Heats of formation in transition intermetallic alloys

Acra mefafl. Vol. 32, No. 7, pp. 1061-1067. Printed in Great Britain. All rights reserved

1984

OOOI-6160/84 $3.00 + 0.00

Copyright 0 1984Pergamon Press Ltd

HEATS OF FORMATION INTERMETALLIC

IN TRANSITION ALLOYS

A. PASTUREL, C. COLINET and P. HICTER Laboratoire de Thermodynamique et Physico-Chimie MCtallurgiques, E.N.S.E.E.G., Domaine Universitaire, B.P. 75, 38402 Saint Martin D’Heres Cedex, France (Received 14 November 1983) Abstract-The heats of formation in intermetallic alloys are calculated within a tight-binding scheme for the d band. We show that the difference in bandwidth between the metals and the difference between their energy levels are two dominant effects in the determination of the formation energy of these alloys. The influence of charge transfer, determined by a self consistent way, on alloy formation is studied. R&um&Les enthalpies de formation des alliages de deux mttaux de I’approximatiqn des liaisons fortes. Nous montrons que la di%rence entre constituants comme la diff&rence entre Ieurs niveaux d’energie jouent un L’infiuence du transfert de charge, dktermin6 de manitre autocohbrente,

transition sont calculteS dans les largeurs de bande des deux rBle important dans ce calcul. est igalement &dike.

Zusammenfaasung-Die Bildungswiirmen intermetallischer Legierungen werden innerhalb eines tightbinding-Schemas fiir das d-Band berechnet. Wir zeigen, daD der Unterschied in den Bandbreiten und die Unterschiede in deren Energieniveaus zwei wesentliche Griil3en bei der Bestimmung der Bildungsenergie dieser Legierungen darstellen. Der EinfluB des auf selbstkonsistente Weise ermittelten Ladungstransfers auf die Legierungsbildung wurde untersucht.

computation of the heats of formation of disordered Ni,&&, alloys (where B is a 3d metal). We showed It is now well established that the cohesive energy of that the experimental variation of AH as a function the pure transition metal constituents arises [l] from of the position of the constituent B in the periodic the strong bonding of the valence d electrons, which table is well reproduced. The aim of this paper is to are well described by the tight-binding approxiinvestigate qualitatively the importance of the mation; the energy in the solid is then given simply different metal parameters; we show that the by the sum of the one-electron band energies, so that difference of bandwidth between the two alloy conthe cohesive energy is stituents can change the order of magnitude and even the sign of AH when this difference is comparable with the difference between the energy levels. In this &oh= EF(E,, - E).n,(E).dE paper, we study also the variation of AH as a function of the composition and the different factors leading where E,, is the atomic d level which broadens, as the to highly asymetrical shape of AH. This model also orbitals overlap, into a band whose density of states provides an estimate of d electrons transfer between is n,(E). This simple formula, used by Friedel [l] and constituents by factoring the alloy density of states Cyrot-Lackmann [2] explains the typical behaviour of into separate constituent partial densities of states. the cohesive energy in transition metals. The partial densities of states are deduced from It is therefore tempting to apply this kind of knowing the moments of the constituents by using tight-binding approach to calculate the heats of for- the continued fraction developed in the first step. This mation AH of transition metal alloys as has in fact development allows to have an analytical selfbeen done by several workers; Van der Rest et al. [3] consistency expression of the charge transfer and to used the C.P.A. with off diagonal disorder to comshow the influence of the effective levels and partial pute the alloy density of states while Cyrot and band parameters. In fact, this determination is not Cyrot-Lackmann [4] used a simple model for this independent of the choice of the parameter U, which density, based on an expansion in its moments. More is some parameter taking into account both the recently, Pettifor [S], Varma [6] and Watson [7] have intra-atomic and intersite Coulomb integrals [l]. However, we show that the determination of the heat applied Friedel’s rectangular d band density of states scheme to the alloy problem and almost identical AH of formation in transition metal alloys is not very sensitive to this choice, the energy due to the charge to these obtained from the more detailed C.P.A. calculations were found. In a recent communication transfer being to small to be a major factor in [8] we presented preliminary result concerning the transition metal alloys. I. INTRODUCTION

s

106

1062

PASTUREL

a al.:

TRANSITION

After recalling and developing some theoretical points, we present and discuss our results in comparison with experimental data.

We have seen that the cohesive energy of transition metals is essentially due to the band energy of d electrons and that the variation of this energy along the transition series can be related to the progressive filling of the d band. Similarly, one can think that it is possible to predict the general behaviour of the heats of formation of transitional alloys from the position of the alloy constituents in the periodic table and from the change in energy associated with the deformation of the d band upon alloying. Neglecting the contribution to AH arising from changes in magnetic interactions, the heat of formation can be written in the Hartree-Fock approximation as [9]

s

band contribution E

ALLOYS

to the cohesive energy.

= -4o*(bp)“2 (1 -A”)jP 8.1 3n

(3)

A: is given by equation (4) can be obtained from the number of d electrons of metal i

2. THEORETICAL ASPECTS

Ef

AH=

INTERMETALLIC

E *n,(E)=dE

So the knowledge of the number of d electrons and of the bandwidth of metal i is sufficient to determine its density of d states and its cohesive energy. For the alloy, the total density of d states is n(E) = c

x,-n,(E)

(7)

i-A,B

where n,(E) is the local density of d states of metal i, which can be also determined as previously by the knowing of Ai and bi parameters; for this last determination, we use the Shiba’s formula [12] to calculate the hopping occurring between different atoms; Ai is still given by Ni=lO*

1/2+l/nsin-iAi+~(l-Af)“2

(8)

I

&i

E.n&(E).dE

“$=-

EF- Ei (9) 2J6

+;(Nf

- N”)

(1) EF, being the Fermi level in the alloy.

where the first and second terms represent the change in the one-electron band energy and double counting energy respectively on going from the pure metals A and B to the alloys A,_,B,; ni.i(E) and Ni are the density of states and total number of electrons respectively associated with atom i; Efi is the corresponding Fermi energy. In the presence of charge transfers, we have to define new effective atomic energy levels, Ei. Ei = E: -I- lJ,AN,

(2) where AN, = Ni - Nf is the charge transfer and Ey is the atomic energy level in pure metal i. Ui is some effective intraatomic Coulomb integral [l] and is assumed to be identical for the two metals. In order to calculate the d band contribution, we use the techniques of moments and continued fraction [ 10, 111. In order to have purely analytical calculations, we will consider the continued fraction developed in the first step. Then the density of d state of metal i is given by n:(E) =*

(1 - Ay)‘j2

The charge transfer, calculated from equation (6) and (8) can be different of the one deduced from the effective level [see equation (2)] and it is necessary to have a self consistent calculation to equalize these two transfers. The effective levels can be written as AE’=Ei-&=EP-Ej+U AE = Ei-&=2,/$.Ai-2,/&Ai

(11)

where Ai is related to N, by equation (8). This self consistent determination of AE is shown in Fig. 1.

(3)

where En-E:

A~=----

2,s:

(4)

by is defined from knowing the two first moments and can be related to the bandwidth of pure metal i [4]. With the same approximation, we can write the d

0

0.1

0.2

AN Fig. I. Self

consistent determination of AE.

. I

PASTUREL ef al.: TRANSITION INTERMETALLIC ALLOYS

6

3

h ‘7

7

6

QN, 5.5 6 6.5

.qq;

5

alloy

(fraction of e)

.;..__-.. 4d‘t \ ‘\\\

\

,//

Table I. Heat of formation and charge transfer of Fe,,Ni,, for various U values

.,._ 5d

/.T.i~=

1063

0.106 0.0975 0.091

-3.33 -4.13 - 5.58

-0.035 - 0.043 -0.058

‘\‘:

II

11

I

I

I

I

I

Fig. 2. Bandwidths for the transition metals.

3. DISCUSSION The determination of the heats of formation quires the knowledge of

re-

-the bandwidths of pure metals, fl -the relative position of the atomic energy levels, EP -the electron-electron effective intra-atomic interaction U determining the charge transfer. The values of the bandwidths and the atomic energy levels, drawn from Herman and Skillman [13] are given in Figs 2 and 3. The number of d electrons in pure metals, NY are taken from standard band calculation. 3.1. Evolution of the parameters 3.1.1. Charge transfer effects. The first point is that the results are not qualitatively dependent on the value of IJ, and the order of magnitude is kept as it is shown in Table 1. The values of the charge transfer, Q, depend also on the choice of U and it is difficult for us to compare our calculated charge transfer with charge transfer computed by other methods. However, we can point out that our value is the same order of the value determined by Alonso and Girifalco [ 141using a loci1 density approximation of energy density functional theory. 3.1.2. Sensitivity of the results to the bandwidths and energy level values. To illustrate the dependence of AH upon the various parameters, i.e. bandwidth and energy levels, we give in Figs 4 and 5 curves showing AH in Fe,,,NiO.s and in Zr,,5Ni,-,J alloys as a function of the difference between the pure energy levels and of the bandwidths of pure metals.

We display so chemical considerations, that is to say heats of formation become more exothermic with increasing separation in energy level, separation which favours the charge transfer, effect which can be related to an electronegativity effect. On the other hand, we can see that an increasing of bandwidths in the diagonal disorder approximation leads to a more endothermic heat of formation; more the cohesion is great in elements and more the alloy is difficult to obtain. 3.1.3. Ofl diagonal disorder. The off diagonal disorder can be very important as it has been already shown [3,8]; Fig. 6 shows that the evolution of AH along the 3d transition series is strongly modified when the off-diagonal disorder is taken into account,

I (a)

5

1

. \

‘\

-5

l \ l \ ‘\

l \.

\

j\

I

1

2

I

1

4 E,,

I

1

L

6

6

IO

-ENi

(eV)

I

,

I

I

1

5

6

7

6

9

W,,

1

(eV)

Fig. 4. Enthalpy of formation of Zr,.,Ni,,, as a function of: (a) the difference between the pure energy levels, (b) the Fig. 3. Atomic energy levels for the transition metals.

values of the bandwidths.

PASTUREL

1064

CI al.:

TRANSITION

INTERMETALLIC

ALLOYS

(a) x

2

x x

x

z

.

s

\

a

3

.

s

\ -2

.

.

. -1.0

\

I

1 2

x -0.5

:

\

’

’

’

4

6

8

’ 10

I . 1

E,-E,i(eV)

TI

(b)

,/’ s a

l

.

W,, =6

.’

W,, (ev)

Fig. 5. Enthalpy of formation of Fe,,NisS as a function of: (a) the difference between the pure energy levels, (b) the values of the bandwidths.

especially for the alloys whose the well separated in the periodic table. difference between energy levels but bandwidths on the determination of fer.)

constituents are (Influene.of the also between the the charge trans-

I

I

V

Cr

I

I

I

Mn

Fo

Co

Fig. 6. Heats of formation for Ni based alloys (a) with diagonal disorder 0 (b) with off diagonal disorder x .

the systematic behaviour of the formation energy in the following way. For two neighbouring elements, the enthalpies of formation are weak, either positive or negative according to the values of their bandwidths; so AH (N~,-,,CO~.~)is equal to zero while AH (Pd,,rRh,& is slightly positive. For one metal at the beginning and one at the end the larger contribution is provided by the alloy effect and the enthalpy is negative. We can also explain the highly negative enthalpies of formation of Zr-(Ni, Co, Fe, Pd, Pt) alloys Ti-(Fe, Co, Ni) and Hf-(Pd, Pt) [ 181alloys; our calculated results are compared with experimental values in Table 2. As a general rule, we can also predict that in absolute value, the heats of formation of Pd-Sd based alloys are larger than the heats of formation of Pd4d based alloys which are also more important than the one of Ni-3d based alloys; this stems from the fact I

I

Pd -4d

3.2. Numerical results 3.2.1. Equiatomic alloys. We now want to show that it is possible to make some predictions about the energy of formation of alloys with components within the same series or between series. The first series is the only one where systematic experimental results exist [15]. The behaviour of Ni-based alloys, shown in Fig. 6, can be qualitatively explained without introducing complications due to magnetism. For example Mn has an anomalously low value for its bandwidth which makes the positive contribution of NiMn very small; on the contrary, NiCr has a maximum positive value in this series. In the last two series, a few experimental values exist and we prefer to compare in Figs 7 and 8 our results with those obtained by Miedema 1161 or by Bennett [ 171. Moreover, we can make predictions for

I

I

1

Zr

Nb

I

1

I

L

MO

Tc

Ru

Rh

Fig. 7. Enthalpy of formation for Pd4f based alloys. (-) Our results, (---) Miedema’s results [la], (-.-.-.) Bennett’s results .1171. a

PhSTlJREl,

c/ cd.:

TRANSITION

INTERMETALLIC

ALLOYS

1065

the values of the charge transfer for other compositions are then given by linear interpolation and we can easily calculate the heats of formation at these compositions from equation (1), with however the use of AE (difference between the effective energy levels) which is given by the average of AE and AE’ to assume the coherence of the calculation. We can see on Fig. 10 the heats of formation for Fe-Ni alloy calculated with the self consistent way and with the assumption of a charge transfer completely linear; from this comparison it may be considered that this assumption is justified. The asymetric shape of AH in function of composition for the Fe-Ni system can be also easily explained by the difference between the

-0.52 : a -1.0-

Fig. 8. Enthalpy of formation for Pt-5d based alloys. (-) Our results. (---) Miedema’s results [16],(-.-.-.) Bennett’s results [ 171.

that the variation of the atomic energy level is more important in the Sd series than in the other series. 3.2.2. Evolution of AH with composition. In our previous paragraph, we studied the variation of AH as a function of the position of constituents in the periodic table and we have obtained some qualitative and quantitative ideas concerning this behaviour. However the analysis of the equiconcentrational alloys does not give a complete description of the variation of AH with alloy composition when the shape of AH is highly asymetric. Figure 9 displays three typical curves of energies of formation in function of composition in transitional alloys and in this section we show that the proposed model is able to reproduce qualitatively an even quantitative@ these different shapes. We first study the Fe-Ni system and the evolution of AH in function of composition. Figure 10 presents the enthalpies of formation AH and the charge transfer QNi calculated with the self consistent way. We notice that the evolution of the charge transfer can be considered as linear in function of composition. This remark yields the calculations still more evident because one self determination of the charge transfer only at a given composition is then necessary; Table 2. Comparison of calculated and experimental results of some exocnergetic alloys (AH in kJ/mol) Com-

pound Zr Zr Zr Zr Zr Ti

Ni co Fe Pd Pt Ni

-AL 52.4 41 35.2 68.2 95. I 69.5

-A& 51.5 42.2 29.1 62 113 40

Compound Ti Co Ti Fe Hf Pd Hf Pt Ti Pd Ti Pt

- AHat 41.4 23.9 87 117 92.1 123

-AK,, 44.3 31 79 113 53 75

0.5

2r

Ni

(cl 4-

,I a

2

E

3 x\

‘; a-2-

‘\ a‘\ i

“\ -4

-

xLx/x I

F*

1

1

I

, 0.5

,

I

1

, Ni

Fig. 9. Experimental values of the heats of formation of (a) Ni-Zr 0 [20] x [lfl, (c). Ni-Fe x [ 181 .. x [18],_ (b) .Ni-Cr___. . alloys as a funcuon 01 Nt composmon.

PASTUREL er al.: TRANSITION INTERMETALLIC ALLOYS

1066

diagonal disorder is taken into account. Since the bandwidths are related to the volume of elements 1191 we display chemical considerations to explain the asymmetry of the thermodynamic data, that is to say the difference of size between the elements. The highly exoenergetic systems can be also represented by the proposed formulation; the negative values of the enthalpies are due to the important difference between the atomic energy levels and the asymmetry of the curves due to the difference between the bandwidths. On Fig. 12 the calculated values for Zr-(Ni, Co, Fe) are compared with experimental values [ 181.

2

7 ij E

2 5

-2

0 -4

-6

(b)

-60

0.5

FO

Ni

(b) Fig. 10. Fe-Ni system. (a) Calculated AH: (-----) self consistent dete~ination of charge transfer, (-) G&Uhted with a linear charge transfer, (b) calculated &.

bandwidths of the two constituents as shown in Fig. 11. The comparison between the two calculated curves, one with the diagonal disorder approximation and the other with the off diagonal disorder gpproximation shows that the evolution of AH as a function of the composition can be modified when the off

2-

-60

I

I

r

B E

2 % \ -6

-

\

/ \

,

a

-40-

/

‘__/’ -60

-

I

Fig. 11. Heats of formation in the FeNi system. (---) Calculated with diagonal disorder: W, = W,, = 4.7 eV. (-) Calculated wi’hoWffdiagonal disorder: W,# = 4.7 eV, pL= 5.7 eV.

zr

I

I

I

L 0.5

,

I

I

I Fe

Fig. 12. Comparison between expetimental Q [to], x [IS], 0 1211and calcuiated values for Zr-(Ni-Co, Fe) alloys.

PASTUREL

et al.:

TRANSITION

ALLOYS

1067

bandwidths of the elements, and the difference between the energy levels are the important parameters of the evolution of the heat of formation of these binary transitional alloys as a function of the position of the elements in the periodic table and as a function

12

_-

INTERMETALLIC

‘-

Yza

of the composition.

2

REFERENCES

-E 4 a

1. J. Friedel, The Physics of Merals (edited by J. M. Ziman), Chap. 8. Cambridge Univ. Press (1969). 2. F. Cyrot-Lackmann, J. Phys. Chem. Solids 29, 1235 ( 1968).

3. J. Van der Res, F. Gamier and F. Brouers, J. Phys. F5,

IIIIlIIll

Cr

0.5

2283

Ni

Fig. 13. Comparison between experimental x [IS] and calculated values for Ni-Cr system.

A qualitative and quantitative agreement In the same way the enthalpies of formation

(1975).

4. M. Cyrot and F. Cyrot-Lackmann,

is found.

of Ni-Cr system is reproduced as shown Fig. 13. In this system, the positive contribution due to the difference between the bandwidths of the elements becomes more and more important with respect to the negative contribution when the chromium composition increases. Finally, we want to show that, if generally the orders of magnitude that this model predicts are in good agreement with the experimental results, some discrepancies remain. With the used assumptions, it is difficult for example to explain the behaviour of Pd-Ni system where the experimental energy of formation changes sign and remains smaller than 400 J [15]. But from the discussion of our results we conclude that when the chemical effects are more important than the magnetic or structural ones, chemical effects which can be either positive like in the case of Ni-Cr system, or negative like in the case. of Zr-Ni system, the model predictions correspond qualitatively and quantitatively to the experimental results. We conclude also that the values of the

J. Phys. F6, 2257

(1976).

5. D. G. Pettifor, Solid St. Commun. 28, 621 (1978). 6. C. M. Varma, Solid St. Commun. 31, 295 (1979). 7. R. E. Watson, L. H. Bennett, Phys. Rev. Mt. 43, 1130 (1979). 8. A. Pasture], P. Hitter and F. Cyrot-Lackmann, Solid St. Commun. 48, 561 (1983).

9. M. C. Desjonqueres and M. Lavagna, J. Phys. F9, 1733 (1979).

10. F. Cyrot-Lackmann, J. Phys., Suppl. Cl, 67 (1970). II. J. P. Gaspard and F. Cyrot-Lackmann. J. Phys. C6, 3077 (1973).

12. H. Shiba, Prog. Theoret. Phys. 46, 77 (1971). 13. F. Herman and S. Skillman, Afomic Sfrucrure Calculations. Prentice-Hall, Englewood Cliffs, NJ (I 963). 14. J. A. Alonso and L. A. Girifalco, J. Phys. Chem. Solids 39, 79

(1978).

IS. R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser and K. K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys. Am Sot. Metals, Metals Park, OH. 16. A. R. Miedema. F. R. de Boer and R. Boom, Calphad 1, 341 (1977). 17. L. H. Bennett and R. E. Watson, CalphadS, 19 (1981). 18. J. C. Gachon and J. Hertz. Cabhad 7. 1 (1983); J. C. Gachon, J. Charles and J. Hertz: Cot&es Calphhd XII, Liege, Belgique (1983). 19. L. Hodges, R. E. Watson and H. Ehrenreich, Phys. Rev. B5, 3953 (1972).

20. M. P. Henaff, C. Colinet, A. Pasture1 and K. H. J. Buschow. J. appl. Phys. In press. 21. A. Schneider, H. Klotz, J. Stendel, G. Strauss, Pure appl. Chem. 2, 13 (1961).