- Email: [email protected]
Energetic effects in some molten alloys of arsenic
Physica B 245 (1998) 330—336
Energetic effects in some molten alloys of arsenic O. Akinlade* International Centre for Theoretical Physics, Trieste, Italy Received 20 August 1997; received in revised form 11 November 1997
Abstract A statistical mechanical theory based on complex formation has been used to study the concentration dependence of the excess free energy of mixing, concentration—concentration fluctuations in the long-wavelength limit and the chemical short-range order parameter for As—Zn and As—Cd molten alloys. From the study of the thermodynamic quantities, we conjecture that the complexes As Zn and As Cd are likely to exist in the melt. The observed concentration dependence 2 3 2 3 of investigated thermodynamic quantities have been explained on the basis of the theory and we infer that a reasonable degree of chemical order, although of a weak nature, persists across the whole concentration range. In conclusion, we state that As—Cd is more ordered than As—Zn molten alloys. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Arsenic alloys; Thermodynamics; As Zn ; As Cd 2 3 2 3
1. Introduction Investigations on the thermodynamic and energetic properties of binary alloys as obtained from structural studies have engaged the attention of many researchers in recent times [1—4]. The underlying idea in the studies is that for molten alloys, their compound, forming or phase separation characteristic are intrinsically related to the longwavelength limit of their concentration—concentration fluctuations, S (0). ## On one hand, a considerable amount of the knowledge gained has come from experiment [4], a significant part has also come by way of empirical
* Correspondence address: Department of Physics, University of Agriculture, P.M.B. 2240, Abeokuta, Ogun State, Nigeria.
models [1—3]. One of this, the quasichemical model (QCM) [1,5], has had a remarkable degree of success. The essential idea of QCM is that the variations with concentration of S (0) can be used to ## study the nature of complex formation in the binary alloys. In strongly interacting liquid alloys for which G /R¹+!3, it is believed [2] that concen. tration-dependent asymmetry and by extension, deviation from regular solution behaviour are indications of the existence of chemical complexes or compound formation in the molten phase. In most alloys, it is observed that S (0) deviates consider## ably from the ideal solution behaviour. It tends to zero for strongly interacting systems while it has intermediate values for weakly interacting systems. The position of the S (0) curve thus yields informa## tion on the nature of the complex formed while its depth signifies strength.
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 8 9 9 - 5
O. Akinlade / Physica B 245 (1998) 330—336
In this article, we have used QCM to investigate the concentration dependence of some thermodynamic properties of mixing such as excess free energy of mixing G94/R¹, S (0) and the chemical short. ## range order parameter a for some arsenic com1 pounds over the whole concentration range thus supplementing available experimental data for the systems. For the present study we have chosen As—Zn and As—Cd for some reasons. In the case of As—Cd, some of its thermodynamic properties has been studied experimentally by Komarek et al. [6] for concentration in the range 0.33)C )0.95 and C$ they observed that its concentration-dependent entropy is anomalous with an S-shape with inflexion at about C +0.57. This anomaly was explained C$ on the basis of the existence of Cd As compound 4 3 in the melt. They conjectured that with increasing temperature, the compound dissociates leading to the disappearance of the anomaly. In the case of As—Zn, not much thermodynamic data is available on it, we are only aware of activity values extrapolated from data on the As—Cd—Zn ternary alloy [7]. Lack of data on it are due to two main reasons; first the high melting point of the intermetallic compound As Zn and second, the 2 3 high vapour pressure of both components of the molten system. From an analysis of phase diagrams [8] for both systems, it is inferred that As Zn and 2 3 As Cd are stable phases in the solid state, we 2 3 assume for the sake of our present calculations that such complexes also persist in the liquid phase. In the next section, we shall present the relevant formalism for calculating thermodynamic properties; in Section 3 we discuss our results of the QCM. We summarize our conclusions at the end of the paper.
2. The quasichemical model (QCM) The general expression for the grand partition function N of a binary alloy AB, which consists of N "Nc atoms of A and N "N(1!c) atoms of A B element B such that N"N #N can be exA B pressed as 1 N"+ qNA(¹)qNB(¹)eb(kANA`kBNB~E) b" , A B K ¹ B E
(1)
331
where the q (¹)s (i"A, B) are the partition funci tions associated with the inner and translational degrees of freedom of the atoms i, k are the chemi ical potentials, E is the configurational energy, K is the Boltzmann constant and ¹ the temperB ature. Bhatia and Singh [5] solved the above equation for a cluster of two lattice sites, by assuming that the energy of an AB, AA or BB bond is a function of whether the bond is a part of the complex A B or not. The details of their derivation can be k l found in their papers but essentially what one does is to express the grand partition function N(2) for a cluster of two lattice sites as N(2)"m2 /2(Z~1)o #m2/2(Z~1)o BB B B AA A A #2m m /Z~1/Z~1o , B AB A B A A m "q (¹)ebk , m "q (¹)ebkB, A A B B where
(2) (3)
(4) o "e~b(eij`PijDeij), i, j"A, B. ij e in Eq. (4) denotes the energy of the free ij bond ij while De is the change in the energy of the ij bond ij in the complex A B , / and / are constants k l A B which are eventually eliminated. P is the probabilij ity that the ij bond is a part of the compound and can be expressed as P "ck~2(1!c)l[2!ck~2(1!c)l], k*2, (5) AA P "ck(1!c)l~2[2!ck(1!c)l~2], l*2, (6) BB and P "ck~1(1!c)l~1[2!ck~1(1!c)l~1]. (7) AB P and P are, respectively, zero for k"l"1. AA BB We note that P as expressed above represents a ij simplistic approximation which may be inappropriate for strongly interacting systems for which G /R¹)!3. . The expression for P could be related to the ij ratio of the two activity coefficients c"c /c , to A B give
A B
Z 2K ¹ B ](P De !P De )#I. AA AA BB BB
ln c"Z ln p#
(8)
332
O. Akinlade / Physica B 245 (1998) 330—336
Here Z is the coordination number and I is a constant which is independent of concentration but may depend on temperature and pressure. In Eq. (8),
C
ln p"1 ln 2
D
(1!c)(b#2c!1) , c(b!2c#1)
(9)
where b"[1#4c(1!c)(g2!1)]1@2
(10)
and g is obtained from g2"
A
exp
B
2u 2P De !P De !P De AA AA BB BB . # AB AB K ¹ ZK ¹ B B (11)
u is the interchange or order energy and is given by u"Z[e !1(e #e )]. AB 2 AA BB We now try to obtain an expression for the excess free energy of mixing defined as (12) G94"G !R¹[c ln c#(1!c) ln(1!c)], . . G being the free energy of mixing. G94 can be . . obtained from c by using the thermodynamic relationship [5]
P
c G94 . " ln c dc R¹ 0 c 1 ln p# (P De !P De ) "Z BB BB 2K ¹ AA AA 0 B ]dc#I (13)
PC
D
The constant I is obtained from the condition that G94"0 at c"0 and c"1. One notes that for De " . ij 0, the expression reduces to the regular solution expression. The concentration—concentration fluctuations in the long-wavelength limit S (0) can be expressed in ## general as: R¹ S (0)" . (14) ## (L2G /Lc2) . T,P,N From Eqs. (12) and (13), the expression for S (0) ## becomes c(1!c) S (0)" , ## W
(15)
where
A B
Zc(1!c) 1 1 W"1# Z !1 # 2bK ¹ 2 b B ][2(1!2c)P@ De #(b!1#2c)P@ De AA AB AB AB #(b!1#2c)P@ De AA AA (16) !(b#1!2c)P@ De ] BB BB In Eq. (16) P@ "dP /dc. The Warren—Cowley shortij ij range order parameter a for the nearest neigh1 bours is defined in terms of the probability that two neighbouring sites are occupied by A and B atoms. Essentially, one states that given an atom A at a lattice site 1, say, let (B/A) denote the probability that a B atom exists at a site 2 which is the nearest neighbour to site 1, then (B/A)"(1!c)(1!a ). (17) 1 Here a "0 for a random alloy, since (B/A)" 1 1!c, the mean concentration of B atoms. From simple probabilistic considerations, it follows that a lies in the range 1 !c 1 )a )1, c) , (18) 1 1!c 2 !(1!c) )a )1, 1 c
1 c* . 2
(19)
By making use of the grand partition function [5] one can obtain expressions for general probability term X (i, j"A or B) (the probability of occupaij tion of the lattice sites by i and j atoms) in terms of the energy terms, thus X X AA BB"g2 (20) X2 AB where the X ’s are related by ij X "c!X ; X "(1!c)!X . (21) AA AB BB AB From Eqs. (20) and (21), one can solve for X to AB obtain 2c(1!c) , X " AB b#1
(22)
where b is as defined in Eq. (10). The conditional probability (A/B) is related to X through the AB
O. Akinlade / Physica B 245 (1998) 330—336
expression, X "(1!c)(A/B). (23) AB By applying Eqs. (21) and (22) in Eq. (17), one can express a in a form suitable for the present calcu1 lations as: b!1 a " 1 b#1
(24)
One notes that the equations stated here can be simplified [9] for a weakly interacting system by assuming ab initio that the interaction parameters are small. From a computational perspective, however, the efforts required are not significantly different either in the simplified form or in the full form as elucidated above, for this reason, the full expression has been used.
3. Results and discussion A detailed discussion of the theory has been given in Section 2 and the formulae there has been applied to calculate the concentration dependence of the free energy of mixing, ratio of activity coefficients and S (0) for the two alloys. For the choice ## of k and l, we have been guided by phase diagrams [8] and a value of k"3 and l"2 has been used for both systems. For Z, we choose a value of 10, since this appears to be a reasonable figure for most binary metallic systems. From experience [3,5,10], one notes that Z does not significantly affect G94, . S (0) and ln c, especially when the energy para## meters are well chosen; its effect is more on a 1 where, although the location is not affected, the magnitude is. The values of the interaction parameter u and De chosen to give a good overall representation of ij
333
G94/R¹ for both systems are given in Table 1. We . note that of all the interaction parameters, the most significantly meaningful one is the interchange or ordering energy u, negative values of which imply that the system is compound forming while the value is positive for a phase-separating system. In general, the values of u/K ¹ for both B systems are quite small compared to Mg—Bi, for example, for which it is equal to !6.5 [1]. The implication is that the two compounds studied here are weakly heterocoordinated ones. This inference is reflected in the values of a for both systems 1 which is about !0.15, compared to a maximum value of !1. At this juncture, we would like to mention that the energy parameters, except u are essentially ‘free’ parameters, chosen to give the best fit to some experimental data (in this case G94/R¹), . are not unique and their effects are more significant in the way they explain other thermodynamic properties. From a perusal of Figs. 1 and 2 for As—Zn and As—Cd, respectively, it is quite clear that the fitted parameters yield a good overall representation of experimental data. We note that all experimental values on thermodynamic properties of As—Zn are obtained from activity data given in Ref. [7], while experimental results for As—Cd are from Ref. [6]. The main objective of this paper is to investigate the energetics of the alloys since these are responsible for the concentration-dependent asymmetries. To this end, the first parameter investigated was S (0); this is significant since any departure from ## ideality of S (0) is useful in studying the nature of ## the molten state. If S (0)(S*$ (0)"c(1!c) (the ## ## ideal value), the system is said to be heterocoordinated and there is tendency for unlike atom pairs to be preferred to like pairs, of course the reverse is the case for S (0)'S*$ (0) since then, homoco## ## ordination (preference for like atoms) is the case.
Table 1 Values of the interaction parameters for As—Zn and As—Cd molten alloys Alloy
Temperature (K)
u K ¹ B
De AA K ¹ B
De BB K ¹ B
De AB K ¹ B
As—Zn As—Cd
1123 983
!2.975 !1.625
!0.278 !0.195
!0.350 !0.147
!0.100 !0.547
334
O. Akinlade / Physica B 245 (1998) 330—336
Fig. 1. Excess free energy of mixing G94/R¹ and activity ratio . ln (c /c ) of As—Zn molten alloys: (solid lines) Theory, (stars) Z/ A4 experimental, G94/R¹. ln (c /c ) (dashed lines) Theory, (points) . Z/ A4 experiment.
S (0) can be derived from excess stability data if ## available, but here we have obtained it from standard thermodynamic relations and Eq. (14) as (1!c)a A , S (0)" ## (La /Lc) A T,P,N
(25)
where a is the thermodynamic activity of atom A A in the melt. From Figs. 3 and 4, we observe that As—Zn and As—Cd are both compound forming systems since for all concentrations, S (0) is less than the ideal ## value. It is also noticed that the peak of S (0) ## moves from Zn-rich in As—Zn to As-rich region of the graph in As—Cd molten alloy. The maximum deviation from ideality in S (0) occurs at about ## equiatomic concentration for As—Zn and for
Fig. 2. Same as for Fig. 1 but for As—Cd.
As—Cd at about C +0.6. As concerns S (0) C$ ## for As—Zn, one observes that the values calculated for the model does not give as good an agreement as that observed for As—Cd and also when the values obtained from the model are compared with those from activity data there is a noticeable deviation. One, however, notes that the most essential features of location of peaks in S (0) ## are given by the model, although the peaks in it for 0.5)C )0.8 is underestimated and there Z/ are some deviations around C +0.4. We obZ/ served that we could obtain a better fit to S (0) ## for As—Zn liquid alloys by assuming a concentration dependence of the interaction parameter, but this would be at the expense of distorting the nature of the model (since it assumes that the interaction parameters are fixed), and is thus not acceptable.
O. Akinlade / Physica B 245 (1998) 330—336
Fig. 3. S (0) for As—Zn molten alloys: experimental data ## (points); theory (solid lines); S*$ (0) (dots): also the chemical ## short-range order parameter a in dashed lines. 1
In order to investigate the energetics of the system quantitatively, we computed the chemical short-range order parameter (CSRO) a . We note 1 that experimentally, (a or Za ) are determined 1 1 from the Fourier transform of S (q) especially in a ## binary alloy where S (q)"0. Although such a deterNC mination is very useful in understanding the local order, it is quite complicated and usually, experimental data is unavailable (as in our case). What we did was to use Eq. (24) to compute it. A plot of values obtained are given in Figs. 3 and 4. Once again, we note that a is negative through1 out for both systems; this indicates a preference for heterocoordination in both systems. The minima in a shifts from about C +0.4 in As—Zn 1 Z/ to C +0.6 in As—Cd, a concentration quite C$
335
Fig. 4. Same as in Fig. 2 but for As—Cd.
close to the Cd As stoichiometric composition 4 3 for the compound that Komarek et al. wrote about [6]. For both systems, a is asymmetric about the 1 equiatomic composition, part of the asymmetry in As—Zn could be due to size effect, X /X +1.5 A4 Z/ although this cannot be the case for As—Cd, because then the ratio is of the order of 1. The two systems are however, quite weakly interacting systems since the calculated minimum values of a are 1 much greater than !1, the value required for complete ordering at the equiatomic composition. From a CSRO point of view, As—Cd is slightly more of a compound-forming system than As—Zn, since a for As—Cd is less. This is in line with the 1 observation for alkali-Pb systems [11] where with increase in atomic number/periodicity, the system becomes more ordered.
336
O. Akinlade / Physica B 245 (1998) 330—336
4. Summary Apart from the excess free energy of mixing which is symmetric about the equiatomic composition, all other thermodynamic quantities of As—Zn and As—Cd are asymmetric about the equiatomic composition. This behaviour has been explained on the basis of the existence of As Zn and As Cd 2 3 2 3 complexes in the melt. Heterocoordination persists throughout the concentration range for both systems although the zinc-rich end is more ordered for As—Zn while its the arsenic region for As—Cd. The two systems are relatively weakly interacting systems though the magnitude of ordering is a little more in As—Zn than that in As—Cd. Acknowledgements The author is grateful to the organizers of the 1996 Research Workshop in Condensed Matter Physics for grants allowing him to participate at the
workshop in Trieste where a significant part of this work was undertaken.
References [1] R.N. Singh, Can. J. Phys. 65 (1987) 309. [2] R.N. Singh, N.H. March, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds — Principle and Practice, Wiley, New York, 1995, p. 661. [3] O. Akinlade, Z. Badirkhan, Physica B 203 (1994) 1. [4] W.H. Young, Rep. Prog. Phys. 55 (1992) 1769. [5] A.B. Bhatia, R.N. Singh, Phys. Chem. Liquids 11 (1982) 285. [6] K.L. Komarek, A. Mikula, E. Hayer, Ber. Bunsengens. 80 (1976) 765. [7] A. Mikula, K.L. Komarek, J. Non. Xtl. Sol. 117/118 (1990) 587. [8] M. Hansen, Constitution of Binary Alloys, McGraw-Hill, New York, 1958, pp. 156—185. [9] A.B. Bhatia, R.N. Singh, Phys. Chem. Liquids 11 (1982) 343. [10] O. Akinlade, J. Phys.: Condens. Matter 6 (1994) 4615. [11] M.-L. Saboungi, S.R. Leonard, J. Ellefson, J. Chem. Phys. 85 (1986) 6072.
Energetic effects in some molten alloys of arsenic O. Akinlade* International Centre for Theoretical Physics, Trieste, Italy Received 20 August 1997; received in revised form 11 November 1997
Abstract A statistical mechanical theory based on complex formation has been used to study the concentration dependence of the excess free energy of mixing, concentration—concentration fluctuations in the long-wavelength limit and the chemical short-range order parameter for As—Zn and As—Cd molten alloys. From the study of the thermodynamic quantities, we conjecture that the complexes As Zn and As Cd are likely to exist in the melt. The observed concentration dependence 2 3 2 3 of investigated thermodynamic quantities have been explained on the basis of the theory and we infer that a reasonable degree of chemical order, although of a weak nature, persists across the whole concentration range. In conclusion, we state that As—Cd is more ordered than As—Zn molten alloys. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Arsenic alloys; Thermodynamics; As Zn ; As Cd 2 3 2 3
1. Introduction Investigations on the thermodynamic and energetic properties of binary alloys as obtained from structural studies have engaged the attention of many researchers in recent times [1—4]. The underlying idea in the studies is that for molten alloys, their compound, forming or phase separation characteristic are intrinsically related to the longwavelength limit of their concentration—concentration fluctuations, S (0). ## On one hand, a considerable amount of the knowledge gained has come from experiment [4], a significant part has also come by way of empirical
* Correspondence address: Department of Physics, University of Agriculture, P.M.B. 2240, Abeokuta, Ogun State, Nigeria.
models [1—3]. One of this, the quasichemical model (QCM) [1,5], has had a remarkable degree of success. The essential idea of QCM is that the variations with concentration of S (0) can be used to ## study the nature of complex formation in the binary alloys. In strongly interacting liquid alloys for which G /R¹+!3, it is believed [2] that concen. tration-dependent asymmetry and by extension, deviation from regular solution behaviour are indications of the existence of chemical complexes or compound formation in the molten phase. In most alloys, it is observed that S (0) deviates consider## ably from the ideal solution behaviour. It tends to zero for strongly interacting systems while it has intermediate values for weakly interacting systems. The position of the S (0) curve thus yields informa## tion on the nature of the complex formed while its depth signifies strength.
0921-4526/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 8 9 9 - 5
O. Akinlade / Physica B 245 (1998) 330—336
In this article, we have used QCM to investigate the concentration dependence of some thermodynamic properties of mixing such as excess free energy of mixing G94/R¹, S (0) and the chemical short. ## range order parameter a for some arsenic com1 pounds over the whole concentration range thus supplementing available experimental data for the systems. For the present study we have chosen As—Zn and As—Cd for some reasons. In the case of As—Cd, some of its thermodynamic properties has been studied experimentally by Komarek et al. [6] for concentration in the range 0.33)C )0.95 and C$ they observed that its concentration-dependent entropy is anomalous with an S-shape with inflexion at about C +0.57. This anomaly was explained C$ on the basis of the existence of Cd As compound 4 3 in the melt. They conjectured that with increasing temperature, the compound dissociates leading to the disappearance of the anomaly. In the case of As—Zn, not much thermodynamic data is available on it, we are only aware of activity values extrapolated from data on the As—Cd—Zn ternary alloy [7]. Lack of data on it are due to two main reasons; first the high melting point of the intermetallic compound As Zn and second, the 2 3 high vapour pressure of both components of the molten system. From an analysis of phase diagrams [8] for both systems, it is inferred that As Zn and 2 3 As Cd are stable phases in the solid state, we 2 3 assume for the sake of our present calculations that such complexes also persist in the liquid phase. In the next section, we shall present the relevant formalism for calculating thermodynamic properties; in Section 3 we discuss our results of the QCM. We summarize our conclusions at the end of the paper.
2. The quasichemical model (QCM) The general expression for the grand partition function N of a binary alloy AB, which consists of N "Nc atoms of A and N "N(1!c) atoms of A B element B such that N"N #N can be exA B pressed as 1 N"+ qNA(¹)qNB(¹)eb(kANA`kBNB~E) b" , A B K ¹ B E
(1)
331
where the q (¹)s (i"A, B) are the partition funci tions associated with the inner and translational degrees of freedom of the atoms i, k are the chemi ical potentials, E is the configurational energy, K is the Boltzmann constant and ¹ the temperB ature. Bhatia and Singh [5] solved the above equation for a cluster of two lattice sites, by assuming that the energy of an AB, AA or BB bond is a function of whether the bond is a part of the complex A B or not. The details of their derivation can be k l found in their papers but essentially what one does is to express the grand partition function N(2) for a cluster of two lattice sites as N(2)"m2 /2(Z~1)o #m2/2(Z~1)o BB B B AA A A #2m m /Z~1/Z~1o , B AB A B A A m "q (¹)ebk , m "q (¹)ebkB, A A B B where
(2) (3)
(4) o "e~b(eij`PijDeij), i, j"A, B. ij e in Eq. (4) denotes the energy of the free ij bond ij while De is the change in the energy of the ij bond ij in the complex A B , / and / are constants k l A B which are eventually eliminated. P is the probabilij ity that the ij bond is a part of the compound and can be expressed as P "ck~2(1!c)l[2!ck~2(1!c)l], k*2, (5) AA P "ck(1!c)l~2[2!ck(1!c)l~2], l*2, (6) BB and P "ck~1(1!c)l~1[2!ck~1(1!c)l~1]. (7) AB P and P are, respectively, zero for k"l"1. AA BB We note that P as expressed above represents a ij simplistic approximation which may be inappropriate for strongly interacting systems for which G /R¹)!3. . The expression for P could be related to the ij ratio of the two activity coefficients c"c /c , to A B give
A B
Z 2K ¹ B ](P De !P De )#I. AA AA BB BB
ln c"Z ln p#
(8)
332
O. Akinlade / Physica B 245 (1998) 330—336
Here Z is the coordination number and I is a constant which is independent of concentration but may depend on temperature and pressure. In Eq. (8),
C
ln p"1 ln 2
D
(1!c)(b#2c!1) , c(b!2c#1)
(9)
where b"[1#4c(1!c)(g2!1)]1@2
(10)
and g is obtained from g2"
A
exp
B
2u 2P De !P De !P De AA AA BB BB . # AB AB K ¹ ZK ¹ B B (11)
u is the interchange or order energy and is given by u"Z[e !1(e #e )]. AB 2 AA BB We now try to obtain an expression for the excess free energy of mixing defined as (12) G94"G !R¹[c ln c#(1!c) ln(1!c)], . . G being the free energy of mixing. G94 can be . . obtained from c by using the thermodynamic relationship [5]
P
c G94 . " ln c dc R¹ 0 c 1 ln p# (P De !P De ) "Z BB BB 2K ¹ AA AA 0 B ]dc#I (13)
PC
D
The constant I is obtained from the condition that G94"0 at c"0 and c"1. One notes that for De " . ij 0, the expression reduces to the regular solution expression. The concentration—concentration fluctuations in the long-wavelength limit S (0) can be expressed in ## general as: R¹ S (0)" . (14) ## (L2G /Lc2) . T,P,N From Eqs. (12) and (13), the expression for S (0) ## becomes c(1!c) S (0)" , ## W
(15)
where
A B
Zc(1!c) 1 1 W"1# Z !1 # 2bK ¹ 2 b B ][2(1!2c)P@ De #(b!1#2c)P@ De AA AB AB AB #(b!1#2c)P@ De AA AA (16) !(b#1!2c)P@ De ] BB BB In Eq. (16) P@ "dP /dc. The Warren—Cowley shortij ij range order parameter a for the nearest neigh1 bours is defined in terms of the probability that two neighbouring sites are occupied by A and B atoms. Essentially, one states that given an atom A at a lattice site 1, say, let (B/A) denote the probability that a B atom exists at a site 2 which is the nearest neighbour to site 1, then (B/A)"(1!c)(1!a ). (17) 1 Here a "0 for a random alloy, since (B/A)" 1 1!c, the mean concentration of B atoms. From simple probabilistic considerations, it follows that a lies in the range 1 !c 1 )a )1, c) , (18) 1 1!c 2 !(1!c) )a )1, 1 c
1 c* . 2
(19)
By making use of the grand partition function [5] one can obtain expressions for general probability term X (i, j"A or B) (the probability of occupaij tion of the lattice sites by i and j atoms) in terms of the energy terms, thus X X AA BB"g2 (20) X2 AB where the X ’s are related by ij X "c!X ; X "(1!c)!X . (21) AA AB BB AB From Eqs. (20) and (21), one can solve for X to AB obtain 2c(1!c) , X " AB b#1
(22)
where b is as defined in Eq. (10). The conditional probability (A/B) is related to X through the AB
O. Akinlade / Physica B 245 (1998) 330—336
expression, X "(1!c)(A/B). (23) AB By applying Eqs. (21) and (22) in Eq. (17), one can express a in a form suitable for the present calcu1 lations as: b!1 a " 1 b#1
(24)
One notes that the equations stated here can be simplified [9] for a weakly interacting system by assuming ab initio that the interaction parameters are small. From a computational perspective, however, the efforts required are not significantly different either in the simplified form or in the full form as elucidated above, for this reason, the full expression has been used.
3. Results and discussion A detailed discussion of the theory has been given in Section 2 and the formulae there has been applied to calculate the concentration dependence of the free energy of mixing, ratio of activity coefficients and S (0) for the two alloys. For the choice ## of k and l, we have been guided by phase diagrams [8] and a value of k"3 and l"2 has been used for both systems. For Z, we choose a value of 10, since this appears to be a reasonable figure for most binary metallic systems. From experience [3,5,10], one notes that Z does not significantly affect G94, . S (0) and ln c, especially when the energy para## meters are well chosen; its effect is more on a 1 where, although the location is not affected, the magnitude is. The values of the interaction parameter u and De chosen to give a good overall representation of ij
333
G94/R¹ for both systems are given in Table 1. We . note that of all the interaction parameters, the most significantly meaningful one is the interchange or ordering energy u, negative values of which imply that the system is compound forming while the value is positive for a phase-separating system. In general, the values of u/K ¹ for both B systems are quite small compared to Mg—Bi, for example, for which it is equal to !6.5 [1]. The implication is that the two compounds studied here are weakly heterocoordinated ones. This inference is reflected in the values of a for both systems 1 which is about !0.15, compared to a maximum value of !1. At this juncture, we would like to mention that the energy parameters, except u are essentially ‘free’ parameters, chosen to give the best fit to some experimental data (in this case G94/R¹), . are not unique and their effects are more significant in the way they explain other thermodynamic properties. From a perusal of Figs. 1 and 2 for As—Zn and As—Cd, respectively, it is quite clear that the fitted parameters yield a good overall representation of experimental data. We note that all experimental values on thermodynamic properties of As—Zn are obtained from activity data given in Ref. [7], while experimental results for As—Cd are from Ref. [6]. The main objective of this paper is to investigate the energetics of the alloys since these are responsible for the concentration-dependent asymmetries. To this end, the first parameter investigated was S (0); this is significant since any departure from ## ideality of S (0) is useful in studying the nature of ## the molten state. If S (0)(S*$ (0)"c(1!c) (the ## ## ideal value), the system is said to be heterocoordinated and there is tendency for unlike atom pairs to be preferred to like pairs, of course the reverse is the case for S (0)'S*$ (0) since then, homoco## ## ordination (preference for like atoms) is the case.
Table 1 Values of the interaction parameters for As—Zn and As—Cd molten alloys Alloy
Temperature (K)
u K ¹ B
De AA K ¹ B
De BB K ¹ B
De AB K ¹ B
As—Zn As—Cd
1123 983
!2.975 !1.625
!0.278 !0.195
!0.350 !0.147
!0.100 !0.547
334
O. Akinlade / Physica B 245 (1998) 330—336
Fig. 1. Excess free energy of mixing G94/R¹ and activity ratio . ln (c /c ) of As—Zn molten alloys: (solid lines) Theory, (stars) Z/ A4 experimental, G94/R¹. ln (c /c ) (dashed lines) Theory, (points) . Z/ A4 experiment.
S (0) can be derived from excess stability data if ## available, but here we have obtained it from standard thermodynamic relations and Eq. (14) as (1!c)a A , S (0)" ## (La /Lc) A T,P,N
(25)
where a is the thermodynamic activity of atom A A in the melt. From Figs. 3 and 4, we observe that As—Zn and As—Cd are both compound forming systems since for all concentrations, S (0) is less than the ideal ## value. It is also noticed that the peak of S (0) ## moves from Zn-rich in As—Zn to As-rich region of the graph in As—Cd molten alloy. The maximum deviation from ideality in S (0) occurs at about ## equiatomic concentration for As—Zn and for
Fig. 2. Same as for Fig. 1 but for As—Cd.
As—Cd at about C +0.6. As concerns S (0) C$ ## for As—Zn, one observes that the values calculated for the model does not give as good an agreement as that observed for As—Cd and also when the values obtained from the model are compared with those from activity data there is a noticeable deviation. One, however, notes that the most essential features of location of peaks in S (0) ## are given by the model, although the peaks in it for 0.5)C )0.8 is underestimated and there Z/ are some deviations around C +0.4. We obZ/ served that we could obtain a better fit to S (0) ## for As—Zn liquid alloys by assuming a concentration dependence of the interaction parameter, but this would be at the expense of distorting the nature of the model (since it assumes that the interaction parameters are fixed), and is thus not acceptable.
O. Akinlade / Physica B 245 (1998) 330—336
Fig. 3. S (0) for As—Zn molten alloys: experimental data ## (points); theory (solid lines); S*$ (0) (dots): also the chemical ## short-range order parameter a in dashed lines. 1
In order to investigate the energetics of the system quantitatively, we computed the chemical short-range order parameter (CSRO) a . We note 1 that experimentally, (a or Za ) are determined 1 1 from the Fourier transform of S (q) especially in a ## binary alloy where S (q)"0. Although such a deterNC mination is very useful in understanding the local order, it is quite complicated and usually, experimental data is unavailable (as in our case). What we did was to use Eq. (24) to compute it. A plot of values obtained are given in Figs. 3 and 4. Once again, we note that a is negative through1 out for both systems; this indicates a preference for heterocoordination in both systems. The minima in a shifts from about C +0.4 in As—Zn 1 Z/ to C +0.6 in As—Cd, a concentration quite C$
335
Fig. 4. Same as in Fig. 2 but for As—Cd.
close to the Cd As stoichiometric composition 4 3 for the compound that Komarek et al. wrote about [6]. For both systems, a is asymmetric about the 1 equiatomic composition, part of the asymmetry in As—Zn could be due to size effect, X /X +1.5 A4 Z/ although this cannot be the case for As—Cd, because then the ratio is of the order of 1. The two systems are however, quite weakly interacting systems since the calculated minimum values of a are 1 much greater than !1, the value required for complete ordering at the equiatomic composition. From a CSRO point of view, As—Cd is slightly more of a compound-forming system than As—Zn, since a for As—Cd is less. This is in line with the 1 observation for alkali-Pb systems [11] where with increase in atomic number/periodicity, the system becomes more ordered.
336
O. Akinlade / Physica B 245 (1998) 330—336
4. Summary Apart from the excess free energy of mixing which is symmetric about the equiatomic composition, all other thermodynamic quantities of As—Zn and As—Cd are asymmetric about the equiatomic composition. This behaviour has been explained on the basis of the existence of As Zn and As Cd 2 3 2 3 complexes in the melt. Heterocoordination persists throughout the concentration range for both systems although the zinc-rich end is more ordered for As—Zn while its the arsenic region for As—Cd. The two systems are relatively weakly interacting systems though the magnitude of ordering is a little more in As—Zn than that in As—Cd. Acknowledgements The author is grateful to the organizers of the 1996 Research Workshop in Condensed Matter Physics for grants allowing him to participate at the
workshop in Trieste where a significant part of this work was undertaken.
References [1] R.N. Singh, Can. J. Phys. 65 (1987) 309. [2] R.N. Singh, N.H. March, in: J.H. Westbrook, R.L. Fleischer (Eds.), Intermetallic Compounds — Principle and Practice, Wiley, New York, 1995, p. 661. [3] O. Akinlade, Z. Badirkhan, Physica B 203 (1994) 1. [4] W.H. Young, Rep. Prog. Phys. 55 (1992) 1769. [5] A.B. Bhatia, R.N. Singh, Phys. Chem. Liquids 11 (1982) 285. [6] K.L. Komarek, A. Mikula, E. Hayer, Ber. Bunsengens. 80 (1976) 765. [7] A. Mikula, K.L. Komarek, J. Non. Xtl. Sol. 117/118 (1990) 587. [8] M. Hansen, Constitution of Binary Alloys, McGraw-Hill, New York, 1958, pp. 156—185. [9] A.B. Bhatia, R.N. Singh, Phys. Chem. Liquids 11 (1982) 343. [10] O. Akinlade, J. Phys.: Condens. Matter 6 (1994) 4615. [11] M.-L. Saboungi, S.R. Leonard, J. Ellefson, J. Chem. Phys. 85 (1986) 6072.