Calphad 60 (2018) 16–28
Contents lists available at ScienceDirect
Calphad
journal homepage: www.elsevier.com/locate/calphad
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: aurelie.jacob@tuwien.ac.at (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/).
Calphad 60 (2018) 16–28
A. Jacob et al.
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
Calphad 60 (2018) 16–28
A. Jacob et al.
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].
18
Calphad 60 (2018) 16–28
A. Jacob et al.
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)
Calphad 60 (2018) 16–28
A. Jacob et al.
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
Calphad 60 (2018) 16–28
A. Jacob et al.
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
Calphad 60 (2018) 16–28
A. Jacob et al.
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
Calphad 60 (2018) 16–28
A. Jacob et al.
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
Calphad 60 (2018) 16–28
A. Jacob et al.
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].
24
Calphad 60 (2018) 16–28
A. Jacob et al.
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
Calphad 60 (2018) 16–28
A. Jacob et al.
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.
26
Calphad 60 (2018) 16–28
A. Jacob et al.
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
Calphad 60 (2018) 16–28
A. Jacob et al.
125–152, http://dx.doi.org/10.1080/10408431003788472.
[14] Q. Chen, B. Sundman, Modeling of thermodynamic properties for BCC, FCC, liquid,
and amorphous iron, J. Phase Equilibria. 22 (2001) 631–644, http://dx.doi.org/10.
1007/s11669-001-0027-9.
[15] R. Naraghi, M. Selleby, J. Ågren, Thermodynamics of stable and metastable structures in Fe-C system, Calphad Comput. Coupling Phase Diagr. Thermochem. 46
(2014) 148–158, http://dx.doi.org/10.1016/j.calphad.2014.03.004.
[16] I. Roslyakova, B. Sundman, H. Dette, L. Zhang, I. Steinbach, Modeling of Gibbs
energies of pure elements down to 0K using segmented regression, Calphad 55
(2016) 165–180, http://dx.doi.org/10.1016/j.calphad.2016.09.001.
[17] ⟨www.matcalc-engineering.com⟩.
[18] A.T. Dinsdale, SGTE data for pure elements, Calphad 15 (1991) 317–425.
[19] A.J. Cook, F. Jones, The brittle constituent of the iron-chromium system (sigma
phase), Iron Steel Inst. - Pap. 217 (1943) 217–226.
[20] R. Vilar, G. Cizeron, Evolutions structurales developpées au sein de la phase sigma
Fe-Cr, Acta Metall. 35 (1987) 1229–1239.
[21] S.M. Dubiel, G. Inden, On the miscibility gap in the Fe-Cr system: a moessbauer
study on long term annealed alloys, Z. Fuer Met. 78 (1987) 544–549.
[22] R. Williams, H. Paxton, The nature of ageing of binary iron-chromium alloys around
500 °C, J. Iron Steel Inst. 185 (1957) 358–374.
[23] T. Koyano, U. Mizutani, H. Okamoto, Evaluation of the controversial s–>(Cr)+(aFe) eutectoid temperature in the Fe-Cr system by heat treatment of mechanically
alloyed powder, J. Mater. Sci. Lett. 14 (1995) 1237–1240.
[24] W.A. Dench, Adiabatic high-temperature calorimeter for the measurement of heats
of alloying, Trans. Faraday Soc. 59 (1963).
[25] F. Müller, O. Kubaschewski, The thermodynamic properties and the equilibrium
diagram of the system chromium-iron, High. Temp. - High. Press. 1 (1969)
543–551.
[26] H. Martens, P. Duwez, Heat evolved and volume change in the alpha-sigma-transformation in Cr-Fe alloys, Trans. Am. Inst. Min. Metall. Eng. 206 (1956) (614–614).
[27] J. Cieślak, M. Reissner, S.M. Dubiel, J. Wernisch, W. Steiner, Influence of composition and annealing conditions on the site-occupation in the σ-phase of Fe–Cr and
Fe–V systems, J. Alloy. Compd. 460 (2008) 20–25, http://dx.doi.org/10.1016/j.
jallcom.2007.05.098.
[28] M. Hillert, The compound energy formalism, J. Alloy. Compd. 320 (2001) 161–176,
http://dx.doi.org/10.1016/S0925-8388(00)01481-X.
[29] A. Watson, F.H. Hayes, Some experiences modelling the sigma phase in the Ni-V
system, J. Alloy. Compd. 320 (2001) 199–206, http://dx.doi.org/10.1016/S09258388(00)01472-9.
[30] I. Ansara, T.G. Chart, A.F. Guillermet, F.H. Hayes, U.R. Kattner, D.G. Pettifor, et al.,
Themodynamic modelling of solutions and alloys, Calphad 21 (1997) 171–218.
[31] A.V. Khvan, B. Hallstedt, C. Broeckmann, A thermodynamic evaluation of the Fe-CrC system, Calphad Comput. Coupling Phase Diagr. Thermochem. 46 (2014) 24–33,
http://dx.doi.org/10.1016/j.calphad.2014.01.002.
[32] A. Jacob, C. Schmetterer, A. Khvan, A. Kondatriev, D. Ivanov, B. Hallstedt, Liquidus
projection and thermodynamic modeling of Cr-Fe-Nb ternary system, Calphad 54
(2016) 1–15, http://dx.doi.org/10.1016/j.jallcom.2008.10.156.
[33] M.H.F. Sluiter, K. Esfarjani, Y. Kawazoe, Site occupation reversal in the Fe-Cr sigma
phase, Phys. Rev. Lett. 75 (1995) 3142–3146.
[34] J. Havránková, J. Vřešťál, L. Wang, M. Šob, Ab initio analysis of energetics of σphase formation in Cr-based systems, Phys. Rev. B. 63 (2001) 1–5, http://dx.doi.
org/10.1103/PhysRevB.63.174104.
[35] E. Kabliman, P. Blaha, K. Schwarz, O.E. Peil, A.V. Ruban, B. Johansson,
Configurational thermodynamics of the Fe-Cr σ phase, Phys. Rev. B. 84 (2011)
184206, http://dx.doi.org/10.1103/PhysRevB.84.184206.
[36] P. Olsson, I. a. Abrikosov, L. Vitos, J. Wallenius, Ab initio formation energies of
Fe–Cr alloys, J. Nucl. Mater. 321 (2003) 84–90, http://dx.doi.org/10.1016/S00223115(03)00207-1.
[37] M. Levesque, E. Martínez, C.-C. Fu, M. Nastar, F. Soisson, Simple concentrationdependent pair interaction model for large-scale simulations of Fe-Cr alloys, Phys.
Rev. B. 84 (2011) 184205, http://dx.doi.org/10.1103/PhysRevB.84.184205.
[38] I. Dopico, P. Castrillo, I. Martin-Bragado, Quasi-atomistic modeling of the
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
[47]
[48]
[49]
[50]
[51]
[52]
[53]
[54]
[55]
[56]
[57]
[58]
[59]
[60]
28
microstructure evolution in binary alloys and its application to the FeCr case, Acta
Mater. 95 (2015) 324–334, http://dx.doi.org/10.1016/j.actamat.2015.05.040.
H. Kuwano, Mössbauer effect study on the miscibility gap of the iron-chromium
binary system, Trans. Jpn. Inst. Met. 26 (1985) 473–481 ⟨http://biz3.bioweb.ne.jp/
jim/journal/e/26/07/473.html⟩.
S.M. Dubiel, J. Zukrowski, Fe-rich border and activation energy of phase decomposition in a Fe-Cr alloy, Mater. Chem. Phys. 141 (2013) 18–21, http://dx.doi.org/
10.1016/j.matchemphys.2013.05.023.
F. Adcock, Alloys of iron research Part X - The chromium-iron constitutional diagram, J. Iron Steel Inst. 129 (1931) 99.
G. Kirchner, T. Nishizawa, B. Uhrenius, The distribution of chromium between
ferrite and austenite and the thermodynamics of the a/g equilibrium in the Fe-Cr
and Fe-Mn system, Mettalurgical Trans. 4 (1973) 167–174.
L. Kaufman, Discussion of thermodynamics of a-y transformation in Fe-Cr alloys,
Mettalurgical Trans. 5 (1974) 1688–1689.
J. Chipman, Thermodynamic of the a-y transformation in the Fe-Cr alloys, Metall.
Trans. 5 (1974) 521–523.
U. Thiedemann, M. Rösner-Kuhn, D.M. Matson, G. Kuppermann, K. Drewes,
M.C. Flemings, et al., Mixing enthalpy measurements in the liquid ternary system
iron-nickel-chromium and its binaries, Steel Res. 1 (1998) 3–7.
Y. Iguchi, S. Nobori, K. Saito, T. Fuwa, A calorimetric study of heats of mixing of
liquid iron alloys Fe-Cr, Fe-Mo, Fe-W, Fe-W, Fe-V, Fe-Nb, Fe-Ta, J. Iron Steel Inst.
Jpn. 68 (1982) 89–96.
G. Batalin, V. Kurach, V. Sudavstova, Enthalpies of mixing of Fe-Cr and Fe-Ti melts,
Zhurnal Fiz. Khimii. 58 (1984) 481–483.
V. Shumikhin, A.K. Biletsky, G. Batalin, V. Anishin, Study of thermodynamic and
kinetic parameters of dissolution of solid materials in ferrocarbonic melts, Arch. Für
Das. Eisenhüttenwes. 52 (1981) 143–146.
I.A. Pavars, B. Baum, P.V. Gel’d, Thermophysical and thermodyanmic properties of
liquid alloys of iron and chromium, Teplofiz. Vysok. Temp. 8 (1970) 72–76.
S. Cui, I.-H. Jung, Thermodynamic modeling of the Cu-Fe-Cr and Cu-Fe-Mn systems,
Calphad 56 (2017) 241–259, http://dx.doi.org/10.1016/j.calphad.2017.01.004.
M.H.F. Sluiter, Ab initio lattice stabilities of some elemental complex structures,
Calphad 30 (2006) 357–366, http://dx.doi.org/10.1016/j.calphad.2006.09.002.
G. Kresse, J. Furthmüller, Efficiency of ab-initio total energy calculations for metals
and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6 (1996)
15–50.
G. Kresse, D. Joubert, From ultrasoft pseudopotentials to the projector augmentedwave method, Phys. Rev. B. 59 11–19. ⟨http://prb.aps.org/abstract/PRB/v59/i3/
p1758_1⟩ (accessed 13 September 2013).
J. Pavlů, J. Vřešťál, M. Šob, Ab initio study of formation energy and magnetism of
sigma phase in Cr-Fe and Cr-Co systems, Intermetallics 18 (2010) 212–220, http://
dx.doi.org/10.1016/j.intermet.2009.07.018.
E.A. Kabliman, A.A. Mirzoev, A.L. Udovskii, First-principles simulation of an ordered sigma phase of the Fe-Cr system in the ferromagnetic state, Phys. Met.
Metallogr. 108 (2009) 435–440, http://dx.doi.org/10.1134/
S0031918×09110027.
J. Vrestal, J. Tousek, A. Rek, Thermodynamic activity of components in the Fe-Cr asolid solution, Kov. Mater. 4 (1978) 393–399.
D. Chandra, L.H. Schwartz, Mössbauer effect study of the 475 °C decomposition of
Fe-Cr, Metall. Trans. 2 (1971) 511–519, http://dx.doi.org/10.1007/BF02663342.
Y. Imai, M. Izumiyama, T. Masumoto, Phase transformation of Fe-Cr binary system
at about 500 °C, Sci. Rep. Res. Inst. Tohoku Univ. Ser. A - Phys. Chem. Metall. S
(1966) 56.
N. Maruyama, S. Ban-Ya, Measurement of activities in liquid Fe-Ni, Fe-Co AND NiCo alloys by a transportation method, Nippon Kinzoku Gakkaishi/J. Jpn. Inst. Met.
42 (1978) 992–999.
R.N. Hajra, R. Subramanian, H. Tripathy, A.K. Rai, S. Saibaba, Phase stability of
Fe–15wt.% Cr alloy: a calorimetry and modeling based approach to elucidating the
role of magnetic interactions, Thermochim. Acta. 620 (2015) 40–50, http://dx.doi.
org/10.1016/j.tca.2015.10.002.