**4. Elastic properties**

Knowledge of elastic constants provides more information about mechanical properties and understanding of the usefulness of the materials. Mechanical stability requires that its independent elastic constants should satisfy the following Born's stability criteria.

For a tetragonal crystal

$$\left| \mathbf{C}\_{44} > \mathbf{0}, \mathbf{C}\_{66} > \mathbf{0}, \mathbf{C}\_{11} > \left| \mathbf{C}\_{12} \right|, \left( \mathbf{C}\_{11} + \mathbf{C}\_{12} \right) \mathbf{C}\_{33} > \mathbf{2}, \mathbf{C}\_{13} \tag{2}$$

For a hexagonal crystal

$$\left| \mathbf{C}\_{44} > \mathbf{0}; \mathbf{C}\_{11} - \left| \mathbf{C}\_{12} \right| > \mathbf{0}; \left( \mathbf{C}\_{11} + 2\mathbf{C}\_{12} \right) \mathbf{C}\_{33} - 2\mathbf{C}^{2}\_{13} > \mathbf{0} \tag{3}$$

For a cubic structure

$$\mathbf{C}\_{11} + \mathbf{2}\,\mathbf{C}\_{12} > \mathbf{0}; \quad \mathbf{C}\_{11} - \mathbf{C}\_{12} > \mathbf{0}; \mathbf{C}\_{44} > \mathbf{0} \tag{4}$$

According to the symmetry of crystalline systems, some elastic constants are null. In this part, **Table 2** gives an overview of the expression of the stiffness tensors corresponding to each crystal system studied (cubic, hexagonal and tetragonal). The cubic structure has the simplest elastic matrix, with only 3 independent


**Table 2.**

*The stiffness tensors of the elastic constants corresponding to the systems: cubic, hexagonal and tetragonal.*

**103**

compound.

*Titanium Aluminide Coating: Structural and Elastic Properties by DFT Approach*

94.90 *74.1* [29] *83* [24]

98.75 *88.7* [30] *83* [31]

constants, while the elastic matrices of the hexagonal and tetragonal structures have

B2-TiAl 76.04 118.26 — — 71.98 —

84.63 *74.4* [29] *84* [24]

78.23 *62.3* [30] *63* [31]

**C11 (GPa) C12 (GPa) C13 (GPa) C33 (GPa) C44 (GPa) C66 (GPa)**

167.96 *178* [29] *168* [24]

237.27 *220* [30] *231* [31]

113.84 *105* [29] *111* [24]

59.30 *62.2* [30] *57* [31]

72.12 *78.4* [29] *75* [24]

> -- -- --

**Table 3** clearly shows that all elastic constants satisfy well the required stability conditions, indicating that both TiAl and Ti3Al compounds are mechanically stable. In addition, our elastic constants calculated fit the experimental and theoretical values. As known, the elastic constants C11 and C33 measure the resistance of alloys under uniaxial stress along *a*- and *c-axis*. The results obtained allow us to predict that these alloys are more resistant to deformation along a and c-axis. Moreover, the C66 value is significantly lower than C44 suggesting; thus, that the (001)[100] deformation would be easier than (010)[100] deformation for these alloys. On the other hand, we can see that the elastic constants of B2 structure do not required the Born-Huang's stability criteria. This indicates that B2-TiAl phase is mechanically

It is well-known that the physical properties of a material are strongly correlated to its electronic structure nature. So, by applying the obtained equilibrium structural parameters, the total and partial densities of states (TDOS and PDOS) were calculated along the principal symmetry directions in the Brillouin zone to further understand the reasonable relationship between the mechanical behavior and bonding characteristics of γ-TiAl based alloys. The total and partial DOS of pure

We can see that Ti-DOS typically Ti-d states play a very important role in the total density of TiAl in L10 structure. In this compound, the total state density has two regions in the valence band: a deep region dominated by Al-s states, the second one is constituted by Al-p and Ti-d states, which are separated by a strong hybridization where Al-p states forming a peak at about −1.5 eV which is more localized contributes to the strong covalence in Ti-d-Al-p bonds. Al the Fermi level, the density of states is not zero, dominated mainly by Ti-d states, attesting to the weak metallic character of this intermetallic class. Hence, the interactions of the strong covalent bond with the weak metallic bond cause an unequal distribution of these forces leading to a cleavage fracture in the direction of the metallic interactions. From **Figure 2**, the interactions between Al-3p and 3d-Ti take place in the Ti-Al binary compounds, leading to the enhancement of the covalent bonding. By the way, it can be found that Al-3p plays an important role in the pseudogap of Ti-Al

Through the further analysis of PDOS, the peaks of the PDOS of Ti-3d and Al-3p are exactly overlapped, indicating the strong covalent bonding originates

unstable, although it is energetically stable as mentioned above.

L10-TiAl, α2-Ti3Al, and B2-TiAl compounds are depicted in **Figure 2**.

*DOI: http://dx.doi.org/10.5772/intechopen.97409*

*183* [29] *173* [24]

*175* [30] *185* [31]

*Elastic constants for B2-TiAl,* γ*-TiAl and* α*2-Ti3Al compounds.*

five and six constants respectively.

γ-TiAl 179.20

α2-Ti3Al 217.36

**Table 3.**

**5. Electronic properties**


*Titanium Aluminide Coating: Structural and Elastic Properties by DFT Approach DOI: http://dx.doi.org/10.5772/intechopen.97409*

#### **Table 3.**

*Transition Metal Compounds - Synthesis, Properties, and Application*

Knowledge of elastic constants provides more information about mechanical properties and understanding of the usefulness of the materials. Mechanical stability requires that its independent elastic constants should satisfy the following Born's

According to the symmetry of crystalline systems, some elastic constants are null. In this part, **Table 2** gives an overview of the expression of the stiffness tensors corresponding to each crystal system studied (cubic, hexagonal and tetragonal). The cubic structure has the simplest elastic matrix, with only 3 independent

**Crystal System Point symmetry group Stiffness Tensor**

m-3 432 -43 m m-3 m

−6 6/m 622 6 mm -62 m 6/mmm

4 mm −42 m 4/mmm 4 -4 4/m

*The stiffness tensors of the elastic constants corresponding to the systems: cubic, hexagonal and tetragonal.*

( ) <sup>2</sup> C 0,C 0,C C , C C C 2C 44 66 11 12 11 12 33 13 >>> + > (2)

( ) <sup>2</sup> C 0;C C 0; C 2C C – 2C 0 44 11 12 > −> + 11 12 33 13 > (3)

C 2C 0; C – C 0;C 0 11 12 +> > > 11 12 44 (4)

11 12 12 11 12 11

11 12 13 11 13 33

11 12 13 11 13 33

*CCC C C C*

*CCC C C C*

*CCC C C C*

44

44

44

*C*

*C*

 

*C*

 

 

44

44

44

*C*

*C*

*C*

44

66

66

*C*

*C*

*C*

**4. Elastic properties**

For a tetragonal crystal

For a hexagonal crystal

For a cubic structure

Cubic 23

Hexagonal 6

Tetragonal 422

stability criteria.

**102**

**Table 2.**

*Elastic constants for B2-TiAl,* γ*-TiAl and* α*2-Ti3Al compounds.*

constants, while the elastic matrices of the hexagonal and tetragonal structures have five and six constants respectively.

**Table 3** clearly shows that all elastic constants satisfy well the required stability conditions, indicating that both TiAl and Ti3Al compounds are mechanically stable. In addition, our elastic constants calculated fit the experimental and theoretical values. As known, the elastic constants C11 and C33 measure the resistance of alloys under uniaxial stress along *a*- and *c-axis*. The results obtained allow us to predict that these alloys are more resistant to deformation along a and c-axis. Moreover, the C66 value is significantly lower than C44 suggesting; thus, that the (001)[100] deformation would be easier than (010)[100] deformation for these alloys. On the other hand, we can see that the elastic constants of B2 structure do not required the Born-Huang's stability criteria. This indicates that B2-TiAl phase is mechanically unstable, although it is energetically stable as mentioned above.

#### **5. Electronic properties**

It is well-known that the physical properties of a material are strongly correlated to its electronic structure nature. So, by applying the obtained equilibrium structural parameters, the total and partial densities of states (TDOS and PDOS) were calculated along the principal symmetry directions in the Brillouin zone to further understand the reasonable relationship between the mechanical behavior and bonding characteristics of γ-TiAl based alloys. The total and partial DOS of pure L10-TiAl, α2-Ti3Al, and B2-TiAl compounds are depicted in **Figure 2**.

We can see that Ti-DOS typically Ti-d states play a very important role in the total density of TiAl in L10 structure. In this compound, the total state density has two regions in the valence band: a deep region dominated by Al-s states, the second one is constituted by Al-p and Ti-d states, which are separated by a strong hybridization where Al-p states forming a peak at about −1.5 eV which is more localized contributes to the strong covalence in Ti-d-Al-p bonds. Al the Fermi level, the density of states is not zero, dominated mainly by Ti-d states, attesting to the weak metallic character of this intermetallic class. Hence, the interactions of the strong covalent bond with the weak metallic bond cause an unequal distribution of these forces leading to a cleavage fracture in the direction of the metallic interactions.

From **Figure 2**, the interactions between Al-3p and 3d-Ti take place in the Ti-Al binary compounds, leading to the enhancement of the covalent bonding. By the way, it can be found that Al-3p plays an important role in the pseudogap of Ti-Al compound.

Through the further analysis of PDOS, the peaks of the PDOS of Ti-3d and Al-3p are exactly overlapped, indicating the strong covalent bonding originates

**Figure 2.** *The total and partial density of states (DOS) for: (a)* α*2-Ti3Al, (b) B2-TiAl, (c)* γ*-TiAl.*

from the interaction between Ti-3d and Al-3p. Sometimes, a high DOS value at Fermi level means unstable structures in some degree. In this work, the values of total DOS at Fermi level of Ti3Al, B2-TiAl, and γ-TiAl compounds are 10.8 eV, 3.4 eV,

**105**

**Author details**

Ouahiba Ouadah

*Titanium Aluminide Coating: Structural and Elastic Properties by DFT Approach*

and 3.0 eV, respectively. From this point of view, Ti3Al compound is considered to

In this chapter, we have studied the structural, elastic, and electronic properties of titanium aluminide intermetallic compounds using first-principles calculations based on density functional theory (DFT). The use of the GGA-PBE approximation for the exchanging correlation potential allowed us to obtain good results of the electronic structure. B2-TiAl, γ-TiAl, and α2-Ti3Al are thermodynamically stable according to both thermodynamical and mechanical criteria. Whereas, B2-TiAl compound is mechanically unstable. Besides, elastic properties showed that these alloys are more resistant to deformation along *a-* and *c-axis*. Moreover, the C66 value is significantly lower than C44 suggesting; thus, that the (001)[100] deformation would be easier than (010)[100] deformation for these alloys. Based on the electronic structure, titanium aluminide binary compounds are composed of both metallic bonds and covalent bonds. α2-Ti3Al shows the strongest metallic bonding

Division of Materials Discovery (DEPM), Unit of Research of Materials and Renewable Energies (URMER), University Abou Bekr Belkaid, Tlemcen, Algeria

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: ouahiba.ouadah@univ-tlemcen.dz

provided the original work is properly cited.

*DOI: http://dx.doi.org/10.5772/intechopen.97409*

be the least stable Ti-Al binary compound.

The authors declare no conflict of interest.

**6. Conclusion**

character.

**Conflict of interest**

*Titanium Aluminide Coating: Structural and Elastic Properties by DFT Approach DOI: http://dx.doi.org/10.5772/intechopen.97409*

and 3.0 eV, respectively. From this point of view, Ti3Al compound is considered to be the least stable Ti-Al binary compound.
