The Fe–Ni system: Thermodynamic modelling assisted by atomistic calculations

Intermetallics 18 (2010) 1148e1162

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The FeeNi system: Thermodynamic modelling assisted by atomistic calculations G. Cacciamani a, *, A. Dinsdale b, M. Palumbo c,1, A. Pasturel d a

Dip. di Chimica e Chimica Industriale, Università di Genova, via Dodecaneso, 31, I-16146 Genova, Italy Materials Centre, National Physical Laboratory, Teddington, UK c Computational Materials Science Center, National Institute for Materials Science, Tsukuba, Japan d Laboratoire de Physique et Modélisation des Milieux Condensés (UMR 5493), CNRS, Grenoble, France b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 January 2010 Received in revised form 23 February 2010 Accepted 24 February 2010 Available online 23 March 2010

The FeeNi system is a key subsystem for several alloys with important applications. It has been thermodynamically assessed according to the CALPHAD methodology and using 0 K enthalpies of formation of the ordered phases resulting from ab-initio atomistic calculations. This allowed us to model both stable and metastable fcc-based ordered phases (L12 Fe3Ni, L10 FeNi and L12 FeNi3) in the framework of the compound energy formalism (CEF) by using a 4-sublattice model. The combined ab-initio and CALPHAD approach enabled us to predict low-temperature stable equilibria which are experimentally not accessible due to an extremely sluggish kinetics. A similar 4-sublattice model has also been used for the bcc-based ordered phases (D03 Fe3Ni, B2 and B32 FeNi, D03 FeNi3), which are metastable in the FeeNi system, but need to be reliably modelled in order to enable extrapolations to higher order systems such as AleFeeNi. Magnetic ordering, which is particularly important in this system, has also been thermodynamically described and the influence of magnetism on phase equilibria evidenced. Ó 2010 Elsevier Ltd. All rights reserved.

Keywords: B. Phase diagram First-principles calculations B. Order/disorder transformations E. Calphad FeeNi

1. Introduction FeeNi is a key system for the development of various alloys of technological interest. The most important FeeNi based alloys are Invar alloys (typically Fe-35%Ni with possible additions of Cr, Cu, Mo). Thanks to their very low coefficient of thermal expansion they are used for thermostatic bimetals, glass sealing, integrated circuit packaging, cathode ray tube shadow masks, composite molds/ tooling, and membranes for liquid natural gas tankers. FeeNi alloys richer in Ni (typically Fee48e50%Ni with possible additions of Co, Cr), because of their soft magnetic properties with very high saturation flux density, are used for read-write heads for magnetic storage, magnetic actuators, magnetic shielding and high performance transformer cores. Moreover Fe and Ni are important addition elements in many materials of technological interest such as maraging steels, shape memory alloys, bulk metallic glasses, alloyed aluminides, etc. The FeeNi system itself is interesting not only for these various applications but also from a fundamental point of view. Despite the large number of investigations, started at the end of the XIX century, some uncertainty still affects low-temperature phase relations * Corresponding author. Tel.: þ39 010 353 6130; fax: þ39 010 362 5051. E-mail address: [email protected] (G. Cacciamani). 1 Present address. ICAMS, STKS, Ruhr University Bochum, Germany. 0966-9795/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.intermet.2010.02.026

mainly because of the difficulty in reaching stable equilibrium at temperatures lower than about 300
C. Moreover FeeNi is a paradigmatic example of how structural and magnetic orderings interact and affect phase equilibria. From this point of view it may be considered one of the best test systems to check the effectiveness and efficiency of the models adopted to describe thermodynamic properties of the different phases and to calculate phase equilibria. To summarise, the aim of the present paper is many-fold. (1) To produce an accurate thermodynamic modelling of the system in the framework of the CALPHAD methodology [1, 2] and the Compound Energy Formalism (CEF) [3] in order to clarify phase equilibria at low temperature, which are still uncertain. This may be obtained on the basis of the available experimental data, but also using phase stability data from atomistic calculations. These are needed especially to get a reliable thermodynamic description of the FeeNi phases at low temperature, where an extremely sluggish kinetics makes experimental investigations impossible or very difficult. (2) To verify effectiveness and efficiency of the 4-sublattice models [4] recently adopted by the Calphad community to describe thermodynamic relations within families of phases related by structural ordering relations. In particular we will focus on the fcc (A1/L12/L10) and bcc (A2/B2/B32/D03) ordering, with special attention not only to the stable phase equilibria, but also to the

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metastable ones, because both stable and metastable equilibria need to be thermodynamically well described in order to be able to use the FeeNi description to calculate reliable phase relations in higher order systems. In this case too atomistic calculations combined with Calphad modelling are especially useful. (3) To highlight the role of magnetism in determining phase equilibria. This is made particularly easy by the CEF modelling, where the magnetic contribution to the Gibbs energy of a phase may be separated from the other contributions. 2. Literature overview The FeeNi phase diagram has been studied for more than a century, the first investigations being carried out by Osmond [5], Guertler and Tamman [6], Hegg [7] and Ruer and Schuz [8]. Critically assessed FeeNi phase diagrams were elaborated by March [9], Kubaschewski [10], Swartzendruber et al. [11], Yang et al. [12], and Kuznetsov [13]. A short discussion of the recent literature on the FeeNi system and an updated phase diagram, here reproduced in Fig. 1a, was reported by Cacciamani et al. [14], who took in particular consideration low-temperature phase equilibria proposed by [12], here reported in Fig. 1b. Owing to the high number of investigations, some uncertainty still affects low-temperature phase relations mainly because of the difficulty in reaching stable equilibria at temperatures lower than about 300
C. The main features of the FeeNi phase diagram are: the g-(Fe,Ni) solid solution, with an fcc (A1) structure, extending over the complete composition range at high temperature and forming a very narrow two-phase field with liquid; the high temperature d-Fe and low-temperature a-Fe solid solutions, with the bcc (A2) structure, which are restricted to a relatively narrow composition range close to pure Fe; the FeNi3 solid solution, with an ordered fcc (L12) superstructure (AuCu3 type), which forms congruently from A1 g-(Fe,Ni) just over 500
C and shows an appreciable solubility range. Both g-(Fe,Ni) and a-Fe undergo para- to ferromagnetic transitions with decreasing temperature and ferromagnetic g-(Fe,Ni)FM forms an eutectoid equilibrium with a-Fe and FeNi3: g-(Fe,Ni)FM $ a-Fe þ FeNi3. More detailed discussion of the available literature data is reported in the following description of the FeeNi thermodynamic assessment. To this end the critical assessment by Swartzendruber [11] will be taken as a starting point.


2.1. High temperature equilibria (T > 1000 C) No recent experimental investigations are available on high temperature FeeNi phase equilibria. Solid-liquid equilibria were investigated by several authors [6,15e21], mainly in the first half of the twentieth century. The g-(Fe,Ni) þ liquid two-phase field is very narrow with a minimum which is most probably located at approximately 1440
C and 66 at% Ni, according to the critical assessment by [11]. Phase boundaries between d-Fe and g-(Fe, Ni) were investigated by [15, 16, 21, 22]. A narrow peritectic equilibrium d-Fe þ liquid $ g-(Fe, Ni) occurs at about 1514
C with phase compositions of 3.5, 4.9 and 4.2 at% Ni, respectively [11]. 2.2. a-Fe þ g-(Fe,Ni) equilibria The a-Fe þ g-(Fe,Ni) two-phase field starts at 912
C, the a to g transformation temperature of pure Fe. The investigation of the a-Fe þ g-(Fe,Ni) phase equilibria is made difficult by the sluggishness of the a to g transformation at

Fig. 1. Critically assessed FeeNi phase diagram: (a) stable phase equilibria according to [14]; (b) low-temperature stable and metastable equilibria according to [12] with Curie temperature (dash-dotted line), metastable continuation of the miscibility gap (dashed lines), boundary of the spinodal region (dotted lines).

temperatures lower than about 800
C. Several experimental techniques were used: XRD [23e26], EPMA [27], STEM [28]. All researchers agree in finding a retrograde solubility of Ni in a-Fe with a maximum of about 5e6 at% Ni at about 500
C. Correspondingly, the Ni-rich boundary of the a-Fe þ g-(Fe,Ni) two-phase field becomes flatter as the temperature is decreased, indicating the presence of a low-temperature metastable miscibility gap in the g-(Fe,Ni) phase. The a-Fe þ g-(Fe,Ni) two-phase field ends at a monotectoid equilibrium g-(Fe,Ni)PM $ a-Fe þ g-(Fe,Ni)FM (where PM refers to a paramagnetic state and FM a ferromagnetic state) first proposed by Nishizawa [29] and subsequently calculated by Chuang et al. [30, 31], but not observed experimentally. 2.3. g-(Fe,Ni)PM þ g-(Fe,Ni)FM equilibria The second order paramagnetic to ferromagnetic transformation in pure Ni occurs at 354.3
C [32, 33]. With increasing Fe concentration the Curie temperature (TC) increases up to a maximum of about 610
C at about 66 at% Ni and then decreases [34e38]. In 1986 Chuang et al. [30, 31] calculated the FeeNi phase diagram by taking into account the magnetic contribution to the Gibbs energy of the

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g-(Fe,Ni) phase. Due to the appreciable slope of the TC curve, they

obtained a tricritical point at about 462
C and 48 at% Ni, followed by a monotectoid equilibrium g-(Fe,Ni)PM $ a-Fe þ g-(Fe,Ni)FM at 389
C with phase compositions of 40, 6 and 46 at% Ni, respectively. The monotectoid equilibrium was not reported by [11], who, however, set to 400
C the TC of g-(Fe,Ni) in equilibrium with a-Fe. No direct experimental information is available on the g-(Fe,Ni)PM þ g(Fe,Ni)FM two-phase field nor on the monotectoid equilibrium. Ref. [14] accepted the monotectoid temperature of 415
C proposed by [13]. 2.4. a-Fe þ g-(Fe,Ni)FM equilibria As a consequence of the monotectoid equilibrium an a-Fe þ g-(Fe, Ni)FM two-phase field should be stable below the invariant equilibrium. Zhang et al. [39] investigated martensitic FeeNi alloys aged in the 300e450
C temperature range and measured a-Fe and g-(Fe, Ni)FM phase compositions in this range. They did not find experimental evidence of the monotectoid equilibrium at temperature lower than 400
C. Combining the results of previous investigations [28, 30, 31, 39] with new measurements on meteoric samples Yang, Williams and Goldstein [12] concluded that the monotectoid equilibrium should occur at about 400
C with phase compositions of about 6 [28], 40 and 46 at% Ni [39] for a-Fe, g-(Fe,Ni)PM and g-(Fe, Ni)FM, respectively. As for equilibrium phase compositions in the a-Fe þ g-(Fe,Ni)FM two-phase field Yang et al. [12] considered the a-Fe compositions determined by [28] and g-(Fe,Ni)FM compositions determined by [39] as most reliable. 2.5. FeNi3 and related equilibria With decreasing temperature g-(Fe,Ni) at compositions around FeNi3 adopts the ordered L12 (AuCu3 type) structure. This ordering transformation has been investigated by several authors. It was first studied by [40e45] using XRD. [40e42] also measured magnetic properties, electrical resistance and hardness of heat treated alloys. More detailed studies were concentrated in the 65e80 at% Ni composition range by [46, 47] using Moessbauer spectroscopy, by [48] using neutron diffuse scattering on single crystals quenched from different temperatures, by [49, 50] using calorimetry, by [51, 52] using dilatometry, by [53] using carbonyl vapour pressure, and by [45,54e72]. On the basis of the literature data it can be concluded that ordering occurs through a first order transformation with a maximum at about 517
C and 72e73 at% Ni, i.e. a slightly offstoichiometric composition. According to [46, 47] there is a 15
C hysteresis in the ordering temperature, which may be explained by magnetic effects. The Ni-rich side of the g-(Fe,Ni) þ FeNi3 region is not well defined. According to [53] it extends between 87  4 and 92  4 at% Ni at about 300
C. On the other side, the existence of the eutectoid equilibrium g-(Fe,Ni) $ a-Fe þ FeNi3 at 345
C with g-(Fe,Ni) at 52 at % Ni was proposed by the same author, based on XRD and SEM observations and on the results by [73]. According to the critical assessment of [11] the eutectoid temperature is 347
C and the corresponding phase compositions are 49, 4.7 and 63 at% Ni, respectively. Slightly different values have been selected by [13]: 345
C and 53, 3.2 and 63.6 at% Ni, respectively. 2.6. Low-temperature equilibria (T < 400
C) Due to the very slow interdiffusion between Fe and Ni, equilibrium can hardly be reached at low temperature (below about 400
C). Phase equilibria in this region have been studied in two

ways: by investigating irradiated samples (in which irradiationinduced crystal defects may increase inter-diffusion) or by investigating meteoric samples which had cooled down very slowly in asteroid bodies over time periods of between 1 and 1000 million of years. It has been estimated [12] that, at temperatures lower than about 200
C, equilibrium cannot be reached even at geological time scale. For this reason [12] considered his observations of meteoric samples to reflect phase equilibria at 200  50
C. A detailed discussion of the stable and metastable FeeNi equilibria at low temperature is reported in the Yang et al.'s [12] assessment. It is mainly based on experimental results obtained by [28, 39, 74, 75] and by [12] himself, together with thermodynamic calculations by [30, 31]. Conclusions by [12] can be summarised as follows. - The tricritical point and the miscibility gap between g-(Fe, Ni)PM and g-(Fe,Ni)FM, proposed by [30, 31] are confirmed and the consequent monotectoid equilibrium occurs at about 400
C. Below this temperature, the miscibility gap and the related spinodal region are metastable. At about 200
C the metastable miscibility gap extends between about 8.5 and 50 at% Ni and the spinodal region between about 27 and 45 at% Ni. - The stability of the ordered L10, (AuCu type) FeNi phase previously proposed by [74, 75] is not confirmed. However, the eutectoid reaction g-(Fe,Ni)FM $ a-Fe þ FeNi3 at about 345
C is confirmed. At about 200
C a-Fe containing 3.5 at% Ni is in equilibrium with FeNi3 at about 64 at% Ni. - L10 FeNi can be obtained as a metastable phase at 50 at% Ni. It has a critical ordering temperature of about 320
C, as determined by [76]. - The existence of a low spin paramagnetic phase at about 25e30 at% Ni proposed as an equilibrium phase by [77] is not confirmed, but cannot be definitely excluded. 2.7. Martensitic transformations Martensitic transformations were reported in the FeeNi system from an austenitic g parent phase to a ferritic a product phase [78e80]. A review of experimental data on the martensitic reaction in several Fe-X systems, including X ¼ Ni, is reported in [81], while a review on the thermodynamic modelling of the martensitic transformation is reported in [82]. Measurements of the transformation temperature under different conditions evidenced that at least two different martensitic transformations take place in the FeeNi system [78, 81, 83]. A lath martensite and a plate martensite can be distinguished in an optical microscope. On a transmission electron microscope the lath martensite appears to be rich in dislocations, while the plate martensite is heavily twinned, suggesting a different formation mechanism. In [83] a third type of martensitic transformation was reported for low Ni content, but its nature is still unclear and controversial. Martensitic transformations are not considered any more in this paper. 2.8. Crystal structures and lattice parameters Crystal structures and lattice parameters of stable FeeNi phases are reported in Table 1 together with crystal structure data about metastable phases considered in this work. Room temperature lattice parameters of the g-(Fe,Ni) and a-(Fe,Ni) solid solutions according to [11] are reported in Fig. 2 as a function of the phase composition. It can be noticed, in particular, that the lattice parameter of g-(Fe,Ni) shows a maximum at about 38e40 at% Ni which corresponds to the composition range where the magnetic moment reaches its maximum value.

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Table 1 FeeNi solid phases considered in this work: crystal structure data, temperature and composition ranges of stability, sublattice models. Phase name, Structurbericht designation

Pearson symbol-prototype, Space group

Lattice parameters (nm) [13]

Temperature range (
C)

Composition range

Database phase name and Sublattice model

g-(Fe,Ni) A1

cF4 e Cu Fme3m

Fe: a ¼ 0.36467 Ni: a ¼ 0.35240 Fe50Ni50: a ¼ 0.3575

Fe: 1394-912 Ni: <1455

0e100 at% Ni

A1 (Fe,Ni), disordered part of FCC4

a-Fe, d-Fe A2

cI2 e W Ime3m

a-Fe: a ¼ 0.28665 d-Fe: a ¼ 0.29315

Fe: 1538e1394 and <912

FeeNi: 0e3 at% Ni at 1514
C 0e5 at% Ni at w500
C

A2 (Fe,Ni), disordered part of BCC4

Fe3Ni L12 FeNi L10 FeNi3 L12

cP4 e AuCu3 Pme3m tP4 e AuCu P4/mmm cP4 e AuCu3 Pme3m

e

Metastable

e a ¼ 0.35523

Metastable above room T <517

Fe3Ni D03 FeNi B2 FeNi B32 FeNi3 D03

cF16 e AlFe3 Fme3m cP2 e CsCl Pme3m cF16 e NaTl Fde3m cF16 e AlFe3 Fme3m

e

Metastable

e

Metastable

e

Metastable

e

Metastable

Fe2Ni C11f FeNi3 C11f

e P4/mmm e P4/mmm

e

Metastable

Not modelled

e

Metastable

Not modelled

FeeNi: 63 e w85 at% Ni

FCC4 (Fe,Ni)1/4 (Fe,Ni)1/4 (Fe,Ni)1/4 (Fe,Ni)1/4 with disordered contribution from A1

BCC4 (Fe,Ni,Va)1/4 (Fe,Ni,Va)1/4 (Fe,Ni,Va)1/4 (Fe,Ni,Va)1/4 with disordered contribution from A2

2.9. Magnetic properties

2.10. Thermodynamics

The Curie temperature has been determined in the 0e30 at% Ni range for A2 a-Fe by [34e36] and for A1 g-(Fe,Ni) in the 30e100 at% Ni by [34e38, 84]. In addition [37] showed that the TC, for the ordered FeNi3 structure was appreciably higher than those for the disordered g-(Fe,Ni) phase. Magnetic moments have been determined, in the abovementioned composition ranges, by [34, 38] for A2 a-Fe and by [34, 38,85e87] for A1 g-(Fe,Ni), respectively.

Thermodynamic measurements in the FeeNi system up to 1989 have been discussed by [11] and more recent investigations have been described in [14]. Two phases have been mainly investigated: liquid and g-(Fe,Ni) at temperatures generally higher than 1000
C. For both phases several authors have measured the integral enthalpy of mixing and activity, or activity coefficient of the component elements. A selection of thermodynamic measurements reported in literature is summarised in Tables 2a and b.

2.11. Atomistic calculations

Fig. 2. Lattice parameters as a function of composition for the bcc (A2) and fcc (A1) phases at room temperature from different authors reported by [11].

The theoretical investigation of magnetic and thermodynamic properties in pure Fe and in binary FeeNi alloys has attracted great interest in the ab-initio community both because of the importance of Fe alloys and the complexity of phenomena involved with magnetic and chemical ordering and the Invar effect. In all calculations magnetism has been found to play a crucial effect and neglecting magnetism in first-principles calculations of these alloys can lead to misleading results. Mohri and co-workers [105e107] have systematically investigated the behaviour of L10 and L12 fcc order-disorder transformations in the FeeNi, Fe-Pd and Fe-Pt systems by means of first-principles calculations, using the full potential linearised augmented plane wave (LAPW) method [108], Cluster Expansion Method (CEM) [109] and Cluster Variation Method (CVM) [110]. In the FeeNi system, they found that, besides the L12 FeNi3 ordered phase, the L10 is also stable at low temperature with a transition temperature around 250
C. The inclusion of vibrational effects has confirmed this result in the most recent work [107], where the calculated transition temperature has been reduced by about 40 K. According to these papers, the L10 ordered phase is thus stable in the FeeNi system, even if a full equilibrium state is never achieved due to sluggish kinetics.

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Table 2 a e list of thermodynamic measurements on the FeeNi liquid phase considered in this work Reference

Measured quantity

Temperature
C

Experimental Method

Lyubimov et al. [88] Speiser et al. [89] Zellars et al. [90] Onillon et al. [91] Belton et al. [92] Predel et al. [93] Tsemekhman et al. [94] Mills et al. [95] Batalin et al. [96] Kubaschewski et al. [97] Conard et al. [98] Maruyama et al. [99] Iguchi et al. [100] Rammensee et al. [101] Waseda et al. [102] Tomiska [103] Thiedemann et al. [104]

Activity of Fe and Ni Activity of Fe and Ni Activity of Fe and Ni Activity of Ni Activity of Fe and Ni Enthalpy of mixing Activity of Fe and Ni Activity of Fe and Ni Enthalpy of mixing Activity of Fe and Ni Activity of Fe and Ni Activity of Fe and Ni Enthalpy of mixing Activity of Fe and Ni Activity of Fe and Ni Activity of Fe and Ni Enthalpy of mixing

1190, 1310, 1430 1510e1600 1600 1552e1657 1550e1650

Knudsen cell þ mass spectrometry Knudsen cell þ vapour analysis Vapour pressure Vapour pressure Knudsen cell þ mass spectrometry Calorimetry Equilibration þ vapour analysis Equilibration þ vapour analysis calorimetry Knudsen cell þ mass spectrometry Knudsen cell þ mass spectrometry equilibration þ vapour analysis calorimetry Knudsen cell þ mass spectrometry X-ray diffraction Knudsen cell þ mass spectrometry Levitation direct calorimetry

1600 1905e2285 1500e1650 Meltinge1630 1600 1566 Meltinge1650 1560 1600 1627, 1640

b e List of thermodynamic measurements on the g-(FeeNi) phase considered in this work Reference

Measured quantity

Temperature
C

Experimental Method

Steiner et al. [142] Dench [143] Kubaschewski et al. [144] Trinel-Dufour et al. [147] Dalvi et al. [148] Robinson et al. [149] Grimsey et al. [145] Kubaschewski et al. [97] Ono et al. [150] Conard et al. [98] Rammensee et al. [101]

Enthalpy of formation Enthalpy of formation Enthalpy of formation Activity of Fe Activity of Fe and Ni Activity of Ni Activity of Fe Activity in the fcc phase Activity of Fe Activity of Fe and Ni Activity of Fe and Ni

850 1050 1300 1000 792e1107 930e1380 1200 1370e1605 750e1150 1230emelting 1100emelting

Sn solution calorimetry Direct calorimetry Direct calorimetry Equilibration in CO þ CO2 atmosphere Equilibration in CO þ CO2 atmosphere Knudsen cell þ mass spectrometry Equilibration in CO þ CO2 atmosphere Knudsen cell þ mass spectrometry e.m.f. Knudsen cell þ mass spectrometry Knudsen cell þ mass spectrometry

Interestingly, a systematic variation in the appearance/disappearance of the L10 and L12 ordered compounds is observed in the Fe-X (X ¼ Ni, Pd, Pt) systems, with the L10 FePd phase appearing to be stable in the Fe-Pd phase diagram as well as the L12 FePd3 phase, while in the Fe-Pt system all L12 Fe3Pt, L10 FePt, and L12 FePt3 ordered phases appear to be stable. Several atomistic computer simulations [111e113] have been performed in the FeeNi system by applying semi-empirical potentials of the embedded-atom method (EAM) [114]. More recently, Mishin et al. [115] have developed a modified EAM method, with angular dependent potentials, for FeeNi alloys based on first-principles calculations of some ordered FeeNi compounds using full potential linearised augmented plane wave (LAPW) method within the PW-91 GGA approximation [116]. They have calculated several competing ordered structures with different magnetic ordering, mostly ferromagnetic, at 0 K. According to their results, both the L12 FeNi3 and L10 FeNi ordered phases appear to be stable at 0 K. In this work, the authors also suggest the possible existence of two more ordered phases with compositions Fe2Ni and FeNi2 and structure C11f (the fcc-based analogue of the bcc-based MoSi2 type structure) obtained by stacking Fe and Ni (002) layers according to the sequences NieFeeFeeNieFeeFe and FeeNieNieFeeNieNi, respectively. Their conventional unit cell is a triple size fcc super-cell (a  a  3a) along the [001] direction. According to the angular EAM potentials developed by Mishin et al., these phases are stable at 0 K. The stability of the L10 FeNi phase in addition to the L12 FeNi3 ordered structure is also confirmed by atomistic calculations carried out by Lechermann et al. [117], who performed DFT calculations in the GGA-PBE approximation using their own ab-initio mixed-basis pseudopotential (MBPP) code. They have also applied the Cluster Expansion Method to the fcc and bcc lattice in the ternary FeeNieAl and all binary subsystems and they have calculated the finite temperature ternary phase diagram by means of the Cluster Variation Method. The Cluster Expansion of the fcc lattice confirms the stability

of L10 FeNi ordered compound at 0 K in relation to the bcc and other ordered fcc phases. A significant number of works in the literature deals with magnetic structures in FeeNi alloys ([118e120, 129, 130] and references therein]). For example, it has been pointed out that at 0 K there is a continuous transition from the high-spin ferromagnetic configuration at large unit cell volumes to a disordered non-collinear configuration at smaller volumes in the fcc FeeNi phase [118, 119]. In [120], several possible magnetic configurations (ferromagnetic, antiferromagnetic, ferromagnetic and spin spirals) have been considered in fcc FeeNi alloys. It has been shown that the difference in energy between different competing magnetic configurations is rather small and the determination of the exact ground state magnetic structure could be beyond present day capability of firstprinciples calculations based on local exchange-correlation functionals. In fact, different results have been obtained not only between LDA and GGA approximations, but also between GGA-PW91 and GGA-PBE potentials for the ground state stability of magnetic structures. Nonetheless, first-principles results have proved to be useful in showing the potential complexity of magnetic structures in FeeNi alloys and identifying some possible trends in magnetic structure transitions. 2.12. CALPHAD thermodynamic assessments Previous CALPHAD-type thermodynamic assessments of the FeeNi system have been obtained by Kaufman and Nesor [121], Chart et al. [122, 123], Chuang et al. [30, 31]. These have been improved by Lee [124], who used a single two-sublattice model to describe both the ordered FeNi3 phase and the disordered fcc phase as one phase. He also reoptimised the liquid parameters. Subsequently Ohnuma et al. [125, 126], re-modelled the magnetic contribution to the Gibbs energy of the ordered FeNi3 phase. The L10 FeNi phase, which appeared to be stable in the first paper [125], was calculated to be

G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

metastable in the latter publication [126]. In an independent thermodynamic calculation of the FeeNi phase equilibria recently published by Howald [127] the L10 FeNi phase is stable in the 100e250
C temperature range. It has to be mentioned, however, that this cannot be considered as a completely reliable assessment because most of the recent FeeNi literature was not considered by the author.

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Gf ðTÞ functions for the pure elements in different stable and metastable phases are available from the Unary database by SGTE [134]. The ideal mixing contribution is: id f

G ¼ RT

X

xi lnðxi Þ

(4)

i

3. Methodology

where R is the gas constant and T the absolute temperature. The excess Gibbs energy is expressed by a RedlicheKister series and, for a generic binary AeB system, it is:

3.1. Atomistic calculations First-principles calculations have been carried out in order to determine the thermodynamic and magnetic properties of several ordered structures in the FeeNi system. A complete list of these structures is given in Table 1. A full structure minimization at the ground state (0 K) has been carried out using density functional theory (DFT) as implemented in the Vienna Ab-initio Simulation Package (VASP) with plane wave basis sets [128,131,132]. The calculations employed the generalized gradient approximation (GGA) of Perdew and Wang [116] and valence electrons were explicitly represented with projector augmented wave (PAW) pseudopotentials. The plane wave cut-off energy was set to 450 eV for all structures. k-points meshes were created by a Monkhorst-Pack scheme [133] and convergence on the number of k-points in the irreducible wedge of the first Brillouin zone has been carefully checked for each structure. The ground state (0 K) relaxed structure was determined by minimizing the Hellmann-Feynman forces with a conjugate gradient algorithm, until all ionic forces were less than 0.01 eV/A. For magnetism, collinear magnetic configurations have been considered, although spin flips were allowed. As pointed out in Section 2.11, differences in energy between possible magnetic structures are small and noncollinear effects are negligible for the purpose of the present work. Harmonic and anharmonic contributions at higher temperature have been neglected in the present calculations. 3.2. Thermodynamic modelling Thermodynamic modelling of the FeeNi phases based on the Compound Energy Formalism (CEF) [3] has been carried out using the CALPHAD method [1, 2]. The molar Gibbs energy of a generic phase is described as the sum of four terms:

Gf ¼ ref Gf þid Gf þex Gf þmag Gf

(1)

f

where ref G is the reference Gibbs energy, id Gf is the ideal mixing ex f mag f

G is the contribution, G is the excess Gibbs energy and magnetic contribution. For the liquid phase and the disordered solid solutions a/d-Fe (A2) and g-(Fe,Ni) (A1) the different terms are discussed in the following. The reference Gibbs energy can be written as: ref

Gf ¼

X

f

xi ,Gi ðTÞ

(2)

i

f

where xi is the mole fraction of component i and Gi ðTÞ the Gibbs energy of the pure component i in the structure of the phase f, referred to the standard element reference (SER) state. Its temperature dependence is expressed as:

Gf ðTÞ  HSER ¼ Af þ Bf T þ C f TlnðTÞ þ Df T 2 þ .

(3)

with A, B, C, D,. empirical parameters evaluated on the basis of the experimental information. Several temperature ranges with different expression of this type can actually be needed.

X

ex f

G ¼ xA x B

i f

L ðTÞ,ðxA  xB Þi

(5)

i ¼ 0.n

where the i Lf ðTÞ functions, which may have the same temperature dependence of Gf ðTÞ reported in Eq. (3), include empirical parameters to be evaluated during the optimisation process. Finally, the magnetic contribution is expressed, for the pure components, according to the model introduced by Inden [135, 136] and subsequently adapted by Hillert and Jarl [137]: mag

Gf ¼ RT,f ðsÞ,lnðb þ 1Þ

(6)

where b is the average magnetic moment per mole of atoms in Bohr magnetons, s is the ratio T/TC (TC ¼ critical temperature for magnetic ordering), and f(s) is a polynomial expression obtained by expanding Inden's description of the magnetic heat capacity into a power series of s. For a binary phase the composition dependence of TC and b is given by a RedlicheKister series expansion: f

f

f

TC ðxÞ ¼ xA TC ðAÞ þ xB TC ðBÞ þ xA xB

bf ðxÞ ¼ xA bf ðAÞ þ xB bf ðBÞ þ xA xB

X

i f

T ,ðxA  xB Þi

(7)

i ¼ 0.n

X

i f

b ,ðxA  xB Þi

(8)

i ¼ 0.n

f

where i T f and i b are expansion parameters to be evaluated on the basis of the experimental information available. In the FeeNi system only one intermetallic phase, L12 FeNi3 (AuCu3 type), seems to be stable at room and higher temperatures. Its structure is an ordered form of A1 g-(Fe,Ni). It could be described in the CEF by a 2-sublattice model (Fe, Ni)(Fe, Ni)3 where both elements are present in each sublattice in order to describe the homogeneity range of the phase, mainly due to the mutual substitution between Fe and Ni. However, other fcc ordered forms of A1 g-(Fe,Ni), namely L12 Fe3Ni (anti-AuCu3 type), and L10 FeNi (AuCu type), compete for equilibrium in this system and must be thermodynamically modelled. Data for these are also necessary when modelling higher order systems (e.g. Al-FeeNieTi) where such phases become stable. This can be achieved by using a 4-sublattice model, as already introduced by Sundman et al. [138] and adopted by De Keyzer et al. [139] while modelling the FeeNieTi ternary system. In the case of multi-sublattice models the Gibbs energy of ðsÞ a phase is expressed as a function of the site fractions yi , the mole fractions of each component i in the sublattice (s). Site fractions obey the conditions:

X

ðsÞ

yi

¼ 1

i

P

ðsÞ

nðsÞ yi  ¼ xi P ðsÞ  ðsÞ n 1  yVa s

s

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G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

where nðsÞ are the stoichiometric coefficients relating the sublattices and s itself indicates the s-th sublattice. Va identifies vacancies. For a 4-sublattice model (A1, ., Ai)1/4 (A1, ., Ai)1/4 (A1, ., Ai)1/4 (A1, ., Ai)1/4 where i different constituents may occupy the four sublattices the different contributions to the Gibbs energy are given by: ref

Gf ¼

XXXX i

j

k

ð1Þ ð2Þ ð3Þ ð4Þ f

yi yj yk yl Gi:j:k:l

(9)

l

1 X X ðsÞ  ðsÞ  G ¼ RT yi ln yi 4 s i

id f

ex f

G ¼

XX i

XX i

ð1Þ ð1Þ

yi yj

XXX

j>i

k

m

l

(10)

k

f

l

ð2Þ ð3Þ ð4Þ f

yk yl ym Li;j:k:l:m ðTÞþ (11)

m

ref Gf

where Gi:j:k:l in are the Gibbs energies of all the so-called end members, i.e. the stoichiometric compounds (either stable, metastable or unstable) formed when only one constituent is present on f each sublattice. Li;j:k:l:m ðTÞ in ex Gf is the interaction parameter, described by a RedlicheKister series expansion, corresponding to the mixing of components i and j on the first sublattice while the other sublattices are singly occupied. It has the same T dependence as the i Lf ðTÞ parameters of Eq. (5). The terms not shown correspond to all possible permutations between the four sublattices. Moreover, additional terms may be added to ex Gf , corresponding to simultaneous mixing on two sublattices while the remaining sublattices are singly occupied. The composition dependence of mag Gf derives by the composition dependence of TC and b, which are given by expressions similar to Eq. (11). This may account for the TC and b variations related to the structural ordering. In the present case L12 FeNi3 results when three of the sublattices have identical occupancy with Ni being the main constituent, while Fe is the main constituent on the fourth sublattice. The opposite occupation corresponds to the L12 Fe3Ni. L10 FeNi results when site occupancy splits into two groups of two sublattices, the first group having identical site occupancy with Ni the main element and the second group similarly with Fe as the main element. When each element occupies all sublattices with the same site fraction the disordered state (A1 structure) is attained and the Eqs. (9e11) become equivalent to Eqs. (2, 4, 5), respectively. As a result of their crystallographic equivalence the four sublattices are also thermodynamically equivalent, which means that groups of selected G functions must be equal to each other. A number of constraints arise: fcc

fcc

fcc

fcc

fcc

fcc

fcc

fcc

GFe:Ni:Ni:Ni ¼ GNi:Fe:Ni:Ni ¼ GNi:Ni:Fe:Ni ¼ GNi:Ni:Ni:Fe ¼ GFe1Ni3 fcc

GFe:Fe:Ni:Ni ¼ GFe:Ni:Fe:Ni ¼ GFe:Ni:Ni:Fe ¼ GNi:Fe:Fe:Ni ¼ GNi:Fe:Ni:Fe ¼ Gfcc ¼ GFe2Ni2 Ni:Ni:Fe:Fe fcc

fcc

fcc

fcc

(13)

where UFeNi is related to the FeeNi bond energy while U'FeNi accounts for its dependence on composition. Even if the above-mentioned approximations are not completely fulfilled it could be expected that GFe1Ni3, GFe2Ni2 and GFe3Ni1 would not deviate too much from these relationships. Similar considerations also lead to similar constraints on the interaction parameters:

Lfcc ¼ Lfcc ¼ Lfcc ¼ Lfcc ¼. Fe;Ni:Ni:Ni:Ni Ni:Fe;Ni:Ni:Ni Ni:Ni:Fe;Ni:Ni Ni:Ni:Ni:Fe;Ni Lfcc ¼ Lfcc ¼ Lfcc ¼ Lfcc ¼. Fe:Fe;Ni:Ni:Ni Fe:Fe;Ni:Ni:Ni Fe:Ni:Fe;Ni:Ni Fe:Ni:Ni:Fe;Ni .

X X X ð1Þ ð3Þ ð4Þ f ð2Þ ð2Þ yi yj yk yl ym Lk:i;j:l:m ðTÞ þ .

j>i

GFe1Ni3 ¼ 3UFeNi þ U'FeNi GFe2Ni2 ¼ 4UFeNi GFe3Ni1 ¼ 3UFeNi  U'FeNi

GFe:Fe:Fe:Ni ¼ GFe:Fe:Ni:Fe ¼ GFe:Ni:Fe:Fe ¼ GNi:Fe:Fe:Fe ¼ GFe3Ni1

ð12Þ

Moreover the three quantities GFe1Ni3, GFe2Ni2 and GFe3Ni1 may be related to each other if a number of approximations is assumed. In particular, if we assume that the same relations which hold between bond energies can also be applied to Gibbs energies, and that bond energies are the result of the sum of all nearest neighbour interactions, the following relations arise:

(14)

These constraints considerably reduce the number of independent empirical parameters to be evaluated during optimisation. See for instance the work of Kussofsky et al. [4] for a detailed discussion of the relations and constraints between parameters in the foursublattice modelling of the fcc ordered phases. The relation between the disordered (A1) and ordered (L12, L10) phases is realised by using a unique Gibbs energy expression for all of them [4, 140]:

  ðsÞ ord yi Gm ¼ Gdis m ðxi Þ þ DGm

(15)

where Gdis m ðxi Þ is the Gibbs energy of the disordered state and is given by Eq. (2, 4, 5) while the ordering contribution is given by:







ðsÞ ðsÞ DGord  Gm y i ¼ xi m ¼ Gm yi



(16)

the difference between two terms, both expressed according to Eq. (9e11): the first term is calculated by using the actual site fractions and the second one by replacing the site fractions by the global mole fractions in order to get the disordered state. The bcc (A2) a/d-Fe phase, which is only stable as a narrow terminal solid solution in the Fe-rich side, is also described by a 4-sublattice model, to enable consideration of its ordered forms B2 (CsCl type), B32 (NaTl type) and D03 (BiF3 type). This would not be strictly necessary for the calculation of the stable FeeNi phase equilibria, but it is necessary when FeeNi based higher order systems (e.g. AleFeeNi) are modelled. Detailed modelling of the

Table 3 Results of the atomistic calculations performed in this work, compared to the available literature data. Formula Structure

Lattice Composition ΔH a0 (nm) M (at% Ni) (kJ/mol at) (mB/at)

Fe Fea Fea Fe3Ni Fe3Ni Fe2Ni FeNi FeNi FeNi FeNi2 FeNi3 FeNi3 Ni Ni

bcc fcc fcc fcc bcc fcc fcc bcc fcc fcc fcc bcc bcc fcc

a

A2 cI2-W A1 cF4-Cu A1 cF4-Cu L12 cP4-AuCu3 D03 cF16-AlFe3 C11f L10 tP4-AuCu B2 cP2-CsCl B32 cF16-NaTl C11f L12 cP4-AuCu3 D03 cF16-AlFe3 A1 cF4-Cu A2 cI2-W

0 0 0 0 25 33.3 50 50 50 66.7 75 75 100 100

0 9.3 16.3 5.2 4.3 0.6 5.5 8.9 2.9 4.6 7.7 2.8 9.2 0

0.2826 0.35 0.37 0.3560 0.5705 0.3504b 0.3547b 0.2844 0.5668 0.3550b 0.35 0.56 0.2800 0.3518

2.156 0.09 2.55 1.834 2.127 1.818 1.606 1.707 1.611 1.256 1.18 1.1 0.537 0.618

two minima have been found as a function of lattice parameter. f.c.c. distorted, c/a ¼ 3.159 for C11f Fe2Ni, c ¼ 0.3585 nm for L10 FeNi, c/a ¼ 2.98 for C11f FeNi2. b

G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

bcc ordered forms is similar to that already expressed for the 4sublattice modelling of the fcc phases. They have recently been illustrated by Sundman et al. [141]. 4. Results and discussion 4.1. Atomistic calculations The results of the present first-principles calculations for several ordered structures are shown in Table 3. The composition of the selected compounds were chosen in order to provide input parameters for thermodynamic modelling and to cover a wide enough composition range in the binary system. For pure Fe the present results are in agreement with previous works [115, 120]. In particular, the energy of the fcc ferromagnetic

1155

pure Fe presents a double minimum as a function of lattice parameter, as pointed out in [120]. A ferromagnetic fcc Fe has been found at higher volumes, while a nearly non-magnetic state is stable at lower volumes. The present ab-initio results for ordered compounds are also in agreement with previous calculations using different methods [115, 117]. In both pure components and ordered compounds, magnetism significantly decreases the energy at 0 K and cannot be neglected. However, as pointed out in [120], the competition between different magnetic structures in fcc pure Fe and FeeNi alloys is rather complex and the exact determination of the magnetic ground state structure could even be not possible within present day local approximated potentials (both LDA and GGA). Fortunately, the energy difference between competing structures is small and, therefore, this issue does not significantly affect the following

Fig. 3. Thermodynamic functions of the liquid phase. Present results (continuous lines) are compared to the literature data (data points) and the assessment by [124] (dashed line). (a) enthalpy of mixing; (b) activity; (c) activity coefficient. [1959Spe] ¼ [89], [1959Zel] ¼ [90], [1967Bel] ¼ [92], [1970Pre] ¼ [93], [1974Bat] ¼ [96], [1977Kub] ¼ [97], [1978Con] ¼ [98], [1978Mar] ¼ [99], [1981Igu] ¼ [100], [1981Ram] ¼ [101], [1998Thi] ¼ [104].

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G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

Fig. 4. Thermodynamic functions of the A1 phase. Present results (continuous lines) are compared to the literature data (data points). (a) enthalpy of mixing; (b) activity coefficient. [1978Con] ¼ [98], [1981Ram] ¼ [101], [1961Ste] ¼ [142], [1963Den] ¼ [143], [1967Kub] ¼ [144], [1977Gri] ¼ [145]. In Fig. 4c thremodynamic functions of the liquid and A1 phases are compared (notice that H and S are temperature independent, while G are calculated at 1000
C).

results for thermodynamic modelling. Thus, for simplicity, only ferromagnetic structures have been calculated for binary compounds. As shown in Table 3, nearly all FeeNi structures have been found to be unstable with respect to A2 Fe and A1 Ni. The most stable structure is the L12 FeNi3, which is consistent with previous calculations [115, 117] and with the assessed phase diagram [11]. According to present results the controversial L10 FeNi phase is stable at 0 K, in agreement with previous first-principles calculations [105, 115, 117]. This is also in agreement with recent findings about the possible stability of the L10 FeNi phase found in meteorite samples [12, 77] and thus the difficulty in achieving this phase in equilibrium is due to sluggish kinetics. However, the energy of the L10 FeNi phase is very close to the convex hull line, the difference being only about 0.4 kJ/mol of atoms. Approximately the same difference has been obtained using both normal and pv PAW pseudopotentials in VASP.

Finally, the C11f phases with compositions Fe2Ni and FeNi2 suggested by Mishin and co-workers [115] as possible stable phases have also been calculated in the present work. According to the present results these structures are not stable at 0 K and their energies lie above the convex hull line. Nonetheless, since the difference is only a few kJ/mol, it may be possible to obtain these phases in metastable equilibrium under certain experimental conditions. However, to the best Authors' knowledge, the existence of these phases has never been confirmed experimentally. The present first-principles results have been used in the thermodynamic modelling of the FeeNi system as discussed in the next section. 4.2. Assessment procedure The thermodynamic optimisation of the system has been achieved by successive refinements. A first set of parameters has been

G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

calculated for the liquid and the A1 phases on the basis of the available thermodynamic data. Then the composition dependence of TC and b for the solid phases A1 and A2 has been modelled. Subsequently liquid, A1 and A2 parameters have been refined by using phase equilibrium data. Finally the FCC4 and BCC4 phases have been added and their parameters optimised using experimental phase equilibria and atomistic calculation results. The final refinement of the optimisation parameters has been obtained by using all the available experimental and calculation results with appropriate weight factors. The temperature range of validity of the present assessed data is, at least, from 100 to 2500
C.

4.3. Calculation results The FeeNi thermodynamic parameters resulting from the present work is reported in the Appendix. 4.3.1. Thermodynamics Thermodynamic functions calculated in this work are presented and compared with experimental data in Fig. 3 for the liquid and in

Fig. 5. (a) Curie temperature and (b) magnetic moment of the g-(Fe,Ni), a-(Fe) and FeNi3 phases versus composition. [1925Pes] ¼ [34], [1929Gos] ¼ [35], [1943Hos] ¼ [36], [1953Wak] ¼ [37], [1963Cra] ¼ [38].

1157

Fig. 4 for the A1 phase. The validity of the Kopp-Neumann rule has been assumed on the basis of the literature data [146]. Good agreement between experimental data and computed functions has been generally obtained. In particular it may be observed that the enthalpy of mixing of the liquid (Fig. 3a) is now less exothermic and less asymmetric than would be expected from the previous assessment done by Lee [124]. The enthalpy of formation of the liquid and the A1 solid solution in the paramagnetic state are very similar to each other (Fig. 4c): this suggests that the average bond energy between Fe and Ni atoms in disordered close packed phases is weakly dependent on the aggregation state or structure. Other thermodynamic quantities, such as element activities (Figs. 3b) or activity coefficients (Figs 3c and 4b) also show that calculated functions are well within the range of experimental uncertainty. At temperatures lower than about 1000 K ferromagnetic ordering occurs in both fcc and bcc phases. The composition dependence of the Curie temperature and the average magnetic moment per atom have been evaluated and experimental data reproduced as close as possible (Figs. 5). The polynomial expansion used to approximate the composition dependence (Eq. (8)) did not allow us to closely reproduce the sharp variation of the magnetic moment around 40 at % Ni. The adoption of the order-disorder relation between the fcc phases, however, allowed us to well reproduce the TC increment and its asymmetric trend around 75 at% Ni (Fig. 5a) related to the L12 ordering. The enthalpy of formation of the solid phases (both stable and metastable) is shown in Fig. 6. It was possible to model the enthalpy of formation of the solid phases, especially the metastable ones, only as a result of the atomistic calculations. Considering the scatter of the literature data reported in the figure, it could be concluded that L10 may be either weakly stable or weakly metastable with respect to A2 Fe and L12 FeNi3. However, our calculations, in agreement with the most recent literature, indicate that it is stable at 0 K. According to our thermodynamic assessment, which takes also into account phase equilibria, we concluded that L10 should

Fig. 6. Enthalpy of formation of stable and metastable FeeNi phases. Results of the present thermodynamic assessment (continuous lines) are compared to results of atomistic calculations performed in this work (T.W.) or obtained from literature (data points). [2005Moh] ¼ [106], [2005Mis] ¼ [115], [2005Lec] ¼ [117], [2006Kis] ¼ [151]. FM and AFM mean ferromagnetic and anti-ferromagnetic, respectively.

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G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

become stable at temperatures below room temperature. Unfortunately this conclusion cannot be verified experimentally, due to the slow diffusivity of Fe and Ni, which makes it impossible to reach equilibrium at such a low temperature. It is interesting to observe that assessed enthalpies of formation are in good agreement with ab-initio results and this is as a result of applying relations (13) between the interaction parameters of the fcc ordered phases. This supports the validity of the assumptions and approximations underlying relations (13). 4.3.2. Stable phase equilibria The stable FeeNi phase equilibria calculated in this work are shown and compared to literature data in Fig. 7 and computed invariant equilibria are summarised and compared to literature data in Table 4. In general there is good agreement between experimental data and calculations. In particular the tricritical point generating the g-(Fe,Ni)PM þ g(Fe,Ni)FM two-phase field, first calculated by Chuang et al. [30], has been confirmed. According to the present assessment the monotectoid equilibrium g-(Fe,Ni)PM $ a-Fe þ g-(Fe,Ni)FM, never observed experimentally, occurs at 428
C. The only appreciable discrepancy between experiments and calculations (in the order of about 8e10 at%) concerns the composition of the L12 FeNi3 phase in equilibrium with a-Fe: at 300
C it is about 65 at% Ni according to [53] while it is about 58 at% Ni according to the present assessment. It was not possible to obtain a better agreement with the experimental results by [53] without decreasing the general good agreement of other equilibria. On the other hand, according to the critical assessment by [12] the solubility limit of L12 FeNi3 at about 200
C is 64 at% Ni, 6e7 at% Ni lower than the experimental observations by [53] and more consistent with our calculations.

Fig. 7. The FeeNi phase diagram computed in this work. (a) invariant temperatures are indicated; (b) present calculation is compared to selected experimental data from literature. [1910Rue] ¼ [8], [1996Yan] ¼ [12], [1923Han] ¼ [15], [1925Kas] ¼ [16], [1931Ben] ¼ [19], [1937Jen] ¼ [20], [1957Hel] ¼ [21], [1965Gol] ¼ [27], [1980Rom] ¼ [28], [1981Van] ¼ [47], [1950Jos] ¼ [51], [1953Gei] ¼ [52], [1963Heu] ¼ [53].

4.3.3. Metastable phase equilibria It is well known that the ability to calculate a stable binary phase diagram in good agreement with the experimental data is not a sufficient condition to conclude that a sound thermodynamic assessment has been obtained which is suitable for reliable extrapolation to higher order systems. This means that metastable phase equilibria are also important and these have to be calculated and discussed. On the other hand, metastable phase equilibria are, in most cases, not experimentally accessible and there is no way to check the soundness of the calculated values by comparison with experiments. In most cases the only possibility is to check their

Table 4 FeeNi invariant equilibria: comparison between literature data and values computed in this work (T.W.). Equilibrium

Equilibrium type

Temperature (
C)

d-Fe þ liquid $ g-(Fe,Ni)

Peritectic

1514 1514

3.5 5.2

4.9 6.4

g-(Fe,Ni) $ liquid

Congruent

1440 1443 1432

66 66 66.3

66 66 66.3

e e e

[11, 13, 14] [11, 14] T.W.

g-(Fe,Ni)PM þ g-(Fe,Ni)FM

Tricritical

462 472

48 48.4

48 48.4

e e

[13, 14] T.W.

g-(Fe,Ni)PM $ a-Fe þ g-(Fe,Ni)FM

Monotectoid

415 400 428

43.5 40 44.4

4.2 6 6.4

47.5 46 47.5

[13, 14] [12] T.W.

FeNi3 $ g-(Fe,Ni)

Congruent

517 514

72.5 72.3

72.5 72.3

e e

[11, 13, 14] T.W.

g-(Fe,Ni)FM $ a-Fe þ FeNi3

Eutectoid

347 345 353

49 53 52.5

4.7 3.2 5.6

63 63.6 55.5

[11] [13, 14] T.W.

First phase

Compositio (at% Ni) second phase

Third phase 4.2 5.7

Reference [11, 14] T.W.

G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

compatibility with stable equilibria in higher order systems, an indirect procedure which is time consuming and itself may not be free from ambiguity. In our case, however, the very slow kinetics of the system makes observable metastable states which may result when the formation of selected phases is kinetically inhibited. Some information on metastable FeeNi equilibria is summarised in the critical assessment by Yang et al. [12], where a semi-quantitative phase diagram including both stable and metastable equilibria, reported here in Fig. 1b, is presented. Moreover atomistic calculations may also be very useful because they allow us to calculate thermodynamic functions (especially enthalpy of formation) of experimentally inaccessible metastable states with uncertainties comparable to the experimental values. In our case the atomistic calculations available in literature combined with those carried out in this work made possible to evaluate thermodynamic functions at 0 K for several metastable ordered phases such as L12 Fe3Ni, L10 FeNi, D03 Fe3Ni and FeNi3, B2 and B32 FeNi, etc. This, in turn, made it possible to simulate metastable phase diagrams where only selected phases are allowed to form. Fig. 8 shows the FeeNi phase diagram resulting when only fccbased phases (A1, L12 and L10) are considered. In this figure several phase equilibria are shown: (1) “stable” fcc phase equilibria by continuous lines; (2) metastable continuation of the g-(Fe,Ni)PM þ g-(Fe,Ni)FM miscibility gap and, at low temperature in the Fe-rich part of the diagram, the metastable miscibility gap occurring in g-(Fe,Ni)PM, both indicated by dashed lines; (3) spinodal boundaries within the g-(Fe,Ni)PM þ g-(Fe,Ni)FM miscibility gap, marked by dotted lines and (4) the Curie temperature by a dash-dotted line. Curves calculated in this work, may be compared to those proposed by Yang et al. [12] (Fig. 1b). Considering the difficulty of the experimental investigations and the consequent uncertainty of the critically assessed equilibria, the agreement between our calculations and Yang et al. [12] is very good. This means that data coming from very different sources, such as phase equilibria observed in meteoric samples, results from 0 K atomistic calculations, thermodynamic measurements at relatively high temperature, are fundamentally consistent with each other. This also means that thermodynamic models adopted in this work and, in particular, constraints here assumed between model parameters of the fcc ordered phases are significant, at least for this system.

Fig. 8. FeeNi phase diagram including only fcc-based phases: stable phase equilibria (continuous lines), metastable phase equilibria (dashed lines), spinodal region boundaries (dotted lines) and Curie temperature (dash-dotted line).

1159

4.3.4. Role of magnetism Finally, it may be interesting to discuss the role of magnetism in determining FeeNi thermodynamic functions and phase equilibria. To this end it is useful to compare thermodynamic functions and phase diagrams calculated with and without magnetic contribution. Entropies of mixing of the A1 and A2 phases calculated at various temperatures with and without magnetic contributions are shown in Fig. 9 (a and b). Similarly, enthalpies of mixing are reported in Fig. 10. Notice that all the functions reported in Figs. 9 and 10 are referred to the pure elements at the stated temperature in a paramagnetic state. From the figures it is evident that,

Fig. 9. Entropy of mixing (referred to the pure elements in the paramagnetic state) at various temperatures (0, 200, 400, 600, 800, 1000
C) calculated for the phase in magnetically ordered (continuous line) and disordered (dashed line, temperature independent) states, respectively, (a) for the A1 phase and (b) for the A2 phase.

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G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

especially at lower temperatures, the contribution due to magnetic ordering exceeds, even largely, the contribution due to element mixing, which has been considered as temperature independent. As a result, in the “paramagnetic” phase diagram (Fig. 11), calculated by neglecting the Curie temperature and the magnetic contribution to the Gibbs energy of the different phases, the A2 phase appears only at high temperature in a very small temperature and composition range (1523e1528
C and 0e0.3 at% Ni, not shown in the figure). Moreover L12 FeNi3 is slightly destabilised and L10 FeNi slightly stabilised with respect to the equilibrium phase diagram (Fig. 7a).

Fig. 11. FeeNi phase diagram including only fcc-based phases: phase equilibria are calculated by neglecting, for each phase, the Curie temperature and the magnetic contribution to the Gibbs energy.

5. Conclusions

Fig. 10. Enthalpy of mixing (referred to the pure elements in the paramagnetic state) at various temperatures (0, 200, 400, 600, 800, 1000
C) calculated for the phase in magnetically ordered (continuous line) and disordered (dashed line, temperature independent) states, respectively. (a) for the A1 phase and (b) for the A2 phase.

The FeeNi system has been thermodynamically assessed according to the CALPHAD methodology, using the compound energy formalism to express the Gibbs energy of FeeNi stable and metastable phases: liquid, A1 and related fcc-based ordered phases (L12 Fe3Ni, L10 FeNi and L12 FeNi3), A2 and related bcc-based ordered phases (D03 Fe3Ni, B2 and B32 FeNi, D03 FeNi3). To this end both experimental data from literature and ab-initio enthalpies of formation have been used. As a result thermodynamic functions and stable phase equilibria experimentally known have been reproduced within the range of the experimental uncertainties. In particular low temperature (below about 300
C) equilibria, hardly accessible experimentally due to an extremely sluggish kinetics, have been calculated and conclusions drawn by [12] on the basis of meteoric samples investigations have been confirmed. Atomistic simulations carried out in this work by means of the VASP software using GGA approximation and PAW pseudopotentials lead to the calculation of 0 K enthalpies of formation, lattice parameters and magnetic moment of several FeeNi phases. In particular the stability of the L10 phase at 0 K was confirmed, while C11f phases resulted to be metastable. Ab-initio results both from literature and from the present work enabled us to model both stable and metastable ordered phases which need to be thermodynamically evaluated in order to make the FeeNi assessment suitable for extrapolations to higher order systems (such as AleFeeNi, where e.g. bcc-based ordered phases are stable). To this end all disordered and related ordered phases were modelled by a single 4-sublattice model for both fccbased (A1, L12 Fe3Ni, L10 FeNi and L12 FeNi3) and bcc-based (A2, D03 Fe3Ni, B2 and B32 FeNi, D03 FeNi3) structures. The use of a single model for a disordered phase and its ordered forms makes it possible to calculate both first and second order transformations. In particular a metastable phase diagram including only fcc-based phases has been calculated, which is in good agreement with metastable phase equilibria experimentally observed in meteorites. That means that thermodynamic functions essentially based on ab-initio data and high temperature phase equilibria are consistent with metastable phase equilibria deducted from meteorite analyses.

G. Cacciamani et al. / Intermetallics 18 (2010) 1148e1162

Magnetic ordering was also taken into account. Unfortunately the RedlicheKister expansion is inadequate to well reproduce the sharp variation of the average magnetic moment versus composition observed in FeeNi, especially for the A1 phase around 30e40 at% Ni. Fortunately this does not appreciably affect stable phase equilibria. It may be interesting to emphasize that, in particular, it has been possible to calculate the TC increase due to the combination of magnetic and structural ordering: this was experimentally known for the FeNi3 phase, but not for the other ordered phases. Finally the general influence of magnetism on the FeeNi thermodynamics and phase equilibria has been evidenced: when magnetism is switched off bcc-based phases are heavily destabilised with respect to fcc and A2 almost completely disappears.

Phase BCC4

Appendix Interaction parameters evaluated in this work with a non-zero value are listed here below. A complete database file (a text file in TDB format) may be obtained from the corresponding author. Phase liquid

Model (FE,NI) G(LIQUID, FE, NI;0) ¼ 18782 þ 3.7011*T G(LIQUID, FE, NI;1) ¼ þ12308e2.7599*T G(LIQUID, FE, NI;2) ¼ þ4457.0e4.1536*T

Phase A1

Model (FE,NI) G(A1,FE,NI;0) ¼ 15500 þ 2.850*T G(A1,FE,NI;1) ¼ þ14000e4.000*T G(A1,FE,NI;2) ¼ 3000 TC(A1,FE,NI;0) ¼ 2200 TC(A1,FE,NI;1) ¼ 700 TC(A1,FE,NI;2) ¼ 800 BMAGN(A1,FE,NI;0) ¼ 10 BMAGN(A1,FE,NI;1) ¼ 8 BMAGN(A1,FE,NI;2) ¼ 4

Phase A2

Model (FE, NI, VA) G(A2,FE,VA;0) ¼ 80*T G(A2,NI,VA;0) ¼ 80*T G(A2,FE,NI;0) ¼ 7500 G(A2,FE,NI;1) ¼ þ8500e5.0*T TC(A2,FE,NI;0) ¼ 1000 TC(A2,FE,NI;1) ¼ 1500 BMAGN(A2,FE,NI;0) ¼ 0.5 BMAGN(A2,FE,NI;1) ¼ 3.5

Phase FCC4

Model (FE,NI)1/4(FE,NI)1/4(FE,NI)1/4(FE,NI)1/4 G(FCC4,NI:FE:FE:FE;0) ¼ G(FCC4,FE:NI:FE:FE;0) ¼ G(FCC4,FE:FE:NI:FE;0) ¼ G(FCC4,FE:FE:FE:NI;0) ¼ 3*UFENIe3000þ0.3*T G(FCC4,NI:NI:FE:FE;0) ¼ G(FCC4,NI:FE:NI:FE;0) ¼ G(FCC4,FE:NI:NI:FE;0) ¼ G(FCC4,NI:FE:FE:NI;0) ¼ G(FCC4,FE:NI:FE:NI;0) ¼ G(FCC4,FE:FE:NI:NI;0) ¼ 4*UFENI G(FCC4,NI:NI:NI:FE;0) ¼ G(FCC4,NI:NI:FE:NI;0) ¼ G(FCC4,NI:FE:NI:NI;0) ¼ G(FCC4,FE:NI:NI:NI;0) ¼ 3*UFENI þ 30000.3*T with UFENI ¼ 2125þ0.625*T TC(FCC4,NI:FE:FE:FE;0) ¼ TC(FCC4,FE:NI:FE:FE;0) ¼ TC(FCC4,FE:FE:NI:FE;0) ¼ TC(FCC4,FE:FE:FE:NI;0) ¼ 155 TC(FCC4,NI:NI:FE:FE;0) ¼ TC(FCC4,NI:FE:NI:FE;0) ¼ TC(FCC4,FE:NI:NI:FE;0) ¼ TC(FCC4,NI:FE:FE:NI;0) ¼ TC(FCC4,FE:NI:FE:NI;0) ¼ TC(FCC4,FE:FE:NI:NI;0) ¼ 200 TC(FCC4,NI:NI:NI:FE;0) ¼ TC(FCC4,NI:NI:FE:NI;0) ¼ TC(FCC4,NI:FE:NI:NI;0) ¼ TC(FCC4,FE:NI:NI:NI;0) ¼ 245 BM(FCC4,NI:FE:FE:FE;0) ¼ BM(FCC4,FE:NI:FE:FE;0) ¼ BM(FCC4,FE:FE:NI:FE;0) ¼ BM(FCC4,FE:FE:FE:NI;0) ¼ 0.115 BM(FCC4,NI:NI:FE:FE;0) ¼ BM(FCC4,NI:FE:NI:FE;0) ¼ BM(FCC4,FE:NI:NI:FE;0) ¼ BM(FCC4,NI:FE:FE:NI;0) ¼ BM(FCC4,FE:NI:FE:NI;0) ¼ BM(FCC4,FE:FE:NI:NI;0) ¼ 0.115 BM(FCC4,NI:NI:NI:FE;0) ¼ BM(FCC4,NI:NI:FE:NI;0) ¼ BM(FCC4,NI:FE:NI:NI;0) ¼ BM(FCC4,FE:NI:NI:NI;0) ¼ 0.115

1161

Model (FE,NI,VA)1/4(FE,NI,VA)1/4(FE,NI,VA)1/4(FE,NI,VA)1/4 G(BCC4,NI:FE:FE:FE:VA;0) ¼ G(BCC4,FE:NI:FE:FE:VA;0) ¼ G(BCC4,FE:FE:NI:FE:VA;0) ¼ G(BCC4,FE:FE:FE:NI:VA;0) ¼ +2*UBNIFE1+UBNIFE2+AD03FENI G(BCC4,NI:NI:NI:FE:VA;0) ¼ G(BCC4,NI:NI:FE:NI:VA;0) ¼ G(BCC4,NI:FE:NI:NI:VA;0) ¼ G(BCC4,FE:NI:NI:NI:VA;0) ¼ +2*UBNIFE1+UBNIFE2+AD03NIFE G(BCC4,NI:NI:FE:FE:VA;0) ¼ G(BCC4,FE:FE:NI:NI:VA;0) ¼ +4*UBNIFE1 G(BCC4,NI:FE:NI:FE:VA;0) ¼ G(BCC4,FE:NI:NI:FE:VA;0) ¼ G(BCC4,NI:FE:FE:NI:VA;0) ¼ G(BCC4,FE:NI:FE:NI:VA;0) ¼ +2*UBNIFE1+2*UBNIFE2 with UBNIFE1 ¼ -1000+1*T; UBNIFE2 ¼ AD03NIFE ¼ -4000+1*T; AD03FENI ¼ +4000-1*T TC(BCC4,NI:FE:FE:FE;0) ¼ TC(BCC4,FE:NI:FE:FE;0) ¼ TC(BCC4,FE:FE:NI:FE;0) ¼ TC(BCC4,FE:FE:FE:NI;0) ¼ TC(BCC4,NI:NI:FE:FE;0) ¼ TC(BCC4,NI:FE:NI:FE;0) ¼ TC(BCC4,FE:NI:NI:FE;0) ¼ TC(BCC4,NI:FE:FE:NI;0) ¼ TC(BCC4,FE:NI:FE:NI;0) ¼ TC(BCC4,FE:FE:NI:NI;0) ¼ TC(BCC4,NI:NI:NI:FE;0) ¼ TC(BCC4,NI:NI:FE:NI;0) ¼ TC(BCC4,NI:FE:NI:NI;0) ¼ TC(BCC4,FE:NI:NI:NI;0) ¼ 250 BM(BCC4,NI:FE:FE:FE;0) ¼ BM(BCC4,FE:NI:FE:FE;0) ¼ BM(BCC4,FE:FE:NI:FE;0) ¼ BM(BCC4,FE:FE:FE:NI;0) ¼ BM(BCC4,NI:NI:FE:FE;0) ¼ BM(BCC4,NI:FE:NI:FE;0) ¼ BM(BCC4,FE:NI:NI:FE;0) ¼ BM(BCC4,NI:FE:FE:NI;0) ¼ BM(BCC4,FE:NI:FE:NI;0) ¼ BM(BCC4,FE:FE:NI:NI;0) ¼ BM(BCC4,NI:NI:NI:FE;0) ¼ BM(BCC4,NI:NI:FE:NI;0) ¼ BM(BCC4,NI:FE:NI:NI;0) ¼ BM(BCC4,FE:NI:NI:NI;0) ¼ 0.1

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