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Aspects of the thermodynamics of liquid alloys of noble metals and transition metals
Journal of Non-Crystalline Solids 61 & 62 (1984) 101-112 North-Holland, Amsterdam
101
ASPECTS OF THE THERMODYNAMICSOF LIQUID ALLOYS OF NOBLE METALS AND TRANSITION METALS O. J. KLEPPA The James Franck I n s t i t u t e and The Department of Chemistry The University of Chicago, Chicago, I l l i n o i s 60637, U.S.A. A review is presented of thermochemical data recently obtained in the author's laboratory for two d i f f e r e n t families of l i q u i d a l l o y s : (1) the binaries formed among the noble metals copper, s i l v e r , and gold, and (2) the l i q u i d solutions of f i r s t row t r a n s i t i o n metals in copper. The results are compared with predictions derived from the semi-empirical model of Miedema et a l . For both families of alloys comparisons between the new enthalpy data and excess Gibbs energies reported in e q u i l i b r i u m or e.m.f, studies provide new i n s i g h t regarding the excess entropy of mixing. 1. INTRODUCTION During recent years Miedema and co-workers I have developed a semi-empirical scheme which allows the prediction of the enthalpies of solution f o r a large number of l i q u i d metals in other l i q u i d metals.
In Miedema's approach, which
represents a s i g n i f i c a n t refinement of e a r l i e r work along s i m i l a r lines by Hildebrand and Scott 2 and by B. W. Mott 3, the enthalpy of mixing is derived from the difference in two semi-empirical parameters AHmix = f ( c ) [ - p ( A ¢ * ) 2 4 Q(Anwsl/3)2].
(i)
In this expression f(c) is a simple function of composition; the f i r s t
(attrac-
t i v e ) term arises from the difference in e l e c t r o n e g a t i v i t y between the two metals; this is measured by the difference in the work function ¢.
The second
(repulsive) term arises from the charge density mismatch between the two metals at the Wigner-Seitz boundary, Anws.
In order to obtain the correct sign of the
enthalpy using the universal constants P and Q Miedema adjusted the work function to new values ¢*. S t r i c t l y speaking, Eq. ( I ) is v a l i d only f o r alloys of two t r a n s i t i o n metals. When one of the two components is a non-transition metal, an additional a t t r a c t i v e term R is introduced AHmix = f(c)[-P(A¢*) 2 + Q(~nwsl/3) 2 - R].
(2)
This term is ascribed to a d-p electron h y b r i d i z a t i o n e f f e c t which changes with the number of p-electrons.
For the noble metals copper, s i l v e r , and gold,
at the very end of t h e i r respective t r a n s i t i o n series, the term R is small. While Miedema's semi-empirical scheme may be c r i t i c i z e d on fundamental 0022-3093/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
102
O.J.
Kleppa / Thermodynamics of liquid alloys
grounds 4, his approach has proved very useful in providing estimates of the enthalpies o f mixing f o r a very wide range of a l l o y s .
Miedema and co-workers
have compared t h e i r predicted enthalpies with experimental data f o r a number of binary systems 5.
However, t h e i r work also has drawn a t t e n t i o n to the fact that
r e l i a b l e experimental data are very scarce for systems in which the a l l o y components are varied in a systematic manner, f o r example w i t h i n a single t r a n s i tion series. During the past few years we have been concerned with obtaining such data. In the present communication we report and discuss the results of some recent enthalpy of mixing measurements for two d i f f e r e n t classes of l i q u i d a l l o y systems. 2. THE SYSTEMS Cu-Ag, Ag-Au, AND Cu-Au
For these alloys a great deal of thermodynamic information from d i f f e r e n t sources already is available in the l i t e r a t u r e .
However, some of this informa-
tion is i n t e r n a l l y inconsistent, which suggested to us that improved thermochemical data should be obtained.
We present below new enthalpy of mixing data
taken from the recent studies of Kleppa and Watanabe6 (Cu-Ag) and of Topor and Kleppa 7 (Ag-Au, Cu-Au). 2.1. Cu-Ag The phase diagram f o r this system8 is of the simple eutectic type.
Hence we
a n t i c i p a t e p o s i t i v e enthalpies of mixing in the l i q u i d mixtures; due to the d i f . ference in atomic size we expect to find larger p o s i t i v e enthalpies for solutions of s i l v e r in copper than f o r copper in s i l v e r .
According to Hultgren's
most recent assessment g this is not the case. We show in Figure I a graph of the recently published enthalpies of mixing 6, the excess Gibbs energies, AGE, from Hultgren 9, and the calculated values of TASE at 1373 K.
I t is p a r t i c u l a r l y noteworthy that the new measurements show 5.0 4.0 3.0 7_ 2.0 LO 0
O ~ l02i l l l ~0.4i l l Ag
Xcu
0.6
0.8
1.0 Cu
FIGURE I Enthalpies, excess Gibbs energies, and excess entropies of mixing in the l i q u i d system Cu-Ag
O.J. Kleppa / Thermodynamics of liquid alloys
103
that the r a t i o of the two l i m i t i n g p a r t i a l enthalpies of solution (17.66/12.2 = 1.45) is e s s e n t i a l l y the same as the r a t i o of the molar volumes, 1.44.
This
indicates that the mixing enthalpies may be represented very well by a volume f r a c t i o n expression.
In this respect the results do not support the Miedema
model from which one predicts that the r a t i o of the enthalpies of solution should be , ~ ) 2 / 3 1.28. tVc u = We also see from Figure I that there is very l i t t l e difference between the enthalpies of mixing and the excess Gibbs energies of mixing, i . e . the excess entropies are very small.
Thus we find that in this system both the en_ergetic
asymmetrE and the entropies of mixing are well described by the usual regular 2 solution expression. 2.2. Ag-Au The phase diagram for this system8 shows a complete range of s o l i d solutions. Furthermore, the course of the solidus and liquidus curves indicates that AGE(s)
-AGE(L) should
have very small negative values.
Although there is no
ordered phase at low temperature, the considerable difference in electronegativity
and the very small difference in atomic size between s i l v e r and gold
makes us expect negative enthalpies of mixing. Figure 2 shows s t i l l
unpublished enthalpy of mixing data for this system7.
The results are compared with the e a r l i e r c a l o r i m e t r i c values given by Oriani and Murphy! O, and by Itagaki and Yazawall; the graph also shows the values recommended by Hultgren et al in 196312 , and in 19739 .
This f i g u r e i l l u s t r a t e s
that even f o r this very simple a l l o y system the published data leave a great deal to be desired.
Note that the enthalpy of mixing curve in this system is
nearly completely symmetrical.
This is consistent with the fact that the atomic
volumes of the two metals are very s i m i l a r . In Figure 3 we compare the new enthalpy data with the excess Gibbs energies; the curve drawn represents our own best estimate of AGE based on the available information 9'12.
For t h i s system we f i n d a somewhat larger difference between
enthalpy and excess Gibbs energy, i . e . small negative excess entropies. This correlates with the observed small negative enthalpies of mixing: deviations from random mixing due to some short range order is indicated. 2.3. Cu-Au The phase diagram8 shows a complete range of s o l i d solutions and the formation of ordered phases.
Since the o r d e r - d i s o r d e r transformations occur near
700 K, we i n f e r that the enthalpy i n t e r a c t i o n parameter for the s o l i d solutions should be of the order of -2RTc : -12 kJ mol - I ,
The phase diagram indicates
that the l i q u i d solutions should be mere exothermic than the s o l i d solutions. Figure 4 gives unpublished c a l o r i m e t r i c results for this system7, along with enthalpy values advanced by several e a r l i e r i n v e s t i g a t o r s . Among these workers
104
O.J. Kleppa / Thermodynamics of liquid alloys
9
+ F
,
Ser~es ,
c Ser~es 3
° Ser,es 2
x Ser,es 4
D ORIANI (i958) -
q
i
0 -~-LZ_ ~ t -I.O
-----[TAGAKI (I97I
--HULTGREN(1963) ---HULTGREN(1973)-
i
o
i
/
'
~
0 Ag
0.2
0.4
AHMfxl1379K)
02
2
0.6
0.8
. _I~
-20 T~ 3.0 -4.0 5.0
, T
0.4
XaU
0.6
0.8
Au
1.0 Au
XAu FIGURE 2 Enthalpy o f mixing and enthalpy i n t e r action parameter in the l i q u i d system Ag-Au at 1379 K. Experimental data from Ref. 7; I t a g a k i , Ref. 11; Oriani and Murphy, Ref. 10; Hultgren (1963), Ref. 12; Hultgren (1973), Ref. 9.
FIGURE 3 Enthalpies, excess Gibbs energies and excess entropies of mixing in the liquid system Ag-Au.
Itagaki and Yazawa13 carried out calorimetric measurements using an adiabatic high temperature calorimeter. The other values in Figure 4 were derived from the temperature dependenceof e.m.f, or vapor pressures (Oriani 14, Edwards and
Brodsky 15, Neckel and Wagner16, Hager, Howard and Jones17). In the upper half of Figure 4 is plotted the enthalpy interaction parameter for this system. By a least squares treatment of the data the following anal y t i c a l expression was derived AHmix/XAuXCu = -28.82-2.47 XAu + 9.54 XAu2 kJ mo1-1.
(3)
We shall return to the significance of this equation below. Since this system shows ordered intermetallic phases at low temperatures, we expect to find negative excess entropies of mixing in the liquid alloys.
How-
even, checking the literature we found that while Oriani 14 and Edwards and Brodsky15 report f a i r l y large negative excess entropies, Neckel and Wagner16 give small positive values.
Hultgren 9 similarly recommendspositive values.
The new enthalpy data allows us to consider this problem again. Surveying the e a r l i e r e.m.f, and equilibrium studies we concluded that the most reliable Gibbs energy data probably are those of Neckel and Wagner16. In
O.J. Kleppa I Thermodynamics o t liquid alloys
105
28~
-
<:]
'
24 i
+ Serles 1 Series 2 c Series 3
*
Series 4
\
0 ~ - ~
I~
E 1 2 I
".
21~2 i
AS =-X (I-X) ~ -
22,
I
J->"710
-'S" / I
........... , ,
.•c•
ITAGAKI(1971,c01}-- --EDWARDS , (1956,~p ) 16[_~ --ORIANI ~956emf) ..... NECKEL L (1~69 v p 1 ~-'"'-HAGER [1970,v p )
-~
]'
....: > /
I
~.
~
E(1449K, Neckel ¢ Wogner) / /
/ Cu
~H~,~Ii379KI 0.2
0.4
0.6
0.8
/ Au
XAu ir
0 (:u
~
I
0.2
,
I
,
0.4
I
0.6
,
I
0.8
XAu
°7 °`
1.0 Au
FIGURE 4 Enthalpy of mixing and enthalpy i n t e r action parameter in the l i q u i d system Cu-Au at 1379 K. Experimental data from Ref. 7; I t a g a k i , Ref. 13; Oriani, Ref. 14; Edwards, Ref. 15; Neckel, Ref. 16; Hager, Ref. 17.
FIGURE 5 Enthalpies, excess Gibbs energies and excess entropies of mixing in the l i q u i d system Cu-Au.
Figure 5 we accordingly compare t h e i r excess Gibbs energies with our own enthalpies.
As expected, we find small negative excess entropies, which suggest some
deviations from random mixing.
Also shown in Figure 5 is an excess entropy
curve calculated from quasi-chemical theory 18.
Although there is some uncer-
t a i n t y regarding what values to adopt for ~ and f o r z in the quasi-chemical expression f o r the excess entropy (see (6) below), there seems to be plausible agreement between the experimental and calculated values. The presence of short range order is probable also because of the f a i r l y large negative enthalpies of mixing.
In this context the c h a r a c t e r i s t i c para-
b o l i c dependence of the enthalpy i n t e r a c t i o n parameter on a l l o y composition probably is s i g n i f i c a n t . In considering this problem we again r e f e r to quasi-chemical theory, from which we obtain the following approximate expression for the molar enthalpy of mixing in a nearly random s o l u t i o n : AHmix ~ X(I-X)X[I-X(1-X)z-~T ].
(4)
O.J. Kleppa / Thermodynamics of liquid alloys
106
Here z + . X = N~[2Vl2-(Vll v22) ] ,
(5)
N is Avogadro's number, v12 is the pair i n t e r a c t i o n energy between atoms I and 2, and z is the nearest neighbor coordination number.
From t h i s theory we also
obtain the expression for the excess entropy of mixing shown in Figure 5: ~2 ASE m _X2(I_X) 2 zRT2 '
(6)
The quasi-chemical theory is based on the s i m p l i f y i n g assumption that the pair i n t e r a c t i o n energies do not vary with a l l o y composition; therefore ~, the enthalpy i n t e r a c t i o n parameter, should be a constant f o r a given binary system. For the copper-gold system this assumption does not hold.
However, i f our
Eq. (3) is r e w r i t t e n to be consistent with (4), we find AHmix = XAu(I-XAu)[a+bXAu+CXAu(I-XAu)].
(7) i
From t h i s we see that we may obtain an average value of X by s e t t i n g X = a+~ (b is the difference between the two l i m i t i n g enthalpies of s o l u t i o n ) . l a r l y , we get
Simi-
2X2 c
:
-
Tiff
"
(8)
In t h i s analysis there remains only a single parameter which is not fixed by the theory, namely the nearest neighbor coordination number, z.
For close-
packed metals, such as pure copper and pure gold, at temperatures j u s t above t h e i r respective melting points, we would expect a value of z s l i g h t l y less than 12.
A comparable value should apply in copper-gold a l l o y s .
From t h e i r
experimental enthalpy data at 1379 K Topor and Kleppa 7 calculated z z 11.7, in remarkably close agreement with these expectations. In Table I we f i n a l l y compare the experimental values of the p a r t i a l enthalpy of solution 6'7 with values predicted by Miedema et al 1'19.
Note that Miedema's
recently revised values 19 of ~* and nws have not improved agreement with experiment in these systems. TABLE I .
Limiting p a r t i a l enthalpies of solution f o r l i q u i d alloys among the noble metals. AH-Solute, kJ mol - I
Solvent-Solute Cu Ag Ag Au Cu Au
Ag Cu Au Ag Au Cu
Experiment +17.66 +12.2 -15.82 -16.35 -28.82 -21.75
Miedema I (1977) + 5 + 4 -20 -18 -29 -20
Miedema 19 (1983) +I0 + 8 -21 -23 -42 -30
O.J. Kleppa / Thermodynamics of liquid alloys
107
3. SOLUTIONS OF THE FIRST ROW TRANSITION METALS IN LIQUID COPPER We r e f e r here f i r s t
to the published c a l o r i m e t r i c data f o r Mn20, for Fe, Co,
and Ni 21, and for Ti 22'23.
A report on Sc, V, and Cr in copper very recently
was submitted f o r publication 24.
The relevant phase diagrams8'25 show that
among these a l l o y systems there are two with extensive s o l i d solutions (Cu-Ni, Cu-Mn), two with quite stable i n t e r m e t a l l i c compounds (Cu-Sc, Cu-Ti), and four eutectic diagrams with l i m i t e d s o l u b i l i t y of the t r a n s i t i o n metal in l i q u i d copper near 1373 K (Cu-Co, ~4 at% Co; Cu -Fe, ~4 at% Fe; Cu-Cr, ~1.5 at% Cr; Cuu-V, ~uO.4 at% V).
Since our experimental c a p a b i l i t y was l i m i t e d to this temp-
erature, most of our c a l o r i m e t r i c data r e f e r to d i l u t e solutions of the t r a n s i tion metal in copper. Figure 6 is a graph of the p a r t i a l enthalpies of solution of the considered t r a n s i t i o n metals in l i q u i d copper near 1373 K. cooled l i q u i d metals.)
(The data r e f e r to the under-
The graph also shows Miedema's o r i g i n a l predictions I ,
as well as his recently revised values 19.
The figure confirms that Miedema's
semi-empirical theory c o r r e c t l y predicts the overall trend of the enthalpies of s o l u t i o n , i . e . p o s i t i v e values of the r i g h t order of magnitude f o r Ni, Co, and Fe, a d i s t i n c t minimum f o r Mn, p o s i t i v e values for Cr and V, and increasing I~'0
[
i
i
*20 / tI
-2 , !
d
f 2 50 ! <:2
-,oo~ .
// I z/ ' /
/
i 7 SC TI
- - Mhedema{L977) x M~edemo{1982~ -'>- Satoand KLeppo
1,3a;;
. . . .
il /
II II
,,
'
-5
1
--.--p,e~oo,wo,, j V Or Mn Fe Co Ni Ou
FIGURE 6 Comparison of experimental p a r t i a l enthalpies of solution of f i r s t row l i q u i d t r a n s i t i o n metals in copper at 1373 K with values predicted by Miedema et a l l , 1 9 .
FIGURE 7 Excess entropies of solution of f i r s t row l i q u i d t r a n s i t i o n metals in copper at about 1400 K.
O.J. Kleppa / Thermodynamics of liquid alloys
108
negative values from Ti to Sc.
Only f o r Mn, for which Sato and Kleppa20 found
a small negative enthalpy of s o l u t i o n , does Miedema obtain the wrong sign. While Miedema predicts the trend of the data very w e l l , we find that most of his values are more exothermic (or less endothermic) than the experimental res u l t s ; this difference is quite large (~30 kJ mo1-1) f o r Ti and V, but much smaller (~i0-15 kJ mol - I ) for Sc, Cr, and Co.
For Mn and Ni the s i t u a t i o n is
reversed and Miedema's values are more endothermic than the epxeriments, while f o r Fe the agreement between prediction and experiment is remarkably close.
On
the whole we find that Miedema's revised values Ig are in considerably b e t t e r agreement with experiment than his e a r l i e r values I. For many of the f i r s t
row solutes in l i q u i d copper excess Gibbs energies of
solution e i t h e r are a v a i l a b l e from e q u i l i b r i u m or e.m.f, studies ( T i , Mn, Ni) 26'27'9, or may be calculated or estimated from the published phase diagrams (V, Cr, Fe, Co) 8'25.
When these Gibbs energies near 1400 K are compared with
the new enthalpies of s o l u t i o n , we gain i n t e r e s t i n g new i n s i g h t regarding the excess entropies.
The relevant data are p l o t t e d in Figure 7.
I t w i l l be seen
that the p a r t i a l excess entropies for l i q u i d nickel and manganese in copper are quite small and negative.
These two metals have the numerically smallest en-
thalpies of solution and both have a wide range of solid s o l u b i l i t y in copper. Among the remaining solutes Co and Fe have p o s i t i v e excess entropies of about 5-10 J K-1 mol - I , while both Cr and V have p o s i t i v e values which are of the order of 20 J K-1 mol - I or more. probably small and p o s i t i v e .
The value f o r Ti is somewhat uncertain, but is There are no data for Sc.
For a beginning we want to take note o f the fact that there is extensive s i m i l a r i t y between Figures 6 and 7; i . e . i t is evident that the excess entropies and enthalpies vary in a roughly s i m i l a r manner when the solute is varied systematically from Ti to Ni.
Since the atomic volumes of the considered metals
f a l l w i t h i n a r e l a t i v e l y narrow range (Vr. = 7.1 cm3; VNi = 6.6 cm3; VCo = 6.6 3 ~u cm3; VFe = 7.1 cm~; VMn = 7.4 cm ; VCr = 7.3 cm3; VV = 8.4 cm3; VTi = 10.6 cm3), i t seems l i k e l y that atomic size plays only a minor role in influencing the enthalpies and entropies of s o l u t i o n .
For t h i s reason we believe that the ob-
served excess entropies probably are predominantly of e l e c t r o n i c and/or magnetic rather than o f v i b r a t i o n a l o r i g i n .
This suggests that any r e a l i s t i c microscopic
theory o f the thermodynamic properties must take into account also the thermal e x c i t a t i o n of the d-electrons which provide the major part of bonding and cohesion in these systems. Let us f i r s t
consider the well-known fact that among the solute elements in-
cluded in Figure 7 the pure metals Cr and Mn show antiferromagnetic ordering at low temperatures, while Fe, Co, and Ni are ferromagnetic. At s u f f i c i e n t l y e l e vated temperatures this interatomic order is destroyed, which gives rise to a
O.J. Kleppa / Thermodynamics of'liquid alloys disordering entropy.
109
With the possible exception of c o b a l t , f o r which the Curie
temperature is 1394 K, the disordering process should be complete at 1400 K. For d i l u t e solutions of these metals in copper near 1400 K t h i s magnetic disorder undoubtedly is also complete.
Hence we conclude t h a t this magnetic e f f e c t
should make no s i g n i f i c a n t c o n t r i b u t i o n to the observed excess entropies. In p r i n c i p l e i t should be possible to r e l a t e a major part of the excess ent r o p i e s in Figure 7 to the d i f f e r e n c e between the e l e c t r o n i c heat capacities yT in the a l l o y and in the pure metal, i . e .
to the d i f f e r e n c e between YMe
~XMe
in the a l l o y and YMe in the pure metal. To a zeroth order of approximation this e l e c t r o n i c entropy term should be of the order to (YMe - YMe)T"
For example, f o r d i l u t e solutions of nickel in cop-
per we compare ¥Ni from the low temperature heat capacity data of Guthrie et a128, 3.1 mJ K-2 mol - I , with the value for pure n i c k e l , YNi = 7.0 mJ K-2 mol-I 29.
This comparison y i e l d s an estimate of the e l e c t r o n i c excess entropy
of nickel in copper at 1400 K o f about -5.5 d K-1 mo1-1, which is in p l a u s i b l e agreement with the experimental value of -2.5 J K- I mel - I . For the o t h e r s o l u t e metals, however, there are complications with this simple scheme.
These complications arise from the f a c t t h a t some of these
solute metals, when dissolved in copper, at low temperature have l o c a l i z e d , quasi-bound magnetic impurity states which have a dramatic e f f e c t on many of the e l e c t r o n i c properties o f the a l l o y s 30. ent study i t
From the point of view o f the pres-
is p a r t i c u l a r l y i n t e r e s t i n g t h a t the thermal d e s t r u c t i o n of these
quasi-bound s t a t e s , which for Cr, Mn, and Fe in Cu l a r g e l y occurs below 10 K, is known to give r i s e to large excess s p e c i f i c heats and entropies 30. is d i f f i c u l t
Although i t
e x p e r i m e n t a l l y to p r e c i s e l y determine the t o t a l entropies of dis-
order, we shall f o r s i m p l i c i t y assume that they may be estimated from the expression Rln (2S+1); f o r Cr and Mn, S is 5/2 (corresponding to the f i v e unpaired spins in the free atom), while for Fe i t is 4/2 = 2.
This " i n t r a - a t o m i c " mag-
netic e f f e c t , which is the r e s u l t of i n t e r a c t i o n between the d-electrons of the solute atoms and the conduction electrons o f the host metal, thus should give r i s e to a p o s i t i v e magnetic entropy term which is about 14.9 J K- I mo1-1 f o r Cr and Mn dissolved in copper and about 13.4 J K- I mol - I f o r Fe. I f t h i s p i c t u r e is c o r r e c t , the experimentally observed excess entropies f o r Cr, Mn, and Fe in copper should in large measure r e f l e c t the d i f f e r e n c e between these magnetic entropy terms, on the one hand, and the e l e c t r o n i c heat capacity terms in the pure metals, y. T, on the o t h e r . For Cr, f o r which yT at 1400 K me r q u i t e low (2.2 J K-1 nlol-1), we thus estimate S~Cr at t h i s temperature to be +12.7 J K- I mol - I .
S i m i l a r l y , f o r Mn, f o r which yT at 1400 K is very high
(17.5 J K- I m o l - l ) , we f i n d ~ n
: 2.7 J K- I n ~ l - l ;
f o r Fe, with yT = 7 J K- I
110
O.Z K l e p p a / T h e r m o d y n a m i ~ o f 6 q u i d a l l o y s
rfo ] - I
we get S~e = 6.4 d K- I mo1-1.
Although any close numerical agreement be-
between these values and the experimental data undoubtedly is f o r t u i t o u s , we see that our estimates c o r r e c t l y reproduce the trend of the data. I t is o f i n t e r e s t to take note here also of the recent experimental and t h e o r e t i c a l study by Gachon et a131 of the Gibbs energies, enthalpies and excess entropies of solution of s o l i d solutes from t i t a n i u m to germanium in s o l i d copper.
I t is not s u r p r i s i n g that some of the conclusions reached by these
authors are s i m i l a r to our own.
They conclude, f o r example, that f o r Fe, Mn,
and Cr as solutes in s o l i d copper i t is necessary to take into account the magnetic decoupling of the impurity states in order to understand the observed excess entropies o f s o l u t i o n . ACKNOWLEDGMENTS This work was supported by the National Science Foundation under Grant CHE8106980, and by the Materials Research Laboratory Program of the NSF at the University o f Chicago under Grant DMR-7924007. REFERENCES I) See, e . g . A . R . Miedema, F. R. de Boer, R. Boom, and J. W. F. D o r l e i j n , Calphad i (1977) 353; A. R. Miedema, P. F. de ChStel, and F. R. de Boer, Physica IOOB (1980) I. 2) J. H. Hildebrand and R. L. Scott, The S o l u b i l i t y of Non-Electrolytes (Reinhold, New York, 1950). 3) B. W. Mott, Phil. Mag. 2 (1957) 259. 4) A. R. Williams, C. D. G e l l a t , J r . , and V. L. Moruzzi, Phys. Rev. Lett. 44 (1979) 429. 5) See, e . g . F . R , de Boer, R. Boom, and A. R. Miedema, Physica I01B (1980) 294; 113B (1981) 18. 6) O. J. Kleppa and S. Watanabe, Met. Trans. 13B (1982) 391. 7) L. Topor and O. J. Kleppa, Met. Trans. B (submitted). 8) M. Hansen, Constitution of Binary Alloys, 2nd ed. (McGraw-Hill, New York, 1958). 9) R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, and K. K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys (American Society for Metals, Metals Park, OH, 1973). i0) R. A. Oriani and W. K. Murphy, J. Phys. Chem. 62 (1958) 199. 11) K. Itagaki and A. Yazawa, J. Met. Soc. Japan 35 (1971) 389. 12) R. Hultgren, R. L. Orr, P. D. Anderson, and K. K. Kelley, Selected Values of Thermadynamic Properties o f Metals and Alloys (Wiley, 1963).
O.Z Kleppa / Thermodynamics of liquid alloys 13) K. Itagaki and A. Yazawa, J. Met. Soc. Japan 35 (1971) 383. 14) R. A. Oriani, Acta Met. 4 (1956) 15. 15) R. K. Edwards and M. 8. Brodsky, J. Am. Chem. Soc. 78 (1956) 2983. 16) A. Neckel and S. Wagner, Ber. Bunsenges. Phys. Chem. 73 (1968) 210. 17) J. P. Hager, S. M. Howard, and J. H. Jones, Metall. Trans. I (1970) 415. 18) See, e . g . E . A .
Guggenheim, Mixtures (Oxford University Press, 1952).
19) A. K. Niessen, F. R. de Boer, R. Boom, P. F. de Ch~tel, W. C. M. Mattens, and A. R. Miedema, Calphad 7 (1983) 51. 20) S. Sato and O. J. Kleppa, Met. Trans. 10B (1979) 63. 21) S. Sato and O. J. Kleppa, Met. Trans. 138 (1982) 251. 22) H. Yokokawa and O. J. Kleppa, J. Chem. Thermodyn. 13 (1981) 703. 23) O. J. Kleppa and S. Watanabe, Met. Trans. 13B (1982) 391. 24) S. Watanabe and O. J. Kleppa, to be published. 25) W. G. M o f f a t t , The Handbook of Binary Phase Diagrams (General E l e c t r i c Company, 1978-83). 26) F. Sommer, K.-H. Klappert, J. Arpshofen, and B. Predel, Z. Metallkunde 73 (1982) 581. 27) P. J. Spencer and J. N. Pratt, Trans. Faraday Soc. 64 (1968) 1470. 28) Guthrie, Friedberg and Goldman (unpublished), quoted by E. W. Pugh, B. R. Coles, A. A r r o t t , and J. E. Goldman, Phys. Rev. 105 (1957) 814. 29) R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, and K. K. Kelley, Selected Values of the Thermedynamic Properties of the Elements (American Society f o r Metals, Metals Park, OH, 1973), p. 20. 30) M. D. Daybell and W. A. Steyert, Rev. Mod. Phys. 40 (1968) 380. 31) J. C. Gachon, M. Notin, C. Cunat, J. Hertz, J. C. Parlebas, G. M o r a i t i s , B. Stuppel, and F. Gautier, Acta Met. 28 (1980) 489.
111
101
ASPECTS OF THE THERMODYNAMICSOF LIQUID ALLOYS OF NOBLE METALS AND TRANSITION METALS O. J. KLEPPA The James Franck I n s t i t u t e and The Department of Chemistry The University of Chicago, Chicago, I l l i n o i s 60637, U.S.A. A review is presented of thermochemical data recently obtained in the author's laboratory for two d i f f e r e n t families of l i q u i d a l l o y s : (1) the binaries formed among the noble metals copper, s i l v e r , and gold, and (2) the l i q u i d solutions of f i r s t row t r a n s i t i o n metals in copper. The results are compared with predictions derived from the semi-empirical model of Miedema et a l . For both families of alloys comparisons between the new enthalpy data and excess Gibbs energies reported in e q u i l i b r i u m or e.m.f, studies provide new i n s i g h t regarding the excess entropy of mixing. 1. INTRODUCTION During recent years Miedema and co-workers I have developed a semi-empirical scheme which allows the prediction of the enthalpies of solution f o r a large number of l i q u i d metals in other l i q u i d metals.
In Miedema's approach, which
represents a s i g n i f i c a n t refinement of e a r l i e r work along s i m i l a r lines by Hildebrand and Scott 2 and by B. W. Mott 3, the enthalpy of mixing is derived from the difference in two semi-empirical parameters AHmix = f ( c ) [ - p ( A ¢ * ) 2 4 Q(Anwsl/3)2].
(i)
In this expression f(c) is a simple function of composition; the f i r s t
(attrac-
t i v e ) term arises from the difference in e l e c t r o n e g a t i v i t y between the two metals; this is measured by the difference in the work function ¢.
The second
(repulsive) term arises from the charge density mismatch between the two metals at the Wigner-Seitz boundary, Anws.
In order to obtain the correct sign of the
enthalpy using the universal constants P and Q Miedema adjusted the work function to new values ¢*. S t r i c t l y speaking, Eq. ( I ) is v a l i d only f o r alloys of two t r a n s i t i o n metals. When one of the two components is a non-transition metal, an additional a t t r a c t i v e term R is introduced AHmix = f(c)[-P(A¢*) 2 + Q(~nwsl/3) 2 - R].
(2)
This term is ascribed to a d-p electron h y b r i d i z a t i o n e f f e c t which changes with the number of p-electrons.
For the noble metals copper, s i l v e r , and gold,
at the very end of t h e i r respective t r a n s i t i o n series, the term R is small. While Miedema's semi-empirical scheme may be c r i t i c i z e d on fundamental 0022-3093/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
102
O.J.
Kleppa / Thermodynamics of liquid alloys
grounds 4, his approach has proved very useful in providing estimates of the enthalpies o f mixing f o r a very wide range of a l l o y s .
Miedema and co-workers
have compared t h e i r predicted enthalpies with experimental data f o r a number of binary systems 5.
However, t h e i r work also has drawn a t t e n t i o n to the fact that
r e l i a b l e experimental data are very scarce for systems in which the a l l o y components are varied in a systematic manner, f o r example w i t h i n a single t r a n s i tion series. During the past few years we have been concerned with obtaining such data. In the present communication we report and discuss the results of some recent enthalpy of mixing measurements for two d i f f e r e n t classes of l i q u i d a l l o y systems. 2. THE SYSTEMS Cu-Ag, Ag-Au, AND Cu-Au
For these alloys a great deal of thermodynamic information from d i f f e r e n t sources already is available in the l i t e r a t u r e .
However, some of this informa-
tion is i n t e r n a l l y inconsistent, which suggested to us that improved thermochemical data should be obtained.
We present below new enthalpy of mixing data
taken from the recent studies of Kleppa and Watanabe6 (Cu-Ag) and of Topor and Kleppa 7 (Ag-Au, Cu-Au). 2.1. Cu-Ag The phase diagram f o r this system8 is of the simple eutectic type.
Hence we
a n t i c i p a t e p o s i t i v e enthalpies of mixing in the l i q u i d mixtures; due to the d i f . ference in atomic size we expect to find larger p o s i t i v e enthalpies for solutions of s i l v e r in copper than f o r copper in s i l v e r .
According to Hultgren's
most recent assessment g this is not the case. We show in Figure I a graph of the recently published enthalpies of mixing 6, the excess Gibbs energies, AGE, from Hultgren 9, and the calculated values of TASE at 1373 K.
I t is p a r t i c u l a r l y noteworthy that the new measurements show 5.0 4.0 3.0 7_ 2.0 LO 0
O ~ l02i l l l ~0.4i l l Ag
Xcu
0.6
0.8
1.0 Cu
FIGURE I Enthalpies, excess Gibbs energies, and excess entropies of mixing in the l i q u i d system Cu-Ag
O.J. Kleppa / Thermodynamics of liquid alloys
103
that the r a t i o of the two l i m i t i n g p a r t i a l enthalpies of solution (17.66/12.2 = 1.45) is e s s e n t i a l l y the same as the r a t i o of the molar volumes, 1.44.
This
indicates that the mixing enthalpies may be represented very well by a volume f r a c t i o n expression.
In this respect the results do not support the Miedema
model from which one predicts that the r a t i o of the enthalpies of solution should be , ~ ) 2 / 3 1.28. tVc u = We also see from Figure I that there is very l i t t l e difference between the enthalpies of mixing and the excess Gibbs energies of mixing, i . e . the excess entropies are very small.
Thus we find that in this system both the en_ergetic
asymmetrE and the entropies of mixing are well described by the usual regular 2 solution expression. 2.2. Ag-Au The phase diagram for this system8 shows a complete range of s o l i d solutions. Furthermore, the course of the solidus and liquidus curves indicates that AGE(s)
-AGE(L) should
have very small negative values.
Although there is no
ordered phase at low temperature, the considerable difference in electronegativity
and the very small difference in atomic size between s i l v e r and gold
makes us expect negative enthalpies of mixing. Figure 2 shows s t i l l
unpublished enthalpy of mixing data for this system7.
The results are compared with the e a r l i e r c a l o r i m e t r i c values given by Oriani and Murphy! O, and by Itagaki and Yazawall; the graph also shows the values recommended by Hultgren et al in 196312 , and in 19739 .
This f i g u r e i l l u s t r a t e s
that even f o r this very simple a l l o y system the published data leave a great deal to be desired.
Note that the enthalpy of mixing curve in this system is
nearly completely symmetrical.
This is consistent with the fact that the atomic
volumes of the two metals are very s i m i l a r . In Figure 3 we compare the new enthalpy data with the excess Gibbs energies; the curve drawn represents our own best estimate of AGE based on the available information 9'12.
For t h i s system we f i n d a somewhat larger difference between
enthalpy and excess Gibbs energy, i . e . small negative excess entropies. This correlates with the observed small negative enthalpies of mixing: deviations from random mixing due to some short range order is indicated. 2.3. Cu-Au The phase diagram8 shows a complete range of s o l i d solutions and the formation of ordered phases.
Since the o r d e r - d i s o r d e r transformations occur near
700 K, we i n f e r that the enthalpy i n t e r a c t i o n parameter for the s o l i d solutions should be of the order of -2RTc : -12 kJ mol - I ,
The phase diagram indicates
that the l i q u i d solutions should be mere exothermic than the s o l i d solutions. Figure 4 gives unpublished c a l o r i m e t r i c results for this system7, along with enthalpy values advanced by several e a r l i e r i n v e s t i g a t o r s . Among these workers
104
O.J. Kleppa / Thermodynamics of liquid alloys
9
+ F
,
Ser~es ,
c Ser~es 3
° Ser,es 2
x Ser,es 4
D ORIANI (i958) -
q
i
0 -~-LZ_ ~ t -I.O
-----[TAGAKI (I97I
--HULTGREN(1963) ---HULTGREN(1973)-
i
o
i
/
'
~
0 Ag
0.2
0.4
AHMfxl1379K)
02
2
0.6
0.8
. _I~
-20 T~ 3.0 -4.0 5.0
, T
0.4
XaU
0.6
0.8
Au
1.0 Au
XAu FIGURE 2 Enthalpy o f mixing and enthalpy i n t e r action parameter in the l i q u i d system Ag-Au at 1379 K. Experimental data from Ref. 7; I t a g a k i , Ref. 11; Oriani and Murphy, Ref. 10; Hultgren (1963), Ref. 12; Hultgren (1973), Ref. 9.
FIGURE 3 Enthalpies, excess Gibbs energies and excess entropies of mixing in the liquid system Ag-Au.
Itagaki and Yazawa13 carried out calorimetric measurements using an adiabatic high temperature calorimeter. The other values in Figure 4 were derived from the temperature dependenceof e.m.f, or vapor pressures (Oriani 14, Edwards and
Brodsky 15, Neckel and Wagner16, Hager, Howard and Jones17). In the upper half of Figure 4 is plotted the enthalpy interaction parameter for this system. By a least squares treatment of the data the following anal y t i c a l expression was derived AHmix/XAuXCu = -28.82-2.47 XAu + 9.54 XAu2 kJ mo1-1.
(3)
We shall return to the significance of this equation below. Since this system shows ordered intermetallic phases at low temperatures, we expect to find negative excess entropies of mixing in the liquid alloys.
How-
even, checking the literature we found that while Oriani 14 and Edwards and Brodsky15 report f a i r l y large negative excess entropies, Neckel and Wagner16 give small positive values.
Hultgren 9 similarly recommendspositive values.
The new enthalpy data allows us to consider this problem again. Surveying the e a r l i e r e.m.f, and equilibrium studies we concluded that the most reliable Gibbs energy data probably are those of Neckel and Wagner16. In
O.J. Kleppa I Thermodynamics o t liquid alloys
105
28~
-
<:]
'
24 i
+ Serles 1 Series 2 c Series 3
*
Series 4
\
0 ~ - ~
I~
E 1 2 I
".
21~2 i
AS =-X (I-X) ~ -
22,
I
J->"710
-'S" / I
........... , ,
.•c•
ITAGAKI(1971,c01}-- --EDWARDS , (1956,~p ) 16[_~ --ORIANI ~956emf) ..... NECKEL L (1~69 v p 1 ~-'"'-HAGER [1970,v p )
-~
]'
....: > /
I
~.
~
E(1449K, Neckel ¢ Wogner) / /
/ Cu
~H~,~Ii379KI 0.2
0.4
0.6
0.8
/ Au
XAu ir
0 (:u
~
I
0.2
,
I
,
0.4
I
0.6
,
I
0.8
XAu
°7 °`
1.0 Au
FIGURE 4 Enthalpy of mixing and enthalpy i n t e r action parameter in the l i q u i d system Cu-Au at 1379 K. Experimental data from Ref. 7; I t a g a k i , Ref. 13; Oriani, Ref. 14; Edwards, Ref. 15; Neckel, Ref. 16; Hager, Ref. 17.
FIGURE 5 Enthalpies, excess Gibbs energies and excess entropies of mixing in the l i q u i d system Cu-Au.
Figure 5 we accordingly compare t h e i r excess Gibbs energies with our own enthalpies.
As expected, we find small negative excess entropies, which suggest some
deviations from random mixing.
Also shown in Figure 5 is an excess entropy
curve calculated from quasi-chemical theory 18.
Although there is some uncer-
t a i n t y regarding what values to adopt for ~ and f o r z in the quasi-chemical expression f o r the excess entropy (see (6) below), there seems to be plausible agreement between the experimental and calculated values. The presence of short range order is probable also because of the f a i r l y large negative enthalpies of mixing.
In this context the c h a r a c t e r i s t i c para-
b o l i c dependence of the enthalpy i n t e r a c t i o n parameter on a l l o y composition probably is s i g n i f i c a n t . In considering this problem we again r e f e r to quasi-chemical theory, from which we obtain the following approximate expression for the molar enthalpy of mixing in a nearly random s o l u t i o n : AHmix ~ X(I-X)X[I-X(1-X)z-~T ].
(4)
O.J. Kleppa / Thermodynamics of liquid alloys
106
Here z + . X = N~[2Vl2-(Vll v22) ] ,
(5)
N is Avogadro's number, v12 is the pair i n t e r a c t i o n energy between atoms I and 2, and z is the nearest neighbor coordination number.
From t h i s theory we also
obtain the expression for the excess entropy of mixing shown in Figure 5: ~2 ASE m _X2(I_X) 2 zRT2 '
(6)
The quasi-chemical theory is based on the s i m p l i f y i n g assumption that the pair i n t e r a c t i o n energies do not vary with a l l o y composition; therefore ~, the enthalpy i n t e r a c t i o n parameter, should be a constant f o r a given binary system. For the copper-gold system this assumption does not hold.
However, i f our
Eq. (3) is r e w r i t t e n to be consistent with (4), we find AHmix = XAu(I-XAu)[a+bXAu+CXAu(I-XAu)].
(7) i
From t h i s we see that we may obtain an average value of X by s e t t i n g X = a+~ (b is the difference between the two l i m i t i n g enthalpies of s o l u t i o n ) . l a r l y , we get
Simi-
2X2 c
:
-
Tiff
"
(8)
In t h i s analysis there remains only a single parameter which is not fixed by the theory, namely the nearest neighbor coordination number, z.
For close-
packed metals, such as pure copper and pure gold, at temperatures j u s t above t h e i r respective melting points, we would expect a value of z s l i g h t l y less than 12.
A comparable value should apply in copper-gold a l l o y s .
From t h e i r
experimental enthalpy data at 1379 K Topor and Kleppa 7 calculated z z 11.7, in remarkably close agreement with these expectations. In Table I we f i n a l l y compare the experimental values of the p a r t i a l enthalpy of solution 6'7 with values predicted by Miedema et al 1'19.
Note that Miedema's
recently revised values 19 of ~* and nws have not improved agreement with experiment in these systems. TABLE I .
Limiting p a r t i a l enthalpies of solution f o r l i q u i d alloys among the noble metals. AH-Solute, kJ mol - I
Solvent-Solute Cu Ag Ag Au Cu Au
Ag Cu Au Ag Au Cu
Experiment +17.66 +12.2 -15.82 -16.35 -28.82 -21.75
Miedema I (1977) + 5 + 4 -20 -18 -29 -20
Miedema 19 (1983) +I0 + 8 -21 -23 -42 -30
O.J. Kleppa / Thermodynamics of liquid alloys
107
3. SOLUTIONS OF THE FIRST ROW TRANSITION METALS IN LIQUID COPPER We r e f e r here f i r s t
to the published c a l o r i m e t r i c data f o r Mn20, for Fe, Co,
and Ni 21, and for Ti 22'23.
A report on Sc, V, and Cr in copper very recently
was submitted f o r publication 24.
The relevant phase diagrams8'25 show that
among these a l l o y systems there are two with extensive s o l i d solutions (Cu-Ni, Cu-Mn), two with quite stable i n t e r m e t a l l i c compounds (Cu-Sc, Cu-Ti), and four eutectic diagrams with l i m i t e d s o l u b i l i t y of the t r a n s i t i o n metal in l i q u i d copper near 1373 K (Cu-Co, ~4 at% Co; Cu -Fe, ~4 at% Fe; Cu-Cr, ~1.5 at% Cr; Cuu-V, ~uO.4 at% V).
Since our experimental c a p a b i l i t y was l i m i t e d to this temp-
erature, most of our c a l o r i m e t r i c data r e f e r to d i l u t e solutions of the t r a n s i tion metal in copper. Figure 6 is a graph of the p a r t i a l enthalpies of solution of the considered t r a n s i t i o n metals in l i q u i d copper near 1373 K. cooled l i q u i d metals.)
(The data r e f e r to the under-
The graph also shows Miedema's o r i g i n a l predictions I ,
as well as his recently revised values 19.
The figure confirms that Miedema's
semi-empirical theory c o r r e c t l y predicts the overall trend of the enthalpies of s o l u t i o n , i . e . p o s i t i v e values of the r i g h t order of magnitude f o r Ni, Co, and Fe, a d i s t i n c t minimum f o r Mn, p o s i t i v e values for Cr and V, and increasing I~'0
[
i
i
*20 / tI
-2 , !
d
f 2 50 ! <:2
-,oo~ .
// I z/ ' /
/
i 7 SC TI
- - Mhedema{L977) x M~edemo{1982~ -'>- Satoand KLeppo
1,3a;;
. . . .
il /
II II
,,
'
-5
1
--.--p,e~oo,wo,, j V Or Mn Fe Co Ni Ou
FIGURE 6 Comparison of experimental p a r t i a l enthalpies of solution of f i r s t row l i q u i d t r a n s i t i o n metals in copper at 1373 K with values predicted by Miedema et a l l , 1 9 .
FIGURE 7 Excess entropies of solution of f i r s t row l i q u i d t r a n s i t i o n metals in copper at about 1400 K.
O.J. Kleppa / Thermodynamics of liquid alloys
108
negative values from Ti to Sc.
Only f o r Mn, for which Sato and Kleppa20 found
a small negative enthalpy of s o l u t i o n , does Miedema obtain the wrong sign. While Miedema predicts the trend of the data very w e l l , we find that most of his values are more exothermic (or less endothermic) than the experimental res u l t s ; this difference is quite large (~30 kJ mo1-1) f o r Ti and V, but much smaller (~i0-15 kJ mol - I ) for Sc, Cr, and Co.
For Mn and Ni the s i t u a t i o n is
reversed and Miedema's values are more endothermic than the epxeriments, while f o r Fe the agreement between prediction and experiment is remarkably close.
On
the whole we find that Miedema's revised values Ig are in considerably b e t t e r agreement with experiment than his e a r l i e r values I. For many of the f i r s t
row solutes in l i q u i d copper excess Gibbs energies of
solution e i t h e r are a v a i l a b l e from e q u i l i b r i u m or e.m.f, studies ( T i , Mn, Ni) 26'27'9, or may be calculated or estimated from the published phase diagrams (V, Cr, Fe, Co) 8'25.
When these Gibbs energies near 1400 K are compared with
the new enthalpies of s o l u t i o n , we gain i n t e r e s t i n g new i n s i g h t regarding the excess entropies.
The relevant data are p l o t t e d in Figure 7.
I t w i l l be seen
that the p a r t i a l excess entropies for l i q u i d nickel and manganese in copper are quite small and negative.
These two metals have the numerically smallest en-
thalpies of solution and both have a wide range of solid s o l u b i l i t y in copper. Among the remaining solutes Co and Fe have p o s i t i v e excess entropies of about 5-10 J K-1 mol - I , while both Cr and V have p o s i t i v e values which are of the order of 20 J K-1 mol - I or more. probably small and p o s i t i v e .
The value f o r Ti is somewhat uncertain, but is There are no data for Sc.
For a beginning we want to take note o f the fact that there is extensive s i m i l a r i t y between Figures 6 and 7; i . e . i t is evident that the excess entropies and enthalpies vary in a roughly s i m i l a r manner when the solute is varied systematically from Ti to Ni.
Since the atomic volumes of the considered metals
f a l l w i t h i n a r e l a t i v e l y narrow range (Vr. = 7.1 cm3; VNi = 6.6 cm3; VCo = 6.6 3 ~u cm3; VFe = 7.1 cm~; VMn = 7.4 cm ; VCr = 7.3 cm3; VV = 8.4 cm3; VTi = 10.6 cm3), i t seems l i k e l y that atomic size plays only a minor role in influencing the enthalpies and entropies of s o l u t i o n .
For t h i s reason we believe that the ob-
served excess entropies probably are predominantly of e l e c t r o n i c and/or magnetic rather than o f v i b r a t i o n a l o r i g i n .
This suggests that any r e a l i s t i c microscopic
theory o f the thermodynamic properties must take into account also the thermal e x c i t a t i o n of the d-electrons which provide the major part of bonding and cohesion in these systems. Let us f i r s t
consider the well-known fact that among the solute elements in-
cluded in Figure 7 the pure metals Cr and Mn show antiferromagnetic ordering at low temperatures, while Fe, Co, and Ni are ferromagnetic. At s u f f i c i e n t l y e l e vated temperatures this interatomic order is destroyed, which gives rise to a
O.J. Kleppa / Thermodynamics of'liquid alloys disordering entropy.
109
With the possible exception of c o b a l t , f o r which the Curie
temperature is 1394 K, the disordering process should be complete at 1400 K. For d i l u t e solutions of these metals in copper near 1400 K t h i s magnetic disorder undoubtedly is also complete.
Hence we conclude t h a t this magnetic e f f e c t
should make no s i g n i f i c a n t c o n t r i b u t i o n to the observed excess entropies. In p r i n c i p l e i t should be possible to r e l a t e a major part of the excess ent r o p i e s in Figure 7 to the d i f f e r e n c e between the e l e c t r o n i c heat capacities yT in the a l l o y and in the pure metal, i . e .
to the d i f f e r e n c e between YMe
~XMe
in the a l l o y and YMe in the pure metal. To a zeroth order of approximation this e l e c t r o n i c entropy term should be of the order to (YMe - YMe)T"
For example, f o r d i l u t e solutions of nickel in cop-
per we compare ¥Ni from the low temperature heat capacity data of Guthrie et a128, 3.1 mJ K-2 mol - I , with the value for pure n i c k e l , YNi = 7.0 mJ K-2 mol-I 29.
This comparison y i e l d s an estimate of the e l e c t r o n i c excess entropy
of nickel in copper at 1400 K o f about -5.5 d K-1 mo1-1, which is in p l a u s i b l e agreement with the experimental value of -2.5 J K- I mel - I . For the o t h e r s o l u t e metals, however, there are complications with this simple scheme.
These complications arise from the f a c t t h a t some of these
solute metals, when dissolved in copper, at low temperature have l o c a l i z e d , quasi-bound magnetic impurity states which have a dramatic e f f e c t on many of the e l e c t r o n i c properties o f the a l l o y s 30. ent study i t
From the point of view o f the pres-
is p a r t i c u l a r l y i n t e r e s t i n g t h a t the thermal d e s t r u c t i o n of these
quasi-bound s t a t e s , which for Cr, Mn, and Fe in Cu l a r g e l y occurs below 10 K, is known to give r i s e to large excess s p e c i f i c heats and entropies 30. is d i f f i c u l t
Although i t
e x p e r i m e n t a l l y to p r e c i s e l y determine the t o t a l entropies of dis-
order, we shall f o r s i m p l i c i t y assume that they may be estimated from the expression Rln (2S+1); f o r Cr and Mn, S is 5/2 (corresponding to the f i v e unpaired spins in the free atom), while for Fe i t is 4/2 = 2.
This " i n t r a - a t o m i c " mag-
netic e f f e c t , which is the r e s u l t of i n t e r a c t i o n between the d-electrons of the solute atoms and the conduction electrons o f the host metal, thus should give r i s e to a p o s i t i v e magnetic entropy term which is about 14.9 J K- I mo1-1 f o r Cr and Mn dissolved in copper and about 13.4 J K- I mol - I f o r Fe. I f t h i s p i c t u r e is c o r r e c t , the experimentally observed excess entropies f o r Cr, Mn, and Fe in copper should in large measure r e f l e c t the d i f f e r e n c e between these magnetic entropy terms, on the one hand, and the e l e c t r o n i c heat capacity terms in the pure metals, y. T, on the o t h e r . For Cr, f o r which yT at 1400 K me r q u i t e low (2.2 J K-1 nlol-1), we thus estimate S~Cr at t h i s temperature to be +12.7 J K- I mol - I .
S i m i l a r l y , f o r Mn, f o r which yT at 1400 K is very high
(17.5 J K- I m o l - l ) , we f i n d ~ n
: 2.7 J K- I n ~ l - l ;
f o r Fe, with yT = 7 J K- I
110
O.Z K l e p p a / T h e r m o d y n a m i ~ o f 6 q u i d a l l o y s
rfo ] - I
we get S~e = 6.4 d K- I mo1-1.
Although any close numerical agreement be-
between these values and the experimental data undoubtedly is f o r t u i t o u s , we see that our estimates c o r r e c t l y reproduce the trend of the data. I t is o f i n t e r e s t to take note here also of the recent experimental and t h e o r e t i c a l study by Gachon et a131 of the Gibbs energies, enthalpies and excess entropies of solution of s o l i d solutes from t i t a n i u m to germanium in s o l i d copper.
I t is not s u r p r i s i n g that some of the conclusions reached by these
authors are s i m i l a r to our own.
They conclude, f o r example, that f o r Fe, Mn,
and Cr as solutes in s o l i d copper i t is necessary to take into account the magnetic decoupling of the impurity states in order to understand the observed excess entropies o f s o l u t i o n . ACKNOWLEDGMENTS This work was supported by the National Science Foundation under Grant CHE8106980, and by the Materials Research Laboratory Program of the NSF at the University o f Chicago under Grant DMR-7924007. REFERENCES I) See, e . g . A . R . Miedema, F. R. de Boer, R. Boom, and J. W. F. D o r l e i j n , Calphad i (1977) 353; A. R. Miedema, P. F. de ChStel, and F. R. de Boer, Physica IOOB (1980) I. 2) J. H. Hildebrand and R. L. Scott, The S o l u b i l i t y of Non-Electrolytes (Reinhold, New York, 1950). 3) B. W. Mott, Phil. Mag. 2 (1957) 259. 4) A. R. Williams, C. D. G e l l a t , J r . , and V. L. Moruzzi, Phys. Rev. Lett. 44 (1979) 429. 5) See, e . g . F . R , de Boer, R. Boom, and A. R. Miedema, Physica I01B (1980) 294; 113B (1981) 18. 6) O. J. Kleppa and S. Watanabe, Met. Trans. 13B (1982) 391. 7) L. Topor and O. J. Kleppa, Met. Trans. B (submitted). 8) M. Hansen, Constitution of Binary Alloys, 2nd ed. (McGraw-Hill, New York, 1958). 9) R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, and K. K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys (American Society for Metals, Metals Park, OH, 1973). i0) R. A. Oriani and W. K. Murphy, J. Phys. Chem. 62 (1958) 199. 11) K. Itagaki and A. Yazawa, J. Met. Soc. Japan 35 (1971) 389. 12) R. Hultgren, R. L. Orr, P. D. Anderson, and K. K. Kelley, Selected Values of Thermadynamic Properties o f Metals and Alloys (Wiley, 1963).
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