Electronic structure and physical properties of hcp Ti₃Al type alloys

PENG Hong-jian(彭红建)^{1, 2}, XIE You-qing(谢佑卿)²

1. School of Chemistry and Chemical Engineering, Central South University, Changsha 410083, China;

2. School of Materials Science and Engineering, Central South University, Changsha 410083, China

Received 20 October 2006; accepted 9 April 2007

Abstract: According to the basic information of sequences of Ti and Al characteristic atoms in hcp Ti-Al system, the compositional variations of the electronic structure, atomic potential energies, atomic volumes, lattice constants and cohesive energies of the ordered hcp Ti₃Al type alloys were calculated by the framework of systematic science of alloys(SSA). The electronic structure of the hcp Ti₃Al compound consisted of and  atoms is 0.75[Ar] (3d_n)^0.573(3d_c)^2.1685(4s_c)^0.972(4s_f)^0.3093+0.25[Ne](3s_c)^1.32? (3p_c)^1.19(3s_f)^0.49. The factors of controlling lattice stability are electronic structure, atomic energies and atomic concentration. The atoms play a determinative role in forming D0₁₉ structure with a=0.287 2 nm, c=0.456 4 nm, atomic cohesive energy ε=4.810 8 eV/atom and heat of formation ΔH=-0.332 8 eV/atom. These calculated values are in good agreement with experimental values (a=0.287 5 nm, c=0.46 0 nm, ?H=-0.27, -0.29 eV/atom). The calculated cohesive energy of the hcp Ti₃Al compound is slightly bigger than that of the fcc Ti₃Al.This is a good sign that makes it feasible to stabilized L1₂ structure of the hcp Ti₃Al compound by ternary element. The new element should have more d_c-electrons than Ti-metal and occupy at the Ti-lattice points.

Key words: Ti₃Al alloy; electronic structure; physical property

1 Introduction

The intermetallic compound hcp Ti₃Al with D0₁₉ structure has been the subject of considerable attention by experimentalists[1-3]. It is considered as a desirable candidate for application in aircraft turbine engines because its static strength and stiffness do not degrade rapidly as temperature increase. The calculations of the Ti-Al phase diagram using a variety of models have been performed[4-8]. But these Gibbs energy functions do not be associated with electronic structure, atomic energies, atomic volumes and lattice constants. The knowledge of the Ti-Al system obtained from Gibbs energy functions is not complete.

According to the framework of systematic science of alloys(SSA)[9], the Ti-Al system has fcc, hcp and bcc lattice systems, besides a liquid system. It has been found that the compositional variations of the electronic structure, atomic potential energies, atomic volumes, lattice constants, cohesive energies and Gibbs energies of these four phases can be calculated from the basic information of the same sequences of the characteristic atoms in the fcc Ti-Al lattice system[10-13].

In this study, the basic information of the sequences of Ti and Al characteristic atoms in the hcp Ti-Al lattice system was determined. The properties of hcp Ti₃Al alloys as a function of composition were obtained and the controlling factors for crystalline stability of hcp Ti₃Al compound were analyzed.

2 Basic information of Ti and Al characteri- stic atoms

According to the basic clusters overlapping(BCO) model in SSA framework, there are two basic cluster sequences that include 13 kinds of Ti and Al basic clusters in hcp Ti-Al system. The disordered and ordered alloys with hcp Ti₃Al type can be formed by arrangements of these 26 kinds of characteristic atoms, or by mixing the following 26 kinds of characteristic crystals. According to information about experimental lattice constants of some hcp Ti_xAl_(1-x) alloys and on the basis of experimental value and theoretical analysis, we have determined the basic information about the electronic structures, volumes, lattice constants, potentialenergies and cohesive energies of the characteristic atoms and corresponding characteristic crystals in the hcp Ti-Al system (Tables 1 and 2).

Table 1 Structural parameters and properties of Ti-characteristic atoms and Ti-characteristic crystals in hcp Ti-Al system

Table 2 Structural parameters and properties of Al-characteristic atoms and Al-characteristic crystals in hcp Ti-Al system

In Tables 1 and 2, s_c, p_c and d_c are respectively the covalent electrons in s, p and d orbitals, s_f and p_f are the near-free electrons in s and p orbitals, and d_n is the nonbonded electrons in d orbital.

As shown in Fig.1, for hcp Ti-Al system, the relationships of the , , and  with the number i can be described by following equation, where Q can respectively denote ε and v.

           (1)

Fig.1 Relationships ofand  (a), and  (b) of  and  atoms with number i of nearest neighbor Al atoms in hcp Ti-Al system

3 Basic information of ordered hcp Ti₃Al type alloys

Since the complex interactions between atoms in the alloys are reflected by the electronic structure, potential energies, atomic volumes, lattice constants, cohesive energies and Gibbs energies of the characteristic crystals, the basic information of the alloys can be obtained by the additive law of characteristic crystals (Eqn.(2)):

  (2)

where and are concentrations of the Ti and Al characteristic atoms in the alloys.

For the ordered hcp Ti₃Al type alloys with maximal ordering degree, the concentrations  and as a function of composition are shown in Fig.2. The basic information about the compositional variations of electronic structure, atomic potential energies, atomic volumes, lattice constants and cohesive energies of the hcpTi₃Al alloys and their components Ti and Al are listed in Table 3. It can be known that the hcp Ti₃Al compound is formed by  and  atoms.

Fig.2 Concentrationsand of ordered Ti_xAl_(1-x) alloys with hcp Ti₃Al-type and maximal ordering degree as functions of composition x_Al

Table 3 Electronic structure and properties of ordered Ti_xAl_1-x alloys with hcp Ti₃Al-type having maximal ordering degree and their components of Ti and Al

4 Crystalline structures of Ti₃Al type alloys

4.1 Relationship of lattice stability with electronic structure for Ti₃Al compound

The factors of controlling lattice stability are electronic structure, atomic energy and atomic concentration. The perfectly ordered Ti_0.75Al_0.25 alloy may be hcp Ti₃Al compound with D0₁₉ structure, or fcc Ti₃Al compound with L1₂ structure.

In order to understand the effect of electronic structure factor on lattice stability, the average electronic structures of hcp Ti₃Al and fcc Ti₃Al[12] compounds are shown as follows:

(3)

(4)

It should be pointed out that the  and  atoms belong in the sequences of Ti- and Al-characteristic atoms in the hcp lattice system, and the and  atoms belong in the sequences of Ti- and Al-characteristic atoms in the fcc lattice system.

The atoms have a tendency to form hcp and bcc structures, because their d_c electrons would rather be in the e_g state with lower potential energy than in the t_2g state with higher potential energy. Only when the Al atoms or other elements have stronger tendency to form fcc structure than the tendency to form hcp and bcc structure of the Ti atoms, can the d_c electrons be in the t_2g state due to strong effect of Al atoms, and can the Ti₃Al compounds have fcc lattice structure.

The atoms have a tendency to form fcc structure. The  atoms in the fcc Ti₃Al compound have less concentration (=25%, mole fraction), less s_c spherical covalent electrons and more p_c directional electrons (1.008 eV/atom) than other atoms. Thereby, the  atoms have weaker role in forming fcc lattice than other atoms in the fcc TiAl₃ compounds.

The  atoms have the most concentration (=75%, mole fraction) and the strongest tendency to form hcp and bcc structures, because their d_c electrons (2.156 eV/atom) are less than that of atoms, and would rather be in the e_g state. The  atoms have the least concentration (=25%) and the weakest tendency to form fcc structure, because their s_c electrons (1.32 electrons/atom) are less than that of  atoms. Therefore the  atoms play the leading role in forming hcp lattice for D0₁₉- Ti₃Al compound.

4.2 Relationship of lattice stability with atomic energies and concentration for Ti₃Al type alloys

The atomic cohesive energies and heat of formation of the hcp and fcc Ti₃Al compounds are as follows:

ε(hcp Ti₃Al)=-(0.75+0.25)=4.810 8 eV/atom (5)

ε(fcc Ti₃Al)=-(0.75+0.25)=4.795 2 eV/atom (6)

?ε=ε(hcp Ti₃Al)- ε(fcc Ti₃Al)=0.015 2 eV/atom     (7)

(8)

  (9)

The ε(hcp Ti₃Al) is slightly larger than ε(fcc Ti₃Al), so the hcp Ti₃Al compound is more stable than the fcc Ti₃Al compound. The calculated heat of formation of the ?H(hcp Ti₃Al) is smaller than its experimental values (-0.27 and -0.29 eV/atom)[14].

From the compositional variation of cohesive energies of the hcp Ti₃Al, fcc Ti₃Al, fcc TiAl and fcc TiAl₃ type ordered Ti_xAl_(1-x)  alloys (see Fig.3), it can be known that the hcp Ti₃Al type ordered Ti_xAl_(1-x)  alloys are more stable than other ordered alloys in the range of 0-40% Al (mole fraction).

Fig.3 Atomic cohesive energies of hcp Ti₃Al(1), fcc Ti₃Al(2), fcc TiAl(3) and fcc TiAl₃(4) type ordered Ti_xAl_(1-x)  alloys and mixed solution [hcp Ti+fcc Al](5)

The calculated energy difference between hcp and fcc Ti₃Al compounds is very small, which makes it feasible to stabilize the L1₂ phase by ternary alloy additions. Considering electronic structure and that the  atoms play the leading role in forming hcp lattice for D0₁₉ Ti₃Al compound, these ternary element additions should occupy the Ti-lattice points, and have much more d_c-electrons than Ti-atoms.

4.3 Relationship of lattice constants and concentra- tion of hcp Ti₃Al type alloys

The average atomic volumes of several ordered Ti_xAl_(1-x) alloys with hcpTi₃Al type are listed in Table 3. Because we still can not establish the function between the ratio c/a and concentration in theory for the ordered Ti_xAl_(1-x) alloys with hcp Ti₃Al type, we have drawn the polynomial equation of the c/a—x_Al curve from the experimental data of ratio c/a:

c/a=1.586 4+0.131 7x_Al-0.085 07            (10)

The lattice constants of several hcp Ti₃Al type Ti_xAl_(1-x) alloys are listed in Table 3. The lattice constants of the hcp Ti₃Al compound are a=0.287 2 nm and c=0.456 4 nm, c/a=1.614, which are in good agreement with experimental values (a=0.287 5 nm, c=0.460 nm)[15].

5 Conclusions

1) It has been proved that the compositional variations of the electronic structure, atomic potential energies, atomic volumes, lattice constants and cohesive energies of the ordered hcp Ti₃Al type alloys can be obtained from the basic information of sequences of Ti and Al characteristic atoms and corresponding characteristic crystals in the Ti-Al hcp lattice system.

2) The hcp Ti₃Al compound is formed by  and atoms. The electronic structure is 0.75[Ar](3d_n)^0.573? (3d_c)^{2.168 5}(4s_c)^0.972(4s_f)^{0.309 3}+0.25[Ne](3s_c)^1.32(3p_c)^1.19 (3s_f)^0.49. The calculated lattice constants and heat of formation of the hcp Ti₃Al compound are in good agreement with experimental values.

3) The factors of controlling crystalline structure are electronic structure, atomic potential energies and atomic concentration. Comparing  and atoms in the hcp Ti₃Al compound with the and atoms in the fcc Ti₃Al compound, it has been found that the  atom has the least d_c-electrons and the strongest tendency to form hcp and bcc structures, and the  atom has the least s_c-electrons and the weakest tendency to form fcc structure. Therefore, the  atoms play the leading role in forming hcp Ti₃Al compound.

4) The calculated cohesive energy of the hcp Ti₃Al compound is slightly bigger than that of the fcc Ti₃Al.This is a good sign that makes it feasible to stabilized L1₂ structure of the hcp Ti₃Al compound by ternary element. The new element should have more d_c-electrons than Ti-metal and occupy at the Ti-lattice points.

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Foundation item: Project(50471058) supported by the National Natural Science Foundation of China; Project(06FJ3133) supported by the Natural Science Foundation of Hunan Province, China

Corresponding author: PENG Hong-jian; Tel: +86-731-8879287; E-mail: phj108@163.com

(Edited by HE Xue-feng)