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L. Gurung et al.  / BIBECHANA 10 (2014) 108-114 : BMHSS, p.108 (Online Publication: Dec., 2013) 

 

 

BIBECHANA 
A Multidisciplinary Journal of Science, Technology and Mathematics 

ISSN 2091-0762 (online) 

Journal homepage: http://nepjol.info/index.php/BIBECHANA 

 

Mixing properties of Ni-Pd liquid alloy 
 

L. Gurung, R. P. Koirala, D. Adhikari
* 

Department of Physics, Mahendra Morang Adarsh Multiple Campus, Biratnagar, Nepal 

*Email: adksbdev@yahoo.com 

.Article history: Received 5 December, 2013 

 

Abstract 

In this work, thermodynamic and microscopic properties of Ni-Pd liquid alloy at 1873 K have been 

studied using a simple theory of mixture. In thermodynamic properties, free energy of mixing, heat of 

mixing and entropy of mixing have been studied. To understand the structural behaviour of the alloy 

concentration fluctuation in long wavelength limit and chemical short range order parameter have 

been computed. The Ni-Pd melt at 1873 K is found to be slightly deviated from regular behaviour. 
The best estimated value of energy parameter is found to be positive. The alloy is found to be weakly 

segregating system. The symmetric behaviour of all functions has been well explained by the simple 

theory of mixing 

 

Keywords: Ideal alloys, regular alloys, non-ideal alloys, Ni-Pd alloy, thermodynamic functions 

  

1. Introduction 
 

The mixing behaviour of binary liquid alloys may be explained in terms of thermodynamic and 

microscopic properties. Thermodynamic behaviour of the alloys is understood by the knowledge of 

free energy of mixing (∆GM), heat of mixing (∆HM), entropy of mixing (∆SM) and activity (a). 

Structural behavior is comprehended by the interpretation of concentration fluctuation in long 
wavelength limit (SCC(0)) and chemical short range order parameter (α1). The thermodynamic 

properties of binary liquid alloys give the information about the maximum stable composition, 

bonding strength and most mixed state of the alloy. The microscopic properties provide the 
information about the local arrangement of atoms.  

It is, therefore, obvious that without having a clear knowledge of the properties of alloys in liquid 

state it is not possible to understand the energetics of the formation of solid alloys.  This requirements 
demand extensive investigations for understanding the properties of alloys in the liquid state.  

Several theoreticians [1-10] have long been carried out to understand various properties of alloys in 

their liquid states. The Ni-Pd alloy is used in hydrogen sensors. This alloy has also glass forming 

properties at eutectic region around 45% of Pd. The knowledge of thermodynamic and structural 

properties at liquid state is essential for investigating the solidification process.  Hence, it is important 

to study different properties of Ni-Pd liquid alloy for its industrial applications. In this work we intend 

to study the thermodynamic and structural properties of Ni-Pd alloy in liquid state at 1873 K by using 

simple theory of mixing.  

In next section the theoretical formalism is presented. Section 3 deals with results and discussion and 
conclusions are outlined in section 4.  

 



 

 

L. Gurung et al./ BIBECHANA 10 (2014) 108-114 : BMHSS, p.109 (Online Publication: Dec., 2013) 

 

2.  Theoretical basis 
 

2.  Theoretical basis 

 

For non-ideal solution we have to take an account of the heat of formation ∆HM. Let’s account for the 

heat of formation ∆HM ≠ 0 using a simple model called statistical or quasi-chemical model. In this 
model the heat of mixing is only related to the bond energies between adjacent atoms. The assumption 

is that the interatomic distances and bond energies are independent of composition. There may be 

three types of bonds:  
 

(i) A-A bonds (Energy EAA):  

(ii) B-B bonds (Energy EBB) 

(iii) A-B bonds (EAB) 

If there are PAA, PBB, PAB bonds of each type, the internal energy of the solution is 

AA AA BB BB AB ABE P E P E P E= + +      (1) 

If z is the coordination number of an atom in a crystal, then the number of atoms NA and NB are 
related to the number of bonds 

   A AA ABN z 2P P= +       (2) 

B BB ABN z 2P P= +                                                              (3) 

A AB
AA

N z P
P

2 2
= −       (4) 

B AB
BB

N z P
P

2 2
= −       (5) 

Using these expressions for PAA and PBB in the expression for energy, we have 

A AB B AB
AA BB AB AB

N z P N z P
E E E P E

2 2 2 2
= − + − +   (6) 

A B AA BB
AA BB AB AB

N z N z E E
E E E P E

2 2 2

+ = + + − 
 

  (7) 

The first two terms of right hand side of Eq. (7) represents the energy of unmixed components. 

Therefore energy of mixing (heat of formation (∆HM)) is given as 

    AA BB
M AB AB

E E
H P E

2

+ ∆ = − 
 

    (8) 

For small differences between EAB and (EAA+EBB)/2 we can still consider a random arrangement of 

atoms in a solution and entropy of mixing of the solution is equal to ideal value 

( [ ]M 1 1 2 2S R X lnX X lnX∆ = − +  ). Such solutions are called regular solutions. Then 

              AB tot 1 2P zN X X=       (9) 

And  

M 1 2H X X∆ = Θ       (10) 

where   AA BB
tot AB

E E
zN E

2

+ Θ = − 
 

    (11) 

called the exchange parameter. It measures how different AB interactions are from the AA and BB 

interactions. If all of the interactions among the components are the same, then and there is no energy 

of mixing. For regular solutions a random arrangement of atoms in a solution is assumed. If there is 

small deviation from regular solution we may have to correct the exchange parameter. For this case let 

us consider Θ is a linear function of T as [12] 



 

 

L. Gurung et al./ BIBECHANA 10 (2014) 108-114 : BMHSS, p.110 (Online Publication: Dec., 2013) 

 

d
T

dT

Ω
Θ = Ω+        (12) 

Here, the term 
d

dT

Ω
 is introduced in order to account the small deviation of entropy from ideal 

behaviour.  Therefore, the heat of mixing and entropy of mixing may be expressed as 

   M 1 2

d
H T X X

dT

Ω ∆ = Ω+ 
 

     (13) 

( )M 1 1 2 2 1 2

d
S R X ln X X ln X X X

dT

Ω
∆ = − + +    (14) 

Then the free energy of mixing of the regular solution is given as 

   [ ]M 1 1 2 2 1 2G RT X ln X X ln X X X∆ = + +Ω    (15) 

The activity of pure components A and B in binary solution may be obtained using a standard 
statistical relation 

M
A M 2

1

G
RT ln a G X

X

∂∆
= ∆ +

∂
      (16) 

Thus, we have 

   
2

A 1 1RT ln a RT ln X (1 X )= +Ω −     (17) 

Similarly the activity of the component B may be obtained as 

   
2

B 2 2RT ln a RT ln X (1 X )= +Ω −     (18) 

The expression for concentration fluctuation in long wavelength limit (SCC(0)) has been introduced 

and derived by Bhatia and March [13]. It provides the idea about ordering and segregating nature of a 

solution. If the computed value of SCC(0) is less than ideal values ( =X1 X2)  there is tendency of 
unlike atoms pairing (ordering) and if it is greater than ideal values ( =X1 X2) there is a tendency of 

like atoms pairing (segregating).  

According Bhatia and March[13] the concentration fluctuation in long wavelength limit may be 
related with the free energy of mixing by the relation 

 

( ) 1
2 2

CC M 1
T,p,N

S (0) RT G x
− = ∂ ∆ ∂  

    (19)  

Therefore SCC(0) for regular solution  is given as 

   1 2
CC

1 2

X X
S (0)

1 2X X
RT

=
Ω

−
      (20) 

A measure of the tendency to compound formation or phase separation in a molten alloy is also given 

by the Warren-Cowley short-range order parameter (SRO), α1 [14,15]. This is an important indicator 

of the degree of ordering in a binary alloy and is given as 

 

         
1

S 1

S(z 1) 1

−
α =

− +
       (21) 

where  

   CC

id

CC

S (0)
S

S
=        (22) 

z being the coordination number of the alloy. We note that the main effect of z on 
1α  is to reduce the 

depth and hence its magnitude, the significant effects of complex formation and phase separation are 

essentially maintained. 

 



 

 

L. Gurung et al./ BIBECHANA 10 (2014) 108-114 : BMHSS, p.111 (Online Publication: Dec., 2013) 

 

 

3. Result and Discussion 

 

We have estimated the best fit value of Ω by the successive approximation using the observed values 

of free energy of mixing [16] and Eq. (15). The value of Ω estimated by this method must satisfy all 

the thermodynamic and microscopic functions. In our case the best fit value of Ω is found to be 7350 

Jmol-1. The value of Ω is found to be positive. This indicates that the Ni-Pd system in liquid state at 

1873 K is segregating system, i.e. like atoms tend to form pairs. We have calculated the free energy of 

mixing of Ni-Pd liquid alloys at 1873 K using Eq. (15) and estimated value of Ω. The computed 

values of free energy of mixing are almost equal to the observed values. Both the computed and 

observed values of free energy of mixing are minimum (computed value = -8951 Jmol-1 and observed 

value = 8928 Jmol-1) at equiatomic composition. The free energy of mixing is negative at all 

composition (Fig. 1). 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 1: Excess free energy of mixing ( XS

MG∆ ) of Ni-Pd liquid alloy at 1873 K versus X1 

(Concentration of Ni); solid lines for theoretical values and circles for experimental values. 
 

 

We have computed the heat of mixing of Ni-Pd liquid alloys at 1873 K on the basis of simple theory 

of mixing (Eq. 13). Before this we estimated the best fit value of  
d

dT

Ω
 using experimental data. Its 

best value is found to be -1.4 Jmol-1K-1.  The theoretical and experimental values are in a very good 

agreement up to X1 = 0.6 ( Fig. 32). Above this region there are some discrepancies. These 

discrepancies may be due to greater deviation of solution from regular behaviour. However, the 

symmetric nature of the experimental results of heat of mixing against the concentration of Ni is not 

violated by the theoretical analysis. Both the experimental and theoretical values of heat of mixing are 

positive at all compositions. The theoretical maximum value of heat of mixing is almost equal to the 

experimental maximum value (Experimental value =1195.5 Jmol-1, Theoretical value = 1182 Jmol-1). 

We have then computed the entropy of mixing of Ni-Pd liquid alloys at 1873 K using Eq. (14). The 

computed entropy of mixing is in good agreement with the observed values at all compositions. The 

entropy of mixing is positive for all compositions. This clearly indicates that the entropy is greater in  



 

 

L. Gurung et al./ BIBECHANA 10 (2014) 108-114 : BMHSS, p.112 (Online Publication: Dec., 2013) 

 

the mixture than that of the pure states of the components. The symmetric behaviour of the observed 

result of entropy of mixing of Ni-Pd liquid alloys at 1873 K is well explained by the theoretical 

analysis. For comparative study the computed and observed values of entropy of mixing of Ni-Pd 

liquid alloys at 1873 are plotted against concentration of Ni (Figure 3). 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

Fig. 2 : Heat of mixing (∆HM) of Ni-Pd liquid alloy at 1873 K versus X1 (Concentration of Ni) ; solid 

lines for theoretical values and circles for experimental values 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 3: Heat of mixing (∆SM) versus X1 (Concentration of Ni)  for different positive values of 
d

dT

Ω
; 

solid lines for theoretical values and circles for experimental values. 
 

 

We have computed the theoretical values of SCC(0) for Ni-Pd liquid alloys at 1873 K using the 

previously determined value of Ω (Figure 4). The computed value of SCC(0) is greater than ideal 

values at all compositions. This shows that the alloy under investigation is segregating in nature. This  
 



 

 

L. Gurung et al./ BIBECHANA 10 (2014) 108-114 : BMHSS, p.113 (Online Publication: Dec., 2013) 

 

result is in agreement with the theoretical and experimental analysis of free energy of mixing and heat 

of mixing.  The symmetric behaviour of SCC(0) is well explained by the theoretical analysis.  

The knowledge of 1α  provides an immediate insight into the nature of the local arrangements of 

atoms in the mixture. At equiatomic composition, one has 1α1 1 ≤≤− . The minimum possible value of  

1α  is 1α
min
1 −=  and that implies complete ordering of unlike atoms paring at nearest neighbours. On 

the other hand the maximum value of 1α  is 1αmax
1 +=  which implies total segregation leading to the 

phase separation and 1α = 0 corresponds to a random distribution of atoms. Here 1α  can be estimated 

from the knowledge of SCC(0) as by the Eq. (21). It is found positive at all compositions. This shows 

that Ni-Pd liquid alloys at 1873 K. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
Fig. 4:  Concentration fluctuation in long wavelength limit (SCC(0)) versus X1 (Concentration of Ni) 

for Ni-Pd liquid alloys at 1873 K; solid lines for theoretical values, circles for experimental values and  

dotted lines for ideal values. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 5: Chemical short range order parameter (α1) versus X1 (Concentration of Ni) for Ni-Pd liquid 

alloys at 1873 K 



 

 

L. Gurung et al./ BIBECHANA 10 (2014) 108-114 : BMHSS, p.114 (Online Publication: Dec., 2013) 

 

4. Conclusions 
 

Based on the above discussion the following conclusions may be drawn about Ni-Pd liquid alloy at 

1873 K: 

(i) It is not a strictly regular solution. It slightly deviates from the regular behaviour. 

(ii) It is a weakly interacting system. 

(iii) It is a segregating alloy, i.e. like atoms tend to pair forming homo-coordination. 

(iv) All the thermodynamic and structural functions of this alloy are symmetric at equiatomic 

composition as a function of concentration. 

 

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