Introduction

The Fe-Cr phase diagram published in handbooks[1,2] shows the existence of four stable solid phases, namely: the body-centred-cubic (bcc) α and α′, respectively rich in iron and in chromium; the face-centred-cubic (fcc) γ, which is confined to a closed region between ~1100 and 1700 K and limited to Cr concentrations, x Cr, up to ~12 at.% (henceforth %)—the so-called γ-loop; and the complex σ-phase, confined around x Cr ~ 50% and above temperatures between ~800 and 1100 K. These phases coexist in different regions: below ~800 K there is a miscibility gap between α and α′, which becomes metastable above this temperature and at equilibrium divides up into the α-σ and σ-α′ regions; in the thin region delimiting the γ-loop, on the other hand, the α and γ phases co-exist. All the rest of the diagram corresponds to a single bcc phase, which is limited by the liquid-solid phase equilibria at very high temperature. This phase diagram is well reproduced by the CALPHAD formulation proposed by Andersson and Sundman[3] (henceforth standard parameterization). The latter is based on experimental data above 700 K and for alloys with x Cr > 10%, but has been shown to appreciably underestimate the Cr solubility below it.[4,5]

The region of Cr concentration around 10% is of special interest for power industry applications at elevated temperature (up to 920 K). As a matter of fact, ferritic-martensitic (FM) steels containing up to 12% Cr have been historically developed to provide high resistance to creep.[6] The addition of 1-2.5% Cr to Mo-alloyed steels doubles the 100,000 h creep rupture strength at 770 K (from 70 to 140 MPa). However, in order to obtain acceptable 100,000 h creep rupture strength (~100 MPa) up to 870 K, a higher Cr content is required and the best performances have been achieved with x Cr between 9 and 12%.

Because of these good high temperature properties, combined with higher thermal conductivity and lower expansion coefficient than austenitic steels, 9-12% Cr FM steels have also been chosen since the 1970s as candidate structural materials for future nuclear reactors, such as Generation IV concepts and fusion systems.[7,8] An additional reason to use these materials for nuclear applications is that experimental studies of their behavior under neutron irradiation showed that they are much less affected by irradiation swelling than austenitic steels.[7,8] Thus, commercial and experimental FM steels containing about 9-12% Cr and few percents of minor alloying elements have been considered or developed for nuclear applications in Russia, the US, Japan, China, and Europe.[9,10] As an illustration, the typical composition of some of these high-Cr steels is summarized in Table 1.

Table 1 Composition (wt.%) of reduced activation FM steels with a favorable combination of physical properties[9,10]
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The target in the power industry, both nuclear and non-nuclear, is to work at high temperatures in order to increase thermal efficiency and, more recently, reduce carbon dioxide emissions. However, the material will, in practice, be exposed to temperatures ranging from ambient to the maximum acceptable by design. The case of liquid-metal-cooled nuclear reactors is especially critical, because the materials will spend long times during maintenance at temperatures that must be high enough to keep the metal liquid. These temperatures correspond to those for which the Fe-Cr standard phase diagram is not correct, by predicting a much lower solubility of Cr in Fe than the real one.[4,5] The actual position of the solubility limit then becomes critical because, if the material happens to be inside the α-α′ miscibility gap, the supersaturation of point-defects produced by irradiation will rapidly lead to phase separation,[11-13] subsequently causing unwanted embrittlement.[14-18] The steel compositions considered for nuclear applications are exactly in the region where the solubility limit is expected to be, therefore small composition variations will determine whether or not the material will be susceptible to embrittlement during operation. It should be noted that in actual steels carbon content and heat treatment have a significant influence on the micro-structure of high-Cr steels. In addition, carbon also influences the equilibrium phase diagram.[19] The present work, however, is limited to the equilibrium phase diagram of the binary Fe-Cr system.

The physical explanation for the higher solubility limit than predicted by the standard phase diagram is the existence of a tendency for Cr atoms to produce short-range order (SRO) in Fe which has been experimentally observed[20-22] and is also supported by both theoretical considerations[23] and density functional theory (DFT) calculations,[24-30] which predict a negative heat of mixing (HOM) below about 10% Cr. However, the standard Calphad parameterization does not account for this at all. Thus, it appears necessary to re-parameterize the equations describing the thermodynamic properties of the Fe-Cr system, as predicted by the standard parameterization, so as to re-adjust the low temperature Fe-rich phase boundary. In this work a new parameterization of the Gibbs free energy for the bcc phase is proposed, based on theoretical considerations and the newly proposed Fe-rich phase boundary.[4] The re-parameterized free energy surface is merged smoothly with the standard parameterization at 1100 K, so that the thermodynamic properties above this temperature remain unchanged. The γ-loop and the liquid-solid equilibrium are beyond the scope of the present work.

Thermodynamic Model

The sub-regular solution model used in the standard parameterization has proven unable to reproduce the change of sign in the mixing enthalpy and the high Cr solubility below 700 K. Since a sub-lattice model seems to be inappropriate to describe the effects of SRO, a solution model including higher order interactions in composition (here up to fifth order) is used.[31]

The total molar Gibbs free energy G bcctot is given as,

$$ G_{\text{tot}}^{\text{bcc}} = x_{\text{Cr}} G_{\text{Cr}}^{\text{bcc}} + x_{\text{Fe}} G_{\text{Fe}}^{\text{bcc}} + RT(x_{\text{Cr}} \,\ln \,x_{\text{Cr}} + x_{\text{Fe}} \,\ln \,x_{\text{Fe}} ) + G_{\text{xc}}^{\text{bcc}} + G_{\text{M}}^{\text{bcc}} . $$
(1)

Here the first two terms give the contribution from the mechanical mixture, the third one is the contribution from the configurational entropy for an ideal solution,[32] the fourth term describes concentration dependent interactions and the last term accounts for the magnetic contribution to the free energy. In addition, in Eq 1, x Cr (x Fe) denotes the molar fraction of Cr(Fe), R denotes the gas constant and T is the absolute temperature.

The free energy of the pure elements, G bccFe and G bccCr is taken from the SGTE (Scientific Group Thermodata Europe) database[33] and is given in the Appendix. The excess free energy is expressed as a Redlich-Kister polynomial,[34]

$$ G_{\text{xc}}^{\text{bcc}} = x_{\text{Fe}} x_{\text{Cr}} \sum\limits_{p = 0}^{N} {L_{p}^{\text{bcc}} (T)(x_{\text{Cr}} - x_{\text{Fe}} )^{p} } , $$
(2)

with N being the order of the composition dependent interaction (here N = 5) and L bcc p (T) the p-th interaction parameter between Cr and Fe, given as,

$$ L_{p}^{\text{bcc}} (T) = l_{0}^{p} + l_{1}^{p} T + l_{2}^{p} T^{2} + l_{3}^{p} T^{3} . $$
(3)

The magnetic term, G bccM , has been defined by Hillert and Jarl[35] following the work of Inden[36,37] and is given in the Appendix.

Parameterization

The optimization of the Gibbs free energy parameters {l p0 , …, l p3 } was performed in the temperature range 298.15-1100 K. The six parameters {l 00 , …, l 50 } were optimized to reproduce the new Fe-rich phase boundary, the standard Cr-rich phase boundary and the DFT calculated mixing enthalpy.[28] For each trial set {l 00 , …, l 50 } the other parameters were determined by the constraints of continuity of Gibbs free energy surface, enthalpy surface, entropy surface and heat capacity in T at all compositions with requirement to blend into the standard parameterization at 1100 K. The latter constraint translates to,

$$ \left. {{\frac{{\partial^{n} G_{\text{xc}}^{\text{new}} }}{{\partial T^{n} }}}} \right|_{T = 1100} = \left. {{\frac{{\partial^{n} G_{\text{xc}}^{\text{standard}} }}{{\partial T^{n} }}}} \right|_{T = 1100} , $$
(4)

for n = 0, 1, 2 and thus uniquely determine the remaining parameters. Note that the temperature 1100 K falls just below the onset of the γ-loop, which starts at 1120 K, so by definition the position of the γ-loop and liquid-solid phase boundaries remains unchanged. The obtained parameters are summarized in the Appendix.

Results and Discussion

The metastable miscibility gap resulting from the new parameterization for the free energy is presented in Fig. 1. In the same figure, the miscibility gap obtained from the standard parameterization is plotted together with the new Fe-rich phase boundary. The new parameterization does not provide an exact fit to the boundary suggested by Bonny et al.,[4] but this is not necessary since the suggested boundary is only indicative. The boundary resulting from the new parameterization, however, does reproduce a large solubility of Cr (~7% Cr) at low temperature (<700 K). Aside from the Fe-rich phase boundary below 700 K, the miscibility gap is almost coincident with the standard one. A comparison of the equilibrium bcc-σ-phases, obtained with the two parameter sets, shows that the phase boundaries are almost coincident. The reason for this lays in the smooth transition of the re-assessed free energy surface with the original one, as explained above. For completeness, we mention that the isotherm at the onset of σ-phase formation is shifted up by 7 K compared to the standard parameterization and that above 750 K the difference in the location of any phase boundary between new and standard parameterization is at most 2%.

Fig. 1
figure 1

Calculated metastable miscibility gap

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Fig. 2
figure 2

Mixing enthalpy at 900 K

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Figure 2 compares the mixing enthalpy at 900 K for the two sets of parameters. Both curves are almost coincident and have a parabolic shape in agreement with experimental observations in the paramagnetic bcc phase,[38] superposed in Fig. 2. The negative HOM at the Fe-rich side has disappeared as the Curie temperature is approached.

The mixing enthalpy of bcc Fe-Cr solution, calculated at 300 K using the new and standard parameter sets, is presented in Fig. 3, together with the data obtained from the DFT calculations[28] (strictly speaking valid only at 0 K for a random alloy). The shape of the mixing enthalpy reproduced by the proposed parameterization set is in good agreement with the DFT curve, and falls well within the error of the different DFT approaches for x Cr > 0.2 (for a detailed discussion and spread of DFT data[28,39]). For x Cr < 0.2 the minimum value is about double the DFT one as is the value of the zero of the curve. This discrepancy is due to the appearance of SRO in Fe-rich Fe-Cr alloys,[20,21] as previously explained.[40] The presence of SRO in the alloy has a negligible effect on the configurational entropy as compared to the random solution, which justifies the use of the ideal solution expression. However, the SRO has a significant impact on the mixing enthalpy, lowering its minimum and shifting it to the Cr-rich side. Keeping in mind that the DFT obtained mixing enthalpy was calculated for random Fe-Cr alloys, the discrepancy can thus be rationalized. The mixing enthalpy obtained from the standard parameterization does not show any change of sign in the whole concentration range and therefore excludes occurrence of SRO, which is also the reason for the low solubility of Cr at temperatures below 700 K predicted by the standard parameterization.

Fig. 3
figure 3

Mixing enthalpy at 300 K

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In Fig. 4, the heat capacity, C p , obtained from the two parameter sets is compared for Fe-0.44Cr in the bcc phase. The two curves are in close agreement and provide the same critical temperature at which the cusp occurs. Note that both calculated curves are also in reasonable agreement with experimental data,[41,42] although such data was not used in the parameterization procedure of either set.

Fig. 4
figure 4

Heat capacity as a function of temperature

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Finally, we emphasize that the use of a high order polynomial in x Cr and T to describe G bccxc did not introduce uncontrolled unnatural oscillations in the thermodynamic function, as illustrated in Figs. 2-4, respectively.

Conclusions

Thermodynamic functions of the bcc phase of Fe-Cr alloys were re-parameterized so as (i) to account for the large solubility of Cr in Fe below 700 K, as suggested by Bonny et al.[4] and (ii) to keep the rest of the phase diagram almost coincident. The mixing enthalpy, calculated with the new parameter set, agrees reasonably with existing DFT data, which provide the physical reason for the high solubility of Cr at low temperature. The higher order derivatives of the Gibbs free energy, such as the heat capacity, were proven to be virtually unaffected by the new parameterization and, at the same time, were found to agree with available experimental results.