BIBECHANA Vol. 21, No. 2, August 2024, 150–158 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, surface and transport properties of Au-Cu melt S. K. Yadav1∗ U. Mehta1, R. K. Gohivar1 1Department of Physics, Mahendra Morang Adarsh Multiple Campus Biratnagar, Nepal. ∗Corresponding author: Email: sashit.yadav@mmamc.tu.edu.np Abstract Thermodynamic, structural, surface and transport properties of Au-Cu liquid alloy were calcu- lated on the basis of different theoretical modeling equations. The thermodynamic properties, such as excess Gibbs free energy of mixing, enthapy of mixing and activity were estimated and compared with the available experimental and literature data on the basis of simple theory of mixing (STM). The best fit values of model parameters were estimated using the experimental values of excess Gibbs free energy of mixing and enthalpy of mixing of the system at 1550 K. Using the same model parameters and frame, the concentration fluctuation in long wave- length limit and Warren-Cowley short range order parameter were calculated and analysed to understand the local arrangement of atoms in the liquid mixture. The surface tension and extent of surface segregation of atoms in the initial melt were computed using Butler’s model. In transport properties, the ratio of mutual to intrinsic diffusion coefficients was calculated using STM and viscosity was calculated using Kaptay equation. The system showed complete ordering tendency at its melting temperature. Keywords Molar surface area, ordering nature, noble metals, transition metals, surface phase. Article information Manuscript received: February 10, 2024; Revised: March 25, 2023; Accepted: May 10, 2024 DOI https://doi.org/10.3126/bibechana.v21i2.66853 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction The important information related to the alloy cat- alytic, wettability, electrical, magnetic, mechanical, etc. behaviours can be obtained by computing and analyzing the thermodynamic, structural, surface and transport properties of liquid alloys. In this re- gard, several researchers [1–11] working in the field of material science and engineering have made nu- merous attempts to access the mixing properties of different binary liquid alloys. Therefore, thermody- namic, structural, surface and transport properties of Au-Cu liquid alloy have been studied in this work on the basis of different theoretical approaches. Topor and Kleppa in 1984 [12] and Okamoto et al. in 1987 [13] have summarized a brief review of the literature related to the thermodynamic properties and phase diagrams of the Au-Cu system. Later, Sundman et al. in 1998 [14] used Calphad ap- proach to access the thermodynamic description of the system using the available experimental results for solid and liquid phases of the system. They successfully reproduced the experimental results by developing a theoretical framework, called Com- 150 http://nepjol.info/index.php/BIBECHANA sashit.yadav@mmamc.tu.edu.np https://doi.org/10.3126/bibechana.v21i2.66853 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ S. K. Yadav et al./ BIBECHANA 21 (2024) 150-158 151 pound Energy Formalism. They further presented optimised values of self-consistent parameters for the excess Gibbs free energy of mixing for the sys- tem. Han et al. in 2004 [15] used molecular dynam- ics simulation techniques to determine the density and specific heat of the system above and below its melting temperature. Their results showed that the density of the system increased linearly with decrease in temperature and specific heat remained constant in the range 900-1900 K. In the same year, Brillo et al. [16] experimentally measured the density and thermal expansion of the system and its pure components. Their results were found to be consistent with those obtained by Han et al. Recently, Kong et al. in 2018 [17] investigated the structural, thermodynamic and elastic properties of the system using first principle calculations based on density functional theory. They found that the metallic complexes, AuCu, Au3Cu and AuCu3 to be mechanically stable. From the above mentioned literature, it can be stated that their appears immense interest to access the mixing properties of Au-Cu system. Therefore, the thermodynamic, structural, surface and trans- port properties of the system have been computed using different theoretical approaches in present work. 2 Formulations 2.1 Thermodynamic properties The thermodynamic properties of the system have been computed using the modeling equations of simple theory of mixing (STM).On the basis of the model, the excess Gibbs free energy of mixing for binary liquid alloy is expressed as [18–20] ∆Gxs M RT = ∫ c 0 log σ2dx = xA log γA + xB log γB (1) where xA (A=Au) and xB (B=Cu) are the mole fractions of atoms A and B respectively in the liq- uid mixture, such that xA + xB = 1. The terms ϕ, γA and γB can be given as ϕ = (ζ + 2xA − 1) exp (−ω/zkBT )/2xA (2) γA = [(ζ − 1 + 2xA)/xA(1 + ζ)]z/2 (3) γB = [(ζ + 1− 2xA)/xB(1 + ζ)]z/2 (4) ζ = 1 + 4xAxB [exp (2ω/zkBT )− 1]1/2 (5) Herein, ω is called the ordering energy, is the model fit parameter and z is the coordination number. If ω < 0, then complex formation or hetero-coordinating tendency in the initial melt is expected, otherwise homo-coordinating tendency is expected. At equiatomic concentration (xA = xB = 0.5), the expression for ∆Gxs M can be given as [18–20] ∆Gxs M RT = ln [2z/2(1 + exp ( −ω zkBT ))−z/2] (6) Once the value of ω is determined at xA = xB = 0.5, ∆Gxs M can be computed in the entire composi- tion range using the relation ∆Gxs M = NkBT (xAxB ω kBT ) (7) The relation between Gibbs free energy of mix- ing (∆GM ) and excess Gibbs free energy of mixing ∆Gxs M is expressed as ∆GM = ∆Gxs M +RT (xA lnxA + xB lnxB) (8) The enthalpy of mixing (∆HM ) is expressed in terms of ∆Gxs M as ∆HM = ∆Gxs M − T ( ∂∆Gxs M ∂T ) P (9) where P is the pressure of liquid mixture. Using Equations (7) in Equation (9), one can obtain ∆HM = NkBT [xAxB ω kBT − 1 kBT xAxB dω dT ] (10) where dω/dT is the temperature derivative term of ordering energy as it is temperature-dependent and concentration independent. Once the values of ∆HM and ∆Gxs M are determined, the values of ex- cess entropy of mixing ∆Sxs M can be calculated using the standard thermodynamic relation ∆Gxs M = ∆HM − T∆Sxs M (11) The activity (ai, i = A,B) of each component of the liquid alloy is expressed in the form as [18–20] ln aA = lnxA + x2 B ω kBT (12) ln aA = lnxA + x2 A ω kBT (13) S. K. Yadav et al./ BIBECHANA 21 (2024) 150-158 152 2.2 Structural properties In structural functions, the expression for con- centration fluctuation in long wavelength limit (SCC(0)) in terms of ∆GM and and ai can be given as [4, 11,18,21–23] SCC(0) = RT ( ∂2GM ∂x2 A )−1 T,P,N = RT ( ∂2GM ∂x2 B )−1 T,P,N (14) and SCC(0) = xBaA( ∂aA ∂xA )−1 = xAaB( ∂aB ∂xB )−1 (15) With the aid of Equation (6), Equation (14) be- comes SCC(0) = xAxB [1 + z 2ζ (1− ζ)]−1 (16) The ideal values of SCC(0) is calculated using the following equation Sid CC(0) = x1x2 (17) Another structural functions, Warren-Cowley short range–order parameter (α1) can be expressed in terms of SCC(0) as [11,21,24,25] α1 = S − 1 [S(Z − 1) + 1] (18) where S = SCC(0) Sid CC(0) (19) 2.3 Surface properties The surface tension (σ) and surface concentration (xS i ) of the system have been calculated in the framework of Butler’s model. According to this model, σ for binary liquid solution is expresses in the form [23,25–28] as σ = σA + RT λA ln xS A xb A + ∆Gxs A,S −∆Gxs A,b λA = σB + RT λB ln xS B xb B + ∆Gxs B,S −∆Gxs B,b λB (20) where σi is the surface tension, λi is molar sur- face area, xS i is the surface concentration, xb i is the bulk concentration, ∆Gxs i,S is the molar surface par- tial excess Gibbs free energy and ∆Gxs i,b is the molar bulk partial excess Gibbs free energy for the compo- nent i in the liquid mixture at the melting temper- ature of the alloy. ∆Gxs i,S and ∆Gxs i,b are correlated by the relation [25,27,28] ∆Gxs i,S = β∆Gxs i,b (21) where the value of β = 0.8181 [25, 28] has been taken for the present calculations. The molar sur- face area (λi) of component i in the liquid mixture can be calculated using the relation λi = fN 1 3 AV 2 3 i (22) 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 [16,28–30] σi = σ0 i + dσ dT (T − T0) (23) and Vi = V 0 i [1 + +κi(T − T0)] (24) where σ0 i is the surface tension, V 0 i is the molar volume and κi [28,29,31] is the temperature coeffi- cient of volume expansion for ith component at its melting temperature (T0). 2.4 Transport properties The viscosity (η) and ratio of mutual to intrinsic dif- fusion coefficients (DM/Did) have been computed in order to analyze the transport properties of the system. On the basis of simple theory of mixing, η of binary liquid alloy can be given as [18,23] η = ηid[1− xAxB( 2ω kBT )] (25) where ηid is the ideal value of viscosity calculated using the relation [18,23,30] ηid = xAηA + xBηB (26) Herein, ηi stands for the viscosity of pure compo- nent i at the melting temperature of the liquid alloy and is calculated using the relation ηi = η0i exp ( E RT ) (27) The term η0i is viscosity and E is activation energy of the component i at its melting temperature. The Kaptay equation for the viscosity of binary liq- uid alloy can be expressed as [10,23,32] η = ( hNA xAVA + xBVB + V E ) x(xAG ∗ A + xBG ∗ B(0.155± 0.015)∆HM RT ) (28) S. K. Yadav et al./ BIBECHANA 21 (2024) 150-158 153 where G∗ i is the Gibbs energy of activation of the viscous flow for pure component i and is given as G∗ i = RT ln ( ηiVi hNA ) (29) where h is the Planck’s constant, V E is excess vol- ume upon alloy formation which can be neglected and the rest of terms carry their usual meanings as stated above. In the framework of STM, the values of DM/Did for binary liquid alloy can be estimated using the relation [18,19,23] DM Did = [ 1− xAxB ( 2ω kBT )] (30) DM = xADB + xBDA; DA and DB are the self- diffusion coefficients of pure components A and B respectively. Equation (30) can also expressed in the form as DM Did = Sid CC(0) SCC(0) (31) Herein, the terms carry their usual meanings as stated in the above sub-sections. 3 Results and Discussion 3.1 Thermodynamic properties The model parameter (ω/kBT ) of simple theory of mixing (STM) is assumed to be temperature- dependent but concentration independent. The ef- fect of model parameter on excess Gibbs free energy of mixing (∆GM ) was analysed by arbitrarily vary- ing the value of ω/kBT in the range ±5. The values of ∆GM were calculated at 1550 K using Equations (7) and (8) in the above mentioned range of model parameter at all concentrations. The compositional dependence of ∆GM/RT so estimated are plotted in Figures 1 and 2. 0.2 0.4 0.6 0.8 1.0 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6  G M /R T x Au /k B T=0 /k B T=1 /k B T=2 /k B T=3 /k B T=4 /k B T=5 Figure 1: Effect of positive values of ω/kBT on ∆GM/RT versus xAu at 1550 K. 0.0 0.2 0.4 0.6 0.8 1.0 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 G M /R T x Au /k B T=-1 /k B T=-2 /k B T=-3 /k B T=-4 /k B T=-5 Figure 2: Effect of negative values of ω/kBT on ∆GM/RT versus xAu at 1550 K. In the preferred model, it is assumed that if ω/kBT < 0, then ordering tendency among the atoms of liquid alloy is expected and if ω/kBT > 0, then segregating tendency is expected. It can be observed that the negative values of ∆GM/RT gradually decreased with increase in positive val- ues of ω/kBT , except for ω/kBT = 3 (Figure 1). Herein, ω/kBT = 0 represents the ideal values of ∆GM/RT . The perusal of Figure 2 corresponds that with increase in negative values of ω/kBT , the negative values of ∆GM/RT gradually increased. Thus, the hetero-coordinating tendency or com- plex forming tendency in the liquid alloy gradu- ally increased with the increase in negative values of ω/kBT . 0.2 0.4 0.6 0.8 1.0 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 S M xs/R H M /RT G M xs/RT G M xs /R T ,  H M /R T ,  S M xs /R x Au This work Experimental Figure 3: Compositional dependence of ∆Gxs M/RT , ∆HM/RT and ∆Sxs M/R for Au–Cu liquid alloy at 1550 K. The value of ω/kBT was first obtained with the help of Equation (6) and experimental value of ∆Gxs M [5] at xAu = 0.5 and 1550 K. Equation (7) was then used to calculate ∆Gxs M for the system at all con- centrations. In due course, the value of ω/kBT was S. K. Yadav et al./ BIBECHANA 21 (2024) 150-158 154 slightly adjusted following the method of succes- sive approximation confirming that the obtained values were consistent with those of experimental. The best fit value of ω/kBT was determined to be −1.900. The compositional dependence of cal- culated and experimental values of ∆Gxs M/RT are depicted in Figure 3. Both the experimental and computed values of ∆Gxs M/RT were found to be consistent with each other at all compositions. The Au–Cu binary sys- tem is a regular alloy and the equilibrium values of ∆Gxs M/RT are −0.47500 (this work) and −0.46611 (experimental [5]) at xAu = 0.5. Therefore, the sys- tem is found to be symmetric with respect to excess Gibbs free energy of mixing and weakly interacting in nature (Figure 3). Moreover, the computed val- ues of ∆Gxs M/RT were found to be negative in the entire concentration range indicating the system to be ordering in nature at its melting temperature, 1550 K. The temperature derivative term of the model pa- rameter (dω/dT ) was estimated using Equation (10) and experimental data of HM [5]. The best fit value of 1 kB dω dT was found to be −0.5500 em- ploying iterative process. The values of HM and Sxs M were then computed at all concentrations using Equations (10) and (11) at 1550 K, and above de- termined model parameters. The computed and ex- perimental values of these thermodynamic param- eters are plotted as a function of concentration in Figure 3. 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 a Cu a Au a A u, a C u x Au This work Experimental Ideal Figure 4: Activities of Au (aAu) and Cu (aCu) ver- sus xAu of Au–Cu liquid alloy at 1550 K. The maximum negative values of HM/RT were −0.33750 (this work) and −0.33864 (experimen- tal [5]) at xAu = 0.5. Meanwhile, the optimum val- ues of Sxs M/R were 0.13750 (this work) and 0.12770 (experimental [5]) at xAu = 0.5. Additionally, the calculated and experimental values of these func- tions were found to be consistent with each other (Figure 3). The small negative value of HM/RT revealed the system to be weakly interacting and ordering in nature at its melting temperature. The activities of the monomers Au (aAu) and Cu (aCu) of the system were estimated using Equa- tions (12) and (13) with the aid of determined value ω/kBT . The experimental and calculated values of aAu and aCu showed negative deviations from their respective ideal values revealing the system to be hetero-coordinating in nature. They agreed well with each other at all concentrations and 1550 K (Figure 4) thereby validating the present iteration process. 3.2 Structural properties The knowledge of structural functions helps to un- derstand the nature of arrangement of atoms in the liquid alloy at microscopic level. The concentration fluctuation in long wavelength limit (SCC(0)) and Warren-Cowley short range order parameter (α1) were calculated in structural properties. At a tem- perature and concentration, if the calculated values of SCC(0) < Sid CC(0), then the alloy shows ordering nature, i.e., there is preferential association among the unlike atoms. Under the similar conditions, if SCC(0) > Sid CC(0), then it shows segregating nature, i.e., the association among the same type of atoms is favoured in the liquid state. Figure 5: Effect of positive values of ω/kBT on SCC(0) for Au-Cu liquid alloy at 1550 K. The effect of ω/KBT on SCC(0) was observed by calculating the physical quantity for ω/KBT = ±5 with the help of Equation (16). The value of Sid CC(0) was estimated using Equation (17) throughout the entire concentration range. It can be observed that the computed values of SCC(0) showed positive de- viation from its ideal values for ω/KBT = 1 and 2 indicating the increase in segregating tendency in the initial melt (Figure 5). Beyond ω/KBT > 2, the S. K. Yadav et al./ BIBECHANA 21 (2024) 150-158 155 estimated values of SCC(0) showed unusual trends. But on increasing the negative values of ω/KBT in the preferred range, the negative deviations of SCC(0) from Sid CC(0) increases (Figure 6) revealing the gradual increase in ordering tendency in the al- loy. These results are in accordance with those pre- dicted by Gibbs free energy of mixing in the above sub-section. 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 Ideal /k B T=-1 /k B T=-2 /k B T=-3 /k B T=-4 /k B T=-5 Figure 6: Effect of negative values of ω/kBT on SCC(0) for Au-Cu liquid alloy at 1550 K. Theoretical, experimental and ideal values of SCC(0) were estimated using Equations (14-17) with help of determined values of model parame- ter and experimental value of activity [5]. Both the calculated experimental and theoretical values of SCC(0) showed negative deviation from Sid CC(0) at all concentrations revealing the system to be com- plete ordering in nature at its melting temperature (Figure 7). Both of these values were in excellent agreement with each other. 0.2 0.4 0.6 0.8 1.0 -0.08 -0.04 0.00 0.04 0.08 0.12 0.16 0.20 0.24 S C C (0 ),  1 a Au S CC (0) (This work) S CC (0) (Ideal) S CC (0) (Experimental)  1 Figure 7: Computed values of SCC(0) and α1 versus xAu for Au–Cu liquid alloy at 1550 K. In case of liquid alloys, the value of α1 can be neg- ative, positive and zero. For a temperature and concentration, the ordering tendency is expected if α < 0, segregating tendency is expected if α1 > 0 and ideal mixing tendency is expected if α1 = 0. The values of α1 were calculated using Equations (17-19) with the aid of estimated values of SCC(0). The calculated values of α1 < 0 at all concentra- tions revealing the system to be complete ordering in nature. 3.3 Surface properties The surface tension (σ) and surface concentrations of components Au (xs Au) and Cu (xs Cu) were com- puted using Equation (20). The required input parameters for the purpose were determined using Equations (21-24) with the help of values in Table 1. The computed values of xs Au, xs Cu and σ are plot- ted as a function of concentration in Figures 8 and 9 respectively. Table 1: Input parameters for surface tension [30] Atom T0 ρ0 ∂ρ/∂T σ0 ∂σ/∂T (K) (kg/m3) (kg/m3.K) (N/m) (N/m.K) Au 1336 17360 -1.50 1.169 -0.00025 Cu 1356 8000 -0.80 1.303 -0.00023 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 xs Cu xs Au xs A u, x s C u x Au This work Ideal Figure 8: Computed values of xs Au and xs Cu versus xAu of Au–Cu liquid alloy at 1550 K. The computed value of xs Au showed positive devi- ation whereas those of xs Cu showed negative devi- ation from their respective ideal values (Figure 8). These findings suggested that Au atoms segregate on the surface phase of the liquid mixture, whereas Cu atoms remain in the bulk phase. The com- puted values of surface tension of individual atoms are σAu = 1.1155 N/m and σCu = 1.2584 N/m at the melting temperature of the liquid alloy, 1550 K. As the surface tension of Au is less than that of Cu, Au atom segregated in the surface phase. The computed values of σ is found to be equal to ideal S. K. Yadav et al./ BIBECHANA 21 (2024) 150-158 156 values in the concentration range xAu < 0.3. But it is found to be less than ideal values in the rest of the concentrations (Figure 9). Moreover, the sur- face tension of the system gradually decreased with the increase in concentration of Au. 0.0 0.2 0.4 0.6 0.8 1.0 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26  [N m -1 ] x Au This work Ideal Figure 9: Computed values of σ versus xAu of Au– Cu liquid alloy at 1550 K. 0.0 0.2 0.4 0.6 0.8 1.0 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 D M /D id x Au Figure 10: DM/Did versus xAu of Au–Cu liquid al- loy at 1550 K. 3.4 Transport properties In transport properties, the ratio of mutual to in- trinsic diffusion coefficients (DM/Did) of the system was calculated in the framework of STM. At a con- stant concentration and temperature, if DM/Did < 1, then it indicates homo-atomic pairing tendency and if DM/Did > 1, then it indicates hetero-atomic pairing tendency among the atoms of liquid mix- ture. Equation (30) was used to calculate DM/Did of the system at 1550 K. From Figure 10, it can be observed that the computed values of the physi- cal quantity is found to be greater than 1 through- out the entire concentration range thereby reveal- ing the system to be complete ordering in nature at its melting temperature. Similar results were also obtained from the investigations of thermodynamic and structural functions in the above sub-sections. Table 2: Input parameters for viscosity [30] Atom η0 E (Nsm−2) (Jmol−1) Au 0.001132 15900 Cu 0.000301 30500 0.0 0.2 0.4 0.6 0.8 1.0 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9  [1 0-3 N sm -2 ] x Au This work Ideal Figure 11: Compositional dependence of η of Au– Cu liquid alloy at 1550 K. The viscosity (η) of the system was calculated on the basis of Kaptay equation [10] (Equations (27- 29)). The necessary parameters required for the process were taken from Table 2. The viscosity of the system showed negative deviation from the ideal value. 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