Phase Stability and Mechanical Properties Analysis of AlCo_xCrFeNi HEAs Based on First Principles

Abstract

With the in-depth research on high-entropy alloys (HEAs), most of the current research uses experimental methods to verify the effects of the main elements of HEAs on the mechanical properties of the alloys. However, this is limited by the long experimental period and the influence of many external factors. The computer simulation method can not only effectively save costs and shorten the test cycle, but also help to discover new materials and broaden the field of materials. Therefore, in this paper, the physical properties (such as lattice constant, density and elastic constant) of AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEAs were calculated based on the first-principles calculation method and virtual crystal approximate modeling method. It is found that AlCo_xCrFeNi HEAs have the best hardness and toughness properties, with a Co content of 0.5~0.7. The research results can provide theoretical guidance for the preparation of HEAs with optimal mechanical properties.

1. Introduction

High-entropy alloys (HEAs) are one of the hotspots in materials research in recent years [1]. They have excellent properties, such as high temperature resistance, corrosion resistance and radiation resistance [2,3], and broad application prospects in the fields of tools, molds and turbine blades [4,5,6]. As one of the most common types of HEAs, the AlCoCrFeNi HEA not only has excellent mechanical properties at room temperature, but also has little performance degradation in high-temperature environments. It is currently used in many fields such as cemented carbide and coated tools [7,8]. The current study found that the phase structure and microstructure of superalloys can be changed by adjusting the content of constituent elements [9,10]. This will in turn affect the mechanical properties of HEAs. On the one hand, the Co content affects the formation of the face-centered cubic (FCC) phase in the AlCo_xCrFeNi HEA. When the Co content is 0, Fe and Cr in HEAs can only generate disordered body-centered cubic (BCC) phase and it is difficult to generate stable FCC phase [11]. This in turn affects mechanical properties such as hardness and strength of the alloy. Therefore, an appropriate amount of Co content can significantly improve the mechanical properties of AlCo_xCrFeNi HEAs. On the other hand, since Co is a scarce resource in nature, it is widely used in metallurgy, the military industry, and the battery and catalysis industries. Therefore, saving Co has become an inevitable resource problem. However, the phase structure and mechanical properties of AlCoCrFerNi HEAs are mainly discussed from the aspects of material composition and preparation technology [12,13]. This experimental verification method has high cost and a long cycle. Using the first-principles calculation method, the ground state properties of the material system can be obtained by solving the Schrödinger equation through density functional theory [14,15,16]. Using this method can not only save costs, but also effectively shorten the test cycle. It greatly increases the discovery speed of new materials and broadens the research scope of existing materials.

In conclusion, the first-principles method was used to analyze the effect of Co content on the phase structure and mechanical properties of AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEAs. The virtual crystal approximation method used in this paper can ensure the calculation accuracy and improve the calculation efficiency. It not only fills the gap in the research field of HEAs, but also has important guiding significance for practical production and application.

2. Materials and Methods

In this study, cambridge sequential total energy package (CASTEP) module in Materials Studio software was adopted, which was developed by Cambridge University in England based on first principles [17]. In Figure 1, the AlCo_xCrFeNi HEA models (FCC, BCC and HCP phase) were established by the virtual crystal approximation (VCA) method, and the AlCoCrFeNi HEA model (FCC phase) was established by the special quasi-random structures (SQS) method. The VCA method can effectively avoid the virtual atom or accumulated error caused by long-range structure in the modeling process. The exchange correlation functional adopted the Perdue–Burke–Ernzerhoff (PBE) functional in the generalized gradient approximation (GGA) [18]. The VCA models were calculated by ultrasoft pseudopotential, while the SQS model was calculated by norm-conserving pseudopotential [19,20,21]. The calculated parameters were obtained by convergence test in the process of geometric optimization. The convergence parameters were set as follows: the total energy tolerance is 10⁻⁵ eV/atom, the force tolerance is 0.03 eV/Å, the maximum stress is 0.05 GPa, and the maximum displacement is 0.001 Å. The cutoff energy of VCA is 630 eV, and that of the SQS model is 540 eV. The values of k points in Brillouin interval are 9 × 9 × 9 and 6 × 3 × 3, and the density parameter is 0.04 1/Å. The initial spin value in spin polarization is 5. After geometric optimization, the calculated lattice constant of the VCA model of the AlCoCrFeNi HEA (FCC phase) is 3.541 Å, while that of the SQS model is 3.54 Å. However, the calculated values are not much different from the lattice constant (3.591 Å) of the AlCoCrFeNi HEA prepared by Y. Garip by the RS method [22]. It can be proved that it is reliable to use the VCA method to study the AlCo_xCrFeNi HEA. Therefore, the calculation parameters in this paper are set reasonably.

3. Results

3.1. Phase Stability

At present, in addition to the traditional XRD pattern to judge the alloy phase structure, the AlCo_xCrFeNi (x= 0, 0.25, 0.5, 0.75, 1) HEAs can also be judged according to the empirical formula including mixed entropy ΔS_mix, mixed enthalpy ΔH_mix, valence electron concentration VEC, atomic radius difference δ, electronegativity difference Δχ and parameter Ω. Based on the obtained empirical parameters, it can provide guidance for the composition design and performance optimization of high-entropy alloys. Thus, blind design of high-entropy alloys is avoided, and the waste of experimental materials is reduced to a certain extent.

The mixed entropy ΔS_mix, mixed enthalpy ΔH_mix, valence electron concentration VEC, atomic radius difference δ, electronegativity difference Δχ and parameter Ω are calculated as follows [23,24,25,26,27]:

$$\Delta S_{\mathit{mix}}=-R{\displaystyle \sum }_{i=1,i\ne j}^n{c}_i{\mathit{lnc}}_i$$

(1)

$$\Delta H_{\mathit{mix}}={\displaystyle \sum }_{i=1,i\ne j}^{n}4 \Delta H_{\mathit{ij}}^{\mathit{mix}}{c}_i{c}_j$$

(2)

$${\mathrm{VEC}}_{}={\displaystyle \sum }_{i=1}^n{c}_i{\left(\mathit{VEC}\right)}_i$$

(3)

$$\delta =\sqrt{{\displaystyle \sum }_{i=1}^n{c}_i{\left({1 -r}_i/\overline{r}\right)}^2}, \overline{r}={\displaystyle \sum }_{i=1}^n{c}_i{r}_i$$

(4)

$$\Delta \chi =\sqrt{{\displaystyle \sum }_{i=1}^n{c}_i{\left({\chi }_i - \overline{\chi }\right)}^2}, \overline{\chi }={\displaystyle \sum }_{i=1}^n{c}_i{\chi }_i$$

(5)

$$\Omega =\frac{T_m\Delta S_{\mathit{mix}}}{\left|\Delta H_{\mathbf{mix}}\right|} , T_m={\displaystyle \sum }_{i=1}^n{c}_i{{(T}_m)}_i$$

(6)

where R is the molar gas constant, C_i and (VEC)_i are the atomic ratio and the number of valence electrons of the i-th component, $\overline{r}$ is the average atomic radius of the HEAs, c_i and r_i are the atomic ratio and atomic radius of the i-th component, $\overline{\chi }$ is the average Pauling electronegativity of HEA, c_i and χ_i are the atomic ratio and Pauling electronegativity of the i-th component, and (T_m)_i is the melting temperature of the i-th component.

The structure rules of alloy phase were predicted by VEC: VEC < 6.87 is the BCC phase; 6.87 < VEC < 8 is the BCC+FCC phase; VEC > 8 is the FCC phase. From

Table 1. The mixed entropy ΔS_mix, mixed enthalpy ΔH_mix, valence electron concentration VEC, atomic radius difference δ, electronegativity difference Δχ, melting point T_m and parameter Ω of AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEA.

Alloys	ΔSmix /[J/(K·mol)]	ΔHmix /(kJ/mol)	VEC	Δ /%	Δχ	Tm /K	Ω
AlCrFeNi	11.56	−13.24	6.75	6.27	0.122	1035.60	0.90
AlCo0.25CrFeNi	12.72	−13.07	6.88	6.14	0.119	1046.55	1.02
AlCo0.5 CrFeNi	13.14	−12.84	7.00	6.01	0.122	1056.31	1.08
AlCo0.75 CrFeNi	13.30	−12.59	7.11	5.89	0.121	1065.03	1.12
AlCoCrFeNi	13.39	−12.32	7.20	5.78	0.121	1072.85	1.17

, VEC > 6.87 indicates that the addition of Co will promote the formation of FCC phase in AlCo_xCrFeNi HEAs [23]. Usually, δ ≤ 6.6 % is considered to form a stable solid solution structure. From Table 1, δ ≤ 6.6 % indicates that AlCo_xCrFeNi HEAs can form a stable solid solution structure and the degree of alloy lattice distortion is gradually reduced [24]. Electronegativity is defined as the ability of an atom to gain or lose electrons [25]. The greater the electronegativity of the element, the stronger the ability of the element to gain and lose electrons, and the stronger its non-metallic properties. Conversely, elements have strong ductility and metallic properties [26]. From Table 1, the small change in Δχ indicates that the ductilities of the HEAs are not very different. Parameter Ω is obtained by thermodynamic calculation of mixing entropy, mixing enthalpy and melting point. When Ω > 1, it can be used to judge the phase stability in the alloy [27]. From Table 1, it is found that the phase structure of the alloy will gradually become stable with the increase in Co content.

In addition, the relative stability of phases in HEAs can be judged by thermodynamics. The CASTEP module is used to calculate the crystal energy of AlCo_xCrFeNi HEA models with different phase structures (FCC, BCC and HCP phase). The stability of the crystal phase structure can be judged by the energy level. In

Table 2. The crystal energies of different phase structures (BCC, FCC and HCP phases) of AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEAs were modeled by VCA method.

Alloys	Energy/eV
	BCC	FCC	HCP
AlCrFeNi	−2229.51	−2229.54	−2229.53
AlCo0.25CrFeNi	−2259.47	−2259.54	−22593.52
AlCo0.5 CrFeNi	−2272.61	−2272.67	−2272.64
AlCo0.75 CrFeNi	−2276.52	−2276.6	−2276.57
AlCoCrFeNi	−2275.32	−2275.38	−2275.34

, when HEA is in the FCC phase, the crystal energy is the lowest. This shows that the FCC phase is easier to form and exists more stably in AlCo_xCrFeNi HEAs than the BCC phase and HCP phase. This also proves the rationality of using the FCC phase to study the mechanical properties of AlCo_xCrFeNi HEAs in this paper.

3.2. Mechanical Properties

Lattice constant is the basic structural parameter of crystal material, which is used to describe the change in atomic space position caused by the change in composition or stress of a crystal material. It not only reflects the interaction between atoms, but also has a certain relationship with the density of the material. CASTEP can calculate the corresponding stress by inputting the strain. The elastic constant is finally obtained by linear fitting of the stress–strain relationship [28,29]. The addition of Co will cause the FCC phase to appear in the AlCo_xCrFeNi (x = 0, 0.5, 1, 1.5, 2) HEAs, and the FCC phase is also an important factor determining the strength of the alloy. Therefore, an AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEA was constructed by the VCA method based on the unit cell (FCC phase). Since the VCA method introduces the concept of mixing in the level of pseudopotential, each atom in the unit cell has the proportion of multiple atoms at the same time. After the geometry optimization, the overall model of the high-entropy alloy has little change, and its lattice constant and density are shown in

Table 3. Lattice constant and density of AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEAs.

Alloys	Lattice Parameter/Å	Mass Density/(g/cm3)	Theoretical Density/(g/cm3)
AlCrFeNi	3.556	6.74	6.6
AlCo0.25CrFeNi	3.55	7.07	6.79
AlCo0.5 CrFeNi	3.541	7.09	6.9
AlCo0.75 CrFeNi	3.538	7.12	7.01
AlCoCrFeNi	3.537	7.14	7.1

. From Table 3, with the increase in Co content, the lattice constant of HEAs gradually decreased and the density increased gradually. The calculated density is not much different from the theoretical density calculated by addition (error < 4 %). This shows that the lattice distortion of AlCo_xCrFeNi (x = 0, 0.5, 1, 1.5, 2) HEAs gradually decreases when Co is added, which is also the same as the δ calculation result.

The size of the material’s elastic constants (C₁₁, C₁₂, C₄₄) determines the bulk modulus K, shear modulus G, Young’s modulus E and Poisson’s ratio υ, which play important roles in determining the mechanical properties of the material. The effect of Co on the mechanical properties of AlCo_xCrFeNi HEAs was further studied. The paper calculated the elastic constants (C₁₁, C₁₂, C₁₃), polycrystalline elastic moduli (K, G, E, υ), anisotropy index A_VR and hardness Hv of AlCo_xCrFeNi HEAs. The calculation formula is shown below [30,31], and the calculation results are listed in

Table 4. Calculated elastic constant, elastic modulus, υ and A_VR of AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEA.

Alloys	C11 /GPa	C12 /GPa	C44 /Gpa	C11-C44 /GPa	K /GPa	G /GPa	E /GPa	K/G /GPa	υ	AVR
AlCrFeNi	172	97	91	82	122	64	162	1.91	0.29	2.41
AlCo0.25CrFeNi	182	97	78	104	125	61	157	2.05	0.28	1.85
AlCo0.5 CrFeNi	180	99	89	90	126	65	166	1.94	0.32	2.21
AlCo0.75 CrFeNi	185	113	74	111	137	56	147	2.45	0.32	2.05
AlCoCrFeNi	188	115	79	110	139	59	153	2.36	0.32	2.14

.

$$K=\frac{C_{11}{+2C}_{12}}{3}$$

(7)

$$G_V=\frac{1}{5} \left(3C_{44}{+C}_{11}{ - C}_{12}\right)$$

(8)

$$G_R=\frac{{5(C}_{11}{ - C}_{12}{)C}_{44}}{{4C}_{44}{+3(C}_{11}{ - C}_{12})}$$

(9)

$$G=\frac{G_V{+G}_R}{2}$$

(10)

$$E=\frac{9\mathrm{KG}}{3K+G}$$

(11)

$$\mathsf{\upsilon }=\frac{3K - 2G}{2(3K+G)}$$

(12)

$$A_{\mathrm{VR}}=\frac{C11 - C12 - 2C44}{C11 - C44}$$

(13)

$$H_v=2 \times {\left(\frac{G^3}{B^2}\right)}^{0.585}- 3$$

(14)

where C_ij is the elastic constant and G_V and G_R are the elastic moduli G obtained by the Voigt and Reuss methods.

In Table 4, Cauchy pressure is positive (C₁₂ − C₄₄ > 0) [32], indicating that the HEA is a ductile material. To facilitate the comparison of mechanical properties, the elastic constants and polycrystalline elastic moduli of AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEAs are plotted in Figure 2. It is generally believed that K reflects the bonding strength between atoms, molecules and ions in the material. On the macroscopic scale, K represents the incompressibility of the material. The larger the bulk modulus K of the material, the stronger its incompressibility and the worse its ductility. The shear modulus G represents the ability of the material to resist shear deformation. The larger the shear modulus G, the better the rigidity of the material and the worse the ductility. Young’s modulus E is a standard used to measure the resistance of materials to deformation. The larger the value, the greater the stress required for the material to elastically deform, that is, the less likely the material is to deform. A material with A_VR = 0 is generally considered to be elastically isotropic, otherwise it is elastically anisotropic. It can be seen from Figure 2 that the bulk modulus (K) increases with the increase in Co content, while the shear modulus (G) does not change significantly [33]. The Young’s modulus is the largest when the Co content is 0.5, and the material has the best resistance to deformation. This shows that AlCo_0.5CrFeNi HEA has the strongest resistance to deformation and better ductility. A_VR > 0 indicates [34] that HEAs are elastic anisotropic materials.

The ductility or brittleness of alloy materials can be judged by the Poisson’s ratio υ and the ratio of bulk modulus to shear modulus K/G [35]. Materials with a Poisson’s ratio υ of about 1/3 are considered ductile materials, while materials in other ranges are called brittle materials and the maximum value cannot exceed 0.5. When the ratio of bulk modulus to shear modulus is greater than 1.75, the material is a ductile material, otherwise it is a brittle material. In Figure 3, the Poisson’s ratio υ of AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEAs is about 1/3 and belongs to ductile materials. K/G > 1.75 indicates that the toughness of the material is gradually enhanced with the addition of Co, and the toughness of AlCo_0.75CrFeNi is the best.

In Figure 4, the calculated microhardnesses of AlCo_xCrFeNi HEAs are 774 Hv, 663 Hv, 777 Hv, 444 Hv and 477 Hv. The experimental microhardnesses of the high-entropy alloy prepared by Faruk Kaya [36] by the suction-cast method are 500 Hv, 575 Hv, 530 Hv, 480 Hv and 520 Hv. The experimental microhardnesses of the high-entropy alloy prepared by the SHS method are 440 Hv, 425 Hv, 510 Hv, 380 Hv and 425 Hv. The microhardness of the alloy is different with different preparation methods. However, HEAs prepared by the suction-cast method have a finer microstructure. They will have higher hardness than HEAs prepared by the self-propagating-high-temperature synhesis (SHS) method. With the increase in Co content, the BCC phase in AlCo_xCrFeNi HEA is gradually transformed into FCC phase. Finally, the BCC+FCC dual-phase structure is formed in the alloy. The analysis results show that with the increase in Co content, the calculated values of hardness are the same as the experimental values obtained by the SHS method. When the content of Co is low, there is a big error between the calculated results and the experimental values. This is due to the low FCC phase content in HEAs. When the Co content increases, the calculated hardness is between the two experimental values. This is because part of the BCC phase in HEAs is converted into the FCC phase. This also proves that the calculation is in line with the reality.

4. Conclusions

In this paper, the AlCo_xCrFeNi (x = 0, 0.25, 0.5, 0.75, 1) HEAs were calculated based on the first-principles-based density functional method, and it is found that the AlCo_xCrFeNi HEAs can form a stable solid solution structure. The addition of Co affects the formation of the FCC phase in HEAs, resulting in the decrease in density and the increase in lattice constant. It also plays an important role in improving the mechanical properties of HEAs. The research shows that the AlCo_0.5CrFeNi HEA has the highest hardness, while the AlCo_0.75CrFeNi HEA has the best toughness. Therefore, the optimal balance point of Co content in AlCo_xCrFeNi HEAs may be between 0.5 and 0.75, which also provides some guidance for specific experiments.