Microsoft Word - BIBECHANA _Final_.doc 1 BIBECHANA Vol. 6, March 2010 Regular Associated Solution Model for the Estimation of Heat of Mixing of Binary Liquid Alloys D. Adhikari a , B.P. Singh a , I.S. Jha b a Univ. Dept. of Physics, T.M.Bhag. University, Bhagalpur, Bihar, India b Dept. of Physics, M.M.A.M. Campus (Tribhuvan University), Biratnagar, Nepal Abstract We have determined the equilibrium constants and pairwise interaction energies between the species and the complexes of liquid CuSn, AgAl and FeSi, alloys on the basis of regular associated solution model. These parameters are then used to estimate the heat of mixing of each alloy. The observed asymmetry in the heat of mixing of each alloy with respect to concentration is well explained. Key Words: Free energy of mixing; Asymmetry; Binary alloys, interaction energy 1. Introduction Several models have been proposed to solve the difficulties and complexities of obtaining thermodynamic parameters. One of the models successfully used for accounts of the thermodynamic characteristics of binary liquid alloy systems, is the model of regular associated solution model. In regular associated solution model, it is assumed that strong associations among the constituent species exist in the liquid phase close to the melting temperature. These associations are given different names such as 'complexes', 'pseudomolecules', 'clusters', 'associations' etc. This assumption has been used by several researchers [1-12] to explain the asymmetry of the properties of mixing for binary alloys. Thus binary alloys in a liquid phase can be considered as a ternary mixture of unassociated atoms of components and complexes, all in chemical equilibrium. Jordan [3] proposed that activity of unassociated atoms and the complexes can be estimated by treating the mixture as a ternary system and termed this mixture as regular associated solution. Jordan [3] applied this idea in the congruently melting semiconductors Zn-Te and Cd-Te and determined thermodynamic equations for liquidus curve (the melting temperature against concentration curve) of these alloys. This model is further extended and applied by other researchers [4, 5, 12] for the determination of thermodynamics and microscopic parameters of different alloy systems in molten state. In present paper, we intend to apply regular associated solution model to obtain the heat of mixing of liquid CuSn, AgAl and FeSi alloys. For this, we have assumed Cu3Sn ,Ag3Al and 2 D. Adhikari et al. Fe2Si complexes in liquid CuSn, AgAl and FeSi alloys respectively by studying their corresponding phase diagram in solid state[13]. Theoretical formalism is given in section 2, section 3 deals with the numerical result and discussion. Conclusion is provided in section 4. 2. Theory Suppose there be three species in equilibrium in the liquid alloy, namely, monomers A, B and ApB molecules in the respective concentrations ,n,n BA and ApBn moles. Considering a solution of 1n atoms of A and 2n atoms of B, the formation of ApBn complex requires ApBA1 pnnn += and ApBB2 nnn += for conservation of mass in the partially associated solution. When there is association, the thermodynamic behaviour of complexes A and B components is governed by their true mole fractions Ax , Bx and ApBx rather than their gross mole fraction 1x and 2x , where )nn/(nx 2111 += etc. and )nnn/(nx ApBBAAA ++= etc. Using above relations, the two sets of mole fractions are related to each other by the relations ApB21A xpxxx −= , ApB22B x)px1(xx −−= (1) In regular associated solution solutions, the gross chemical potentials of components 1 and 2 are equal to the chemical potentials of the monomeric species A and B [14]. The activity coefficients Aγ , Bγ and ApBγ of monomers and complex can be expressed in terms of pairwise interaction energies through [3] )ωωω(xxωxωxγlnRT 132312ApBB13 2 ApB12 2 BA +−++= (2a) )ωωω(xxωxωxγlnRT 121323ApBA12 2 A23 2 ApBB +−++= (2b) )ωωω(xxωxωxγlnRT 231213ApBB23 2 B13 2 ApBA +−++= (2c) where 12ω , 13ω and 23ω are interaction energies for the species A, B ; A, ApB and B, ApB respectively, T the temperature and R stands for the universal gas constant. The equilibrium constant in a regular associated can be obtained [6] as ]x)px1(x[ RT ]x)x1(px[ RT ]x)x1(px[ RTx xx lnkln BBApB 23 AAApB 13 ABB 12 ApB B P A −− ω +−− ω ++− ω +         = (3) 3 BIBECHANA Vol. 6, March 2010 Now using the equations listed above the integral excess free energy xsG∆ is given by klnRT )px1( x )xlnxxlnx(RT)xlnxxlnxxlnx( )px1( RT )xxxxxx( )px1( 1 G ApB ApB 2211ApBApBBBAA ApB 23ApBB13ApBA12BA ApB xs + ++−++ × + +ω+ω+ω + =∆ (4) Once the expressions for G∆ )]xlnxxlnx(RTG[ 2211 xs ++∆= is obtained, heat of mixing can be found using standard thermodynamic relation P,T T G∆ TG∆H∆         ∂ ∂ −= (4a) The pairwise interaction energies and equilibrium constant are determined by the following method: In a regular associated solution AA11 γxγx = and BB22 γxγx = , where 1γ and 2γ are respective gross activity coefficients of components 1 and 2. Thus 1 A A1 x x lnlnln +γ=γ (5a) and 2 B B2 x x lnlnln +γ=γ (5b) the pairwise interaction energies, the equilibrium constants and the activity coefficients at infinite dilution can be written as [9] RT ln 12 0 1 ω =γ (6a) o 2 o 1 o 2 o 1 13 γγ γγ )RT/ωexp(k − = (6b) where oγ1 and o γ 2 are activity coefficients of component A and that of B at zero concentrations. Solving equations (2a) and (2b) we obtain 2 ApB 12 BB A 1 B B 2 B 13 x RT )x1(x x a ln)x1( x a lnx RT ω −−        −+        = ω (7) 2 ApB 12 AA B 2 A A 1 A 23 x RT )x1(x x a ln)x1( x a lnx RT ω −−        −+        = ω (8) Using equations (6), (16) and (17), we can derive         +         −        +                + =+ ApB 2 p 112 B 2 ApB B A 1 ApB A13 x aa ln RT ω x a ln x x x a ln x x1 RT ω kln (9) 4 D. Adhikari et al. 3. Results and Discussion To find heat of mixing of binary alloys, we have found the complex concentration in a regular associated solution of respective alloys by using equations (6), (9) and observed data of corresponding activities [13]. The complex concentration of liquid alloys in molten state are presented in table 1. We have found the equilibrium constants and the pairwise interaction energies between the species and the complexes using equations (3), (4), (6), (7), (8), and observed data of corresponding integral excess free energies of mixing. The equilibrium constants and pairwise interaction energies of different binary alloys are listed in table 2. 3.1 CuSn alloys at 1400K It is found from the analysis that the heat of mixing is negative at all concentration. Our theoretical calculation shows that the minimum value of the heat of mixing is -4.95 kJ at Cux = 0.8 which exactly matches with the experimental value [1]. Further it is observed that the concentration dependence of asymmetry in H∆ can be explained only when one considers the temperature dependence of the pairwise interaction energies. The agreement between the calculated and experimental values is also good. The calculated and observed values of heat of mixing are compared in figure 1. Figure-1: Free energy of mixing (∆HRT) versus xCu of liquid CuSn solution (1400K) ;(–––) theory, (○○○) experiment [13] 5 BIBECHANA Vol. 6, March 2010 3.2 AgAl alloys at 1273K It is found from the analysis that the enthalpy of mixing is negative at all concentration, being minimum around stoichiometric composition ( H∆ = -0.611RT at xAg = 0.72). Further it is observed that the concentration dependence of asymmetry in H∆ can be explained only when one considers the temperature dependence of the pairwise interaction energies. The calculated and observed values heat of mixing is compared in figure 2. Figure-2: Free energy of mixing (∆H/RT) versus xAg of liquid AgAl solution (1273K); (––––) theory, (○○○) experiment [1] 3.3 FeSi alloys at 1873K It is found from the analysis that the heat of mixing is negative at all concentration. Our theoretical calculation shows that the minimum value of the heat of mixing is -2.585 kJ at Fex = 0.55. Further it is observed that the concentration dependence of asymmetry in H∆ can be explained only when one considers the temperature dependent of the 6 D. Adhikari et al. pairwise interaction energies. Figure 3 show the comparison between the experimental and calculated values of heat of mixing and entropy of mixing. Figure-3: Free energy of mixing (∆G/RT) versus xFe of liquid FeSi solution (1873K); (––––) theory, (○○○) experiment [13] Table 1 xCu/ xAg/ xFe SnCux 3 AlAgx 3 SiFex 2 0.1 0.001095 0.008257 0.01635 0.2 0.008378 0.03075 0.05294 0.3 0.02658 0.0656 0.1086 0.4 0.05146 0.1138 0.1869 0.5 0.08541 0.1803 0.2903 0.6 0.1240 0.2517 0.3337 0.7 0.1525 0.3075 0.3720 0.8 0.1650 0.2355 0.2968 0.9 0.1192 0.1056 0.1098 -3 -2.5 -2 -1.5 -1 -0.5 0 0.2 0.4 0.6 0.8 1 R T / H ∆ Fex 7 BIBECHANA Vol. 6, March 2010 Table 2 Liquid alloy Systems Parameter CuSn at 1400K AgAl at 1273K FeSi at 1873K k 0.1652 0.0494 0.004139 12ω (Jmol -1 ) -13500 -11382 -64170 13ω (Jmol -1 ) -24900 -617 -14720 23ω (Jmol -1 ) -16500 -28064 -45500 4. Conclusion The regular associated solution model is found to be suitable to estimate the heat of mixing for both weakly and strongly interacting binary liquid alloys. The observed asymmetries in the heat of mixing of binary liquid alloys with respect to the concentrations are well explained on the basis of regular associated solution models. Acknowledgement One of the authors (D. Adhikari) is thankful to University Grant Commission (UGC), Nepal, for providing financial support to pursue the research. References [1] Bhatia, A.B. and Hargoove, W.H. 1974.Phys. Rev. B-10:316 . [2] Singh, R.N. 1987. Can J. Phys. 65: 309. [3] Jordan, A.S. 1970. Metall. Trans. 1:239. [4] Lele, S.and Ramchandra Rao, P. 1981. Metall. Trans. 12 B: 659. [5] Osmura, K. and Predel , B. 1977. Tans. J. Phys. Inst. Met. 18 :765 . [6] K. Hoshino and W.H. Young, J. of Phys. F: Met. Phys. 10,1365 (1980). [7] McAlister, S.P. and Crozier, E.D. 1974. J. of Phys. C-7:3509. 8 D. Adhikari et al. [8] Jha, I.S. ,Singh, R. N. Shrivastava ,P.L. and Mitra, N.R. 1990.Phil. Mag. 61:8445. [9] Singh, R. N. , Jha, I.S. and Pandey, D.K. 1993.J. Condens. Matter 5: 2469 . [10] Bhatia,A.B. and Singh, R. N. 1980.Phys. Letters A-78:460 . [11] Gerling, U. Pool ,M. J. and Predel, 1983.B. Z. Metallkde 74:616 . [12] Adhikari, D., Jha, I.S. and Singh, B. P. 2010.Physica B- 405 :1861 . [13] Hultgren, R. Desai, P. D.,Hawkins, D.T., Gleiser, M. and Kelley, K.K. (ASM,Metal Park,1973).Selected Values of the Thermodynamic Properties of Binary Alloys [14] Prigogine, I. and Defay, R. ,( Longmans Green and Co.London, 1974) Chem. Thermodynamicsp.257.