Electronic and elastic effects in the interaction of impurities in ternary metallic alloys

1. Phys. Chem. So/ids Vol. 46. No. IO, pp. 1147-l 151, 1985 Printed in Great Britain.

COZZ-3697/U $3.00 + .GU Q 1985 Pcixamon Press Ltd.

ELECTRONIC AND ELASTIC EFFECTS IN THE INTERACTION OF IMPURITIES IN TERNARY METALLIC ALLOYS J. A. ALONSO Departamento de Fisica Teorica, Universidad de Valladolid, Valladolid, Spain

T. E. CRANSHAW AERE Harwell, OXI I ORA, England

and N. H. MARCH TheoreticalChemistryDepartment,University of Oxford, I South Parks Road, Oxford, OX13 3TG, England (Received 8 November 1984; accepted 2 I March 1985) Abstract-A comparison is made between interaction energies of a variety of impurities in transition and noble metal hosts and Miedema’s semiempirical formula of heats of alloy formation. For interactions between impurities on neighbouring sites, Miedema’s formula reproduces the gross trends of the experimental data. The error in the Miedema predictions is shown to correlate with atomic volume mismatch effects. Though less easy to interpret, impurities at next near-neighbour distances in an iron host interact in a way that is again shown to correlate with size effects. Keywords: alloy, ternary, impurity, interactions.

I. INTRODUCTION

The motivation for the present study lies in the availability of a substantial amount of hyperfine data from which interaction energies between impurities in some transition and noble metals can be extracted [l-6]. In particular, Fe-based alloys, and Cu-, Ag-, and Au-based alloys will be considered in detail. The majority of the Fe-based alloys contain Sn as an impurity. More generally, in most of the alloys one impurity has an sp electronic structure (e.g. Sn, In, Ga) while the other is a transition metal atom. The body of data already referred to will be presented and discussed in Section 3 below. In analysing this data, we shah, in Section 2, consider the predictions made by the Miedema semiempirical model of heats of alloy formation [7, 81. Though our tabulation of the Miedema results is more extensive, such an approach has already been implemented by Krolas [5]. In fact, independently of the use of the Miedema formula, Krolas approximately expressed the interaction energies between impurities in ternary alloys in terms of properties of the binary systems. 2. PREDICTIONS

OF MIEDEMA’S

FOR INTERACTION

ENERGIES

(-P(AtiY + Q(An”3)2 - R).

2.1 Ternary alloys Turning now to the case of a ternary alloy, the binding energy Eb of the two impurity complex BC (at nearest neighbour sites) in host A is defined as the difference between the energy of the system when the impurities are nearest neighbours and the energy of the system when the two impurities are far apart. As Krolas [5] pointed out, as a result of the impurity pair formation, a bond with a host atom is broken for both impurities, while a bond between impurities is generated instead. If we neglect atomic size difference effects, then the results of Miedema model can

FORMULA

To establish a notation, the chemical part of the heat of solution of metal B in a solid transition metal A in Miedema’s treatment can be written [7, 81:

AH:,, = 2 (,,)_l,‘y;*B)_,X

In this equation, A4 = 4” - &, is the electronegativity difference between A and B, Anit is the difference of electron density (to the one-third power) at the boundary of bulk Wigner-Seitz cells, V, is the atomic volume of metal B and P, Q and R are empirical constants. All the parameters needed to evaluate AH:,, are given in Miedema’s articles [7, 81. The electronegativity difference leads to electron transfer between similar atomic cells, which favours the formation of the alloys. On the other hand, smoothing out An leads to an increase in the electronic kinetic energy, which opposes alloy formation. Finally, R is a term representing the hybridization of the p electrons of an sp impurity and the d electrons of the host. Additional contributions to the heat of solution (in the solid state) exist, not contained in eqn (I), the most important one being an elastic contribution due to the difference in atomic volume between the solute and solvent [9-141.

(I) II47

J. A. ALON~Oet ul.

1148

be brought into contact with the ternary system by writing Eb as: E

=

b

AH,,. -----

AH,,

AH,,.

N

N

N

(2)

*

The last two terms represent the energies of the impurity-host bonds broken. These bond energies have been written as a heat of solution divided by the coordination number N of the host lattice (that is, the number of bonds between one impurity and the host atoms). On the other hand, since one of the impurities (the one labelled B) is a transition or a noble metal, the energy of the BC bond formed has been written as the heat of solution of sp impurity in B, also divided by the coordination number of the A metal lattice (to account for the fact that the BC complex is in the A lattice). This point is elaborated a little bit more below. Predictions along these lines for Eb in the ternary alloys systems under consideration are collected in Tables 1 and 2. Two sets of theoretical binding energies are given for the noble metal alloys of Table I. The first set was obtained from eqn (2), whereas the second one used a slightly different approximation for the BC bond energy, involving the average of the heats of solution of B in C and of C in B, namely: AHis E = 1 (AH!<. + AH&) -___ b N N 2

AH;<. N *

(3)

N is equal to 12 for the noble metal alloys. For the

case. of Fe alloys of Table 2, we give results from eqns (2) and (3) with N = 8, plus another set of predictions corresponding to the definition: Eb=---_-

AH;<. 12

AHiB 8

AH& 8 ’

(4)

This equation is intended to take account of the fact that Fe is a bee metal, whereas Co, Ni, Cu, Zn, Pd and Pt are closed-packed metals. Before summarising the experimental data, we emphasize that the input for the Miedema predictions is essentially designed to account for electronic redistribution effects, no direct account being taken of elastic size mismatch effects. 3. DISCUSSION

OF INTERACTION

ENERGY

DATA AS EXTRACTED FROM MOSSBAUER

EXPERIMENTS

The data of Fb for the ternary alloys under consideration are collected in Tables I and 2. They have been taken from Krolas’ compilation [5] and from recent experiments by one of the authors [6]. As a starting point for the discussion, let us focus on the behaviour of the systems Fe(Sn - X), where X is one element of the set Fe through Ge. Figure 1 shows that the general trend of the experimental data is reproduced by the Miedema predictions. In particular, whether the impurities attract or repel is predicted correctly in every case. A more complete comparison is given in Tables 1 and 2 for noble metal-based and Fe-based alloys respectively. There is good semiquantitative agreement, either using eqns (2) or (3) (or eqn (4) for the Fe alloys). Therefore, we were encouraged to seek the origin of the difference between the Miedema curve and the experimental results shown in Fig. 1. Since Fig. 1 represented one specific class of alloys, it will be advantageous to consider together all the alloys of Tables 1 and 2. As already indicated above, from previous work on binary alloys [9-141, it is natural to seek a correlation of the difference between Miedema’s predictions and experiment with atomic size mismatch. In host A, the variable chosen to represent

Table I, Binding energies of impurities (as nearest-neighbours) in noble metal hosts. A negative sign means attractive interaction. Alloys are numbered for the purposes of identification in Fig. 2. Host impurities

Eb (meY)

A

B-C

Au

Pd -In (1) Pd - Sn (2) Ag - In (3) Pt - In (4)

Ag

cu

Pd Pd Pd Pd Au

-

Cd In Sn Sb In Pt - In Pt-Sn Rh Pd Pd Pd Pd Pt-In

(5) (6) (7) (8) (9) (10) (11)

In (12)

Ga (13) Ge (14) Sn (15) Sb (16) (17)

Ew (2)

Ew (3)

Expt

-124 -142 +61 -97

-96 -110 +63 -76

-130 -170 +25 -118

4 22 I 4

-102 -148 -161 -146 -29 -127 -142

-87 -I20 -128 -114 -22 -106 -117

-77 -135 -192 -182 -25 -171 -240

18 18, 2. 4 18 18 2 23

-76 -127 -79.5 -137 -120 -100

-61 -110 -74.5 -104 -88 -79

-86 -90 -75 -120 -90 -62

4 22 22 22 22 4, 21

Ref

I

Electronic and elastic effects in the interaction of impurities

1149

Table 2. Binding energies of impurities (as nearest-neighbours) in Fe. A negative sign means attractive interaction. Alloys are numbered for the purposes of identification in Fig. 2.

Host impurities A

Eb (mev)

B-C

Fe

Co - Sn (18)

Eqn (2)

Eqn (3)

Eqn (4)

-70

Expt

Ref

-51

-42

-20

- Sn - Sn -Sn - Sn - Sn

(19) (20) (21) (22) (23)

-95 -106 t36

-69 -98 t35 +90 +94

-57 -93 +34

-100 -140 +130 +170 +145

19 6 6 6 6

Pd - Sn Pt - Sn v-v

(24) (25) (26)

-265 -156 +78

-215 -118 +78

-165 -76

-280 -0 +100

19 3 17

Mn - Mn (27)

-2

-2

+40-90

20

Ni Cu Zn Ga Ge

the size mismatch contribution to Eb must obviously involve both (VA - V,) and (VA - Vc), the Vs denoting atomic volumes. The product of these two quantities seems the simplest plausible variable to use in seeking the above correlation. If Vc. < VA < VB, then the individual size mismatch effects of B and C with respect to A tend to cancel each other when B and C form a nearest neighbour complex. Consequently, size mismatch provides an attractive contribution to the binding energy, which correlates with the negative sign of (V,, - V,)(V, - Vc.). The opposite effect can be expected when V, and Vc are smaller or both larger than VA. While the choice of (V, - V8)(V, - I’,.), in the end, must rest on its practical usefulness, it is perhaps worth adding that, in an elasticity treatment such as that of Eshelby [ 151,the contribution from size mismatch to the heat of solution is quadratic in the volume difference, that is: AH~;LY(V,, - I’,#,

(5)

while the proportionality constant is largely determined by the matrix. Then, the size mismatch contribution to Eb can be written:

300

6, 3

where AHyi and AH2;F have an obvious meaning and AH?&, is due to the difference between two atomic volumes of matrix A and the volume of the BC complex (which we take as the sum of the atomic volumes of B and C). Using (5) in (6) with the admittedly drastic assumption that each of the three terms involved has the same proportionality factor multiplying (A I’)*, the above variable (VA - V,)( VA - Vc) emerges. That this is a useful variable is clear from Fig. 2, where the Miedema discrepancy is plotted against (V, - VB)(VA - V,), with atomic volumes taken from Gschneidner [ 161. A marked correlation is apparent. Only in five alloys (numbered 5, 6, 12, 18 and 24) does the sign of the Miedema discrepancy clearly disagree with the sign of the elastic interaction. Error bars, corresponding to the reported experimental uncertainty [5] have been plotted for Ag(Pd - Cd) (5), Ag(Pd - In) (6), and Cu(Rh - In) (12). While the line shown in the figure is only intended as a guide to the eye, such a correlation should occasion no great surprise and is at least partly anticipated by earlier work in binary alloys [914]. Certainly, more quantitative work is needed, along the lines already developed for binary alloys [9, 10. 141, that is, making quantitative use of Eshelby’s “inclusion in hole” model for the elastic size mismatch effects; but the main problem resides in describing the effective compressibility of the BC complex, to which quantity the results are very sensitive.

1

4. NEXT NEAREST-NEIGHBOUR

INTERACTION ENERGIES

-200’

, Fe

CO

N,

C”

zn

GO

Gc

Fig. I. First neighbour binding energies for impurities in the ternary alloys Fe(Sn - X) with X indicated in the figure. Experimental data is from Cranshaw [6]. Miedema prediction is from eqn (3).

Having dealt above, in some detail, with impurities at near-neighbour distances, we shall briefly discuss the sparser data on second nearest-neighbour interaction energies. Figure 3 shows data for the systems Fe(Sn - X), with X the same as in Fig. 1. The experimental results clearly correlate with volume mismatch. In other words, at this increased separation, electronic redistribution effects are now dominated by size mismatch. This is consistent with the Miedema point of view that interactions are proportional to the surface area of contact between atomic cells.

1150

J. A. ALONSO et al.

o

Au

alloys

n

Ag

alloys

A

Cu

alloys

Eb f Mledem&EbfeXpf

I 100

(mV)

0’1

Fig. 2. The difference between experimental binding energy and its Miedema counterpart (compare Fig. 1) correlated with size parameter (VA - V,)( VA - Vc). V is in cm3/mole. The Miedema result was calculated from eqn (2) except for the alloys Fe(SnGa) and Fe(SnCie), for which eqn (3) was used instead. The numbers denote the specific ternary alloys (see Tables 1 and 2). 5. SUMMARY AND CONCLUSIONS The main conclusion from this work is that the Miedema treatment is successful in giving insist into whether impurities in ternary alloys will attract or repel at nearest-neighbour separations. To explain the data quantitatively, there can be little doubt from

the present work that size effects must be carefully incorporated. For next-near neighbour interaction energies, there is an imm~iate correlation with size mismatch. It is encouraging to see that a simple analysis provides some insight into a problem which, ap-

350-

300 -

2503 E” & z 2 :

zoo-

150-

0 c m loo-

/ Fe.-.

Fig. 3. Second neigh~ur

Co .e .--we

Ni ,c@Cu

_01__

Experiment

---t--

AV

I

I

I

tn

Go

Ge

intemction energies correlated with size mismatch for Fe(Sn - X) alloys, with X indicated in the Figure. Data ref. [6].

Electronic and elastic effects in the interaction of impurities

II51

from However, it is necessary, in conclusion, to note that our work does not throw light on the quantitative relation between the interaction energies at the two different distances. In particular, it remains somewhat surprising to us that the second neighbour interaction energies have such substantial magnitude. An analysis of the relation between first and second nearest neighbour interactions is made difficult by two factors: the first one is the accuracy of the experimental measurements and the second one is our empirical estimation of the elastic contribution to nearestneighbour interaction, that is, the horizontal axis in Fig. 2. This estimation is influenced by the choice

5. Krolas K., Phys. Let&.85A, 107 (198 I ). 6. Cranshaw T. E., (In press). 7. Miedema A. R., de Chatel P. F. and de Boer F. R., Physica, lOOB, I (I 980). 8. Niessen A. K., de Boer F. R., Boom R., de Chatel P. F.. Mattens W. C. M. and Miedema A. R., CALPHAD, 7,51 (1983). 9. Miedema A. R. and Niessen A. K., CALPHAD. 7, 27 (1983). 10. Niessen A. K. and Miedema A. R., Ber. Bunsenges. Phys. Chem. 87, 7 I7 ( 1983). II. Alonso J. A. and Simozar S., Phys. Rev. B22, 5583 ( 1980). 12. Alonso J. A., Lopez J. M., Simozar S. and Girifalco L. A., Acta Metall. 30, 105 (1982). 13. Lopez J. M. and Alonso J. A., Physica, 113B, 103

between

14. Lopez J. M. and Alonso J. A.. Phvs. Stat. Solidi. A76. 675 (1983). 15. Eshelby J. D., Solid Slate Physics, (Edited by F. Seitz and D. Turnbull), Vol. 3. p. 79. Academic Press. New York, (1956). 16. Gschneidner, Jr. A. K., Solid State Physics, Vol. 16 (Edited by F. Seitz and D. Turnbull), Vol. 16, p. 275. Academic Press, New York (1964). 17. Pierron V. and Cadeville M. C.. J. Phys. F. Mer. Phys.

eqns (2) or (3), (or eqn (4) for Fe alloys).

Acknow/edgements-One of us (J.A.A.) thanks the British Council for supporting a short visit to Oxford in February of 1983, at which time this work was started. Two of us (J.A.A. and N.H.M.) wish to acknowledge that their contribution to this work was finalised at the ICTP. Trieste Condensed Matter Workshop in the summer of 1684.

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