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Effect of Solid Solution Elements on Solubility Products of Carbides and Nitrides in Austenite: Thermodynamic Calculations

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Abstract

To provide guidance on composition design of steel through control of the fraction of carbide or nitride precipitates in austenization, the effects of solid solution elements on solubility products of titanium, niobium, and vanadium carbides and nitrides in austenite were investigated by thermodynamic calculations using the two sublattice model. Solubility products of the carbides and nitrides in pure austenite were calculated, and the results were highly consistent with findings of previous studies. Solubility products of the carbides and nitrides in austenite containing different solid solution elements were evaluated and, in general, were also consistent with the reports of previous studies. Solubility products of titanium, niobium, and vanadium carbides and nitrides in austenite containing solid solution elements, on the condition that the total alloy content is relatively small, are given as follows:

$$ \log^{\gamma } K_{\text{TiC}} = 3.77 - \frac{8339}{T} + \left( { -\, 0.002 + \frac{14.2}{T}} \right)\left[ {\text{wt pct Mn}} \right] + \left( { -\, 0.009 + \frac{84.6}{T}} \right)\left[ {\text{wt pct Ni}} \right] + \left( {0.019 + \frac{12.3}{T}} \right)\left[ {\text{wt pct Cr}} \right] + \left( { -\, 0.017 + \frac{52.1}{T}} \right)\left[ {\text{wt pct Mo}} \right] $$
$$ \log^{\gamma } K_{\text{NbC}} = 3.72 - \frac{8534}{T} + \left( {0.004 + \frac{3.0}{T}} \right)\left[ {\text{wt pct Mn}} \right] + \left( {0.013 + \frac{35.1}{T}} \right)\left[ {\text{wt pct Ni}} \right] + \left( {0.010 + \frac{41.2}{T}} \right)\left[ {\text{wt pct Cr}} \right] + \left( { -\, 0.013 + \frac{40.4}{T}} \right)\left[ {\text{wt pct Mo}} \right] $$
$$ \log^{\gamma } K_{\text{VC}} = 4.70 - \frac{7041}{T} + \left( { -\,0.004 + \frac{18.6}{T}} \right)\left[ {\text{wt pct Mn}} \right] + \left( { - \,0.004 + \frac{37.0}{T}} \right)\left[ {\text{wt pct Ni}} \right] + \left( {0.003 + \frac{52.8}{T}} \right)\left[ {\text{wt pct Cr}} \right] + \left( { - \,0.036 + \frac{63.4}{T}} \right)\left[ {\text{wt pct Mo}} \right] $$
$$ \log^{\gamma } K_{\text{TiN}} = 4.37 - \frac{15029}{T} + \left( {0.009 + \frac{36.5}{T}} \right)\left[ {\text{wt pct Mn}} \right] + \left( { -\, 0.014 + \frac{73.4}{T}} \right)\left[ {\text{wt pct Ni}} \right] + \left( {0.004 + \frac{122.9}{T}} \right)\left[ {\text{wt pct Cr}} \right] + \left( { -\, 0.002 + \frac{37.3}{T}} \right)\left[ {\text{wt pct Mo}} \right] $$
$$ \log^{\gamma } K_{\text{NbN}} = 3.94 - \frac{10036}{T} + \left( {0.015 + \frac{25.2}{T}} \right)\left[ {\text{wt pct Mn}} \right] + \left( {0.008 + \frac{23.9}{T}} \right)\left[ {\text{wt pct Ni}} \right] + \left( { - \,0.004 + \frac{151.8}{T}} \right)\left[ {\text{wt pct Cr}} \right] + \frac{34.1}{T}\left[ {\text{wt pct Mo}} \right] $$
$$ \log^{\gamma } K_{\text{VN}} = 3.33 - \frac{8278}{T} + \left( {0.007 + \frac{40.6}{T}} \right)\left[ {\text{wt pct Mn}} \right] + \left( { - \,0.009 + \frac{26.0}{T}} \right)\left[ {\text{wt pct Ni}} \right] + \left( { - \,0.012 + \frac{163.5}{T}} \right)\left[ {\text{wt pct Cr}} \right] + \left( { - \,0.023 + \frac{57.1}{T}} \right)\left[ {\text{wt pct Mo}} \right] $$

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Abbreviations

\( X_{i} ,X_{{i^{\prime}}} \) :

i-th and \( i^{\prime} \)-th (\( i^{\prime} < i \)) solid solution elements (not compound-formed elements) in austenite, the atoms of which occupy the first sublattice

\( M \) :

Binary compound-formed element, the atoms of which occupy the first sublattice

\( N \) :

Binary compound-formed element, the atoms of which occupy the second sublattice

\( {\text{Va}} \) :

Vacancy, which occupies the second sublattice

\( G^{\gamma } \) :

Gibbs energy of γ phase (J/mol)

\( ^{\text{mg}} G^{\gamma } \) :

Magnetic contribution to Gibbs energy of γ phase (J/mol)

\( j,j^{\prime} \) :

Element types in first sublattice (in the order of \( {\text{Fe}} \),\( X_{1} \), \( X_{2} \), \( \cdots \) and \( M \), where the sequence number of \( j^{\prime} \) is smaller than that of \( j \))

\( k,k^{\prime} \) :

Element types in second sublattice (in the order of \( N \) and \( {\text{Va}} \), where the sequence number of \( k^{\prime} \) is smaller than that of \( k \))

\( ^\circ G_{j:k}^{\text{fcc}} \) :

Gibbs energy of face-centered cubic (fcc) phase where the first and second sublattices are filled with j atoms and k atoms, respectively (J/mol)

\( \mu_{j:k}^{\text{fcc}} \) :

Chemical potential of fcc phase where the first and second sublattices are filled with j atoms and k atoms, respectively (J/mol)

\( \mu_{j}^{\text{fcc}} \) :

Chemical potential of j element in fcc phase

\( \mu_{k}^{\text{fcc}} \) :

Chemical potential of k element in fcc phase

\( L_{ }^{\text{fcc}} \) :

Interaction parameter of fcc phase (in the subscripts of \( L_{ }^{\text{fcc}} \), components in different sublattices are separated by a colon and components in the same sublattice are separated by a comma) (J/mol)

\( y_{j}^{\gamma } \) :

Site fraction of j atoms in the first sublattice in austenite

\( y_{k}^{\gamma } \) :

Site fraction of k atoms in the second sublattice in austenite

\( T \) :

Temperature (K)

\( R \) :

Universal gas constant (J/mol K)

\( x_{m}^{ } \) :

Mole fraction of m element in the system

\( \left[ {{\text{wt pct }}m} \right] \) :

Mass percentage of m element in austenite

\( A_{m} \) :

Relative atomic mass of m element

\( { \log }^{\gamma } K_{{MN}}^{0} \) :

Logarithmic solubility product of binary compound \( {{MN}} \) in pure austenite in \( {\text{Fe}} - {{M}} - {{N}} \) system

\( { \log }^{\gamma } K_{{MN}}^{{X_{i} }} \) :

Logarithmic solubility product of binary compound \( {{MN}} \) in austenite in \( {\text{Fe}} - X_{i} - {{M}} - {{N}} \) system

\( \Delta { \log }^{\gamma } K_{{MN}}^{{X_{i} }} \) :

Increment of logarithmic solubility product of binary compound \( {{MN}} \) in austenite upon addition of solid solution element \( X_{i} \)

\( { \log }^{\gamma } K_{{MN}} \) :

Logarithmic solubility product of binary compound \( {{MN}} \) in austenite in \( {\text{Fe}} - X_{1} - X_{2} - \ldots - {{M}} - {{N}} \) system

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Acknowledgments

This work was supported by a start-up fund for new researchers of Jiangxi University of Science and Technology (Grant No. jxxjbs18037), the department of Science and Technology of Jiangxi Province (Post-doctoral fund under Grant No. 3205700012 and High-level talent fund under Grant No. 3401223254), and National Natural Science Foundation of China (Grant No. 51871114 and 51804138). We thank the editing team from Liwen Bianji, Edanz Editing China, for editing the English text of this manuscript.

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Authors and Affiliations

  1. School of Materials Science and Engineering, Jiangxi University of Science and Technology, Ganzhou, 341000, People’s Republic of China

    Xuan-Wei Lei, Xue-Hui Zhang & Tong-Xiang Liang

  2. Bohai Shipyard Group Co., Ltd, Huludao, 125000, People’s Republic of China

    Da-Yong Li

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Manuscript submitted December 23, 2018.

Appendix

Appendix

See Tables AI and AII.

Table AI Thermodynamic Parameters of Gibbs Energies
Full size table
Table AII Thermodynamic Interaction Parameters
Full size table
$$ \frac{{\partial G^{\gamma } }}{{\partial y_{{{\text{Fe}}}}^{\gamma } }} = y_{{{N}}}^{\gamma } \,^{^\circ } G_{{{\text{Fe}}:{{N}}}}^{{{\text{fcc}}}} + y_{{{\text{Va}}}}^{\gamma } \,^{^\circ } G_{{{\text{Fe}}:{\text{Va}}}}^{{{\text{fcc}}}} + + ~RT\left( {{\text{ln}}y_{{{\text{Fe}}}}^{\gamma } + 1} \right) + y_{{{N}}}^{\gamma } y_{{{\text{Va}}}}^{\gamma } L_{{{\text{Fe}}:{{N}},{\text{Va}}}}^{{{\text{fcc}}}} + y_{{{M}}}^{\gamma } y_{{{N}}}^{\gamma } L_{{{\text{Fe}},{{M}}:{{N}}}}^{{{\text{fcc}}}} + y_{{{M}}}^{\gamma } y_{{{\text{Va}}}}^{\gamma } L_{{{\text{Fe}},{{M}}:{\text{Va}}}}^{{{\text{fcc}}}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } y_{N}^{\gamma } L_{{{\text{Fe}},X_{i} :N}}^{{{\text{fcc}}}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } y_{{{\text{Va}}}}^{\gamma } L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{{{\text{fcc}}}} $$
(A1a)
$$ \frac{{\partial G^{\gamma } }}{{\partial y_{{X_{i} }}^{\gamma } }} = y_{{N}}^{\gamma } \;^{\circ } G_{{X_{i} :{{N}}}}^{\text{fcc}} + y_{\text{Va}}^{\gamma } \;^{\circ } G_{{X_{i} :{\text{Va}}}}^{\text{fcc}} + RT\left( {\ln y_{{X_{i} }}^{\gamma } + 1} \right) + y_{{N}}^{\gamma } y_{\text{Va}}^{\gamma } L_{{X_{i} :{{N}},{\text{Va}}}}^{\text{fcc}} + y_{\text{Fe}}^{\gamma } y_{{N}}^{\gamma } L_{{{\text{Fe}},X_{i} :{{N}}}}^{\text{fcc}} + \mathop \sum \limits_{{i^{\prime}}} y_{{X_{{i^{\prime}}} }}^{\gamma } y_{{N}}^{\gamma } L_{{X_{{i^{\prime}}} ,X_{i} :{{N}}}}^{\text{fcc}} + y_{{M}}^{\gamma } y_{{N}}^{\gamma } L_{{X_{i} ,{{M}}:{{N}}}}^{\text{fcc}} + y_{\text{Fe}}^{\gamma } y_{\text{Va}}^{\gamma } L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{{i^{\prime}}} y_{{X_{{i^{\prime}}} }}^{\gamma } y_{\text{Va}}^{\gamma } L_{{X_{{i^{\prime}}} ,X_{i} :{\text{Va}}}}^{\text{fcc}} + y_{{M}}^{\gamma } y_{{N}}^{\gamma } L_{{X_{i} ,{{M}}:{\text{Va}}}}^{\text{fcc}} . $$
(A1b)
$$ \frac{{\partial G^{\gamma } }}{{\partial y_{{M}}^{\gamma } }} = y_{{N}}^{\gamma } \;^{\circ } G_{M:{{N}}}^{\text{fcc}} + y_{\text{Va}}^{\gamma } \;^{\circ } G_{{M:{\text{Va}}}}^{\text{fcc}} + RT\left( {{ \ln }y_{{M}}^{\gamma } + 1} \right) + y_{{N}}^{\gamma } y_{\text{Va}}^{\gamma } L_{{M:{{N}},{\text{Va}}}}^{\text{fcc}} + y_{\text{Fe}}^{\gamma } y_{{N}}^{\gamma } L_{{{\text{Fe}},M:{{N}}}}^{\text{fcc}} + y_{\text{Fe}}^{\gamma } y_{\text{Va}}^{\gamma } L_{{{\text{Fe}},M:{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } y_{{N}}^{\gamma } L_{{X_{i} ,M:{{N}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } y_{\text{Va}}^{\gamma } L_{{X_{i} ,M:{\text{Va}}}}^{\text{fcc}} . $$
(A1c)
$$ \frac{{\partial G^{\gamma } }}{{\partial y_{{N}}^{\gamma } }} = y_{\text{Fe}}^{\gamma } \;^{\circ } G_{{{\text{Fe}}:{{N}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } \;^{\circ } G_{{X_{i} :{{N}}}}^{\text{fcc}} + y_{{M}}^{\gamma } \;^{\circ } G_{M:{{N}}}^{\text{fcc}} + RT\left( {{ \ln }y_{{N}}^{\gamma } + 1} \right) + y_{\text{Fe}}^{\gamma } y_{\text{Va}}^{\gamma } L_{{{\text{Fe}}:{{N}},{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } y_{\text{Va}}^{\gamma } L_{{X_{i} :{{N}},{\text{Va}}}}^{\text{fcc}} + y_{{M}}^{\gamma } y_{\text{Va}}^{\gamma } L_{{M:{{N}},{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{\text{Fe}}^{\gamma } y_{{X_{i} }}^{\gamma } L_{{{\text{Fe}},X_{i} :{{N}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } \mathop \sum \limits_{{i^{\prime}}} y_{{X_{{i^{\prime}}} }}^{\gamma } L_{{X_{i} ,X_{{i^{\prime}}} :{{N}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } y_{{M}}^{\gamma } L_{{X_{i} ,M:{{N}}}}^{\text{fcc}} + y_{\text{Fe}}^{\gamma } y_{{M}}^{\gamma } L_{{{\text{Fe}},{{M}}:{{N}}}}^{\text{fcc}} $$
(A1d)
$$ \frac{{\partial G^{\gamma } }}{{\partial y_{\text{Va}}^{\gamma } }} = y_{\text{Fe}}^{\gamma } \;^{\circ } G_{{{\text{Fe}}:{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } \;^{\circ } G_{{X_{i} :{\text{Va}}}}^{\text{fcc}} + y_{{M}}^{\gamma } \;^{\circ } G_{{{{M}}:{\text{Va}}}}^{\text{fcc}} + RT\left( {{ \ln }y_{\text{Va}}^{\gamma } + 1} \right) + y_{\text{Fe}}^{\gamma } y_{{N}}^{\gamma } L_{{{\text{Fe}}:{{N}},{\text{Va}}}}^{\text{fcc}} + \mathop \sum \nolimits_{i} y_{{X_{i} }}^{\gamma } y_{{N}}^{\gamma } L_{{X_{i} :{{N}},{\text{Va}}}}^{\text{fcc}} + y_{{M}}^{\gamma } y_{{N}}^{\gamma } L_{{{{M}}:{{N}},{\text{Va}}}}^{\text{fcc}} + \mathop \sum \nolimits_{i} y_{\text{Fe}}^{\gamma } y_{{X_{i} }}^{\gamma } L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } \mathop \sum \nolimits_{{i^{\prime}}} y_{{X_{{i^{\prime}}} }}^{\gamma } L_{{X_{i} ,X_{{i^{\prime}}} :{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} y_{{X_{i} }}^{\gamma } y_{{M}}^{\gamma } L_{{X_{i} ,{{M}}:{\text{Va}}}}^{\text{fcc}} + \mathop \sum \nolimits_{i} y_{{X_{i} }}^{\gamma } y_{{M}}^{\gamma } L_{{X_{i} ,{{M}}:{\text{Va}}}}^{\text{fcc}} + y_{\text{Fe}}^{\gamma } y_{{M}}^{\gamma } L_{{{\text{Fe}},{{M}}:{\text{Va}}}}^{\text{fcc}} $$
(A1e)
$$ \begin{aligned} \log {}_{~}^{\gamma } K_{{{MN}}}^{~} - \log {}_{~}^{\gamma } K_{{{MN}}}^{0} = \left[{{\mathop \sum \nolimits_{i} x_{{X_{i} }}^{~} (^\circ G_{{{\text{Fe}}:{{N}}}}^{\text{fcc}} - ^\circ G_{{{\text{Fe}}:{\text{Va}}}}^{\text{fcc}} ) - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} (^\circ G_{{X_{i} :{{N}}}}^{\text{fcc}} - ^\circ G_{{X_{i} :{\text{Va}}}}^{\text{fcc}} ) + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{{\text{Fe}},{{M}}:{\text{Va}}}}^{\text{fcc}} - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{X_{i} ,{{M}}:{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} - \left( {\mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{X_{i} :{{N}},{\text{Va}}}}^{\text{fcc}} - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{{\text{Fe}}:{{N}},{\text{Va}}}}^{\text{fcc}} } \right) - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \left( {L_{{{\text{Fe}},X_{i} :{{N}}}}^{\text{fcc}} - L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} } \right) - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \left( {L_{{{\text{Fe}},X_{i} :{{N}}}}^{\text{fcc}} - L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} } \right)}}\right]/{{RT\ln 10}}+ \log \frac{{A_{{{\text{Fe}}}} ^{2} }}{{\left[ {A_{{{\text{Fe}}}} \left( {1 - \sum\nolimits_{i} {x_{{X_{i} }} } } \right) + \sum\nolimits_{i} {x_{{X_{i} }} A_{{X_{i} }} } } \right]^{2} }} = \left[ {\mathop \sum \limits_{i} x_{{X_{i} }}^{~} (^\circ G_{{{\text{Fe}}:{{N}}}}^{\text{fcc}} - ^\circ G_{{{\text{Fe}}:{\text{Va}}}}^{\text{fcc}} ) - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} (^\circ G_{{X_{i} :{{N}}}}^{\text{fcc}} - ^\circ G_{{X_{i} :{\text{Va}}}}^{\text{fcc}} ) + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{\text{Fe}},{M}:{\text{Va}}}^{\text{fcc}} - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{X_{i} ,{{M}:{\text{Va}}}}}^{\text{fcc}} + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} - \left( {\mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{X_{i} :{{N}},{\text{Va}}}}^{\text{fcc}} - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{{\text{Fe}}:{{N}},{\text{Va}}}}^{\text{fcc}} ) - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} (L_{{{\text{Fe}},X_{i} :{{N}}}}^{\text{fcc}} - L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} ) - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} x_{{X_{i} }}^{~} L_{{\text{Fe}},X_{i} :{\text{Va}}}^{\text{fcc}} + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} x_{{X_{i} }}^{~} (L_{{{\text{Fe}},X_{i} :{{N}}}}^{\text{fcc}} - L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} } \right)} \right]/RT\ln 10 + \mathop \sum \limits_{i} \log \frac{{A_{{\text{Fe}}} ^{2} }}{{\left[ {A_{{{\text{Fe}}}} \left( {1 - x_{{X_{i} }}^{~} } \right) + x_{{X_{i} }}^{~} A_{{X_{i} }} } \right]^{2} }} + \hfill \\ \left[ {\mathop \sum \limits_{i} x_{{X_{i} }}^{~} x_{{X_{i} }}^{~} L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} x_{{X_{i} }}^{~} (L_{{{\text{Fe}},X_{i} :{{N}}}}^{\text{fcc}} - L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} ) - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \left(L_{{{\text{Fe}},X_{i} :{{N}}}}^{\text{fcc}} - L_{{{\text{Fe}},X_{i} :{\text{Va}}}}^{\text{fcc}} \right)} \right]/RT\ln 10 - \log \frac{{A_{{{\text{Fe}}}} ^{{2i - 2}} \left[ {A_{{\text{Fe}}} \left( {1 - \mathop \sum \nolimits_{i} x_{{X_{i} }}^{~} } \right) + \mathop \sum \nolimits_{i} x_{{X_{i} }}^{~} A_{{X_{i} }} } \right]^{2} }}{{\mathop \prod \nolimits_{i} \left[ {A_{{\text{Fe}}} \left( {1 - x_{{X_{i} }}^{~} } \right) + x_{{X_{i} }}^{~} A_{{X_{i} }} } \right]^{2} }} = \Delta \log {}_{~}^{\gamma } K_{{MN}}^{{X_{i} }} + \Delta \log {}_{~}^{\gamma } K_{{MN}}^{{X_{2} }} + \cdots + \left[ {\mathop \sum \limits_{i} x_{{X_{i} }}^{~} x_{{X_{i} }}^{~} L_{{Fe,X_{i} :{\text{Va}}}}^{\text{fcc}} - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} x_{{X_{i} }}^{~} (L_{{Fe,X_{i} :{{N}}}}^{\text{fcc}} - L_{{Fe,X_{i} :{\text{Va}}}}^{\text{fcc}} ) - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \mathop \sum \limits_{i} x_{{X_{i} }}^{~} L_{{Fe,X_{i} :{\text{Va}}}}^{\text{fcc}} + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} \mathop \sum \limits_{i} x_{{X_{i} }}^{~} (L_{{Fe,X_{i} :{{N}}}}^{\text{fcc}} - L_{{Fe,X_{i} :{\text{Va}}}}^{\text{fcc}} )} \right]/RT\ln 10 - \log \frac{{A_{{Fe}} ^{{2i - 2}} \left[ {A_{{Fe}} \left( {1 - \mathop \sum \limits_{i} x_{{X_{i} }}^{~} } \right) + \mathop \sum \limits_{i} x_{{X_{i} }}^{~} A_{{X_{i} }} } \right]^{2} }}{{\mathop \prod \limits_{i} \left[ {A_{{Fe}} \left( {1 - x_{{X_{i} }}^{~} } \right) + x_{{X_{i} }}^{~} A_{{X_{i} }} } \right]^{2} }} \cong \Delta \log {}_{~}^{\gamma } K_{{MN}}^{{X_{i} }} + \Delta \log {}_{~}^{\gamma } K_{{MN}}^{{X_{2} }} + \cdots \hfill \\ \end{aligned} $$
(A2)

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Lei, XW., Li, DY., Zhang, XH. et al. Effect of Solid Solution Elements on Solubility Products of Carbides and Nitrides in Austenite: Thermodynamic Calculations. Metall Mater Trans A 50, 4445–4461 (2019). https://doi.org/10.1007/s11661-019-05295-w

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