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Construction of liquidus curves of simple-eutectic binary alloys from Miedema theory
Physica B 150 (1988) 369-377 North-Holland, Amsterdam
CONSTRUCTION OF LIQUIDUS CURVES OF SIMPLE-EUTECTIC BINARY ALLOYS FROM MIEDEMA THEORY J.M. LOPEZ*,
J.A. ALONSO*,
L.J. GALLEGO**
and M. SILBERT
School of Mathematics and Physics, University of East Anglia, Norwich NR4 7TJ.
Received 29 June 1987 Revised version received 21 September
UK
1987
The liquidus curves of twenty five simple-eutectic binary alloys are constructed using the semi-empirical theory of heats of formation developed by Miedema and coworkers. Overall the predictions of this theory are quite correct. In cases where discrepancies exist, it is possible to improve the results by retaining the formal expression of the heat of formation proposed by Miedema. but modifying the prescribed values of the heats of solution.
1. Introduction Notwithstanding recent advances in the computations of the electronic structure of metals and alloys, the theoretical determination of alloy phase diagrams of binary aloys is based, in most cases, on empirical or semiempirical phenomenological descriptions [l-5]. This is mainly due to the difficulties involved in calculating the free energies of the competing phases as a function of concentration and temperature. Several authors have developed useful empirical correlations between some salient features of the phase diagram and combinations of “atomic” or “solid state” coordinates of the individual components of the binary alloy (see, e.g., refs. [6] to [ll]). The most successful of these correlations have been recently put forward by Villars [12]. This author was able to separate systems that form compounds from those that do not by means of two three-dimensional diagrams. The three coordinates used by Villars are: (i) the ratio TA/ T, of the melting temperatures (T, > TB) of components A and B; (ii) the magnitude * Permanent address: Department0 de Fisica Teorica, Universidad de Valladolid, 47005 Valladolid, Spain. ** Permanent address: Department0 de Fisica de la Materia Condensada, Universidad de Santiago de Compostela, La Coruna, Spain.
= [(r, + r,)‘, - (r, - I~):[ of the difA(r, + QLAB ference between the Zunger atomic pseudopotential radii sums [6]; and (iii) the magnitude [AVE,, 1 of the difference between the number of valence electrons. The separation achieved was 96% accurate. Moreover, the systems which do not form compounds can be separated into four different types - solubility, insolubility, eutectic, and peritectic types - with an accuracy of 90%. The remarkable success achieved by Villars has been the main motivation for this work. Unfortunately, we have been so far unable to relate directly his coordinates to phenomenological or more refined descriptions of the free energies of the binary alloys. Consequently we opted for the somewhat related approach described below which, we believe, goes a long way in explaining the main features of the phase diagrams of a particular class of binary alloys. The recent work of Alonso and coworkers (for a review, see [13]) on substitutional solid solubility is an attempt to go from empirical maps to semiempirical theories. In this approach, two coordinates are chosen: the heat of mixing in the liquid alloy AHAB, taken from the semiempirical model of Miedema and coworkers [14], and the atomic volume ratio VA/V,. It then follows that in a solubility map constructed for alloys based on a given host metal, the elements showing
0378-4363/88/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
370
J.M.
Lopez et al. I Liquidus curves of simple-eutectic binary alloy
large solid solubility in that host cluster in the map in the close neighbourhood of the point of coordinate values corresponding to the host [15,16]. More recently the empirical coordinates were incorporated into a theory which allowed the calculation of the free energy of the competing phases and then the prediction of the extent of solid solubility [17]. A similar procedure has been used to go from empirical maps constructed to predict glass-forming ability of binary alloys [l&20] to a theory capable of predicting the glass-forming concentration range in transition metals alloys [21]. In this paper we continue with our programme of predicting the main features of the phase diagram of binary alloys by using the semiempirical theory of Miedema (lot. cit.). Henceforth we concentrate on a well-defined class of alloys, namely those showing simple eutectics with null
or very low solid solubility at both ends of the concentration range and without intermediate compounds. Our aim is to predict the liquidus curves for these systems. The limitation to systems with negligible solid solubility and without intermediate compounds leads to an important simplification in the thermodynamics of the problem. However, both features may be incorporated into the theory, and we hope to be in a position to study these more complex systems in the near future. The layout of the paper is as follows. In section 2 we briefly survey the basic thermodynamic relations relevant to our problem. and the main ideas of Miedema’s theory of heats of formation. In section 3 we apply the theory to calculate the liquidus curves for a representative number of eutectic-type systems, and discuss the results of our calculations.
2. Thermodynamic equations The liquidus curves. at constant pressure, the equation [22] bY -In(x,y,)
= hi+
- 1) + +(In
of simple eutectic-type
$ + 1 - $1
binary mixtures can be described
(i=A,B),
I
(1)
where x, and y, are, respectively. the composition and activity coefficient of component i in the mixture, R is the gas constant. f., the latent heat of melting at the freezing temperature T, of the pure liquid i, and AC,., the difference between the heat capacities of the pure component i in the liquid and solid state of the same temperature. which is assumed to be constant in the temperature range of interest. Eq. (1) requires the knowledge of the activity coefficients y,. We use the values obtained from the theory of mixtures discussed below. This model, which has proved very useful in predicting the thermodynamic properties of liquid binary alloys (see, e.g., refs. [23] and [24]), assumes that the Gibbs free energy of mixing may be written as AC = RT(x,
In (pA+ xn In (pB) + W(A in B)x,xi
In eq. (2) .x; is the atomic cell surface-area
.
concentration
(2)
of component
B in the mixture defined by
(3) VA and V, are the molar volumes of the components by volume, i.e.
in the mixture; and (pAand (pg the concentrations
J. M. Lopez et al.
(PA =
i Liquidus curves of simple-eutectic binary alloys
371
(pB=l-qA.
XAVA + XBVB ’
The first term of eq. (2) is - T times Flory’s entropy of mixing of a mixture of molecules with different sizes [25], while the second term is the heat of mixing, as given by the semiempirical theory developed by Miedema and coworkers [14]. According to this theory, the heat of mixing is controlled by electronic interactions of magnitude proportional to the total area of contact between dissimilar atomic cells. Therefore the atomic surface-area concentration xi appears in eq. (2). W(A in B) is the heat of solution of A in B, and is given by Miedema’s theory as W(A in B) = 2v;3[-P(@A
- GB)* + Q(n;’
- n;“)‘-
R]
-1’3 + G1’3
7
nA
(5)
where Gi and ni are, respectively, the electronegativity and the electron density at the boundary of the bulk atomic cells of the metal i. P and Q are constants, and R is different from zero only for combinations of a transition metal and a polyvalent non-transition element. Extensive tables of W have been published by Miedema and coworkers [26], from which we have taken the values listed in column 2 of Table I. In using eqs. (2), (3) and (4) we assume VA and V, to be the pure metal volumes. The assumption ignores the changes in atomic volume that arise upon alloying. Miedema and coworkers [26] have introduced approximate corrections to the volumes which take into account the effects due to charge transfer. As far as this work is concerned, we believe that these corrections have only a minor effect because the electronegativity difference between the components of the alloys studied here is rather small. From eq. (2) the activity coefficients may be found to be given by VA
In yA = In XAVA
+
+ ‘BVB
dvB XAVA
+
‘A)
+
WAR;
+
WB
B)
(6)
‘BVB
and +
v*
In ya = In ‘A’,
+
XBVB
xA(vA ‘AVA
+
v,) ‘BvB
in
RT
A)
(a2
9
(7)
where W(B in A), the heat solution of B in A, is related to W(A in B) by
Using eqs. (6) and (7), in conjunction curves:
T= TA
and
with eq. (l), we obtain the following expressions for the liquidus
J. M. Lopez er al.
372
I Liquidus curves of simple-eutectic binary alloys
I + W(B in A) (x”,)? I T=
T, I-&(lnrp,+q,f,B
where AS, i = L,/T, ;.At the eutectic point pure solids A and B, and the liquid mixture, coexist in equilibrium and the‘two equations are simultaneously satisfied. In the following section we present the results of the calculations carried out for an illustrative sample of eutectic-type binary alloys.
Results and discussion We have used the formulation presented above to calculate the liquidus curves in twenty five binary alloys. The experimental liquidus curves have been taken from Hansen et al. [27] and Moffatt [28]. All the systems considered here are of simple eutectic type, with null or very small solid solubility and without intermetallic compounds. The liquidus curves have been firstly calculated using directly the heats of solution given by Miedema’s theory as inputs in eqs. (9) and (10). These values are given in the second column of table I. For convenience, we show in table II the experimental R/AS,., and ACp,jIASf., values used in our calculations [29]. The results obtained correspond to the curves denoted by M in figs. 1 and 2. From these results the twenty five systems studied fall into two groups. In the first group, which includes GeSb, ZnGe, GeSn, InZn, AuMo, AuSi, AuW, AuGe. AgPb, CdBi, AuTl, AlSn, AlBe, AlGe, AlHg, and AlSi, the theoretical and experimental curves are in close agreement; this is not the case with the second group, which includes ThTi, NaRb, AgBi, NbSc, ZnSn, CrU, CrTh, CrSc, and CrCu. The discrepancies between the predicted and experimental results which exist in the second group of systems can, in principle, be attributed either to unaccurancies in the values of W(A in B), as given by Miedema’s theory, or to a possible concentration dependence of this quantity. Obviously, in the latter case the equations of the liquidus curves would be different from eqs. (9) and (10). To elucidate this point we have calculated again the liquidus curves of all the systems using eqs. (9) and (lo), but taking now W(A in
B) as a parameter to be fitted to experiment. The resulting values of this parameter are listed in the second column of table I. In these calculations we have conserved the ratio W(A in B)/ W(B in A) as given by eq. (8). The results obtained are also shown in figs. 1 and 2 and correspond to the curves denoted by F. We note Table I Values of the enthalpy of solution of metal A in B given by Miedema’s theory [26]. obtained by fitting eqs. (9) and (10) to the experimental liquidus curves [27.28], and experimental values taken from Hultgren et al. 1301(see text). Units are KJ/mol. System A B
W(A in B) Miedema
W(A in B) Fit
Ag Bi Ag Pb Al Be Al Ge Al Hg Al Si Al Sn Au Ge Au MO Au Si AuTl AuW Bi Cd Cr Cu Cr SC Cr Th CrU Ge Sb Ge Sn Ge Zn In Zn Na Rb Nb SC Th Ti Sn Zn
6 9 31 -8 14 -9 14 -39 14 -57 -6 48 5 51 2 7 -9 0 0 -12 14 6 66 41 3
4 7 33 -12 13 - 14 12 -30 22 -57 -3 55 2 31 13 22 8 0 0 -4 12 4.6 30 27 8
W(A in B) Exp 4 11 -14
13
0 3
1.5 4
15
373
J. M. Lopez et al. 1 Liquidus curves of simple-eutectic binary alloys
that the liquidus curves calculated with the fitted values of W(A in B) agree quite well in all cases with the experimental results. This means that the concentration dependence of the heat of formation given by Miedema’s theory is, in general, correct although the values of W(A in B) are not accurate enough in some cases. This point has also been made by Miedema and coworkers [26]. In support of the above conclusions we have calculated the values of W(A in B) in several of the systems we have studied making use of the equation AH W(A in B) = 7
(11)
XAXB
and the available experimental enthalpies of formation at xA = 0.5 in the liquid state [30]. The values thus obtained, which are also listed in table I, agree better, in general, with the fitted values than with those provided by Miedema’s theory. The systems AgPb and InZn are the only exceptions to this rule. To close this paper, a few more comments are in order. In some of the systems considered here, the observed discrepancies between the theoretical and experimental curves can be attributed to a small solid solubility which has not been taken into account in the calculations. This happens in the system AgPb and AgBi (see fig. l(i) and 2(c)). It is also worth noting that in the formalism presented in this paper we have assumed that the heat capacity differences AC,,,; are constant in the temperature range of interest, In those systems for which experimental information is available, as it is the case for the system CdBi [31], we have also calculated the liquidus curves taking into account the variation of ACp,i with temperature. The results are similar to those obtained above. This appears to show that such a correction would not be crucial in a possible improvement of the results obtained here. In this connection, we also note that a correction in the temperature dependence of Acp,i would only modify the factor of AC,,ilASf,i m eqs. (9) and (lo), leaving unchanged the term in R/A&.
Table II Experimental values of R/AS,,, and ACp,,/AS,,, after Hultgren et al. [29]. Metal i
RIG.<
AC,.,/4
Ag
1.27 1.37 1.15 1.70 2.39 1.25 0.81 1.15 3.15 1.17 0.91 1.15 0.84 0.88 0.95 0.90 1.02 3.57 1.70 0.80 0.88 0.88 0.88 1.15 1.28
0.17 -0.24 0.45 -0.33 0.08 0.02 -1.35 0.08 -0.24 -0.001 -0.10 -2.17 -0.015 -0.22 0.15 0.24 0.0 -0.34 -0.12 0.0 0.03 -0.36 1.10 -0.09 0.22
Al Au Be Bi Cd Cr CU Ge Hg In MO Na Nb Pb Rb SC
Si Sn Th Ti Tl U w Zn
However, the latter is the dominant term in the denominator of these equations, as it can be appreciated by comparing the values of R/AS,,, and ACp,ilASl.i given in table II. With the exceptions of Cr, MO and U, the first quantity is always, at least, one order of magnitude greater than the second.
Acknowledgements This work was supported by the CAICYT of Spain (Grant 326583) and the British/Spanish Joint Research Programme (Acciones Integradas). Two of us are grateful for the award of a NATO Research Fellowship (JML), and a British Council/Ministerio de Education y Ciencia (Spain) Fleming Fellowship (LJG) to spend the academic year 1986187 at the University of East Anglia.
374
J.M.
Lopez et al.
I Liquidus curves of simple-eutectic binary allo_ys
200.
Ge
20
40
Atomic
per
60 cent
60
Sb
Zn
20
40
Atomx
Sb
I
r--(d)
60
60
per cent
Ge
Ge
20
Ge
40 Atomtc
60 per cent
60
Sn
Sn
3000
;
w
12001
In
20
40
Atomic
60
IX?, cent
60
‘“Ii Au
Zn
20
40
60
60
A”
MO
20
40
60
60
I SI
Zn w
3600
1
12oLl
-;
”
(h)
800
t’ I
; 600 E ,” 400
Au
20
40
60
60
I
W
Au
I
I
L
20
40
60
60
Ge
Ag
20
40
60
Ato”vc
‘ter cent
60
Ph
60
Stl
Pb
1200 (*)
0 ? 600 2 e “0
\
FL
i‘\, 20
40 Atomtc
60 per cent
60 81
61
\ E ‘\ M\
E 400
Cd
\
\
\ Al
20
40 Atomsc
60 ,,er
cent
Sn
.I. M. Lopez et al. I Liquidus curves of simple-eutectic binary alloys
375
(ml 1200(n)
QOO-
0
500 -
700 0 m a
2 500
.t t
600.
E +
I-” 300 400. Al
20
40
60
80
Se
Al
20
Atomic Per cent Se
40
60
80
GE Atomic per
Atomic Per cent Ge 1800.
cent Hg
J (P)
Ui Al
20
40
SO
80
Atomic per cent Si 1. Experimental (see refs. (271 and [28]) and theoretical liquidus curves of binary alloys. The experimental curves are shown by solid continuous lines, and indicated by E. The theoretical curves (this work), obtained by using in eqs. (9) and (10) the heats of solution predicted by Miedema’s theory (see text and the second column of table I), are shown by broken lines and indicated by M. The theoretical curves (this work), obtained by fitting the heats of solutions to experiment (see text, and the third column of table I), are shown by dots and indicated by F. In those figures where the theoretical curves are absent it indicates that there is practically no difference between either or both of them with the experimental results. In those figures where there is practically no difference between the F and M curves, these are shown by broken lines followed by dots. (a) GeSb; (b) ZnGe; (c) GeSn; (d) InZn; (e) AuMo; (f) AuSi; (g) AuW, (h) AuGe; (i) AgPb; (j) CdBi; (k) Audi; (I) Al%; (m) AIBe; (n) AIGe; (0) AIHg; (p) AISi.
Fig.
J. M. Lopez et al.
376
I Liquidus curves of simple-eutectic binary alloys
,
1 (a) ~
(b)
120 i
I
I-
looo_
,
Th
, 1
,
20
60
40 Atomic
per
cent
60
-n)/
Ti
, , , ,
Na
20
40 Atomic
TI
, / 1
60 per
80
cent
300
Ll
Ab
ae
40
20 Atomic
Rb
80
60 per
cent
SI
Bi
3000
2500
0
2400
1
2000
1
(1)
i
2000
t? 2 E 1500 E” c”
‘1: ,,,,,,,,, 1 / Nb
20
40
Atom,c
2400
60
per
Cent
60
Cr
w
20 Atomic
40 per
60
80
20
40 Atomic
60 per
cent
80
Sll
Sn
1
SC
SC
1
U
I
Zn
Cr 1900
-
700
-
(h)
20
-
40 Atomic
60 IX?, cent
80
U
U
-
Th
Cr
cent Th
Fig. 2. Same as in fig. I. (a) ThTi; (b) NaRb;
20
40 Atomic
(c) AgBi;
60 per
cent
80
SC
SC
(d) NbSc; (e) ZnSn:
Cr
20
40 Atomic
60 per cent
80 Cu
(f) CrU; (g) CrTh; (h) CrSc; (i) CrCu
Cu
J. M. Lopez et al. I Liquidus curves of simple-eutectic binary alloys
References ill N. Saunders, CALPHAD 9 (1985) 297. PI A.B. Bathia and N.H. March, Phys. Lett. A 41 (1972) 397. 131 A.B.
Bathia and N.H. March, Phys. Chem. Liq. 4 (1975) 279. [41 A.B. Bathia and N.H. March, Phys. Chem. Liq. 5 (1976) 45. 151W. Geertsma, Physica 132B (1985) 337. I61 A. Zunger, in: Structure and Bonding in Crystals, vol. 1, M. O’Keefe and A. Navrotsky, eds. (Academic Press, New York, 1981), p. 73. A.N. Bloch and G. Schatteman, in ref. [6] above, p. 49. D.G. Pettifor, J. Phys. C: Solid State Phys. 19 (1986) 285. 191A. Bieber and F. Gautier, Acta Metall. 34 (1986) 2291. PI J.C. Phillips, Helv. Phys. Acta 58 (1985) 209. WI P. Villars, J. Less-Common Met. 92 (1983) 215; 99 (1984) 33. PI P. Villars, J. Less-Common Met. 109 (1985) 93. 1131J.A. Alonso, Rev. Latinam. Met y Mat. 5 (1985) 3. 1141A.R. Miedema, P.F. de Chltel and F.R. de Boer, Physica 1OOB(1980) 1. (151 J.A. Alonso and S. Simozar, Phys. Rev. B 22 (1980) 5583. WI J.M. Lopez and J.A. Alonso, Physica 113B (1982) 103. 1171J.M. Lopez and J.A. Alonso, Z. Naturforsch. A 40 (1985) 1199. [18] B.C. Giessen and S. Whang, J. Phys. (Paris) 41 (1980) C8-95.
377
[19] J.A. Alonso and S. Simozar. Solid State Commun. 48 (1983) 765. PO1 J.A. Alonso and J.M. Lopez, Mat. Lett. 4 (1986) 316. WI J.M. Lopez. J.A. Alonso and L.J. Gallego, Phys. Rev. B., to be published. PI I. Prigogine and R. Defay, Chemical Thermodynamics (Longmans, London, 1969) u31 J.A. Alonso, J.M. Lopez and N.H. March. J. Physique Lett. 43 (1982) L441. u41 L.J. Gallego, J.M. Lopez and J.A. Alonso. Z. Naturforsch. A 39 (1984) 842. v51 P.J. Flory, J. Chem. Phys. 10 (1942) 51. WI A.K. Niessen, F.R. de Boer, R. Boom, P.F. de ChPtel, W.C.M. Mattens and A.R. Miedema, CALPHAD 7 (1983) 51. v71 M. Hansen and K. Anderko. Constitution of Binary Alloys (McGraw-Hill, New York, 1958). PI W.G. Moffatt, The Handbook of Binary Phase Diagrams (General Electric Co., Schenectady, New York, 1977). 1291R. Hultgren, P.D. Desay, D.T. Hawkins, M. Gleiser, K.K. Kelley and D.D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (Am. Sot. for Met., Metals Park, Ohio, 1973). 1301R. Hultgren, P.D. Desay, D.T. Hawkins, M. Gleiser and K.K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys (Am. Sot. for Met., Metals Park, Ohio, 1973). [311 J.H. Perepezco and J.S. Paik, J. Non-Cryst. Solids 61-62 (1984) 113.
CONSTRUCTION OF LIQUIDUS CURVES OF SIMPLE-EUTECTIC BINARY ALLOYS FROM MIEDEMA THEORY J.M. LOPEZ*,
J.A. ALONSO*,
L.J. GALLEGO**
and M. SILBERT
School of Mathematics and Physics, University of East Anglia, Norwich NR4 7TJ.
Received 29 June 1987 Revised version received 21 September
UK
1987
The liquidus curves of twenty five simple-eutectic binary alloys are constructed using the semi-empirical theory of heats of formation developed by Miedema and coworkers. Overall the predictions of this theory are quite correct. In cases where discrepancies exist, it is possible to improve the results by retaining the formal expression of the heat of formation proposed by Miedema. but modifying the prescribed values of the heats of solution.
1. Introduction Notwithstanding recent advances in the computations of the electronic structure of metals and alloys, the theoretical determination of alloy phase diagrams of binary aloys is based, in most cases, on empirical or semiempirical phenomenological descriptions [l-5]. This is mainly due to the difficulties involved in calculating the free energies of the competing phases as a function of concentration and temperature. Several authors have developed useful empirical correlations between some salient features of the phase diagram and combinations of “atomic” or “solid state” coordinates of the individual components of the binary alloy (see, e.g., refs. [6] to [ll]). The most successful of these correlations have been recently put forward by Villars [12]. This author was able to separate systems that form compounds from those that do not by means of two three-dimensional diagrams. The three coordinates used by Villars are: (i) the ratio TA/ T, of the melting temperatures (T, > TB) of components A and B; (ii) the magnitude * Permanent address: Department0 de Fisica Teorica, Universidad de Valladolid, 47005 Valladolid, Spain. ** Permanent address: Department0 de Fisica de la Materia Condensada, Universidad de Santiago de Compostela, La Coruna, Spain.
= [(r, + r,)‘, - (r, - I~):[ of the difA(r, + QLAB ference between the Zunger atomic pseudopotential radii sums [6]; and (iii) the magnitude [AVE,, 1 of the difference between the number of valence electrons. The separation achieved was 96% accurate. Moreover, the systems which do not form compounds can be separated into four different types - solubility, insolubility, eutectic, and peritectic types - with an accuracy of 90%. The remarkable success achieved by Villars has been the main motivation for this work. Unfortunately, we have been so far unable to relate directly his coordinates to phenomenological or more refined descriptions of the free energies of the binary alloys. Consequently we opted for the somewhat related approach described below which, we believe, goes a long way in explaining the main features of the phase diagrams of a particular class of binary alloys. The recent work of Alonso and coworkers (for a review, see [13]) on substitutional solid solubility is an attempt to go from empirical maps to semiempirical theories. In this approach, two coordinates are chosen: the heat of mixing in the liquid alloy AHAB, taken from the semiempirical model of Miedema and coworkers [14], and the atomic volume ratio VA/V,. It then follows that in a solubility map constructed for alloys based on a given host metal, the elements showing
0378-4363/88/$03.50 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
370
J.M.
Lopez et al. I Liquidus curves of simple-eutectic binary alloy
large solid solubility in that host cluster in the map in the close neighbourhood of the point of coordinate values corresponding to the host [15,16]. More recently the empirical coordinates were incorporated into a theory which allowed the calculation of the free energy of the competing phases and then the prediction of the extent of solid solubility [17]. A similar procedure has been used to go from empirical maps constructed to predict glass-forming ability of binary alloys [l&20] to a theory capable of predicting the glass-forming concentration range in transition metals alloys [21]. In this paper we continue with our programme of predicting the main features of the phase diagram of binary alloys by using the semiempirical theory of Miedema (lot. cit.). Henceforth we concentrate on a well-defined class of alloys, namely those showing simple eutectics with null
or very low solid solubility at both ends of the concentration range and without intermediate compounds. Our aim is to predict the liquidus curves for these systems. The limitation to systems with negligible solid solubility and without intermediate compounds leads to an important simplification in the thermodynamics of the problem. However, both features may be incorporated into the theory, and we hope to be in a position to study these more complex systems in the near future. The layout of the paper is as follows. In section 2 we briefly survey the basic thermodynamic relations relevant to our problem. and the main ideas of Miedema’s theory of heats of formation. In section 3 we apply the theory to calculate the liquidus curves for a representative number of eutectic-type systems, and discuss the results of our calculations.
2. Thermodynamic equations The liquidus curves. at constant pressure, the equation [22] bY -In(x,y,)
= hi+
- 1) + +(In
of simple eutectic-type
$ + 1 - $1
binary mixtures can be described
(i=A,B),
I
(1)
where x, and y, are, respectively. the composition and activity coefficient of component i in the mixture, R is the gas constant. f., the latent heat of melting at the freezing temperature T, of the pure liquid i, and AC,., the difference between the heat capacities of the pure component i in the liquid and solid state of the same temperature. which is assumed to be constant in the temperature range of interest. Eq. (1) requires the knowledge of the activity coefficients y,. We use the values obtained from the theory of mixtures discussed below. This model, which has proved very useful in predicting the thermodynamic properties of liquid binary alloys (see, e.g., refs. [23] and [24]), assumes that the Gibbs free energy of mixing may be written as AC = RT(x,
In (pA+ xn In (pB) + W(A in B)x,xi
In eq. (2) .x; is the atomic cell surface-area
.
concentration
(2)
of component
B in the mixture defined by
(3) VA and V, are the molar volumes of the components by volume, i.e.
in the mixture; and (pAand (pg the concentrations
J. M. Lopez et al.
(PA =
i Liquidus curves of simple-eutectic binary alloys
371
(pB=l-qA.
XAVA + XBVB ’
The first term of eq. (2) is - T times Flory’s entropy of mixing of a mixture of molecules with different sizes [25], while the second term is the heat of mixing, as given by the semiempirical theory developed by Miedema and coworkers [14]. According to this theory, the heat of mixing is controlled by electronic interactions of magnitude proportional to the total area of contact between dissimilar atomic cells. Therefore the atomic surface-area concentration xi appears in eq. (2). W(A in B) is the heat of solution of A in B, and is given by Miedema’s theory as W(A in B) = 2v;3[-P(@A
- GB)* + Q(n;’
- n;“)‘-
R]
-1’3 + G1’3
7
nA
(5)
where Gi and ni are, respectively, the electronegativity and the electron density at the boundary of the bulk atomic cells of the metal i. P and Q are constants, and R is different from zero only for combinations of a transition metal and a polyvalent non-transition element. Extensive tables of W have been published by Miedema and coworkers [26], from which we have taken the values listed in column 2 of Table I. In using eqs. (2), (3) and (4) we assume VA and V, to be the pure metal volumes. The assumption ignores the changes in atomic volume that arise upon alloying. Miedema and coworkers [26] have introduced approximate corrections to the volumes which take into account the effects due to charge transfer. As far as this work is concerned, we believe that these corrections have only a minor effect because the electronegativity difference between the components of the alloys studied here is rather small. From eq. (2) the activity coefficients may be found to be given by VA
In yA = In XAVA
+
+ ‘BVB
dvB XAVA
+
‘A)
+
WAR;
+
WB
B)
(6)
‘BVB
and +
v*
In ya = In ‘A’,
+
XBVB
xA(vA ‘AVA
+
v,) ‘BvB
in
RT
A)
(a2
9
(7)
where W(B in A), the heat solution of B in A, is related to W(A in B) by
Using eqs. (6) and (7), in conjunction curves:
T= TA
and
with eq. (l), we obtain the following expressions for the liquidus
J. M. Lopez er al.
372
I Liquidus curves of simple-eutectic binary alloys
I + W(B in A) (x”,)? I T=
T, I-&(lnrp,+q,f,B
where AS, i = L,/T, ;.At the eutectic point pure solids A and B, and the liquid mixture, coexist in equilibrium and the‘two equations are simultaneously satisfied. In the following section we present the results of the calculations carried out for an illustrative sample of eutectic-type binary alloys.
Results and discussion We have used the formulation presented above to calculate the liquidus curves in twenty five binary alloys. The experimental liquidus curves have been taken from Hansen et al. [27] and Moffatt [28]. All the systems considered here are of simple eutectic type, with null or very small solid solubility and without intermetallic compounds. The liquidus curves have been firstly calculated using directly the heats of solution given by Miedema’s theory as inputs in eqs. (9) and (10). These values are given in the second column of table I. For convenience, we show in table II the experimental R/AS,., and ACp,jIASf., values used in our calculations [29]. The results obtained correspond to the curves denoted by M in figs. 1 and 2. From these results the twenty five systems studied fall into two groups. In the first group, which includes GeSb, ZnGe, GeSn, InZn, AuMo, AuSi, AuW, AuGe. AgPb, CdBi, AuTl, AlSn, AlBe, AlGe, AlHg, and AlSi, the theoretical and experimental curves are in close agreement; this is not the case with the second group, which includes ThTi, NaRb, AgBi, NbSc, ZnSn, CrU, CrTh, CrSc, and CrCu. The discrepancies between the predicted and experimental results which exist in the second group of systems can, in principle, be attributed either to unaccurancies in the values of W(A in B), as given by Miedema’s theory, or to a possible concentration dependence of this quantity. Obviously, in the latter case the equations of the liquidus curves would be different from eqs. (9) and (10). To elucidate this point we have calculated again the liquidus curves of all the systems using eqs. (9) and (lo), but taking now W(A in
B) as a parameter to be fitted to experiment. The resulting values of this parameter are listed in the second column of table I. In these calculations we have conserved the ratio W(A in B)/ W(B in A) as given by eq. (8). The results obtained are also shown in figs. 1 and 2 and correspond to the curves denoted by F. We note Table I Values of the enthalpy of solution of metal A in B given by Miedema’s theory [26]. obtained by fitting eqs. (9) and (10) to the experimental liquidus curves [27.28], and experimental values taken from Hultgren et al. 1301(see text). Units are KJ/mol. System A B
W(A in B) Miedema
W(A in B) Fit
Ag Bi Ag Pb Al Be Al Ge Al Hg Al Si Al Sn Au Ge Au MO Au Si AuTl AuW Bi Cd Cr Cu Cr SC Cr Th CrU Ge Sb Ge Sn Ge Zn In Zn Na Rb Nb SC Th Ti Sn Zn
6 9 31 -8 14 -9 14 -39 14 -57 -6 48 5 51 2 7 -9 0 0 -12 14 6 66 41 3
4 7 33 -12 13 - 14 12 -30 22 -57 -3 55 2 31 13 22 8 0 0 -4 12 4.6 30 27 8
W(A in B) Exp 4 11 -14
13
0 3
1.5 4
15
373
J. M. Lopez et al. 1 Liquidus curves of simple-eutectic binary alloys
that the liquidus curves calculated with the fitted values of W(A in B) agree quite well in all cases with the experimental results. This means that the concentration dependence of the heat of formation given by Miedema’s theory is, in general, correct although the values of W(A in B) are not accurate enough in some cases. This point has also been made by Miedema and coworkers [26]. In support of the above conclusions we have calculated the values of W(A in B) in several of the systems we have studied making use of the equation AH W(A in B) = 7
(11)
XAXB
and the available experimental enthalpies of formation at xA = 0.5 in the liquid state [30]. The values thus obtained, which are also listed in table I, agree better, in general, with the fitted values than with those provided by Miedema’s theory. The systems AgPb and InZn are the only exceptions to this rule. To close this paper, a few more comments are in order. In some of the systems considered here, the observed discrepancies between the theoretical and experimental curves can be attributed to a small solid solubility which has not been taken into account in the calculations. This happens in the system AgPb and AgBi (see fig. l(i) and 2(c)). It is also worth noting that in the formalism presented in this paper we have assumed that the heat capacity differences AC,,,; are constant in the temperature range of interest, In those systems for which experimental information is available, as it is the case for the system CdBi [31], we have also calculated the liquidus curves taking into account the variation of ACp,i with temperature. The results are similar to those obtained above. This appears to show that such a correction would not be crucial in a possible improvement of the results obtained here. In this connection, we also note that a correction in the temperature dependence of Acp,i would only modify the factor of AC,,ilASf,i m eqs. (9) and (lo), leaving unchanged the term in R/A&.
Table II Experimental values of R/AS,,, and ACp,,/AS,,, after Hultgren et al. [29]. Metal i
RIG.<
AC,.,/4
Ag
1.27 1.37 1.15 1.70 2.39 1.25 0.81 1.15 3.15 1.17 0.91 1.15 0.84 0.88 0.95 0.90 1.02 3.57 1.70 0.80 0.88 0.88 0.88 1.15 1.28
0.17 -0.24 0.45 -0.33 0.08 0.02 -1.35 0.08 -0.24 -0.001 -0.10 -2.17 -0.015 -0.22 0.15 0.24 0.0 -0.34 -0.12 0.0 0.03 -0.36 1.10 -0.09 0.22
Al Au Be Bi Cd Cr CU Ge Hg In MO Na Nb Pb Rb SC
Si Sn Th Ti Tl U w Zn
However, the latter is the dominant term in the denominator of these equations, as it can be appreciated by comparing the values of R/AS,,, and ACp,ilASl.i given in table II. With the exceptions of Cr, MO and U, the first quantity is always, at least, one order of magnitude greater than the second.
Acknowledgements This work was supported by the CAICYT of Spain (Grant 326583) and the British/Spanish Joint Research Programme (Acciones Integradas). Two of us are grateful for the award of a NATO Research Fellowship (JML), and a British Council/Ministerio de Education y Ciencia (Spain) Fleming Fellowship (LJG) to spend the academic year 1986187 at the University of East Anglia.
374
J.M.
Lopez et al.
I Liquidus curves of simple-eutectic binary allo_ys
200.
Ge
20
40
Atomic
per
60 cent
60
Sb
Zn
20
40
Atomx
Sb
I
r--(d)
60
60
per cent
Ge
Ge
20
Ge
40 Atomtc
60 per cent
60
Sn
Sn
3000
;
w
12001
In
20
40
Atomic
60
IX?, cent
60
‘“Ii Au
Zn
20
40
60
60
A”
MO
20
40
60
60
I SI
Zn w
3600
1
12oLl
-;
”
(h)
800
t’ I
; 600 E ,” 400
Au
20
40
60
60
I
W
Au
I
I
L
20
40
60
60
Ge
Ag
20
40
60
Ato”vc
‘ter cent
60
Ph
60
Stl
Pb
1200 (*)
0 ? 600 2 e “0
\
FL
i‘\, 20
40 Atomtc
60 per cent
60 81
61
\ E ‘\ M\
E 400
Cd
\
\
\ Al
20
40 Atomsc
60 ,,er
cent
Sn
.I. M. Lopez et al. I Liquidus curves of simple-eutectic binary alloys
375
(ml 1200(n)
QOO-
0
500 -
700 0 m a
2 500
.t t
600.
E +
I-” 300 400. Al
20
40
60
80
Se
Al
20
Atomic Per cent Se
40
60
80
GE Atomic per
Atomic Per cent Ge 1800.
cent Hg
J (P)
Ui Al
20
40
SO
80
Atomic per cent Si 1. Experimental (see refs. (271 and [28]) and theoretical liquidus curves of binary alloys. The experimental curves are shown by solid continuous lines, and indicated by E. The theoretical curves (this work), obtained by using in eqs. (9) and (10) the heats of solution predicted by Miedema’s theory (see text and the second column of table I), are shown by broken lines and indicated by M. The theoretical curves (this work), obtained by fitting the heats of solutions to experiment (see text, and the third column of table I), are shown by dots and indicated by F. In those figures where the theoretical curves are absent it indicates that there is practically no difference between either or both of them with the experimental results. In those figures where there is practically no difference between the F and M curves, these are shown by broken lines followed by dots. (a) GeSb; (b) ZnGe; (c) GeSn; (d) InZn; (e) AuMo; (f) AuSi; (g) AuW, (h) AuGe; (i) AgPb; (j) CdBi; (k) Audi; (I) Al%; (m) AIBe; (n) AIGe; (0) AIHg; (p) AISi.
Fig.
J. M. Lopez et al.
376
I Liquidus curves of simple-eutectic binary alloys
,
1 (a) ~
(b)
120 i
I
I-
looo_
,
Th
, 1
,
20
60
40 Atomic
per
cent
60
-n)/
Ti
, , , ,
Na
20
40 Atomic
TI
, / 1
60 per
80
cent
300
Ll
Ab
ae
40
20 Atomic
Rb
80
60 per
cent
SI
Bi
3000
2500
0
2400
1
2000
1
(1)
i
2000
t? 2 E 1500 E” c”
‘1: ,,,,,,,,, 1 / Nb
20
40
Atom,c
2400
60
per
Cent
60
Cr
w
20 Atomic
40 per
60
80
20
40 Atomic
60 per
cent
80
Sll
Sn
1
SC
SC
1
U
I
Zn
Cr 1900
-
700
-
(h)
20
-
40 Atomic
60 IX?, cent
80
U
U
-
Th
Cr
cent Th
Fig. 2. Same as in fig. I. (a) ThTi; (b) NaRb;
20
40 Atomic
(c) AgBi;
60 per
cent
80
SC
SC
(d) NbSc; (e) ZnSn:
Cr
20
40 Atomic
60 per cent
80 Cu
(f) CrU; (g) CrTh; (h) CrSc; (i) CrCu
Cu
J. M. Lopez et al. I Liquidus curves of simple-eutectic binary alloys
References ill N. Saunders, CALPHAD 9 (1985) 297. PI A.B. Bathia and N.H. March, Phys. Lett. A 41 (1972) 397. 131 A.B.
Bathia and N.H. March, Phys. Chem. Liq. 4 (1975) 279. [41 A.B. Bathia and N.H. March, Phys. Chem. Liq. 5 (1976) 45. 151W. Geertsma, Physica 132B (1985) 337. I61 A. Zunger, in: Structure and Bonding in Crystals, vol. 1, M. O’Keefe and A. Navrotsky, eds. (Academic Press, New York, 1981), p. 73. A.N. Bloch and G. Schatteman, in ref. [6] above, p. 49. D.G. Pettifor, J. Phys. C: Solid State Phys. 19 (1986) 285. 191A. Bieber and F. Gautier, Acta Metall. 34 (1986) 2291. PI J.C. Phillips, Helv. Phys. Acta 58 (1985) 209. WI P. Villars, J. Less-Common Met. 92 (1983) 215; 99 (1984) 33. PI P. Villars, J. Less-Common Met. 109 (1985) 93. 1131J.A. Alonso, Rev. Latinam. Met y Mat. 5 (1985) 3. 1141A.R. Miedema, P.F. de Chltel and F.R. de Boer, Physica 1OOB(1980) 1. (151 J.A. Alonso and S. Simozar, Phys. Rev. B 22 (1980) 5583. WI J.M. Lopez and J.A. Alonso, Physica 113B (1982) 103. 1171J.M. Lopez and J.A. Alonso, Z. Naturforsch. A 40 (1985) 1199. [18] B.C. Giessen and S. Whang, J. Phys. (Paris) 41 (1980) C8-95.
377
[19] J.A. Alonso and S. Simozar. Solid State Commun. 48 (1983) 765. PO1 J.A. Alonso and J.M. Lopez, Mat. Lett. 4 (1986) 316. WI J.M. Lopez. J.A. Alonso and L.J. Gallego, Phys. Rev. B., to be published. PI I. Prigogine and R. Defay, Chemical Thermodynamics (Longmans, London, 1969) u31 J.A. Alonso, J.M. Lopez and N.H. March. J. Physique Lett. 43 (1982) L441. u41 L.J. Gallego, J.M. Lopez and J.A. Alonso. Z. Naturforsch. A 39 (1984) 842. v51 P.J. Flory, J. Chem. Phys. 10 (1942) 51. WI A.K. Niessen, F.R. de Boer, R. Boom, P.F. de ChPtel, W.C.M. Mattens and A.R. Miedema, CALPHAD 7 (1983) 51. v71 M. Hansen and K. Anderko. Constitution of Binary Alloys (McGraw-Hill, New York, 1958). PI W.G. Moffatt, The Handbook of Binary Phase Diagrams (General Electric Co., Schenectady, New York, 1977). 1291R. Hultgren, P.D. Desay, D.T. Hawkins, M. Gleiser, K.K. Kelley and D.D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (Am. Sot. for Met., Metals Park, Ohio, 1973). 1301R. Hultgren, P.D. Desay, D.T. Hawkins, M. Gleiser and K.K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys (Am. Sot. for Met., Metals Park, Ohio, 1973). [311 J.H. Perepezco and J.S. Paik, J. Non-Cryst. Solids 61-62 (1984) 113.