Nitrogen is an element that plays a significant role in steel production [1]. In recent decades, special attention has been given to the production and application of high-nitrogen steels [2; 3], as well as to the behavior of nitrogen in various manufacturing processes [4 – 6]. The solubility of nitrogen in liquid steel is of primary importance [7 – 10]. Research in this area aims to improve the accuracy of solubility predictions and the potential for nitride formation.

Let us consider the thermodynamics of nitrogen solutions in liquid binary Fe – j alloys, where j represents nickel and chromium. The concentrations of elements in the Fe – j – N solution, expressed in mole fractions, will be denoted as cFe , cj и cN , respectively. In practical metallurgy, these concentrations are typically expressed as mass percentages and designated as [% Fe], [% j] and [% N].

We will proceed from the concept of the absolute [11] activity of nitrogen in the solution, denoted as aN . The rational activity coefficient will be represented as γN (γN = aN /cN ). The mass-percent activity coefficient of nitrogen will be denoted as fN (fN = aN /[% N]). The activity coefficients in an infinitely nitrogen-diluted solution (cN → 0; [% N] → 0) will be denoted as \(\gamma _{\rm{N}}^0\) and \(f_{\rm{N}}^0\), respectively. These coefficients will be normalized based on the condition: \(\gamma _{\rm{N}}^0\) → 1 as cFe → 1; \(f_{\rm{N}}^0\) → 1 as [% Fe] → 100.

The fundamental concept of the phenomenological thermodynamics of multicomponent dilute alloys was introduced by Wagner [12]. Wagner’s idea was applied to the calculation of nitrogen solubility in steel by Langenberg [13]. A significant development of Wagner’s approach was presented in [14]. According to this study, within a certain convergence interval, the following expansion holds

where Jn is the rational n-th order interaction coefficient.

A similar expansion can be written.

where \(\widetilde {{J_n}}\) is the mass-percent n-th order interaction coefficient.

For first- and second-order interaction coefficients, there exist more specific notations [12 – 14]: J1 = \(\varepsilon _{\rm{N}}^j\); J2 = \(\rho _{\rm{N}}^j\); \(\widetilde {{J_1}}\) =  \(e_{\rm{N}}^j\); \(\widetilde {{J_2}}\) = \[r_{\rm{N}}^j\], where \(\varepsilon _{\rm{N}}^j\) and \(e_{\rm{N}}^j\) are the Wagner and Langenberg interaction coefficients, respectively.

Next, we will limit our consideration to the condition of constant temperature T = const.

Under this condition, from the invariance of the differential of the logarithm of the activity coefficient, the following relationship was obtained [15]

where AFe is the atomic mass of iron, and Aj is the atomic mass of the alloying element j. Similarly, the following relationship was derived [14]

To measure the thermodynamic parameters of nitrogen interaction with element j in liquid steel, it is, in principle, sufficient to experimentally study the dependence of nitrogen solubility in a binary Fe – j alloy on the concentration of element j. The solubility of nitrogen in liquid iron was first measured in 1938 [16]. Shortly thereafter, research on nitrogen solubility in liquid binary iron alloys began [17]. The first-order thermodynamic interaction parameters of nitrogen with alloying elements were thoroughly investigated in studies [18 – 21]. The results of the main stage of experimental research on these parameters were summarized in a review article [22]. Such studies continue to this day [23].

Truncating the power series expansions (1) and (2) while retaining only the linear terms does not allow for an adequate description of the concentration dependence of nitrogen solubility in liquid binary iron alloys for a number of systems. Therefore, it is necessary to account for at least the quadratic terms as well. The technical capabilities of experimental methods allow for the reliable determination of second-order thermodynamic interaction parameters of nitrogen with an alloying metal only in specific cases. Therefore, a theoretical approach capable of predicting these values would be useful for assessing the reliability of the obtained experimental results.

In this study, we propose a simple model of nitrogen solutions in liquid Fe – j (j = Ni, Cr) alloys that allows expressing the second-order thermodynamic interaction coefficient \(\rho _{\rm{N}}^j\) in terms of the Wagner interaction coefficient \(\varepsilon _{\rm{N}}^j\). The theory is based on a lattice model of Fe – j solution. The model assumes a face-centered cubic (FCC) lattice, where the lattice sites are occupied by iron atoms and atoms of element j. Nitrogen atoms are located in octahedral interstices. A nitrogen atom interacts only with metal atoms occupying the neighboring lattice sites, and this interaction is pairwise. It is assumed that the energy of this interaction is independent of the alloy composition and temperature. The liquid solutions in the Fe – j system are considered ideal. Within the framework of the proposed theory, we express the logarithm of the nitrogen activity coefficient ln\(\gamma _{\rm{N}}^0\), in an infinitely nitrogen-diluted solution as a function of the concentration cj . In doing so, we use the result obtained in studies [24; 25]:

where δ is the number of FCC lattice sites surrounding an octahedral interstice (δ = 6).

Next, we use the logarithmic expansion

\[\ln (1 + x) = \sum\limits_{n = 1}^\infty  {{{( - 1)}^{n + 1}}\frac{{{x^n}}}{n}} .\]

It follows that

The radius of convergence for these expansions is equal to 1.

From expressions (5) and (6), we obtain

Thus, for the n-th order interaction coefficient of nitrogen with element j, we have

\[{J_n} = \frac{1}{n}\frac{{{{\left( {\varepsilon _{\rm{N}}^j} \right)}^n}}}{{{\delta ^{n - 1}}}}\]

or

\[{J_n} = \frac{{{6^{1 - n}}}}{n}{\left( {\varepsilon _{\rm{N}}^j} \right)^n}.\]

The radius of convergence of expansion (7) is \(\frac{6}{{\left| {\varepsilon _{\rm{N}}^j} \right|}}.\)

For the second-order interaction coefficient \(\rho _{\rm{N}}^j\) = J2 we obtain

As an experimental validation of equation (8), this study utilizes the results of nitrogen solubility measurements in liquid binary Fe – Ni and Fe – Cr alloys at a temperature of 1873 K and partial nitrogen pressures \({P_{{{\rm{N}}_2}}}\) up to 100 bar [26]. A comparison between the theoretical predictions and experimental data is presented in the Table. The results obtained in [26] appear to be more reliable than the data reported in [7 – 10].

From the Table, it follows that the theoretical calculations based on equation (8) show satisfactory agreement with the experimental data from [26].

Conclusions

A model-based theory of structure and interatomic interaction has been applied to nitrogen solutions in liquid alloys of the Fe – Ni and Fe – Cr systems.

Equation (8) has been derived, expressing the second-order thermodynamic interaction coefficients \(\rho _{\rm{N}}^{{\rm{Ni}}}\) and \(\rho _{\rm{N}}^{{\rm{Cr}}}\) in liquid steel in terms of the Wagner interaction coefficients \(\varepsilon _{\rm{N}}^{{\rm{Ni}}}\) and \(\varepsilon _{\rm{N}}^{{\rm{Cr}}}\) in liquid iron-based alloys. The equation is given by \(\rho _{\rm{N}}^j = \frac{1}{{12}}{\left( {\varepsilon _{\rm{N}}^j} \right)^2}\), where j = Ni, Cr.

Theoretical values of the second-order interaction coefficients in liquid steel at T = 1873 K were obtained: \(\rho _{\rm{N}}^{{\rm{Ni}}}\) = 0.56 and \(\rho _{\rm{N}}^{{\rm{Cr}}}\) = 8.67, which show satisfactory agreement with the experimental estimates: \(\rho _{\rm{N}}^{{\rm{Ni}}}\) = 0.8 and \(\rho _{\rm{N}}^{{\rm{Cr}}}\) = 6.3, as reported in [26].