An evaluation of the configurational and non-configurational entropies of some binary alloys

Scripta METALLURGICA

Vol. 6, pp. 277-286, 1972 Printed in the United States

Pergamon Press, Inc.

AN EVALUATION OF THE CONFIGURATIONAL AND NON-CONFIGURATIONAL ENTROPIES OF SOME BINARY ALLOYS

R. Crombie and D.B. Downie Department of Metallurgy, University of Strathclyde Glasgow,

(Received October

14,

1971;

Scotland

Revised

February

18,

1972)

Introduction In published literature there exist values for both long range and short range order p a r a m e t e r s for a number of binary metallic systems but in only a few cases have the data been used to devise configurational entropies of mixing.

This publication is concerned with

(i) using all the available order data to obtain configurational e n t r o p i e s , (ii) combining these with the best available total entropies of mixing to obtain the thermal entropies of mixing, (ill)confirming the thermal entropies of mixing by integration of ~ C p data where these are available, (iv) deriving mathematical relationships between enthalpies of mixing and the configurational and thermal entropies of mixing using data from the systems investigated. Evaluation of Cou.fi~urational Entropies

Long Range Order Fowler and Guggenheim (1) have shown that the number of ways of arranging A and B atoms on the available sites in a binary alloy which forms a superlattice consisting of two equivalent sub-lattices

(eg /~- CuZn) is given by In g(L)

z

(XA+XAL) In (XA + XAL) + ( % - XAL) In ( % - XAL) +(XA

XAL) In (XA -XAL)

+(XB + XAL) In (XB + XAL)}. . . . . . . . . . . . . . . . . . . . (1)

277

278

ENTROPIES OF SOME BINARY ALLOYS

Vol.

where XA and XB are the atomic fractions of A and B respectively,

6, No.

4

L is the long range order p a r a m e t e r

of Bragg and Williams (2, 3), and N is the total number of atoms present.

The expression for super-

lattices not having equivalent sub-l~ttices (eg Cu3Au) is given by Fowler and Guggenheim as: lng

LI -- N { x A In x A •

In

-XA(X A + XB L ) In XA(XA + XB L ) - 2XAXB(I- L) In XAXB ( l - L ) ÷

In

÷ XA

..........

(2)

The configurational entropy of both types of system can be obtained using the statistical relationship S

=

COIl

k In g (L)

.....................

(3)

In the p r e s e n t a s s e s s m e n t substitution of measured p a r a m e t e r s into equations (1) or (2) and combination with (3) gave values for the configurattonal entropies of mixing.

Subtraction of the ideal

configurational entropy viz. s.,

--

- a ( x A in x A + x B in XBI

.................

gave the corresponding values of the excess couftgurattonal entropies. using the above method are shown in column 6 of Table 1.

(41

The values obtained for ~ Sxs

COIl

The alloys to which these values refer,

together with the values of L, the sources of these values and the temperatures to which they apply are given in columns l to 4 of the same table. Short Rathe Order In binary alloy systems short range order and clustering can be defined in t e r m s of the probability of finding an A-atom on a neighbouring site to a B-atom in the f i r s t co-ordination shell of atoms, viz ~1

=

1 -

~AB.

. .............

(5)

XA !

where PAB is the probability of the AB bond and ~1 is the order p a r a m e t e r of Cowley (4).

From quasi-

chemical theory, considering only interactions between n e a r e s t neighbours, it can be shown that ~1 is related to the number of AB bonds ( PAB) by PAB ZN

=

XA X B (1 -~.11

...............

where Z is the co-ordination number of the atomic configuration.

(6 I

Takagi (5) has related the p a r a m e t e r

PAB to the free energy of a system consisting of A and B atoms and from the equation (8.2) in the above reference it can be shown that

Vol. 6, No. 4

ENTROPIES OF SOME BINARY ALLOYS

~.S

con

=

R ( Z - 1 ) ( X A l n X A + XB I n

279

XB)

_ ~.~Z~2PzNAB In pZNAB+(XA- PzNAB)In(XA-PzNAB + (XB - P A B ) In (XB -

PAB )} . . . . . (7) ZN In this a s s e s s m e n t reported values o f ~ 1 have been used along with equation (6), (7) and (4) to

ZN

evaluate ~ Sxs for a number of alloys. The r e s u l t s are shown in column 6 of Table 2 and the alloy coll compositions, the ~ l values, the sources of these and the temperatures at which they apply are shown in columns 1 to 4 of the same table. Evaluation of Non-coufi~urationa.1 Entropies Contribution to the entropies of mixing from sources other than coufiguratioual have been obtained by subtracting the excess configurational entropies shown in columns 6 of Tables 1 and 2 from the best total excess entropies available. sources indicated in the footnotes.

The values used are shown in columns 7 of Tables 1 and 2 and the The values obtained for the non-coufigurational contributions are

shown in columns 5 of Tables 1 and 2 and are designated "~S

" thermal "

It is desirable to confirm the ~ Stherma 1 values by integration of ~ Cp data.

Due to the lack of

Cp data and, in some iustances, to the occurrence of duplex phase fields at low temperatures this is only possible for the Cu Au and Cu3Au alloys.

Since these alloys are subject to o r d e r / d i s o r d e r transformat-

ion, the question also a r i s e s as to whether the disordering peaks in the Cp curves should be included or whether a smooth curve, representing only the non-cooperative heat capacities should be used. authors consider there are theoretical arguments in favour of the latter method.

The

Consequently, values

using both methods have been found andare reported in columns 9 and 10 of Tables 1 and 2. It will be noted that only in one instance, viz.the value for Cu3Au at 678°K, using non-cooperative heat capacities, is satisfactory agreement found.

The AStherma 1 value for the alloy A10.9 Ago. 1 reported in column 9

of Table 1 is that derived by Simerska (6) using an X - r a y technique and also shows good agreement with the subtracted value in column 8. Relationships between Excess Entropies and Heats of Formation C oufi~urational Eutrupies Quasi-chemical theory indicates that high numerical (positive or negative) values of • H

should m correspond to high negative values of i~Sco n, indicating the teudency to cluster or order respectively. xs

Thus a plot of AScXcoSnv e r s u s ~ H m should show a maximum near to ~ H = 0 and possibly be symmetrical about this ~ H m value.

Also, since

A S xs has a minimum value of - 1.38 cal/deg g.a, r e p r e s e n t con lug complete order, the curve would be come asymptotic to the ~I-I axis. This indicates a relationm ship of the "cosb" type.

-0.66 -0.78 -1.27 -1.38 -0.92

0.71 0.60 0. II 0.00 0.20

0.80 0.82 0.98 1.0 0. 944

623

573

573

573

9

I0

II

12

Cu Zn

Cu Au

Au Zn

Cu3Au

AC

data neglecting P a r e a under d i s o r d e r i n g peak.

(c) Calculated f r o m

(b) F r o m Hultgren et al (19)

(a) Values a s s e s s e d by B l a i r and Downie (18)

XS

Sto t

7 ZX S t h e r m a 1

I 9

-0.05

- 0 . II

-1.38(b)

- 0 . 9 7 (b)

-

-0.15

- 0 . 9 3 (a)

+0.19

+0.21

-

-O. 27

By diff l~rom (b) erence ~Cp

8

- 0 . 9 3 (a)

cal/deg g. a.

COl'}.

~S xs

623

L

COD-

8

Temp oK

S

5

Cu0.52Zn0.4~

Ref

L . R . O . p a r a m e t e r data

21 3 ] 4

+0.13

+0.19

See ~c) footnote

I0

12

573

-1.67

-6.17

717 20 19

-2.16

-2.64 623 18

573

-2.62 623 18

19

Value k e a l / g , a.

13

Temp. oK

~m

I

Ref.

ll

Order parameters, entropies and enthalpies of mixin~ for L. R.O. allo~fs

TABLE 1

z o

O

<

o-1

0

>

>

0

o

P~

o

r~ z

oo o

673

678

573

813

623

14

-

15

16

17

15

15

-Ag Zn

Al Zn

Cu3Au

Ag3Au

-0.15

+0.77

+0.38

-0.22

+0.19

+0.46

+0.3~*)

+0.61

+0.16

19

23

19

19

20

19

18

19

623

820

800

678

673

324

603

700

+0.71

+0.19

-0.86

-1.08

+0.87

-1.54

-1.21

-1.29

-2.35

(d) Evaluated fro m e. m.f. data of Hilliard et al (22) and calorimetric data of Conned (20)

(g) Calculated from A C data neglecting area under disordering peak. P

+0.73 (d)

+0.26 (e)

-0.26 (b)

-0.14(b)

-0.31(d)

+0.18

816

(c) Assessed by Balir and Downie (18)

-0.04

-0.12

-0.04

-0.33

0.92

19

13

Toemp Value kcal/g, a. K

12

(f) Simerska (6) using x - r a y technique

0.60

0.53

1.08

0.79

1.24

-0.10

+0.76

+0.01

-0.44

Ref.

ll

(b) From Hultgren et al (19)

+0.09

+0.15

-0.05

-0. 218

+0.16

+0" 74(b)

0.7~ a) -0.64

-0.08 (b)

-0.57 (b)

+0.37 (c)

-0.09

-0.13

-0.39

0.99

1.29

1.25

Stherma 1

10

(e) Hillert, Averbach and Cohen at 798°K (23)

324

-0.31

-0. 123

-0.18

E

9

By d i l l - I From(b) See (g) erenee [ Zh Cp footnote

8

cal/deg g. a.

~sX~t

7

(a) Orr and Rovel (21)

AI0.9Zn O.1

A10.9Ag0.1

698

10

~-Ag Zn

603

816

13

oK

Weml~

Cu Au

Ref.

con

z~sxs

AS

S. R. O. p a r a m e t e r con

6

5

4

3

C u Zn

Alloy

2

Order p a r a m e t e r s , entropies and enthalpies of mixin[~ for S.R.O. alloys

TABLE 2

,-<

r-D GO

C~

0

3>

t~ ~=~ Z

O

C)

b-d t'~

m z 7o o

4~

o

Ox

O

282

ENTROPIES OF SOME BINARY ALLOYS

Vol.

6 , No. 4

At p r e s e n t there is a lack of order p a r a m e t e r data for systems showing positive heats of formation, thus limiting specualtion on overall trends but when the AS xs values from Tables 1 and 2 con are plotted against appropriate LkH values (Fig. 1) a cosh relationship is suggested. The ~ H m

m

values used have been chosen as the best available for the temperatures at which the order p a r a m e t e r s apply and are shown in the last three columns of Tables I and 2 along with the sources and temperatures of determination.

The curve drawn in Fig. 1 to r e p r e s e n t the r e s u l t s was deduced assuming the

maximum to occur at ~ H

m The relationship is

g.a.

= A Sxs con

Sxs

=

0, and that the minimum value of

=

A Sxs

con

is

-1.38 cal/deg

...........

con

sh(

(8)

3.10AHm ) ~ktI m + 6.17

where I ~ H

is in

m

Sxs

cal/g, a. and

is in

cal/deg g. a.

con

It is clear from Fig. 1 ~hat the above

relationship is only a fair representation of the plotted points, but it is worthy of consideration. Thermal .Entrop~r It is to be expected that the thermal entropies of mixing In the alloys considered would be d e t e r m i n ed mainly by the change in lattice vibrations.

Thus, high negative values of ~ S t h e r m a 1 would be associated

with high negative values of ~ H m and high positive values of

~ S t h e r m a I with high positive values of ~ H m.

The r e s u l t s from Tables 1 and 2 are plotted in Fig. 2 and indinate a well-defined linear relationship, ff the point representing /~-Ag Zn is neglected.

The equation of the line shown was calculated by the method

of least squares (omitting ~ - Ag Zn) and is ~X S t h e r m1a where

=

0.25 ~ H m

+

~ H m is in k cal/g, a. and ~ S is in cal/deg g. a.

0.34

..................

(9)

The correlation coefficient is 0.886.

The reason for the anomalously high ~ , S t h e r m1a value for the (~-Ag Zn alloy is not clear but it may be that the thermodynamic data used to obtain A ~ t either

A H m or

An e r r o r of 600 cal/g, a. in

are in e r r o r .

~ G m would account for the discrepancy. Discussion

Addition of equations (8) and (9) gives an expression for the total excess entropy of mixing in t e r m s of

~ Hm viz ~ s

t

= cosh

1.38 ~ H m 3.10

) ~tHm+ 6,17

O. 25 ~ - I m - 1.04 . . . . . . (10) +

Kubasche vski (7) obtained a linear relationship between maximum total excess entropy of mixing and the maximum enth~.lpy of mixing for a large number of binary systems when allowance was made for the relative bond stabilities of the alloying elements.

His expression is

Vol. 6, No. 4

ENTROPIES OF SOME BINARY ALLOYS

•

•

I

•

•

d

!

'

F

l:%,~r~ad/f i I~ ,.-0"I

oi .I

-41,'0

-@0

,41,0

d.,

0

K (,m~$4

AM

Figure l Relationship of excess configurational entropy of mixing to heat of mixing : X L.R.O. alloys,

Q

S.R.O. alloys

+C~I

.IIoAs~. I

*AILo.4 Z m . . ,

/2.

4+0.4

~Svlb ¢4~G K 0

_

"AS+ ,~,-,

. i i i '4~o, 4 l oO-d

¢

¢

-4-0

. |

I

i

-~

,AH

i 0

i

i

*&O

X cal/~,k

Figure 2 Relationship of non-configurational entropy of mixing to heat of mixing: X L. R.O. alloys •

S.R.O. alloys

283

284

ENTROPIES

OF SOME BINARY ALLOYS

LX S x s

=

O. 64

~

Vol.

Hmax

6, No.

4

(11)

m a x

° ' "

. . . . . . .

"°

½(Te1 + Te2) where Te I and Te 2 are the boiling points of the component metals.

In the p r e s e n t work, however,

allowance for the relative bond stabilities using Kubaschewski's factor gave a linear relationship with ~ S t h e r m a 1 with a slightly lower correlation factor, viz. 0.871.

Thus the relative bond stabilities

of the pure metals is not of significance in the relationship between A S v i b and ~ H m.

Also it seems

probable that the linear relationship obtained by Kubaschewski r e s u l t s from the non-coufigurational component of excess entropy of mixing alone.

Since most of his data refer to molten alloys, excess

configurational entropies would be near zero and of less significance in determining the trend of values than in the solid state.

Certainly, a number of his alloys which "show excess entropies between zero

and +0.60 e.u. must have considerable non-configurational contributions. Equation (9) and Fig. 2 indicate that it is possible to have a positive is still negative.

A S v i b value while A H m

This apparent loosening of the alloy structure with respect to the pure metals,even

when there is an inherent attraction between them,may be due to either a size-factor influence or i n t e r action of sub-valence electrons, or both.

References 1. R.H. Fowler and E.A. Guggenheim, "Statistical Thermodynamics" p 568 and p. 598, Cambridge University P r e s s (1939). 2. W. L. Bragg and E.J. Williams, Proc. R. Soc. A145, 899 (1934). 3. W.L. Bragg and E.J. Williams, Proc. R. Soc. A]51,

540 (1935).

4. J M. Cowley, J. App, Phys. 21, 24 (1950). 5. Y. Takagi, Proc. Phys. Math. Soc. (Japan)2__33, 44 (1941). 6. M. Simerska, Acta Met.

1__33,113 (1965).

7. O. Kubaschewski, "Phase Stability of Metals and Alloys", p 63, Edited by P.S. Rudman, J. Stringer and R . I . Jeffries, Batelle Memorial Inst., McGraw-Hill (1967). 8. R. Crombie and D.B. Downie, Acta Met,, 19 ,

1227 (1971).

9. D. Chipman and B.E. Warren, J. App. P h y s . , 21, 10. B.W. Roberts, Acta Met,

696 (1950)0

2, 597 (1954).

11. H. Iwasaki and T. Uesugi, J. Phys. Soc. of Japan, 25.._., (2), 1640 (1968). 12. Do T. Keating and B.E. Warren, J. App, P h y s . , 22_, 286 (1951). 13. C.B. Walker and D.T. Keating, Phys. Rev. 130, 1726 (1963). 14. E. Suoninen and B . E . W a r r e n , Acta Met. 6_, 172 (1958). 15. P.S. Rudman and B.L. Averbach, Acta Met. 2, 576 (1954) 16. S.C. Moss, J. App. Phys, 35_, 3547 (1964).

Vol.

6, No.

4

ENTROPIES

OF SOME BINARY A L L O Y S

17. N. Norman and B°E. Warren, J. App. Phys. 22, 483 (1951) 18. G.R. Blair and D.B. Dow~ie, Metal Sci. J. 4, 1 (1970) 19. R. Hultgren, R L. Orr, P . D . Anderson and K.K. Kelly, "Selected Values of Thermodynamic P r o p e r t i e s of Metals and Alloys".

John Wiley (1963)

20. R A. Connell, Ph.D. Thesis, University of Strathclyde, 1971 21. R . L . O r r and J . M . Rovel, Aeta Met. 10, 935 (1962) 22. J . E . Hil!iard, B . L . Averbach and M. Cohen, Acta Met. 2, 621 0954) 23. M. Hilert, B . L . Averbach and M. Cohen, Acta Met. 4, 31 (1956)

285