First Principles Study of Bonding Mechanisms at the TiAl / TiO 2 Interface

: The adhesion properties of the TiAl / TiO 2 interface are estimated in dependence on interfacial layer composition and contact conﬁguration using the projector augmented wave method. It is shown that a higher value of the work of separation is obtained at the interface between the Ti-terminated TiAl(110) surface and the TiO 2 (110) O one than at that with the Al-terminated alloy. An analysis of structural and electronic factors dominating the chemical bonding at the interfaces is carried out. It is shown that low bond densities are responsible for low adhesion at both considered interfaces, which may a ﬀ ect the spallation of oxide scale from the TiAl matrix.


Introduction
Among the big family of intermetallic compounds, Ti-Al alloys attract much attention from both experimental and theoretical researchers. In most studies, γ-TiAl is investigated due to its excellent combination of mechanical properties such as low density, high specific strength, and stiffness, as well as good high-temperature creep resistance [1,2]. However, potential application of the alloy in aeronautical and space technologies is restricted by the poor high-temperature oxidation resistance, which is connected with the growth of non-protective Al 2 O 3 + TiO 2 mixed oxide layers on the internal scale [3,4]. To improve the oxidation resistance of the γ-TiAl alloy, numerous treatment techniques, including the addition of alloying elements and surface modification, have been applied [1,2]. There is an option that the oxidation resistance of γ-TiAl might be improved by avoiding the formation of TiO 2 and promoting the formation of an Al 2 O 3 protective dense layer. However, the Al chemical activity is reduced in Ti-Al alloys with increasing Ti-content. Therefore, this fact, in combination with the thermodynamic characteristics of the oxides, leads to a larger stability of the interfaces with TiO and TiO 2 rather than with Al 2 O 3 . Experiments [5][6][7][8][9][10] have shown that Nb, Mo, W, and Re benefit, whereas V, Mn, and Cu deteriorate the oxidation resistance of γ-TiAl. Since the experimental measurements are very sensitive to many factors, including temperature, partial pressure of oxygen, composition variation, and so forth, the obtained conclusions are often contradictory. It is believed that the alloying with impurities such as Nb, Mo, W, and so forth can increase the activity and diffusivity of aluminum and therefore enhance the formation of the alumina layer [1,11]. It was demonstrated in [12,13] that The energy cut-off for the plane waves was set to 550 eV. For interface calculations, we adopted a k-points mesh of 2 × 8 × 1 obtained by the Monkhorst-Pack method [37]. The total energies were converged up to 10 -5 eV. The theoretical lattice parameters of the bulk TiO 2 with rutile structure (a = 4.663 Å, c = 2.969 Å) and the γ-TiAl alloy (a = 3.977 Å, c = 4.081 Å) are in good agreement with experimental ones [38,39]. More details about structural parameters of both TiAl(110) and TiO 2 (110) surface and interface structures will be discussed in the corresponding section.
The work of separation (W sep ) or the ideal adhesion energy at the alloy-oxide interface was calculated as where E(TiAl/TiO 2 ) is the total energy of a supercell containing the multilayered slabs, E(TiAl) and E(TiO 2 ) are the total energies of the same supercell containing a single slab of the alloy or the oxide, respectively, and S is the area of the interface. In order to estimate the surface energy of stoichiometric surfaces the following formula was used: where E slab is the total energy of the supercell containing a slab and vacuum region, E bulk -the reference energy of the bulk compound per formula unit, N-the number of formula units in the slab, S-the surface area, and factor 2 corresponds to two identical surface on both slab sides.
In the case of nonstoichiometric surfaces of an A n B m compound, σ is a function of the chemical potential of any component and can be calculated as follows: where N A (N B ) is the number of atoms of element A (B), Γ A -the excess of element A at the surface with respect to B one: µ bulk A -the chemical potential of element A in elementary substance under bulk conditions, ∆µ A -he change of chemical potential of element A in A n B m compound with respect to the elementary substance bulk. It may change in the range: where ∆H f A n B m is the formation enthalpy of the compound per formula unit. More details can be found in our earlier paper [22]. Often authors draw σ as function of ∆µ/∆H f instead of ∆µ. This allows us to reach uniformity of the surface stability diagrams for different compounds.

Surface Energy
The surface energy of low-index surfaces of the γ-TiAl alloy as a function of the Ti chemical potential is shown in Figure 1a. The atomic structures of the γ-TiAl(001), (100), (110), and (111) surfaces were modeled by 11-layer thin films, whereas slabs of the TiO 2 with rutile structure contained 21-35 atomic layers in dependence on surface orientation. It is seen that the TiAl(100) surface with the stoichiometric composition is the most stable one in the Ti-rich region. At the same time, the TiAl(110) surface with Al termination has the lowest surface energy in the Al-rich and near-stoichiometry regions. In the limit of high aluminum concentrations, the surface energies of both TiAl(001) Al and TiAl(110) Al surfaces are almost the same. The values of σ for other considered surfaces are higher. One can see from Figure 1b that the stoichiometric TiO 2 (110) O surface is stable for almost the whole interval of ∆µ Ti . Only in the Ti-rich limit, the TiO 2 (110) TiO surface with TiO termination has the smallest value of σ. The surface energy of (100) O is only higher by 0.16 J/m 2 than that of (110) O .
Metals 2020, 10, x FOR PEER REVIEW 4 of 14 termination has the smallest value of σ. The surface energy of (100)O is only higher by 0.16 J/m 2 than that of (110)O. The obtained surface energies are in line with earlier data from [40][41][42][43]. For example, the surface energy of TiAl(100) is 1.70 J/m 2 , that is, higher by ~5.0% than the value of 1.62 J/m 2 [40] and lower by ~11.0% than 1.91 J/m 2 obtained in [41]. It should be noted that the latter value was calculated with the local density approximation for an exchange-correlation functional (PAW-LDA). The value of 0.46 J/m 2 in the case of the TiO2(110)O surface agrees well with 0.48 J/m 2 [43]. All calculated values of σ for TiAl and TiO2 stoichiometric surfaces in comparison with available theoretical results are summarized in Table 1. One can see that values obtained within LDA are higher by 0.3-0.5 J/m 2 than those within GGA. Thus, based on thermodynamic findings it is reasonable to consider the interface between TiAl(110)Al and TiO2(110)O surfaces. As the self-diffusion rate in the γ-TiAl alloy [44] is higher than in TiO2 [45], a Ti segregation from the alloy toward the oxide can change the composition of interfacial layers. In this connection, we consider the TiAl(110)Ti/TiO2(110)O interface as well.

Work of Separation
The TiAl(110) surface cell has parameters a1 × b1 = c × a 2 / 2 = 4.081 × 2.812 Å 2 while TiO2(110)-a2 × b2 = a 2 × c = 6.594 × 2.969 Å 2 , where a and c are lattice parameters of the corresponding compounds. It is clear that cells of (3a1 × b1) and (2a2 × b2) have enough good agreement. In this case, the misfit is δ1 = 2(2a2 − 3a1)/(2a2 + 3a1)•100% = 7.4% and δ2 = 2(b2 − b1)/(b2 + b1)•100% = 5.4%. As the bulk modulus of TiO2 is higher by ~62% than that of γ-TiAl, we used the parameters of the former to construct the interface structure. Therefore, the alloy structure was expanded by 7.7% and 5.  The obtained surface energies are in line with earlier data from [40][41][42][43]. For example, the surface energy of TiAl(100) is 1.70 J/m 2 , that is, higher by~5.0% than the value of 1.62 J/m 2 [40] and lower bỹ 11.0% than 1.91 J/m 2 obtained in [41]. It should be noted that the latter value was calculated with the local density approximation for an exchange-correlation functional (PAW-LDA). The value of 0.46 J/m 2 in the case of the TiO 2 (110) O surface agrees well with 0.48 J/m 2 [43]. All calculated values of σ for TiAl and TiO 2 stoichiometric surfaces in comparison with available theoretical results are summarized in Table 1. One can see that values obtained within LDA are higher by 0.3-0.5 J/m 2 than those within GGA. Thus, based on thermodynamic findings it is reasonable to consider the interface between TiAl(110) Al and TiO 2 (110) O surfaces. As the self-diffusion rate in the γ-TiAl alloy [44] is higher than in TiO 2 [45], a Ti segregation from the alloy toward the oxide can change the composition of interfacial layers. In this connection, we consider the TiAl(110) Ti /TiO 2 (110) O interface as well.

Work of Separation
The TiAl(110) surface cell has parameters 2 × c = 6.594 × 2.969 Å 2 , where a and c are lattice parameters of the corresponding compounds. It is clear that cells of (3a 1 × b 1 ) and (2a 2 × b 2 ) have enough good agreement. In this case, the misfit is δ 1 = 2(2a 2 − 3a 1 )/(2a 2 + 3a 1 )·100% = 7.4% and δ 2 = 2(b 2 − b 1 )/(b 2 + b 1 )·100% = 5.4%. As the bulk modulus of TiO 2 is higher by~62% than that of γ-TiAl, we used the parameters of the former to construct the interface structure. Therefore, the alloy structure was expanded by 7.7% and 5.  For the modeling of the TiAl/TiO2(110) interface, two configurations (hollow and top) of contacts with the Al-and Ti-terminated γ-TiAl(110) surface were considered. Both configurations before relaxation are presented in Figure 3. In case of the hollow configuration, one interfacial O atom (O2I in Figure 3a) locates above the TiI-1 atom of the second layer. It is a so-called H or hollow site for O adsorption on the TiAl(110) surface (see, e.g., [22]) that was a reason for the configuration denotation. The other interfacial O atom (O1I in Figure 3a) occupies the short-bridge position between two Al1I atoms. The top configuration can be obtained from the hollow one by a shift of the oxide film along the [001] direction by b2/2. As a result, the O2I atom locates now in the long-bridge position between two Al2I atoms and O1I-in the top one ( Figure 3b). The denotation of the top configuration is due to this position of the O1I atom. It should be noted that in the case of the Ti-terminated alloy surface, Al and Ti atoms of its film are swapped ( Figure S1). The initial interface distance was chosen so that the interatomic distance between the nearest interfacial atoms is slightly greater than the sum of their covalent radii.  For the modeling of the TiAl/TiO 2 (110) interface, two configurations (hollow and top) of contacts with the Al-and Ti-terminated γ-TiAl(110) surface were considered. Both configurations before relaxation are presented in Figure 3. In case of the hollow configuration, one interfacial O atom (O2 I in Figure 3a) locates above the Ti I-1 atom of the second layer. It is a so-called H or hollow site for O adsorption on the TiAl(110) surface (see, e.g., [22]) that was a reason for the configuration denotation. The other interfacial O atom (O1 I in Figure 3a) occupies the short-bridge position between two Al1 I atoms. The top configuration can be obtained from the hollow one by a shift of the oxide film along the [1] direction by b 2 /2. As a result, the O2 I atom locates now in the long-bridge position between two Al2 I atoms and O1 I -in the top one ( Figure 3b). The denotation of the top configuration is due to this position of the O1 I atom. It should be noted that in the case of the Ti-terminated alloy surface, Al and Ti atoms of its film are swapped ( Figure S1). The initial interface distance was chosen so that the interatomic distance between the nearest interfacial atoms is slightly greater than the sum of their covalent radii.  For the modeling of the TiAl/TiO2(110) interface, two configurations (hollow and top) of contacts with the Al-and Ti-terminated γ-TiAl(110) surface were considered. Both configurations before relaxation are presented in Figure 3. In case of the hollow configuration, one interfacial O atom (O2I in Figure 3a) locates above the TiI-1 atom of the second layer. It is a so-called H or hollow site for O adsorption on the TiAl(110) surface (see, e.g., [22]) that was a reason for the configuration denotation. The other interfacial O atom (O1I in Figure 3a) occupies the short-bridge position between two Al1I atoms. The top configuration can be obtained from the hollow one by a shift of the oxide film along the [001] direction by b2/2. As a result, the O2I atom locates now in the long-bridge position between two Al2I atoms and O1I-in the top one (Figure 3b). The denotation of the top configuration is due to this position of the O1I atom. It should be noted that in the case of the Ti-terminated alloy surface, Al and Ti atoms of its film are swapped ( Figure S1). The initial interface distance was chosen so that the interatomic distance between the nearest interfacial atoms is slightly greater than the sum of their covalent radii.   The calculated values of the work of separation and the interfacial distance (d) for all considered structures are given in Table 2. Note that for the d estimation, the averaged values of positions of the interfacial atoms in the oxide and the alloy were used. One can see that for the Al-terminated alloy, the obtained values of W sep are within 0.74-1.27 J/m 2 that is in line with 0.58-1.64 J/m 2 [28]. The highest value of the work of separation was calculated for the hollow configuration. In work [28], five interface configurations were considered. As the authors of [28] matched TiO 2 (110)−(1 × 1) and TiAl(110)−(2 × 1) surface cells (which results in too large misfit), a direct comparison of the present results with earlier ones is impossible. Nevertheless, in the case of the most stable configuration B(Ti-Al) in [28], there are bonds between interfacial O I and subinterfacial Ti I-1 atoms as well as between O I+1 and Al I ones. Our hollow configuration demonstrates almost similar chemical bonding. The difference in W sep for these configurations is 0.37 J/m 2 . Some discussion will be given in Section 3.3. The values of the work of separation for the interface with the Ti-terminated alloy film increase to almost twice in comparison with the previous case, but they remain substantially lower than the data from [30] (Table 2). It should be noted that the authors of [30] used almost the same surface cells as in [28], they were just doubled along the shortest distance, i.e., TiO 2 (110)−(1 × 2) and TiAl(110)−(2 × 2) cells were matched. In the present paper as well as in [30], the formation O I -Al I-1 and O I -Ti I bonds is observed. At the same time, in work [30] a huge distortion of the atomic structure of TiO 2 near the interface takes place. The latter allows us to assume that the authors adopted the alloy lattice parameters for interface modeling. As a rule, the parameters of oxide surface cell are used in such calculations [46,47]. It should be noted that enough low values of the work of separation were calculated earlier in several papers, if oxide is terminated by one oxygen layer. For example, W sep equal to 1.37 J/m 2 was obtained at the Nb(110)/Al 2 O 3 (1120) O interface [48] while the value is one order more at the Nb(110)/Al 2 O 3 (1120) 2O one [49]. In the case of the U(110)/Al 2 O 3 (1120) O interface, the value of 1.90 J/m 2 was calculated, but 11.5 J/m 2 was obtained at the U(110)/Al 2 O 3 (1120) 2O interface [50]. Analysis of both TiAl/TiO 2 interface characteristics will be performed in the next section.  Figure 3c. We emphasize that ∆ρ(r) = ρ ox + ρ me − ρ me/ox , where ρ me/ox is the total charge density of a supercell containing both alloy and oxide slabs; ρ ox and ρ me are the total charge densities of the same supercell containing a single slab of the oxide or the alloy. The section planes were chosen to give a better insight into the charge redistribution near interface bonds.

Atomic and Electronic Factors
It is seen in Figure 4a that the formation of O1 I -Al1 I bonds results in weakening of the bonds between this O1 I atom and the nearest Ti I+1 ones in the oxide. The latter can be seen by the appearance of pronounced charge depletion regions at the Ti1 I+1 -O1 I bonds that leads to charge redistribution around Ti1 I+1 atoms. In accordance with the analysis of charge states performed within DDEC6 method [51][52][53], Ti I+1 atoms lose a smaller charge than those at the clean surface (1.99 el. instead of 2.13 el.), whereas the O1 I atom gets additional 0.05 el. In comparison with that on the free oxide surface (Table 3). Moreover, the overlap population of the Ti1 I+1 -O1 I bonds decreases significantly at the TiAl(110) Al /TiO 2 (110) O interface. We emphasize that the overlap population is used for the estimation of the bond strength. It should be noted that there is a charge depletion region between Al1 I atoms in the alloy. Charge states of these atoms change from -0.24 el. on the clean surface to +0.69 el. at the interface.
Metals 2020, 10, x FOR PEER REVIEW 7 of 14 oxide surface (Table 3). Moreover, the overlap population of the Ti1I+1-O1I bonds decreases significantly at the TiAl(110)Al/TiO2(110)O interface. We emphasize that the overlap population is used for the estimation of the bond strength. It should be noted that there is a charge depletion region between Al1I atoms in the alloy. Charge states of these atoms change from -0.24 el. on the clean surface to +0.69 el. at the interface.    Figure 4b demonstrates that all Ti atoms surrounding the oxygen at the interface lose the charge: in particular, Ti2 I+1 atoms have a charge smaller by 0.12 el. than on the free surface; Ti I-1 atoms lose 0.13 el., whereas the charge of O2 I atoms increases by 0.04 el. It is seen that a large charge accumulation region occurs at O2 I -Ti I-1 bonds, which are responsible for the interface strength. The appearance of both accumulation and depletion regions at these bonds indicates a large covalent contribution in the chemical bonding. Indeed, one can see from Table 3 that the overlap population of the O2 I -Ti I-1 bond is higher than that of the O1 I -Al1 I one by 0.11 el. At the same time, ionicity of the latter bond is higher due to the large positive charge of the Al1 I atoms. It is necessary to emphasize that valence p-states of Al are less localized than Ti d-states, which allows them to be more easily involved in the interaction with oxygen. In addition, the in-plane average <∆ρ(z)> = ∆ρ(z)/S is given in Figure S2a, where ∆ρ(z) is a result of in-plane integration of ∆ρ(r) over the plane and S is the area. It illustrates where the maximum of charge accumulation/depletion is located.
Interfacial O I+1 -Al2 I bonds (Figure 4c) also have an ionic character, and a charge depletion region occurs mainly near Al2 I atoms. In spite of that, the O I+1 -Al2 I bond length is almost the same as O1 I -Al1 I ; the overlap population of the former is larger by 0.078 el. At the same time, Al2 I atoms lose a smaller charge in comparison with Al1 I (Table 3). All these features indicate that O I+1 -Al2 I has a less ionic but more covalent character and the large charge accumulation region located near O I+1 atoms is conditioned primarily by charge redistribution rather than by charge transfer from the alloy atoms.
All mentioned peculiarities of O-Ti and O-Al bonds are still valid in the case of the TiAl(110) Ti /TiO 2 (110) O interface. Formation of the O1 I -Ti1 I interfacial bonds (Figure 4d) leads to a weakening of Ti1 I+1 -O1 I bonds in the oxide (overlap population is almost one and a half times less (Table 3)), to the appearance of charge redistribution around Ti1 I+1 atoms and localized charge Metals 2020, 10, 1298 8 of 14 depletion regions near Ti1 I atoms. The charge state of Ti1 I atoms changes from +0.16 el. on the clean alloy surface to +0.82 el. at the interface ( Table 3). Formation of the O2 I -Al I-1 bond (Figure 4e) leads to a decrease of charge transfer from Ti2 I+1 atoms in the oxide to O2 I by 0.24 el. As a result, the charge at Al I-1 atoms decreases by 0.28 el. (Table 3). Figure 4f demonstrates O I+1 -Ti2 I interfacial bonds, which have also pronounced ionic-covalent character. Although the O I+1 -Ti2 I bond length is larger by 0.06 Å than the O1 I -Ti1 I one, the overlap population changes insignificantly (by 0.02 el.). Additionally, Ti2 I atoms have a smaller positive charge in comparison with Ti1 I . In general, Figure 4 demonstrates that all formed interfacial bonds have an ionic-covalent character and the formation of the interfacial bonds leads to a decrease of the chemical bonding in both oxide and alloy. The integrated ∆ρ(z) ( Figure S2b) shows that the charge depletion region associated with Al interfacial atoms is more widespread through the slab than that connected with Ti interfacial atoms. This is connected with the nature of valence electrons in Al and Ti. Additionally, it is seen that charge transfer to O interfacial atoms is larger in the case of the TiAl(110) Ti /TiO 2 (110) O interface. Table 3. Charge states of some interfacial and subinterfacial atoms (Q in el.), bond length between them (d in Å) and averaged overlap population (θ in el.) for pairs of the atoms in the case of the hollow interface configuration. The corresponding data for free surfaces of the alloy and the oxide are given also.  Figure 5 demonstrates the local densities of states (DOSs) of the interfacial atoms for both interfaces in the case of the hollow configuration. It is seen that sharp peaks of subinterfacial O I+1 atoms lying at -7.2, -6.9, and -5.6 eV coincide well with similar peaks of Al2 I (Figure 5a). Similarly, the thin structure of O1 I DOS (position of some peaks) agrees well with that of the Al1 I atom. The difference from the previous case is a wider spread of O1 I and Al1 I states toward negative energies. It should be noted that there is small density of states on both DOS curves for O I+1 and O1 I atoms in the band gap, which is typical for metallic oxide due to the presence of the interface states induced by interaction with interfacial alloy atoms. It is known that the valence band of Ti atoms is almost unoccupied. The interaction of Ti I−1 with O2 I results in the appearance of low-lying states, which spread in the region from -7.1 eV up to -3.4 eV. Additionally, the shift of the valence band of all O atoms, involved in the formation bonds through interface, toward negative energy is a consequence of charge transfer from nearest metal atoms. The increase of Ti unoccupied states in the region 0.4-0.9 eV is connected with the charge transfer from Ti to oxygen due to their interaction.

Atom
Metals 2020, 10, 1298 9 of 14 oxygen than Ti d-states, which is expressed in a stronger change of Al DOSs than that of Ti. As a result, at the interface with the Ti-terminated alloy surface (Figure 5b), the shift of occupied states of OI+1 and O1I atoms is less pronounced than that in the previous case. However, the states of O2I atoms interacting with AlI-1 atoms spread by 0.7 eV more toward negative energies. In general, DOS curves confirm strong interaction between interfacial atoms. Note that detailed comparison of DOS thin structures can be found in Figure S3. Now we will discuss the difference in the structural characteristics of the present interfaces with hollow configuration and the most stable interfaces from papers [28,30]. The comparison of present interfacial bond lengths with those from paper [28] is given Figure 6a. Within our interface model, there are two O-Al bonds of 1.89 Å (1.78 Å [28]) and two of 1.90 Å (1.85 Å [28]), as well as one O-Ti bond of 1.92 Å (two bonds of 2.08 Å [28]), which are distributed over the area of 39.155 Å 2 (~22.95 Å 2 [28]). Thus, the total density of O-Al and O-Ti bonds (the number of bonds per unit area) is 0.102 Å −2 and 0.026 Å −2 within the present model and 0.174 Å −2 and 0.087 Å -2 according to [28], respectively. It is seen in Figure 6a that our interface model demonstrates about half of the bond density (0.128 Å −2 in comparison with 0.261 Å −2 in [28]) and weaker O-Al bonds (larger bond length than in [28]). At the same time, the O-Ti interaction is substantially stronger than that in [28]. The competition of these factors leads to a smaller value of the work of separation than that in [28], but the difference is not large. It is necessary to recall that Al s,p-states involved much more easily in the interaction with oxygen than Ti d-states, which is expressed in a stronger change of Al DOSs than that of Ti. As a result, at the interface with the Ti-terminated alloy surface (Figure 5b), the shift of occupied states of O I+1 and O1 I atoms is less pronounced than that in the previous case. However, the states of O2 I atoms interacting with Al I-1 atoms spread by 0.7 eV more toward negative energies. In general, DOS curves confirm strong interaction between interfacial atoms. Note that detailed comparison of DOS thin structures can be found in Figure S3. Now we will discuss the difference in the structural characteristics of the present interfaces with hollow configuration and the most stable interfaces from papers [28,30]. The comparison of present interfacial bond lengths with those from paper [28] is given Figure 6a. Within our interface model, there are two O-Al bonds of 1.89 Å (1.78 Å [28]) and two of 1.90 Å (1.85 Å [28]), as well as one O-Ti bond of 1.92 Å (two bonds of 2.08 Å [28]), which are distributed over the area of 39.155 Å 2 (~22.95 Å 2 [28]). Thus, the total density of O-Al and O-Ti bonds (the number of bonds per unit area) is 0.102 Å −2 and 0.026 Å −2 within the present model and 0.174 Å −2 and 0.087 Å -2 according to [28], respectively. It is seen in Figure 6a that our interface model demonstrates about half of the bond density (0.128 Å −2 in comparison with 0.261 Å −2 in [28]) and weaker O-Al bonds (larger bond length than in [28]). At the same time, the O-Ti interaction is substantially stronger than that in [28]. The competition of these factors leads to a smaller value of the work of separation than that in [28], but the difference is not large.
In the case of the interface with the Ti-terminated TiAl(110) surface, the bond density is the same (only Al is replaced by Ti and vice-versa): O-Ti-0.102 Å -2 with bond lengths of 1.98 and 2.04 Å; O-Al-0.026 Å -2 with bond length of 2.07 Å. At the same time, the situation is quite different in [30]. As the TiO 2 atomic structure is much distorted in [30], formation of additional interfacial bonds is possible. Based on figures of atomic structure and tables with bond lengths from [30] (Figure 6b). One can see that the bond density within the model from [30] is more than twice as high in comparison with the present model. Moreover, the lengths of some O-Ti and O-Al bonds in [30] are shorter than those in the present paper. As a result, a lower value of W sep than that in [30] was obtained. It is known that compression in the interface plane results in an increase of interlayer distances and weakening of interatomic interaction inside the slab. In turn, that results in strengthening of interatomic interactions at the interface. This can explain the overestimation of the adhesion energy at the interface. [28]). Thus, the total density of O-Al and O-Ti bonds (the number of bonds per unit area) is 0.102 Å −2 and 0.026 Å −2 within the present model and 0.174 Å −2 and 0.087 Å -2 according to [28], respectively. It is seen in Figure 6a that our interface model demonstrates about half of the bond density (0.128 Å −2 in comparison with 0.261 Å −2 in [28]) and weaker O-Al bonds (larger bond length than in [28]). At the same time, the O-Ti interaction is substantially stronger than that in [28]. The competition of these factors leads to a smaller value of the work of separation than that in [28], but the difference is not large. It should be noted that cleavage between the oxide slab with the TiO-terminated (110) surface and TiAl(110) Ti with O layer (violet plane in Figure S1a) leads to a value of 1.77 J/m 2 , whereas it was equal to 1.53 J/m 2 in [30]. In accordance with the Griffith fracture theory [54,55], the fracture work (G) of bulk materials along a certain lattice plane can be estimated as double surface energy. If G < W sep , the fracture will occur in the bulk material; otherwise, the fracture will occur at the interface. Our estimation of the surface energy of TiO 2 (Table 1) allows to conclude that the cleavage energy along the (110) plane between two stoichiometric surfaces needs only 0.92 J/m 2 , which is lower than W sep obtained for both considered interfaces with the hollow configuration. This means that mechanical failure may be initiated in the oxide.
Finally, we will not discuss the interface with the top configuration. The distribution of the charge density difference shown in Figure 7 allows us to understand the peculiarities of the chemical bonding at the interfaces and together with Table S1 the lower values of W sep at them in comparison with the hollow configuration. In the case of the interface with the Ti-terminated TiAl(110) surface, the bond density is the same (only Al is replaced by Ti and vice-versa): O-Ti-0.102 Å -2 with bond lengths of 1.98 and 2.04 Å; O-Al-0.026 Å -2 with bond length of 2.07 Å. At the same time, the situation is quite different in [30]. As the TiO2 atomic structure is much distorted in [30], formation of additional interfacial bonds is possible. Based on figures of atomic structure and tables with bond lengths from [ (Figure 6b). One can see that the bond density within the model from [30] is more than twice as high in comparison with the present model. Moreover, the lengths of some O-Ti and O-Al bonds in [30] are shorter than those in the present paper. As a result, a lower value of Wsep than that in [30] was obtained. It is known that compression in the interface plane results in an increase of interlayer distances and weakening of interatomic interaction inside the slab. In turn, that results in strengthening of interatomic interactions at the interface. This can explain the overestimation of the adhesion energy at the interface.
It should be noted that cleavage between the oxide slab with the TiO-terminated (110) surface and TiAl(110)Ti with O layer (violet plane in Figure S1a) leads to a value of 1.77 J/m 2 , whereas it was equal to 1.53 J/m 2 in [30]. In accordance with the Griffith fracture theory [54,55], the fracture work (G) of bulk materials along a certain lattice plane can be estimated as double surface energy. If G < Wsep, the fracture will occur in the bulk material; otherwise, the fracture will occur at the interface. Our estimation of the surface energy of TiO2 (Table 1) allows to conclude that the cleavage energy along the (110) plane between two stoichiometric surfaces needs only 0.92 J/m 2 , which is lower than Wsep obtained for both considered interfaces with the hollow configuration. This means that mechanical failure may be initiated in the oxide.
Finally, we will not discuss the interface with the top configuration. The distribution of the charge density difference shown in Figure 7 allows us to understand the peculiarities of the chemical bonding at the interfaces and together with Table S1 the lower values of Wsep at them in comparison with the hollow configuration.

Conclusions
A comparative study of atomic and electronic structures of the TiAl/TiO 2 (110) interface and its adhesion properties in dependence on the alloy surface termination was performed using the projector-augmented wave method. Within a more commensurable interface model than in [28,30], with a smaller misfit between the alloy and the oxide surface cells, the adhesive properties were clarified. The most preferred contact configuration was found to be the hollow one, in which one interfacial O atom locates above a Ti(Al) atom of the alloy subinterfacial layer and another interfacial O atom occupies the short-bridge position between two Al(Ti) interfacial atoms. The work of separation of 2.44 J/m 2 was obtained at the TiAl(110) Ti /TiO 2 (110) O interface, whereas a lower value of 1.27 J/m 2 was calculated at the TiAl(110) Al /TiO 2 (110) O one. The trend in the W sep lowering in dependence on the alloy termination is consistent with that in [28,30]. Although the covalent contribution to chemical bonding is slightly higher at the interface with the Al-terminated TiAl(110) surface, charge transfer from metal to oxygen is higher at the interface with the Ti-terminated one. The overlap population analysis demonstrates that O-Ti bonds are stronger than O-Al ones at both interfaces. The interaction of titanium dioxide with the interfacial atoms of the alloy gives rise to the restoration of bonding in the bulk oxide. In general, the obtained small values of the work of separation at both interfaces are explained by the low bond density. In addition, it should be noted that the predicted adhesion at the TiAl/TiO 2 interface is considerably lower than that based on earlier obtained values of 10.43 J/m 2 at the TiAl(111)/Al 2 O 3 (0001) O [56] and 11.02 J/m 2 at the Ti 3 Al(0001)/Al 2 O 3 (0001) O [57] interfaces. We hope that the interfaces with the oxide termination by an oxygen double layer will demonstrate much better adhesive properties as in the case of NiTi/TiO 2 (100) 2O [58] because of the increase of both ionic-covalent contribution to chemical bonding and bond density. In general, the obtained results allow us to get a better insight into the mechanisms of oxide scale formation on the TiAl surface. Furthermore, it can be useful for the simulation of impurity effects on the adhesion at the interface, which is a subject of our forthcoming work.
Supplementary Materials: The following are available online at http://www.mdpi.com/2075-4701/10/10/1298/s1: Figure S1: Atomic structures of unrelaxed interfaces with hollow and top configuration in the case of the Ti-terminated alloy surface, Figure S2: In-plane average charge density difference at the interface with the hollow configuration; Table S1: Charge states of some interfacial and subinterfacial atoms (Q in el.), bond length between them (d in Å) and averaged overlap population (θ in el.) for pairs of the atoms in the case of the top interface configuration.