Phase Stability of Carbides and Nitrides in Steel
Marcel Sluiter
MSE-3ME, TU Delft, Mekelweg 2, Delft, 2628 CD, Netherlands
ABSTRACT
Carbides, nitrides, and carbonitrides have great influence on the properties of steel but are
relatively little studied at a fundamental level: experimentally because of small dimensions of the
precipitates and other difficulties, theoretically because of complex structures and poorly defined
compositions. Modern density functional calculations reveal significant trends that can be partially
summarized in terms of atomic size, carbon affinity, and hybridization de-magnetization concepts. Phase
stability and order-disorder phenomena in widely occurring Cr-based M23C6 carbides and Mo-based M6C
carbides are used to illustrate predictions relevant to low alloyed steel.
INTRODUCTION
Transition metal carbides and carbonitrides appear in many technologically important materials,
in steels, both the alloyed and the unalloyed varieties, in hard refractory materials and coatings. Many of
the general features of carbides apply also to carbonitrides so that from here on I will refer carbonitrides
implicitly when mentioning carbides. The structures of binary carbide phases exhibit certain trends:
carbides of the group IV transition metals (TiC, ZrC, HfC) are cubic NaCl type, group V transition metals
(V, Nb, Ta) also form carbides with the NaCl structure, but in addition hexagonal M2C carbides occur.
Group VI transition metals (Cr, Mo, W) form several carbide phases, some of them forming under special
conditions only, such as the various crystal structures of MoC. Carbides become progressively less stable
with transition metals of the VII (Mn) and VIII (Fe,Co) groups. Theoretical studies based on electronic
density functional approaches correctly calculate the stable structures and their lattice parameters.
However, so far in the literature these studies have been limited mostly to the simplest carbide phases,
typically those of the NaCl type [1] and of the MoC varieties [2,3] where the effect of carbon vacancies,
and the stacking of hexagonal planes are the main issues. Only very recently have studies dealing with
cementite (Fe3C) and other iron carbides appeared in the literature [4,5,6]. The carbides that occur in
low-alloyed steels usually involve structurally much more complex phases involving more than a single
metallic species. Some carbide phases in steel, such as M6C are not known in any binary system M-C,
but occur only in ternary or higher order systems such as Fe-Mo-C as e.g. Fe3Mo3C. Until quite recently
theoretical calculations on multicomponent carbides such as M6C with 112 atoms in the face centered
cubic cell were unpractical, but now with advances in computational methodology and computer
hardware these phases can be calculated with good precision. An intermediary approach, where ab initio
calculated cohesive energies vs lattice parameter are used to obtain an approximate pair potential through
an in principle exact Möbius-Chen inversion [7], has been used for several carbide phases with intermixed
metal species, such as (CrxFe1-x)23C6 [8], (CrxMo1-x)23C6 and (CrxW1-x)23C6 [9], and (CrxCo1-x)7C3 [10].
Modeling of the carbide phase stability can thus be expected to move from fitting thermo-dynamic
potentials to a variety of experimental data, such as in the conventional CALPHAD approach [11,12],
towards more predictive density functional based approaches. The stability of carbides and the
competition between different carbide phases is an area of significant research activity [13,14,11,12]. The
general aim is to better predict and model 1) the phases present in steels, 2) the microstructures of steel
[15,16], and 3) the resulting physical properties of steel [17,18], during and after processing [19].
Addressing these questions has many practical implications, not just for the usage of the final product, but
also for the optimizing the production and processing aspects of steels. Increased predictive capabilities
through ab initio modeling can thus be anticipated to have significant practical benefits.
Below I will focus on predictions for carbide formation in a series of M-C systems (M= Cr, Fe, Nb, Mo)
and subsequently examine more closely the intermixing with Fe of the Cr-type M23C6 and the Mo-type
M6C carbides.
THEORY
Initial guesses for the various carbide structures were obtained from the prototypes as listed in the
literature [20]. The electronic calculations used the projector augmented wave (PAW) method [21] as
implemented in the Vienna ab initio simulation package (VASP) [22-24]. The exchange correlation
potential was of the generalized gradient approximation (GGA) type as formulated by Perdew and Wang
[25]. Integrations in reciprocal space used grids, as homogeneously distributed as possible, with the
product of the number of grids points (k-points) in the first Brillouin zone and the number of atoms in the
cell being about 5000. Precision was set to ``high", and convergence criteria for energy, stress and force
were 0.02 meV, 0.5 Kbar, 50 meV/nm, respectively. Structural optimizations were re-initiated at least 4
times. The first structural optimization was always over the volume only so that cell shape and relative
atomic positions, i.e. the internal degrees of freedom, were retained. In the second structural optimization
the cell shape was allowed to relax, but relative atomic positions were held fixed, and in subsequent
structural optimizations all degrees of freedom were optimized simultaneously. Calculations were started
with a spin-polarized state in which initially one more electron was put in the majority band. In some
cases the calculations consistently reverted quickly to a non-spin polarized state and then structural
optimizations were continued without spin polarization. It must be emphasized that the procedure
followed here does not guarantee finding the most stable magnetic ordering for structures with multiple
inequivalent atomic positions. In some cases the magnetic state was checked by starting with specific
local moments. Occasionally this gave rise to high moment states that had been overlooked previously.
Only collinear magnetic states were considered. After stresses and forces were within tolerances, all
structures were examined with regard to symmetry and coordination number for each site in the structure.
Coordination numbers were computed with a heuristic method that was tuned to give the correct
coordination numbers for all the prototypes, details of which are published elsewhere [26].
Carbide structures considered where those commonly found in carbides of Ti, V, Cr, Mn, Fe, Nb, Mo as
well as a few other structures: M6C (cF112), M4C (cP5), M23C6 (cF116), M3C (oP16) “cementite”, M5C2
(mS28), M7C3 (hP80), M7C3 (oP40), M7C3 (hP20), M2C (hP3), M2C (oP12), M3C2 (oP20), M3C2 (oS20),
M4C3 (cP7), MC (hP4) “wurzite”, MC (hP4) “anti-NiAs”, MC (hP4) “NiAs”, MC (hP2) “WC”, MC (hP8)
“gamma’ ”, MC (hP12) “eta”, MC (cF8) “NaCl”, where the Pearson classification has been given
between braces and other commonly used names of the structures are given in quotation marks. The M4C
(cP5) structure is a hypothetical structure with metal atoms on fcc positions and a carbon atom in a
tetrahedral interstice. The M4C3 (cP7) structure is the well-known NaCl structure with one carbon
vacancy per cubic cell. It was considered because many NaCl type carbides are known to occur with
carbon deficient compositions.
Fig. 1: Common interstitial positions of carbon in transition metals, from left to right: octahedral, trigonal
prism, square antiprism.
For carbon, diamond cubic (cF8) and hexagonal graphite structures (hP4) were considered. For the pure
metals the bcc (cI2), hcp (hP2), and alpha-Mn (cI58) where taken into account. For the metals considered
here this limited selection reproduced the experimentally known ground states.
Carbon takes a variety of (interstitial) positions in transition metals, such as octahedral, trigonal prism,
square antiprism, see fig. 1. Larger transition metal atoms allow high concentrations of C in 6 fold
coordinated interstices such as the octahedron (NaCl type structures) and the trigonal prism (e.g.
hexagonal Mo2C), while smaller late transition metals can allow C in such interstices only up to low C
concentrations (e.g. bcc Fe).
RESULTS
The enthalpy difference between the two carbon modifications is calculated to be about 137
meV/atom in favor of graphite, which is considerably larger than the enthalpy difference in the SGTE
database. The SGTE database extrapolates to about 17 meV/atom at T=0 K, and takes a value of about 30
meV/atom at room temperature [27]. It is possible that the weak van der Waals type bonding between the
graphene sheets is overestimated in density functional calculations, so that the graphite structure stability
may be exaggerated. Assuming that the enthalpy of diamond is correct, and “adjusting” the enthalpy of
graphite to be closer to that of diamond, the known metastable Fe carbides become stable when graphite
is only 25 meV/atom lower in enthalpy than diamond (see fig. 2). As it is known that the Fe carbides are
in fact metastable, this presents an upper limit to how much the enthalpy of graphite can be “adjusted”.
For other M-C systems adjusting the graphite enthalpy would change the formation enthalpies a little, but
would not change the stability from metastable to stable. As adjusting the graphite enthalpy has a rather
minor influence and as it does not substantially alter conclusions, no attempt was made to adjust the
graphite-diamond enthalpy difference.
In fig. 2 the calculated enthalpies of formation in the Fe-C and the Cr-C system are displayed. It is
apparent that in Fe-C all carbides are metastable. Among the low carbon carbides Fe3C and Fe5C2 stand
out as being relatively more stable. Indeed Fe3C is the commonly observed cementite phase in steel and
various types of cast iron. That I find Fe5C2 slightly more stable than Fe3C is at variance with
experimental observations. It might be due to various neglected factors such as vibrational effects. As
could be expected on the basis of Fe being a rather small atom, high carbon carbides are unstable. This is
particularly evident at the composition FeC.
Cr exhibits several stable carbides, such as M23C6 (see fig. 3) and M7C3. M7C3 occurs in at least three
different structures (hP20, oP40, hP80) which are energetically all very close.
Fig. 2: Enthalpy of formation of a series of carbide structures for the binary Fe-C and Cr-C systems. Note
that all carbides in the Fe-C system are predicted to be metastable wrt to bcc-Fe and graphite C with
cementite Fe3C and some higher carbon structures being almost stable, whereas in Cr-C several wellknown carbides are stable. (1 eV/atom = 96 kJ/mol)
The M23C6 structure (cF116, displayed in fig. 3) is characterized by 4 symmetrically distinct metal sites
with coordination numbers 12 (brown), 13 (red),14 (yellow), and 16 (dark red), and 8-fold coordinated
carbon sites of square antiprism type (small dark blue spheres). The 12- and 16-fold coordinated sites
have no C nearest neighbors, while the 13- and 14 fold coordinated sites have 2 and 3 C nearest neighbors
respectively. It is somewhat surprising that one metal species would take so many diverse atomic
environments and coordination numbers. Possibly this is related to the magnetic state, which can vary
greatly for Cr and Mn, the only two elements for which M23C6 has been observed.
The formation enthalpies for the Mo-C and Nb-C system are displayed in fig. 4. It is apparent that Mo
has similar formation enthalpies as Cr, and therefore is a carbide former of comparable strength. The
formation enthalpies with Nb are much larger which is in keeping with the known strong carbide (and
nitride) forming tendencies of Nb. The M6C carbide, in steel often associated with Mo, does not occur as
Fig. 3: Structures of M23C6 (left) and M6C (right). Large spheres represent M, small blue spheres C
atoms.
Fig. 4: Enthalpy of formation of a series of carbide structures for the binary Mo-C and Nb-C systems. In
Mo-C the stable carbides are Mo2C (oP12) and MoC (WC prototype, hP2). In Nb-C NbC with the NaCl
structure dominates, the carbon deficient NaCl derived structure Nb4C3 (cP7) is stable also.
a stable phase in the Mo-C system. A close examination of the structure reveals that straightforward
atomic size arguments can explain why it would occur in a ternary such as Fe-Mo-C, and not in a binary
such as Mo-C. As fig. 3 illustrates, there are three metal sites in the M6C structure, 2 types of sites with
12 fold coordination and no C nearest neighbors (yellow and red spheres in fig. 3), and 16-fold
coordinated sites with 2 C nearest neighbors (brown spheres). Apparently, in the Mo-C system, the large
fraction of Mo on the small 12-fold coordinated sites without any C neighbors is energetically
unattractive. However, when a rather small atom with but weak C affinity, such as Fe, takes these
positions while the large Mo atom occupies the 16 fold, C neighboring sites a highly favorable situation
arises. This is evident in fig. 5 where the enthalpy of formation of hypothetical mixtures of Fe6C - Mo6C
and Fe6C - Cr6C are shown, where metal sites of a given type are either all occupied by the first or the
second metal species. In the first case where atomic sizes differ considerably, a strong stabilization
occurs near the composition Fe3Mo3C. The blue circle indicates the structures where Fe is not on the 12but on the 16 fold coordinated sites – this is clearly very unfavorable as these sites are much too large.
However, a small amount of Mo can be tolerated on the 12 fold coordinated sites. The red and black
Fig. 5: Enthalpy of formation in eV/atom relative to the pure elements (C-graphite, Fe-bcc, Mo-bcc, Crbcc) for the quasi-binary Fe6C-Mo6C and Fe6C-Cr6C systems.
datapoints also illustrate that several magnetic states can be defined, with appreciably different enthalpies.
In the case of Fe6C - Cr6C atomic size differences are much smaller which is reflected in the enthalpy
scale. Similar tendencies can be recognized but it appears that Cr is not sufficiently larger than Fe to
result in stabilization of this structure.
CONCLUSIONS
Ab initio calculations on carbides in transition metal systems relevant to alloyed steel reproduce
experimentally known tendencies. Correct ground states are predicted among a series of alternate
structures. Through computational experimentation, i.e. by performing calculations on hypothetical
structures, it is possible to extract explanations for the phase stability tendencies in carbides that could not
be obtained through other means. As such, ab initio calculations can greatly contribute not only to the
accurate representation of the thermodynamics of these systems, but also actually enhance our
understanding and ultimately our engineering of Fe-C based alloys.
ACKNOWLEDGMENTS
This work is part of the research program 02EMM32 of the Stichting voor Fundamenteel
Onderzoek der Materie (Foundation for Fundamental Research of Matter), and was made possible by
financial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek NWO
(Netherlands Organization for Scientific Research) and the Netherlands Institute for Metals Research
(NIMR).
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