Effects of Alloying Elements on the Stacking Fault Energies of Ni₅₈Cr₃₂Fe₁₀ Alloys: A First-Principle Study

Abstract

Ni₅₈Cr₃₂Fe₁₀-based alloys, such as Alloy 690 and filler metal 52 (FM-52), suffer from ductility dip cracking (DDC). It is reported that decreasing the stacking fault energy (SFE) of these materials could improve the DDC resistance of Alloy 690. In this work, the effects of alloying elements on the stacking fault energies (SFEs) of Ni₅₈Cr₃₂Fe₁₀ alloys were studied using first-principle calculations. In our simulations, 2 at.% of Ni is replaced by alloy element X (X=Al, Co, Cu, Hf, Mn, Nb, Ta, Ti, V, and W). At a finite temperature, the SFEs were divided into the magnetic entropy (SFE^mag) and 0 K (SFE⁰) contributions. Potentially, the calculated results could be used in the design of high-performance Ni₅₈Cr₃₂Fe₁₀-based alloys or filler materials.

1. Introduction

Ni–Cr–Fe alloys have excellent mechanical and anti-corrosion properties. Thus, they are extensively used in the aerospace, power generation, and transportation industries. Nuclear energy plays a critical role in modern society because fossil fuels (like coal, oil, and gas) might run out shortly. Nuclear reactors’ construction materials should have excellent anti–corrosion properties to the intergranular stress corrosion cracking (IGSCC). In the 1950s, Ni-based Alloy 600 was developed to replace 304 stainless steels in nuclear reactor constructions. Then, researchers found that Alloy 690 has even better resistance to IGSCC [1,2,3,4,5,6].

Filler metal 52 (FM-52), having excellent resistance to stress corrosion cracking at moderate temperatures, is widely used for joining Alloy 690. However, FM-52 is susceptible to ductility dip cracking (DDC) at temperatures ranging from about 0.5 to 0.7 melting temperature. Researchers have found that the addition of alloying element Nb can improve the DDC resistance of FM-52 [1,6,7,8,9,10]. It is commonly known the following reasons could associate with DDC in the Alloy 690 weld joints: grain size, grain boundary migration, grain boundary segregation, precipitation behavior, chemical composition, and so forth [1,2,3,4,5,6]. Recent studies revealed that DDC resistance of Alloy 690 weld joints could be improved by tailoring the stacking fault energy (SFE) [11,12,13,14].

Fundamentally, DDC is a solid-state intergranular hot cracking phenomenon caused by local exhaustion of ductility at stress concentration area. A more homogeneous deformation process (or a more homogeneous distribution of dislocations) is required to achieve a better DDC resistance. Stacking fault energy (SFE) is an essential physical parameter for alloys because it intimately relates to the behavior of dislocations. In face-centered cubic (fcc) alloys, a full dislocation could dissociate into two partial dislocations separated by an intrinsic stacking fault. The dissociated dislocation cannot climb or cross-slip until a recombination process is fulfilled. Generally, a lower stacking fault energy (SFE) value leads to a broader stacking fault. As a consequence, the recombination process in low SFE materials is harder than that in high SFE materials. This condition leads to a more homogeneous distribution of dislocations and facilitates more homogeneous deformed microstructures compared with high SFE materials [11,12,13,14]. It should be pointed out that decreasing the SFE may not increase the global ductility (or macro ductility), but could potentially lead to a more homogeneous distribution of dislocations.

To get a better understanding of the dislocation behavior in Ni₅₈–Cr₃₂–Fe₁₀-based alloys, the SFEs should be characterized in a wide group of alloying elements. Experimental measurements of SFE require delicate techniques. Moreover, the experimental values of SFEs might be influenced by residual stress, which arises from the sample preparation process [15]. Fortunately, first-principle studies based on density functional theory (DFT) have been considered as effective computational methods to calculate the SFE. Available calculated results are limited to Ni-continuing binary alloys [16,17,18], ternary alloys [17], and austenitic stainless steels [19,20,21].

In this work, the effects of alloying elements on the stacking fault energies (SFEs) of Ni_60.5Cr_29.6Fe_9.9 (wt.%) or Ni₅₈Cr₃₂Fe₁₀ (at. %) are studied using first-principle simulations. The selected composition is similar to Alloy 690 and FM-52. Here and throughout this paper, the compositions are expressed in at.%. In the simulations, 2 at.% of Ni is replaced by alloy element X (X=Al, Co, Cu, Hf, Mn, Nb, Ta, Ti, V, and W). Potentially, the simulated results could be used in the design of Ni₅₈Cr₃₂Fe₁₀-based high-performance alloys or filler materials.

2. Computational Details

The SFEs were calculated by the axial next-nearest neighbor Ising model (ANNNI) and derived as follows [21,22,23]:

$$SFE=\left(E_{\mathrm{hcp}}+2E_{\mathrm{dhcp}}-3E_{\mathrm{fcc}}\right)/A_{2\mathrm{D}},$$

(1)

where E_hcp, E_dhcp, and E_fcc are the energies of hexagonal close-packed (hcp), double hexagonal close-packed (dhcp), and face-centered cubic (fcc) structures, respectively. A_2D is the area of the stacking fault. At finite temperature, the SFE in the Ni–Fe–Cr alloys could be roughly divided into the magnetic entropy (SFE^mag) and 0 K (SFE⁰) contributions [20,21], as the contributions of electronic entropy and phonon were demonstrated to be relatively small [24]. SFE^mag is derived as follows:

$$SF{E}^{\mathrm{mag}}=-\mathrm{T}(S_{\mathrm{hcp}}^{\mathrm{mag}}+2S_{\mathrm{dhcp}}^{\mathrm{mag}}-3S_{\mathrm{fcc}}^{\mathrm{mag}})/A_{2\mathrm{D}},$$

(2)

where S^mag is derived as follows [25]:

$$S^{\mathrm{mag}}={{\displaystyle \sum }}_{\mathrm{i}}{k}_{\mathrm{B}}{c}_{\mathrm{i}}\log ({\mu }_{\mathrm{i}}+1),$$

(3)

where k_B is the Boltzmann constant, c_i is the concentration of atom i, and μ_i is the local magnetic moments of atom i. Previous studies demonstrated that the thermal lattice expansion is the main parameter influencing the temperature dependence of SFE [24,26]. In present work, SFEs⁰ were calculated at 0 K using the equilibrium volume obtained from energy-volume calculations, and SFEs^mag under finite temperatures were calculated with thermal-expanded cells. The thermal expansion coefficient was adopted from the data of Alloy 690 [27].

The energies of Ni₅₈Cr₃₂Fe₁₀-based random alloys were calculated with the exact muffin-tin orbitals (EMTO) [28,29]. Together with the coherent potential approximation (CPA) [30,31], the EMTO–CPA method properly describes the chemical and magnetic disorder systems [21,29]. The k-mesh for dhcp, hcp, and fcc structures was carefully tested and set as 17 × 25 × 7, 17 × 25 × 13, and 17 × 17 × 17, respectively. During the self-consistent calculations, frozen core approximation, disordered local magnetic moment (DLM) [32], and the local density approximation (LDA) [33] were adopted. It should be noted that the critical temperature for alloy 690 is 370 K [34]. Considering that DDC is a high temperature phenomenon, it is reasonable to discuss the SFEs within the DLM scheme. The total energy is converged within 10⁻⁷ Ry per atom. Then, the total energy was evaluated with the full charge density technique (FCD) approach [35]. In the FCD calculation, the exchange-correlation interactions were described by the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) form [36]. This combination of approximations has been successfully used to calculate the elastic properties and SFE of Fe–Cr–Ni alloys [20,21,37]. The equilibrium unit cell volume (V), bulk modulus (B), shear modulus (G), and stacking fault energies (SFEs) of Ni and Ni₅₈Cr₃₂Fe₁₀ are listed in

Table 1. The equilibrium unit cell volume (V) in 10⁻¹⁰m³, bulk modulus (B) and shear modulus (G) in GPa, and stacking fault energies (SFEs) in mJm⁻² of Ni and Ni₅₈Cr₃₂Fe₁₀. Values without a superscript are calculated at 0 K.

Alloy	V	B	G	SFE
Ni	10.97	198.4	102.7	140.9
	10.982 a	196.5 a	103.2 a	131.5 b
	10.95 c	187.6 c	101.1 c	120–130 d
Ni58Cr32Fe10	11.59	180.1	82.1	20.5 21.2 e
Alloy 690			79.3 f
Ni59.3Cr30.7Fe10	11.37 f			61.0 g

^a First principle data from the literature [38]; ^b first principle data from the literature [17]; ^c experimental data from the literature [39]; ^d experimental data from the literature [40]; ^e first principle data at 293 K; ^f experimental data from (https://www.specialmetals.com/assets/smc/documents/alloys/inconel/inconel-alloy-690.pdf); ^g experimental data from the literature [13].

. The calculated SFEs and equilibrium unit cell volumes (V) of Ni₅₆Cr₃₂Fe₁₀X₂ are listed in

Table 2. Equilibrium cell volumes (V) in 10⁻¹⁰m³ and stacking fault energies (SFEs) in mJm⁻² of Ni₅₆Cr₃₂Fe₁₀X₂. The SFEs are divided into the 0 K contribution (SFEs⁰) and the magnetic entropy contribution (SFEs^mag). SFEs^mag are calculated with thermal-expanded cells (at 298, 673, 973, and 1173 K).

Alloy	V	SFEs0	SFEsmag (298 K)	SFEsmag (673 K)	SFEsmag (973 K)	SFEsmag (1173 K)
Ni58Cr32Fe10	11.60	20.5	0.7	1.3	3.4	4.7
Ni56Cr32Fe10Al2	11.66	21.9	0.6	1.0	1.4	6.9
Ni56Cr32Fe10Co2	11.59	18.6	0.6	1.3	2.1	3.6
Ni56Cr32Fe10Cu2	11.63	21.8	0.5	0.8	2.0	10.3
Ni56Cr32Fe10Hf2	11.93	11.5	0.5	2.5	7.1	16.2
Ni56Cr32Fe10Mn2	11.63	20.4	0.6	1.5	2.6	6.2
Ni56Cr32Fe10Mo2	11.76	18.5	0.5	1.1	1.7	8.7
Ni56Cr32Fe10Nb2	11.83	18.5	0.5	1.9	4.4	14.4
Ni56Cr32Fe10Ta2	11.80	15.3	0.5	1.7	3.8	9.9
Ni56Cr32Fe10Ti2	11.79	18.9	0.4	0.9	1.7	6.3
Ni56Cr32Fe10V2	11.66	19.6	0.5	0.9	1.8	3.3
Ni56Cr32Fe10W2	11.74	20.7	0.6	0.9	1.7	5.2

.

3. Results and Discussions

3.1. The Elastic Properties and SFEs of Ni and Ni₅₈Cr₃₂Fe₁₀

To evaluate the reliability of our simulations, let us compare the simulated equilibrium unit cell volumes, elastic constants, and SFEs of Ni and Ni₅₈Cr₃₂Fe₁₀ with the available data. The calculated equilibrium unit cell volume (V), bulk modulus (B), shear modulus (G), and SFE of Ni are 10.97 10⁻¹⁰m³, 198.4 GPa, 102.7 GPa, and 140.9 mJm⁻², respectively. These values coincide well with the previous experimental and simulation values [17,38,39,40], as listed in Table 1.

The calculated values of equilibrium unit cell volume (V), bulk modulus (B), shear modulus (G), and SFE of Ni₅₈Cr₃₂Fe₁₀ are 11.59 10⁻¹⁰m³, 180.1 GPa, 82.1 GPa, and 20.5 mJm⁻², respectively. A detailed simulation process of the elastic constants can be found in previous work [37]. The calculated values of shear modulus (G) and equilibrium cell volume (V) coincide well with the experimental results tested from alloy with a similar composition. The calculated SFE at 0 and 298 K is 20.5 and 21.2 mJm⁻², respectively (the influence of magnetic entropy on the SFEs will be discussed in Section 3.3). The difference between the calculated SFE and experimental SFE (61.0 mJm⁻²) published by Jimy et al. is obvious [13]. It should be noted that Jimy et al. reported that the addition of 0.56 at.% Ti reduces the SFE from 61.0 to 20.8 mJm⁻². The reduction scale is ~40 mJm⁻². Zhang et al. simulated the SFE of Ni-based binary alloy. They reported that the addition of 1.4 at.% Ti (the solute construction on the doped plane is 8.3 at.%) only achieves a reduction of ~19 mJm⁻² [16]. We suggest that the SFE (61.0 mJm⁻²) of Alloy 690 reported by Jimy et al. might be overestimated, because the experimental values of SFE might be influenced by residual stress, which arises from the sample preparation process [15].

3.2. SFEs at 0 K

As shown in Figure 1a, the SFE⁰ for Ni₅₈Cr₃₂Fe₁₀ is 20.5 mJm⁻². The addition of Mn and W leaves the SFE⁰ almost unchanged. Al and Cu increase the SFE⁰ at scales of 1.4 and 1.3 mJm⁻², respectively. All other elements decrease the SFE and these elements can be ordered in terms of SFE decreasing scales: Hf (9.0 mJm⁻²) > Ta (5.2 mJm⁻²) > Mo (2 mJm⁻²) = Nb (2 mJm⁻²) > Co (1.9 mJm⁻²) > Ti (1.6 mJm⁻²) > V (0.9 mJm⁻²).

Previously, Shang et al. simulated the SFEs of Ni-based binary alloy using supercells Ni₇₁X [16]. Shang et al. suggested that the SFEs of Ni₇₁X decrease about linearly with an increasing equilibrium volume of Ni₇₁X. As shown in Figure 1b, the addition of Hf causes the most significant increase in equilibrium volume and the most apparent decrease in SFE. However, the addition of Co decreases the equilibrium volume and SFE simultaneously, and the addition of Al and Cu increase the equilibrium volume and SFE simultaneously. Thus, we suggest that, in Ni₅₆Cr₃₂Fe₁₀X₂, the alloying effects on the SFEs are determined not only by the equilibrium volume, but also by the interaction between electrons. There is another discrepancy between Ni-based binary alloys and Ni₅₆Cr₃₂Fe₁₀X₂. Shang et al. reported that all alloying elements (Al, Co, Cr, Cu, Fe, Hf, Ir, Mn, Mo, Nb, Os, Pd, Pt, Re, Rh, Ru, Sc, Si, Ta, Tc, Ti, V, W, Y, Zn, and Zr) considered in their work decrease the SFE of Ni. However, our results reveal that Al and Cu increase the SFE of Ni₅₆Cr₃₂Fe₁₀X₂. Lu et al. demonstrated that the alloying effects on SFE are associated with the initial composition of the matrix [20].

3.3. SFEs at Finite Temperature

At a finite temperature, the SFE in the Ni–Fe–Cr alloys could be roughly divided into the magnetic entropy (SFE^mag) and 0 K (SFE⁰) contributions [20,21]. The thermal lattice expansion has been demonstrated as the main reason influencing the temperature dependence of SFE^mag. It has been demonstrated that the contributions of electronic entropy and phonon were demonstrated to be relatively small. Although electronic entropy and phonon do influence the SFEs, the computational errors can be cancelled to some extent owing to the same group of cells being used [24]. For all of the systems considered in this study, we recalculated the SFE⁰ with thermal expanded cells. The influence of thermal expansion on the SFE⁰ is very small (for all of the systems, the influence is smaller than 0.3 mJm⁻²). Although thermal expansions do increase the energy of cells, the energy difference between fcc, hcp, and dhcp structures remained almost unchanged. Thus, only the SFEs⁰ at 0 K are listed in Table 2 and used in the following discussions. On the basis of Equation (2), one can deduce that the SFE^mag is larger in alloys where the local magnetic moments from the fcc (μ_fcc), hcp (μ_hcp), and dhcp (μ_dhcp) structures exhibit a significant difference.

In Ni₅₈Cr₃₂Fe₁₀ alloy, the local magnetic moments mainly develop on the Fe atom at 298 K. As listed in

Table 3. Local magnetic moments (μ_B) on atom Fe and SFE^mag (mJm⁻²) in Ni₅₈Cr₃₂Fe₁₀ at 298 K.

Alloy	μfcc	μhcp	μdhcp	SFEmag
Ni58Cr32Fe10	2.04	1.94	2.01	0.7
Fe70.5Cr17.5Ni12 a	1.62	0.00	–	36.2

^a First principle data at 300 K [21].

, the local magnetic moments (on the Fe atom) in the fcc, hcp, and dhcp structures are 2.04, 1.94, and 2.01 μ_B, respectively. The difference of these magnetic moments is so insignificant that the SFE^mag is very small, only 0.7 mJm⁻². Meanwhile, Vitos et al. reported that the local magnetic moments (on the Fe atom) in the fcc and hcp structures are 1.62 and 0.0 μ_B, respectively. This significant difference results in a considerable value of SFE^mag (36.2 mJm⁻²) in their study.

In Figure 2, the SFEs (sum of chemical and magnetic contributions) at 298 K are exhibited with red rectangles, and the SFE value of Ni₅₈Cr₃₂Fe₁₀ is marked out with a red line. At 298 K, the SFEs^mag are so small (<1 mJm⁻²) in all of the alloys considered in this paper that the SFEs exhibit almost the same trend as the SFEs⁰. The addition of Mn and W leaves the SFE almost unchanged, while the addition of Al and Cu increases the SFE at scales of 1.3 and 1.1 mJm⁻², respectively. All other elements decrease the SFE and these elements can be ordered in terms of SFE decreasing scales: Hf (9.2mJm⁻²) > Ta (5.4 mJm⁻²) > Mo (2.2 mJm⁻²) = Nb (2.2 mJm⁻²) > Co (2 mJm⁻²) > Ti > (1.9 mJm⁻²) > V (1.1mJm⁻²). Experiments at room temperature demonstrated that Hf, Mo, Nb, Ti, and V could decrease the SFEs of Alloy 690 [13].

The SFEs at 673 K are exhibited with blue rectangles and the SFE value of Ni₅₈Cr₃₂Fe₁₀ is marked out with a blue line, in Figure 2. The addition of Mn and W leaves the SFE almost unchanged, while the addition of Al and Cu increases the SFE at scales of 1.0 and 0.6 mJm⁻², respectively. Other elements can be ordered in terms of SFE decreasing scales: Hf (7.8 mJm⁻²) > Ta (4.8mJm⁻²) > Mo (2.2 mJm⁻²) > Ti (2.0 mJm⁻²) > Co (1.9 mJm⁻²) > Nb (1.4 mJm⁻²) > V (1.3 mJm⁻²).

The SFEs at 973 K are exhibited with green rectangles and the SFE value of Ni₅₈Cr₃₂Fe₁₀ is marked out with a green line, in Figure 2. All of the elements considered in this paper decrease the SFE and these elements can be ordered in terms of SFE decreasing scales: Hf (5.3 mJm⁻²) > Ta (4.8 mJm⁻²) > Mo (3.7 mJm⁻²) > Ti (3.3 mJm⁻²) > Co (3.2 mJm⁻²) > V (2.5 mJm⁻²) > W (1.5 mJm⁻²) > Nb (1.0 mJm⁻²) > Mn (0.9 mJm⁻²) > Al (0.6 mJm⁻²) > Cu (0.1 mJm⁻²).

The SFEs at 1173 K are exhibited with purple rectangles and the SFE value of Ni₅₈Cr₃₂Fe₁₀ is marked out with a purple line, in Figure 2. The addition of Ta and Ti leaves the SFE unchanged, while the addition of Co and V decreases the SFE at scales of 3.0 and 2.3 mJm⁻², respectively. Other elements can be ordered in terms of SFE increasing scales: Nb (7.7 mJm⁻²) > Cu (6.9 mJm⁻²) > Al (3.6 mJm⁻²) > Hf (2.5 mJm⁻²) > Mo (2.0 mJm⁻²) > Mn (1.4 mJm⁻²) > W (0.7 mJm⁻²).

In Figure 3, Ni₅₆Cr₃₂Fe₁₀Hf₂ was chosen as an example, as the addition of Hf causes the most significant SFE reduction in the temperature range from 0 to 973 K, but causes a significant SFE increase at 1173 K. Previous study on austenitic stainless steels revealed that SFE^mag exhibits a linear feature in the temperature range from 0 to 400 K [20]. Our results show that, in Ni₅₈Cr₃₂Fe₁₀, SFE^mag exhibits an almost linear feature in the temperature range from 0 to 1173 K, as shown in Figure 3a. However, SFE^mag in Ni₅₆Cr₃₂Fe₁₀Hf₂ only exhibits a linear feature in the temperature range from 0 to 673 K. Considering that SFE^mag is a local magnetic moment-related parameter, we plotted the local magnetic moments of Ni₅₈Cr₃₂Fe₁₀ and Ni₅₆Cr₃₂Fe₁₀Hf₂, as shown in Figure 3b.

In Ni₅₈Cr₃₂Fe₁₀, the local magnetic moments mainly develop on the Fe atom in the temperature range from 0 to 1173 K. However, in Ni₅₆Cr₃₂Fe₁₀Hf₂, the local magnetic moments also develop on the Cr atom in the temperature range from 673 to 1173 K. We suggest that the nonlinear feature of SFE^mag could be caused by local magnetic moments on Cr atoms. It should be noted that DLM adopts a static picture, meaning that no thermal spin fluctuations are taken into account. However, it is suggested that the computational errors could be cancelled to some extent owing to the same group of cells being used.

During a deformation process, low SFE materials could achieve a more homogeneous distribution of dislocations and facilitate more homogeneous deformed microstructures compared with high SFE materials. The homogeneous deformation process means better DDC resistance [11,12,13,14]. On the basis of our simulation, the addition of Nb significantly decreases the SFE at room temperature. Thus, one might deduce that, at room temperature, the addition of Nb could increase the DDC resistance of Ni₅₈Cr₃₂Fe₁₀-based alloys. However, room temperature experiments reveal that the improvement of DDC resistance only exists when the Nb concentration is bigger than ~1.48 at.% [6], because lower Nb concentration means more M₂₃C₆ exists in the alloy. As a consequence, micro-cracks could generate aside the M₂₃C₆, which increases the susceptibility of DDC. Besides the SFE, many other factors (such as grain size, grain boundary migration, grain boundary segregation, precipitation behavior, chemical composition, and so forth [1,2,3,4,5,6]) should be taken into account to increase the DDC resistance of Ni₅₈Cr₃₂Fe₁₀-based alloys or filler materials. It should be pointed out that decreasing the SFE may not increase the global ductility (or macro ductility), but could potentially lead to a more homogeneous distribution of dislocations.

4. Conclusions

In summary, the effects of alloying elements on the stacking fault energies of Ni₅₈Cr₃₂Fe₁₀ alloys were studied using first-principle calculations. In these simulations, 2 at.% of Ni is replaced by alloy element X (X = Al, Co, Cu, Hf, Mn, Nb, Ta, Ti, V, and W). At finite temperature, the SFEs were divided into the magnetic entropy (SFE^mag) and 0 K (SFE⁰) contributions. The results reveal that only Co and V decrease the SFE in the temperature range from 0 to 1173 K. Hf, Mo, and Nb decrease the SFE in the temperature range from 0 to 973 K, but increase the SFE at 1173 K. Ta and Ti decrease the SFE in the temperature range from 0 to 973 K, but keep the SFE unchanged at 1173 K.