Titanium Aluminide Coating: Structural and Elastic Properties by DFT Approach

Abstract

The stability, elastic and electronic properties of titanium aluminide compounds have been systematically studied by the first-principles calculation. The calculated lattice parameters are consistent with the results found in the literature. The three Ti-Al binary compounds are thermodynamically stable intermetallics depending on their negative formation enthalpy. It has been found that the Ti-Al binary compounds are composed of both metallic and covalent bonds. Elastic properties revealed that these alloys are more resistant to deformation along the a- and c-axis. Besides, the (001)[100] deformation would be easier than (010)[100] deformation for these alloys. The results found in this chapter give a reliable reference for the design of novel Ti-Al binary alloys.

1. Introduction

Titanium aluminide-based alloys (TiAl) have attracted more attention as a potential coating material for high-temperature structural applications. They have many applications in engineering due to their high melting points, high tensile strength, good stiffness, low density, high corrosion, and oxidation resistance at elevated temperature [1, 2, 3]. TiAl has an ordered tetragonal face-centered γ-TiAl phase and a hexagonal α₂ (Ti₃Al) phase DO₁₉. Experimental works highlighted that the deformation mechanism (α₂ + γ) binary phase is mainly made by ordinary dislocation, super-dislocations, and twinning following the compact plane (111) [4, 5, 6]. However, TiAl intermetallic compounds suffer from low ductility and toughness at room temperature, which seriously limits their broad range of applications [7].

To understand the bonding and materials characteristics of this type of system, many experimental and theoretical studies have been investigated. A study of fundamental properties such as the nature of interatomic bonding, the stability of crystal structures, elastic properties, dislocations, grain boundaries, interfaces, as well as point defects and diffusion are therefore warranted to gain more insight into the behavior of these intermetallic alloys under high temperatures [8]. Sun et al. [9] have synthesized a new multilayer TiAl intermetallic by spark plasma sintering, which provides a novel possibility to improve fracture toughness. It is believed that TiAl intermetallic compounds with different Al contents play important roles in controlling the microstructure and mechanical properties.

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, 11, 12, 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 TiAl intermetallic compounds using density functional density (DFT).

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 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.

Figure 1.

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

3. Structural properties

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.

	a (Å)	c (Å)	c/a	B0 (GPa)	B′	ΔEf (eV/at.)
B2-TiAl	3.185; 3.189 [21]; 3.196 [22]	—	—	111.46	4.04	−0.265
γ-TiAl	3.993; 3.997 [23]; 3.996 [24]	4.068; 4.062 [23]; 4.075 [24]	1.018; 1.016 [23]; 1.020 [24]	113.88; 113.29 [23]; 112.82 [24]	3.95; 3.76 [24]	−0.405; −0.367 [25]
α2-Ti3Al	5.746; 5.765 [26]; 5.760 [27]	4.666; 4.625 [26]; 4.659 [27]	0.812; 0.802 [26]; 0.809 [27]	114.81; 113 [26]; 114.39 [27]	3.83; 3.61 [27]	−0.275; −0.290 [28]

Table 1.

Lattice parameters, bulk modulus, and formation enthalpy for B2-TiAl, γ-TiAl, and α₂-Ti₃Al compounds.

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

$$\Delta \mathrm{H}={\mathrm{E}}_{\mathrm{total}}\left(\mathrm{TiAl}\right)-{\mathrm{E}}_{\mathrm{solid}}\left(\mathrm{T}\mathrm{i}\right)-{\mathrm{E}}_{\mathrm{solid}}\left(\mathrm{A}\mathrm{l}\right)$$E1

We can see that the enthalpies of formation E_f 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, 22, 23, 24, 25, 26, 27, 28].

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

$${\mathrm{C}}_{44}>0,{\mathrm{C}}_{66}>0,{\mathrm{C}}_{11}>\left|{\mathrm{C}}_{12}\right|,\left({\mathrm{C}}_{11}+{\mathrm{C}}_{12}\right){\mathrm{C}}_{33}>2{{\mathrm{C}}^2}_{13}$$E2

For a hexagonal crystal

$${\mathrm{C}}_{44}>0;{\mathrm{C}}_{11}-\left|{\mathrm{C}}_{12}\right|>0;\left({\mathrm{C}}_{11}+2{\mathrm{C}}_{12}\right){\mathrm{C}}_{33}–2{{\mathrm{C}}^2}_{13}>0$$E3

For a cubic structure

$${\mathrm{C}}_{11}+2{\mathrm{C}}_{12}>0;{\mathrm{C}}_{11}–{\mathrm{C}}_{12}>0;{\mathrm{C}}_{44}>0$$E4

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

Crystal System	Point symmetry group	Stiffness Tensor
Cubic	23; m-3; 432; -43 m; m-3 m	$\left(\begin{array}{cccccc}\hfill C_{11}\hfill & \hfill C_{12}\hfill & \hfill C_{12}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill C_{11}\hfill & \hfill C_{12}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill C_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{44}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{44}\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{44}\hfill \end{array}\right)$
Hexagonal	6; −6; 6/m; 622; 6 mm; -62 m; 6/mmm	$\left(\begin{array}{cccccc}\hfill C_{11}\hfill & \hfill C_{12}\hfill & \hfill C_{13}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill C_{11}\hfill & \hfill C_{13}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill C_{33}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{44}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{44}\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{66}\hfill \end{array}\right)$
Tetragonal	422; 4 mm; −42 m; 4/mmm; 4; -4; 4/m	$\left(\begin{array}{cccccc}\hfill C_{11}\hfill & \hfill C_{12}\hfill & \hfill C_{13}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill C_{11}\hfill & \hfill C_{13}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill C_{33}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{44}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{44}\hfill & \hfill 0\hfill \\ \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill C_{66}\hfill \end{array}\right)$

Table 2.

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

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.

	C11 (GPa)	C12 (GPa)	C13 (GPa)	C33 (GPa)	C44 (GPa)	C66 (GPa)
B2-TiAl	76.04	118.26	—	—	71.98	—
γ-TiAl	179.20; 183 [29]; 173 [24]	94.90; 74.1 [29]; 83 [24]	84.63; 74.4 [29]; 84 [24]	167.96; 178 [29]; 168 [24]	113.84; 105 [29]; 111 [24]	72.12; 78.4 [29]; 75 [24]
α2-Ti3Al	217.36; 175 [30]; 185 [31]	98.75; 88.7 [30]; 83 [31]	78.23; 62.3 [30]; 63 [31]	237.27; 220 [30]; 231 [31]	59.30; 62.2 [30]; 57 [31]	--; --; --

Table 3.

Elastic constants for B2-TiAl, γ-TiAl and α₂-Ti₃Al compounds.

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.

Figure 2.

The total and partial density of states (DOS) for: (a) α₂-Ti₃Al, (b) B2-TiAl, (c) γ-TiAl.

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

6. Conclusion

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 character.

Conflict of interest

The authors declare no conflict of interest.