Wagner interaction coefficient between nitrogen and cobalt in liquid steel

To predict the solubility of nitrogen in liquid steel, understanding the nitrogen solubility in liquid iron is crucial, along with at least obtaining the first order interaction coefficients between nitrogen and alloying elements. Typically, these coefficients are derived from experimental investigations of nitrogen solubility in the melts of binary metal systems like Fe – j, where iron serves as the solvent and j represents the alloying element. Nevertheless, values obtained through this method often encompass experimental uncertainties, occasionally of significant magnitude. This scenario also holds true for the interaction between nitrogen and cobalt.

Currently, cobalt finds diverse technological applications, including its use in alloying special steels known for their high-speed, magnetic, and heat-resistant properties. The significance of the Wagner interaction coefficient between nitrogen and cobalt in liquid steel lacks consensus. Hence, an intriguing avenue lies in investigating this matter from a theoretical perspective.

To delve into the thermodynamics of nitrogen solutions in the Fe – Co. system’s liquid alloys, we denote the concentrations of the components in molar fractions as c_Fe, c_Co and c_N. Alternatively, expressing these concentrations in mass percentages yields [% Fe], [% Co] and [% N]. Let a_N means present the thermodynamic activity of nitrogen in the solution, \({\gamma _{\rm{N}}} = \frac{{{a_{\rm{N}}}}}{{{c_{\rm{N}}}}} - \) the rational coefficient of nitrogen activity in the solution, \({f_{\rm{N}}} = \frac{{{a_{\rm{N}}}}}{{[\% {\rm{ N}}]}} - \) the mass-percentage coefficient of nitrogen activity. The thermodynamic first order interaction coefficients between nitrogen and cobalt in liquid iron-based alloys of the Fe – Co – N systems are determined by the following formulas

\[\begin{array}{c}\varepsilon _{\rm{N}}^{{\rm{Co}}} = \frac{{\partial \ln {\gamma _{\rm{N}}}}}{{\partial {c_{{\rm{Co}}}}}}{\rm{at}}{c_{{\rm{Fe}}}} \to 1;\\e_{\rm{N}}^{{\rm{Co}}} = \frac{{\partial \lg {f_{\rm{N}}}}}{{\partial [\% {\rm{ Co}}]}}{\rm{at}}[\% {\rm{ Fe}}] \to 100,\end{array}\]

where \(\varepsilon _{\rm{N}}^{{\rm{Co}}}\) is the Wagner interaction coefficient, while \(e_{\rm{N}}^{{\rm{Co}}}\) is the Langenberg interaction coefficient. A correlation between these parameters is presented in [1]:

where A_Fe and A_Co are the atomic masses of the corresponding elements.

The solubility of nitrogen in liquid alloys of the Fe – Co system, expressed in mass percentage, is denoted as [% N]\(^*\). At a partial pressure of nitrogen in the liquid phase P_N2 approaching zero P_N2 → 0, the square root law, also known as Sieverts law, applies:

\[{[\% {\rm{ N}}]^*} = K'\sqrt {\frac{{{P_{{{\rm{N}}_2}}}}}{{{P_0}}}} ,\]

where P₀ is the standard pressure (P₀ = 1 atm ≈ 0.101 MPa); K′ is the Sieverts law constant. Let K′ = K′(Fe) at c_Fe = 1 and K′ = K′(Co) at c_Co = 1.

Following the proposed simple theory regarding the thermodynamic properties of liquid nitrogen solutions in Fe – Co alloy systems, an alignment is observed with the theoretical framework governing nitrogen solutions in Fe – Cr and Ni – Cr alloy systems [2]. The abstract of this paper outlines the theoretical model. Utilizing the findings from [2], we arrive at the model presented below:

At a temperature of T = 1873 K K′(Fe) = 0.044 wt. % [3] and K′(Co) = 0.0047 wt. % [4]. With A_Fe = 55.847 and A_Co = 58.9332 applying formula (2) yields the theoretical Wagner interaction coefficient between nitrogen and cobalt in liquid steel at T = 1873 K as \(\varepsilon _{\rm{N}}^{{\rm{Co}}}\) = 1.8. Subsequently, equation (1) provides the corresponding value of the Langenberg interaction coefficient  \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.0076.

Consideration of experimental values of the \(e_{\rm{N}}^{{\rm{Co}}}\) coefficient in liquid steel at T = 1873 K reveals various findings. In [5], nitrogen solubility in Fe – Co alloys was studied by quenching samples to a concentration of [% Co] = 24 wt. %, resulting in an estimated \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.0072. Continuation of this study in [6] up to [% Co] = 100 wt. % produced an estimate for nitrogen solubility in liquid cobalt K′(Co) = 0.0044 wt. %, which closely aligns with the value used in this paper K′(Co) = 0.0047 wt. %.

In [7], an experimental value of \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.007 was reported.

Additionally, [8] investigated the nitrogen solubility in melts of the Fe – Co system using the Sieverts method, determining an experimental estimate of the interaction coefficient at 1873 K as \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.011.

Comparing these estimates of the Langenberg interaction coefficient at T = 1873 K: \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.0072 [5] and \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.007 [7] are closer to the theoretical estimate \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.0076 than the experimental one \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.011 [8]. Consequently, based on the theory presented in this paper, the estimates from [5] and [7] appear more plausible than the one described in [8].

Conclusions

Theoretical estimates for the thermodynamic first-order interaction coefficients between nitrogen and cobalt in liquid steel at T = 1873 K: \(\varepsilon _{\rm{N}}^{{\rm{Co}}}\) = 1.8; \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.0076.

The experimental estimates of the Langenberg interaction coefficient \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.0072 [5] and \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.007 [7] appear more credible or reliable compared to the estimate \(e_{\rm{N}}^{{\rm{Co}}}\) = 0.011 [8].