First-Principles Study of Vacancies in Ti3SiC2 and Ti3AlC2

MAX phase materials have attracted increased attention due to their unique combination of ceramic and metallic properties. In this study, the properties of vacancies in Ti3AlC2 and Ti3SiC2, which are two of the most widely studied MAX phases, were investigated using first-principles calculations. Our calculations indicate that the stabilities of vacancies in Ti3SiC2 and Ti3AlC2 differ greatly from those previously reported for Cr2AlC. The order of the formation energies of vacancies is VTi(a) > VTi(b) > VC > VA for both Ti3SiC2 and Ti3AlC2. Although the diffusion barriers for Ti3SiC2 and Ti3AlC2 are similar (~0.95 eV), the properties of their vacancies are significantly different. Our results show that the vacancy–vacancy interaction is attractive in Ti3AlC2 but repulsive in Ti3SiC2. The introduction of VTi and VC vacancies results in the lattice constant c along the [0001] direction increasing for both Ti3SiC2 and Ti3AlC2. In contrast, the lattice constant c decreases significantly when VA are introduced. The different effect of VA on the lattice constants is explained by enhanced interactions of nearby Ti layers.


Introduction
The MAX phases form a large family of ternary carbides/nitrides with the general formula Mn+1AXn, where n varies from 1 to 3, M is an early transition metal, A is an A-group element, and X is C or N [1][2][3]. The MAX phases have a unique combination of the properties of ceramics and metals. Similar to metals, they are electrically and thermally conductive, easy to machine, ductile at high temperatures, and exceptionally resistant to damage and thermal shock. Like ceramics, they are elastically rigid, lightweight, and oxidation resistant.
On one hand, defects can be unintentionally introduced into Ti 3 SiC 2 and Ti 3 AlC 2 during their synthesis. These materials are refractory ceramics and considerable concentrations of vacancies and impurities are introduced during their multi-component nanolaminate formation. On the other hand, Ti 3 SiC 2 and Ti 3 AlC 2 are potential structural materials for nuclear applications. Defects are created in the lattices of Ti 3 SiC 2 and Ti 3 AlC 2 by irradiation when the incident particles displace atoms from their substitutional positions. As mentioned above, their unique properties make Ti 3 SiC 2 and Ti 3 AlC 2 suitable candidates to be adopted in applications where materials are subject to extreme environments, such as nuclear reactors [11][12][13][14]. Amorphization is an important factor to evaluate the irradiation-resistant of a material. The resistance of amorphization is dependent on the competing effects between the defect production and annihilation rate. Vacancies are the simplest defects in MAX phases. A deeper knowledge of the properties of them in Ti 3 SiC 2 and Ti 3 AlC 2 is crucial for the understanding of the defect production, annihilation process, and phase stability [15][16][17][18][19][20][21][22][23].
A large number of experimental studies [11][12][13][14][24][25][26][27][28][29][30][31] have investigated the properties of Ti 3 SiC 2 and Ti 3 AlC 2 when subjected to irradiation with heavy ions and neutrons. Nappé et al. [11] studied the defect properties of Ti 3 SiC 2 under Au-, Kr-, and Xe-ion irradiation. They observed generation of many defects in the structure and an expansion along the c axis. Liu et al. [12] reported that atomic disorder appeared in Ti 3 (Si,Al)C 2 after Kr-ion irradiation, but the typical layered structure was preserved. Whittle et al. [13] reported that Ti 3 AlC 2 and Ti 3 SiC 2 showed high tolerance to damage from Xe-ion irradiation. Hoffman et al. [14] compared the neutron irradiation tolerances of Ti 3 SiC 2 and Ti 3 AlC 2 with those of SiC and alloy 617. They concluded that Ti 3 SiC 2 and Ti 3 AlC 2 have good irradiation tolerances.
Compared with experimental methods, first-principles calculations have the advantage of enabling the study of materials at the atomic scale. Such calculations have frequently been used to predict the crystal structures and stabilities of MAX phases, and to model their defects and related properties. Wang et al. [15][16][17] systematically studied the effects of vacancies and impurities in the Ti 2 AlC phase. They calculated the stabilities of Ti 2 AlC samples with different types of vacancies. Music et al. [18] studied the vacancies in Ti 4 AlN 3 , and reported that the introduction of about 25% N vacancies in Ti 4 AlN 3 is energetically favorable. Tan et al. [19] studied vacancy diffusion in Ti 2 AlC and its impurity phase Ti 3 AlC. Du et al. [20] studied the C vacancies in Ta 4 AlC 3 , and suggested that the introducing of C vacancies decreases the phase stability. Han et al. [21] studied defect stabilities in Cr 2 AlC under different magnetic orderings.
However, although Ti 3 SiC 2 and Ti 3 AlC 2 are the most extensively studied MAX phases experimentally, theoretical investigations of their defect properties are rare. Medvedev at al. [22] studied the influence of disorder associated with the presence of vacancies on the electronic properties of Ti 3 SiC 2 . They found that the presence of C vacancies in Ti 3 SiC 2 caused local perturbations of the electronic structures. Zhao et al. [23] studied the formation energies of different defects in Ti 3 SiC 2 and Ti 3 AlC 2 . They found that replacement of Ti by Al in Ti 3 AlC 2 was more energetically favorable than replacement of Ti by Si in Ti 3 SiC 2 . These previous theoretical works mainly investigated point defect stabilities. In this work, we focus on the formation, stability, geometry, and diffusion properties of vacancies in Ti 3 SiC 2 and Ti 3 AlC 2 .

Theoretical Method
Our calculations were performed under the framework of density functional theory as implemented in the Vienna ab initio simulation package (VASP) [32,33]. The projected augmented wave method (PAW) [34] and the generalized gradient approximation (GGA) [35] were used. According to our previous study on MAX phases [19,21], the exchange and correlation energies were calculated using the Perdew−Burke−Ernzerhof (PBE) functional [36]. The wave functions were expanded in a plane-wave basis set with an energy cutoff of 400 eV. The lattice constants and internal freedom of the unit cell were fully optimized until the Hellman-Feynman forces on the atoms were less than 0.01 eV/Å. The effective charge for each atom (charge difference after bonding) is given using Bader charge analysis [37].
In order to simulate a single vacancy structure, we employed a 2 × 2 × 1 supercell, which contains 48 atoms. According to our previous studies on defects properties of MAX phases [19,21,38], the supercell has been proved to be sufficient to reproduce the defect structures. The special k-point sampling integration was used over the Brillouin zone by using the Γ-centered 5 × 5 × 5 for this supercell [39]. All these calculation setups were checked using a larger energy cutoff and k-mesh; the results of total energy and Hellmann-Feynman forces are convergent within 0.01 eV and 0.01 eV/Å, respectively.
To evaluate the energy barrier of an Al-vacancy, the climbing image nudged elastic band method (cNEB) [40,41] was employed to investigate the saddle points and minimum energy paths for vacancy diffusion from the initial state to the final state. In all transition state search calculations performed in this paper, a total of eight images were used (not including the initial and final images of each transition).

Properties of Perfect Ti 3 SiC 2 and Ti 3 AlC 2
Ti 3 SiC 2 and Ti 3 AlC 2 are both belonging to 312 phases with the same crystal symmetries, as shown in Figure 1. They are based on layers of hexagonally close-packed Ti and Al/Si layers with C occupying octahedral centers between the Ti layers. The structures of Ti 3 SiC 2 and Ti 3 AlC 2 can also be regarded as alternating stacks of two layers of edge-sharing Ti 6 C octahedra and a planar close-packed Al/Si layer. The Si/Al atoms are located in the Wyckoff 2b (0, 0, 1/4) positions and the C atoms are in 4f (1/3, 2/3, z C ) positions. There are two types of non-equivalent Ti atoms, denoted by Ti(a) and Ti(b), which are located at 4f (1/3, 2/3, z Ti ) and 2a (0, 0, 0), respectively. The calculated structural parameters for Ti 3 SiC 2 and Ti 3 AlC 2 are listed in Table 1; the experimental results are also listed for comparison. The differences between the calculated and experimental values of the lattice constants are all smaller than 1%, indicating reliable predictions by our PBE calculations. less than 0.01 eV/Å. The effective charge for each atom (charge difference after bonding) is given using Bader charge analysis [37]. In order to simulate a single vacancy structure, we employed a 2 × 2 × 1 supercell, which contains 48 atoms. According to our previous studies on defects properties of MAX phases [19,21,38], the supercell has been proved to be sufficient to reproduce the defect structures. The special k-point sampling integration was used over the Brillouin zone by using the Γ-centered 5 × 5 × 5 for this supercell [39]. All these calculation setups were checked using a larger energy cutoff and k-mesh; the results of total energy and Hellmann-Feynman forces are convergent within 0.01 eV and 0.01 eV/Å, respectively.
To evaluate the energy barrier of an Al-vacancy, the climbing image nudged elastic band method (cNEB) [40,41] was employed to investigate the saddle points and minimum energy paths for vacancy diffusion from the initial state to the final state. In all transition state search calculations performed in this paper, a total of eight images were used (not including the initial and final images of each transition).

Properties of Perfect Ti3SiC2 and Ti3AlC2
Ti3SiC2 and Ti3AlC2 are both belonging to 312 phases with the same crystal symmetries, as shown in Figure 1. They are based on layers of hexagonally close-packed Ti and Al/Si layers with C occupying octahedral centers between the Ti layers. The structures of Ti3SiC2 and Ti3AlC2 can also be regarded as alternating stacks of two layers of edge-sharing Ti6C octahedra and a planar close-packed Al/Si layer. The Si/Al atoms are located in the Wyckoff 2b (0, 0, 1/4) positions and the C atoms are in 4f (1/3, 2/3, zC) positions. There are two types of non-equivalent Ti atoms, denoted by Ti(a) and Ti(b), which are located at 4f (1/3, 2/3, zTi) and 2a (0, 0, 0), respectively. The calculated structural parameters for Ti3SiC2 and Ti3AlC2 are listed in Table 1; the experimental results are also listed for comparison. The differences between the calculated and experimental values of the lattice constants are all smaller than 1%, indicating reliable predictions by our PBE calculations.   After optimization of the crystal structures, the mechanical property parameters were calculated. In the Voigt-Reuss-Hill approximation [43][44][45], the bulk modulus B, and the shear modulus G are the average of the values obtained by Voigt and Reuss approximations [43]. The Young's modulus (E), the Poisson ratio (v), the transverse (V t ), longitudinal (V l ), and average (V a ) acoustic wave velocities, and the Debye temperature (Θ D ) can be obtained. Experimental values for Ti 3 AlC 2 have not been reported, therefore only the calculated values for Ti 3 SiC 2 are listed in Table 2 and compared with the experimental values. The results show that the calculated values are reasonably consistent with the experimental results. Table 2. Calculated elastic properties of Ti 3 SiC 2 , including the bulk modulus B, the shear modulus G, the Young's modulus E, the Poisson ration v, the acoustic wave velocities (V l , V t , V a ), and the Debye temperature Θ D . The experimental values [46] are also listed for comparison. The stabilities of vacancies at different atomic sites in crystals can be evaluated by the vacancy formation energy, which is defined as follows: where E vac (V X ) is the vacancy formation energy of atom X (X = Ti, Al, C), E tot (V X ) is the calculated total energy of a cell with defect X, E tot (perf) is the total energy of a perfect crystal without defects, and µ X is the chemical potential of X. Here, µ X is chosen as the energy of an isolated X atom for simplicity. As shown in Figure 2, for both Ti 3 SiC 2 and Ti 3 AlC 2 , A-group element vacancies have the lowest formation energies, indicating that they are easily formed. The non-equivalent Ti(a) and Ti(b) atoms have different vacancy formation energies. Figure 1 shows that the Ti(a) atoms are located between Al and C layers. Ti(a) forms covalent bonds with C atoms, but forms weak metallic bonds with Al atoms. In contrast, the Ti(b) atoms are located at the center of [Ti 6 C] octahedra, and have stronger interactions with surrounding atoms. The vacancy formation energies of the Ti(b) atoms are therefore larger than those of the Ti(a) atoms. The order of the vacancy formation energies is V Ti(a) > V Ti(b) > V C > V A . These results for Ti 3 SiC 2 and Ti 3 AlC 2 differ greatly from our previously reported results for Cr 2 AlC, in which the Al vacancies were predicted to have high formation energies and the Cr vacancies were predicted to have low formation energies [21]. The formation energy of V Al is 0.9 eV lower than that of V Si , indicating that an A-group element mono-vacancy is more easily formed in Ti 3 AlC 2 .

Vacancy-Vacancy Interactions of VA
These above calculation results indicate that VA vacancies are easily formed when Ti3SiC2 and Ti3AlC2 are in oxidizing, corrosive, and irradiation environments. The effects of VA vacancies on the phase stabilities of Ti3SiC2 and Ti3AlC2 were explored by introducing more vacancies and calculating their vacancy formation energies: where n is the concentration of VA vacancies in Ti3SiC2 and Ti3AlC2. Figure 3 shows that for Ti3AlC2 the vacancy formation energy decreases as the number of vacancies increases, indicating that existing vacancies can accelerate the formation of new vacancies. Therefore, decomposition of Ti3AlC2 can be caused by formation of a large number of vacancies in the Al layers. In contrast, the relationship between the VA content and the vacancy formation energy is different for Ti3SiC2; the vacancy formation energy increases significantly with the increasing number of vacancies. This indicates that it is difficult to introduce a new VSi near the original one because of the increased vacancy formation energy. Based on these results, it is reasonable to conclude that the interactions between nearby vacancies in Ti3AlC2 are attractive, but are repulsive in Ti3SiC2. The vacancies therefore tend to disperse in Ti3SiC2 but are accommodated in Ti3AlC2. In order to verify this conclusion, we calculated and compared the vacancy formation energies for three configurations with two vacancies introduced at different locations. The results are shown in Figure 4. For Ti3SiC2, the configuration with two vacancies located in different layers has a low formation energy. Vacancy pair formation (config.1) increases the energy by ~0.2 eV compared with

Vacancy-Vacancy Interactions of V A
These above calculation results indicate that V A vacancies are easily formed when Ti 3 SiC 2 and Ti 3 AlC 2 are in oxidizing, corrosive, and irradiation environments. The effects of V A vacancies on the phase stabilities of Ti 3 SiC 2 and Ti 3 AlC 2 were explored by introducing more vacancies and calculating their vacancy formation energies: where n is the concentration of V A vacancies in Ti 3 SiC 2 and Ti 3 AlC 2 . Figure 3 shows that for Ti 3 AlC 2 the vacancy formation energy decreases as the number of vacancies increases, indicating that existing vacancies can accelerate the formation of new vacancies. Therefore, decomposition of Ti 3 AlC 2 can be caused by formation of a large number of vacancies in the Al layers. In contrast, the relationship between the V A content and the vacancy formation energy is different for Ti 3 SiC 2 ; the vacancy formation energy increases significantly with the increasing number of vacancies. This indicates that it is difficult to introduce a new V Si near the original one because of the increased vacancy formation energy. Based on these results, it is reasonable to conclude that the interactions between nearby vacancies in Ti 3 AlC 2 are attractive, but are repulsive in Ti 3 SiC 2 . The vacancies therefore tend to disperse in Ti 3 SiC 2 but are accommodated in Ti 3 AlC 2 .

Vacancy-Vacancy Interactions of VA
These above calculation results indicate that VA vacancies are easily formed when Ti3SiC2 and Ti3AlC2 are in oxidizing, corrosive, and irradiation environments. The effects of VA vacancies on the phase stabilities of Ti3SiC2 and Ti3AlC2 were explored by introducing more vacancies and calculating their vacancy formation energies: where n is the concentration of VA vacancies in Ti3SiC2 and Ti3AlC2. Figure 3 shows that for Ti3AlC2 the vacancy formation energy decreases as the number of vacancies increases, indicating that existing vacancies can accelerate the formation of new vacancies. Therefore, decomposition of Ti3AlC2 can be caused by formation of a large number of vacancies in the Al layers. In contrast, the relationship between the VA content and the vacancy formation energy is different for Ti3SiC2; the vacancy formation energy increases significantly with the increasing number of vacancies. This indicates that it is difficult to introduce a new VSi near the original one because of the increased vacancy formation energy. Based on these results, it is reasonable to conclude that the interactions between nearby vacancies in Ti3AlC2 are attractive, but are repulsive in Ti3SiC2. The vacancies therefore tend to disperse in Ti3SiC2 but are accommodated in Ti3AlC2. In order to verify this conclusion, we calculated and compared the vacancy formation energies for three configurations with two vacancies introduced at different locations. The results are shown in Figure 4. For Ti3SiC2, the configuration with two vacancies located in different layers has a low formation energy. Vacancy pair formation (config.1) increases the energy by ~0.2 eV compared with In order to verify this conclusion, we calculated and compared the vacancy formation energies for three configurations with two vacancies introduced at different locations. The results are shown in Figure 4. For Ti 3 SiC 2 , the configuration with two vacancies located in different layers has a low formation energy. Vacancy pair formation (config.1) increases the energy by~0.2 eV compared with the other two configurations (config.2 and config.3). In contrast, config.1 is energetically more favorable for Ti 3 AlC 2 . Therefore, Ti 3 SiC 2 should be more stable than Ti 3 AlC 2 in a corrosive environment. the other two configurations (config.2 and config.3). In contrast, config.1 is energetically more favorable for Ti3AlC2. Therefore, Ti3SiC2 should be more stable than Ti3AlC2 in a corrosive environment.

Diffusion of VA Vacancies
It is well known that the Al/Si atoms move in MAX phases predominantly by vacancy-mediated diffusion [19,21]. To ensure that the supercell was sufficiently large to avoid the influence of adjacent cells, a √3 × 2√3 × 1 supercell was used to calculate the diffusion barrier. The obtained values are consistent with those obtained using a 2 × 2 × 1 supercell.
The calculated diffusion barriers (Bdiff) for Si/Al in Ti3SiC2 and Ti3AlC2 are less than 1 eV; these are close to the self-diffusion barriers of many metals, as shown in Table 3. The diffusion of vacancies along the (0001) plane can therefore occur frequently in these two materials. As mentioned previously, the interactions of vacancies in Ti3SiC2 are repulsive, whereas they are attractive in Ti3AlC2. A new vacancy will therefore diffuse away from an existing vacancy in Ti3SiC2; this does not greatly affect the stability of the material. In contrast, the low diffusion barrier indicates that vacancies in Ti3AlC2 tend to be accommodated. A large number of vacancies may therefore lead to decomposition of the material. The diffusion of atoms in the corresponding free-standing Si/Al layers was also studied using the same method shown in Figure 5. The diffusion barriers in free-standing layers (~0.2 eV) are clearly different from those in the Si/Al layers of MAX phases (~0.95 eV). These results indicate that the main contribution to the barrier is the interaction between the Al/Si and Ti layers, rather than the interaction in the Si/Al layers.

Diffusion of V A Vacancies
It is well known that the Al/Si atoms move in MAX phases predominantly by vacancy-mediated diffusion [19,21]. To ensure that the supercell was sufficiently large to avoid the influence of adjacent cells, a √ 3 × 2 √ 3 × 1 supercell was used to calculate the diffusion barrier. The obtained values are consistent with those obtained using a 2 × 2 × 1 supercell.
The calculated diffusion barriers (B diff ) for Si/Al in Ti 3 SiC 2 and Ti 3 AlC 2 are less than 1 eV; these are close to the self-diffusion barriers of many metals, as shown in Table 3. The diffusion of vacancies along the (0001) plane can therefore occur frequently in these two materials. As mentioned previously, the interactions of vacancies in Ti 3 SiC 2 are repulsive, whereas they are attractive in Ti 3 AlC 2 . A new vacancy will therefore diffuse away from an existing vacancy in Ti 3 SiC 2 ; this does not greatly affect the stability of the material. In contrast, the low diffusion barrier indicates that vacancies in Ti 3 AlC 2 tend to be accommodated. A large number of vacancies may therefore lead to decomposition of the material. The diffusion of atoms in the corresponding free-standing Si/Al layers was also studied using the same method shown in Figure 5. The diffusion barriers in free-standing layers (~0.2 eV) are clearly different from those in the Si/Al layers of MAX phases (~0.95 eV). These results indicate that the main contribution to the barrier is the interaction between the Al/Si and Ti layers, rather than the interaction in the Si/Al layers.

Effects of Vacancies on Lattice Constants
Defects in a material can lead to changes in the lattice constants. For example, irradiation of nuclear graphite increases the lattice constant c along the [0001] direction, and decreases the lattice constants a and b in the (0001) plane. This is because of the large numbers of interstitial carbons in the graphite interlayers. In this work, the effects of vacancies on the lattice constants of Ti3SiC2 and Ti3AlC2 were investigated. Figure 6 shows the trends in the changes in the lattice constants of Ti3SiC2 and Ti3AlC2 with increasing the number of vacancies in the supercell. The introduction of vacancies increases the lattice constant a and decreases c. The change in a is negligible, but a significant change in c is observed along the [0001] direction. The lattice constant changes for Ti3SiC2 are larger than those for Ti3AlC2. The lattice constant changes induced by other types of vacancies were also calculated. The results for VTi and VC are the opposite of those for VA. As shown in Figure 7, when VTi and VC vacancies are introduced, the lattice constant a decreases and c increases. The effects of VA and VTi/VC on the lattice constants differ because the interactions between the corresponding atoms and their surrounding atoms are different. In the formation of VTi and VC, the strong Ti-C covalent bond is broken; this is the driving force behind the decrease in the lattice constant in the (0001) plane. In the

Effects of Vacancies on Lattice Constants
Defects in a material can lead to changes in the lattice constants. For example, irradiation of nuclear graphite increases the lattice constant c along the [0001] direction, and decreases the lattice constants a and b in the (0001) plane. This is because of the large numbers of interstitial carbons in the graphite interlayers. In this work, the effects of vacancies on the lattice constants of Ti 3 SiC 2 and Ti 3 AlC 2 were investigated. Figure 6 shows the trends in the changes in the lattice constants of Ti 3 SiC 2 and Ti 3 AlC 2 with increasing the number of vacancies in the supercell. The introduction of vacancies increases the lattice constant a and decreases c. The change in a is negligible, but a significant change in c is observed along the [0001] direction. The lattice constant changes for Ti 3 SiC 2 are larger than those for Ti 3 AlC 2 .

Effects of Vacancies on Lattice Constants
Defects in a material can lead to changes in the lattice constants. For example, irradiation of nuclear graphite increases the lattice constant c along the [0001] direction, and decreases the lattice constants a and b in the (0001) plane. This is because of the large numbers of interstitial carbons in the graphite interlayers. In this work, the effects of vacancies on the lattice constants of Ti3SiC2 and Ti3AlC2 were investigated. Figure 6 shows the trends in the changes in the lattice constants of Ti3SiC2 and Ti3AlC2 with increasing the number of vacancies in the supercell. The introduction of vacancies increases the lattice constant a and decreases c. The change in a is negligible, but a significant change in c is observed along the [0001] direction. The lattice constant changes for Ti3SiC2 are larger than those for Ti3AlC2. The lattice constant changes induced by other types of vacancies were also calculated. The results for VTi and VC are the opposite of those for VA. As shown in Figure 7, when VTi and VC vacancies are introduced, the lattice constant a decreases and c increases. The effects of VA and VTi/VC on the lattice constants differ because the interactions between the corresponding atoms and their surrounding atoms are different. In the formation of VTi and VC, the strong Ti-C covalent bond is broken; this is the driving force behind the decrease in the lattice constant in the (0001) plane. In the The lattice constant changes induced by other types of vacancies were also calculated. The results for V Ti and V C are the opposite of those for V A . As shown in Figure 7, when V Ti and V C vacancies are introduced, the lattice constant a decreases and c increases. The effects of V A and V Ti /V C on the lattice constants differ because the interactions between the corresponding atoms and their surrounding atoms are different. In the formation of V Ti and V C , the strong Ti-C covalent bond is broken; this is the driving force behind the decrease in the lattice constant in the (0001) plane. In the formation of V A , the bonds between Al/Si atoms and the surrounding atoms are broken. According to our previous analysis of diffusion barriers, the interactions between the Al/Si layer and the two neighboring Ti layers are stronger than the in-plane interactions for V A . The formation of V A therefore contracts the materials along the [0001] direction.
formation of VA, the bonds between Al/Si atoms and the surrounding atoms are broken. According to our previous analysis of diffusion barriers, the interactions between the Al/Si layer and the two neighboring Ti layers are stronger than the in-plane interactions for VA. The formation of VA therefore contracts the materials along the [0001] direction. To verify this conclusion, the interactions between atoms in Ti3SiC2 with vacancies were analyzed based on the deformation charge densities. As shown in Figure 8, unlike the electron density distributions in the configurations of VTi and VC, there is an electron accumulation area around the two Ti atoms neighboring VA. The electron accumulation of these two Ti atoms along the [0001] direction indicates that the interaction between them is enhanced by the Si vacancy. The effects of VSi on the lattice constants are therefore different from those of VTi and VC.  To verify this conclusion, the interactions between atoms in Ti 3 SiC 2 with vacancies were analyzed based on the deformation charge densities. As shown in Figure 8, unlike the electron density distributions in the configurations of V Ti and V C , there is an electron accumulation area around the two Ti atoms neighboring V A . The electron accumulation of these two Ti atoms along the [0001] direction indicates that the interaction between them is enhanced by the Si vacancy. The effects of V Si on the lattice constants are therefore different from those of V Ti and V C . formation of VA, the bonds between Al/Si atoms and the surrounding atoms are broken. According to our previous analysis of diffusion barriers, the interactions between the Al/Si layer and the two neighboring Ti layers are stronger than the in-plane interactions for VA. The formation of VA therefore contracts the materials along the [0001] direction. To verify this conclusion, the interactions between atoms in Ti3SiC2 with vacancies were analyzed based on the deformation charge densities. As shown in Figure 8, unlike the electron density distributions in the configurations of VTi and VC, there is an electron accumulation area around the two Ti atoms neighboring VA. The electron accumulation of these two Ti atoms along the [0001] direction indicates that the interaction between them is enhanced by the Si vacancy. The effects of VSi on the lattice constants are therefore different from those of VTi and VC.

Conclusions
In this study, the properties of vacancies in Ti 3 AlC 2 and Ti 3 SiC 2 , which are two of the most widely studied MAX phases, were investigated using first-principles calculations. Our results show that an A-group element vacancy (V A ) has the lowest formation energy, therefore the vacancy-vacancy interactions, the effects of V A on the lattice constants, and the charge redistribution of V A were studied. The formation energy of V Al is 0.9 eV lower than that of V Si , indicating that an A-group element mono-vacancy is more easily formed in Ti 3 AlC 2 . Although the diffusion barriers for Ti 3 SiC 2 and Ti 3 AlC 2 are similar (~0.95 eV), the vacancy properties are different. Our results show that the vacancy-vacancy interaction is attractive in Ti 3 AlC 2 but repulsive in Ti 3 SiC 2 . The vacancies therefore tend to disperse in Ti 3 SiC 2 but are accommodated in Ti 3 AlC 2 . Based on these results, we conclude that Ti 3 SiC 2 should be more stable than Ti 3 AlC 2 in a corrosive environment. The introduction of V Ti and V C vacancies causes the lattice constant c along the [0001] direction to increase for both Ti 3 SiC 2 and Ti 3 AlC 2 . The changes in the lattice constants caused by V A are opposite. The effect of V A on the lattice constants is explained by enhanced interactions of nearby Ti layers.