First-Principles Calculations on Stabilization of Iron Carbides (Fe₃C, Fe₅C₂, and η-Fe₂C) in Steels by Common Alloying Elements

We used the spin-polarized generalized gradient approximation to density functional theory (DFT)[36,37] and a plane wave basis with a kinetic energy cut-off of 400 eV. The Kohn-Sham equations were solved using the Vienna ab initio simulation package (VASP, version 4.6.36 Computational Materials Physics, Vienna, Austria).[38‐40] The valence electron and core interactions were described using the projector augmented wave method.[41] The first-order Methfessel-Paxton method was used with a smearing width of 0.1 eV. The PW91 exchange correlation functional[42] with the Vosko–Wilk–Nusair interpolation[43] for the correlation part was used. Structural relaxations were considered converged when the energy in two consecutive ionic relaxation steps differed by less than 10 μeV and the maximum force (worst case) on any atom in the supercell was less than 40 meV/Å. Both volume and ionic positions were relaxed in all supercells considered. For accurate bulk energies, a final calculation was done without any relaxation using the linear tetrahedron method including the Blöchl corrections.[44] Integrations in reciprocal-space employed evenly spaced Monkhorst-Pack sampling[45] such that the product of the number of k-points in the first Brillouin zone and the number of atoms in the supercell equaled approximately 10,000. Both the k-point density and energy cutoff were verified to give total energy convergence of 1 meV/supercell or better. Pure elements, except Fe, were modeled using the unit cells (or primitive cells when possible) of their respective crystal structures. Co and Ni were considered FM, and Cr was considered antiferromagnetic. It is well known that current DFT exchange-correlation functionals do not model graphite accurately. To overcome this shortcoming, the enthalpy of diamond was computed and a correction of −17 meV was added to account for the diamond to graphite transformation.[46] Pure bcc-Fe was modeled with a 128-atom supercell Fe₁₂₈ consisting of 4 × 4 × 4 bcc-Fe unit cells. We used a 3 × 3 × 3 bcc-Fe supercell with 54 atoms in our previous work.[31] The \(1 \times 1 \times 1\;\hbox{Fe}_{3}\hbox{C}\) supercell is the same as its unit cell with 16 atoms, it is modeled as \(\hbox{Fe}_{12}\hbox{C}_{4},\) whereas its 2 × 2 × 2 supercell with eight unit cells was modeled as \(\hbox{Fe}_{96}\hbox{C}_{32}. \ \hbox{Fe}_{5}\hbox{C}_{2}\) was modeled using its unit cell with 28 atoms, \(\hbox{Fe}_{20}\hbox{C}_{8}. \ \eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) was modeled with 2 × 2 × 3 unit cells with 72 atoms, \(\hbox{Fe}_{48}\hbox{C}_{24}.\) Alloying-atom-substituted supercells were modeled by replacing one Fe (or C) atom in the unit cell with the alloying element. We considered the alloying atom substituting the Fe atom on all possible Fe-occupied Wyckoff sites. Al, Si, P, and S were also considered on the C site of all the carbides. Alloying-element-substituted iron (ferrite solid solution) was modeled with \(\hbox{Fe}_{127}\hbox{M}.\) Similarly, alloying-element-substituted \(\hbox{Fe}_{3}\hbox{C}, \hbox{Fe}_{5} \hbox{C}_{2},\) and \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) were modeled with \(\hbox{Fe}_{11}\hbox{MC}_{4}\;(\hbox{or}\; \hbox{Fe}_{12}\hbox{C}_{3}\hbox{M}\) or for the 2 × 2 × 2 supercell as \(\hbox{Fe}_{95}\hbox{MC}_{4}\) or \(\hbox{Fe}_{96}\hbox{C}_{31}\hbox{M}), \hbox{Fe}_{19}\hbox{MC}_{8}\) (or \(\hbox{Fe}_{20}\hbox{C}_{7}\hbox{M}\)), and \(\hbox{Fe}_{47}\hbox{MC}_{24}\) (or \(\hbox{Fe}_{48}\hbox{C}_{23}\hbox{M}\)), respectively.

\(\hbox{Fe}_{3}\hbox{C}\)[47] and \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\)[4,48‐50] crystallize in an orthorhombic unit cell with 16 and 6 atoms, respectively, whereas \(\hbox{Fe}_{5}\hbox{C}_{2}\)[51‐55] crystallizes in a monoclinic unit cell with 28 atoms. The first-principles structurally optimized crystal structure parameters along with the experimental results are given in Table I. The crystal structures of the carbides with the fractional coordinates of the atoms on various Wyckoff sites agree well with experiments and previous first-principles calculations. The formation enthalpies of pure \(\hbox{Fe}_{3}\hbox{C}, \hbox{Fe}_{5}\hbox{C}_{2},\) and \(\hbox{Fe}_{2}\hbox{C}\) are 16 meV/atom, 13 meV/atom, and 5 meV/atom, respectively. Our results are in good agreement with previous first-principles results.[6,13,23,26‐28,33,56,57] Formation enthalpies, which we consider too high for this class of carbides, of 1.6 eV/atom and 1.3 eV/atom for \(\hbox{Fe}_{5}\hbox{C}_{2}\) and \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C},\) respectively, have been reported.[7] A positive formation enthalpy implies that the compound is not stable compared with the elements in their standard states (Eq. [2]). Accordingly, the carbides in the order of decreasing stability are \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}, \hbox{Fe}_{5}\hbox{C}_{2},\) and \(\hbox{Fe}_{3}\hbox{C},\) which is consistent with earlier calculations.[56,57] It is interesting to note that these carbides also occur in the same sequence during precipitation in steels.[6] Although the description of the properties of the pure carbides is of importance, here we are interested in the effect of alloying elements on the carbides.

To investigate the effect of increasing supercell size on partitioning enthalpies and other properties, we use supercells of different sizes for both bcc-Fe and cementite. In the case of bcc-Fe, we use 3 × 3 × 3 and 4 × 4 × 4 supercells, whereas for cementite, we use 1 × 1 × 1 and 2 × 2 × 2 supercells. In computing the partitioning enthalpies, the use of either a 3 × 3 × 3 or a 4 × 4 × 4 supercell of bcc-Fe does not change the partitioning enthalpies except for a few meV in the worst cases, indicating that the relaxation effects in a larger the smaller bcc-Fe supercell. When a larger 2 × 2 × 2 cementite supercell is used instead of a 1 × 1 × 1 supercell, the partitioning enthalpies changed significantly (Figure 1). Although the formation enthalpy and the crystal structure of pure cementite do not show any perceivable changes, the increase in supercell size leads to considerable changes in the partitioning enthalpies of the impurity-substituted carbides. This change is clearly noticeable in the case of the alloying elements that lead to large changes in volume of the alloying element substituted supercells, namely P, S, Nb, Mo, and W, along with Al and Si when substituted on the C site (Figure 2). The supercell effect is less pronounced in the case of Ti, V, Cr, Mn, Co, Ni, and Cu where the volume change is much less compared with the former group of elements. The increase in carbide supercell size, thus, generally leads to a less stable carbide and, hence, a stronger preference of the bcc-Fe phase. This is easily rationalized: In the small supercell, oversized atoms are accommodated mostly by expanding the volume, whereas in larger supercells, the distortion of the carbide lattice predominates. The expansion of the small supercell does not take into account that this volume expansion will lead to elastic strains at larger lengths scales, and therefore, small supercell calculations energetically might be biased toward the carbide phase unless the actual alloying element concentrations in the carbide are representative for the supercell compositions. In the subsequent discussion, we refer to the results calculated using the larger 4 × 4 × 4 and the 2 × 2 × 2 bcc-Fe and cementite supercells.

Site preferences can be deduced either from formation enthalpies (Figure 3 and Table II) or partitioning enthalpies (Figure 4 and Table III). In \(\hbox{Fe}_{3}\hbox{C},\) when substitution of the alloying elements is considered only on the metal site, all alloying elements prefer to occupy the 8d site.[20,31] When P and S are substituted on the metal sites, it is observed that there is a major reorganization of the nearest neighbor atoms. Fe atoms are observed to move closer to the P and S atoms, both of which have p valence electrons. No such reorganization was observed when the transition alloying elements with only d valence electrons occupied the Fe site. These observations, along with the fact that the C site has the maximum number of Fe neighbors in the carbide structures, prompted us to consider P and S along with Al and Si on the C site. As might be expected, P and S, on the basis of the strong bonding with the Fe atoms, preferred the C site over the Fe sites (Figure 4 and Table III). The case of Si is somewhat ambiguous: In the 1 × 1 × 1 cementite supercell, there is a clear preference for the C site, whereas in the larger 2 × 2 × 2 supercell, the Fe2(8d) and C sites are almost degenerate (Figure 1). It should be remarked, however, that Si strongly favors dissolution in the ferrite phase rather than in cementite. Al is most stable on the Fe2(8d) site. The preference of Al, Si for the Fe site and P, S for the C site from our 2 × 2 × 2 supercell calculations agree well with previous results,[22] which were carried out at a lower kinetic energy cutoff (350 eV).

In \(\hbox{Fe}_{5}\hbox{C}_{2},\) alloying elements prefer various sites (Figure 4 and Table III). The symmetry of the crystal structure is completely lost only on substitution of the alloying element on the Fe1(8f) or the Fe2(8f) sites. This loss in symmetry leads the atoms constituting the structure to have more degrees of freedom to relax, and hence, most of the alloying elements prefer to occupy either the Fe1(8f) or the Fe2(8f) site. All carbide formers like Ti, V, Nb, and Mo seem to prefer the Fe1(8f) site, which has four close C neighbors, two at approximately 2 Å, third at 2.4 Å, and a fourth at 2.8 Å. Although Al and Cu are not considered good carbide formers, they also prefer to occupy this site. Co occupies the Fe2(8f) site with the least number of C neighbors, two at approximately 2 Å and the third at approximately 2.4 Å. Ni and W do not show any preference between the Fe1(8f) or the Fe2(8f) sites. The Fe3(4e) site is preferred by Cr and Mn only. Cr and Mn have a stronger affinity for carbon than Fe and tend to be soluble in most carbide phases. The somewhat unique site preference of Cr and Mn may be caused by the rather small atomic size difference with Fe, which makes relaxation effects less important. Simultaneously, the Fe3 site provides 4 close C nearest neighbors at approximately 2 Å. Si, P, and S prefer to occupy the C site.

In \(\hbox{Fe}_{2}\hbox{C},\) P, and S prefer to occupy the C site, whereas Al prefers the Fe site. Although Si prefers the C site in \(\hbox{Fe}_{3}\hbox{C}\) and \(\hbox{Fe}_{5}\hbox{C}_{2},\) it prefers the Fe site \(\hbox{Fe}_{2}\hbox{C}.\)

When fully relaxing the impurity substituted supercell, we make the implicit assumption that the supercell under investigation experiences no external stress. Although this assumption is valid for massive bulk materials, the assumption must be considered carefully when dealing with precipitate phases such as the ones being considered in this article. The precipitates are embedded in a ferrite matrix with which at least partial coherency exists, which leads to a strained impurity substituted precipitate phase. Changes in volume of the precipitate phase can increase or decrease the strain. In light of this, it is worthwhile to examine the volume changes brought about by the substitution of the alloying elements on various sites. Situations can be envisaged where although it might be energetically favorable to occupy a certain site, the strain effects might not actually allow such preference. Such a situation seems to manifest clearly in the case of Si. Although the preference in energy between the Fe sites and the C site is little, the volume change brought about by its substitution on the C site is much higher than its substitution on the Fe sites (10 to 15 Å³ vs  −2.5 to 2.5 Å³) (see Figure 2). It is harder to make such an argument in the case of P and S because although the volume changes on the C site are higher compared with the changes on the Fe site, the preference to the C site is much larger compared with Si.

In our previous work,[31] we showed how using \(\Updelta H_{f}\) to determine stabilization of cementite gives results that do not agree with experiments in the case of Al, Mo, and W. The case of Al is striking as it is predicted to stabilize cementite almost twice as much as Mn. Al, of course, does not stabilize cementite, whereas Mn is experimentally known to partition to, and hence stabilize, cementite[58‐60] (for more references, Reference 31). Despite using a 4 × 4 × 4 bcc-Fe and a 2 × 2 × 2 cementite supercell in the current work instead of the 3 × 3 × 3 and 1 × 1 × 1 supercells, respectively, as used in Reference 31, our conclusions about the partitioning and stabilization of the alloying elements between cementite and ferrite remain qualitatively the same when considering partitioning on the Fe site. Please note that we list H _f in Table III of Ref. 31, while in Table II of this article, we list \(\Updelta H_{f}.\)

In the case of \(\hbox{Fe}_{5}\hbox{C}_{2},\) examining \(\Updelta H_{f}\) might suggest that Al, Ti, V, Cr, Mn, and Nb stabilize the carbide phase, whereas the rest of the alloying elements destabilize it (Table II and Figure 3). As in the case of cementite, this is a misinterpretation of the data.

However, when we consider the reference states of the alloying elements correctly via the partitioning enthalpies instead of formation enthalpies, both Al[61,62] and Si[59,60,63‐65] destabilize the formation of not only cementite but also all the carbide phases (Figure 4 and Table III) regardless of whether Fe or C site substitution is considered. This finding shows that the conclusions about stabilization of carbides with respect to ferrite based on partitioning enthalpies are more reliable than the ones based on formation enthalpies. Not only do both Al and Si destabilize all three carbide phases, but also Si destabilizes them more than Al (Figure 4).

P, Co, Ni, Cu, Mo, and W also destabilize the three carbides considered in this article, although not nearly as strongly as Si and Al. Surprisingly, S on the C site stabilizes \(\hbox{Fe}_{3}\hbox{C}\) and \(\hbox{Fe}_{5}\hbox{C}_{2}\) while destabilizing \(\hbox{Fe}_{2}\hbox{C}.\) Ti and Mn stabilize the three carbide phases while Nb stabilizes cementite and Hägg carbides but not \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}.\) V and Cr destabilize \(\hbox{Fe}_{3}\hbox{C}\) and \(\hbox{Fe}_{5}\hbox{C}_{2}\) by a negligible amount while V destabilizes \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}.\)

However, the ab initio computed partitioning enthalpies do not agree with all experimental observations. Cr is known to partition, and hence stabilize, \(\hbox{Fe}_{3}\hbox{C}.\)[59,60,63] In fact, more so than Mn. Alas, Cr is computed to destabilize slightly and hence partition away from cementite, contrary to experimental evidence.[59,60,63] Possibly, Cr-rich carbides form prior to Fe-based carbides and act as nucleation sites for the cementite and other Fe-based carbides. If those initial Cr-rich carbides are small enough they might not be recognized as distinct phases. We do compute that Mn stabilizes \(\hbox{Fe}_{3}\hbox{C}\) in agreement with experimental observations.[58‐60,63] Mn is computed to stabilize \(\hbox{Fe}_{5}\hbox{C}_{2}\) and \(\hbox{Fe}_{2}\hbox{C}\) even a little more. Our computed partitioning enthalpies do not reflect correctly the fact that Mo[59,60,63] and W[63] also partition to cementite, although the enthalpies involved are rather small. It should be noted that all our calculations are based on enthalpies obtained at 0 K. The neglect of entropy (S) changes at finite temperature is less likely to be tenable when enthalpy changes are of order TS, where S could be mainly the result of alloy element-induced magnetic (dis)ordering, which could be of order k _B . Given that experimental measurements are typically in the neighborhood of the ferrite-austenite transition temperature, it follows that partitioning enthalpies less than 0.1 eV are rather inconclusive.

Si, P, S, and Al destabilize \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) much more than cementite and Hägg carbide. All alloying elements except Mn destabilize \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) relative to \(\hbox{Fe}_{3}\hbox{C}\) and \(\hbox{Fe}_{5}\hbox{C}_{2}.\) At this juncture, it is interesting to note that Mn also stabilizes \(\epsilon{\text{-}}\hbox{Fe}_{2}\hbox{C},\) a carbide closely related to \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C},\) over cementite.[27] The competition between cementite and Hägg carbide is not nearly as strongly affected by alloying additions as the competitions involving \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}.\) Si, Mo, and W are found to disfavor cementite less than Hägg carbide, whereas Ti, Mn, and Nb promote Hägg carbide at the expense of cementite. This is in line with the complete intermixing between \(\hbox{Fe}_{5}\hbox{C}_{2}\) and \(\hbox{Mn}_{5}\hbox{C}_{2},\)[58] whereas such thermodynamically favorable dissolution does not exist for cementite.[26] Our results indicate that Mn stabilizes \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) and \(\hbox{Fe}_{5}\hbox{C}_{2}\) approximately equally.