Microsoft Word - B.P Singh.doc B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.81 BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Thermodynamic and structural properties of Mg-Tl liquid alloy B.P. Singh 1 , D. Adhikari 1 , I.S. Jha 2* , B.C. Kumar 1 , S.K. Chaudhary 1 , S.K. Jayaswal 1 , R.P. Koirala 2 1 Univ. Dept. of Physics, T.M.Bhag. University, Bhagalpur, Bihar, India 2 Dept. of Physics, M.M.A.M. Campus (Tribhuvan University), Biratnagar, Nepal (*Email address of corresponding author: indusekharjha@yahoo.com) Article history: Received 25 June, 2011; Accepted 24 August, 2011 Abstract The concentration dependent asymmetry in mixing properties of Mg-Tl liquid alloys at 923 K has been investigated on the basis of regular associated solution model. The concentration of ApB type complex in a regular associated solution of Mg and Tl have been determined. We have then used the concentration of complex to calculate the free energy of mixing, enthalpy of mixing, entropy of mixing, activity, concentration fluctuations in long wavelength limit SCC(0) and the Warren Cowley short-range parameter 1α .The analysis suggests that heterocoordination leading to the formation of chemical complex Mg2Tl is likely to exist in the melt. The analysis reveals that there is a tendency of unlike atom pairing (Mg-Tl) in Mg-Tl alloy whole range of concentration. Keywords: Mg-Tl alloy; microscopic structure; pairwise interaction energy; chemical short range order 1. Introduction Physics and chemistry of liquid alloys, in which the atomic arrangement is not spatially periodic in contrast to the case of crystalline materials, is well-recognized as important and promising area for research. Structural and thermodynamic properties of the initial melt play important role in the formation of alloy. Thus the properties of alloys in the melt are helpful to understand the alloying behaviour of alloys in solid state. Growing technological interest to the nonperiodicity in the atomic arrangement of disordered materials has led to an increasing need for a better description of their atomic scale structures. In this paper, we intend to study the thermodynamic properties of Mg-Tl alloys in liquid state at 923 K on the basis of regular associated solution model. In regular associated solution model strong interaction among the constituent species of the alloy is assumed [1-5]. Such assumptions have been used in different models [6-10] by the investigators. The phase diagram of Mg-Tl alloys shows the existence of Mg-Tl and Mg2Tl intermetallic compounds in its solid state [11].In this paper; we have assumed Mg2Tl complex is energetically favoured in the liquid state of Mg-Tl alloy. Assuming Mg2Tl complex, we have determined free B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.82 energy of mixing (GM), heat of mixing (HM), entropy of mixing (SM), concentration fluctuation in long wavelength limit (SCC(0)) and chemical short range order parameter (α1) of Mg-Tl liquid alloy at 923 K. Basic formalism of regular associated solution model is given in Section 2 and Section 3 deals with the numerical results and discussion. Conclusion is provided in section 4. 2. Basic Formalism Let us consider one mole of binary solution comprising of x1 mole of A atoms and x2 moles of B atoms. The presence of ApB type complex in the solution results in a depletion of concentration of free atoms of the components of A and B. The liquid solution is thus composed of three species namely free atoms A and B and the complex ApB. As a result of associations, the thermodynamic behaviour of the components A and B is governed by the true mole fractions xA and xB rather than the gross mole fraction x1 and x2. Thus it is convenient to operate with two frames of references, one referring to gross mole fractions x1 and x2 and other referring to actual mole fractions of each species (xA, xB and xApB). Further, it is assumed that there are n1 moles of species A, n2 moles of species B and n3 moles of species ApB per mole of the binary solution. From the conservation of mass, the two frames of reference can be interrelated as follows: 311 pnxn −= 322 nxn −= and 3321 pn1nnnn −=++= (1) 3 1 321 1 A pn1 n nnn n x − = ++ = 3 2 321 2 B pn1 n nnn n x − = ++ = and 3 3 321 3 ApB pn1 n nnn n x − = ++ = (2) Here, ApB 3 3 3 px1 pn1 pn 1 pn1 1 n 1 += − += − = (3) Now using Eq. (2.3) in Eq. (2.2), we have, ApB A 1 px1 x n + = , ApB B 2 px1 x n + = , ApB ApB 3 px1 x n + = (4) For the sake of convenience one or more of these frames of reference may be used. Now xA, xB and xApB can be inter-related with each other as follows 1221 xxxx = Using 311 pnxn −= and 322 nxn −= , we get B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.83 132231 x)nn(x)pnn( +=+ After performing some algebraic operations and rearranging the terms, we obtain )nnn(xnpx)xx(n 321132211 ++=++ and ApB21A xpxxx −= (5a) Similarly we can obtain ApB22B x)px1(xx −−= (5b) The equilibrium constant for the reaction ApB ⇔ BpA + is given ApBApB B p AB p A x xx k γ γγ = (6) where Aγ , Bγ and ApBγ are activity coefficients of monomers A, B and complex ApB. Following Lele and Ramchandrarao [1] the free energy of mixing MG is given by M A B 12 A ApB 13 B ApB 23 ApB ApB ApB A A B B ApB ApB ApB 1 RT G (x x x x x x ) (1 px ) (1 px ) x (x ln x x ln x x ln x ) RT ln k (1 px ) = ω + ω + ω + × + + + + + + (7) 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. Progonine and Defay [12] have shown that in associated solutions, the gross chemical potentials of components 1 and 2 are equal to the chemical potentials of the monomeric species A and B. Following Jordan [2] the activity coefficients Aγ , Bγ and ApBγ of monomers and complex can be expressed in terms of pairwise interaction energies through )ωωω(xxωxωxγlnRT 132312ApBB13 2 ApB12 2 BA +−++= (8a) )ωωω(xxωxωxγlnRT 121323ApBA12 2 A23 2 ApBB +−++= (8b) 2 2 ApB A 13 B 23 B B 13 12 23 RT ln x x x x ( )γ = ω + ω + ω −ω +ω (8c) Thus, using equation (5), (6) and (7), one gets ]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 +        = (9) Once the expressions for MG is obtained, other thermodynamic and microscopic functions follow readily. Heat of mixing, entropy of mixing and concentration fluctuations in the long-wavelength limit are related to MG through standard thermodynamic relations B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.84 M M M T,P G H G T T ∂ = −  ∂  (9) M M M H G S T − = (10) 2 2 1 CC M T,PS (0) RT( G C )−= ∂ ∂ (11a) 1 P,T22 1 P,T11CC ))C1(a(Ca )Ca(a)C1()0(S − − −∂∂= ∂∂−= (11b) where C (= Mgx ) is concentration of A component in the alloy. Equation (7) is used in equation (9) and (11), we obtained expressions for MH and )0(SCC as M A B 12 A ApB 13 B ApB 23 ApB ApB ApB 213 2312 A B A ApB B ApB ApB 1 T H (x x x x x x ) (1 px ) (1 px ) x dln k x x x x x x RT T T T (1 px ) dT = ω + ω + ω − × + + ∂ω ∂ω∂ω + + − ∂ ∂ ∂ +  (12) =)0(SCC 1 ApB 2/ ApB B 2/ B A 2/ A 23 / ApB / B13 / ApB / A12 / B / A ApB x x x x x x )xxxxxx( RT 2 )px1( 1 −                         +++ω+ω+ω + (13) Here, 2 2 C G∆ ∂ ∂ > 0 for 0 C G∆ = ∂ ∂ where prime denotes the differentiations with respect to concentration and / Ax and / Bx are determined by using equation (5). / ApBx is determined using the equation (9) and the condition 0 dC klnd = [3]. The concentration fluctuation in long wavelength limit (SCC(0) ) can be determined from measured activity data following equations (11b) [13]. This is usually considered as the experimental value. In order to fit the degree of order in the liquid alloy, Warren- Cowley short-range parameter 1α [14, 15] can be estimated from the knowledge of concentration-concentration structure factor S CC(q) and the number-number structure factor SNN(q). However, in most diffraction experiments these quantities are not easily measurable for all kinds of binary liquid alloy [16, 17]. On the other hand 1α can be estimated from the knowledge of SCC(0) [18,19] )0(S )0(S S, 1)1Z(S 1S id CC CC 1 = +− − =α ; )1C(CSid CC −= (14) where Z is coordination number and Z =5 is taken for our calculation. We note that varying the value of Z does not have any effect on the position of the minima of 1α ; the effect is to vary the depth while the overall feature remains unchanged. B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.85 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 (15a) and 2 B B2 x x lnlnln +γ=γ (15b) Following the technique of Lee and Ramchandrarao [1] the pairwise interaction energies, the equilibrium constants and the activity coefficients at infinite dilution can be written as RT ln 120 1 ω =γ (16a) o 2 o 1 o 2 o 1 13 γγ γγ )RT/ωexp(k − = (16b) where o 1γ and o 2γ are activity coefficients of component A and that of B at zero concentrations. Solving equations (8a) and (8b) 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 (17) 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 (18) where 1a and 2a are respective activities of Mg and Tl atoms in the liquid alloys. Using equations (9), (17) and (18), 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 (19) 3. Results and Discussion The mole fraction of complex Mg Tl− is determined using experimental data of activity [9] and equations (16) and (19) employing the iterative procedure. The best fit values of equilibrium constant and pairwise interaction energies for the alloy Mg2Tl in liquid state at 923 K are found to be k = 0.113, 12ω = -20940 J mol -1 , 13ω = -7870 J mol -1 and 23ω = -29050 J mol -1 All the interaction energies are negative and show that Mg and Tl atoms are attracted to each other and to the complex. B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.86 The compositional dependence of various species (Fig. 1) shows that the maximum association occurs at 60 at. pct. of Mg. At this composition and 923 K, about 21 mol pct. of the liquid alloy is associated. Theoretical calculation of free energy of mixing for Mg-Tl liquid alloy shows that it is moderately interacting system. Fig. 2 shows excellent agreement between the experimental and calculated free energies. The free energy of mixing is minimum (= -12.6 kJ) at xMg = 0.5 which is almost equal to the experimental result [1]. Fig. 2 shows an excellent agreement between the experimental and calculated free energies. On using equation (12) and observed values of MH [1], we have chosen the following values for the given parameters as the best fit values for the heat of formation of Mg-Tl complex. 1 112 10Jmol K T − −∂ω = − ∂ , 1 113 22.5Jmol K T − −∂ω = + ∂ , 1 123 2Jmol K T − −∂ω = + ∂ and 2 ln k R T T ∂ = ∂ 14250 ± 800 J mol -1 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 -6.87 kJ at Mgx = 0.6. Further it is observed that the concentration dependence of asymmetry in MH can be explained only when one considers the temperature dependence of the pairwise interaction energies. Theoretical values of heat of mixing are in very good agreement with observed values. Fig.-1 : Compositional dependence of mole fractions xA (A=Mg), xB (B=Tl) and xApB (ApB= Mg2Tl) versus xMg (concentration of Mg) at 923K. B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.87 Fig.-2 : Upper part : Entropy of mixing (SM) versus xMg, Lower part : free energy of mixing (GM) and heat of mixing (HM) versus xMg (concentration of Mg) at 923K; (––––) theory, (οοο) experiment [1]. We have calculated entropy of mixing of Mg-Tl alloy in liquid state using equation (10). The calculated values always match in sign with observed values. The calculated values and experimental values are in good agreement in all concentration of Mg. The concentration dependence of asymmetry in MS is well explained (Fig. 2). The basic inputs for the calculations of activity are pairwise interaction energies and mole fractions of the unassociated atoms of component 1, called A, unassociated atoms of component 2, called B and the complex, ApB. We have used the same values of mole fractions and pairwise interaction energies for the evaluation of activity which were used for the evaluation of free energy of mixing, heat of mixing and entropy of mixing. The agreement between observed and calculated values of activity of Mg and Tl is also good as shown in Fig. 3. Fig. 4 shows the computed and experimental values of Scc(0) as well as ideal values. The calculated values for Scc(0) shows excellent agreement with the experimental values. The Scc(0) can be used to understand the nature of atomic order in the binary liquid alloys. At a given composition, if Scc(0) < )0(Sid CC , ordering in liquid alloy is expected and if Scc(0) > )0(Sid CC , there is tendency of segregation. Our theoretical analysis clearly indicates that, there is unlike atoms are pairing as nearest neighbours in full range of concentration of Mg, i.e., Mg-Tl alloy in the liquid state behaves like unlike atoms ordering pair in whole concentration range. The knowledge of 1α provides an immediate insight into the nature of the local arrangements of atoms in the mixture. At equiatomic composition, one has 1α1 1 ≤≤− . The minimum possible value of 1α is 1αmin 1 −= and that implies complete ordering of unlike atoms paring at nearest neighbours. On the other hand the maximum value of 1α is 1αmax 1 += which implies total segregation leading to the phase separation and 1α = 0 corresponds to a random distribution of atoms. The variation of 1α (Fig. 4) clearly strengthens the result obtained from the study of SCC(0). B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.89 Fig.-3 : Activity (a) of Mg and Tl in liquid Mg-Tl liquid alloy at 923K versus xMg ; (––––) theory, (○○○) experiment [1]. Fig.-4 : Upper part : Concentration fluctuations in long wavelength limit (Scc(0)) versus xMg (concentration of Mg) at 923K ; Lower part : short range ordering parameter (α1) versus xMg at 923K; (––––) theory, (○○○) experiment, (----) ideal values. B.P.Singh et al. / BIBECHANA 8 (2012) 81-89 : BMHSS, p.89 4. Conclusion The thermodynamic properties and microscopic structure of Mg-Tl alloy in liquid state are explained assuming Mg2Tl complex in the melt on the basis of regular associated solution model The analysis suggests that there is a tendency of unlike atom pairing (Mg-Tl) in Mg-Tl alloy whole range of concentration. The result also indicates that there exist Mg2Tl complex in the liquid state of Mg-Tl alloy. Acknowledgement D. Adhikari gratefully acknowledges University Grant Commission (UGC), Nepal for providing financial support to pursue this research. References [1] S. Lele and P. Ramchandra Rao, Metall. Trans. 12B (1981) 659. [2] A.S. Jordan, Metall. Trans. 1 (1970).239. [3] B.P. Singh , D. Adhikari and I.S. Jha , J. Non-Crystalline Solids 356 (2010)1730. [4] D. Adhikari,, B.P. Singh, I.S. Jha , B.K. Singh, J. Mol. Liq., 156 (2010)155. [5] B.P. Singh, I.S. Jha, D. Adhikari, B.K. Singh, BIBECHANA, 7(2011) 1. [6] I.S. 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