174 Physica l14B (1982) 174-180 North-Holland Publishing Company

ENTROPY OF MIXING OF LIQUID METAL ALLOYS

K.N. K H A N N A and P. S INGH Department of Physics, V.S.S.D. College, Kanpur, India

Received 1 March 1982

An analysis of the dependence of entropy of mixing on the concentration of the constituents in an alloy has been presented, based on the hard-sphere reference system. The parameters of the hard-sphere system, as calculated by fitting the excess entropy at equiatomic concentration, provide results comparable with those obtained by the variational method. The alloying properties are found to be sensitive to the atomic composition of the alloy.

1. Introduction

The hard-sphere model has been successfully used to calculate several thermodynamical pro- perties of liquid metals and their alloys [1-7]. Recently, the variational method based on the Gibbs-Bogolyubov inequality has been applied by several workers [7-10] to study the entropy of mixing of equiatomic binary alloys. Hoshino and Young [11] have developed a theory for the entropy of mixing of a pseudo binary alloy in which the formation of molecules has been assumed. The basic quantities are the packing fraction, hard-sphere diameters and effective volume of the binary liquid mixtures. These quantities, and hence the thermodynamical pro- perties, change significantly when the atomic volumes of alloying elements are far from equal- ity. In the literature the hard-sphere parameters such as packing fraction and excess volume are determined by minimising the free energy with respect to the hard-sphere diameters as (OF/tgo.i)a,v = 0. The present work examines an alternative procedure of fitting these parameters to match the experimental value of the excess entropy at equiatomic concentration and using an empirical relation to get the parameters for general concentrations. Following the success of our earlier calculations [12] for monovalent

liquid alloy systems, we now intend to examine the applicability of the new fitting procedure to a wider class of homovalent and heterovalent liquid binary mixtures and to elucidate the con- centration dependence of various contributions to the entropy of mixing.

2. Hard-sphere description

A liquid binary mixture with components of atomic concentrations Cl and c2 comprises ClN hard spheres with diameter o-~ and c2N with diameter o-2. Following Umar et al. [2] the expression for the entropy in the hard-sphere reference system is given by

Shs = ~gas -I'- ~7 q'- ~c "q'- ~a , (1)

where Sc is the ideal entropy of mixing and $~ corresponds to the mismatch between the hard spheres with different diameters. With O as the atomic volume of the system (V/N), the packing fraction ~7 is given by

77" 3 '1~ = ~ (CIO" 1 Q- C20r23) .

The expressions for various S contributions are

0378-4363/82/0000-0000/$02.75 O 1982 North-Holland

K.N. Khanna and P. Singh / Entropy of mixing of liquid metal alloys 175

then given by [6, 7]

NkBS~= 2.5+ In [~(2 (-m ~ B T ) 3/2] , (2)

s0 NkB -(c l In c1 + c2 In c2) (3)

1 /VkBS~ = _(so _ 1)(s e + 3), with ~: = - - ' 1 - 7/ (4)

s~ Nka = clc2(trl - tr2)2{(Xl + x2)[se(~: - 1) - In ~]

+ 3x,(~ - 1)}, (5)

with

Thus knowing the value of excess entropy at any concentration, the entropy of mixing can be cal- culated straightforwardly by including the term Sc- strictly concentration-dependent term.

In the binary liquid mixture, formed by mixing constituents of different atomic sizes, the change in the packing fraction and volume on alloying plays a significant role in the determination of thermodynamical properties, such as excess entropy. In the present computation 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

x, = (~1 + ~ 2 ) / ( c ~ + c 2 ~ ) ,

X 2 = 0r10"2(C10"21 + C20"2)/(C10"31 + C20"3) 2 .

The hard-sphere formula for the entropy of mix- ing is given by

AShs = Shs-- C l S l - c2S2, (6)

where

Si = Sgas,i + S~,i.

Thus the entropy of mixing becomes [6]

A S m = ASgas + A S , / + So- + S e , (7)

where

ASgas = S g a s - ClSgas,1 -- C2Sgas,2

12

and

AS~ = S, - clS, ,- c2S~. (9)

/2 = J-~ideal + ClC2 AJ-~o, (11)

where

f~ideal = Cl~'~I + C2~'~2

and the fractional change in the volume on mix- ing is defined as

m/2 = ( /2 -- /2ide, al)/~'~ . (12)

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

7/" 77 = ~ (ClOt31 + C2 O'3)

O1 02 = C I ' O I ' ~ - + C2 '02"~ , (13)

using the packing fraction for the separate liquids as ~/i = 7r~3/6/2i.

In eq. (7) Sc is the ideal entropy of mixing. Thus, the excess entropy can be written as

AS e NkB AS~ + AS, + So-. (10)

3. Results and discussion

In the present work the atomic volume /2 of the liquid mixture at equiatomic concentration is so fitted as to satisfy eq. (10) with the experi-

176 K.N. Khanna and P. Singh / Entropy of mixing of liquid metal alloys

mental excess entropy [13]. That leads to an evaluation of AO0 from eq. (11) and the same value of A/20 is then used to calculate 12 and 77 at different concentrations, and thereby to evaluate the three contributions on the right-hand side of eq. (10) at different concentrations. These are plotted in fig. l(a to f) for six binary liquid systems. While S~ is always positive (see eq. (5)), the present computation predicts that the con- tributions AS~ and AS~ to excess entropy can be

"o2

' o l .

?° ~ - "o |

d~ <3 - "05

I . .o 4

-- "05"

- ' @ 7

- .OI~

- ' 1 0

- .1l

S r

Fig. 1. (a)

~ . . s ..6 ~ ,.o

t+ z

+'1 <3

I

' lO

1.,8

"e2

.i 'R "3 'A "s .& "7 "8 "q i.o

CI----~.

Fig. 1. (c)

either positive or negative. The term ASh which depends upon the relative magnitude of the packing fractions of the constituent elements, takes the positive or negative values as follow- ing: AS~ > 0 for ~7 < ~7c and AS~ < 0 for ~7 > ~1c, where the critical packing fraction is determined by S,c = c]S m + c2S~. Thus AS~ becomes negative in PbSn, AIMg and MgZn and positive in InZn, CdTI and Cdln. It is interesting to note that the plot of AS~ with respect to concentration is

",4

"lg

"1o

"o8

"o6

' o6o

. o ~ $

" 0 ~ 0

' o65

l 'oAo

o '~

.'~,,,.~+ 'o~0

I 'ogo

.oIS

'Olo

'00~

' *4

'o2

o

o "1 '2 '5 '4 '~ '~, "7 '8 '9 Po - - C I It

Fig. 1. (b)

Ca.'t't.

• +. .2 "5 'A .5 "G "T "8 .,I I.o

Cf- - ,"

Fig. 1. (d)

K.N. Khanna and P. Singh / Entropy of mixing of liquid metal alloys 177

"2

- 2

.oo'5

A ~.0 t~

L-.o¢

Pb S~

Fig. 1. (e) Fig. 1. (f)

Fig. 1. (a)-(f)Concentration dependences of AS~, A S ~ and S,, which contribute to the entropy of mixing for six liquid binary mixtures. Contribution of So in CdTI is very small and is not shown in the figure.

asymmetric about the mean concentration value (c1 = c2 = 0.5). This phenomenon is closely asso- ciated with the volume of the constituent ele- ments. With suffixes chosen such that /21//22 > 1 the maximum occurs at cl <0 .5 and the shift becomes more dominant with the increase of volume ratio. This is particularly so in the case of NaCs where the ratio is as large as three [12]. Table II exhibits the behaviour of the shifting in the AS~ curve. The term AS~ depends upon the relative magnitudes of the atomic volumes of the constituent elements. In most of the cases ASg, remains positive even for heterovalent alloys, contradicting the views of Hafner [3], but becomes negative for large volume contraction such as in A1Mg and MgZn. Here, too, the shifting in the maximum of the ASg~ curve is observed as described earlier but is less affected in comparison to AS,. Examples of numerical results are 12% for AIMg and 4% for CdIn. The mismatch term S~ increases with increasing Io-2- O-ll and remains positive at all concentrations for

all alloys. In this case also the shifting depends upon the ratio of the two volumes but is again small as in ASs,. As regards the comparison of the three contributions, we find AS~ > ASg-* > S~ in CdTl and CdIn alloys and AS~ > S~ > ASs,, in InZn, PbSn, A1Mg and MgZn. Thus, as remar- ked by other workers [3, 10], AS~ dominates in the determination of excess entropy, contrary to pure metals where ASs-* dominates. Our packing fractions [eq. (13)] at different concentrations, as shown in fig. 2, are in reasonable agreement with those derived by Hoshino [6] only for the alloys

Table I Parameters used in the calculations

Alloy ,(21(a.u.) /22(a.u.) 7/1 172 AS*/kB

ALMg 127.99 171.49 0.527 0.483 -0.099 CdIn 158.8 188.86 0.451 0.446 0.066 InZn 188.86 111.59 0.446 0.489 0.144 CdTl 158.8 205.21 0.451 0.375 0.122 MgZn 171.49 111.59 0.483 0,489 -0 .3 PbSn 222.10 194.6 0.421 0.431 0.0

178 K.N. Khanna and P. Singh / Entropy of mixing of liquid metal alloys

Table II Effective volume, packing fraction and volume of mixing for six binary alloys at equiatomic concentration

Alloy

Effective volume Packing fraction Excess volume Shifting in ASh A(trA + o'a) /2(a.u.) ~7 A~(%) Shifting (trn + o'a) Ours Ref. [10] Ours Ref. [10] Ours Ref. [10] (%) (~Q1/122) (%)

AIMg 147.196 146.78 0.5104 0.504 -1.7 -2.7 +20 0.746 -0.50 (-0.22) ")

Cdln 174.826 175.78 0.4457 0.441 0.6 0.7 +6 0.840 0.2 InZn 149.26 151.02 0.4649 0.466 -0.6 0.5 -14 1.69 -0.09 CdTI 181.67 183.82 0.4089 0.412 -0.18 1.0 +10 0.774 0.07 MgZn 137.22 134.82 0.5006 0.486 -3.15 -6.3 -12 1.54 -0.9 PbSn 208.07 - 0.4262 - -0.13 - -4 1.14 -0.036

") From ref. [3].

having smal l excess vo lume , bu t dev i a t e con-

s ide rab ly for the a l loys possess ing la rge v o l u m e

con t r ac t ion such as A I M g a n d M g Z n . This is due

to the se r ious a s sumpt ion O = Oid,~l c o n s i d e r e d

.~! ~ ~ s ~

'4g

• ~,s e a . I ~ . . . . . . -

. /~

'42

.41

./,.

~ C 4 r- Fig. 2. Concentration dependence of the packing fraction (~/) of binary liquid alloys. Full and broken curves refer to the values calculated by using eq. (13) (our result) and eq. (14) [6], respectively.

in the de r iva t i on of A*/ in the fo l lowing equa t ion

[61:

* / - (C1./1 + c2./2) = clc2(*/2- */1)(02- 01) (14) C101 "F C202

If we cons ide r the ac tua l v o l u m e (/2), we find

* / - (Cl*/1 + c2./2) = [CLC2(*/2 - */1)(O2 - O1)

- (c1./1 + c2./2) A O ' ] / O . (15)

In view of ou r empi r i ca l r e l a t ion of excess

v o l u m e (eq. (11)), eq. (15) b e c o m e s

* / - (c1./1 + c2./2) = c,c2[(*/2 - rh)(O2 - 01)

- */iae~l AOo]/g2. (16)

N o w it shou ld b e n o t e d tha t 7/ can possess a

va lue > o r <*/ideal d e p e n d i n g upon the sign and

m a g n i t u d e of ADo, as is obv ious f rom eq. (16),

and thus in ou r analysis t he pack ing dens i ty

dec reases in I n Z n , CdT1 and C d l n and increases

in A I M g and M g Z n . Thus , the cond i t ion 7 / <

Cl*/1 + C2./2 for */1 > */2, ~~1 < ~-~2 and */1 < *12, J~l > O2 s ta ted by H o s h i n o [6] is not fo l lowed in A1Mg

and M g Z n . T h e va r ia t ion b e t w e e n */ and Cl is

o b s e r v e d to be m a x i m u m for CdTI and m i n i m u m

for CdIn , and is p r o p o r t i o n a l to ]*/1 - */2]. A m o r e

i m p o r t a n t f e a tu r e of the pack ing f rac t ion is tha t

the c onc e n t r a t i on d e p e n d e n c e of A*/ exhib i t s a

non l inea r behav iou r , as d i sp l ayed in fig. 3. H e r e

K.N. Khanna and P. Singh / Entropy of mixing o/ liquid metal alloys 179

'olk

'or2

.o|o

-o~j

"OOG.

'004

~S,,t

Cl---~ /

cd.

c-&TL

Fig. 3. Concent ra t ion dependence of A17 of six binary liquid alloys.

again the shift in the maximum value from mean concentration depends upon the ratio of the two volumes.

We now turn to the hard-sphere diameters. In a homovalent alloy like MgZn, the diameter of the lighter element is increased whereas the diameter of the heavier element is reduced, as was also reported by other workers [9, 10]. For the heterovalent alloys our analysis reveals that the hard-sphere diameters of a more electrone- gative element expands and that of a more elec- tropositive element shrinks except in AIMg, which is in agreement with other theoretical results [3, 10]. The change in hard-sphere diameters is observed to be minimum at about 50-50% composition that are reported in table II and increases on either direction. Fig. 4 explains the change in hard-sphere diameters with con- centration; only three alloys have been shown in order to avoid the complexity of the figure.

The excess volume, an important feature in the determination of thermodynamical proper- ties [14, 15], is plotted at different concentrations in fig. 5. Alloys AIMg and MgZn show a large

6.0

I °

.I "2 '~ "/~ '5 ,¢,, .7 "8 "*1 I.o

Fig. 4. Concentration dependence of hard spheres of liquid metal alloys; CdTI (open circles), CdIn (black circles), A1Mg (crosses); A, B, C, D, E represent Cd, In, T1, AI, Mg, respectively.

.o0~

l "o61

~ -'°°4

_,oo 8

-'ol~.

- - 'o l~

--'0~. 0

--" 0~.1

Ca. I~.

Fig. 5. Concentration dependence of volume of mixing (AD) of six liquid metal alloys.

volume contraction, in agreement with other results [10, 16]. This behaviour of AIMg in com- parison to alloys CdTI and CdIn possessing ap- proximately same volumes ratio may indicate that hard-sphere mixture of AIMg is more den- sely packed than the other two alloys. This argument is also supported by the result ~TA~g >

cam or ~Tc, m. Again the maximum value of the A/2 curve shifts from the mean concentration in the manner described earlier.

In table II some of the quantities at 50-50%

18o K.N. Khanna and P. Singh / Entropy of mixing of liquid metal alloys

c o m p o s i t i o n a re c o m p a r e d with t he r ecen t cal-

cu la t ion [10] using a va r i a t i ona l t echnique . T h e

c o m p a r i s o n shows r e a s o n a b l e a g r e e m e n t .

4. Conclusion

T h e p r e s e n t s tudy exp la ins the change in

excess e n t r o p y with concen t r a t i on , f rom al loy to

a l loy; a d e c r e a s e in excess e n t r o p y in the a l loy

w h e r e t he con t r i bu t i ons in A S , and S~ t e n d to

cance l each o t h e r a n d vice versa . T h e ra t io of the

vo l umes of the cons t i t uen t e l e m e n t s is a

s ignif icant f e a tu r e tha t changes the ther -

m o d y n a m i c a l p r o p e r t i e s of an a l loy sharply . T h e

va r i a t ion of A/2 and A,/ wi th concen t r a t i on and the shift of t he m a x i m u m va lue f rom m e a n

c o n c e n t r a t i o n show the assoc ia t ive t e n d e n c y be-

tween un l ike a t o m s is s t rong at t he pa r t i cu l a r

concen t r a t i on .

Acknowledgement

T h e au tho r s a r e gra te fu l to P ro fe s so r D.P.

K h a n d e l w a l for s t imula t ing d iscuss ions and

he lp fu l c o m m e n t s .

References

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34 (1976) 535. [3] J. Hafner, Phys. Rev. A16 (1977) 351. [4] W.H. Young, Proc. Int. Conf. on Liquid Metals (Bristol,

1976), Inst. Phys. Conf. Ser. 30, p. 1. [5] E.G. Visser, W. van der Lugt and J. Th. M. de Hosson,

F10 (1980) 1681. [6] K. Hoshino, J. Phys. F10 (1980) 2157. [7] I. Yokoyama, A. Meyer, M.J. Stott and W.H. Young,

Phil. Mag. 35 (1977) 1021. [8] N.W. Ashcroft and D. Stroud, Solid State Phys. 33

(1978) 1. [9] R.N. Singh, J. Phys. F10 (1980) 1411.

[10] R.N. Singh and R.B. Choudhary, J. Phys. F l l (1981) 1577.

[11] Kozo Hoshino and W.H. Young, J. Phys. F10 (1980) 1365.

[12] K.N. Khanna and D.P. Khandelwal, Phys. Stat. Sol. (b) 106 (1981) 715.

[13] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser and K.K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys (1973), (Metals Park, Ohio: American Society of Metals).

[14] A.B. Bhatia and R.N. Singh, Phys. letters 78 (1980) 460. [15] M. Shimoji, Liquid Metals (Academic Press, New York,

1977). [16] J. Harrier, Inst. Phys. Conf. Ser. No. 30 (Bristol, 1976) p.

107.