3.1. Stability

In order to study the thermodynamic stability of Cr–Si binary compounds, their enthalpies of formation and cohesion are calculated and the two energy formulas are defined as follows:

where E_coh(Cr_xSi_y) and (Cr_xSi_y) represent the cohesive energy and formation enthalpy of the chrome-silicon compound, respectively; E_tot(Cr_xSi_y), E_iso(Cr), and E_iso(C) denote the total energy of the Cr_xSi_y phase, a single Cr atom and a single Si atom, respectively; E_bin (Cr) and E_bin (Si) refer to the cohesive energy of Cr atom and Si atom, respectively.^[15,16]

Table 1 shows the calculated parameters about thermodynamic stability, such as cell parameters, cohesive energy, and formation enthalpy, and their comparison with other researchers’ results. It can be seen that the crystal parameters of the Cr–Si system match well with other experimental and calculated values.^{[7,13,16–24]} Furthermore, the tiny deviation among them can be attributed to the different calculation methods and experimental conditions. According to Table 1, the cohesive energies of the Cr–Si compounds decrease from −10.207 eV/atom to −7.406 eV/atom, but the Si content in the crystal cell increases from 0.25 to 0.67, which correspond to those of Cr₃Si, Cr₅Si₃, CrSi, and CrSi₂ as shown in Fig. 3. In the meantime, the formation enthalpy varies with the Si content increasing similarly.^[25] The lowest formation enthalpy is −0.355 eV/atom for Cr₃Si, while the highest one is −0.284 eV/atom for CrSi₂. All values of cohesive energy and formation enthalpy are negative, which means that the thermodynamic properties of all Cr–Si compounds are stable.

According to the order of the formation enthalpy and the bonding energy of the Cr–Si compounds, the order of thermal stability of Cr–Si compounds is . Therefore, Cr₃Si is the most stable in thermal properties among CrSi binary compounds.

3.2. Electronic structures

In order to analyze the chemical bond properties of chrome-silicon compounds and the electron density distribution of the electronic structure, the total state of atoms (TDOS) and the partial state of charge (PDOS) are shown in Fig. 4.

The nature of the magnetic characteristic can be understood from the spin-polarized total density of states. Generally speaking, the up and down states are symmetric, but the up and down states of CrSi are dissymmetric, which indicates the magnetic characteristic.^[26,27] Actually, the low and high valence bands are almost symmetric, and the up and down states are dissymmetric with respect to the Fermi level but very close to the Fermi level. Comparing with other compounds, we can guess they have no magnetic characteristics, because the down and up states are symmetric.

The electron density values are greatly larger than zero even in the interstitial regions, which presents the strong metallic characteristic of Cr–Si compounds.^[28,29] For all chromium silicides, the d band of Cr dominates the total state density, because the variation of d bond corresponds most to the total state density. Their Fermi level lies at the shoulder of the TDOS peak, which means that they are all stable. It is consistent with the results of binding energies and formation enthalpy.

Mulliken population analysis is one parameter to reveal the material interior chemical bonding characteristics. The average bonding length and average population can be calculated from the following equations:^[30]

Here is the average bond length and is the mean bond population. The and represent the total number of i bonds in the cell and the bond length of i type, respectively. The population analysis results are listed in Table 2. For all of the Cr–Si compounds, Si carries the positive charges and the values vary from 0.19e to 0.42e. The Cr₅Si₃ shows the stronger ionicity due to the fact that the largest negative charges are carried by Cr atoms of it. In the Cr–Si compounds, there are three kinds of Si–Cr, Cr–Cr, and Si–Si bonds. All compounds have Si–Cr bonds, which have positive values for the average population. The average population of Si–Cr bonds is larger in each of Cr₃Si and CrSi, implying the Si–Cr bond is strong. In the case of CrSi₂, both the Si–Cr and Si–Si bonds have positive values for the overlap population, indicating a covalent bonding character. In addition, the Cr–Cr bond has a negative overlap population in the Cr–Si compound, which implies that there exists an anti-bonding effect in the chromium-silicon compound.

From Fig. 4, it can be seen that although the Cr-d orbit plays a major role in TDOS from −4 eV to 4 eV, the Cr-p orbit plays a major role in a range from 8 eV to 16 eV. The p band of Si located near the Fermi level overlaps with the orbit of Cr to some extent, which implies a covalent interaction between Si and Cr atoms p–d orbital hybridization. Therefore, strong covalent bonding exists among these compounds, which is advantageous for improving their high-temperature creep strength, shear strength, and hardness. The valence band of the orbit and the valence band of the top of the orbit are mainly composed of the p orbit and the d orbit of the Cr atom, but the contribution of the s and p orbits of the Cr atom cannot be ignored. The s orbit clearly illustrates that the evolutions of the silicon content forbidden bands of metal-rich compounds are formed in the Si-s orbit from −8 eV to −7 eV, whereas in disilicides from −12 eV to −4 eV, Si-s band with p bands shows the continuity of the TDOS. In addition, the s orbit and p orbit of Si contribute significantly to TDOS in the conduction band for Cr₅Si₃, indicating that its conductivity is higher than that of other Cr_xSi_y compounds.

Figure 5 shows the electron density distributions in Cr–Al binary compounds. As can be seen from Figs. 5(a), 5(c), 5(e), and 5(g), all Cr and Si atoms in the core region have smaller gap regions and larger values in color bar. In these figures, the Cr and Si atoms show the covalent bond and elongated outline along the Si–Si bond axis, indicating that they have covalent interactions. From Cr₃Si to CrSi₂, there are covalent bonds between Si atoms. We can also find more detailed information about the chemical bond properties from the different distributions of electron density in Figs. 5(b), 5(d), 5(f), and 5(h). Some non-localized electrons pass through the circular profile of the region between Cr and Si atoms, which means that they have covalent and metallic bonds. The blue areas surrounding the Cr and Si atoms suggest the existence of covalent bonds. Chromium silicon compounds possess covalent bonds and metal bonds, which makes the metal silicide atoms have good plasticity and high atomic binding. In addition, we also find the anti-bonding effects of chromium-silicon compounds, that is, chromium-chromium bonding. Based on the above discussion, we conclude that the combination of these silicides is a combination of covalent and metal bonds, resulting in a high melting point, high hardness, and high temperature oxidation resistance.

3.3. Mechanical stability and elastic properties

Based on the generalized Hook law, the elastic constants of Cr–Si compounds are calculated by using the stress-strain method. The structure of the crystal is calculated by several different strain modes, and the corresponding Cauchy stress tensor is evaluated.

According to these strain–stress relationships in Eq. (5),^[31] the related elastic constants are determined.

The symmetry of crystal has a dominant role in affecting the total number of independent elastic constants. In the equation, represents the elastic constant, the normal stress, the shear stress, the shearing strain, and the corresponding normal. When the system is highly symmetric, the required calculations can greatly reduce the different strain modes.

The calculated elastic constants of the Cr–Si system are summarized in Table 2. The stabilities of these compounds are studied from the mechanical properties.^[32] Based on Born-Huang’s lattice dynamics theory, the strain energy of the mechanical stability of a structure must be positive in one of the conditions for any uniform elastic deformation as indicated below.^[30,33,34]

For cubic system (for Cr₃Si, CrSi):

for tetragonal system (for Cr₅Si₃):

for hexagonal system (for CrSi₂):

From Table 3, we find that the calculated elastic constant values are consistent with the above criteria, indicating that all Cr–Si compounds are elastically stable. Moreover, Table 3 shows the comparisons among the results obtained in this paper, the results obtained by other methods, and the experimental results. In Cr–Si binary compounds, the Cr₃Si obtained , and (the same value 480.8 GPa) are larger elastic constants than other compounds’, showing that they are very incompressible under uniaxial stress along the , or () axis. The , and represent the shear moduli on the (100), (010), and (001) crystal plane, respectively. In Table 2, the values for CrSi₂ have the largest , and C₆₆ value of 170.2 GPa, 170.2 GPa, and 183.2 GPa, respectively.

The bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (σ) are calculated using Eqs. (9)–(12), and the polycrystalline crystals are estimated with independent single crystal elastic constants according to the Viogt–Reuss–Hill (VGH) approximation.^[39]

The value of VRH refers to the average of the Reuss and Viogt, and the value of Reuss and the value of Viogt represent the maximum and minimum value of the elastic modulus, respectively (Eqs. (6)–(11)).^[41,42,42] The calculated values of bulk modulus (B), shear modulus (G), Young’s modulus (E), and Poisson’s ratio (σ) of the Cr–Si compounds are shown in Table 4. To the best of our knowledge, the bulk modulus reflects the ability to compress a solid under hydrostatic pressure. All the Cr_xSi_y compounds each present a large bulk modulus. Therefore, they can potentially be superhard materials. The bulk modulus of the Cr₃Si compound with the lowest Si content is highest (251.0 GPa). The shear modulus and Young’s modulus of CrSi₂ with the highest Si content are largest (169.5 GPa and 394.9 GPa, respectively). Generally speaking, the values of hardness and stiffness of the compounds are positively correlated with the value of shear modulus and Young’s modulus. As shown in Table 3, the Poisson ratios of the Cr–Si system’s values are calculated, which range from 0.164 to 0.239, illustrating that the covalent bonds in these structures are more directional. The results show that the brittle Cr–Si compounds have weak metal properties. The Poisson ratio greater or less than 0.25 indicates the toughness or brittleness of the compounds. The value of B/G is used to illustrate the brittleness and toughness of the compound.^[43] The brittle behaves as the Cr–Si binary compound when the B/G ratio is less than the critical value of 1.75. Furthermore, Cr₃Si with the lowest Si content has the greatest B/G value of 1.59, while CrSi₂ with the highest Si atom content has the smallest B/G value of 1.16 of the Cr–B compound. The B/G value decreases with the increase of Si content, which indicates that the compounds exhibit more brittle behaviour.

In order to depict the mechanical properties of the Cr–B binary compounds, the elastic constant curves with the change of Si content are listed in Figs. 6 and 7. It can be seen that and shear modulus (G) exhibit the same change trend, but the bulk modulus and have no clear relationship. The Young modulus and () present a similar variation tendency as shown in Fig. 7.

3.4. Debye temperature

The specific heat, elastic constant, melting point, and chemical bonding properties of the compound can be reflected by the Debye temperature.^[44] In this subsection, Debye temperature (HD) and average sound velocity (v_m) can be described by the following relations:^[31,45,46]

where is the Debye temperature; h and k_B are the Planck and Boltzmann constant, respectively; N_A and n are the Avogadro constant and the total number of atoms per formula, respectively; M represents the molecular weight per formula; v_s, v_l, and ρ are the transverse sound velocity, the longitudinal sound velocity, and the theoretical density, respectively; G is the shear modulus, and B represent the bulk modulus values.^[47–49] The calculated results of sound velocity and Debye temperature for Cr–Si compounds are shown in Table 5. The Debye temperature can be reflected by the strength of covalent bond in the solid.^[50] Table 5 shows that the CrSi₂ has the largest value (262.0 K) and Cr₅Si₃ has the lowest value (216.8 K). Therefore, it is concluded that the CrSi₂ has greater covalent bonds than other silicides and the Cr₅Si₃ has the strongest metallic character because of the highest B/G ratio and lowest value. In addition, as shown in Fig. 8, the value of silicides have an opposite trend to the Poisson ratio, which may be due to the fact that the compounds bonding features exhibit the effect trend of Poisson’s ratio and Debye temperature. Moreover, the calculated average sound velocities of the compounds are larger, which are affected by their large elastic modulus and low density as well as the density, and the bulk modulus and shear modulus affect the v₁ and v_s values.

3.5. Mechanical anisotropy

The mechanical anisotropy is essential in the applications of Cr–Si materials. Micro cracks form not only on the surface, but also inside of the material. The formation and propagation of micro cracks relate to anisotropy. Cr–Si compounds act as one of the structural materials used in elevated temperature, and their mechanical anisotropy must be investigated. The following equations are combined to describe the degree of anisotropy, a number of parameters, including the anisotropic index and ), the shear anisotropic factors (, , and ) as well as universal anisotropic index ():^[42,45]

where G_V and B_V are the shear modulus and bulk modulus estimated by the Voigt method, and G_R and B_R are the shear modulus and bulk modulus estimated by the Reuss method, respectively. The unity value of shear anisotropy indicates the isotropic extent in the crystal. For example, B and G each have an approximate value, which expresses that the value of A^U is zero. The highly mechanical anisotropic properties are due to the greater anisotropy index (A^U).

Table 6 shows that Cr₅Si₃ has the largest value and Cr₃Si has the smallest AU value. The strong anisotropy is due to the fact that the value is large. The universal anisotropic index () can provide the corresponding results for the mechanical anisotropy. Therefore, the order of elastic anisotropy of Cr–Si compounds is as follows: . Furthermore, the largest value of is only 0.002% for all compounds, which implies that the bulk modulus anisotropy of a compound is weak. The shear modulus is more dependent on the direction which is confirmed by , , , and , but their functions are different. In different crystal planes, the values of , , and show the anisotropies of the shear modulus. The CrSi₂ ( and value are both 1.11) has greater anisotropic indices of the shear modulus in the (100) and (010) plane respectively and the Cr₅Si₃ ( value is 1.05) has the strongest anisotropic index of the shear modulus in the (001) plane.

Young’s modulus curved surface of three-dimensions (3D) would explain why the anisotropic properties are so much simpler. The Young modulus is calculated on the basis of the following formulas:^[51–53]

for the cubic crystal (for Cr₃Si, CrSi):

for the tetragonal crystal (for Cr₅Si₃) and hexagonal crystal (for CrSi₂):

where the values are the elastic compliance constants, and , and are the directional cosines. The contour plots of the Young modulus of these Cr–Si compounds are illustrated in Fig. 9. We find that all the compounds show a strong anisotropic character in Young’s modulus. According to the 3D curved surface and the color bar, we can see that the stronger the color, the greater the Young modulus is. However, the 3D curved surface of CrSi₂ is close to an ellipsoid that is different from others. It indicates that the Young modulus of CrSi₂ has weaker anisotropy than other compounds. Furthermore, plots of the Young modulus at different crystal planes of a crystal in more detail show the Cr–Si compounds’ anisotropic properties as shown in Fig. 10. On the (100), (001), and (110) planes. The Young modulus is highly dependent on the direction. On the (100) plane direction, CrSi₂ exhibits the maximum value of Young’s modulus in the [010] and [001] directions.

Moreover, in the [010] and [100] directions, the Cr₃Si exhibits the maximum value of Young’s modulus, and Cr₅Si₃ shows the smallest Young’s modulus on the (001) plane. In addition, the largest values of Young’s modulus for Cr₅Si₃ and CrSi₂ deviate from their principal axes at the (110) plane. In addition, it is found that the CrSi shows the smallest Young’s modulus in the [010] and [100] directions on the (001) plane. On the (001) and (110) planes, the planar contours of CrSi₂ seems to be a spherical shape, which means that the Young modulus of CrSi₂ has a weaker anisotropy than other compounds’.