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

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 1–2–3 rich in component 1, 18 interaction parameters are required for defining the activity coefficients of both solutes 2 and 3 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.