**2. Computational details**

Ab initio calculation is a useful method to predict and explore the interrelationships among structure properties. In this chapter, the first-principles modeling

**101**

**Table 1.**

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

based on the density functional theory (DFT) was implemented in the Vienna Ab-initio Software Package (VASP) [14, 15]. Projector Augmented Wave (PAW) [16] pseudopotential is used to describe the interactions between electrons and ions [17]. The exchange and correlation energy have been performed using the Generalized Gradient Approximation (GGA) [18]. Plane-wave cutoff energy of 400 eV was used for all calculations. In this work, the k-point method has been adopted for sampling the Brillouin zone and selected to 18 x 18 x 18. The Brillouin zone integration was executed using the Methfessel-Paxton technique [19] with a 0.1 eV smearing of the electron levels. The fully structural relaxations were performed by minimizing the ionic Hellman-Feynman [20] force until the maximum

Titanium aluminides crystallized in three major phases: Hexagonal Ti3Al, cubic

The total energy of TiAl and Ti3Al intermetallic compounds was calculated at many different volumes around equilibrium fitted to Murnaghan's equation of state from which we obtained the equilibrium structural parameters. The computed equilibrium lattice parameters and the bulk modulus of these compounds are listed in **Table 1**. The calculated lattice parameters and the bulk modulus (B) for these compounds are in good agreement with those measured experimentally. Hence, we can see that the computation parameters and conditions selected in the present

To examine the stability of these systems, the formation enthalpy (ΔH) was

We can see that the enthalpies of formation Ef of this compound crystallizing

in the three phases take negative values. This indicated that the three phases of titanium aluminide are energetically stable. From these values, we can

see that all three phases can exist with the γ-TiAl phase as the most energetically stable. These findings are in good agreement with the previous published

> 1.018 *1.016* [23] *1.020* [24]

0.812 *0.802* [26] *0.809* [27]

*Lattice parameters, bulk modulus, and formation enthalpy for B2-TiAl,* γ*-TiAl, and* α*2-Ti3Al compounds.*

∆= − − H E TiAl E Ti E Al total ( ) solid ( ) solid ( ) (1)

**a (Å) c (Å) c/a B0 (GPa) B**′ Δ**Ef (eV/at.)**

— — 111.46 4.04 −0.265

113.88 *113.29* [23] *112.82* [24]

114.81 *113* [26] *114.39* [27] 3.95 *3.76* [24]

3.83 *3.61* [27] −0.405 *−0.367* [25]

−0.275 *−0.290* [28]

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

forces achieved less than 0.02 eV.

**3. Structural properties**

work should be suitable.

calculated by

works [21–28].

B2-TiAl 3.185

γ**-TiAl** 3.993

α**2-Ti3Al** 5.746

*3.189* [21] *3.196* [22]

*3.997* [23] *3.996* [24]

*5.765* [26] *5.760* [27] 4.068 *4.062* [23] *4.075* [24]

4.666 *4.625* [26] *4.659* [27]

TiAl, and tetragonal TiAl, as shown in **Figure 1**.

**Figure 1.** *Crystal structures of titanium aluminides. (a) DO19, (b) L10, and (c) B2.*

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

based on the density functional theory (DFT) was implemented in the Vienna Ab-initio Software Package (VASP) [14, 15]. Projector Augmented Wave (PAW) [16] pseudopotential is used to describe the interactions between electrons and ions [17]. The exchange and correlation energy have been performed using the Generalized Gradient Approximation (GGA) [18]. Plane-wave cutoff energy of 400 eV was used for all calculations. In this work, the k-point method has been adopted for sampling the Brillouin zone and selected to 18 x 18 x 18. The Brillouin zone integration was executed using the Methfessel-Paxton technique [19] with a 0.1 eV smearing of the electron levels. The fully structural relaxations were performed by minimizing the ionic Hellman-Feynman [20] force until the maximum forces achieved less than 0.02 eV.

Titanium aluminides crystallized in three major phases: Hexagonal Ti3Al, cubic TiAl, and tetragonal TiAl, as shown in **Figure 1**.

#### **3. Structural properties**

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

Several ab-initio studies of the structural stability, elastic properties, and the nature of bonding have been reported for γ-TiAl, α2-Ti3Al, and B2-TiAl [10–13]. To optimize ion beam treatment of TiAl based intermetallic alloys for better performance. It is essential to gain a deeper insight into radiation effects in these materials. From the fundamental perspective, TiAl intermetallic compounds represent a good model system for studying radiation effects in ordered metallic alloys for future engineering applications. To understand some of the physical properties of these compounds, knowledge of the phase stability and elastic properties of TiAl binary compounds is required. In this chapter, we summarized the calculated results of

Ab initio calculation is a useful method to predict and explore the interrelationships among structure properties. In this chapter, the first-principles modeling

TiAl intermetallic compounds using density functional density (DFT).

**2. Computational details**

**100**

**Figure 1.**

*Crystal structures of titanium aluminides. (a) DO19, (b) L10, and (c) B2.*

The total energy of TiAl and Ti3Al intermetallic compounds was calculated at many different volumes around equilibrium fitted to Murnaghan's equation of state from which we obtained the equilibrium structural parameters. The computed equilibrium lattice parameters and the bulk modulus of these compounds are listed in **Table 1**. The calculated lattice parameters and the bulk modulus (B) for these compounds are in good agreement with those measured experimentally. Hence, we can see that the computation parameters and conditions selected in the present work should be suitable.

To examine the stability of these systems, the formation enthalpy (ΔH) was calculated by

$$\mathbf{A}\mathbf{H} = \mathbf{E}\_{\text{total}}\left(\text{Trial}\right) - \mathbf{E}\_{\text{solid}}\left(\text{Ti}\right) - \mathbf{E}\_{\text{solid}}\left(\text{Al}\right) \tag{1}$$

We can see that the enthalpies of formation Ef of this compound crystallizing in the three phases take negative values. This indicated that the three phases of titanium aluminide are energetically stable. From these values, we can see that all three phases can exist with the γ-TiAl phase as the most energetically stable. These findings are in good agreement with the previous published works [21–28].


**Table 1.**

*Lattice parameters, bulk modulus, and formation enthalpy for B2-TiAl,* γ*-TiAl, and* α*2-Ti3Al compounds.*
