Regular solution model [14] deals with the A-B alloy in liquid state in which the constituent atoms A and B are sufficiently similar in size and shape so that they are interchangeable on the lattice or quasi-lattice, and the configuration is no longer independent of the mutual disposition of the two or more kinds of molecules. We suppose a liquid binary mixture A-B having c_A ( ≡ c) mole of A and c_B { ≡ ( 1 − c ) } mole of B respectively, where c_A and c_B are the mole fractions of A ( ≡ In) and B ( ≡ Tl) in the binary liquid alloy A-B.

The expression for free energy of mixing (G_M) can be derived on the basis of regular soultion model. It is given as [14]

G M = R T [ c ln c + ( 1 − c ) ln ( 1 − c ) ] + c ( 1 − c ) ⋅ ω (1)

where R is the universal gas constant, T the temperature, and ω the interaction energy parameter to be determined.

The term c ( 1 − c ) ⋅ ω represents the heat of mixing in the frame work of regular solution model, i.e.

H M = ω c A c B (2)

The activity a_A of the element A in the binary alloys A-B is given by the standard relation

R T ln a A = G M + ( 1 − c ) ∂ G M ∂ c (3)

Using Equations (1) and (3) we get the activity of element A as

ln a A = ln c + ω R T ( 1 − c ) 2 (4)

Similarly, the activity of the element B is given by

ln a B = ln ( 1 − c ) + ω R T c 2 (5)

The temperature derivative of G_M provides an expression for integral entropy of mixing (S_M) which is given by

S M = − ∂ G M ∂ T (6)

From Equations (1) and (6), we get

S M R = − [ c ln c + ( 1 − c ) ln ( 1 − c ) ] − c ( 1 − c ) ⋅ 1 R ∂ ω ∂ T (7)

The relation between free energy of mixing (G_M), entropy of mixing (S_M), and heat of mixing (H_M) is given by a standard thermodynamic relation,

H M R T = S M R + G M R T (8)

Using Equations (1), (7) and (8), we get

H M R T = c ( 1 − c ) ⋅ ω R T + c ( 1 − c ) 1 R ⋅ ∂ ω ∂ T (9)

The concentration structure factor or concentration fluctuation in the long wavelength limit (S_cc(0)) which has been used to study the nature of atomic order in binary liquid alloy [17] can also be deduce using regular solution model and is given by using relation,

S c c ( 0 ) = R T ( ∂ 2 G M ∂ c 2 ) T , P , N (10)

From Equations (1) and (10), we get

s c c ( 0 ) = c A c B 1 − 2 c A c B ⋅ ω R T (11)

The observed value of concentration fluctuation in the long wavelength limit S_cc(0) is obtained from observed data of the activities of the constituent species of the binary liquid alloys from the relation

s c c ( 0 ) = ( 1 − c ) a A ( ∂ a A ∂ c ) T , P , N − 1 = c a B ( ∂ a B ∂ c ) T , P , N − 1 (12)

where, a_A and a_B are the activities of the component of A and B respectively.

The degree of local order in the liquid mixture is quantified by the Warren-Cowley [18] [19] short range order parameter (α₁). It provides insight into the local arrangement of the atoms in the molten alloys. The theoretical values of (α₁) can be evaluated as

α 1 = s − 1 s ( Z − 1 ) + 1 (13)

where, s = S c c ( 0 ) S c c i d ( 0 ) , s c c i d ( 0 ) = c A c B and Z is the coordination number, which is taken 10 (= Z) for our purpose.

Butler derived an expression for the surface tension of liquid binary mixture on the basis of the assumption that there exists a mono atomic layer at the surface of a liquid solution, called surface monolayer, as a separate phase that is in thermodynamic equilibrium with the bulk phase [15] [20] . The expression for surface tension, Γ on the basis of Butler equation for binary A-B solution at temperature T is given by

Γ = Γ 1 + 1 A 1 ( G 1 E , s − G 1 E , b ) + R T A 1 [ ln ( 1 − c 2 s ) − ln ( 1 − c 2 b ) ] = Γ 2 + 1 A 2 ( G 2 E , s − G 2 E , b ) + R T A 2 [ ln ( c 2 s ) − ln ( c 2 b ) ] (14)

where, R is universal gas constant. c i s and c i b are mole fraction of component “ i ” in the surface and bulk respectively. Γ₁ and Γ₂ are the surface tension of the pure component 1 and 2 respectively. G i E , s and G i E , b ( i = 1.2) are partial excess free energy of component “ i ” in the surface and the bulk respectively. The molar surface area of the component “ i ” can be computed by using the relation

A i = K ⋅ N A 1 / 3 ⋅ V i 2 / 3 (15)

where, K (=1.091) is geometrical factor for the liquid alloy, N_A is Avogadro’s number, V_i (= V I n , V T l ) is the molar volume of the component i.

where, V I n = V M , I n [ 1 + α P , I n ( T − T M , I n ) ] and V T l = V M , T l + [ 1 + α P , T l ( T − T M , T l ) ]

V M , I n = atomic volume of In at melting point

V M , T l = atomic volume of Tl at melting point

T M , I n = melting temperature of In

T M , T l = melting temperature of Tl

α P , I n = volume coefficient of In at constant temperature

α P , T l = volume coefficient of Tl at constant temperature

The diffusion coefficients also provide the insight into the mixing behavior of an alloy in microscopic level. The relation between diffusion coefficient and concentration fluctuation is given as [21]

D M D i d = S C C i d ( 0 ) S C C ( 0 ) (16)

With

D M = c 1 D 2 + c 2 D 1 (17)

where, D_M is the mutual diffusion coefficient and D_id is the intrinsic diffusion coefficient for an ideal mixture; D₁ and D₂ are the self-diffusivities of pure components A and B respectively.

In term of energy order parameter ω, the diffusion coefficient can be expressed as [22]

D M D i d = [ 1 − 2 ω R T S c c i d ( 0 ) ] (17) If D M / D i d > 1 , then it indicates compound formation and if D M / D i d < 1 then there is tendency of phase separation. For ideal mixing, D M / D i d approaches 1.

The mixing behavior of liquid alloys at microscopic level can also be understood in terms of viscosity. We have employed the Moelwyn-Hughes equation [16] in order to observe the atomic transport behavior of In-Tl alloy. The Moelwyn-Hughes equation for viscosity of liquid mixture is given as

η = ( c 1 η 1 + c 2 η 2 ) ( 1 − c 1 c 2 ⋅ H M R T ) (18)

where, η K is the viscosity of pure component K and for most liquid metals, it can be calculated from Arrhenius type equation [21] at temperature T as

η K = η O K exp [ E n R T ] (19)

where, η O K is constant (in unit of viscosity) and E_n is the energy of activation of viscous flow for pure metal (in unit of energy per mole).

The model parameters which can be used to optimize thermodynamic, structural and transport properties of the liquid mixture can be obtained by thermodynamic description based on statistical thermodynamics or polynomial expressions. The adjustable coefficients, used in the process, are estimated by least square method which gives an idea to extrapolate into temperature and concentration region in which the direct observed determination is unavailable.

Redlich-Kister polynomial equation gives the composition dependence of excess free energy of mixing and is given by

G M X S ( c , T ) = c ( 1 − c ) ∑ l = 0 m K l ( T ) [ c − ( 1 − c ) ] l (20)

with

K l ( T ) = A 1 + B 1 T + C 1 T ln T + D 1 T 2 + ⋯ (21)

The coefficients K_l are the function of the temperature.

The values for the free energy of mixing (G_M) of In-Tl liquid alloy at different temperatures can be calculated from Equation (1) by knowing the values of ordering energy parameter (ω) at different temperatures from the relation [23]

ω ( T K ) = ω ( T ) + d ω d T ( T K − T ) (22)

where, ω(T) is the order energy parameter at the temperature 723 K, ω(T_K) is order

energy parameter at required temperature T_K, and d ω d T represents temperature

derivative of order energy parameter which has already been estimated from observed data of entropy of mixing (S_M) of In-Tl liquid alloys at 723 K from Hultgren et al. 1973 [24] .

The value of main input parameters i.e. interchange energy (ω) and 1 R ∂ ω ∂ T

used for the calculation of thermodynamic properties of In-Tl liquid alloy at 723 K were computed from Equations (1) and (9) by using observed values [24] of G_M and H_M respectively. The best fit parameters were found to be

ω R T = 0.52 ,     1 R ∂ ω ∂ T = 0.15

The positive value of interaction energy indicates that In-Tl is segregating in nature. Using above interchange energy (ω), we computed free energy of mixing (G_M), activity (a) entropy of mixing (S_M), heat of mixing (H_M), concentration fluctuation (S_cc(0)), ratio of diffusion coefficient (D_M/D_id), short-range order parameter (α₁), surface tension (Γ₁) and viscosity (η) at 723 K using regular solution model.

The plot of observed and computed values of free energy of mixing with respect to the concentration of Tl is shown in Figure 1. The computed and observed values [24] of free energy of mixing are in good agreement in the entire concentration range. Both the computed and observed value of free energy of mixing are minimum at C T l = 0.5 , which indicates that In-Tl alloys in liquid state at 723 K is symmetric at equiatomic composition.

We computed the activities of In and Tl in the liquid In-Tl alloys using the same energy parameter (ω) as used in Equation (1), and then compared them with observed values as depicted in Figure 2. The plot shows that there is a good agreement between the sets of computed and observed results. Activity coefficient represents the measure of tendency of a component to leave the solution and it provides correlation of the behavior of the systems.

The heat of mixing and entropy of mixing of In-Tl binary liquid alloy at 723K were computed from Equations (9) and (7) respectively. To determine heat of mixing (H_M) and entropy of mixing (S_M) we need temperature derivatives of energy parameters. The observed values of heat of mixing [24] were utilized to obtain the temperature derivatives by the successive approximation. The best fit

parameters was found to be 1 R ∂ ω ∂ T = 0.15 . Both computed and observed values

of heat of mixing and entropy of mixing are shown in Figure 3 and Figure 4. Both the computed and observed value of heat of mixing and entropy of mixing are maximum at C T l = 0.5 . The positive value of heat of mixing indicates the segregating nature of the alloy which is in agreement with the sign of the interaction energy ω. The value of entropy of mixing is positive in whole range of concentration as depicted in Figure 4.

We computed concentration fluctuation in the long wavelength limit S_cc(0) using Equation (11) with the help of same energy parameter ω for the consistency as used for the computation of free energy of mixing, activity and heat of mixing. The observed S_cc(0) were computed from Equation (12). It was found that the computed and observed values of S c c ( 0 ) > S c c i d ( 0 ) at all compositions. This indicates that segregation is favored in the In-Tl alloy at 723 K. The computed values of S_cc(0) are in good agreement with the observed values in entire concentration range. The theoretical value is maximum at C T l = 0.5 (i.e. S c c ( 0 ) = 0.3378 ) as depicted in Figure 5.

By using the best fit value of d ω d T and ω(T) at the given temperature T = 723 K

in Equation (22), the value of ω(T_K) at temperature T_K = 623 K, 800 K, 923 K are estimated and listed in Table 1.

The values of free energy of mixing (G_M) of In-Tl liquid alloy at different temperatures (i.e. 623,723,800 and 923 K) computed by using the corresponding values of ω(T) in Equation (22) over the entire range of concentration and then they are used to calculate the corresponding excess free energy of mixing ( G M X S ) of the alloy at temperature of study by using Equation (20) conjugation with Equations (21) and (22).

The least-square method was used to calculate the parameters involved in Equation (21) and then the optimized coefficients for the alloy are computed which are listed in the Table 2.

We used the parameters i.e. k₀, k₁, k₂ and k₃ to obtain partial excess free energy. The partial excess free energy of mixing ( G M − X S ) of the components A (=In) and B (=Tl) were computed and tabulated below using equations [22]

G M , A − X S ( C , T ) = ( 1 − C ) 2 ∑ l = 0 m K l ( T ) [ ( 1 + 2 l ) c − ( 1 − c ) ] ( 2 c − 1 ) l − 1 (23)

and

G M , B − X S ( C , T ) = C 2 ∑ l = 0 m K l ( T ) [ c − ( 1 + 2 l ) ( 1 − c ) ] ( 2 c − 1 ) l − 1 (24)

The partial excess free energy of mixing of both the components In and Tl involved in In-Tl liquid alloy at different temperatures (i.e. = 623 K, 723 K, 800 K, 923 K) were calculated separately over the entire concentration range by Equations (23) and (24) with the help of optimized coefficients. With the help of this optimized partial excess free energy of mixing of both the components in the alloy have been used to calculate the corresponding excess free energy of the alloy at different temperatures over the entire range of concentration from the relation

G M X S = C G M , A − X S + ( 1 − C ) G M , B − X S (25)

The optimized values of excess free energy of mixing for the alloys at different temperatures (i.e. = 623 K, 723 K, 800 K, 923 K) over the entire concentration range are shown in Figure 6.

Now, the activity coefficients (Υ_i), (i = In or Tl) at different temperature over the entire range of concentration for In or Tl components have been computed

from the relation G ¯ M , i X S = R T ln ϒ i , with ϒ i = a i c i , where, a_i and c_i be the activity

and concentration of the component respectively of In-Tl liquid alloy at corresponding temperature.

The calculated optimized values partial excess free energy of mixing, the corresponding activity coefficients and corresponding activity of both the components involved in In-Tl liquid alloy in the entire concentration range at the temperature T = 623 K, 723 K, 800 K, 923 K are shown in Table 3, Table 4, Table 5 and Table 6.

The concentration fluctuations in long wavelength limit S c c ( 0 ) of the alloy at different temperatures in entire concentration range have been calculated using Equation (14) with the help of the optimized values of the activity of both the components which is shown in Figure 7. The values of concentration fluctuation decreases as the temperature of the alloy increases correspond to the concentration of Tl component. Also, S c c ( 0 ) > S c c i d ( 0 ) at entire concentration range of Tl at all temperature of investigation which indicates segregating nature of the alloy.

The natural logarithms of optimized values of activity of both the components in the alloy at different temperatures in the entire concentration range are shown in Figure 8.

The Warren-Cowley chemical short-range order parameter α₁ is used to explain the degree of local arrangement of atoms in the mixture. The chemical short-range order parameter was computed from Equation (13) using coordination number, Z = 10 for liquid the alloy at 723 K. The value of short range order parameter has been found positive in all concentration range which indicates that the alloy is segregating at all compositions. The value of short range order parameter has been found maximum at C_Tl = 0.5 at 723 K as shown in Figure 9. We have computed chemical short-range order parameter at temperatures 723 K, 623 K, 800 K and 923 K. The value of short range order parameter has been found maximum at C_Tl = 0.5 at all temperature of study as shown in Figure 9. The plot depicts that α₁ is positive in the entire range of concentration showing that α₁ in In-Tl is segregating.

The surface tension of the liquid alloys can be computed using Equation (14) conjugation with Equations (15) and (16) using Buttler’s model [15] [20] [25] . The ratio of partial excess Gibbs energy in the bulk and that in the surface can be

expressed as β = G i E , s G i E , b , β has been taken as 0.83 [26] [27] . We have taken the

surface tension of In-Tl and temperature coefficients for pure In and Tl components from the reference [28] . The surface tension of the pure component at temperature of investigation has been calculated using the relation.

Γ ( T ) = Γ m + ∂ Γ ∂ T ( T − T m ) , where, ∂ Γ ∂ T (= −0.09 mNm⁻¹K⁻¹ for In, −0.08 mN

m⁻¹K⁻¹ for Tl) is temperature coefficient of surface tension, Tm (= 430 K for In and 577 K for Tl) is melting temperature and T = 723 K.

The partial excess free energy of mixing of the pure components was taken from Hultgren et al. [24] . It was found from the analysis that the computed surface tension for In-Tl system at 723 K is less than ideal value (= X₁Γ₁ + X₂Γ₂) at all the concentration of In i.e. there is negative departure of surface tension from ideality at 723 K. It was found that surface tension increases with increase in the concentration of component Tl on the alloy as shown in the Figure 10.

To find the surface tension (Γ) of the alloy at different temperature we have used the values of optimized partial excess free energy i.e. G ¯ M X S from the Tables 3-6, and using Buttler’s Equation (16) conjugation with Equation (17). The calculated values of surface tension at different temperatures are shown in Figure 10. The surface tension of the alloy in the concentration range of Tl component showed that as the temperature increases, the surface tension decreases.

Viscosity of pure component In and Tl at 723 K are calculated using Equation (19) which is required to compute viscosity of In-Tl alloy at 723 K with the help of the value of the constants E and η_ok for the metals [28] . To compute viscosity using Moelwyn-Hughes equation [16] , heat of mixing H_M is required which is taken from regular solution model calculations. The result of viscosity at 723 K of the alloy with the ideal values ( η = c A η A + c B η B ) is shown in Figure 11. The graph shows that with the increase in concentration of Tl component in the alloy, the viscosity of the alloy decreases (Figure 11).

The optimized viscosity of the alloys is calculated by the Moelwyn-Hughes Equation (22) conjugation with Equation (23). The values of optimized H_M/RT is used in whole concentration range obtained using Equation (11) with the help of

optimized values of ω(T) and d ω d T at temperature of study. The values of

parameters on Equation (23) is taken from [28] . The plot exhibits that the viscosity show small negative deviation from ideality at all compositions at 723 K. The computed values of viscosity at different temperatures is as shown in Figure 11.

The diffusion coefficients D M D i d of In-Tl liquid alloy at temperatures of study

in the entire concentration range can also be calculated using the optimized data from the relation (19) which are shown in the Figure 12. The D M / D i d < 1 which indicates that there is tendency of phase separation.