The Uniqueness of a Correction to Interaction Parameter Formalism in a Thermodynamically Consistent Manner
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rties of a multicomponent dilute solution such as molten steel, copper, manganese are most well practically characterized by partial excess Gibbs energies of solutes: in other words, activity coefficients of solutes. For practical applications, Wagner proposed a Taylor series expansion for the partial excess Gibbs energy, and it has been most widely used in the metallurgical community.[1] Although this formalism is strictly valid only at infinite dilution, thus thermodynamically inconsistent at a finite concentration as discussed by Darken,[2,3] the formalism has been still used at finite concentrations. Depending on the system used, however, the formalism can cause significant errors.
YOUN-BAE KANG is with the Graduate Institute of Ferrous Technology, Pohang University of Science and Technology, Pohang, 37673, Republic of Korea. Contact e-mail: [email protected] Manuscript submitted October 4, 2019.
METALLURGICAL AND MATERIALS TRANSACTIONS B
Darken corrected the Wagner’s formalism to be thermodynamically consistent at finite concentration, but it has not been well appreciated. In 1986, Pelton and Bale proposed a modified form of the Wagner’s formalism,[4] which has been further discussed in their subsequent articles.[5,6] The basic idea of their modification is a correction to the Wagner’s formalism by adding activity coefficient of solvent to activity coefficients of all solutes. This makes the Gibbs–Duhem relation for the activity coefficients among the solvent and all the solutes be obeyed, even at finite concentration. They also stated that Maxwell’s relation for partial excess Gibbs energy should also be respected.[4–6] Their proposed formalism exactly reduces to Wagner’s formalism at infinite dilution, to Darken’s quadratic formalism at finite concentration, and also could be reduced to Lupis and Elliott’s formalism[7] as well as that of Margules.[8] They named it as a Unified Interaction Parameter Formalism (UIPF), and the formalism has been successfully applied to multicomponent metallic solutions.[9–13] Some other approaches also have been proposed to make the Wagner’s formalism be thermodynamically consistent.[14–18] Srikanth and Jacob[14] attempted to keep the original expression of Wagner for the activity
coefficient of solute, while the activity coefficient of solvent can be obtained by path-dependent integration of the Gibbs–Duhem equation. This is due to the thermodynamic inconsistency of the original expression of Wagner’s formalism. Hajra and coworkers suggested using the activity coefficients of solutes derived from a Maclaurin infinite series of integral excess Gibbs energy.[15–17] While the approaches by Srikanth and Jacob[14] and Hajra et al.[15–17] keep the original expression of Wagner for the activity coefficient of solute, the proposal by Darken,[2,3] Pelton and Bale[4–6] uses corrected forms of the activity coefficient of solutes. All these approaches proposed one to consider the activity coefficient of solvent to represent thermodynamic consistency between all components through the Gibbs– Duhem equation, which
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