Structural and energetic anomaly in liquid Na–Sn alloys

Journal of Molecular Liquids 167 (2012) 52–56

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Structural and energetic anomaly in liquid Na–Sn alloys D. Adhikari a, b,⁎, B.P. Singh a, I.S. Jha b a b

Univ. Dept. of Physics, T. M. Bhag. University, Bhagalpur, Bihar, India Dept. of Physics, M.M.A.M. Campus (Tribhuvan University), Biratnagar, Nepal

a r t i c l e

i n f o

Article history: Received 28 September 2011 Received in revised form 14 November 2011 Accepted 23 December 2011 Available online 9 January 2012 Keywords: Binary alloy Na–Sn alloy Complexes Interaction energies

a b s t r a c t The alloying behaviour of Na–Sn liquid alloys at 773K has been studied by using regular associated solution model. This model has been utilized to determine the complex concentration in a regular associated solution of Na and Sn. We have then used the complex concentration to calculate the free energy of mixing (GM), enthalpy of mixing (HM), entropy of mixing (SM), concentration fluctuations in long wavelength limit (SCC(0)), the Warren Crowley short-range parameter (α1) and ratio of mutual and intrinsic diffusion coefficients (DM/Did). The analysis suggests that heterocoordination leading to the formation of complex Na3Sn is likely to exist in the liquid and is of a strongly interacting nature. The theoretical analysis reveals that the pairwise interaction energies between the species depend considerably on temperature and the alloys are more ordered towards intermediate region. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The study of the mixing behaviour of liquid alloys is of immense importance for physicists, chemists and engineers for designing and exploring new materials. Thus determination of different properties of liquid alloys, such as thermodynamic, surface, structural, electrical, magnetic properties has been the subjects of active research in metallurgical science for many years. But understanding the properties of liquid alloys is much more difficult than that of crystals due to the presence of strong interactions among the particles and their state of disorder in liquid state. Several theoretical models [1–12] have long been employed to solve the complexities of obtaining different properties of binary liquid alloys. In this work we have studied the thermodynamic and structural properties of Na–Sn liquid alloy at 773K on the basis of regular associated solution model. In regular associated solution model, strong associations among the constituent species are assumed to exist in the liquid phase of binary alloys close to the melting temperature. Due to the strong associations present in the solution, complexes are formed. Thus the binary alloys in a liquid phase can be considered as a ternary mixture of unassociated atoms of components and complexes, all in chemical equilibrium. But the interactions between both the unassociated atoms and the complex are considered no longer equal and hence unassociated atoms do not interact equally with the complex. Several workers [13–21] have theoretically and experimentally tried to understand different properties of Na–Sn system in liquid

⁎ Corresponding author at: Univ. Dept. of Physics, T. M. Bhag. University, Bhagalpur, Bihar, India. E-mail address: [email protected] (D. Adhikari). 0167-7322/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2011.12.010

state. Asymmetry in various properties of mixing of molten Na–Sn alloys is noticed around equiatomic composition. The size factor (ΩNa/ ΩSn = 1.45; Ω being the atomic volume) and electronegativity difference ( ESn − ENa = 1.03) are not large enough to account for the anomalous behaviour of mixing properties. The phase diagram shows the existence of several intermediate phases in the liquid state of Na–Sn alloys which has been confirmed by several workers [19,22,23]. Several pieces of experimental evidence clearly demonstrate that the asymmetric behaviour for a large number of liquid alloys occur at or near the stoichiometric composition where stable intermetallic compound exist in the solid phase. It is, therefore, natural to propose that the ‘chemical complexes’ or psedomolecules' exist in the liquid phase near the melting temperature. From the Na–Sn phase diagram (see refs. suggested above), the Na9Sn4 intermetallic compound exists up to 478 °C, and it melts congruently at that temperature. The curve of the enthalpy of mixing of Na–Sn solutions at 500 °C exhibits the minimum value at 43 at.% Sn [22,23]. These findings are corroborated by neutron diffraction measurements [18].Taking into account that in the liquid phase the irregularities (due to strong interactions in the system in questions) on the propertycurves sometimes are often shifted with respect to the exact composition of an energetically favoured intermetallic compound, the Na9Sn4 (31 at.%Sn) can be approximated by A3B stoichiometry (Na3Sn, with 25 at.%Sn), because none of the models used takes into account the stoichiometry A9B4 (always the stoichiometric coefficients are small integers). Thus we consider Na3Sn phase to describe the thermodynamic and structural properties of Na–Sn liquid alloy at 773K. The layout of the paper is as follows. In Section 2, the theoretical basis of our work is presented. Section 3 gives the results and discussion of this work. Finally, the conclusions are outlined in Section 4.

D. Adhikari et al. / Journal of Molecular Liquids 167 (2012) 52–56

wavelength limit (SCC(0)) are related to GM through standard thermodynamic relations

2. Theory Let the solution of the binary alloy A–B (= Na–Sn) consists of n1 atoms of species A and n2 atoms of species B. Following Lele and Ramchandrarao [24], it is assumed that chemical complexes ApB (ApB⇔pA + B) exist in the melt, where p is a small integer which is usually determined from the compound –forming concentration (= p / (p + 1)) in the solid state. Because of the existence of the compound, liquid alloys are supposed to be composed of three species, namely, monomers A (=Na), B (=Sn) and complex ApB (=Na3Sn) in equilibrium, where p = 3. Let the concentration of A, B and ApB species be nA, nB, and nApB moles respectively. In the partially associated solution the formation of nApB complex requires n1 = nA + pnApB and n2 = nB + nApB for conservation of mass. When there is association, the thermodynamic behaviour of complexes A and B components is governed by their true mole fractions xA,xB and xApB (where xA ¼ nA þnnBAþnApB etc.) rather than their gross mole fraction x1 and x2, 1 (where x1 ¼ n1nþn etc.). 2 Using above relations the two sets of mole fractions are related to each other by the relations xA ¼ x1 −px2 xApB ; xB ¼ x2 −ð1−px2 ÞxApB

ð1Þ

Progonine and Defay [25] 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 [26] the activity coefficientsγA, γB and γApB of monomers and complex can be expressed in terms of pairwise interaction energies through 2

2

RT lnγ A ¼ xB ω12 þ xApB ω13 þ xB xApB ðω12 −ω23 þ ω13 Þ

2

2

RT lnγ B ¼ xApB ω23 þ xA ω12 þ xA xApB ðω23 −ω13 þ ω12 Þ

2

2

RT lnγ ApB ¼ xA ω13 þ xB ω23 þ xB xA ðω13 −ω12 þ ω23 Þ

p

ð2bÞ

ð2cÞ

p

xA xB γ A γB : xApB γApB

H M ¼ GM −T

SM ¼

ð3Þ

Thus, using Eqs. (1)–(3), one gets ! P i xA xB ω ω h þ 12 ½pxB ð1−xA Þ þ xA  þ 13 pxApB ð1−xA Þ−xA xApB RT RT i ω h ð4Þ þ 23 xApB ð1−pxB Þ−xB : RT

ln k ¼ ln

  ∂GM ∂T P

ð6Þ

H M −GM T

ð7Þ

  2 2 −1 SCC ð0Þ ¼ RT ∂ GM =∂C

ð8aÞ

T;P

−1

SCC ð0Þ ¼ ð1−C Þa1 ð∂a1 =∂C ÞT;P

¼ Ca2 ð∂a2 =∂ð1−C ÞÞ−1 T;P

ð8bÞ

where C (= xNa) is concentration of A component in the alloy. Eq. (5) is used in Eqs. (6) and (8a), we obtained expressions for HMand SCC(0) as   1 T  xA xB ω12 þ xA xApB ω13 þ xB xApB ω23 −   HM ¼  1 þ pxApB 1 þ pxApB   xApB ∂ω12 ∂ω13 ∂ω23 d lnk  RT 2 þ xA xApB þ xB xApB − xA xB dT ∂T ∂T ∂T 1 þ pxApB

ð9Þ ( SCC ð0Þ ¼

ð2aÞ

whereω12, ω13 and ω23 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 for the reaction ApB⇔pA + B is given k¼

53

"  1 2  = = = = = =   xA xB ω12 þ xA xApB ω13 þ xB xApB ω23 RT 1 þ pxApB =2 !#)−1 =2 =2 xApB x x : þ A þ B þ xA xB xApB

ð10Þ

M ¼ 0where prime denotes the differentiations Here, ∂∂CG2M > 0 for ∂G ∂C with respect to concentration and xA/ and xB/ are determined by using / lnk Eq. (1). xApB is determined using the Eq. (4) and the condition ddC ¼0 [27,28]. It may be noted that the factor (1 + pxApB) − 1 which appears as a coefficient of all terms containing xA,xBand xApB in the Eqs. (5), (9) and (10), is a result of the change in the basis for expressing mole fractions of species A, B and ApB from that used for x1 and x2. The SCC(0) can be directly determined using Eq. (8b) [29]. This is usually considered as the experimental value. In order to quantify the degree of local order in the liquid alloy, Warren-Cowley short-range parameter α1 [30,31] 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. On the other hand α1 can be estimated from the knowledge of SCC(0) [32,33] 2

α1 ¼

S−1 ; SðZ−1Þ þ 1

S¼

SCC ð0Þ id ; SCC ¼ x1 x2 Sid CC ð0Þ

ð11Þ

Now using the equations listed above the free energy GM is given by   RT  xA xB ω12 þ xA xApB ω13 þ xB xApB ω23 þ   1 þ pxApB 1 þ pxApB   xApB  RT lnk: xA lnxA þ xB lnxB þ xApB lnxApB þ  ð5Þ 1 þ pxApB

GM ¼ 

1

Once the expressions for GM is obtained, other thermodynamic and microscopic functions follow readily. Heat of mixing (HM), entropy of mixing(SM) and concentration fluctuations in the long-

where Z is coordination number and Z = 10 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 α1; the effect is to vary the depth while the overall feature remains unchanged. The mixing behaviour of the alloys forming molten metals can also be studied at the microscopic level in terms of the coefficient of diffusion. The SCC(0) and diffusion coefficients can be related using Darken thermodynamic equation for diffusion [34,35] as follows, DM x x ¼ 1 2 Did SCC ð0Þ

ð12Þ

D. Adhikari et al. / Journal of Molecular Liquids 167 (2012) 52–56

DM ¼ Did

∂ lnaA ∂c

ð13Þ

with DM ¼ x1 DB þ x2 DA

ð14Þ

where DA and DB are the self-diffusion coefficients of pure components A and B respectively. DM/Did indicates the mixing behaviour of alloy, i.e., DM/Did > 1 indicates the tendency for compound formation and DM/Did b 1 indicates phase separation. For ideal mixing, DM/ Did approaches 1. The model parameters are determined by follows method: In a regular associated solution x1γ1 = xAγA andx2γ2 = xBγB, where γ1 and γ2 are respective gross activity coefficients of components 1 and 2. Thus lnγ 1 ¼ lnγA þ ln

xA x1

ð15aÞ

xB x2

ð15bÞ

and lnγ 2 ¼ lnγB þ ln

the pairwise interaction energies, the equilibrium constants and the activity coefficients at infinite dilution can be written as [24] 0

lnγ 1 ¼

ω12 RT

ð16aÞ

k expðω13 =RT Þ ¼

γ o1 γo2 γ o1 −γo2

ð16bÞ

whereγ1o and γ2o are activity coefficients of component A and that of B at zero concentrations. Solving Eqs. (2a) and (2b) we obtain ω13 xB ln ¼ RT

 

ω23 xA ln ¼ RT

a2 xB

  þ ð1−xB Þ ln ax1 −xB ð1−xB Þ ωRT12 A

x2ApB   a1 xA

þ ð1−xA Þ ln

  a2 ω12 x −xA ð1−xA Þ RT B

x2ApB

ð17Þ

ð18Þ

Using Eqs. (4) and (16a and 16b) we can derive ω lnk þ 13 ¼ RT

1 þ xA xApB

!

      a x a ω ln 1 þ B ln 2 − 12 x xApB xB RT ! A

ap a þ ln 1 2 : xApB

ð19Þ

3. Results and discussion To calculate the free energy of mixing, we require mole fraction of various species, pairwise interaction energies between the species formed in the molten state and equilibrium constant. The mole fraction xNa3Snof complex Na3Snis determined using experimental data of activity [22] and Eqs. (16) and (19) employing the iterative procedure. The compositional dependence of various species (Fig. 1) shows that the maximum association occurs at 70 at. pct. of Na. At this composition and 773K, about 56 mol pct. of the liquid alloy is associated.

The equilibrium constant and pairwise interaction energies are determined from the Eqs. (4), (16a), (17), (18) and the computed values are slightly adjusted using observed data of integral excess free energy of mixing [22]. The values of pairwise interaction energies and the equilibrium constant are slightly adjusted using the expression for free energy of mixing as a function of concentration and the experimental values of GM. The best fit values of equilibrium constant and interaction energies for the alloy Na–Sn in liquid state at 773 K are found to be k ¼ 0:0062; ω12 ¼ −32632Jmol ¼ −102778Jmol

−1

−1

; ω13 ¼ −11948Jmol

−1

and ω23

:

All the interaction energies are negative and show that unassociated-Na and unassociated -Sn atoms are attracted to each other and to the complex Na3Sn. Theoretical calculation of free energy of mixing for Na–Sn liquid alloy shows that the minimum value of free energy of mixing is −18885 Jmol − 1 at xNa = 0.6. Fig. 1 shows a very good agreement between the experimental and calculated free energies. The uncertainty in the experimental data in free energy of mixing is ±313 Jmol − 1 at xNa = 0.57[22]. Theoretical calculation of free energy of mixing for Na–Sn liquid alloy shows that Na–Sn alloy in liquid state is strongly interacting system. We have observed that if the interaction energies are supposed to 12 be independent of temperature, i.e., ∂ω ¼ 0 etc., then HM and SM so ∂T obtained are in very poor agreement with experimental data. This simply suggests importance of the dependence of interaction energies on temperature. We have thus assumed that the pairwise interaction energies ω ij are temperature dependent. On using Eq. (9) and observed values of HM[22], the best fit values of heat of dissocialnk tion R T 2 ∂∂T and other temperature dependent parameters are found to be ∂ω12 −1 −1 ∂ω 13 −1 −1 ∂ω23 −1 −1 ¼ −2 Jmol K ; ¼ þ15 J mol K ; ¼ −28:5 J mol K ∂T ∂T ∂T and R T

2

∂ lnk −1 ¼ 56000Jmol ∂T

lnk The best fit value of R T 2 ∂∂T lies within the uncertainty of ±1500 Jmol − 1. The dependence of energy parameters on temperature can be observed from the study of HM and SM. It is found from the analysis that the heat of mixing is negative at all concentration. Our theoretical values of HM agree well with the experimental values obtained from direct reaction calorimetry methods

xA , x B , xA p B

where DM is the chemical or mutual diffusion coefficient and Did is the intrinsic diffusion coefficient for an ideal mixture given as,

xA

xB xA B p

Sn

Na x Na

GM/RT

54

Fig. 1. Upper part: Compositional dependence of mole fractions xA (A = Na), xB (B = Sn) and xApB (APB = Na3Sn) versus xNa (concentration of Na); Lower part: free energy of mixing (GM/RT) versus xNa in the liquid Na–Sn solution (773K); (––––) theory, (○○○) experiment [22].

SM/R

D. Adhikari et al. / Journal of Molecular Liquids 167 (2012) 52–56

55

Sn

x Na

Na

x Na

Na

HM/RT

α1

Sn

Fig. 2. Upper part: Entropy of mixing (SM /R) versus xNa, Lower part: heat of mixing (HM /RT) versus xNa of liquid Na–Sn solution (773K); (––––) theory, (○○○) experiment [22].

[22]. Our calculation shows that the minimum value of the heat of mixing is −18863 Jmol − 1 at xNa = 0.6 which is close to the experimental value [22]. The uncertainty in the experimental value of HM is ±104 Jmol − 1 atxNa = 0.57 [22]. Further it is observed that the concentration dependence of asymmetry in HMcan be explained only when one considers the temperature dependence of the pairwise interaction energies. The entropy of mixing is found to be S-shaped which is in agreement with its experimental behaviour. The asymmetries in HM and SM are well explained (Fig. 2). Fig. 3 shows the computed and experimental values of Scc(0) as well as ideal values. We have found that the calculated value of Scc(0) is less than the ideal value of Scc(0) at all concentration except xNa = 0.1. At xNa = 0.1, the Scc(0) is slightly greater than ideal value. The Scc(0) can be used to understand the nature of atomic order in id the binary liquid alloys. At a given composition, if Scc(0) b SCC (0), orid dering in liquid alloy is expected and if Scc(0) > SCC (0), there is tendency of segregation. Our theoretical analysis shows that the order exists for Na-Sn alloy in the liquid state at almost whole concentration range. The better insight into the degree of atomic ordering in molten Na–Sn alloy have been extended by calculating the Warren-Cowely short range ordering parameter, α1 using Eq. (11). Plot of chemical short range order (α1) with respect to concentration of Na (= xNa) is displayed in Fig. 4 which shows that α1is negative throughout the

concentration range except at xNa = 0.1. At equiatomic composition, one has − 1 ≤ α1 ≤ 1. The minimum possible value of α1 is α1min = − 1 and that implies complete ordering of unlike atoms paring at nearest neighbours. On the other hand the maximum value of α1 is α1max = + 1 which implies total segregation leading to the phase separation and α1= 0 corresponds to a random distribution of atoms. The negative value of α1 is the confirmation of heterocoordination in Na–Sn liquid alloy as indicated by the result SCC(0). The calculated values of SCC(0) are used in Eq. (12) to evaluate the ratio of the mutual and self-diffusion coefficients , DM/Did. The value of DM/Did is greater than 1 in the entire range of concentration except at xNa = 0.1 (Fig. 5) which is indicative for the presence of chemical order in the alloy. At intermediate regions DM/Did > > 1 which suggests that the degree of order in this liquid alloy is strong at intermediate region, as earlier evident in the calculation of SCC(0) and α1. 4. Conclusion In present work, we have used regular associated solution model to obtain equilibrium constant, pairwise interaction energies of NaSn liquid alloy at 773K. The knowledge of nature and extent of interaction energies between the components and complex have been used for the estimation of thermodynamic properties of Na–Sn alloy in liquid state. Computed results suggest that Na-Sn is ordering system of unlike atoms in whole range of concentration except at

SCC(0)

DM/Did

id S CC (0)

Fig. 4. Short range ordering parameter (α1) of liquid Na–Sn solution (773K) versus xNa.

S CC (0)

Sn

Na xNa

Fig. 3. Concentration fluctuations in long wavelength limit (Scc(0)) versus xNa of liquid Na–Sn solution (773K); (––––) theory, (○○○) experiment, (––) ideal values.

Na

Sn

xNa Fig. 5. Ratio of mutual and self-diffusion coefficients (DM/Did) of liquid solution (773K) versus xNa.

56

D. Adhikari et al. / Journal of Molecular Liquids 167 (2012) 52–56

xNa = 0. At this concentration the system is slightly segregating. The analysis suggests that the alloy is more ordered at intermediate region. The analysis also shows that the Na-Sn liquid alloys represent a strongly interacting system as can be deduced from normalised form of the Gibbs energy of mixing (GM /RT is close to -3 ) and the pairwise interaction energies of this system depend strongly on temperature. Our theoretical analysis shows that there exist complexes Na3Sn in Na–Sn alloy in molten state at 773K. Acknowledgement D. Adhikari gratefully acknowledges the support of University Grant Commission (UGC), Nepal, for providing financial support to pursue the research.

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