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

Abstract

The significance of local equilibria (LE) models in understanding phase transformations in multicomponent steels has been revisited and the necessity for a firsthand computational tool capable of estimating metastable ferrite–austenite and cementite-austenite phase boundaries along with possible LE modes has been realized. For this purpose, thermodynamically consistent Unified Interaction Parameter Formalism (UIPF) has been adopted owing to its computational simplicity. All the necessary parameters required for UIPF-based simulations in Fe–C–Mn system, have been evaluated in the present work. The UIPF-based framework has been shown to be a computationally simple yet an efficient tool for prediction of metastable phase boundaries at low temperatures and wide composition range in the Fe–C–Mn system, where most of the available commercial thermodynamic simulation packages have prediction limitation. The phase boundaries predicted using UIPF have been validated with those predicted by ThermoCalc®, wherever possible/available. Additionally, existing experimental data on growth rates, interlamellar spacing, and partitioning coefficients prevalent during pearlitic transformation in Fe–C–Mn systems have been used to validate the predictions by different LE models adopting UIPF, which in turn affirms the potential of UIPF in predicting thermodynamics and kinetics of phase transformation in Fe–C–Mn systems.

Appendices

Appendix A. Darken’s Quadratic Formalism

The expression for the logarithm of activity coefficients extended to ternary system by Darken using Quadratic formalism has been given by Equations [A1] through [A3] where ${N}_{2}$, ${N}_{3}$ refer to atom fractions of solutes 2 and 3, while ${\alpha }_{12}$, ${\mathrm{\alpha }}_{13}$, ${\mathrm{\alpha }}_{23}$ are the coefficients of the Quadratic formalism.

$$\mathrm{log}{\upgamma }_{1}={\mathrm{\alpha }}_{12}{\mathrm{N}}_{2}^{2}+{\mathrm{\alpha }}_{13}{\mathrm{N}}_{3}^{2}+\left({\mathrm{\alpha }}_{12}+{\mathrm{\alpha }}_{13}-{\mathrm{\alpha }}_{23}\right){\mathrm{N}}_{2}{\mathrm{N}}_{3}$$

(A1)

$$\mathrm{log}\left(\frac{{\gamma }_{2}}{{\gamma }_{2}^{0}}\right)=-2{\alpha }_{12}{N}_{2}+\left({\alpha }_{23}-{\alpha }_{12}-{\alpha }_{13}\right){N}_{3}+{\alpha }_{12}{N}_{2}^{2}+{\alpha }_{13}{N}_{3}^{2}+\left({\alpha }_{12}+{\alpha }_{13}-{\alpha }_{23}\right){N}_{2}{N}_{3}$$

(A2)

$$\mathrm{log}\left(\frac{{\gamma }_{3}}{{\gamma }_{3}^{0}}\right)=-2{\alpha }_{13}{N}_{3}+\left({\alpha }_{23}-{\alpha }_{12}-{\alpha }_{13}\right){N}_{2}+{\alpha }_{12}{N}_{2}^{2}+{\alpha }_{13}{N}_{3}^{2}+\left({\alpha }_{12}+{\alpha }_{13}-{\alpha }_{23}\right){N}_{2}{N}_{3}$$

(A3)

Appendix B. Comparison of Model by Lupis and Elliot

Lupis and Elliot expressed the logarithm of activity coefficient of solute and solvent in the form of Taylor series expansion about the point of infinite dilution. The expression in the form of Taylor series expansion sets forth the assumption that all the derivatives of $ln\gamma$ are finite resulting in an implied assumption that the solute follows Henry’s Law. The Taylor series expansion for $ln{\gamma }_{1}$, $ln{\gamma }_{2}$ and $ln{\gamma }_{3}$ have been given by the Equations [B1] through [B3]. The superscript and subscript of the coefficient ‘${J}_{i,j}^{k}$’ represents the solute under consideration (i.e. k) and the order of differential of its logarithm of activity coefficient with respect to solutes (i and j) respectively.

$$ln{\gamma }_{1}={J}_{\mathrm{1,0}}^{(1)}{x}_{2}+{J}_{\mathrm{0,1}}^{(1)}{x}_{3}+{J}_{\mathrm{2,0}}^{(1)}{x}_{2}^{2}+{J}_{\mathrm{1,1}}^{(1)}{x}_{2}{x}_{3}+{J}_{\mathrm{0,2}}^{(1)}{x}_{3}^{2}+ \cdots $$

(B1)

$$ln{\gamma }_{2}={J}_{\mathrm{0,0}}^{(2)}+{J}_{\mathrm{1,0}}^{(2)}{x}_{2}+{J}_{\mathrm{0,1}}^{(2)}{x}_{3}+{J}_{\mathrm{2,0}}^{(2)}{x}_{2}^{2}+{J}_{\mathrm{1,1}}^{(2)}{x}_{2}{x}_{3}+{J}_{\mathrm{0,2}}^{(2)}{x}_{3}^{2}+ \cdots $$

(B2)

$$ln{\gamma }_{3}={J}_{\mathrm{0,0}}^{(3)}+{J}_{\mathrm{1,0}}^{(3)}{x}_{2}+{J}_{\mathrm{0,1}}^{(3)}{x}_{3}+{J}_{\mathrm{2,0}}^{(3)}{x}_{2}^{2}+{J}_{\mathrm{1,1}}^{(3)}{x}_{2}{x}_{3}+{J}_{\mathrm{0,2}}^{(3)}{x}_{3}^{2}+ \cdots $$

(B3)

The ${J}_{\mathrm{1,0}}^{(1)}$, ${J}_{\mathrm{1,0}}^{(2)}$ and ${J}_{\mathrm{1,0}}^{(3)}$ terms represent first order differential of logarithm of solvent (1), solute (2) and (3) with respect to solute (2) respectively as given in Equations [B4] through [B6]. Similarly ${J}_{\mathrm{0,1}}^{(1)}$, ${J}_{\mathrm{0,1}}^{(2)}$ and ${J}_{\mathrm{0,1}}^{(3)}$ represent first order differential of logarithm of solvent (1), solute (2) and (3) with respect to solute (3) respectively as given in Equations [B7] through [B9].

$$J_{1,0}^{\left( 1 \right)} = \left( {\frac{{\partial \ln \gamma_{1} }}{{\partial x_{2} }}} \right)_{{x_{1} \to 1}}$$

(B4)

$$J_{1,0}^{\left( 2 \right)} = \left( {\frac{{\partial \ln \gamma_{2} }}{{\partial x_{2} }}} \right)_{{x_{1} \to 1}} = \varepsilon_{2}^{2}$$

(B5)

$$J_{1,0}^{\left( 3 \right)} = \left( {\frac{{\partial \ln \gamma_{3} }}{{\partial x_{2} }}} \right)_{{x_{1} \to 1}} = \varepsilon_{3}^{2}$$

(B6)

$$J_{0,1}^{\left( 1 \right)} x_{3} = \left( {\frac{{\partial \ln \gamma_{1} }}{{\partial x_{3} }}} \right)_{{x_{1} \to 1}}$$

(B7)

$$J_{0,1}^{\left( 2 \right)} = \left( {\frac{{\partial \ln \gamma_{2} }}{{\partial x_{3} }}} \right)_{{x_{1} \to 1}} = \varepsilon_{2}^{3}$$

(B8)

$$J_{0,1}^{\left( 3 \right)} = \left( {\frac{{\partial \ln \gamma_{3} }}{{\partial x_{3} }}} \right)_{{x_{1} \to 1}} = \varepsilon_{3}^{3}$$

(B9)

The ${J}_{\mathrm{2,0}}^{(1)}$, ${J}_{\mathrm{2,0}}^{(2)}$ and ${J}_{\mathrm{2,0}}^{(3)}$ terms represent the second order differential of logarithm of activity coefficient of solvent (1), solute (2) and (3) with respect to solute (2) respectively as given in Equations [B10] through [B12]. Similarly ${J}_{\mathrm{1,1}}^{(1)}$, ${J}_{\mathrm{1,1}}^{(2)}$ and ${J}_{\mathrm{1,1}}^{(3)}$ represent the second order differential of logarithm of activity coefficient of solvent (1), solute (2) and (3) with respect to solute (2) and (3) respectively, as given in Equations [B13] through [B15]. Furthermore ${J}_{\mathrm{0,2}}^{(1)}$, ${J}_{\mathrm{0,2}}^{(2)}$ and ${J}_{\mathrm{0,2}}^{(3)}$ represent the second order differential of logarithm of activity coefficient of solvent (1), solute (2) and (3) with respect to solute (3) respectively as given in Equations [B16] through [B18].

$$J_{2,0}^{\left( 1 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{1} }}{{\partial x_{2}^{2} }}} \right)_{{x_{1} \to 1}}$$

(B10)

$$J_{2,0}^{\left( 2 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{2} }}{{\partial x_{2}^{2} }}} \right)_{{x_{1} \to 1}} = \rho_{2}^{2}$$

(B11)

$$J_{2,0}^{\left( 3 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{3} }}{{\partial x_{2}^{2} }}} \right)_{{x_{1} \to 1}} = \rho_{3}^{2}$$

(B12)

$$J_{1,1}^{\left( 1 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{1} }}{{\partial x_{2} \partial x_{3} }}} \right)_{{x_{1} \to 1}}$$

(B13)

$$J_{1,1}^{\left( 2 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{2} }}{{\partial x_{2} \partial x_{3} }}} \right)_{{x_{1} \to 1}} = \rho_{2}^{2,3}$$

(B14)

$$J_{1,1}^{\left( 3 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{3} }}{{\partial x_{2} \partial x_{3} }}} \right)_{{x_{1} \to 1}} = \rho_{3}^{2,3}$$

(B15)

$$J_{0,2}^{\left( 1 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{1} }}{{\partial x_{3}^{2} }}} \right)_{{x_{1} \to 1}}$$

(B16)

$$J_{0,2}^{\left( 2 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{2} }}{{\partial x_{3}^{2} }}} \right)_{{x_{1} \to 1}} = \rho_{2}^{3}$$

(B17)

$$J_{0,2}^{\left( 3 \right)} = \left( {\frac{{\partial^{2} \ln \gamma_{3} }}{{\partial x_{3}^{2} }}} \right)_{{x_{1} \to 1}} = \rho_{3}^{3}$$

(B18)

The coefficient ${J}_{\mathrm{0,0}}^{(2)}$ and ${J}_{\mathrm{0,0}}^{(3)}$ in Equations [B2] and [B3] are the value of $ln{\gamma }_{2}$ and $ln{\gamma }_{3}$ at infinite dilution of all solutes, represented by $ln{\gamma }_{2}^{0}$ and $ln{\gamma }_{3}^{0}$ respectively (with respect to Raoultian standard state). Further, replacing the ${J}_{i,j}^{k}$ terms in Equations [B2] and [B3] with the interaction parameters (${\varepsilon }_{2}^{2}$,${\varepsilon }_{2}^{3}$,${\varepsilon }_{3}^{2}$, ${\varepsilon }_{3}^{3}{\rho }_{2}^{2}$,${\rho }_{2}^{\mathrm{2,3}}$,${\rho }_{2}^{3}$,${\rho }_{3}^{2}$, ${\rho }_{3}^{3}$ and ${\rho }_{3}^{\mathrm{2,3}}$) lead to new set of Equations given by Equations [B19] and [B20].

$$ln{\gamma }_{2}={ln{\gamma }_{2}^{0}+\varepsilon }_{2}^{2}{x}_{2}+{\varepsilon }_{2}^{3}{x}_{3}+ {\rho }_{2}^{2}{x}_{2}^{2}+{\rho }_{2}^{\mathrm{2,3}}{x}_{2}{x}_{3}+{\rho }_{2}^{3}{x}_{3}^{2}+ \ldots \ldots \ldots \ldots .$$

(B19)

$$ln{\gamma }_{3}={ln{\gamma }_{3}^{0}+\varepsilon }_{3}^{3}{x}_{3}+{\varepsilon }_{3}^{2}{x}_{2}+{\rho }_{3}^{2}{x}_{2}^{2}+{\rho }_{3}^{\mathrm{2,3}}{x}_{2}{x}_{3}+{\rho }_{3}^{3}{x}_{3}^{2}+ \ldots \ldots \ldots \ldots \ldots$$

(B20)

Substituting the expression of $ln{\gamma }_{1}$, $ln{\gamma }_{2}$ and $ln{\gamma }_{3}$ given in Equations [B19] and [B20] in the Gibbs Duhem relation given by Equations [B21] and [B22] yields Equations [B23] and [B24] which can further be simplified resulting in Equations [B25] and [B26]. The term ${J}_{\mathrm{0,0}}^{(1)}$ in the expansion of $ln{\gamma }_{1}$ is equal to zero (${\left(ln{\gamma }_{1}\right)}_{{x}_{1\to 1}}=0)$ as the Raoultian standard state has been considered for solvent.

$$\left(1-{x}_{2}-{x}_{3}\right)\frac{\partial {\overline{G} }_{1}^{ex}}{\partial {x}_{2}}+ {x}_{2}\frac{\partial {\overline{G} }_{2}^{ex}}{\partial {x}_{2}}+ {x}_{3}\frac{\partial {\overline{G} }_{3}^{ex}}{\partial {x}_{2}}=0$$

(B21)

$$\left(1-{x}_{2}-{x}_{3}\right)\frac{\partial {\overline{G} }_{1}^{ex}}{\partial {x}_{3}}+ {x}_{2}\frac{\partial {\overline{G} }_{2}^{ex}}{\partial {x}_{3}}+ {x}_{3}\frac{\partial {\overline{G} }_{3}^{ex}}{\partial {x}_{3}}=0$$

(B22)

$$\left(1-{x}_{2}-{x}_{3}\right)\left({J}_{\mathrm{1,0}}^{(1)}+{2J}_{\mathrm{2,0}}^{(1)}{x}_{2}+{J}_{\mathrm{1,1}}^{(1)}{x}_{3}+ \cdots \cdots .\right)+{x}_{2}\left({\varepsilon }_{2}^{2}+2{\rho }_{2}^{2}{x}_{2}+{\rho }_{2}^{\mathrm{2,3}}{x}_{3}+ \cdots \cdots .\right)+{x}_{3}\left({\varepsilon }_{3}^{2}+{2\rho }_{3}^{2}{x}_{2}+{\rho }_{3}^{\mathrm{2,3}}{x}_{3}+ \ldots \ldots.\right)=0$$

(B23)

$$\left(1-{x}_{2}-{x}_{3}\right)\left({J}_{\mathrm{0,1}}^{(1)}+{2J}_{\mathrm{0,2}}^{(1)}{x}_{3}+{J}_{\mathrm{1,1}}^{(1)}{x}_{2}+ \cdots \cdots .\right)+{x}_{2}\left({\varepsilon }_{2}^{3}+{\rho }_{2}^{\mathrm{2,3}}{x}_{2}+2{\rho }_{2}^{3}{x}_{3}+ \cdots \cdots .\right)+{x}_{3}\left({\varepsilon }_{3}^{3}+{\rho }_{3}^{\mathrm{2,3}}{x}_{2}+2{\rho }_{3}^{3}+{x}_{3}+ \ldots \ldots .\right)=0$$

(B24)

$$J_{1,0}^{\left( 1 \right)} + x_{2} \left( {2J_{2,0}^{\left( 1 \right)} - J_{1,0}^{\left( 1 \right)} + \varepsilon_{2}^{2} } \right) + x_{3} \left( {J_{1,1}^{\left( 1 \right)} - J_{1,0}^{\left( 1 \right)} + \varepsilon_{3}^{2} } \right) + \ldots \ldots . = 0$$

(B25)

$$J_{0,1}^{\left( 1 \right)} + x_{2} \left( {J_{1,1}^{\left( 1 \right)} - J_{1,0}^{\left( 1 \right)} + \varepsilon_{2}^{3} } \right) + x_{3} \left( {2J_{0,2}^{\left( 1 \right)} - J_{0,1}^{\left( 1 \right)} + \varepsilon_{3}^{3} } \right) + \ldots \ldots . = 0$$

(B26)

The Equations [B25] and [B26] are of the form

$$A+B{x}_{2}+C{x}_{3}+D{x}_{2}^{2}\cdots ..=0$$

(B27)

For any value of ${x}_{2}$ and ${x}_{3}$ the expression in B27 will yield a positive value. The only way for the expression in B27 to be zero is that the coefficients A, B, C, D etc. should all be individually zero.

The first order and the second order terms in the expansion of $ln{\gamma }_{1}$ are obtained only by equating the coefficients A, B and C with zero which yields relations given by Equations [B28] through [B33]. Relation in Equation [B28] implies that the Raoults law is obeyed for the binary system 1– 2.

$$J_{1,0}^{\left( 1 \right)} = 0 = J_{0,1}^{\left( 1 \right)}$$

(B28)

$$J_{2,0}^{\left( 1 \right)} = - \frac{1}{2}\varepsilon_{2}^{2}$$

(B29)

$$J_{1,1}^{\left( 1 \right)} = - \varepsilon_{2}^{3}$$

(B30)

$$J_{1,1}^{\left( 1 \right)} = - \varepsilon_{3}^{2}$$

(B31)

$$\varepsilon_{3}^{2} = \varepsilon_{2}^{3}$$

(B32)

$$J_{0,2}^{\left( 1 \right)} = - \frac{1}{2}\varepsilon_{3}^{3}$$

(B33)

On substituting the values of ${J}_{\mathrm{0,0}}^{(1)}$, ${J}_{\mathrm{1,0}}^{(1)}$, ${J}_{\mathrm{2,0}}^{(1)}$, ${J}_{\mathrm{1,1}}^{(1)}$ and ${J}_{\mathrm{0,2}}^{(1)}$ in equation [B1] the expression for $ln{\gamma }_{1}$ can be written as shown in Equation [B34].

$${\mathrm{ln}}{\gamma }_{1}=-\frac{1}{2}{{\varepsilon }_{2}^{2}x}_{2}^{2}-{\varepsilon }_{2}^{3}{x}_{2}{x}_{3}-\frac{1}{2}{{\varepsilon }_{3}^{3}x}_{3}^{2}+ \cdots ..$$

(B34)

Once the series expression for ${\mathrm{ln}}{\gamma }_{1}$ in terms of ${\varepsilon }_{i}^{j}\mathrm{s}$ is obtained, overall excess Gibbs energy of a phase in a ternary system can be evaluated by adopting the expression for integral molar excess Gibbs energy, ${G}_{\text{m}}^{xs}$ as shown in Equation [B35].

$$\frac{{G}_{\text{m}}^{xs}}{RT}=\left(1-{x}_{2}-{x}_{3}\right){\mathrm{ln}}{\gamma }_{1}+{x}_{2}{\mathrm{ln}}{\gamma }_{2}+ {x}_{3}{\mathrm{ln}}{\gamma }_{3}$$

(B35)

Substituting the expression of $ln{\gamma }_{1}$, $ln{\gamma }_{2}$ and $ln{\gamma }_{3}$ in Eq. [B35] yields Eq. [B36].

$$\frac{{G}_{\text{m}}^{xs}}{RT}={x}_{2}ln{\gamma }_{2}^{0}+{x}_{3}ln{\gamma }_{3}^{0}+\frac{1}{2}{{\varepsilon }_{2}^{2}x}_{2}^{2}+\frac{1}{2}{{\varepsilon }_{3}^{3}x}_{3}^{2}+{\varepsilon }_{2}^{3}{x}_{2}{x}_{3}\cdots \cdots \cdots$$

(B36)

Equations [B37] through [B39] represent the relationship of partial molar excess Gibbs energy of each component with the integral molar excess Gibbs energy in a ternary system. Adopting Equations [B37] through [B39], expression for $ln{\gamma }_{1}$, $ln{\gamma }_{2}$ and $ln{\gamma }_{3}$ can be evaluated using the second order truncated expansion of integral molar excess Gibbs energy as given by Equation [B36]. These expressions for $ln{\gamma }_{1}$, $ln{\gamma }_{2}$ and $ln{\gamma }_{3}$ are given by Equations [B40] through [B42], which are similar to the ones given by Pelton and Bale for UIPF.

$$\overline{G}_{1}^{xs} = RT\ln \gamma_{1} = G_{\text{m}}^{xs} - x_{2} \frac{{\partial G^{xs} }}{{\partial x_{2} }} - x_{3} \frac{{\partial G^{xs} }}{{\partial x_{3} }}$$

(B37)

$$\overline{G}_{2}^{xs} = RT\ln \gamma_{2} = G_{\text{m}}^{xs} + (1 - x_{2)} \frac{{\partial G^{xs} }}{{\partial x_{2} }} - x_{3} \frac{{\partial G^{xs} }}{{\partial x_{3} }}$$

(B38)

$$\overline{G}_{3}^{xs} = RT\ln \gamma_{3} = G_{\text{m}}^{xs} - x_{2} \frac{{\partial G^{xs} }}{{\partial x_{2} }} + \left( {1 - x_{3} } \right)\frac{{\partial G^{xs} }}{{\partial x_{3} }}$$

(B39)

$$ln{\gamma }_{1}=-\frac{1}{2}{{\varepsilon }_{2}^{2}x}_{2}^{2}-{\varepsilon }_{2}^{3}{x}_{2}{x}_{3}-\frac{1}{2}{{\varepsilon }_{3}^{3}x}_{3}^{2}$$

(B40)

$${\mathrm{ln}}{\gamma }_{2}={ln{\gamma }_{2}^{0}+\varepsilon }_{2}^{2}{x}_{2}+{\varepsilon }_{2}^{3}{x}_{3}-\frac{1}{2}{{\varepsilon }_{2}^{2}x}_{2}^{2}-{\varepsilon }_{2}^{3}{x}_{2}{x}_{3}-\frac{1}{2}{{\varepsilon }_{3}^{3}x}_{3}^{2}$$

(B41)

$${\mathrm{ln}}{\gamma }_{3}={ln{\gamma }_{3}^{0}+\varepsilon }_{2}^{3}{x}_{2}+{\varepsilon }_{3}^{3}{x}_{3}-\frac{1}{2}{{\varepsilon }_{2}^{2}x}_{2}^{2}-{\varepsilon }_{2}^{3}{x}_{2}{x}_{3}-\frac{1}{2}{{\varepsilon }_{3}^{3}x}_{3}^{2}$$

(B42)

On comparing Equations [B39] and [B40] with Equations [B19] and [B20] respectively, yields the relations given in Equations [B43] through [B45].

$${\rho }_{2}^{2}=-\frac{1}{2}{\varepsilon }_{2}^{2}$$

(B43)

$${\rho }_{3}^{\mathrm{2,3}}=-{\varepsilon }_{2}^{3}={\rho }_{2}^{\mathrm{2,3}}$$

(B44)

$${\rho }_{3}^{3}=-\frac{1}{2}{\varepsilon }_{3}^{3}$$

(B45)

Appendix C. The Compositions Across the Selected Tie–Lines for Austenite-Cementite and Austenite-Ferrite Equilibria Considering PLE and NPLE Mode for Fe–0.8C–1.08Mn and Fe–0.69C–1.8Mn Systems Respectively

See Tables CI, CII, CIII.

Table CI Equilibrium Composition Across tie Lines for Fe–0.8C–1.08Mn Considering PLE

Table CII Equilibrium Composition Across tie Lines for Fe–0.69C–1.8Mn Considering PLE

Table CIII Equilibrium Composition for Fe–0.8C–1.08Mn Considering NPLE