Physica 124B (1984) 369-374

North-Holland, Amsterdam

ENTROPY OF MIXING CALCULATIONS FOR COMPOUND FORMING LIQUID ALLOYS IN THE HARD SPHERE SYSTEM

P. SINGH and K.N. KHANNA

Department of Physics, V.S.S.D. College, Kanpur, India

Received 9 September 1983

Revised 20 December 1983

It is shown that the semi-empirical model proposed in a previous paper for the evaluation of the entropy of mixing of

simple liquid metals alloys leads to accurate results for compound forming liquid alloys. The procedure is similar to that

described for a regular solution. Numerical applications are made to NaGa, KPb and KTl alloys.

1. Introduction

The theoretical study of the entropy of mixing of binary liquid alloys is a topic of current inter- est, motivated by the vast amount of experimen- tal information. The hard sphere system is one of the recent techniques to deal with the problem which has been proved to be successful for sim-

ple liquid metals mixtures [l-5]. For compound forming alloys, Bhatia and coworkers [6, 71,

Hoshino and Young [8], and Hoshino [9] developed a theory of pseudobinary alloy in

which formation of molecules is assumed. The existence of molecules with finite lifetime in compound forming liquid binary alloys is widely believed but it has not yet been confirmed definitely by experiments [lo] and the actual form of the associates is still in question [ll]. However, this associative tendency is always ac-

companied by large volume contraction. In the

present paper, we illustrate that the usual for- mula [5] for the entropy of mixing of binary hard spheres mixture applicable to simple liquid alloys can also be used to describe the composition dependence of the entropy of mixing for com- pound forming alloys. In a recent paper [12] this semi-empirical model was applied to study the concentration dependence of the entropy of mixing of liquid LiPb mixture across the whole composition diagram. This provides only a limited test to the usual theory of the hard spheres mixture. Therefore we present in this

paper a calculation of the entropy of mixing for NaGa, KPb and KTl compound forming alloys using our procedure in [5]. The method provides a way to calculate the separate contribution of

each component to the entropy of mixing by

fitting the entropy of mixing to its experimental value at the equiatomic concentration.

The calculation of the entropy of mixing also puts emphasis on the accurate knowledge of the atomic volume of the liquid solution on mixing which is generally obtained from the experimen- tal density data. But information on such data is very meagre and scattered. Another advantage

of the present model is that the volume of mixing can be calculated across the whole composition diagram.

2. Entropy of mixing of binary mixture of hard

spheres

The expression for the entropy of a binary hard sphere system is well known and is given in the work of Umar et al. [l] as follows:

&=sgas+s~+sC+s~. (I)

The expressions for various contributions are

given by

(2)

0378-4363/84/$03.00 @ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

370 P. Singh and K.N. Khanna / Entropy of mixing of compound forming alloys

s A= -(Ciln C1+ C2ln C2), Nka

$=-(I-1)(5+3) (4)

1 with 6 =l_rl,

S (r = GC*(o1- a&(x1 + x2) NkB

x [5(5 - I)- In 51+ 3x166 - 1)) (5)

modynamical properties. In the present com- putation, we have considered that the actual volume of an alloy is concentration dependent and is different from the ideal volume of an alloy. The effective volume has been considered as

0 = aideal+ CiC2 A& 7 (IO)

where

f&deal = c,n, + c,n, . with

Xl = (a1 + ai)l(Cd + C,a”,) ,

x2 = CT1 * u2(Cd + C2a3/(cid + c2ay .

The hard sphere formula for the entropy of mixing is given by

AS,,, = S,,s - C,S, - C2S2, (6)

where

Si = S*m,i + Sv,i *

Thus the entropy of mixing becomes [3]

A&=AS,,+AS,,+S,+S,, (7)

where

AS,, = S,, - GS,,,, - C2S,,,2

= NkB1n(&) and

AS,, = S,, - C,S,,, - C,S,,, . (9)

The concept of association in liquid alloys has proved very important for the description of the thermodynamic functions of mixing [13]. Thus in a binary liquid mixture, formed by mixing con- stituents of different atomic sizes, the change in packing fraction and volume on alloying plays a significant role in the determination of ther-

Thus, the effective volume R of an alloy differs from the ideal volume by

fl - f&deal = +C,C2 AL+, , (lOa)

where AL& is a constant for a given pair of constituents and is obtained by the fitting pro- cedure described later. This expression for volume of mixing is symmetrical about the equi- atomic concentration. The single parameter equation (lOa) has been found to give fairly good agreement with the experimental results for the volume of mixing in several cases, particularly for NaCs [14] and NaHg [15] alloys.

The fractional change in the volume on mixing is defined as

+f = (0 - i2i&d)/i2. (11)

The same value of the effective volume has been used in the determination of packing fraction as that used to determine AS, and S, for the binary mixture, viz.

01 rl = &(crU:+ C2a:) = cm,+ C2772 fl 3 3 (12)

where

using the effective packing fraction for the separate liquids as

P. Sin& and K.N. Khanna / Entropy of mixing of compound forming alloys 371

9 = G7?;+czgl. (13)

Now, 7; and 7; can be used to calculate the effective diameters of the mixture.

3. Results and discussion

The entropy of mixing calculation requires the volume of mixing, a concentration dependent value, as input information. In the present work the atomic volume 0 of the liquid mixture at equiatomic concentration is fitted with the experimental value of the entropy of mixing [16, 171 so as to satisfy eq. (7). That leads to an evaluation of the volume of mixing (CC2 A&) from eq. (10) and the same value of AL& is then used to calculate 0 and 77 at different concen- trations and thereby to evaluate the three con- tributions AS,,, AS,, and S, of the entropy of mixing at different concentrations. The fourth term, the ideal entropy of mixing, S,, is a purely concentration dependent term. In choosing the appropriate hard spheres data, we have taken care to include the temperature effect on the parameters. We determine the hard sphere diameters of pure elements at a particular tem- perature by using the experimental data [18] for the entropy at that temperature. That leads to the evaluation of packing fraction 7 which, in turn, gives the value of hard sphere diameter. The parameters obtained in this way are presen- ted in table I. The computed values of the entropy of mixing along with its components have been plotted in figs. l(a-c) for NaGa, KPb and KTl binary liquid systems. While S, is al-

ways positive, the present computation predicts that AS,, will be negative for most of the com- pound forming alloy systems and its magnitude increases with increasing packing fraction difference and atomic volume difference between the components. The curve AS,, shows an asymmetry as a function of concentration and the asymmetry increases with the increase of atomic volume difference between the com- ponents. The term A&, a volume dependent term, plays a significant role in compound form- ing alloys. In view of our previous [5, 121 and present studies, we conclude that the AS,, term generally remains positive for simple alloys hav- ing either excess volume of mixing or very small volume contraction but becomes negative for large volume of mixing, i.e. in compound form- ing alloys. The influence of mixing on the pack- ing fraction follows the relation [5]

87) = rl - (Cl711 + C2172)

= F [(‘I2 - 7/402- 01) - %deal AL&,] .

All Aq values are significantly positive. The results have been plotted in fig. 2 which actually shows the increase in packing fraction, hence compactness, on alloying the constituent elements of compound forming alloys and the relative deviation of A7 among these three alloys shows the maximum compactness in KPb alloy. The shifting of the peak value in the Aq curve from the mean concentration is also quite noticeable. This and a large volume contraction (fig. 3) with a shifting far from the mean concen- tration are the characteristic features of the

Table I

Input parameters used in the entropy of mixing calculations. 01, 02, ~1 and 72

are calculated as described in the text

Alloy 2) ::.) 4

(a.u.) rll 72

Exp. value of A& at equiatomic concentration [16,17]

NaGa 843 316.4258 139.5994 0.298 0.379 -0.1957 KPb 848 609.0739 224.7001 0.264 0.405 -0.3754 KTI 798 601.3116 208.8012 0.275 0.41 -0.0156

372 P. Singh and K.N. Khanna / Entropy of mixing of compound forming alloys

Na @I

SC

(4

KPb

6)

Fig. l(a-c). Concentration dependence of the entropy of mixing, A&, of the liquid alloys NaGa, KPb and KTI at 843, 848 and 798 K, respectively. The experimental data (from ref. 16 for KPb and KTI, and from ref. 17 for NaGa) are shown by crosses.

compound forming alloys. On mixing, the size of the hard sphere of each of the constituent ele- ments changes; it shrinks for the heavier element and expands for the lighter element. The con- centration dependence of hard sphere diameters

shows a nonlinear variation, displayed in fig. 4, indicating a different nature than that observed in simple alloy systems. Similarly, the change in hard sphere diameters does not show the mini- mum at 50-50% composition as is reported for

P. Singh and K.N. Khanna I Entropy of mixing of compound forming alloys 373

0 .4 k0

Fig. 2. Concentration dependence of An for NaGa, KPb and KTI alloys.

---CI-w .a ,4 .6

Fig. 3. Concentration dependence of the fractional change in volume (AL@/@ for NaGa, KPb and KTI alloys.

0 .2 9 -6 .a Lo -ct-

Fig. 4. Concentration dependence of hard spheres (mixture) of liquid NaGa and KPb alloys. The hard spheres of KTI alloy have similar features and are not shown here in order to avoid the complexity of the figure.

simple alloy systems [5]. Our results for the entropy of mixing are in good agreement with the experiments [16, 171 except for the anomaly observed for KTl below 0.5 concentration range. Our results show the possibility of existence of KTL, as also reported by Bruzzone [19] but later this phase was not confirmed by Thiimmel and Klemm [20]. However, two peaks on the positive side as shown in the experimental curve cannot be achieved by simple hard spheres mixture with single variable parameter.

Now, we collect the informations drawn from all the figures and think in terms of compound formation. From the maximum magnitude of Aq at the concentration corresponding to the mini- mum value of entropy of mixing one can con- clude that the associative tendency to form a stable compound is strong at that particular concentration. This shows the possibility of chemical complex formation, more or less similar to those observed for solids [16]. In a simple alloy system, the net core change (shrink- age/expansion) to the total volume change is so much that the overall packing is little altered. In compound forming alloys, in the particular region of concentration corresponding to the negative value of the entropy of mixing, the net core expands and the total volume contracts so that the packing fraction is considerably in- creased. This is obvious from the small change in hard sphere diameters and the sharp fall in atomic volume, resulting in a sharp increase in An near the compound forming concentration. Thus our results manifest that the concept of association can be viewed in the simple hard sphere theory and encourage one to estimate the entropy of mixing of compound forming alloys from the known volume contraction. A more precise method to evaluate the volume of mixing at different concentrations may lead to better results.

4. Conclusion

Our results are largely consistent with the experimental results, thus we can conclude that the thermodynamic properties of several com-

374 P. Singh and K.N. Khanna / Entropy of mixing of compound forming alloys

pound forming alloys can be described using the hard sphere system, nevertheless, we do not claim that all the compound forming alloys can be well described using this technique. Further investigations are needed to make the distinction between molecular forming and non-molecular forming compounds. The work is in progress in this direction.

Acknowledgements

We would like to thank Prof. D.P. Khandelwal for useful discussions and the University Grants Commission (India) for financial support.

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