BIBECHANA Vol. 21, No. 3, December 2024, 328–335 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 Cu–Zr liquid alloy at different temperatures: A theoretical approach Shashit Kumar Yadav*, Dipak Rohita Yadav, Ramesh Kumar Gohivar, Upendra Mehta Department of Physics, Mahendra Morang Adarsh Multiple Campus, Tribhuvan University Biratnagar, Nepal. ∗Corresponding author. Email: yadavshashit@yahoo.com Abstract Quasi-lattice model has been employed to study the thermodynamic and microscopic struc- tural properties of Cu–Zr liquid alloy in the temperature range 1400-1700 K. The model fit parameters required for the purpose have been optimised using the available literature data of thermodynamic properties at the melting temperature of the system, 1400 K. These model parameters have been then computed at different temperatures assuming them to be linear temperature-dependent. The surface properties of the system have been computed using But- ler model at above mentioned temperatures. Keywords Quasi–lattice test, ordering nature, segregating nature, surface tension. Article information Manuscript received: August 10, 2024; Revised: September 27, 2024; Accepted: October 1, 2024 DOI https://doi.org/10.3126/bibechana.v21i3.70438 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction Pure metals are inappropriate for the most of indus- trial uses as they are often soft and ductile. As a re- sult, alloying is done to produce desirable composite material with low to high melting points, increased tensile strength, improved corrosion resistance and greater cost accessibility [1]. They are used in culi- nary utensils, automobile parts, aerospace parts, cellphone parts, medicines, military, nuclear reac- tors, etc. Therefore, alloying phenomenon is incred- ibly crucial in our daily lives [2]. Alloys of Cu–Zr have a relatively weak interaction [3] and tend to pass into the glassy state over a wide composition range. In this regard, they are used as the founda- tion for a wide range of bulk amorphous materials. The long-term operation of nuclear reprocessing plants is jeopardized by the management of zirco- nium base alloy wastes as cladding hulls [4]. Hence, the system has been studied by several researchers employing different experimental techniques and 328 http://nepjol.info/index.php/BIBECHANA yadavshashit@yahoo.com https://doi.org/10.3126/bibechana.v21i3.70438 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Shashit Kumar Yadav et al./ BIBECHANA 21 (2024) 328-335 329 theoretical approaches. The Butler Model was used to study the surface characteristics of the system above its melting point [5]. In 2003, Zaitsev et al. produced a full ex- perimental thermodynamic description of Cu-Zr in- termetallic complexes [3]. The knowledge of phase diagram and thermodynamic parameters includ- ing enthalpy of mixing, enthalpy of formation and heat contents are required for the purpose. The information for glass forming propensity requires knowledge of the thermodynamic and kinetic pa- rameters of Cu–Zr alloys [4]. In 2008, Yamaguchi et al. evaluated the standard enthalpy of formation of Cu9Zr2, Cu51Zr14, Cu8Zr3, Cu10Zr7, and CuZr2 complexes at 298.15 K [2]. The present literature review shows that the complete assessment of mix- ing properties of the system in liquid state is not available to date. Therefore, the thermodynamic and structural prop- erties of the Cu–Zr liquid alloy at 1400 K have been studied in the work on the basis of quasi-lattice test [6–10]. The model parameters required for the process have been optimised taking the thermody- namic data of ref. [11] as reference. The calculated values of thermodynamic and structural properties have been then compared with the reference data to check the validity of best fit values of so obtained model parameters. The surface tension and sur- face concentrations of components in the alloy have been computed using Butler model [10, 12–14]. In order to analyse the mixing tendency of the system at higher temperatures, above mentioned proper- ties have been calculated in the temperature range 1400-1700 K. 2 Formalism 2.1 Thermodynamics properties The complex having the stiochiometric composition AµB,µ = 2, i.e., Zr2Cu [11] has been assumed to be energetically stable in the alloy. According to the quasi-lattice model, the expression for excess free energy of mixing (∆Gxs M ) for the preferred complex is given as [6–10] ∆Gxs M = N [∆ωϕ+∆ωABϕAB +∆ωAAϕAA] (1) where ϕ = x1x2 (2a) ϕAB = ( 1 6 x1 + x1 2 − 5 3 x1 3 + 1 2 x1 4) (2b) ϕAA = (−1 4 x1 + 1 2 x1 2 − 1 4 x1 4) (2c) And the expression for free energy of mixing is given by ∆GM = ∆Gxs M +RT [x1 lnx1 + x2 lnx2] (3) Herein, ∆ω and ∆ωij ; i, j = A,B are the interaction energy parameters, termed as model parameters. R is the real gas constant and T is the absolute tem- perature. xi, i = A,B is the mole fraction of pure component i in the liquid alloy. Let aA and aB denote the activities of the elements A (=Zr) and B(=Cu) in the alloy. aA and aB can be expressed as RT ln aA = ∆GM + x2 ( ∂∆GM ∂x1 ) T,P,N (4) and RT ln aB = ∆GM + x1 ( ∂∆GM ∂x2 ) T,P,N (5) The expression for the excess entropy of mixing (∆Sxs M ) can be expressed in terms of ∆Gxs M with the help of standard thermodynamic relation as ∆Sxs M = − ( ∂∆Gxs M ∂T ) P (6) Using Equation (2) in Equation (6), one can obtain ∆Sxs M = −N [ ∂∆ω ∂T ϕ+ ∂∆ωAB ∂T ϕAB + ∂∆ωAA ∂T ϕAA ] (7) where ∂∆ω ∂T and ∂∆ωij ∂T are the temperature deriva- tive terms of interaction energy parameters. The enthalpy of mixing (∆HM ), ∆Sxs M and ∆Gxs M can be related as ∆HM = ∆Gxs M + T∆Sxs M (8) 2.2 Structural properties The arrangement of atoms in the liquid alloys can be studied by calculating characteristics functions, such as concentration fluctuation in the long wave length limit (SCC(0)), Warren-Cowley short range order parameter (α1) and the ratio of mutual to in- trinsic diffusion coefficients (DM/Did) [6, 7, 15, 16]. SCC(0) can be expressed in terms of ∆GM as SCC(0) = RT [ ∂2∆GM ∂xi 2 ]−1 T,P,N (9) With the aid Equations (2 and 3) in Equation (2.2), SCC(0) can be expressed as SCC(0) = x1x2 ∗ P−1 Shashit Kumar Yadav et al./ BIBECHANA 21 (2024) 328-335 330 with P = 1+x1x2RT (∆ωϕ” + ∆ωABϕ”AB +∆ωAAϕ”AA) (10) Moreover, the expression for ideal value of concen- tration fluctuation in long wavelenth limit Sid cc(0) as Sid cc(0) = x1x2 (11) As sated above, another structural function, α1 can be expressed as [15,17,18] α1 = S − 1 [S(Z − 1) + 1] (12) Here, Z represents the coordination number and its value has been taken to be 10 for the work. The term S in above expression is defined as S = SCC(0) Sid CC(0) (13) The ratio of mutual to intrinsic diffusion coefficients is expressed as [9, 19] DM Did = Sid CC(0) SCC(0) (14) 2.3 Surface properties The knowledge of surface tension and surface seg- regation allows us to understand and explain the mechanical behavior, phase change, catalytic activ- ity, and other physical behaviours of an alloy . The expression for the surface tension (σ) for binary liq- uid alloy is given as [12,20–22] σ = σ0 1(T ) + RT A1 ln ( xs 1 x1 ) + ( ∆Gxs,s 1 −∆Gxs,b 1 A1 ) σ = σ0 2(T ) + RT A2 ln ( xs 2 x2 ) + ( ∆Gxs,s 2 −∆Gxs,b 2 A2 ) (15) where Ai = molar surface area of ith component, xs i = surface mole fraction of the ithcomponent, xi= bulk mole fraction of ith component, ∆Gxs,s i = par- tial excess free energy of the ith component in the surface phase of liquid solution and ∆Gxs,b i = par- tial excess free energy of the ith component in the liquid solution. ∆Gxs,s i and ∆Gxs,b i is related by the expression ∆Gxs,s i = β∆Gxs,b i (16) The value of β depends on the ratio of coordination number of atoms in surface phase and bulk phase. The value of β is taken to be 0.8181 [10, 13, 20] in the work. In Equation (12), σ0 i (T ) is the temperature depen- dent surface tension of pure ith component in the mixture that can be expressed as [14,23] σ0 i (T ) = σ0 i + (T − T0) dσ dT (17) The terms σ0 i is the surface tension of pure compo- nent at its melting temperature (T0), σ0 i (T ) is the surface tension at the required temperature T , and dσ dT is the temperature derivative term of surface tension. The expression of the molar surface area of the ith component can be given by [10,13] Ai = f(V 0 i ) 2/3(NAV ) 1/3 (18) The terms Vi and NAV are the molar volume of ith component at temperature T and Avogadro’s num- ber respectively. The term f is geometrical constant and its value is 1.000 that can be calculated using the relation [21,24] f = ( 3fb 4 ) π1/3 fs (19) Herein,fb and fs are the volume and surface pack- ing fractions. The ideal surface tension of the bi- nary alloy is the weighted sum of surface tension of individual components present in the alloy. Math- ematically, σid = σ0 1(T )x1 + σ0 2(T )x2 (20) where the terms carry their usual meanings as stated above. 3 Results and discussion 3.1 Thermodynamic properties The model parameters required for the quasi-lattice test, have been obtained by considering the thermo- dynamic database of Cost 507 [11] as reference. The self-consistent parameters for ∆Gxs M is expressed in terms of coefficients of Redlich-Kister polynomial [25] in the ref. [11], Table 1. Table 1: Interaction energy parameters for ∆Gxs M Parameters [Jmol−1] Ref. L0 = −61685.53 + 11.29235 ∗ T L1 = 8830.66 + 5.045658 ∗ T [11] ∆ω = −43271.6 + 15.680 ∗ (T − 1400) ∆ωAB = 3776.065 + 1.200 ∗ (T − 1400) This wok ∆ωAA = 65496.02 + 0.010 ∗ (T − 1400) ∆Gxs M for the system in terms of coefficients of R-K polynomial can be expressed as [9, 10,25] ∆Gxs M = x1x2 (L0 + L1(x1 − x2)) (21) where Li (in Jmol−1is the coefficient of R-K poly- nomial depending on temperature but independent Shashit Kumar Yadav et al./ BIBECHANA 21 (2024) 328-335 331 of concentration. Li = ai + bi ∗ T , where ai is the contribution of enthalpy of mixing ∆HM and bi is the contribution of entropy of mixing (∆Sxs M ) on ∆Gxs M . The expressions of ∆Sxs M and ∆HM then can be obtained using the Equations (6 and 8) as [9] ∆Sxs M = x1x2 [b0 + b1(x1 − x2)] (22) and ∆HM = x1x2 [a0 + a1(x1 − x2)] (23) The reference values of ∆Gxs M , ∆HM and ∆Sxs M have been calculated using Equations (21-23) with the help of parameters in Table 1. The values of model parameters ∆ω, ∆ωAB and ∆ωAA and their temperature derivative terms (∂∆ω ∂T , ∂∆ωAB ∂T and ∂∆ωAA ∂T ) at 1400 K have been then optimised by using Equations (1,2, 7 and 8) and calculated reference values of ∆Gxs M , ∆HM and ∆Sxs M . The best best fit values of these model parameters are presented in Table 1. These model parameters are also assumed to depend only on temperature. The calculated values of these thermodynamic functions are plotted in Figure 1. 0.0 0.2 0.4 0.6 0.8 1.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 H M /RT S M xs G M xs/RT G M xs /R T ,  S M xs ,  H M /R T x Zr Cost 507 This work Figure 1: Plot of ∆Gxs M versus xZr for Cu–Zr liquid alloy at 1400 K. The values of ∆Gxs M , ∆HM and ∆Sxs M calculated us- ing the optimised parameters of the work and Cost 507 [11] are found to be consistent with each other at its melting temperature, 1400 K. The negative values of ordering energy parameter, ∆ω, signifies the compound forming behavior of the system at its melting temperature. The optimum value of ∆Gxs M = −11612.8 Jmol−1 and ∆HM = −17104.0 Jmol−1 at xxs = 0.4, indicating the system to be asymmetric with respect to these functions, Figure 1. Moreover, the system is found to be weakly in- teracting in nature. The values of ∆Gxs M of the system have also been computed at higher temperatures, 1400 K, 1500 K, 1600 K and 1700 K following the similar proce- dure. The calculated optimum values of ∆Gxs M at xZr are −11612.8 Jmol−1 at 1400 K, − − 11220.6 Jmol−1 at 1500 K, −10828.4 Jmol−1 at 1600 K and −10436.1 Jmol−1 at 1700 K. Thus, the negative val- ues of ∆Gxs M are found to gradually decrease with an increase in the temperature of the system, Figure 2. These results indicate that the compound form- ing tendency in the liquid alloy gradually decreases with an increase in its temperature. 0.0 0.2 0.4 0.6 0.8 1.0 -12 -10 -8 -6 -4 -2 T=1400 K T=1500 K T=1600 K T=1700 K  G M xs [ kJ m o l-1 ] x Zr Figure 2: Calculated values of aZr and aCu versus xZr for Cu–Zr liquid alloy at 1400 K. Activity is another important thermodynamic pa- rameter which is greatly influenced by the values of model parameters and is reflection of ∆Gxs M . The activities of components Zr (aZr) and Cu (aCu) of liquid alloy have been calculated using Equations (4, 5 and 21) with the aid of parameters in Table 1. 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 Zr a Z r, a C u x Zr Cost 507 This work Ideal values Figure 3: Calculated values of aZr and aCu versus xZr for Cu–Zr liquid alloy at 1400 K. The perusal of Figure 3 conveys that the calculated values of aZr and aCu using the model parameters of the work are found to be consistent with those cal- culated using the data of Cost 507 [11]. The activ- ities of these components show negative deviation from Raoult’s law or their respective ideal values Shashit Kumar Yadav et al./ BIBECHANA 21 (2024) 328-335 332 at lower concentrations of Zr indicating the system to be ordering in nature. However, values of aZr approaches ideal values at higher concentration of Zr. 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 Zr a Z r, a C u x Zr T=1400 K T=1500 K T=1600 K T=1700 K Ideal values Figure 4: Calculated values of aZr and aCu as a function of concentration of Zr for Cu–Zr liquid al- loy at different temperatures. The activity of the system has also been computed at above mentioned different temperatures. The calculated values of aZr and aCu are found to gradu- ally increase at higher temperatures, Figure 4. The deviation of calculated values from the ideal values gradually decreases. The activity of the compo- nent Zr shows positive deviation from its respective ideal value at 1500 K and above in the concentra- tion range xZr > 0.8. These results correspond the transformation from ordering to segregating nature of the system. Thus, the present investigations re- veal that the compound forming tendency in the alloy decreases at elevated temperatures. 3.2 Structural properties 0.2 0.4 0.6 0.8 1.0 -0.2 -0.1 0.0 0.1 0.2 0.3  1 S CC (0) S C C (0 ),  1 x Zr This work Cost 507 Ideal values Figure 5: Calculated values of SCC(0) and α1 ver- sus xZr for Cu–Zr liquid alloy at 1400 K. The ordering nature or hetero-atomic pairing and segregating nature or homo-atomic pairing tenden- cies in the liquid alloys can be better understood with the knowledge of microscopic structural func- tions. For the purpose, concentration fluctuation in long wavelength limit (SCC(0)), Warren-Cowley short-range order parameter (α1) and ratio of mu- tual to intrinsic diffusion coefficients (DM/Did) have been computed in the work using Equations (9-14) and parameters in Table 1. 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 Zr Ideal values T=1400 K T=1500 K T=1600 K T=1700 K Figure 6: Calculated values of SCC(0) versus xZr for Cu–Zr liquid alloy at different temperatures. 0.2 0.4 0.6 0.8 1.0 -0.25 -0.20 -0.15 -0.10 -0.05 0.00  1 x Zr T=1400 K T=1500 K T=1600 K T=1700 K Figure 7: Calculated values of α1 versus xZr for Cu– Zr liquid alloy at different temperatures. At a given concentration and temperature, if SCC(0)>Sid CC , then it indicates ordering tendency in the alloy for which α1 > 1 and DM/Did < 1. Likewise, if SCC(0) 1. If α1 = 0, then ideal mixing tendency in the liquid alloy is expected. The cal- culated values of SCC(0) 0.8 at T>1400 K. Thus, it can be stated that the system changes from ordering to segregating nature at these condi- tions. Likewise, the calculated values of α1 is found to be greater than 1 in the above mentioned condi- tions revealing the similar mixing tendency. 0.0 0.2 0.4 0.6 0.8 1.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 D M /D id x Zr T=1400 K T=1500 K T=1600 K T=1700 K Figure 8: Compositional dependence of DM/Did for Cu–Zr liquid alloy in the temperature range 1400- 1700 K. The calculated values of DM/Did are found to be greater than 1 at all concentrations at 1400 K, melt- ing temperature. This result reveals that the sys- tem shows hetero-coordinating tendency. Mean- while, the calculated values of DM/Did decreases with gradual increase in temperature of the system. The calculated values of DM/Did < 1 in the range xZr > 0.8 and T>1400 K indicating the transfor- mation from ordering to segregating nature, Figure 8. Thus, the nature of temperature-dependent mix- ing tendency of the system predicted by thermody- namic and structural functions are in accordance with each other. 3.3 Surface properties The surface tension (σ) and surface concentra- tion (xs i ) of the system have been computed using Equations (15-21) and the parameters in Table 2. The above determined values of thermodynamic parameters have been used for the purpose. The calculated values of xs i are plotted as a function of concentration in Figures 9. It can be observed that the calculated values of xs ZrxCu in the entire composition range at 1400 K, Figure 9. These results have been occurred as the surface tension of Cu (σCu = 1.293 Nm−1) is less than that of Zr (σZr = 1.633 Nm−1) at this temperature. Hence, Cu atoms segregate in the surface phase whereas Zr atoms remain in the bulk phase of the liquid alloy. Table 2: Surface tension and density of each com- ponent of liquid alloy [23] Surface tension [Nm−1] σ0 Zr = 1.459− 0.00024(T − 2128) σ0 Cu = 1.303− 0.00023(T − 1356) Density [kgm−3] ρ0Zr = 6240− 0.29(T − 2128) ρ0Cu = 8000− 0.8(T − 1356) 0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 x Cu s x Zr s x Z rs , x C us x Zr This work Ideal values Figure 9: Compositional dependence of calculated values of xs Zr and xs Cu for Cu–Zr liquid alloy at 1400 K. Shashit Kumar Yadav et al./ BIBECHANA 21 (2024) 328-335 334 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 Zr xs Z r, xs C u x Zr T=1400 K T=1500 K T=1600 K T=1700 K Ideal values Figure 10: Calculated values of xs Zr and xs Cu for Cu– Zr liquid alloy at different temperatures. The values of xs Zr and xs Cu have also been computed in the temperature range 1400-1700 K. It can be observed that the calculated values of xs Zr are found to increase while those of xs Cu are found to decrease at higher temperatures. Both of these values get closure to their respective bulk concentrations or ideal values as the temperature of the system is in- creased, Figure 10. Thus, Cu atoms move towards the bulk phase whereas Zr atoms move towards the surface phase at elevated temperatures. 0.0 0.2 0.4 0.6 0.8 1.0 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65  [N m -1 ] x Zr This work Ideal values Figure 11: Calculated values of σ for Cu–Zr liquid alloy at 1400 K. The computed values of σ of the system is found to be less than its ideal values. Moreover, the sur- face tension of the system gradually increases with increase in concentration of Zr, Figure 11. The temperature-dependent variation of σ of the sys- tem is as shown in Figure 12. The surface tension of the system decreases gradually and linearly with an increase in temperature of the system. 1400 1450 1500 1550 1600 1650 1700 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60  [N m -1 ] Temperature [K] x Zr =0.1 x Zr =0.2 x Zr =0.3 x Zr =0.4 x Zr =0.5 x Zr =0.6 x Zr =0.7 x Zr =0.8 x Zr =0.9 Figure 12: Calculated values of σ for Cu–Zr liquid alloy at different temperatures. 4 Conclusion The computed values of the thermodynamic prop- erties of the system using the best fit values of the model parameters of the work are found to be con- sistent with the reference database. The Cu-Zr sys- tem is found to be weakly interacting in nature and shows complete ordering tendency at its melting temperature, 1400 K. With an increase in temper- ature of the system, it shows transformation from ordering to segregating nature in the concentrations range xZr > 0.8 and T>1400 K. The results pre- dicted by the thermodynamic and structural func- tions are in accordance with each other. 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Ther- modynamic, structural, transport and surface properties of Pb-Tl liquid alloy. Bibechana, 13:100, 2016. [25] 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. Introduction Formalism Thermodynamics properties Structural properties Surface properties Results and discussion Thermodynamic properties Structural properties Surface properties Conclusion