The Effects of Co and W on Structural Stability and Mechanical Properties of Austenitic Heat-Resistant Steel Sanicro 25: A First-Principle Study

Abstract

Sanicro 25 austenitic heat-resistant steel is expected to be used in superheaters and reheaters for ultra-supercritical power plants above 600 °C due to its excellent structural stability and high temperature mechanical properties. In this paper, the effects of Co and W on the structural stability, thermodynamic stability and mechanical properties of Sanicro 25 steel are analyzed by calculating the formation energy, binding energy, Gibbs free energy, elastic constant, Peierls stress and generalized stacking fault energy (GSFE) with first-principles calculation method. By calculating the formation energy, binding energy and Gibbs free energy, it concludes that alloying elements Co and W in Sanicro 25 steel can improve the structural stability and thermodynamic stability. It indicates that W and a small amount of Co can improve the plasticity and ductility of Sanicro 25 steel by calculating the bulk modulus (B), shear modulus (G), Young’s modulus (E), the B/G ratio, Poisson’s ratio and Peierls stress. It is found that when Co and W are far from the stacking fault region, it will promote the formation of partial dislocations and twins in the system, thereby improving its plastic deformation ability and mechanical properties.

1. Introduction

As a highly efficient and clean power generation technology, ultra-supercritical technology is the key development direction of coal-fired thermal power stations [1]. In order to improve efficiency and reduce coal consumption, the 630–650 °C ultra-supercritical unit is the next important objective of the thermal power construction. Austenitic heat-resistant steels such as HR3C, which are used in 600 °C ultra-supercritical units, are subjected to carbide precipitation and coarsening when operating under higher steam parameters. The structural stability is deteriorated, and the high-temperature mechanical properties and its service life will also be seriously affected [2]. New materials that can withstand high temperature loads for a long time need to be developed in order to improve efficiency of the thermal power plants [3].

Sanicro 25 steel is the main candidate material for austenitic heat-resistant steel of 630–650 °C ultra-supercritical boilers due to its high creep strength, oxidation resistance, structural stability and good processing ability [4]. By comparing the chemical composition and mechanical properties of HR3C [5] and Sanicro 25 [6] steel, it can be found that alloying elements such as Co and W play an important role in structural stability and high temperature mechanical properties. The creep rupture strength of Sanicro 25 steel is 45% higher than HR3C, and the corrosion resistance to high temperature steam of Sanicro 25 steel is better than HR3C [7,8]. Moreover, Zhao [9] studied the effects of W content on the microstructures and properties of Sanicro 25 steel, and concluded that the higher W content exhibits excellent thermal deformation resistance and deformation activation energy. Jang [10,11] studied the effect of W addition on the high temperature properties and microstructures of austenitic heat-resistant steels. W increased the volume fraction of the Laves phase and enhanced precipitation strengthening, thereby improving its mechanical properties. Furthermore, the addition of alloying elements affects deformation degree of the metal. An improved HR3C steel with 5 wt.% Co, V and B has been developed, which has good impact toughness and fracture strength. The main reason is that the addition of Co can reduce the stacking fault energy of steels and promote the formation of twins, thus improving the mechanical properties of steels [12]. In fact, it can be found that there are indeed dislocations and twins in Sanicro 25 steel [7,13,14]. The stacking fault energy is an important parameter for the plastic deformation of the material. By reducing the stacking fault energy of the material, the plastic deformation ability of the material can be improved, thereby improving the strength and processing performance of the material. It has been studied to calculate the stacking fault energy of high-entropy alloys and nickel-based alloys by first principle, and it is found that Co can significantly reduce the stacking fault energy and improve its ductility and plastic deformation ability [15,16]. Similarly, the stacking fault energy of Co and W containing Sanicro 25 steel can be calculated by first principle calculation, and then the plastic deformation mechanism can be analyzed.

At present, the effects of alloying elements on the high temperature mechanical properties of austenitic heat-resistant steel is studied mainly from the phase of precipitation by experimental methods. However, the mechanism of Co and W impact on the structural stability and mechanical properties of Sanicro 25 steel only by experimental methods is very limited. The knowledge on how Co and W, single or common, affect the stability and mechanical properties of Sanicro 25 steel is lacking and has been in dispute; thus, more detailed analyses at the atomic-scale are still required.

In this work, we employed first-principles method to investigate the effects of Co and W addition on the structural stability and mechanical properties of Sanicro 25 austenitic heat-resistant steel. Firstly, the effects of Co and W on the structural stability of Sanicro 25 were analyzed by the formation energy, binding energy and Gibbs free energy. Secondly, the effects of alloying elements Co and W on the mechanical properties of Sanicro 25 were discussed by means of elastic modulus, Poisson’s ratio, B/G ratio, Peierls stress and stacking fault energy. Finally, we simply compared the calculated results with the experimental results and found that the calculated results were basically consistent with the experimental results.

2. Computational Details

2.1. Models

At temperatures around 1200 °C, Co and W play the role of solid solution strengthening in Sanicro 25 [17]. In order to study the solid solution strengthening of Co and W in Sanicro 25 steel, we built the calculation models based on the sample after solution treatment. The research system of this paper is Sanicro 25 steel. Therefore, γ-Fe with face-centered cubic structure was chosen as the ideal crystal. The steel mainly contains 21.5–23.5 wt.% Cr, 23.5–26.5 wt.% Ni, 1.0–2.0 wt.% Co, 2.0–4.0 wt.% W in addition to a number of other minor alloying constituents such as Nb, Cu, Si, B and N. Based on the main chemical composition of Sanicro 25 steel, considering the number of atoms and the actual calculation efficiency, 2 × 2 × 2 supercells containing 32 Fe atoms were selected as the matrix, and 16 Fe atoms were replaced by 8 Cr atoms and 8 Ni atoms respectively to form the Fe₁₆Cr₈Ni₈ system (approximately 22 wt.% Cr, 25 wt.% Ni and 53 wt.% Fe). Subsequently, a small fraction of Fe atoms was substituted by Co (3.1 wt.% and 6.3 wt.%) and W (9.8 wt.% and 19.5 wt.%), as shown in Figure 1. Wherein Fe, Cr, Ni, Co and W atoms were blue, green, pink, orange and yellow colored balls, respectively.

Fe is the close-packed face-centered cubic structure with an arrangement of …ABCABC… along the <111> direction. In order to calculate the generalized stacking fault energy (GSFE), a 9-layer Fe₃₆Cr₁₈Ni₁₈ structure was built by cleaving surface along the <111> direction. The GSFE values were calculated by shearing the perfect crystal structure along the <112> slip direction in two processes, as shown in Figure 2. The stacking fault model was created by fixing the bottom 5 layers of atoms and displacing the above 4 layers of atoms along <112> over a distance a/√6. A vacuum layer of 10⁻⁹ m was created to eliminate layer-to-layer interactions. The atomic arrangement was changed from ABCABCABC to ABCABABCA, where indicates the position of the intrinsic stacking fault. Similarly, based on the stacking fault model, the bottom 6 layers of atoms are fixed and the above 3 layers of atoms are moved a/√6 to obtain the two-layer twin model.

The GSFE curves by replacing Co and W on the basis of Fe₃₆Cr₁₈Ni₁₈ system were calculated to study the effects of Co and W on Sanicro 25 steel stacking fault energy and plastic deformation mechanism. Two positions were substituted for Co and W atoms in the Fe₃₆Cr₁₈Ni₁₈ system, one of which was far from the stacking fault region (3 layer) and the other located in stacking fault region (5 layer). The mass fraction of Co in the model is 1.4 wt.% and 2.8 wt.%, then W is 4.3 wt.% and 8.6 wt.%.

2.2. Calculation Methods

The density functional theory calculations were carried out with the plane-wave based Cambridge Serial Total Energy Package (CASTEP) code within the generalized gradient approximation (GGA) in the parametrization by John Perdew, Kieron Burke and Matthias Ernzerhof (PBE). The plane waves have been included up to a cutoff energy of 500 eV which yields converged results. As an austenitic stainless steel, Fe crystallizes in the fcc structure, so the magnetic effects were not considered and calculations were non-spinpolarized. The convergence parameters are as follows: total energy tolerance 10⁻⁵ eV/atom, force tolerance 0.03 eV/Å, stress component less than 0.03 GPa and displacement less than 10⁻⁴ nm. During geometry optimization, the Brillouin zone sampling was carried out with a 4 × 4 × 7 Monkhorst–Pack grid of k-point [2]. Using the equilibrium structure after structural optimization, the Gibbs free energy and elastic constants of different systems are calculated in the absence of possible microstructure variations. When calculating the GSFE curves, considering the calculation model and calculation accuracy, the plane wave cutoff energy value is chosen to be 330 eV. The Brillouin zone sampling was performed using a 5 × 3 × 1 Monkhorst–Pack grid of k-point [18].

3. Results and Discussions

3.1. Structural Stability

The structural constants of the above six models after structural optimization are shown in

Table 1. Calculated structural constants of different systems after optimization.

Structure	α = β = γ/°	a = b/10−10 m	c/10−10 m	V/10−30 m3
Fe16Cr8Ni8	90	6.956	7.042	340.768
Fe15Cr8Ni8Co	90	6.954	7.059	341.345
Fe14Cr8Ni8Co2	90	6.948	7.083	341.908
Fe15Cr8Ni8W	90	7.010	7.058	346.817
Fe14Cr8Ni8W2	90	7.079	7.063	353.940
Fe14Cr8Ni8CoW	90	7.003	7.082	347.330

. After Co and W are dissolved in the Fe₁₆Cr₈Ni₈ system, these systems are still tetragonal systems, but all of them have different degrees of volume expansion. Because the atomic radius of Co (1.253 × 10⁻¹² m) and the atomic radius of Fe (1.241 × 10⁻¹² m) are close, the lattice distortion of the systems are relatively small and the volume of the systems containing Co change little; while the atomic radius of W (1.37 × 10⁻¹² m) is quite different from that of Fe, the lattice distortion of the systems are relatively larger and the volume of the systems containing W increase obviously. With the increase of Co and W content, the volume expansion is more obvious. Moreover, the volume expansion of the Fe₁₄Cr₈Ni₈CoW system is larger than the Fe₁₅Cr₈Ni₈Co system and the Fe₁₅Cr₈Ni₈W system.

The degree of difficulty of dissolving Co and W in the Fe₁₆Cr₈Ni₈ system to form new systems can be measured by the formation energy, which is the energy difference before and after the formation of the system [19]. The calculation formula is as follows:

$$E_f=\frac{1}{\Sigma N_i}\left[E_{total}-\Sigma \left(N_i{E}_{atom}^i\right)\right]$$

(1)

where E_total is the total energy of each system after structural optimization, N_i is the number of i atoms (i = Fe, Cr, Ni, Co or W) in the system and $E_{atom}^i$ is the monatomic energy of atom i in its elemental state. The same calculation conditions as the structural optimization are used to calculate the monatomic energy. When the formation energy is negative, it indicates that the system is easy to form. On the contrary, the system is difficult to form when the formation energy is positive [19]. The formation energy of each system can be shown in Figure 3a. It can be seen that the formation energy values of all systems are between 0.125 and 0.20 eV/atom, indicating that all systems are relatively easy to form. When Co is dissolved into the Fe₁₆Cr₈Ni₈ system, its formation energy decreases, indicating that the system is relatively easy to form. Then the formation energy decreases with the increase of Co content, and the system is more easily formed. Conversely, when W is dissolved into the Fe₁₆Cr₈Ni₈ system, the increase of formation energy indicates that the system is not easy to form. Furthermore, with the increase of W content, the formation energy increases, indicating that the system is relatively difficult to form. Furthermore, the value of the formation energy also indicates that the Fe₁₄Cr₈Ni₈CoW system is more difficult to form than the Fe₁₅Cr₈Ni₈Co system, but it is easier to form than the Fe₁₅Cr₈Ni₈W system.

In order to analyze the structural stability of Co and W dissolved in the Fe₁₆Cr₈Ni₈ system, it is necessary to calculate the binding energy. The binding energy refers to the energy that is released upon the creation of a bound state. Therefore, the greater the absolute value of the binding energy, the better the structural stability [2,20]. The calculation formula of binding energy is as follows:

$$E_b=\frac{1}{\Sigma N_i}\left[E_{total}-\Sigma \left(N_i{E}_{iso}^i\right)\right]$$

(2)

where $E_{iso}^i$ is the energy of i atom (i = Fe, Cr, Ni, Co or W) in isolated state, which is the energy calculated by fully relaxing the i atom at the center of a simple cubic structure with a lattice constant of 10⁻⁹ m. The binding energy of each system is shown in Figure 3b. It can be seen that the binding energy values of all systems are negative, indicating that all systems have good structural stability. When Co is dissolved in the Fe₁₆Cr₈Ni₈ system, the structural stability of the system decreases with the increase of Co content. In contrast, when W is dissolved in the Fe₁₆Cr₈Ni₈ system, the increase of the absolute value of the binding energy means that the system has good structural stability. Furthermore, increasing W content can improve the structural stability of the system. Moreover, the structural stability of the Fe₁₄Cr₈Ni₈CoW system is lower than that of the Fe₁₅Cr₈Ni₈W system, but its structural stability is higher than that of the Fe₁₅Cr₈Ni₈Co system.

In order to explore the mechanism of Co and W in Sanicro 25 steel from the perspective of microscopic electrons, charge density distribution maps were drawn to analyze the electronic properties of different systems. As shown in Figure 4, the electrons are concentrated around the atoms in a spherical distribution and the electron distribution between atoms is uniform, which indicates that there are metal bonds in systems. When Co and W are dissolved in the Fe₁₆Cr₈Ni₈ system, the charge distribution around atoms change little, which shows that all systems are stable. However, through careful comparison, it can be found that when Co is dissolved in the Fe₁₆Cr₈Ni₈ system, the surrounding charge distribution is slightly reduced, and the bonds between Co and surrounding atoms are slightly worse, so the structural stability of the system is also reduced. Conversely, when W is dissolved in the Fe₁₆Cr₈Ni₈ system, the surrounding charge distribution increases, and the bonds between W and surrounding atoms are closer, so W will enhance the structural stability of the system. In addition, compared to the Fe₁₆Cr₈Ni₈ system, the charge distribution around the Fe₁₄Cr₈Ni₈CoW system is increased, so the structural stability of the Fe₁₄Cr₈Ni₈CoW system is also improved. The results of charge density analysis are consistent with those of binding energy calculation.

The Gibbs free energy curve is often used to analyze the structural stability of materials at different temperatures [21]. The smaller the Gibbs free energy value is, the higher the thermodynamic stability is [22,23]. In order to study the structural stability of Co and W dissolved in the Fe₁₆Cr₈Ni₈ system at 630–650 °C, the curve of Gibbs free energy with temperature is plotted, as shown in Figure 5. At −273 to 727 °C, the Gibbs free energy decreased with increasing temperature in all systems, that is, all systems become stable with the increase of temperature. Especially at 630–650 °C, all systems have good thermodynamic stability. When Co and W are dissolved into the Fe₁₆Cr₈Ni₈ system, the Gibbs free energy decreases more sharply and the thermodynamic stability is stronger. Moreover, it can also be found that W can stabilize the Fe₁₆Cr₈Ni₈ system more than Co. Figure 3b, Figure 4 and Figure 5 consistently illustrate that Co and W can be stably present in the Fe-Cr-Ni system.

3.2. Elastic Constants Calculation

In addition to structural stability and thermodynamic stability, mechanical stability is also an important criterion to estimate whether the crystal structure is stable or not. The elastic constants reflect the ability of the crystal material to cope with external strain and are closely related to the mechanical properties of the material [24]. Using the equilibrium structure after structural optimization, the elastic constants and modulus of different systems are calculated through the stress–strain method based on a generalized Hooke’s Law [25]: σ_i = C_ijε_j, where σ_i is the tensile stress and ε_j is the longitudinal strain. In our calculations, the elastic constants C_ij can be directly obtained from the results after calculating the “Elastic Constants” in CASTEP code. The criteria of mechanical stability can be evaluated by single-crystal elastic constants according to the Born–Huang criterion, which is dependent on the crystal system [26]. The systems studied in this paper are tetragonal crystal systems with six independent elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, C₄₄ and C₆₆) and the elastic constants calculated by the six models are shown in

Table 2. Calculated single-crystal elastic constants for different systems.

Elastic Constants	C11/GPa	C12/GPa	C13/GPa	C33/GPa	C44/GPa	C66/GPa
Fe16Cr8Ni8	379.505	222.616	190.723	429.976	199.253	207.725
Fe15Cr8Ni8Co	399.361	219.855	201.078	431.200	193.911	208.585
Fe14Cr8Ni8Co2	392.879	216.273	192.282	407.783	193.120	206.080
Fe15Cr8Ni8W	400.161	217.589	208.935	406.525	190.279	206.765
Fe14Cr8Ni8W2	410.875	242.675	246.390	366.543	182.459	193.174
Fe14Cr8Ni8CoW	399.863	226.603	214.530	383.379	187.364	204.087

. The condition for the stable mechanical properties of the tetragonal system is C₁₁ > 0, C₃₃ > 0, C₄₄ > 0, C₆₆ > 0, C₁₁ – C₁₂ > 0, C₁₁ + C₃₃ – 2C₁₂ > 0 and 2(C₁₁ + C₁₂) + C₃₃ + 4C₁₃ > 0 [24]. It can be found by calculation that all systems satisfy the mechanical stability criterion of tetragonal system, and the mechanical properties of these systems are stable.

Many elastic properties of crystals can also be obtained by calculating elastic constants, such as bulk modulus B, shear modulus G, Young’s modulus E and Poisson’s ratio $\nu$, et cetera. The elastic modulus of the polycrystalline system can be obtained using the Voight–Reuss–Hill average scheme, the calculation formulas of B, G, E and $\nu$ are as follows [27,28]:

$$B=\frac{1}{9}\left(C_{11}+C_{22}+C_{33}\right)+\frac{2}{9}\left(C_{12}+C_{13}+C_{23}\right)$$

(3)

$$G=\frac{1}{15}\left(C_{11}+C_{22}+C_{33}-C_{12}-C_{13}-C_{23}\right)+\frac{1}{5}\left(C_{44}+C_{55}+C_{66}\right)$$

(4)

$$E=\frac{9GB}{G+3B}$$

(5)

$$\nu =\frac{3B-2G}{2\left(G+3B\right)}$$

(6)

The bulk modulus B, shear modulus G and Young’s modulus E of the different systems obtained by calculation are shown in Figure 6. The bulk modulus characterizes the ability of the material to resist deformation under the applied stress, and the greater the value of the bulk modulus, the stronger the resistance to deformation. In general, the large value of bulk modulus indicates that the material has high compressive strength [29]. Shear modulus is the ratio of shear stress to shear strain that characterizes the material’s ability to resist shear strain. The larger the values of the shear modulus, the stronger the shear strain resistance. Young’s modulus is an important parameter to characterize the stiffness of a material. The smaller the value of the Young’s modulus, the lower the stiffness of the material.

It can be seen from Figure 6a that as the Co content increases, the bulk modulus first increases and then decreases, and the compressive strength also first increases and then decreases. With the increase of W content dissolved in the Fe₁₆Cr₈Ni₈ system, it can be seen that bulk modulus gradually increases, indicating that its resistance to volume deformation is improved and the compressive strength is increased. It can be seen from Figure 6b that the shear modulus decreases with the increase of Co and W content, indicating that the resistance to shear strain is decreased. In Figure 6c, as the Co content increases, the stiffness of the system increases first and then decreases. Besides, the stiffness decreases as the W content increases. Finally, compared to the Fe₁₆Cr₈Ni₈ system, the Fe₁₄Cr₈Ni₈CoW system has higher compressive strength and lower shear deformation resistance and stiffness.

Poisson’s ratio is often used to estimate the stability of the material’s crystal shear resistance. The greater the value, the better the plasticity of the material [23]. The ratio of bulk modulus to shear modulus, B/G, is often used to distinguish a material is brittle or ductile. When B/G ≥ 1.75, the ductility of the material is great and the larger the B/G, the better the ductility [30,31,32]. It can be seen from Figure 7 that when Co is dissolved in the Fe₁₆Cr₈Ni₈ system, as the Co content increases, the Poisson’s ratio and B/G value of the system first increases and then decreases, indicating that its plasticity and ductility also first increase and then decrease. It can also be found that when W is dissolved in the Fe₁₆Cr₈Ni₈ system, the increases of Poisson’s ratio and B/G means that the system have great plasticity and ductility. Furthermore, with the increase of W content, the Poisson’s ratio and B/G value increases, indicating that the plasticity and ductility of the system is better. In addition, the Fe₁₄Cr₈Ni₈CoW system has higher plasticity and ductility compared to the Fe₁₅Cr₈Ni₈Co and the Fe₁₅Cr₈Ni₈W systems. This shows that Co and W jointly improve the plasticity and ductility.

In order to study the effect of single crystal elastic anisotropy on its mechanical properties, we use the Zener’s anisotropy parameter A_Z to calculate the elastic anisotropy of different systems, the formula of A_Z is as follows [33]:

$$A_Z=\frac{2C_{44}}{C_{11}-C_{12}}$$

(7)

The calculated Zener’s anisotropy parameters of different systems are shown in Figure 8. Generally, the A_Z of isotropic materials is equal to 1. Therefore, the greater the degree of A_Z deviation, the greater the degree of elastic anisotropy in the system. When Co and W are dissolved in the Fe₁₆Cr₈Ni₈ system, the degree of elastic anisotropy decreases.

3.3. Peierls Stress Calculation

The Peierls stress can also estimate the plastic deformation ability of these systems. During the process of dislocation movement (before reaching the equilibrium position), the dislocation center will deviate from the equilibrium position to increase the crystal energy to form an energy barrier, which is the dislocation motion resistance caused by the crystal lattice, also called Peierls stress. The smaller the values of the Peierls stress, the easier the dislocations are to slip. Its formula is as follows [34]:

$${\tau }_{P-N}\approx \frac{2G}{1-\nu }{e}^{-\left[\frac{2\pi p}{\left(1-\nu \right)q}\right]}\approx \frac{2G}{1-\nu }{e}^{-\left(\frac{2\pi W}{q}\right)}$$

(8)

where p is the surface spacing of the slip surface, q is the atomic spacing in the slip direction, W = p/(1 – $\nu$) is the width of the dislocation. This paper selects the (111) and (001) crystal planes to calculate the Peierls stress of different systems, as shown in Figure 9. The slip directions of (111) and (001) crystal planes are both the easiest slip direction (110). It can be found that the calculation results of Peierls stress in different crystal planes are consistent with the code that the face-centered cubic structure preferentially slides along its (111) close-packed surface. It can be seen from Figure 8 that Co has little effect on the Peierls stress of the (111) and (001) crystal plane. On the contrary, alloying elements W dissolved in the Fe₁₆Cr₈Ni₈ system reduces Peierls stress, thereby promoting the dislocations slip. Furthermore, with the increase of W content, the Peierls stress decreases, indicating that the dislocations motion resistance of the system is reduced. Compared with the Fe₁₅Cr₈Ni₈Co and Fe₁₅Cr₈Ni₈W systems, the Fe₁₄Cr₈Ni₈CoW system has smaller dislocations motion resistance on the (111) and (001) crystal plane. This indicates that the interaction of Co and W can significantly reduce the dislocations motion resistance of the Fe₁₄Cr₈Ni₈CoW system.

3.4. Stacking Fault Energy Calculation

To further explain the effects of Co and W on plastic deformation in Sanicro 25 steel from the perspective of dislocations and twins, we analyzed the GSFE. In this paper, the GSFE value is calculated by [35]:

$$\gamma =\frac{E_u-E_0}{S}$$

(9)

where E_u is the total energy of the supercell with the fault vector (varies from 0.0 to 2.0 with a step of 0.1), E₀ is the total energy of the perfect fcc supercell and S is the area of the stacking fault region. Although the calculation is assumed to be at 0 K, the results of GSFE would reflect the effect tendency of W and Co at high temperature [36,37]. Figure 10 shows the first-principles calculated GSFE curves of different systems. The GSFE curve of the Fe₃₆Cr₁₈Ni₁₈ system is similar to the Fe₁₃Cr₅Ni₆ in reference [38]. It can be seen that Co will reduce unstable SFE (γ_usf) and when Co is far from the stacking fault region, the intrinsic SFE (γ_isf) will decrease, which promotes the system to slip in <112> direction. Furthermore, the Fe₃₅Cr₁₈Ni₁₈W and the Fe₃₄Cr₁₈Ni₁₈W₂ curves are mostly above the Fe₃₆Cr₁₈Ni₁₈ curve. When Co and W locate in the stacking fault region, the Fe₃₄Cr₁₈Ni₁₈CoW curve is above the Fe₃₆Cr₁₈Ni₁₈ curve.

Related studies showed that the GSFE values, including intrinsic SFE (γ_isf), unstable SFE (γ_usf), twin SFE (γ_tsf) and unstable twin SFE (γ_utf) all affect the plastic deformation mechanism. The large value of γ_usf/γ_isf indicates that the system is easy to form partial dislocations from full dislocations [39]. It can be found from Figure 11a that the γ_usf/γ_isf values increase when Co and W are far from the stacking fault region. It indicates that Co and W are prone to generate partial dislocations when they are far from the stacking fault region and the higher the content of Co and W, the easier it is to generate partial dislocations. When Co and W locates in the stacking fault region, the decrease of γ_usf/γ_isf value shows that all different systems tend to generate full dislocations.

Slip and twinning are two ways of plastic deformation. The way in which metal is plastically deformed is determined by ${\delta }_{\mathrm{us}}^{ut}$ (${\delta }_{\mathrm{us}}^{ut}$ ≡ γ_utf – γ_usf) [40]. If ${\delta }_{\mathrm{us}}^{ut}$ > 0, it indicates that during the plastic deformation process, after the formation of the stacking fault, it is more inclined to produce deformation by slipping; if ${\delta }_{\mathrm{us}}^{ut}$ < 0, after the formation of stacking faults, it is more likely to produce deformation by twinning. It can be seen from Figure 11b that the values of ${\delta }_{\mathrm{us}}^{ut}$ decrease when Co is far from the stacking fault region, which increases the possibility that the Fe-Cr-Ni system is deformed by twinning. When Co locates in the stacking fault region, the increase of ${\delta }_{\mathrm{us}}^{ut}$ indicates that Co can promote the ability of the Fe-Cr-Ni system to deform in a slipping manner. A decrease in the values of ${\delta }_{\mathrm{us}}^{ut}$ when W is dissolved in the Fe₃₆Cr₁₈Ni₁₈ system indicates that W promotes deformation by twinning. Additionally, the higher the Co and W content, the higher the tendency to produce deformation by twinning. When Co and W are far from the stacking fault region, the ${\delta }_{\mathrm{us}}^{ut}$ of the Fe₃₄Cr₁₈Ni₁₈CoW system is lower than the Fe₃₆Cr₁₈Ni₁₈ system, indicating that the interaction of Co and W promotes deformation by twinning. When Co and W locates in the stacking fault region, the values of ${\delta }_{\mathrm{us}}^{ut}$ increase indicating that the Fe₃₄Cr₁₈Ni₁₈CoW system is more easily deformed by slipping.

In addition, there are many criteria to estimate the tendency of twinning. Tadmor and Hai [41] found that γ_utf/γ_usf can reflect the twinning tendency. When this value is smaller, the twinning is more likely to appear in the system. Tadmor and Bernstein [42,43] used a statistical average of the twinning tendency under all orientation conditions in a polycrystalline system to define the twinning ability of a homogeneous polycrystalline material, called crack tip twinning criterion:

$${\tau }_a=\left[1.136-0.151\frac{{\gamma }_{isf}}{{\gamma }_{usf}}\right]\sqrt{\frac{{\gamma }_{usf}}{{\gamma }_{utf}}}$$

(10)

τ_a is the tendency of the material to undergo deformation twinning. When τ_a < 1, the deformation is dominated by dislocation slipping; when τ_a > 1, the deformation is dominated by twinning, and the larger the τ_a, the stronger the twinning ability.

Another criterion is the grain boundary twinning criterion deduced by Asaro and Suresh [44], denoted by τ. The formula is as follows:

$$\tau =\sqrt{\left(1+2\beta \right)\frac{{\gamma }_{usf}}{{\gamma }_{utf}}},\beta =1-{\gamma }_{isf}/{\gamma }_{usf}$$

(11)

A larger τ indicates an obvious tendency to twinning. However, since crack tips or grain boundaries are assumed, criteria τ_a and τ cannot be used to reveal the inherent twinnabilities of perfect fcc metals. The inherent twinnabilities are obtained by the following formula [45]:

$$P=\frac{6}{\pi }arc tan[\frac{2\left({\gamma }_{usf}-{\gamma }_{isf}\right)}{\sqrt{3}\left({\gamma }_{utf}-{\gamma }_{isf}\right)}-\frac{1}{\sqrt{3}}]$$

(12)

A larger P value will promote the appearances of twins. All in all, these criteria show a similar situation. It can also be seen from Figure 11b,c that the values of ${\delta }_{\mathrm{us}}^{ut}$ and γ_utf/γ_usf decrease when alloying elements are far from the stacking fault region. The values of τ_a, τ and P increase when Co and W are far from the stacking fault region, as shown in Figure 11d–f. It indicates that all systems are more likely to twinning when alloying elements Co and W are far from the stacking fault region. When W locates in the stacking fault region, the Fe₃₅Cr₁₈Ni₁₈W and the Fe₃₄Cr₁₈Ni₁₈W₂ systems may be more susceptible to twinning. Conversely, when Co locates in the stacking fault region, the Fe₃₅Cr₁₈Ni₁₈Co system and the Fe₃₄Cr₁₈Ni₁₈Co₂ system have little tendency to twinning. Furthermore, the tendency of the Fe₃₄Cr₁₈Ni₁₈CoW system to deform is similar to the Fe₃₅Cr₁₈Ni₁₈Co system when Co and W locate in the stacking fault region. It is noted that the calculated twinning abilities depends on the GSFE values, and the GSFE are directly linked with the distribution among the atoms. In conclusion, when Co and W are far from the stacking fault region, the GSFE values indicate that alloying with Co and W atoms are expected to bring a significant increase in the tendency for formation of partial dislocations and mechanical twinning. Plastic deformation by twinning will increase the amount of plastic deformation and promote grain refinement, so that the plasticity of the material can be increased [46]. Therefore, W can improve the plasticity of the system, which is consistent with the conclusions drawn in Figure 7.

3.5. Experimental Results

In order to verify the accuracy of the calculation results, we conducted a room temperature tensile test on Sanicro 25 steel with different W contents (1.5 wt.% and 2.5 wt.%). The chemical composition of these two steels is shown in

Table 3. Chemical compositions in weight % (wt.%) of two Sanicro 25 steels.

Steels	Cr	Ni	C	Si	Mn	W	Co	Cu	Nb	P	S	N	Fe
Sanicro 25-1.5 W	21.66	24.54	0.065	0.27	0.74	2.31	4.62	2.84	0.45	0.01	0.003	0.14	Bal.
Sanicro 25-2.5 W	21.47	24.36	0.060	0.25	0.76	1.42	4.62	2.74	0.44	0.01	0.002	0.13	Bal.

. The mechanical tensile tests performed were according to the GB/T228.1-2010 standard. The size of test samples is shown in Figure 12. The dog-bone shaped specimens with a gauge length of 23 mm, thickness of 2 mm and width of 6 mm were cut from the Sanicro 25 steels after solution-treated at 1220 °C for 1 h using a wire-cut electrical discharge machine (DK7720, Huadong Automation Co., Ltd., Taizhou, China), and then the surface was polished to a mirror-like finish. The tensile tests were carried out on an Instron 5582 testing machine (Instron, Boston, MA, USA) at a strain rate of 2 × 10⁻⁴ s⁻¹. Moreover, three identical specimens were measured to ensure accurate experimental data were obtained. Figure 13 shows the result of tensile testing at room temperature of two different W content Sanicro 25 steels. It can be seen from the stress–strain curve that increasing the W content increases its tensile strength, yield strength and plasticity. The calculation results of bulk modulus show that W will increase the compressive strength of the system, and the calculation results of Poisson’s ratio and generalized stacking fault energy show that W will increase the plastic deformation ability of the system, which is consistent with the experimental stress–strain curve results and our previous work [47]. Although the calculation model systems are highly anisotropic, while the experimental samples after solution treatment are isotropic, the results of solid solution strengthening effects of Co and W in Sanicro 25 steel neglecting isotropic effects were of better credibility.

4. Conclusions

In this work, we have revealed the effects of Co and W on structural stability and mechanical properties of Sanicro 25 austenitic heat-resistant steel through first principles calculations. The results of the present study are summarized as follows:

Based on the forming energy, the binding energy, charge density distribution maps and the Gibbs free energy, it can be concluded that Co and W can improve the structural stability and thermodynamic stability of Sanicro 25, that is, Co and W are stably present in Sanicro 25 steel. It is obvious that W has a great influence on the structural stability and thermodynamic stability of Sanicro 25 steel. We also calculate the elastic modulus, B/G and Poisson’s ratio, and we find that W and a small amount of Co can improve the plasticity and ductility of Sanicro 25 steel. Furthermore, Peierls stress and evaluation of the GSFE values based on γ_usf/γ_isf, ${\delta }_{\mathrm{us}}^{ut}$, γ_utf/γ_usf, τ_a, τ and P, concludes that the dominant deformation mechanisms of Sanicro 25 are extended partial dislocations and twinning. Moreover, the position of Co in the system has a great influence on the formation of partial dislocations and twinning. These results imply that alloying with Co and W atoms are expected to bring a significant increase in the tendency for formation of partial dislocations and twinning when Co and W are far from the stacking fault region. At two positions, W can significantly increase the possibility that the Fe-Cr-Ni system is deformed by twinning, thereby improving its plastic deformation ability. In short, W can significantly improve the structural stability and mechanical properties of Sanicro 25 compared to Co. Finally, combining experiment and calculation results, it is found that W can improve the strength and plasticity of the system.