Optimization method for the study of the properties of Al-Sn binary liquid alloys

Author’s Accepted Manuscript Optimization method for the study of the properties of Al-Sn binary liquid alloys G.K. Shrestha, B.K. Singh, I.S. Jha, B.P. Singh, D. Adhikari www.elsevier.com/locate/physb

PII: DOI: Reference:

S0921-4526(17)30107-2 http://dx.doi.org/10.1016/j.physb.2017.03.005 PHYSB309849

To appear in: Physica B: Physics of Condensed Matter Received date: 3 January 2017 Accepted date: 2 March 2017 Cite this article as: G.K. Shrestha, B.K. Singh, I.S. Jha, B.P. Singh and D. Adhikari, Optimization method for the study of the properties of Al-Sn binary liquid alloys, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.03.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Optimization method for the study of the properties of Al-Sn binary liquid alloys G. K. Shrestha1,2, B. K. Singh1, I. S. Jha3, B.P. Singh1, D. Adhikari3* 1

University Department of Physics, T.M. Bhagalpur University, Bhagalpur, India 2

Pulchowk Campus, IOE, Lalitpur, Tribhuvan University, Nepal 3

M.M.A.M. Campus, Biratnagar, Tribhuvan University, Nepal *

Corresponding author: [email protected]

Abstract The best fit value of order energy parameter (W) has been estimated over the entire range of concentration in Al-Sn binary liquid alloy at a specified temperature to determine the thermodynamic properties and concentration fluctuations, obtained by a theoretical formalism in which the combined effect of size ratio, entropic and enthalpic effect is considered. The values of W at different temperatures have been determined by finding the temperature derivative of W which are then used for the optimization procedure in order to determine the corresponding values of excess free energy of mixing, partial excess free energy of mixing and activity of the components involved in the alloy. These parameters have been used to calculate the concentration fluctuations in long wavelength limit {Scc(0)} at different temperatures over the entire range of concentration which predict the stability of the alloy at different temperatures. Keyword: Concentration Fluctuations, Excess free energy, Order energy parameter, Optimization procedure, Stability.

1

INTRODUCTION

Aluminium based alloys have wide applications in modern age in many fields such as engineering infrastructure, aerospace industries, automobile industries, marine industries, even in daily-life uses, etc. So, the production of aluminium in present days is the second highest after the iron in the world. Fortunately, it is the most abundant metal in the Earth's crust so that we can easily fulfill the present demand. Aluminium is a relatively soft, durable, lightweight, nonmagnetic, ductile, and malleable metal. It has good thermal and electrical conductivity but it does not easily ignite. It is non-toxic so that it may be used in our daily use and even in the replacement of lead, used in solder, in future [1]. One of the important structural limitation of aluminium is its fatigue strength, which can be improved markedly by choosing a suitable component in aluminium alloy. In the other hand, tin is a soft, ductile, malleable and highly crystalline silvery-white metal which has low melting temperature. Tin is mostly used in making alloys with lead which is used as solder but the lead is going to be replaced by appropriate metal, may be aluminium, because of toxic nature of lead. A heavy amount of tin is used in making other alloys like brass, bronze, etc. It is also used in tin plating for coating lead, zinc and steel containers to prevent corrosion which are widely used for food preservation. Al-Sn alloys may be used as solder. It can also be used as bearing alloy which requires bearing material properties like conformability, fatigue strength, wear resistance, corrosion resistance, cavitation resistance, etc. But, Aluminium tin alloys have a wide miscibility gap in the molten state and are virtually insoluble in each other during solidification because of huge difference in density and melting temperature of the two components used in this alloy [2]. Therefore, investigations in manufacturing process as well as theoretical investigations to study of the thermodynamic and structural properties of alloys in liquid state are very important for the explanation of mixing behaviors of the constituents and subsequently the corresponding properties of crystalline materials. Several theoretical models [3]–[17] have long been used by metal physicists [18]–[25]to understand the mixing properties of alloys in molten state. In this paper we have used quasi-lattice theory by considering the combined effect of size ratio, entropic and enthalpic effect to predict the values of thermodynamic and structural properties of Al-Sn liquid alloys at different temperatures [4], [26] which will be discussed in section-3. We have used the optimization procedure [26] to compute the excess free energy of mixing and optimized coefficients of Al-Sn liquid alloys which are then used to determine the partial excess free energy of mixing as well as activity of both the components and hence concentration fluctuations in long wave-length limit {Scc(0)} at different temperatures over the entire range of concentration. They are again used to predict the stability of Al-Sn liquid alloys at different temperatures with the help of excess stability function.

2

Formalism

2.1 Thermodynamic and structural properties We consider one mole binary liquid mixture of concentration cA= c of constituent A ( Al) and cB = (1-c) of constituent B ( Sn) so that cN = NA(No. of constituent A) and (1-c)N = NB(No. of constituent B). Suppose the constituent atoms A and B of the binary mixture have different shape and size. If we consider the entropic as well as enthalpic effect of the binary mixture (i.e. HM ≠ 0), then Quasi- Lattice Theory (QLT) of liquid mixture as explained by Guggenheim [4] in the limit Z(the coordination number) at a temperature, T provides an expression for free energy of mixing (G M) as ln ψ + c ln with

ψ + c ψ W]

------- (1)

=

and W

γ (

------- (2)

)

------- (3)

which represents order energy parameter, γA is the number of a group of lattice sites, occupying by N A atoms of the constituent A and ω is the interchange energy, given by =( Here,

)

------- (4)

are i-j bond strength.

Also, R is the molar gas constant, KB is the Boltzmann constant and Ω ( =

) is the size ratio. In equation (2),

the first two terms on right hand side exist due to entropic contribution and last term is due to enthalpic contribution. Again, the concentration fluctuations in long wavelength limit {S cc(0)} at a temperature (T) as defined by Bhatia and Thornton [7] is given by (

)

------- (5)

On solving equation (1) and (5), we obtain ------- (6)

–

with

-------(7)

By using the concept of activity, the concentration fluctuations can also be determined from the relation S

ca (

)

where ci and aj are the concentration and activity of component i, and j (i, j

------- (8) ) respectively.

Also, the ideal value of concentration fluctuations is obtained from the relation c

------- (9)

The concentration fluctuations in long wavelength limit{Sc c(0)} is the structural property of binary liquid alloy which represents the reciprocal of stability of the alloy so that it is very useful to know the nature of liquid alloy. Though it is very difficult to measure the value of Sc c(0) directly from experiment, it can be calculated from the experimental value of activity by using equation (8), which represents its experimental value. The mixing behavior, i.e. the nature of interaction of components in binary liquid alloys can be analyzed with the help of deviation of S cc(0) from (0). For the given composition, the binary liquid alloy of ordering nature is expected if S (0), but the alloy of segregating nature is expected if Sc c(0) (0). From the standard thermodynamic relation, the Entropy of mixing for a binary liquid alloys is given by

On solving this equation by using equation (1), we obtain ln

+ ln

+

+

]

-------- (10)

The heat of mixing for a binary liquid alloys is given by =

+

-------- (11)

The activity of a constituent i of the binary alloys is given by ln

+

(

)

------- (12)

where ai and cj are the activity and concentration of component i and j ( i, j

A, B) respectively.

On solving, equation (1) and (12) provide ln a

ln

ψ + ψ{

ln

ln

}+ ψ W

------ (13)

and +

------ (14)

where aA and aB are the activity of the constituent A and B respectively. 2.2 Optimization of free energy of mixing, activity and concentration fluctuations The process in which statistical thermodynamics and polynomial expressions are used for the thermodynamic description is known as optimization procedure [26]. It has good potentiality to obtain an accordant set of model parameters in an analytical approach which provides idea to extrapolate into temperature and concentration region in which the direct experimental determination is not available. The adjustable coefficients, used in this process, are estimated by least-square method. The various thermodynamic properties, described by a power-series law whose coefficients are A, B, C, D, E, …….(say), are determined by least-square method. The heat capacity can be expressed as 𝐶

𝐶

2𝐷

2𝐸

… … … … ..

------- (16)

From the thermodynamic relation, the enthalpy is given by 𝐻

𝐻 𝐴

∫ C dT 𝐶

𝐷

+ 2𝐸

………………

-------- (17)

Also, the entropy is given by +∫ 𝐵

𝐶

dT + ln

2𝐷 + 𝐸

…………..

-------- (18)

On using equation(15) and (16), the standard thermodynamic relation, G = H – TS, provides the temperature(T) dependent free energy as 𝐴 + 𝐵 + 𝐶 ln

+𝐷

+𝐸

+ …………..

-------- (19)

The composition dependence of excess free energy of mixing is given by Redlich-Kister polynomial equation as ∑

c

c ]

-------- (20)

with

𝐴 + 𝐵

+ 𝐶

ln

+ 𝐷

+ …………

-------- (21)

The coefficients depend upon the temperature same as that of G in equation (19). The least-square method can be used to obtain the parameters involved in equation (20). For this purpose, we require the excess free energy of mixing ( ) of the binary liquid alloys at different temperatures. The values of can be determined by the relation

ln +

ln

c ]

------- (22)

Thus, we need the values for the free energy of mixing (GM) of binary liquid alloys at different temperatures which can be computed from equation (1) for different temperatures by knowing the values of order energy parameter (W) at different temperatures from the relation +

T

T

-------- (23)

where W(T) is the order energy parameter at the given temperature T, W(Tk) is order energy parameter at required temperature Tk and represents temperature derivative of order energy parameter for the given liquid alloy. 3 RESULTS AND DISCUSSION We require two parameters i.e. size ratio (Ω) and order energy parameter (W) to calculate the free energy of mixing (G M) and activity of both the components (a i) in binary liquid alloys at a specified temperature. The parameter Ω for a binary liquid alloy has been determined by knowing the atomic volumes of the constituent i at the specified temperature, T from the relation +

]

where i = A; B, is the atomic volume of the constituent i at the melting temperature (T m) and of thermal expansion of the constituent i.

------ (15) is the coefficient

On taking the values of α p i and of Al-Sn [13] for Al-Sn binary liquid alloy at 973K, the value of size ratio, Ω is found to be 1.5598. The value of order energy parameter, W has been estimated from equation (1), (13) and (14) to reproduce simultaneously an overall fit for the experimental values of free energy of mixing (G M), and activity (ai) of the component i (i= A, B) in Al-Sn liquid alloys at 973K [28] over the entire concentration range from 0.1 to 0.9. The best fit value of W for Al-Sn liquid alloys at 973K is found to be 1.25. The theoretical values of GM/RT as computed by using equation (1) on taking the above values of Ω and W, and experimental values of G M /RT of Al-Sn liquid alloys at 973K in the entire concentration range from 0.1 to 0.9 are shown in Fig-1, which are in excellent agreement. The experimental value of GM/RT at 973K is minimum at the concentration, c = 0.4, which is found to be -0.350838 and the theoretical value of GM/RT at 973K is minimum at the same concentration, c= 0.4 which is found to be -0.345581. The theoretical values of concentration fluctuations in long wave-length limit {Scc(0)} for Al-Sn binary liquid alloys at 973K over the entire concentration range have been calculated by taking the above mentioned value of (i.e. 1.5598) and W (i.e. 1.25) in equation (6), while the experimental values of S c c(0) have been calculated by taking the experimental values of activity in equation (8). The theoretical and experimental values S c c(0) of Al-Sn liquid alloys at 973K in the entire concentration range are shown in Fig-2, which are in excellent agreement. The experimental value of Sc c(0) is maximum at concentration, c = 0.6 which is found to be 0.942175 while the theoretical value of S c c(0) is maximum at the same concentration, c = 0.6 which is found to be 0.944859. It is obvious that the concentration fluctuations in long wave-length limit {Sc c(0)} of Al-Sn liquid alloy at 973K is greater than the value of (0) ( c (1-c) ) at each concentration which indicates the segregating nature of Al-Sn liquid alloy. By taking the above mentioned values of Ω and W for Al-Sn liquid alloys at 973K, the value of temperature derivative of order energy parameter, i.e. , has been estimated from equation (10) or (11) to reproduce simultaneously an overall fit for the experimental values of entropy of mixing (SM) or heat of mixing (SM) for Al-Sn liquid alloy at 973K [28] over the entire concentration range from 0.1 to 0.9. The best fit value of for Al-Sn liquid alloys at 973K is found to be– 0.0017K-1. The theoretical values of SM/R as computed by using equation (10) and experimental values of SM /R of Al-Sn liquid alloys at 973K in the entire concentration range from 0.1 to 0.9 are shown in Fig-3, which are in excellent agreement. The experimental value of SM/R at 973K is maximum at the concentration, c = 0.5, which is found to be 0.840322 and the theoretical value of SM/R at 973K is maximum at the same concentration, c = 0.5 which is found to be 0.840768.

Again, from the best fit value of for Al-Sn liquid alloy, i.e.– 0.0017K-1, the theoretical values of HM/RT as computed by using equation (11) and experimental values of H M /RT of Al-Sn liquid alloys at 973K [28] in the entire concentration range from 0.1 to 0.9 are shown in Fig-4, which are in excellent agreement. The experimental value of H M/RT at 973K is maximum at the concentration, c = 0.6, which is found to be 0.503488 and the theoretical value of HM/RT at 973K is maximum at the same concentration, c = 0.6 which is found to be 0.505928. The theoretical values of ln (i = A, B) as computed by using equation (13) and (14) (on taking Ω = 1.5598 and W = 1.25), and experimental values of ln (i = A, B) of Al-Sn liquid alloys at 973K [28] over the entire concentration range are shown in Fig-5, which are in excellent agreement. It is obvious that the present theoretical model successfully explain the free energy of mixing (G M), entropy of mixing (SM), heat of mixing (HM), activity coefficient ( ai) of both the components and concentration fluctuations in long wave-length limit {Sc c(0)}of Al-Sn liquid alloys at 973K, i.e. their theoretical and experimental values are in excellent agreement. Thus, the estimated values of Ω = 1.5598, W= 1.25 and = – 0.0017K-1for Al-Sn liquid alloys at 973K. These values may be used in optimization procedure to predict the stability of Al-Sn liquid alloys at different temperatures. By using the best fit value of and W(T) at the temperature T= 973K (as obtained in section-3) in equation (23), the values of W(Tk) at temperature Tk = 1073K, 1123K, 1123K are estimated and listed in Table-1. It is to be noted that the change in temperature may change the size ratio of Al-Sn liquid alloy but the consequent change in the values of various thermodynamic properties of the alloys is only negligible. Thus, for entire calculation of thermodynamic properties, the value of size ratio is taken to be constant (i.e. Ω = 1.5598 for Al-Sn liquid alloy at all temperatures). The values of free energy of mixing (GM) of Al-Sn liquid alloys at different temperatures (i.e. at 973K,1073K, 1123K and 1173K) have been computed by using the corresponding values of W in equation (1) over the entire range of concentration and then they are used to calculate the corresponding excess free energy of mixing ( ) of AlSn liquid alloys at different temperatures (i.e. at 973K,1073K, 1123K and 1173K) by using equation (22). The least-square method has been used to calculate the parameters involved in equation (20) and then the optimized coefficients for Al-Sn liquid alloys are computed which are listed in the Table-2. Again, the partial quantities, i.e. the partial excess free energy of mixing of the components A( liquid alloys are given by ̃

∑

+ 2l c

c ] 2c

Al) and B( Sn) in Al-Sn

---------(24)

And ̃

∑

c

+ 2l

c ] 2c

-------- (25)

These equation (24) and equation (25) have been used to compute the partial excess free energy of mixing for Al and Sn component separately of Al-Sn liquid alloys at different temperatures (i.e. 973K, 1073K, 1123K and 1173K) with the help of optimized coefficients. Also, the optimized values of excess free energy of mixing for Al-Sn liquid alloys at a temperature (T) over the entire concentration range can be determined by using the corresponding partial excess free energy of mixing of the two components involved in the alloy from the relation {̃

}+

̃

---------- (26)

The optimized values of excess free energy of mixing for Al-Sn liquid alloys at the temperature, T= 973K, 1073K, 1123K and 1173K over the entire concentration range have been computed by using equation (26) which are shown in Fig-6. Further, the activity coefficients (γi), (i Al or Sn), over the entire range of concentration for Al and Sn component separately at corresponding temperatures have been computed from the relation ̃

ln

with where ai and ci be the activity and concentration of the component i (i corresponding temperature.

-------- (26) -------- (27) Al or Sn) respectively of Al-Sn liquid alloy at

The optimized values of partial excess free energy of mixing, the corresponding activity coefficients and hence corresponding activity of both the components present in Al-Sn liquid alloys in the entire range of concentration at the temperature T=973K, 1073K, 1123K and 1173K are listed in Table-3, Table-4, Table-5 and Table-6 respectively. The optimized values of the natural logarithms of activity of both the components in Al-Sn liquid alloys at different temperatures in the entire concentration range are shown in Fig-7. Finally, the optimized values of the activity of both the components have been used to calculate the concentration fluctuations in long wave-length limit {Scc(0)} of Al-Sn liquid alloys at different temperatures by using equation (8) which are shown in Fig.-8. It is obvious that the concentration fluctuations in long wavelength limit of Al-Sn liquid alloys at the temperature above 973K decreases and shift towards the ideal value of concentration fluctuations ( ) at each concentration as the temperature increases but remains greater than the ideal value of concentration fluctuations which indicates that the Al-Sn liquid alloy has segregating nature but tendency of segregation decreases as the temperature increases. It is also to be noted that the critical temperature (Tc) for Al-Sn liquid alloy is 746.92K and the critical concentration fluctuations is 20378.1231 which occurs at the concentration, c = 0.7. From the relation for excess stability function 𝐸

(

)

[

] [26] the excess stability function at

the temperature T = 973K, 1073K, 1123K and 1173K in the entire range of concentration have been computed from optimized data which are shown in Fig-9 and they give the idea to predict stability of Al-Sn liquid alloys at different temperatures. From the optimized data, the diffusion coefficient

of Al-Sn liquid alloy at different temperatures in the entire

concentration range can also be computed from the relation

[26] , which are shown in Fig-10.

5 CONCLUSION From this theoretical investigation in Al-Sn liquid alloys, following conclusions may be drawn:    

The free energy of mixing (GM), entropy of mixing (SM), heat of mixing and concentration fluctuations {Sc c(0)} of Al-Sn liquid alloy are symmetrical or very close to it at the equiatomic composition of its constituents (i.e. c = 0.5) so that it is symmetric alloy. The values of concentration fluctuations over the temperature range 973K to 1173K in Al- Sn liquid alloys in the entire concentration range are always greater than the ideal value of concentration fluctuations which indicate the segregating nature of the alloys in this range of temperature. The order energy parameter (W) in Al-Sn liquid alloy is temperature dependent. The values of W between temperature range 973K and 1173K in Al-Sn liquid alloys are +ve but its value decreases as the temperature increases which indicates that the segregating nature of the alloy decreases as the temperature increase. The concentration fluctuations in Al-Sn liquid alloy at each concentration decreases as the temperature increases However, the rate of decrement in concentration fluctuations near the lower range of temperature is high than that at higher range of temperature so that the rate of increment in stability near the lower range of temperature is high than that at higher range of temperature.

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Fig-1: Graph for Fig-2: Graph for

versus

of Al-Sn liquid alloys at 973K

versus

of Al-Sn liquid alloys at 973

Fig-3: Graph for

versus

of Al-Sn liquid alloys at 973K

Fig-4: Graph for

versus

of Al-Sn liquid alloys at 973K

Fig-5: Graph for

𝐴

versus

Fig-6: Graph for optimized values of Temperatures

of Al-Sn liquid alloys at 973K

versus

of Al-Sn liquid alloys at different

Fig-7: Graph for optimized values of different Temperatures

𝐴

Fig-8: Graph for optimized values of Temperatures

versus

Fig-9: Graph for optimized values of Temperatures Fig-10: Graph for optimized values of

versus

versus

versus

of Al-Sn liquid alloys at

of Al-Sn liquid alloys at different

of Al-Sn liquid alloys at different

of Al-Sn liquid alloys at different

Temperatures

Table-1: Estimated values of order energy parameter (W) at different temperatures in Al-Sn liquid alloy. Temperature (Tk) in K

Order Energy Parameter, W(Tk)

973

1.250

1073

1.080

1123

0.995

1173

0.910

Table-2: Calculated values of optimized coefficients A l , B l , C l , D l (l = 0 to 3) for Al-Sn liquid alloy Values of l

A l (J mol -1)

B l (J mol -1 K - 1)

C l (J mol-1K-1)

D l (Jmol-1K- 2)

0

-0.000000104

28.609008544

0.000000000

-0.017224128

1

0.000000072

6.315294715

0.000000000

-0.003766727

2

0.000000073

1.414737590

0.000000000

-0.000839807

3

-0.000000198

0.310246572

0.000000000

-0.000183657

Table-3: Optimized values of partial excess free energy of mixing, activity coefficients and activity of both the components present in Al-Sn liquid alloys at 973K.

𝐶

Al- component

̃

ln

Sn-component

̃

ln

̃

ln

̃

ln

0.1

8223.479

1.017

2.764

0.276

-1.2860

63.642

0.008

1.008

0.907

-0.097

0.2

7022.152

0.868

2.382

0.476

-0.7414

277.350

0.034

1.035

0.828

-0.189

0.3

5825.280

0.720

2.055

0.616

-0.4839

678.454

0.084

1.087

0.761

-0.273

0.4

4653.239

0.575

1.778

0.711

-0.3411

1312.431

0.162

1.176

0.706

-0.349

0.5

3527.174

0.436

1.547

0.773

-0.2571

2237.818

0.277

1.319

0.659

-0.417

0.6

2473.920

0.306

1.358

0.815

-0.2050

3531.125

0.437

1.547

0.619

-0.480

0.7

1530.914

0.189

1.208

0.846

-0.1674

5291.755

0.654

1.924

0.577

-0.550

0.8

751.109

0.093

1.097

0.878

-0.1303

7646.914

0.945

2.574

0.515

-0.664

0.9

207.893

0.026

1.026

0.923

-0.0797

10756.529

1.330

3.780

0.378

-0.973

Table-4: Optimized values of partial excess free energy of mixing, activity coefficients and activity of both the components present in Al-Sn liquid alloys at 1073K. 𝐶

Al- component

̃

ln

Sn-component

̃

ln

̃

ln

̃

ln

0.1

7746.759

0.868

2.383

0.238

-1.434

59.820

0.007

1.007

0.906

-0.099

0.2

6616.968

0.742

2.100

0.420

-0.868

260.809

0.029

1.030

0.824

-0.194

0.3

5490.738

0.615

1.851

0.555

-0.588

638.249

0.072

1.074

0.752

-0.285

0.4

4387.300

0.492

1.635

0.654

-0.424

1235.129

0.138

1.148

0.689

-0.372

0.5

3326.600

0.373

1.452

0.726

-0.320

2106.815

0.236

1.266

0.633

-0.457

0.6

2333.963

0.262

1.299

0.779

-0.249

3325.711

0.373

1.452

0.581

-0.543

0.7

1444.758

0.162

1.176

0.823

-0.195

4985.923

0.559

1.749

0.525

-0.645

0.8

709.061

0.079

1.083

0.866

-0.144

7207.921

0.808

2.243

0.449

-0.801

0.9

196.315

0.022

1.022

0.920

-0.083

10143.204

1.137

3.117

0.312

-1.166

Table-5: Optimized values of partial excess free energy of mixing, activity coefficients and activity of both the components present in Al-Sn liquid alloys at 1123K. 𝐶

Al- component

̃

ln

Sn-component

̃

ln

̃

ln

̃

ln

0.1

7416.002

0.794

2.213

0.221

-1.508

57.184

0.006

1.006

0.906

-0.099

0.2

6335.609

0.679

1.971

0.394

-0.931

249.391

0.027

1.027

0.822

-0.196

0.3

5258.236

0.563

1.756

0.527

-0.641

610.462

0.065

1.068

0.747

-0.291

0.4

4202.314

0.450

1.568

0.627

-0.466

1181.646

0.127

1.135

0.681

-0.384

0.5

3186.954

0.341

1.407

0.703

-0.352

2016.080

0.216

1.241

0.621

-0.477

0.6

2236.430

0.240

1.271

0.762

-0.271

3183.279

0.341

1.406

0.563

-0.575

0.7

1384.661

0.148

1.160

0.812

-0.208

4773.615

0.511

1.667

0.500

-0.693

0.8

679.702

0.073

1.076

0.860

-0.150

6902.806

0.739

2.095

0.419

-0.870

0.9

188.224

0.020

1.020

0.918

-0.085

9716.399

1.041

2.831

0.283

-1.262

Table-6: Optimized values of partial excess free energy of mixing, activity coefficients and activity of both the components present in Al-Sn liquid alloys at 1173K. 𝐶

Al- component

̃

ln

Sn-component

̃

ln

̃

ln

̃

ln

0.1

7023.647

0.720

2.055

0.205

-1.582

54.066

0.006

1.006

0.905

-0.100

0.2

6001.740

0.615

1.850

0.370

-0.994

235.873

0.024

1.024

0.820

-0.199

0.3

4982.247

0.511

1.667

0.500

-0.693

577.553

0.059

1.061

0.743

-0.297

0.4

3982.652

0.408

1.504

0.602

-0.508

1118.275

0.115

1.122

0.673

-0.396

0.5

3021.071

0.310

1.363

0.682

-0.383

1908.524

0.196

1.216

0.608

-0.497

0.6

2120.527

0.217

1.243

0.746

-0.293

3014.365

0.309

1.362

0.545

-0.607

0.7

1313.217

0.135

1.144

0.801

-0.222

4521.714

0.464

1.590

0.477

-0.740

0.8

644.787

0.066

1.068

0.855

-0.157

6540.613

0.671

1.956

0.391

-0.939

0.9

178.597

0.018

1.018

0.917

-0.087

9209.500

0.944

2.571

0.257

-1.358