1 2¯ 〉. The saddle points γus1 (2201 mJ/m2) and γus2 (2603 mJ/m2) of the GSFE curve in this slip system implies the
8
8
2
1 8
2 1
1
slip plane 2
A
1.0 bp
B
(b)
(c)
C 8
1 1
2
2
(a) (d)
(e)
Fig. 4. Structural models representing coordination polyhedrons under a normalized displacement of (a) 0 bp, (b) 0.35 bp, (c) 0.5 bp, (d) 0.7 bp, and (e) 1.0 bp. A, B, and C represents a coordination polyhedron, containing central atoms (C), bonds, and apex atoms (Cr), respectively. (The pink atoms 1, 2, 8 represents Cr 1, 2, 8 in Fig. 3, respectively. The Cartesian coordinate system represents crystallographic axes of a: [1 0 1¯], b: [0 1 1¯], and c: [1 1 1].) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 24
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dislocations within Cr23C6 need to overcome a higher energy barrier than typical fcc metals and alloys, thus explaining the relatively brittle nature of Cr23C6 precipitates within Ni-based superalloys. (3) The dominant factor governing the dislocation glide of Cr23C6 is clarified to be the chemical bonding configuration near the slip plane. Under shear deformation, the stable six-membered bonding characteristics of atoms are disrupted, and then a five-membered bonding ring is reformed at the displacement of 0.5 bp, resulting in the related stacking fault energy displaying a local minimum value. Subsequent shear displacement breaks the five-membered rings until the intrinsic stacking fault is reached at a deformation of 1.0 bp, where both six- and five-membered bonding configurations are observed. (4) The quantified coordination polyhedrons distorted index further supports the analysis of bonding characteristics in Cr23C6. The coordination number and the bonding length of atoms near the slip plane at 0.5 bp are much closer to the values calculated at displacements of 0 bp and 1.0 bp. However, the saddle points show a large polyhedron distortion compared with the local stable points of 0 bp, 0.5 bp, and 1.0 bp. CRediT authorship contribution statement Fangfang Xia: Methodology, Formal analysis, Writing - original draft. Weiwei Xu: Methodology, Formal analysis, Writing - original draft, Funding acquisition. Lijie Chen: Conceptualization, Writing review & editing, Resources, Supervision, Project administration, Funding acquisition. Shunqing Wu: Validation, Formal analysis. Michael D. Sangid: Conceptualization, Validation, Writing - review & editing, Resources. Acknowledgements The authors would like to thank the support of the National Natural Science Foundation of China (Grant Nos. 51475396 and 51601161) and the Fundamental Research Funds for the Central Universities (Grant No. 20720170048). This work originated during Prof. Chen’s visit to Purdue University as a visiting scholar. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.commatsci.2018.11.006. References [1] Y.Y. Song, Thermodynamic study on b and fe substituted Cr23C6 using first-principles calculations, Ph. D. thesis, 2010. [2] S. Suresh, Fatigue of Materials, Cambridge University Press, Cambridge, England, 1998. [3] K. Tanaka, T. Mura, A theory of fatigue crack initiation at inclusions, Metall. Trans. A 13 (1982) 117–123. [4] A.A. Malekbarmi, S. Zangeneh, A. Roshani, Assessment of premature failure in a
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Generalized stacking fault energies of Cr23C6 carbide: A first-principles study Fangfang Xia
a,1
, Weiwei Xu
a,1
a,⁎
b,c
, Lijie Chen , Shunqing Wu , Michael D. Sangid
d,⁎
T
a
School of Aerospace Engineering, Xiamen University, Xiamen 361005, China Collaborative Innovation Center for Optoelectronic Semiconductors and Efficient Devices, Department of Physics, Xiamen University, Xiamen 361005, China Key Laboratory of Low Dimensional Condensed Matter Physics (Department of Education of Fujian Province), Xiamen University, Xiamen 361005, China d School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, USA b c
A R T I C LE I N FO
A B S T R A C T
Keywords: Nickel-based superalloys Cr23C6 carbide First-principles calculations Generalized stacking fault energy
In many high temperature-resistant Ni-based superalloys, M23C6 carbides form during processing and remain as non-metallic inclusions during the materials use in application. M23C6 type inclusions are principal sites for potential fatigue crack initiation, thus the accurate quantification of their energy barriers for deformation is necessary for in-depth understanding of the overarching mechanical behavior of the material system. In this study, the generalized stacking fault energies (GSFE) are theoretically determined for chromium-rich types of M23C6 carbides (i.e., Cr23C6) in two prominent slip systems: {1 1 1}〈1 1 2¯ 〉 and {1 1 1}〈1 0 1¯〉. The results present GSFE curves with double saddle points in both slip systems. The double saddle points of the GSFE curves are explained via the electronic structure of atoms near the slip plane. At deformation quantified by half of a Burgers vector (i.e., 0.5 bp), a pentacyclic charge density distribution appears representing a local minimum energy configuration, which results in a valley of the GSFE curve. Furthermore, the effective coordination number (n) and the distortion index (Δd) of the coordination polyhedrons are employed to characterize the structural change. The analysis reveals that the values of the indexes n and Δd at a displacement value of 0.5 bp are close to the values of a perfect Cr23C6 crystal, while comparatively the distortion index is higher at the saddle points. These results provide new insights into the deformation pathways of M23C6 carbides.
1. Introduction Despite incredible advances in material cleanliness through processing improvements, carbides are inherent within many structural materials. In many superalloys, MC carbides (typically rich in Ti, Ta, or Hf) form at high temperatures and are metastable at low temperatures, while M23C6 carbides (e.g. Cr23C6) form at low temperatures and remain as the majority carbide inclusion phase, in equilibrium and volume fraction, during time-dependent deformation at elevated temperatures [1]. M23C6 type carbides, as prototypical inclusions prefer to precipitate at grain boundaries, are known to distinctly influence the fatigue properties of materials [2]. During fatigue loading, strain rapidly accumulates in the form of a dislocation pile-up at M23C6 inclusions [3], which is a precursor for crack initiation [4,5]. In order to explore the fatigue behavior of Ni-based superalloys, the mechanical and thermodynamic properties of M23C6 carbides, especially Cr23C6 (which is as a representation of M23C6 in most Ni-based superalloys [6]), are desired to identify the mechanism of
heterogeneous deformation and crack initiation. Unfortunately, there are limited experimental reports on these mechanical properties of M23C6. Using first-principles calculations, various foundational properties have been theoretically investigated for Cr23C6 carbides based on density functional theory (DFT) analysis [7–10]. For example, Jiang [7] studied the structural, elastic, and electronic properties of Cr23C6, it is found that Cr23C6 exhibits a high elastic moduli and low Poisson’s ratio, which is indicative of a hard material. Yet, further investigation is needed to study the heterogeneous deformation and the failure mechanism of Cr23C6. The generalized stacking fault energy (GSFE) is inherently related to the strength of materials and is used to determine the mechanical properties of material constituents in the form of a comprehensive description of the energy barrier to deformation or dislocation motion. The GSFE represents the interplanar potential energy required to displace one elastic half space of the crystal along a specific crystallographic direction with respect to the remaining lattice [11]. However, the complete GSFE is experimentally not accessible, hence the GSFE
⁎
Corresponding authors at: School of Aerospace Engineering, Xiamen University, Xiamen 361005, China (L.J. Chen). School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-0307, USA (M.D. Sangid). E-mail addresses: [email protected] (L. Chen), [email protected] (M.D. Sangid). 1 Fangfang Xia and Weiwei Xu contributed equally to this work. https://doi.org/10.1016/j.commatsci.2018.11.006 Received 19 June 2018; Received in revised form 1 November 2018; Accepted 2 November 2018 0927-0256/ © 2018 Published by Elsevier B.V.
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studies with first-principles have achieved considerable attentions in the simulations of plasticity of bulk crystalline materials [12–14]. In previous research, Sangid et al. [15,16], modeled GSFEs of Ni and Ni3Al within Ni-based superalloy and used this information to study the stability of a persistent slip band with respect to dislocation motion as failure criterion for fatigue crack initiation. Cr23C6 carbide is an interstitial compound with a complex facecentered cubic (fcc) structure, for which the conventional cell has 116 atoms [17]. Due to this complex structure, the GSFE of Cr23C6 is investigated to a far less extent, compared to fcc pure metals or binary alloys, both experimentally and theoretically. It is of the believe that M23C6 carbides are generally intermetallic particles that are brittle, however, Rong et al. [18] observed dislocations, stacking faults, and microtwins composing the defect structure of M23C6 type carbides in Nickel-based superalloys based on transmission electron microscopy. Furthermore, Xu et al. [19] reported stacking faults along the boundaries and extended stacking faults along 〈1¯ 1 1〉 directions within M23C6 type carbides in high-manganese steel. Although these studies [18,19] reported the existence of stacking faults in M23C6 type carbides along the {1 1 1} planes indicative of crystallographic slip, there is little information about the energy barriers to achieve these structures, and the associated GSFE of the M23C6. The incomplete description pertaining to the mechanical behavior of such material motivates the present study. In this scenario, the GSFEs of Cr23C6 carbide are investigated in two typical {1 1 1}〈1 1 2¯ 〉 and {1 1 1}〈1 0 1¯〉 slip systems by employing firstprinciples calculations. Through the analysis of atomic structures, electron density distributions, and coordination polyhedrons near the slip plane, the present theoretical results are further discussed.
constructing a supercell model with 6 layers, as shown in Fig. 1(a). The GSFEs are calculated by rigidly shearing the upper layers of the cell, numbers 4–6 relatively to the lower layers number 1–3, along typical 〈1 1 2¯ 〉 and 〈1 0 1¯〉 directions in the (1 1 1) plane. 2.2. Computational details The present calculations were performed by using the projector augmented wave (PAW) method within the density functional theory (DFT) [20,21] as implemented in the Vienna Ab initio Simulation Package (VASP) [22,23]. The exchange-correlation functional was treated within the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PBE) [24,25]. Wave functions were expanded in a plane wave, up to a cutoff energy of 550 eV. The non-spin polarized calculations were considered to perform the calculations, since previous studies indicate that Cr23C6 exhibits paramagnetic characteristic with an almost zero magnetic moment [8,26]. Brillouin zone integrations were approximated by using a special kpoint sampling of Monkhorst-Pack scheme [27] with Γ-centered 15 × 15 × 15 and 5 × 5 × 1 grids for lattice constant and GSFE calculation, respectively. The convergence threshold of total energy is 10−4 eV. All the atoms were relaxed in the direction perpendicular to the glide plane until the force on each atom is smaller than 10−2 eV/Å. In order to reduce the effect of periodic arrangements of atoms on the GSFE, as much as possible, a vacuum layer of 10 Å is added on the top of the (1 1 1) supercell. 3. Results and discussion
2. Computational methodology
3.1. Generalized stacking fault energy
2.1. Modeling methodology
The obtained reference lattice constant a for Cr23C6 is 10.54 Å, which agrees well with experimental (10.64 Å [28]) and theoretical values (10.53 Å [7,8], 10.56 Å [26]). To confirm the convergence for the number of (1 1 1) stacking layers, both 6- and 10-layer supercells are used to calculate the GSFE for Cr23C6. It is found that the difference in the γ values based on the 6- and 10-layer model is extremely small (less than 2%). Therefore, a 6-layer supercell is constructed, considering the computational efficiency. Fig. 2 shows the Burgers vectors of possible slip and the γ curve as a function of the Burgers vector for a bulk Cr23C6 carbide in two typical slip systems: {1 1 1}〈1 1 2¯ 〉 and {1 1 1}〈1 0 1¯〉. Some critical γ values are listed in Table 1. Note that bp represents the Burgers vector for a partial dislocation along 〈1 1 2¯ 〉 directions at {1 1 1} crystal planes; while b represents the Burgers vector for a full dislocation along a 〈1 1 0〉 directions. As we know, the GSFE curve of fcc metals and alloys, such as Ni and Ni3Al, exhibits a typical bell-shaped profile with one peak at 0.5 bp (or 0.5b). However, the GSFE curves of Cr23C6 show double saddle points along 〈1 0 1¯〉 and 〈1 1 2¯ 〉 directions (as shown in Fig. 2(b)). Table 1 presents the saddle points γus1 (2201 mJ/m2) and γus2 (2603 mJ/m2) are nine orders of magnitude larger than those of Ni (277 mJ/m2) and Ni3Al (257 mJ/m2) in {1 1 1}〈1 1 2¯ 〉 slip system (the detailed data for the GSFE of Ni and Ni3Al is discussed in the Supplementary material). As a higher γus value represents a greater energy barrier for emission and associated plastic deformation, it implies that the dislocations within Cr23C6 in this slip system are difficult to form. This extremely large value of the γus within Cr23C6 is possibly due to the complex topological structure of carbides. Although such GSFE curve has not been observed for any fcc structure, it was previously reported for some inorganic materials, such as Mg2SiO4-forsterite [29]. The valley of the GSFE curve along 〈1 1 2¯ 〉 implies there is a local stable structural configuration at 0.5 bp (m position shown in Fig. 2(a)). At position m, there are two potential slip paths for the next step: (i) slip continues along the 〈1 1 2¯ 〉 (m → M) or (ii) slip occurs along the 〈2 1¯ 1¯〉 (m → h), as shown in Fig. 2(a). In Fig. 2(b), the energies associated with slip along m → h reach a stable high value at 0.6 bp, while for the slip
As previous investigations [18,19] reported that the stacking faults of M23C6 were observed along the {1 1 1} planes and there is little information along other planes, we can focus on the stacking faults on the (1 1 1) plane. A stacking fault is formed if the regular fcc stacking of {1 1 1} planes is altered, thus it is possible to generate a stacking fault on {1 1 1} planes by shifting the upper half of the crystal relative to the lower half by a displacement vector u. The GSFE (γ) can be computed as a function of displacement u, defined as γ(u), and the local maximum energy of the GSFE curve is termed the unstable stacking fault energy (γus) [11]. The GSFE can be calculated by:
γ (u ) =
E (u ) − E0 A
(1)
where E (u) is the total energy of the supercell for a given fault vector, u, E0 is the total energy of the perfect supercell, and A is the area of the fault plane. In fcc structures, the calculated supercells for the GSFEs are built up from (1 1 1) layers via an ABC stacking to produce a disruption in the stacking sequence and a single layer intrinsic stacking fault (ISF) is formed when the stacking of the {1 1 1} planes is …ABC|BCA…, where | indicates the position of the fault. The complex fcc crystal structure of Cr23C6 contains 92 chromium and 24 carbon atoms. Cr is located at positions of 4a (0 0 0) (Cr1), 8c (0.25 0.25 0.25) (Cr2), 32f (0.385 0.385 0.385) (Cr3), 48 h (0 0.165 0.165) (Cr4) Wyckoff sites, and C is located at 24e (0.275 0 0) Wyckoff site, respectively [17]. Although Cr23C6 possesses a complex structure, the stacking sequences along the {1 1 1} planes can be defined by repeated ABC, where each unit A (B or C) consists of a “superatom” with 29-atom basis (i.e., primitive cell of Cr23C6). Fig. 1 displays the generation of the computational supercell used in the present calculations. The supercell is created from a primitive cell of pure Cr23C6 in [1 0 1¯], [0 1 1¯] and [1 1 1] directions as illustrated in Fig. 1(c). Then, the primitive cell is used as a “superatom” for 21
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C B
A C
c
B
b t a
(c)
c
Cr (4a)
Cr (8c)
Cr (32f)
Cr(48h)
C
t
A (a)
(b)
Fig. 1. Generation of a 174-atom supercell for Cr23C6 along (1 1 1) planes. (a) A two-dimensional crystal consisting of 6-layer stacking form (ABCABC) along the (1 1 1) planes for Cr23C6 with the 29-atom basis as a “superatom”. (b) The Bravais lattice is specified by two repeat vectors t ([1 1 2¯ ]) and c ([1 1 1]). (c) The “superatom” contains 29 atoms. (4a, 8c, 32f and 48 h represent Wyckoff sites. a: [1 0 1¯], b: [0 1 1¯]).
along the path m → M, the energies reach a smaller value of 1248 mJ/ m2 (as seen in Table 1). The energy barrier for the slip along m → M is slightly bigger than slip along m → h, as seen in Fig. 2(b), which implies the slip along possible path m → h is not energetically favorable compared to slip along 〈1 1 2¯ 〉. Therefore, slip along path 〈1 1 2¯ 〉 at 0.5 bp will continue in the same direction, which can be expressed as: O → m → M. Comparing the GSFE curves of the two prominent slip systems: {1 1 1}〈1 1 2¯ 〉 (O → M) and {1 1 1}〈1 1 2¯ 〉 (O → R), except near 0.7 bp (or 0.7b), the energies associated with slip along the 〈1 0 1¯〉 from 0.1b to 0.8b is always greater than those of slip along the 〈1 1 2¯ 〉 direction correspondingly (detailed data listed in Table 1). As shown in Fig. 2(b), there is only a small gap between the energies required for slipping along O → M and O → R near 0.7b (or 0.7 bp), which implies that slip along 〈1 0 1¯〉 is more energetically difficult than 〈1 1 2¯ 〉. Therefore, the most prominent slip system for dislocation activity in Cr23C6 is {1 1 1} 〈1 1 2¯ 〉, thus we focus on this slip system for further analysis.
Table 1 GSFE values (γ) of Cr23C6 at distinct crystallographic displacements (mJ/m2). ¯
γ
¯
¯
¯
¯
¯
< 101 > γ0.5 b
< 101 > γ0.7 b
< 112 > γ0.35 bp
< 112 > γ0.5 bp
< 112 > γ0.7 bp
< 112 > γ1.0 bp
2434
2315
2496
2201
1660
2603
1248
To clarify the influence of atomic redistribution on the GSFE curve of Cr23C6 during the process of crystallographic slip, the atoms were relaxed and held fixed (termed as no-relaxed in Fig. 2(b)) in the direction perpendicular to the glide plane, respectively. Noteworthy that, if the GSFE for Cr23C6 is calculated directly without the relaxation of atoms, then the obtained GSFE for Cr23C6 contains artificially large values and does not possess a local minimum energy, as shown in Fig. 2(b). The existence of an energy valley at 0.5 bp may be attributed to the stabilization of a local cluster of atoms during the slip process. To 10000
Ȗus2
Cr23C6 2500
Top layer Middle layer
m O
h
r
h
h
Ȗus1
8000
2000
R
enlarge
6000
m
1500
Ȗisf M
Ȗ(m-m2)
Ȗ(m-m2)
Bottom layer
M
¯
< 101 > γ0.4 b
4000
1000
2@ [11m
O
M
[2
1 r 1@
R h
RHOax a2 101!(OĺR) RHOax a6 112!(OĺM) RHOax a6 112!(Oĺm)a6 211!(mĺh) 1oRHOax a6 112!(OĺM)
500
0 0
O
0.1
0.2
0.3
0.4
0.5
[101@
ux(b or bp)
(a)
(b)
0.6
0.7
0.8
2000
R 0.9
0 1.0
Fig. 2. (a) Illustration of three successive (1 1 1) planes of Cr23C6 and the Burgers vectors forming three possible slip paths. The circles in the figure present a “superatom” of Cr23C6 on different planes. m, h, and r represent the midpoints of the segment OM, OR, and MR, respectively. (b) GSFE (γ) as a function of the Burgers vector for Cr23C6 in the {1 1 1} crystal plane along different slip directions. Note that the relaxed energies correspond to the y-axis on the left and unrelaxed energies correspond to the energies on the right y-axis. a stands for lattice constant. In x-axis label, bp = a/6{1 1 1}〈1 1 2¯ 〉 and b = a/2{1 1 1}〈1 0 1¯〉. 22
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(a) 7
2500
(d)
Ȗ(m-m2)
2000
8
(b)
1500
(c)
1000
(e)
7
(a) 0
0.1
Relax <11-2>
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
C
Cr
0.026
(c)
7
(d)
5
6
4
2
7 8 3
2
1 7
5
6
8 1
7
4 2
2
4
4
8 3
9
2
1
3
4
1
3
2
7
9
6
8
4
1
5
6
9
5
6
9
3
3
9
(e)
8 1
2
8
9 Cr 5
6
4
11
4
1
5
6
8
5
7
99 99
3
7
9
C
8
-0.074
8
3
2 6
8
ux(bp)
4
11
slip plane
500
0
6
(b)
5
2 7
4
3 6
5 4
4
9 55
8
4 4
1 2
3
9
Fig. 3. Atomic structure near the slip plane and the relevant contour plots of charge density differences (CDD) (the unit in eV/Å3) in the (1 1¯ 0 ) plane under a normalized displacement of (a) 0 bp, (b) 0.35 bp, (c) 0.5 bp, (d) 0.7 bp, and (e) 1.0 bp. (bp = a/6{1 1 1}〈1 1 2¯ 〉).
among the Cr atoms 1, 2, and 6 after a slip distance of 0.35 bp (Fig. 3(b)). The observed charge distributions change the overall structural configuration from stable to metastable, and thereby lead to a relative large energy value of the GSFE at 0.35 bp and an associated saddle point, as shown in Fig. 2(b). Upon consecutively shearing, the stable six-membered charge density among the Cr atoms 1, 2, 3, 4, 5 and 6 would be more significantly distorted and lead to a larger GSFE value. Interestingly, after shearing to a value of 0.5 bp as illustrated in Fig. 3(c), the Cr atom 2 jumps from its original position to an equilibrium position to accommodate the region that was previously chargefree and participate in the electronic interaction with the Cr atoms 1 and 6. In addition, the neighboring Cr atoms 7 and 8 also participate in the electronic interaction with the Cr atoms 1, 2, and 6, re-forming a local stable structural configuration with the geometry of the fivemembered ring from the distorted six-membered ring. The formation of such five-membered ring remarkably decreases the total energy of Cr23C6, and it results in the local minimal of the GSFE curve at a displacement of 0.5 bp. For a shearing displacement of 0.7 bp (Fig. 3(d)), a similar configuration is observed as the case of 0.35 bp. The local stable five-membered ring is distorted again, which implies the local stable structural configuration is changed from stable back to metastable, leading to the second saddle point of the GSFE curve at 0.7 bp. As the crystal structure is sheared to a displacement value of 1.0 bp, the stacking sequences change from ABC|ABC to ABC|BCA, implies there may be a stable structural configuration. As shown in Fig. 3(e), with more neighboring Cr atoms participating in the bonding interaction corresponding to atoms 1, 2, 3, 4, 5, 6, 7, 8 and 9, both six- and fivemembered rings are formed, which stabilizes the structural configuration of Cr23C6 by reducing its corresponding value of the GSFE, as plotted in Fig. 2(b).
gain deeper insight into the double saddle of GSFE curves, the electron density distributions and the change of coordination polyhedrons near the slip plane in the {1 1 1}〈1 1 2¯ 〉 slip system are analyzed. 3.2. Chemical bonding characteristics The GSFEs are related to the atomic bonding characteristics and a fundamental understanding of the characteristics could reveal the reasons for the double saddle configuration of the GSFE curves. Charge density differences (CDD), which denote charge redistributions, represent a means to further study the bonding characteristics of the material. To gain additional insights about the bonding structure during the slip mediate process of Cr23C6, we plot the structural and CDD maps for atoms near the slip plane under a normalized displacement, as shown in Fig. 3. The scale for CDD is displayed on the left of Fig. 3. The blue color represents the maximum delocalization of electrons and the red color indicates the maximum localization of electrons. For a perfect crystal structure, without a normalized displacement (0 bp), Fig. 3(a) shows that the electrons are oriented among the Cr atoms indicated by 1, 2, 3, 4, 5 and 6 numbering, forming a localized charge distribution with the geometry of a six-membered ring. This electronic hybridization indicates the non-polar covalent bonding characteristics, which is believed to be beneficial to the phase stability of Cr23C6. From the CDD field distributions with a normalized displacement of 0.35 bp as shown in Fig. 3(b), the stable six-membered charge density is distorted and the electronic charge effectively localized between the Cr atoms 5 and 6 in the upper part of the plane, as well as the Cr atoms 2 and 3 in the lower part of the plane. The bond length between Cr atoms 1, 6 increases from 2.42 Å to 3.01 Å with a elongation of 24%, and between Cr atoms 3, 4 increases from 2.42 Å to 3.38 Å with a elongation of 40%. Based on the Lindemann criterion [30], this large elongation results in the breaking of chemical bonds. Meanwhile, an almost charge-free region is observed 23
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3.3. Coordination polyhedrons
Table 2 Effective coordination number (n) and distortion indexes (Δd) of the coordination polyhedrons at distinct crystallographic displacements of the {1 1 1} 〈1 1 2¯ 〉.
To quantify the structural changes during shear deformation, we also investigate the coordination polyhedrons of Cr23C6 near the fault plane of the {1 1 1}〈1 1 2¯ 〉 slip system. In the polyhedral model, crystal structures are represented by coordination polyhedrons where the central atoms, bonds, and apex atoms are included [31]. The coordination number denotes the number of atoms coordinated to a central atom in a coordination polyhedron. In general, the coordination number correlates with electronic interaction and thereby is associated with the structural stability. However, it is difficult to state the coordination of the central atom as a simple number in relatively distorted coordination polyhedron. Hence, the effective coordination number (n) [32,33] is determined by adding the surrounding atoms with a weighting scheme, which is adopted in this paper. The distortion in a coordination polyhedron can be quantified with the distortion index (Δd), which based on the bond lengths is defined as
Δd =
1 n
n
∑ i=1
|di −d¯| d¯
n Δd
0 bp
0.35 bp
0.5 bp
0.7 bp
A, B, C
A, B, C
A
B, C
A
B, C
A, B, C
7.9 0.04
6.8 0.12
6.6 0.067
7.7 0.072
6.7 0.076
6.6 0.077
7.8 0.057
indicate the structural configuration at 0.5 bp form a local stable structural configuration as shown in Fig. 3(c). Upon further continuous shearing to a displacement of 0.7 bp, the coordination polyhedrons appear similar to the structure at a displacement of 0.35 bp, as shown in Fig. 4(b) and (d), indicating the local stable structural configuration is destroyed. However, with shearing to the ISF configuration corresponding to a displacement of 1.0 bp, as shown in Fig. 4(e), the coordination polyhedrons look similar to the perfect structure (Fig. 4(a)). In addition, the values of Δd at a displacement of 1.0 bp are close to the values at 0 bp. Therefore, it reveals the ISF configuration (1.0 bp) is more stable than structural configuration at 0.5 bp. In other words, upon consecutively shearing, atoms are redistributed and re-form stable structural configuration at 0.5 bp and 1.0 bp.
(2)
where di is the bond lengths of Cr-Cr atoms in a coordination polyhedron and d¯ is the average di value. Fig. 4 shows the coordination polyhedrons for the structural model of Cr23C6 under a series of normalized displacements. The coordination polyhedrons n and Δd near the slip plane are calculated and summarized in Table 2. The value of n is the reflection of the structural stability, and for a perfect structure, the value of n is about 8, as shown in Fig. 4(a). A smaller Δd exhibits a smaller distortion of the atomic structure; as expected the smallest Δd value corresponds to a perfect crystal structure (see Table 2). Fig. 4(b) shows that the values of n for the coordination polyhedrons A, B, C are about 7, under the shearing displacement of 0.35 bp. Meanwhile, the polyhedrons values of Δd significantly increase; thus implying that the stable structural configuration at 0 bp is significantly destroyed, and the result is consistent with the features of the CDD map shown in Fig. 3(b). Upon consecutively shearing of the crystal structure to a displacement of 0.5 bp, due to the Cr atoms 1 and 2 drastic swing (shown in Fig. 4(c)), the n of coordination polyhedrons B and C are about 8, while their values are about 7 at 0.35 bp. The values of Δd for the polyhedrons A, B, C at a displacement of 0.5 bp are smaller than those at 0.35 bp. While the value of n for coordination polyhedron A is similar between a displacement of 0.35 bp and 0.5 bp, the values of n for coordination polyhedrons B and C represent a more stable structural configuration for a displacement of 0.5 bp. These numerical values
4. Conclusions By constructing a 174-atom supercell, the present work quantified the GSFE of Cr23C6 for two typical slip systems in a fcc–type crystal structure: {1 1 1}〈1 1 2¯ 〉 and {1 1 1}〈1 0 1¯〉 using first-principles calculations. The charge density difference and the coordination polyhedron around the slip plane were analyzed to rationalize the unique features of the GSFE curves for Cr23C6. Based on the present calculations, results of this study can be summarized as follows: (1) Unlike Ni and Ni3Al, Cr23C6 exhibits a double bell-shaped GSFE curve, which represents a series of saddle points. A valley point representing a stable stacking fault energy is found at the shearing displacement value of 0.5 bp. (2) Energetically, the most favorable slip system for dislocation activity in Cr23C6 is {1 1 1}〈1 1 2¯ 〉. The saddle points γus1 (2201 mJ/m2) and γus2 (2603 mJ/m2) of the GSFE curve in this slip system implies the
8
8
2
1 8
2 1
1
slip plane 2
A
1.0 bp
B
(b)
(c)
C 8
1 1
2
2
(a) (d)
(e)
Fig. 4. Structural models representing coordination polyhedrons under a normalized displacement of (a) 0 bp, (b) 0.35 bp, (c) 0.5 bp, (d) 0.7 bp, and (e) 1.0 bp. A, B, and C represents a coordination polyhedron, containing central atoms (C), bonds, and apex atoms (Cr), respectively. (The pink atoms 1, 2, 8 represents Cr 1, 2, 8 in Fig. 3, respectively. The Cartesian coordinate system represents crystallographic axes of a: [1 0 1¯], b: [0 1 1¯], and c: [1 1 1].) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 24
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dislocations within Cr23C6 need to overcome a higher energy barrier than typical fcc metals and alloys, thus explaining the relatively brittle nature of Cr23C6 precipitates within Ni-based superalloys. (3) The dominant factor governing the dislocation glide of Cr23C6 is clarified to be the chemical bonding configuration near the slip plane. Under shear deformation, the stable six-membered bonding characteristics of atoms are disrupted, and then a five-membered bonding ring is reformed at the displacement of 0.5 bp, resulting in the related stacking fault energy displaying a local minimum value. Subsequent shear displacement breaks the five-membered rings until the intrinsic stacking fault is reached at a deformation of 1.0 bp, where both six- and five-membered bonding configurations are observed. (4) The quantified coordination polyhedrons distorted index further supports the analysis of bonding characteristics in Cr23C6. The coordination number and the bonding length of atoms near the slip plane at 0.5 bp are much closer to the values calculated at displacements of 0 bp and 1.0 bp. However, the saddle points show a large polyhedron distortion compared with the local stable points of 0 bp, 0.5 bp, and 1.0 bp. CRediT authorship contribution statement Fangfang Xia: Methodology, Formal analysis, Writing - original draft. Weiwei Xu: Methodology, Formal analysis, Writing - original draft, Funding acquisition. Lijie Chen: Conceptualization, Writing review & editing, Resources, Supervision, Project administration, Funding acquisition. Shunqing Wu: Validation, Formal analysis. Michael D. Sangid: Conceptualization, Validation, Writing - review & editing, Resources. Acknowledgements The authors would like to thank the support of the National Natural Science Foundation of China (Grant Nos. 51475396 and 51601161) and the Fundamental Research Funds for the Central Universities (Grant No. 20720170048). This work originated during Prof. Chen’s visit to Purdue University as a visiting scholar. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.commatsci.2018.11.006. References [1] Y.Y. Song, Thermodynamic study on b and fe substituted Cr23C6 using first-principles calculations, Ph. D. thesis, 2010. [2] S. Suresh, Fatigue of Materials, Cambridge University Press, Cambridge, England, 1998. [3] K. Tanaka, T. Mura, A theory of fatigue crack initiation at inclusions, Metall. Trans. A 13 (1982) 117–123. [4] A.A. Malekbarmi, S. Zangeneh, A. Roshani, Assessment of premature failure in a
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