1 Introduction
In advanced low-alloy steels, it is important to retain austenite to ambient temperature, and in this regard, C acts as an efficient austenite stabilizer. The precipitation of carbides in steel depletes the amount of C available for austenite stabilization, hence, it is desirable to suppress the formation of most carbide phases in advanced steels. The most commonly observed carbides in low-alloy steels are cementite \((\hbox{Fe}_{3}\hbox{C}),\) Hägg carbide \((\hbox{Fe}_{5}\hbox{C}_{2}),\) and eta-carbide \((\eta{\text{-}}\hbox{Fe}_{2}\hbox{C})\). The easiest way to suppress carbide phases is by adding alloying elements that destabilize them. Therefore, it is of interest to know quantitatively to what degree various alloying elements affect carbide stability. Although experimentally it might be difficult to control and observe the occurrence of very small precipitates of the three carbide structures in steels, it will be shown that it is rather straightforward to compute the main enthalpic contribution of alloying elements to carbide stability by first-principles methods.
Experimentally, it was found that
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) forms first in quenched steels at temperatures between 370 K (100 °C) and 470 K (200 °C).[
1,
2] But, it was also noticed that
\(\epsilon{\text{-}}\hbox{Fe}_{2}\hbox{C}\) is the only carbide forming up to 520 K (250 °C) and forms along with cementite till 600 K (330 °C) and that it acts as a precursor for the formation of
\(\hbox{Fe}_{5}\hbox{C}_{2}.\)[
3,
4] A long aging study at 300 K (30 °C) followed by a brief 405 K (130 °C) anneal showed the presence of both
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) and
\(\epsilon{\text{-}}\hbox{Fe}_{2}\hbox{C}.\)[
5]
\(\epsilon{\text{-}}\hbox{Fe}_{2}\hbox{C}\) is a nonstoichiometric carbon deficient structure of
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}.\) It was recently shown that
\(\epsilon{\text{-}}\hbox{Fe}_{2}\hbox{C}\) is only slightly more unstable than
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) and that it can relax to the latter structure.[
6] Above 720 K (450 °C), it has been observed that
\(\hbox{Fe}_{3}\hbox{C}\) forms exclusively.[
1,
2] Roughly, the carbides seem to precipitate in the order
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}, \hbox{Fe}_{5}\hbox{C}_{2},\) and
\(\hbox{Fe}_{3}\hbox{C}\) with increasing temperature[
6,
7] with the appearance of
\(\epsilon{\text{-}}\hbox{Fe}_{2}\hbox{C}\) preceding
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}.\) Both kinetic and thermodynamic factors could be responsible for this observation. But the predominance of each of the carbide in a definite temperature range has been attributed to the lowering of its free energy (and hence stabilization) with temperature.[
6] The precipitation sequence can be altered by the application of a magnetic field thereby showing that the magnetic free energy plays an important role in the stabilization of the carbide phases.[
6,
8,
9]
Several
ab initio studies on pure and impurity substituted cementite have been already performed. Electronic, structural, and magnetic properties of pure cementite were described in many previous communications.[
7,
10‐
12] Furthermore, there are detailed studies of thermodynamic properties of pure cementite,[
9,
13] elastic properties,[
14‐
17] point defects, and possible C diffusion paths.[
12] The energetics and electronic structure of impurity substituted cementite have also been the focus of a considerable number of previous studies.[
11,
18‐
30] The partitioning behavior of alloying elements between cementite and ferrite has been described,[
31] and the stabilization of cementite by various alloying elements has been studied.[
20‐
30] In most previous computational work on the stabilization of carbide phases by alloying elements, conclusions were based on enthalpies of formation with respect to the pure carbide phase. Recently, the authors of this article argued that carbide stabilization must be evaluated based on partitioning enthalpies instead of formation enthalpies.[
31]
Relatively less attention has been paid to the carbides
\(\hbox{Fe}_{5}\hbox{C}_{2}\) and
\(\hbox{Fe}_{2}\hbox{C},\) for both pure and impurity substituted phases. The electronic, magnetic, and structural properties of
\(\hbox{Fe}_{3}\hbox{C},\hbox{Fe}_{5}\hbox{C}_{2},\) and
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) have been reported.[
7] Formation enthalpies,[
7,
33] surface properties,[
34] thermodynamic properties, and formation enthalpies[
6] of
\(\hbox{Fe}_{5}\hbox{C}_{2}\) have also been described. Comparable work, excluding the surface properties, has been done on
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}.\)[
6,
7,
15] Calculations have been interpreted to show that whereas Mn and Al stabilize
\(\epsilon{\text{-}}\hbox{Fe}_{2}\hbox{C},\) Si destabilizes it.[
27] To the best of our knowledge, no work has been done on the stabilization of either
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) or
\(\hbox{Fe}_{5}\hbox{C}_{2}\) by alloying elements.
In this article, adding to our previous work on alloying-element-substituted
\(\hbox{FeC}_{3},\)[
31] we calculate the stabilization of
\(\hbox{Fe}_{5}\hbox{C}_{2}\) and
\(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) by various alloying elements. Therefore, we can now comment on relative stabilization and address the question whether alloying species (dis)favor one carbide in relation to another. Moreover, although we have not considered substitution of the alloying elements on the C site in our previous communication,[
31] in this article we consider Al, Si, P, and S on the C site of the carbides. To investigate supercell effects on stabilization, we consider two supercells, 1 × 1 × 1 and 2 × 2 × 2, in the case of impurity-substituted cementite. We first describe the crystal structures of the carbides, then elucidate the calculation methodology of carbide (de)stabilization, and finally, we describe our results on the role of alloying elements on (de)stabilization of the carbides with respect to ferrite and the competition between carbides.
2 Methodology
Representing
\(\hbox{Fe}_{km}\hbox{C}_{kn}\) as the pure carbide supercell, where
k is the number of formula units used to model the pure
\(\hbox{Fe}_{m}\hbox{C}_{n}\) carbide, and
\(\hbox{Fe}_{km-1}\hbox{MC}_{kn}\) as the alloying-element-substituted carbide supercell, the balance for the formation of alloying-element-substituted carbide from the elements is given as
$$(km-1)\hbox{Fe}+\hbox{M}+kn\hbox{C}\rightleftharpoons \hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}$$
(1)
The formation enthalpy of the impurity substituted carbide is given as
$$H_{f}[\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}]=H[\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}] -(km-1)H[\hbox{Fe}]-H[\hbox{M}]-knH[\hbox{C}]$$
(2)
where
\(H[\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn} ] \) is the enthalpy of the alloying-element-substituted cementite.
\(H [ \hbox{Fe} ] , H [ \hbox{M} ] ,\) and
\(H [ \hbox{C} ] \) are the enthalpies of the elements (used as reference phases) at their respective room temperature and pressure crystal structures. A similar balance and formation enthalpy applies to the pure carbide
\(\hbox{Fe}_{km}\hbox{C}_{kn}\) and the C site substituted carbide
\(\hbox{Fe}_{km}\hbox{C}_{kn-1}\hbox{M}.\)
The stabilization of a carbide by an alloying element is usually given[
20,
27] by the change in formation enthalpy of the alloying-element-substituted carbide with respect to the pure carbide as
$$\Updelta H_{f}[\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}]= H_{f}[\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}]-H_{f}[\hbox{Fe}_{km}\hbox{C}_{kn}]$$
(3)
or, in terms of compound enthalpies as,
$$\Updelta H_{f}[\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}] = H[\hbox{Fe}] + H[\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}] - H[\hbox{M}] - H[\hbox{Fe}_{km}\hbox{C}_{kn}]$$
(4)
Similar equations apply for the C site substituted carbide.
To overcome the shortcoming of using the alloying element in its ambient temperature and pressure crystal structure as the reference state, we use another quantity defined as the partitioning enthalpy.[
31] The partitioning enthalpy looks at stabilization of the carbide phase by the alloying element as a competition for the alloying element between the carbide phase and the ferrite phase. In the carbide phase, the alloying element can either occupy the Fe site or the C site. Depending on which site the alloying element occupies, we have two balances that determine the partitioning enthalpy. To compute the partitioning enthalpy for Fe substitution, the balance is given as
$$\hbox{Fe}_{p-1}\hbox{M}+\hbox{Fe}_{km}\hbox{C}_{kn}\,\leftrightharpoons \,\hbox{Fe}_{p} +\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}$$
(5)
The partitioning enthalpy for Fe substitution is given as
$$H_{p}^{(\hbox{Fe})}=H[\hbox{Fe}_{p}]+H[\hbox{Fe}_{km-1}\hbox{M}\hbox{C}_{kn}] -H[\hbox{Fe}_{p-1}\hbox{M}]-H[\hbox{Fe}_{km}\hbox{C}_{kn}])$$
(6)
where
\(\hbox{Fe}_{p}, \hbox{Fe}_{p-1}\hbox{M}, H [\hbox{Fe}_{p} ] ,\) and
\(H [\hbox{Fe}_{p-1}\hbox{M} ] \) represent pure body-centered cubic(bcc-)Fe, the dilute solid solution of M in bcc-Fe, and their enthalpies, respectively.
Similar equations for C substitution are given as
$$\hbox{Fe}_{p-1}\hbox{M}+\frac{nk-1}{nk}\hbox{Fe}_{km}\hbox{C}_{kn}\,\leftrightharpoons\,\frac{np-(n+m)}{np}\hbox{Fe}_{p}+\hbox{Fe}_{km}\hbox{C}_{kn-1}\hbox{M}$$
(7)
$$H_{p}^{(C)} =\, \frac{np-(n+m)}{np}H[\hbox{Fe}_{p}] + H[\hbox{Fe}_{km}\hbox{C}_{kn-1}\hbox{M}] - H[\hbox{Fe}_{p-1}\hbox{M}] - \frac{nk-1}{nk}H[\hbox{Fe}_{km}\hbox{C}_{kn}]$$
(8)
A negative value for the partitioning enthalpy implies a stabilization of the carbide, whereas a positive value indicates stabilization of bcc-Fe. The partitioning enthalpy has been recognized as the main driving force for partitioning elsewhere also,
e.g., in Eqs. [11] and [14] in the work by Benedek
et al.[
35] We will show in this article, that the stabilization of an alloying element substituted carbide with respect to ferrite can be wrongly predicted when using
\(\Updelta H_{f}\) instead of
H
p
. For the first-principles calculations, we consider the (alloying-element-substituted) carbides and (alloying-element-substituted) bcc-Fe in their 0 K ferromagnetic (FM) state.
2.1 Computational Details
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 Fe
128 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.
4 Conclusions
First-principles calculations on the alloying element substituted carbides \(\hbox{Fe}_{3}\hbox{C}, \hbox{Fe}_{5}\hbox{C}_{2},\) and \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) show that Si and Al destabilize the formation of carbides, and Si is the most effective. P and S prefer to occupy the C site in all the carbides, whereas Si weakly prefers to occupy the C site in two of them, \(\hbox{Fe}_{3}\hbox{C}\) and \(\hbox{Fe}_{5}\hbox{C}_{2}.\) On a per-atomic-fraction basis, Si is approximately twice as effective as Al for carbide suppression. All alloying elements considered, except Mn, destabilize \(\eta{\text{-}}\hbox{Fe}_{2}C\) relative to \(\hbox{Fe}_{3}\hbox{C}\) and \(\hbox{Fe}_{5}\hbox{C}_{2}.\) The competition between Fe3C and \(\hbox{Fe}_{5}\hbox{C}_{2}\) is not so strongly affected by alloying elements. Si, Mo, and W disfavor \(\hbox{Fe}_{5}\hbox{C}_{2}\) more than \(\hbox{Fe}_{3}\hbox{C},\) whereas Ti, Mn, and Nb stabilize \(\hbox{Fe}_{5}\hbox{C}_{2}\) over \(\hbox{Fe}_{3}\hbox{C}.\) Mn stabilizes both \(\hbox{Fe}_{5}\hbox{C}_{2}\) and \(\eta{\text{-}}\hbox{Fe}_{2}\hbox{C}\) to a comparable degree over \(\hbox{Fe}_{3}\hbox{C}.\) At a finite temperature the observed partitioning behaviors of Cr, V, Mo, and W are not explained satisfactorily based on first-principles zero-temperature partitioning enthalpies. It is to be borne in mind that experimental observations pertain to carbides in the paramagnetic state, whereas the first-principles calculations pertain to the ferromagnetic state at zero temperature. Although configurational entropy effects can be shown to play a minor role at the temperatures of interest in relation to the computed enthalpy changes, the same cannot be said of magnetic entropies. Possibly, by considering the carbides in a disordered local moment state and by explicitly considering magnetic entropy contributions to the free energy, a better agreement with experiment might be found. Experimentally, there is a possibility for misinterpretations if Cr-, V-, Mo-, or W-rich carbides form prior to Fe-based carbides and then subsequently act as nucleation sites for cementite or other Fe-based carbides. If those initial alloy element-rich carbides are small enough, they might not be recognized as distinct phases.