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Intermetallics 5 (1991) 467469 0 1997 Elsevier Science Limited PII: SO966-9795(97)00018-6
Printed in Great Britain. All rights reserved 0966-9795/97/%17.00 + 0.00
Vacancies in FeAl and NiAl
A. H. Cottrell
Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street, Cambridge, UK, CB2 3QZ (Received 5 March 1997; accepted 5 March 1997)
The B2 compound NiAl extends substantially into Al-rich compositions where it forms constitutional vacancies abundantly, but the corresponding FeAl does not.
However, FeAl forms thermal vacancies abundantly. This difference is traced to the smaller heat of formation of FeAl, so that the enthalpy of formation of con- stitutional vacancies, from the stoichiometric compound in the presence of excess Al, is favourable for NiAl but slightly unfavourable for FeAl. At high tempera- tures, the effect of mixing entropy enables FeAl to overcome this small enthalpy disadvantage, despite the competition of the next phase (FeAl2) for the excess Al.
A corresponding calculation of the NiAl and Ni*Als equilibrium enables the Al-rich phase limit of NiAl to be estimated. The energy of formation of triple defects is calculated to be appreciably lower in FeAl than in NiAl, so that these become abundant at high temperatures in stoichiometric FeAl. 0 1997 Elsevier Science Limited
Key words: A. iron aluminides (based on FeAl), D. defects: point defects, site occupancy, E. defects theory, phase stability, prediction.
1. INTRODUCTION
The B2 compounds, FeAl and NiAl, although similar in many respects, differ strikingly in their formation of vacancies. In Al-rich compositions, the nickel compound forms constitutional vacancies on the nickel sublattice, whereas the iron one does not, at least to any appreciable extent. On the other hand, the iron compound forms thermal vacancies abundantly, indicating a low formation energy.
About this difference, Cahn’ has speculated that there may be a reciprocal relation between ease of formation of the two kinds of vacancies. The pur- pose of this note is to explore such a relationship.
The method follows one used recently2 to explain the existence of constitutional vacancies in NiAli +&, where x<l and 0 represents vacancies on the Ni sublattice, in terms of the many-atom tight-binding character (of the bonding. When such a vacancy is formed, the loss of cohesion in its cage of nearest-neighbour atoms is proportional only to the square root of the ‘effective’ local co-ordination number, z, of the cage atoms. This value of z is
‘effective’ because int.rinsically stronger bonds make effectively greater contributions to it than weaker ones.3 Because the loss goes only as the square root, there is a net energy gain when extra Al is added to NiAl and the Al atoms all then
467
form, albeit only partially, strong bonds with neighbouring Ni atoms; moreover, this gain is only slightly reduced by the loss of Ni-Ni bonds-due to the vacation of some of the sites on the Ni sub- lattice-because these bonds are weak at the large spacing of this sublattice. The repulsive interac- tions between nearest and next-nearest neighbours also play a minor part.
All energies will be given in eV per formula unit.
The calculations below are for the Fe-Al system.
Previously published2 values for the Ni-Al system are given in brackets.
2. ENERGY OF CONSTITUTIONAL VACANCIES IN FeAl
The previous method2 for NiAl will be used to deduce the energy change of the reaction
FeAl + xA1 -+ FeAli+.o,, (1) with appropriate changes to the values of some input parameters.4 In particular, we take the cohe- sive energy of Fe as -4.28 (-4.44) the bulk modu- lus as 1060 eV/nmp3 (1100) and the lattice constant of FeAl as 0.290nm (O-289). (As before, we take the cohesive energy of Al as -3.43). The major
468 A. H. Cottrell change is in the heat of formation, 0.52 (1.22).
Thus the cohesive energy of FeAl is -4.28 -3.43 -0.52 = -8.2 (-9.1) and the bonding component of this is - 11 (- 12) made up of the following individual contributions: Al-Al, -1.97 (-182);
Fe-Al, -7.58 (-8.75); Fe-Fe, -1.49 (-1.37). We notice that the weaker Fe-Al bonding, by com- parison with Ni-Al, is accompanied by slightly stronger Al-Al and Fe-Fe bondings, which is a consequence of their many-body character.
The same method of calculation as for NiAII +,O, then leads to the following x-depen- dent contributions to the energy of the reaction in eqn (1): -3.43 (xA1) and -3.28 (FeAli +XO,), the latter being made up from the following bonding energies: Al-Al, -2.69; Fe-Al; -2.38; Fe-Fe, + 1.32; together with a repulsive energy, + 0.47.
The reaction is thus disfavoured by +0.15x. This contrast with the corresponding value (-0.39~) which favours the reaction to NiAlr +.&I, is due mainly to the weaker affinity of Fe for Al.
3. CONSTITUTIONAL AND THERMAL VACANCIES
Because the reaction of eqn (1) is energetically dis- favoured, constitutional vacancies are not formed in FeAl, in contrast to NiAl. However, this unfavour- able energy, being small, might be overcome easily by thermal energy leading, as Cahn surmised, to an abundance of thermal vacancies. To estimate this, we have first to take account of alternative states of existence for the system FeAl + xA1.
The phase diagram5 shows that the Al-rich phase in equilibrium with FeAl is FeA12, with a heat of formation of 0.82 per formula unit.4 Thus the reaction
FeAl + Al + FeA12 (2) is favoured by 0.82 - 0.52 = 0.3. Starting from a two-phase FeAl + FeA12 mixture, the defect phase FeAli +,I-& could be formed from the reactions
( 1 - x)FeAl + xFeAl2 --+ FeAl + xA1
--f FeAl1+,0,, (3)
which involve a total unfavourable energy change of O-3 + 0.15 = 0.45 per vacancy formed. Regard- ing this as the enthalpy of vacancy formation and ignoring the vibrational entropy of formation, it would give a concentration of thermal vacancies on the iron sublattice of about 1.5 x 10V2 at 1300K, which is comparable with observed values.6
It also follows that the value of x in FeAli +X0, must always remain small, as shown in the phase diagram. By contrast, the favourable energy change for the formation of NiAli +,O, should enable x here to become large in an Al-rich envir- onment. This happens, but when there is a high concentration of vacancies on the Ni-sublattice these interact to form an ordered structure which, with an accompanying minor crystallographic change, becomes the thermodynamically distinct phase, Ni2A13, with a heat of formation of 3.00 per formula unit.4 The reaction
2NiAl+ Al --+ NizA13 (4) is thus favoured by 3.00 -(2 x l-22) = 0.56. It fol- low that the sequence
(1 - 2x)NiAl+ xNiIA13 +NiAl + xA1
--t NiAli +XO, (5)
is disfavoured by 0.56-0.39= 0.17 per vacancy.
Since the concentration of vacancies on the Ni- sublattice in NiAli +Xtl, is x/(1 +x), the mixing entropy for a random distribution of these is
S = -k
[ ln( &) + $n(&)] (6) per vacancy. At 1200K this gives TS = 0.17 at x = 0.24, so that the Al-rich boundary of NiAl is deduced to occur at 55 at% Al, as observed.7
4. TRIPLE DEFECTS
The abundance of thermal vacancies in FeAl is not confined to Al-rich compositions. For example, it has been measured6 to be 0.0092 per alloy atom in stoichiometric FeAl at 1173 K, which implies that the formation energy of a triple defect (i.e. two vacancies on the Fe-sublattice and one antisite Fe atom on the Al-sublattice) is small. This is consis- tent with the phase diagram in which the wide extension of the B2 phase, from FeAl through Fe3A1 to pure bee Fe, reflects the ease with which Fe atoms can be accepted on the Al-sublattice. In contrast with NiAl, the relatively small affinity of Fe for Al implies that a correspondingly small energy penalty is paid when an Fe atom is trans- ferred from the Fe-sublattice, where its immediate neighbours are Al atoms, to the Al-sublattice with Fe neighbours.
Vacancies in FeAl and NiAl 469
Following again the previous method,* consider the reaction
2 FeAl -+ (1 - O.Sx)(Fet+,Al + FeAlt,,), (7) which produces x/S triples per atom of alloy, The calculation, as before, gives the enthalpy of a triple as 1.36. The difference between this value and that for NiAl (1.7) is mainly due to the smaller cohesive energy of FeAl (-8.2) compared with that (-9.1) of NiAl, due mainly to the smaller heat of formation.
The same calculation of mixing entropy as for tri- ples in NiAl then shows that the free energy change favours the formation of triples in FeAl at 1200K, up to a concentration of O-01 per alloy atom, which is comparable with the observed value.6
ACKNOWLEDGEMENTS
The author is grateful to Professor A. H. Windle for making available the facilities of the Depart-
ment of Materials Science and Metallury during the course of this work, and to Professor R. W.
Cahn for raising the interesting question of the differences between FeAl and NiAl.
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