BIBECHANA Vol. 20, No. 1, April 2023, 86-91 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 Mixing properties of liquid Al–Au alloys Shashit Kumar Yadav, Upendra Mehta Ram Prasad Koirala, Ramesh Kumar Gohivar∗ Department of Physics, Mahendra Morang Adarsh Mulpiple Campus Tribhuvan University, Biratnagar, Nepal *Corresponding author. Email: ramesh.gohibar@mmamc.tu.edu.np Abstract The thermodynamic and structural properties of liquid Al–Au alloy have been studied in frame- work of R-K polynomial using temperature-dependent energy interaction parameters at differ- ent temperatures. Thermodynamic properties, excess free energy of mixing and activity, and in structural properties, concentration fluctuation in long wave-length limit have been com- puted at temperatures 1338 K, 1500 K and 1600 K. The properties, such as surface tension and surface concentration of the system have been computed at above mentioned temperatures using Butler model. The system shows transformation from segregating to ordering in nature with increase in concentration of Au. Keywords R-K polynomial, Al–Au alloys, Thermodynamic, Structural and Surface properties. Article information Manuscript received: March 21, 2023; Accepted: April 2, 2023 DOI https://doi.org/10.3126/bibechana.v20i1.53776 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction The majority of semiconductor packages are as- sembled via wire bonding, which is typically re- garded as the most affordable and efficient connec- tivity approach. The intermetallic compounds of Al-Au alloys are profoundly used in these metal- lization processes [1–4]. Hence, the mixing prop- erties of the system has been extensively investi- gated by several researchers [3,5–18]. The details of the literature related to the chronological investiga- tions of phase structures and thermodynamic prop- erties of the system till 2004 are presented by Li et al. [3]. They accessed the thermodynamic proper- ties of the system using Thermo-Calc (computer- based software) along with other theoretical ap- proaches. They also presented the self-consistent temperature-dependent interaction parameters for excess Gibbs free energy of mixing. Later, Olajire and Musari used different the- oretical models to investigate the thermodynamic and structural properties of the system [10]. Peng et al. [19] experimentally measured the composi- tional and temperature dependence of density, tem- perature coefficient of density, molar volumes, self- and inter diffusion and viscosity of Al-Au liquid al- loy at 1400 K. They also developed a molecular dynamics (MD) model in order to computed the above mentioned properties using the experimen- tal results [19]. Later Brillo and Kolland [20] ex- perimentally measured the surface tensions of the system and found that the surface tension (σ) de- 86 http://nepjol.info/index.php/BIBECHANA ramesh.gohibar@mmamc.tu.edu.np https://doi.org/10.3126/bibechana.v20i1.53776 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Krishna Prasad Kandel et al./ BIBECHANA 20 (2023) 86-91 87 creases linearly with temperature. Further, they found that σ of the system decreased monotonically with increase in Al content at 1400 K. From above literature survey, it can be stated that Al-Au liquid alloy has been extensively studied. Therefore, an attempt has been made in this work to reassess the thermodynamic properties of the system using optimised coefficients Li et al. [3] of Redlich-Kister (R-K) polynomial [21]. These coefficient are used to compute thermodynamic, structural and surface properties of the system at different temperatures in frame work R-K poly- nomial. The thermodynamic properties, such as Gxs M , activity (a) and structural properties, such as concentration fluctuation in long wavelength limit (SCC(0)) have been computed. The surface proper- ties of have been computed using Butler model [22] with the help of determined values of partial excess Gibbs free energy of individual component of the al- loy. The different modeling equations used for the present investigations are presented in the Section 2, the results and discussion are mentioned in the Section 3 and important conclusions of the work are highlighted in the Section 4. 2 Methods and methodology According to R-K polynomial, the excess free en- ergy of mixing (Gxs M ) is expressed as [21,23] Gxs M = xAlxAu[L0 + L1(xAl − xAu)] (1) where L0 and L1 are coefficients of R-K polyno- mial, also called interaction energy parameters and are assumed to depend linearly on temperature and xAl and xAu are moalr fraction of constituents of liquid Al–Au alloy. The energy interaction parameters is linear tem- perature dependent and the excess free energy of mixing is expressed as Gxs M = xAlxAu[(a0 + b0T ) + (a1 + b1T )(xAl − xAu)] (2) The partial excess free energy (Gxs Al, G xs Au) of each component of liquid Al–Au alloy can be calculated using the relation [24] Gxs Al = Gxs M + xAu ( ∂Gxs M ∂xAl − ∂Gxs M ∂xAu ) Gxs Au = Gxs M + xAl ( ∂Gxs M ∂xAl − ∂Gxs M ∂xAu ) (3) Once the values of partial excess free energy of mix- ing are computed, the values of activity of each com- ponent (aAl, aAu) in the liquid alloy is obtained by following expression aAl = xAl exp ( Gxs Al RT ) , aAu = xAu exp ( Gxs Au RT ) (4) The structural function in long wave-length limit (SCC(0)) of binary liquid Al–Au alloy can be ex- pressed in the following form [25] SCC(0) = RT ( ∂2GM ∂x2 Al )−1 T,P,N = RT ( ∂2GM ∂x2 Au )−1 T,P,N (5) The analytical expression for SCC(0) of liquid Al– Au alloy [26] is SCC(0) = RT [−2L0 + (−12xAl + 6)L1 +(−48x2 Al + 48xAl − 10)L1 + RT xAl(1− xAl) ]−1 (6) The information related to the arrangement of atoms in the nearest neighbourhoods of initial metallic solution can be obtained by computing pa- rameters like concentration fluctuation in the long wave length limit (SCC(0)) [25,26]. The expression for ideal value of SCC(0) is calculated using the re- lation Sid CC(0) = xAlxAu (7) The surface properties of the system have been studied using Butler model [22, 27]. This approx- imation is based on the assumption of hypotheti- cal monoatomic surface layer being in equilibrium with the bulk phase of the metallic solution. Ac- cordingly, the surface tension (σ) of the solution in terms of molar surface area (λ1) of constituent atoms is expressed as σ = σAl + 1 λAl (Gxs Al,s −Gxs Al,b) + RT λAl ln( xs Al xb Al ) = σAu + 1 λAu (Gxs Au,s −Gxs Au,b) + RT λAu ln( xs Au xb Au ) (8) where σAl and σAu are surface tension of pure com- ponents , and Gxs Al and Gxs Au are the partial excess free energy of component in the surface phase and bulk phase respectively, and xs Al, x s Au and xb Al, x b Au are the surface and bulk concentrations of Al and Au respectively. Herein, Gxs i,s = βGxs i,b, i = Al,Au with β = 0.8181 [22] and λiis expressed as λi = 1.00N 1/3 A ( Mi ρi )2/3 (9) where NA is the Avogadro’s number, Mi and ρi are the mass and density in the Al–Au alloy respec- tively. The expression for σi and ρi can be given as σi = σ0 i + ∂σi ∂T (T − T 0 i ) (10) ρi = ρ0i + ∂ρi ∂T (T − T 0 i ) (11) where T 0 i is the melting temperature of the ith com- ponent and T is the required temperature at which these parameters are to be calculated. Krishna Prasad Kandel et al./ BIBECHANA 20 (2023) 86-91 88 3 Results and Discussion 3.1 Excess free energy of mixing The energy interaction parameters of R-K polyno- mial for liquid Al–Au alloy were taken from Ref. [3] are presented in Table 1. The interaction param- eters of Table 1 have been used in Equation (1) to compute Gxs M of Al–Au alloy at 1338 K, 1500 K and 1600 K. The computed values are plotted in Figure 1. The negative value of Gxs M of the system decreases with increase in temperature. The mini- mum value at 1338 K is found to be -19.505 kJ/mol at equi-atomic composition. Table 1: Energy interaction parameters of liquid Al–Au alloy Interaction Parameters (J/mol) [3] L0 = −131996.19 + 36.42T L1 = 40781.83− 1.896T Table 2: Physical quantities of liquid Al–Fe alloy [28] Quantities Element Al Element Au Mi (kg) 0.2698154 0.19696665 T0 i 933 K 1336 K ρ0i (kg/m3) 2385 17360 dρi dT (kgm−3T−1) -0.28 -1.5 σi(mN/m) 914 1140 dσi dT (mN/(mT)) -0.35 -0.52 0.2 0.4 0.6 0.8 1 −25 −20 −15 −10 −5 0 xAu G x s M (k J /m ol ) Al-Au alloy 1338 K 1500 K 1600 K Figure 1: Excess free energy of mixing of liquid Al–Au alloy at different temperatures. 3.2 Activity The parameters of Table 1 have been used in Equa- tion (3) to calculated the partial excess free energy (Gxs i , i=Au, Al) of each component at above men- tioned temperatures. The activity of components Al and Au (aAu, aAl) have been calculated using Equation (4) and these values are plotted in Fig- ure 2. The activity of component Al in the re- gion of high concentration of Al (xAu < 0.2) shows positive deviation from Raoult’s law and negative deviation at remaining concentrations. This indi- cates the system shows ordering nature in the re- gion xAu < 0.2 and segregating nature at remaining concentrations. Both the components have a very small value of activities in the concentration range xi = 0.6 which indicates strong ordering behaviour of the system (Figure 2). Krishna Prasad Kandel et al./ BIBECHANA 20 (2023) 86-91 89 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 aAl aAu xAu A ct iv it y Al–Au alloy 1338 K 1500 K 1600 K Ideal Figure 2: Activity of liquid Al–Au alloy at different temperatures. 3.3 Concentration fluctuation in long wavelength limit (SCC(0)) The structural function (SCC(0)) of liquid Al–Au alloy gives an idea about the local arrangement of components Al and Au in the system. The func- tion SCC(0) have been calculated using energy in- teraction parameters of Table 1 in Equation (6) at above mentioned temperatures, 1338 K, 1500 K and 1600 K. The computed values are plotted in Fig- ure 3. The calculated value of SCC(0) at lower concentrations of Au (xAu < 0.2) exceeded ideal values showing segregating nature in those region. However, these values at remaining concentration are less than ideal values indicating the ordering nature. Hence, the system shows transformation from segregating to ordering nature. As the tem- perature of the system is gradually increased, the computed values of structural function gradually in- creases (Figure 3). 0.2 0.4 0.6 0.8 1 0 5 · 10−2 0.1 0.15 0.2 0.25 xAu S c c (0 ) Al–Au alloy 1338 K 1500 K 1600 K Ideal Figure 3: SCC(0) of liquid Al–Au alloy at different temperatures. 3.4 Surface tension (σ) Surface tension (σ) of liquid Al–Au alloy has been calculated at above mentioned temperatures us- ing physical parameters of Table 2 and Equations (8,9,10,11). The calculated values of (σ) are plot- ted in Figure 4 as a function of concentration of Au. The surface concentrations xs Al and xs Au of the alloy have been optimised during the computations of (σ) and are plotted in Figure 5. It can be observed that the surface tension (σ) of the system at its melting temperature (1338 K) gradually increased with increase in the bulk con- centration of Au. As mentioned earlier, σ was also computed at different temperatures. With increase in the temperature, the values of σ gradually de- creased with increase in the content of Au (Figure 4) and these results are similar to those obtained by [19], [20]. Krishna Prasad Kandel et al./ BIBECHANA 20 (2023) 86-91 90 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 xAu S u rf ac e te n si o n (N / m ) Surface tension of Al–Au alloy 1338 K 1500 K 1600 K Figure 4: Surface tension of liquid Al–Au alloy at different temperatures. 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 aAu aAl xAu S u rf ac e C on ce n tr a ti on Surface Concentration of Al–Au alloy 1338 K 1500 K 1600 K Figure 5: Surface concentration of liquid Al–Au alloy at different temperatures. The surface concentrations of Au and Al (xs Au and xs Al) have the same value at/about xAu = 0.7. With increase in xAu, xs Au gradually increases whereas xs Al gradually decreases. At higher temper- atures, the computed values of xs Al gradually de- creases whereas those of xs Au gradually increases (Figure 5). These results indicate that there is exchange of atoms between the surface and bulk phases of the liquid alloy in order to maintain equi- librium [29]. 4 Conclusion Present theoretical investigations show that the liq- uid Al–Au is found to be the most interacting at its melting temperature. The system shows tranfor- mation from segregating to ordering nature. With increase in temperature, its mixing tendency grad- ually decreases. References [1] Vaclav Valenta, Thomas Spreng, Shuai Yuan, Wolfgang Winkler, Volker Ziegler, Dragos Dancila, Anders Rydberg, and Hermann Schu- macher. Design and experimental evalua- tion of compensated bondwire interconnects above 100 GHz. International Journal of Microwave and Wireless Technologies, 7(3- 4):261–270, 2015. [2] J. M. Vandenberg and R. A. Hamm. A contin- uous x-ray study of the interfacial reaction in Au–Al thin-film couples. Journal of Vacuum Science and Technology, 19(1):84–88, 1981. [3] Mei Li, Changrong Li, Fuming Wang, Degui Luo, and Weijing Zhang. Thermodynamic as- sessment of the Al–Au system. Journal of al- loys and compounds, 385(1-2):199–206, 2004. [4] H Piao, N. S. McIntyre, G Beamson, M-L Abel, and J. F. Watts. Electronic structures of Au– Al thin-film alloys by high-energy XPS and XANES. Journal of electron spectroscopy and related phenomena, 125(1):35–45, 2002. [5] Charles Thomas Heycock and Francis Henry Neville. V. Gold-aluminium alloys. Philo- sophical Transactions of the Royal Society of London. Series A, Containing Papers of a Krishna Prasad Kandel et al./ BIBECHANA 20 (2023) 86-91 91 Mathematical or Physical Character, 194(252- 261):201–232, 1900. [6] B Gunther, O Kanert, and W Tietz. In situ NMR study of the two-phase equilibrium in Au-Al alloys. Journal of Physics F: Metal Physics, 16(1):L27, 1986. [7] J. L. Murray, H Okamoto, and T. B. Massalski. The Al- Au (aluminum-gold) system. Bulletin of Alloy Phase Diagrams, 8(1):20–30, 1987. [8] Klaus-Jürgen Range and Harald Büchler. Hochdrucksynthese und Kristallstruktur von Al3Au8. Journal of the Less Common Metals, 154(2):251–260, 1989. [9] H Okamoto. Al-Au (aluminum-gold). Journal of Phase Equilibria and Diffusion, 26(4):391, 2005. [10] B. A. Olajire and A. A. Musari. Thermody- namic and structural properties of liquid Al– Au alloys. International Journal of Modern Physics B, 31(20):1750134, 2017. [11] M. E. Straumanis and K. S. Chopra. Lattice parameters, expansion coefficients and extent of the Al2Au phase. Zeitschrift für Physikalis- che Chemie, 42(5_6):344–350, 1964. [12] M. H. Francombe, A. J. Noreika, and W. J. Takei. Thin film and bulk structures of phases in the system gold-aluminum. Thin Solid Films, 1(5):353–366, 1968. [13] Bruno Predel and Udo Schallner. Ther- modynamische untersuchung der systeme aluminium-antimon und aluminium-gold. Ma- terials Science and Engineering, 5(4):210–219, 1970. [14] Akira Yazawa and Yong Keun Lee. Thermo- dynamic studies of the liquid aluminum alloy systems. Transactions of the Japan Institute of Metals, 11(6):411–418, 1970. [15] G Piatti and G Pellegrini. The structure of the unidirectionally solidified Al-Al2Au eutec- tic. Journal of Materials Science, 11(5):913– 924, 1976. [16] L Erdelyi, J Tomiska, A Neckel, G Rose, E. S. Ramakrishnan, and D. J. Fabian. Thermody- namic parameters of liquid gold-aluminum al- loys. Metallurgical Transactions A, 10:1437– 1443, 1979. [17] Rodney P Elliott and Francis A Shunk. The Al- Au (Aluminum-Gold) system. Bulletin of Alloy Phase Diagrams, 2(1):70–75, 1981. [18] R Hultgren, P. D. Desai, D. T. Hawkins, M Gleiser, and K. K. Kelley. Selected Values of the Thermodynamic Properties of the Ele- ments, ASM, Metals Park, OH (1973). Search in. [19] H. L. Peng, Th Voigtmann, G Kolland, H Ko- batake, and J Brillo. Structural and dynamical properties of liquid Al-Au alloys. Physical Re- view B, 92(18):184201, 2015. [20] J Brillo and G Kolland. Surface tension of liq- uid Al–Au binary alloys. Journal of Materials Science, 51:4888–4901, 2016. [21] Otto Redlich and A. T. Kister. Algebraic rep- resentation of thermodynamic properties and the classification of solutions. Industrial & En- gineering Chemistry, 40(2):345–348, 1948. [22] John Alfred Valentine Butler. The thermody- namics of the surfaces of solutions. Proceedings of the Royal Society of London. Series A, Con- taining Papers of a Mathematical and Physical Character, 135(827):348–375, 1932. [23] R. K. Gohivar, S. K. Yadav, R. P. Koirala, and D Adhikari. Assessment of thermo-structural properties of Al-Fe and Fe-Si alloys at high temperatures. Physics and Chemistry of Liq- uids, 59(5):679–689, 2021. [24] S. B. Zhang and D. Z. Li. Phase Dia- gram—Principle, Calculation and Application in Metallurgy, 1986. [25] A. B. Bhatia, W. H. Hargrove, and N. H. March. Concentration fluctuations in confor- mal solutions and partial structure factor in al- loys. Journal of Physics C: Solid State Physics, 6(4):621, 1973. [26] R. K. Gohivar, S. K. Yadav, R. P. Koirala, and D Adhikari. Study of artifacts in thermody- namic and structural properties of Li–Mg al- loy in liquid state using linear and exponential models. Heliyon, 7(3):e06613–e06621, 2021. [27] S. K. Yadav, U Mehta, R. K. Gohivar, A Dhun- gana, R. P. Koirala, and D Adhikari. Reassess- ments of thermo-physical properties of Si-Ti melt at different temperatures. Bibechana, 17:146–153, 2020. [28] Eric A Brandes and G. B. Brook. Smithells metals reference book. Elsevier, 2013. [29] S. K. Yadav, L. N. Jha, and D Adhikari. Modeling equations to predict the mixing be- haviours of Al- Fe liquid alloy at different tem- peratures. Bibechana, 15:60–69, 2018. Introduction Methods and methodology Results and Discussion Excess free energy of mixing Activity Concentration fluctuation in long wavelength limit (SCC(0)) Surface tension () Conclusion