Thermodynamics of the Fe-N and Fe-N-C Systems: The Fe-N and Fe-N-C Phase Diagrams Revisited

For the binary system Fe-N, the data as assembled in Reference 12[25,35,36,66‐85] was chosen according to the recommendations given there. The binary thermodynamic descriptions of the α and γ phase were taken from Reference 22; for the choice of parameters to be optimized, see Section IV. Thus, only equilibrium data including the phases γ′ or ε have been used, in particular the data for the α + γ′, γ + γ′, γ + ε, and γ′ + ε two-phase equilibria. The available data are compositions and activities at the phase boundaries. If instead of activities, the nitriding potential, a (technical) process parameter,[1] was given, the activities were calculated using the Gibbs-energy equations for various gas species given in Reference 86. Newer data for the γ\(\rightleftarrows \)α + γ′ invariant equilibrium and the α + γ′ two-phase equilibrium[13,87,88] and the γ + γ′ and γ + ε equilibria[15,87] were also included. As an additional information, the activity curves for γ′ from References 25 and 82 and for ε in References 17 and 89 were used. However, these so-called absorption isotherms obtained from gaseous nitriding of Fe specimens are affected by the establishment of steady states instead of true metastable equilibrium at the surface of the specimen at higher temperatures and N contents,[1,13,14] making it impossible to use all data above 823 K (550 °C). Already at lower temperatures, but high N contents (>30 at. pct), a steady state instead of an equilibrium prevails at the surface. Therefore, the affected data have not been used during the optimization. Furthermore, during the optimization process, agreement of the model with the activity data has been considered less important than agreement with the information on solid-solid equilibria.

Based on this experimental information, the parameters \({{^\circ}{G}}^{\upgamma ^\prime }_{\text{Fe:N}}\) and \({{^\circ}{G}}^{\upgamma ^\prime }_{\text{Fe:Va}}\) of γ′ and the parameters \({^{0}}{L}^{\upvarepsilon }_{\text{Fe:N,Va}}\) and \({^{1}}{L}^{\upvarepsilon }_{\text{Fe:N,Va}}\) of ε were optimized.

For the optimization process of the model parameters for the ternary Fe-N-C system, primarily recently published data was used. During the optimization, care was taken that the resulting invariant temperatures comply with the ranges as determined experimentally in References 15 and 53. The second source of data was the location of the phase boundaries at 853 K and 893 K (580 °C and 620 °C) as determined experimentally in Reference 15. The experimental information that in the considered C and N content ranges the off-diagonal components of the thermodynamic factor of ε are positive[51,58] was used as a constraint for the model of the ε phase. Additionally, the N-solubility data in C-containing γ from Reference 90 was used.

For the C content of γ′, no reliable equilibrium data is available. EPMA investigations on specimens produced for the investigations in References 51 through 53 and 58 showed C contents in γ′ which were always below 1 at. pct. Therefore, during the optimization care was taken that this level of C content was not exceeded considerably. In θ only trace amounts of N have been found at temperatures ≤1073 K (800 °C).[91] At lower temperatures, θ layers can be produced on Fe substrates by heat treatment in an atmosphere containing CO, NH₃, and H\(_2\).[92] During the treatment, the substrate is gradually saturated with N that has diffused through θ,[93] eventually leading to formation of an ε layer underneath the θ layer.[50] θ produced under these conditions has been investigated by atom probe tomography,[94] revealing, nevertheless, a maximum total impurity content of only 0.01 at. pct, also including N. This supports the above experimental results of Reference 91 and is in contrast with the prediction of N contents of >1 at. pct resulting from the thermodynamic description of Reference 22.

On the basis of these ternary experimental data, the binary parameters \({{^\circ}{G}}^{\upgamma ^\prime }_{\text{Fe:C}}\) of γ′ and \({^{0}}{L}^{\upvarepsilon }_{\text{Fe:C,Va}}\) and \({^{1}}{L}^{\upvarepsilon }_{\text{Fe:C,Va}}\) of ε, which are only relevant for the ternary system, and the ternary parameters \({^{0}}{L}^{\upgamma }_{\text{Fe:C,N}}\) of γ and \({^{0}}{L}^{\upvarepsilon }_{\text{Fe:C,N}}\) and \({^{1}}{L}^{\upvarepsilon }_{\text{Fe:C,N}}\) of ε were optimized.

The temperatures and the compositions of the phases at the invariant equilibria in the Fe-N system, as predicted by the thermodynamic description resulting from the present work, can be compared with the experimental data from References 12, 13, and 15 and the previous predictions from References 22 and 24 in Table III. The agreement of these features of the Fe-N phase diagram with the experimental data is comparably good for the new and old[22,24] thermodynamic descriptions.

The Fe-N phase diagram as calculated using the model parameters from the present work is shown in Figure 1. Various enlarged sections of the phase diagram are shown in Figure 2 to allow a more detailed comparison with both the experimental data[13,15,25,35,36,66,67,70‐76,78‐82,84,87,88] and the previous predictions from References 22 and 24.

The homogeneity range of α agrees well with the experimental data and with the homogeneity range resulting from the thermodynamic description of Reference 22 (see Figure 2(a)). The newer experimental data from Reference 13 are better described by the thermodynamic description from Reference 24. This description, however, shows an α + ε equilibrium below approximately 580 K (310 °C), a temperature at which the α + γ′ equilibrium is observed experimentally (see also below) and, therefore, disagrees with the experimental phase-boundary data at low temperatures (cf. Figure 2(a)).

The calculated phase boundaries of the γ single-phase field agree well with the few available experimental data, see Figure 2(b). Experimental data on the phase boundaries γ/γ + γ′ and γ/γ + ε is somewhat contradictory: recent investigations obtained by EPMA measurements on nitrided specimens[15] showed N contents of up to 1 at. pct less than given in older works.[67,70,74,83] For the phase boundary γ/γ + γ′, the predictions by the thermodynamic description from the present work lie between these values. For the phase boundary γ/γ + ε, the data from Reference 15 is described better than the data from References 67, 70, 74, and 83. Overall, a good representation of all data employed in the optimisation is given, also accounting for the error margins of the usually applied EPMA method to determine these phase-boundary compositions.

The γ′/γ′ + ε phase boundary agrees well with the experimental data from Reference 87, whereas the N content at the phase boundary α + γ′/γ′ is lower than indicated by most of the experimental data and by the phase boundary as calculated using the thermodynamic description from Reference 22, but still agrees within less than 0.1 at. pct (see Figure 2(c)). Better agreement could be achieved by introducing a more advanced model considering N disorder, see Section V–C.

The thermodynamic description from the present work reproduces the phase boundary γ′ + ε/ε significantly better than the previous descriptions[22,24] (see Figure 2(d)). The data point from Reference 66 and semi-quantitative investigations in Reference 97 suggest that the phase boundary γ′ + ε/ε might extend to lower N contents in the low-T range. In this low-T range, the agreement of the experimental data with the phase boundary from Reference 98, based on a thermodynamic description considering ordering of N on its sublattice, is better, which is also shown in Figure 2(d). The thermodynamic description of Reference 98, however, gives multiple expressions for the phase boundary in order to cover the whole temperature range (see the overlap of the two curves in Figure 2(d)). Moreover, in the high-T range, the agreement with the experimental data is poor and there is a maximum in the proposed phase boundary. This is thermodynamically only possible if the congruent transition ε \(\rightleftharpoons \) γ′ occurs at N contents as high as 21.2 at. pct, which is impossible according to the model from the present work (maximum N content of γ′ is 20 at. pct) and also incompatible with the prediction according to the model for γ′ from Reference 98.

A “potential phase diagram” using the activity of N (reference state N₂ gas at 1 × 10⁵ Pa and at the considered temperature) as a variable is shown in Figure 3(a), allowing comparison of the phase boundaries as calculated using the thermodynamic description from the present work with the respective phase boundaries as calculated using the datasets from References 22 and 24, the phase boundary γ′/ε as calculated using the expressions given in Reference 98, and the phase boundaries as indicated by the experimental data.[13,25,68,69,71,76,77,79,82,85] The same diagram using as a variable the often applied nitriding potentiala (technical process) parameter used for gaseous nitriding, which is a measure for the activity of N, but multiplied with \({\,^{\circ }}{p}^{1/2}\) (where \({\,^\circ }p \,=\, 1 \times 10{^5}\) Pa is the pressure of the reference state, to obtain a dimensionless variable[1]), is given in Figure 3(b). For the phase boundary α/γ′, the thermodynamic dataset of the present work describes the experimental data equally as well as the dataset published in Reference 22. For the γ range, the thermodynamic descriptions from the present work and from References 22 and 24 reproduce the experimental data well. However, the phase boundary γ′/ε as calculated using the thermodynamic dataset of the present work agrees significantly better with the experimental data than the phase boundaries resulting from the previous descriptions of References 22 and 24. The phase boundary γ′/ε is even better described with the expressions given in Reference 98. However, in that work direct least-squares fitting of the phase boundary was performed, yielding several expressions for different temperature ranges. The thermodynamics of both the binary and ternary γ′ and ε phases are discussed in a comparative manner in Sections V–C and V–D, respectively.

Including the thermodynamic description of the liquid Fe-N phase from Reference 21 peritectic melting of ε is predicted in the binary Fe-N system at 1654 K (1381 °C). A similar prediction is obtained from the thermodynamic descriptions of the Fe-N system of References 21 and 22. The thermodynamic descriptions from References 20 and 24, however, predict a congruent transition γ \(\rightleftharpoons \) ε. Since there is no corresponding experimental data available, no conclusion can be drawn which variant is correct. Nevertheless, the shape of the γ phase field as predicted by References 20 and 24 seems unrealistic. Thus, the thermodynamic description involving peritectic melting is the preferred one, at least until more experimental data is available.

At low temperatures, an α + ε equilibrium is predicted by the presently obtained thermodynamic description, which disappears upon heating at 443 K (170 °C, see the dotted lines in Figure 1). The same feature at low temperature is predicted using the thermodynamic descriptions from References 20, 22 and 24. The appearance of an α + ε equilibrium at low temperatures is not necessarily a modeling artifact: experimental investigations[99‐101] suggested that ε was in equilibrium with α at low temperatures [approximately ≤550 K (280 °C)]. Another work[97] excluded the possibility of the α + ε equilibrium in the binary Fe-N system at low temperatures and discussed the possible formation of the cubic α″-Fe₁₆N₂ nitride as an equilibrium phase, which was not included in the present assessment. At least at 623 K (350 °C) (and at higher temperatures), precipitation of γ′ from ε was still observed.[102] However, due to the very slow kinetics at those low temperatures it is difficult to reach a genuine equilibrium state and this prohibits to draw a final conclusion. Finally, as a fine point, recent ab initio calculations pertaining to 0 K (−273 °C)[103,104] indicate that a mechanical mixture of pure α-Fe and ε-Fe₃N with a gross N content of 20 at. pct has a lower enthalpy than pure γ′-Fe₄N, supporting the occurrence of an α + ε equilibrium at low temperature. It is noted that this point was not addressed specifically in these works.[103,104]

A Scheil reaction scheme[105‐107] illustrating the sequence of invariant reactions as resulting from the present thermodynamic description of the Fe-N-C system is shown in Figure 4. The temperatures of the invariant reactions in the Fe-N-C system as calculated using the thermodynamic description of the present work can be compared in Table IV with the corresponding experimental data and the predictions as obtained using the thermodynamic descriptions from References 22 and 24, which are the two thermodynamic descriptions giving the best agreement with experimentally determined invariant temperatures[15,53] according to the detailed discussion in Reference 53. Both Figure 4 and Table IV use the designations for the invariant reactions introduced in Reference 53. The possibilities for the sequence of invariant reactions in the system Fe-N-C below 853 K (580 °C) have been discussed in detail in Reference 53, offering two possibilities realized by the previous various thermodynamic descriptions for the Fe-N-C system.[20,22,24,55‐57] In the first case, upon cooling, the α + ε equilibrium is replaced by the γ′ + θ equilibrium via a single transition reaction U\(_2\), α + ε \(\rightleftharpoons \) γ′ + θ. In the second case, upon cooling, the γ′ + θ equilibrium first appears via the pseudo-binary eutectoid reaction e₄, ε \(\rightleftharpoons \) γ′ + θ, dividing the ε single phase field into two separate ε single phase fields. Subsequently, the second ε single phase field vanishes via the ternary eutectoid reaction E\(_2\), ε \(\rightleftharpoons \) α + γ′ + θ. The thermodynamic description of the present work reproduces the first sequence of invariant reactions (see Figure 4).

The temperature of the transitional reaction U\(_2\) as calculated using the dataset from the present work [839 K (566 °C)] is only slightly below the value of the temperature for this reaction as determined experimentally [842 ± 2 K (569 ± 2 °C)].[53] The here predicted temperatures of the U₁ and E₁ invariant reactions are within the boundaries determined experimentally for these reactions in Reference 15. The previous thermodynamic descriptions for the system Fe-N-C[20,22,24,54‐57] only describe a part of the invariant temperatures correctly and give significantly deviating values for other ones (see the discussion in Reference 53 and the examples in Table IV). In contrast, the new thermodynamic description predicts correctly the values of all invariant temperatures, as recently determined experimentally.

Isothermal sections of the Fe-N-C phase diagram as calculated using the thermodynamic dataset derived in the present work at the (technologically relevant) temperatures of 853 K and 893 K (580 °C and 620 °C) are shown in Figure 5 together with the phase boundaries as proposed in Reference 15 on the basis of EPMA investigations on nitrocarburizing Fe specimens. These experimental data have been used in the optimization process. It was not possible to obtain an even better fit of the phase boundaries without allowing the formation of a large miscibility gap in the ε phase. The agreement with the phase boundaries from Reference 15 at 853 K (580 °C) is significantly better than as obtained by the predictions from References 22 and 55 and comparable to the phase boundaries resulting from the prediction from Reference 24 as follows from Figure 8d in Reference 15. At the same temperature, according to the descriptions from References 56 and 57, the α + ε equilibrium is non-existent or just disappearing. Therefore, the phase boundaries resulting from these descriptions cannot be compared to experimental data or to the phase boundaries resulting from the description from the present work. Further experimental data points have been given in Figure 5(a): (i) the single data point from Reference 41 for the phase boundary α + ε/ε shows higher N and lower C contents than predicted here; (ii) the data points for the same phase boundary from Reference 46 (for 120 and 240 minutes), however, agree well with the calculations from the present work; (iii) recent experimental data measured at 853 K (580 °C) for either the α + ε/ε or the α + ε + θ/ε equilibrium[51] also agree well with the predictions from the present work.

At 893 K (620 °C), the agreement with the experimental data is also good, especially for the γ single phase field, with the deviations between the data from Reference 15 and the calculated phase boundaries being close to the accuracy of EPMA. There is a clear deviation at 893 K (620 °C) for the phase boundary γ′ + ε/ε. However, the phase boundary given there is an estimation which is compatible with EPMA data presented in the same work but not based on a thermodynamic model. In principle, the phase boundary as predicted by the thermodynamic description from the present work is compatible with the EPMA data from Reference 15, see their Figure 8(b). Thermodynamic calculations at the same temperature using the description from Reference 22 showed significantly higher N contents an lower C contents in ε for the equilibria with γ and especially θ than determined in Reference 15 and predicted by the thermodynamic description from the present work.

Both at 853 K and 893 K (580 °C and 620 °C), the phase boundaries of the γ′ phase show excellent agreement with those shown in Reference 15. The ternary γ′ single phase field given there is, however, not based on quantitative experimental data, but has only estimative character. The γ′ single phase field resulting from the model from the present work agrees well with the EPMA data mentioned in Section III–B.

The priority in the present work was to obtain a reasonable representation of the phase boundaries of γ at 893 K (620 °C) and to describe correctly the invariant temperatures. On this basis, the parameter \({^{0}}{L}^{\upgamma }_{\text{Fe:C,N}}\) was optimized. In order to obtain a better description of γ in the range of higher temperatures, a temperature dependence of \({^{0}}{L}^{\upgamma }_{\text{Fe:C,N}}\) could be introduced as soon as more experimental data is available. Even though systematically too low, the here determined prediction for the N solubility in carbon containing γ is already good; see the comparison of experimental data[90] and the values predicted by the thermodynamic description from the present work in Table V.

In the present work, a different approach for modeling the thermodynamics of γ′ than in Reference 22 has been used. Instead of setting \({{^\circ}{G}}^{\upgamma ^\prime }_{\text{Fe:Va}}\) equal to the Gibbs energy of γ-Fe and introducing interaction parameters, \({{^\circ}{G}}^{\upgamma ^\prime }_{\text{Fe:Va}}\) has been used as a model parameter. Its value should, therefore, not be interpreted as the Gibbs energy of a hypothetical compound. The physical meaning of the value of \({{^\circ}{G}}^{\upgamma ^\prime }_{\text{Fe:Va}}\)−4\({{^\circ}{G}}^{\upalpha }_{\text{Fe}}\) can be understood as the Gibbs energy of N-vacancy formation according to the formal reaction \((1-x)\) Fe₄N + 4x α-Fe \(\rightleftharpoons \) Fe₄N_1−x , as similarly described in References 6 and 7. As mentioned in Section IV, the parameter \({{^\circ}{G}}^{\upgamma ^\prime }_{\text{Fe:C}}\) was adjusted in a way that the solubility of C in γ′ agrees well with the experimental equilibrium values. Therefore, a physical interpretation is even more difficult. Ab initio calculations performed in Reference 108 indicated a positive enthalpy of formation of γ′-Fe₄C as it is the case in the present work.

Experimentally obtained data for the relationship of the activity of N and the N content of γ′,[25] for the binary Fe-N system, were discussed in detail in Reference 26. In that work modeling was performed using three different approaches and the results were compared with the thermodynamic description from Reference 22. The “Langmuir-type approach” in Reference 26 is identical to the model applied in the present work. The other two models (“Wagner–Schottky (WS) approach” and “Gorsky–Bragg–Williams (GBW) approach”) allow for (dis)order of N. In Figure 1 in Reference 26, a function characterizing the deviation of the thermodynamic data from the expected values according to a Langmuir-type model (yielding a constant value for this function), is plotted showing the good fit of the WS and GBW models to the experimental data from Reference 25. Thus, it was concluded[26] that a model allowing for disorder is needed in order to give a meaningful description of the thermodynamics of the γ′ phase, with the “WS approach” and the “GBW approach” giving equally meaningful descriptions.

Using the expressions for the Gibbs energy of γ′ in Reference 109 it can be shown that also the WS and GBW models, allowing for disorder of N, can be expressed in the compound energy formalism[59] using a sublattice model indicated by the formula unit Fe₄(N,Va)(N,Va)₃, i.e., as compared to the sublattice model applied in the present work with the formula unit Fe₄(N,Va), with a second interstitial sublattice, and ideal (WS) or regular (GBW) interactions. Following the conclusion from Reference 26, we then tried to use a model equivalent to the “WS approach” from Reference 26 in the binary Fe-N system. The optimization of the model parameters, however, gave unreasonable results with e.g., γ′ replacing the γ phase or no disorder in γ′ at all. This is caused by a lack of any direct experimental data quantifying disorder in γ′. Other approaches to the thermodynamics of the γ′ phase adopt the cluster variation method,[27‐29,110] but the experimental data do not allow to prefer one or the other model:

If realistic errors for the N-content determination in Reference 25 are assumed (approximately \(5\,{\text{pct}}\)), the resulting deviations from the experimental results of the model used in the present work can be ascribed to this experimental uncertainty. Therefore, and also because the solubility of C in γ′ is considered, it was decided to adopt the two-sublattice model in the present work, giving a reasonable agreement of the predicted activity curves and the data from References 25 and 82 (see Figure 6).

The values of the Gibbs energy of the end-members of ε, \({{^\circ}{G}}^{\upvarepsilon }_{\text{Fe:C}}\) and \({{^\circ}{G}}^{\upvarepsilon }_{\text{Fe:N}}\) have been taken from previous descriptions.[22,96] The enthalpies of formation following from these Gibbs-energy functions are compatible with recent ab initio data from Reference 111, predicting negative values close to the one following from the applied Gibbs-energy function for various ordering states of nitrides with the formula Fe\(_2\)N. The values for the corresponding carbides with the formula Fe\(_2\)C from Reference 111 are positive as it is the case for the Gibbs-energy function applied in the present work, whereas they are considerably smaller.

The relationship of the activity of N and the N content in ε for the binary Fe-N system as obtained from the present thermodynamic description is shown in Figure 7. The prediction agrees well with the (rather inaccurate) experimental data (errors in the range of 5 to 10 at. pct) from Reference 89, as follows from Figure 7. Considering the more accurate data from Reference 17, shown in Figure 7 as well, good agreement occurs in the region of low N content and low N activity; at higher N activities lower N contents are predicted than experimentally observed. It was not possible to reproduce these data better without losing the good agreement with the two- and three-phase equilibrium data in which ε participates. In the present work, the thermodynamics of the ε phase have been described focusing on correct description of the available solid-solid equilibrium data. The description of the activity of N in ε on the basis of the gas-solid equilibrium data from References 17 and 89 could be better described by using models more explicitly considering the state of order in ε than that in the present case.

Several approaches have been presented in the literature to describe the thermodynamic behavior of ε in the range of high N content, i.e., close to the maximum N content of \(x_{\text N}=1/3\). Descriptions on the basis of a long-range order, GBW model of ε have been presented in References 14, 30, and 31, similar to the approach mentioned above for γ′,[26] and descriptions on the basis of the cluster variation method have been presented in References 27 and 32. Finally, ordering in ε and also the equilibrium with orthorhombic \(\zeta \)-Fe\(_2\)N (not considered in the present work), which can formally be described as ordered ε, have been investigated recently by first-principles calculations.[33] Recognizing the large homogeneity ranges of N and C in ε and the necessary extension of the binary model into the ternary Fe-N-C system, it is reasonable to apply a sub-regular solution model for ε indicated by the formula unit Fe(C,N,Va)\(_{1/2}\), and therefore, handle all non-ideal (e.g., ordering) effects by introducing interaction parameters instead of applying a physically more meaningful, but disproportionally complex long-range order model. This approach has been adopted in the present work, thereby enabling successful description of the equilibria in both the binary Fe-N system and especially the ternary Fe-N-C system.

In the present work, the function for \({{^\circ}{G}}^{\upvarepsilon }_{\text{Fe:N}}\) has been taken from Reference 22. Comparison of the Gibbs energy of ε-Fe\(_2\)N as predicted by this function and by the thermodynamic description for ε from Reference 24, i.e., setting \(y_{\text{N}}^{\upvarepsilon }=1/2\) for the 1:1 model applied there, shows that the corresponding Gibbs-energy values are virtually identical over a large temperature range.

The relationships of the N and C activities and the N and C contents of ε at 843 K (570 °C) have been plotted in Figure 8 together with the experimental data from Reference 112. The agreement is very good for N. Compared to the experimental data, the predicted C contents are too low for high N activities; for low N activities, the predicted values are closer to the experimentally determined ones. Similar experimental investigations have been performed in Reference 113, but are not included here recognizing the application of technical steels, i.e., a high impurity content, for the specimens used in Reference 113.

A general trend visible in Figure 8 is the obvious mutual influence of N and C: an increase in the C activity leads to a decrease of the N content and vice versa. This can be discussed as follows:

Simultaneous interstitial diffusion of N and C in ε is governed by the thermodynamic factorwith the chemical potential of component i (=N,C), \(\mu _i\), being the proportional constant between the intrinsic diffusion coefficients and the corresponding self-diffusion coefficients.[51,58,114] It has been found that at both 823 K and 853 K (550 °C and 580 °C) the off-diagonal components of \(\vartheta _{ij}\) are positive.[51,58] This information has been used as a constraint during the optimization: the thermodynamic description from the present work results in positive values of the off-diagonal components of the thermodynamic factor over a large composition range of ε. Only at low N and C contents, not covered by the experimentally observable homogeneity ranges of ε, negative values of those off-diagonal components of \(\vartheta _{ij}\) occur.

The present thermodynamic description predicts that a small miscibility gap occurs in ε below approximately 855 K (582 °C) close to the line connecting Fe\(_2\)N and Fe\(_2\)C. No experimental data exist to (in)validate this result.