Estimation of point defect formation energies in the D019-type intermetallic compound Ti3Al

Solid State Sciences 4 (2002) 1113–1117 www.elsevier.com/locate/ssscie

Estimation of point defect formation energies in the D019-type intermetallic compound Ti3 Al Olga Semenova, Regina Krachler, Herbert Ipser ∗ Institut für Anorganische Chemie, Universität Wien, Währingerstrasse 42, A-1090 Wien, Austria Received 22 April 2002; received in revised form 17 June 2002; accepted 24 June 2002

Abstract A statistical-thermodynamic model for binary nonstoichiometric intermetallic A3 B compounds with D019 -structure was developed based on a mean-field approximation. Vacancies and anti-structure atoms are allowed on the two sublattices as possible point defects. Due to the identical stoichiometry and the analogous coordination around A and B atoms it turned out that the same approach is valid as for A3 B compounds with L12 -structure, and identical expressions were obtained, both for the concentrations of the different point defects and for the thermodynamic activities. The energies of formation of the four types of point defects were used as parameters. The model equations were applied to the intermetallic compound Ti3 Al using experimental aluminum activities from the literature. By a simple curve fitting procedure the following defect formation energies were obtained: E f (VTi ) = E f (VAl ) = 1.5 eV, and E f (TiAl ) = E f (AlTi ) = 0.6 eV. This results in very low vacancy concentrations which means that the thermal disorder and the deviation from stoichiometry in Ti3 Al is caused almost entirely by anti-structure atoms. Their concentrations (referred to the total number of lattice sites) are found to be about 0.0009 at 1123 K, i.e., 0.12% of the Ti sites are occupied by Al atoms and 0.36% of the Al sites by Ti atoms.  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Keywords: Statistical-thermodynamic model; D019 -phases; Point defects; Thermodynamic activities; Ti3 Al

1. Introduction

Alloys based on intermetallic compounds of the light elements Ti and Al are promising candidates for applications as high temperature engineering materials due to some very interesting properties like low density, high melting point, high strength, and an adequate creep resistance [1–4]. Of particular importance is γ -TiAl (with L10 -structure), very frequently in combination with α2 -Ti3 Al (with D019 -structure) and with B2-compounds which are stabilized by ternary additions [5,6]. By this combination of phases it has been attempted to optimize the corresponding microstructure of the alloys. However, there are also several two-phase alloys under development which are based on α2 -Ti3 Al as the main component [2]. α2 -Ti3 Al (which will be designated simply Ti3 Al from now on) crystallizes in the ordered hexagonal D019 -type

* Correspondence and reprints.

E-mail address: [email protected] (H. Ipser).

Fig. 1. Part of a perfectly ordered crystal with the D019 -structure (space group P 63 /mmc; prototype Ni3 Sn): white balls are A-atoms (Ti), black balls are B-atoms (Al). This illustration shows the close relationship with the hcp A3-structure.

structure (Fig. 1) which can be derived from the hexagonally close-packed (hcp) solid solution (α-Ti) by ordering of the Al and Ti atoms: looking at the close-packed layers of hypothetical perfectly ordered Ti3 Al, every other position in alternate rows of atoms is occupied by Al, whereas all other positions are occupied by Ti. This leads to a coordination where every Al atom is surrounded by twelve

1293-2558/02/$ – see front matter  2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved. PII: S 1 2 9 3 - 2 5 5 8 ( 0 2 ) 0 1 3 7 7 - 8

1114

O. Semenova et al. / Solid State Sciences 4 (2002) 1113–1117

Ti nearest neighbors, and every Ti atom is surrounded by four Al and eight Ti nearest neighbors. In this respect, the D019 -structure is analogous to the cubic L12 -structure where identical coordination numbers are observed, the only difference being the stacking sequence. Whereas in the hexagonal D019 -type an ABAB... stacking of the hexagonal layers occurs (as in the hcp A3 structure), the characteristic ABCABC. . . stacking is observed in the L12 -type (as in the fcc A1 structure). Therefore it is not surprising that several A3 B intermetallic compounds are known where L12 → D019 transformations occur, as for example in Fe3 Ge, Fe3 Ga, Ga3 Tb, BiTl3 , and MnZn3 [7,8]. Although the exact phase relationships around Ti3 Al are still not clarified unequivocally (compare [9–14]) it is undisputed that the phase exhibits a considerable range of homogeneity (up to about 15 at%) [14] which is due to constitutional point defects in the crystal lattice. Additionally, there will be thermal (or intrinsic) defects at any temperature above 0 K which cause a certain disorder even at the stoichiometric (3 : 1) composition. Since various physical properties are closely related to the amount of disorder in an intermetallic compound it is of interest to learn more about the type and concentration of the point defects present in Ti3 Al. It is the aim of the present study to compare thermodynamic activities that had been calculated by a statisticalthermodynamic model with experimental data and to estimate in this way the energies of formation of the possible point defects. With these energy values it will be possible to calculate the concentration of the defects as a function of temperature and composition.

2. Statistical-thermodynamic model Fig. 1 shows a small part of the hexagonal D019 -structure (space group: P 63 /mmc; prototype: Ni3 Sn) in perfect order illustrating the close relationship with the hcp A3-structure. For the derivation of the statistical-thermodynamic model the crystal lattice is divided into two different sublattices: the α-sublattice is preferentially occupied by A (Ti) atoms, the β-sublattice by B (Al) atoms. If N  is the total number of lattice sites, then the numbers of sublattice sites, N α and N β , are given by 3 1 N α = N  and N β = N  4 4 thus reflecting the ideal (3 : 1) stoichiometry. Four types of point defects are allowed in the lattice, both as thermal defects that are present at finite temperatures T > 0 K even in the strictly stoichiometric crystal, and also as constitutional defects which are responsible for deviation from stoichiometry. These are anti-structure atoms and vacancies on the two sublattices, and their numbers are given by β

β

NBα , NA , NVα , NV ,

where the subscript indicates the species, A atoms, B atoms, or vacancies (V), and the superscript indicates the sublattice, α or β. The possibility of interstitial defects is not considered. As discussed above, the nearest neighbor coordination numbers in the D019 -lattice are identical to those in the L12 -structure. Additionally, we have the same numbers of sublattice sites in the two structure types. This means, that from a statistical-thermodynamic point of view, the two crystal structures are analogous and can be treated with the same formalism. Consequently, the model equations for L12 -phases that were derived recently by Krachler et al. [15] can be likewise applied to compounds crystallizing in the D019 -structure. It should suffice here to point out that a grand-canonical approach is used which describes an open system that is allowed to exchange both energy and matter with its surrounding. Therefore the grand partition function Ξ is used  
(NA )n (NB )n Ξ= (1) exp µA + µB − En /kT NL NL n and correspondingly the grand potential Ω Ω = −kT ln Ξ.

(2)

In these equations µA and µB are the chemical potentials of A and B, resp.; (NA )n and (NB )n are the numbers of A- or B-atoms in the crystal, and En is its energy when it exists in the state n; NL is Avogadro’s number. In order to simplify the derivation, several assumptions are introduced: (i) the introduction of defects into the lattice does not cause any changes in the vibrational entropy, (ii) it does not cause any volume changes, and (iii) the defects are assumed to be non-interacting as suggested originally by Wagner and Schottky [16]. All other details of the derivation can be found in Ref. [15]. At the end we obtain equations for the concentrations of the four types of point defects on the two sublattices. With these it is then possible to calculate the chemical potentials of the two components as a function of temperature and composition. Since absolute values of the chemical potentials µi cannot be obtained, they are frequently referred to their values µi,0 at the stoichiometric composition A3 B for which xB = 0.25: ai µi − µi,0 = ln . RT ai,0

(3)

This activity ratio can be calculated and is conveniently plotted as a function of composition. Thus a comparison of the theoretical activity curve with experimental data points is easily possible because different experimental thermodynamic methods yield directly such activities.

O. Semenova et al. / Solid State Sciences 4 (2002) 1113–1117

As in previous investigations [15,17–19], the energies of formation of the four different types of point defects at the stoichiometric composition (as defined, for example, by Gao et al. [20]) are used as variable parameters:         Ef Aβ , Ef Bα , Ef Vα , and Ef Vβ , where Ef (Aβ ) is the energy of formation of an anti-structure A-atom on the β-sublattice, with all other definitions correspondingly. If such defect formation energies have been determined (e.g., by theoretical methods), the defect concentrations and the thermodynamic activities can be calculated in a straightforward way and can be compared with experimental data. On the other hand, if these defect formation energies are not available in the literature, as in the case of Ti3 Al, the statistical-thermodynamic formalism provides a tool to at least estimate their values from experimental thermodynamic data by a simple curve-fitting procedure.

3. Experimental aluminum activities for Ti3 Al

(4)

where GAl is the partial Gibbs energy of mixing of aluminum in Ti3 Al, z is the number of electrons exchanged in the galvanic cell, F is Faraday’s constant, and E is the measured emf; R and T have their usual meaning. Eckert et al. [23], on the other hand, reported aluminum partial pressures, p Al , as a function of temperature which allowed to obtain the corresponding values for 1123 K; they were o , combined with the vapor pressure of pure aluminum, pAl at the same temperature from Kubaschewski et al. [30] to calculate the activity values according to o . aAl = pAl /pAl

Table 1 Experimental aluminum activities in Ti3 Al according to Refs. [21–23], converted to 1123 K x Al

ln a Al

ln(aAl /aAl,0 )

Reference

0.224 0.250 0.268 0.289 0.329 0.365

−7.769 −7.179 −6.564 −6.722 −5.973 −5.279

−0.589 0.000 0.616 0.458 1.206 1.899

Samokhval et al. [21]

0.250 0.300 0.350

−9.544 −8.479 −6.626

0.000 1.064 2.918

Eckert et al. [23]

0.313

−5.870

1.309

Reddy et al. [22]

[22] the value ln aAl,0 = −7.179 by Samokhval et al. [21] was used since they used the same experimental method at approximately the same temperature.

4. Results and discussion

Thermodynamic activities of aluminum in the composition range of Ti3 Al were determined experimentally by Samokhval et al. [21] and Reddy et al. [22] (emf method with CaF2 as solid electrolyte, about 820 to 1010 K), and by Eckert et al. [23] (Knudsen cell-mass spectrometric method, about 1180 to 1430 K). In order to be able to compare the corresponding experimental data, an average temperature of 1123 K was chosen and all activity values were converted to this temperature. Since the emf values in Refs. [21,22] were given as functions of temperature the calculation of aluminum activities, a Al , at 1123 K was straightforward using the relationship RT ln aAl = GAl = −zF E,

1115

(5)

Unfortunately, the absolute values are in severe disagreement which is probably due to the different temperature ranges of the experiments and the errors incurred by converting them. However, if the data sets are normalized with respect to the activity at the exactly stoichiometric composition, i.e., if values of aAl /aAl,0 are calculated, the data become more or less consistent. The results are listed in Table 1. For the single data point of Reddy et al. at 31.3 at% Al

The experimental aluminum activities are shown in Fig. 2. For the calculation of the theoretical activity curve it was assumed that the energies of formation of vacancies on the two sublattices would be rather high compared to those for anti-structure atoms. This is certainly justified since Ti3 Al exhibits an order-disorder transition into α-Ti, i.e., the hcp solid solution of aluminum in titanium [9–14] (even if some particularities of this part of the phase diagram are still disputed), which is caused by a random distribution of Ti and Al atoms over both sublattices. Thus it is to be expected that any deviation from the stoichiometric composition is also caused by anti-structure atoms whereas vacancies will play a minor role as point defects. These conclusions are further supported by the results of positron lifetime spectrometric experiments by Würschum et al. [24] and Shirai et al. [25] which showed that the deviation from stoichiometry must be mainly caused by anti-structure defects with negligible contributions by constitutional vacancies. Similar results were obtained by Rüsing et al. [26, 27] and Herzig et al. [28] from their diffusion measurements using 44 Ti and 69 Ga as tracers: due to the very small composition dependence of the self-diffusion coefficient of Ti and Ga (as an Al-substituting element) it was concluded that the concentrations of constitutional vacancies must be negligibly small. From their positron annihilation study Würschum et al. derived a value for the vacancy formation enthalpy of 1.55 ± 0.2 eV [24]. Therefore it was assumed that Ef (Vα ) = Ef (Vβ ) = 1.5 eV would be a reasonable value for the vacancy formation energies in our calculations. In Fig. 2 three theoretical curves are drawn through the experimental activity data at 1123 K using the following parameter sets,     Ef Vα = Ef Vβ = 1.5 eV;

1116

O. Semenova et al. / Solid State Sciences 4 (2002) 1113–1117

Fig. 2. Natural logarithm of the aluminum activity in Ti3 Al at 1123 K as a function of composition, calculated with the defect formation energies Ef (VTi ) = Ef (VAl ) = 1.5 eV, and Ef (TiAl ) = Ef (AlTi ) = 0.6 eV; experimental data points are from Table 1.

Fig. 3. Defect concentrations in Ti3 Al at 1123 K calculated with the defect formation energies Ef (VTi ) = Ef (VAl ) = 1.5 eV, and Ef (TiAl ) = Ef (AlTi ) = 0.6 eV; the lines for the concentrations of vacancies on the two sublattices coincide with the axis.

Ef (Aβ ) = Ef (Bα ) = 0.5 eV (curve 1) Ef (Vα ) = Ef (Vβ ) = 1.5 eV; Ef (Aβ ) = Ef (Bα ) = 0.6 eV (curve 2) Ef (Vα ) = Ef (Vβ ) = 1.5 eV; Ef (Aβ ) = Ef (Bα ) = 0.7 eV (curve 3). The phase boundaries indicated in the figure (about 22.3 to 37.3 at% Al at 1123 K) were taken from Zhang et al. [14]. Although the scatter of the data points is considerable it is probably safe to conclude that the energies of formation of anti-structure defects should lie within the indicated range, i.e., between 0.5 and 0.7 eV. There is only one single data point at 35 at% Al by Eckert et al. [23] with a large deviation. However, one has to keep in mind that the simplifying assumptions of the statistical-thermodynamic model become increasingly less valid with large deviations from stoichiometry where the defect concentrations become large and the defects are certainly not isolated anymore. Thus we suggest a value of Ef (Aβ ) = Ef (Bα ) = 0.6±0.1 eV for the energies of formation of anti-structure atoms. As far as the vacancy formation energies are concerned, Würschum et al. [24] reported (based on nearest-neighbor bond model calculations) that vacancy formation occurs predominantly on the Ti sublattice, indicating a higher value of Ef (VAl ) compared to Ef (VTi ). Due to the scatter of the experimental activity values in Fig. 2 a more detailed evaluation of the energy parameters by our model is certainly not justified. Yet a higher value of Ef (VAl ) ≈ 2.0 eV would appear reasonable considering the experience from the intermetallic compound Ni3 Al with L12 -structure [15,29]. Fig. 3 shows the variation of the defect concentrations with the composition, calculated with the parameter set Ef (VTi ) = Ef (VAl ) = 1.5 eV, and Ef (TiAl) = Ef (AlTi ) = 0.6 eV, as suggested above. It can be seen that nonstoichiometry is practically exclusively caused by antistructure defects, as had to be expected from the chosen

parameter set. The vacancy concentrations are several orders of magnitude smaller so that the lines describing their concentrations coincide with the corresponding axis in Fig. 3. It turns out that the concentration of thermal antistructure atoms at the stoichiometric composition, referred to the total number of lattice sites, is  α=

Al NTi N



 = stoich

Ti NAl N

 = 0.0009

(6)

stoich

according to our evaluation. This means that at 25 at% Al, i.e. at the exactly stoichiometric composition, 0.12% of the Ti sublattice sites are occupied by Al atoms and 0.36% of the Al sublattice sites are occupied by Ti. As it was pointed out earlier by Krachler et al. [15], it is rather the average value of the enthalpies of formation of the anti-structure atoms, 1/2[Ef(Aβ ) + Ef (Bα )], than the individual values that determines the slope of the activity curve in Fig. 2. Therefore it was assumed here that Ef (Aβ ) and Ef (Bα ) are equal (see above). On the other hand, it is clear that with the considerably higher energy of formation of vacancies their concentrations will be much lower than those of the anti-structure atoms (cf. Fig. 3). As a consequence, the numbers of (thermal) anti-structure atoms at the exactly stoichiometric composition, xAl = 0.25, must be more or less equal in order to retain stoichiometry. Therefore it is not unreasonable to expect the values of Ef (Aβ ) and Ef (Bα ) to be very similar if not actually equal, as it was found for the L12 -compound Ni3 Al from firstprinciple calculations by Schweiger et al. [29]. With the application of the statistical-thermodynamic model we were able to estimate, for the first time, reasonable values for the energies of formation of the point defects in Ti3 Al. It is hoped that in the future it will be possible to calculate the corresponding values by first principle methods as has been done for the L12 -compound Ni3 Al [29].

O. Semenova et al. / Solid State Sciences 4 (2002) 1113–1117

Acknowledgement Financial support of this study by the Austrian Science Foundation (under project No. P14519-CHE) is gratefully acknowledged. References [1] R.W. Cahn, Metals Mater. Processes 1 (1989) 1. [2] F.H. Froes, C. Suryanarayana, D. Eliezer, J. Mater. Sci. 27 (1992) 5113. [3] Y.-W. Kim, J. of Metals 46 (1994) 30. [4] E.A. Loria, Intermetallics 9 (2001) 997. [5] R. Kainuma, Y. Fujita, H. Mitsui, I. Ohnuma, K. Ishida, Intermetallics 8 (2000) 855. [6] R. Kainuma, I. Ohnuma, K. Ishikawa, K. Ishida, Intermetallics 8 (2000) 869. [7] Q.Z. Chen, A.H.W. Ngan, B.J. Duggan, Intermetallics 6 (1998) 105. [8] Q.Z. Chen, A.H.W. Ngan, B.J. Duggan, J. Mater. Sci. 33 (1998) 5405. [9] U.R. Kattner, J.-C. Lin, Y.A. Chang, Metall. Trans. A 23A (1992) 2081. [10] R. Kainuma, G. Inden, Z. Metallkd. 88 (1997) 429. [11] D. Veeraraghavan, U. Pilchowski, B. Natarajan, V.K. Vasudevan, Acta Mater. 46 (1998) 405. [12] H.M. Flower, J. Christodolou, Mater. Sci. Technol. 15 (1999) 45. [13] I. Ohnuma, Y. Fujita, H. Mitsui, K. Ishikawa, R. Kainuma, K. Ishida, Acta Mater. 48 (2000) 3113.

1117

[14] F. Zhang, S.L. Chen, Y.A. Chang, U.R. Kattner, Intermetallics 5 (1997) 471. [15] R. Krachler, O.P. Semenova, H. Ipser, Phys. Stat. Sol. (b) 216 (1999) 943. [16] C. Wagner, W. Schottky, Z. Physik. Chem. B 11 (1931) 163. [17] O.P. Semenova, W. Yuan, R. Krachler, H. Ipser, Scripta Mater. 42 (2000) 567. [18] W. Yuan, O. Diwald, A. Mikula, H. Ipser, Z. Metallkde. 91 (2000) 448. [19] H. Ipser, O. Semenova, R. Krachler, J. Alloys Comp. 338 (2002) 20. [20] F. Gao, D.J. Bacon, G.J. Ackland, Phil. Mag. A 67 (1993) 275. [21] V.V. Samokhval, P.A. Poleshchuk, A.A. Vecher, Zh. Fiz. Khim. 45 (1971) 2071; V.V. Samokhval, P.A. Poleshchuk, A.A. Vecher, Russ. J. Phys. Chem. 45 (1971) 1174. [22] R.G. Reddy, A.M. Yahya, L. Brewer, J. Alloys Comp. 321 (2001) 223. [23] M. Eckert, L. Bencze, D. Kath, H. Nickel, K. Hilpert, Ber. Bunsenges. Phys. Chem. 100 (1996) 418. [24] R. Würschum, E.A. Kümmerle, K. Badura-Gergen, A. Seeger, Chr. Herzig, H.-E. Schaefer, J. Appl. Phys. 80 (1996) 724. [25] Y. Shirai, T. Murakami, N. Ogawa, M. Yamaguchi, Intermetallics 4 (1996) 31. [26] J. Rüsing, Chr. Herzig, Intermetallics 4 (1996) 647. [27] J. Rüsing, Chr. Herzig, Scripta Met. Mater. 33 (1995) 561. [28] Chr. Herzig, M. Friesel, D. Derdau, S.V. Divinski, Intermetallics 7 (1999) 1141. [29] H. Schweiger, O. Semenova, W. Wolf, W. Püschl, W. Pfeiler, R. Podloucky, H. Ipser, Scripta Mater. 46 (2002) 37. [30] O. Kubaschewski, E.Ll. Evans, C.B. Alcock, Metallurgical Thermochemistry, Pergamon Press, Oxford, 1967.