BIBECHANA 18 (1) (2021) 149-158

BIBECHANA ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Department of Physics, Mahendra Morang A.M. Campus, TU, ,

Investigation on thermo-physical properties of liquid In-Tl alloy

I. B. Bhandari1,2, N. Panthi1,3 , I. Koirala1* 1Central Department of Physics, , Kirtipur, Nepal 2Department of Applied Sciences, Purwanchal Campus, Tribhuvan University, , Nepal 3Department of Physics, Patan Multiple Campus, Tribhuvan University, Lalitpur, Nepal *E-mail: [email protected]

Article Information: ABSTRACT Received: June 29, 2020 This research explores mixing behaviour of liquid In – Tl system through Accepted: August 8, 2020 thermodynamic and the structural properties on the basis of Complex Formation Model. The properties like surface tension and viscosity have been analyzed through simple statistical model and Moelwyn – Hughes equation. The Keywords: interaction parameters are found to be positive, concentration independent and Mixing properties temperature dependent. Theoretical results are in a good agreement with the Ordering energy corresponding literature data which support homo-coordinating tendency in the Complex formation model liquid In-Tl alloy. Segregation DOI: https://doi.org/10.3126/bibechana.v18i1.29531 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons.org/licenses/by-nc/4.0/

1. Introduction Indium is also used in nuclear reactor control rod alloys, low pressure sodium lamps and alkaline Indium is a substance which is used in solder alloys batteries. Additions of indium to lead–tin bearings which are applied in electronics for assembling are utilized in piston type aircraft engines, high semiconductor chips to a base and hybrid integrated performance automobile engines and in turbo– circuits and to seal glass to metal in vacuum tubes. diesel truck engines. The addition of indium to gold Fusible indium alloys are used to bend thin walled dental alloys recuperates their mechanical tubes without wrinkling the wall or changing the properties and increases resistance to discoloring. original cross-section. These alloys are not only Small amount of indium is used to improve the used in fire control system, restraining links that machinability of gold alloys for jewelry [1].The hold alarm, water valve and door operating indium–thallium alloy is a classic type of shape mechanism but also used as temperature indicators memory alloy with a low melting temperature. It in situations where other methods of temperature has wide range of practical applications in the field measurements are impracticable and infeasible [1]. of metallurgy which includes the use in

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thermostats, hydraulic lines and electrical circuits total number of atoms is N = NA + NB. When [2]. Thallium alone is improper for direct use components A and B are amalgamated together to because of its properties of toxicity, unfavorable form a binary A-B solution, thermodynamic mechanical properties and significant tendency to properties are changed. The liquid alloy is oxidize. Thallium contains the most constant considered to be ternary mixture of three species; A atomic vibration so far experimented. This property atom, B atom and chemical complex AuBv, is also of Tl preceded it to be used in atomic clocks. called conformal solution. The number of free Thallium immediately forms alloys with most other atoms will be reduced due to compound formation metals. There is incomplete mutual insolubility in the melt. Now for n1g atoms of A, n2g atoms of with iron and limited solubility in the liquid state B and n3g atoms of AuBv, with copper, aluminum, zinc, arsenic, manganese 푛1 = 푥 − 푢푛3 and 푛2 = (1 − 푥) − 푣푛3 (1) and nickel. Gold, silver, cadmium and tin formulate simple eutectic point with thallium. Thallium also The total number of atoms after mixing can be forms binary alloys with antimony, barium, given as calcium, cerium, cobalt, germanium, lanthanum, 푛 = 푛1 + 푛2 + 푛3 = 1 − (푢 + 푣 − 1)푛3 (2) lithium, magnesium, strontium, tellurium, bismuth The free energy of mixing of the binary A-B and indium. The ternary alloys of thallium Tl–Pb– mixture can be written as [23],

Bi, Tl–Al–Ag, In–Hg–Tl, Sn–Cd–Tl, Bi–Sn–Tl and ′ 퐺푀 = −푛3푔 + 퐺 (3) Bi–Cd–Tl are used as semiconductors in ceramic compounds. Thallium has good wear resistance Here,−푛3푔 stands for lowering of free energy due when it is used in bearing shafts. Thallium to compound formation, g is the formation energy containing alloys are frequently recommended for of complex. 퐺′ is the free energy of mixing of the bearings, electronics industry such as in solid state ternary mixture of A, B and AuBv. If the ternary rectifiers, electrical fuses and soldering materials mixture is an ideal solution,

[1]. The study of In-Tl alloy is also useful to 푛 퐺′ = 푅푇 ∑ 푛 푙푛 ( 푖) (4) 푖 푛 investigate the corresponding higher order alloy through different approaches [3]. There is difficulty If the effects of differences in sizes of the various in studying the properties of alloys in liquid state constituents in the mixture cannot be ignored and due to lack of long range atomic order. Therefore, the interaction 휔푖푗 are small but not zero, the theoreticians have exercised different models to theory of regular solutions in the zeroth understand the properties of various binary liquid approximation [25] or the conformal solution alloys [4–22]. The different properties were studied approximation [26] is valid. For regular solution at fixed temperature of 723 K through different 푛 푛 푛 퐺′ = 푅푇 ∑ 푛 푙푛 ( 푖) + ∑ 휔 ( 푖 푗) (5) models. In present work, we have explored the 푖 푛 푖푗 푛 energetic of In – Tl alloy at a temperature of 723K This equation is concerned to conformal solution using complex formation model [23]. The approximation. Where 휔푖푗= 0 (for i = j) are termed outcomes are analyzed and compared with as the interaction energies and by definition are literature data [24] to explicate the accuracy of this independent of concentration, although they are method in thermodynamic and structural depended upon temperature and pressure. description of the presented binary system. Now the expression for free energy of mixing GM

for the compound forming binary alloy is 2. Theory 푛 퐺 = −푛 푔 + 푅푇 ∑3 푛 푙푛 ( 푖) + Thermodynamic properties 푀 3 푖=1 푖 푛

푛푖푛푖 If a binary alloy contains NA = x number of A ∑ ∑ ( ) 휔 (6) 푖<푗 푛 푖푗 atoms and NB = (1-x) number of B atoms, so that

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The expression for heat of mixing HM is given of atomic order. The concentration fluctuation at by[23] long wavelength limit is related with free energy of 휕퐺푀 mixing by the expression [27], 퐻푀 = 퐺푀 − 푇 ( ) (7) 휕푇 푃 푅푇 푆푐푐(0) = 2 (13) Substituting for GM, 휕 퐺푀 2 3 휕푐 푛푖 퐻푀 = −푛3푔 + 푅푇 ∑ 푛푖푙푛 ( ) 푅푇 푛 ( ) 1 푆푐푐 0 = 2 (푛′) ′ 2 ′ 푛 ′ 푛 푛 3 푖 (푛 ) 푛푖 푗 푖 푗 푅푇 ∑푖=1( − )+2푛 ∑푖<푗 ∑ 휔푖푗( ) ( ) + ∑ ∑ ( ) 휔 푛푖 푛 푛 푛 푛 푖푗 푖<푗 (14) 3 휕 푛 Theoretically computed values of 푆푐푐(0) can be − 푇 [−푛 푔 + 푅푇 ∑ 푛 푙푛 ( 푖) 휕푇 3 푖 푛 compared with the observed values computed from 푖=1 activity data by the expression, −1 −1 푛푖푛푗 휕푎퐴 휕푎퐵 + ∑ ∑ ( ) 휔 ] 푆푐푐(0) = (1 − 푥)푎퐴 ( ) = 푥푎퐵 ( ) 푛 푖푗 휕푐 푇,푃,푁 휕푐 푇,푃,푁 푖<푗 (15) 휕푔 1 퐻 = −푛 [푔 − 푇 ( ) ] + ∑ ∑(푛 푛 ) [휔 − The ideal value of Scc(0) can be expressed as, 푀 3 휕푇 푛 푖<푗 푖 푗 푖푗 푃 푖푑 휕휔 푆푐푐 (0) = 푥(1 − 푥) (16) 푇 ( 푖푗) ] (8) 휕푇 푃 The Warren–Cowley short range order parameter The expression for entropy of mixing SM can be quantify the degree of local order in the binary obtained as [23] alloy [28,29]. The theoretical values of this 휕푔 3 푛푖 푛푖푛푗 휕휔푖푗 parameter can be calculated as 푆푀 = 푛3 − 푅 ∑푖=1 푛푖푙푛 − ∑푖<푗 ∑ 휕푇 푛 푛 휕푇 (s−1) Scc(0) α1 = ,S = id (17) (9) s(z−1)+1 Scc(0) The equilibrium value of 푛3 at a given pressure and where z is coordination number, which is taken as temperature is given by[23] 10 for our calculation. 휕퐺 ( 푀) = 0 (10) Transport Properties 휕푛 3 푇,푃,푁,퐶

Substituting the value of GM from Equation (6) and The mixing behaviour of the alloys forming molten after some algebraic calculation alloy can also be studied at the microscopic level in 푔 푙푛(푛 푛푢+푣−1푛−푢푛−푣) + 푌 = (11) terms of coefficient of diffusion. The mutual 3 1 2 푅푇 The Equation (11) is called equilibrium equation, diffusion coefficient (DM) of binary liquid alloys where can be expressed in terms of activity (ai) and self- 푛 푛 푛 푛 휔 diffusion coefficient (Did) of pure component with 푌 = [ 1 2 (푢 + 푣 − 1) − 푢 2 − 푣 1] 12 + 푛2 푛 푛 푅푇 the help of Darken’s equation [30] 푛2푛3 푛3 푛2 휔23 푑푙푛푎 [ (푢 + 푣 − 1) − 푣 + ] + 퐷 = 퐷 푥 푖 (18) 푛2 푛 푛 푅푇 푀 푖푑 푑푥 푛1푛3 푛3 푛1 휔13 [ (푢 + 푣 − 1) − 푢 + ] (12) with 퐷푀 = 푐퐴퐷퐵 + 푐퐵퐷퐴 푛2 푛 푛 푅푇 where DA and DB are the self – diffusion Structural Properties coefficients of pure components A and B

The concentration fluctuation at long wavelength respectively, limit is of good interest because any deviation from The expression for DM in terms of Scc(0) can be 푖푑 given as ideal value 푆푐푐 (0) is significant in describing the 퐷 푆푖푑(0) nature of ordering and phase segregation in molten 푀 = 푐푐 (19) 퐷 푆 (0) alloys. This has been used to investigate the nature 푖푑 푐푐

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2/3 The mixing behaviour of liquid alloys at 훺푖 휉푖 = 1.102 ( ) (25) microscopic level can also be understood in terms 푁퐴 of viscosity. The Moelwyn – Hughes equation for where 훺 is the molar volume of the component i viscosity of liquid alloy [31] is 푖 and 푁 represents Avogadro number. Equating 2푔 퐴 휂 = 휂 [1 − 푥 푥 ( )] (20) 푠 푖푑 퐴 퐵 푅푇 equations (22) and (23), we can solve it for 푥 as with the function of x and hence compositional

휂푖푑 = 푥휂퐴 + (1 − 푥)휂퐵 dependence of surface tension can be evaluated. where 휂푖 is the viscosity of pure component i. At where 휏퐴 and 휏퐵 are the surface tensions of pure temperature T, it is given by [32] components A and B respectively, 푥 and 푥푠 are the 퐸 bulk and surface concentration of the components 휂푖 = 휂푖0exp ( ) (21) 푅푇 of alloy, p and q are called coordination fractions, Here 휂 a constant in the unit of viscosity and E is 푖0 which are defined as the fraction of the total the activation energy. number of nearest neighbors made by atom within Surface Properties its own layer and that in the adjoining layer. The

The surface properties of the liquid mixture give coordination fractions p and q are related to each insight into the metallurgical phenomenon, such as other by the relation crystal growth, wielding, gas absorption and 푝 + 2푞 = 1, for closed packed structure, 푝 = 0.5 nucleation of gas bubbles [33]. The expressions for and 푞 = 0.25 surface tension proposed by Prasad et al., [34,35], The expression for the mean atomic surface area 휉 has been reduced in the simple form using zeroth is approximation as 휉 = ∑ 푐푖휉푖 (24) The atomic surface are for each component is 푘 푇 푥푠 푔 휏 = 휏 + 퐵 푙푛 + [푝(1 − 푥푠)2 + 2/3 퐴 휉 푥 휉 훺푖 휉푖 = 1.102 ( ) (25) (푞 − 1)(1 − 푥)2] (22) 푁퐴 푠 where 훺푖is the molar volume of the component i 푘퐵푇 (1−푥 ) 푔 푠 2 2 휏 = 휏퐵 + 푙푛 + [푝(푥 ) + (푞 − 1)(푥) ] 휉 (1−푥) 휉 and 푁퐴represents Avogadro number. Equating (23) equations (22) and (23), we can solve it for 푥푠 as where 휏퐴 and 휏퐵 are the surface tensions of pure the function of x and hence compositional components A and B respectively, 푥 and 푥푠 are the dependence of surface tension can be evaluated. bulk and surface concentration of the components 3. Results and Discussion of alloy, p and q are called coordination fractions, which are defined as the fraction of the total Thermodynamic properties number of nearest neighbors made by atom within The experimental data on the thermodynamic its own layer and that in the adjoining layer. The properties as well as phase diagram information coordination fractions p and q are related to each [24] have been used for the calculation of order other by the relation energy parameters for liquid phase In–Tl system. 푝 + 2푞 = 1, for closed packed structure, 푝 = 0.5 The data set of the Gibbs energy of mixing (GM) and 푞 = 0.25 were taken as input data to calculate by the CFM

The expression for the mean atomic surface area 휉 the interaction energy parameters, i.e. g, ω12, ω13, is ω23. The starting values of g/RT and ωij/RT were obtained as suggested in ref. [23]. Equilibrium 휉 = ∑ 푐푖휉푖 (24) Equation (11) along with Equations (1) and (2) The atomic surface are for each component is were applied to compute the number of complexes,

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n3, as a function of concentration. The values of maximum value of entropy of mixing is 0.6107 at interaction energy parameters were adjusted to give xIn = 0.5 (Fig.4). There is excellent agreement the concentration dependence of free energy of between experimental and calculated values of HM. mixing which fits well with the corresponding The calculated values of SM deviates from thermodynamic data. From the phase diagram [24] experimental values by maximum percentage of

In-Tl alloy is expected to aggregate with 8.31 at xIn = 0.8 and by minimum percentage of stoichiometry In-Tl ( u = 1, v = 1).The calculations 5.56 at xIn = 0.4. This deviation is because of the were done at temperature of 723 K. The interaction propagation of error from previous calculation. energy parameters for In–Tl liquid alloys are found to be g = 0.755 RT, ω12 =0.476 RT, ω13=1.610 RT and ω23= 1.481 RT. The positive interaction energies imply the repulsion between the 1.0 n n2 1 corresponding species. 0.8

The concentration dependence of the equilibrium 0.6

3

n , values of chemical complexes, n3, (Fig. 1) displays 2

n3 n

, 0.4 1

the symmetry with the maximum value of 0.4178 at n equiatomic composition. The curve describing the 0.2 Gibbs free energy of mixing of the In–Tl liquid 0.0 phase is symmetric with respect to the equiatomic 0.0 0.2 0.4 0.6 0.8 1.0 xIn composition (Fig. 2). Theoretical calculation of free Fig.1 energy of mixing for In-Tl liquid alloy shows that

In - Tl alloy in liquid state is weakly interacting or Fig.1: Number of complexes (n , n , n ) vs. homo-coordinating system. There is an excellent 1 2 3 concentration x of liquid In - Tl alloy at 723K. agreement between the experimental and calculated In integral free energies. Very poor agreement of xIn 0.0 0.2 0.4 0.6 0.8 1.0 calculated values of HM and SM with experimental 0.0 simply indicates the importance of the dependence -0.1 of interaction energies on temperature. To account -0.2 this, we have used Equations (8) and (9) to

-0.3 / RT /

determine the variation in energy parameters with M G respect to temperature from experimental values of -0.4

HM and SM [24]. The temperature dependent -0.5 Experimental interaction energies at T=723K are found to be -0.6 Theoretical 1 휕푔 1 휕휔12 = 0.845, = 0.383, Fig.2 푅 휕푡 푅 휕푡 1 휕휔 13 = 1.493, 푅 휕푡 Fig. 2: Free energy of mixing (GM/RT) vs. 1 휕휔23 = 1.430 concentration (xIn) of liquid In - Tl alloy at 723K. 푅 휕푡 It is found from the present analysis that the heat of mixing and entropy of mixing both are positive at Structural Properties all concentrations. Our theoretical calculation It has been reported that when S (0) < Sid(0), the shows that the maximum value of the heat of cc cc existence of chemical ordering leading to complex mixing is 0.0512 RT at xIn = 0.6 (Fig.3) and the

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id formation is expected while Scc(0) > Scc(0), is an DM/Did against concentration of indium. It can be indication of segregation. The same interaction observed that the ratio DM/Did is less than 1 parameters used in the calculation of the throughout the whole concentration range. This thermodynamic properties were employed in the indicates the homo – coordinating tendency in In– calculations of Scc(0) using Equation (14) while the Tl alloys at the temperature of investigation. It is experimental values of Scc(0) were obtained from also noticed that DM/Did exhibits maximum peak at Equation (15) using the experimental activity data. around the equi-atomic composition. The result The results obtained from the above computations predicted by DM/Did is in agreement with the results are plotted in Fig. (5). It is found that Scc(0) > obtained from the free energy of mixing, id concentration fluctuations and CSRO parameter. Scc(0) throughout the entire concentration range, this also confirms the presence of chemical The viscosity of the In–Tl liquid alloy has been segregation or a preference for like atoms to pair. calculated numerically using Equation (20). From The value of short range order parameter is positive the plot of η verses bulk concentration of indium through the whole concentration range which (Fig. 8) small negative deviation from the linear indicates that the alloy is segregating at all law (Raoult's law) in viscosity isotherms η(x) has compositions. The value of short range order been inspected. parameter has been found maximum (= 0.02438) at Surface Properties xIn = 0.5 at 723 K (Fig. (6)). The surface concentrations and surface tension of

0.06 In–Tl alloy have been computed numerically using Experimental Equations (22) and (23). The values of densities 0.05 Theoretical and surface tension at melting temperature (T0) of 0.04 pure atoms are taken from ref. [32]. These values

0.03 have been optimized at required temperature (T) by

/ RT / M

H using the expressions 0.02 0 0 휌푖(푇) = 휌푖 + (푇 − 푇푖 ) 푑휌푖⁄푑푡 (26)

0.01 0 0 휏푖(푇) = 휏푖 + (푇 − 푇푖 ) 푑휏푖⁄푑푡 (27)

0.00 0.0 0.2 0.4 0.6 0.8 1.0 where 푑휌푖⁄푑푡 and 푑휏푖⁄푑푡 represent temperature Fig.3 xIn coefficients of density and surface tension respectively for the components of the metal alloys. Fig. 3: Heat of mixing (HM) vs. concentration of The computed values of surface concentration for indium (xIn) in the liquid In - Tl alloy at 723K. molten In–Tl alloys at 723 K are depicted in Fig.

(9). Surface concentration of indium in In–Tl alloys is found to increase with the increase of bulk Transport Properties concentration of In. The computed surface tension The calculated values of Scc(0)) by Equation (17) for In–Tl alloys at 723 K is less than ideal values at can be applied to evaluate the ratio of the mutual all concentrations of indium; i.e., there is negative and intrinsic-diffusion coefficients (DM/Did) using departure of surface tension from ideality (τ = Equation (19), against the concentration of indium. τAx+τB(1−x)) throughout the bulk concentrations of We note that the ratio of diffusivities can also be indium in In–Tl alloys (Fig. 10). For the In-Tl melt, used to indicate levels of order in the liquid binary the surface tension of Tl is smaller than that of In alloys. The presence of chemical order is indicated atom. Therefore, Tl atoms having lower surface by DM/Did>1. Similarly, DM/Did<1 suggests the tension segregates on the surface phase but In tendency for segregation. Fig. (7) shows the plots

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0.020

0.7 Experimental 0.015 Theoretical 0.6 1 a 0.5 0.010

0.4

/ R 0.005 M S 0.3 0.2 0.000 0.0 0.2 0.4 0.6 0.8 1.0 xIn 0.1 Fig.6

0.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 6: Chemical short range order (α1) vs. Fig.4 xIn concentration of indium (xIn) in the liquid In - Tl alloy at 723K. Fig. 4: Entropy of mixing (SM) vs. concentration of indium (xIn) in the liquid In - Tl alloy at 723K.

0.92

0.35 Experimental Theoretical Ideal 0.30 0.88

0.25

0.84 id

(0) 0.20

cc S

0.15 / D M

D 0.80 0.10

0.05 0.76 0.00 0.0 0.2 0.4 0.6 0.8 1.0 Fig.5 xIn 0.0 0.2 0.4 0.6 0.8 1.0

Fig.7 xIn Fig. 5: Concentration fluctuation at long wavelength limit (Scc(0)) vs. concentration of indium (xIn) in the liquid In - Tl alloy at 723K. Fig. 7: Ratio of mutual and intrinsic diffusion

coefficients (DM/Did) vs. concentration of indium

(xIn) in the liquid In - Tl alloy at 723K.

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concentration fluctuation in long wavelength limit and CSRO show that there is tendency of phase separation in In –Tl liquid alloy. Negative deviation 0.0016 Theoretical alues of viscosity isotherms from Raoult law is observed. Ideal values Viscosity of the alloys decreases with increase in 0.0014 the concentrations of indium. The ratio of diffusion

coefficients (DM/Did) is found to be greater than one -2 0.0012 at all compositions which also indicates segregating

Nsm tendency of the system. At the temperature of h

0.0010 investigation, the surface tension increases with the increase in the bulk concentration of In. The surface tension of the liquid In–Tl alloy is found to 0.0008 be smaller than ideal values.

0.0 0.2 0.4 0.6 0.8 1.0 Fig.8 xIn

0.52 Fig. 8: Viscosity (η) vs. concentration of indium

(xIn) in the liquid In - Tl alloy at 723K.

0.50

-1 Nm

1.0 t 0.48

Theoretical 0.8 Ideal

0.46

0.6

In 0.0 0.2 0.4 0.6 0.8 1.0 s Fig.10 x xIn 0.4 Fig. 10: Surface tension (τ) vs. bulk concentration 0.2 Theoretical values Ideal values of indium (xIn) in the liquid In - Tl alloy at 723K.

0.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig.9 xIn References

s [1] F. Habashi, ed., Alloys Preparation, Properties, Fig. 9: Surface concentration of indium ( xIn) vs. Applications, First, Wiley-VCH (1998). bulk concentration of indium (xIn) in the liquid In - Tl alloy at 723K. [2] Z.P. Luo, An Overview on the Indium-Thallium (In-Tl) Shape Memory Alloy Nanowires, Metallogr. Microstruct. Anal. 1 (2012) 320. 4. Conclusions https://doi.org/10.1007/s13632-012-0046-4 [3] U. Mehta, S.K. Yadav, I. Koirala, D. Adhikari, The theoretical analysis of the thermodynamic Thermo-physical properties of ternary Al–Cu–Fe properties reveals that there is a tendency of like alloy in liquid state, Philos. Mag. 0 (2020) 1. atom pairing in the liquid In–Tl alloys at all https://doi.org/10.1080/14786435.2020.1775907 concentrations. The ordering energy is found to be [4] D. Adhikari, A Theoretical study of thermodynamic properties of Cd-Bi liquid alloy, positive and temperature dependent. The study of BIBECHANA 8 (2012) 90.

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