Papers in Physics, vol. 14, art. 140005 (2022) Received: 7 July 2021, Accepted: 18 February 2022 Edited by: A. Goñi Licence: Creative Commons Attribution 4.0 DOI: https://doi.org/10.4279/PIP.140005 www.papersinphysics.org ISSN 1852-4249 Thermophysical behavior of mercury-lead liquid alloy N. Panthi1,2∗, I. B. Bhandari1, I. Koirala1† Thermophysical properties of compound forming binary liquid mercury-lead alloy at tem- perature 600 K have been reported as a function of concentration by considering HgPb2 complex using different modelling equations. The thermodynamic properties such as the Gibbs free energy, enthalpy of mixing, chemical activity of each component, and micro- scopic properties such as concentration fluctuation in long-wavelength limit and Warren- Cowley short range order parameter of the alloy are studied by quasi-chemical approxima- tion. This research paper places additional emphasis on the interaction energy parameters between the atoms of the alloy. The theoretical and experimental data are compared to determine the model’s validity. Compound formation model, statistical mechanical tech- nique, and improved derivation of the Butler equation have all been used to investigate surface tension. The alloy’s viscosity is investigated using the Kozlov-Ronanov-Petrov equation, the Kaptay equation, and the Budai-Benko-Kaptay model. The study depicts a weak interaction of the alloy, and the theoretical thermodynamic data derived at 600 K are in good agreement with the experimental results. The surface tension is slightly different in the compound formation model than in the statistical mechanical approach and the Butler equation at greater bulk concentrations of lead. The estimated viscosities in each of the three models are substantially identical. I Introduction The knowledge of thermophysical characteristics of alloys is regarded as a necessary foundation for de- veloping novel materials. The creation of an al- loy is linked to changes in the structure of a sys- tem as well as bonding between the constituent atoms. The subject is more intricately understood by studying the interaction and structural rear- rangement of constituent atoms during alloy for- mation. The electrochemical effect, atom size, and constituent element concentration all influence the ∗narayan.755711@cdp.tu.edu.np †iswar.koirala@cdp.tu.edu.np 1 Central Department of Physics, Tribhuvan University, Kirtipur, Kathmandu, Nepal. 2 Department of Physics, Patan Multiple Campus, Trib- huvan University, Nepal. alloy’s mixing properties, causing atoms of particu- lar elements to align in either a self-coordinated or strong ordering pattern [1–4]. The alloying prop- erties of liquid alloys vary as a function of compo- sition, temperature, and pressure, all of which are important for the materials’ strength, stability, and electrical resistivity. As a result, metallurgists and physicists have been interested in understanding the mixing behavior of metals that produce alloys. However, due to experimental difficulties as well as time limitations, the study of various alloys’ charac- teristics is still incomplete. Different theoreticians have produced numerous concentration-dependent theoretical models to comprehend the alloying be- havior of compound forming binary alloys in order to address such challenges and facilitate study as well as speed up the investigation process [5–7]. Because of their direct impact on human health, mercury and lead are the most studied metals. Our 140005-1 Papers in Physics, vol. 14, art. 140005 (2022) / N. Panthi et al. study focuses on one of the lead alloys, Hg-Pb, to theoretically determine various properties at 600 K, assuming HgxPby (x = 1, y = 2) complex in the melt, by using compound formation models [6]. Lead, being very soft and ductile, is often used com- mercially as lead alloys [8]. Zabdyr [9] explored phase diagram, crystal structure and lattice param- eter by varying atomic weight percentage of Hg but the detailed thermophysical investigation is incom- plete. The properties under investigation include the Gibbs free energy of mixing, enthalpy of mix- ing, chemical activity, concentration fluctuation in long-wavelength limit and Warren–Cowley short- range order parameter of the alloy. Similarly, concentration-dependent surface tension and vis- cosity of binary liquid alloys are investigated, these being the most desirable in metallurgical research for specifying the surface and transport properties of liquid mixtures: as such, scientists are attempt- ing to investigate these aspects by proposing several models [10–16] . Furthermore, surface segregation, which primarily refers to the concentration dispar- ity between the alloys’ surface and bulk materials, is one of the most essential elements to be investi- gated in metallurgical research. The difference in surface energy between the alloy’s constituent ele- ments is the fundamental source of this disparity, the element with lower surface energy tending to segregate on the surface [17]. Theoretical study in- dicates that the atom with a larger size tends to segregate on the surface of liquid alloy [18]. The present work also aims to study the surface tension of the alloy with a compound formation model [13]. Due to a lack of experimental data, the computed result is compared with two other models: a statistical mechanical approach [12] and an improved derivation of the Butler equation [16]. For the study of viscosity, this study employs three models; the Kozlov-Ronanov-Petrov equation [11], the Kaptay equation and the Budai-Benko-Kaptay model [10]. II Theoretical formulation i Thermodynamic functions Let a binary alloy contain NA and NB number of A and B atoms respectively. The model assumes the existence of complexes AxBy in such a way that xA + yB = AxBy (1) where x and y are small integers. With this assumption, the grand partition function in terms of configurational energy [6] is solved and excess free energy of mixing is obtained as given in Eq. (2) by which various properties are obtained. GXSM = RT ∫ C 0 γdC (2) where γ is the activity coefficient ratio of atom A to B, C is the concentration of A atom and R is universal gas constant. After simple mathematical calculation, the solution of Eq. (2) is given as GXSM = N[θω+θAB∆ωAB+θAA∆ωAA+θBB∆ωBB] (3) where θ = C(1 − C) and θj,k’s are the simple poly- nomials in C that depend on the values of integers x and y, ω is interchange energy, and ∆ωjk are the interaction energy parameters. For A =Hg, B =Pb, x = 1, y = 2, the values of θj,k’s are found to be [6,19] θAA(C) = 0 (4) θAB(C) = 1 6 C + C2 − 5 3 C3 + 1 2 C4 (5) θBB(C) = − 1 4 C + 1 2 C2 − 1 4 C4 (6) The Gibbs free energy of mixing for complex for- mation is given by GM = G XS M + G ideal M = GXSM + RT(C ln C + (1 − C) ln(1 − C)) = RT [ θ ω kBT + θAB ∆ωAB kBT + θAA ∆ωAA kBT +θBB ∆ωBB kBT + C ln C + (1 − C) ln(1 − C) ] (7) 140005-2 Papers in Physics, vol. 14, art. 140005 (2022) / N. Panthi et al. Here θAA is taken as zero because, according to the model used, the value of x is 1. In this case, the probability of A and A pair to be part of the complex is zero, such that the coefficient of ∆ωAA kBT in Eq. (6) also tends to zero. If there are no complexes in the alloy, then ∆ωjk is zero. In such a case, the above equation takes the form as given below: GM = RT [ θ ω kBT + C ln C + (1 − C) ln(1 − C) ] (8) The enthalpy of mixing is calculated with the stan- dard thermodynamic relation: HM RT = GM RT − [ dGM RdT ] C,N,P = θ [ ω kBT − 1 kB dω dT ] + θAB [ ∆ωAB KBT − 1 kB d∆ωAB dT ] + θBB [ ∆ωBB KBT − 1 kB d∆ωBB dT ] (9) The activity of each constituent element of the alloy is revealed following the standard thermodynamic relation, RT ln aj(j = A, B) = GM + (1 − C) [ ∂GM ∂Cj ] T,P,N (10) Now, by solving Eqs. (7) and (10), the theoretical value of activity of each constituent component is given as follows: ln aA = GM RT + 1 − C kBT [(1 − 2C)ω + θ′AB∆ωAB + θ′BB∆ωBB + ln C 1 − C ] (11) ln aB = GM RT − C kBT [(1 − 2C)ω + θ′AB∆ωAB + θ′BB∆ωBB + ln C 1 − C ] (12) where, θ′AB, θ ′ AA and θ ′ BB, respectively, are deriva- tives of θAB, θAA and θBB with respect to concen- trations. ii Microscopic Functions The concentration fluctuation in the long- wavelength limit SCC(0)for the alloy is given from the relation as [20], SCC(0) = RT [ ∂2GM ∂C2 ] T,P,N (13) The value of SCC(0) can be obtained by using ex- perimentally observed activities with the help of the following Eq. (14). Hence the values of SCC(0) obtained from this equation are called experimental values. SCC(0) = aA(1 − C) [ ∂aA ∂CA ]−1 T,P,N = aBC [ ∂aB ∂CB ]−1 T,P,N (14) where aA and aB are observed activities of elements A and B respectively. For simplicity, we can write C and 1-C in place of CA and CB, respectively. Solving Eqs. (7) and (14), the value of SCC(0) is found as, SCC(0) = C(1 − C) [ 1 + C(1 − C) ( − 2 ω KBT + θ”AB ∆ωAB KBT + θ”BB ∆ωBB KBT )]−1 (15) Where θ”jk are second concentration derivatives of θjk. The Warren-Cowley short-range order parameter (α1) is related to concentration fluctuation in the long-wavelength limit [21,22] as: α1 = (S − 1)[S(Z − 1) + 1]−1 (16) where Z is coordination number and S = SCC(0) SidCC(0) (17) 140005-3 Papers in Physics, vol. 14, art. 140005 (2022) / N. Panthi et al. iii Transport property: viscosity At the microscopic level, the mixing nature of molten alloy may be examined in terms of viscosity, which provides basis for some of the most funda- mental theories concerning atomic transport qual- ities. It is regarded as one of the most important thermophysical qualities in metallurgical research, which primarily deals with industrial processes and a variety of natural occurrences. It is influenced by factors such as the liquid’s composition, cohesion energy, and molar volume [23, 24]. The composi- tion dependence of viscosity at 600 K is computed to examine the atomic transport features of the Hg-Pb alloy. But due to the lack of experimen- tal data, viscosity is compared using three differ- ent models: the Kozlov-Ronanov-Petrov equation, the Kaptay equation, and the BBK (Budai-Benko- Kaptay) model. a Kozlov-Ronanov-Petrov equation In liquids, viscous flow depends on cohesive inter- action, this interaction results from geometric and electronic shell effects [25]. The KRP equation has been developed to incorporate cohesion interaction in terms of enthalpic effect in order to consider the viscous flow in a liquid alloy. At temperature T, the equation is given as: ln η = C ln C ln ηA + (1 − C) ln ηB − HM 3RT (18) where η and ηj are viscosity of the alloy and vis- cosity of individual elements A and B, respectively. For the metals, the variation of viscosity with tem- perature is given as [26] ηj = η0 exp ϵ RT (19) where η0 and ϵ are constants of each metal’s units of viscosity and energy per mole. b Kaptay equation Kaptay developed an equation by considering the theoretical relationship between the cohesive en- ergy and activation energy of the viscous flow. At temperature T, the equation is: η = hNAv CVA + (1 − C)VB + V E × exp ( CGA + (1 − C)GB − ΦHM RT ) (20) where h is Plank’s constant, NAv is Avogadro’s number, Vj(j = A, B) is the molar volume of pure metal, V E is excess molar volume upon alloy for- mation, Gj is Gibb’s energy of activation of the viscous flow in pure metals, and ϕ is a constant whose value is (0.155±0.015) [27]. The Gibb’s en- ergy of activation of pure metal is calculated by the following equation: Gj = RT ln ( ηjVj hNAv ) (21) c BBK (Budai-Benko-Kaptay) model The BBK model is used for the viscosity of multi- component alloys. At temperature, it is given as: η = LT 1/2(CMA + (1 − C)MB)1/2 × (CVA + (1 − C)VB + V E)−2/3 × exp [ (CTm,A + (1 − C)Tm,B − HM χR ) I T ] (22) where L and I are constants whose values are (1.80±0.39)×10−8j/Kmol1/3) 1/2 and (2.34±0.02), respectively, and χ is a semi-empirical parameter having a value equal to 25.4. Similarly Mj and TmJ are, respectively, molar mass and the effective melting temperature of constituent elements of the alloy. iv Surface property In metallurgy and industry, the surface properties (surface tension and surface concentration) of liq- uid alloy or liquid metal are considered to be prime factors for the processing, as well as for the pro- duction, of new materials due to their relation with both surface and interface in the liquid metal pro- cess [28,29]. The interfacial motion caused by the surface tension of liquid plays a major role in many industrial phenomena, hence the importance given 140005-4 Papers in Physics, vol. 14, art. 140005 (2022) / N. Panthi et al. to the surface and interfacial behaviors of liquid metals in the metallurgical process for solidifica- tion, controlling the processes of welding and cast- ing [30]. a Compound Formation model The model assumes that there is a compound form- ing tendency in the binary liquid alloy similar to that of the compound forming tendency in the solid state, in the form of short-ranged volume elements, due to the formation of intermetallic compound AxBy in the melt. The equation in this model is developed by using the grand partition function similar to the quasi chemical approximation. The equation at temperature T is given below: σ = σA + kBT ρ ln Cs C + ω ρ [p(fS − f) − qf] + ∆ωAB ρ [p(fSAB − fAB) − qfAB] + ∆ωBB ρ [p(fSBB − fBB) − qfBB] (23) = σB + kBT ρ ln 1 − Cs 1 − C + ω ρ [p(φS − φ) − qφ] + ∆ωAB ρ [p(φSAB − φAB) − qφAB] + ∆ωBB ρ [p(φSBB − φBB) − qφBB] (24) where φ, f, φjk and fjk are bulk concentration functions. Similarly, φS, fS, φSjk and f S jk are sur- face concentration functions, and ρ is the mean area of the surface per atom. For x = 1 and y = 2, the bulk concentration functions are [13,31]: φ = C2 (25) φAB = 1 6 + 2(1 − C) − 6(1 − c)2 + 16 3 (1 − C)3 − 3 2 (1 − C)4 (26) φBB = − 1 4 + (1 − C) − 1 2 (1 − C)2 + (1 − C)3 − 3 4 (1 − C)4 (27) f = (1 − C)2 (28) fAB = (1 − C)2 + 10 3 (1 − C)3 − 3 2 (1 − C)4 (29) fBB = −(1 − C)2 + 3 4 (1 − C)4 (30) The functions φs, φSjk, f s and fsjk can be obtained from Eqs. (26) to (30) by replacing bulk concentra- tion C with surface concentration CS, while p and q are surface coordination fractions that indicate the fraction of the number of nearest neighbors of an atom within its own layer and in the adjoining lay- ers, respectively, and and are related as p + 2q = 1. In a simple cubic crystal, p = 2/3 and q = 1/6. In a bcc crystal, p = 3/5 and q = 1/5, and in close packed crystal, p = 1/2 and q = 1/4. The mean atomic surface area is given by following relation [13]: ρ = ∑ j Cjρj (31) The atomic surface area of each component is given as ρj = 1.012 ( Vj NAv )2/3 (32) b Statistical mechanical approach This method is mainly based on the concept of lay- ered structure near the interface. The model con- nects the surface tension to thermodynamic prop- erties through the activity coefficient (γj) and the interchange energy between the components of an alloy. The equation at temperature T is given as below: 140005-5 Papers in Physics, vol. 14, art. 140005 (2022) / N. Panthi et al. σ = σj + KBT ρ ln CSj Cjγj + [ p(1 − CSj ) 2 − q(1 − Cj)2 ] ω ρ (33) c Improved Derivation of the Butler equation According to this model, there exists a monoatomic layer, called surface monolayer, at the surface of a liquid as a separate phase, and it is in thermody- namic equilibrium with the bulk phase. The sur- face tension (σ) of binary alloy at temperature T is given by the improved Butler equation as: σ = S0j Sj σ0j + RT Sj ln CSj Cbj + G S,XS j − G b,XS j Sj (34) where, σ0j , S 0 j , Sj are surface tension of pure liq- uid metal, molar surface area of pure liquid metal, and partial molar surface area of jth component, respectively. G S,Xs j and G b,Xs j are partial excess free energy of mixing in the surface and bulk of constituent elements of the alloy, respectively. The molar surface area of pure component is given as: S0j = δ ( M0j λ0j )2/3 N 1/3 Av (35) where δ, M0j , λ 0 j, δ and NAv are geometrical con- stant, molar mass, density of each constituent el- ement at its melting temperature, and Avogadro’s number, respectively. The value of geometrical con- stant is expressed as, δ = ( 3fv 4 )2/3 π fs (36) where fv is volume packing fraction and fs is sur- face packing fraction. For liquid metal, the values of fv and fs are 0.66 and 0.906 respectively [33]. III Results and discussion i Thermodynamic and microscopic proper- ties Generally, the properties of binary liquid alloys de- pend on temperature, composition, and pressure. -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.2 0.4 0.6 0.8 1 G M /R T CPb theoretical experimental Figure 1: Gibbs free energy of mixing versus bulk con- centration of Pb. Our study of the binary alloy Hg-Pb is carried out at fixed atmospheric pressure and fixed tempera- ture of 600 K as a function of the composition of the alloy. During the study, we assumed complex- ity with x = 1 and y = 2 and computed different thermodynamic and structural properties for com- pound forming molten alloys. The different results thus obtained from the study are outlined in the sections below. a Thermodynamic Properties For the analysis of the thermodynamic properties, we consider Eqs. (7), (9), (11), and (12), as men- tioned above. For the Gibbs free energy of mixing, the interaction energy parameters are determined by the method of successive approximation for sev- eral concentrations, following stoichiometry of the HgPb2 alloy with the help of experimental values in the concentration range (0.1 to 0.9) [34]. The approximated values of energy parameters are as follows: ω kBT = 2.139, ∆ωAB kBT = −2.264, ∆ωBB kBT = 0.392 To calculate the interaction energy parameters, no statistical methods such as mean square deviation were used to decide the best fit values, hence the 140005-6 Papers in Physics, vol. 14, art. 140005 (2022) / N. Panthi et al. -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 H M /R T CPb theoretical experimental Figure 2: Enthalpy of mixing versus bulk concentration of Pb parameters thus obtained are considered reasonable for analysis and have been considered throughout the study of different mixing properties. The com- puted values of GM/RT are in good agreement with experimental values as shown in Fig. 1. The the- oretically computed value of free energy of mixing is a minimum of −0.533RT at 0.6 concentration of Pb. The calculation of free energy of mixing indi- cates that the alloy HgPb at molten state is weakly interacting. Similarly, being asymmetric at 0.5 con- centration, it is classified as an irregular alloy. The temperature derivatives of interaction energy parameters which are used for the theoretical cal- culation of enthalpy of mixing are obtained by the method of successive approximation. The best fit approximated values of such parameters are: 1 kB dω dT = 0.767, 1 kB d∆ωAB dT = −0.3128, 1 kB d∆ωBB dT = 0.429 The plot of enthalpy of mixing versus concentra- tion of lead is shown in Fig. 2. It is positive be- low 0.6 concentration of lead, while above this con- centration it is negative, and both computed and experimental values of enthalpy of mixing are in agreement, with small discrepancies. The deviation of alloy from ideal behavior can be examined by chemical activity, a measure of effec- 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 Pb Hg a c ti v it y CPb theoretical experimental Figure 3: Chemical activity versus bulk concentration of Pb tive concentration in the mixture, as its magnitude depends on the interaction of constituent binary components of the alloy. Equations (11) and (12) are used for the calculation of the chemical activity of elements of alloy Hg-Pb. Figure 3 plots exper- imental and theoretically computed values of the chemical activity of the components of the alloy, showing good agreement between the experimental and theoretical activities of the components in the alloy at temperature 600 K at all concentrations of Pb. b Microscopic Properties It is difficult to perform diffraction experiments on materials at high temperatures. Thus, to make the study of local arrangement of atoms in the bi- nary alloy more effective, the concentration fluc- tuations in the long-wavelength limit (SCC(0)) are 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 S C C (0 ) CPb theoretical ideal experimental Figure 4: Concentration fluctuation in long-wavelength limit versus bulk concentration of Pb 140005-7 Papers in Physics, vol. 14, art. 140005 (2022) / N. Panthi et al. -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.2 0.4 0.6 0.8 1 α 1 CPb Figure 5: Warren-Cowley short-range order parameter versus bulk concentration of Pb considered some of the most important microscopic functions [20, 35]. For any given concentration, if SCC(0) < S id CC(0), the alloy is expected to have complex formation in nature and if SCC(0) > SidCC(0), the expected nature of the alloy is segre- gating. The graph of experimental, theoretical and ideal values of SCC(0) versus concentration of Pb is shown in Fig. 4. In the figure, both the experimen- tal and theoretical values of SCC(0) lie above the ideal value for lead concentration values below 0.6 , indicating that the alloy has a segregating nature below this concentration of lead, while above this concentration it exhibits an ordering nature. The Warren-Cowley short range parameter (α1) is considered one of the most powerful parameters for information regarding the arrangement of atoms in the liquid alloys. It provides quantitative informa- tion about the degree of local arrangement of atoms in the alloys. Its value lies between +1 and -1. The positive value of α1 is considered an indication of a segregating nature, which is complete for α1 = 1, whereas its negative value indicates an ordering na- ture, and is complete for α1 = −1. Similarly the value α1 = 0 indicates the random arrangements of atoms in the liquid mixture. The value of α1 com- puted as a function of the concentration of Pb using Eq. (16) is shown in Fig. 5 where we took coordi- nation number Z = 10. It is observed that the α1 is positive up to 0.6 concentration range of lead, with highest values at a concentration of 0.2, indicating the strong segregating tendency of the alloy. But above a 0.6 concentration of lead, the value of α1 goes on decreasing, showing the ordering tendency of the alloy. 0.0008 0.001 0.0012 0.0014 0.0016 0.0018 0.002 0.0022 0.0024 0.0026 0 0.2 0.4 0.6 0.8 1 η ( p a s) CPb KRP Kaptay BBK Figure 6: Viscosity versus bulk concentration of Pb ii Viscosity For the theoretical calculation of viscosity of Hg- Pb alloy at 600 K, the viscosities of each compo- nent (Pb and Hg) are required for KRP and Kaptay models. These values are obtained from Eq. (19) after substituting the values of η0 and ϵ of the met- als as given in reference [26]. The value of enthalpy for different concentrations is used as obtained from Eq. (9) and the Gibbs energy of activation of each pure metal is obtained from Eq. (21). Due to the lack of an experimental value for V E, it is taken as zero. In fact, the value of V E is non-zero for a non-ideal alloy, but the contribution of this term is very small for the determination of viscosity [15]. The results obtained from three models are com- pared as shown in Fig. 6. In the models, the viscosity of the liquid alloy increases with the in- crease in concentration of lead. The figure shows that there is a small deviation of the viscosity com- puted by BBK model as compared to the others. Due to the inability to compare theoretically com- puted results with experimental results, it becomes difficult to draw conclusions based on the models for the concentration dependence of the viscosity of Hg-Pb liquid alloy at temperature 600 K. iii Surface segregation and surface tension To calculate the surface tension of the alloy Hg-Pb, the densities and surface tension of individual met- als for all models required at 600 K are calculated by using the relations given in reference [26]. For the compound formation model, the same in- teraction parameters ω and ωjk used in thermody- 140005-8 Papers in Physics, vol. 14, art. 140005 (2022) / N. Panthi et al. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.2 0.4 0.6 0.8 1 C S CPb Butler Statistical Compound Figure 7: Surface concentration of lead versus bulk Pb concentration namic properties are used. Now, writing these val- ues and values of other quantities of both metals in Eqs. (23) and (24) and solving them simultane- ously, we first obtain surface concentrations of both metals, and then using each surface concentration of the corresponding metals, the surface tension is obtained. A similar method is applied to the other models. For statistical mechanical approach inter- change energy, ω = 0.699, obtained from Eq. (8), is used. For the improved derivation of the Butler model, the bulk and partial excess free energy of mixing of individual lead and mercury in a liquid state at 600 K are taken from reference [34]. The geometrical constant and the ratio of surface excess energy to the bulk excess energy (G S,Xs i /G b,Xs i ) are respec- tively considered as 1.061 and 0.818 [33]. Kaptay suggested that, in the case of negligible or unknown excess molar volume of the mixture, the partial mo- lar volume can be replaced by the molar volume of the same component. In such a situation, the partial surface area (Si) is replaced by the surface area (S0i ) of the same pure component [16,36]. The computed values of surface concentrations and sur- face tensions from all three models are compared in Figs. 7 and 8 respectively. Figure 7 shows the increasing pattern of the sur- face concentration of Pb with the increase in bulk concentration of Lead in all models. At 600 K the surface tension of mercury is less than the surface tension of lead. This suggests the surface segre- gating tendency of Hg. Thus, at higher bulk con- centration of Pb, two different atoms of the alloy are involved in the formation of chemical complexes 400 410 420 430 440 450 460 470 0 0.2 0.4 0.6 0.8 1 s ( N /m ) CPb Butler Statistical Compound Figure 8: Surface tension versus bulk Pb concentration or intermetallic compounds assumed to be HgPb2, but at lower bulk concentration of Pb, the surface of the alloy is enriched with Hg atoms. In Fig. 8, the surface tension of alloy Hg-Pb in- creases gradually with the increase in bulk concen- tration of Pb. The variation of surface tension in the compound formation model at higher bulk con- centration of Pb than in the other two models is believed to be the cause of consideration of set of the interaction energy parameters because, as we already mentioned, there is the possibility of com- pound formation at higher bulk concentrations of Pb. The compound formation model is expected to give better results than the other two models due to the presence of interaction parameters. However, due to the lack of experimental results, computed results cannot be compared. IV Conclusions The present study is a theoretical analysis for the understanding of thermodynamic, structural, transport and surface behavior of the binary liq- uid alloy Hg-Pb at 600 K under the assumption of the existence of the HgPb2 complex in the liquid mixture by compound formation model. The study explains the asymmetric behavior of the thermody- namic properties as a function of concentration as well as of a weakly interacting alloy. The theoret- ical study shows that the alloy has the nature of segregating at a lower concentration of Pb, but it shows an ordering nature at higher concentration of Pb at 600 K. 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Colloid Interface Sci. 283, 102212 (2020). 140005-11 https://doi.org/10.3139/ijmr-2005-0004 https://doi.org/10.3139/ijmr-2005-0004 https://doi.org/10.1080/09507119409548692 https://doi.org/10.3126/sw.v12i12.13564 https://doi.org/10.3126/sw.v12i12.13564 https://doi.org/10.1016/0021-9991(92)90240-Y https://doi.org/10.1016/S0925-8388(01)01684-X https://doi.org/10.1016/S0925-8388(01)01684-X https://doi.org/10.1016/0039-6028(76)90010-8 https://doi.org/10.1016/j.msea.2007.10.112 https://doi.org/10.1016/j.molliq.2014.09.053 https://doi.org/10.1016/j.cis.2020.102212 Introduction Theoretical formulation Thermodynamic functions Microscopic Functions Transport property: viscosity Kozlov-Ronanov-Petrov equation Kaptay equation BBK (Budai-Benko-Kaptay) model Surface property Compound Formation model Statistical mechanical approach Improved Derivation of the Butler equation Results and discussion Thermodynamic and microscopic properties Thermodynamic Properties Microscopic Properties Viscosity Surface segregation and surface tension Conclusions