312 MAX Phases: Elastic Properties and Lithiation

Interest in the Mn+1AXn phases (M = early transition metal; A = group 13–16 elements, and X = C or N) is driven by their ceramic and metallic properties, which make them attractive candidates for numerous applications. In the present study, we use the density functional theory to calculate the elastic properties and the incorporation of lithium atoms in the 312 MAX phases. It is shown that the energy to incorporate one Li atom in Mo3SiC2, Hf3AlC2, Zr3AlC2, and Zr3SiC2 is particularly low, and thus, theoretically, these materials should be considered for battery applications.

material, as it exhibits better cycles than graphite [27,28]. However, in order to create the MXenes, the method of hydrofluoric acid (HF) etching is used, and although it removes the A layer, it also affects the bonding between the M and X layers. This creates some unwanted characteristic changes, as it sometimes affects the elastic properties of the material. Moreover, although in most cases MXenes have unique electrical characteristics, sometimes when the etching removes the A layer, the resulting structure becomes a semiconductor [29], and as a result, this MXene cannot be used as an anode material. Recently, the first 2D MAX phases (MAXenes) were demonstrated that consist of a 2D structure, which, unlike the 2D MXenes, keeps the A layer [30]. In order to examine the application of MAXenes in Li-ion batteries, it is evident that the interactions of Li in the MAX phases should be examined first in order to predict how the lithiation is affected by the A layer. Lastly, as has been indicated in previous studies, the creation of MXenes is a high-cost method [29]. From all of the above, we believe that our research is important, as the Li-doped MAX phases could have better performances in the Li-ion batteries technology than MXenes, and moreover, they have never been investigated before. The aim of the present work is to study the elastic properties and Li formation in the M3AX2 phases (M = Hf, Nb, Ta, Ti, V, Zr, and Mo; A = Si, Al, Sn, Ga, and In; X = C) using the density functional theory (DFT), and to compare our results with other related experimental and theoretical studies. Although Mo3SiC2, Hf3AlC2, Zr3AlC2, and Zr3SiC2 are of interest, it should be noted that not all of these compounds have been synthesized yet, although there are many reports that provide some of them in stable forms. Specifically, Zhou et al. [31] have synthesized an Zr3AlC2 together with Solvas-Zapata et al. [32], who synthesized Zr3(Al1−xSix)C2. Regarding Mo3SiC2, there are many theoretical reports [33] about its properties, like self-healing. However, it has not been synthesized. There are also experimental reports on stable forms of Zr3SnC2 and Hf3AlC2 from Lapauw et al. [34]. There are also some trends in thin film MAX phases [35], as well as in sol-gel methods of creating the MAX phases compounds [36], that are examined by experimentalists for the synthesis of new MAX phases. To conclude, of the four 312 materials that we propose should be examined for potential lithiation, two of them have already been synthesized in stable forms (Zr3SnC2 and Hf3AlC2) and one has been synthesized as part of a mixed structure (Zr3SiC2); accordingly, we believe that their lithiation abilities could be investigated both experimentally and theoretically. We will then propose some experimental works with the stable synthesized forms of the 312 MAX phases that could be examined, in order to see their performances as Li-ion battery anodes.

Computational Methods
CASTEP, a plane-wave DFT code, was employed for all the calculations [37,38]. The generalized gradient approximation, ultra-soft pseudopotentials [39], and the Purdew, Burke, and Ernzerhof (PBE) [40] exchange-correlation function were used. To optimize the geometry, the Broyden-Fletcher- The aim of the present work is to study the elastic properties and Li formation in the M 3 AX 2 phases (M = Hf, Nb, Ta, Ti, V, Zr, and Mo; A = Si, Al, Sn, Ga, and In; X = C) using the density functional theory (DFT), and to compare our results with other related experimental and theoretical studies. Although Mo 3 SiC 2 , Hf 3 AlC 2 , Zr 3 AlC 2, and Zr 3 SiC 2 are of interest, it should be noted that not all of these compounds have been synthesized yet, although there are many reports that provide some of them in stable forms. Specifically, Zhou et al. [31] have synthesized an Zr 3 AlC 2 together with Solvas-Zapata et al. [32], who synthesized Zr 3 (Al 1−x Si x )C 2 . Regarding Mo 3 SiC 2 , there are many theoretical reports [33] about its properties, like self-healing. However, it has not been synthesized. There are also experimental reports on stable forms of Zr 3 SnC 2 and Hf 3 AlC 2 from Lapauw et al. [34]. There are also some trends in thin film MAX phases [35], as well as in sol-gel methods of creating the MAX phases compounds [36], that are examined by experimentalists for the synthesis of new MAX phases. To conclude, of the four 312 materials that we propose should be examined for potential lithiation, two of them have already been synthesized in stable forms (Zr 3 SnC 2 and Hf 3 AlC 2 ) and one has been synthesized as part of a mixed structure (Zr 3 SiC 2 ); accordingly, we believe that their lithiation abilities could be investigated both experimentally and theoretically. We will then propose some experimental works with the stable synthesized forms of the 312 MAX phases that could be examined, in order to see their performances as Li-ion battery anodes.

Computational Methods
CASTEP, a plane-wave DFT code, was employed for all the calculations [37,38]. The generalized gradient approximation, ultra-soft pseudopotentials [39], and the Purdew, Burke, and Ernzerhof (PBE) [40] exchange-correlation function were used. To optimize the geometry, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimizer was employed and implemented in the CASTEP. The supercells contained 108 atomic sites, with a plane-wave basis set cut-off of 450 eV, 3 × 3 × 1 Monkhorst-Pack (MP) [41]. The Li interstitial was placed at all possible sites. After an extensive search for all the possible sites, we found all the minimum energy positions of the Li interstitials. The minimum energy sites are presented in Figure 2. For the elastic properties, a unit cell was considered, with a plane-wave energy cut-off of 550 eV and with 18 × 18 × 2k-point mesh. Goldfarb-Shanno (BFGS) minimizer was employed and implemented in the CASTEP. The supercells contained 108 atomic sites, with a plane-wave basis set cut-off of 450 eV, 3 × 3 × 1 Monkhorst-Pack (MP) [41]. The Li interstitial was placed at all possible sites. After an extensive search for all the possible sites, we found all the minimum energy positions of the Li interstitials. The minimum energy sites are presented in Figure 2. For the elastic properties, a unit cell was considered, with a planewave energy cut-off of 550 eV and with 18 × 18 × 2k-point mesh.

Elastic Properties
The mechanical behavior of materials is dependent on their elastic constants. Moreover, the elastic properties of the materials are linked to the bonding characteristics, so the information that the elastic constants provide is also connected to the chemical bonds of the atoms of the solid. The MAX compounds have hexagonal crystal structures [5,42]. Therefore, the 312 MAX phases have six different elastic constants: c11, c12, c13, c33, c44, and c66. Only the first five of them are independent, taking into account that c66 = (c11 − c12)/2. In order for the MAX compounds to be dynamically stable, the following conditions must be met [43]: The calculated results for the selected 312 MAX phases have been investigated in previous studies [44,45]. In Τable 1, we present the elastic properties of the 312 MAX phases. It is seen that the above conditions are met, and so the studied MAX phases are mechanically stable.
The elastic stiffness of a solid, regarding the (100) <100> strain, is calculated by the c11 constant. Thus, V3AlC2 is the stiffest. On the other hand, Zr3AlC2 and Ti3InC2 are the least stiff. The c12 elastic constant is a measure of the deformation of the material in the (110) plane along the <100> direction. Therefore, Ti3AlC2 is the most easily deformed. The c12 and c13 values indicate that when force is applied along the a-crystallographic axis, Ti3AlC2, Ti3InC2, Ti3GaC2, and V3AlC2 are easier to shear along the b and c axes than the other MAX compounds in Table 1. Lastly, the lower value of c33 for Zr3SnC2 results in the conclusion that it is easiest to deform via <001> compression under uniaxial stress.
Focusing on the bulk elastic parameters, in Table 1, the bulk modulus B, Young's modulus Y, and the shear modulus G have been calculated. It is evident that the Zr3SnC2 has the lowest value of B, and, as a consequence, it has the lowest resistance under compression. Conversely, Mo3SiC2, which has the highest value, has the highest resistance to compression. The shear modulus G has the lowest value in Zr3SnC2, and so this MAX phase is more prone to shape change than the others. The Young's modulus Y, which is a measure of the stress required for deformation, has the lowest value in the Zr3SnC2 MAX phase, compared to the other MAX phases in Table 1 (also refer to [44][45][46][47][48][49][50][51][52]).

Elastic Properties
The mechanical behavior of materials is dependent on their elastic constants. Moreover, the elastic properties of the materials are linked to the bonding characteristics, so the information that the elastic constants provide is also connected to the chemical bonds of the atoms of the solid. The MAX compounds have hexagonal crystal structures [5,42]. Therefore, the 312 MAX phases have six different elastic constants: c 11 , c 12 , c 13 , c 33 , c 44, and c 66 . Only the first five of them are independent, taking into account that c 66 = (c 11 − c 12 )/2. In order for the MAX compounds to be dynamically stable, the following conditions must be met [43]: c 11 > 0, c 33 > 0, c 44 > 0, (c 11 + c 12 ) c 33 > 2(c 13 ) 2 , and (c 11 − c 12 ) > 0 (1) The calculated results for the selected 312 MAX phases have been investigated in previous studies [44,45]. In Table 1, we present the elastic properties of the 312 MAX phases. It is seen that the above conditions are met, and so the studied MAX phases are mechanically stable.
The elastic stiffness of a solid, regarding the (100) <100> strain, is calculated by the c 11 constant. Thus, V 3 AlC 2 is the stiffest. On the other hand, Zr 3 AlC 2 and Ti 3 InC 2 are the least stiff. The c 12 elastic constant is a measure of the deformation of the material in the (110) plane along the <100> direction. Therefore, Ti 3 AlC 2 is the most easily deformed. The c 12 and c 13 values indicate that when force is applied along the a-crystallographic axis, Ti 3 AlC 2 , Ti 3 InC 2 , Ti 3 GaC 2, and V 3 AlC 2 are easier to shear along the b and c axes than the other MAX compounds in Table 1. Lastly, the lower value of c 33 for Zr 3 SnC 2 results in the conclusion that it is easiest to deform via <001> compression under uniaxial stress.
Focusing on the bulk elastic parameters, in Table 1, the bulk modulus B, Young's modulus Y, and the shear modulus G have been calculated. It is evident that the Zr 3 SnC 2 has the lowest value of B, and, as a consequence, it has the lowest resistance under compression. Conversely, Mo 3 SiC 2, which has the highest value, has the highest resistance to compression. The shear modulus G has the lowest value in Zr 3 SnC 2, and so this MAX phase is more prone to shape change than the others. The Young's modulus Y, which is a measure of the stress required for deformation, has the lowest value in the Zr 3 SnC 2 MAX phase, compared to the other MAX phases in Table 1 (also refer to [44][45][46][47][48][49][50][51][52]). In order to gain information about the brittle or ductile failure of the MAX phases, the Pugh's modulus (B/G) is used [53]. More analytically, when the Pugh's modulus exceeds 1.75, the material is characterized as ductile, which means that a crack progresses slowly when plastic deformation occurs. Conversely, in brittle materials, cracks extend rapidly with little applied stress. According to the results of Table 1, all of the MAX phases studied are brittle, except for Mo 3 SiC 2 and Ta 3 SiC 2 . Another important parameter is the anisotropy factor k c /k a = (c 11 + c 12 − 2c 13 )/(c 33 − c 13 ), which indicates whether the MAX phase is more compressible along the a-or c-axis. It is obvious that Ti 3 SiC 2 , Mo 3 SiC 2 , Hf 3 SnC 2 , Hf 3 SiC 2 , V 3 AlC 2 , and Ta 3 SiC 2 are the only MAX compounds of those studied where compression on the a-axis has almost the same value as on the c-axis.
The Poisson's ratio is another important constant that informs us if the material is a central-force solid or a non-central-force solid [48], and also classifies the materials as brittle or ductile [54,55]. If the Poisson's ratio is between 0.25 and 0.50, then the material is a central-force solid. Otherwise, it is a non-central-force solid. Furthermore, if the Poisson's ratio is more than 0.26, then the solid is ductile, and if it has a lower value, it is brittle. The calculated results show that all the studied MAX phases in Table 1 are non-central-force and brittle, except for Mo 3 SiC 2 and Ta 3 SiC 2 , which are central-force.
The elastic anisotropy, A, is an important description meaning that a body cannot develop the same strain independently of the direction in which the stress is applied. The elastic anisotropy factor indicates how the elastic properties of a solid are dependent on the direction of the stress. Additionally, the elastic anisotropy is connected with the thermal expansion and the crystal microcracks [56]. For the MAX phase systems that are hexagonal, the elastic anisotropy factor is calculated from the equation A = 4c 44 /(c 11 + c 33 − 2c 13 ), and if A = 1, the crystal is isotropic. The results of Table 1 characterize Mo 3 SiC 2 and Ta 3 SiC 2 as being more elastically anisotropic than the other MAX phases, and because the value of the elastic anisotropy factor of Hf 3 SnC 2 is almost 1, this MAX phase is elastically isotropic.
As regards the elastic properties of the MXenes, focusing on the research of Ge et al. [57] on the superconducting and high hardness of Mo 3 C 2 , it has been proved that it is a brittle material with a B/G of 2.35. We calculated that the hypothetical Mo 3 SiC 2 is a brittle material, and we found that the B/G is slightly lower than the similar MXene, with a value of 2.11. As a result, it is seen that the Mo 3 SiC 2 is not as brittle as Mo 3 C 2 , which makes it more difficult to crack during "diffusion". As regards the Ti 3 C 2 , Borysiuk et al. [58] used the molecular dynamics method in order to predict the elastic properties of the Ti n+1 C n MXenes, and they calculated, in the case of Ti 3 C 2, a value of 502 GPa for the Young's modulus, while Bai calculated a c 11 constant equal to 523 GPa [59]. Compared to our results, it is evident that for every one of the Ti 3 AC 2 MAX compounds, the Young modulus and the elastic constant c 11 have much lower values. Focusing on the Zr 3 C 2 MXene, Xie et al. [60] made a theoretical study on the elastic properties of that MXene, and compared to our 312 MAX phases with Zr-A-C, it is seen that only the Zr 3 SnC 2 is less stiff. As a result, it will be softer and more easily machinable than the Zr 3 C 2 . From all of the above, it is evident that the majority of our MAX phases are performing with better elasticity characteristics and can be more easily manipulated than the similar MXenes, in order to be used as anodes in Li-ion batteries (LIBs).

Lithiation
The formation energy to incorporate a Li atom in the MAX phase, with ∆H for Li-intercalated systems, is defined by the following equation: where E (with xLi) and E (without Li) are the energies of the system with and without Li atoms. Herein, we used one Li atom as an interstitial, and in result, x = 1. Also, E (Li) is the total energy of a single Li atom (here, it is 192.029 eV). In order to calculate the energy of the one Li atom, we performed a calculation for a supercell consisting of 67 Li atoms, and we performed geometry relaxation. We thus calculated the energy of the supercell, and we divided it with 67 in order to find the energy of the one atom. In Table 2, the formation energies of the lithiated 312 MAX phases are provided. From Table 2, it is evident that the lithiation of the 312 MAX phases studied here is endothermic, which reflects instability. However, Mo 3 SiC 2 , Hf 3 AlC 2 , Zr 3 AlC 2 , and Zr 3 SiC 2 exhibit Li formation energies less than 0.5 Ev, making the incorporation of Li in the MAX lattice feasible, and they are even lower than those of MAX materials considered in previous works [61][62][63]. This is extremely important, as MXenes are generally considered better candidates for battery applications, compared to the MAX phases. Nevertheless, the present work identifies four materials (Mo 3 SiC 2 , Hf 3 AlC 2 , Zr 3 AlC 2 , and Zr 3 SiC 2 ) that are potentially important for such applications. It should be stressed that previous experimental works have only identified oxygen-doped Ti 3 SiC 2 as having high Li-ion storage capacity, and hence, as being potentially important as an anode material for Li-ion batteries [64]. In the present study, though, we found that Ti 3 SiC 2 has a high formation energy for lithiation (refer to Table 2). This implies that doping could further decrease the Li-intercalation formation energy of Mo 3 SiC 2 , Hf 3 AlC 2 , Zr 3 AlC 2 , and Zr 3 SiC 2 , thus making them appropriate candidates for battery applications. MAX phases present potential advantages over MXenes, exhibiting better material properties (high thermal-shock resistance, elastic stiffness, melting temperatures, and electrical and thermal conductivity). They are less-complicated structures, as they do not need functional groups to be stabilized [61]. We searched the literature about the formation energy of similar lithiated MXene structures, but we could only find a report about Ti 3 C 2 where the formation energy for the lithiated structure was calculated at 4.40 eV [65]. It is obvious that this value is higher than our theoretical results, so the lithiation in the 312 MAX phases needs less energy than the above-mentioned MXene.
In an effort to link the elastic properties with the Li formation energies in the 312 MAX phases, we considered Figure 3. This was motivated by initial work on the Ti 3 AC 2 (A = Sn, Si, Ge, Ga, Al, and In) MAX phases where there is a decrease of Li formation energy with respect to the C 11 elastic constant (refer to Figure 3a). Nevertheless, when considering the whole range of the 312 MAX phases, there is no specific trend (refer to Figure 3b-d). Future theoretical work should include thermodynamic models to investigate further if the bulk properties impact the formation energies of Li in these systems [66][67][68].

Conclusions
To summarize, the mechanical behavior and the formation energy for the lithiation of the 312 MAX phases has been calculated using the density functional theory. From our calculations, it is evident that Zr3SnC2 is more prone to shape change along the b-and c-axes when stress along the aaxis is applied. Moreover, Zr3SnC2 does not need high stress in order to deform, and has low resistance to deformation under compression.
The energy to incorporate Li-ions in the 312 MAX phases is considerably high, with the exception of Mo3SiC2, Hf3AlC2, Zr3AlC2, and Zr3SiC2, for which formation energies of Li intercalation less than 0.5 eV were calculated. The Li formation energies in Mo3SiC2 and Zr3SiC2 are particularly low. However, they have not yet been synthesized. Regarding the other compounds, it could be proposed that their potential as LIBs or supercapacitor anodes should be examined.
To conclude, from the four 312 materials that we propose should be examined for potential lithiation, two of them have already been synthesized in stable forms (Zr3SnC2 and Hf3AlC2) and one has been synthesized as part of a mixed structure (Zr3SiC2), and so we believe that their lithiation ability could be investigated both experimentally and theoretically. There have been, compared to the MXenes, very few studies of the MAX phases for battery applications. Obviously, the incorporation of Li is only part of the picture, and future studies should focus on the diffusion of Li from both an experimental and theoretical viewpoint. Doping strategies should also be employed to lower the formation and migration energies of Li in the MAX phases.

Conclusions
To summarize, the mechanical behavior and the formation energy for the lithiation of the 312 MAX phases has been calculated using the density functional theory. From our calculations, it is evident that Zr 3 SnC 2 is more prone to shape change along the b-and c-axes when stress along the a-axis is applied. Moreover, Zr 3 SnC 2 does not need high stress in order to deform, and has low resistance to deformation under compression.
The energy to incorporate Li-ions in the 312 MAX phases is considerably high, with the exception of Mo 3 SiC 2 , Hf 3 AlC 2 , Zr 3 AlC 2 , and Zr 3 SiC 2 , for which formation energies of Li intercalation less than 0.5 eV were calculated. The Li formation energies in Mo 3 SiC 2 and Zr 3 SiC 2 are particularly low. However, they have not yet been synthesized. Regarding the other compounds, it could be proposed that their potential as LIBs or supercapacitor anodes should be examined.
To conclude, from the four 312 materials that we propose should be examined for potential lithiation, two of them have already been synthesized in stable forms (Zr 3 SnC 2 and Hf 3 AlC 2 ) and one has been synthesized as part of a mixed structure (Zr 3 SiC 2 ), and so we believe that their lithiation ability could be investigated both experimentally and theoretically. There have been, compared to the MXenes, very few studies of the MAX phases for battery applications. Obviously, the incorporation of Li is only part of the picture, and future studies should focus on the diffusion of Li from both an experimental and theoretical viewpoint. Doping strategies should also be employed to lower the formation and migration energies of Li in the MAX phases.
Funding: This research received no external funding.