The Effect of Reduced Co Content in CrMnFeCoNi Alloys: A First Principles Study

Abstract

This study investigates high-entropy CrMnFeCoNi alloys with reduced Co content using density functional theory. The muffin-tin orbital method and coherent potential approximation successfully predict experimental values for volume, magnetic moment, and elastic constants. Thermodynamic properties, analyzed using the Debye–Gruneisen model, emphasize the need to consider both electronic and magnetic contributions to the free energy. The alloys exhibit anti-Invar behavior, with a significant increase in the linear thermal expansion coefficient with increased temperature. This effect is slightly more pronounced for reduced Co content, leading to a larger lattice parameter and a decrease in elastic constants. However, the changes are small, suggesting that similar mechanical properties can be achieved with lower Co content.

1. Introduction

High-entropy alloys (HEAs) are composed of five or more elements in nearly equal concentrations [1]. The high entropy associated with these alloys suppresses the formation of ordered intermetallic compounds, resulting in disordered phases with relatively simple lattice structures, such as face-centered cubic (fcc) or body-centered cubic (bcc). Many HEAs exhibit outstanding mechanical properties, including high yield strength [2], superior fatigue resistance [3], and improved wear resistance [4]. From an engineering perspective, HEAs are particularly intriguing because their atomic composition can be tuned to achieve specific desired properties. Computational methods play a crucial role in this field, enabling the exploration of various atomic configurations.

The chemical disorder inherent in these alloys is typically modeled using the coherent potential approximation (CPA) [5,6,7,8,9], which is considered a reasonable approximation, since the effect of short-range order (SRO) is minimal by definition. In particular, calculations that use the exact muffin-tin orbitals (MTO) method combined with CPA have been widely employed [6,7,8]. A similar approach can be used to model paramagnetism (PM), which is known as the disordered local moment (DLM) method [10,11]. The DLM state is defined as a random arrangement of two distinct magnetic moments pointing either up or down of the same atomic species in a metallic system. Using the DLM model, it is possible to consider the effect of finite-temperature spin fluctuations [12], which can be crucial when describing the thermal properties of transition metals in their PM state [13]. This approach has proven successful in recent years [13,14,15,16,17], providing an accurate description of spin fluctuations at finite temperatures in the PM state.

One of the most extensively studied HEAs is the CrMnFeCoNi alloy, also known as the Cantor alloy [18]. This alloy consists of a disordered fcc structure with equal concentrations of the magnetic elements Cr, Mn, Fe, Co, and Ni. The magnetic properties of these components vary: Fe, Co, and Ni are ferromagnetic (FM); Mn exhibits a multi-magnetic state; and Cr is antiferromagnetic (AFM). The CrMnFeCoNi alloy displays a diverse range of magnetic properties depending on sample preparation [19], similar to other closely related HEAs [20,21,22]. As a result, numerous studies have attempted to optimize the properties of these alloys by adjusting the concentrations of specific components. For example, the Cr concentration has been studied in AlFeNiCoCr [23], while the Fe and Ni concentrations were explored in CoCrFeNiTi [24].

Computational studies of the NiCrCoFe alloy have shown that varying the Co content significantly affects both the lattice parameter and the elastic constants [5]. However, the influence of Co content in the CrMnFeCoNi alloy remains less well explored. Reducing or replacing Co with alternative elements is of considerable interest for economic, ethical, and health-related reasons [25,26,27,28,29]. Consequently, it has attracted a lot of attention in a wide range of different applications such as batteries, catalysts, superalloys, and cemented carbides [30,31,32,33,34,35]. Given the favorable combination of yield strength and ductility already exhibited by the CrMnFeCoNi alloy, further investigations of near-equal concentrations are highly valuable. Hence, in this study, the temperature-dependent magnetic and elastic properties of the ${\text{(CrMnFeNi)}}_{1-x}$${\text{Co}}_x$ alloy for $0\le x\le 0.2$ are investigated.

2. Methods

In order to model thermodynamic properties, the quasiharmonic approximation within the Debye–Grüneisen model was applied. The free energy of the Debye–Grüneisen model as described in Ref. [36] is given by

$$\begin{array}{ccc}\hfill F(V,T)& =& E_{\mathrm{el}}\left(V\right)+F_{\mathrm{el}}(V,T)+F_{\mathrm{mag}}(V,T)\hfill \\ & & -k_{\mathrm{B}}T(D({\Theta }_{\mathrm{D}}/T)+3ln\left(1-e^{-{\Theta }_{\mathrm{D}}/T}\right))+{\displaystyle \frac{9}{8}}{k}_{\mathrm{B}}{\Theta }_{\mathrm{D}},\hfill \end{array}$$

(1)

where $E_{\mathrm{el}}\left(V\right)$ is the energy of the static lattice at a given volume V; $F_{\mathrm{el}}(V,T)$ is the thermal electronic contribution to the free energy at temperature T; $F_{\mathrm{mag}}(V,T)$ is the magnetic contribution to the free energy; and $D({\Theta }_{\mathrm{D}}/T)$ and ${\Theta }_{\mathrm{D}}$ are the Debye function and the Debye temperature respectively.

The volume-dependent thermal electronic free energy can be expressed as

$$\begin{array}{c}\hfill F_{\mathrm{el}}(V,T)=E_{\mathrm{el}}(V,T)-T{S}_{\mathrm{el}}(V,T),\end{array}$$

(2)

where $E_{\mathrm{el}}(V,T)$ corresponds to the energy due to thermal electronic excitations and $T{S}_{\mathrm{el}}$ to the electronic entropy. The former can be expressed as

$$\begin{array}{c}\hfill E_{\mathrm{el}}(V,T)=\int N(ϵ,V)f(ϵ,T)ϵdϵ-{\int }^{{ϵ}_{\mathrm{F}}}N(ϵ,V)ϵdϵ,\end{array}$$

(3)

where $N(ϵ,V)$ is the electronic density of state, $ϵ$ the energy eigenvalues, ${ϵ}_{\mathrm{F}}$ is the Fermi energy, and $f(ϵ,T)$ is the Fermi–Dirac distribution function

$$\begin{array}{c}\hfill f(ϵ,T)={\left(1+e^{(ϵ-\mu )/k_{\mathrm{B}}T}\right)}^{-1},\end{array}$$

(4)

in terms of the electronic chemical potential $\mu$ and Boltzmann’s constant $k_{\mathrm{B}}$. The electronic entropy $S_{\mathrm{el}}$ can be calculated as follows:

$$\begin{array}{cc}\hfill S_{\mathrm{el}}(V,T)=& -k_{\mathrm{B}}\int N(ϵ,V)(f(ϵ,T)ln(f(ϵ,T))\\ & +\left(1-f(ϵ,T)\right)ln\left(1-f(ϵ,T)\right))dϵ.\end{array}$$

(5)

Two different approaches were used to describe the paramagnetic state at finite temperatures. In the first approach, the completely random configuration of magnetic moments is represented by the DLM model [10,11]. In the second approach, the magnetic free energy due to thermal spin fluctuations is included in the DLM model as in Ref. [13]. The magnetic free energy was approximated by

$$\begin{array}{c}\hfill F_{\mathrm{mag}}(V,T)=-k_{\mathrm{B}}Tln(m+1),\end{array}$$

(6)

where m is the absolute value of the spin moment.

The first-principles calculations presented in this study are based on density functional theory (DFT) [37] and the exact muffin-tin orbital (EMTO) method [38,39,40]. Random substitutional disorder is modeled using an effective medium, represented by a coherent potential, as is done in the CPA approach [41,42]. The additional electrostatic contributions to the one-electron potentials of the alloy components and the total energy should be taken into account [43,44]. In the present formalism, the on-site screened Coulomb interaction, $V_i^{\mathrm{scr}}$, which contributes to the one-electron potential and the corresponding electrostatic energy $E_i^{\mathrm{scr}}$ [43,44], is given by

$$V_i^{\mathrm{scr}}=-e^2{\alpha }_i^{\mathrm{scr}}{\displaystyle \frac{q_i}{r_{\mathrm{ws}}}},E_i^{\mathrm{scr}}=-e^2{\displaystyle \frac{1}{2}}{\alpha }_i^{\mathrm{scr}}{\beta }_i^{\mathrm{scr}}{\displaystyle \frac{q_i^2}{r_{\mathrm{ws}}}}.$$

(7)

In Equation (7), $q_i$ is the net charge of the atomic sphere of the ith alloy component, $r_{\mathrm{ws}}$ the Wigner–Seitz radius, and ${\alpha }_i^{\mathrm{scr}}$ and ${\beta }_i^{\mathrm{scr}}$ the on-site screening constants. The screening parameters are commonly determined from supercell calculations using the locally self-consistent Green’s function technique [45,46]. This was done in Ref. [47] for the equiatomic CrMnFeCoNi alloy. However, we only found very small changes in the studied properties with respect to the screening parameters, and we have used ${\alpha }^{\mathrm{scr}}=0.796$ and ${\beta }^{\mathrm{scr}}=1.0$ in all calculations similar to Ref. [48].

In order to perform the self-consistent loop, the local density approximation (LDA) [49] was used for the exchange-correlation potential. All self-consistent EMTO-CPA calculations were performed using an orbital momentum cutoff of $l_{\max }=3$ for the partial-wave expansion. The Monkhorst–Pack grid [50] with subdivisions along each reciprocal lattice vector 31 × 31 × 31 was used for integration over the Brillouin zone in the self-consistent LDA calculations. The generalized gradient approximation (GGA) [51] and the full charge density formalism [39] were used to calculate the total energy.

3. Results and Discussion

The Co concentration was reduced by increasing the concentrations of the other constituent elements equally, i.e., from CrMnFeCoNi ($x=0.2$) to CrMnFeNi ($x=0$). This was done in order to maintain the disordered state as well as possible, where the CPA method remains valid. The calculations were performed using the primitive fcc cell, as the fcc phase is most commonly observed in the CrMnFeCoNi alloy [18,52]. However, theoretical calculations actually predict the hcp phase to be the most stable ground state [7]. This is likely due to the underestimated lattice parameter associated with the semi-local GGA approach [7]. Interestingly, the hcp phase has been experimentally observed under high-pressure conditions [53,54].

Recent experimental results suggest two ordering temperatures: 93 K, where the system forms a spin glass state, and 38 K, where it becomes FM [19]. Hence, for most applications, it is reasonable to assume a PM state. To model this, the DLM approach was employed. The magnetic contribution to the free energy can have a significant effect and must also be considered [55,56]. This is evident in the behavior of the local magnetic moment as a function of the Wigner–Seitz radius, as shown in Figure 1, where only the Mn and Fe moments are non-zero near equilibrium (∼2.6 a.u.). Including the magnetic contribution to the free energy eliminates the discontinuous jump from a nearly zero Mn moment to a high-spin state for Mn. More importantly, without this contribution, the Mn moment is zero at the equilibrium volume.

3.1. Equilibrium Lattice Parameter and Thermal Expansion Coefficient

To explore the thermodynamic properties, free energy calculations for different lattice parameters were fitted to a Birch–Murnaghan equation of state [57,58]. This approach enables the estimation of the equilibrium lattice constant and the bulk modulus. Furthermore, by calculating the volume derivative of the bulk modulus, the Grüneisen constant can be determined, allowing for the description of the volume dependence of the Debye temperature [36,59] and, by extension, the temperature dependence of the free energy. However, fitting an equation of state to free energy calculations presents challenges for these systems. This is primarily due to the Mn moment transitioning from a low to a high spin state around equilibrium, which causes a kink in the volume-free energy curve. This behavior is characteristic of anti-Invar materials and has been observed, for example, in fcc-Fe [60] and FeNiCr alloys [61].

Due to the kink in the volume-free energy curve, selecting the appropriate fitting interval and number of data points requires careful attention. In particular, the second derivative of the bulk modulus is highly sensitive, which makes determining the Grüneisen constant a challenging task. Consequently, performing thermodynamic calculations using Equation (1) proves to be difficult. In this study, the fitting interval was chosen to be $0.98r_0<r_{\mathrm{ws}}<1.05r_0$, where $r_0$ represents the equilibrium Wigner–Seitz radius. Below the lower bound of this range, the Fe moment disappears, while above the upper bound, a non-zero Co moment is observed. The Ni moment was found to be nearly zero, except for very large radii. It should be noted that, in the FM state, Co also exhibits a non-zero moment at equilibrium.

The equilibrium Wigner–Seitz radius, $r_0$, is shown in

Table 1. The calculated equilibrium Wigner–Seitz radius, $r_0$ (in a.u.), of the ${\text{(CrMnFeNi)}}_{1-x}$${\text{Co}}_x$ alloy for different concentrations (atomic fractions) x. The equilibrium radii were calculated in the PM state (0 K, 300 K, and 900 K). Note that $r_0$ is directly proportional to the lattice parameter.

x	0.00	0.04	0.08	0.12	0.16	0.20
PM (0 K)	2.6070	2.6066	2.6059	2.6051	2.6040	2.6027
PM (300 K)	2.6292	2.6276	2.6257	2.6249	2.6217	2.6198
PM (900 K)	2.6795	2.6757	2.6725	2.6690	2.6655	2.6619

. As expected, $r_0$ increases with temperature. However, it is slightly underestimated when compared to experimental data, such as the recent measurements of the equiatomic CrMnFeCoNi alloy by Schneeweiss et al., who report values of 2.6507 a.u. (0 K) and 2.6583 a.u. (300 K) [19]. It is well-established that GGA generally provides a more accurate representation of the equilibrium volume for 3d metals compared to LDA [62]. The slight underestimation of the Wigner–Seitz radius at low temperatures is not unexpected for Fe-based alloys calculated using GGA [63,64]. In fact, experimental studies have shown that the zero-temperature lattice parameter for pure bcc-Fe (with zero-point vibrations removed) is 2.853 Å [64], while the corresponding theoretical value is somewhat smaller, at 2.837 Å (as reported in this work). The discrepancy between calculations and experimental results is typically small when using the GGA approach (here, it is less than 2% for $x=0.2$).

The lattice parameter is observed to decrease with increasing Co concentration. Although the change is small (less than 0.2%), the trend is clear. Furthermore, the variation in lattice parameter with concentration becomes more pronounced at elevated temperatures. The temperature dependence of the lattice parameter is significant, and to explore this behavior further, we calculate the linear thermal expansion coefficient

$$\begin{array}{c}\hfill \alpha \left(T\right)={\displaystyle \frac{1}{3V}}{\displaystyle \frac{dV}{dT}}.\end{array}$$

(8)

As mentioned above, the linear thermal expansion coefficient, $\alpha$, is highly sensitive to the fitting procedure, as it depends on the derivative of the bulk modulus. Therefore, an accurate determination of $\alpha$ would validate the method used.

The temperature dependence of $\alpha$ is shown in Figure 2 for $x=0$, $x=0.12$, and $x=0.2$. The calculated results are compared with existing experimental data for $x=0.2$ [19,65]. These calculations were carried out for the PM state, both with and without accounting for the magnetic contribution to the free energy. Including the magnetic contribution significantly improves the results, leading to a much better agreement with experimental data above room temperature. The significance of the magnetic contribution to the free energy in determining mechanical properties has been demonstrated in previous studies, such as those of austenitic steels [66,67].

The significant increase in the linear thermal expansion coefficient with temperature reflects the anti-Invar behavior of CrMnFeCoNi, similar to that observed in fcc-Fe [68]. This property has been utilized to increase the ordering temperature of CrMnFeCoNi to $T_{\mathrm{C}}=400$ K by filling the octahedral sites of the fcc structure with carbon, which represents a considerable enhancement [68]. Reducing the Co concentration results in a slight decrease in the valence electron density, from 8 $(e/a)$ to 7.75 $(e/a)$, causing the thermal expansion curve to shift to higher values. This shift is evident in the three cases $x=0$, $x=0.12$, and $x=0.2$, as shown in Figure 2. Hence, the anti-Invar effect is somewhat enhanced with decreasing Co concentration. As a result, lower Co concentrations lead to a greater increase in the lattice parameter with temperature, as shown in Table 1.

3.2. Elastic Constants

The elastic properties of the cubic lattice are described by three independent elastic constants: $c_{11}$, $c_{12}$, and $c_{44}$. These constants were determined by calculating the total free energy as a function of volume-conserving strains, following the method outlined by Mehl [69,70]. The elastic constant $c_{44}$ was obtained by fitting the free energy-strain relationship, $F\left(x\right)=2V{c}_{44}{x}^2$, using a monoclinic strain x applied to the base fcc lattice. Similarly, the tetragonal shear modulus $c^{\prime }$ was derived from $F\left(x\right)=V{c}^{\prime }{x}^2$ by applying an orthorhombic strain. Using the bulk modulus, determined from the equation of state described earlier, the constants $c_{11}$ and $c_{12}$ were calculated from $c^{\prime }$ using the relationships $B=(c_{11}+2c_{12})/3$, and $c^{\prime }=(c_{11}-c_{12})/2$.

Figure 3 shows the temperature dependence of the bulk modulus and elastic constants for the CrMnFeCoNi alloy $(x=0.2)$. The calculated values are presented together with existing fitted experimental results from Teramoto et al. [71]. Overall, the temperature dependence is seen to be relatively small, and, despite the various approximations involved, there is generally good agreement between the theoretical and experimental data. The calculated values for the bulk modulus and $c_{44}$ constant are seen to be slightly overestimated compared to the experimental values. The deviation for the bulk modulus is more significant at low temperatures but remains within a few percent. The error is larger for the $c_{44}$ constant. At low temperatures, the error is less than 10% but increases to more than 30% at 1200 K.

Although the exact cause of these discrepancies remains unclear, they likely indicate that our theoretical treatment of the paramagnetic state is too simple. However, this does not appear to affect the $c^{\prime }$ constant derived from the orthorhombic strain, which agrees excellently with the experimental data. It should be noted that the calculated results are very similar to those of Ma et al. [48]. It should also be mentioned that calculations performed without a complete charge description yield comparable results in most cases. The only exception is for the cell with monoclinic distortion, where the total free energy is significantly overestimated. This issue is well known [72] and leads to a corresponding overestimation of the $c_{44}$ constant.

Figure 4 shows the calculated results of the bulk modulus and elastic constants for different types of Co content. The elastic constants are seen to increase with the Co content. This is consistent with the reduction in the lattice constant with increasing Co content. However, the change is very small. The traditional mechanical stability conditions for cubic crystals $(c_{44}>0$, $c_{11}>\left|c_{12}\right|$ and $c_{11}+2c_{12}>0)$ are satisfied throughout the entire concentration range. In particular, the $c_{11}$ constant increases with Co content, a trend also observed in the CrMnNi alloy [7]. Both $c_{12}$ and $c_{44}$ remain positive, and the Cauchy pressure $(c_{12}-c_{44})$ is negative for all alloys considered, becoming more negative as the Co content increases at room temperature. Negative Cauchy pressures have been associated with the covalent nature of the metallic bond and are characteristic of brittle alloys [73]. In general, the impact of the Co content on the mechanical properties is found to be relatively small.

3.3. Stacking Fault and Mixing Energy

Determining the exact stacking fault energy (SFE) and understanding the effects of the alloying elements on the SFE remains a challenging task, with several unresolved questions about what is actually measured in experiments [74]. In this study, only intrinsic stacking fault defects are considered, which are the simplest planar defects in the crystal lattice. These are characterized by a fault in the usual ABC planar stacking sequence of the fcc structure and resemble the stacking sequence of the hcp structure. For this type of defect, the stacking fault energy ${\gamma }_{\mathrm{SFE}}$ can be determined from the total energy calculations of the fcc, hcp, and dhcp structures by considering the interactions between the next nearest neighbor stacking planes. This is known as the axial next-nearest neighbor Ising (ANNNI) model:

$${\gamma }_{\mathrm{SFE}}=(F_{\mathrm{hcp}}+2F_{\mathrm{dhcp}}-3F_{\mathrm{fcc}})/A,$$

(9)

where $F_{\mathrm{fcc}}$, $F_{\mathrm{hcp}}$, and $F_{\mathrm{dhcp}}$ represent the free energies of the fcc, hcp, and dhcp structures, respectively, and $A={\displaystyle \frac{\sqrt{3}}{4}}{a}^2$ is the area of a hexagon that defines a unit cell in a single layer [75]. The ANNNI model has shown to be a reasonable choice, offering a good balance between accuracy and computational cost for ab initio calculations of fcc alloys [55,76]. However, the theoretical results from DFT calculations are known to underestimate the stacking-fault energy [66,67], primarily due to the underestimation of the lattice constant.

The calculated SFE results are presented in Figure 5 for the two limiting cases: CrMnFeCoNi $(x=0.2)$ and CrMnFeNi $(x=0)$. For both cases, the SFE is negative below 500 K, indicating that the hcp phase is more stable than the fcc phase. This finding is consistent with several other theoretical studies, such as [7]. The stability of the hcp phase is probably due to the underestimation of the lattice parameter resulting from the semi-local GGA approach. In fact, it was shown that increasing the lattice parameter makes the fcc phase more stable [7]. The hcp phase has also been observed experimentally under high-pressure conditions [53,54]. Reducing the Co content is seen to increase the stacking fault energy, thereby stabilizing the fcc phase. Therefore, given that the fcc phase is commonly observed in the CrMnFeCoNi alloy, these results suggest that reducing the Co content will also favor the fcc phase.

In order to further investigate the stability aspect of the CrMnFeCoNi alloy with reduced Co content, the mixing energy was calculated. The mixing energy was obtained by

$$F_{\mathrm{mix}}=F_{\mathrm{a}}-\sum _i{x}_i{F}_i.$$

(10)

In Equation (10), $F_{\mathrm{a}}$ represents the total free energy per atom of the alloy, while $x_i$ and $F_i$ denote the atomic fraction and the free energy per atom of component i, respectively. The total free energy of the individual elements was calculated using the relaxed lattice parameters for each phase: the bcc phase for Fe and Cr, the fcc phase for Ni and Mn, and the hcp phase for Co (see

Table 2. Calculated lattice parameter in Å (300 K) for Cr, Mn, Fe, Co, and Ni in the crystal structures found at room temperature (the complex structure of Mn is approximated with the fcc structure). Experimental results are given at room temperature. VASP calculations are given as reference (0 K). For hcp-Co, the lattice parameter is reported as $a/c$.

	bcc-Cr	fcc-Mn	bcc-Fe	hcp-Co	fcc-Ni
	AFM	AFM	FM	FM	FM
EMTO	2.86	3.62	2.86	2.49/4.07	3.54
VASP	2.87	3.58	2.83	2.49/4.02	3.52
Exp. [77]	2.88	-	2.87	2.51/4.07	3.52

for results at room temperature).

The results, shown in Figure 6, reveal that $F_{\mathrm{mix}}$ generally changes only slightly with Co content (less than 1 mRy). From a mixing energy perspective, the most stable phase is observed for the CrMnFeCoNi $(x=0.2)$ configuration, regardless of temperature. The mixing energy is positive, indicating that the compounds are less stable than their individual components. However, multiple mechanisms contribute to the overall stability of a system. In particular, the contribution to the free energy from the configurational entropy of mixing is considered a major stabilizing factor for high-entropy alloys [18], and this contribution would be of a similar order of magnitude at elevated temperatures. Therefore, it is likely that compounds with reduced Co concentration will remain stable in the pure fcc phase. This has been confirmed in recent experiments [52] for $x=0.05$, $0.10$, and $0.20$, all of which exhibit a single-phase fcc structure in the as-cast state.

4. Conclusions

The ${\text{(CrMnFeNi)}}_{1-x}$${\text{Co}}_x$ high-entropy alloy was investigated for $0\le x\le 0.2$ using DFT calculations. Guided by experimental results for the CrMnFeCoNi alloy, all calculations were performed in the paramagnetic fcc phase. The results demonstrate that the exact muffin-tin formalism, combined with the coherent potential approximation, provides reliable predictions for the equilibrium volume and magnetic moment.

Thermodynamic properties were analyzed using the Debye–Grüneisen model. To match experimental data for the linear thermal expansion coefficient and lattice parameter, both electronic and magnetic contributions to the free energy must be considered. The investigated alloys exhibit anti-Invar behavior, with a significant increase in the thermal expansion coefficient with temperature, leading to a reduction in the elastic constants. For alloys with lower Co content, both the linear thermal expansion coefficient and the lattice parameter increase, resulting in a further decrease in the elastic constants.

To assess the stability of the ${\text{(CrMnFeNi)}}_{1-x}$${\text{Co}}_x$ alloy, the intrinsic stacking fault energy and mixing energy were calculated. The stacking fault energy increases with decreasing Co content, suggesting a stronger tendency toward the fcc phase. However, the mixing energy also increases as the Co content decreases, implying that the CrMnFeCoNi alloy with equal component concentrations is more stable.

Although the combination of the CPA method and the DLM approach has proven effective in modeling paramagnetic multi-component alloys, questions remain regarding the impact of local relaxations and short-range order on the mechanical properties, especially when deviating from near-equal concentrations of the components. Therefore, further supercell simulations, along with additional experimental measurements, are required to fully understand the Co concentration dependence in these alloys.