Insights to the fracture toughness, damage tolerance, electronic structure, and magnetic properties of carbides M2C (M = Fe, Cr)

The fracture toughness, damage tolerance, electronic structure, and magnetic properties of M2C (M = Fe, Cr) carbides were analyzed using first-principles calculations. Calculations of formation energy and modulus of elasticity indicate that a Cr/Fe ratio of 1/3 is a critical threshold which triggers a significant increase in the corresponding stability and related mechanical properties. Cr atomic content enhances the crack resistance, while Cr has a significantly detrimental effect on damage resistance. The electronic properties demonstrated that the Cr atom content enhances the metallic, ionic and covalent bonding. Furthermore, the reduction in the coordination number of Fe atoms is the main reason for the reduction in the local magnetic moment of the low-spin Cr atoms, which is strongly supported by the electronic structure. These studies provide detailed insights into Cr-containing carbides, providing valuable theoretical and technological information for the knowledge-based design and prediction of the mechanical properties of chromium-containing iron-based materials.


Introduction
There has been extensive research in recent years on the effects of microalloy elements on fine-grain strengthening and precipitation strengthening to improve the performances of iron-based materials [1][2][3][4]. Carbides, such as sub-stable M 2 C and M 5 C 2 and stable state M 3 C, M 7 C 3 , and M 23 C 6 , (where M = Fe, Cr, Mo, etc) are essential strengthening phases in steel [5][6][7]. The type, size, and quantity of carbides can dramatically affect the mechanical properties (such as hardness, toughness, wear resistance, creep resistance, and other unique properties) of steel materials [8][9][10]. As an important precipitation phase for strengthening the wear and fatigue properties [11] of steel materials in low temperature heat treatment, M 2 C is widely present in many steel materials [12,13]. The experimental results show that M 2 C has both orthorhombic and hexagonal crystal structures [14,15]. The ground state properties of η-Fe 2 C and ε-Fe 2 C have been extensively investigated using first-principles calculations [16,17], showing that η-Fe 2 C is mechanically and energetically more stable than ε-Fe 2 C.
In the context of the era of microalloying to strengthen the mechanical properties of materials, a variety of complex carbides represented by (Fe, M) x C y have arisen. In particular, the addition of the transition metal element Cr, which is a strong carbide forming element [18], has led to the formation of a variety of complex carbides with Fe (M 3 C, M 7 C 3 , M 23 C 6 , etc). Although the effect of the transition metal element Cr [19] on the stability of η-Fe 2 C has been investigated using first-principles calculations. However, only the case of complete substitution has been studied and the evolution of the stability of M 2 C at different Cr concentrations has not been analysed in a comprehensive and systematic way. In particular, the mechanism of the effect of chromium concentration on the fracture toughness and magnetic evolution of M 2 C remains elusive. Therefore, a theoretical study of the effect of Cr concentration on the properties and structure of M 2 C through first principles calculations is essential.
Disorder calculations, a current hot topic in theoretical calculations, are an important theoretical calculation tool for modelling solid state alloying processes [20,21]. This paper compares the computational results of orthogonal M 2 C in both disordered and ordered substitutions. A systematic study of M 2 C in the ordered state is carried out using first principles calculations. Firstly, the effect of Cr on the stability of the M 2 C phase was examined based on calculated values and formation energy formulae. Secondly, the mechanical properties of M 2 C at various Cr concentrations were investigated using the classical Voigt-Reuss-Hill method combined with elastic constant calculations. And, the differential charge, density of states, charge density, Bader charge, and Debye temperature analysis methods were used to investigate the bonding in M 2 C. Finally, the magnetic characteristics of M 2 C were evaluated at different Cr concentrations.

First-principles calculations
The first-principles calculations are based on density functional theory (DFT) using the Vienna ab initio simulation package (VASP) [22][23][24]. The exchange-correlation energy is the Perdew-Burke-Ernzerhof (PBE) functional method under generalized gradient approximation (GGA) [25], and the interaction between valence electrons and atoms is described by ultra-soft pseudopotential. There are two different crystal structures of Fe 2 C: orthorhombic and hexagonal [14], with the ground state energy being lower in an orthorhombic lattice. Therefore, the orthorhombic Fe 2 C lattice of space group P1 was chosen as the base configuration for this study. The Fe 2 C single cell in the orthogonal crystal system has four Fe atoms and two C atoms at Wyckoff positions 4g and 2a, respectively. In the model formula Fe 4−x Cr x C 2 (x = 0, 0.25, 0.5, 1, 2, 3, 4), x is the number of Fe atoms replaced by Cr atoms. For example, when x is 0.25 and 0.5, the chemical formula is Fe 3.25 Cr 0.25 C 2 and Fe 3.5 Cr 0.5 C 2 , corresponding to the supercells 1 × 2 × 2 and 2 × 1 × 1, respectively. When x is 0, 1, 2, 3, and 4, it corresponds to the number of Fe atoms or Cr atoms replacing single cells (Fe 4 C 2 , Fe 3 Cr 1 C 2 , Fe 2 Cr 2 C 2 , Fe 1 Cr 3 C 2 , and Cr 4 C 2 ). In the supercell corresponding to Fe 3.25 Cr 0.25 C 2 and Fe 3.5 Cr 0.5 C 2 , the K-point grid of the simple Brillouin zone using the Monkhorst-Pack method [26] is 11 × 6 × 9 and 5 × 12 × 18, respectively. By contrast, the K-point grid based on the single-cell calculation is 11 × 12 × 18. The crystal structures with the lowest ground state energies at different Cr concentrations are selected after sufficient structural relaxation (figure 1).
The cut-off energy of the plane wave expansion is set to 500 eV, the convergence accuracy of the maximum force acting on each atom is 1 × 10 −4 eV Å, and the convergence criterion of the relaxed energy is 1 × 10 −7 eV/ atom. The valence electron atomic orbitals of Fe, Cr, and C are 3d 5 4s 1 , 3d 7 4s 1 , and 2s 2 p 2 , respectively. After the unit cell is completely relaxed, all atoms are completely released to their equilibrium position, and the 'stressstrain' method is then used to calculate the elastic constants. The symmetry of the orthorhombic system [27] is very high and there are nine independent stiffness matrix elements. All calculations are completed under the condition of spin polarization (ISPIN = 2).

Calculation of ATAT disorder structures based on cluster expansion methods
The Automated Toolbox for Alloy Theory (ATAT) [28] integrates a variety of calculation codes, often in conjunction with first-principles calculations, to calculate the thermodynamic properties of solid-state alloys. The cluster expansion (CE) method [29, 30] is a disorder calculation method in ATAT [28] that is capable of describing ordered and disordered states during alloying in a single framework.
In order to simulate real Fe-Cr alloys, this paper constructed orthogonal M 2 C supercells with 96 atoms. The flow of the disorder calculation based on the CE method is illustrated in figure 2 below. The Massachusetts Institute of Technology ab-initio phase stability (MAPS) code [30] is integrated in ATAT as a subsystem that is completely independent of the rest of the computing environment. This method is mainly used to improve the accuracy of cluster expansion and to find the best cluster expansion. In figure 2. MAPS generates a random state lat.in file based on the provided M 2 C lattice geometry, which contains a variety of lattice information such as lattice constants and the cell basis vectors. The Ab-initio code parameters generate the first principles calculation parameter file (vasp.wrap file). The other subsystems generate the VASP input files from the lat.in file and the vasp.wrap file. MAPS reads the energy of the structure calculated by VASP and generates a new structure for the next round of calculations. In the course of each calculation, MAPS saves the last successfully executed crystal structure in a folder. After the entire program has been run, the best disordered configurations of Fe Cr C 3.75 0. 25 2 and Fe Cr C 3.5 0.5 2 can be identified from a wide range of configurations.

Phase stability calculation formula
The formation energy is often used as a measure of the stability of alloyed carbides and represents the energy absorbed or released into the compound from the elemental state by the corresponding atom during the formation of the compound [31]. The energy of formation of M 2 C at different chromium concentrations is calculated as follows: is the total energy of the unit cell at various Cr concentrations, x is the number of Fe atoms substituted by Cr, and E(Fe) E(Cr) and E(C) are the standard chemical potentials of Fe, Cr and C atoms, respectively.

Results and discussion
3.1. Structural properties and phase stability Based on the disorder calculation in section 2.2, the total energy of alloy carbides at Fe 3.75 Cr 0.25 C 2 and Fe 3.5 Cr 0.5 C 2 with different Cr concentrations has been obtained. In addition, the total energy of Fe 3.75 Cr 0.25 C 2 and Fe 3.5 Cr 0.5 C 2 under ordered substitution was also calculated and the results are shown in table 1. In terms of the total energy (table 1) presented for both ordered and disordered substitution methods for orthogonal M 2 C, the ordered substitution result is slightly better than that in the disordered structure. Based on the above analysis, this paper will eventually follow an ordered substitution approach to develop a systematic study of orthogonal M 2 C at different Cr concentrations.
Since the Fe atoms in Fe 2 C occupy only Wyckoff position 4g in the orthorhombic crystal system, there is no need to consider the positional preference of the atoms when the Fe is replaced by Cr. The equilibrium lattice constants, atomic positions, and cell volumes of compounds under complete relaxation were calculated to elucidate the ground state properties of Fe 4−x Cr x C 2 at different Cr concentrations. Previous calculations and experimental results were also taken into account (table 2). The maximum difference between the current analyses and previous studies is 2.8%, which is consistent with other reports [15,16]. The cell volume gradually increases and the lattice is slightly distorted after more Cr atoms enter the cell (table 2). This phenomenon is easily explained by the larger atomic radius of Cr (118 pm) replacing the smaller radius of Fe (117 pm) and squeezing the surrounding atoms, resulting in the volume expanding. In addition, the electronegativity of Cr atoms (χ Cr = 1.6) is greater than Fe atoms (χ Fe = 1.83), which will directly affect the type and strength of chemical bonds between the atoms, leading to lattice distortion during the hybridization process [32].
The calculated values of formation energy (ΔH) for the M 2 C phase are shown in figure 3. As more Cr atoms are added to the system, the formation energy of the compound decreases slowly at first (Cr/Fe < 1/3) and then sharply (Cr/Fe > 1/3). This suggests that Cr atoms can improve thermodynamic stability and that high Cr concentrations significantly affect the rate of change of formation energy.

Elastic properties and fracture toughness
3.2.1. Effect of Cr concentration on the elastic properties of M 2 C Elastic constants are essential parameters for calculating the elastic properties of a material. They characterize a material's mechanical stability, dislocation movement, and atomic bonding force. They are also used to calculate various modulus values of crystal structures, which provide a basis for understanding and predicting the mechanical properties of materials, such as wear resistance and brittleness [35,36]. The stress-strain method was used to calculate the nine independent elastic constants [27] of the various orthorhombic M 2 C carbides. Table 3 shows these are consistent with previously published values for the elastic constants of Fe 2 C [16][17][18].
Mechanical stability under the orthogonal crystal system [27] was determined from the data in table 3 according to the criteria in Supplementary Note 1. All these carbides meet the criteria for mechanical stability. The elastic constants C 11 , C 22 , and C 33 of M 2 C decrease in turn. These represent the ability of the crystal structure to resist linear compression in the directions [100], [010], and [001] respectively, so the ability of M 2 C to resist linear compression is strongest in the direction [100]. C 44 represents the ability of the crystal structure to Cell parameters  Figure 4 shows that the number of Cr atoms has a negligible effect on the elastic moduli at Cr/Fe ratios below 1/3, which remain essentially constant. At Cr/Fe ratios above 1/3, Cr has a significant effect on the moduli, which increase rapidly as the number of Cr atoms increases. Thus, the Young's, bulk, and shear modulus of M 2 C are only significantly impacted when Cr content reaches a certain level.
Poisson's ratio (v) is an indicator of the plasticity, toughness, and brittleness of a material [41]. When v is above the critical value of 0.26, the material is ductile; when v is below 0.26, it is brittle [42]. Poisson's ratio is greater than 0.26 for all the M 2 C compounds, indicating the excellent ductility of the M 2 C compounds. The addition of Cr atoms decreases the v-value of M 2 C, indicating that varying Cr content does not improve the plasticity of M 2 C. The Pugh's ratio was above the critical value [43] of 1.75 for all compounds, with Fe 2 C exhibiting the highest (2.28). This is consistent with the Poisson's ratio data (table 4).

Effect of Cr concentration on the fracture toughness
Fracture toughness is one of the critical mechanical properties which reflects the ability of a material to resist crack expansion [44]. The brittleness index M is used to investigate the damage resistance [45]. The magnitude of    x Cr x C 2 calculated using the Voigt-Reuss-Hill approximation method [39]. the damage tolerance of M 2 C is excellent compared to the damage tolerance of c-BN (13.2 μm −1/2 ) [47]. In summary, Cr content improves the fracture toughness and reduces the damage resistance of M 2 C carbides.

Effect of Cr concentration on the hardness of M 2 C
Hardness is often used to evaluate the wear resistance of materials: the larger the value, the better the resistance [48]. The Vickers hardness values for M 2 C with various Cr concentrations were calculated using the Voigt-Reuss-Hill approximation method. Hardness rises with increasing Cr concentration (figure 5), the lowest value being 9.45 GPa for Fe 2 C and the highest 16.57 GPa for Cr 2 C. The rate of increase in hardness increases significantly at higher Cr concentrations (Cr/Fe > 1/3).

Effect of Cr concentration on the anisotropy of M 2 C elasticity
Anisotropic elasticity is used to describe the mechanical properties of materials in different directions. Internal directional differences in mechanical properties explain phase transformation and the generation and propagation of microcracks [43]. Several indices used to evaluate the anisotropy of elasticity were calculated for M 2 C. The indices A 1 , A 2 , and A 3 , which evaluate the (100), (010), and (001) plane shear anisotropy [49,50], were calculated based on the elastic constants. Based on the bulk and shear modulus data, the anisotropy percentages A B for compression and A G for shear [51] and the generic elastic anisotropy [52] index A U were calculated (table 6). The influence of Cr concentration on these anisotropy indices is illustrated in figure 6. A shear anisotropy index (A 1 , A 2 , and A 3 ) of 1 proves there is no difference in the mechanical properties of M 2 C in different directions. A significant divergence from 1 implies high shear anisotropy. Figure 6 shows that the shear anisotropy of M 2 C in the (100) and (010) planes is small (close to 1), but it is large in the (001) plane (far from 1). The effect of adding Cr atoms on the shear anisotropy of M 2 C in the (100) and (010) planes is also not significant, since the values of A 1 and A 2 are unchanged. However, addition of Cr significantly decreases the A 3  value, implying that Cr reduces the shear anisotropy of M 2 C in the (001) plane. These data provide guidance for future experiments on the shearing of M 2 C along the (001) plane. The anisotropic percentage of compression and shear and the general elastic anisotropy index evaluate anisotropy relative to a value of zero: the more significant the divergence from zero, the more pronounced the anisotropy [45,53]. A B is smaller than A G for all the carbides (figure 6), indicating that M 2 C is more prone to shear than compression. The addition of Cr atoms significantly reduces the shear anisotropy and compression anisotropy of these carbides. This is supported by the effect of Cr on the general elastic anisotropy (A U ), which gradually decreases and tends toward zero as Cr atoms increase.
The effect of Cr atoms on elastic anisotropy is most significant at low Cr concentrations (Cr/Fe < 1/3). There is a slight decrease at high Cr concentrations (Cr/Fe > 1/3). That is, Cr atoms have the greatest effect on the various anisotropies of M 2 C at low concentrations, and further addition of Cr atoms does not lead to improvement of elastic anisotropy.
To visualize the changes in elastic anisotropy, 3D Young's modulus diagrams [54] of the compounds, and their projections in the (100), (010), and (001) planes are shown in figures 7(a)-(g) and 8(a)-(c), respectively. The deviation of a 3D Young's modulus from a sphere is often used to describe the elastic anisotropy of materials. Figure 7 shows that the 3D Young's modulus for M 2 C tends to become more spherical as the number of Cr atoms increases. This is consistent with the calculations of the A U values, and confirms that Cr atoms reduce the elastic anisotropy of M 2 C. The 2D Young's modulus diagrams of the three planes (figure 8) are consistent with the analysis of the shear anisotropy coefficients A 1 , A 2 , and A 3 . Figure 9 plots the differential charge density in the (001) plane for Fe 2 C, Fe 2 Cr 2 C 2 , and Cr 2 C. Differential charges can be used to analyze the electron transfer between atoms in a compound. The red regions represent electron aggregation, the blue regions represent the absence of electrons, and the yellow regions represent electron excursion. In Fe 2 C ( figure 9(a)), a large number of electrons gather near the C atoms and electron transfer is evident near the Fe atoms, indicating covalent bonding between Fe and C. In Fe 2 Cr 2 C 2 ( figure 9(b)), as Cr atoms are incorporated into M 2 C, more electrons gather between the Fe and C atoms, while the free electrons between Cr and Fe indicate the presence of metallic bonds between these atoms [55]. In Cr 2 C (figure 9(c)), as more Cr atoms are incorporated, the aggregation of electrons between the C and Cr atoms becomes more pronounced and the Cr atoms are linked by metallic bonds. Notably, the electron distribution between the Fe/Cr and C atoms in M 2 C is not symmetrical [56], indicating the presence of ionic bonds. It is clear there is a mixture of covalent, metallic, and ionic bonds in M 2 C, and the addition of Cr atoms enhances the covalent bonding. Figure 10 shows the charge density distribution along the (002) plane for carbides with various Cr concentrations. The electron density between the metal atoms in figures 10 (b) and (c) increases significantly with increasing Cr atom concentration, and this enhances the strength of the interatom metal bonds due to the low localization [57]. In Cr 2 C ( figure 10(c)), metal-carbon-metal chains are formed. This is consistent with a previous study of Cr 7 C 3 , where Cr-C-Cr chains were detected in the Fermi electron gas. The high electron density in Cr 2 C (highlighted) contributes to the formation of more robust electron chains between the metal atoms [53,58]. Thus, higher Cr concentrations enhance the strength of metal bonds in M 2 C.   Table 7 shows the electron transfer between C and metal atoms in M 2 C at different Cr concentrations. The Bader charge calculation was used, which helps to reveal the ionic properties of atoms, since the greater the electron transfer between atoms, the more pronounced their ionic bonding properties. The electrons gained by the C atom in M 2 C increased from 1.13e to 1.55e as the Cr content increased. The Cr atoms lost more electrons to the C atoms than did the Fe atoms, which means that the Cr-C bond has more ionic bonding characteristics than the Fe-C bond. This result is consistent with the results that the difference in electronegativity between atoms contributes to the formation of ionic bonds (χ Fe = 1.83, χ Cr = 1.6, χ C = 2.55) [58].

Effect of Cr atoms on chemical bonding in M 2 C
In addition to describing thermal conductivity and lattice vibrations, the Debye temperature can also characterize the strength of covalent bonds. To further examine the effects of Cr concentration on the chemical Figure 9. The differential charge density in the (001) plane of (a) Fe 2 C, (b) Fe 2 Cr 2 C 2 , and (c) Cr 2 C. Red represents electron aggregation, blue represents absence of electrons, and yellow represents the presence of free-state electrons. Electron distribution is measured between −0.01 and 0.01 e/Bohr 3 . Figure 10. Charge density (e/Å) along the (002) plane of (a) Fe 2 C, (b) Fe 2 Cr 2 C 2 , and (c) Cr 2 C. High electron densities between metal atoms are highlighted by an ellipse.

Magnetic properties
To gain insight into the magnetic characteristics of M 2 C at different Cr concentrations, the partial density of states (PDOS) and total density of states (TDOS) of M 2 C were calculated under the conditions of spin polarization ( figure 10). Fe 3.75 Cr 0.25 C 2 and Fe 3.5 Cr 0.5 C 2 are composed of four and two single cells, respectively, and have higher densities of state than the other compounds. The vertical dotted line at 0 eV in figure 10 represents the Fermi energy level. The TDOS at the Fermi energy level for all compounds is formed primarily by the strong hybridization of Fe-3d and Cr-3d orbitals. A gap of zero for the TDOS at the Fermi energy level proves that all the compounds have metallic properties [61].
The TDOS of the M 2 C compounds is composed mainly of three parts. The low valence band with energy ranging from −15 to −11.5 eV is filled primarily by C-2s orbitals. The strong hybridization of Fe (Cr)-3d and C-2p results in valence bands with energy ranging from −8 to −4 eV. Fe (Cr)-3d orbitals play a vital role in the conduction band with energy ranging from −4 to 0 eV. Therefore, the PDOS between the C-2p orbital and M-3d orbital of the transition metal atom overlap, which leads to formation of covalent bonds between the transition metals and C atoms (Consistent with the results of the analysis in section 3.3). Figures 11(a)-(g), shows that the upper and lower spin channels of the TDOS of all the compounds are asymmetric, indicating that these six compounds are magnetic. By contrast, the TDOS upper and lower spin channels of Cr 2 C in figure 11(g) are symmetrical, indicating that Cr 2 C is nonmagnetic. In addition, the central region of the PDOS corresponding to the C-2p is not the same as for M-3d, indicating that the M 2 C compound exhibits characteristics of ionic bonding [54]. Table 9 shows the calculated single cell magnetic moment and atomic magnetic moment of the M 2 C compounds, and literature values for Fe 2 C [17,50,62]. To verify the reliability of these calculations, the densities of state (integration range below the Fermi plane) of all 3d orbits of the carbide metal atoms were integrated. Integration shows that the difference between the spin rising and spin falling electrons is similar to the calculated local magnetic moments of the metal atoms. As the Cr atoms increase, the magnetic properties of the single cell of the compounds decrease from 3.15 B m (Fe 2 C) to 0 B m (Cr 2 C). The magnetic properties of the compounds originate mainly from Fe atoms, and the incorporation of Cr atoms decreases these properties. The decrease in the local magnetic moment of Fe atoms is due to the opposing magnetic properties of C and Fe atoms. The magnetic properties of Fe atoms are influenced by the Fe-C bond length; that is, the shorter the bond length, the smaller the Fe local magnetic moment [63]. Furthermore, the magnetic moment of the C atoms decreases from −0.11 B m for Fe 2 C to 0 B m for Cr 2 C due to the decreased polarization of C atoms adjacent to Cr atoms.
To analyze the connection between the local magnetic moments of Cr atoms and magnetic Fe atoms, the concept of atomic coordination numbers was introduced [64], such that CN = CN (MMA) + CN (NMMA) + CN (NMA) , where CN is the total number of atomic coordination numbers, CN (MMA) is the number of magnetic metal atomic coordination numbers, CN (NMMA) is the number of non-magnetic metal atomic coordination numbers, and CN (NMA) is the number of non-metal atomic coordination numbers. Due to the non-magnetic character of Cr atoms, only the effect of coordination of Fe and C atoms on the magnetic properties of Cr is analyzed. CN is 15 and CN (NMA) is 3 for Cr atoms in all the carbides (table 9). CN (MMA) gradually decreases from 12 to 0 with increasing Cr content. The local magnetic moment of Cr atoms decreases to 0 .    Fe 3.5 Cr 0.5 C 2 and Fe 3 CrC 2 is 10. The magnetic properties of the Cr atoms of (0.61 B m ) are smaller than those of the Cr atoms of Fe 3 Cr 1 C 2 (0.66 B m ). This is due to the inverse parallelism of the magnetic moments of the Cr atom and the C atom, while the Cr-C bond length of Fe 3.5 Cr 0.5 C 2 is much shorter.

Conclusions
The formation energy, fracture toughness, damage tolerance, electronic structure, and magnetic properties of M 2 C compounds containing various concentrations of Cr atoms were systematically investigated using first principles calculations. The bonding properties of M 2 C were also analyzed to elucidate the mechanism of the influence of Cr atoms on the properties of M 2 C. The main conclusions are as follows: (1) A Cr/Fe ratio of 1/3 is a critical threshold for M 2 C stability and elastic constant transition. When the Cr/Fe ratio is below 1/3, the effect of Cr atoms on stability is not significant, but significantly reduces the elastic anisotropy. In contrast, at Cr/Fe ratios above 1/3, Cr atoms significantly increase stability while have little effect on elastic anisotropy. In addition, the Cr atom contributes to the crack resistance, but it has an opposite effect on the damage resistance.
(2) The addition of Cr atoms enhances the bonding strength of metallic, ionic, and covalent, respectively. The comparisons provide direct evidence that all the modulus (bulk modulus, shear modulus, and Young's modulus) increases with the addition of Cr atom.
(3) The local magnetic moments of low spin Cr atoms in M 2 C decrease with increasing Cr concentration. Higher Cr concentrations significantly reduce the coordination of the Fe atoms of Cr, thus reducing the local magnetic moments of the low spin Cr atoms.