Materials Sciences and Applications, 2011, 2, 605-608
doi:10.4236/msa.2011.26081 Published Online June 2011 (http://www.SciRP.org/journal/msa)
605
Microscopic Structure of NaCd Compound
Forming Binary Molten Alloys
Sujeet Kumar Chatterjee1, Lokesh Chandra Prasad2, Ajaya Bhattarai1
1Department of Chemistry, Mahendra Morang Adarsh Multiple Campus, Tribhuvan University, Biratnagar, Nepal; 2 Department of
Chemistry, Tilka Manjhi Bhagalpur University, Bhagalpur, India.
Received March 14th, 2011; revised April 15th, 2011; accepted May 5th, 2011.
ABSTRACT
A simple statistical mechanical theory based on compound formation model (compound Aμ Bν formed by the preferen-
tial arrangement of A and B constituent atoms of the alloy AB, μA + νB ⇌ Aμ Bν) has been used to investigate the phe-
nomena of compound formation in NaCd liquid alloys through the study of concentration fluctuation in the long wave-
length region [SCC(0)] and chemical short range order parameter (α1). The study explains the observed asymmetry suc-
cessfully and suggest the existence of Cd2Na complex in the melt. The tendency of compound has been found to be weak.
A tendency of inversion from segregation to order although very small, has been observed, in NaCd liquid alloys.
Keywords: A Simple Statistical Mechanical Theory, Compound Formation Model, Concentration Fluctuation in the
Long Wave Length Region [SCC(0)] and Chemical Short Range Order Parameter (α1)
1. Introduction
Although the value of ex cess free energy of mixing XS
M
G
0.66 for NaCd liquid alloys is small (–4.49 KJ at cd C

),
still it shows asymmetry around equiato mic composition.
This interesting aspect led the authors to undertake a
theoretical investigation of NaCd liquid alloys. Theoreti-
cians have worked with various theoretical models [1-10],
to understand the alloying behavior of compound form-
ing binary molten alloys. All the theoretical model led to
the fact that interatomic interaction plays an important
role in comound formation. Due to the presence of strong
interatomic interaction, these alloys form intermetallic
compounds at one or more stoichiometric composition,
which is also manifested in their phase diagram.
The formation of compound in solid state led many
theoreticians to believe in the existence of chemical
complexes, pesudomolecules and privileged group of
atoms near the melting temperature in binary liquid al-
loys.
Many theoretical models based on the above assump-
tions have been used to investigate the alloying behavior
of such type of binary molten alloys. The microscopic
properties such as [SCC(0), α1] show maximum deviation
from ideal behavior around compound forming concen-
tration and α1 has large negative values indicating chem-
ical order essential for compound formation. The phase
diagram of NaCd alloy shows the existence of Cd2Na
compound in the solid state.
Hence, in the present investigation, a simple theoreti-
cal model [2,7-11] of compound formation has been used
to investigate the asymmetry in the property of mixing
[Scc(o), α1] of Cd2Na compound forming liquid alloys.
2. Theoretical Consideration
Let the binary solution contains in all NA = NC and NB =
N(1 – C) gm moles of A and B atoms, respectively, C
being the atomic fraction of A atoms. Following Bhatia
and Hargrovem [1] and Bhatia et al. [3], only on e type of
complex Aμ Bν (μ and ν are small integers) is assumed to
be formed. If, in the solution n1 g mole A atoms and n2 g
mole of B atoms and n3 g mole of Aμ Bν are formed, then
from the law of conservation of atoms,


13
23
123
1
1
nNC n
nN Cn
nn nnNn
3






 
(1)
where n is the total number of atoms in the case of com-
pound formation. Under the frame work of above model,
the free energy of mixing GM can be expressed as
3M
Gng

G

 (2)
where (–n3g) is lowering of free energy due to the forma-

Microscopic Structure of NaCd Compound Forming Binary Molten Alloys 606
tion of the complex and “g” is the free energy of mixing
of ternary mixture of fixed n1, n2 and n3 whose constitu-
ents A, B an d Aμ Bν are supposed to interact weakly with
one another. Flory [12] gave an expression of G

taking
into account the difference of size between the unassoci-
ated species given as
 
3
33
1ln ln
ii
i
ij ij
GRTn nNnnN
nn V
NRT
















(3)
where Vij(i, j = 1, 2, 3) are the interaction energies
through which the left over A atom, B atom and the com-
plex Aμ Bν interact with one another. The energies Vij are
independent of concentration but may depend upon tem-
perature and pressure. It is convenient to write the ex-
pression for GM from Equations (2) and (3):
 
3
33
1ln ln
Mii
i
ij ij
GngRTnnNn nN
nn V
NRT



 
 










3

(4)
The assignment of the value n3 for the evaluation of
GM, through Equation (4) is the main problem which can
be solved by using equilibrium relation given as
3,,,
0
M
TPnC
G
n




 (5)
Equations (4) and (5) give
 
1
12 3exp 1nnnNgRT Z
 
 






(6)
where
 

1
13 132323
2112
Z
NRTnnVnnV
nnV





 

(7)
Once the expression for GM obtained, other thermody-
namic and microscopic functions, which are related to
GM through standard thermodynamic relation follows
readily as,
,,
M
MM
TPN
G
HGT
T






(8a)

MMM
SHGT
(8b)



22
1
,,
0
1
CC M
AA TPN
SNRTGC
Caa C

  (8c)






22
1
,,
0
1
CC M
BB TPN
SNRTGC
CaaC 

  (8d)
where aA, aB are the activities of A and B components
respectively. When Equation (4) is used in Equations (8c)
and (8d), the expression for Scc(0) is obtained as

1
2
311
2
1
2
0i
CCij ij
ii
n
SN nnv
nNRT
















(9)
The prime denotes differentiation with respect to con-
centration i.e. 1
1
1
23
1
1
nn
nn
3



 (10)
Differentiation of Equation (6) gives
1
3
n
The determination of partial structure factor at q = 0,
i.e. Scc(0) poses much more difficult experimental prob-
lem and has never been accomplished successfully. Till
now Scc(0) can not be evaluated experimentally by dif-
fraction experiment. It can be determined experimentally
from the measured activity data following two equalities
of Equation (8d). This is usually considered as experi-
mental data.
The chemical short range order parameter (α1) quanti-
fies the degree of order. It provides the immediate insight
into local arrangement of atoms in the alloy. Preference
for unlike atoms pairing or heterocoordination is indi-
cated by α1 < 1, whereas like atoms pairing or self coor-
dination is indicated by α1 > 1, α1 = 0 indicates complete
disorder of the atom in the alloys.
From probability approach one can show the limiting
range of α1 easily as


1
1
1,1 2
1
11,1 2
cc
c
cc
c






(11)
The minimum possible value of , gives
complete order, whereas , gives complete
segregation of atoms in the alloy.
max
11
 
max
11
 
The experimental data of α1 is also very difficult be-
cause it requires the knowledge of SCC(q) and SNN(q)
which are concentration-concentration structure factor
and number - number structure factor, respectively.
However, in practice it is difficult to determine SNN(q)
and SCC(q) for all kinds of binary alloys.
The theoretical relation between α1 and SCC(0) for the
first neighbour shell exists as

1
1
1
0
CC
S
SZS
 



 (12)

Microscopic Structure of NaCd Compound Forming Binary Molten Alloys607


Here Z = coordination number, whose value is taken as
10 and S1 = SCC(0) – SCC(0,id)
 
0, 1
CC
SidCC
 is the ideal value for


0
CC
S
(13)
3. Results and Discussion
3.1. Concentration Fluctuation in the Long
Wavelength Limit and CSRO (α1)
Equations (9) and (12) have been used to compute the
value of SCC(0 ) and α1, respectively. The basic inputs for
the calculation are n3g and Vij. Equilibrium relation (6)
which involves g and Vij are used to calculate n3, the me-
thod suggested by Bhatia and Hargrove [1] and Bhatia et
al. [3], which used to obtain the starting value of g and
Vij. The value of μ and ν, usually selected from the com-
positional dependence of physical properties or from the
phase diagram, are taken for Cd + Na system, μ = 2, ν =
1. As discussed earlier, the tendency of compound for-
mation is weak in CdNa system, the value of g should be
small. Therefore for the starting value one can take

M
g
G   .
The values of g and Vij are adjusted to obtain the con-
centration dependence of GM through Equations (4) and
(6). The above process repeated until a good fit for GM or
XS
M
G is obtained. The value of interaction energy pa-
rameters are chosen as
13 23
12
1.5,3.0,0.9, 1.7
VV V g
RT RTRTRT







(14)
The above set of energy parameters are utilized in Eq-
uation (6) to obtain n3 as a function of concentration. The
quantities n1 and n2 can be obtained by substitution of n3
in Equation (1). The computed and experimental values
are given in Table 1 and which are plotted in Fig-
ures 1 and 2 as a function of concentration.
No distinct minimum is observed around compound
forming concentration Ccd = 0.66, but the maximum de-
viation of SCC(0) from SCC(id, 0), [SCC(id, 0) – SCC(0)] is
sufficient to indicate the presence of Cd2Na complex in
NaCd liquid alloy at Ccd = 0.66 which is compound
forming concentration c
C











.
SCC(0) is greater than SCC(id, 0) in the concentration
range 0 < Ccd < 0.58, which indicates the presence of self
coordination leading to segregation, being maximum
around Ccd = 0.8. So an inversion from segregation to
order is exhibited in NaCd liquid alloys around Ccd =
0.58.
For the computation of α1, Equation (12) is used,
where the value of Z is taken equal to 10. For want of
experimental values α1 could not be compared. In order
to quantify the degree of heterocoordination α1 versus Ccd
plotted in Figure 2. The positive value of α1 in the con-
centration range 0 < Ccd < 0.58 indicates segregation or
like atoms pairing (Na-Na, Cd-Cd) as nearest neighbour.
The negative value of α1 beyond this range indicates ex-
istence of atomic order in the melt. The minimum value
of α1 (–0.176) has been found near compound forming
concentration Ccd = 0.66, which is very low indicating a
weak order. The asymmetry in α1 is distinctly visible in
Figure 2.
4. Conclusions
The theoretical model presented by authors, which as-
sumes the existence of Cd2Na compound in NaCd molten
alloys explain the concentration dependence asymmetry
of mixing properties satisfactorily, computed result show
that the tendency of unlike atom pairing (Na-Cd) de-
pends very much on concentration, maximum deviation
Table 1. Concentration fluctuation at the long wavelength
limit SCC(0) and chemical short range order parameter (α1)
of NaCd liquid alloys at 673 K.
Ccd S
CC(0) TheoryExpt.* SCC(id, 0) α1 Theory
0.1 0.210 0.170 0.09 0.060
0.2 0.409 0.366 0.16 0.064
0.3 0.512 0.586 0.21 0.060
0.4 0.470 0.516 0.24 0.048
0.5 0.332 0.308 0.25 0.024
0.6 0.211 0.171 0.24 –0.021
0.7 0.101 0.103 0.21 –0.099
0.8 0.048 0.064 0.16 –0.176
0.9 0.037 0.035 0.09 –0.110
*Hultgren et al. [6].
Figure 1. Scc(0) vs Ccd for NaCd liquid alloys at 673 K (Solid
lines): theory; (Closed circle): experiment.

Microscopic Structure of NaCd Compound Forming Binary Molten Alloys 608
[3] A. B. Bhatia, W. H. Hargrove and D. E. Thorntron, “Con-
centration Fluctuations and Partial Structure Factors of
Compound-Forming Binary Molten Alloys,” Physical
Review B, Vol. 9, No. 2, 1974, pp. 435-444.
doi:10.1103/PhysRevB.9.435
[4] L. C. Prasad, R. N. Singh, V. N. Singh and S. K. Chatter-
jee, “Compound Formation in Sn-Based Liquid Alloys,”
Physica B: Physics of Condensed Matter, Vol. 215, No.
2-3, 1995, pp. 225-232.
doi:10.1016/0921-4526(95)00393-N
[5] L. C. Prasad, S. K. Chatterjee and V. N. Singh, “Inter-
metallic Associations in AlMg Liquid Alloys,” Physica B:
Physics of Condensed Matter, Vol. 217, No. 3-4, 1996, pp.
285-291. doi:10.1016/0921-4526(95)00885-3
[6] R. Hultgren, et al., “Selected Values of the Thermody-
namic Properties of Binary Alloys,” The American Soci-
ety of Metals, Ohio, 1973.
Figure 2. α1 vs Ccd for NaCd liquid alloys at 673 K (Solid
lines).
[7] L. C. Prasad and R. N. Singh, “A Quasi-Lattice Model for
the Thermodynamic Properties of Au-Zn Liquid Alloys,”
Physics and Chemistry of Liquids, Vol. 22, No. 1-2, 1990,
pp. 1-9. doi:10.1080/00319109008036406
is observed around Ccd = 0.66, which is compound form-
ing concentration, indicates existence of Cd2 Na complex
in the melt.
The low negative value of α1 is sufficient to indicate
the tendency of compound formation is very weak. An
inversion from segregation to order is also clearly visible
in NaCd liquid alloys.
[8] K. Hoshino and W. H. Young, “Entropy of Mixing of
Compound Forming Liquid Binary Alloys,” Journal of
Physics F: Metal Physics, Vol. 10, No. 7, 1980, p. 1365.
doi:10.1088/0305-4608/10/7/006
[9] N. K. P Singh, R. N. Singh and R. B. Choudhary, “Ther-
modynamic Investigation of Atomic Order in AlMg Liq-
uid Alloys,” Journal of Physics: Condensed Matter, Vol.
3, No. 20, 1991, p. 3635.
doi:10.1088/0953-8984/3/20/024
5. Acknowledgements
The authors are grateful to Prof. R. N. Singh, presently at
the department of Physics, Sultan Quaboos University,
Muscat, OMAN for his valuable suggestions and discus-
sions. [10] R. N. Singh and F. Sommer, “Temperature Dependence
of the Thermodynamic Functions of Strongly Interacting
Liquid Alloys,” Journal of Physics: Condensed Matter,
Vol. 4, No. 24, 1992, p. 5345.
doi:10.1088/0953-8984/4/24/004
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