Stability and structures of the ε-phases of iron nitrides and iron carbides from first principles

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Scripta Materialia 64 (2011) 296–299 www.elsevier.com/locate/scriptamat

Stability and structures of the e-phases of iron nitrides and iron carbides from first principles C.M. Fang,a,b,⇑ M.A. van Huisb,c and H.W. Zandbergenb a

Materials Innovation Institute (M2i), 2628 CJ Delft, The Netherlands Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands c EMAT, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium

b

Received 2 August 2010; revised 24 August 2010; accepted 25 August 2010 Available online 31 August 2010

First-principles calculations were performed for the e-phases and other iron carbides/nitrides with hexagonal close-packed Fe sublattices. Although these nitrides/carbides have similar crystal structures, they exhibit different chemical and physical properties. Relative to a-Fe, graphite and N2, all the e-type nitrides are stable, while all the carbides are metastable. The lattice parameters of the e-iron nitrides vary differently from those of the e-carbides, as a function of the concentration of X (X@N, C). The structural relationships of e-Fe2X with g-Fe2X and f-Fe2X are discussed. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Iron nitrides and carbides; Precipitates in steels; Formation enthalpy; Density functional theory calculations

Iron nitrides and carbides play an important role in iron alloys and steels [1–3]. Nitrogen has been introduced into stainless steels to reduce the Ni content for application in both humans and animals [1,3,4]. Tanelke and co-workers [5] reported improvement in creep resistance through boundary pinning effects of carbonitride precipitates. In transformation-induced plasticity (TRIP) aided steels, fine, hard particles, such as iron nitride and carbide precipitates, are crucial to increase their hardness [6–8]. Therefore, knowledge about the formation and stability of iron nitride and -carbide precipitates is useful for the further development of stronger and tougher steels [1,2,5,9,10]. Around 1950, Jack reported his study of the crystal structures of e-iron nitrides and proposed models of phase transitions from e-phases to g-Fe2C and f-Fe2N [11,12]. Since then, many experimental and theoretical studies have been made on the crystal structures, and the electronic and magnetic properties, of the carbides and nitrides of the e-phases [13–22]. Liapina and coworkers [13] investigated the structural properties of the e-Fe3N1+x phases. Nagakuri and Tanehashi [14]

⇑ Corresponding

author at: Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands. Tel.: +31 15 2781536; fax: +31 15 2786600; e-mail: [email protected]

studied the e-Fe2N and f-Fe2N phases and proposed different ionic models for f-Fe2N and e-Fe2N. On the other hand, there is little information about the crystal structures of the e-phase iron carbides. Using first-principles methods, Eck and co-workers [17] investigated structures and electronic properties of binary 3d transition metal nitrides, while Shang et al. [18,19] studied the structural behavior of the e-nitrides using first-principles techniques. Recently, we performed first-principles calculations for the Fe2C phases and showed how e-Fe2C transforms into g-Fe2C, and addressed their structural relationships with the v-Fe5C2, h-Fe3C and Fe7C3 phases [22]. In the present letter we report our systematic study on the stability and structural properties of the e-Fe2X1+x, and related g- and f-Fe2X phases, as well as v-Fe5X2, h-Fe3X and Fe7X3 phases. Their relative stability and mutual structural relationships are addressed. A small hexagonal unit cell was first used by Jack and co-workers to describe the structure of the e-phases (Fig. 1a) [2,11,12,14]. We refer to this model here as Jack-1. Its space group is P 3m1 (nr. 164), with lattice ˚ and c0  4.35 A ˚ [1,2,22,23]. In parameters a0  2.75 A that unit cell, there are two Fe atoms and interstitial (octahedral) sites partially occupied by X atoms (see Fig. 1a). p Jack also used a supercell model (Jack-2) with ah = 3 a0, ch = c0 (where a0 and c0 are the lattice parameters of Jack-1), which contains six Fe atoms [12,18,19,23]. In this model there are several possible

1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.08.048

C. M. Fang et al. / Scripta Materialia 64 (2011) 296–299

Figure 1. Structural models for e-Fe3X1+x (X = N or C). The black spheres represent iron atoms and the blue (or red or green) spheres represent X atoms. (a) Jack-1 model. (b) Jack-2 model and g-Fe2X with red lines for the unit cell. (c) (0 0 1) projection of Jack-3 and (1 0 0) projection of f-Fe2X phase. The blue spheres are the 4 X atoms at the b-sites and the green spheres are the 8 X atoms at the c-sites.

ways in which the carbon atoms can be arranged in the interstitial sites (Wyckoff sites: 2b, 2c, 2d) in the hexagonal close-packed (hcp) Fe sublattice. Note that the 2cand 2d-sites are structurally equivalent (Fig. 1b). Jack-2 is generally used to describe the structure of e-phases [12,17–23]. Furthermore, Jack proposed anotherpsupercell (Jack-3) with lattice parameters ah = 2 3 a0, ch = c0 (again, a0 and c0 are the lattice parameters of Jack-1) in order to account for the weak diffraction lines, which could not be indexed within the Jack-2 model [11,12,22,23]. Jack-3 contains 24 Fe atoms with X atoms at the 8c-sites and the 8b-sites (Fig. 1c). Note that, different from the averaged occupation model used before, the arrangements of the X atoms in Jack-3 may

Figure 3. The calculated formation energies of the e-phases (connected with the dotted lines), and other members of the hcp family for nitrides (left) and carbides (right). Two extreme cases of distributions of X at the b-sites are connected with the dotted lines in order to guide the eye.

in some cases break the symmetry of the Jack-2 model; however, we still employ the same symbols of the Wyckoff sites to describe the Fe hcp sheets and interstitial sites for carbon and nitrogen atoms because the Fe sublattice is almost unchanged. The formation energy (DEf) per atom of an iron nitride or carbide (FenXm) from the elements (a-Fe, graphite and N2) can be described as [20–23,24]: DEf ¼ fEðFen Xm Þ  ½nEðFeÞ þ mEðXÞg=ðn þ mÞ

ð1Þ

At T = 0 K and p = 0 Pa, the formation enthalpy equals the calculated formation energy, i.e. DH(FenXm) = DE(FenXm), when the zero-vibration contribution is ignored, since it is much smaller than the formation energies [24]. The formation enthalpy/energy defined in this way can be used to measure the stability of the iron nitrides and carbides with respect to the a-Fe phase, graphite and molecular N2. First-principles calculations were performed for the ephases using the density functional theory method1 within the generalized gradient approximation (DFTGGA) [25]. The computational details used here, together with the results of a-Fe and graphite, have been described in earlier publications [22,24,27]. Calculations for the N2 molecule were performed in a cube with an ˚ . A cut-off energy of the wave funcaxis length a = 12 A tions of 1000.0 eV was employed to describe the strongly localized 2p bonds of the N2 molecule. The bond length ˚ , which is close to the calculated in this way is 1.11 A ˚. experimental value of 1.10 A Our calculated results are shown in Figures 2 and 3. Figure 2 shows the calculated lattice parameters of the

1

Figure 2. The calculated lattice parameters of the e-phases for nitrides and carbides. The dotted lines are for guidance of eyes.

297

We use the VASP code, employing the density functional theory within the projector-augmented wave method. The generalized gradient approximation was employed for the exchange and correlation energy terms [24]. The cut-off energy of the wave functions was 550/500 eV for nitrides/carbides. Reciprocal space integrations were carried out using dense k-meshes, e.g. a 12  12  12 grid with 84 kpoints in the irreducible Brillouin zone of h-Fe3X using the Monkhorst and Pack method [26]. Details of the settings are described in Refs. [22,24,27].

298

C. M. Fang et al. / Scripta Materialia 64 (2011) 296–299

e-phases of the nitrides (a) and carbides (b). The lattice parameters of nitrides follow a different trend from those of carbides with increasing X concentration. With the increase in nitrogen concentration, the a-axis and caxis both increase in magnitude, in agreement with experimental observations [12–17]. For the iron carbides, though, the a-axis increases while the c-axis decreases in magnitude. Unfortunately, only few reliable experimental data are available on the chemical composition and lattice parameters. This is probably because multiple phases are formed simultaneously due to their very similar formation energies (see below). Figure 3 shows the calculated formation energies for the iron nitrides and carbides. The crystal structure of e-Fe3X has been well established both experimentally and theoretically [11,12,14– 23]. All the nitrogen and carbon atoms occupy 2c (or 2d) sites for the Jack-2 model and 8c (or 8d) sites in the Jack-3 supercell. This structure can be regarded the basis structure for the e-Fe3X1+x (0.0 < x 6 0.50) series. In the structure of e-Fe3X1+x (0.0 < x 6 0.50), the added X atoms occupy the 8b-sites [11,12,20–22]. Former calculations [22] show that the X atoms at the bsites are arranged in such a way that the stacking of two X atoms on top of each other (along the c-axis) is avoided. However, even if this condition is satisfied, there are still numerous ways in which the X atoms can be distributed over the b-sites, especially for the compositions close to Fe2X (Table 1). Figure 3 shows the calculated formation energies of the iron nitrides (left) and carbides (right). Different X atom orderings at the b-sites for the e-Fe3X1+x with x > 0.125 are considered in two extreme cases: in one extreme, all the X are occupying the b-sites of one layer, which yields the highest formation energies, and in the other extreme case the X atoms are distributed evenly over the b-sites of the two layers. The latter configurations have the lowest formation energies, as shown in Figure 3. The other mixed arrangements have intermediate formation

energies. The formation energies of the other members (g-Fe2X, f-Fe2X, h-Fe3C, v-Fe5X2 and Fe7X3 phases) of the hcp family, which exhibit hcp-Fe sublattices, are included for comparison. For all compositions, the formation energies of the e-Fe3N1+x (x = 0.0–0.50) are negative, so that these phases are stable relative to a-Fe and N2. All the carbides have positive formation energies, so that they are metastable relative to a-Fe and graphite. Next we discuss the structural relationships between e-Fe2X and g- and f-phases. Table 1 lists the structural parameters of the orthorhombic g- and f-Fe2X phases, and of the hexagonal e-Fe2X phase. Both g- and fFe2X are more stable than the corresponding e-phases on one hand. On the other hand, f-Fe2N is much more stable than g-Fe2N, while the carbides have the opposite relationship (Table 1). The lattice parameters of the orthorhombic phases can be compared directly with those of the hexagonal e-Fe2X phases (Fig. 1). Table 1 also shows the calculated results for different configurations in Jack-3. The calculated lattice parameters for eFe2X vary slightly due to different X ordering. In fact, both g- and f-Fe2X originate from different X orderings in the e-Fe2X configurations. The Fe sublattices of the g- and f-Fe2X phases are just slightly distorted from the perfect hcp lattice. Their relationships are shown in Figure 1 and Table 1. In 1968, Nagakuri and Tanehashi published their study on the Fe2N phases [14]. They proposed ionic models with the formulae (Fe+3/2)2N3 and (Fe+3/4)2N3/2 for the e- and f-phase, respectively [14]. However, since the difference between these two phases is mainly the ordering of the X atoms at the b-sites, the chemical bonds in f-Fe2X and e-Fe2X are expected to be very similar. Therefore, the difference between the ionic models in Ref. [14] is unlikely. Here we briefly discuss the origin of the difference of the relative stability of iron carbides and nitrides. The electronegativity of N (3.04) is much larger than that of C (2.55) [28]. Therefore, iron nitrides are more ionic

Table 1. The calculated results for Fe2X and Fe3X phases using DFT-GGA. Formula

Symmetry/X positions

Nitrides lattice ˚) parameters (A

Nitrides DE (meV atom1)

Carbides lattice ˚) parameters (A

Carbides D E (meV atom1)

Fe2X phases e-Fe2X-a

X: 8c and 4b in a plane

42.0

X: 8c, 3b + 1b in two planes

e-Fe2X-c

X: 8c, 2b + 2b in two planes

g -Fe2X

Pnnm (58)

f-Fe2X

Pbcn (60)

a* = 4.8025 c = 4.2951 a* = 4.8027 c = 4.2921 a* = 4.7991 c = 4.2908 a = 4.7066 b = 4.2796 c = 2.8242 a = 4.2997 b = 5.4810 c = 4.8511

+35.3

e-Fe2X-b

a* = 4.7474 c = 4.4167 a* = 4.7634 c = 4.3273 a* = 4.8672 c = 4.3229 a = 4.7038 b = 4.3143 c = 2.7686 a = 4.3406 b = 5.4480 c = 4.7544

Fe3X phases e-Fe3X

X: 8c

78.0

h-Fe3X

Pnma (62)

a = 4.6559 c = 4.3180 a = 4.9230 b = 7.0491 c = 4.4020

44.9 46.0 32.5

46.1

+45.7

a = 4.6572 c = 4.3143 a = 5.0268 b = 6.7203 c = 4.4818

+28.9 +27.9 +17.4

+24.3

+25.1 +20.6

For the sake of comparison, the length of the a-axis of the Jack-3 supercell for the e-Fe2X phases is reduced to half, a* = aJack3/2.

C. M. Fang et al. / Scripta Materialia 64 (2011) 296–299

than the carbides. The charged nitrogen ions tend to be as far apart as possible, while C atoms in steel are more covalent and thus tend to cluster. Consequently, for phases of the same chemical composition, iron nitrides with less distorted hcp-Fe sublattices, such as the ephases, are more stable than those with strongly distorted Fe sublattices, such as h-Fe3N or g-Fe2N, as shown in Table 1. Carbides show the opposite behavior, as discussed previously [22,27]. The details of the formation energies and distortion of the hcp-Fe sublattices for the iron carbides are discussed in Ref. [22]. The chemical bonds and electronic properties of the iron carbides and nitrides are beyond the scope of the present letter but are worth further exploration. In summary, first-principles calculations were performed for hexagonal e-Fe3X1+x (X = N or C, 0 6 x 6 0.5) phases, as well as for other iron nitrides and carbides with hcp-type Fe sublattices: the g-Fe2X, f-Fe2X, v-Fe5X2, h-Fe3X and Fe7X3 phases. The total energy calculations show that iron nitrides with less distorted hcp-Fe sublattices (e-phases) are more stable than those with more distorted Fe sublattices (e.g. g- and h-), while the opposite is true for the iron carbides. f-Fe2N structures can be obtained from the e-Fe2N phase whereby two-thirds of the N atoms occupy the c- or dsites and the other third is distributed over the b-sites of two layers. g-Fe2C is formed from the e-Fe2C phase whereby two-thirds of the C atoms occupy the c- or dsites and the other third is distributed over the b-sites in only one layer. Our calculations also show that, as the X concentration increases in the e-Fe3X1+x (X = N or C, 0 6 x 6 0.5) phases, in the case of nitrides the magnitudes of both the a- and c-axes increase, while in the case of carbides the magnitude of the a-axis increases and the magnitude of the c-axis decreases. The present results provide an understanding of the transformation mechanisms and the thermal evolution of the precipitates in steels, and show that iron nitrides and iron carbides, although both acting as hardening precipitates, also exhibit very different physical properties. We thank Dr. D. Hanlon and Dr. S. Celotto (Corus RDT). The authors acknowledge financial support from the Materials Innovation Institute (M2i, Project No. MC5.06280), The Netherlands. [1] K.H. Lo, C.H. Shek, J.K.L. Lai, Mater. Sci. Eng. R65 (2009) 39. [2] L.J.E. Hofer, E.M. Cohn, W.C. Peebles, J. Am. Chem. Soc. 71 (1949) 189.

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