Sundman AlFediagram

7/28/2019 Sundman AlFediagram

1/13

An assessment of the entire AlFe system including D03 ordering

Bo Sundman a,*, Ikuo Ohnuma b, Nathalie Dupin c, Ursula R. Kattner d, Suzana G. Fries e,f

a CIRIMAT, UPS/CNRS/ENSIACET, 116 Route de Narbonne, 31077 Toulouse, Franceb Department of Materials Science, Tohoku University, Sendai, Japan

c Calcul Thermodynamique, FR 63670 Orcet, Franced National Institute of Standards and Technology, Gaithersburg, MD, USA

e SGF Scientific Consultancy, Aachen, GermanyfICAMS, Ruhr University Bochum, Bochum, Germany

Received 10 December 2008; received in revised form 15 December 2008; accepted 27 February 2009Available online 9 April 2009

Abstract

The AlFe system is important for many alloys and the new interest in iron aluminides makes it necessary to improve the modeling ofthe different ordered forms on the body-centered cubic lattice in this system. This has now been done using a four-sublattice model basedon the compound energy formalism, which can describe disordered A2 and the B2, D03 and B32 ordering. The chemical and ferromag-netic ordering transitions can be both first and second-order and they have a strong interaction. Almost all available experimental andtheoretical data for all phases in the system have been fitted within estimated uncertainties. 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Calphad; Ordering; Compound energy formalism; Phase diagram; AlFe

1. Introduction

Many assessments of important binary systems still usedin commercial databases were typically derived 1015 yearsago, when the computer facilities were new for mostresearchers in materials science. The thermodynamicdescriptions of many of these systems have often beenamended, sometimes without proper publication. Newexperimental data and ab initio calculations, especially formetastable states, have also been published and the model-

ing technique has improved, which makes it important topresent a complete review of many of these binary systems.This has to be done even if it is not possible to integratethese new assessments directly into existing commercialdatabases because all higher-order systems which havebeen assessed using the previous assessment will also haveto be revised.

The emphasis of this paper is on modeling issues; theevaluation of the experimental data is not much differentfrom earlier assessments and is only discussed very briefly.For only a few cases, for example when new conflictingdata exist, will an evaluation of the available experimentaldata be presented.

The AlFe system is important both for commercialalloys and for theoretical studies of ordering transforma-tions. It forms a part of many commercial alloys, whichmeans that the extrapolations to higher-order systems of

the stable and metastable ordered forms require accuratemodeling of the binary system itself. As there are no exper-imental data for phases that are not stable in the binary,one must rely on ab initio calculations or backwardextrapolation from ternary systems.

The generally accepted assessed phase diagram for AlFe shown in Fig. 1 is from an assessment by Kattner andBurton [1]. The different phases are listed in Table 1. TheAlFe (B2) and AlFe3D03 phases are ordered forms onthe body-centered cubic (bcc) lattice and are separatedfrom the disordered A2 phase by first- or second-order

1359-6454/$36.00 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.actamat.2009.02.046

* Corresponding author. Tel.: +33 684330509.E-mail address: [email protected] (B. Sundman).

www.elsevier.com/locate/actamat

Available online at www.sciencedirect.com

Acta Materialia 57 (2009) 28962908
mailto:[email protected]:[email protected]


2/13

transitions, the latter dash-dotted. There is also a dash-dot-ted line for the Curie temperature denoted TC. Some of thesolubility lines are dashed because they are not welldetermined.

Kattner and Burton [1] did not present a set of parame-ters to calculate the whole diagram as the modeling tech-nique at the time did not allow an acceptablethermodynamic model for the B2 and D03 ordering. Thethermodynamic assessment most used in databases is bySeiersten [2] from the COST 507 project [3]. However,her assessment has never been published together with acomparison with experimental data, and the two-sublatticemodel used can describe only the B2 ordering, not the D03one. There have been several amendments of the assess-ment by Seiersten to improve various features, but withoutproper documentation or publication.

In a recent revision of the Seiersten description, Jacobsand Schmid-Fetzer [4] introduced the formation of vacan-cies in addition to the anti-site defects into the two-sublat-

tice model and found that the use of vacancy defects had

no significant effect on the calculated A2/B2 transition.Jacobs and Schmid-Fetzer could not find a description thataccurately reproduced the vacancy concentrationsobserved by Kerl et al. [5] and they used a two-sublatticemodel for the ordering in the bcc lattice.

The current paper presents a complete revision of themodeling using a four-sublattice model based on the com-pound energy formalism (CEF) and taking all experimentaland theoretical data into account. The calculated phasediagram is shown in Fig. 2. The invariant temperaturesare listed in Table 2.

The transitions between the disordered and the differentordered structures can be of first and second order, and thiscan only be achieved using a single Gibbs energy function.The B2 ordering has the ideal stoichiometry AlFe and theD03 ordering has the ideal stoichiometry AlFe3 but, ascan be seen in the phase diagrams, there are considerableranges of solubility in both phases. This has been modeledusing anti-site defects and, for compatibility with higher-

order systems like AlFeNi, vacancy defects have also

Fig. 1. The AlFe phase diagram assessed by Kattner and Burton [1]. See the text for details.

Table 1Phases and structures.

Phase Label in Fig. 1 Label in Fig. 2 Struktur Bericht Pearson symbol Space group Prototype

Liquid L Liquid fcc cFe or (Al) A1 A1 cF4 Fm3m Cubcc aFe A2(pm) or A2(fm) A2 cI2 Im3m WAlFe FeAl B2(pm) B2 cP8 Pm3m CsClAlFe3 Fe3Al D03 D03 cF16 Fm3m BiF3

Al8Fe5 Al8Fe5 D82 cI52 I43m Cu5Zn8Al2Fe FeAl2 Al2Fe aP18 P1 FeAl2Al5Fe2 Fe2Al5 Al5Fe2 oC? Cmcm Al13Fe4 FeAl3 Al13Fe4 mC102 C2/m

B. Sundman et al. / Acta Materialia 57 (2009) 28962908 2897


3/13

been introduced. The modeling of the different orderedforms of the bcc lattice is discussed in Section 3.1 and thefit to various experimental data is discussed in Section 4.

2. Experimental data

The assessment by Kattner and Burton gives a completereview of all experimental data up to 1992. In most casesthe same selection of data to be fitted has been followedexcept for the liquidus in equilibrium with the Al-rich com-pounds, where the more recent assessment by Du et al. [7]has been used. The new data on the phase diagram havegenerally confirmed previous data; however, significantlylower enthalpies of formation were found for the thermo-dynamic data, as shown in Fig. 3 and discussed below.

2.1. New phase diagram data

The experimental data by Ikeda et al. [8] established theshape of the region around AlFe3. Stein and Palm [6] mea-sured the liquidus and solidus and the second-order transi-tions in bcc, mainly confirming existing data. Althoughthey measured only a few liquidus temperatures in theregion with Al fractions between 0.65 and 0.8, their temper-

atures are in agreement with the assessment by Du et al. [7].

The crystal structure of the high temperature phasewas determined by Vogel et al. [9] to have the gamma-brassstructure D82 with prototype Cu5Zn8.

2.2. New thermochemical data

The calorimetric measurements of Breuer et al. [10] give

significantly lower enthalpies of formation of the AlFe B2phase than the earlier experiments by Oelsen and Middel[11] and Kubaschewski and Dench [12]. The calculatedenthalpies of the previous thermodynamic descriptions ofthe bcc-based phases were lower than the data from theseexperiments and, in a way, the new experimental data fromBreuer et al. confirm what had been found by modeling.

The calculated enthalpies of formation of bcc and thecompounds stable at 300 K are shown in Fig. 3. The morerecent experiments and the ab initio data from Lechermanet al. [13] are included. There are also calculated curves forthe metastable A2 and B32 structures, and dashed curvesfor the A1 and its ordered forms L12 and L10.

Kattner and Burton [1] noted that the data for the chem-ical potential of Al in the A2 phase can be divided into twogroups with different slopes. Newer data from Jacobsonand Mehrotra [14] agree with the group of data havingthe flatter slope, while the data from Kleykamp and Glas-brenner [15] and Bencze et al. [16] agree with the data withthe steeper slope. Huang et al. [17] determined the chemicalpotential of Al in the D03 phase.

2.3. New theoretical data

All ab initio calculations have so far found that the sta-

ble D03 ordered form on bcc lattice is less stable than the

400

600

800

1000

1200

1400

1600

Temperatu

reoC

0 20 40 60 80 100

Atomic percent Al

liquid

A1

A2(pm)

A2(fm) D03

B2(pm)

Al8Fe

5

Al2Fe

Al5Fe2

Al13

Fe4

A1

Fig. 2. The calculated AlFe phase diagram from the current assessment.The phase regions are labeled according to Table 1. The lines representingsecond-order transitions between chemically ordered states are short

dashed and those between ferro- and paramagnetic states are long dashed.

Table 2Invariants.

Transformation T C[1]

T C[3]

T C[6]

T C[7]

T C (thiswork)

liq B2 ! Al8Fe5 1232 1222 1231 1222 1226liq ! Al8Fe5 Al5Fe2 1169 1157 1155 1155 1154Al8Fe5 Al5Fe2 ! Al2Fe 1 165 1155 1146 1155 1153liq Al5Fe2 ! Al13Fe4 1160 1151 1149 1153 1151Al8Fe5 ! B2 Al2Fe 1102 1115 1092 1094 1089liq ! A1 Al13Fe4 655 654 654 654

-35

-30

-25

-20

-15

-10

-5

0

Enthalpyk

J/mol

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Al

A2

B2

D03

B32

A1L12

L10

Fig. 3. The full lines are calculated enthalpy of formation at 300 K ofseveral ordered forms of bcc (A2) and the symbols for the calculated Al-rich compounds are for Al2Fe,} for Al5Fe2 and M for Al13Fe4. Thereference state is bcc for Fe and fcc for Al. The enthalpy curves for face-centered cubic (fcc) (A1) and the ordered L12 and L10 are shown as dashedlines. Experimental data are from + [11], [12], [18], [19], g [20], ffl[10] and ab initio data from [13] for D03 and B2 structures.

2898 B. Sundman et al. / Acta Materialia 57 (2009) 28962908


4/13

L12 ordered form on fcc lattice and, therefore, the ab initioresults cannot be used uncritically. The calculated curvesfor A1 (disordered) and its ordered forms are included asdashed lines in Fig. 3. It should be noted that this assess-ment gives a minimum difference in the Gibbs energybetween D03 and L12 of 150 J/mol at 700 K.

The ab initio enthalpy value for the metastableD03 Al3Fe is less negative than that for the B2 phaseimplying that the D03 ordered Al3Fe should not be stableeven when only the bcc-based phases are considered.

3. Modeling

All the modeling has been based on the CEF asdescribed by Lukas et al. [21]. The molar Gibbs energy isdivided into four terms:

Gm srfGm

cfgGm mgnGm

EGm 1

The first two terms are the surface of reference and con-

figurational contributions, and for a substitutional regularsolution model these are:

srfGm X

i

xiGiT

cfgGm RTX

i

xi lnxi2

where xi is the mole fraction of component i and Gi is theGibbs energy of the component i as a function of temper-ature. The configurational term represents the ideal entro-py of mixing, the factor R is the gas constant and T is theabsolute temperature. The third term in Eq. (1) is the mag-

netic contribution described in Section 3.2. The fourth termis the excess part and is described by a RedlichKisterseries:

EGm xAlxFeXm0

xAl xFem m

LAl;Fe 3

The mLAl;Fe can be temperature dependent. The bcc and fccphases have been modelled with four sublattices for order-ing. The model for bcc ordering is described in Section 3.1.No stable ordered fcc phases exist in the binary AlFe, butthese are formed with the addition of Ni, Ti or C. Themodel for the fcc ordering in AlFe is similar to that de-

scribed here for bcc ordering. Details can be found in thepaper on the AlCFe system by Connetable et al. [22].

The three intermediate phases stable at low tempera-tures on the Al-rich side have been modelled the sameway as by Seiersten [2]. The Al5Fe2 and Al2Fe phases aretreated as stoichiometric with no solubility ranges, their

Gibbs energies depend only on temperature. The Al13Fe4phase is described with three sublattices: one sublattice isonly occupied by Al, the second one is only occupied byFe and mixing of Al and vacancies occurs on the third.

3.1. Modeling ordering in bcc

The B2 ordering on the bcc lattice can easily be mod-elled with the two-sublattice CEF model, each sublatticecorresponding to a site of crystallographic structure, asshown in Fig. 4(a). These two sites are crystallographicallyequivalent. Exchanging the occupation of the two sublat-tices should thus induce an identical energy i.e.GA:B GB:A and mLA;B:i mLi:A;B.

When modeling the D03 ordering, shown in Fig. 4(b), afour-sublattice CEF model is needed. The different sublat-tices correspond to the edges of the tetrahedra constitutingthe bcc lattice. These tetrahedra are irregular as theyinvolve first and second nearest neighbors. The four sublat-tices are thus not equivalent. As a consequence, the idealorder at equiatomic composition can correspond to theB2 ordering where the bonds between different elementsare only between first nearest neighbors or to the B32-typeordering, shown in Fig. 4(c), where such bonds are betweenboth first and second nearest neighbors.

In the B2 phase of the AlFe system the dominatingdefects are anti-site atoms, i.e. Al and Fe atoms canoccupy all sublattices. Close to the equiatomic composi-tion a small number of the lattice sites are occupied byvacancies. In the binary AlNi system, this type of defectis responsible for the extension of the B2 phase up to

xAl 0:58. In order to extend the description to the AlFeNi system, it is thus important to include vacanciesin the model, but this has to be done in the simplest pos-sible way as there are almost no data available for thisbinary system. Therefore, no ordering between the vacan-cies and the atoms is considered.

Fig. 4. The B2, D03 and B32 structures [23].



5/13

The molar Gibbs energy for a phase with four sublattic-es is divided into the same four parts as Eq. (1) and the firsttwo parts can be written as:

srfGm X

i

Xj

Xk

Xl

y1i y

2j y

3k y

4l

Gijkl

cfgGm

RTXs

asX

i

ys

iln

y

s

i

4

The site fraction ys

i gives the fractions of the constituent iin sublattice s and as is the number of sites on sublattice s.When the phase is disordered the site fractions of all con-stituents are equal in all sublattices. The parameter Gijklis the energy of the configuration ijkl in the four sublatticesrepresenting the bcc tetrahedron. There are four bonds be-tween nearest neighbors and two between next nearestneighbors in this tetrahedron. Many configurations thushave the same energy, for example:

GAlAlAlFe GAlAlFeAl

GAlFeAlAl GFeAlAlAl:

Assuming that the next nearest neighbor bonds arebetween atoms on sublattices 1 and 2 and between atomson sublattices 3 and 4, one has

GAlAlFeFe GFeFeAlAl and

GAlFeAlFe GAlFeFeAl

GFeAlAlFe GFeAlFeAl:

These parameters are related to the ordered structures:GAlFeFeFe corresponds to D03;

GFeFeAlAl to B2 andGFeAlFeAl to B32.

The excess Gibbs energy term for the sublattice phasecan be expanded in RedlichKister parameters as for the

substitutional model in Eq. (3). In the present work, theseterms were taken independent of the occupation in theother sublattices except that where the interaction isconsidered.

EGm y1Al y

1Fe

Xm0

y1Al y

1Fe

mmLAl;Fe:::

y2Al y

2Fe

Xm0

y2Al y

2Fe

mmL:Al;Fe::

y3Al y

3Fe

Xm0

y3Al y

3Fe

mmL::Al;Fe:

y4Al y

4Fe

X

m0

y4Al y

4Fe

mmL:::Al;Fe 5

The character * in a sublattice means that the param-eter is independent of the occupation of that sublattice. Theinteractions are identical in all sublattices:mLAl;Fe:::

mL: Al;Fe:: mL:: Al;Fe:

mL:::Al;Fe:

The Gibbs energy for the ordered and disordered formsof bcc are calculated from the same set of parameters. Thecontribution from the ordering part is added to the descrip-tion of the disordered state using the following relations.

Gtotalm Gdism x DG

ordm 6

DGord

m

Gsublm

y Gsublm

y x 7

In Eq. (7) DGordm will always be zero when the phase isdisordered, i.e. when all site fractions are the same on allsublattices, because the four sublattice Gibbs energy,Gsubl, is calculated twice with the same set of model param-eters: once with the original set of site fractions, y, and asecond time with all site fractions equal to the overall com-

position, x. However, this also means that if care is nottaken in assessing the parameters, ordering may occur forthe wrong reason, i.e. that Gsublm y x is more negativethan Gsublm y. The Thermo-Calc

1 software, for example,will give a warning when this happens.

Recent modifications to the Thermo-Calc software [24]include automatic handling of the permutations of thesame parameter for four-sublattice modeling of fcc, hcpand bcc ordering, thus further reducing the possibility ofincorrect use of the model.

3.2. Magnetic model

The magnetic model was proposed by Inden [25] and itcontributes to the Gibbs energy according tomgnGm RT fs lnb 1 8

where R is the gas constant, T is the absolute temperatureand f (s) is the integral of a function fitted to the contribu-tion to the heat capacity due to the magnetic transition. s isT=TC where TC is the Curie temperature and b is the aver-age Bohr magneton number per atom. The compositionand ordering dependence of the Curie temperature andthe Bohr magneton number is described in the same wayas for the Gibbs energy, i.e.

TtotalC TdisC x DTordC 9

DTordC TsublC y T

sublC y x 10

The use of RedlichKister series are not well suited todescribe the Curie temperature and Bohr magneton num-ber as these may very rapidly with composition and canbe zero for an extended composition range. However, inorder to calculate chemical potentials it is necessary to havea smooth function and thus they are used.

4. Assessment procedure and results

The first AlFe phase diagram including D03 orderingwas assessed by Ohnuma [26] but was never publishedbecause it had some unsatisfactory features. However,Ohnuma obtained a better A2/B2 transition line thanSeiersten [2] and fitted the composition dependence of theCurie temperature better. In order to avoid the miscibilitygap between D03 and B2 on the Al-rich side of D03, asfound by Ohnuma, the present assessment enforced a con-tinuous increase of the chemical potential of Al for xAl

1 Commercial products are referenced in this paper as examples. Suchidentification does not imply recommendation or endorsement by the

National Institute of Standards and Technology.

http://-/?-http://-/?-


6/13

between 0.25 and 0.5 at several temperatures. During theassessment, which was attempted on and off during severalyears, unrealistic shapes of the ordering transformationsand changes from second- to first-order A2/B2 transitionshad to be corrected by adding constraints on the experi-mental data and the optimizing parameters in order to

force the optimization to reproduce the available dataand give reasonable extrapolations.The current assessment is a balanced result between cor-

rectly describing the experimental data and using a reason-able number of adjustable parameters in the models. Forexample, the steep decrease in the Curie temperature inthe D03 region would require many coefficients to keepthe Curie temperature small for xAl > 0:4. Instead, a lesssteep slope was accepted and a single coefficient was used;see also Section 5.

All parameters for all phases have been reassessed in thepresent work and are listed in the Appendix A. In the finalset, the parameters for disordered bcc were so close to

those by Seiersten [3] that her parameters were retainedbecause this reduces the number of changes needed inhigher-order systems. Due to the use of Eq. (7) to expressthe ordering contribution, there are an infinite number ofequivalent expressions that can be used for the parametersof G sublm y; those listed were chosen in order to minimizethe number of parameters.

4.1. Equilibria with the liquid

In Fig. 5(a) the Fe-rich part of the liquidus and solidus isshown together with experimental data. The agreement is

good and the maximum difference between the calculatedcurve and the most recent experimental data by Stein andPalm [6] is 10 K. The calculated liquidus has a maximumat xAl 0:05 and 1814 K.

In the center of the phase diagram, shown in Fig. 5(b),there are four intermediate phases and two of these havebeen modelled as stoichiometric compounds, Al2Fe andAl5Fe2. The structure of the phase that is only stableat high temperature, often called or Al5Fe4, hasrecently been determined by Vogel et al. [9] as Cu5Zn8-type (Strukturbericht D82) and has been modelled withtwo sublattices with anti-site defects in both,Al; Fe

8

Fe;Al5

.In agreement with the experiments, the Al8Fe5 phase

forms peritectically at 1226C and in the assessment theAl2Fe phase forms 1 degree below the eutectic betweenAl8Fe5 and Al5Fe2. The Al5Fe2 has congruent meltingpoint. The invariants are listed in Table 2 and are in closeagreement with the recent assessment by Du et al. [7]. Theliquidus curve between Al13Fe4 and pure Al is well fitted tothe experimental data, as shown in Fig. 5(c).

For the liquid, experimental data on the enthalpy ofmixing are shown in Fig. 6(a). The agreement with the cal-culated line is excellent. In Fig. 6(b) the chemical potentialsat 1873 K are compared to experimental data; the agree-

ment is again very good.

1450

1500

1550

1600

1650

1700

1750

1800

1850

Temperature(K)

0 0.1 0.2 0.3 0.4 0.5 0.6

Mole fraction Al

liquid

A2

A1 B2

1350

1400

1450

1500

1550

Temperature(K)

0.50 0.55 0.60 0.65 0.70 0.75 0.8

Mole fraction Al

liquid

B2

Al8Fe

5

Al2Fe

Al5Fe2

Al13

Fe4

800

900

1000

1100

1200

1300

1400

1500

Tem

perature(K)

0.70 0.75 0.80 0.85 0.90 0.95 1.0

Mole fraction Al

A1

liquid

Al13

Fe4

Fig. 5. The calculated phase diagram for the liquid in equilibrium with bcccompared with experimental data. In (a) the dashed line is the second-orderA2/B2. The invariant temperatures are listed in Table 2. The experimentaldata forthe liquidus arefromM [27], [28], } [29], [30],O [31], +[32],g [6]and for the solidus are from ffl [28],z [29], [30], [44], [31],I [32],f [6].

http://-/?-http://-/?-http://-/?-http://-/?-


7/13

4.2. Equilibria with the fcc phase

The fcc phase is only stable close to the pure elements:on the Fe side as a so-called c loop and on the Al side witha very small solubility of Fe. These are shown in Fig. 7,together with experimental data. The metastable orderingin the fcc phase is discussed together with the bcc phasein Section 6.

4.3. Equilibria with ordered and disordered bcc phases

The bcc phase has ordering transformations to B2 andD03, both of which can be first- or second-order, and a fer-romagnetic transition that depends on the composition, allof which are shown in Fig. 8(a)(c). The full lines are sol-ubility lines, the second-order chemical transitions are

short dashed and the ferromagnetic transitions are long

dashed. The second-order A2/B2 transition extends up tothe melting temperature and the fit of the calculated curveto the experimental data is very good, as shown inFig. 8(a).

The ferromagnetic transition is shown in Fig. 8(b),together with experimental data. The ferromagnetic transi-tion temperature is considerably decreased when chemicalordering occurs. The fit to the data for the disorderedregion is excellent, but the calculated Curie temperaturefor the D03 phase does not decrease as steeply as indicatedby the experimental data. All attempts to improve the fit tothe Curie temperature in the D03 region failed because asteeper slope makes it impossible to fit the phase diagramaround the A2/B2/D03 first- and second-order transitions.A better magnetic model may be needed to improve this, asdiscussed in Section 5, but the present assessment is limited

to what is available in the software.

-25

-20

-15

-10

-5

0

EnthalpykJ/mol

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Al

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

104

ChemicalpotentialJ/mol

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Al

Fig. 6. The calculated enthalpy of mixing and chemical potential of thecomponents in the liquid at 1873 K, compared with experimental datafrom: M [33], [34], } [35], [36], O [37]. Chemical potentials for Al arefrom: + [38], [39], [40], [41], I [42], and for Fe are from g [41].

1100

1200

1300

1400

1500

1600

1700

Temperature(K)

0 5 10 15 20 25 30

10-3 Mole fraction Al

600

650

700

750

800

850

900

950

1000

Temperature(K)

0 5 10 15 20 25 30

10-5 Mole fraction Fe

Fig. 7. The calculated c loop and solubility of Fe in Al compared withexperimental data from: M [29], [43], } [44], [45], O [46], + [47], [48], [49], [50], H [51], f [52].



8/13

In Fig. 8(d) the average Bohr magneton number peratom is shown for the whole composition range and thedrastic effect of chemical ordering is evident. Two coeffi-cients were used to describe the composition dependenceof the Curie temperature in the disordered bcc and forthe ordered state. In the disordered state the Bohr magne-ton number varies linearly with composition, while in theordered state four coefficients were used to describe thesharp decrease.

Fig. 8(c) is a magnification of the region where complexfirst- and second-order transitions occur between the differ-ent ordered states A2, B2 and D03, as well as the ferro- andparamagnetic states. The fit to the experimental data isgood and even by using models with a better description

of the short-range order contribution to the Gibbs energy,

like CVM or Monte Carlo [67,68], the fit has not been anybetter. The basic features are well reproduced, i.e. the sec-ond-order transitions for A2/B2 turns into first-order tran-sitions just before meeting the ferromagnetic transitionline. The calculated D03/B2 second-order transition fol-lows the experimental data, but has a maximum at aslightly higher temperature than the experimental value.The shape of the solubility lines when the second-ordertransitions opens up to first-order transitions is also closeto the experimental data, and the two-phase regionbetween A2 and D03 extends down to 0 K. However, thesolubility of Al in A2 at temperatures below 700 K ishigher than the experimental data.

The chemical potential of Al in the bcc phases has been

measured at 1273 K for low Al content and is shown in

800

900

1000

1100

1200

1300

1400

1500

1600

Temperature(K)

0.20 0.25 0.30 0.35 0.40 0.45 0.5

Mole fraction Al

B2(pm)

A2(pm)

400

500

600

700

800

900

1000

1100

Temperature(K)

0 0.05 0.10 0.15 0.20 0.25 0.30 0.3

Mole fraction Al

A2(fm)

A2(pm)

D03(fm)

B2(pm)

550

600

650

700

750

800

850

900

950

1000

Temperature(K)

0.15 0.20 0.25 0.3

Mole fraction Al

A2(fm)

B2(pm)A2(pm)

D03(fm)

D03(pm)

0

0.5

1.0

1.5

2.0

2.5

Bohrmagnetonnumber

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Al

Fig. 8. The calculated first- and second-order transition lines for A2/B2, A2/D03, B2/D03 and ferromagnetism together with experimental data. Themagnetic transition lines are long dashed, the chemical second-order transition lines are short dashed. Experiments of the A2/B2 ordering are from: 4 [53], [54], } [55], [56], O [57], + [58], [59], ffl [6]. The B2/D03 ordering experiments are from: z [8], g [6], [60], [61], I [62]. The ferromagnetictransitions experiments are from: [63], N [64],j [65],. [66], . [59], / [61], H [62], [6]. In (d) the calculated Bohr magneton number for A2, B2 and D03is shown over the whole composition range. Experimental data are from M [63], [64] and } [65].



9/13

Fig. 9. There are two distinct sets, one showing a rather flatshape [20], while the values in the other set are continu-ously increasing [69,19]. The calculated curve is closest tothe latter. 1273 K is above the ordering temperature forD03 and there is no reason to have a flat curve for thechemical potential around xAl 0:25; rather, the contrarywould be the case if there were any remaining short-rangeorder in the bcc phase.

The variation in the site fractions representing different

ordered states in bcc as a function of the composition fora fixed temperature and with the temperature for a fixedcomposition is shown in Fig. 10. In Fig. 10(a) the diagonalrepresents the disordered state with the same fractions inall four sublattices. When the D03 phase forms, the Al frac-tion in sublattice 1 increases rapidly and then decreases,while in sublattice 2 the Al fraction first decreases but thenincreases. The two remaining sublattices have the same lowfraction of Al until 50% Al. Around 40% Al the D03 trans-forms to B2 when sublattices 1 and 2 are of the same frac-tion. On the Al-rich side there is no D03 ordering. InFig. 10(b) the D03 ordering is stable at 0 K, but around800 K it becomes B2 and is finally disordered.

5. Influence of the ordering on the Curie temperature

According to the experimental information, the Curietemperature changes drastically with the ordering transfor-mation, as shown in Fig. 8(b). It has not been possible to fitsuch a drastic change; one reason for this may be that themodel used for the ferromagnetic transition overestimatesthe magnetic contribution to the Gibbs energy when theCurie temperature is small. If the Curie temperature isassumed to be independent of ordering, i.e. setting DTordCin Eq. (9) equal to zero, the phase diagram around

xAl 0:25 and T 700 K changes drastically, as shown

in Fig. 11. The same region using the assessed value ofDTordC is shown in Fig. 12(a).

6. Phase diagram with only bcc- or fcc-based phases

In Fig. 12 the phase diagram with different orderedforms based on the bcc lattice is shown, all otherphases being suppressed. On the Al-rich side there isonly B2 ordering, in accordance with ab initio results[13].

A persistent problem during this assessment was that,even when the experimental data were well fitted, first-order transitions occurred between the B2 and D03 orderedphases for mole fractions of Al between 0.35 and 0.45.Even in the present assessment it has been impossible to

avoid such phenomena completely, but the remaining nar-

-10

-9

-8

-7

-6

-5

-4

-3

-2

104

Chemicalpote

ntialofAl

0 0.1 0.2 0.3 0.4 0.5

Mole fraction Al

Fig. 9. The calculated chemical potential of Al, referred to fcc, at 1273 K,together with experimental data in A2 from M [69], [19], } [20] and in B2from [19], O [20], and with salt from + [20].

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Sitefract

ions

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Al

1=2=3=4

3=4

1

2

1=2

3=4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Sitefractions

0 500 1000 1500

Temperature (K)

1

2

3=4

1=2

1=2=3=4

Fig. 10. The site fractions in the ordered and disordered bcc phases: (a)for 300 K across the whole system; (b) for xAl 0:25 from 0 to 1500 K.The numbers represent the sublattices.



10/13

row first-order transition between B2 and D03 on the highAl side shown in Fig. 12 is probably impossible to verify orrefute by experiments or theory.

There are no experimental data for the fcc phase exceptat the limits of the phase diagram; however, for extrapola-tions to higher-order systems it is important to have knowl-edge of the properties of the fcc phase also at higher Alcontent. Additionally, as there is ordering transformationsin the bcc phase it is likely that there are ordering tenden-cies also in the metastable fcc phase. This is confirmed by

ab initio calculations.A reassessment of the fcc phase in AlFe, including the

ordering, became necessary during a recent assessment ofthe AlCFe system [22]. This description will also be usedin the present work. The metastable phase diagram for thefcc phase is shown in Fig. 12(b). No L12 ordering but amiscibility gap occurs on the Al-rich side. A four-sublatticeCEF model was used and the energy of formation of theL12 and L10 phases were fitted to ab initio data by Lecher-man et al. [13] and Connetable et al. [70].

7. Using a two-sublattice model for bcc-based phases

The present model can be reduced to two sublattices,but then the D03 ordering is not described. The phase dia-gram for this case is shown in Fig. 13. This model is suffi-cient for calculations for temperatures above 900 K andcould be easier to apply in more complex calculations,e.g. diffusion simulations.

It is interesting to note that the Gibbs energies of forma-tion of the two D03 compounds (Al3Fe and AlFe3) enteras regular parameters in the two-sublattice model, andtheir difference to the bond energies assessed for B2 andB32 appear in the reciprocal parameter (see the bcc-2SL

phase in the Appendix A).

8. Conclusion

A four-sublattice CEF-based model has been used todescribe the stable and metastable equilibria in the AlFesystem. The overall fit to experimental and theoretical datais very good. The extrapolations of the fcc and bcc phasesin the metastable ranges are reasonable. This confirms thatCEF can describe very complex transitions between first-and second-order transformations. The new set of param-eters has been tested in several ternary systems, such asAlCFe, AlFeNi and AlFeTi, and gives reasonableextrapolations using existing assessments of the other bin-

ary systems and small ternary parameters.

0

200

400

600

800

1000

1200

1400

1600

1800

Temperature(K)

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Al

A2(fm)

A2(pm)

D03(fm)

B2(pm)

0

200

400

600

800

1000

1200

1400

1600

1800

Temperature(K)

0 0.2 0.4 0.6 0.8 1.0

Mole fraction Al

A1

L12

L10

Fig. 12. The calculated phase diagrams for bcc and fcc with all otherphases suspended. (a) All phases except the disordered and orderedvariants of bcc lattice down to 0 K. The second-order transitions are shortdashed and the ferromagnetic transitions are long dashed. Note that thesmall first-order transition along the B2/D03 second-order transition existsbetween 200 and 400 K only. (b) The different fcc-based ordered forms.The L12 phase should be almost stable as ab initio calculations alwaysfinds it more stable than the D03 phase.

200

300

400

500

600

700

800

900

1000

1100

1200

Temperatu

re(K)

0 0.1 0.2 0.3 0.4 0.5

Mole fraction Al

A2(pm)

A2(fm)

B2(pm)

D03(fm)

D03(pm)

B2(fm)

Fig. 11. The calculated phase diagram when ignoring the effect on theCurie temperature by the chemical ordering.

http://-/?-http://-/?-


11/13

Acknowledgement

The authors are grateful for inspiration and many usefuldiscussions within the COST 535 action THALU.

Appendix A

List of parameters for the AlFe assessment. See Section3 for the relevant equations. Note that a star as a con-

stituent in a sublattice means that the parameter is indepen-dent of the constituents in this sublattice. The unaryparameters are from SGTE [71].

liquid (Al,Fe)

GliquidAl

Hfcc-A1Al GALLIQ

GliquidFe

Hbcc-A2Fe GFELIQ

0LliquidAl;Fe 88; 090 19:8 T

1LliquidAl;Fe 3800 3 T

2LliquidAl;Fe 2000

fcc-A1 (Al,Fe)

Gfcc-A1Al H fcc-A1Al GHSERAL

Gfcc-A1Fe H bcc-A2Fe GFEFCC

Tfcc-A1CFe 201

bfcc-A1Fe 2:10Lfcc-A1Al;Fe 104; 700 30:65 T

1Lfcc-A1Al;Fe 30; 000 7 T

2Lfcc-A1Al;Fe 32; 200 17 T

bcc-A2 (Al,Fe).

This phase is the disordered contribution for the bcc-based phases. It has to be combined to the bcc-4SL orbcc-B2 contribution as indicated by Eqs. (6), (7) to describethe ordered phases.

Gbcc-A2Al Hfcc-A1Al GALBCC

Gbcc-A2Fe Hbcc-A2Fe GHSERFETbcc-A2CFe 1043

bbcc-A2Fe 2:220Lbcc-A2Al;Fe 122; 960 32 T

1Lbcc-A2Al;Fe 2945:2

0Tbcc-A2CAl;Fe 438

1Tbcc-A2CAl;Fe 1720

Adding substitutional vacancies to the bcc-A2 phaserequires the following three parameters. The first is an esti-mated energy to create vacancy defects in the bcc lattice;the other two have been estimated from the AlFeNisystem.

Gbcc-A2Va 30 T0Lbcc-A2Al;Va 46; 912

0Lbcc-A2Fe;Va 150; 000

Al13Fe4Al:6275Fe:235 Al;Va:1375G

Al13Fe4Al:Fe:Al 0:765

Hfcc-A1Al 0:235Hbcc-A2Fe

30; 680 7:4 T :765GHSERAL :235 GHSERFE

GAl13Fe4Al:Fe:VA 0:6275Hfcc-A1Al 0:235Hbcc-A2Fe 28; 100 7:4 T :6275GHSERAL :235 GHSERFE

Al2FeAl2Fe1G

Al2FeAl:Fe 2

Hfcc-A1Al Hbcc-A2Fe 104; 000 23 T

2GHSERAL GHSERFE

Al5Fe2Al5Fe2G

Al5Fe2Al:Fe 5

Hfcc-A1Al 2Hbcc-A2Fe

235; 600 54 T 5GHSERAL 2GHSERFE

Al8Fe5 D82Al;Fe8Al;Fe5G

Al8Fe5D82Al:Al 13

Hfcc-A1Al 13GALBCC

GAl8Fe5D82Fe:Al 5

Hfcc-A1Al 8H bcc-A2Fe

200; 000 36 T 5GALBCC 8 GHSERFE

GAl8Fe5D82Al:Fe 8

Hfcc-A1Al 5H bcc-A2Fe

394; 000 36 T 8GALBCC 5GHSERFE

GAl8Fe5D82Fe:Fe 13

Hbcc-A2Fe 13GHSERFE 13; 000

0LAl8Fe5D82Al:Al;Fe 100; 000

0LAl8Fe5D82Al;Fe:Fe 174; 000

bcc 4SL Al;Fe:25Al;Fe:25Al;Fe:25Al;Fe:25

200

400

600

800

1000

1200

1400

1600

1800

2000

Temperature(K)

0 0.1 0.2 0.3 0.4 0.5

Mole fraction Al

A1

A2(pm)

A2(fm)

B2(fm)

B2(pm)

Fig. 13. The phase diagram up to 50 at% Al with the two-sublattice modelfor bcc. The rest is the same as previously. Note that the A2/B2 first-ordertransition now bends towards Al, which is as expected because removing

the D03 phase should extend the A2 region.



12/13

Note that this phase has a contribution from the disor-dered part bcc-A2. Parameters that are identical for sym-metry reason are listed only once.Gbcc-4SLAl:Al:Al:Al ZEROGbcc-4SLAl:Al:Al:Fe GD03ALFE

G

bcc-4SL

Al:Fe:Al:Fe GB32ALFEGbcc-4SLAl:Fe:Fe:Fe GD03FEALGbcc-4SLAl:Al:Fe:Fe GB2ALFEGbcc-4SLFe:Fe:Fe:Fe ZEROTbcc-4SLCAl:Al:Al:Fe: 4 UTCALFE

Tbcc-4SLCAl:Al:Fe:Fe 6 UTCALFE

Tbcc-4SLCAl:Fe:Al:Fe 5 UTCALFE

Tbcc-4SLCAl:Fe:Fe:Fe 4 UTCALFE

bbcc-4SLAl:Al:Al:Fe UBMALFE 3 BM0ALFE

bbcc-4SL

Al:Al:Fe:Fe 2 UBMALFE 4 BM0ALFEbbcc-4SLAl:Fe:Al:Fe UBMALFE 4 BM0ALFE

bbcc-4SLAl:Fe:Fe:Fe UBMALFE 3 BM0ALFE1Lbcc-4SLAl;Fe::: L1ALFE

2Lbcc-4SLAl;Fe::: L2ALFE

1bbcc-4SLAl;Fe::: BM1ALFE

2bbcc-4SLAl;Fe::: BM2ALFE

fcc 4SL Al;Fe:25Al;Fe:25Al;Fe:25Al;Fe:25Note that this phase has a contribution from the disor-

dered part fcc-A1. Parameters that are identical for symme-

try reason are listed only once.Gfcc-4SLAl:Al:Al:Al ZEROGfcc-4SLFe:Al:Al:Al GAL3FEGfcc-4SLFe:Fe:Al:Al GAL2FE2Gfcc-4SLFe:Fe:Fe:Al GALFE3Gfcc-4SLFe:Fe:Fe:Fe ZERO0Lfcc-4SLAl;Fe:Al;Fe:: SFALFE

bcc 2SL Al;Fe:5Al;Fe:5This phase can be used instead of bcc-4SL if D0 3 order-

ing can be ignored. Note that this phase has a contribution

from the disordered part, bcc-A2, and that all permuta-tions of identical parameters are listed.

Gbcc-2SLAl:Al ZEROGbcc-2SLFe:Al 2 GB2ALFEGbcc-2SLAl:Fe 2 GB2ALFEGbcc-2SLFe:Fe ZEROTbcc-2SLCFe:Al 4 UTCALFE

Tbcc-2SLCAl:Fe 4 UTCALFE

bbcc-2SLFe:Al 4 UBMALFE 2 BM0ALFE

bbcc-2SLAl:Fe 4 UBMALFE 2 BM0ALFE0Lbcc-2SLAl;Fe:Al 2 GD03ALFE

0Lbcc-2SLAl:Al;Fe 2 GD03ALFE

0Lbcc-2SLFe:Al;Fe 2 GD03FEAL

0Lbcc-2SLAl;Fe:Fe 2 GD03FEAL

0Lbcc-2SLAl;Fe:Al;Fe 4 AD03ALFE 4 AD03FEAL

1Lbcc-2SLAl;Fe: 2 L1ALFE

1Lbcc-2SL:Al;Fe 2 L1ALFE

2Lbcc-2SL:Al;Fe 2 L2ALFE

2Lbcc-2SLAl;Fe: 2 L2ALFE

1bbcc-2SLAl;Fe: 2 BM1ALFE

1bbcc-2SL:Al;Fe 2 BM1ALFE

2bbcc-2SLAl;Fe: 2 BM2ALFE

2

b

bcc-2SL

:Al;Fe 2 BM2ALFE

Functions

R 8:31451

GALLIQ

298:15 6 T < 933:47 : 11005:029 11:841867 T

7:934 1020 T7 GHSERAL

933:47 6 T < 6000:00 : 10482:382 11:253974 T

1:231 1028 T9 GHSERAL

8>>>>>:

GFELIQ

298:15 6 T < 1811:00 : 12040:17 6:55843 T

3:6751551 1021 T7 GHSERFE

1811:00 6 T < 6000:00 : 10839:7 291:302 T 46 T ln T

8>:

GHSERAL

298:15 6 T < 700:00 : 7976:15 137:093038 T 24:3671976 T ln T

0:001884662 T2 8:77664 107 T3 74092 T1

700:00 6 T < 933:47 : 11276:24 223:048446 T 38:5844296 T ln T

0:018531982 T2 5:764227 106 T3 74092 T1

933:47 6 T < 2900:00 : 11278:378 188:684153 T

31:748192 T ln T 1:230524 1028 T9

8>>>>>>>>>>>>>>>:

GFEFCC

298:15 6 T < 1811:00 : 1462:4 8:282 T 1:15 T ln T

6:4 104 T2 GHSERFE

1811:00 6 T < 6000:00 : 27097:396 300:25256 T

46 T ln T 2:78854 1031 T9

8>>>>>:

GALBCC 10083 4:813 T GHSERAL

GHSERFE

298:15 6 T < 1811:00 : 1225:7 124:134 T

23:5143 T ln T :00439752 T2

5:8927 108 T3 77359 T1

1811:00 6 T < 6000:00 : 25383:581 299:31255 T

46 T ln T 2:29603 1031 T9

8>>>>>>>>>>>:

UBALFE1 4023 1:14 T

UBALFE2 1973 2 TAD03ALFE 3900

AD03FEAL 70 0:5 T

GB2ALFE 4UBALFE1

GB32ALFE 2UBALFE1 2UBALFE2

GD03ALFE 2UBALFE1 UBALFE2 AD03ALFE

GD03FEAL 2UBALFE1 UBALFE2 AD03FEAL

L1ALFE 634 0:68 T

L2ALFE 190

UTCALFE 125

UBMALFE 1:36

BM0ALFE 0:3

BM1ALFE 0:8

BM2ALFE 0:2

UFALFE 4000 T

GAL3FE 3UFALFE 9000



13/13

GAL2FE2 4UFALFE

GALFE3 3UFALFE 3500

SFALFE UFALFE

ZERO 0:0

Appendix B. Supplementary data

Supplementary thermodynamic database file associatedwith this article can be found, in the online version, atdoi:10.1016/j.actamat.2009.02.046.

References

[1] Kattner UR, Burton BP. In: Okamoto H, editor. Phase diagrams ofbinary iron alloys. Materials Park, OH: ASM International; 1993. p.1228.

[2] Seiersten M. SINTEF report STF-28F93051, Oslo, Norway, 1993.[3] Seiersten M. In: Ansara I, Dinsdale AT, Rand MH. editors. COST

507, Thermochemical Database for Light Metal Alloys, vol. 2.Luxembourg: Office for Official Publications of the European

Communities; 1998.[4] Jacobs MHG, Schmid-Fetzer R. Calphad 2009;33:1708.[5] Kerl R, Wolff J, Hehenkamp Th. Intermetallics 1999;7:3018.[6] Stein F, Palm M. Int J Mat Res 2007;98:5808.[7] Du Y, Schuster JC, Liu Z-K, Hu R, Nash P, Sun W, et al.

Intermetallics 2008;16:55470.[8] Ikeda O, Ohnuma I, Kainuma R, Ishida K. Intermetallics

2001;9:75561.[9] Vogel SC, Eumann M, Palm M, Stein F. COST 535 (THALU) final

meeting, Interlaken Switzerland, 2008.[10] Breuer J, Grun A, Sommer F, Mittemeijer EJ. Metall Mater Trans B

2001;32B:9138.[11] Oelsen W, Middel W. Eisenforschung 1937;19:126.[12] Kubaschewski O, Dench WA. Acta Metall 1955;3:33946.[13] Lechermann F, Fahnle M, Sanchez JM. Intermetallics

2005;13:1096109.[14] Jacobson NS, Mehrotra GM. Metall Trans B 1993;24B:4816.[15] Kleykamp H, Glasbrenner H. Z Metallkd 1997;88:2305.[16] Bencze L, Raj DD, Kath D, Oates WA, Herrmann J, Singheiser L,

et al. Metall Mater Trans A 2003;34A:240919.[17] Huang Y, Yuan W, Qiao Z, Semenova O, Bester G, Ipser H. J Alloy

Compd 2008;458:27781.[18] Ferro R. Atti Acad Naz Lincei Rend Classe Sci Fis Mat Nat

1963;36:6538.[19] Radcliffe SV, Averbach BL, Cohen M. Acta Metall 1961;9:16976.[20] Eldridge J, Komarek KL. Trans AIME 1964;230:22633.[21] Lukas HL, Fries SG, Sundman B. Computational thermodynamics.

Cambridge: Cambridge University Press; 2007. ISBN 978-0521868112.

[22] Connetable D, Lacaze J, Maugis P, Sundman B. Calphad

2008;32:36170.[23] Crystal lattice website. Available from: http://cst-www.nrl.navy.mil/

lattice.[24] Thermo-Calc Software AB, version S. 2008. Available from: http://

www.thermocalc.se.

[25] Inden G. Physica 1981;103B:82100.[26] Ohnuma I. 2000 [unpublished work].[27] Kurnakov N, Urasov G, Grigorjew A. Z Anorg Chem

1922;125:20727.[28] Gwyer AGC, Phillips HWL. J Inst Met 1927;38:2983.[29] Isawa M, Murakamim T. Kinsoku-no-Kenkyu 1927;4:46777.[30] Lee JR. J Iron Steel Inst 1960;194:2224.[31] Schurmann E, Kaiser H-P. Arch Eisenhuettenwes 1980;51:3257.[32] Wachtel E, Bahle J. Z Metallkde 1987;78:22932.[33] Wooley F, Elliott JF. Trans AIME 1967;239:187283.[34] Petrushevsky MS, Esin YuO, Geld PV, Sandakov VM. Russ Metall

1972;6:14953.[35] Dannohl H-D, Lukas HL. Z Metallkde 1974;65:6429.[36] Lee JJ, Sommer F. Z Metallkde 1985;76:7504.[37] Mathieu JC, Jounel B, Desre P, Bonnier E. Thermodyn Nucl Mater

Symp Vienna 1967; 1968. p. 76776.[38] Chipman J, Floridis TP. Acta Metall 1955;3:4569.[39] Coscun A, Elliott JF. Trans AIME 1968;242:2535.[40] Mitani H, Nagai H. J Jpn Inst Met 1968;32:7525.[41] Belton GR, Fruehan RJ. Trans AIME 1969;245:1137.[42] Batalin GI, Beloborodova EA, Stukalo VA, Goncharuk LV. Russ J

Phys Chem 1971;45:113940.[43] Wever F, Muller A. Z Anorg Allg Chem 1930;192:33745.[44] Lesnik AG, Skvorchuk YP. Dopovidi Akad Nauk Ukr RSR

1960:140812.[45] Rocquet P, Jegaden G, Petit JC. J Iron Steel Inst 1967;205:43741.[46] Hirano K, Hishinuma A. J Jpn Inst Met 1968;32:51621.[47] Rocquet P, Petir JC, Urbain G. J Iron Steel 1971;209:6970.[48] Roth A. Z Metallkde 1939;31:299301.[49] Crussard C, Aubertin F. Revue Metall 1949;46:66175.[50] Edgar JK. Trans AIME 1949;180:2259.[51] Nishio M, Nasu S, Murakami Y. J Jpn Inst Met 1970;34:11737.[52] Oscarsson A, Hutchinson WB, Ekstrom H-E, Dickson DPE, Simen-

sen CJ, Raynaud GM. Z Metallkde 1988;79:6004.[53] Sykes C, Evans H. J Iron Steel Inst 1935;131:22547.[54] Taylor A, Jones RM. J Phys Chem Solids 1958;6:1637.[55] McQueen HJ, Kuczynski GC. Trans AIME 1959;215:61922.

[56] Lawley A, Cahn RW. J Phys Chem Solids 1961;20:20421.[57] Rassmann G, Wich H. Arch Eisenhuettenwes 1962;33:11522.[58] Davies RG. J Phys Chem Solids 1963;24:98592.[59] Eguchi T, Matsuda H, Oki K, Kioto S-I, Yasutake K. Trans Jpn Inst

Met 1967;8:1749.[60] Swann PR, Duff WR, Fisher RM. Trans AIME 1969;245:8513.[61] Okamoto H, Beck PA. Met Trans 1971;2:56974.[62] Koster W, Godecke T. Z Metallkde 1980;71:7659.[63] Fallot M. Ann Phys Paris 1936;6:30587.[64] Sucksmith W. Proc R Soc A 1939;170:55160.[65] Parsons D, Sucksmith W, Thompson JE. Philos Mag 1958;3:117484.[66] Rimlinger L. C R Acad Sci Paris. 1965;261(Gr.7):40903.[67] Schon CG, Inden G, Eleno LTF. Z Metallkd 2004;95:45963.[68] Gonzales-Ormeno PG, Petrilli HM, Schon CG. Scr Mater

2006;54:12716.

[69] Gross P, Levi DL, Dewing EW, Wilson GL. Phys Chem ProcessMetal I Proc Pittsburgh Symp AIME 1959:40312.[70] D. Connetable. 2006. [unpublished work].[71] Dinsdale AT. Calphad 1991;15:317425.

http://dx.doi.org/10.1016/j.actamat.2009.02.046http://cst-www.nrl.navy.mil/latticehttp://cst-www.nrl.navy.mil/latticehttp://www.thermocalc.se/http://www.thermocalc.se/http://www.thermocalc.se/http://www.thermocalc.se/http://cst-www.nrl.navy.mil/latticehttp://cst-www.nrl.navy.mil/latticehttp://dx.doi.org/10.1016/j.actamat.2009.02.046

Documents

Sundman AlFediagram