Diffusion in nickel-based intermetallic compounds taking the L12 structure

Journal of Materials Processing Technology 118 (2001) 82±87

Diffusion in nickel-based intermetallic compounds taking the L12 structure G.E. Murch*, I.V. Belova Department of Mechanical Engineering, The University of Newcastle, Callaghan, NSW 2308, Australia

Abstract In this paper, we review the present status of understanding of diffusion in the three compounds Ni3Al, Ni3Ge and Ni3Ga. We discuss the six-jump-cycle mechanism, vacancy pair mechanism, intersublattice mechanism, majority atom sublattice mechanism, and the mixed inter/ intrasublattice mechanisms, in the light of the available tracer diffusion data. We also present Monte Carlo results for the computer diffusion coef®cients using available migration and defect formation energies in Ni3Al. We show that overall the data are consistent with the majority atom species diffusing largely on its own sublattice with the minority atom species probably diffusing rapidly on the majority atom sublattice as antistructural atoms. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Diffusion; Intermetallics

1. Introduction The L12 structure illustrated in Fig. 1 is a common structure taken by intermetallic compounds, e.g. Ni3Al, Ni3Ge and Ni3Ga the subjects of this paper. In contrast with the B2 or CsCl structure, much less theoretical diffusion work has been done in this structure: this is largely a result of the greatly added complication in the analysis when intrasublattice atomic jumps coexist with intersublattice jumps. In the present paper, we review the present understanding of diffusion in this structure by focusing on the possible diffusion mechanisms and their compatibility with the available tracer diffusion data in Ni3Al, Ni3Ge and Ni3Ga. 2. Analysis When the L12 structure is taken by an A3B alloy, the major constituent A mainly occupies the a sublattice, whilst the B atoms mainly occupies the b sublattice. The A atoms are surrounded by eight a sites and four b sites, but a B atom is surrounded by 12 a sites, see Fig. 1. It seems reasonable for A atoms to diffuse primarily on the a sublattice. B atoms either make next nearest neighbour jumps to the b sublattice or make excursions to the a sublattice, where they diffuse freely or they diffuse by jumping between the b and a sublattices.

* Corresponding author. Tel.: ‡61-2-4921-6191; fax: ‡61-2-4921-6946. E-mail address: [email protected] (G.E. Murch).

Now let us consider the experimental tracer diffusion data given in Fig. 2 (taken from the review [1]) for the three compounds Ni3Al, Ni3Ga, and Ni3Ge. In Ni3Al, the Ni tracer diffusion coef®cient has been measured reliably. On the other hand, the Al diffusion coef®cient data are generally accepted as being unreliable, since the Ni data measured in the same series of experiments are much higher than that obtained in later work suggesting a substantial component for both the species from short-circuit diffusion. The Al diffusion coef®cient, however, can be estimated from the Darken±Manning relation which relates the two tracer diffusion coef®cients, DA and DB, the interdiffusion coef®~ and the thermodynamic activity coef®cient g: cient D   ~ ˆ …cB DA ‡ cA DB † 1 ‡ @ln g V; D (1) @ln c where the vacancy wind factor V is given by V ˆ1‡

…1 f0 †cA cB …DA DB † ; f0 …cA DA ‡ cB DB †…cA DB ‡ cB DA †

(2)

where f0 is the geometric tracer correlation factor for the structure. Although the Darken±Manning relation was derived originally for the random alloy [2], more recently it has been derived for the B2 structure (intersublattice jumps only) by Belova and Murch [3]. Very recent computer simulations by Belova and Murch [4] have shown that the relation is a very good approximation for the L12 structure where intrasublattice jumps also occur. There is, therefore, no reason not to use this relation to estimate the Al diffusivity in Ni3Al.

0924-0136/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 1 ) 0 0 8 7 2 - X

G.E. Murch, I.V. Belova / Journal of Materials Processing Technology 118 (2001) 82±87

83

Fig. 1. (a) The L12 structure; (b) the nearest neighbour coordination of an A atom; (c) the nearest neighbour coordination of a B atom.

When this is done we ®nd that at 1400 K DAl is about 0.2± 0.4 of the Ni diffusion coef®cient (the ratio DNi/DAl is then between 4.0 and 2.5). This estimated Al diffusion coef®cient is shown as a point in Fig. 2. In Ni3Ga, both diffusion coef®cients have been measured: they are quite close over the investigated temperature range, but nonetheless DNi is always >DGa. In Ni3Ge, it is found that DNi @ DGe and exempli®es the so-called `Cu3Au rule' which states that the majority atom species will have a much larger diffusion coef®cient than the minority species in such structures as the L12. There is no doubt that diffusion in these three compounds is mediated generally by vacancies. It is the exact path taken by these vacancies that has been the subject of considerable discussion. In particular, the reason why in Ni3Al and Ni3Ga the minority atom can have a diffusivity comparable to that of the majority atom has been especially perplexing. It became obvious quite early in the study of diffusion in intermetallic compounds generally that vacancies could not jump randomly as they do in pure metals. If a material started as

Fig. 2. Self diffusion in Ni3Al, Ni3Ga and Ni3Ge normalised to the melting temperatures 1726, 1383 and 1405 K, respectively. Data compiled in Ref. [1].

very highly ordered, a randomly moving vacancy would soon leave a large amount of disorder in its wake. To avoid this, a vacancy must follow a limited set of energy paths which preserve order. The best known of these is the six-jump-cycle mechanism, ®rst suggested for the B2 structure [5]. Here a vacancy makes three disordering jumps followed by three ordering jumps. Hancock [6] adapted the concept to the L12 structure taken by certain A3B alloys. There are two cycles (using intersublattice jumps only): bent and straight. These were later analysed to show that the limits of the tracer diffusion coef®cients DA and DB of the components are [7] DA 0:1064 < < 0:8524: (3) DB Thus B atoms are required to diffuse faster than A atoms. This is not actually observed in any of Ni3Al, Ni3Ge and Ni3Ga. Thus we can rule out the six-jump-cycle itself. It has been suggested that vacancy pairs may be responsible for diffusion in these compounds [8]. This would require that B atoms diffuse by next nearest neighbour jumps on the b sublattice using one end of the vacancy pair, whilst A atoms diffuse by nearest neighbour jumps on the a sublattice using the other end. Like all vacancy pair mechanisms, this results in a fairly close coupling of the tracer diffusion coef®cients of A and B. As an extreme example, if the vacancy at one end is immobile, the vacancy at the other end will be caged. There seems to be no direct evidence of vacancy pairs in these compounds, moreover, energy calculations, admittedly of sole vacancies, suggest that the migration energy for next nearest neighbour jumps of Al in Ni3Al is quite high. Furthermore, the limits for the ratio of the diffusion coef®cients are roughly 0.1 and 10.0 which are well outside those found in Ni3Ge. Without compelling evidence in its favour, it seems prudent, therefore, to put this mechanism to one side for the time being. It is useful at this juncture to write out the formal expressions for the tracer diffusion coef®cients for an A3B alloy taking the L12 structure and where diffusion takes place by isolated vacancies [9]. They are DA ˆ

a2 caA aa a!a a!b fA …p w ‡ pab Av wA †; 2 cA Av A

(4)

DB ˆ

a2 caB aa a!a a!b fB …p w ‡ pab †; Bv wB 2 cB Bv B

(5)

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G.E. Murch, I.V. Belova / Journal of Materials Processing Technology 118 (2001) 82±87

where fA and fB are the tracer correlation factors for A and B atoms; caA , caB the occupations of A and B atoms on the a sublattice; paa Av the probability of a vacancy being available to an A atom for an A atom undergoing an a ! a jump; pab Av the corresponding probability for an A atom undergoing an ab a!a a ! b jump (paa the Bv and pBv are defined by analogy); wA atom±vacancy exchange frequency for A atoms undergoing an a ! a jump; wa!b the atom±vacancy exchange freA quency for A atoms undergoing an a ! b jump (wa!a B and wa!b are defined by analogy). B Eqs. (4) and (5) are perfectly general in the sense that they refer to a six-frequency model for diffusion in the L12 structure by nearest neighbour jumps. This covers all possible jumps by isolated vacancies at least when the structures are fairly highly ordered. The two `extra' frequencies wb!a A and wb!a which do not feature explicitly in Eqs. (4) and (5) B are nonetheless implied there because of detailed balance [10,11]. The relative importance of the terms in parentheses in Eqs. (4) and (5) and, more subtly, the coupling of the tracer correlation factors fA and fB has been the subject of much discussion. Although the vacancy probability factors and the sublattice occupations can be fairly readily expressed in terms of the atom±vacancy exchange frequencies [10,11], the calculation of the tracer correlation factors represents a very serious stumbling block. Although Manning's kinetic analysis for the random alloy has been adapted to the task [10,11], signi®cant shortcomings remain which are especially deleterious when considering the ratio of the diffusion coef®cients. To proceed in a discussion of DA/ DB, the approaches have been to consider parts of Eqs. (4) and (5) either analytically or using Monte Carlo simulation. First, there has been a suggestion of a ``sublattice mechanism'' for the minority component B [12]. In this, A atoms largely diffuse on the a sublattice (the ®rst term in parentheses in Eq. (4) predominates), and B atoms largely diffuse on the a sublattice (®rst term in parentheses in Eq. (5) predominates). A minor re®nement was also made in the latter case, such that vacancies on the a sublattice may be attracted to B atoms on the a sublattice [12]. Thus the diffusion process for Al in Ni3Al is that a given Al atom spends time on the b sublattice (where it is immobile) and then occasionally jumps to the a sublattice where it is suf®ciently mobile that in, say, Ni3Al, the overall Al diffusion coef®cient becomes comparable with that for Ni. The formal expression for the minority atom diffusion coef®cient for diffusion on the a sublattice is 2 w4 DB ˆ a2 cav wa!a f B pa ; 3 w3 B

pa ˆ

caB ; cB

(6)

where cav is the site fraction of vacancies on the a sublattice, w3 and w4 the frequencies of dissociation and association of the antistructural B atom and a vacancy on the a sublattice and fB now refers to the correlation factor for B atoms on the a sublattice. At low values of caB , i.e. the site fraction of antistructural B atoms, ideas from the well-known fivefrequency model for impurity diffusion (B is the `impurity'

on the a sublattice) may be usefully employed. The chief difficulty with the idea of B atoms diffusing on the a sublattice, i.e. Al atoms diffusing on the Ni sublattice in Ni3Al is the size of pa (which is expected to be very small): this results from a high value of the formation energy for antistructural Al atoms [13]. A second possibility is that both the species diffuse by intersublattice jumps only (the second term in parentheses in Eq. (4) predominates and the second term in parentheses in Eq. (5) predominates). In the limit of perfect order, this will automatically result in the six-jump-cycle mechanism and we have already seen that the limits put on the ratio of DA/ DB (Eq. (3)) put that mechanism out of contention. However, it would be quite wrong nonetheless to rule out intersublattice jumps in general. In the B2 structure where intersublattice jumps clearly predominate, at least in intermetallic compounds with low ordering energies it has recently been shown that the limits (0.5 and 2.0) set for the ratio of diffusion coef®cients by the six-jump-cycle mechanism widen very greatly as soon as some slight disorder or deviation from stoichiometry is allowed [14]. Calculations by the present authors in the L12 structure show similar behaviour and certainly would encompass the diffusion ratio data in Ni3Al and Ni3Ga [15]. We remarked above that it is widely accepted that Ni probably diffuses predominantly by intrasublattice jumps. Yet energy calculations in Ni3Al (admittedly they can be rather unreliable in intermetallic compounds) give this jump as having the highest migration energy. For example, many-body potential calculations gave the migration energy for the a ! a jump for Ni as 1.0855 eV, the migration energy for the a ! b jump for Ni as 0.6868 eV and the migration energy for the b ! a jump for Ni as 0.5645 eV [16]. An embedded atom method calculation gave the ®rst of these energies as 1.02 eV and the second as 0.91 eV [17]. Semi-empirical potential calculations gave these two energies as 1.06 and 0.76 eV, respectively [13]. It should be remembered, however, that the ®nal diffusivity always depends basically on the product of the correlation factor, the vacancy site fraction and the atom± vacancy exchange frequency with the latter containing the migration energy, see Eq. (4) say. The correlation factor for intersublattice jumps is not well known, but is expected to be low because of jump reversals [10]. The vacancy concentrations differ by about an order of magnitude between the sublattices (a is the higher) [12]. A low value of the correlation factor for intersublattice jumps and a higher vacancy site fraction on the a sublattice may ultimately make Ni diffuse faster by intrasublattice jumps despite the higher calculated migration energy for that type of jump. This brings us to the ®nal possibility where B diffuses only by intersublattice jumps (the second term in parentheses in Eq. (5) predominates) and A diffuses by both intraand intersublattice jumps (both terms in parentheses in Eq. (4) contribute, though at least in Ni3Al the ®rst probably dominates overall because the vacancy concentration on the Ni sublattice is more than an order of magnitude greater than

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85

Fig. 3. The RASB mechanism illustrated for the triangular lattice, see text for details.

on the Al sublattice). Monte Carlo computer simulations [18] reveal that a coupled diffusion mechanism then results, which is shown in two-dimensional analogue form in Fig. 3. If the B atom on the b1 site exchanges with the vacancy on the a1 site (this is an intersublattice jump for the B atom) and this is followed closely by the antistructural B atom jumping from the a2 site, the vacancy will have migrated from a1 to a2. If the vacancy now exchanges with an A atom at a3, this vacancy is now available for exchange with the B atom, still on the b2 site. It is then clear that the B atom has, in effect, migrated from a2 to a3 by way of the b sublattice, i.e. without making a direct a ! a intrasublattice jump. This new mechanism termed the rotational antistructural bridge (RASB) mechanism provides an easier path for the migration of Al atoms in Ni3Al without the requirement of formal jumps of Al atoms on the Ni sublattice. The mechanism also requires that the Ni atom mobility on the Ni sublattice effectively determines the mobility of Al atoms by providing a more random distribution of vacancies with respect to a given Al atom. Without this, many Al jumps would be reversed since the vacancy left immediately after a jump would not be transported away. While it has not been possible so far to investigate all combinations of the six atom±vacancy exchange frequencies, the RASB mechanism certainly would encompass the values of DA/DB observed in Ni3Al and Ni3Ga, i.e. the diffusion coef®cient of the minority atoms can take a value within an order of magnitude less than the diffusion coef®cient of the majority atoms [15]. It is clear then that the closeness of the diffusion coef®cients of the components in Ni3Al and Ni3Ga can be explained in several different ways from a diffusion kinetics point of view. It is known from studies notably on the B2 structure [19] that when intersublattice jumps alone occur, there should be a minimum in the diffusion coef®cient as a function of composition at the stoichiometric composition. Although there is quite a lot of scatter, no such minimum appears to occur for Ni diffusion in Ni3Al at 1400 and

1200 K, see Ref. [12] for a compilation of the data. This strongly suggests that Ni must diffuse largely by intrasublattice jumps in Ni3Al, since that mode of diffusion for the majority atom is not sensitive to the concentrations of antistructural atoms. On the other hand, the Al diffusion coef®cient must be especially sensitive to composition if we assume either the `sublattice mechanism' or the intersublattice mechanism or the RASB mechanism. To estimate DA/DB, we have taken the following set of migration energies and antistructural defect formation energies provided in Ref. [13] to determine the `reordering' migration energies (this assumes that there is no vacancy± antistructural atom binding): Em …Nia!b † ˆ 0:76 eV; Em …Nia!a † ˆ 1:06 eV; Ef …NiAl † ˆ 0:31 eV;

Em …Alb!a † ˆ 1:57 eV; Em …Ala!a † ˆ 0:76 eV; Ef …AlNi † ˆ 1:13 eV;

where the subscripts `m' and `f' signify migration and formation, respectively. We soon find the two `reordering' energies of migration: Em …Ala!b † ˆ 0:44 eV;

Em …Nib!a † ˆ 0:45 eV:

If we assume the same attempt frequencies for both the species and ignore the entropies of migration, we can form the set of six atom±vacancy exchange frequencies wi . If the fraction of antistructural Al atoms is high enough, the values taken would tend towards a `sublattice diffusion mechanism' for Al (the exchange frequencies are all scaled to the highest Ð for the jump of an Al atom from a to b sublattice): wa!b ˆ 1; wb!a ˆ E1:569 ; wa!a ˆ E0:375 ; wb!a ˆ E0:0139 ; Al Al Al Ni a!b 0:444 a!a 0:861 wAl ˆ E ; wAl ˆ E , where   EE E ˆ exp (7) ; EE ˆ 0:72 eV: kT With this value of E in hand, we soon find that for the range of temperature from 1400 to 1000 K, the following values

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G.E. Murch, I.V. Belova / Journal of Materials Processing Technology 118 (2001) 82±87

Fig. 4. Simulated values of DA/DB at 1400 K as a function of composition cB using vacancy-exchange frequencies derived from the energies given in Ref. [13].

of caAl and pa are obtained: At T ˆ 1400 K : E ˆ 0:00225; 0:75E caAl ˆ p ˆ 0:001; 3 ‡ E2 ‡ 2E pa ˆ 0:004: At T ˆ 1000 K : E ˆ 0:0002; pa ˆ 0:00034:

(8) caAl ˆ 0:000085; (9)

Note that at 1400 K, pa was estimated to be 0.046 in Ref. [12], i.e. 10 times higher than from the energy calculations. This set of frequencies was employed in a Monte Carlo simulation to estimate DNi/DAl as a function of cAl and as a function of pa using standard techniques [20]. The result at 1400 K is shown in Fig. 4. It is clear that the ratio is much higher than observed at the stoichiometric composition in Ni3Al, but importantly decreases rapidly for Al-rich compositions. This suggests that real stoichiometric material may have a higher concentration of antistructural Al atoms than is possible to generate by thermal means. It is also interesting to note that in Ni3Ge real material is always slightly Ge-de®cient. The resulting diffusion behaviour where DNi @ DGe seems also to be consistent with Fig. 4. 3. Summary In this paper, we have reviewed the present status of understanding of diffusion in Ni3Al, Ni3Ge and Ni3Ga. A clear picture is starting to emerge in which Ni diffuses on the

Ni sublattice as well as being involved in intersublattice jumps, with the former probably being more important. On the other hand, the minority species has a small population on the Ni sublattice and these atoms diffuse rapidly. The minority species also diffuse by intersublattice jumps, but the intrasublattice jumps probably are responsible for most of the diffusion. The minority atom will then have a diffusivity highly dependent on composition (because of the fraction of its atoms on the Ni sublattice) in contrast incidentally with the behaviour of the majority atoms. Acknowledgements We wish to thank the Australian Research Council (Large Grants Scheme) for its support of this research. One of us (IVB) wishes to thank the Australian Research Council for the award of a Queen Elizabeth II Fellowship. References [1] H. Mehrer, F. Wenwer, Diffusion in metals, in: J. KuÈrger, P. Heitjans, R. Haberlandt (Eds.), Diffusion in Condensed Matter, Vieweg, Wiesbaden, 1998 (Chapter 1). [2] J.R. Manning, Correlation factors for diffusion in nondilute alloys, Phys. Rev. B 4 (1971) 1111±1121. [3] I.V. Belova, G.E. Murch, The manning relations for atomic diffusion in a binary ordered alloy, Phil. Mag. A 75 (1997) 1715±1723. [4] I.V. Belova, G.E. Murch, Test of the validity of the darken/manning relation for diffusion in ordered alloys taking the L12 structure, Phil. Mag. A 78 (1998) 1085±1092.

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[14] I.V. Belova, G.E. Murch, Limits of the tracer diffusion coefficient ratio in highly ordered alloys and intermetallic compounds taking the B2 structure, Phil. Mag. A 79 (1998) 193±202. [15] I.V. Belova, G.E. Murch, Analysis of diffusion mechanisms in Ni3Al Ni3Ge and Ni3Ga, Defect and Diffusion Forum 177 (1999) 59± 68. [16] H. Shenyang, W. Tianmin, L. Yulan, The stable configurations of small vacancy clusters in Ni3Al, Modell. Simul. Mater. Sci. Eng. 4 (1996) 493±499. [17] S.M. Foiles, M.S. Daw, Application of the embedded atom method to Ni3Al, J. Mater. Res. 2 (1987) 5±15. [18] I.V. Belova, G.E. Murch, Percolation and the anti-structural bridge mechanism for diffusion in ordered alloys of the L12 type, Intermetallics 6 (1998) 403±411. [19] G.E. Murch, Diffusion in crystalline solids, in: P. Haasen (Ed.), Materials Science and Technology, VCH, Weinheim, 1991 (Chapter 2). [20] G.E. Murch, Simulation of diffusion kinetics, in: G.E. Murch, A.S. Nowick (Eds.), Diffusion in Crystalline Solids, Academic Press, Orlando, 1984 (Chapter 7).