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Phase Equilibria and Thermodynamic Properties in the Fe-Cr SystemWei Xiongab; Malin Sellebya; Qing Chenc; Joakim Odqvista; Yong Dub

a Department of Materials Science and Engineering, Royal Institute of Technology (KTH), Stockholm,Sweden b State Key Laboratory of Powder Metallurgy, Central South University, Changsha, P.R. Chinac Thermo-Calc Software AB, Stockholm Technology Park, Stockholm, Sweden

Online publication date: 08 June 2010

To cite this Article Xiong, Wei , Selleby, Malin , Chen, Qing , Odqvist, Joakim and Du, Yong(2010) 'Phase Equilibria andThermodynamic Properties in the Fe-Cr System', Critical Reviews in Solid State and Materials Sciences, 35: 2, 125 — 152To link to this Article: DOI: 10.1080/10408431003788472URL: http://dx.doi.org/10.1080/10408431003788472

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Critical Reviews in Solid State and Materials Sciences, 35:125–152, 2010Copyright c© Taylor and Francis Group, LLCISSN: 1040-8436 print / 1547-6561 onlineDOI: 10.1080/10408431003788472

Phase Equilibria and Thermodynamic Properties in theFe-Cr System

Wei Xiong,1,2,∗ Malin Selleby,1 Qing Chen,3 Joakim Odqvist,1 and Yong Du2

1Department of Materials Science and Engineering, Royal Institute of Technology (KTH), Stockholm,Sweden2State Key Laboratory of Powder Metallurgy, Central South University, Changsha, P.R. China3Thermo-Calc Software AB, Stockholm Technology Park, Stockholm, Sweden

Phase equilibria and thermodynamic properties in the Fe-Cr system have been reviewed com-prehensively based on experimental information and available computer simulations in dif-ferent scales. The evaluated phase equilibria show significant differences from the currentlyaccepted thermodynamic description by CALPHAD (calculation of phase diagram) approach.The thermodynamic properties of the Fe-Cr system, such as heat capacity, enthalpy, and ac-tivity, have been evaluated in reported experiments. The experiments on phase separation inthe Fe-Cr system have also been critically reviewed with a focus on spinodal decomposition.The reported data are concentrated in the temperature range from 673 to 823 K. In addition,there is a transition region between spinodal decomposition and nucleation regimes within thecomposition limit from 24 to 36.3 at.% Cr and the temperature range between 700 and 830 K.In view of the importance of magnetism in the Fe-Cr system, some inadequacies of the currentlyused thermodynamic description are pointed out in addition to some problems with the currentmagnetic model. Remaining issues on the thermodynamics of the Fe-Cr system have been elab-orated for future refinement of the thermodynamic description of the Fe-Cr system. Accordingto the present review, the melting temperature of Cr is recommended to be about 2136 K, whichis 44 K lower than the value adopted in the research community on thermodynamics, such asthe Scientific Group Thermodata Europe.

Keywords phase diagram, CALPHAD, ab initio, irradiation, melting temperature, spinodal de-composition

Table of Contents

1. INTRODUCTION ................................................................................................................................................. 126

2. PHASE EQUILIBRIA ........................................................................................................................................... 1262.1. High-Temperature, with Liquid Phase .............................................................................................................. 1262.2. Medium-Temperature, γ -Loop ........................................................................................................................ 1282.3. Low-Temperature, with α and σ Phases ............................................................................................................ 128

2.3.1. Phase Equilibria Containing σ Phase .................................................................................................... 1282.3.2. Bcc Miscibility Gap ............................................................................................................................ 1302.3.3. Spinodal Decomposition ...................................................................................................................... 133

3. MAGNETISM IN FE-CR ...................................................................................................................................... 138

4. THERMODYNAMIC PROPERTIES .................................................................................................................... 1424.1. Heat Capacity ................................................................................................................................................ 1424.2. Enthalpy of Mixing ......................................................................................................................................... 1434.3. Activities ....................................................................................................................................................... 145

∗Email: [email protected] or [email protected]

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126 W. XIONG ET AL.

5. CONCLUDING REMARKS ................................................................................................................................. 145

ACKNOWLEDGMENTS ........................................................................................................................................... 146

REFERENCES .......................................................................................................................................................... 146

1. INTRODUCTIONThe binary Fe-Cr system is the basis for a large class of im-

portant engineering materials known as stainless steels. Stain-less steels combine good corrosion resistance1 with attractivemechanical properties. However, the ferrite in ferritic, marten-sitic and duplex stainless steels is susceptible to the notorious“475 ◦C embrittlement” at intermediate temperatures, whichis detrimental to the mechanical properties, especially impacttoughness.2,3 The embrittlement stems from a phase separationreaction where the ferrite decomposes into regions which arerich in iron or chromium.

Some other features of the Fe-Cr alloys are also of in-terest, such as spin glass formation,4,5 sluggish formation ofthe σ phase,6,7 and strong swelling resistance in an irradiativeenvironment.8 All of these call for a thorough understanding ofthe Fe-Cr system.

So far, including computed phase diagrams,9−19 at least 18versions of the Fe-Cr phase diagram has been proposed.9−27

The evaluated phase diagram from the ASM handbook27 is re-produced in Figure 1. However, this phase diagram needs tobe revised according to recent ab initio studies at low temper-atures and new experimental data, which will be discussed inSection 2.3.

Nowadays, the CALPHAD (CALculation of PHAse Dia-grams) approach based on a scrupulous evaluation is a pow-erful tool to construct phase diagrams.28,29 There are at least

FIG. 1. Evaluated phase diagram of the Fe-Cr system accordingto Okamoto.27

10 versions of CALPHAD modeling of the Fe-Cr system.9−19

Among these the one by Andersson and Sundman,11 which waslater slightly modified by Lee19 for the liquid phase, and is thecommonly accepted description of the Fe-Cr system. Unfortu-nately, according to the present evaluation, none of the abovedescriptions constitute a reliable thermodynamic description ofthe Fe-Cr system.

This work provides a comprehensive evaluation on the phaseequilibria and thermodynamic properties of the Fe-Cr systembased on a thorough literature survey. The work is also a pre-requisite for a reassessment of the thermodynamic descriptionof this binary. The low-temperature phase equilibria concerningthe miscibility gap in the bcc phase will be refined in accordanceto the available atomistic methods, e.g., ab initio calculations,which can be used as “experimental data” at the low tempera-tures, where the phase equilibria are too hard to obtain via nor-mal thermal aging experiment from a practical point of view.In addition, some of the inadequacies of the existing thermo-dynamic description are noted. This work shows how urgentlya new thermodynamic description is needed for this importantbinary. The assessment done by Andersson and Sundman,11 andlater revised partially by Lee,19 has been chosen for comparison,since their thermodynamic description has been used in manymulticomponent databases. However, one should keep in mindthat the assessment performed at that time11 focused more onthe CALPHAD methodology itself. Besides, ab initio methodswere not readily available for CALPHAD then.

In the present work, since the α phase will decompose intothe α′ and α′′ phases and form a miscibility gap, the (α′-Fe) isconsidered as equal to (α-Fe) in the temperature range of the mis-cibility gap. Meanwhile, (α′′-Cr) denotes the Cr-rich bcc phase.

2. PHASE EQUILIBRIA2.1. High-Temperature, with Liquid Phase

All experimental information and the assessed phase equi-libria concerning the liquidus and solidus are summarized inFigure 2. As easily seen, there are large discrepancies betweenexperimental and assessed results over a large part of the wholecomposition range. It seems that the melting point adopted forpure Cr is too high in the assessment work. As a matter of fact,the values selected in compilations30−34 for the melting pointof pure Cr lie between 2103 and 2193 K. It may be noted thatthe conversion into the IPTS-90 (International Practical Tem-perature Scale year 1990)35−37 does not give any substantialchange. According to the most recent pyrometric experimentsby Josell et al.38 from NIST (National Institute of Standards andTechnology), the melting point for pure Cr coated in W, which


PHASE EQUILIBRIA AND THERMODYNAMIC PROPERTIES IN THE FE-CR SYSTEM 127

FIG. 2. Phase equilibria at high-temperature range of the Fe-Cr system.

provide a good oxidation resistance, is determined as 2115 ± 20K. This value is in accordance with 2133 ± 6 K given by Rudyand Windisch39 using the Pirani method40 which is a pyrometrymethod with high accuracy for refractory melts.

The melting temperature, 2180 K, of pure Cr used in An-dersson and Sundman’s assessment and later accepted in theSGTE (Scientific Group Thermodata Europe) unary database41

is based on the evaluation by Andersson42 who chose thevalue from Gurvich et al.’s compilation,33 where the workby Rudy and Windisch39 was not considered. A more widelyknown compilation is from Hultgren et al.31,32 They first31 ac-cepted the value of 2176 K from the experiment by Bloomet al.,43 and later32 recommended 2133 K on account of moreexperiments.44−46 In the phase diagram compilation publishedby ASM International,27,47 a value of 2136 K is suggested forthe melting point of pure Cr. Considering all the facts mentionedabove, we believe, in future CALPHAD assessment, the meltingpoint of pure Cr should be changed into 2136 K, which con-forms with the values selected in well known compilations27,32

and the reliable experimental data by Rudy and Windisch39 andJosell et al.38 More importantly, as can be easily seen in Figure2, this value makes it possible to account for the experimen-tal liquidus and solidus data in the binary Fe-Cr system. OtherCr-containing binary and higher order systems have similar sit-uations, for example, the calculated liquidus at the Cr-rich sideare higher than the experimental results in the CALPHAD mod-eling of the Cr-C48 and Cr-Hf49 systems.

The first experimental investigation of Fe-Cr phase equilibriawas carried out by Oberhoffer and Esser.50 Although the rawmaterials were poor in purity, the general shape of the liquiduswas concave pointing upwards at about 15.9 at.% Cr.

Another early experimental work for the high-temperaturephase equilibria in the Fe-Cr system was contributed byAdcock.51 Thermocouples were used only for the alloys with Cr

content up to 70 at.%. The samples with higher Cr content weredetermined by optical pyrometer. Adcock51 stated clearly thatthere was no indication of oxidation but Cr volatilization wasdetected above 1923 K in the samples with more than 65 at.%Cr. This means that the liquidus points for alloys containingmore than 65 at.% Cr should shift slightly toward left in Figure2, which will then give an excellent extrapolation to the melt-ing point of pure Cr recommended above. However, since thesamples used in the work of Adcock51 have not been annealedsufficiently (12 hours, at 1623 K), the segregation of the samplesmay lower the accuracy of the solidus constructed by Adcock.51

In their mass spectrometric measurement of activities, Beltonand Fruehan52 extrapolated also the liquidus and solidus, whichare lower than the experimental data by Adcock.51 Since Beltonand Fruehan’s data52 were not determined directly, we believeit should not be attached with any significant weight in futurethermodynamic modeling work.

It should be mentioned that, in the work by Hertzman andSundman,18 the experimental liquidus determined by Adcock51

was accepted. But, for some conjectural reasons regarding pos-sible oxidation of the samples, it was rejected in the laterthermodynamic assessment by Andersson and Sundman.11

Therefore, their thermodynamic description11 shows a largetemperature differences between calculation and experimentseven for the parts with Cr content lower than 65 at.%. More-over, Lee19 found that the minimum of the liquidus calculatedby Andersson and Sundman11 should be shifted towards higherCr content, from 16.98 to 21.17 at.% Cr, due to difficultieswhen extrapolating into some ternary systems, e.g., Fe-Cr-C.However, no major changes were made in the later assessmentby Lee.19

Putman et al.53 determined the liquidus temperature for sevenFe-Cr alloys by thermal analysis using thermal pyrometry. Dueto the high volatility of Cr, chemical analysis was done after


128 W. XIONG ET AL.

thermal analysis for each sample. Their data agree more or lesswith those reported by Adcock51 and should be accepted inthermodynamic assessments.

In summary, the present evaluation of the Cr-rich side of theliquidus is close to those by Muller and Kubaschewski22 and byHertzman and Sundman.18

The Fe-rich part of the liquidus region has been measuredby several research groups.51,53−56 As shown in Figure 3, theexperimental data from Putman et al.53, Hellawell and Hume-Rothery54, and Shurmann and Brauckmann55 are consistent,showing that the liquidus and solidus will pass through a min-imum in the region of 20 to 21 at.% Cr at about 1782 K. Asmentioned above, the experiments by Adcock51 have larger un-certainties for the solidus points in the Cr-rich alloys. This is truealso for the Fe-rich alloys in his experimental work. In addition,the experimental data by Kundrat et al.56 show a liquidus 10 Khigher than the others.53−55

2.2. Medium-Temperature, γ -LoopMany measurements have been performed to construct the

γ -loop of the Fe-Cr system,50,57−67 and the experimental datahave been plotted in Figure 4. The γ -loop in the Fe-Cr sys-tem was firstly found by Bain57 and confirmed by Oberhofferand Esser50 using a combination of XRD (X-ray diffraction)and thermal analysis. It was further confirmed by Kinzel58 whoused chemical analysis and a telescopic dilatometer which he in-vented. The scatter of the experimental results50,58 is largely dueto the low purity of the raw materials at that time. Using thermalanalysis, the γ -loop was also constructed by Adcock51 showinga similar shape but with a smaller area. Parts of the γ -loop wereconstructed by Roe and Fishel59 during their dilatation studieson Fe-Cr alloys. They determined the temperature minima ofthe γ -loop to be around 1073 K, which is lower than most otherexperimental data as shown in Figure 4. The lower temperaturepart of the γ -loop has also been determined by Bungardt et al.60

via dilatometric methods.A comprehensive experimental work was carried out by

Nishizawa62 and was later confirmed by another work on theγ -loop by Kirchner et al.65 who not only performed the experi-ments but also evaluated the former experiments on the γ -loop.Normanton et al.67 later performed a calorimetric and mass-spectrometric study of solid Fe-Cr alloys. However, their sam-ples contained considerable C, Si, and Al impurities. Taking intoaccount the experiments with different contamination of C andNi, Poyet et al.64 proposed that the γ -loop would shrink with Caddition, which is consistent with the conclusion of Baerleckenet al.61

Thermodynamic analyses based on experimental data havealso been carried out, especially for the γ -loop by manygroups.65,68−73 All of these investigations show good agree-ment with the experimental data by Nishizawa62 and Kirchneret al.,65 on which the thermodynamic description by Anderssonand Sundman11 is mainly based. The good agreement can beseen in Figure 4.

In conclusion, the maximum solubility of Cr in (γ -Fe) isevaluated to be at 11.9 at.% Cr, and it is then in equilibriumwith (α′-Fe) with 13.6 at.% Cr. The minima of the γ -loop liesat about 1119 K with 7 at.% Cr, which is in accordance withthe description provided by Andersson and Sundman.11 It isnoteworthy that the temperature corresponding to the solubilitylimit is actually very hard to measure precisely. According toAndersson and Sundman,11 the model predictions are 1252 and1261 K at 11.9 and 13.6 at.% Cr, respectively.

2.3. Low-Temperature, with α and σ Phases2.3.1. Phase Equilibria Containing σ Phase

Phase equilibria at low temperatures in the Fe-Cr systemare the most challenging for experimental investigations. In thissection, the stable phase equilibria involving the σ phase willbe discussed first, followed by a thorough examination of themiscibility gap in the bcc phase. Finally, based on the analysisof the available experimental data, a possible region for thespinodal decomposition will be outlined.

There is only one intermetallic compound, σ , in the Fe-Cr system and it forms very slowly as a brittle phase. Thecrystal structure of the σ phase can be described by space groupP42/mmm, and Strukturbericht D8b.

74 A sublattice model withthe formula (Cr)4(Fe,Cr)18(Fe)8 was first used for the σ phaseby Andersson and Sundman.11 Later, in the Schloß Ringbergworkshop in 1997,75 the model for the σ phase was suggestedto be modified to (Cr)4(Fe,Cr)16(Fe)10 which is more reasonableaccording to the crystal structure. The reason for using thismodified three sublattice model has been discussed again inanother review paper by Ferro and Cacciamani.76 Although thecomplete disordered state of the σ phase does not exist, theorder-disorder transition model has been tried for the σ phasein the Co-Cr system by Hallstedt et al.77

Recently, a systematic review of the crystal chemical prop-erties of the σ phase was presented by Joubert,74 who sug-gested to use a simpler two sublattice model (Fe,Cr)20(Fe,Cr)10

to describe the σ phase in multicomponent thermodynamicdatabases.

Phase equilibria at low temperatures are difficult to determineexperimentally51,78 due to slow kinetics which in turn requireslong term annealing. Static phase identifications, e.g., XRD andmetallography, rather than dynamic methods, e.g., thermal anal-ysis and electrical resistivity measurement, are preferential todetermine the σ phase formation, since the latter ones some-times are not sensitive enough to determine the phase transitiontemperatures due to the sluggish formation of the σ phase. In theexperimental part of the work by Hertzman and Sundman,18 thenumber of nuclei of the σ phase was still low after an isothermalaging at 973 K for 18 months. On the other hand, it is interestingto know that during the investigation on the oxidation of Fe-Cralloys by Frattini et al.,79 the formation time of the σ phasewas shortened significantly if some heterogeneities phases, likeoxides, were present.



FIG. 3. Magnified Fe-rich side of the phase equilibria at high-temperature range of the Fe-Cr system.

Because of the short term annealing used, the σ phase was notfound in the pioneering investigation by Adcock et al.51 Later,Cook and Jones78 studied the phase transformation between thebcc and σ phases, showing the possible stability range of σ

by XRD and metallography. The highest temperature for the σ

phase was expected to be at about 1093 K with approximately 47at.% Cr,78 which was later confirmed by Pomey and Bastien80

who used thermal analysis and metallography. In addition, alater work by Yukawa et al.81 found that the homogeneity rangeof sigma phase between 863 and 1013 K will be in the vicinityof 50 at.% Cr, slightly shifted to the Fe side.

The experiments performed by Dubiel and Inden82 couldbe regarded as the most reliable phase equilibrium data using

conventional alloy methods due to the extremely long annealingtimes (4 to 11 years) for five Fe-Cr samples. The experimentsshowed that the temperature range for the decomposition of theσ phase is expected to be between 773 and 783 K, which is closeto the estimated value of 793 K by Williams et al.6 who studiedthe kinetics of σ phase formation. The alloy with 48.6 ± 0.06at.% Cr showed pure σ phase even after the long-time annealingat 783 K. Therefore, the decomposition of the σ phase may beexpected to be at around 49 at.% Cr. As shown in Figure 5, thephase boundary of the miscibility gap has also been determinedat 773 and 783 K, respectively,82 and confirmed recently byNovy et al.83 with the tomographic atom probe analysis for aFe-20 at.% Cr alloy annealed at 773 K for 44 days.

FIG. 4. γ -loop of the Fe-Cr system.


130 W. XIONG ET AL.

FIG. 5. Stable phase diagram of the Fe-Cr system below 1100K, without showing the γ -loop.

Chandra and Schwartz84 used Mossbauer spectroscopy tostudy the decomposition of Fe-Cr alloys at 748 K. The solubilityof Cr in (α′-Fe) was determined to 12 at.% Cr, in accordancewith the measurement by Dubiel and Inden.82

The only exception is from the experiment by De Nys andGielen85 who also used Mossbauer spectropy, where the σ phaseonly appeared after ageing at 973 K. This is obviously in con-tradiction to all the other experiments including the work byDubiel and Inden.82 One possible reason could be attributed toinsufficient annealing for the quenched alloys.

The above information has here been used to construct thephase diagram between 700 and 1100 K as shown in Figure 5. Inaddition, a modification of the low temperature phase equilibriais needed for the Fe-rich solvus according to the recent studieson the bcc miscibility gap.

2.3.2. Bcc Miscibility GapSo far, there are two possible ways to construct the low-

temperature phase diagram. Firstly, classical equilibrium alloyscould be used to determine the phase transition temperatureor phase boundary. However, the efficiency of this method isreduced by the need of long-term heat treatment because of thelow atomic mobility at low temperatures. Secondly, the stateof art of ab initio calculations allows us to make reasonablepredictions of the low temperature phase equilibria. Actually,for the Fe-Cr binary system, there are a huge amount of abinitio calculations predicting different solubility limits of Cr in(α’-Fe) ranging from 2 to 7 at.% Cr at 0 K. Some of these resultswill be presented and discussed later in Section 4.2.

Most recently, an evaluation of the low-temperature phaseequilibria at the Fe-rich side was performed by Bonny et al.86

The authors presented a good agreement of the phase bound-ary between atomistic calculation performed by their group andtheir evaluation based on reported irradiative experiments. Intheir analysis of the irradiative experimental data they madesome assumptions: Firstly, they assumed that Cr precipitationunder irradiation is simply due to radiation-enhanced diffusion.Second, they supposed that the ordering process can only beaccelerated by irradiation but not induced by it. Third, they as-sumed that Cr precipitation detected in alloys of commercialpurity and steels is a consequence of accelerated phase sepa-ration. As a result, Bonny et al.86 believed that the position ofthe phase boundary is not expected to be affected by irradiationand they presented a phase boundary as the one indicated inFigure 6.

Unfortunately, none of the assumptions put forward byBonny et al.86 are reasonable. It is surprising to see that only theradiation-enhanced diffusion was mentioned but not the otheropposing process of ballistic mixing which will dissolve theprecipitation simultaneously. It should be kept in mind thatmaterials under irradiation can only reach steady state but notequilibrium.87 In the discussion of the phase diagram modifi-cation under irradiation by Wilker,88 the behavior of nucleationand growth under irradiation was illustrated and shown to betotally different from the situation under thermal equilibriumconditions. Moreover, the morphology of the precipitate will bechanged during the process approaching steady state.88 Actu-ally, none of the irradiative experiments cited by Bonny et al.86

were aimed at determining the Cr solubility in (α’-Fe). There-fore, the precipitates observed in these experiments have notbeen checked to see whether they are formed during steadystate or not. For example, in the work of Gelles89 used in theanalysis by Bonny et al.,86 Gelles89 realized that, during theexperiment, the steady state may not have been achieved forthe sample Fe-19.08Cr (at.%). Actually, the irradiation studieson the Fe-Cr and the commercial steels were mainly focusingon the swelling resistance and mechanical behavior rather thanprecipitation.87,89−91

The second assumption by Bonny et al.86 cannot be supportedby a shred of evidence. The radiation can induce disorderingin the ordered phase.92 Although Bonny et al.86 claimed thatthey have not found the mechanism of induced non-equilibriumordering in metals, the SRO (Short Range Order) degree ofAu-Ag alloys could be increased by irradiation as reported byPoerschke and Wollenberger.93

Evidently, it is impossible to determine binary phase dia-grams using data for multi-component steels even for phaseequilibrium states. Moreover, it is very hard to believe that,in the work of Bonny et al.,86 any consistency can be foundfrom using irradiation results for steels, commercially purealloys and binary Fe-Cr alloys. Bonny et al.86 claimed thatthe Cr precipitation was observed to be finely dispersed un-der Transmission Electron Microscopy (TEM). Indeed, in the



FIG. 6. Magnified Fe-rich side of the Fe-Cr phase diagram between 200 and 900 K. The shaded area shows the possible locationof the Fe-rich solvus.

samples, especially for the low-purity ones, the carbides arevery easily formed and mixed with the Cr precipitates. There-fore, it is impossible to make any definite judgment. In con-trast to Bonny’s statement, using TEM, energy dispersive x-raymicroanalysis, and electron energy loss spectroscopy, Ohnukiet al.94 showed clearly that irradiation produced Cr precip-itates were enriched at grain boundaries and voids underion irradiation up to 118 dpa (displacements per atom) at798 K.

Bonny et al.86 also found recent experimental data by Wakaiet al.91 who revealed the completely different phenomena inFe-Cr alloys with two different purities at the same irradia-tive environment. However, the difference is simply ascribedwith a reason of radiation-induced segregation which was usedto explain all of the other experimental phenomena under ir-radiation afterwards.86 Actually, radiation-induced segregationis complex and related to interstitial binding, solute size effect,temperature, barrier strength of dislocation loops, dose rate, etc.,not only intrinsic impurities of materials.

The choice of using irradiation experimental data by Bonnyet al.86 seems to indicate some lack of judgment. For instance,the work by Gelles and Schaublin95 was taken into account dur-ing the analysis by Bonny et al.86 As a matter of fact, Gelles andSchaublin95 found precipitates and ordered clusters at mechan-ically deformed areas after the irradiation.

To the best of our knowledge, no one has drawn such a conclu-sion before Bonny et al.86 that the position of the phase bound-ary is not expected to be affected by irradiation. The Mossbauerstudy of the Fe-Cr alloys by Kuwano and Hamaguchi96 usingproton irradiation with the damage of 0.2 to 0.4 dpa at 748K could be regarded as another experimental proof against theconclusion of Bonny et al.86 As shown in Figure 6, Kuwano andHamaguchi96 found that Cr enrichment can be found at 5.03

at.% but not at 7.90 at.% Cr. Compared with the former ther-mal experimental results (9.8 at.% Cr) by the same author,97

significant compositional shift could be found. These data arereproduced in Figure 6, which are the same as the symbols, redC and green F, in Figure 1 of the paper by Bonny et al.86

Another serious mistake made by Bonny et al.86 is the con-ceptual confusion on the SRO type change and phase transition.In the Fe rich part, there are two types of SRO effects. Withincreasing Cr concentration, the pairs of unlike atoms in thetendency of a SRO arrangement (named local long range order)will be changed into another type of SRO where the Cr atomstend to be together forming Cr-Cr clusters. However, the oc-currence of SRO with Cr-Cr clustering does not correspond tothe threshold of phase separation.98 In the work by Filippovaet al.,99 the change in the type of SRO at the Fe rich side hasbeen studied under irradiation. As mentioned above, the com-position of the SRO effect change measured under irradiationcannot be taken into account for phase equilibrium. Unfortu-nately, the composition threshold of the SRO effect change hasbeen regarded as the position of phase boundary in the workof Bonny et al.86 as indicated in Figure 6. Consequently, evenif Bonny’s assumptions were correct, the estimated solubilitylimit of Cr in (α’-Fe) could not be the one as they plotted (seedashed line in Figure 6).

Furthermore, using diffuse neutron scattering and resistivitymeasurements, the pioneer investigation by Mirebeau et al.100

found that the SRO parameter will change its sign at 750, 730and 717 K, with 5, 15 and 10 at.% Cr, respectively. As shown inFigure 6, one of them (750 K, 5 at.% Cr) is even far away fromthe solvus of both stable and metastable phase boundary.

After the publication by Bonny et al.,86 another experimentalwork running under irradiation was carried out for Fe-Cr alloyswith industrial purity at 573 K by Bergner et al.101 was reported.


132 W. XIONG ET AL.

FIG. 7. Bcc miscibility gap (a) Comparison between atom-istic simulations, as cited in Bonny et al.,86 thermodynamicmodeling.11 (b) Comparisons among experiments,7,82,97,113−115

thermodynamic modeling,11 and the present evaluation. The ex-perimental data are in some cases connected by lines of differenttypes to guide the eyes.

The solubility limit was estimated using a so-called ‘iterativeapproach’ based on the irradiation experiments together with theconcept of phase equilibrium during the analysis. Furthermore,the estimation was based on the former CALPHAD model-predicted solubility of Fe in (α”-Cr)11 which is just a simpleextrapolation from higher temperatures due to nonexistence ofexperimental data for the Cr-rich solvus.

In the paper by Bonny et al.,86 some calculations for thesolvus at the Fe-rich side performed in their group by semi-empirical cohesive models have been presented as well (see Fig-ure 7), but the original ab initio data has not been interpreted indetail, and cannot be found elsewhere before this publication.86

Consequently, no judgment can be made for these atomisticsimulations. However, in view of the conceptual problem of theSRO found in the analysis by Bonny et al.,86 the validity of thesedata could also be questioned.

Up till now, the possibility of significant solubilities of Cr in(α’-Fe) at the temperatures lower than 400 K is only predictedby atomistic simulations for ferromagnetic states,102−111 suchas, ab initio calculations and Monte Carlo (MC) simulations.Since the sign change of the enthalpy of formation for randombcc alloys in the ferromagnetic state in the Fe-rich region hasbeen found in the range between 4 and 12 at.% Cr at 0 K indifferent ab initio calculations,110−112 the solubility of Cr in(α’-Fe) could be expected to be within the range of 2 ∼ 7 at.%after using the common-tangent construction for Gibbs energycurves.

As a result, in Figure 6, a green shaded area has been high-lighted as the possible location for Fe-rich solvus. This could beconsidered as a simple and smooth connection between point A(see Figure 6) and the solubility limit of Cr in (α’-Fe) at 0 Kpredicted by ab initio calculations. Apparently, the line evalu-ated by Bonny et al.86 is similar to our choice, but the argumentsare completely different. It should be emphasized again that thepresent evaluation of the solvus at the Fe-rich side is the con-sequence of ab initio predictions for the solubility limit of Crin (α’-Fe) at 0 K. However, at the moment, there are still someunresolved problems with the ab initio calculations for the Fe-Cr binary. The available ab initio calculations will be furtherdiscussed in Section 4.2.

At the Cr-rich side in the Fe-Cr phase diagram, neither ex-perimental data nor atomistic calculations are available for thesolubility of Fe in (α′′-Cr) at temperatures lower than 1043 K.Therefore, the Cr side is temporary more or less the same as theprediction by the former CALPHAD assessment,11 which showsno solubility of Fe in (α′′-Cr) at 0 K. It should be mentionedthat the ab initio calculations on the enthalpy of formation per-formed for the ferromagnetic states cannot be the cause for thelow solubility of Fe in (α"-Cr) with antiferromagnetic propertiesunder 311.5 K (the Neel temperature for pure Cr).41

In some experiments, attempts were made to measure themetastable miscibility gap of the bcc phase, i.e., for regionswhere the σ phase is stable.6,7,97,113−115 Binary alloys werefirst melted and prepared at high temperature and then annealed



FIG. 8. Spinodal decomposition region in the Fe-Cr system.

in the bcc state for specified periods. After that the alloys werequickly quenched so as to suppress the formation of the σ phase.Different kinds of characterization were performed eventually.During the whole process, phase identification was essential tomake sure that the σ phase did not precipitate.

In spite of numerous investigations, the determined locationof the metastable bcc miscibility gap seems still ambiguous,especially for the critical consolution temperature of the misci-bility gap. As shown in Figure 7(b), different experiments showdifferent location of the miscibility gap. In fact, the highest tem-perature of the critical consolution temperature according to aMossbauer spectroscopic study by Kuwano97 is around 950 K.The author used Mossbauer spectroscopy and electrical resis-tivity measurements simultaneously. As can be seen in Figure7(b), two sets of data are different in temperature by more than100 K. It is worthwhile to point out that the higher temperaturedata points measured by Mossbauer study were confirmed byXRD in the same work. The author found that the Mossbauer ef-fect is more effective and sensitive than the electrical resistivitymeasurement to detect the composition change of phase decom-position. According to Kuwano,97 the lower temperature datasetare supposed to be a general demarcation for the phase stability,although they are in accordance with the experiments done byWilliams and Paxton,6,7 who used hardness and electrical resis-tivity measurements. However, it should be mentioned that, inthe experiment by Williams and Paxton,6,7 the measurements forthe miscibility gap were only performed at temperatures lowerthan 943 K, which might be substantially lower than the truephase boundary. The miscibility gap obtained in the work ofImai et al.113 and Miller et al.115 is also below 830 K. Moreover,

on the one hand, the difference between coherent and incoherentmiscibility gaps will be subtle if the lattice mismatch is small,as in the case in the Fe-Cr binary. On the other hand, one wouldexpect a noticeable difference between the bcc miscibility gapswith and without magnetism. This may be found easily duringthe computational simulation, like CALPHAD modeling.

In several investigations, Monte Carlo simulations have beenused to predict the miscibility gap, but the results still need to beimproved.115,116 The main difficulties may be attributed to theproblem of finding more realistic interatomic potentials. Owingto the phase separation and magnetism, this pair potential prob-lem becomes more severe in the case of Fe-Cr. Actually, in thework of Miller et al.115 and Turchi et al.116 the predicted bccmiscibility gap by the CALPHAD approach has been regardedas the criterion since the authors115,116 believed that the availablethermodynamic description11 could fully represent the experi-mental data. Unfortunately, as discussed above, this is not true.

A new thermodynamic description is necessary for this im-portant binary. Although the CALPHAD modeling cannot beexpected to represent fully the complex short range orderingat low temperatures for the Fe-rich side, from the viewpointof recent ab initio calculations, it would be nice to introducea higher solubility of Cr in (α’-Fe) compared to the formerdescription.11

2.3.3. Spinodal DecompositionThe spinodal decomposition of the bcc phase in the Fe-Cr sys-

tem has attracted much attention as well.3,84,85,113,117−148 In Fig-ure 8, the predicted chemical spinodal line by thermodynamic


134 W. XIONG ET AL.

TABLE 1Summary of the experimental studies of spinodal decomposition in the Fe-Cr system

Solutionizing historyAlloy composition, aging temperature,

Ref. No. Temperature (K) Time and time for study Methods

3 1323 0.5 hrs. 30 at.% Cr, 748 and 823 K, from 1 to 2000 hrs. HT, TEM84 1273 144 hrs. 24, 30, 37, 44 and 60 at.% Cr, 748K, from 30 to

2476 hrs.MS

85 1323 and 1423 8.5 hrs. 20, 30, 40 and 50 at.% Cr, 743 and 813 K, from160 to 1050 hrs.

MS

117 1073 ∼ 1223 12 min. 31.5 and 46.8 at.% Cr, 773 K, up to 28 hrs. HT, TEM, XRSAD,MM, MT

118 1273 2 hrs. 32 at.% Cr, 743 K, 8 ∼ 11000 hrs. HT, FIM, APA119 1373 120 hrs. 34 at.% Cr, 773 K, 20 hrs. NSAS120 1123 and 1473 unknown 32, 52 at.% Cr, 773 K, 10 ∼ 200 hrs. NSAS124 1273 2 hrs. 46.8 at.% Cr, 673 and 773 K, up to 8760 hrs. APA125 1173 1 hr. 6.4, 12.3, 14.9, 16.0, 19.2, 22.5, 26.3, 29.5, 35.3,

40.5 and 45.5 at.% Cr, 688 K, up to 19 hrs.in situ MS

126 1273 24 hrs. 30 and 50 at.% Cr, 748, 773, 798 and 823 K, SANS127 1273 and 1673 24 and 20 hrs. 47.8 at.% Cr, 773 K, up to 9650 hrs. HT, XRD, TEM128 1373 11 hrs. 21.3, 31.4, 40.1 and 48.3 at.% Cr, 773 K, 150 hrs. MS129 873, 973, 1073,

1173, 1273, 1373,1473 and 1573

1 hr. 28 and 40 at.% Cr, 773 K, 3 and 20 hrs. NSAS

130 1273 48 hrs. 20, 30, 40 and 60 at.% Cr, 773, 788 and 813 K, upto 3 hrs.

NSAS

131 1373 1 hr. 24, 32 and 40 at.% Cr, 723, 773 and 798 K, up to20 hrs.

NSAS

132 1373 120 hrs. 34 at.% Cr, 773 K, 500 hrs. NSAS133 1300 unknown 21, 28.7, 31.5, 36.3, 41.6, 50.3, 55.6, 59.0 and 63.9

at.% Cr, 748 K, up to 2.8 hrs.MS

134 1300 unknown 10.2, 28.7, 36.3, 55.6, 62.6 at.% Cr, 748 K, up to 3hrs.

MS, TEM

135 1300 1 hr. 55.6 at.% Cr, 748 K, up to 25 hrs. MS, TEM136 1273 24 hrs. 20, 30, 40 and 60 at.% Cr, 773, 788 and 813 K, up

to 51 hrs.NSAS

137 1473 and 1123 0.5 hr. 31.5 at.% Cr, 748, 773 and 793 K, up to 2000 hrs. MT, TEM, SANS138, 139 1273 unknown 45.7 and 53.5 at.% Cr, 748 K, up to 878 hrs. MS140 1473 unknown 28 at.% Cr, 748 K, up to 0.17 hrs. MS141 1173+ unknown 20, 35 and 50 at.%, 773 K, up to 500 hrs. NSAS144 1300 1 hr. 25 at.% Cr, 773 K, 400 and 2600 hrs. APA145 1273 2 hrs. 32 at.% Cr, 563 K, 743 and 773 K, 500 ∼ 10800

hrs.FIM

NSAS = Neutron Small Angle Scattering; FIM = Field Ion Microscopy; AP = Atom Probe Analysis; HT = Hardness Testing; XRSAD =X-ray Small Angle Diffraction; TEM = Transmission Electron Microscopy; MS = Mossbauer spectroscopy; MT = Mechanical Testing; MM= Magnetization Measurement.+ There are two sets of samples, one set is furnace cooling, and homogeneity has been checked by a reflexion scanning electron microscope.The other set is annealed at 1173 K for a period of time (unspecified duration).



FIG. 9. Curie temperatures in the single bcc states of the Fe-Cr system. S: Spin glass; T: Transverse spin density wave; L:Longitudinal spin density wave; C: Commensurate spin density wave.

modeling in the work of Andersson and Sundman11 has beengiven in order to make a comparison with the experimentaldata.3,84,85,113,117−120,124−141,144−145 As can be seen, the reportedexperimental data are concentrated in the range from 673 to 823

K, and there is no clear boundary corresponding to the so-calledspinodal line as indicated by thermodynamic modeling.11 Inpractice, there will be a transition region from nucleation andgrowth to spinodal regime at the position near to the spinodal line

FIG. 10. Magnified part of the Curie temperature curve at the Cr-rich side.


136 W. XIONG ET AL.

FIG. 11. Mean magnetic moment in the single bcc states of theFe-Cr system.

which has been interpreted in the theoretical work by Binder.149

In the experimental work by Kuwano133 it was found that thenucleation and growth process will take place in the spinodalregion at 36.3 at.% Cr at 748 K, which is about 9 at.% overthe experimental spinodal boundary determined in his earlierwork.97

Nevertheless, it seems that the predicted chemical spinodalregion by Andersson and Sundman11 is too narrow according tothe different regions sketched by Binder.149 The present workmakes no attempt to perform a theoretical analysis on the spin-odal decomposition in the Fe-Cr system, but some importantfeatures and experimental findings must be noted for futurestudies. Detailed information on the related experimental infor-mation is summarized in Table 1.

First, it is interesting to find that all the related experimentsare limited to the temperature range between 673 and 823 K.The experimental spinodal decomposition is too slow at lowertemperatures due to low atomic mobility.145 In addition, attemperatures higher than 773 K, especially at the Fe-rich side,the spinodal curve is getting close to the magnetic transitioncurve which may play a significant role in the evolution of thespinodal structure.

Second, it should be kept in mind that not all of the re-ported experimental results can reflect the true regime in thecorresponding composition range. The wavelength of the de-composition in Fe-Cr is found to be at the nanometer scale.Therefore, the possible early stages of the spinodal decomposi-tion may be neglected unintentionally during the experiments.For example, the nucleation dominating process may be deter-mined incorrectly as the pure spinodal regime during neutronscattering measurement, which easily overlooks the nucleationand growth process of the precipitates.

Third, the magnetism in the Fe-Cr system could also bringtrouble to some extent during the spinodal decompositionstudy. Moreover, the solutionizing temperature and continu-ous quenching history will also affect the spinodal decompo-sition. For instance, in the work by LaSalle and Schwartz,120

FIG. 12. Summary of the available experiments on heat capacity. The dashed lines are the ones suggested to be ignored due tothe low accuracy as evaluated, while the solid lines represent the reliable data which should be considered in thermodynamicmodeling.



FIG. 13. Comparison between model-predicted results11 andexperimental data.191,192 The circles are from the phase transi-tion signal in the experiment191 which should not be consideredduring the comparison.

FIG. 14. Comparison between model-predicted results11 andexperimental data by Kendall et al.193

FIG. 15. Comparison between model-predicted results11 andexperimental data by Inden.171

the intensity of the scattering pattern with the solutionizingtemperature of 1473 K is much lower than that of 1123 K.In addition, the authors120 claimed that the experiments forthe Fe68Cr32 alloy performed between 873 and 973 K exhib-ited an erratic results for the spinodal region, which may bedue to the influence of the magnetic properties. It is worth tomention that the pioneering experimental work on the relation

FIG. 16. Comparison between model-predicted results11 andexperimental data by Normanton et al.67 For each successivecurve the scale is displaced upward by 20 units.


138 W. XIONG ET AL.

between heat treatment history and early stages of spinodal de-composition was carried out by Vintaikin et al.129 One plausiblereason is that there might be a high degree of short range sep-aration during the solutionizing process, and such a state canbe easier frozen with lower solutionizing temperature ratherthan higher one after subsequent quenching. This frozen statewill later facilitate phase decomposition during thermal aging.A related discussion was presented in the paper by Carmesinet al.150

Theoretically, it was found that the CHC (Cahn-Hillard-Cook) and LBM (Langer-Bar-on-Miller) models are not suitableto describe the spinodal phenomena in the Fe-Cr system.120,142

According to the CHC model, in the early stages, the compo-sition amplitude grows exponentially with a dominating smallrange of wavelengths, which means the wavelength will ap-proximately keep its constant value. In fact, the experimentresults have shown that the wavelength grows even in the earli-est stage.142 The dynamic Ising model, such as Penrose modelbased on the mean-field approximation seems more suitable toinvestigate the spinodal decomposition at the atomic scale 121 inthis system.

It is interesting to note that, since year 2002, there is almostno attention paid to the spinodal decomposition in the Fe-Crsystem.

3. MAGNETISM IN FE-CRThere are many measurements of the magnetic transition

in the Fe-Cr system.4,5,50,51,67,100,113,114,128,130,136,151−174 In thiswork, the experimental information on the Curie/Neel tempera-ture and mean magnetic moment needed for CALPHAD mod-eling is the main focus.

The SDW (spin density wave) antiferromagnetism ofpure Cr175−177 brings more complex issues into the magneticproperties of the present system. Together with CDW (ChargeDensity Wave), two SDW states were found in the Fe-Crsystem as shown in Figure 9: ISDW (Incommensurate SDW),CSDW (Commensurate SDW), which stems from pure bccCr.176−177 Using electrical resistivity measurements, a sketchof the magnetic property diagram was proposed by Mitchelland Goff,161 and an overlapping region of antiferromagnetismand ferromagnetism was found between 81 and 84.6 at.%Cr below 40 K. Similarly, using an integrated approach ofchemical analysis, electrical resistivity and magnetizationmeasurement, a subsequent work carried out by Leogel163 alsofound an analogous supermagnetic region between 71 and 80at.% Cr below 30 K. Later, the low temperature magnetism inFe-Cr has been studied in the work by Burke et al.4,5,173 vianeutron time-of-flight spectroscopy. The spin glass region wasfound to be surrounded by paramagnetic, ferromagnetic, andantiferromagnetic regions4 which can be seen in Figure 10.

Generally, some features should be mentioned about theCurie temperature curve. Firstly, as shown in Figure 9, the de-termined Curie temperature vs. the Cr composition has a smallpeak at around 2.1 at.% Cr, which is 10 to 15 K higher than

FIG. 17. Comparison for the enthalpy of mixing of the bccphase in the paramagnetic state between experiments192,200 andsimulations11,107,111 at different temperatures.

FIG. 18. Comparison for the enthalpy of mixing of the bccphase in the ferromagnetic state among different calculations byCALPHAD11 and ab initio method107,111 at low temperatures.



FIG. 19. Comparison of �Emag between ab initiocalculations111 and thermodynamic modeling.11

the Curie temperature for pure Fe (1043 K).51 Secondly, a spinglass formation region was found, as mentioned above, between80 to 90 at.% below 30 K (see Figure 10). Last but not least, theregion of antiferromagnetism can be divided into three parts.One is the ISDW region, and the other two are TSDW (Trans-

FIG. 20. Comparison of MOEM between ab initiocalculations107,111 and thermodynamic modeling.11

TABLE 2Summary of the experimental methods for enthalpy of mixing

of the liquid phases

Ref. Method Temperature (K) Note*

203 Vapor pressure method 1923 o204 High temperature

isothermal Calorimetry1873 +

205 High temperatureisothermal Calorimetry

1863 −


1960 +

208 Levitation alloyingcalorimetry

2223 o


1973 +

∗“+” stands for the positive deviation, “-” denotes the negative de-viation, “o” indicates that the experimental data are close to zero.

verse SDW) and LSDW (Longitudinal SDW), which belong toCSDW. The measurements of the spin-flip transition temper-atures were performed by several groups158,172 with mutuallyconsistent results, as shown in Figure 10.

The only exception in the Curie temperature measurementis the early experimental work by Fallot,151 who determineda Curie temperature curve systemically higher than the otherexperimental results. Unfortunately, it is this work that has beenadopted in the widely accepted CALPHAD modeling.11 Thereare at least five reassessments after the work by Andersson andSundman,11 but all of them overlooked the misfit of the Curietemperature. The evaluated Curie temperature curve is shownin Figures 9 and 10. As can be seen, the experimental data byAdcock et al.51 also show higher values in the composition rangebetween 40 and 60 at.% Cr.

As to the mean magnetic moment, the experimental resultsare plotted in Figure 11. The magnetic moment for pure Cr isrecommended to be 0.4 µB in the early work by Chin et al.,178,179

which is the same as the results from the neutron diffractionmeasurements by Shull and Wilkinson.180 However, it shouldbe mentioned that the Neel temperature determined by Shulland Wilkinson180 is 475 K which is higher than the acceptedvalue of 311.5 K.41 After that, another neutron diffraction studyfor pure Cr by Bacon181 revised the Bohr magnetic momentfor pure Cr to be 0.47 µB, with a Neel temperature of 312K. During the experimental investigation of Fe-Cr alloys, Arrottet al.158 reported another value of 0.62 µB for the Bohr magneticmoment of Cr, while Ishikawa et al.159 determined it to be0.59 µB. The value determined by Ishikawa et al. has beenconfirmed by a recent work using magnetic XRD combinedwith XRD focusing optics for the studies of SDW evaluationin pure Cr.177 In fact, the value of 0.6 µB was considered tobe a maximum value due to the amplitude of SDW.177 In thepresent paper, Bohr magnetic moment of Cr is accepted to be


TAB

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FIG. 21. Comparison of the enthalpy of mixing of the liquid phase between experiments203−206,208−209 and thermodynamicmodeling.11,19

FIG. 22. Comparison of the activities of Cr and Fe at 1173 K between experimental data221,223 and calculated results fromCALPHAD modeling.11

FIG. 23. Comparison of the activities of Cr and Fe from 1273 to 1573 K between experimental data67,212,215,219,221 and calculatedresults from CALPHAD modeling.11


142 W. XIONG ET AL.

FIG. 24. Comparison of the activities of Cr and Fe from 1600 to 1667 K between experimental data213−214,219 and calculatedresults from CALPHAD modeling.11

0.60 µB. It is worth noting that unlike ferromagnetism, thecomplex antiferromagnetic ground state is difficult to deal within ab initio calculations. A comprehensive discussion on theSDW antiferromagnetism in Cr can be found in the review paperby Fawcett.176

As shown in Figure 11, good consistency could be foundin the work by Arrott et al.158 and Ishakawa et al.159 on themagnetic moment determined in the antiferromagnetic state ofdilute Fe-Cr alloys.

It should be pointed out that the current CALPHAD modelfor the magnetic transition is based on the work by Inden182 andHillert and Jarl.183 This model can be used for the reproduction,rather than prediction, of the Curie/Neel temperature and mag-netic entropy. It is based on the assumption that the magneticordering is an independent effect of the specific heat changein the ferromagnetic transition. More severely, in the currentlyused description of pure Cr,42 a small artificial value (0.008)was assigned to the magnetic moment to be able to describe theweak magnetic first-order transition at the Neel temperature.The need of such simplifications shows that the description ofmagnetism at the Cr-rich side is beyond the competence of theCALPHAD approach. Therefore, the Curie/Neel temperaturesbut not mean magnetic moment can be correctly reproduced inthe Fe-Cr system by the currently used magnetic model.183

Referring to the magnetism of the σ phase, the early workdone by Read et al.184 and Sumimoto et al.185 revealed a lowtemperature weak ferromagnetism (<50 K) with the maximumaverage magnetic moment per Fe atom of 0.25 µB. It should bepointed out that in recent experiments186−188 it is shown that theσ phase has a maximum average Curie temperature of 38.9 Kat Fe55Cr45. In addition, the mean magnetic moment was foundto decrease linearly with the Cr content.188

4. THERMODYNAMIC PROPERTIES4.1. Heat Capacity

The experimental information from different measurementson Cp (heat capacity) is summarized in Figure 12. The firstsystematic measurements on Cp for the Fe-Cr system were con-tributed by Matsumoto et al.189,190 Although their experimental

FIG. 25. Comparison of the activities of Cr and Fe from 1673to 1773 K between experimental data216 and calculated resultsfrom CALPHAD modeling.11



results for pure Fe are fairly reasonable compared to the cur-rently accepted ones,41 the insufficiently annealed Fe-Cr binarysamples with appreciable impurities of Mn (0.37 to 0.44 wt.%)and Si (0.31 to 0.95 wt.%) predestine the low accuracy of theexperimental data.

Later, Cp measurement for an Fe54.2Cr45.8 (at.%) alloy wascarried out by Backhurst191 using adiabatic calorimetry. Ander-sson and Sundman11 made a comparison between their calcu-lated results and the experimental data191 as presented in Figure13. In fact, the experimental data by Backhurst191 are not veryuseful to make such a comparison. Firstly, the related experi-mental data for the bcc phase were achieved from the coolingrather than heating procedure. Secondly, it should be noticed thatthe determined data for the phase transition in their adiabaticcalorimetry is not accurate enough to determine the magnetictransition since a peak can be detected in the Cp curve for theα ↔ β transformation of nonmagnetic pure Ti in the same workby Backhurst.191 According to the measurement by Backhurst,the magnetic transition temperature determined during coolingis 833 ± 10 K, the calculated value from the previous CAL-PHAD modeling11 is 820 K, but the suggested value in thiswork is about 670 K according to the evaluated Curie temper-ature curve in Figure 9. However, the experimental data above1100 K in the pure bcc state above the Curie temperature re-ported by Backhurst191 seem reasonable. It has been confirmedby another measurement for an Fe53Cr47 (at.%) alloy192 using ahigh-temperature calorimeter.

The heat capacity of an Fe21.5Cr78.5 at.% (alloy) was mea-sured by Kendall et al.,193 and collected by Touloukian andBuyco.194 The samples were homogenized at 1623 K for 4 daysunder helium atmosphere, and then air cooled to room tempera-ture. As shown in Figure 14, the experimental results 193 can bereproduced using the assessment by Andersson and Sundman11

except for the lower temperature part since the Curie tempera-tures were not well described.

Inden171 reported a set of experimental Cp data for anFe79Cr21 (at.%) alloy provided by Pepperhoff by using adia-batic calorimetry195 through private communication. Accordingto Inden,171 the experimental data correspond to the single bccstate. Therefore, these experimental data are used for compari-son as shown in Figure 15. Good agreement between the exper-iments and previous modeling11 can be found only below themagnetic transition.

Another systematic adiabatic calorimetry was carried out byNormanton et al.67 measuring Cp on Fe-Cr alloys containing upto 16 at.% Cr over the temperature range from 700 to 1500 K.The comparison between the previous assessment11 and theseexperiments is shown in Figure 16. Despite the contaminationof the samples, the agreement between the experimental dataand the modeling is generally good.

Downie and Martin196 measured the heat capacities for alloysFe52.5Cr47.5 and Fe56.6Cr43.4 (at.%) using a combination of adi-abatic calorimetry and differential scanning calorimetry. Theirexperimental data is not accurate enough, since the determined

Curie temperatures for both α and σ phases are 50 to 100 Khigher than the accepted temperatures.

A recent measurement on heat capacity of the alloyFe48.2Cr51.8 (at.%) was performed using adiabatic scanningcalorimetry by Abiko and Kato.197 Since the sample prepa-ration, like hot/cold rolling, was performed in air, it could beexpected that the accuracy will probably be not high due tooxide contamination. Unfortunately, no chemical analysis wasperformed for the samples.

Low-temperature Cp between 133 and 623 K was determinedby Schroder198 by means of adiabatic calorimetry. AlthoughSchroder claimed that the accuracy of the experimental datawould be low only above 573 K, the experimental data aboveroom temperature for pure Fe deviates remarkably from therecommended data.41,199

4.2. Enthalpy of MixingEmploying adiabatic high-temperature calorimetry, the en-

thalpy of mixing for the bcc phase at 1529 K has been determinedin the work of Dench.200 The experimental data for an Fe53Cr47

alloy has been confirmed later by Malinskey and Claisse192

using high-temperature calorimetry at 1550 K. The foregoingexperiments are well described by Andersson and Sundman.11

Moreover, a lot of atomistic simulations for the enthalpy of mix-ing of the bcc phase in both ferromagnetic and paramagnetic(or Disordered Local Moment (DLM)) states, such as ab initiocalculations,102,108,111,112,201 have been performed as well. Asshown in Figures 17 and 18, the ab initio calculations performedin DLM states present a symmetrical curve for the enthalpy ofmixing of the bcc phase111 which has a similar shape as themeasured data by Dench200 despite the absolute values at differ-ent temperatures. In contrast, as shown in the same figure, thecalculated enthalpy of mixing of the bcc phase at ferromagneticstates shows a change of sign being positive over around 7 at.%Cr.111 In fact, this is the strongest evidence so far to model such ahigh solubility of Cr in (α’-Fe) as shown in Figure 5. Besides, nothermal aging experimental data are available at such low tem-peratures approaching 0 K. The calculated enthalpies of mixingfor the bcc phase by CALPHAD modeling in both paramag-netic and ferromagnetic states are also shown in Figures 17 and18. It is clear that there are significant disagreements betweenab initio calculations107,111 and CALPHAD modeling11 for theferromagnetic state while they agree well with experimentalobservation for the paramagnetic state.192,200 From Figure 18,It is interesting to note that the Cr content at the sign changefor enthalpy of mixing of the bcc phase calculated using SQS(Special Quasirandom Structure) by PAW-VASP (the projectoraugmented wave method implemented in the Vienna ab initiosimulation package) is lower than the ones by EMTO-CPA (theexact muffin-tin orbitals theory within the coherent potentialapproximation).

Let us now focus on the energy of magnetic ordering, whichis the difference between the energies of ferromagnetic and para-magnetic states. As reproduced in Figure 19 from the work of


144 W. XIONG ET AL.

FIG. 26. Comparison of the activities of Cr and Fe from 1813 to 1873 K between experimental data52,203,211,216−218,220,224 andcalculated results from CALPHAD modeling.11

Korzhavyi et al.,111 the ab initio energy of magnetic ordering inthe Fe-Cr system, �Emag = EDLM − EFM , where EDLM andEFM have been calculated using LDA (Local Density Approxi-mation) and GGA (Generalized Gradient Approximation) for afixed lattice parameter of 2.87 A, disagree even for pure Fe withthe CALPHAD modeling. Besides, the ab initio results showa lager energy difference between ferromagnetic bcc and anti-ferromagnetic fcc iron compared to CALPHAD modeling.202

It is interesting to see from Table 4 in the paper of Chen andSundman202 that the results of ab initio calculations are show-ing less and less discrepancy with that of CALPHAD over time.

From this aspect, the ab initio calculations need to be improvedfurther for pure ferromagnetic Fe.

If we plot the difference between the heat of mixing of theparamagnetic and ferromagnetic states shown in Figures 17 and18 or redraw Figure 19 with the values for the pure elementsas reference, we obtain an interesting quantity, which we shallcall MOEM (magnetic ordering energy of mixing) and is ameasurement of the relative alloying effect on the energy ofmagnetic ordering. Figure 20 shows the MOEM values fromvarious calculations.11,102,111 The ab initio calculations show afeature with the maximum value of �Emag at about 9 at.% Cr

FIG. 27. Comparison of the activities of Cr and Fe from 1903 to 2100 K between experimental data68,203,216,222 and calculatedresults from CALPHAD modeling.11



while CALPHAD modeling does not, and the absolute valuesof ab initio results are much larger than that of CALPHAD as-sessment. Obviously, it is the large positive MOEM near pureFe end that causes the negative enthalpy of mixing in the fer-romagnetic state. Since the present CALPHAD modeling doesnot describe the composition dependence of the experimentalCurie temperature very well, we cannot rule out the possibilityof a positive MOEM in Fe-rich alloys from the viewpoint ofCALPHAD modeling. However, the absolute values of ab initioresults might again be too large to accommodate. It should befurther noted that, in the presently available ab initio calcula-tions of the enthalpy of formation at 0 K, the Cr-rich side isregarded as ferromagnetic or paramagnetic. In reality, it is an-tiferromagnetic and SDW which is at present quite difficult totreat reasonably with the ab initio calculations. Consequently,one should be careful to make judgment on the ab initio results,even though the atomistic model could be sophisticated on de-scribing the short range ordering on the Fe-rich side to someextent.

The enthalpies of mixing of the liquid phase at high tem-peratures have been determined by several groups.203−209 Theemployed methods are summarized in Table 2. Unfortunately,the experimental data are scattered between −4000 and 5000J/mol as shown in Figure 21. A recent measurement by Thiede-mann et al.208 together with the earliest measurement done byPavars et al.203 show that the solution in the liquid state is closeto ideal. However, the CALPHAD modeling done by Anders-son and Sundman11 and Lee19 agree well with the experimentaldata reported by Iguchi et al.205 In the work of Muller andKubaschewski,22 the enthalpy of mixing in the liquid state has,on the contrary, been evaluated to be symmetrical and positivewith a maximum of 6276 J/mol·atom at 50 at.% Cr, seeminglywithout any evidence. In principle, the CALPHAD modelingwill play a key role in judging the quality of the experimentaldata. However, since the liquidus has not been described well inthe previous assessments, there has been no opportunity to testthe accuracy of the reported experiments.

4.3. ActivitiesThere are a large number of measurements on the Fe-Cr sys-

tem at different temperatures.203,210−227 The experimental infor-mation, including experimental temperatures, reference states,and experimental methods, are summarized in Table 3. In orderto make a distinct comparison between different experimen-tal activities, the experimental data were plotted in Figures 22to 28. Although the thermodynamic description by Anderssonand Sundman11 needs to be improved to fit the Curie temper-ature, the calculated activities are being used here to make acomparison with the experimental data. It should be noted thatthere is no comprehensive comparison performed in the previousassessment.11

In Figures 22 to 24, the experimental activities show posi-tive deviation from Raoult’s law. In Figure 24, the determinedactivities at 1667 K by Vintaikin213 show large positive de-

FIG. 28. Comparison of the activities of Cr and Fe at 2200K between experimental data222 and calculated results fromCALPHAD modeling.11

viation from the ideal solution which seems too high and in-consistent with the measurements by Kubaschewski et al.214 InFigures 26 and 27, scattered experimental data can be foundin the temperature range between 1813 and 2100 K. However,when the temperature reaches 2200 K, at which both Fe andCr are in the liquid state, the experimental activities222 as givenin Figure 28 seem to obey the Raoult’s Law with negligibledeviation.

It is easy to find that the experimental activities is scattered.For instance, in Figure 25, if the experimental data at 1673 and1773 K each were connected with lines, the two lines wouldcross which is of course not reasonable. This means that theexperimental data have large uncertainties. It is expected thatthe judgments on these scattered experimental activities couldbe assisted by an improved thermodynamic description of theFe-Cr system. Besides, systematic activity measurements areneeded as well.

5. CONCLUDING REMARKSThe Fe-Cr system needs a new thermodynamic description

with correct representation of the Curie temperatures. As shownin this work, the evaluation of the Curie temperature and meanmagnetic moment at the metastable phase region are importantbut easily neglected in CALPHAD modeling in order to describecorrectly the magnetic contribution to the Gibbs energy.

Melting temperature of pure Cr has been evaluated to be2136 K, which is 44 K lower than the one proposed bySGTE, and used consequently in some research communities on


146 W. XIONG ET AL.

thermodynamics, like the CALPHAD community. Therefore, anew description of pure Cr is needed. Recently, a publicationon a new description for pure Fe was presented by Chen andSundman202 by using models that contain some parameters ofphysical significance, such as the Einstein temperature and elec-tron density of states at the Fermi level. The magnetic modelhas also been refined in the same work however keeping theformulation by Hillert and Jarl.183 This could serve as a goodexample for further work on pure Cr. As a consequence, high-temperature phase equilibria with the liquid phase have beenrefined.

According to the available ab initio calculations for enthalpyof mixing at 0 K, the low-temperature phase equilibria in the Fe-Cr system have been modified by introducing a larger solubilitylimit of Cr in (α’-Fe) compared to the previous work.11 However,in order to make a reasonable extrapolation of the solubilitylimit to higher temperatures the ab initio calculations need to beimproved even for the description of pure Fe.

It has been proved that experimental data under irradiationcannot be used to assist determination of phase diagram. Thedetermined SRO effect change under irradiation has no relationwith the phase transition under equilibrium.

The phase separation in the bcc miscibility gap needs to bestudied further. Some modern techniques, such as, Local Elec-trode Atom Probe tomography, may facilitate the understandingof the dominant mechanisms in the transition range from nucle-ation and growth to spinodal decomposition.

The measurements on thermodynamic properties of the Fe-Cr alloys are difficult to perform. Moreover, because of the smallvalues, the enthalpies of mixing of the liquid phase are difficultto measure with high accuracy. Similarly, the small negativevalues of the enthalpy of mixing at the Fe rich side predictedby ab initio calculations will also be difficult, if not impossible,to confirm by experiments. The activities in the Fe-Cr systemare very scattered and need to be determined systematically andprecisely. The difficulties of the experiments could be due to theeasy oxidation and evaporation of Fe-Cr alloys.

In conclusion, the Fe-Cr system is actually challenging forboth theoretical calculations and experimental investigations.

ACKNOWLEDGMENTSThis work was performed within the VINNEX centre HERO-

M, financed by VINNOVA, the Swedish Government Agency ofInnovation Systems, Swedish Industry and the Royal Institute ofTechnology (KTH). The financial support from Chinese Schol-arship Council (Grant No. 2008637018) is acknowledged. Theauthors are grateful to Drs. P.A. Korzhavyi, A. Ruban, Profs. M.Hillert and J. Agren (KTH, Sweden) for helpful discussions.

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