A cubic formalism for linking dilute and concentrated regions of ternary and multicomponent solutions

A cubic formalism for linking dilute and concentrated regions of ternary and multicomponent solutions K. T. Jacob*, B. Konar and G. N. K. Iyengar A cubic formalism is proposed for representation of thermodynamic properties of ternary and higher order systems that connect data on interaction parameters for solutes in metallic solvents with available thermodynamic data on binary systems. Since most metallic alloys have asymmetric excess Gibbs energies of mixing, a cubic expression in mole fraction is the minimum requirement for representation of binary data. Subregular solution model provides such a minimum framework. To link seamlessly with binary data, activity coefficients of solutes in metallic solvents have to be represented by third-order interaction parameter formalism. Although this requires values for a large number of interaction parameters, it is shown that these parameters are interrelated and many parameters can be obtained from data on the constituent binaries; only a limited number have to determined by measurement on ternary systems. For example, for a ternary 1 rich in component 1, 18 interaction parameters are required for defining the activity coefficients of both solutes and when third-order formalism is used. However, when interaction parameter formalism is made consistent with the Gibbs Duhem relation, 11 separate relations are obtained between interaction parameters, reducing the number of independent parameters to seven. Six of these parameters can be obtained from the properties of the constituent binaries and only one parameter need to be determined from measurements in the ternary system. The number of independent parameters required becomes explicit when the excess Gibbs energy of mixing of the ternary system is represented by a subregular type model with an additional ternary parameter. Relations between coefficients of the subregular model and interaction parameters are derived. Thus, the use of the cubic formalism does not necessarily require additional measurements. The advantage is better representation of data at higher concentrations compared to the quadratic formalism of Darken or the unified interaction parameter formalism, which are essentially identical. Keywords: Multicomponent solutions, Thermodynamics, Activity, Activity coefficient, Interaction parameters, Analytical representation Introduction A large amount of data on interactions between two solutes in liquid metallic solvents have been measured over three decades starting from 1960s to assist physiochemical analysis of refining and deoxidation processes for ferrous and non-ferrous metals and alloys. Most of these data are reported as Wagner (196) firstorder interaction parameters, although in a few cases second- and third-order parameters have also been Department of Materials Engineering, Indian Institute of Science, Bangalore 56001, India *Corresponding author, email katob@materials.iisc.ernet.in evaluated. Using these interaction parameters, activity coefficient of a solute in multicomponent dilute solutions can be estimated. Measurements of interaction parameters in several metallic solvents have been assessed and tabulated (Sigworth and Elliot, 1974a, b; Sigworth et al., 1976, 1977). A theoretical approach for calculating interaction parameters has been proposed based on the pseudopotential formalism and free energy of the hard sphere system (Ueno, 1988). At the present time, the theoretical predictions are qualitative. Unfortunately the interaction parameter formalism in its simplest forms is thermodynamically inconsistent and some methods for removing this inconsistency have been discussed in the literature (Lupis and Elliot, 1966; Darken, 1967; Schuhmann Jr, 1985; Pelton and Bale, ß 01 Institute of Materials, Minerals and Mining and The AusIMM Published by Maney on behalf of the Institute and The AusIMM Received January 010; accepted 11 August 011 48 DOI 10.1179/17485511Y.000000006 Mineral Processing and Extractive Metallurgy (Trans. Inst. Min. Metall. C) 01 VOL 11 NO 1

1986, 1990; Srikanth and Jacob, 1988, 1989a; Nagamori, 1989). Analysis of thermodynamic measurements in dilute solutions is constrained by the formalism selected for the representation of data. In parallel with this development, phase diagrams and thermodynamic properties of a large number of binary and some ternary alloys have been measured to assist alloy development, understand phase transformations, and provide a rationale for the development of thermal and thermomechanical processes. With the advent of computer coupling of phase diagrams and thermodynamics, a large body of information has been integrated and compiled across the full range of composition. Based on this database, properties of multicomponent systems can be generated, thus reducing the amount of experimentation required for the development of complex alloys for a variety of applications. It would be useful if the information available in chemical metallurgy on interactions between solutes in liquid metallic solvents can be coupled with information on full range of alloy compositions available in data banks. The purpose of this article is to show that a cubic formalism for dilute solutes in metallic solvents is the minimum requirement for achieving integration with thermodynamic data for the full range of binary and ternary alloys. Requirement for cubic formalism Since most alloy systems exhibit asymmetric behaviour with respect to composition, the simplest model that can be used with reasonable generality for a binary system is the subregular model of Hardy (195). The representation of partial thermodynamic properties of a component in the subregular model requires terms involving the cube of composition. Binary subregular solutions can be combined to form higher order systems, with or without ternary parameters. The need for higher order terms is dictated by the nature of the system and the accuracy of data. Any treatment of dilute solutions must include at least the cubic terms if they have to be incorporated into a larger framework extending across a larger composition range. With the advent of concentrated alloys for corrosion and erosion resistance in high temperature technology, the study of solute interactions in multicomponent systems to higher solute compositions has become necessary. A cubic formalism would fulfil this requirement. Before developing the cubic formalism, a short review of the interaction parameter formalism and methods of making it thermodynamically consistent, and Darken s quadratic formalism (Darken, 1967a, b) will be presented to provide historical continuity and outline the evolution of the subject. Interaction parameter formalism Wagner (196) suggested the interaction parameter formalism based on Maclaurin (Taylor) series expansion for the activity coefficient of a solute expanded at the composition corresponding to the pure solvent. The formalism has been cast in a more general framework and extended by Lupis (198). The series expansion for the activity coefficient of solute in a ternary system 1 can be written as ln c ~ ln c 0 X L ln c L ln c X LX LX X 1 L ln c LX X 1 L ln c LX X L ln c X X X 1 L ln c LX LX 6 LX 1 L ln c LX LX X X 1 L ln c LX LX 1 L ln c 6 LX... 1 n ½X n n L n ln c (n{m)m LX n{m X n L n ln c LX n LX m Š L n ln c LX n X n{m X m X X where components, are the solutes and component 1 is the solvent. The infinite Maclaurin series expansion is exact (Lupis and Elliot, 1966). The coefficients of the Maclaurin series expansion are termed as free-energy interaction parameters. Current limitations on experimental accuracy make it difficult to measure interaction parameters beyond the third-order. To justify a truncation, the remainder term arising out of the truncation should be less than the estimated experimental error. The truncated expressions are inexact (Srikanth and Jacob, 1988). The first-order free-energy interaction parameters defined by Wagner (196) are e ~ L ln c LX (1), X e ~ L ln c L. The self-interaction represents the effect of concentration of parameter e solute on its own activity coefficient. The parameter can be obtained from binary thermodynamics of system 1. The parameter e represents the effect of concentration of solute on the activity coefficient of solute. To derive a value for this parameter, measurements are required on the ternary system 1, where components and are in the dilute domain. The second-order free-energy interaction coefficients defined by Lupis and Elliot (1966) are r ~ 1 L ln c LX r ~ L ln c LX LX, r ~ 1 L ln c LX where r represents the second-order self-interaction parameter obtained from measurements on the binary system 1. The parameter r, representing the secondorder effect of component on activity coefficient of component, is obtained from measurements in ternary system 1. The parameter r, which represents the combined effect of components and on activity coefficient of, is again obtained from measurements on the 1 system, with components and in dilute range., Mineral Processing and Extractive Metallurgy (Trans. Inst. Min. Metall. C) 01 VOL 11 NO 1 49

The third-order free-energy interaction coefficients are h ~ 1 L ln c 6 LX, h ~ 1 L ln c 6 LX, h ~ 1 L ln c LX LX, h ~ 1 L ln c LX LX where h represents the third-order self-interaction parameter obtained from measurements on the binary system 1. The third-order parameters h, h and h have to determined by measurements on ternary alloys rich in component 1. Thus for a ternary system 1 rich in component 1, there are nine interaction parameters defining the activity coefficient of each solute (, ). Thus, 18 interaction parameters are required to describe the system if third-order terms are included. The Maxwell s relation L Ln i LG Ln j ~ L Ln j LG Ln i can be used to provide a link between interaction parameters. A universally valid relation, often referred to as the reciprocal relation (Lupis, 198) between firstorder parameters obtained from Maxwell s relation, valid irrespective of the nature of truncation, is () e j i ~ei j () Lupis and Elliot (1966) have derived other general recurrence relations between interaction parameters of the Maclaurin infinite series expansion using the Gibbs Duhem relation. The recurrence relations between solute interaction parameters in the second-order representation for a solution with m1 components are r ij i ej i ~ri j ei i (4) r jk i e j k ~rki j e k i ~rij k ei j (5) for all values of i, j, k from to m where i?j?k. The Wagner (196) first-order interaction parameter formalism has been extensively used to compute activity coefficients of solutes in multicomponent solutions; for example, oxygen in steels. For a multicomponent solution, the activity coefficient of the solute i can be written as ln c i ~ ln c o i Xm j~ e j i X j (6) where X j (j5 m) are the mole fraction of the solutes. The advantage of the first-order interaction parameter formalism is its simplicity and availability of an extensive range of first-order parameters. The reciprocal relationship between first-order interaction parameters reduces the number of independent parameters that have to be obtained from experiment. At higher concentrations of solutes, higher order interaction coefficients are required. It is also realised now that the first-order representation is thermodynamically inconsistent at finite solute concentrations (Lupis and Elliot, 1966; Darken, 1967; Schuhmann Jr, 1985; Pelton and Bale, 1986, 1990; Srikanth and Jacob, 1988) and its continued use is unjustified except for rough estimates. The second-order interaction parameter formalism is also used in the literature for the representation of activity coefficient or excess partial free energies in the multicomponent solutions. The second-order interaction parameter formalism as introduced by Lupis (198) is given by ln c i ~ ln c 0 i Xm X m j~ j~ e j i X j r j i X j Xm{1 X m j~ kwj r jk i X j X k... (7) where i, j and k indicate solutes, 1 denotes the solvent in system of m1 components and r denotes the secondorder free-energy interaction parameter. The secondorder formalism can be used at higher concentrations. However, it does not satisfy the condition for exactness (Maxwell s relation) (Srikanth and Jacob, 1988). Thus, the formalism is inexact except at infinite dilution of the solute. Only path dependant integrals can be applied to solve for the activity coefficient of the solvent, when the activity coefficient of the solutes is defined by Maclaurin series expansion truncated to the second-order terms. Special relationships between the interaction parameters (Srikanth and Jacob, 1988, 1989a), are required to ensure consistency with the Gibbs Duhem equation e i i ~{ri i (i~... m), e j i ~{rij i (i, j~... m, i=j), r jk i ~r jk q (i, j, k, q~... m and j=k), r j i ~rj q (i, j, q~... m) (8) Special relations are not equivalent to the recurrence relations proposed by Lupis (198) (equations (4) and (5)), though they satisfy them. The special relations give more information than the recurrence relations. As a matter of consequence on imposing these relations, the condition for exactness is restored. In a ternary system 1 rich in component 1, five interaction parameters are required for each solute in the second-order formulation. For both solutes together, 10 interaction parameters are needed. Since seven special relations between interaction parameters are required to make the formalism consistent with Gibbs Duhem relation, only three independent parameters are needed, of which two (for example, e and e ) can be obtained from binary data on 1 and 1 systems. Only one parameter (for example, e ) needs to be obtained accurately by measurement on the ternary system. Darken (1967b) showed in the terminal regions of many binary systems, extending in many cases up to y0 at-% of the solute, the solvent obeys regular solution type behaviour (ln c 1 ~g 1 X ). The activity coefficient of the solute in the terminal region is then given by ln (c =c o )~g 1({X X ). This quadratic formalism of Darken (1967b) is different from the regular solution model since ln c o is not generally equal to the g 1. Darken has shown that the extension of this the formalism to ternary systems provides a thermodynamically consistent and adequate framework for representation of data for relatively concentrated solutions. Srikanth and Jacob (1989b) have extended 50 Mineral Processing and Extractive Metallurgy (Trans. Inst. Min. Metall. C) 01 VOL 11 NO 1

Darken s quadratic formalism to multicomponent systems. The corresponding expressions (Srikanth and Jacob, 1989b; Srikanth et al., 1989) for the activity coefficients for a solution with m1 components, where solvent is denoted as 1 and i, j, k m are the solutes, are given as ln c 1 ~ Xm jvk ln c i ~ log c 0 i {g 1iX i { Xm X m jvk X (g1j g 1k {g jk )X j X k Xm j~ j=i j~ (g 1i g 1j {g ij )X j X (g1j g 1k {g jk )X j X k Xm j~ g 1j X j (9) g 1j X j (10) where the constants g 1j and g 1k are obtained from the corresponding binary systems 1 j and 1 k. The constant g ij is obtained from ternary 1 i j solution rich in solvent 1. The quadratic formalism is identical to the secondorder interaction parameter formalism with the special relations (Srikanth and Jacob, 1988). Pelton and Bale (1986, 1990) have proposed a unified interaction parameter formalism which on careful examination is found to be identical to Darken s quadratic formalism, and the truncated Maclaurin series expansion including terms up to second-order along with the special relations suggested by Srikanth and Jacob (1989b) which makes the formalism thermodynamically consistent. The asymmetrical nature of experimental data for integral thermodynamic excess properties of most binary systems makes it difficult to integrate them with the quadratic formalism (Darken, 1967b; Srikanth and Jacob, 1989b) or by the second-order interaction parameter formalism with special relations (Srikanth and Jacob, 1988). The quadratic formalism is restricted to the terminal regions, whereas the cubic formalism can express the thermodynamic properties of solutions to more extended regions. Cubic formalism for ternary systems The basic equation for the excess Gibbs free-energy of mixing of a ternary system can be written as G E t ~ X i=j=k a ij X i X j X i=j=k b ij X i X j (X i {X j )a t X i X j X k (11) where a ij, b ij are subregular solution parameters of i j binary (Hardy, 195) and a t is a parameter characteristic of the ternary. The subregular solution parameters can be obtained from the thermodynamic data for the binary system G E b ~X ix j fa ij b ij (X i {X j )g. It is advantageous to start with an expression for the integral property since the partials derived there from will obviously satisfy the Gibbs Duhem equation and Maxwell s relation. For a ternary system, there are seven parameters in equation (7), of which six come from the binaries and only one from the ternary. The partial excess Gibbs energy of component i can be obtained from equation (11) G E i ~GE (1{X i )(LG E =LX i ) Xj =X k Among the three components of the ternary, let component 1 be the solvent and, the solutes. The partial Gibbs free-energy of solvent 1 can be expressed as G E 1 ~GE (1{X 1 )(LG E =LX 1 ) X =X (1) Equations (11) and (1) are combined to obtain an equation for G E 1 in terms of a ij, b ij, a t, X 1, X and X. G E 1 ~f½a 1X 1 X (1{X 1 )a 1 (L(X 1 X )=LX 1 )Š ½a X X (1{X 1 )a (L(X X )=LX 1 )Š ½a 1 X 1 X (1{X 1 )a 1 (L(X 1 X )=LX 1 )Š ½b 1 X 1 X (X 1 {X )(1{X 1 )b 1 (L(X 1 X (X 1 {X ))=LX 1 )Š ½b X X (X {X )(1{X 1 )b (L(X X (X {X ))=LX 1 )Š ½b 1 X 1 X (X {X 1 )(1{X 1 )b 1 (L(X 1 X (X {X 1 ))=LX 1 )Š ½a t X 1 X X (1{X 1 )a t (L(X 1 X X )=LX 1 )Šg (1) Simplifying G E 1 ~f½a 1X 1 X (1{X 1 )(a 1 X a 1 X 1 dx )Š ½a X X (1{X 1 )(a X dx a X dx )Š dx ½a 1 X 1 X (1{X 1 )(a 1 X a 1 X )Š ½b 1 X 1 X (X 1 {X )(1{X 1 )b 1 (X 1 X X1 dx {X dx {X dx 1X )Š 1 ½b X X (X {X )(1{X 1 )b ( dx (X X {X dx ) dx (X 1 dx {X X ))Š 1 ½b 1 X 1 X (X {X 1 )(1{X 1 )b 1 (X X dx 1X {X 1 X {X dx 1 )Š ½a t X 1 X X a t (1{X 1 ) (X X X 1 X dx X 1 X dx )Šg (14) In order to further simplify equation (14), differentials of X and X with respect to solvent composition X 1 need to be evaluated. For convenience, let (X /X )5r. Since the sum of mole fractions is unity, X 1 X X 51, X 1 X (1r)51. Expression for dx / and dx / can be obtained as dx ~ {1 1r ~ {1 1(X =X ) ~ {X ~ {X (15) X X 1{X 1 Similarly dx ~ {X (16) 1{X 1 Substituting equations (15) and (16) in equation (14), one obtains Mineral Processing and Extractive Metallurgy (Trans. Inst. Min. Metall. C) 01 VOL 11 NO 1 51

G E 1 ~fa 1X a 1X (a 1a 1 {a )X X b 1 ½X 1 X (X 1 {X )X 1 X {X {X 1 X X 1 X Š b ½X X (X {X ){X X X X Š b 1 ½X 1 X (X {X 1 ){X 1 X X1 X X {X 1X Š a t ½X 1 X X X X {X 1 X X Šg (17) Equation (17) can be rewritten in terms of activity coefficient as RTlnc 1 ~½a 1 X a 1X (a 1a 1 {a )X X Š b 1 ½{X X 1X {X 1 X X 1 X Š b ½X X {X X Š b 1 ½X {X 1X X1 X {X 1 X Š a t X X ½1{X 1 Š (18) The activity coefficient of solvent 1 in terms of two composition variables of components and can be written as RTlnc 1 ~½X (a 1b 1 )X (a 1{b 1 ) X X (a 1 a 1 {a b 1 {b 1 {a t ) X ({4b 1)X (4b 1) X X ({6b 1 {b b 1 a t ) X X ({b 1b 6b 1 a t ) (19) As X,X R0, c 1 R1, equation (19) is consistent with Raoult s law for the solvent. Similarly, the activity coefficient for the components and can be written as RTlnc ~½a 1 X 1 a X (a 1a {a 1 )X 1 X Š b ½{X X X X X X {X Š b 1 ½X 1 X {X 1 X Š b 1 ½X1 {X 1X {X1 X X 1 X Š a t X 1 X ½1{X Š (0) RTlnc ~½a 1 X 1 a X (a 1a {a 1 )X 1 X Š b 1 ½{X 1 X 1X {X 1 X X 1 X Š b 1 ½X 1 X {X 1 X Š b ½X {X X {X X X X Š a t X 1 X ½1{X Š (1) The activity coefficient of solutes expressed in terms of composition variables X and X becomes RTlnc ~½(a 1 b 1 )X ({a 1 {6b 1 ) X (a {a 1 {a 1 {b 1 b 1 a t ) X (a 19b 1 )X (a 1b 1 {b {6b 1 {a t ) X X (a 1 a 1 {a 8b 1 b {4b 1 {a t ) X ({4b 1)X (4b 1)X X ({6b 1 {b b 1 a t ) X X ({b 1b 6b 1 a t ) () When X, X R0, the activity coefficient of component is represented as c 0 and from equation (), its value is defined by RT ln c 0 ~(a 1b 1 ). Equation () is written such that the first-, second- and third-order free-energy interaction parameters can be readily identified by comparison with the corresponding expression for the activity coefficient in the third-order interaction parameter formalism RTlnc ~RTlnc 0 RT½e X e X r X r X r X X h X h X h X X h X X Š () where e represents the first-order interaction parameters, r the second-order interaction parameters and h the third-order interaction parameters. For example, the first-order interaction parameters (Wagner, 196) are RTe ~({a 1{6b 1 ) and RTe ~(a {a 1 {a 1 {b 1 b 1 a t ) The expressions for all the interaction parameters in terms of the coefficients of the cubic formalism are summarised in Table 1. Table 1 Expressions for interaction parameters of solutes and in solvent 1 in terms of parameter of cubic formalism* Composition term Coefficient in the expression for component (equation ()) IP() Coefficient in the expression for component (equation (5)) IP() X (a 1 6b 1 ) RT e (a a 1 a 1 b 1 b 1 a t ) RT e X (a a 1 a 1 b 1 b 1 a t ) RT e (a 1 6b 1 ) RT e X (a 1 9b 1 ) RT r (a 1 6b 1 b b 1 a t ) RT r X (a 1 b 1 b 6b 1 a t ) RT r (a 1 9b 1 ) RT r X X 5X X (a 1 a 1 a 8b 1 b 4b 1 a t ) RT r (a 1 a 1 a 4b 1 b 8b 1 a t ) RT r X (4b 1 ) RT h (4b 1 ) RT h X (4b 1 ) RT h X X (6b 1 b b 1 a t ) RT h X X (b 1 b 6b 1 a t ) RT h *IP()5interaction parameter for component ; IP()5interaction parameter for component. (4b 1 ) RT h (6b 1 b b 1 a t ) RT h (b 1 b 6b 1 a t ) RT h 5 Mineral Processing and Extractive Metallurgy (Trans. Inst. Min. Metall. C) 01 VOL 11 NO 1

Following an identical procedure, the activity coefficient of component in terms of composition variables X and X is given by RTlnc ~½(a 1 {b 1 ) X (a {a 1 {a 1 {b 1 b 1 a t ) X ({a 1 6b 1 )X (a 16b 1 b {b 1 {a t ) X (a 1{9b 1 ) X X (a 1 a 1 {a 4b 1 {b {8b 1 {a t ) X ({4b 1)X (4b 1)X X ({6b 1 {b b 1 a t ) X X ({b 1b 6b 1 a t ) (4) When X, X R 0, activity coefficient of component is represented asc 0 ; RTlnc0 ~(a 1{b 1 ). By comparison with the corresponding expression for the activity coefficient in the third-order interaction parameter formalism RTln c ~RT ln c 0 RT½e X e X r X r X r X X h X h X h X X h X X Š (5) the first-, second- and third-order free-energy interaction parameters for component can be readily identified. Expressions for all the interaction parameters in terms of the coefficients of the cubic formalism are listed in Table 1. From an inspection of the corresponding terms, it is seen that the reciprocal relation between first-order interaction parameters is satisfied e ~(a {a 1 {a 1 { b 1 b 1 a t )=RT~e. The recurrence relations between first- and second-order interaction parameters suggested by Lupis (198) are also satisfied. For example r e ~r e ~(6b 1b {b 1 {a t )=RT r e ~r e ~(b 1{b {6b 1 {a t )=RT Other recurrence relations involving third-order interaction parameters obtained from the general relation proposed by Lupis (198) and simplified by Srikanth and Jacob (1988) h {h ~r {r,h {h ~r {r are also satisfied. Further, additional special relations between interaction parameters suggested by Srikanth and Jacob (1989a) are also satisfied h ~h, h ~h, h ~h, h ~h, e r h ~e r h ~ e r h ~e r h ~0 The 10 relations listed above along with the universally valid reciprocal relation between first-order interaction parameters reduces the number of independent parameters required to describe the behaviour of two solutes in the third-order formalism from 18 to 7, of which six (e, r, h and e, r, h ) can be obtained from the constituent binary systems and only one parameter has to be determined from careful measurements in the ternary. Expressions for lnc 1,lnc,lnc and Gt E given by equations (1), (), (4) and (11) respectively, when substituted satisfy the relation RT½X 1 lnc 1 X lnc X lnc Š~Gt E, thus providing additional confirmation of the correctness of the derived equations. Discussion The cubic formalism developed in this article provides expressions for the activity coefficients of solutes in metallic solvents which are analogous to the more familiar Maclaurin (Taylor) series expansion truncated to third-order. Although the third-order interaction parameter formalism looks formidable with 18 interaction parameters to describe the behaviour of two solutes in a ternary system, they can all be related to the seven coefficients of the cubic formalism, six of which are directly obtained from data on the three constituent binary systems and only one parameter needs to be established by measurements on ternary alloys. Thus, the cubic formalism does not add any extra burden of measurement. The main advantage is better representation of data over a more extended composition range compared to either the quadratic formalism of Darken (1967b) or the unified interaction parameter formalism (Pelton and Bale, 1990), which are essentially identical. The cubic formalism provides a thermodynamically consistent representation of both solute and solvent properties. It integrates seamlessly data available in the literature on activity coefficients of solutes in metallic solvents with data on binary alloys. It provides a marked improvement over Darken s quadratic formalism. In the traditional approach to the study of solute interactions in ternary and higher order systems, the activity coefficient of a solute is measured as a function of composition of all other solutes. Depending on the accuracy of data obtained and composition range of solutes covered, the first- and second-order, and in some rare cases third-order interaction coefficients can be evaluated. Only the self-interaction parameters (six for a ternary system in the third-order formalism) come from binary data; all others (twelve) are to be derived from measurements on ternary alloys if special relations between interaction parameters (Srikanth and Jacob, 1988) are not invoked. This is a formidable experimental task. If the cubic formalism is adopted, there are only seven coefficients to be evaluated, six of which come from the three binaries. Only one parameter needs to be measured in the ternary, which is relatively easy. All the interaction parameters up to third-order can be derived from the seven coefficients. The approach in this new formalism is to rely more on binary parameters available in the literature and use interrelations between solute interaction parameters to reduce the burden of measurements on ternary alloys. The validity of the cubic formalism for representation of data on selected ternary systems will be presented in a separate communication. It is conceivable that even the subregular format with cubic terms may not be adequate to represent some binaries. More complex polynomial expressions for excess Gibbs energy of mixing may be necessary. The consequences of more complex expressions for Gibbs energy on the relations between interaction parameters will explored separately if required. Experimental data on the variation of activity coefficient of solutes with composition in metallic solutes can be reanalysed using the cubic formalism and available binary data to generate Mineral Processing and Extractive Metallurgy (Trans. Inst. Min. Metall. C) 01 VOL 11 NO 1 5

coefficients or interaction parameters, which can represent the data more accurately over more extended ranges of composition. The cubic formalism can be readily extended to multicomponent systems. For a system of n components, the number of constituent binaries is given by n/ {(n)}, and the number of constituent ternaries by n/{(n)}. Each binary requires two coefficients and each ternary requires one coefficient. Hence, the total number of coefficients required to describe all the activity coefficients in the system can be easily assessed. For example for a quaternary system, 16 coefficients are needed, 1 coefficients that can be obtained from the six binaries and four ternary parameters that can be assessed from each constituent ternary. If the activity coefficients have to be expressed as a function of temperature, each free energy coefficient or interaction parameter has to be decomposed into an enthalpy and entropy term. Hence twice the number of coefficients will be needed to express the activity coefficients as a function of temperature. Acknowledgements The authors wish to record the support of Indian National Academy of Engineering (INAE), New Delhi, for grant of INAE Distinguished Professorship to K. T. Jacob and Summer Research Fellowship to Bikram Konar at the Indian Institute of Science. References Darken, L. S. 1967a. Thermodynamics of ternary metallic solutions. Trans. Metall. Soc. AIME, 9, 90 96. Darken, L. S. 1967b. Thermodynamics of binary metallic solutions. Trans. Metall. Soc. AIME, 9, 80 89. Hardy, H. K. 195. A sub-regular solution model and its application to some binary alloy systems. Acta Metall., 1, 0 09. Lupis, C. H. P. 198. Chemical thermodynamics of materials, New York, Elsevier Science Publishing Co. Lupis, C. H. P. and Elliot, J. F. 1966. Generalied interaction coefficients Part II: Free energy terms and the quasi-chemical theory. Acta Metall., 14, 59 58. Nagamori, M. 1989. Discussion of Thermodynamic consistency of the interaction parameter formalism Metall. Trans. B, 0, 44 47. Pelton, A. D. and Bale, C. W. 1986. A modified interaction parameter formalism for non-dilute solutions, Metall. Trans. A, 17, 111 115. Pelton, A. D. and Bale, C. W. 1990. The unified interaction parameter formalism thermodynamic consistency and applications, Metall. Trans. A, 1, 1997 00. Schuhmann Jr, R. 1985. Solute interactions in multicomponent solutions. Metall. Trans. B, 16, 807 81. Sigworth, G. K. and Elliot, J. F. 1974a. The thermodynamics of liquid dilute iron alloys, Met. Sci., 8, 98 10. Sigworth, G. K. and Elliot, J. F. 1974b. The thermodynamics of dilute liquid copper alloys, Can. Metall. Q., 1, 455 461. Sigworth, G. K. and Elliot, J. F. 1976. Thermodynamics of dilute liquid copper alloys. Can. Metall. Q., 15, 1. Sigworth, G. K., Elliot, J. F., Vaughan, G. and Geiger, G. H. 1977. The Thermodynamics of dilute liquid nickel alloys. Trans. Metall. Soc. CIM, 16, 104 110. Srikanth, S. and Jacob, K. T. 1988. Thermodynamic consistency of the interaction parameter formalism. Metall. Trans. B, 19, 69 75. Srikanth, S. and Jacob, K. T. 1989a. Thermodynamic consistency of the interaction parameter formalism. Metall. Trans. B, 0, 47 49. Srikanth, S. and Jacob, K. T. 1989b. Extension of Darken s quadratic formalism to multicomponent solutions. Trans. Iron Steel Inst. Jpn, 9, (), 171 174. Srikanth, S., Jacob, K. T. and Abraham, K. P. 1989. Representation of thermodynamic properties of dilute multicomponent solutions - options and constraints. Arch. Eisenhuttenwes., 60, 6 11. Ueno S., Waseda, Y., Jacob, K. T. and Tamaki, S. 1988. Theoretical treatment of interaction parameters in multicomponent metallic solutions, Steel Res., 59, (11), 474 48. Wagner, C. 196. Thermodynamics of alloys. Cambridge, MA, Addison-Wesley. 54 Mineral Processing and Extractive Metallurgy (Trans. Inst. Min. Metall. C) 01 VOL 11 NO 1