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Bulk and surface properties of liquid In–Cu alloys
Journal of Alloys and Compounds 333 (2002) 84–90
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Bulk and surface properties of liquid In–Cu alloys 1
O. Akinlade , R.N. Singh* Physics Department, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Oman Received 3 May 2001; accepted 30 May 2001
Abstract In–Cu is a weak interacting system (the excess free energy of mixing G xs M ¯ 2 0.2RT at equiatomic composition) but the composition dependence of the thermodynamic properties exhibit interesting features. G xs M is negative in the Cu-rich region and positive for the In-rich end. Similarly other thermodynamic properties are asymmetric about the equiatomic composition. Such an interesting feature is explained here on the basis that a complex of the form Cu 4 In exists in the bulk phase. Its energetics are used to study the surface composition of the In–Cu liquid alloys. Our study indicates that the surface is enormously rich with indium concentrations throughout the composition of the alloys. 2002 Elsevier Science B.V. All rights reserved. Keywords: Liquid alloys; Entropy; Enthalpy; Thermodynamic modelling
1. Introduction The liquid In–Cu alloy is interesting for several reasons, such as the fact that its excess free energy of mixing G xs M, heat of mixing HM and entropy of mixing SM are distinctly asymmetrical around the equiatomic composition [1,2], so xs much that the excess entropy S xs M is S-shaped. Both G M and HM change sign from negative to positive values towards the In-rich end. Its thermodynamic activity shows pronounced negative deviations at low indium compositions and positive departure from Raoult’s law for higher compositions of indium [1]. Furthermore, In–Cu liquid alloys have been very useful [1] in investigating the properties of ternary alloys namely, In–Bi–Cu and In–Sb– Cu. The composition dependence of the concentration fluctuations Scc (0) computed from thermodynamic activity data are less than the ideal S id cc (0) 5 cA c B for 0 # c In ¯ 0.5 and greater than the ideal in the region c In . 0.5. The computed values of the Warren–Cowley short range order parameter (a1 ) is thus negative in the Cu-rich region and positive in the In-rich region. This indicates a preference for order (unlike atoms as nearest neighbour) in the range of composition 0 # c In # 0.5 and a transformation to a *Corresponding author. E-mail address: [email protected] (R.N. Singh). 1 Permanent address: Department of Physics, University of Agriculture, Abeokuta, Nigeria
segregating state (preference for like neighbours) for composition c In $ 0.5. The size factor (VIn /VCu 52.21, V is atomic volume), is quite large, but the electronegativity difference (¯0.28 on the Pauling scale) is rather too small to explain the composition-dependent asymmetry. In the current paper, we consider the energetics of the interaction energy of the constituent species to explore the composition dependence of the order–disorder transformation in In–Cu liquid alloys. It has been assumed that the complex Cu 4 In exists in the liquid phase and the associated energetics can be discussed with the quasi-lattice model (QLM) [3,4]. QLM has been successful [4–6] in discussing the energetics of the complexes and the composition-dependent asymmetry of the thermodynamic functions. The thermodynamic activity of the component elements act as input data and this is available from the recent thermodynamic measurements [1]. The calculated values of GM , HM , SM and Scc (0) are in very good agreement with the experimental observation and our theory successfully explains the composition-dependent asymmetry of the thermodynamic functions. Within our theoretical model, it is possible to obtain a relation between surface (c si ) and bulk (c bi ) compositions. (c si ) and (c bi ) are connected through surface tensions of the pure components, energetics of the complex and the surface area. Our investigations suggest that the surface of In–Cu liquid alloys is enormously rich with In atoms b (c sIn 4 c In ) throughout the composition of the alloys. In the following section we give an overview of the
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 01 )01733-9
O. Akinlade, R.N. Singh / Journal of Alloys and Compounds 333 (2002) 84 – 90
quasi-lattice theory. This is followed by results and discussion on the free energy of mixing and activity in Section 3. In Section 4 the results for concentration fluctuations, Warren–Cowley short range order parameter and the chemical diffusion are presented. The enthalpy and entropy of mixing are discussed in Section 5. We discuss our results for the surface properties in Section 6 followed with a summary and conclusion in Section 7. 2. The quasi-lattice theory Bhatia and Singh [3] showed that the thermodynamic properties of compound-forming binary alloys could be explained using a quasi-lattice picture in which an A–B alloy is treated as a pseudo ternary mixture of A atoms, B atoms and Am Bn complex ( m and n are small integers). It provides useful insights for the bonding of complexes which can be used to solve the configurational partition function analytically. These authors explicitly showed that their general formalism reduces to Flory’s and conformal solution approach used in the Bhatia and Hargrove work [4] for coordination number Z 5 ` and 2, respectively. The basic assumption is that an alloy in the molten state consists of NA 5 Nc and NB 5 N(1 2 c) g moles of A and B atoms, and a chemical complex of the type Am Bn . ( mA 1 n BáAm Bn ). Here, c is the atomic fraction of A atoms. As the binary alloy contains in all N atoms, it can be shown from conservation of atoms that: n 1 5 Nc 2 m n 3 , n 2 5 N(1 2 c) 2 n n 3 , (1) n 5 n 1 1 n 2 1 n 3 5 N 2 ( m 1 n 2 1)n 3 . The value of n 3 at a given temperature and pressure is obtained from the equilibrium condition for the free energy of mixing GM i.e.
S D ≠GM ]] ≠n 3
50
(2)
T,P,N,c
where GM for the binary alloy can be written as: GM 5 2 n 3 g 1 DG.
F OO
S
Here, R is the molar gas constant. The Vij s (i, j 5 1, 2, 3) are average interaction energies among the species i and j. From Eqs. (2) and (4), the equilibrium value of n 3 is given by the equation n m1 n n2 5 ( m 1 n )n 3 e 2( m 1 n 21) e (Y 2g / RT ) N m 1 n 21 ,
(5)
where 1 Y 5 ]] [(n 1 2 m n 3 )V13 1 (n 2 2 n n 3 )V23 NRT 2 ( m n 2 1 n n 1 )V12 ].
(6)
Eq. (5) provides numerically the equilibrium values of n 3 . It requires m and n (number of atoms in the complex) and the energy parameters g and Vij . In order to correlate it to the directly observed thermodynamic functions, we consider activity, a, that is,
S D
≠GM RT ln a k 5 ]] ≠Nk
.
(7)
T, p,N
In the above equation, k refers to the components A or B, from which one can write
S D
n1 n 1 ln aA 5 1 2 ] 1 ln ] 1 ]] (n 3V13 1 n 2V12 ) N N NRT 1 2 ]] n i n jVij N 2 RT i ,j
S D
OO
(8)
and
S D
n2 n 1 ln a B 5 1 2 ] 1 ln ] 1 ]] (n 2V12 1 n 3V23 ) N N NRT 1 2 ]] n i n jVij . N 2 RT i ,j
S D
OO
(9)
Eqs. (8) and (9) can be safely used in conjunction with Eq. (5) to estimate the energy parameters Vij and g provided that a good set of data for activities aA and a B are available from the experiment.
(3)
Here g is the formation energy of the complex. The first term in Eq. (3) represents the lowering of the free energy due to the formation of complexes. DG represents the free energy of mixing of the ternary mixture of fixed n 1 , n 2 and n 3 whose constituents A, B and Am Bn are assumed to be interacting weakly with each other. The strong bonding between the individual elements are taken into account via the formation of the chemical complex. Using DG as originally formulated [4] we can express the free energy of mixing as GM 5 2 n 3 g n1 n2 ( m 1 n )n 3 1 RT n 1 ln ] 1 n 2 ln ] 1 n 3 ln ]]] N N N 1 1] n i n jVij . N i ,j
85
DG (4)
3. Free energy of mixing and activity The foremost task for numerical calculation is to fix the number of atoms m and n of the complex ( m Cu1 n In5 Cum Inn ). From the phase diagram [2], it is evident that Cu 4 In ( m 54, n 51) exists as a solid intermediate phase which melts at 983 K. It is likely that such associates in some form also exist in the liquid phase close to the melting temperature. Once m and n are found, then Eqs. (5) and (8) can be solved self consistently to determine g and Vij so that the experimentally observed activity [1] as a function of composition of the alloy can be reproduced. The values of the energy parameters obtained for Cu–In at T 5 1073 K are: g /RT 5 1.90 V12 /RT 5 0.30 V13 /RT 5 2 0.40 V23 /RT 5 2 0.80.
(10)
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4. Concentration fluctuations, Warren Cowley shortrange order parameter and chemical diffusion The long wavelength limit (Scc (0)) of the concentration– concentration structure factor [7] is of considerable importance [8,9] to study the nature of atomic order in binary liquid alloys. The basic advantage of Scc (0) is that one can determine it making use of the thermodynamic relations
S D
Fig. 1. Composition dependence of activity at 1073 K, 3 3 3 (experimental points are from Ref. [1]) and theory is obtained using Eq. (8).
Theoretically fitted and the experimentally observed activity are plotted in Fig. 1. It may be noted that, once these energy parameters are selected, they remain the same for all the other investigations. It is quite obvious that g /RT is comparatively smaller in magnitude than that for strongly interacting systems e.g. K–Te [5], Na–Sn [6], Mg–Bi and Tl–Te alloys [4] for which the g /RT values are 47.8, 22.75, 16.7 and 10.84, respectively. In addition, one can deduce from the evaluated interaction energies (i.e. the Vij s) that some of the species have an attractive interaction as implied by the negative values of V13 and V23 . xs The theoretical and observed values of G M /RT as functions of indium concentration are given in Fig. 2. The xs GM is quite asymmetric. It turns from negative in the Cu-rich end to positive at the In-end. Such a strong composition-dependent asymmetry has been successfully explained here within the complex formation model. From our theoretical calculations, we obtain a minimum at c In ¯ 0.2 which is the stoichiometric composition c c 5 m /( m 1 n ) 5 0.2. The free energy of mixing also suggests that In–Cu (G min M 5 2 0.96RT ) is not such a strongly interacting system as Mg–Bi (G min M 5 2 3.38RT ) and liquid min amalgams such as Hg–K (G M 5 2 3.35RT ) and Hg–Na min (G M 5 2 3.23RT ).
Fig. 2. Composition dependence of the free excess energy of mixing for liquid In–Cu alloys at 1073 K. Solid lines, calculated values; 3 3 3, experimental points as determined from activity in Ref. [2].
≠ 2 GM 21 ≠aA Scc (0) 5 RT ]] 5 (1 2 c)aA ] 2 ≠c T,P,N ≠c 21 ≠a B 5 ca B ]]] . ≠(1 2 c) T,P,N
F
S D
G
21 T,P,N
(11)
The last two equalities of Eq. (11) can be used directly to compute Scc (0) from observed numerical activity data. This is usually known as experimental values of Scc (0). It is possible to use the variation of Scc (0) with concentration to understand the nature of atomic order in liquid alloys. Basically, the deviation of this quantity from its ideal values given by S id cc (0) 5 cA c B is significant in explaining the interaction between the components of the binary mixture. The basic result is that Scc (0) , S id cc (0) implies a tendency for heterocoordination (preference of unlike id atoms to pair as nearest neighbours), while Scc (0) . S cc (0) implies homocoordination (preference of like atoms as nearest neighbours). For a demixing system Scc (0) 4 id S cc (0). The analytical expressions for Scc (0) in terms of n i and energy parameters Vij can be obtained using Eqs. (11), (8) and (9), the desired expression is: 2 sn 9d 1 ] O O n 9 n 9V G FO]] RT sn d 3
Scc (0) 5
2
i
i
i 51
i
j
ij
21
,
(12)
i ,j
here n 9i refers to the derivative of n i with respect to c. The computed values of Scc (0) from Eq. (12) and those from activity data [1] are plotted in Fig. 3. From the figure one can infer that the computed Scc (0) values are in reasonable agreement with those obtained
Fig. 3. Scc (0) versus composition for liquid In–Cu alloys. Dots, ideal values; solid lines, calculated values. From Eq. (12), 3 3 3 (as derived from activity data using Eq. (11)).
O. Akinlade, R.N. Singh / Journal of Alloys and Compounds 333 (2002) 84 – 90
directly from activity data and are quite asymmetric as a function of concentration. The two sets of data exhibit a weak deepening around the concentration Cu 4 In. The Scc (0) of In–Cu alloys plotted in Fig. 3 clearly indicates that in the concentration range, 0 # c In # 0.5, there is a preference for heterocoordination (i.e. preference for unlike Cu–In atoms as nearest neighbours) whereas in the concentration range 0.5 , c In # 1.0, there is homocoordination, i.e. like atoms Cu–Cu and In–In tend to pair as nearest neighbours. In order to measure the degree of order in the liquid alloy, the Warren–Cowley short range order parameter [10,11] a1 can be computed. Experimentally, a1 can be determined from knowledge of the concentration–concentration Scc (q) and the number–number structure factors SNN (q). However, in most diffraction experiments, these quantities are not easily measurable. On the other hand, a1 can be estimated from knowledge of Scc (0) [3]. Knowledge of a1 provides an immediate insight into the nature of the local arrangement of atoms in the mixture. a1 5 0 corresponds to a random distribution, a1 , 0 refers to unlike atoms pairing as nearest neighbours whereas a1 . 0 corresponds to like atoms pairing in the first coordination shell. From a simple probabilistic approach [12], one can show that the limiting values of a1 lie in the range 2c 1 ]] # a1 # 1, c # ] 2 (1 2 c)
(13)
Fig. 4. Calculated Warren–Cowley short-range order parameter a1 using Eq. (16).
level of atomic interactions in a binary liquid alloy can be further gained from the knowledge of the dynamic properties of the alloys. Using the Darken [14] thermodynamic equation for diffusion, an expression can be found that relates diffusion and Scc (0) [15], thus one can write, DM c(1 2 c) ] 5 ]]]. Did Scc (0)
(14)
For equiatomic composition the above relations simply reduce to 2 1 # a1 # 1 1.
(15)
The minimum possible value of a1 is a min 1 5 21 implies complete ordering of unlike atoms as nearest neighbours. On the other hand the maximum value a max 51 implies 1 total segregation leading to phase separation. Singh et al. [13] have suggested that a1 can be computed from Scc (0): Scc (0) S21 a1 5 ]]]], S 5 ]]] S(Z 2 1) 1 1 c(1 2 c)
(17)
Here, DM is the chemical or interdiffusion coefficient and Did is given as: Did 5 cDB 1 (1 2 c)DA
2 (1 2 c) 1 ]]] # a1 # 1, c $ ]. c 2
87
(18)
where, DA and DB are self diffusion coefficients. With regards to DM /Did , we note that it approaches 1 for ideal mixing, is greater than 1 for an ordered alloy and is less than 1 for a segregating system. A perusal of Fig. 5 shows the existence of a maximum at c In ¯ 0.10. This is quite close to c In ¯ 0.2 which is the value that would be required for the formation of a Cu 4 In complex that was shown to be the more stable complex from structural measurements. Again the nature of the order segregation transformation at c In ¯ 0.5 is evidenced from DM /Did .
(16)
Z being the coordination number of the alloy. For the present calculations, Z is taken as 10 and the Scc (0) values are as evaluated above. The computed value of the a1 2 c variations are plotted in Fig. 4. This indicates that a1 is negative in the region 0 # c In # 0.53 and is positive for 0.56 # c In # 1.0. The minimum value (a1 5 2 0.2155) occurs at c 5 0.1. The maximum value, (a1 5 0.0455) is observed at c In 5 0.86. Both Scc (0) and a1 suggest that the order-segregation transformation occurs around 50 at.% In. A great amount of understanding at the microscopic
Fig. 5. Computed values of the ratio DM /Did vs. composition for liquid In–Cu liquid alloy.
O. Akinlade, R.N. Singh / Journal of Alloys and Compounds 333 (2002) 84 – 90
88
5. Enthalpy and entropy of formation Within the complex formation model, the evaluation of the enthalpy of mixing is very important because it involves the temperature dependence of the interaction parameters. The enthalpy of formation HM can be obtained from the standard thermodynamic expression
S D.
≠GM HM 5 GM 2 T ]] ≠T
(19)
p
From the expression for GM in Eq. (4), we obtain
S
≠v D D OO n n Sv 2 T ] ≠T
≠g HM 5 2 n 3 g 2 T ] 1 ≠T
ij
i j
ij
(20)
i ,j
Fig. 7. Computed and experimental values of the excess entropy of mixing for liquid In–Cu at 1073 K. Solid lines, calculated values; 3 3 3, experimental points from Ref. [2].
and thus the entropy of mixing SM 5 (HM 2 GM ) /T.
(21)
From the expression for HM , it is obvious that one requires the temperature dependence of the interaction parameters. Using Eq. (20) and the observed [2] values of HM , we get ≠V12 ≠g ] 5 2 1.20R, ]] 5 2 1.1, ≠T ≠T ≠V13 ≠V23 ]] 5 2 0.6R, ]] 5 1.2R. ≠T ≠T
(22)
These values suggest that apart from V23 , all the energy parameters have negative temperature coefficients. The fitted results for HM /RT and the resulting value of (the xs excess entropy) S M ( 5 SM 1 R o i251 c i ln c i ) are shown in Figs. 6 and 7, respectively. The heat of formation and the excess entropy are both S-shaped. It is interesting to observe that the concentration-dependent asymmetry can be explained from the model-based calculations. In addition, the computed values of the excess entropy of mixing from the present study yield a negative value at c In ¯ 0.2 which is quite close to the stoichiometric composition.
xs The agreement of the computed value of S M with experiment is not as good as that of HM . A possible reason is that errors in SM are a combination of errors in GM and HM . However, the experimental and theoretical values obtained using the interaction parameters and their temperature dependence are in quite good agreement with experimental data to produce the composition-dependent asymmetry.
6. Surface properties From the point of view of theoretical modeling of surface properties, a statistical mechanical approach, which derives from the concept of a layered structure [16] near the interface is very useful. This has been used with great success to model the surface tension in liquid binary alloys [17–19]. The grand partition functions set up for the surface layer and that of the bulk [20] provides a relation between surface (c si ) and bulk (c i ) compositions. The resulting expressions [20] are:
s
KB T c SA KB T g SA 5 sA 1 ]] ln ] 1 ]] ln ] a cA a gA KB T c SB KB T g SB 5 sB 1 ]] ln ] 1 ]] ln ] a cB a gB
(23)
where si (i5A or B) is the surface tension of the pure components. The mean atomic surface area a can be calculated from the relation:
S D
Vi ai 5 1.102 ] NA
Fig. 6. Computed and observed values of the enthalpy of formation for liquid In–Cu at 1073 K. Solid lines, calculated values; 3 3 3, experimental points from Ref. [2].
2/3
(24)
NA is the Avogrado’s number and Vi is the atomic volume. The mean surface area a of the alloy was calculated from the relation a 5 o i c i ai . gi ( 5 a i /c i ) and g si ( 5 a si /c si ) are activity coefficients. In conformity with the concept of
O. Akinlade, R.N. Singh / Journal of Alloys and Compounds 333 (2002) 84 – 90
Fig. 8. Surface concentration c sIn vs. bulk concentration c In for Cu–In liquid alloys. Solid lines, theory; dotted lines, ideal values.
89
The surface composition calculated from Eq. (23) as a function of bulk concentration is shown in Fig. 8. Due to lack of experimental data it could not be compared. Nevertheless, our investigations indicate that the surface becomes enormously rich in indium concentrations even for small bulk composition of In. Also, In atoms segregate to the surface. The computed values of s are presented in Fig. 9. The general observable feature is that the surface tension sharply decreases with increasing In content. ds / dcu c In →0 , 0 and is quite sharp in the composition range c In # 0.4, however, the variation of s with c is almost flat for higher contents of In in the In–Cu melt.
7. Conclusions s i
layered atomic structure near the interface, gi and g are assumed to be related as: ln g si 5 p(ln gi (c si )) 1 q ln gi .
(25)
In the above equation gi (c si ) implies that we used the expression for gi , with c i replaced by c si . p and q are termed as surface coordination functions. They are fractions of total number of nearest neighbours made by an atom within the layer in which it lies and that in the adjoining layer, respectively, so that p 1 2q 5 1. For closed packed structures one has p 5 1 / 2 and q 5 1 / 4. We have solved the simultaneous Eq. (23) numerically to compute the concentration dependence of the surface composition (c si ) of In–Cu liquid alloy as a function of bulk composition at 1073 K. The interaction energy parameters used are the same as in Eq. (10). In order to solve Eq. (23), we require the surface tension for the pure species, these are taken from Ref. [21] and the values at 1073 K are:
sCu 5 1.368 Nm 21 sIn 5 0.498 Nm 21 .
(26)
Fig. 9. Calculated value of the surface tension of In–Cu system at 1073 K.
1. The thermodynamic properties of In–Cu liquid alloys exhibit significant deviation from the ideal solution behaviour, though, thermodynamically, it is a weak interacting system. 2. The strong concentration-dependent asymmetries observed in thermodynamic properties can be successfully explained on the basis of a complex formation model assuming the existence of heterocoordinated species of the form Cu 4 In in the liquid phase. 3. The concentration fluctuations, chemical short range order parameter and the chemical diffusion all indicate that the liquid alloy undergoes a transition from an ordered (Cu-rich end) to a segregated (In-rich end) state. 4. Our study reveals that the surface of Cu–In liquid alloys is enormously rich with In atoms.
References [1] S. Itabashi, K. Kameda, K. Yamaguchi, T. Kon, J. Jpn. Inst. Metals 63 (1999) 817. [2] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys, ASM, Metal Park, OH, 1973. [3] A.B. Bhatia, R.N. Singh, Phys. Chem. Liq. 13 (1984) 177. [4] A.B. Bhatia, W.H. Hargrove, Phys. Rev. B 10 (1974) 3186. [5] O. Akinlade, J. Phys. Condens. Matter 6 (1994) 4615. [6] O. Akinlade, Phys. Chem. Liq. 29 (1995) 9. [7] A.B. Bhatia, D.E. Thornton, Phys. Rev. B2 (1970) 3004. [8] P. Chieux, H. Ruppersberg, J. Phys. Coll. C8 (1980) 41. [9] C.N.J. Wagner, in: S. Steeb, H. Warlemount (Eds.), Rapidly Quenched Metals, North-Holland, Amsterdam, 1985, p. 405. [10] B.E. Warren, X-Ray Diffraction, Addison Wesley, Reading, MA, 1969. [11] J.M. Cowley, Phys. Rev. 77 (1950) 669. [12] R.N. Singh, Can. J. Phys. 65 (1987) 309. [13] R.N. Singh, D.K. Pandey, S. Sinha, N.R. Mitra, P.L. Srivastava, Physica B 145 (1987) 358. [14] L.S. Darken, Trans. AIME 175 (1948) 184. [15] R.N. Singh, F. Sommer, Z. Metallkd. 83 (1992) 7. [16] E.A. Guggenheim, Mixtures, Oxford University, Oxford, 1952.
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[17] L.C. Prasad, R.N. Singh, V.N. Singh, G.P. Singh, J. Phys. Chem. B102 (1998) 921. [18] B.C. Anusionwu, O. Akinlade, L.A. Hussain, J. Alloys Comp. 278 (1998) 175. [19] L.C. Prasad, R.N. Singh, Phys. Rev. B 44 (1991) 13768.
[20] L.C. Prasad, R.N. Singh, G.P. Singh, J. Phys. Chem. Liq. 27 (1994) 179. [21] T. Iiida, The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1988.
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Bulk and surface properties of liquid In–Cu alloys 1
O. Akinlade , R.N. Singh* Physics Department, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Oman Received 3 May 2001; accepted 30 May 2001
Abstract In–Cu is a weak interacting system (the excess free energy of mixing G xs M ¯ 2 0.2RT at equiatomic composition) but the composition dependence of the thermodynamic properties exhibit interesting features. G xs M is negative in the Cu-rich region and positive for the In-rich end. Similarly other thermodynamic properties are asymmetric about the equiatomic composition. Such an interesting feature is explained here on the basis that a complex of the form Cu 4 In exists in the bulk phase. Its energetics are used to study the surface composition of the In–Cu liquid alloys. Our study indicates that the surface is enormously rich with indium concentrations throughout the composition of the alloys. 2002 Elsevier Science B.V. All rights reserved. Keywords: Liquid alloys; Entropy; Enthalpy; Thermodynamic modelling
1. Introduction The liquid In–Cu alloy is interesting for several reasons, such as the fact that its excess free energy of mixing G xs M, heat of mixing HM and entropy of mixing SM are distinctly asymmetrical around the equiatomic composition [1,2], so xs much that the excess entropy S xs M is S-shaped. Both G M and HM change sign from negative to positive values towards the In-rich end. Its thermodynamic activity shows pronounced negative deviations at low indium compositions and positive departure from Raoult’s law for higher compositions of indium [1]. Furthermore, In–Cu liquid alloys have been very useful [1] in investigating the properties of ternary alloys namely, In–Bi–Cu and In–Sb– Cu. The composition dependence of the concentration fluctuations Scc (0) computed from thermodynamic activity data are less than the ideal S id cc (0) 5 cA c B for 0 # c In ¯ 0.5 and greater than the ideal in the region c In . 0.5. The computed values of the Warren–Cowley short range order parameter (a1 ) is thus negative in the Cu-rich region and positive in the In-rich region. This indicates a preference for order (unlike atoms as nearest neighbour) in the range of composition 0 # c In # 0.5 and a transformation to a *Corresponding author. E-mail address: [email protected] (R.N. Singh). 1 Permanent address: Department of Physics, University of Agriculture, Abeokuta, Nigeria
segregating state (preference for like neighbours) for composition c In $ 0.5. The size factor (VIn /VCu 52.21, V is atomic volume), is quite large, but the electronegativity difference (¯0.28 on the Pauling scale) is rather too small to explain the composition-dependent asymmetry. In the current paper, we consider the energetics of the interaction energy of the constituent species to explore the composition dependence of the order–disorder transformation in In–Cu liquid alloys. It has been assumed that the complex Cu 4 In exists in the liquid phase and the associated energetics can be discussed with the quasi-lattice model (QLM) [3,4]. QLM has been successful [4–6] in discussing the energetics of the complexes and the composition-dependent asymmetry of the thermodynamic functions. The thermodynamic activity of the component elements act as input data and this is available from the recent thermodynamic measurements [1]. The calculated values of GM , HM , SM and Scc (0) are in very good agreement with the experimental observation and our theory successfully explains the composition-dependent asymmetry of the thermodynamic functions. Within our theoretical model, it is possible to obtain a relation between surface (c si ) and bulk (c bi ) compositions. (c si ) and (c bi ) are connected through surface tensions of the pure components, energetics of the complex and the surface area. Our investigations suggest that the surface of In–Cu liquid alloys is enormously rich with In atoms b (c sIn 4 c In ) throughout the composition of the alloys. In the following section we give an overview of the
0925-8388 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0925-8388( 01 )01733-9
O. Akinlade, R.N. Singh / Journal of Alloys and Compounds 333 (2002) 84 – 90
quasi-lattice theory. This is followed by results and discussion on the free energy of mixing and activity in Section 3. In Section 4 the results for concentration fluctuations, Warren–Cowley short range order parameter and the chemical diffusion are presented. The enthalpy and entropy of mixing are discussed in Section 5. We discuss our results for the surface properties in Section 6 followed with a summary and conclusion in Section 7. 2. The quasi-lattice theory Bhatia and Singh [3] showed that the thermodynamic properties of compound-forming binary alloys could be explained using a quasi-lattice picture in which an A–B alloy is treated as a pseudo ternary mixture of A atoms, B atoms and Am Bn complex ( m and n are small integers). It provides useful insights for the bonding of complexes which can be used to solve the configurational partition function analytically. These authors explicitly showed that their general formalism reduces to Flory’s and conformal solution approach used in the Bhatia and Hargrove work [4] for coordination number Z 5 ` and 2, respectively. The basic assumption is that an alloy in the molten state consists of NA 5 Nc and NB 5 N(1 2 c) g moles of A and B atoms, and a chemical complex of the type Am Bn . ( mA 1 n BáAm Bn ). Here, c is the atomic fraction of A atoms. As the binary alloy contains in all N atoms, it can be shown from conservation of atoms that: n 1 5 Nc 2 m n 3 , n 2 5 N(1 2 c) 2 n n 3 , (1) n 5 n 1 1 n 2 1 n 3 5 N 2 ( m 1 n 2 1)n 3 . The value of n 3 at a given temperature and pressure is obtained from the equilibrium condition for the free energy of mixing GM i.e.
S D ≠GM ]] ≠n 3
50
(2)
T,P,N,c
where GM for the binary alloy can be written as: GM 5 2 n 3 g 1 DG.
F OO
S
Here, R is the molar gas constant. The Vij s (i, j 5 1, 2, 3) are average interaction energies among the species i and j. From Eqs. (2) and (4), the equilibrium value of n 3 is given by the equation n m1 n n2 5 ( m 1 n )n 3 e 2( m 1 n 21) e (Y 2g / RT ) N m 1 n 21 ,
(5)
where 1 Y 5 ]] [(n 1 2 m n 3 )V13 1 (n 2 2 n n 3 )V23 NRT 2 ( m n 2 1 n n 1 )V12 ].
(6)
Eq. (5) provides numerically the equilibrium values of n 3 . It requires m and n (number of atoms in the complex) and the energy parameters g and Vij . In order to correlate it to the directly observed thermodynamic functions, we consider activity, a, that is,
S D
≠GM RT ln a k 5 ]] ≠Nk
.
(7)
T, p,N
In the above equation, k refers to the components A or B, from which one can write
S D
n1 n 1 ln aA 5 1 2 ] 1 ln ] 1 ]] (n 3V13 1 n 2V12 ) N N NRT 1 2 ]] n i n jVij N 2 RT i ,j
S D
OO
(8)
and
S D
n2 n 1 ln a B 5 1 2 ] 1 ln ] 1 ]] (n 2V12 1 n 3V23 ) N N NRT 1 2 ]] n i n jVij . N 2 RT i ,j
S D
OO
(9)
Eqs. (8) and (9) can be safely used in conjunction with Eq. (5) to estimate the energy parameters Vij and g provided that a good set of data for activities aA and a B are available from the experiment.
(3)
Here g is the formation energy of the complex. The first term in Eq. (3) represents the lowering of the free energy due to the formation of complexes. DG represents the free energy of mixing of the ternary mixture of fixed n 1 , n 2 and n 3 whose constituents A, B and Am Bn are assumed to be interacting weakly with each other. The strong bonding between the individual elements are taken into account via the formation of the chemical complex. Using DG as originally formulated [4] we can express the free energy of mixing as GM 5 2 n 3 g n1 n2 ( m 1 n )n 3 1 RT n 1 ln ] 1 n 2 ln ] 1 n 3 ln ]]] N N N 1 1] n i n jVij . N i ,j
85
DG (4)
3. Free energy of mixing and activity The foremost task for numerical calculation is to fix the number of atoms m and n of the complex ( m Cu1 n In5 Cum Inn ). From the phase diagram [2], it is evident that Cu 4 In ( m 54, n 51) exists as a solid intermediate phase which melts at 983 K. It is likely that such associates in some form also exist in the liquid phase close to the melting temperature. Once m and n are found, then Eqs. (5) and (8) can be solved self consistently to determine g and Vij so that the experimentally observed activity [1] as a function of composition of the alloy can be reproduced. The values of the energy parameters obtained for Cu–In at T 5 1073 K are: g /RT 5 1.90 V12 /RT 5 0.30 V13 /RT 5 2 0.40 V23 /RT 5 2 0.80.
(10)
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4. Concentration fluctuations, Warren Cowley shortrange order parameter and chemical diffusion The long wavelength limit (Scc (0)) of the concentration– concentration structure factor [7] is of considerable importance [8,9] to study the nature of atomic order in binary liquid alloys. The basic advantage of Scc (0) is that one can determine it making use of the thermodynamic relations
S D
Fig. 1. Composition dependence of activity at 1073 K, 3 3 3 (experimental points are from Ref. [1]) and theory is obtained using Eq. (8).
Theoretically fitted and the experimentally observed activity are plotted in Fig. 1. It may be noted that, once these energy parameters are selected, they remain the same for all the other investigations. It is quite obvious that g /RT is comparatively smaller in magnitude than that for strongly interacting systems e.g. K–Te [5], Na–Sn [6], Mg–Bi and Tl–Te alloys [4] for which the g /RT values are 47.8, 22.75, 16.7 and 10.84, respectively. In addition, one can deduce from the evaluated interaction energies (i.e. the Vij s) that some of the species have an attractive interaction as implied by the negative values of V13 and V23 . xs The theoretical and observed values of G M /RT as functions of indium concentration are given in Fig. 2. The xs GM is quite asymmetric. It turns from negative in the Cu-rich end to positive at the In-end. Such a strong composition-dependent asymmetry has been successfully explained here within the complex formation model. From our theoretical calculations, we obtain a minimum at c In ¯ 0.2 which is the stoichiometric composition c c 5 m /( m 1 n ) 5 0.2. The free energy of mixing also suggests that In–Cu (G min M 5 2 0.96RT ) is not such a strongly interacting system as Mg–Bi (G min M 5 2 3.38RT ) and liquid min amalgams such as Hg–K (G M 5 2 3.35RT ) and Hg–Na min (G M 5 2 3.23RT ).
Fig. 2. Composition dependence of the free excess energy of mixing for liquid In–Cu alloys at 1073 K. Solid lines, calculated values; 3 3 3, experimental points as determined from activity in Ref. [2].
≠ 2 GM 21 ≠aA Scc (0) 5 RT ]] 5 (1 2 c)aA ] 2 ≠c T,P,N ≠c 21 ≠a B 5 ca B ]]] . ≠(1 2 c) T,P,N
F
S D
G
21 T,P,N
(11)
The last two equalities of Eq. (11) can be used directly to compute Scc (0) from observed numerical activity data. This is usually known as experimental values of Scc (0). It is possible to use the variation of Scc (0) with concentration to understand the nature of atomic order in liquid alloys. Basically, the deviation of this quantity from its ideal values given by S id cc (0) 5 cA c B is significant in explaining the interaction between the components of the binary mixture. The basic result is that Scc (0) , S id cc (0) implies a tendency for heterocoordination (preference of unlike id atoms to pair as nearest neighbours), while Scc (0) . S cc (0) implies homocoordination (preference of like atoms as nearest neighbours). For a demixing system Scc (0) 4 id S cc (0). The analytical expressions for Scc (0) in terms of n i and energy parameters Vij can be obtained using Eqs. (11), (8) and (9), the desired expression is: 2 sn 9d 1 ] O O n 9 n 9V G FO]] RT sn d 3
Scc (0) 5
2
i
i
i 51
i
j
ij
21
,
(12)
i ,j
here n 9i refers to the derivative of n i with respect to c. The computed values of Scc (0) from Eq. (12) and those from activity data [1] are plotted in Fig. 3. From the figure one can infer that the computed Scc (0) values are in reasonable agreement with those obtained
Fig. 3. Scc (0) versus composition for liquid In–Cu alloys. Dots, ideal values; solid lines, calculated values. From Eq. (12), 3 3 3 (as derived from activity data using Eq. (11)).
O. Akinlade, R.N. Singh / Journal of Alloys and Compounds 333 (2002) 84 – 90
directly from activity data and are quite asymmetric as a function of concentration. The two sets of data exhibit a weak deepening around the concentration Cu 4 In. The Scc (0) of In–Cu alloys plotted in Fig. 3 clearly indicates that in the concentration range, 0 # c In # 0.5, there is a preference for heterocoordination (i.e. preference for unlike Cu–In atoms as nearest neighbours) whereas in the concentration range 0.5 , c In # 1.0, there is homocoordination, i.e. like atoms Cu–Cu and In–In tend to pair as nearest neighbours. In order to measure the degree of order in the liquid alloy, the Warren–Cowley short range order parameter [10,11] a1 can be computed. Experimentally, a1 can be determined from knowledge of the concentration–concentration Scc (q) and the number–number structure factors SNN (q). However, in most diffraction experiments, these quantities are not easily measurable. On the other hand, a1 can be estimated from knowledge of Scc (0) [3]. Knowledge of a1 provides an immediate insight into the nature of the local arrangement of atoms in the mixture. a1 5 0 corresponds to a random distribution, a1 , 0 refers to unlike atoms pairing as nearest neighbours whereas a1 . 0 corresponds to like atoms pairing in the first coordination shell. From a simple probabilistic approach [12], one can show that the limiting values of a1 lie in the range 2c 1 ]] # a1 # 1, c # ] 2 (1 2 c)
(13)
Fig. 4. Calculated Warren–Cowley short-range order parameter a1 using Eq. (16).
level of atomic interactions in a binary liquid alloy can be further gained from the knowledge of the dynamic properties of the alloys. Using the Darken [14] thermodynamic equation for diffusion, an expression can be found that relates diffusion and Scc (0) [15], thus one can write, DM c(1 2 c) ] 5 ]]]. Did Scc (0)
(14)
For equiatomic composition the above relations simply reduce to 2 1 # a1 # 1 1.
(15)
The minimum possible value of a1 is a min 1 5 21 implies complete ordering of unlike atoms as nearest neighbours. On the other hand the maximum value a max 51 implies 1 total segregation leading to phase separation. Singh et al. [13] have suggested that a1 can be computed from Scc (0): Scc (0) S21 a1 5 ]]]], S 5 ]]] S(Z 2 1) 1 1 c(1 2 c)
(17)
Here, DM is the chemical or interdiffusion coefficient and Did is given as: Did 5 cDB 1 (1 2 c)DA
2 (1 2 c) 1 ]]] # a1 # 1, c $ ]. c 2
87
(18)
where, DA and DB are self diffusion coefficients. With regards to DM /Did , we note that it approaches 1 for ideal mixing, is greater than 1 for an ordered alloy and is less than 1 for a segregating system. A perusal of Fig. 5 shows the existence of a maximum at c In ¯ 0.10. This is quite close to c In ¯ 0.2 which is the value that would be required for the formation of a Cu 4 In complex that was shown to be the more stable complex from structural measurements. Again the nature of the order segregation transformation at c In ¯ 0.5 is evidenced from DM /Did .
(16)
Z being the coordination number of the alloy. For the present calculations, Z is taken as 10 and the Scc (0) values are as evaluated above. The computed value of the a1 2 c variations are plotted in Fig. 4. This indicates that a1 is negative in the region 0 # c In # 0.53 and is positive for 0.56 # c In # 1.0. The minimum value (a1 5 2 0.2155) occurs at c 5 0.1. The maximum value, (a1 5 0.0455) is observed at c In 5 0.86. Both Scc (0) and a1 suggest that the order-segregation transformation occurs around 50 at.% In. A great amount of understanding at the microscopic
Fig. 5. Computed values of the ratio DM /Did vs. composition for liquid In–Cu liquid alloy.
O. Akinlade, R.N. Singh / Journal of Alloys and Compounds 333 (2002) 84 – 90
88
5. Enthalpy and entropy of formation Within the complex formation model, the evaluation of the enthalpy of mixing is very important because it involves the temperature dependence of the interaction parameters. The enthalpy of formation HM can be obtained from the standard thermodynamic expression
S D.
≠GM HM 5 GM 2 T ]] ≠T
(19)
p
From the expression for GM in Eq. (4), we obtain
S
≠v D D OO n n Sv 2 T ] ≠T
≠g HM 5 2 n 3 g 2 T ] 1 ≠T
ij
i j
ij
(20)
i ,j
Fig. 7. Computed and experimental values of the excess entropy of mixing for liquid In–Cu at 1073 K. Solid lines, calculated values; 3 3 3, experimental points from Ref. [2].
and thus the entropy of mixing SM 5 (HM 2 GM ) /T.
(21)
From the expression for HM , it is obvious that one requires the temperature dependence of the interaction parameters. Using Eq. (20) and the observed [2] values of HM , we get ≠V12 ≠g ] 5 2 1.20R, ]] 5 2 1.1, ≠T ≠T ≠V13 ≠V23 ]] 5 2 0.6R, ]] 5 1.2R. ≠T ≠T
(22)
These values suggest that apart from V23 , all the energy parameters have negative temperature coefficients. The fitted results for HM /RT and the resulting value of (the xs excess entropy) S M ( 5 SM 1 R o i251 c i ln c i ) are shown in Figs. 6 and 7, respectively. The heat of formation and the excess entropy are both S-shaped. It is interesting to observe that the concentration-dependent asymmetry can be explained from the model-based calculations. In addition, the computed values of the excess entropy of mixing from the present study yield a negative value at c In ¯ 0.2 which is quite close to the stoichiometric composition.
xs The agreement of the computed value of S M with experiment is not as good as that of HM . A possible reason is that errors in SM are a combination of errors in GM and HM . However, the experimental and theoretical values obtained using the interaction parameters and their temperature dependence are in quite good agreement with experimental data to produce the composition-dependent asymmetry.
6. Surface properties From the point of view of theoretical modeling of surface properties, a statistical mechanical approach, which derives from the concept of a layered structure [16] near the interface is very useful. This has been used with great success to model the surface tension in liquid binary alloys [17–19]. The grand partition functions set up for the surface layer and that of the bulk [20] provides a relation between surface (c si ) and bulk (c i ) compositions. The resulting expressions [20] are:
s
KB T c SA KB T g SA 5 sA 1 ]] ln ] 1 ]] ln ] a cA a gA KB T c SB KB T g SB 5 sB 1 ]] ln ] 1 ]] ln ] a cB a gB
(23)
where si (i5A or B) is the surface tension of the pure components. The mean atomic surface area a can be calculated from the relation:
S D
Vi ai 5 1.102 ] NA
Fig. 6. Computed and observed values of the enthalpy of formation for liquid In–Cu at 1073 K. Solid lines, calculated values; 3 3 3, experimental points from Ref. [2].
2/3
(24)
NA is the Avogrado’s number and Vi is the atomic volume. The mean surface area a of the alloy was calculated from the relation a 5 o i c i ai . gi ( 5 a i /c i ) and g si ( 5 a si /c si ) are activity coefficients. In conformity with the concept of
O. Akinlade, R.N. Singh / Journal of Alloys and Compounds 333 (2002) 84 – 90
Fig. 8. Surface concentration c sIn vs. bulk concentration c In for Cu–In liquid alloys. Solid lines, theory; dotted lines, ideal values.
89
The surface composition calculated from Eq. (23) as a function of bulk concentration is shown in Fig. 8. Due to lack of experimental data it could not be compared. Nevertheless, our investigations indicate that the surface becomes enormously rich in indium concentrations even for small bulk composition of In. Also, In atoms segregate to the surface. The computed values of s are presented in Fig. 9. The general observable feature is that the surface tension sharply decreases with increasing In content. ds / dcu c In →0 , 0 and is quite sharp in the composition range c In # 0.4, however, the variation of s with c is almost flat for higher contents of In in the In–Cu melt.
7. Conclusions s i
layered atomic structure near the interface, gi and g are assumed to be related as: ln g si 5 p(ln gi (c si )) 1 q ln gi .
(25)
In the above equation gi (c si ) implies that we used the expression for gi , with c i replaced by c si . p and q are termed as surface coordination functions. They are fractions of total number of nearest neighbours made by an atom within the layer in which it lies and that in the adjoining layer, respectively, so that p 1 2q 5 1. For closed packed structures one has p 5 1 / 2 and q 5 1 / 4. We have solved the simultaneous Eq. (23) numerically to compute the concentration dependence of the surface composition (c si ) of In–Cu liquid alloy as a function of bulk composition at 1073 K. The interaction energy parameters used are the same as in Eq. (10). In order to solve Eq. (23), we require the surface tension for the pure species, these are taken from Ref. [21] and the values at 1073 K are:
sCu 5 1.368 Nm 21 sIn 5 0.498 Nm 21 .
(26)
Fig. 9. Calculated value of the surface tension of In–Cu system at 1073 K.
1. The thermodynamic properties of In–Cu liquid alloys exhibit significant deviation from the ideal solution behaviour, though, thermodynamically, it is a weak interacting system. 2. The strong concentration-dependent asymmetries observed in thermodynamic properties can be successfully explained on the basis of a complex formation model assuming the existence of heterocoordinated species of the form Cu 4 In in the liquid phase. 3. The concentration fluctuations, chemical short range order parameter and the chemical diffusion all indicate that the liquid alloy undergoes a transition from an ordered (Cu-rich end) to a segregated (In-rich end) state. 4. Our study reveals that the surface of Cu–In liquid alloys is enormously rich with In atoms.
References [1] S. Itabashi, K. Kameda, K. Yamaguchi, T. Kon, J. Jpn. Inst. Metals 63 (1999) 817. [2] R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K. Kelley, Selected Values of the Thermodynamic Properties of Binary Alloys, ASM, Metal Park, OH, 1973. [3] A.B. Bhatia, R.N. Singh, Phys. Chem. Liq. 13 (1984) 177. [4] A.B. Bhatia, W.H. Hargrove, Phys. Rev. B 10 (1974) 3186. [5] O. Akinlade, J. Phys. Condens. Matter 6 (1994) 4615. [6] O. Akinlade, Phys. Chem. Liq. 29 (1995) 9. [7] A.B. Bhatia, D.E. Thornton, Phys. Rev. B2 (1970) 3004. [8] P. Chieux, H. Ruppersberg, J. Phys. Coll. C8 (1980) 41. [9] C.N.J. Wagner, in: S. Steeb, H. Warlemount (Eds.), Rapidly Quenched Metals, North-Holland, Amsterdam, 1985, p. 405. [10] B.E. Warren, X-Ray Diffraction, Addison Wesley, Reading, MA, 1969. [11] J.M. Cowley, Phys. Rev. 77 (1950) 669. [12] R.N. Singh, Can. J. Phys. 65 (1987) 309. [13] R.N. Singh, D.K. Pandey, S. Sinha, N.R. Mitra, P.L. Srivastava, Physica B 145 (1987) 358. [14] L.S. Darken, Trans. AIME 175 (1948) 184. [15] R.N. Singh, F. Sommer, Z. Metallkd. 83 (1992) 7. [16] E.A. Guggenheim, Mixtures, Oxford University, Oxford, 1952.
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[17] L.C. Prasad, R.N. Singh, V.N. Singh, G.P. Singh, J. Phys. Chem. B102 (1998) 921. [18] B.C. Anusionwu, O. Akinlade, L.A. Hussain, J. Alloys Comp. 278 (1998) 175. [19] L.C. Prasad, R.N. Singh, Phys. Rev. B 44 (1991) 13768.
[20] L.C. Prasad, R.N. Singh, G.P. Singh, J. Phys. Chem. Liq. 27 (1994) 179. [21] T. Iiida, The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1988.