Interaction between vacancies and interstitial solutes (C, N, and O) in $\alpha \text{−}\mathrm{Fe}$: From electronic structure to thermodynamics

The interplay between vacancies ($V$) and interstitial solutes $X$ ($X=\text{C}$, N, and O) and its impact on thermodynamic properties of $\alpha \text{−}\mathrm{Fe}$ solid solutions are studied, starting from first principles calculations. A systematic comparison between the three solutes is performed, investigating $X–\mathrm{Fe}$, $X$–$X$, and $V$–$X$ interactions. In the $\alpha \text{−}\mathrm{Fe}$ lattice, the strength of $X–\mathrm{Fe}$ interactions is found to govern the dissolution properties. Next to vacancies, the competition between solute volume effects and $X–\mathrm{Fe}$ interactions results in the preference of all the solutes to occupy off-centered sites. Low-energy configurations of small $V_n{X}_m$ clusters are calculated for $n$ and $m$ up to 4. They are used to parametrize lattice interaction models at the atomic scale. A detailed analysis of the cluster properties suggests the relevance of many-body terms in these models. The accuracy of the resulting models is verified through their satisfactory prediction of vacancy-solute cluster properties beyond the fitting database. From these models, an entire set of $V_n{X}_m$ clusters is generated with a new configurational space exploration method. Statistical treatment of the solid solution including these clusters is then achieved by means of low-temperature expansions, checked against Monte Carlo simulations in some specific conditions. Based on the calculation of equilibrium cluster distributions, it is shown that the solubility limit of oxygen in Fe, hardly measurable experimentally, is largely affected by the presence of small $V_n{\mathrm{O}}_m$ clusters.

Figure 1

List of nearest-neighbors ($nn$) sites of an atom sitting on either an octahedral site (left) or a substitutional site (right) of the bcc Fe lattice. Circles and squares denote respectively octahedral sites and substitutional sites. Note that there may be two nonequivalent 4th, 7th, and 8th $nn$ positions. For instance, there is a substitutional site between solute “0” and its 4th $nn$ “F,” while there is not between solute “0” and its 4th $nn$.

Figure 2

Schematic representation of a solute at (a) an octahedral site, (b) a tetrahedral site, (c) a substitutional site, and (d) an off-centered site where the solute is midway between a substitutional and an octahedral site. Solute atoms are represented by blue spheres, Fe atoms by black spheres, and $V$ by grey cubes.

Figure 3

Differential charge density distributions (in eV/Å${}^3$) for (a) C, (b) N, and (c) O at an octahedral site in the (010) plane. They show the charge distributions of the solutes and their 1st $nn$ Fe atoms.

Figure 4

Projected density of states of a solute $X$ at an octahedral site (red curve) for $X=\mathrm{C}$, N, and O. The PDOS of one 1st $nn$ Fe atom of the solute (black curve) and of a perfect Fe lattice atom (dashed black curve) are also plotted.

Figure 5

Differential charge density distributions (in eV/Å${}^3$) for (a) N and (b) O at a substitutional site, (c) $V\mathrm{C}$, (d) $V\mathrm{N}$, and (e) $V\mathrm{O}$ in the (110) plane. It shows the charge distribution of a solute near its 1st $nn$ Fe atom compared to a perfect lattice Fe atom.

Figure 6

Binding energy (in eV) of $VX$ pairs as a function of the $nn$ distance between $X$ and $V$ for $X=\mathrm{C}$ (red squares), N (green triangles), and O (blue diamonds). Note that there are two points for each solute in the 8th $nn$ configuration.

Figure 7

Binding energy (in eV) of $X_2$ pairs as a function of the $nn$ distance between the solutes, as described in Fig. 1, for $X=\mathrm{C}$ (red squares), N (green triangles), and O (blue diamonds). Note that there are two points for each solute in the 4th, 7th, and 8th $nn$ configurations. Only the two 4th $nn$ configurations are distinguishable.

Figure 8

Schematic representation of the lowest-energy clusters configurations found for: (a) $V{\mathrm{C}}_2$, (b) $V{\mathrm{O}}_2$ (and $V{\mathrm{N}}_2$), (c) $V_3\mathrm{C}$ (and $V_3\mathrm{N}$), (d) $V_3\mathrm{O}$, (e) $V_4{\mathrm{C}}_2$, (f) $V_4{\mathrm{N}}_2$, and (g) $V_4{\mathrm{O}}_2$. Solute atoms are represented by blue spheres, Fe atoms by black spheres, and $V$ by grey cubes.

Figure 9

Binding energies of the clustering reactions $V_n{X}_{m-1}+X\to V_n{X}_m$ for $X=\mathrm{C}$, N, and O and for $n$ and $m$ up to four. Positive values indicate exothermic reactions.

Figure 10

Differential charge density distributions (in eV/Å${}^3$) in the (010) plane of (a) $V{\mathrm{C}}_2$ and (b) $V{\mathrm{O}}_2$ at their lowest-energy configuration and (c) $V{\mathrm{C}}_2$ at the lowest-energy configuration of $V{\mathrm{O}}_2$ and (d) $V{\mathrm{O}}_2$ at the lowest-energy configuration of $V{\mathrm{C}}_2$.

Figure 11

Schematic view of an unit cell of a coherent ${\mathrm{Fe}}_3{\mathrm{O}}_4$ structure, a possible nuclei of the magnetite. Solute atoms are represented by blue spheres, Fe atoms by black spheres, and $V$ by grey cubes.

Figure 12

Relative error between our model and DFT calculations of $V_n{X}_m$ cluster binding energies plotted against the DFT binding energy. Blue dots are the data that were used to fit the interaction parameters. Orange squares are the predictions of our Hamiltonian from which the parametrization is validated. The upper figure (Fe-V system) also includes comparison with vacancy formation energy in pure Fe, Fe cohesive energy, and low-index surface energies.

Figure 13

Cluster distribution in the Fe-C solid solution predicted by MC simulations (symbols) and LTEs (lines), with nominal concentrations $\left[{\mathrm{C}}_{\mathrm{total}}\right]\sim {10}^{-4}$ and $\left[V_{\mathrm{total}}\right]\sim 3\times {10}^{-6}$.

Figure 14

Fe-rich part of the Fe-N phase diagram. Solubility limit of N in the Fe solid solution in equilibrium with ${\mathrm{Fe}}_4\mathrm{N}$ is given for two fitting procedures of ${\mathcal{S}}^f\left(\mathrm{N}\right)$, “with $V$” (black curves) and “without $V$” (blue curves). Dashed lines correspond to solubility limit boundaries deduced from a sensitivity study and solid lines show the “mean phase diagram.” Experimental data (squares) are taken from Ref. [11] The green circle is the experimental point used to fit the solute formation entropy. The insert shows the average monomer proportion among the total N population.

Figure 15

Fe-rich part of the Fe-O phase diagram. Solubility limit of O in the Fe solid solution in equilibrium with magnetite (${\mathrm{Fe}}_3{\mathrm{O}}_4$) for $T<843$ K and with wüstite (${\mathrm{Fe}}_{0.947}\mathrm{O}$) for $T>843$ K is given for three fitting procedures of ${\mathcal{S}}^f\left(\mathrm{O}\right)$: “with $V$” (black curves), “without $V$” (blue curves), and “mixed” (orange curves). Dashed lines correspond to solubility limits boundaries deduced from a sensitivity study and solid lines show the “mean phase diagram.” The insert shows the average monomer proportion among the total O population.