Diffusion mechanism and defect concentrations in β′-FeAl, an intermetallic compound with B2 structure

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Diffusion mechanism and defect concentrations in P’-FeAl, an intermetallic compound with B2 structure R. Krachler, H. Ipser,* Institut fir

Anorganische

Chemie,

Univ. Wien, Wiihringerstrasse

42, Austria

B. Sepiol & G. Vogl Institut fir Festkarperphysik, (Received

Univ. Wien, Strudlhofgusse

28 February

1994; accepted

4, A-1090

25 March

Wien, Austria

1994)

A statistical thermodynamic model of the Wagner-Schottky type is applied in order to determine the defect pattern in the P’-FeAI phase. Model calculations based on the concentration dependence of the thermodynamic activity of Al yield the following defect concentrations for the composition Feso 5A1495 at 1338 K: 1.76 + 0.20% of all lattice sites are vacant, 3.36 f 0~40% of the Al-sublattice sites are occupied by Fe atoms, and 0.84 + 0.40% of the Fe-sublattice sites are occupied by Al atoms. The obtained concentration of anti-structure Fe atoms is close to the number derived from Miissbauer spectroscopy studies which were carried out to determine the diffusion jump mechanism of Fe atoms in the ordered alloy. Experimental results and model calculations imply vacancies and anti-structure atoms on both sublattices. Therefore, although P’-FeAl in principle belongs to the triple-defect B2-type, a certain degree of ‘hybrid behaviour’ must be taken into consideration. Key words: FeAl; nonstoichiometry in @‘-FeAl; defect concentrations in P’-FeAI; vacancy concentrations in /?‘-FeAl; statistical thermodynamic model for P’-FeAl; diffusion of Fe atoms in P’-FeAl.

1

INTRODUCTION

Fe-sublattice and the Al-sublattice. The movement from one Fe-sublattice site to another is not direct but is a combination of a jump into a nearest-neighbour vacancy, creating an antistructure Fe atom, and a jump back into a vacancy on the Fe-sublattice. From the ratio of the residence times of the Fe atom on the two different sublattices, it was possible to derive a concentration of anti-structure Fe atoms of about 3% in a single crystal with 49.5 at% Al at 1338 K, i.e. about 3% of the Al-sublattice sites are occupied by Fe atoms. The aim of the present work is to compare this value with expectations from a statistical thermodynamic treatment of the vapour pressure measurements by Eldridge and Komarek6 in order to determine the full defect pattern (antistructure atoms and vacancies) in p-FeAl. The results of field ion-microscopy investigations and dilatometric measurements by Paris et al.,’ the vacancy concentrations which were obtained by pycnometric density measurements by Riviere,* and

From measurements of lattice parameters and densitiese3 it is known that the nonstoichiometric B2-phase in the Fe-Al system can contain large numbers of vacancies. Vacancy concentration and diffusivity are correlated phenomena, since an atom needs a nearest-neighbour or next-nearestneighbour vacancy for changing place in the crystal lattice. This has recently been discussed for a number of B2-phases by Kao and Chang.4 The exact diffusion mechanism in P’-FeAl has been studied by Mossbauer spectroscopy’ which allows the elementary diffusion jumps of Fe atoms to be deduced by measuring the angular dependence of the diffusional line broadening. A model for the jump mechanism of Fe atoms has been constructed based on the results of this investigation, which implies vacancies on both sublattices, the *

To whom correspondence

should

be addressed. 83

84

R. Krachler, H. Ipser, B. Sepiol, G. Vogl

the Fe anti-structure atom concentrations resulting from magnetic measurements by Haberkern” are compared with our results. The only point defects occurring in B2-phases are vacancies and anti-structure atoms, as discussed by Chang and Neumann.’ At higher temperatures thermodynamic equilibrium is established as a consequence of the rapid movement of the atoms and vacancies in the crystal lattice. The vacancy content in P’-FeAl at 1338 K is sufficiently high so as to guarantee several atomic jumps during one Miissbauer lifetime of 10e7 seconds. With this high jump-frequency it can be assumed that thermodynamic equilibrium is established in a short time. In thermodynamic equilibrium the point defect concentrations on the two sublattices are functions of the temperature and the composition of the alloy. It is therefore possible to calculate point defect concentrations from thermodynamic considerations on the basis of thermodynamic data.

2 STATISTICAL MECHANISM

MODEL AND DEFECT

If a random distribution of the point defects over the crystal lattice sites is assumed, simple statistical models of the Bragg-Williams type”-‘” or of the Wagner-Schottky typei4.” can be applied to calculate the equilibrium concentrations of the defects for all compositions within the homogeneity range. The Bragg-Williams theory is based on the approximation that the energy of the crystal can be represented as the sum of the interaction energies between the atoms (or vacancies) situated on nearest-neighbour lattice sites. The interaction energies of individual pairs are assumed to be independent of the alloy composition, degree of long-range order, temperature and the type of atoms surrounding a given pair. The Wagner-Schottky approximation works under the assumption that at constant temperature the internal energy, volume and vibrational entropy of the crystal are linear functions of the numbers of atoms or vacancies, respectively, in the different sublattices. In both cases, geometrical distortions of the crystal are neglected, as are the effects of the variation of the lattice constants with composition, degree of order and temperature on the interaction energies. The B2-type lattice consists of two interpenetrating simple cubic sublattices, the (Y-and the psublattices. In a completely ordered state, the A

atoms are situated on the a-sublattice and the B atoms on the @sublattice. Traditionally, B2-phases have been considered to belong to one of two idealized types. Antistructure type B2-phases contain anti-structure atoms on both sublattices, but the vacancy concentrations are very small. In triple-defect type B2-phases, vacancies are found on the cr-sublattice and anti-structure atoms on the P-sublattice, but the two other defect concentrations (vacancies on the @sublattice and anti-structure atoms on the (Ysublattice) are negligible. This has been discussed by Chang and Neumann.’ For B2-phases commonly regarded as belonging to the triple-defect type like, for example, CoAl, NiAl, FeAl and CoGa, the lattice parameter tends to increase with the concentration of the larger B component until the stoichiometric composition is approached. However, when the content of B becomes higher than 50 at’% the lattice parameter starts to decrease due to the formation of vacancies on the cY-sublattice. In contrast, for B2-phases attributed to the anti-structure type, the lattice parameter continues its increasing tendency beyond the stoichiometric composition, although a change of slope is usually observed. From this point of view, P’-FeAl seems to be a typical triple-defect BZphase, since the lattice parameter passes through a maximum at 50 at% A1.r~~On the other hand, the diffusion mechanism deduced by Sepiol and Vogl’ implies vacancies on both sublattices. This corresponds with the ideas of NeumannI who found a relationship between the enthalpy of formation and the defect mechanism in B2-phases. Small enthalpies of formation are correlated with anti-structure defects, large ones with tripledefects. For P’-FeAl, Neumann’s model predicts a hybrid behaviour, with vacancies and antistructure atoms on both sublattices. Although different statistical models9~10~‘4have been presented in the past, they were designed to describe either typical anti-structure or typical triple-defect B2-phases. None of them could deal with a situation, as expected in P’-FeAl, where both defect mechanisms seem to occur simultaneously. Therefore, Krachler et ~1.‘~ derived a set of equations based on the Wagner-Schottky approximation” in order to describe the composition dependence of the point defect concentrations and of the thermodynamic activities of the alloy components in B2-phases with vacancies and anti-structure atoms on both sublattices, as required by Neumann. Any degree of ‘hybrid behaviour’ is allowed in this model.

D#usion mechanism and defect concentrations in p’-FeAI

The complete derivation of the corresponding equations is given by Krachler et al. I5 Three ‘disorder parameters’, which are the point defect concentrations at the stoichiometric composition, are introduced:

where N’ = total number of lattice lattices) Nl = number of anti-structure Ng = number of anti-structure N,” = number of vacancies on NVp= number of vacancies on

sites (both subA atoms B atoms the cu-sublattice the P-sublattice.

According to Wagner and Schottky,‘7 there are five variables for the statistical treatment. These are: N’, Ng, N/, N,“, N{. The Gibbs energy can be expressed as: G = G(p, T, N, N:, N/, NJ,“,N,p> The condition for thermodynamic (SG),,T.N.,,,Y, =O

(1)

equilibrium is:

85

vacancy, creating a substitutional defect (corresponding to a pure anti-structure defect mechanism). In this case the vacancy concentration is rather small and nearly independent of the composition. For a z b, a certain contribution of triple defects exists which depends on the difference between these parameters. The vacancy concentrations of the two sublattices at the equiatomic composition are different in this case. As shown by the model calculations, this different behaviour of the two sublattices results in a rise of the total vacancy concentration with increasing mole fraction of the component whose sublattice exhibits the lesser tendency to create vacancies. The results of the Mossbauer studies5 have suggested that the vacancies might be arranged in a sort of ordered structure (DO,-type structure) caused by repulsive interactions. However, no interactions between point defects are considered in our model;15 rather all defects are assumed to be distributed randomly over the corresponding sublattice sites.

(2)

where N, = total number of A atoms N, = total number of B atoms. Considering the basic feature of the B2-(CsCI-) structure (two sublattices with equal numbers of lattice sites) eqns (1) and (2) result in a relation where the number of variables is reduced to three. If it is assumed that the probability of an A atom (B atom, respectively) changing places with a nearest-neighbour vacancy in the ‘wrong’ sublattice is independent of composition (for not-toolarge deviations from stoichiometry) two additional independent relations result. With this system of equations, the five unknowns can be calculated for given parameters a, b and c and given alloy composition. In this way we get the equilibrium point defect concentrations as functions of the alloy composition. Finally, the activities of the alloy components can be derived from the defect concentrations.‘5 The parameters u. h and c are connected by the following condition: ((c 2 la - bl X 2)). In cases where experimental values for the disorder parameters are not available, these values can be obtained from the shape of activity vs composition curves resulting from thermodynamic measurements, as will be shown below. For u = b, both A and B atoms have the same tendency to change place with a nearest-neighbour

3 COMPARISON EXPERIMENTAL

OF THE DATA

MODEL

WITH

Activities of aluminium in /Y-FeAI were determined between 1100 and 1400 K by Eldridge and Komarek.6 They used an isopiestic method in which iron specimens, heated in a temperature gradient, were equilibrated in a closed all-ceramic system with aluminium vapour. Experimental data points are shown as so-called ‘equilibrium curves’, i.e. plots of specimen composition vs specimen temperature. The final results (partial molar thermodynamic properties of aluminium) are given later in Table 1, as smoothed values at certain compositions. Unfortunately. the spacing of these compositions in the PI-phase is rather large, so subtle effects might be overlooked. Therefore, the original experimental data points were re-evaluated very carefully, resulting in partial molar enthalpies of aluminium which differ somewhat from the published smoothed values (see Fig. 1). In Figs 2 and 3, aluminium activities are shown as functions of the alloy composition, calculated for the temperatures 1338 K and 1073 K, respectively, by use of the partial molar enthalpies in Fig. 1 (re-evaluation). Theoretical activity curves were fitted through the experimental data, and the best agreement was obtained with the parameters: n = 0.014, b = 0.005, c = 0.02 at 1338 K (Fig. 2) and a = 0.012, b = 0.004, c = 0.017 at 1073 K

R. Krachler, H. Ipser, B. Sepiol, G. Vogl

86

I

I

I

Fe-AI

I

I

I

I

K

1073

- 60

-70

- 80

I

I

I

I

I

I

I

36

30

40

42

44

46

48

-I

I at%

- 4.0

Al

0

Fig. 1. Aluminium partial molar enthalpies in /3’-FeAl according to Eldridge and Komarek. Open circles: originally tabulated values; full circles: re-evaluation.

(Fig. 3). The error limits for these disorder parameters are estimated to be approximately kO.002. For large deviations from stoichiometry (i.e. for compositions below about 44 at% Al) a discrepancy between the theoretical activity curve and the experimental data begins to develop (Fig. 3). This may be explained by the influence of second-nearest-neighbour interactions,” which are not considered in the present model calculations,‘5 or even by possible changes in the crystal structure, as suggested by Koster and Godecke.” The parameters a, b and c, obtained from the application of the statistical model to the results of the Al activity measurements, can now be used to calculate the concentrations of all point defects as functions of the composition within the P’-FeAl phase. In this way we obtain the following values for a composition Fe5,-,.5Al,,, at 1338 K: N;/Na = 0.0084 (NV”+ N&/N’ = 0.0176. 1

0

ap

- 4.4

00 -

N,&Np= 0.0336

3.6

/

40

8

I

I

I

I

I

40

42

44

46

48

I at %

AL

Fig. 3. Natural logarithms of aluminium activities in p’-FeAl at 1073K according to Eldridge and Komarek. Reference state is liquid aluminium. The theoretical curve was calculated with a = 0,012, b = 0.004, c = 0.017 and In u(A~),,.,~,, = -3.54.

Thus, including estimated error bars, 3.36 f 0.40% of the Al-sublattice sites are occupied by Fe atoms, 0.84 f 0.40% of the Fe-sublattice sites are occupied by Al atoms, and 1.76 + 0.20% of all lattice sites are empty. It must be pointed out that the value of 3.36% Fe anti-structure atoms compares very well with the result estimated from the Miissbauer measurements, i.e. approximately 3%.5 Paris et cd7 determined densities of Fe-Al alloys, quenched from 1273 K, by a dilatometric technique and derived from these the total vacancy concentration. For an alloy Fe,.SA1,,, they obtained a value of 1.85%. Although this is slightly higher than our values (1.76% at 1338 K and 1.46% at 1073 K), and despite the difference in temperature, the agreement would seem to be I

2

I

I

I

I

I

50

52

I

0.04 Fe-AI

1073

K

_

0.03

0.02 i

4,

0.0 1

40

42

44

46

40

at %

Al 60

Fig. 2. Natural logarithms of aluminium activities in P’-FeAl at 1338 K according to Eldridge and Komarek. Reference state is liquid aluminium. The theoretical curve was calculated with a = 0,014, b = 0.005, c = 0.02 and In a(Al),, at0L, = -2.20.

42

,, 44

46

40

at %

Al

Fig. 4. Experimental vacancy concentrations in P’-FeAl according to Riviere. The theoretical curve was calculated with a = 0.012, b = 0,004 and c = 0.017 The vacancy concentration is expressed as: z = (N,” + N&IN’.

D@ision mechanism and defect concentrations in /T-FeAl Table 1. Point defect concentrations in p-FeAI, Reference number

Method

T(R)

at% AI

referred to the total number of lattice sites Vacancy Total

Dilatometry

Quenched

(1123) Displacement method

Mossbauer spectroscopy Field ionmicroscopy Dilatometry

Quenched

1338 Quenched (1273) Quenched (1273)

The different Fe-sublattice

455 47.0

0.44 0.56

40.5 44.0 47.0 50.5

0.3 0.5 0.8 1.9

concentration

%I

Anti-structure Fe atoms concentration

1.5

49.5 49.5

1123

45.0 46.0 47.0 48.0 49.0 50.0

Present work Model calculations, parameters fitted to vapour pressure measurements

1073

40.5 44.0 45.0 45.5 46.0 47.0 48.0 49.0 49.5 50.0 50.5 44.0 46.0 48.0 49.5 50.5

1.2 1.85 5.17 4.32 3.52 2.75 2.02 I.35 0.48 0.63 0.69 0.74 0.79 0.90 I .05 1.32 I .46 1.70 1.94 0.83 1.02 1.32 1.76 2.22

9.76 6.39 5.48 4.89 4.44 3.57 2.73 1.82 1.50 I .20 0.90 6.50 4.57 2.89 I .68 1.10

0.46 0.60 0.66 0.71 0.75 0.85 1.00

1.26 1.40 1.65 I.91 0.76 0.93

1.22 1.65 2.15

concentrations are expressed as: total vacancy concentration = lOO(N,” + N&N’; = 100 N,“N’; anti-structure Fe atoms concentration = 100 NjIN’.

satisfactory. For the same alloy (same composition, same quenching conditions), the vacancy concentration on iron sites was found to be 1.2% by field ion microscopy,’ a result very close to our values of 1.65% (1338 K) and 1.40% (1073 K). In Fig. 4, the theoretical variation of the vacancy concentration with the alloy composition for 1073 K is shown, calculated with the parameter values from Fig. 3. The theoretical curve is compared with experimental vacancy concentrations obtained from pycnometric density measurements by Riviere.2 These measurements were carried out at room temperature, apparently on quenched samples. According to Chang and Neumann9 vacancies in @‘-FeAl alloys seem to become frozen-in at about 1000 K. Consequently, one could assume that the vacancy concentrations determined by Riviere2 correspond to a tempera-

%

Fe-sublattice

49.5

Magnetic measurements

1338

87

vacancy

concentration

on the

ture of around 1000 K. Considering the experimental problems encountered in pycnometric measurements on quenched samples, the agreement of the theoretical curve and the experimental data points in Fig. 4 is quite encouraging. Magnetic measurements can also be used to determine the presence of Fe atoms on Al-sublattice sites. This method was employed by Haberkern’ to derive the concentration of anti-structure Fe atoms in P’-FeAl at 1123 K. The results are shown in Fig. 5 and are compared with our theoretical curve for 1073 K (again computed with the parameters obtained from the application of the model to the activity data in Fig. 3). It can be seen that, despite a small difference in temperature of 50 K, the curve describes the experimental data points very well. Table 1 compares defect concentrations ob-

88

R. Krachler,

H. Ipser,

B. Sepiol, G. Vogl

obtain satisfactory agreement between theoretical predictions and experimental observations. The excellent agreement between the results of the different experimental methods considerably increases the confidence in the reported defect concentrations in the p’-FeAl phase at high temperatures.

ACKNOWLEDGEMENT

1 45

I 46

I Ll

I

I

I

I

48

49

50

at %

A I

This work was supported by the Austrian ‘Fonds zur Fiirderung der wissenschaftlichen Forschung’ (project S5601).

Fig. 5. Experimentally

determined Fe anti-structure atom concentrations in P’-FeAl at 1123 K according to Haberkern. The theoretical curve was calculated with a = 0.012, b = 0.004 and c = 0.017. The concentration is expressed as: C, = 100 Nl/N’.

tained by different experimental methods with those obtained from the statistical thermodynamic model.

4 CONCLUSIONS The present study suggests that the results of very different experimental methods can be treated with the help of a statistical theoretical model leading to a clear and consistent picture of the defect mechanism in the P’-FeAl phase. Thermodynamic measurements, diffusion measurements using Mossbauer spectroscopy, field ion-microscopy, density measurements by dilatometry and pycnometric methods, and magnetic measurements yield comparable results for the concentrations of the different point defects. Although /3’-FeAI had been considered previously as a typical triple defect B2-phase,’ it is obvious from the values of the model parameters a and b in Figs 2 and 3 that it exhibits a certain Anti-structure atoms and ‘hybrid behaviour’. vacancies must be taken into consideration on both sublattices by the statistical treatment to

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K. L., Trans. TMS-AZME,

230

(1964) 226.

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G. G. & Lightstone,

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15. Krachler,

R., Ipser, H. & Komarek, K. L., J. Phys. Chem. Solids, 50 (1989) 1127; 51 (1990) 1239. 16. Neumann, J. P., Acta Metal/., 28 (1980) 1165. 17. Wagner, C. & Schottky, W., Z. Physik. Chemie B, 11 (1931) 163. 18. Ipser, H., Neumann, J. P. & Chang, Y. A., Monatsh. Chem., 107 (1976) 1471. 19. Kbster, W. & Giidecke, T., Z. Metallkd., 71 (1980) 765.