## 3 Results and Discussion  
### Structural Stability  
To understand the structural properties of Co-Cr-Fe-Ni HEAs with different contents of each component, the VEC, \(\delta\), \(\Delta H_{mix}\), \(\Delta S_{mix}\), \(\Omega\), and tetragonal shear modulus (\(c^{\prime}\)) were calculated and are shown in Figure 2. The VEC, \(\delta\), \(\Delta H_{mix}\), \(\Delta S_{mix}\) and \(\Omega\) for the equimolar ratio of the CoCrFeNi alloy are consistent with previous works [40,42]. According to previous results [40], HEAs prefer to form a single BCC phase when the VEC is 4.33-7.55, whereas the structure is FCC for VEC between 7.80 and 9.50. As shown in Figure 2a, all calculated VEC values are higher than 7.8, indicating that Co-Cr-Fe-Ni HEAs may maintain a single-phase FCC structure. This result is consistent with previous works [9, 10, 11, 38]. The VEC of Cr and Fe is lower than the average valence electron concentration of the four elements, which causes the VEC of the alloys to decrease with increasing Cr and Fe contents. On the contrary, the VEC of the alloys increases with Co and Ni contents because the VEC of Co and Ni are higher. The \(\delta\) is related to the atomic radius. Co has the largest atomic radius, and Fe has the smallest. This indicates that \(\delta\) increases with Co and Fe contents, and decreases with Cr and Ni contents are shown in Figure 2b. HEAs form solid solutions with \(-15\leq\Delta H_{mix}\leq\)5 kJ mol\({}^{-1}\) and \(11\leq\Delta S_{mix}\leq\) 19.5 J K\({}^{-1}\) mol\({}^{-1}\)[40]. As shown in Figure 2c,d, the values of \(\Delta H_{mix}\) and \(\Delta S_{mix}\) are in the above range, indicating that Co-Cr-Fe-Ni HEAs may be a solid-solution phase. This was confirmed by previous works [25, 36, 37, 38]. Only when the absolute value of \(\Delta H_{mix}\) is close to zero can the solid solution be stable in the solid phase. The results show that the solid-solution phase stability of the alloys decreases with increasing Cr and Ni contents and increases with increasing Co and Fe contents. This is consistent with previous work [38]. However, the entropy of mixing result shows that only the alloys with an equal molar ratio have the largest entropy of mixing, which is not affected by the component. For HEAs, \(\Omega>1\) means that the contribution of \(T\Delta S_{mix}\) will be higher than \(\Delta H_{mix}\), corresponding to the formation of the solid-solution phase in the alloy [42]. As shown in Figure 2e, all alloys are solid solutions and become more stable with decreasing Cr and Ni content and increasing Co and Fe content. According to a previous report [42], a random solid solution is stable when \(\delta\leq 6.6\%\) and \(\Omega\geq 1.1\). Our results indicate that the Co-Cr-Fe-Ni HEAs studied in this work may be a random solid solution. This result is consistent with previous works [25, 36, 37, 38]. In addition, the parameter \(\Omega\) shows a similar trend to that of the \(\delta\). The tetragonal shear modulus (\(c^{\prime}=(C_{11}-C_{12})/2\)) is related to the energy difference of different crystal structures and also can reflect the structural stability [56]. As shown in Figure 2f, \(c^{\prime}\) decreases with increasing Cr and Ni contents and increases as the contents of Co and Fe increase. For HEAs composed of \(3d\) transition elements, \(c^{\prime}\) is determined by the energy difference between the FCC and BCC structures of the multicomponent alloy system, which, in turn, is determined by the \(d\) orbital electrons [56]. Therefore, the energy difference of the alloys increases with Co and Fe contents and decreases with Cr and Ni contents. This result is consistent with the  
Figure 1: Co-Cr-Fe-Ni HEAs’ structures of supercell solid-solution alloy models. **(a)** The values of \(x\) are 0.2, 0.56, 1 and 1.57; **(b)** the value of \(x\) is 2.  
enthalpy of mixing, indicating that the structure is more stable as Co and Fe increase and Cr and Ni decrease.