Revised thermodynamic description of the Fe-Cr system based on an improved sublattice model of the σ phase

Calphad 60 (2018) 16–28

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Revised thermodynamic description of the Fe-Cr system based on an improved sublattice model of the σ phase

T

⁎

Aurélie Jacob , Erwin Povoden-Karadeniz, Ernst Kozeschnik TU Wien, Institute of Materials Science and Technology, Getreidemarkt 9, 1060 Vienna, Austria

A R T I C L E I N F O

A B S T R A C T

Keywords: Calphad σ phase Crystal structure DFT

The Fe-Cr system is re-assessed, focusing on an improved modeling of σ phase. The three sublattice model (Cr,Fe)10(Cr,Fe)4(Cr,Fe)16 is parameterized to model the σ phase, solving discrepancies between computed and experimental site fractions of previous descriptions. Taking into account relative metastability trends of endmember compounds from first-principles analysis, only two additional interaction parameters of the σ-FeCr description were required for the reproduction of the chemical structure, thermodynamic properties and phase boundaries.

1. Introduction A considerable amount of studies has been dedicated to the Fe-Cr system due to its technological importance, often related to the occurrence of the intermetallic topologically close-packed Frank-Kasper σ phase. Due to its brittleness and wide solubility potential for various elements, formation of σ phase is known to be crucial for materials properties and, thus, technological usability of many high-alloyed steel grades. Therefore, understanding and prediction of its temperature- and composition-dependent stability is desirable. Robust phase prediction within the Calphad spirit requires physically correct thermodynamic modeling from low-order binary and ternary systems to multi-components. A proper thermodynamic phase description for this aim starts with correct reproduction of known crystal structural chemistry, and more precisely, fractions of different sorts of atoms on different crystallographic sites, which often reveal a distinct coordination number. In Calphad modeling of intermetallic solid solutions, this kind of data is “translated” to sublattice descriptions. Their experimental identification, however, is exceptionally challenging even by the use of highresolution analysis [1,2]. As a consequence, previous sublattice suggestions for the σ-FeCr phase [3–6] have often represented only a vague approximation of its real crystal chemistry. Whereas rough model approximations may reproduce phase boundaries in binary diagrams well, they become particularly crucial at extensions to multi-component systems and applied calculations in technological alloys. For instance, due to insufficiently precise calculated chemical potentials associated with inaccurately modeled site fractions, application for thermo-kinetic precipitation simulations often tends to inadequate particle evolution. Only recently, ⁎

first-principles data have become an invaluable source of crystal structural chemical data [7], allowing for a revision of the thermodynamic description of the σ phase in order to achieve a better agreement of phase equilibria, compositions and structural chemistry. For the model revision of σ phase in the present study, all to the authors´ knowledge available phase stability and crystal chemistry data are collected and critically reviewed. A modified sublattice description is based on structural-chemical data [1,2,8] and new first-principles analyses of σ phase compound enthalpies. Parameterization of the Fe-Cr alloy phases as well as liquid is re-adjusted, correspondently. The resulting improvements of the calculated phase diagram and thermodynamic data in comparison with previous Calphad assessments [3–6,9–12] are presented. 2. Summary and assessment of previous studies of the Fe-Cr system Xiong et al. [13] gave an extensive review about the thermodynamics of this system, which has been available until 2010, including more than ten Calphad assessments. The most accepted version of the calculated phase diagram is given by Andersson and Sundman [3] together with a later modification of the liquid description from Lee [12]. Their thermodynamic modeling [3,12] is commonly used in multicomponent thermodynamic database. These widely-used assessments [3,12] put up with the following problems, however. The sublattice description for the σ phase [3] does not reflect the site occupancies of elements correctly [8]. Moreover, their calculated Cr solubility in the low temperature α-Fe phase is too small with respect to the miscibility gap (see Fig. 7 of [13]), which

Corresponding author. E-mail address: [email protected] (A. Jacob).

http://dx.doi.org/10.1016/j.calphad.2017.10.002 Received 20 July 2017; Received in revised form 15 September 2017; Accepted 1 October 2017 0364-5916/ © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

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Paxton [22] and Dubiel and Inden [21], the eutectoid temperature can be accepted to be between 773 and 802 K. The enthalpy of formation of the σ phase at T=923 K has been determined from adiabatic calorimeter by Dench [24]. Assessed enthalpy data by Müller et al. [25] took into account also other collected data [26] and suggested slightly higher values than the results by Dench [24].

additionally affects the spinodal decomposition. Further, the proposed magnetic parameters of Fe-Cr BCC do not reproduce the experimental Curie temperature towards higher Cr-alloying (see Fig. 9 from Xiong et al. [13]). Xiong et al. [4] provide a better description of the miscibility gap as well as of the Curie temperature and the magnetic moment. Their assessment of the Fe-Cr system is based on the revised lattice stability of pure iron from Chen and Sundman [14], providing a physically improved description of magnetism and no thermal discontinuities of the Gibbs energy function. So far, the revised lattice stabilities have only been used in the thermodynamic modeling of the Fe-Cr [4] and Fe-C [15] systems. For proper extension into ternary and multi-component systems, refinement of all involved lattice stabilities would be required. However, associated research activities are still ongoing [16], and combination to full descriptions of higher-order systems and implementations into more comprehensive thermodynamic databases, such as MatCalc-compatible [17] databases being developed by the authors of this study, is not performed at present. Consequently, we are still relying on the first-generation lattice stabilities from Dinsdale [18]. Xiong et al. [4] used the simplified two-sublattice model suggestion [8] for the σ phase, which, they claimed, would however be unlikely appropriate in multi-component extensions. The re-calculated phase diagram of the Fe-Cr system based on the present study is shown in Fig. 1. As expected, the FCC γ-loop is observed on the Fe-rich side. The BCC phase has a large miscibility gap between Fe-rich (BCC) and Cr-rich (BCC’) alloy compounds. σ phase is stable at elevated temperatures.

2.1.2. Crystal structure of σ phase The σ phase crystallizes as a tetragonal structure in the P42/mnm space group. 30 atoms per unit cell are distributed on five different Wyckoff positions [8]. Thus, the formula A212B415C814D812E814 can be given, where each letter represents a Wyckoff position, the subscript the number of atoms per position and the higher indices the coordination number. The crystal structure of the σ phase in the Fe-Cr system was investigated by Yakel [1] by XRD measurements of single crystals. He confirmed the space group P42/mnm and determined that each Wyckoff position is mixed occupied by Fe and Cr. Details of the crystal structure and information on the atomic positions are given in Table 1. From structural-chemical investigation of samples annealed at different temperatures up to 1400 h for the lower temperature, Yakel [1] concluded that temperature does not influence the site preference significantly. Cieślak et al. [27] studied the site preferences for different compositions in samples at 973 K annealed for up to 6650 h. The results of these two studies are shown in Fig. 2. The studies [1,27] of site occupancies (Fig. 2) in σ-FeCr reveal that Fe prefers the A and D positions, whereas Cr sits preferentially on B, C and E. The two studies were done at different temperatures, T=923 K [1] and T=973 K [27], respectively, but provide similar results, confirming the suggestion made by Yakel [1].

2.1. σ phase in the Fe-Cr system 2.1.1. Phase stability of σ phase The σ phase was discovered for the first time in 1907 in the Fe-Cr system. It is nowadays known to be present in more than forty binary systems [8]. In the Fe-Cr system, the σ phase is stable from ~44 to ~50 at% Cr [8]. It is formed by a congruent reaction BCC ↔ σ at temperatures between 1093 and 1098 K [19,20] and decomposes by a eutectoid reaction at T=773 K [21] into BCC+BCC’. Reported experimental temperatures of the eutectoid reaction are controversial [21–23]. Williams and Paxton [22] determined this value to be between 783 and 823 K. According to Dubiel and Inden [21], this temperature lies between 773 and 805 K, whereas Koyano et al. [23] determined this value to be below 783 K, but traces of carbides may have blurred the results. Dubiel and Inden [21] annealed their C-free Fe-Cr samples for four years, and, thus, likely approached the equilibrium state. Considering the temperature range given by Williams and

2.1.3. Thermodynamic modeling Several thermodynamic models [3–5,11], distinguished by the number of sublattices and atoms per sublattice, have been proposed for the σ phase in the Fe-Cr system since the establishment of the compound energy formalism (CEF) [28] for non-stoichiometric phases. The CEF allows for the energetic description of interactions among different sorts of atoms, sharing the same sublattice, as well as the combination of different sublattices. This made the CEF an essential pre-requisite for the physically proper modeling of intermetallic phases with relatively complex crystallographic order, such as it is the case for the group of topologically close-packed phases including σ phase. In the CEF, the mathematical model divides the phase into different sublattices. The molar Gibbs energy of a phase can be formulated as:

Gm = G ref + Gid + Gex .

(1)

The first term, G ref , represents the surface of reference. In the σ phase, with full five-sublattice description, each of the five sublattices represent distinct Wyckoff positions, and G ref is written

G ref =

∑

σ yA2a yB4f yC8i yD8i ′ yE8j GABCDE ,

ABCDE

(2)

x

where yi expresses the site fraction of each constituent on each sublattice. The second term Gid represent the ideal entropy of mixing and is defined as

Gid = RT ∑ yi ln (yi ). i

(3)

The last term represents the excess Gibbs energy, i.e. the term summing up all non-ideal mixing contributions of the phase. This part represents the focus during the thermodynamic optimization process. The thermodynamic modeling of σ-FeCr given by Hertzman [11] and Anderson and Sundman [3] was proposed at a time where the quantitative analysis of site distributions in complex intermetallic

Fig. 1. Equilibrium phase stability diagram of Fe-Cr as calculated from the present parameters.

17

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Table 1 Crystal structure of σ-CrFe as given by Yakel [1].

Space group P42/mnm (no. 136) Pearson symbol tP30 Lattice parameters a= 8.7961Å [1] c=4.5605 Å [1] Model: FeCr

Wyckoff position

x

y

z

CN

2a (blue) – A

0

0

0

12

4f (red) – B

0.39864

x

0

15

8i (orange) – C

0.46349

0.13122

0

14

8i’ (green) – D

0.73933

0.0661

0

12

8j (grey) – E

0.18267

x

0.25202

14

describe the σ phase from 13.3% to 66.7% of A atoms only. Herein, this model cannot fully reflect the full homogeneity range of the σ phase in other systems where this phase is also present. Anyway, the model was applied in Fe-Cr by Chuang et al. [5] and by Westman [6]. The model has also been used to model σ phase in Ni-V [29] and Mo-Re [7]. Chuang et al.'s [5] data were not used in ternary extensions due to unclear formulations of model parameters and Westman did not publish his results. Nevertheless, his parameterization has already been used in ternary extensions, such as, Fe-Cr-C [31] and Fe-Cr-Nb [32]. Joubert [8] published a two sublattice alternative, which solved the modeling problem of σ phase composition ranges. Xiong et al. [4] used this model, i.e. (Cr,Fe)10(Cr,Fe)20, for σ-FeCr. However, Joubert [8] also noted the simplifications of the model in terms of site distribution. Xiong et al. [4] also stated some doubt on the model extension to multicomponent systems. Calculated phase boundaries using Andersson and Sundman [3], Westman [6] and Xiong et al. [4] descriptions of σ-FeCr are shown in Fig. 3. Associated site occupancies of Fe and Cr on the different sublattices are also displayed in the Fig. 4. Clearly, results using (Fe)8(Cr)4(Cr,Fe)18 and (Fe)10(Cr)4(Cr,Fe)16

Fig. 2. Site occupancies of Fe on the different Wyckoff positions in the σ phase according to Yakel [1] and Cieslak [27].

phases was far less accurate than today. Thus, Andersson and Sundman [3] used, “for convenience”, the three sublattices model (Fe)8(Cr)4(Cr,Fe)18. This sublattice model allows Fe on the D site, Cr on the B site and a mixing on A, C and E sites (See Table 1). While the model properly reproduces the homogeneity range of σ-FeCr, Andersson [3] already speculated that it might be inappropriate for Nicontaining σ phase. Furthermore, this model wrongly assigned the moles of atoms in the different sublattices [8]. Joubert [8] revealed that in several systems (e.g. Ni-V [29]), the homogeneity range of σ phase cannot be represented by (D)8(B)4(A,C,E)18. Nevertheless, the set of parameters from [3] have been widely used in extensions to ternary [8] and multicomponent thermodynamic databases (e.g., TCFE, mc_fe). In 1997, a scientific group [30] proposed a new sublattice model for the σ phase, (Fe)10(Cr)4(Cr,Fe)16. This description indeed also represented a simplification compared to the real crystallographic case. Nevertheless, it obeys the rule to combine experimentally determined Wyckoff positions of same coordination numbers, i.e. (2A+8D)(4B) (8C+8E). While, in the binary Fe-Cr system, this description requires only two end-members, none of them lies within the composition range of the observed stability of σ phase. Moreover, this model allows to

Fig. 3. Comparison of calculated σ-FeCr stability region according to Andersson and Sundman [3], Westman [6] (i.e. only modification of the σ phase compared to Andersson and Sundman [3]) and Xiong et al. [4] compared to Cook and Jones [19].

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Fig. 4. Calculated site occupancies of Fe and Cr on different sublattices according to the modeling of a) Andersson and Sundman [3], b) Westman [6] and c) Xiong et al. [4] compared to experimental site occupancy [1,27].

2.1.4. DFT calculations of σ phase Sluiter et al. [33] reported calculated enthalpies of formation for σFeCr within the Ising model in the framework of density functional theory (DFT). All of their calculations were done in a non-magnetic state. Later, Havránková et al. [34], by using DFT, confirmed the results of Sluiter et al. [33]. Both studies reveal that the enthalpy of formation of the different configurations are positive and that the most stable configurations are - following the sequence of Wyckoff position A212B415C814D812E814 as given in Table 1 - FeCrCrFeCr and FeFeCrFeCr. Site occupancies in σ-FeCr have been studies by DFT calculations taking into account magnetism by Kabliman et al. [35] as a function of temperature for the composition Fe0.5Cr0.5. Their results are consistent with the experimental data from Yakel [1] and Cieślak et al. [27].

[3,6] are close to each other, whereas the σ-FeCr phase boundaries using (Cr,Fe)10(Cr,Fe)20 [4] deviate significantly. It was mentioned that the slight “lining off” to the Fe-rich side is an intrinsic consequence of the model and does not reproduce an experimental feature [19]. None of the up to now available model parameterizations can represent the crystal chemistry of the σ-Fe-Cr phase properly, as indicated in Fig. 4 by the deviations of site occupancies from the experimental ones [1,27]. Previous modeling alternatives of σ phase can be summarized as suitable approaches for specific questions in a distinct low-order system, either the reproduction of experimental phase boundaries, or that of site occupancies. For the proper extension of σ phase modeling to multicomponent systems, it is relevant to cope with both of these features. A proper approach will be presented in Section 3 of this paper. This, in turn, will reflect a higher physical meaningfulness of the assessed description and guarantee the usefulness of the concerning databases for kinetic simulations of σ phase evolution during complex heat treatments of technological alloys [17].

2.2. Solution phases The total Gibbs energy of the solution phase φ is given the same as in Eq. (1). Its Gibbs energy of reference is φ φ G ref , φ = x Cr GCr + xFe GFe ,

19

(4)

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Table 2 Calculated enthalpy of formation at T=0 K as obtained from DFT. Compound according to Wyckoff position A212B415C814D812E814

Compound according to Calphad model

Ferromagnetic

Non magnetic

References

ΔfH (kJ.mol−1.atom−1)

Lattice parameter (Å)

ΔfH (kJ.mol−1.atom−1)

Lattice parameter (Å)

– a=8.735 c=4.55 a=8.69 c=4.43 a=8.64 c=4.39 a=8.6 c=4.58 a=8.66 c=4.58 a=8.44 c=4.54 – –

– 14.04

– a=8.71 c=4.54 a=9.14 c=6.04 a=8.64 c=4.44 a=8.43 c=4.62 a=8.48 c=4.51 a=8.42 c=4.44 – –

Cr2Cr4Cr8Cr8Cr8 Cr2Fe4Cr8Cr8Cr8

Cr10Cr4Cr16 (Cr30) Cr10Fe4Cr16 (Cr26Fe4)

13.2 14.11

Fe2Cr4Cr8Fe8Cr8

Fe10Cr4Cr16 (Cr20Fe10)

5.62

Fe2Fe4Cr8Fe8Cr8

Fe10Fe4Cr16 (Cr16Fe14)

4.62

Cr2Cr4Fe8Cr8Fe8

Cr10Cr4Fe16 (Cr14Fe16)

12.35

Cr2Fe4Fe8Cr8Fe8

Cr10Fe4Fe16 (Cr10Fe20)

14.89

Fe2Cr4Fe8Fe8Fe8

Fe10Cr4Fe16 (Cr4Fe26)

15.27

Fe2Fe4Fe8Fe8Fe8 Fe10Fe4Fe16 (Fe30)

Fe10Fe4Fe16 (Fe30)

17.3 –

5.69 4.94 13.52 19.74 18.40 – 25.8

[51] Present work Present work Present work Present work Present work Present work [51] [54]

attempts to fit both low and high temperature ΔHmix [24] resulted in a wrong solubility of Cr in α-Fe and a too high stability of the miscibility gap α-α’. The thermodynamic modeling of Xiong et al. [4] is, in principle, in agreement with experimental data [21,39,40] and computed results from Eich et al. [10]. The results from [10] only deviate from the experimental data of Kuwano [39] at high Cr content for the composition range between 0.65 < x(Cr) < 0.95. Bonny et al. [9] could only reproduce the mixing enthalpy at 0 K and the miscibility gap by using an excessively complex Redlich-Kister (RK) model up to 5th order interactions and excess heat capacities of mixing, which lack any experimental fundament and are unlikely reflecting a physical base. We found that producing proper phase stabilities using the DFT enthalpy of mixing at 0 K directly by Calphad modeling was not possible. We speculate that this discrepancy is associated with approximations made for the magnetic contributions in DFT. The same problem occurred in the work of Xiong et al. [4]. 2.2.2. FCC The austeno-ferritic transformation in Fe-Cr has been investigated experimentally in refs. [41,42]. Based on these results, several thermodynamic assessment have been carried out [3,4,11,43,44]. In the present work, the experimental data of refs. [41,42] are shown to result in the best optimization of phase boundaries.

Fig. 5. Calculated enthalpies of formation of σ phase at 0 K by DFT compared to previous studies [33,50,52,53].

Where xCr and xFe represent the molar fraction of Cr and Fe, respectively. The Gibbs energy of mixing describing the excess Gibbs energy is:

2.2.3. Liquid Solidus and liquidus data have been measured by thermocouple and optical pyrometer techniques at high temperature up to 2100 K by Adcock [41]. The enthalpy of mixing of liquid was measured several times [45–49]. Among the existing data, huge scatter exists. One study predicted negative enthalpy [46] of mixing, suggesting a strong interaction between Fe and Cr, whereas others [47,48] suggest a repulsion of the atoms in liquid Fe-Cr. These differences are most likely the result of impurities in the samples influencing the measurements and the experimental technique used by Batalin et al. [47]. The studies of Thiedemann et al. [45] and Pavars et al. [49] suggest an almost ideal behavior of the enthalpy of mixing, which is consistent with the experimental results for the liquidus [47] and the phase diagram. Thus, the results of Thiedemann et al. [45] and Pavars et al. [49] are accepted in the present work. The liquid phase was recently re-modeled using the description of solid phases as described by Andersson and Sundman [3] by Cui et al. [50]. Their description of liquid was done with the modified quasi-chemical model, which is not compatible with the software program MatCalc and cannot be implemented in our databases for thermokinetic computations of multi-component systems.

n i, φ Gex ,φ = x Cr xFe ∑ LCr , Fe i=1

(5)

i, φ The interaction parameters LCr , Fe among Cr and Fe can be temperature dependent.

2.2.1. BCC Various atomistic studies [10,36–38] have determined the enthalpy of mixing of BCC Fe-Cr at 0 K. Consistently, a negative enthalpy of mixing below 10 at. % Cr and a maximum around 50 at. % Cr were obtained. The DFT calculations of Levesque [37] and the results of embedded-atom method from Eich et al. [10] were used in Monte Carlo simulations [10,37] to determine the miscibility gap α/α’ consistent with available experimental data [39]. The miscibility gap has also been determined experimentally by Mössbauer spectroscopy by Kuwano [39], Dubiel and Inden [19] and Dubiel and Zukrowski [21,40]. The miscibility gap of the Fe-Cr BCC phase [21,40] has been reproduced by Xiong et al. [4] using newly suggested lattice stabilities of pure Fe from Chen and Sundman [14]. Xiong et al. [4] reproduced the experimental mixing enthalpy at 1529 K [22] well, whereas further 20

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Table 3 Re-optimized model parameters of the Fe-Cr system. Phase

Model

Liquid

(Cr,Fe)

BCC

(Cr,Fe:Va)

FCC

(Cr,Fe:Va)

σ phase

(Cr,Fe)10(Cr,Fe)4(Cr,Fe)16

Parameters (J.mol−1) 0 Liquid LCr , Fe

= −5.257 × T

1 Liquid LCr , Fe = −5419.8 0 BCC_A2 LCr , Fe = 24600 − 14.98 × T 1 BCC_A2 LCr , Fe = 500 − 1.5 × T 2 BCC_A2 LCr , Fe = −14000 + 9.15 × T 0 TC Cr , Fe = +932.5 1 TC Cr , Fe = −548.2

Bmag Cr , Fe = +2.15 0 FCC_A1 LCr , Fe = 28767 − 21 × T 1 FCC_A1 LCr , Fe = 33057 − 20.7 × T 0 σ BCC_A2 GCr:Cr:Cr = +132000 + 30× 0GCr 0 σ BCC_A2 BCC_A2 GCr:Fe:Cr = +140486 + 26× 0GCr +4× 0GFe BCC_A2 0 σ BCC_A2 GFe:Cr:Cr = +56200 + 20× 0GCr +10× 0GFe 0 BCC_A2 0 BCC_A2 0 σ GFe:Fe:Cr = +46200 + 16× GCr +14× GFe 0 σ BCC_A2 BCC_A2 GCr:Cr:Fe = +123500 − 110 × T + 14× 0GCr +16× 0GFe 0 σ 0 BCC_A2 0 BCC_A2 GCr:Fe:Fe = 148800 + 10× GCr +20× GFe 0 σ BCC_A2 BCC_A2 GFe:Cr:Fe = 152700 + 26× 0GCr +4× 0GFe 0 σ 0 BCC_A2 GFe:Fe:Fe = 173333+30× GFe 0 σ LFe:Cr:Cr , Fe = −170400 − 70 × T 0 σ LFe:Fe:Cr , Fe = −330839 − 10 × T

Fig. 7. Phase stability of σ phase of the present work compared with previous thermodynamic [3,4,6] modeling and experimental data [18].

Fig. 6. Gibbs energy of σ phase calculated at T=1000 K obtained with the DFT enthalpies at 0 K, modified DFT, and from previous thermodynamic modeling [3,4,6]. The symbols show the chemical composition of the end-members according to the model used by each authors.

contributions of non-ideal mixing to the molar Gibbs energy of the phase. This will particularly be hampering the model extension to multicomponent systems. Here, we thus follow the convention of the Ringberg meeting [26], i.e. combining Wyckoff positions of same coordination number in the sublattice description (Cr,Fe)10(Cr,Fe)4(Cr,Fe)16. This model is rather similar to the one from Westman [6]. Yet, whereas Westman described σ phase with a (Fe)10(Cr)4(Fe,Cr)16 model, we allow Fe and Cr on each sublattice in accordance with its structural-chemical analysis [1]. Importantly, this allows to cover the composition range of phase stability in closer agreement to the real physico-chemical nature of the σ phase. The total Gibbs energy with the three sublattice model (Cr,Fe)10(Cr,Fe)4(Cr,Fe)16 is given by

A complete literature review was provided by Xiong et al. [13]. The interested reader is invited to refer to this paper for further details. 3. Revisions of thermodynamic modeling 3.1. σ phase 3.1.1. Sublattice description: Our literature survey of Section 2 suggests that there is a need for an improved thermodynamic description of the σ phase in terms of its proper computed phase stability and site occupancies. Indeed, a full sublattice description, considering all of the five Wyckoff positions in the form of independent sublattices, will result in a large number of end-member compounds and even more potential interactions for 21

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Fig. 8. Site occupancy of Fe in σ phase as calculated from the present thermodynamic modeling compared with experimental data [1,26].

Several authors reported the enthalpy of formation at 0 K calculated with DFT. Depending on the study, different DFT codes have been used. The reported enthalpies of formation as function of the chemical composition are shown in Fig. 5 together with the present results. In our modeling, the atoms are constrained to the chosen sublattice models and certain atomic positions, which are not necessarily the same as the ones from previous DFT calculations [34,54,55]. Thus, the previous calculations [34,54,55] cannot be directly used as input for the Calphad parameter optimizations and are, consequently, not listed in Table 2. The DFT calculations show that the Fe10Cr4Cr16 and Fe10Fe4Cr16 are the two most favorable configurations of all the end-members in the CEF. These results suggest that Fe on the first sublattice and Cr on the third sublattice are preferred, which is in agreement with experimental data [1,27]. All the data are within the same range of magnitude except for the data given by Havránková et al. [34], which show slightly higher values. Fig. 9. Calculated enthalpy of formation at T=923 K compared to previous thermodynamic modeling [3–5] and experimental data from Dench [23] and assessed data [24].

3.1.3. Parameter optimization: All optimized model parameters of this work are summarized in Table 3. Several difficulties are encountered during the optimization process of σ phase. Clearly, a “reduced” sublattice description relative to the complex nature of TCP phases does not fully take into account all nonideal mixing contributions. In Calphad modeling, the Gibbs energy curve of a phase with a certain homogeneity range is obtained by the summation of the Gibbs energies of the different end-members (See Eq. (6)). The end-members represent line compounds at a certain composition. By using the CEF[28], the different elements are placed in the different sublattices. The model can then represent the whole composition range of the system and, thus, of the phase. However, even if the sublattice model chosen covers the full homogeneity range of the phase, it nevertheless cannot represent the complexity of the crystal structure of TCP phases in full extend [7]. In the present work, a three sublattice model combining sublattices with the same coordination number and both elements is chosen, with Fe and Cr allowed on each sublattice in order to represent the full homogeneity range of the phase. The different end-members obtained by this sublattice model represent a fixed composition. In no matter, these compounds can represent the true nature of the crystal structure of the σ phase, which is reflected by their positive enthalpies of formation. Interaction parameters modeled with R-K polynomials, are used to provide reasonable consistency with the physical nature of the σ phase. Modeled excess Gibbs energies of mixing

σ 1 2 3 1 2 3 1 2 3 σ σ Gσ = yCr yCr yCr GCr : Cr : Cr + yCr yFe yCr GCr : Fe : Cr + yFe yFe yCr GFe : Fe : Cr σ 1 2 3 1 2 3 1 2 3 σ σ + yCr yCr yFe GCr : Cr : Fe + yCr yFe yFe GCr : Fe : Fe + yFe yCr yFe GFe : Cr : Fe 1 2 3 σ 1 1 1 1 + yFe yFe yFe GFe : Fe : Fe + RT [10(yCr lnyCr + yFe lnyFe ) 2 2 2 2 3 3 3 3 +4(yCr lnyCr + yFe lnyFe )+16(yCr lnyCr + yFe lnyFe )] 1 2 3 3 1 2 3 3 + yFe yCr yCr yFe LFe : Cr : Cr , Fe + yFe yFe yCr yFe LFe : Fe : Cr , Fe

(6)

Superscripts in Eq. (6) denote sublattices 1–3. L denotes the impact of non-ideal mixing between atoms separated by comma. Only interactions, which have shown to be relevant for the optimized σ-FeCr description, are given. 3.1.2. DFT data of compound energies In our model parameterization, at first, newly obtained compound enthalpies of formation at 0 K (see Table 2), analyzed by DFT, and the ones from Sluiter [52] for the pure elements, are used to parameterize the end-members of (Cr,Fe)10(Cr,Fe)4(Cr,Fe)16 σ-FeCr. DFT calculations were done with VASP [53] using the GGA:PBE approximation and the projector augmented waves (PAW) [54]. The ground state energy for each compound was obtained as follows: 1) the volume and ions were relaxed; 2) then the stresses and forces were relaxed; 3) in the last step, the tetrahedron smearing method was used for accurate calculations. 22

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Fig. 10. Calculated miscibility gap of the Fe-Cr system.

Fig. 11. Calculated spinodal decomposition in Fe-Cr compared with previous calculations from Calphad [4], molecular dynamics [10] and experiments [4,56,57].

with a difference of around 3 kJ, which is, as far as our experience goes, a typical deviation, found likewise for assessed intermetallic phases in many systems. For the less stable compounds, the Calphad assessment leads to large deviation of −8 to −10 kJ. Thereby, the σ phase is indeed significantly stabilized and temperature dependence was only necessary for the compound Cr2Cr4Fe8Cr8Fe8 (Cr10Cr4Fe16 in the 3SL model, at x(Cr)=0.467) to stabilize the σ-FeCr on the Cr rich side. The difference between the data obtained by DFT and Calphad is in the order of 8–10 kJ mol−1. This difference, which is higher than an expected error (~5 kJ mol−1), is probably due to the problem of handling magnetism by DFT (see ΔHmix calculated by DFT [10,36–38]) and due to the simplification made in the Calphad modeling in the CEF (i.e. the constraint to a certain lattice, mixing on the same sublattice), leading to an error of magnitude of about 10 kJ mol−1.

may be considered as an approximation to the partial disordering of the phase. At first, we model the σ phase by using the energy of the endmembers as obtained by DFT (we chose the energy for FM state from Table 2). This however leads to a too unstable σ phase. Any attempt to stabilize it entropically, by adding some negative temperature dependence to the end-members provoke inevitably the creation of miscibility gaps, which is physically not correct and was experimentally not observed. The prevention of miscibility gaps and stabilization of σ phase could only be obtained with a relatively low weight given to the energy of all end-members as obtained by DFT during the optimization. As a consequence, the Calphad-assessed enthalpies deviate from the DFT data. For the two most stable end-members, Fe2Cr4Cr8Fe8Cr8 and Fe2Fe4Cr8Fe8Cr8, best reproduction of phase boundaries was obtained 23

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Fig. 12. Calculated Curie temperature compared to experiments [4] and previous thermodynamic modeling [3,4].

Fig. 14. a) Calculated ΔfH (T=1529 K) compared to experimental data [23] (open triangles), thermodynamic modeling [4] and computation from molecular dynamics and Monte Carlo [10]. b) Calculated ΔHmix at 300 K from the present work and previous thermodynamic modeling [3,4] compared to enthalpy of mixing obtained by DFT at 0 K [10,35–37].

Fig. 13. Calculated heat capacity at x(Cr)=0.13 compared to experiments [4,59,60] and previous thermodynamic calculations [3,4].

To illustrate the instability of the σ phase as obtained with DFT used for the end-members, the Gibbs energy of σ phase at T=1000 K is shown in Fig. 6 and compared with previous thermodynamic modeling and the present optimization. The data as obtained by DFT and used in the thermodynamic modeling is approximately 10 kJ higher than in all thermodynamic assessments. In the model optimization, interaction parameters were chosen such as to distribute the atoms over the sublattice for a close match to the sublattice occupations [1]. According to the calculation of the site occupancies compared to experiments [1], Calphad data yield to small fractions for Fe in the first sublattice, whereas the third sublattice is completely mixed, which is correct. The second sublattice is correctly represented with an ideal mixing of species. The third sublattice with highest occupation (i.e. a fraction of 0.533) has a high influence on the stability of the σ phase. For closest agreement of computed and experimental phase boundaries, interaction parameters L(Fe:Cr:Cr,Fe) and L(Fe:Fe:Cr,Fe) were parameterized. This keeps the σ phase particularly stable around ~50%, and reproduces the phase boundaries [19] and the invariant reactions [4,21]. The phase stability of the σ phase, as calculated in the present work, is shown in Fig. 7 and compared with previous thermodynamic modeling and experimental data [19]. Andersson and Sundman [3] as well

Fig. 15. Activities of Fe and Cr calculated at T=1173 K from the present thermodynamic modeling compared to experiment [55] and previous modeling [3,4].

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to proper phase boundaries, calculated site occupancies are in good agreement to the experimental evidence. This is shown in Fig. 8. Any attempt to get even closer to the data of the enthalpy from Dench [24] in the parameter optimization leads to worse computation of phase boundaries and invariant reactions. Thus, the deviation by our thermodynamic optimization of 1 kJ mol−1, see Fig. 9, is likely indicating the experimental uncertainty range and was decided to be acceptable. Compared to previous thermodynamic modeling of the σ-FeCr [3–6], our description can reproduce the phase stability and the crystal chemistry of the phase. Correct computed site fractions are of fundamental importance when using thermodynamic modeling in thermokinetic precipitation simulations as associated chemical potentials are also influencing the kinetic evolution of a steel matrix-precipitate system. 3.2. Alloy phases Fig. 16. Calculated FCC-BCC phase boundary according to the present thermodynamic modeling.

The BCC phase is the predominant alloy phase in the Fe-Cr system and occurs in direct equilibrium with all the other phases. Thus, we optimized its model parameters first. Since the BCC phase has adjacent phase boundaries to σ-FeCr, modifications of the thermodynamic description of the latter also require adjustments of the description of the former. This, in turn, requires re-adjustments of the FCC and liquid phase descriptions in the Fe-Cr system.

Table 4 Invariant reactions as obtained from the present thermodynamic modeling compared with experiments and their respective references. Reaction

Calculated

Experiments

C: L ↔ BCC

1782 K x(Cr)=0.21 1126 K x(Cr) = 0.073 1096 K x(Cr)=0.46 788 K x(Cr,BCC)=0.15 x(Cr,σ)=0.49 x(Cr,BCC’)=0.85

1783 K, x(Cr)=0.21 [41] 1783 K, x(Cr)=0.21 [13]a 1119 K, x(Cr)=0.07 [13]a

C: FCC ↔ BCC C: BCC ↔ σ Eutectoid: σ ↔ BCC + BCC’

a

3.2.1. BCC phase The BCC phase was considered for re-modeling since the thermodynamic description of Andersson and Sundman [3] does not allow Cr to dissolve into BCC α-Fe and leads to a high Curie temperature compared to experiment [4,13]. Even though Xiong et al. [4] presented an accurate description of the alloy phases, their parameters are not being adopted in the present study due to our focus of inserting our present system description into an existing multi-component database. Thus, drifting away from conventional pure element lattice stabilities [14] would necessitate enormous efforts of further Fe-base system re-optimizations. Anyway, we will show, in the following, that proper phase stabilities can be obtained in the Fe-Cr system with lattice stabilities from Dinsdale [18]. In order to reproduce the experimental solubilities of Cr and Fe in the BCC-phase, or, in other words, its miscibility gap [21,39,40], the excess parameters of mixing of Cr and Fe from Xiong et al. [4] were used as a starting point for the necessary re-adjustments. Recent molecular dynamics calculations of BCC-FeCr from Eich et al. [10] were also considered for the optimization, as well as thermodynamic [24,56] and solidus-liquidus data [41]. For the final description of the BCC phase, three interactions parameters were required, all of them with temperature dependence. The miscibility gap between BCC+BCC’ is associated with the occurrence of a spinodal decomposition reaction. The calculated miscibility gap obtained from the present thermodynamic modeling is given in Fig. 10 and the calculated curves of spinodal decomposition are drawn in Fig. 11. There is a good agreement between the experimental data of the miscibility gap [21,39,40] and the present calculations. The present thermodynamic modeling leads to a slight deviation in the higher temperature range of the miscibility gap. Any attempt to improve this part of the phase diagram, adjusting temperature-dependences of 1st and 2nd order interactions in BCC, lead to erroneous destabilization of the liquid phase. With the present thermodynamic parameters, a maximum difference of about 50 K from the experimental data of Kuwano [39] is obtained. The recent computational determination of the miscibility gap by combination of molecular dynamics and Monte Carlo simulation [10] shows a good agreement with experiments [21,39,40] in comparison to previous computation [37]. The work of Eich et al. [10] considered the experimental determination of the enthalpy of

1093 K, x(Cr)=0.47 [19] 1103 K, x(Cr)=0.47 [13]a 783 K [21] x(Cr,BCC)=0.143 x(Cr,σ)=0.49 x(Cr,BCC’)=0.88

-according to literature assessment.

Fig. 17. Calculated ΔHmix of liquid Fe-Cr at T=1960 K compared with experimental data, T=1960 K [44], T=1863 K [45], T=1873 K [46], T=1973 K [47], T=1873-1973 K [48] and previous thermodynamic modeling [3,4,12,49].

as Westman [6] obtained a too small single-phase region compared to the experimental data of Cook and Jones [19], whereas the present thermodynamic modeling as well as Xiong et al. [4] are in closer agreement with [19]. The advantage of our revised model, with the discussed combination rules for sublattices, over previous assessments is that, in addition 25

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Fig. 18. Calculated activity of the Fe-Cr liquid at T=1873 K referring to pure liquid Fe and pure BCC Cr compared to previous thermodynamic modeling and experiments [48,58].

mixing at high temperature given by Dench [24] and not the one obtained by their own DFT, which was also problematic for the Calphad reproduction of the phase diagram [4]. With the present thermodynamic modeling, more Cr compared to Andersson and Sundman [3], but slightly less than Xiong [4] and Eich et al. [10] is dissolved in the α-Fe at low temperature. As for the time being, there is no experimental solubility data available in this part of the phase diagram, it is difficult to judge about the best optimization. In any case, it is important that a reasonable amount of Cr is allowed to dissolve into α-Fe as this will influence important phase stability features such as BCC → FCC transformation and spinodal decomposition in multi-component, i.e. technologically relevant extensions. For instance, ferrite decomposition to a Fe- and Cr-rich solid solution BCC+BCC’ is in part responsible for embrittlement of Cr-rich steels at low temperatures. Therefore, in the context of model extensions and their use in phase stability simulations of steels, it is important that the spinodal decomposition is correctly described by the modeling as shown in Fig. 11. The spinodal, as calculated in the present work (Fig. 11), is in good agreement with experiments [4,57,58]. The model parametrization for the ferromagnetic to paramagnetic transition in Fe-Cr BCC was reconsidered to obtain an improved reproduction of the experimental Curie temperatures as a function of Cr-alloying. Consistency with the experiment was obtained, as shown in Fig. 12, with the parametrization include in Table 2. For the magnetic moment we chose +2.15 leading to close agreement with the experimental Cp peak shape (see Fig. 13).

Fig. 19. Calculated phase boundary Liquid – BCC compared to experimental works [40].

3.2.1.1. Thermodynamic properties. The enthalpy of mixing at high temperature is calculated in Fig. 14 and compared to Dench [24]. For comparison, results of Xiong [4] and Eich et al. [10] are also given. The results of the calculated enthalpies at high temperature are consistent with experiments and the previous Calphad assessment [4]. The calculations of several studies [10,36–38] show a small negative enthalpy of mixing at the Fe-rich side. By considering a negative value in their assessment, Xiong et al. [4] were also not successful to reproduce a proper phase diagram. The same problem occurred in the present work, and the negative value at the Fe-rich side was discarded in the optimization procedure. The calculated ΔHmix at 300 K of the Calphad modeling (present work and [3,4]) thus does not reproduce a negative ΔHmix at low Cr content as given by the different DFT studies [10,36–38]. Even though, experimental thermodynamic properties could be well reproduced as shown by the calculated activities at 1173 K compared to [56] in Fig. 15.

Fig. 20. Calculated phase diagram according to the present work.

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3.2.2. FCC The thermodynamic modeling of Andersson and Sundman [3] provide the best agreement among various Calphad assessments. The experimental data on the austeno-ferrite transformation from Adcock [41] and Kirchner [42] were reproduced (see Fig. 16) with a relative small re-adjustment of interaction parameters relative to the previous suggestion [4].

technological alloys, we believe that this effort will be highly worthwhile. It should be mentioned that the revised σ phase modeling does not produce problems for the calculated phase stabilities of the alloy phases in the Fe-Cr system. Only relatively slight re-adjustments of the model parameterizations are necessary for the reproduction of all phase boundaries.

3.2.3. Liquid In the present work, the enthalpy of mixing, as given by Thiedemann et al. [45] and Pavars et al. [49], and experimental solidus and liquidus data and the congruent melting temperature [41] were considered for the re-optimization of the liquid phase. The enthalpy of mixing of liquid on the Fe-rich side is close to zero J.mol−1, thus, the liquid phase follows almost Raoult's law, i.e. ideal behavior of mixture. Nevertheless, as a consequence of the considerable non-ideality of the BCC-description, excess Gibbs energy of mixing is required in the Calphad description in order to reproduce the experimental solidus and liquidus data. A small temperature-dependence for the 0th order R-K polynomial was used, as well as a 1st order interaction parameter without temperature dependence. This, in turn, required optimization of temperature the dependence of the 1st and 2nd order R-K polynomials of the BCC phase for the correct reproduction of the hightemperature section of the system (see Table 4). The results for the enthalpy of mixing of liquid at T=1960 K are given in Fig. 17 and compared with experiments [45–49] and previous thermodynamic modeling [3,4,12,50]. Our calculated enthalpies of mixing are in good agreement with the experimental work of Thiedeman [45] and Pavars et al. [49] as well as the activity calculated at T=1873 K and compared with experiments [49,59] (the calculation refers to pure liquid of Fe and Cr as given experimentally), see Fig. 18. The respective solidus and liquidus obtained from the present thermodynamic modeling is shown in Fig. 19 and compared with the experimental results of Adcock [41]. In principle, good agreement is found. Only at higher Cr content, the calculations deviate from the measurements, similar as for previous modeling. Indeed, at high temperature, Cr volatizes, which affects the chemical composition during the measurement. Thus, the experimental data in this part of the phase diagram likely contain considerable uncertainty and are not given high weight in the parameter optimization.

4. Conclusion Thermodynamic model descriptions of the Fe-Cr system have been revised with the focus of physico-chemically correct site occupancies of solute atoms in the model description of σ phase. Wyckoff positions and first-principles compound energies are considered for the optimization of model parameters. The presented and discussed three sublattice model (Cr,Fe)10(Cr,Fe)4(Cr,Fe)16 shows to be more respectful to the structuralchemical nature of the σ-phase compared to previous assessments. This is reflected by proper trends of calculated site occupancies as function of composition and temperature. The consistency of the present model is further reflected in the close agreement between experimental and calculated phase diagram data. Moreover, the model could be kept relatively simple; only two interactions parameters for the non-ideal mixing of Fe and Cr in Fe-Cr σ have been adapted. Acknowledgements The computational results presented in this work have been achieved using the Vienna Scientific Cluster (VSC). Funding by the Austrian Science Fund (FWF): P13 (SFB ViCoM) is gratefully acknowledged. Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at doi:10.1016/j.calphad.2017.10.002. References [1] H.L. Yakel, Atom distributions in sigma phases. I. Fe and Cr atom distributions in a binary sigma phase equilibrated at 1063, 1013 and 923 K, Acta Crystallogr. Sect. B. 39 (1983) 20–28, http://dx.doi.org/10.1107/S0108768183001974. [2] H. Yakel, Atom distributions in sigma phases. II. Estimations of average site-occupation parameters in a sigma phase containing Fe, Cr, Ni, Mo and Mn, Acta Crystallogr. Sect. B Struct. Sci. 39 (1983) 28–33, http://dx.doi.org/10.1107/ S0108768183001986. [3] J.-O. Andersson, B. Sundman, Thermodynamic properties of the Cr-Fe system, Calphad 11 (1987) 83–92, http://dx.doi.org/10.1016/0364-5916(87)90021-6. [4] W. Xiong, P. Hedström, M. Selleby, J. Odqvist, M. Thuvander, Q. Chen, An improved thermodynamic modeling of the Fe–Cr system down to zero kelvin coupled with key experiments, Calphad 35 (2011) 355–366, http://dx.doi.org/10.1016/j. calphad.2011.05.002. [5] Y. Chuang, J.-C. Lin, Y.A. Chang, A thermodynamic description and phase relationships of the Fe-Cr system: Part I The bcc phase and the sigma phase, Calphad 11 (1987) 57–72. [6] S. Westman, Unpublished work, 2000. [7] R. Mathieu, N. Dupin, J.C. Crivello, K. Yaqoob, A. Breidi, J.M. Fiorani, et al., CALPHAD description of the Mo-Re system focused on the sigma phase modeling, Calphad Comput. Coupling Phase Diagr. Thermochem. 43 (2013) 18–31, http://dx. doi.org/10.1016/j.calphad.2013.08.002. [8] J.-M. Joubert, Crystal chemistry and Calphad modeling of the σ phase, Prog. Mater. Sci. 53 (2008) 528–583, http://dx.doi.org/10.1016/j.pmatsci.2007.04.001. [9] G. Bonny, D. Terentyev, L. Malerba, New contribution to the thermodynamics of FeCr alloys as base for ferritic steels, J. Phase Equilibria Diffus. 31 (2010) 439–444, http://dx.doi.org/10.1007/s11669-010-9782-9. [10] S.M. Eich, D. Beinke, G. Schmitz, Embedded-atom potential for an accurate thermodynamic description of the iron-chromium system, Comput. Mater. Sci. 104 (2015) 185–192, http://dx.doi.org/10.1016/j.commatsci.2015.03.047. [11] S. Hertzman, B. Sundman, A thermodynamic analysis of the Fe-Cr system, Calphad 6 (1982) 67–80. [12] B.-J. Lee, Revision of thermodynamic descriptions of the Fe-Cr & Fe-Ni liquid phases, Calphad 17 (1993) 251–268. [13] W. Xiong, M. Selleby, Q. Chen, J. Odqvist, Y. Du, Phase equilibria and thermodynamic properties in the Fe-Cr system, Crit. Rev. Solid State Mater. Sci. 35 (2010)

3.2.4. Equilibrium phase transformations The full phase diagram of Fe-Cr was calculated from the present thermodynamic parameters (Table 3) and is presented in Fig. 1 and compared to previous assessments and experimental phase diagram data in Fig. 20. The calculated invariant reactions obtained from the present thermodynamic modeling are given in Table 4 in comparison to experiment data. To summarize, the agreement of the present calculated phase diagram (Fig. 20), special points and transformations compared to experiments [19,21,41,42] are good. The present thermodynamic modeling allows for a higher amount of Cr being dissolved in α-Fe at low temperature compared to Andersson et al. [3]. The calculated decomposition of BCC agrees with the experimental observation [21]. The recalculated phase diagram is also in agreement with the previous one given by Xiong et al. [4]. 3.2.5. Consequences of thermodynamic modeling The present improvement of the sublattice model for σ-FeCr is the first step towards more physically-based steel databases. Indeed, to change the thermodynamic modeling of σ phase in various other systems in accordance will be a challenging task. Nevertheless, in view of the associated improvements of the usability of multicomponent databases for predictive simulations of phase transformations in 27

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