BIBECHANA Vol. 20, No. 3, December 2023, 316–325 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher:Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University)Biratnagar Thermodynamic, structural and surface properties of rare earth metallic alloys: Au-La liquid system S. K. Yadav∗ Department of Physics, Mahendra Morang Adarsh Multiple Campus Biratnagar, Nepal. ∗Corresponding author: Email: sashit.yadav@mmamc.tu.edu.np Abstract A complete information related to the mixing behaviours of Au alloyed with rare earth metals or lanthanides is very scarce. Therefore, an attempt has been made in this work to com- pute and study the temperature and concentration dependent thermodynamic, structural and surface properties of Au-La liquid alloy using different theoretical approaches. The thermo- dynamic properties, such as excess Gibbs free energy of mixing, enthalpy of mixing, excess entropy of mixing and activity of the system were computed using available coefficients of interaction energy parameters in the framework of Redlich-Kister polynomial. Taking these as reference values, model parameters for quasi-lattice model were optimised at 1473 K. The model parameters were then determined at higher temperatures assuming them to be linear temperature-dependent. The thermodynamic and structural properties were then computed in the temperature range 1473 K-1773 K. The surface properties of the system were computed using Bulter’s model using determined values of partial excess Gibbs free energy of its com- ponents. Present investigations revealed that the compound forming tendency of the system gradually decreased with increase in temperature of the system. Keywords Ordering energy, hetero-coordinating nature, segregating nature, surface segregation, surface tension. Article information Manuscript received: March 10, 2023; Accepted: March 25, 2023 DOI https://doi.org/10.3126/bibechana.v20i3.59896 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction Development of lead free solders has been a great task for the researchers working in the field of ma- terials design and fabrication. One of the most promising alternative for the purpose has been found to be the use of Sn-based alloys, more pre- cisely Sn-Ag-Cu ternary alloys. Addition of small amount of rare earth (RE) elements to these alloys, increases the mechanical behaviour, creep-fatigue resistance and wetting properties [1–4]. In due course, knowledge related to the mixing behaviours of RE-based liquid alloys are important. But a very few information regarding the thermo-physical properties of metallic alloys having RE metals, also called lanthanide, as ingredient is available to date. In order to develop the thermodynamic database of RE-based liquid alloy, the thermodynamic and structural properties of Au-La liquid alloy was stud- ied in the previous work [5]. To further enhance the procedure, the mixing properties of Au-La liquid alloy have been computed and studied at different 316 http://nepjol.info/index.php/BIBECHANA sashit.yadav@mmamc.tu.edu.np https://doi.org/10.3126/bibechana.v20i3.59896 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 317 temperatures in the present work. Dong et al. in 2011 [6] have summarized the de- tails of literature associated with the works carried out by different researchers to assess the mixing properties of Au-La alloy. In their work, phase relations in Au-La and Au-Er alloys have been thermodynamically studies using CALPHAD tech- nique incorporated with the ab initio calculations. They calculated enthalpy of mixing of the system at 1473 K and compared them with the experimental values of Fitzner et al. [7]. Further, they presented the self consistent parameters of Redlich-Kister polynomial [8] for the excess Gibbs free energy of mixing for different phases present in the alloy. They calculated the existence of different stable phases, such as AuLa2, AuLa, Au2La, Au51La14 and Au6La. These results were found to be in good agreement with those obtained by Massalski [9], ex- cept for the reaction temperature and composition involving the constitution of Au2La and Au51La14 phases. They found the reaction temperature to be 75 K higher and close to xAu = 0.04 than observed by Massalski. In the knowledge of the author of this work, the complete description of the mixing properties of the system is lacking to date. Therefore, an attempt has been made in this work to study and explain the thermodynamic, structural and surface properties of the liquid alloy at different temperatures. In thermodynamic properties, such as excess Gibbs free energy of mixing (∆Gxs M ), en- thalpy of mixing (∆HM ), entropy of mixing (∆Sxs M ) and activity (ai; i = Au,La) were computed us- ing modeling equations of quasi-lattice model. In the same frame, the structural properties, such as concentration fluctuation in long wavelength limit (SCC(0)), Warren-Cowley short range order param- eter (α1) and ratio of mutual to intrinsic diffusion coefficients (DM/Did) were calculated. The surface tension and surface concentrations of the system were calculated using Butler’s model [10–13]. 2 Formulations 2.1 Quasi–lattice model This theoretical approach assumes that when two elements A and B are mixed in liquid state, the formation of the complex of the type AµBν takes place. The values of µ and ν depend upon stio- chiometric compositions at which the stable phase is present and are determined from the phase dia- gram of the system. In this work, the existence of the complex Au2La [6, 7] was considered as ener- getically favoured . In this regard, the expression for excess Gibbs free energy of mixing (∆Gxs M ) can be given as [14–16] ∆Gxs M = N [Φω +ΦABωAB +ΦAAωAA] (1) where ω, ωAB and ωAA are the interaction energy or model parameters. They are assumed to be temperature-dependent but concentration indepen- dent. Φ, ΦAB and ΦAA are the simple polynomials in x1 and x2, such that x1 + x2 = 1. Φ, ΦAB and ΦAA are expressed as [14–16] Φ = x1x2 ΦAB = x1 6 + x2 1 − 5x3 1 3 + x4 1 2 ΦAA = −x1 4 + x2 1 2 − x4 1 4 (2) ∆Gxs M can be expressed in terms of Gibbs free en- ergy of mixing (∆GM ) as ∆GM = ∆Gxs M +RT [x1 lnx1 + x2 lnx2] (3) where R (in J/(mol.K) is the real gas constant and T (in K) is the absolute temperature. The excess entropy of mixing (Sxs M ) is related to ∆Gxs M as ∆Sxs M = − ( ∂∆Gxs M ∂T ) P (4) From Equations (1) and (4), one can obtain ∆Sxs M = −N [ ∂∆ω ∂T Φ+ ∂∆ωAB ∂T ΦAB + ∂∆ωAA ∂T ΦAA ] (5) Herein, ∂∆ω ∂T ,∂∆ωAB ∂T and ∂∆ωAA ∂T are the tempera- ture derivative terms of interaction energy param- eters. The enthalpy of mixing (∆HM ) can be related to ∆Gxs M and ∆Sxs M by the well known thermodynamic expression as ∆HM = ∆Gxs M + T∆Sxs M (6) Using Equation (4) in Equation (6), yields ∆HM = ∆Gxs M − T ( ∂∆Gxs M ∂T ) P (7) The activity of component i (ai; i = Au,La) in the binary solution can be expressed in terms of ∆GM as RT ln ai = ∆GM + (1− xi) ( ∂∆GM ∂xi ) T,P,N (8) Using Equations (1) and (3) in Equation (8), one can obtain the expression for ( ∂∆GM ∂xi ) T,P,N as Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 318 ( ∂∆GM ∂xi ) T,P,N = ∆ωΦ ′ +∆ωABΦAB ′ +∆ωAAΦ ′ AA + ln ( xi 1− xi ) (9) where Φ ′ and Φ ′ ij are the first order derivatives of respective parameters (in Equation (2)) with re- spect to concentration of ith element. To study and understand the arrangement of atoms at atomic level in the initial melt, the structural functions have become an essential tool. Among them, the expression for concentration fluctuation in long wavelength limit (SCC(0)) is expressed as [17–20] SCC(0) = RT ( ∂2GM ∂x2 1 )−1 T,P,N = RT ( ∂2GM ∂x2 2 )−1 T,P,N (10) Using Equations (1) and (3) in Equation (10), yields SCC(0) = x1x2[1 + x1x2RT (∆ωΦ′′ +∆ωABΦ ′′ AB +∆ωAAΦ ′′ AA)] −1 (11) where ϕ′′ and ϕij ′′ are the second order derivatives of respective parameters with respect to concentra- tion (xi) and can be obtained from Equation (2). The ideal values of SCC(0) is obtained by the fol- lowing relation Sid CC(0) = x1x2 (12) The structural functions, Warren-Cowley short range–order parameter (α1) and the ratio of mutual to intrinsic diffusion coefficients (DM/Did) can be expressed in terms of SCC(0) as [13,15,18,21] α1 = S − 1 [S(Z − 1) + 1] (13) with S = SCC(0) Sid CC(0) (14) and DM Did = Sid CC(0) SCC(0) (15) 2.2 Redlich-Kister (R-K) polynomial R-K polynomial is used to model the temperature- dependent thermodynamic properties of liquid al- loys. It is extensively used as modeling equations in theoretical calculations, software based compu- tations and experimental measurements [6, 8]. In this approach, ∆Gxs M is expressed as [5, 6, 8, 22] ∆Gxs M = x1x2 n∑ q=0 Lq(x1 − x2) q (16) where Lq are the linear T–dependent coefficients or interaction energy parameters of R-K polynomial. They are expressed in the form Lq = aq + bqT , where aq (in J/mol) are contributions due to ∆HM and bq (in J/mol-K) are contributions of ∆Sxs M to ∆Gxs M equivalent to Equations 6. The partial excess Gibbs free energy (∆Gxs i ) of the component i in the binary liquid alloy can be given as [22] ∆Gxs i = ∆Gxs M + (1− xi) ( ∂∆Gxs M ∂xi − ∂∆Gxs M ∂(1− xi ) (17) The activity coefficient of component i in the binary solution is related to ∆Gxs i as RT ln γi = ∆Gxs i (18) After the computations of γi, the activity of com- ponent i can be obtained by the relation ai = xiγi (19) The values of Sxs M and HM can be obtained using Equations (4), (7) and (16). Using Equations (10) and (16), SCC(0) for this system having q = 0, 1 can be obtained as [5, 23] SCC(0) = RT [−2L0 + (−12x1 + 6)L1 + RT x1(1− x1) ]−1 (20) The values of other structural functions in this framework can also be obtained using Equations (13-15). 2.3 Butler model The surface properties, surface tension (σ) and sur- face concentrations (xS i ) of the system have been calculated using Butler model. According to this model, σ for binary liquid solution can be given as [10–13] Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 319 σ = σ1 + RT A1 ln xS 1 xb 1 + ∆Gxs 1,S −∆Gxs 1,b A1 = σ2 + RT A2 ln xS 2 xb 2 + ∆Gxs 2,S −∆Gxs 2,b A1 (21) where σi is the surface tension, Ai is molar surface area, xS i is the surface concentration, xb i is the bulk concentration, ∆Gxs i,S is the molar surface partial excess Gibbs free energy and ∆Gxs i,b is the molar bulk partial excess Gibbs free energy for the com- ponent i of the liquid mixture at the melting tem- perature of the alloy. ∆Gxs i,S and ∆Gxs i,b related by the relation ∆Gxs i,S = β∆Gxs i,b (22) where β = 0.8181 [11, 13] for simple liquid metals. The molar surface area (Ai) of component i in the liquid mixture can be calculated using the relation Ai = fN 1 3 AV 2 3 i (23) where NA is the Avogadro’s number and Vi(= mi ρi ) is the molar volume of element i in the liquid mix- ture. The values of σi and Vi are calculated with the help of the following relations [11,24] σi = σ0 i + dσ dT (T − T0) (24) and Vi = V 0 i [1 + +λi(T − T0)] (25) where σ0 i is the surface tension, V 0 i is the molar vol- ume and λi [11,24] is the temperature coefficient of volume expansion for ith component at its melting temperature (T0). 3 Results and Discussion 3.1 Thermodynamic properties The self-consistent parameters for excess Gibbs free energy of mixing (∆Gxs M ) of Au–La liquid alloy were taken from Dong et al. [6] (Table 1). These pa- rameters were used to calculate ∆Gxs M of the sys- tem at 1473 K in the frame work of R-K polyno- mial [8] (Equation (16)). The values so computed were considered as reference data and then quasi– lattice model was employed to determine the model fit parameters using Equations (1-3). The best fit values of the model parameters were estimated as- suming the existence of Au2La complex and are presented in Table 1. Table 1: Interaction energy parameters for ∆Gxs M of Au–La liquid alloy Parameters [J/mol] Reference L0 = −254446.24 + 6.00025 ∗ T L1 = −85046.07 + 2.89377 ∗ T [6] ∆ω = −264279.94476 + 8.97912 ∗ dT ∆ωAB = −3042.03606 + 0.083140 ∗ dT This work ∆ωAA = −345964.24650 + 1.66280 ∗ dT The determined values of model parameters ∆ω, ∆ωAB and ∆ωAA were found to have negative val- ues (Table 1). Among them, ∆ω is called ordering energy parameter and its value was estimated to be −264279.94476 indicating the system to be strongly interacting in nature. The value of ∆Gxs M was com- puted at 1473 K using the parameters from Table 1 and Equations (1-3). The compositional depen- dence of ∆Gxs M of the this work along with the values computed using the parameters of Dong et al. are displayed in Figure 1. 0.0 0.2 0.4 0.6 0.8 1.0 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 S M xs/R G M xs/RT H M /RT G M xs /R T , H M /R T , S M /R x Au Dong et al. [6] This work Fitzner et al. [7] Figure 1: Compositional dependence of ∆Gxs M/RT , ∆HM/RT and ∆Sxs M/R of Au–La liquid alloy at 1473 K. The computed values of ∆Gxs M were found to be in consistent with each other at composi- tions thereby validating the present optimisation procedure (Figure 1). The maximum negative value of ∆Gxs M = −5.12014RT (this work) and ∆Gxs M = −5.12991RT (Dong et al.) [6] at xAu = 0.6. The high negative value of ∆Gxs M/RT indicates the system to be strongly interacting in nature at its melting temperature, 1473 K. Moreover, the system is found to be asymmetric with respect to ∆Gxs M . The enthalpy of mixing (∆HM/RT ) and excess entropy of mixing ∆Sxs M/R of the system were cal- culated with the help of parameters from Table 1 and Equations (4-7). These computed values are portrayed as a function of composition in Figure 1. Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 320 The extremum negative value of ∆HM/RT = −5.37922RT (this work) and ∆HM/RT = −5.38665RT (Dong et al. [6]) at xAu = 0.6. The high negative value of HM/RT corresponds the strong association between Au and La in the as- sumed complex at 1473 K. Further, the computed values of ∆HM/RT of this work and those of Dong et al. were consistent with each other and the system is found to be asymmetric with respect this physical function. The values of ∆HM/RT were also found to be in agreement with the ex- perimental values of Fitzner et al. [7] at higher compositions of Au. Likewise, the maximum neg- ative value of ∆Sxs M = −0.26781R (this work) and ∆Sxs M = −0.26744R (Dong et al.) at xAu = 0.5. The Au-La liquid alloy is found to be symmetric with respect to ∆Sxs M . Thus, the above theoretical investigations reveal that the model parameters de- termined in the frame work of quasi-lattice model are fruitful in explaining the thermodynamic prop- erties of the system. To further validate the present optimisation pro- cess, the activity of the system was computed using the same determined model parameters. The ac- tivities of the individual atoms (aAu and aLa) of the initial melt were calculated on the basis of R- K polynomial (Equations (16-19)) and quasi-lattice model (Equations (8 and 9)) using the parameters from Table 1. The calculated and ideal values of aAu and aLa are plotted as a function of concentra- tion in Figure 2. 0.0 0.2 0.4 0.6 0.8 1.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 a La a Au a A u, a L a x Au Dong et al. [6] This work Ideal Figure 2: Computed values of aAu and aLa versus concentration of Au (xAu) of Au–La liquid alloy at 1473 K. The computed values of aAu and aLa were found to be much less than their respective ideal values at all compositions indicating the system to be strongly interacting in nature. Both of these values, com- puted using the parameters of present work and Dong et al., were found to be consistent with each other at all concentrations (Figure 2). Thus, it can be stated that the preferred model is capable of explaining the thermodynamic functions and re- producing the activity of the system. Therefore, these model parameters were considered for the computations of structural and surface properties. In thermodynamic properties, ∆Gxs M , ∆HM and ai; i = Au,La were computed at different temper- atures assuming the model parameters of quasi- lattice model to be linear temperature-dependent. The model parameters ∆ω, ∆ωAB and ∆ωAA in correlation with their respective tempera- ture derivatives terms ∂∆ω/∂T , ∂∆ωAB/∂T and ∂∆ωAA/∂T can be expressed as [5, 13] ∆ω(T ) = ∆ω(T0) + ∂∆ω ∂T (T − T0) (26) ∆ωij(T ) = ∆ωij(T0) + ∂∆ωij ∂T (T − T0) (27) where T0 (= 1473 K) is the melting temperature of the system and T is the temperature of inter- est at which thermodynamic properties are to be calculated. The determined values of temperature- dependent model parameters for this liquid alloy are portrayed in Table 1. 1500 1600 1700 1800 -60 -50 -40 -30 -20 G M xs [ kJ m o l-1 ] Temperature [K] Au 10 La 90 Au 20 La 80 Au 30 La 70 Au 40 La 60 Au 50 La 50 Au 60 La 40 Au 70 La 30 Au 80 La 20 Au 90 La 10 Figure 3: Temperature dependence of excess Gibbs free energy of mixing (∆Gxs M ) for Au–La liquid al- loy. The values of ∆Gxs M were computed using Equations (1 and 2), ∆HM were computed using Equations (1 and 7) and ai were computed using Equations (17-19) at different temperatures with the aid of parameters in Table 1. The temperature variation of ∆Gxs M and ∆HM at fixed composition ratios of Au and La are presented in Figures 3 and 4. The compositional dependence of ai at different tem- peratures are plotted in Figures 5 and 6. Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 321 The negative values of ∆Gxs M and ∆HM decreased linearly with increased in temperature of the sys- tem. 1500 1575 1650 1725 1800 -70 -60 -50 -40 -30 -20 H M [ kJ m o l-1 ] Temperature [K] Au 10 La 90 Au 20 La 80 Au 30 La 70 Au 40 La 60 Au 50 La 50 Au 60 La 70 Au 70 La 30 Au 80 La 20 Au 90 La 10 Figure 4: Variation of ∆HM with temperature of Au–La liquid alloy. 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 a A u x Au T=1473 K T=1573 K T=1673 K T=1773 K Ideal Figure 5: Activity of Au (aAu) versus xAu in tem- perature range 1474–1773 K. 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 a L a x Au T=1473 K T=1573 K T=1673 K T=1773 K Ideal Figure 6: Compositional dependence of activity of La (aLa) at different temperatures. These results revealed the decrease in the complex formation tendency or strength of interaction be- tween the monomers. Indeed, the activities of Au and La gradually increased with increase in tem- perature. The deviation of ai from ideal values gradually decreased with increase in temperature indicating the decrease in mixing tendency of the system (Figures 5 and 6). 3.2 Structural properties The information related to the local arrangement of atoms in the initial melt can be obtained form the knowledge of structural functions. In structural functions, concentration fluctuation in long wave- length limit (SCC(0)), Warren-Cowley short range order parameter (α1) and ratio of mutual to intrin- sic diffusion coefficients (DM/Did) were computed in the frame work of quasi-lattice model using Equa- tions (10-15) and parameters from Table 1. Like- wise, Equations (12-15) and (20) were used to cal- culate these values on the basis of R-K polynomial with the aid of parameters of Dong et al. (Table 1). The compositional dependence of SCC(0), α1 and DM/Did are plotted in Figures 7 and 8. 0.2 0.4 0.6 0.8 1.0 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 alpha 1 S CC (0) a lp h a 1, S C C (0 ) x Au Dong et al. [6] This work Ideal Figure 7: Computed values of SCC(0) and α1 of Au–La liquid alloy at 1473 K. At a temperature and concentration, if SCC(0) 1, then complex formation tendency or hetero-coordinating tendency in the alloy is expected. At the same con- straints, if SCC(0)>Sid CC(0), α1 > 0 and DM/Did < 1, then self association among the atoms of alloy or homo-coordinating tendency is expected. The com- puted values of SCC(0) showed negative deviation with respect to its ideal values in the entire concen- tration range indicating the system to be ordering in nature. This finding is further supported by the negative values of α1 and DM/Did > 1 (Figures 7 and 8). Moreover, the values of SCC(0) and α1 computed using the parameters of this work and Dong et al. [6] were found to be consistent with each other at all concentrations. Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 322 Following the similar procedure as mentioned above, the values of SCC(0), α1 and DM/Did > 1 were computed in the temperature range 1473-1773 K. The compositional and temperature dependence of these functions are displayed in Figures 9-11. 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8 10 12 14 D M /D id x Au This work Figure 8: Computed values of DM/Did versus xAu of Au–La liquid alloy at 1473 K. 0.0 0.2 0.4 0.6 0.8 1.0 0.05 0.10 0.15 0.20 0.25 S C C (0 ) x Au T=1473 K T=1573 K T=1673 K T=1773 K Ideal Figure 9: SCC(0) versus xAu of Au–La liquid alloy at different temperatures. The computed values of SCC(0) gradually increased and got closure to its ideal values with the rise in temperature of the system above its melting tem- perature (Figure 9). At equiatomic composition (xAu = 0.5), the deviation of SCC(0) from Sid CC(0) were found to be −0.22529, −0.22370, −0.22211 and −0.22054 at 1473 K, 1573 K, 1673 K and 1773 K respectively. These decrease in negative deviations correspond the decrease in in compound forming tendency in the initial melt at high tem- peratures. This result is further supported by the decrease in negative values of α1 and decrease in positive val- ues of DM/Did with increase in temperature of the system (Figures 10 and 11). At xAu = 0.5, the computed values of α1 were found to be −0.47692, −0.45959, −0.44331 and −0.42814 and those of DM/Did were computed to be 10.11753, 9.50450, 8.96344 and 8.48676 at 1473 K, 1573 K, 1673 K and 1773 K respectively. These results further stands in favour of those revealed by SCC(0). Thus, the re- sults obtained from the investigations of structural functions are in accordance with those indicated by the thermodynamic functions in the above section. 0.0 0.2 0.4 0.6 0.8 1.0 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 a lp h a 1 x Au T=1473 K T=1573 K T=1673 K T=1773 K Figure 10: α1 versus xAu of Au–La liquid alloy at different temperatures. 1500 1575 1650 1725 1800 2 3 4 5 6 7 8 9 10 11 12 13 D M /D id Temperature [K] Au 10 La 90 Au 20 La 80 Au 30 La 70 Au 40 La 60 Au 50 La 50 Au 60 La 40 Au 70 La 30 Au 80 La 20 Au 90 La 10 Figure 11: Computed values of DM/Did of Au–La liquid alloy at different temperatures. 3.3 Surface properties In surface properties, surface tension (σ) and ex- tent of surface segregation (surface concentrations, xs i ) were computed in the frame work of Butler model [10, 24] using Equations (26-25) and param- eters from Table 2. The compositional dependence of xs i and σ are portrayed in Figures 12 and 13 re- spectively. Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 323 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 xS La xS Au xS A u, x S L a x Au This work Ideal Figure 12: Computed values of xS Au and xS La versus xAu of Au–La liquid alloy at 1473 K. The surface concentration of La (xS La) showed pos- itive deviation whereas that of Au (xS Au) showed negative deviation from their respective ideal val- ues in the concentration range xS Au < 0.9 (Figure 12). Among the two components, La has the lower surface tension (σLa = 0.63360 N/m) and that of Au is higher (σAu = 1.13475 N/m) at 1473 K, melting temperature of the system. Therefore, La segregated in the surface phase and Au remained in the bulk phase of the initial melt in the range xS Au < 0.9. xS Au gradually increased and xS La grad- ually decreased with the increase in the bulk con- centration of Au (xAu). They exceed their respec- tive ideal values beyond xS Au > 0.9 indicating ex- change of their positions in the two phases of melt. The computed values σ of the system gradually in- creased with increased in the bulk concentration of Au. It showed negative deviation from its ideal value beyond xAu < 0.6 while showed positive de- viation in the remaining range (Figure 13). 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.7 0.8 0.9 1.0 1.1 1.2 S u rf a ce t e ns io n [N m -1 ] x Au This work Ideal Figure 13: Compositional dependence of σ of Au– La liquid alloy at 1473 K. Table 2: Input parameters for surface tension [25] Atom T0 ρ0 ∂ρ/∂T σ0 ∂σ/∂T (K) (kg/m3) (kg/m3K) (N/m) (N/mK) Au 1336 17360 -1.50 1.169 -0.00025 La 1203 5955 -0.24 0.720 -0.00032 The surface tension and surface concentrations of the components of the system were also calculated at different temperatures with the aid of Equations (26-25) and parameters from Table 2. The compo- sitional and temperature dependence of estimated values are presented in Figures 14-16. 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 xs A u x Au Ideal T=1473 K T=1573 K T=1673 K T=1773 K Figure 14: Computed values of xS Au versus xAu of Au–La liquid alloy at different temperatures. 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 xS L a x Au Ideal T=1473 K T=1573 K T=1673 K T=1773 K Figure 15: Computed values of xS La versus xAu of Au–La liquid alloy at different temperatures. The surface concentrations of Au gradually in- creased and those of La gradually decreased with increase in temperature of the liquid mixture. The negative deviations of xAu and positive deviation of xLa from their respective ideal values eventually decreased at elevated temperature (Figure 14 and 15). Therefore, Au atoms move from bulk to sur- face phase while those of La moves from surface Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 324 to bulk phase to maintain the equilibrium condi- tion in the liquid. Moreover, the surface tension of the mixture gradually decreased linearly with the increase in temperature beyond its melting temper- ature (Figure 16). These results are as expected since the cohesive force between the components of the liquid mixture gradually decreases with the in- crease in temperature. 1500 1575 1650 1725 1800 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 S u rf ac e t e n si o n [N m -1 ] Temperature [K] Au 10 La 90 Au 20 La 80 Au 30 La 70 Au 40 La 60 Au 50 La 50 Au 60 La 40 Au 70 La 30 Au 80 La 20 Au 90 La 10 Figure 16: Variation of σ with temperature of Au– La liquid alloy. 4 Conclusions The preferred theoretical models, quasi-lattice and Butler’s models, successfully explained the compo- sitional and temperature dependence of thermody- namic, structural and surface properties of Au-La liquid alloy. The system was found to be the most interacting at its melting temperature whereas this tendency gradually decreased with increase in tem- perature. The surface tension of the system de- creased linearly at elevated temperatures. The re- sults obtained from the theoretical investigations of thermodynamic, structural and surface properties were in accordance with each other. References [1] D. Q. Yu, J. Zhao, and L. Wang. Improve- ment on the microstructure stability, mechan- ical and wetting properties of Sn–Ag–Cu lead- free solder with the addition of rare earth el- ements. Journal of alloys and compounds, 376(1-2):170–175, 2004. [2] WeiMin Xiao, YaoWu Shi, GuangChen Xu, Ren Ren, Fu Guo, ZhiDong Xia, and YongPing Lei. Effect of rare earth on mechanical creep– fatigue property of SnAgCu solder joint. Jour- nal of alloys and compounds, 472(1-2):198–202, 2009. [3] Z. G. Chen, Y. W. Shi, Z. D. Xia, and Y. F. Yan. Study on the microstructure of a novel lead-free solder alloy SnAgCu-RE and its sol- dered joints. Journal of electronic materials, 31:1122–1128, 2002. [4] C. M. L. Wu, D. Q. Yu, C. M. T. Law, and L. Wang. Improvements of microstruc- ture, wettability, tensile and creep strength of eutectic Sn–Ag alloy by doping with rare- earth elements. Journal of materials research, 17(12):3146–3154, 2002. [5] S. K. Yadav. Assessment of Thermodynamic and Structural Properties of Al–Er liquid Al- loy at Different Temperatures. The Journal of Knowledge and Innovation, pages 66–74, 2023. [6] H. Q. Dong, X. M. Tao, H. S. Liu, T. Lau- rila, and M. Paulastro-Kröckel. Thermody- namic assessment of Au–La and Au–Er binary systems. Journal of alloys and compounds, 509(13):4439–4444, 2011. [7] K. Fitzner, W. G. Jung, and O. J. Kleppa. Thermochemistry of binary alloys of transition metals: the Me-Sc, Me-Y, and Me-La (Me= Ag, Au) systems. Metallurgical Transactions A, 22:1103–1111, 1991. [8] O. Redlich and A. T. Kister. Algebraic repre- sentation of thermodynamic properties and the classification of solutions. Industrial & Engi- neering Chemistry, 40(2):345–348, 1948. [9] T. B. Massalski. Binary alloy phase diagrams. ASM International, 1990. [10] J. A. V. Butler. The thermodynamics of the surfaces of solutions. Proceedings of the Royal Society of London. Series A, Containing Pa- pers of a Mathematical and Physical Charac- ter, 135(827):348–375, 1932. [11] George Kaptay. A unified model for the co- hesive enthalpy, critical temperature, surface tension and volume thermal expansion coeffi- cient of liquid metals of bcc, fcc and hcp crys- tals. Materials Science and Engineering: A, 495(1-2):19–26, 2008. Shashit Kumar Yadav/ BIBECHANA 20 (2023) 316-325 325 [12] George Kaptay. Partial surface tension of com- ponents of a solution. Langmuir, 31(21):5796– 5804, 2015. [13] S. K. Yadav, M. Gautam, and D. Adhikari. Mixing properties of Cu–Mg liquid alloy. AIP Advances, 10(12):125320, 2020. [14] A. B. Bhatia and R. N. Singh. A quasi–lattice theory for compound forming molten alloys. Physics and Chemistry of Liquids an Interna- tional Journal, 13(3):177–190, 1984. [15] D. Adhikari, S. K. Yadav, and L. N. Jha. Thermo-physical properties of Mg-Tl melt. Journal of Basic and Applied Research Inter- national, 9:103–110, 2015. [16] S. K. Yadav, S. Lamichhane, L. N. Jha, N. P. Adhikari, and D. Adhikari. Mixing behaviour of Ni–Al melt at 1873 K. Physics and Chem- istry of Liquids, 54(3):370–383, 2016. [17] A. B. Bhatia and W. H. Hargrove. Concentra- tion fluctuations and thermodynamic proper- ties of some compound forming binary molten systems. Physical Review B, 10(8):3186, 1974. [18] S. K. Yadav, L. N. Jha, and D. Adhikari. Thermodynamic and structural properties of Bi-based liquid alloys. Physica B: Condensed Matter, 475:40–47, 2015. [19] S. K. Yadav, L. N. Jha, and D. Adhikari. Segregating to ordering transformation in In– Sn melt. Physics and Chemistry of Liquids, 53(4):443–454, 2015. [20] S. K. Yadav, L. N. Jha, and D. Adhikari. Ther- modynamic, structural, transport and surface properties of Pb-Tl liquid alloy. Bibechana, 13:100–113, 2016. [21] D. Adhikari, S. K. Yadav, and L. N. Jha. Thermo-physical properties of Al–Fe melt. Journal of the Chinese Advanced Materials So- ciety, 2(3):149–158, 2014. [22] S. K. Yadav, U. Mehta, and D. Adhikari. Opti- mization of thermodynamic and surface prop- erties of ternary Ti–Al–Si alloy and its sub- binary alloys in molten state. Heliyon, 7(3), 2021. [23] R. K. Gohivar, S. K. Yadav, R. P. Koirala, G. K. Shrestha, and D. Adhikari. Tempera- ture dependence of interaction parameters for Al–Li liquid alloy. Philosophical Magazine, 101(2):179–192, 2021. [24] George Kaptay. A coherent set of model equa- tions for various surface and interface energies in systems with liquid and solid metals and al- loys. Advances in colloid and interface science, 283:102212, 2020. [25] Eric Adolph Brandes and G. B. Brook. Smithells metals reference book. Elsevier, 2013. Introduction Formulations Quasi–lattice model Redlich-Kister (R-K) polynomial Butler model Results and Discussion Thermodynamic properties Structural properties Surface properties Conclusions