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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
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