140
Ind. Eng. Chem. Fundam. 1985, 2 4 , 140-147
Kennard, M. L. Ph.D. Thesis, Unlversity of British Coiumbla, Vancouver, BC,
1983.
Kennard, M. L.; Meisen. A. J . Chromtcgr. W83, 267,373. Kennard, M. L.; W e n , A. l-@drocarbon Process. 1980, 59(4), 103. Kennard, M. L. Melsen, A. J . Chem. Eng. Data 1984, 29(3),309. Nonhebel, G. “Gas Pwlficatlon Processes for Air Pollution Control”, 2nd ed.; Newness-Butterworths: London, 1972. Polderman, L. D.; Steele, A. B. Oil Gas J . 1958, 54(65), 206.
Scheirman, W. L. Hydrocarbon Process. 1073, 53(7),95. Smlth, R. F.;Younger, A. H. Roc. Gas Cod. Conf. 1972, 22(E),1. Smlth, R. F.; Younger, A. H. Hydrocarbon Process. 1972, 57(7),98. Younger, A. H. Roc. Gas Cond. Conf. 1973, 23(E),1.
Received for review September 13, 1984 Accepted July 9, 1984
Nonideal Behavior in Liquid Metal Solutions. 1 Physical Theory Model Thomas Stolcos and Charles A. Eckert’ Department of Chemical Engineering, University of Illinois, Urbana, Iilinois 6 180 7
A “physical” model of liquid metal solutions, based on the concept of electron transfer, has been used to calculate mixture propertles for systems with positive or small negative deviations from Raouit’s law. I t gives very good quantitative results for excess Oibbs energies and excess enthalpies, and qualitatively correct resuits for excess volumes. I t has been used successfully to predict solid-liquid and liquid-liquid equilibria, and the extension to multicomponent systems is discussed. The physical significance of the charge transfer parameter is described.
Introduction The thermodynamics of liquid metal mixtures is rapidly becoming more important as an increasing number of metallurgical processes depend heavily on solution chemistry. For example, in some cases aluminum metal is used to deoxidize steel and often pig iron is desulfurized with caustics; in both cases the process depends on activities in liquid solutions. Some processes depend on very large positive deviations from Raoult’s law, to give separations by liquid-liquid equilibria. These include the lead-copper system, which exhibits a relatively small miscibility gap, and the lead-zinc system, where the gap is substantially larger. There are also a number of processes which depend on large negative deviations from ideality, such as the carbothermic and carbonitrothermic solvent metal reductions of reactive metal oxides (Anderson and Parlee, 1974, 1976) or the liquid tin repurification of spent uranium fuel elements (Anderson and Parlee, 1971). Liquid metal solutions tend to have relatively larger nonidealities than the mixtures of organic chemicals with which chemical engineers are familiar. Despite the higher temperatures, the interatomic forces are a great deal stronger for metals, and few mixtures exhibit anything like ideality. Mixtures can show such varied and complex behavior that there exists no physical explanation which is extensive enough for the general description of all the actual deviations from ideality. The utility of different expressions based on semiempirical interpretations such as the Wohl expansion, the Scatchard formula, the van Laar equation, the Wilson equation, and the NRTL equation are discussed by Tomiska (1980) and Eckert et al. (1982) and were shown to be generally not applicable for regression of experimental data. Depending on the nature and the degree of the nonideality, the metallic solutions can be classified into two main categories. First, there are systems which show large or small positive or small negative deviations from Raoult’s law, resulting from relatively weaker, nonspecific interactions between the components, which are characterized as physical forces. This paper deals with the physical model for such systems. Second, there are solutions which 01 96-431 3/85/1024-0 140$01.50/0
show pronounced negative deviations from ideality and very abrupt changes of transport properties such as viscosity and density at specific compositions. These deviations are attributed to strong specific forces which lead to the formation of new molecular species; such forces are called chemical forces for which a more extensive discussion follows in part 2 (Stoicos and Eckert, 1985). Attempts to calculate the excess thermodynamic properties of simple liquid alloys have been made from firstprinciple methods. Christman (1967),Tamaki and Shiota (1968),Umar et al. (1974), and Hafner (1976) have all used some form of the pseudopotential approach, coupled with an energy minimization procedure for calculating the necessary parameters. These calculations have been applied to a few systems only, mainly liquid alkali alloys, and have not been extended to a wide category of metallic solutions. Paulaitis and Eckert (1981) proposed a physical model which combined corresponding-states theory with a perturbed hard-sphere representation for predicting mixture properties. Another model based on an electron theory which includes both structure-dependent and structureindependent contributions has been developed by Cox (1979). Both of these models contain one parameter which was obtained from mixture data. The models have been successfully used for describing the thermodynamic behavior of many binaries and have some generality. The physical model developed and presented here is based on an electron theory coupled with an empty-core pseudopotential form. As in the two previously described approaches,it contains one model parameter. Positive and small negative deviations from Raoult’s law are attributed to a certain amount of charge transferred between dissimilar metals upon mixing. The important elements in the proposed model are that all the equations have a theoretical significance, and the charge-transfer parameter is based on a phenomenon of plausible physical importance. By and large, experimental data on liquid metal solutions are very difficult to take, because of the high temperature, or they are fragmentary, because of freezing or 0 1985 American Chemlcal Society
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
solid intermetallic compound formation. In fact, the common techniques (such as EMF) generally measure activity of only one species in the mixture. Thus a major application for any theory is not only predicting mixture properties from pure-component properties but also the extension of what often are very limited data to wider ranges of temperature and composition. This will be discussed further in part 2. Pure Liquid Metals The theory developed here treats a liquid metal system as a collection of positively charged ions immersed in an electron gas at a density of 2 electrons per atom, where 2 is the conventional valence of the metal. Considerable progress has been made with this type of model in interpreting and evaluating successfully the internal energy of a pure liquid metal. The internal energy of such a system can be expressed as the sum of the kinetic energy of the conduction electrons, the potential energy due to the electron-ion and electron-electron interactions, the potential energy from the interatomic interactions, and the contribution of the kinetic energy of the ions. The conduction electrons in metals form a nearly free gas and are responsible for most physical and chemical metallic properties, indicating that the interactions of these electrons with the ions have only a small influence on the electronic energies. This is attributed to the fact that, in addition to the strong Coulombic attraction between the conduction electrons and the ions, there are other contributions giving rise to repulsive potentials, and the net effective interaction turns out to be very small. This net effective interaction is called the pseudopotential. If the electrons were not interacting and were moving freely, their kinetic energy would be the only contribution to the total energy. If the electron-electron interaction is taken into account, then the energy associated with the electron gas is given by (Nozigres and Pines, 1958; Heine and Weaire, 1970) uel = 2.212 0.9162 ---Z(0.115 - 0.031 In R,) ryd/atom (1) RE2 RE where R, is the radius of a sphere which on the average contains one electron and is related to the ionic density through
Equation 1 includes the average electronic kinetic energy and the exchange and correlation contributions. The exchange effect per atom is given by -0.9192/R, and arises from the fact that when two electrons exchange positions they are indistinguishable. The third term is the correlation energy and represents the long-range and shortrange coulomb interactions between the valence electrons. The next contribution to the cohesive (internal) energy of the metal is the electron-ion interaction. In order to use the nearly-free electron picture in which the electron gas is considered a nearly separate fluid, one must seek small net potentials. This problem has been solved by introducing a model potential for the ion, through which the conduction electrons interact weakly with the parent ion. In the development of the present theory, we select an empty-core model potential, similar to the one proposed by Ashcroft (1966, 1968) which has the form W(r)= 0 when r < R, (3a) W(r)= - Z / r when r > Rc (3b)
141
1
Figure 1. Potential energy diagram for the electron-ion interaction.
The ionic radius of the metal is chosen as the model radius for the model potential. This assumption is in accord with the cancellation theorem (Heine, 1970). The upper section of Figure 1 represents the potential of the electron gas, and the empty core model potential is shown in the lower part. When these contributions are added, they cancel each other outside of the metallic sphere of radius R,. The average value of the remaining terms inside the atomic volume yields the effective total potential seen by the electrons within a metal atom, the pseudopotential, and is given by -0.62513 32R: Uel-ion = ryd/atom (4) RE R,3 The potential energy resulting from the interaction between the metal atoms is the sum of the direct Coulomb repulsion and an indirect interaction associated with the electron gas uniformly distributed between the ions. These two terms have opposite effects, and in the formulation of the present model it is assumed that the electron gas screens out the ions completely. This is considered a plausible, fully justified assumption (Heine and Weaire, 1970; Shimoji, 19771, and it greatly simplifies the calculation of the internal energy of the system. Substituting for RE = [3/4?rpZ)]1/3 and changing to the more conventional units of MJ/mol, the expression giving the internal energy of the system is u = 1 . 5 0 5 ~ ~ /-~02. ~8 /6 ~5 ~ ' / ~ 2 ~0.1372 / ~ - 0.01352 In 3 ( p Z ) - 1.561~'/~2? 1.468pR:P + 2 RT (MJ/mol)
+-
+
(5) Experimental values of u can be obtained by adding to the energy of vaporization the ionization potential for the valence electrons and changing the sign u = -u,,p
- Uion
(6)
Thus it is possible to compare the calculated internal energy with the corresponding experimental value and test the validity of the model chosen. Figure 2 shows such a comparison,which indicates that the theoretical equations give good results. The agreement is less successful for liquid Bi and Sb, which both have five valence electrons and hence extremely large ionization energies. The discrepancy may be attributed to the fact that these two elements are less metallic in nature than the other metals considered. The entropy of the liquid metal is described as that of a hard-sphere system which has the form = sgas + Spack (7)
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!O Internal Energy (calculated), MJImole
Entropy (calculated). J/mle-OK
Figure 2. Experimental and calculated internal energies of liquid metals at the melting point.
Figure 3. Experimental and calculated entropies of liquid metals at the melting point.
where sgas is the expression for the ideal monatomic gas given by
Since the atoms are statistically distributed, the change of the molar volume of A on mixing is taken to be proportional to the mole fraction xB. It is also directly related to the magnitude of the charge transfer A2 and the ratio of the molar volumes of the pure elements uBo/uAo. This ratio is introduced into the expression in order to emphasize the effect of the difference of the molar volumes of the two constituents. When charge flows from B to A and uB0 > uA0, there will be a greater tendency for the volumes to change than if uB0 < uAo. A similar expression exists for the second component
sgas= -R 5 2
+ R In
[ -(
1 -m )k IB2 T]
P
2Th2
(8)
The second term, Spa&, arises from the finite sizes of the atoms and depends only on the packing density, 5, a quantity which denotes the fraction of the total volume occupied by these atoms. A convenient form of Spa&, consistent with the Percus-Yevick solutions for the hard-sphere system, was proposed by Carnahan and Starling (1969)
At the melting point, the packing density is chosen equal to 0.46 for all metals, as suggested by Faber (1972). Theoretical entropy values calculated from this formulation are compared with experimental results obtained from the tables of Hultgren et al. (1973). Even though the contribution of the valence electrons to the entropy of the system was neglected, Figure 3 indicates that good quantitative agreement is attained when the hard-sphere model is employed. Liquid Metal Mixtures In solutions we seek to express the thermodynamic excess functions, which denote behavior differences relative to the pure species. The model presented above is readily extended to mixtures by assuming that some electron transfer occurs between unlike atoms. Since the nonspecific interactions are generally relatively weaker than "chemical forces" (part 2), we neglect the order-disorder problem and assume random mixing. Then the valence of metal A in the mixture is 2, = ZA+ X g A z (10) where 2, is the valence of the pure metal A, xE is the mole fraction of metal B, and A 2 is the amount of charge transferred to each metal atom A. The charge transfer between the two different metals will have, as a result, not only the change in the valence of the constituent elements but their partial molar volumes as well. If electrons move from metal B to metal A it is reasonable to expect that the size of A may increase while that of B will decrease. Thus, the partial molar volume of A can be written as
..
0
Combining eq 11 and eq 12, we obtain an expression for the excess volume of mixing for the A-B alloy which is given by
It is evident that U E has a parabolic dependence on composition and vanishes when the two molar volumes are identical. During the mixing process, the charge transfer originates from the conduction electrons. Hence, it is reasonable to assume that the ionic core radii remain invariant. In view of the charge transferred, and knowing how the valence and molar volume of the components change upon mixing, it is possible to evaluate the excess molar internal energy of mixing i
i
0.865[(piZi)4/3/pi- ( p i Z ) 4 / 3 / p i ] - 0.137[Zi - Zi] 0.0135[ZiZn ( p i Z i ) - ZiIn (piZi)]1.561[pi1/3Z: - pi1/3Z?]+ 1.468R;[piZF - p i Z F ] ) (14) The entropy of the metallic mixture is =
sgm
+ Spa& + SId
(15)
Since there is a small volume change involved upon mixing, we carry out a Taylor expansion of the excess molar entropy with respect to volume
If we assume that the packing density of each component does not change with composition, the excess molar entropy at constant volume is sE = s - xAsAO
-
xBsBO
- sid
(17)
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
Invoking the appropriate Maxwell relation, the derivative (ds/dv)p is given by
The quantity represented by dP/dT)u is the thermal pressure coefficient. Using the Carnahan-Starling equation and assuming that 5 = 0.46 throughout the composition range
143
Table I. Comparison of the Charge-Transfer Parameter Determined from the Eutectic Point Method and Activity Coefficient Data Fitting alloy (A-B) AZ(ACDF)" AZ(EPM)* 0.022 Sn-A1 0.023 0.0040 Sn-Cd 0.0025 0.0050 Ga-Zn 0.0039 0.026 0.024 Cu-Ag 0.012 Zn-Cd 0.013 0.091 In-Zn 0.084 0.015 TI-Cd 0.013 0.0071 Pb-Cd 0.0067 Sn-Zn 0.0086 0.0084 Activity coefficient data fitting. Eutectic point method. 3.5-
Hultgren etul. (1973) Activity coefficient dato
0 \
Then, the excess molar entropy is equal to
Next, the excess molar Gibbs energy is gE = UE - TsE
(22)
c
a
and the activity coefficient of component i is given by yi = g i e / R T
(23)
1.0 -
0.2
Oa50
0.4
0.6
0,8
1.0
Mole Froction of Zn
with
Figure 4. Activity coefficient curves for the zinc-cadmium system a t 800 K. Hultgren elol. 119731 Acti& Coefficient Data Fitting Eutectic Point Method
0
3*5K--
The temperature dependence of the internal energy and entropy is introduced by considering the variation of the density with respect to temperature
\-
1
P(T) = P(Tm) (1- P AT)
(25) where T, is the melting temperature, AT = T - T,, and j3 is the volume thermal expansion coefficient. Applications The equations describing the excess Gibbs energy depend on pure component properties except for the charge transfer parameter AZ, which is characteristic of the particular alloy and assumed independent of temperature. This parameter must be obtained for each binary from some mixture data. Clearly if gE data exist at one temperature, even if they are fragmentary, these may be used to evaluate A2 by the usual fitting techniques (Prausnitz et al., 1980). Fortunately for liquid metal mixtures, there exists virtually always a phase diagram showing a eutectic point, and this is so-to-speak a "free" datum which almost always is available and can easily be used to evaluate a mixture parameter (Paulaitis and Eckert, 1981). This eutectic point method has been tested on a number of metallic systems for which the liquids forms a simple eutectic. The values of the charge transfer parameter thus obtained are compared in Table I with the corresponding parameter values obtained by fitting the equations to existing experimental activity coefficients. For the majority of the systems, this comparison shows remarkable agreement. Figures 4 and 5 compare the results of the two different methods for determining AZ with data for the systems Zn-Cd (Hultgren et al., 1973) and Cd-Pb (Hultgren et al., 1973); the differences ate minute. The significant advantage of the eutectic point method is that it gives unequivocal results and does not involve any adjustable parameter.
1
b YCd
u
0850
02
0.4
0.6
0,s
I,O
Mole Froction of Cd
Figure 5. Activity coefficient curves for the cadmium-lead system at 773 K.
The application of the phpical model to binary solutions over a range of composition and temperature shows excellent results. For the system Cu-Pb studied in this work (see part 2 for experimentalmethod), the results are shown, along with data reported by Hultgren et al. (1973) in Figure 6. The charge-transfer parameter AZ was fit to activity coefficients a t one temperature, and it is clear that this parameter is temperature-independent. The eutectic methods give identical results. In the A1-Sn system, EMF data exist for yA1at 973 K (Hultgren et al., 1973) and data for +ySn were reported at a much higher temperature by Cox (1979) using an effusion technique. A comparison with the physical model is shown in Figure 7. To show the versatility of this model, results for the Pb-Bi system (Niwa et al., 1961) which exhibits negative deviations from Raoult's law, are shown in Figure 8; again agreement is very good. A strictly regular solution expression of the form A log y L = -(1 - x , ) 2 (26)
RT
with A = a
+ bT can be used to represent systems like
144
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
-
u'dh
0
' 0
6.5
war:.
This Work, 1276'K :his 1373OK Hultgren dd(19731.
I
-Physicol Theory Model
ac8 Y
c
fU
-
4,5
a
2,5
21
I
0.5;
0.2
I 0.4
I
0,6
I
0#8
J 1.0
2
8
6
4
Excess Gibbs Energy (colculoled), kJ/rnole
Figure 6. Activity coefficient data of lead in the copper-lead system at three temperatures and the corresponding curves calculated with the physical theory model. 4.
0
: 4 - 2
Mole Froction of Pb
Figure 9. Comparison of experimental excess gibbs energies for equimolar liquid alloys with the corresponding values calculated with the pphysical theory model.
Hultgren efUL 11973),
0
_r Pb-Cu
10
Excess Entholpy (colculoledl, kJ/rnole
Figure 10. Comparison of experimental excess enthalpies for equimolar liquid alloys with the corresponding values calculated with the physical theory model. 0
Niwo eta/. (1961)
- Physic01 Theory
400
Model
350 Y
0
300 250
200
'
0*20
0,2
0.4
06
0,8
1.0
150
u
'O0O
Figure 8. Physical theory model fit of lead activity coefficient data in the lead-bismuth system at 700 K.
Pb-Bi with symmetrical activity coefficients. The advantage of our approach is that the present model is not as restrictive and can account for the behavior of binary alloys with unsymmetrical excess functions (see Figure 7). Furthermore, it contains only one parameter, AZ,with physical significance, while eq 26 has two arbitrary constants, a and b. Data for gE for many systems at equimolar composition are compared with experimental results in Figure 9, and agreement is generally very good. In Figure 10, the predicted equimolar excess enthalpy is compared with available data, and the agreement is surprisingly good. Often solution theories predict excess Gibbs energies far better than the excess entropy and enthalpy. Phase Diagrams The good temperature dependence of the physical theory model renders it applicable to the calculation of phase
0.2
0.4
0.6
0,8
1.0
Mole Froction of No
Mole Fraction of Pb
Figure 11. Equilibrium phase diagram of the sodium-potassium system.
,
2000
0.2
0.4
0,6
0,s
IO
Mole Fraction of AI
Figure 12. Equilibrium phase d i a g r d of the aluminum-tin system.
diagrams. The calculational method is straightforward and has been described recently by Eckert et al. (1982), as the
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
145
Table 11. Physical Theory Model Parameters for Selected Liquid Alloys One Liquid Phase
1200
Ll2-A-l
8OO0
0,2
0,4
0.6
0.8
LO
Mole Fraction of AI
Figure 13. Liquid-liquid coexistence curve for the aluminum-indium system calculated with the physical theory model.
1
o Predel m i Sandip'(1969)
u
IIoo0
02
0.4
06
0.8
I,O
Mole Fraction of TI
Figure. 14. Liquid-liquid coexistence curve for the thallium-copper system calculated with the physical theory model.
activity coefficient model is used with an iterative algorithm to determine the solid-liquid curve. Resulta for the Na-K system are shown in Figure 11and resulta for the Al-Sn system are shown in Figure 12. The excellent agreement is not surprisingly since the calculation of the solid-liquid curve is relatively insensitive to small errors in the Gibbs energy model. On the other hand, liquid-liquid equilibria are very sensitive to the Gibbs energy model and thus provide a stringent test of the theory. Systems showing very strong positive deviations from Raoult's law exhibit a miscibility gap, and the solution theory forms the basis for an iterative calculation of the phase equilibria (similar to that of Prausnitz et al., 1980). Such calculations were made for two typical systems (see Figures 13 and 14) and the agreement is very good. Certainly some uncertainty in the vicinity of an upper critical solution point can be attributed to the random mixing approximation used in deriving the theory. Near the consolute point the order-disorder effect is certainly not negligible (Eckert, 1968) and could have some effect on the calculation. Although no rigorous solution is now known, the quasi-chemical method is a reasonable approximation and would result in a lower critical temperature, which in both cases would be a correction in the right direction.
Multicomponent Mixtures The physical theory presented here is readily extended to multicomponent mixtures if pairwise additivity is assumed. As is so often the case in solution statistical mechanics, the three-body problem is at this stage unsolvable, and only the assumption of pairwise additivity renders the problem tractable. Such an assumption is exact for second virial coefficients, but it introduces some errors in third virial coefficients (Sherwood et al., 1966). Fortunately, there seems to be some cancellation of errors in liquid
alloy (A-B) Pb-Cu Na-K Sn-A1 Sn-Cd A1-Ga Ga-Zn Bi-Sn Pb-Cd Zn-Cd Sn-T1 Al-Zn In-Cd Pb-In Sn-Zn Ga-Cd In-TI TI-Cd In-Zn TI-CU AI-In Sn-Pb Au-Ag Pb-Bib In-Bib T1-Sbb Cu-Znb Fe-Alb
valences 4 1 1 1 4 3 4 2 3 3 3 2 5 4 4 2 2 2 4 3 3 2 3 2 4 3 4 2 3 2 3 3 3 2 3 2 3 1 3 3 4 4 1 1 4 5 3 5 3 5 1 2 2 3
CT, U , electron fraction 0.027 0.041 0.023 0.0025 0.0018 0.0039 0.0025 0.0067 0.013 0.0044 0.0067 0.0056 0.0046 0.0084 0.0074 0.014 0.013 0.084 0.055 0.065 0.060 0.075 0.0065 0.0050 0.0064 0.016 0.10
(uE/u0) X 100" 0.82 (-11.8) -0.32 (-1.2) 0.29 0.012 (1.8) 0.
0.022 0.010 (1.1) 0.052 (0.8) -0.15 (0.6) -0.0050 0.026 (0.0) 0.025 (0.7) 0.015 0.14 (1.5) -0.048 -0.024 0.084 (1.0) 1.4 (0.5) 1.5 -0.80
-0.16 (0.1) -0.034 -0.0080 (0.3) -0.021 -0.0030 -0.19 (-3.6) -0.025
"At ZA = ZB = 0.5. The numbers in parentheses denote experimental values from Wilson (1965). bActivity Coefficients of these systems show negative deviations from Raoult's law.
mixtures, so that the pairwise additivity assumption is reasonable here (Prigogine, 1957). The theory presented above is essentially a two-fluid theory (Scott, 1956);thus in all the mixture equations (such as the excess energy, eq 14) the summation may be extended over all species. Unfortunately, adequate data on multicomponent systems are not available for comparison. The Charge Transfer Parameter The single model parameter hz represents the average charge transfer between two dissimilar metal atoms. There is no flow of charge between identical atoms. Moreover, it is assumed that the charge transferred originates from the valence electrons and there is no contribution from the core electrons. In the present formulation, this redistribution of charge is considered to be the cause for the deviations from Raoult's law. Charge transfer (CT) parameter values have been estimated for a number of binary liquid alloys and appear in Table 11. The general picture that emerges is that the charge transfer is very small, of the order of fractions of an electron, even between metals with different valences. The table has been arranged so that it indicates charge flow from the metal A to the left toward the metal B at the right. We would like to determine which factors and physical parameters determine the direction and magnitude of the charge transfer. If it is possible to make such predictions from pure component properties, one would then be able to predict qualitatively and quantitatively the excess thermodynamic functions. For most of the systems which show positive deviations from Raoult's law the charge flows from the metal of higher valence to that of lower valence. On the other hand, for systems with negative deviations, the movement of the charge is always in the opposite direction. The charge does not necessarily move to the more electronegative element, which may indicate that there are many factors leading
148
Ind. Eng. Chem. Fundam., Vol. 24, No. 2, 1985
Table 111. Comparison of the Charge-Transfer Parameter for Selected Alloys with the Corresponding Differences in Conduction Electron Density between the Constituent Metals alloy (A-B) A2 X lo2 (e-)a Innw. X 10'1 (e-/au3)* AI-In 6 55 1.08 0.43 0.19 A1-Ga 1.19 Au-Ag 7.55 0.558 4.07 0.33 T1-Cd 1.35 0.51 0.34 Ga-Cd 0.74 0.31 In-Cd 0.56 0.39 Pb-Cd 0.67 Sn-Cd 0.25 0.00 Sn-Zn 0.84 0.39 0.00 Sn-Cd 0.25 2.31 0.77 Sn-AI 0.44 0.51 Sn-T1 Charge transferred from metal A to metal B. *Difference in conduction electron density between metals A and B.
to charge flow, but with varied effects. Unfortunately, there is no direct experimental evidence to provide a means of comparison for the calculated values of the charge transfer. There have been a number of theoretical treatments attempting to estimate the charge transfer in alloys. All these studies (Hodges and Stott, 1972; Pratter et al., 1977; Alonso and Girifalco, 1977; Ratti and Ziman, 1974) are involved with the determination of the electron density in the ion cores and the interstitial regions. The authors use various assumptions and definitions of charge transfer and consequently arrive at different results. They generally follow a process during which the two components become equivolume before any charge transfer takes place. Even though the orders of magnitude of charge transfer for many systems studied in the current work are comparable to those of the aforementioned authors, the directions of the flow do not always agree. One pure component property that gives an insight to the dependence of the charge transfer is the conduction electron density. Am empirical formula giving this parameter was proposed by Miedema et al., (1973) nws= 0.82
X
(B/u)'12
(27)
where B is the bulk modulus and u is the molar volume of the metal. This property, nws,represents the number of conduction electrons per unit volume in the space where they are allowed to move. Table I11 includes a series of groups of metal binary mixtures with individual constituents having the same valence or the same difference in valence. The charge transfer for each alloy is compared with the difference in conduction electron density between the two metals, and there appears to be a trend of increasing charge transfer with increasing difference in conduction electron density. As discussed above, the charge transfer will cause an unequal change in the partial molar volumes of the metal components, and hence a nonzero excess volume. The percent excess molar volumes at equimolar composition appear in Table I1 next to the charge-transfer values. A common feature of all systems is that the excess volumes obtained are small. This observation is reasonable, since all the metal binaries examined in this chapter are characterized by weak interatomic forces. Also, every system that exhibits negative deviation from Raoult's law has negative excess volume. In general, the estimated excess volumes are smaller than the corresponding experimental
values given by Wilson (1965); nevertheless, the majority of them agree as far as the sign of the excess volume is concerned. The discrepancies between the experimental and predicted values are not unexpected in view of the characteristics of the model. Acknowledgment
The authors are very grateful for the financial support of the Standard Oil Company (Indiana). Nomenclature A = constant a = constant B = bulk modulus b = constant g = molar Gibbs energy gE = excess molar Gibbs energy glE= excess partial Molar Gibbs energy of component i h = Planck's constant divided by 2s k B = Boltzmann's constant nws = conduction electron density N A = Avogadro's number P = pressure of system R = gas constant R, = metallic radius R, = metallic core radius R, = radius of a sphere which on the average contains one conduction electron r = distance from the center of a metallic atom s = molar entropy T = absolute temperature T,,, = melting temperature u = molar internal energy uIo = molar volume of pure component i 0, = partial molar volume of component in the mixture W = model potential n = bulk mole fraction 2 = metallic valence A 2 = charge transfer
Greek Letters /3 = volume thermal expansion coefficient y = bulk activity coefficient
= packing density p =
metallic density
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Received f o r review July 7, 1982 Revised manuscript received July 2, 1984 Accepted September 21, 1984
Nonideal Behavior in Liquid Metal Solutions. 2. Chemical-Physical Theory Model Thomas Stolcos and Charles A. Eckert” Department of Chemical Engineering, Universe of Illlnols, Urbana, Illinois 6 180 1
The “physicai” model developed in part 1 is coupled with a “chemical” model to account for the very strong negative deviations from Raouit’s law in systems forming intermetallic compounds. The resulting chemicai-physical model represents well both Gibbs energies and enthalpies of highly solvated metal systems. New data are presented for two titanium systems, for which only a limited composition range is experimentally accessible. The theory permits facile extension of such limited data.
Introduction
Many binary liquid alloys have chemical and physical properties with pronounced minima or maxima, and such behavior cannot be explained in terms of weak and nonspecific interactions between the components. Experimental results show that such anomalies usually occur at characteristic compositions where stable intermetallic compounds exist in the solid state and they show very large negative deviations from Raoult’s law. These observations have led many investigators to propose the existence of metallic clusters in the melt and incorporate them in solution models. Dolezalek (1908) first used the concept of compound formation and attempted to interpret solution nonidealities solely in terms of chemical forces. This idea was applied to metallic systems by Hildebrand and Eastman (1915), who assumed the presence of the compound HgT12 in thallium amalgams in order to explain deviations of these solutions from ideal behavior. In order to account for both chemical forces which lead to the formation of intermetallic compounds, and the nonspecific physical forces between the different species present in a mixture, several authors have developed chemical-physical models. Jordan (1970) has proposed a regular associated model based on the theory of Prigogine and Defay (1954). His model assumes that the free atoms in solution interact equally strongly with the AB type compounds. Osamura and Predel(l977) have followed a similar analysis, but their formulation accounts for unequal interactions. Cox (1979) has also modeled the combined effects of both chemical and physical contributions. This paper is devoted to the development of a solution model involving both chemical and physical forces. This is achieved by combining the physical theory model used in part 1 (Stoicos and Eckert, 1985) with the appropriate expressions associated with compound formation. The number of parameters, all of which are physically signif-
icant, are kept to a minimum. This model is shown to correlate successfully many results on strongly solvated liquid metal solutions. In addition, the method is applied to systems where only limited data are available and it is used to extend such data to experimentally inaccessible regions. Evidence of Existence of Intermetallics in t h e Liquid State Substantial evidence has been presented indicating that in many liquid alloys, the atomic configuration deviates from random mixing and intermetallic compounds are formed. The magnesium-tin binary is considered as an example. The phase diagram of this system, depicted in Figure 1, clearly shows the existence of a solid compound with the empirical chemical formula Mg2Sn,which melts congruently at 1051 K. The stoichiometry of the intermetallic compound is deduced from the location of the maximum of the liquidus curve, which occurs at 66.7 at. % Mg. Viscosity measurements (Gebhardt et al., 1955) a few degrees above the melting temperature of the solid compound shown in Figure 2 indicate that this structure-sensitive quantity reaches a relatively broad maximum at the same position in the concentration range as that corresponding to the stoichiometric composition of the compound. The viscosity increases due to an increase in the binding forces and the formation of structural units with directional bonds. The electrical resistivity also displays a similar maximum at the same composition (Steeb and Entress, 1966). From the shape of these curves, it appears that local order persists in the liquid phase not only in the very narrow homogeneity region of the solid intermetallic but over a large portion of the composition range. Results of X-ray structure studies for the same alloy by the latter authors are also attributed to the fact that part of the melt is solvated. In addition to the physical properties which provide evidence about the existence of intermetallic compounds, 0 1985
American Chemical Society
