Unified Interaction Parameter Formalism Research Articles

On the Unified Interaction Parameter Formalism and Its Application in Critical Reassessment of Pearlitic Transformation in Fe–C–Mn System

On the Unified Interaction Parameter Formalism and Its Application in Critical Reassessment of Pearlitic Transformation in Fe–C–Mn System

Development of Higher‐Order Interaction Parameter Formalisms for a Ternary Solution in a Thermodynamically Consistent Manner

Wagner's interaction parameter formalism (WIPF), being thermodynamically inconsistent (on truncation) and limited in applicability to a very dilute solution, is modified, ever so often, to extend its utility. In WIPF, the activity coefficients of the solutes present in a solution are represented by Taylor's series. Herein, the analogy between the activity coefficient expression obtained by two different ways, that is, 1) by partially differentiating the (Redlich–Kister) polynomial representing the excess molar free energy and 2) by expressing the truncated Taylor's series (WIPF) along with the special relationships obtained between the interaction parameters by applying Gibbs–Duhem equation, is exhibited. WIPF truncated after the second order along with the special relations obtained by applying Gibbs–Duhem equation and the unified interaction parameter formalism (UIPF) truncated after the first order are identical to the Darken's quadratic formalism which is thermodynamically consistent and restricted to a limited concentration range. Therefore, these formalisms cannot be universal. Higher‐order terms must be included in Redlich–Kister polynomials or in WIPF in a thermodynamically consistent manner to extend its utility for a wide concentration range. Herein, a general quadratic and higher‐order formalisms such as cubic and quartic are developed for a ternary system.

Application and assessment of aluminium deoxidation equilibria in liquid steel using various formalisms based on some statistical thermodynamic models

ABSTRACT Aluminium, is being added to liquid steel in a wide concentration range. Wagner's interaction parameter formalism (WIPF) has been successfully used to fit the experimental data reported by several researchers at 1873 K in a low Al content range, however, a huge variation was observed beyond a few wt.% of Al. In the present work, to calculate Al-O equilibria, besides (i) WIPF, (ii) Lupis and Elliot's formalism has been recalculated and (iii) Unified interaction parameter formalism (UIPF) has been developed for the first time on a wt.% scale. Thermodynamic analysis has also been carried out to account for short range order in Fe-Al-O system using (iv) associate model while (v) modified quasi-chemical model (MQM) is summarized from the literature. The present thermodynamic analysis assesses the suitability of various formalisms based on regular and quasichemical model for a wide Al concentration range in liquid steel.

The Uniqueness of a Correction to Interaction Parameter Formalism in a Thermodynamically Consistent Manner

Polynomial representation of partial excess Gibbs energy (i.e., activity coefficient) in multicomponent dilute solution has been widely used after Wagner. Although Wagner’s Interaction Parameter Formalism has been known to be only strictly valid at infinite dilute concentration, it has been used often, even at finite concentration, due to its mathematical simplicity. Nevertheless, several attempts have been made to correct the formalism to be thermodynamically consistent at finite concentration. Among those, Unified Interaction Parameter Formalism proposed by Pelton and Bale, which may be considered as an extension of Darken’s quadratic formalism, has obtained much attention. However, there have been much confusion and debate about the way of the correction. Recently, a thermodynamic analysis was reported that there are infinite numbers of ways to correct Wagner’s formalism to be thermodynamically consistent, which may prevent one from using the Unified Interaction Parameter Formalism with confidence. In the present article, the correction to the Wagner’s formalism is discussed by revisiting Darken’s condition of the thermodynamic consistency. It is shown that the correction to the Wagner’s formalism can be made uniquely. It is pointed out that to ensure the thermodynamic consistency among the activity coefficients of all components, Gibbs–Duhem relation and Maxwell relation among all components including solvent–solute, must be obeyed. Derived expressions for activity coefficients of all components by path-independent integration are also shown to be the same as those obtained by differentiating a corresponding integral excess partial Gibbs energy.

Assessment of Mn-Fe-Si-C Melt in Unified Interaction Parameter Formalism

The solubility of C in Mn-Fe-Si-C melt decreases with increasing Si content, and its decrease becomes greater when the phase in equilibrium with Mn-Fe-Si-C melt changes from carbon to SiC. Such behavior has an industrial implication for low carbon products and processes of steels and Mn alloys. Li and Morris assessed the solution properties of Mn-Fe-Si-C system in the UIP formalism, but the effectiveness of their assessment was limited in applicable ranges of composition to Mn-rich solution and of temperature to around 1673 K (1400 °C). This study develops the information of activity coefficients of C and Si of Mn-Fe-Si-C system from the consistent experimental solubility of C and assesses them in the UIP formalism. This assessment of Mn-Fe-Si-C system describes confidently the behavior of solution properties in a wide range of composition and temperature.

Prediction of All Component Activities in Iron-Based Liquid Ternary Alloys Containing Phosphorus, Titanium, and Vanadium

To carry out thermodynamic calculation of removing P, Ti, and V from liquid iron, their activity data are indispensable. The molecular interaction volume model (MIVM) was used to predict them. Its effectiveness and reliability in the Fe-Cr-P and Fe-Mn-P alloys were verified by comparing the predicted values of activity of all components with their experimental data. All component activities in the Fe-Ti-P, Fe-V-P, Cr-Mn-P, and Fe-Ti-V liquid alloys were predicted. Their thermodynamic behavior showed some interesting tendencies from the predicted iso-activity curves in the ternary alloys. The results indicated that the MIVM was superior to the unified interaction parameter formalism (UIPF) and might replace the Wagner interaction parameter formalism (WIPF) or the Wagner consistent formalism (WCF) theoretically because it is of clearer physical meaning and its extrapolation is more credible.

Thermodynamic Assessment of Liquid Fe–Si–C System by Unified Interaction Parameter Model

The present study assesses the available experimental data and proposes a model based on the Unified Interaction Parameter Formalism to describe the solution properties of the Mn-Fe-C system. The experimental information developed from the recent works by Katsnelson et al., Fenstad, and Kim et al. defines the solution properties as the ratio of the activity of C to Mn in the Mn-Fe-C system. These data were used to determine the activity coefficient of carbon at infinite dilute solution of Mn in the Mn-C system at various temperatures. In λc°(Mn)=0.32-2 679/T (K) (1 628-1 773K) K) A determination of the individual activity of C and Mn from experimental data requires additional information. As it meets the necessary condition for the task by virtue of satisfying the Gibbs-Duhem relationship, the Unified Interaction Parameter (UIP) model was correlated with the experimental data of the Mn-Fe-C system. The interaction parameters of the UIP model were determined by multiple regression analysis of the correlated equations. The activity coefficients of carbon and manganese in reference to graphite and liquid Mn as respective standard states in the Mn-Fe-C system are determined as follows: In γc= In γ c ° (Mn)+e cc (x c -1/2 x c 2 ]+ e cFe [x Fe -x c x Fe ]+ e FeFe [-1/2 x 2 Fe ] e ccc [x 2 c -2/3 x c 3 ] +E CCFe [2X C X Fe -2X2 c X Fe ]E CFeFe [X Fe -2XC X 2 Fe ]+E FeFeFe [-2/3 X 3 Fe ] In γ Mn =e CC [-1/2 x2 c ]+ e CFe [-X C X Fe ]+ e FeFe [-1/2 X2 Fe ]+ e CC [-2/3 X 3 C ]+ e CCFe [ -2X 2 C X Fe ] +E CFeFe [-2X C X Fe ]+ E FeFeFe [-2/3X3 Fe ] where E cc =9.24-16 060/T, E ccc = -51.8+157 800/T, £ cFe =7.52 -7 250/T, E CCFe =-8.39 +16 190/T, E CFeFe = -9.93+12 790/T, e FeFe =0, and e FeFeFe =0.

Prediction Expressions of Component Activity Coefficients in Si-Based Melts

Prediction expressions of component activity coefficients of the molecular interaction volume model (MIVM) are suggested for making up for absence of Wagner interaction parameters in Si-based melts. Their effectiveness is verified in liquid Fe-Si-B system at 1823 K (1550 °C). In comparison with experimental data, all errors of ±22 to 56 pct and deviations of ±0.0132 to 0.0318 predicted by MIVM are smaller than those (±53 to 94 pct and ±0.0759 to 0.1010) by the unified interaction parameter formalism. This indicates that the former is better than the later in the system. Accordingly, some interested thermodynamic diagrams and parameters at 1687 K and 1823 K (1414 °C and 1550 °C) are predicted in liquid Si-Al-Fe system, for instance, \( \varepsilon_{\text{Al}}^{\text{Al}} = 2.318 \), \( \varepsilon_{\text{Fe}}^{\text{Fe}} = 4.297 \), and \( \varepsilon_{\text{Fe}}^{\text{Al}} = \varepsilon_{\text{Al}}^{\text{Fe}} = - 2.443 \) in the dilute solution at 1687 K (1414 °C). The method of MIVM is able to expand to Si-based multicomponent melt if its sub-binary activity data are available. The reliability of predicted results for the melt is closely dependent upon that of component activities or infinite dilute activity coefficients in its sub-binary systems.

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.

Eyring Functions of Activation for the Viscous Flow of Some Higher Alcohols in Cyclohexane Solutions

The viscosities of solutions of 1-octanol, 1-nonanol, and 1-decanol in cyclohexane up to concentrations of 1.56 molċkg−1 were determined in the temperature range from 293.15 to 333.15 K. Applying Eyring’s transition state theory, the Eyring functions of activation for the viscous flow of binary mixtures, i.e. the molar Gibbs free energy, entropy, and enthalpy were estimated. The correlation coefficients β Y (Y=G, H, or S) were derived and used to discuss the initial effect of the solute on the activation parameters for the viscous flow. From the partial molar thermodynamic functions of activation and the partial molar volumetric data of solute and solvent at infinite dilution, the viscosity coefficient B was determined. For the interpretation of the experimental data, the Unified Interaction Parameter Formalism was used.

Thermodynamic properties of alumina‐saturated iron‐carbon‐bismuth alloys

Using direct and indirect equilibration methods, the solubilities of iron in molten bismuth and those of bismuth in molten iron‐carbon alloys have been measured over the temperature range 1473‐1873 K. The Henrian behaviour of bismuth in molten ferrous alloys allows the calculation of its activity coefficient, which in turn facilitates modelling as a function of temperature and carbon content, using the unified interaction parameter formalism developed by Pelton and Bale. The modelling results generate the expression: .A model has also been developed expressing bismuth content in molten iron‐carbon alloys as a function of temperature and composition: .

Thermodynamics of the iron-carbon-zinc system

A two-zone isopiestic experimental technique was used to determine the solubility of zinc vapor in liquid and solid iron-carbon alloys as a function of zinc partial pressure (0.1 to 1 atm), carbon content (0 to 4.6 wt Pct), and temperature (1473 to 1873 K). The solubility of zinc at a given partial pressure decreases with both increasing temperature and carbon content in both liquid alloys and solid austenite; its activity in these solutions, and in pure δ-ferrite, deviates more positively from ideality than previous model-based predictions have suggested. The Bale-Pelton unified interaction parameter formalism was successfully applied to the results of liquid-alloy experiments, but the degree of experimental scatter in the austenite equilibrations was too great to allow its application in the calculation of solid-solution iron-carbon-zinc thermodynamic parameters. Using the available results, values were calculated for the equilibrium partition coefficientK zn in solidifying iron-carbon alloys as a function of alloy carbon content; the results suggest that significant segregation of zinc between solid and liquid phases is not likely.

Solid-solute phase equilibria in aqueous solution: VII. A re-interpretation of magnesian calcite stabilities

It is proposed to model magnesian calcites thermodynamically as dilute solid solutions of MgCO 3 in CaCO 3 according to the Unified Interaction Parameter Formalism ( Bale and Pelton, Met. Trans. A 21A, 1997–2002, 1990). In this case both stoichiometric saturation and precipitation data of synthetic magnesian calcites taken from literature can consistently be explained.

The unified interaction parameter formalism: Thermodynamic consistency and applications

In a recent article, we proposed modifications to the standard interaction parameter formalism. The modified formalism, known as the “Unified Interaction Parameter Formalism,” is discussed in the present article with respect to thermodynamic consistency at finite concentrations in binary, ternary, and multicomponent systems. A new method, which is independent of integration paths, is proposed to derive the equations of the formalism by differentiation of the integral Gibbs energy expression. It is shown that the formalism is thermodynamically exact in both dilute and nondilute composition regions. It is also shown that the formalism reduces to Wagner’s formalism at infinite dilution and to Darken’s quadratic formalism in dilute solutions. Examples are presented and methods are discussed for determining the parameters of the formalism from thermodynamic data.