Structural stability of Fe-based topologically close-packed phases

Intermetallics 59 (2015) 59e67

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Structural stability of Fe-based topologically close-packed phases A.N. Ladines*, T. Hammerschmidt, R. Drautz €t Bochum, D-44801 Bochum, Germany ICAMS, Ruhr-Universita

a r t i c l e i n f o

a b s t r a c t

Article history: Received 10 October 2014 Received in revised form 2 December 2014 Accepted 15 December 2014 Available online

Precipitates of topologically close-packed (TCP) phases play an important role in hardening mechanisms of high-performance steels. We analyze the influence of atomic size, electron count, magnetism and external stress on TCP phase stability in Fe-based binary transition metal alloys. Our density-functional theory calculations of structural stability are complemented by an analysis with an empirical structure map for TCP phases. The structural stability and lattice parameters of the FeeNb/Mo/V compounds are in good agreement with experiment. The average magnetic moments follow the Slater-Pauling relation to the average number of valence-electrons and can be rationalized in terms of the electronic density of states. The stabilizing effect of the magnetic energy, estimated by additional non-magnetic calculations, increases as the magnetic moment increases with band filling for the binary systems of Fe and early transition metals. For the case of Fe2Nb, we demonstrate that the influence of magnetism and external stress is sufficiently large to alter the energetic ordering of the closely competing Laves phases C14, C15 and C36. We find that the A15 phase is not stabilized by atomic-size differences, while the stability of C14 is increasing with increasing difference in atomic size. © 2014 Elsevier Ltd. All rights reserved.

Keywords: B. Density functional theory B. Electronic structure E. Phase stability E. Ab-initio calculations E. Mechanical properties, theory E. Phase stability, prediction

1. Introduction Precipitates of secondary phases play a central role in the mechanical properties of microstructured materials. A prominent class of precipitates are the topologically close-packed (TCP) phases that are used in precipitation hardening [53] and grain-boundary strengthening [23]. On the other hand, the TCP phases may degrade the properties of the material due to their brittleness [22], the weak host-TCP interface that may fracture during tensile loading [34], and the depletion of alloying elements in the host material [13]. The crystallography of TCP phases was established by Frank and Kasper [14] who showed that it is possible to pack atomic spheres such that only tetrahedral voids are present. This requires icosahedral atomic environments (12-fold coordination) combined with higher coordinated polyhedra. The resulting polyhedra with coordination numbers (CN) 12, 14, 15 and 16 are referred to as Frank-Kasper (FK) polyhedra. The different ways to arrange FK polyhedra in a periodic way leads to the TCP phases A15, s, c m, R, M, P, d, C14, C15, and C36 (see e.g. Ref. [45]). The c phase is also considered a TCP phase although it includes the CN-13 polyhedron. * Corresponding author. E-mail addresses: [email protected] (A.N. Ladines), thomas. [email protected] (T. Hammerschmidt), [email protected] (R. Drautz). http://dx.doi.org/10.1016/j.intermet.2014.12.009 0966-9795/© 2014 Elsevier Ltd. All rights reserved.

The structural stability of TCP phases in transition metal (TM) alloys is to a large degree driven by the interplay of atomic size and band filling [15,51,31,42,33,48]. In an alloy, the smaller constituent atoms are centered in CN-12 polyhedra while the larger ones tend to occupy the centers of CN-15 or CN-16 polyhedra. This atomicsize difference contributes significantly to the stabilization of the structures C14, C15, C36, also known as Laves phases. The CN-14 polyhedra can be associated with either atom type which implies a more flexible site occupation [12]. The influence of band filling has been observed in calculations of TCP phase stability across the TM series using density-functional theory (DFT) [3,46] as well as approximate electronic-structure methods [41,18,44,17,16,37]. The latter studies also highlighted two distinct groups of TCP phases that exhibit a similar trend in stability across the TM series: Laves and m phases are stabilized by size mismatch while band filling is shown to drive s, A15 and c phase formation. A structure map [43] incorporates these trends in terms of the valence-electron concentration N and atomic-size differences DV=V. Magnetism can play a significant role in the structural stability of Fe-based systems. While this is intensely studied for bcc and fcc based alloys, little is known about the impact of magnetism on TCP stability. For Febased TCP phases, the influence of magnetism has been investigated in detail only for few selected compounds, particularly the C14 phase for Fe2Nb [49,19], for (Fe1exMnx)2Nb [35], for

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(Fe1exCox)2Nb [36], and for Fe2TM across the 4d and 5d TM series [28], as well as the s phase for the FeeCr system [39,21]. In this study we carry out DFT calculations of TCP stability in Febased binary TM alloys. We analyze the effects of band filling, atomic-size difference, magnetism and external stress, for FeeNb/ Mo/V and consider the TCP phases A15, s, c, C14, C15, C36, and m in all occupancies of the Wyckoff sites. The influence of atomic-size differences on the stabilization of Laves phases across the TM series is then analyzed by comparing A15 and C14 as representatives of the distinct groups of TCP phases for all binary systems of Fe with earlier TM. We verify our findings by comparison to FeeOs and FeePd that do not form TCP phases. The outline of the paper is as follows: In Sec. 2, we briefly describe the relevant aspects of the DFT calculations and the structure map analysis. In Sec. 3, we compare the systems FeeNb/ Mo/V with respect to the stability, volume and elastic properties of the TCP phases A15, s, c, m, C14, C15 and C36. In Sec. 4 we discuss the influence of magnetic ordering and external pressure for the case of the closely competing Laves phases in Fe2Nb and analyze the influence of atomic-size differences by comparing A15 and C14 in binaries of Fe with earlier TM. Our findings are summarized in Sec. 5.

individually. The discussed volume expansion/contraction is based on the deviation from Vegard's law computed as

DVf ¼ VFex TMy  x$VbccFe  ð1  xÞ$VTM

(2)

with the volume of bcc-Fe and the TM ground-state structure as reference volumes. 2.2. Structure map Our DFT calculations are complemented by an analysis with the structure map [43] for TCP phases in TM alloys, shown in Fig. 1. The dashed regions demarcate experimentally observed TCP phases in terms of their corresponding valence-electron concentration N and atomic-size difference DV=V [43]. A compound with a chemical composition outside any TCP region is not expected to form a TCP phase. A system that forms TCP phases is expected to form those TCP phases that are intersected by its chemical composition. Considering a particular binary system A-B and varying its chemical composition from A to B corresponds to a parabola in the TCP structure map with

N ¼ xNA þ ð1  xÞNB

and

(3)

2. Methodology

DV=V ¼ 4xð1  xÞ

2.1. Density-functional theory For our self-consistent DFT total-energy calculations we used the VASP code [26,24,25] and PAW pseudopotentials [5] with p semi-core states for all elements with additional s semi-core states for Sc, Y and Zr. The generalized gradient approximation of the exchange correlation energy parametrized by Perdew, Burke and Ernzerhof [40] was employed. The formation energies are converged to approximately 1 meV/atom using a cut-off energy of 450 eV and a k-point mesh of 0.02/Å. We carried out spin-polarized calculations starting from a ferromagnetic spin configuration. Uncertainties should be expected due to the neglect of the orbital moment which can contribute as much as 10% on the magnetic moments of TCP phases [28]. A full relaxation of the unit cell was first performed using the method of Methfessel and Paxton [30] for Brillouin zone integration. The linear tetrahedron method with €chl corrections [6] was employed for subsequent total-energy Blo calculations around the relaxed unit cell volume. The BircheMurnaghan equation-of-state was implemented to determine the equilibrium energy, volume and bulk modulus. The elastic constants of the predicted stable phases of FeeNb/Mo/V (see Sec. 3) are obtained from a least-squares fit of the stress for a suitable set of deformations. All DFT calculations were carried out using our highthroughput DFT environment described in Ref. [16]. For the computation of the magnetic energy (Sec. 3.3), we used the relaxed structures from the spin-polarized calculations and employed them in additional non-spin-polarized calculations without further relaxation. The composition of the TCP phases is varied by permutating the occupation of each Wyckoff site with an Fe or TM atom. To compare with experimentally observed phase stability, we also included bcc, fcc and hcp-based ordered phases. We determined the stability of the compounds from the formation energy

DEf ¼ EFex TMy  x$EbccFe  ð1  xÞ$ETM

(1)

with respect to the elemental ground-state structures based on the DFT total energies after relaxation. To illustrate the competition of phases across the range of chemical composition, we determined the convex hull for all formation energies and for each phase

jVA  VB j : VA þ VB

(4)

The parabolas in Fig. 1 indicate the binary systems of Fe with earlier TM that are discussed in the following. We investigate the binary systems FeeNb/Mo/V in detail. The focus on these binary systems is motivated by their importance for steels for hightemperature applications [20,4]. In addition we considered the representative TCP phases A15 and C14 in order to analyze the influence of atomic size differences for all binaries of Fe with earlier TM. 3. Comparison of FeeNb/Mo/V 3.1. Structural stability For the binary systems FeeNb/Mo/V we considered the TCP phases A15, s, c, C14, C15, C36, and m. For each TCP phase in each binary we computed the formation energy DEf in all occupations of the inequivalent sublattices. The DFT results are compiled in Fig. 2, together with bcc, fcc and hcp-based ordered phases and the range of experimentally observed phase stability. The FeeNb system with a large value of DV=V intersects the Laves/m region while the large variation of N avoids the A15/c/s region as can be read from the structure map (Fig. 1). The trend in phase stability of bcc / Laves / m / bcc with increasing Nb content is in agreement with experiment, also with regard to the chemical composition of the stable phases as can be seen from Fig. 2(a). Our calculations indicate a close competition of the Laves phases with differences in DEf of only a few meV/atom. We find the C15 phase to be most stable for Fe2Nb while experiments report the C14 to be stable [38]. This discrepancy is discussed in detail in Sec. 4.1. The FeeV system with only a small size difference ðDV=V Fe0:5 V0:5 ¼ 0:08Þ is below the Laves/m region in the structure map, i.e., the Laves and the m phases are not stable. Instead, this system intersects the A15/c/s regions for nearly the full range of N. This is in line with the negative values of DEf that we obtain in DFT calculations for these phases across a broad range of chemical composition reported in Fig. 2(b). However, the competing bcc solid solutions rule out the TCP phases except for the s phase at

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61

s that are less than 50 meV/atom above the convex hull that could be overcome by entropy at high-temperature. However, to the best of our knowledge, no such TCP phases have been observed experimentally.

Fig. 1. Structure map of TCP phases [43] in terms of atomic-size differences DV=V and valence-electron concentration N of Fe-TM systems with stable TCP phases from DFT. The dashed lines indicate the boundaries of the stability region of the TCP phases. The symbols denote the predicted stable structures from our DFT calculations, i.e. A15 (△), C14 (▽), C15 (8), C36 (9), c (+), m (,), s ( ).

⋄

approximately 66 at.% V and the A15 phase at around 75 at.% V. While the s phase is found to be stable in experiment at the same stoichiometry as predicted by our calculations, the A15 does not appear in the FeeV equilibrium phase diagram. Our DFT calculations also show that the s phase is very close to the convex hull of the stable bcc for all Fe-rich compositions which explains the experimentally observed broadening of the s phase stability to 70 at.% Fe with increasing temperature. The FeeMo system, in contrast to FeeNb and FeeV, intersects both regions of TCP phase stability in the structure map. With a value of DV=V comparable to FeeNb it reaches the Laves/m phase stability while the lower bound of N positions the parabola in the A15/c/s regions for Fe-poor concentrations. This is in line with the Laves and m phases observed experimentally at low temperatures and the s phase found at temperatures above 1000 K. The R phase of FeeMo is included in the original TCP structure map [43] but not shown in Fig. 1. We excluded this phase from our discussion as it would require to consider 211 binary compositions in the DFT calculations. The calculated DEf for the FeeMo compounds are shown in Fig. 2(c). The chemical composition at the minima of the Laves and m phases are in agreement with the experimental ranges of stability although the corresponding values of DEf are close to zero. The s phase is reported to be stable at approximately 56 at.% Mo while our DFT calculations show the lowest value of DEf for the s phase at 63 at.% Mo. The positive value of DEf and the experimental finding of a high-temperature stability suggests that the Mo-rich s phase is stabilized by entropy. In fact, the convex hull of the s phase is close to the competing bcc solid solution. This explains the experimentally observed shift of the s phase stability towards the Fe-rich side with increasing temperature, similar to FeeV. Comparing the binary systems, we find that the increase of DV=V from FeeV to FeeNb at the same N decreases DEf of Laves/m phases considerably which makes them more stable than the A15/ c/s phases. The increase of atomic-size difference in going from FeeMo to FeeNb decreases the DEf of the Laves/m phases significantly and stabilizes them over the competing bcc solid solutions. The different range of N decreases DEf of A15/c/s but are still higher in energy than the competing Laves and m phases. Our investigations show several TCP structures such as the Fe-rich A15/c/

Fig. 2. Formation energy of TCP phases and bcc/fcc/hcp solid solutions in (a) FeeNb, (b) FeeV, and (c) FeeMo. The solid lines connect the low-energy compounds for each phase, the dotted line represents the convex hull.

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3.2. Equilibrium volume and elastic properties In order to compare our DFT calculations to the available experimental data, we compiled the lattice parameters of the stable TCP phases and find very good agreement, see Table 1. Part of the deviations may arise from the use of off-stoichiometric samples in the experiments and the approximations inherent in our DFT calculations. From our calculations we additionally obtain the atomic volumes of all low-energy compounds of the individual TCP phases (Fig. 2) that we compiled in Fig. 3. The volume of the TCP compounds follows approximately a linear relation with chemical composition. Overall, we also find a weak increase of the density of the TCP compounds with decreasing size difference between Fe and TM atom. From fitting the DFT calculations to the BircheMurnaghan equation-of-state we also obtain the bulk moduli that are shown for all low-energy compounds of the individual TCP phases (Fig. 2) in Fig. 4. The deviation from a linear relation is much more pronounced for the bulk modulus and also subject to considerable scatter. The observed trends in the atomic volumes and bulk moduli are related to the varying magnetic behavior of the TCP alloys. The deviation of the bulk moduli of the TCP phases from the elements is greater in the FeeV system. As discussed in Sec. 3.3, magnetism becomes weaker as we go from (Fe-)Nb / Mo / V alloys which is consistent with our observations of the trends in the volumes and bulk moduli. This influence of magnetism varies with structure and hence leads to comparably large scatter especially for Fe-rich alloys. 3.3. Magnetic energy: FeeNb/Mo/V We isolate the magnetic contributions to different TCP phases in FeeNb/Mo/V with varying chemical composition. The values of the average magnetic moment as a function of band filling are summarized in Fig. 5 only for the low-energy compounds for each structure. The overall effect is an increasing number of up-spin d electrons and hence an increasing magnetic moment with increasing Fe content as expected. The average magnetic moments are in line with the trend suggested by the generalized Slater-Pauling relation [52,27], also indicated in Fig. 5. This relation is motivated by rigid-band arguments with a different treatment of the DOS (see Fig. 6) for a socalled (i) split-band character with the Fermi level just above the d-states (as observed for >75 at% Fe) and (ii) a common-band character with the Fermi level in the minority d-states (as observed for <75 at% Fe). Therefore, in the Fe-rich part, the magnetization M in the generalized Slater-Pauling relation is given by M ¼ 2N[  Z ¼ Zm þ 0.6. For less than 75 at% Fe it is described by M ¼ Z6 if we take NY ¼ 3 since the Fermi level is fixed at the middle of the minority d-band states. Here, Zm is the magnetic valence defined as the negative chemical valence for the nonmagnetic element and a value of 2 for Fe.

Fig. 3. Atomic volume (symbols) of TCP phases and bcc/fcc/hcp solid solutions for all low-energy FeeNb/Mo/V compounds of the individual phases (Fig. 2) and Vegard's law (line). The circled symbols correspond to the stable phases.

The deviations from the Slater-Pauling relation can be traced to the sp-contribution to N[ that is not exactly 0.3 as assumed in the generalized Slater-Pauling relation and to the Fe-induced antiferromagnetic coupling as shown in Fig. 7. The magnetic moments depend on the degree of hybridization between Fe and Nb/Mo/V states that is approximately proportional to the inverse of the energy difference [28]. For the 3d element V, we expect the overlap of the Fe and V states to be larger than for the 4d elements Nb and Mo. This results in larger induced moments on V and therefore explains the slightly smaller net moment for FeeV alloys shown in Fig. 5. In order to relate the magnetic energy contribution to TCP phase stability, we performed additional non-spin polarized (NSP) calculations for the structures as obtained after relaxation with spinpolarized (SP) calculations. We take the magnetic energy as the difference between the total energies of the NSP and SP case. The results for the low-energy TCP compounds of each structure are shown in Fig. 8. The variation of the magnetic energy with band filling can be understood in terms of the Stoner model of ferromagnetism. Spin-polarization results from the intricate balance between two energy contributions arising from the transfer of

Table 1 Lattice parameters of experimentally observed FeeMo/Nb/V TCP phases in comparison to our DFT calculations. System

Structure

a(Å)

c(Å)

Reference

FeeNb

C14

4.798 4.841 4.891 4.928 4.720 4.744 4.760 4.751 8.985 9.015

7.919 7.893 26.770 26.830 7.639 7.725 25.829 25.680 4.587 4.642

This work Expt [38] This work Expt [38] This work Expt [1] This work Expt [1] This work Expt [47]

m FeeMo

C14

m FeeV

s

Fig. 4. Bulk modulus of TCP phases and bcc/fcc/hcp solid solutions for the low-energy FeeNb/Mo/V compounds on the DEf convex hull of the individual phases. The circled symbols correspond to the stable phases.

A.N. Ladines et al. / Intermetallics 59 (2015) 59e67

Fig. 5. Average magnetic moment of FeeNb/Mo/V alloys. The dashed lines correspond to the two branches of the generalized Slater-Pauling curves. The line with the larger (smaller) slope corresponds to the equation M ¼ Zm þ 0.6(M ¼ Z  6) where Zm (Z) is the magnetic (chemical) valence. The circled symbols correspond to the stable phases.

electrons from the minority spin bands to the majority spin bands. The exchange part of the potential energy lowers the energy of the system favoring parallel spins. At the same time the single-particle kinetic energy is increased as states above the Fermi level are occupied. The Stoner model relates the associated magnetic energy to the square of the magnetic moment. As the magnetic moments roughly vary linearly with the average band filling (see Fig. 5), we expect the magnetic energy to be roughly proportional to the square of the band filling. This is consistent with our DFT results as indicated in Fig. 5. The apparent deviations can be attributed to the approximations of the Stoner model [29] as well as its poor description of the small magnetic moments of the Laves phases and large magnetic moments of the A15 phase. For m, s and c, we find that the magnetic energy for the same composition decreases as Nb / Mo / V which indicates the influence of the TM size. 4. Stability of Laves phases 4.1. Magnetic ordering in Fe2Nb

63

Fig. 7. Average magnetic moment projected on Fe and Nb/Mo/V sites in FeeNb/Mo/V alloys as function of the average band filling. Positive(negative) moments belong to the Fe(Nb/Mo/V) sites. The solid line represents a linear fit to the Fe moments in the m phase.

The near degenerate values of the formation energies are attributed to the similarities of the crystal structures, shown in Fig. 9. C15 (shown in the hexagonal representation along the <111> direction) has a cubic structure while C14 and C36 are both hexagonal. The three Laves phases are composed of the same set of FK polyhedra (75% CN-12 and 25% CN-16) but vary only on the stacking sequence: aþb in C14, aþbþcþ in C15 and aþbacþ in C36 where þ and  denote the orientation of the D elements of the Kagome net while a and b refer to the main layer configurations. For the case of Fe2Nb with the most pronounced Laves phase stability, we investigate the energetic competition of C14, C15 and C36 in more detail. We analyzed the influence of magnetic ordering on the relative stability by DFT calculations with different arrangements of the spin orientations in addition to the ferromagnetic ordering used in Sec. 3. In particular, we considered all permutations of up/down spin on the symmetry inequivalent Wyckoff sites for the Laves (Fig. 9) and m phases. A further analysis of non-collinear magnetism, however, is beyond the scope of this work. The resulting formation energies from DFT for the different considered magnetic configurations of Fe2Nb Laves phases are shown in Fig. 10, together with the values of

Our results for the formation energies (Fig. 2) suggest a very close energetic competition of the Laves phases in Fe2(Nb/V/Mo).

Fig. 6. Local density of states of A15-Fe3X (top) and C14-Fe2X (bottom) for FeeNb(left)/ Mo(middle)/V(right) projected on the Wyckoff sites.

Fig. 8. Magnetic energy as function of the average band filling. The solid lines correspond to a quadratic fit to the data. The circled symbols correspond to the stable phases.

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A.N. Ladines et al. / Intermetallics 59 (2015) 59e67

Fig. 9. Crystal structure of Laves phases with colored Wyckoff positions.

the Fe7Nb6 m phase for comparison. The values of DEf of the Laves phases are very sensitive to magnetic ordering in contrast to the m phase with a nearly degenerate energy for different magnetic orderings. In our DFT calculations, the antiferromagnetic configuration stabilizes C14 and C36 with respect to the ferromagnetic case by approximately 15 meV/atom. Previous LAPW results [49] found a larger value of 40 meV/atom for the maximum energy difference between magnetic orderings in C14. For C15, in contrast, the AFM configuration is nearly 30 meV/atom less stable than the FM case. Although C14 is stabilized by magnetism, it is not the ground-state structure in our DFT calculations. The energy shifts due to magnetism rather lead to a change of the energetic ordering of the Fe2Nb Laves phases from C15 < C36 < C14 to C36 < C14 < C15. The minimum energy spin configuration of C14 obtained from our DFT calculations is shown in Fig. 11. We find the ferrimagnetic spin configuration for C14 to be more stable than the ferromagnetic case in agreement with previous LMTO [19] and LAPW [49] calculations, albeit with slightly larger values for the magnetic moments. The same spin configuration also yielded the minimum energy for C36 but extending the set of spin configurations for C14 to include the case where the spins on the 6 h sites are anti-parallel lowers the energy by about 3 meV/atom. This spin configuration was also investigated in Ref. [49] but they found the spins on the 2a sites to vanish. Moreover, we find that the moment on the 6 h site is 1.6 mB compared to their result of 1.1 mB. In fact, the Fe2Nb Laves phase C14 is known to exhibit a complex magnetic structure due to details of the Fermi surface [2]: while C14-Fe2Nb is paramagnetic above approximately 70 K, at T ¼ 0 K it is close to a quantum-critical point with a spin-density wave at stoichiometric composition and stable ferromagnetic ordering for few at.% off-stoichiometric composition [8,32]. C14 phases with complex magnetic structure have also been observed experimentally in a broad range of compositions x in alloyed (Fe1exMnx)2Nb [35] and (Fe1exCox)2Nb [36] systems.

Fig. 10. Formation energy of Fe2Nb Laves and Fe7Nb6 m for different spin configurations as a function of the total magnetic moment normalized by the number of Fe atoms in the unit cell. The different spin configurations were generated by assigning up (U) or down (D) spin for each inequivalent Wckoff site. In the case of C14, other spin configurations were considered such as C14.(UD) (UD)U which correspond to the case where the 2a atoms are antiferromagnetically ordered and the spins of the 6 h atoms on adjacent planes are oriented in opposite directions. The formation energies of nonmagnetic (NM) calculations are included for comparison.

that the ordering of Laves phases is altered already at moderate hydrostatic pressure. The C14 phases becomes more stable compared to C36 at z600 MPa of pressure. This phase transition with pressure can be attributed to the lower bulk modulus of Fe2Nb in the C14 phase (168 GPa) compared to the C36 phase (188 GPa) in their corresponding lowest energy magnetic configurations. The difference in the bulk moduli can also be seen in Fig. 12 where a larger change in the volume with pressure can be observed for C14. The relative stability of the different magnetic orderings of C14 changes with pressure. The energetically most stable structure changes from antiferromagnetic ordering of the 2a atoms at no pressure to parallel alignment of the 2a spins at 500 MPa. However,

4.2. External pressure in Fe2Nb We further examined the effect of hydrostatic pressure on the structural ordering of the energetically closely competing Fe2Nb Laves phases. In particular, we computed the formation enthalpy H as a function of external hydrostatic pressure P by

HðPÞ ¼ EðVðPÞÞ þ P*VðPÞ

(5)

using the parameterized BircheMurnaghn equations-of-state. The resulting variation of formation enthalpy with pressure for the Fe2Nb Laves phases C14, C15 and C36 is shown in Fig. 12. We find

Fig. 11. Magnetic configuration with lowest formation energy of C14- and C36-Fe2Nb from our DFT calculations. The length of the arrows are scaled with size of the magnetic moment. The moments on the Nb atoms are not visible as these are around 0.4 mB. The C15 phase (not shown) is found to be ferromagnetic.

A.N. Ladines et al. / Intermetallics 59 (2015) 59e67

Fig. 12. Effect of hydrostatic pressure P on the formation enthalpy (left) and volume (right) of Fe2Nb Laves phases with different spin orientations.

the value of the transition pressure may be altered by the magnetic configuration and by pressure-dependencies of the elastic constants reported, e.g., for C14-Cr2Nb [50]. Although we did not exhaust all possible magnetic ordering, our findings seem to suggest that the Fe2Nb C14 is stabilized by pressure with respect to the other Laves phases. The elastic constants of the experimentally observed FeeNb/ Mo/V TCP phases are compiled in Table 2, together with the values of C15 and C36 for FeeNb for comparison. To the best of our knowledge, there are no available experimental data for these systems. However, other DFT calculations for the chemically similar C14-Cr2Nb observed similar values of the elastic constants [50]. In addition, the value of the Debye temperature derived from the elastic constants for FeeV s is in agreement with the value reported in Refs. [10,9]. The values are relatively large which is consistent with the fact that these phases have very low deformability. Using the Born-Huang criteria for mechanical stability [7], we expect that these alloys are mechanically stable.

65

very small values of DEf only in the range of approximately N ¼ 5:0  7:5. The corresponding range of (nearly) stable C14 alloys is larger with approximately N ¼ 4:6  7:7. This is consistent with the range of N in the corresponding stability regimes in the structure map (Fig. 1). The variation of DEf with DV=V shows a similar trend across the 4d and 5d TM series while in the case of 3d many systems are shifted to smaller DV=V due to the smaller atomic size. For A15, the value of DEf across the 4d and 5d is fairly constant with increasing DV=V. For the size-compound C14, however, the formation energy is systematically deceasing with increasing difference in atomic size DV=V. This size-stabilization of C14 vanishes for approximately DV=V < 0:15 (DV=V < 0:05 for 3d TM) and for approximately DV=V > 0:43. The bounds of the resulting region of possible Laves phase stability that we obtain from our DFT calculations (4:6 < N < 7:7 and 0:1 < DV=V < 0:43) are in very good agreement with the structure map. This is also in agreement with the observation of binary Laves phase formation for 1.05 < rA/ rB < 1.67 [54] which correspond to roughly 0:06 < DV=V < 0:57. The different influence of DV=V on A15 and C14 is also apparent in the equilibrium volume as plotted in Fig. 14 as a function of composition. We see a significant contraction of the stable C14 Fe2TM and a volume expansion of the unstable TM2Fe compounds. The A15 compounds, in contrast, show a nearly constant volume across the composition range. The value of the c/a ratio for C14 varies around the ideal ratio of 1.63 between the minimum value of 1.59 (Fe2Mn) and the maximum value of 1.67 (Fe2Zr). For C36 we find values of

4.3. Trend across TM series In order to understand the influence of atomic size differences on the stabilization of Laves phases across the TM series, we compare A15 and C14 as representatives of the two distinct groups of TCP phases A15/s/c and Laves/m that are separated by the value of DV=V in the structure map for all binary compounds of Fe with earlier TM. We computed the formation energy DEf for all occupations of the Wyckoff sites and we compiled the values of DEf for the low-energy compounds as a function of N and DV=V in Fig. 13. The values of DV=V are determined with the atomic volumes of the respective TM ground-state structure as obtained by our DFT calculations. We find that the variation of DEf with N is similar for the 3d, 4d and 5d TM series for C14 while lower values of DEf are observed for A15 with 3d TM. The A15 phase exhibits negative or

Table 2 Elastic constants (in GPa) of stable ferromagnetic FeeX alloys. System

Structure

C11

C12

C13

C33

C44

C66

FeeNb

C14 C15 C36

268 276 297 313 445 335 364

100 144 130 136 160 111 157

116

339

41

106 122 171 124 147

336 322 371 354 360

59 109 58 44 80 59 78

m FeeMo

C14

FeeV

m s

42 14 72 18 100

Fig. 13. Formation energy of lowest energy A15 (left) and C14 (right) compounds for all TCP-forming Fe-TM systems as a function of the average electron count, N and the relative volume difference, DV=V.

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A.N. Ladines et al. / Intermetallics 59 (2015) 59e67

Fig. 14. Equilibrium volume of A15 and C14 low-energy alloys for all TCP-forming FeTM binaries as a function of increasing TM ratio x. Symbols denote the A15 (△) and C14 (;) phases.

the c/a ratio between 3.25 (Fe2Mn) and 3.29 (Fe2W). In both cases we find no systematic trend of the c/a ratio with band filling, size difference or heat of formation. The structure map suggests that no TCP phases should be stable for Na8, i.e. for Fe compounds with later TM. We verify this exemplarily by DFT calculations for FeeOs that is just outside the region of TCP phase stability in the structure map and for FeePd as an example of a system further away from the TCP regions. Both systems are known to be free of TCP phases. In order to quantify the energy difference to the stable phases we determine the formation energy of all TCP phases across the full range of chemical composition. Our results confirm the absence of a stable TCP phase in FeeOs by positive values of the formation energy for all TCP phases. In FeePd we find negative values of DEf for the s, c and m phase. While these are superseded by the more stable fcc and bcc solid solutions, they are unexpectedly close to the convex hull, only 10 meV/atom for the Fe7Pd6 m-phase. A rough estimate of the sequence of phase stability in these systems is obtained by additional calculations of DEf for the remaining TCP phases at the ideal1 occupancies together with several bcc, fcc and hcp ordered structures. We find that nearly all stable phases predicted by DFT are within the stability regions specified by the structure map as indicated in Fig. 1. Only the predicted FeeTa and FeeNb m phases are located slightly outside the m region in the structure map which we attribute to the fact that we only considered the 7:6 composition for the Fe-TM m phase that is most frequently observed. Our DFT calculations do not show the s phase to be stable in the FeeRe and FeeTc systems although it is observed experimentally. This matches previous DFT calculations [11] which showed that configurational entropy can stabilize the s phase in FeeRe. 5. Conclusions Employing DFT calculations and an empirical structure map we analyze topologically close-packed phases in all TCP-forming Febased binary transition metal compounds. In particular, we

1 Since the Fe atom is always the smaller atom, we surmise that the lowest energy configuration is always the case when the Fe atom atoms occupy the CN-12 sites while the CN-15 and CN-16 sites are occupied by the early TM. In cases where CN-14 sites are present, both cases were considered.

determine the influence of atomic size, electron count, magnetism and external stress on TCP phase stability. For the FeeNb/Mo/V we considered the TCP phases A15, s, c, m, C14, C15 and C36. We find that the predicted stable alloys are consistent with experiment. The Laves and m phases were found to be stable or close to the convex hull in both the FeeNb and FeeMo systems. In the FeeV system, the s and A15 phase were predicted to be stable. A linear variation of the volume was observed as a function of the concentration of the refractory element with a slight volume contraction of the stable TCP phases. The trends of the bulk moduli was shown to be related to magnetism in TCP phases. The analysis of the magnetic moment indicates that the influence of magnetism on the phase stability of TCP phases is strongly affected by the interplay of band filling and crystal structure. We further investigated the stabilizing factors for the energetically closely competing Laves phases. For Fe2Nb, we observed that the influence of antiferromagnetism is absent for the cubic Laves but considerable for the hexagonal Laves phases. The influence of external pressure is sufficiently large to alter the most stable Laves phase from C36 to C14 for pressures of only a few hundred MPa. The C14 phase as a representative Laves phase is furthermore contrasted to the A15 phase in order to pinpoint the influence of atomic size on TCP phase stability. Our DFT calculations of these phases for all binary systems of Fe with earlier transition metals show that the A15 phase is not stabilized by atomic-size differences while the stability of C14 is increasing with increasing difference in atomic size as expected. The region of negative formation energies of Laves phases in terms of N and DV=V is in good agreement with the corresponding region in the structure map.

Acknowledgments We acknowledge fruitful discussion with Stephan Huth and Werner Theisen. We acknowledge financial support through ThyssenKrupp AG, Bayer MaterialScience AG, Salzgitter Mannesmann Forschung GmbH, Robert Bosch GmbH, Benteler Stahl/ Rohr GmbH, Bayer Technology Services GmbH and the state of North-Rhine Westphalia as well as the EU in the framework of the ERDF. ANL acknowledge the Philippine Department of Science and Technology Science Education Institute for support. This work was carried out within the International Max-Planck Research School SurMat.

Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.intermet.2014.12.009.

References [1] Sinha AK, Buckley RA, Hume-Rothery W. Equilibrium diagram of the irone molybdenum system. J Iron Steel Inst 1966;205:191e5. [2] Alam A, Johnson DD. Chemically mediated quantum criticality in NbFe2. Phys Rev Lett 2011;107:206401. [3] Berne C, Pasturel A, Sluiter M, Vinet B. Ab initio study of metastability in refractory metal based systems. Phys Rev Lett 1999;83(8):1621e3. [4] Bewlay BP, Jackson MR, Subramanian PR. Processing high-temperature refractory-metal silicide in-situ composites. J Oncol Manag 1999;51(4):32e6. €chl PE. Projector augmented-wave method. Phys Rev B 1994;50(24): [5] Blo 17953e79. €chl PE, Jepsen O, Andersen OK. Improved tetrahedron method for Brillouin[6] Blo zone integrations. Phys Rev B 1994;49(23):16223e33. [7] Born M, Huang K, Born M, Born M, Britain G. Dynamical theory of crystal lattices. Clarendon Press Oxford; 1954. [8] Brando M, Duncan WJ, Moroni-Klementowicz D, Albrecht C, Grüner D, Ballou R, et al. Logarithmic fermi-liquid breakdown in NbFe2. Phys Rev Lett 2008;101(2):026401.

A.N. Ladines et al. / Intermetallics 59 (2015) 59e67 [9] Cieslak J, Dubiel S. The Debye temperature of sigma-phase Fe-Mo compounds €ssbauer spectroscopy. Intermetallics 2013;34(0): as determined with Mo 159e61. [10] Cieslak J, Reissner M, Dubiel SM, Costa BF, Steiner W. On the Debye temperature in sigma-phase Fe-V alloys. Intermetallics 2010;18(8):1695e8. [11] Crivello J-C, Breidi A, Joubert J-M. Chi and sigma phases in binary rhenium transition metal systems: a systematic first-principles investigation. Inorg Chem 2013;52(7):3674e86. [12] Crivello J-C, Joubert J-M. First principles calculations of the sigma and chi phases in the Mo-Re and W-Re systems. J Phys Condens. Matter 2010;22(3): 035402. [13] Dimmler G, Weinert P, Kozeschnik E, Cerjak H. Quantification of the Laves phase in advanced 9-12% Cr steels using a standard SEM. Mater Charact 2003;51(5):341e52. [14] Frank FC, Kasper JS. Complex alloy structures regarded as sphere packings. I. Definitions and basic principles. Acta Crystallogr Mar 1958;11(3):184e90. [15] Hafner J. Structure, bonding, and stability of topologically close-packed intermetallic compounds. Phys Rev B 1977;15(2):617e30. [16] Hammerschmidt T, Bialon AF, Pettifor DG, Drautz R. Topologically closepacked phases in binary transition-metal compounds: matching highthroughput ab initio calculations to an empirical structure map. New J Phys 2013;15(11):115016. [17] Hammerschmidt T, Seiser B, Cak M, Drautz R, Pettifor DG. Structural trends of topologically close-packed phases: understanding experimental trends in terms of the electronic structure. In: Huron ES, Reed RC, Hardy MC, Mills MJ, Montero RE, Portella PD, et al., editors. Superalloys 2012. The Metals, Minerals and Materials Society; 2012. p. 135. [18] Hammerschmidt T, Seiser B, Drautz R, Pettifor DG. Modelling topologically close-packed phases in superalloys: valence-dependent bond-order potentials based on ab-initio calculations. In: Reed RC, Green K, Caron P, Gabb T, Fahrmann M, Huron E, et al., editors. Superalloys 2008. The Metals, Minerals and Materials Society; 2008. p. 847. [19] Haynes T, Maskery I, Butchers MW, Duffy JA, Taylor JW, Giblin SR, et al. Ferrimagnetism in Fe-rich NbFe2. Phys Rev B 2012;85:115137. v V, Leonte va-Smirnova M, Votinov S, Shamardin V, [20] Ioltukhovsky A, Kondrate Povstyanko A, et al. Metallurgical aspects of possibility of 9-12% chromium steel application as a structural material for first wall and blanket of fusion reactors. J Nucl Mater 1996;233:299e304. [21] Kabliman E, Blaha P, Schwarz K, Peil O, Ruban A, Johansson B. Computational thermodynamics of the FeeCr s phase. Phys Rev B 2011;84:184206. [22] Kazantzis A, Aindow M, Jones I, Triantafyllidis G, De Hosson J. The mechanical properties and the deformation microstructures of the C15 Laves phase Cr2Nb at high temperatures. Acta Mater 2007;55(6):1873e84. [23] Kobayashi S, Kimura K, Tsuzaki K. Interphase precipitation of Fe2Hf Laves phase in a Fe9Cr/Fe9CrHf diffusion couple. Intermetallics 2014;46:80e4. [24] Kresse G, Furthmüller J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comp Mater Sci 1996a;6(1):15e50. [25] Kresse G, Furthmüller J. Efficient iterative schemes for ab-initio total-energy calculations using a plane-wave basis set. Phys Rev B 1996b;54(16): 11169e86. [26] Kresse G, Hafner J. Ab initio molecular dynamics for liquid metals. Phys Rev B 1993;47(1):558e61. [27] Kübler J. First principle theory of metallic magnetism. Phys BþC 1984;127(13):257e63. [28] Kumar P, Kashyap A, Balamurugan B, Shield JE, Sellmyer DJ, Skomski R. Permanent magnetism of intermetallic compounds between light and heavy transition-metal elements. J Phys Condens. Matter 2014;26(6): 064209. [29] Marcus PM, Moruzzi VL. Stoner model of ferromagnetism and total-energy band theory. Phys Rev B 1988;38:6949e53.

67

[30] Methfessel M, Paxton AT. High-precision sampling for Brillouin-zone integration in metals. Phys Rev B 1989;40(6):3616e21. [31] Morinaga M, Yukawa M, Adachi H, Ezaki H. In: Gell M, Kortovich CS, Bricknell RH, Kent WB, Radavich JF, editors. Superalloys 1984. The Metals, Minerals and Materials Society; 1984. p. 523. [32] Moroni-Klementowicz D, Brando M, Albrecht C, Duncan WJ, Grosche FM, Grüner D, et al. Magnetism in Nb(1-y)Fe(2þy): composition and magnetic field dependence. Phys Rev B 2009;79(22):224410. [33] Ohta Y, Pettifor DG. Size versus electronic factors in transition metal Laves phase stability. J Phys Condens Matter 1990;2(41):8189. [34] Padilha AF, Machado IF, Plaut RL. Microstructures and mechanical properties of Fe15% Cr15% Ni austenitic stainless steels containing different levels of niobium additions submitted to various processing stages. J Mater Proc Tech 2005;170(12):89e96. [35] Paduani C, Ardisson JD, Schaf J, Takeuchi AY, Yoshida MI. Structure and magnetic properties of the C14 laves phase (Fe1-xMnx)2Nb intermetallics. Intermetallics 2008;16:384. [36] Paduani C, Schaf J, Persiano AIC, Ardisson JD, Takeuchi AY, Raposo MT. Evidence of weak ferromagnetism in the C14 Laves phase (Fe1exCox)2Nb. J Alloys Comp 2009;474:34. [37] Palumbo M, Fries S, Hammerschmidt T, Abe T, Crivello J-C, Breidi A, et al. Firstprinciples-based phase diagrams and thermodynamic properties of TCP phases in Re-X systems (X ¼ Ta, V, W). Comp Mater Sci 2014;81:433. [38] Paul E, Swartzendruber LJ. The FeNb (iron-niobium) system. Bull Alloy Phase Diag 1986;7(3):248e54.  l J, Sob [39] Pavlu J, Vresta M. Ab-initio study of formation energy and magnetism of sigma phase in Cr-Fe and Cr-Co systems. Intermetallics 2010;18(2):212e20. [40] Perdew JP, Burke K, Ernzerhof M. Generalized gradient approximation made simple. Phys Rev Lett 1996;77(18):3865e8. [41] Pettifor DG, Podloucky R. Microscopic theory of the structural stability of pdbonded AB compounds. Phys Rev Lett 1984;53:1080e3. [42] Phillips R, Carlsson AE. Universal stabilizing mechanism for transition-metal polytetrahedrally packed phases. Phys Rev B 1990;42(6):3345e50. [43] Seiser B, Drautz R, Pettifor D. TCP phase predictions in Ni-based superalloys: structure maps revisited. Acta Mater 2011a;59(2):749e63. [44] Seiser B, Hammerschmidt T, Kolmogorov AN, Drautz R, Pettifor DG. Theory of structural trends within 4d and 5d transition metal topologically close-packed phases. Phys Rev B 2011b;83(22):224116. [45] Sinha AK. Topologically close-packed structures of transition metal alloys. Prog Mater Sci 1972;15(2):81e185. [46] Sluiter M. Lattice stability prediction of elemental tetrahedrally close-packed structures. Acta Mater 2007;55(11):3707e18. [47] Smith JF. The FeV (iron-vanadium) system. Bull Alloy Phase Diag 1984;5(2): 184e94. [48] Stein F, Palm M, Sauthoff G. Structure and stability of Laves phases. Part I. critical assessment of factors controlling Laves phase stability. Intermetallics 2004;12(79):713e20. [49] Subedi A, Singh DJ. Band structure and itinerant magnetism in quantum critical NbFe2. Phys Rev B 2010;81:024422. [50] Sun F, Zhang J, Mao S, Han X. Structural, electronic and elastic properties of the C14 NbCr2 laves phase under hydrostatic pressure. Solid State Comm ̇ 2013;174:46. [51] Watson R, Bennett L. Transition-metal alloy formation. The occurrence of topologically close packed phases-I. Acta Metall 1984;32(4):477e89. [52] Williams AR, Moruzzi V, Malozemoff A, Terakura K. Generalized Slater-Pauling curve for transition-metal magnets. IEEE Trans Magn 1983;19(5):1983e8. [53] Yamamoto K, Kimura Y, Wei F-G, Mishima Y. Design of Laves phase strengthened ferritic heat resisting steels in the FeCrNb(Ni) system. Mat Sci Eng A 2002;329:249e54. [54] Zhu J, Liu C, Pike L, Liaw P. Enthalpies of formation of binary Laves phases. Intermetallics 2002;10(6):579e95.