A binary system, A-B is supposed to be consisted of N A = N C A as well as N B = N C B atoms of metal A and metal B respectively where C A = C and C B = 1 − C denote the concentrations of constituents A and B, respectively. In SAM model, it is considered that the atoms are situated on equivalent lattice sites with Z nearest neighbors. The effective interactions are short-ranged and exist only between nearest neighbors. Further, consider that sizes of A and B are comparable and there exist like-atom clusters or self-associates like A i and B j so that

where i and j = number of atoms in self-associates types A and type B matrices respectively.

In SAM model [4] assuming Z → ∞ [15], the expression for G M is given by

where,

with R and T = universal gas constant and absolute temperature respectively, W = i ω , denotes interchange energy while n = j i , ( j > i ) , ratio of number of two atoms in self-associates. It is pointed out that n and W are independent of composition and dependent on the pressure and temperature.

The equation of activity [4] for a binary melt is represented by

Equations (2) and (4) provide

Again, S M for a binary melt is represented by [4]

Equations (2) and (7) provide

where, ∂ W ∂ T = temperature derivative of W .

The H M for a binary melt is expressed as

Equations (2), (8) and (9) provide

The structural functions like S C C ( 0 ) [16] and α 1 [17], [18] are very useful microscopic functions for acquiring the knowledge of the nature as well as degree of local arrangement of atoms in the binary melts. The S C C ( 0 ) and G M are related as [16]

Expressions (2) and (10) yield

where,

For f ( n , W ) = 0 , expression (12) reduces to

which represents the ideal value of S C C ( 0 ) .

The activity and S C C ( 0 ) are related as [16]

The Equation (14) provides the experimental data of S C C ( 0 ) for the binary melts if the experimental data for a A and a B are utilized.

Again, α 1 in the nearest-neighbor shell for a binary melt [4], [17], [18] and S C C ( 0 ) are related as

where, Z denotes the coordination number, which is taken to be 10.

The S C C ( 0 ) is an important parameter used to describe the stability of a binary melt. In terms of S C C ( 0 ) , excess stability function, E X S [19] is represented by [4], [20], [21]

The properties like C i s , σ as well as surface concentration fluctuations, S C C s ( 0 ) describe the characteristics like energy of phase alternation, mechanical behavior, thin film, etc. Hence, the present investigation is concerned with the study of the surface properties of binary melts by Butler model [7].

According to Butler model [7], the surface tension, σ of a binary melt is expressed as

where, σ i ( i = A , B ) = surface tension of i^th component in molten state, G ˜ i E , s = partial excess free energy at the surface for component i, G ˜ i E , b = partial excess free energy in the bulk for component i, S i ( i ≡ A , B ) = molar surface area of i t h component, and C i s and C i = surface and bulk compositions respectively. Now, S i ( i ≡ A , B ) may be expressed as [9]

where, V i = atomic volume of component i, N A = Avogadro’s number.

Now, the expression for G ˜ i E , b is given by [4]

where, γ i = activity coefficient of i^th component.

Again, G ˜ i E , b and G ˜ i E , s ( i ≡ A , B ) are related as [22]

with, λ = Z S Z [23] where Z S = surface coordination number and Z = bulk coordination number. It is necessary to proclaim that the value of λ lies between 0.50 to 0.84 [24]. In present study, the value of λ is taken to be 0.818 [25].

Now, S C C s ( 0 ) which deals with the arrangement of atoms at the surface of binary solution. The expression for S C C s ( 0 ) is given by [26]

with,

where,

and

The ideal S C C s ( 0 ) is represented by [26]

The diffusivity ratio and S C C ( 0 ) is related as [4], [14]

where, D M denotes the mutual diffusion coefficient while D i d represents the intrinsic diffusion coefficient of an ideal mixture.

Again, D i d is represented by

where, D A and D B = self-diffusion coefficients of pure elements A and B respectively.

The viscosity is helpful to explain the interatomic interactions between the elements of a binary melt. From Moelwyn-Hughes approach [8], η can be expressed as

where, η i d = viscosity of the ideal melt which is represented as

where, η A and η B = viscosities of pure elements A and B respectively.

Now, H M may be represented by [4]

Equations (28) and (30) provide

Equation (31) indicates that η depends upon the magnitude of H M R T .

The model parameters like n and W are required for the theoretical investigation of the bulk properties of InTl melts at 723 K. The values of n and W are estimated by the method of successive approximation approach [11], [13], [14] on taking the experimental data of G M [2] in Equation (2) for InTl melts at 723 K. The suitable values are

The theoretical data for G M R T of InTl melts at 723 K are presented in Figure 1 along with the correlated experimental data [2] relative to the composition of In which displays well agreement in the whole composition region of In, C I n = 0.1 to 0.9. The values of G M R T are negative in the whole composition range with minima at C I n = 0.5 i.e. G M R T ( min m ) = − 0.5664 (Theory) and −0.5669 (Experiment). Thus, the symmetry in G M for InTl melts at 723 K is well explained theoretically.

Figure 1. G M X S R T and G M R T versus C I n for InTl melts at 723 K.

The G M X S for InTl melts at 723 K is computed by the relation

A well agreement is observed between theory and experiment [2] for G M X S of InTl melts at 723 K as depicted in Figure 1. The positive values of G M X S are found in whole region, C I n = 0.1 to 0.9 having maxima i.e. G M X S R T ( max m ) = 0.1268 (Theory) and 0.1267 (Experiment) both at C I n = 0.5 . Hence, the symmetry in G M X S for InTl melts at 723 K is explained satisfactorily. Again, the magnitude as well as sign of G M X S R T reveal that InTl melts at 723 K is a feebly segregating alloy [4].

Now, the activities a I n and a T l of In and Tl respectively, computed via Equations (5) and (6) are displayed in Figure 2 along with the corresponding experimental results [2] for InTl melts at 723 K. An excellent harmony is noticed. Again, the small positive departures from Raoult’s law are noticed for a I n and a T l in the complete composition range i.e. C I n = 0.1 to 0.9, which signifies the feebly segregating nature [4] of InTl melts at 723 K. Thus, the well agreement between theory and experiment for a I n as well as a T l provides the authenticity of the evaluated values of n and W.

Figure 2. a i ( i = I n , T l ) versus C I n for InTl melts at 723 K.

For the computations of S M and H M for InTl melts 723 K via Equations (8) and (9), the values of ∂ β ∂ T and ∂ W ∂ T are required. The evaluated values of ∂ β ∂ T and ∂ W ∂ T by least square method [13], [14] on employing the experimental values of S M [2] in expression (8) are −1 × 10⁻⁵ K⁻¹ and 0.333 R, respectively.

Figure 3 shows the comparison between the theoretical result of S M determined from Equation (8), and the correlated experimental data [2] for InTl melts at 723 K and exhibits excellent agreement in the region, C I n = 0.1 to 0.9. The peak values are observed at C I n = 0.5 i.e. S M R ( max m ) = 0.6595 (Theory) and 0.6596 (Experiment). This successfully reproduces the symmetry in S M for InTl melts at 723 K.

Again, the theoretical values of H M in terms of the composition of In for InTl melts at 723 K, computed from Equation (9) are in reasonable agreement with the corelated experimental data [2] as mentioned in Figure 3. However, a minor discrepancy is noticed i.e. H M R T ( max m ) = 0.093 (Theory) at C I n = 0.55 and 0.0947 (Experiment) at C I n = 0.6 , which shows the asymmetric behaviour of InTl melts at 723 K. Further, the values of H M are positive in the complete composition region of In, which indicates the demixing nature of InTl melts at 723 K [4].

Figure 3. S M R and H M R T versus C I n for InTl melts at 723 K.

The theoretical data of S C C ( 0 ) for InTl melts at 723 K in terms of composition of In, computed via Equations (11) and (12) are mentioned in Figure 4. The experimental data of S C C ( 0 ) [2] and ideal S C C ( 0 ) are also displayed in Figure 4. A well agreement is noticed between theory and experiment with some departures in the region 0.4 ≤ C I n ≤ 0.6 so that maximum error is about 5.38% at C I n = 0.5 . In the complete range of composition, 0.0 ≤ C I n ≤ 1.0 , S C C ( 0 ) > S C C i d ( 0 ) having the peak values of 0.315 (Theory) and 0.318 (Experiment) both at C I n = 0.5 . This advocates the segregating nature of InTl melts at 723 K [4]. Further, the difference between the peak values of S C C ( 0 ) and S C C i d ( 0 ) i.e. S C C ( 0 ) − S C C i d ( 0 ) for InTl melts at 723 K is found to be 0.085 which is very close to 0.0846 as available in the literature [3].

Equation (15) is utilized to compute the values of α 1 relative to the composition of In for InTl melts at 723 K. Figure 4 also displays the α 1 − C isotherm for InTl melts at 723 K. It should be pointed out that no experimental data of α 1 for InTl melts at 723 K are available in the literature for comparison. It is obvious from Figure 4 that α 1 is positive at each composition throughout the composition range of In form 0.1 to 0.9. The sign and magnitude of α 1 indicate that InTl melt is a feebly segregating alloy.

Figure 4. S C C ( 0 ) and α 1 versus C I n for InTl melts at 723 K.

Figure 5. E X S R T versus C I n for InTl melts at 723 K.

The excess stability function is useful to study the nature of atomic interactions in a binary melt. The negative value of E X S signifies the segregating nature of binary melts because in this case, S C C ( 0 ) > S C C i d ( 0 ) . Similarly, E X S becomes positive for ordered alloys due to S C C ( 0 ) < S C C i d ( 0 ) [4]. The theoretical values of S C C ( 0 ) are utilized in Equation (16) to explain the concentration dependence of E X S for InTl melts at 723 K. Figure 5 displays the E X S R T − C isotherm for InTl

melts at 723 K. Obviously, InTl melt at 723 K represents the segregating behaviour because its value is negative in the complete composition region of In i.e. C I n = 0.1 to 0.9. Again, the magnitude of the negative values of E X S R T advocates the weakly segregating nature of InTl melts at 723 K.

The simultaneous solution of Equations (17a) and (17b) provides the values of C I n s and σ relative to the bulk concentration of In i.e. C I n b ( = C I n ). The necessary parameters are evaluated from Equations (18), (19) and (20). The values of σ for pure In and pure Tl at 723 K are achieved from the expression given below

T m = melting temperature;

σ i ( T ) = surface tension of molten component i in pure state at T K;

σ i ( T m ) = surface tension of pure component i at T m K;

∂ σ i ∂ T = derivative of surface tension for i^th component with respect to temperature.

The values of input parameters are incorporated in Table 1 [27].

Table 1. Surface tension and temperature coefficient of surface tension for Tl and In [27].

The values of σ I n and σ T l at 723 K are found to be 529.590 mN·m⁻¹ and 452.212 mN·m⁻¹, respectively.

The values of V i ( i = I n , T l ) for pure components i.e. In and Tl at 723 K, are determined from the equations given below

with

where,

ρ i ( T ) = density of the i^th component at 723 K;

ρ i ( T m ) = density of the i^th component at melting temperature;

M i = atomic mass of the i^th component;

∂ ρ i ∂ T = temperature coefficient of the density for the i^th component.

The required physical quantities are presented in Table 2 [27].

Table 2.Data for the physical properties of In and Tl [27].

The theoretical values of C I n s relative to the bulk concentration of In i.e. C I n are depicted in Figure 6. It is clear that C I n s rises when the bulk composition of In increases. Again, in the complete composition range of In, C I n = 0.1 to 0.9, C I n s denotes the negative deviation from linearity which advocates that atoms of Tl segregate on the surface for InTl melts at 723 K.

Figure 7 displays the theoretical values of σ in terms of the bulk composition of In for InTl melts at 723 K. Obviously, the value of σ increases when concentration of In increases from 0.0 to 1.0. This arises due to the fact that pure In has larger value of surface tension than pure Tl.

Now, equations (21) and (22) are utilized to study the composition dependency of S C C s ( 0 ) relative to the bulk composition of In for InTl melts at 723 K. The theoretical result is illustrated in Figure 8 along with the ideal values of S C C s ( 0 ) . Obviously, S C C s ( 0 ) exhibits positive deviation from S C C s , i d ( 0 ) in the entire bulk composition of In. The S C C s ( 0 ) has maximum value of 0.3301 at C I n = 0.7 which indicates the maximum possibility of phase separation in InTl melts at 723 K close to C I n = 0.7 [13].

Figure 6. C I n s versus C I n for InTl melts at 723 K.

Figure 7. σ versus C I n for InTl melts at 723 K.

Figure 8. S C C s ( 0 ) versus C I n for InTl melts at 723 K.

Figure 9 displays the theoretical values of D M D i d relative to the composition of In for InTl melts at 723 K. For computations, the theoretical values of S C C ( 0 ) for InTl melts at 723 K are used in Equation (26). It is evident from Figure 9 that D M D i d is less than 1 in the complete composition region of In with the minimum value of 0.7468 at C I n = 0.5 , which is very close to 0. 7465 as available in the literature [5]. This indicates that S C C ( 0 ) is greater than S C C i d ( 0 ) in the region, C I n = 0.1 to 0.9. The minimum value of D M D i d shows that the InTl melt at 723 K is a weakly segregating alloy [28].

The computed data of H M R T are employed to compute the values if η relative to the composition of In for InTl melts at 723 K. The values of η for pure In and pure Tl melts at 723 K are evaluated from the data available in the literature [29], which are found to be 0.9872 (mPa s) and 1.8678 (mPa s) respectively. Figure 10 illustrates the composition dependency of η for InTl melts at 723 K in the region 0.0 ≤ C I n ≤ 1.0 . Obviously, the viscosity of InTl melts at 723 K declines with the elevation of bulk composition of In.

Figure 9. D M D i d versus C I n for InTl melts at 723 K.

Figure 10. η versus C I n for InTl melts at 723 K.