zn-fe-al phase diagram low temp

Computer Coupling of Phase Diagrams and Thermochemistry 31 (2007) 125–140www.elsevier.com/locate/calphad

A full thermodynamic optimization of the Zn–Fe–Al system within the420–500 ◦C temperature range

Jinichiro Nakanoa,∗, Dmitri V. Malakhovb,1, Shu Yamaguchic, Gary R. Purdyb,2

a Departement des Sciences des Materiaux et des Procedes, Universite catholique de Louvain, IMAP, Place Sainte Barbe 2, B-1348 Louvain-la-Neuve, Belgiumb Department of Materials Science and Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L7

c Department of Materials Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan

Received 9 May 2006; received in revised form 23 September 2006; accepted 23 September 2006Available online 26 October 2006

Abstract

The Zn–Fe–Al system was assessed by using the CALPHAD technique. The optimization was based on three cornerstones. Firstly,crystallographically consistent sublattice models recently proposed by Nakano et al. for all intermetallic phases in the Zn–Fe system made itpossible to predict which sublattice(s) would be most capable of hosting aluminum. Secondly, a careful analysis of all available phase diagramdata allowed identifying those investigations in which the equilibrium state of the system was unquestionably achieved. Only the results of thosestudies were employed for the assessment. Thirdly, the PARROT module of Thermo-Calc, which was used for the optimization, was fed withan extensive array of accurate and reliable activities of Al. The activities were derived from electromotive forces measured by Yamaguchi et al.in various two- and three-phase regions. In the past, only a fraction of data collected was published; a complete compilation is presented in thiscontribution.

By comparing calculated quantities and their experimental counterparts, it was demonstrated that the thermodynamic model proposed forthe Zn–Fe–Al system could be relied upon within the 420–500 ◦C range, i.e. within the region important for galvanizing and galvannealing. It isdifficult to speculate whether the model remains workable outside this temperature interval since almost all experimental data reported in literaturewere obtained within the limits specified.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Sublattice model; Zn–Fe–Al system; Crystallographic information; Galvanizing; Inhibition layer

1. Introduction

Millions of tons of steel are galvanized and galvannealedannually.3 In the former case, a Zn bath may contain0.15–0.2 wt% of Al; in the latter case aluminum concentrationusually ranges from 0.1 to 0.4 wt%. The intended outcomeof this immersion is the formation of a uniform, adherent,protective coating on the steel surface. Many years of practical

∗ Corresponding author. Tel.: +32 10 47 23 38; fax: +32 10 47 40 28.E-mail addresses: [email protected] (J. Nakano),

[email protected] (S. Yamaguchi).1 Tel.: +1 905 525 9140x24308; fax: +1 905 528 9295.2 Tel.: +1 905 525 9140x24785; fax: +1 905 528 9295.3 Galvanizing baths are maintained within specific ranges of Al levels in

order to suppress the formation of Zn–Fe based phases. The galvannealingprocess adds a subsequent annealing stage designed to control the formationof these phases.

0364-5916/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.calphad.2006.09.003

experience4 clearly indicate that quality of the protectivemulti-layered structure is greatly influenced by many factors,with temperature (usually varying from 450 to 470 ◦C), steelcomposition and Al concentration in a bath being the mostimportant of them [1,2]. The purpose of Al additions to moltenZn is to suppress the formation of a multi-layered coating. Ananalysis of an immense array of experimental evidence suggeststhat thermodynamics per se cannot be used for modeling andoptimizing the hot-dip galvanizing process. Kinetics of steeldissolution in the bath and kinetics of a subsequent phaseformation on the substrate surface is equally if not moreimportant. However, there are essential questions, answers to

4 Although batch galvanizing was patented in 1837, continuous hot-dipgalvanizing was invented by Tadeusz Sendzimir who constructed the firstindustrial-scale galvanizing unit in Poland in 1931.

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http://dx.doi.org/10.1016/j.calphad.2006.09.003

126 J. Nakano et al. / Computer Coupling of Phase Diagrams and Thermochemistry 31 (2007) 125–140

Nomenclaturei aphase

, i bphase, i cphase variables in the i th order interac-

tion parameter of a phaseGphase

m molar Gibbs energy of a phase, J/mol0Gphase

component,0Gphase

end member standard Gibbs energies of acomponent or an end member in a phase (e.g.,lattice stabilities of elements’ phases), J/mol

exGphasem excess molar Gibbs energy of a phase, J/mol

i Lphaseconstituents i th order interaction parameter of a phase

between constituents in a sublattice, J/molR gas constant, J/mol/KT temperature, Kxphase

component mole fraction of a component in a phaseycomponent site fraction of a component

which can be sought on purely thermodynamic grounds. Forexample:

1. What is the upper limit of metastable solubility of Fe in abath of a given composition, i.e. the solubility governed by ametastable equilibrium between the liquid and BCC phases?How does this limit depend on temperature?

2. How much dross (i.e. unwanted particles of intermetallicphases accumulating in a bath) will form from thissupersaturated solution? What is the nature of precipitatingintermetallics?

3. What are the interfacial concentrations of components inadjacent layers of intermetallics assuming that the localequilibrium (stable or metastable) is reached?

4. How is a total amount of Al in a bath distributed betweenintermetallic particles and the liquid phase?

5. What are the driving forces for the onset of precipitation ofvarious solid phases at the steel/melt interface?

If the thermodynamic properties of phases participating ingalvanizing processes are so important, then one might expectthat a self-consistent and reliable thermodynamic model of theZn–Fe–Al system, which is most important for the theory andpractice of galvanizing and galvannealing, has been alreadyproposed. Although attention of many researchers was focusedon this ternary system and especially on its Zn-rich corner[3–10], the whole Zn–Fe–Al system has never been assessedby using the CALPHAD method [11–13]. This system wasconsidered as challenging for optimization due to the followingcircumstances:

1. In contrast to abundant phase diagram data, little reliablethermodynamic information on activities or enthalpies hasbeen published in scientific periodicals.

2. An abundance of experimental data on conditions of phaseequilibria is accompanied by serious contradictions. Notonly were different coordinates of corners of non-varianttriangles reported in the literature, but dissimilar phaseportraits within similarly located triangles were reported aswell.

3. It is well known that all binary phases in the Zn–Fesystem and a majority of binary phases in the Fe–Alsystem can host Al and Zn, correspondingly. An absenceof crystallographically correct sublattice models for thesebinary phases made the application of the CALPHADmethod to the ternary system very difficult.

Recently, the situation has changed for the better. Firstly,a significant and very extensive collection of raw EMFmeasurements of aluminum activity in various two-phaseand three-phase regions (see Appendix) became available.Secondly, it was realized that contradictions between phasediagrams reported in literature could be attributed toinsufficiently long homogenization times, and that the resultsof those studies [7,8,10] in which those times were long enoughare not very dissimilar. Consequently, only these sources wereused in the assessment. It is worth mentioning that only in thesethree investigations the ternary Γ2 phase, which undoubtedlyexists near the Zn-rich corner, was observed. Thirdly, for thecrystallographically consistent models of Zn–Fe intermetallicsproposed in [14], it was possible to point to sublattices capableof accommodating Al. Modifications to the sublattice modelsof Fe4Al13, FeAl2 and Fe2Al5 allowed the description of thesolubility of Zn in these phases. The objective of this work is tobuild a complete thermodynamic description of the Zn–Fe–Alsystem, taking a full advantage of the new situation.

2. Previous thermodynamic modeling of a Zn-rich portionof the Zn–Fe–Al system

As Table 1 indicates, previous endeavors in modeling theZn–Fe–Al system were primarily focused on its Zn-rich corner.In a majority of cases, an approach based on the solubilityproduct was utilized to model the liquidus surface.

3. Phase models

None of the phases was modeled as a stoichiometriccompound in this contribution. All phases were handled assolutions with either one lattice (substitutional solutions) orseveral sublattices. The following reference states were used:HCP for Zn, BCC for Fe, and FCC for Al.

3.1. Substitutional solutions

The thermodynamic models used for describing theproperties of five substitutional solutions in the Zn–Fe–Alsystem are presented in Table 2. In the absence of anyexperimental information about its behavior in the ternarysystem, the model of the Fe4Al5 phase suggested in [15] wasused without any modifications. In particular, it was postulatedthat zinc could not dissolve in this phase.

3.2. Phases with several sublattices

3.2.1. Phases originating from Zn–Fe binary phasesThe Zn–Fe binary system is shown in Fig. 2. All sublattice

models for Γ , Γ1, δ and ζ phases proposed in [14] were claimed

J. Nakano et al. / Computer Coupling of Phase Diagrams and Thermochemistry 31 (2007) 125–140 127

Table 1Previous applications of computational thermodynamics to the Zn–Fe–Al system

T (◦C) Fe (wt%) Al (wt%) Entity modeled Method of modeling Data used Reference

447–480 0–0.08 0–0.5 Liquidus Solubility product [26–29] [33]450–485 0–0.1 0–0.3 Liquidus Solubility product [18] [38]450–465 0–1 0–10 Liquidus Extrapolation from binaries [15,30,31] [35]450–480 0–0.08 0–0.14 Liquidus Solubility product [20] [20]450–480 0–0.06 0.1–0.3 Liquidus CALPHAD [15,20,31,32] [39]450 0–0.04 0–0.4 Liquidus CALPHAD [5,7,8,19,33] [40]450 0–100 0–100 Whole system Extrapolation from binaries [15,31,32] [41]450–480 0–0.06 0–0.25 Liquidus Solubility product [21] [21]450, 460 0–0.04 0–0.4 Liquidus CALPHAD [5,7,8,19,32–35] [42]430–480 0–0.06 0–0.25 Liquidus Kohler’s “geometric” model [23,36,37] [43]

Table 2Models used for substitutional solutions

Phase Model

LIQUID (Al, Fe, Zn)1BCC (Al, Fe, Zn)1(Va)3FCC (Al, Fe, Zn)1(Va)1HCP (Al, Fe, Zn)1(Va)0.5Fe4Al5 (Al, Fe)1

Fig. 1. A calculated 450 ◦C section of the Zn–Fe–Al phase diagram.

to be crystallographically consistent. Based on coordinationnumbers and point symmetries, some of the sublattices wereidentified as potential hosts for aluminum. For instance, thefollowing sublattice model was suggested for the Γ phase [14]:

(Fe, Zn)0.154(Fe, Zn)0.154(Fe, Zn)0.231Zn0.461. (1)

Based on crystallographic data, fragmentary information onsite occupancies and an experimentally seen width of thehomogeneity region demonstrated by the Γ phase, it wasspeculated that the first and the third sublattices would likelyaccommodate aluminum [14]:

(Al, Fe, Zn)0.154(Fe, Zn)0.154(Al, Fe, Zn)0.231Zn0.461. (2)

The total number of end-members in the model (2) is equal to3 × 2 × 3 = 18. Since a total number of end-members in themodel (1) is equal to 2×2×2 = 8, as many as 18−8 = 10 new

Fig. 2. The Zn–Fe phase diagram [14].

Gibbs energies of formation of compounds are to be found inthe course of optimization. In addition to these unknown quan-tities, at least seven interaction parameters should be consid-ered: 0LAl,Fe:∗:∗:Zn,

0LAl,Zn:∗:∗:Zn,0LFe,Zn:∗:∗:Zn,

0L∗:Fe,Zn:∗:Zn,

0L∗:∗:Al,Fe:∗:Zn,

0L∗:∗:Al,Zn:∗:Zn and 0L

∗:∗:Fe,Zn:∗:Zn. It is worthremembering that the Gibbs energies of formation and the in-teraction parameters may depend on temperature and that a totalnumber of adjustable parameters in the expression for the totalGibbs energy of the Γ phase to be evaluated may be forbid-dingly large. This large number of coefficients does not consti-tute a problem if one has representative arrays of reliable ther-modynamic data (activities, enthalpies, etc.) and conditions ofphase equilibria (tie-lines, locations of invariant triangles, etc.)the Γ phase participates in, and if these arrays are mutually con-sistent. In our case, available thermodynamic data are not soaccurate, and agreement between the various phase diagramsproposed is not so perfect that one can hope to derive a largenumber of adjustable coefficients without strong correlationsamong them. But if the model’s parameters are strongly corre-lated, then an alternative model with a fewer number of param-eters should be considered. Let us analyze what happens withthe thermodynamic model for the Γ phase if either the first sub-lattice or the third sublattice (but not both!) hosts aluminum.

Version 1: (Al, Fe, Zn)0.154(Fe, Zn)0.154(Fe, Zn)0.231Zn0.461

(3)


Table 3Sublattice models of ternary phases originating from Zn–Fe binary phases

ΓModel proposed in [14] (Al, Fe, Zn)0.154(Fe, Zn)0.154(Al, Fe, Zn)0.231Zn0.461Model used in this work (Fe, Zn)0.154(Fe, Zn)0.154(Al, Fe, Zn)0.231Zn0.461

Γ1Model proposed in [14]

Fe0.137(Al, Fe, Zn)0.118Zn0.745Model used in this work

δModel proposed in [14] Fe0.058(Al, Fe, Zn)0.180Zn0.525(Al, Zn)0.237Model used in this work Fe0.058(Al, Fe, Zn)0.180Zn0.525Zn0.237

ζModel proposed in [14] (Al, Fe, Va)0.072(Al, Zn, Va)0.072Zn0.856Model used in this work (Fe, Va)0.072(Al, Zn, Va)0.072(Al, Zn)0.856

Version 2: (Fe, Zn)0.154(Fe, Zn)0.154(Al, Fe, Zn)0.231Zn0.461.

(4)

The number of new Gibbs energies of compounds is equal to 4for both models. The models (3) and (4) have the same minimalnumber of interaction parameters. An important distinctionbetween these models is that they predict different upper limitsof the Al solubility: 15.4 at.% for the model (3) and 23.1at.% for the model (4). However, both upper limits by farexceed an experimentally established solubility of Al in theΓ phase (∼1.4 at.% Al [7]). Which model – (3) or (4) –should preference be given to? The assessment of the Zn–Fe–Alsystem was tried with both models. After it had been foundthat the model (4) provided a better match with experimentalquantities, this model was adopted as the model for the Γphase. That choice should not be misinterpreted! It is notclaimed that if the optimization is easier to carry out witha particular phase model, then this model is more physicallyfeasible in comparison with other possible models. Let us recallthat the models (3) and (4) have the same number of adjustableparameters. Although the upper limits of Al solubility predictedby the models are different (see above), this difference isimmaterial because the solubility of aluminum in the Γ phase isonly a few percent. Moreover, a lack of data on site occupanciesdoes not allow one to make a choice: before the optimization,both models are equally reasonable. A quality of assessment isa non-physical criterion, but at least it is a criterion.

The same reasoning was employed for choosing the sublat-tice models for the rest of ternary phases originated from thebinary Zn–Fe phases. These models are presented in Table 3.

A possibility of having two modifications of the δ phase [16]deserves a special experimental inquiry. When and if evidencethat two polymorphic modifications exist, the thermodynamicdescriptions of the Zn–Fe and Zn–Fe–Al systems will need tobe refined. The present assessment was performed assumingthat a single δ phase existed.

3.2.2. Phases stemming from Fe–Al binary phasesAll phases in the Fe–Al system except Fe4Al5 were

“allowed” to dissolve zinc. Consequently, the phase modelssuggested by Seiersten [15] are to be modified. An alterationmade is based on experimental observations that FeAl2 andFe2Al5 are line phases originating from corresponding binaryFe–Al phases and always aiming at the Zn corner of theconcentration triangle [7]. To reflect this behavior, the modelsproposed for FeAl2 and Fe2Al5 in [15] were artlessly modified

Table 4Sublattice models of ternary phases stemming from Fe–Al binary phases

Phase Seiersten’s model Model used in this work

Fe4Al13 Al0.6275Fe0.235(Al, Va)0.1375 Al0.6275Fe0.235(Al, Zn, Va)0.1375FeAl2 Fe1Al2 Fe1Al2(Zn, Va)0.035Fe2Al5 Fe2Al5 Fe2Al5(Zn, Va)3

Fig. 3. The Fe–Al phase diagram [15].

Fig. 4. The Zn–Al phase diagram [31].

by using an additional sublattice accommodating Zn andvacancies (Table 4). The phase diagram of the Fe–Al systemresulting from the Seiersten model is presented in Fig. 3.

As seen from Fig. 4, there are no intermediate phases in theZn–Al system [31].


5 Scientific Group Thermodata Europe (http://www.sgte.org).

3.3. The ternary Γ2 phase

It is pertinent to start this section with addressing aterminological issue. Perrot et al. [7] named the ternary phaseexisting in the Zn–Fe–Al system the Γ2 phase. In contrast,Tang [10] named this phase the T phase, which seems morejustified since the phase is a ternary one and since the phasediscovered in the Zn–Fe–Ni system with the same crystalstructure had been named the T phase. In this work, it wasdecided to use the Γ2 designation for emphasizing that thisternary phase is isomorphic with the binary Γ1 phase in theZn–Fe system. Another reason for giving preference to Γ2is that a crystallographically different ternary phase in theAl–Fe–Si system was referred to as the tau phase [44]. Ifa thermodynamic description of the Zn–Fe–Al–Si system isconstructed, it would be disadvantageous to have two differentphases having the same name.

Although the Γ2 phase is presumably isomorphic with theΓ1 phase, they are always separated by a two-phase region,i.e. they never form a solid solution whatever the temperature(420–500 ◦C). It is justifiable to deem that Γ2 and Γ1 aretwo composition sets of the same phase having a miscibilitygap. Consequently, one can try to construct a single analyticalexpression for the Gibbs energy of this phase instead of twoseparate models, one for Γ2 and another for Γ1. Let us recallthat in the Zn–Fe system, the Γ1 phase was described by themodel Fe0.137(Fe, Zn)0.118Zn0.745 and that an excess term wasnot needed in the expression for its Gibbs energy. An absenceof the excess term means that the Γ1 phase does not tend todecompose. When Γ2 was considered as an individual phaseand when the model (Al, Fe, Zn)0.255Zn0.745 was ascribed to it,optimization results clearly indicated that this phase was alsonot prone to decomposition since all eigenvalues of the Hessianof its Gibbs energies were positive within the xFe − xAl − Tvolume enveloping Γ2. Since the Γ2 and Γ1 phases are situatednot far from each other, it is difficult to model them as twocomposition sets of one phase without making use of absurdlyhuge positive interaction parameter(s). Due to these reasons,the Γ2 phase is considered in the present work as an individualphase, whose sublattice model resembled that of the Γ1 phase.

It should be mentioned that there is no agreement in theliterature as to whether the Γ2 phase is formed in a bathor not. A difference of opinion may be caused by the factthat the composition of the Γ2 phase is not very differentfrom the composition of the δ phase when it dissolves asignificant amount of Al. In other words, an identificationbased on a chemical analysis is not very suitable for specifyingintermetallics presented in the bath. If it is believed that theΓ2 phase does not appear due to kinetic reasons, then it shouldbe excluded from a list of possible phases before startingthermodynamic calculations. For example, the equilibriumdiagram shown in Fig. 14 corresponds to the case when the Γ2phase participates in equilibria. Isothermal sections presentedin Figs. 9–12 are metastable in the sense that the Γ2 phasewas not taken into account. If the formation of this phase iskinetically impeded, then the latter diagrams are more usefulfrom the technological viewpoint.

4. Mathematical model

4.1. Substitutional solutions

The molar free energy of each substitutional phase wasdescribed on the same basis as the binary case [14] except thatin the Zn–Fe–Al system, ternary interaction parameters maybe required. For example, the molar Gibbs free energy for theternary LIQUID phase can be described as

GLIQm = xLIQ

Al0GLIQ

Al + xLIQFe

0GLIQFe + xLIQ

Zn0GLIQ

Zn

+ RT (xLIQAl ln xLIQ

Al + xLIQFe ln xLIQ

Fe

+ xLIQZn ln xLIQ

Zn ) +exGLIQ

m (5)

where

exGLIQm = xLIQ

Fe xLIQZn

n∑i=0

i LLIQFe,Zn(xLIQ

Fe − xLIQZn )i

+ xLIQAl xLIQ

Zn

n∑i=0

i LLIQAl,Zn(xLIQ

Al − xLIQZn )i

+ xLIQAl xLIQ

Fe

n∑i=0

i LLIQAl,Fe(xLIQ

Al − xLIQFe )i

+ xLIQAl xLIQ

Fe xLIQZn LLIQ

Al,Fe,Zn. (6)

The interaction parameter L may be temperature dependent:

i LLIQFe,Zn =

i aLIQ+

i bLIQ T +i cLIQ T ln T . . . . (7)

The lattice stabilities of elements’ phases were taken fromSGTE5 data for pure elements [45]. The order of elementsin (xLIQ

Fe − xLIQZn )i , (xLIQ

Al − xLIQZn )i and (xLIQ

Al − xLIQFe )i must

be treated carefully as the sign of Eq. (6) is affected wheni = odd integer. The ternary interaction parameter LLIQ

Al,Fe,Zncan be expressed as

LLIQAl,Fe,Zn = xLIQ

Al0LLIQ

Al,Fe,Zn + xLIQFe

1LLIQAl,Fe,Zn

+ xLIQZn

2LLIQAl,Fe,Zn (8)

where k LLIQAl,Fe,Zn may be temperature-dependent.

During the later stages of the optimization process, it wasfound that the ternary interaction parameters were not necessaryfor any of the substitutional phases; they are therefore notincluded in the present model.

4.2. Intermediate phases

The same basic procedures described in [14] werefollowed except that additional terms were required for thethird component. For example, with the sublattice model(Fe)0.137(Al, Fe, Zn)0.118(Zn)0.745, the Γ1 phase in Zn–Fe–Alis described as

http://www.sgte.org


GΓ1m = yAl

0GΓ1Fe:Al:Zn +yFe

0GΓ1Fe:Fe:Zn +yZn

0GΓ1Fe:Zn:Zn

+ 0.118RT (yAl ln yAl + yFe ln yFe + yZn ln yZn)

+ yAl yFe

n∑i=0

i LΓ1Fe:Al,Fe:Zn(yAl − yFe)

i

+ yAl yZn

n∑i=0

i LΓ1Fe:Al,Zn:Zn(yAl − yZn)

i

+ yFe yZn

n∑i=0

i LΓ1Fe:Fe,Zn:Zn(yFe − yZn)

i (9)

where the total number of all the sublattice sites is equal tounity.6 The excess terms may or may not be needed dependingon complexity of related equilibria and a model description ofthe phase. This can only be determined during optimization.

The molar Gibbs energies of the rest of the intermediatephases were described in the same fashion. For phases thatwere approximated as stoichiometric (FeAl2 and Fe2Al5), nointeraction parameters were required. It is now worth recallingthat one of the major objectives of this modeling exercisewas to keep the number of interaction parameters as smallas possible by utilizing physical consistency of the model.As demonstrated with the binary assessment in the previouschapter, this is effectively achievable by basing sublatticemodels on crystallographic information.

5. Survey of experimental data used in the optimization

5.1. Phase diagram data

Among all the phase diagram evaluations previouslyreported for the Zn–Fe–Al system [3–10], only three – Perrotet al., Yamaguchi et al. and Tang et al. [7,8,10] – includedthe ternary phase Γ2 at the Zn-rich corner. In each case, theseinvestigators utilized long equilibration times: 1000 h, 1000 hand 360 h, respectively. Since these investigators seem morelikely to attain equilibrium, their works were primarily utilizedfor this study.

Perrot et al. and Tang et al. investigated the entire isothermsat 450 ◦C and 435 ◦C, respectively. Perrot et al. pursuedtheir investigation by immersing Al-containing Fe samples inZn–Al baths. Three-phase equilibria are reported after EnergyDispersive Spectroscopy (EDS) analyses with an error of±0.2 at.% for compositions. Tang et al. carried out their workby synthesizing each alloy of interest. Both two-phase andthree-phase equilibria are reported from EDS analyses. Errorsassociated with compositions are estimated as being from 0.1to 0.8 at.%.

The liquidus has been studied for decades, particularlyin the temperature range relevant to galvanizing practices[17–21]. In the work of McDermid et al. [21], dross particleswere physically filtered out and then the liquid portion waschemically analyzed. Their data were primarily used foroptimization related to liquidus, along with others [17–20].

6 0.137 + 0.118 + 0.745 = 1.

Additional liquidus data [22] with more temperature variations(450–480 ◦C) were also utilized. The composition of thesolution was varied from 0.09 to 0.24 wt% Al and 0.006 to0.042 wt% Fe. The errors in these compositions were estimatedto be 5% of the amount present. It should be noted that thisliquidus was measured after 2 h of equilibration, which wasmuch longer than the typical galvanizing practice (∼3 s) butnot long enough to attain true global equilibrium where theΓ2 phase nucleates in the system. The data were thereforetreated as metastable and Γ2 was suppressed during the relatedoptimization.

5.2. EMF data

Raw EMF data were obtained using the Al sensortechnique. The activities were measured both by varying Alcontent in LIQUID while keeping the temperature, and byvarying temperature while keeping the composition of LIQUIDconstant. The detailed experimental procedures are explainedin the previous reports [8,23]. Phase diagram data were alsomeasured along with activities under the same conditions. Thisset of activities and compositional data is therefore uniquelyvaluable and it was primarily used for optimization related toany equilibrium with LIQUID. EMF and activity values arepresented in Appendix.

6. Optimization: Results and discussion

The optimization was carried out by using the PARROTmodule of Thermo-Calc. It is worth mentioning that statisticalweights assigned to experimental points remained intact in thecourse of assessment.

Since this work is not devoted to particularities andpeculiarities of using the PARROT module of Thermo-Calc, a detailed description of steps leading to a successfuloptimization of the Zn–Fe–Al system would not be appropriate.Instead, let us briefly discuss how activities of aluminumcalculated from EMF data were made use of. Let us recall thatthese activities were measured in either two- or three-phaseregions. For the latter case, an overall composition is immaterialsince aAl remains constant within an invariant triangle. If itis known which three phases are in equilibrium, then thefollowing should be added to a POP file, i.e. the file containingexperimental data:

@@ Example: LIQUID, ZETA@@ and GAMMA2 are in equilibriumCREATE NEW EQUILIBRIUM 1 1CHANGE STATUS PHASE LIQUID = FIXED 1CHANGE STATUS PHASE ZETA = FIXED 1CHANGE STATUS PHASE GAMMA2 = FIXED 1SET REFERENCE STATE AL FCC * 1E5SET CONDITION P=1E5SET CONDITION T=698.75:1EXPERIMENT ACR(AL)=0.0144:0.005

In addition to EMF data, Yamaguchi et al. estimated theconcentration of aluminum in the liquid phase coexisting with


two other phases. This important extra piece of information canand should be reflected:

EXPERIMENT W(LIQUID,AL)=0.094E-2:0.06E-2

The situation becomes quite different in a two-phase region,in which a measurement of the electromotive force is uselessif nothing is known about the compositions of two coexistingphases or about the overall composition of the system. Since theconcentration of aluminum in the liquid phase was evaluated byYamaguchi et al., the following fragment in the POP file reflectsexperimental conditions and results:

@@ Example: LIQUID and ZETA are in equilibriumCREATE NEW EQUILIBRIUM 2 1CHANGE STATUS PHASE LIQUID = FIXED 1CHANGE STATUS PHASE ZETA = FIXED 1SET REFERENCE STATE AL FCC * 1E5SET CONDITION P=1E5SET CONDITION T=698.75:1SET CONDITION W(LIQUID,AL)=0.0005:0.0006EXPERIMENT ACR(AL)=0.0133:0.005

Sometimes, not only Al concentration, but Fe concentrationas well was measured at the phase boundary separating theliquid phase from a two-phase region. While this additionalknowledge cannot be used in setting conditions (for a ternarysystem exactly five conditions are needed), it can be reflectedin an extra EXPERIMENT line.

In the course of the assessment, significant difficulties wereencountered. More specifically, post-optimization artifactswere observed. Since the problem of “unexpected equilibria” isimportant from both methodological and practical angles, it isdeemed pertinent to discuss the complications the optimizationstumbled upon in some detail. The first artifact is an invertedmiscibility gap in the liquid phase predicted by the model at el-evated temperatures. This shortcoming is inherited from a ther-modynamic model for the binary Zn–Fe melt proposed in [14].According to that description, the minimum of an actually non-existing inverted two-phase liquid+liquid cupola is located atxZn ≈ 0.835 and T ≈ 1802 K. Although the presence of thisartifact is awkward, it does serve to caution that the thermody-namic model of the Zn–Fe–Al system presented in this workshould never be used so far beyond the 420–500 ◦C range.

An artifact of another kind was detected when athermodynamic description of the whole Zn–Fe–Al systemwas used for constructing three binary diagrams. It is worthemphasizing that from a precision viewpoint (i.e. from aformal viewpoint), the assessment of the ternary system wasa success: almost all experimental observations had beenaccurately reproduced. In spite of this confirmation of accuracy,the Γ2 phase unexpectedly appeared in the Zn–Fe and Zn–Alsystems. Moreover, at low temperatures, pure Zn having theΓ2 structure was found to be more thermodynamically stablethan Zn possessing the HCP structure! To understand this, letus recall that the sublattice model used for this phase was(Al, Fe, Zn)0.255Zn0.745. Let us also realize that statisticallyoptimal Gibbs energies of three end-members and interactionparameters were derived from experimental information related

exclusively to the interior of the concentration triangle. Itshould also be remembered that experimental data wereobtained within the 420–500 ◦C region. The optimizationprocedure resulted in 0GΓ2

Zn:Zn =0GHCP

Zn:Zn −3967.04 + 6Tas the best temperature dependency of the Gibbs energy ofthe Zn0.255Zn0.745 end-member. With this 0GΓ2

Zn:Zn (T ) andother parameters the procedure led to, various isothermal andisoplethal section can be calculated. Inside the 420–500 ◦Crange, they all look very reasonably and conform well to whatis known from experiments. However, 0GΓ2

Zn:Zn < 0GHCPZn:Zn if

T < 388.02 ◦C, which is absurd. The interaction parameter0LΓ2

Zn,Fe:Zn = −9502.85 suggested by the optimizationprocedure is negative, which implies that Γ2 would possessa noticeable homogeneity region in the Zn–Fe system, i.e.in the system in which it does not exist! After the problemhad been detected and its implications had been realized, thethermodynamic model of the Zn–Fe–Al system was refined. Itis now guaranteed that the only remaining artifact is the invertedmiscibility gap transferred from a thermodynamic descriptionof the liquid phase in the Zn–Fe system.

The optimized parameters for the Zn–Fe–Al system areshown in Table 5. Note that compound energies and interactionparameters not shown in Table 5 are zero. A Thermo-Calccompatible database (a TDB file) that the assessment resultedin is available upon request.

The calculated activities of Al in three-phase regions withrespect to temperature are presented in Fig. 5 along withexperimental measurements. Boundaries between two-phasefields correspond to three-phase equilibria. Evidently, thepresent model accurately reproduces all three-phase equilibriaseen in reality.

EMF values measured in various two-phase regions werereproduced with a great deal of accuracy. The maximal absolutevalue of the weighted deviation is less than 0.12. Weightedresiduals for all EMF measurements (i.e., corresponding to bothtwo- and three-phase fields) are shown in Fig. 6. Although thenegative tail is a bit heavy, the histogram has a characteristicbell shape pointing to the normal distribution of weighteddeviations.

The whole isothermal section of the Zn–Fe–Al system at450 ◦C is computed and shown in Fig. 1. Experimentally foundcoordinates of corners of invariant triangles are compared withcomputed quantities in isothermal sections presented in Figs. 7and 8. In these figures, each calculated corner is marked with aletter. In the figure and in a corresponding legend, experimentalpoints related to a particular corner are shown by using easilydistinguishable symbols. In the legend, these symbols aresituated below the letter used for marking the corner. The closerthe symbols are located to the letter, the better the accuracythe position of the corner is reproduced by the model. Dueto an abundance of experimental data and complexity of theisothermal sections, Figs. 7 and 8 are not easy to analyze.Note that points “evaluated” from experimental phase diagramassessment [10] are also shown in the figures. However, alongwith Table 6, these figures suggest that phase equilibria weredescribed reasonably accurately. However, several exceptionsdeserve explanation.


Table 5Analytical descriptions of the thermodynamic properties of phases in theZn–Fe–Al system (0G and i L are in J/mol, quantities evaluated in this workare in bold)

LIQUID: (Al, Fe, Zn)1

0LLIQFe,Zn = +58 088 − 23.665T

1LLIQFe,Zn = +92 219 − 55.584T

2LLIQFe,Zn = +13 570

0 LLIQAl,Fe = −91 976.5 + 22.1314T a

1 LLIQAl,Fe = −5672.58 + 4.8728T a

2 LLIQAl,Fe = +121.9a

0 LLIQAl,Zn = +10 465.55 − 3.39259T b

BCC: (Al, Fe, Zn)1(Va)3

0LBCCFe,Zn:Va = −10 494 + 18.299T

1LBCCFe,Zn:Va = +15 513 − 12.608T

TcBCCFe:Va = +1043.85

TcBCCFe,Zn:Va = +504.3

0 LBCCAl,Fe:Va = −122 960 + 7.9972T a

1 LBCCAl,Fe:Va = +2945.2a

T cBCCAl,Fe:Va = +504a

FCC: (Al, Fe, Zn)1(Va)1

0LFCCFe,Zn:Va = +6934.7 + 4.212T

1LFCCFe,Zn:Va = +691

0 LFCCAl,Fe:Va = −76 066.1 + 18.6758T a

1 LFCCAl,Fe:Va = +21 164.4 + 1.3398T a

0 LFCCAl,Zn:Va = +7297.48 + 0.47512T b

1 LFCCAl,Zn:Va = +6612.88 − 4.5911T b

2 LFCCAl,Zn:Va = −3097.19 + 3.30635T b

HCP: (Al, Fe, Zn)1(Va)0.5

0LHCPFe,Zn:Va = +12 786

0 LHCPAl,Fe:Va = −106 903 + 20T a

0 LHCPAl,Zn:Va = +18 820.95 − 8.95255T b

1 LHCPAl,Zn:Va = +1000 000b

2 LHCPAl,Zn:Va = +1000 000b

3 LHCPAl,Zn:Va = −702.79b

Table 5 (continued)

Γ : (Fe, Zn)0.154(Fe, Zn)0.154(Al, Fe, Zn)0.231Zn0.461

0GΓFe:Fe:Al:Zn − 0.231 0GFCC

Al − 0.308 0GBCCFe − 0.461 0GHCP

Zn = 0

0GΓFe:Zn:Al:Zn − 0.231 0GFCC

Al − 0.154 0GBCCFe − 0.615 0GHCP

Zn

= −2000 − 3.5T0GΓ

Fe:Zn:Fe:Zn − 0.385 0GBCCFe − 0.615 0GHCP

Zn = −5900.8 + 2.406T

0GΓFe:Zn:Zn:Zn − 0.154 0GBCC

Fe − 0.846 0GHCPZn = −2959.6 − 0.448T

0GΓZn:Fe:Al:Zn − 0.231 0GFCC

Al − 0.154 0GBCCFe − 0.615 0GHCP

Zn = 0

0GΓZn:Zn:Al:Zn − 0.231 0GFCC

Al − 0.769 0GHCPZn = 0

0GΓZn:Zn:Fe:Zn − 0.231 0GBCC

Fe − 0.769 0GHCPZn = +793 + 4.782T

0GΓZn:Zn:Zn:Zn −

0GHCPZn = +6602.65 − 8.157T

0LΓFe:Zn:Al,Fe:Zn = −15 000 − 4T

0LΓFe:Zn:Al,Zn:Zn = −100 − 12T

0LΓFe:Zn:Fe,Zn:Zn = −10 394.77 + 12.1876T

Γ1: Fe0.137(Al, Fe, Zn)0.118Zn0.745

0GΓ 1Fe:Al:Zn − 0.118 0GFCC

Al − 0.137 0GBCCFe − 0.745 0GHCP

Zn = −6900

0GΓ 1Fe:Fe:Zn − 0.255 0GBCC

Fe − 0.745 0GHCPZn = −8609.4 + 5.4T

0GΓ 1Fe:Zn:Zn − 0.137 0GBCC

Fe − 0.863 0GHCPZn = −5089.67 + 1.898T

0LΓ 1Fe:Al,Fe:Zn = −5300

0LΓ 1Fe:Al,Zn:Zn = −2500

Γ2: (Al, Fe, Zn)0.255Zn0.745

0GΓ 2Al:Zn − 0.255 0GFCC

Al − 0.745 0GHCPZn = +7370.454

0GΓ 2Fe:Zn − 0.255 0GBCC

Fe − 0.745 0GHCPZn = 0

0GΓ 2Zn:Zn −

0GHCPZn = +2665.3728

0LΓ 2Al,Fe:Zn = −32 689.0355

0LΓ 2Al,Zn:Zn = −16 069.1316 + 0.781T

0LΓ 2Fe,Zn:Zn = −12 561.3751

0LΓ 2Al,Fe,Zn:Zn = −15 588.2333

δ: Fe0.058(Al, Fe, Zn)0.180Zn0.525Zn0.237

0GδFe:Al:Zn:Zn − 0.18 0GFCC

Al − 0.058 0GBCCFe − 0.762 0GHCP

Zn

= +10 919 − 10.5T0Gδ

Fe:Fe:Zn:Zn − 0.238 0GBCCFe − 0.762 0GHCP

Zn = −3886 + 1.365T

0GδFe:Zn:Zn:Zn − 0.058 0GBCC

Fe − 0.942 0GHCPZn = −3072 + 0.893T

0LδFe:Al,Fe:Zn:Zn = −23 514

0LδFe:Al,Zn:Zn:Zn = −12 317


Table 5 (continued)

1LδFe:Al,Zn:Zn:Zn = −4318

0LδFe:Fe,Zn:Zn:Zn = −5742.666 + 3.755T

ζ : (Fe, Va)0.072(Al, Zn, Va)0.072(Al, Zn)0.856

0GζFe:Al:Zn − 0.856 0GHCP

Zn − 0.072 0GBCCFe − 0.072 0GFCC

Al = +1023−7.6T

0GζFe:Va:Zn − 0.072 0GBCC

Fe − 0.856 0GHCPZn = +700.31 − 2.562T

0GζFe:Zn:Zn − 0.072 0GBCC

Fe − 0.928 0GHCPZn = −3861.9 + 1.152T

0GζVa:Al:Zn − 0.072 0GFCC

Al − 0.856 0GHCPZn = +100

0GζVa:Va:Zn − 0.856 0GHCP

Zn = +14T

0GζVa:Zn:Al = +2000

0GζVa:Zn:Zn − 0.928 0GHCP

Zn = +808.7 − 0.102T

Fe4Al13: Al0.6275(Fe, Zn)0.235(Al, Zn, Va)0.1375c

0GFe4Al13Al:Fe:Zn − 0.6275 0GFCC

Al − 0.235 0GBCCFe − 0.1375 0GHCP

Zn = −23 500

0GFe4Al13Al:Zn:Al − 0.765 0GFCC

Al − 0.235 0GHCPZn = +3147

0GFe4Al13Al:Zn:Va − 0.6275 0GFCC

Al − 0.235 0GHCPZn = +2000

0GFe4Al13Al:Zn:Zn − 0.6275 0GFCC

Al − 0.3725 0GHCPZn = 0

0GFe4Al13Al:Fe:Al − 0.765 0GFCC

Al − 0.235 0GBCCFe = −30 714.3 + 7.44T a

0GFe4Al13Al:Fe:Va − 0.6275 0GFCC

Al − 0.235 0GBCCFe = −27 781.3 + 7.2566T a

Fe2Al5: Fe2Al5(Zn, Va)3c

0GFe2Al5Fe:Al:Zn − 5 0GFCC

Al − 2 0GBCCFe − 3 0GHCP

Zn = −277 947 + 121.95T

0GFe2Al5Fe:Al:Va − 5 0GFCC

Al − 2 0GBCCFe = −228 576 + 48.99503T a

FeAl2: Fe1Al2(Zn, Va)0.035c

0GFeAl2Fe:Al:Zn − 2 0GFCC

Al −0GBCC

Fe −0GHCP

Zn = −96 068 + 16T

0GFeAl2Fe:Al:Va − 2 0GFCC

Al −0GBCC

Fe = −98 096.9 + 18.7503T a

Fe4Al5: (Al, Fe)1c

0GFe4Al5Al −

0GFCCAl = +12 178.9 − 4.813T a

0GFe4Al5Fe −

0GBCCFe = +5009.03a

0 LFe4Al5Al,Fe = −131 649 + 29.4833T a

1 LFe4Al5Al,Fe = −18 619.5a

a Parameters adopted from [15].b Parameters adopted from [31].c Sublattice models adopted or extended from [15].

Firstly, the solubility of Zn in Fe2Al5 slightly exceeds thatexperimentally observed. This can likely be attributed to the

Fig. 5. Calculate activities of aluminum in three-phase regions (lines) withexperimental data superimposed.

Fig. 6. Weighted residuals for EMF measurements in two- and three-phaseregions.

fact that the sublattice model used for this phase is not acrystallographically consistent one. This excessive solubilityhas virtually no effect upon equilibria in the Zn-rich corner ofthe diagram.

Secondly, a homogeneity region of Fe4Al13 has an unusualshape, which, however, is consistent with the Schreinemakers’rule [24].

Thirdly, the BCC position of the invariant δ + Fe2Al5 +

BCC triangle does not conform well to its experimentallyfound location. This is a direct consequence of adoptingthe description of the Fe–Al system proposed in [15]. Abetter match with experiment can hardly be achieved withoutrevisiting the Seiersten’s model, which is beyond the scopeof the present contribution. The present model is focused on


Fig. 7. A zinc-rich portion of a calculated 450 ◦C section with experimentaldata points superimposed.

Fig. 8. A portion of a calculated 450 ◦C section with experimental data pointssuperimposed.

the Zn-rich corner of the Zn–Fe–Al system at galvanizingtemperatures, which has been well reproduced.

It is worth emphasizing that the Γ2 phase behaves exactlyas experimental observations suggest: being an entity whosepresence is crucial for the analysis of phase equilibria in theZn-rich corner at relatively low temperatures, this phase ceasesto exist slightly above 460 ◦C.

Isothermal sections depicted in Figs. 9–12 demonstrate thatliquidus data were reproduced with a great deal of accuracyfrom 450 to 480 ◦C. For comparing experimental and calculatedquantities, it was necessary to disallow the Γ2 phase toparticipate in equilibria. The reason is that Γ2 was not seenin those investigations [18,20–22] that are compared with theresults of present modeling. With the existence of Γ2, the δ

phase is no longer in equilibrium with Fe2Al5 in LIQUID(Fig. 14).

Fig. 9. A near-Zn fragment of a calculated 450 ◦C section with experimentaldata superimposed.

Fig. 10. A zinc-rich corner of a 460 ◦C section with experimental datasuperimposed.

A projection of the liquidus surface with areas of primarysolidification identified is shown in Fig. 13. This figurestipulates that even a small fluctuation of process parameters(e.g., temperature) is important for fine-tuning the galvanizingprocess.

Despite some minor disagreements with phase diagramdata, the thermodynamic description of the Zn–Fe–Al systemproposed in this work and summarized in Table 5 conformsvery well to an overwhelming majority of experimentalobservations. It is worth reiterating that the model reproducesvery accurately activities of Al in two- and three-phasemixtures. Although it has not been proven or even demonstratedthat the model remains workable above 500 ◦C, it is definitelyvalid within a region important for the theory and practice ofgalvanizing and galvannealing. From this perspective, it is notsurprising that the model was successfully used for finding


Table 6Characteristics of invariant equilibria in the Zn–Fe–Al system

Equilibrium Phase (letters in Figs. 7 and8)

Reference Fe (wt%) Al (wt%) T (◦C)

Exp. Cal.

δ-Fe2Al5–BCC

Present 11.3 1.8 450

δ [5] 8.4 ± 1 3.4 ± 1 450

(b) [6] 9.0 ± 0.6 2.8 ± 0.3 450

[7] 10.4 ± 0.2 1.3 ± 0.2 450

Present 35.1 42.4 450

Fe2Al5 [5] 42.9 ± 1 43.73 ± 1 450

(p) [6] 36.0 ± 0.6 43.1 ± 1.1 450

[7] 36.3 ± 0.2 45.0 ± 0.2 450

Present 74.3 25.0 450

BCC [5] 69.1 ± 2 18.9 ± 2 450

(l) [6] 63.7 ± 1.5 34.3 ± 0.6 450

[7] 63.4 ± 0.2 32.39 ± 0.2 450

Γ2–δ–Fe2Al5

Present 7.0 2.8 450

Γ2 Present 6.9 2.8 435

(g) [7] 8.6 ± 0.2 2.9 ± 0.2 450

[10] 8.6 ± 0.4 3.2 ± 0.3 435

Present 7.3 1.5 450

δ Present 7.0 1.3 435

(c) [7] 8.9 ± 0.2 1.7 ± 0.2 450

[10] 10.4 ± 0.4 1.5 ± 0.4 435

Present 33.7 40.7 450

Fe2Al5 Present 33.3 40.3 435

(p) [7] 36.3 ± 0.2 45.0 ± 0.2 450

[10] 33.6 ± 0.5 45.9 ± 0.6 435

BCC–Γ1–δ

BCC Present 74.7 24.7 438

(k) [10] 79.1 ± 0.8 11.2 ± 0.6 435

Γ1Present 15.3 2.5 438

[10] 13.6 ± 0.5 2.8 ± 0.1 435

δ Present 11.3 1.7 438

(a) [10] 10.4 ± 0.4 1.5 ± 0.4 435

BCC–Γ–Γ1BCC Present 85.30 14.1 450

(j) [7] 91.15 ± 0.2 4 ± 0.2 450

Fe4Al13–Fe2Al5–LIQFe4Al13 Present 37.3 54.0 450

(q) [7] 37.6 ± 0.2 54.8 ± 0.2 450

Γ2–Fe2Al5–LIQ

Present 6.6 2.8 450

Γ2 Present 6.3 2.8 435

(i) [7] 6.4 ± 0.2 3.8 ± 0.2 450

[10] 6.9 ± 0.4 3.2 ± 0.3 435

Present 33.6 40.5 450

Fe2Al5 Present 33.2 40.1 435

(p) [7] 36.3 ± 0.2 45.0 ± 0.2 450

[10] 34.8 ± 0.5 44.8 ± 0.6 435

(continued on next page)


Table 6 (continued)

Equilibrium Phase (letters in Figs. 7 and8)

Reference Fe (wt%) Al (wt%) T (◦C)

Exp. Cal.

Γ2–δ–LIQ

Γ2 Present 6.7 2.8 450

(h) [7] 6.6 ± 0.2 2.1 ± 0.2 450

δ Present 6.8 1.4 450

(d) [7] 6.6 ± 0.2 1.8 ± 0.2 450

δ–ζ–LIQ

δ Present 6.7 1.1 450

(e) [7] 6.8 ± 0.2 1.0 ± 0.2 450

ζ Present 5.8 0.6 450

(f) [7] 6.0 ± 0.2 0.8 ± 0.2 450

Present 0.032 0.057 460

LIQ [20] 0.03 ± <0.0010 0.10 ± <0.0020 460

[43] 0.037 ± 5% 0.067 ± 5% 460

δ-Fe2Al5–LIQ (metastable)

Present 0.024 0.137 460

LIQ [20] 0.03 ± <0.0010 0.135 ± <0.0020 460

[43] 0.027 ± 5% 0.136 ± 5% 460

Fig. 11. A comparison of a calculated 470 ◦C section with experimental data.

ways to minimize dross formation in galvanizing baths, for athermodynamic explanation of the inhibition layer breakdownas well as for revealing the effect of the silicon content in thebath on the properties of the final product [25].

7. Conclusion

With the aid of ternary activity data and new information onthe ternary liquidus surface, a thermodynamic description forthe system Zn–Fe–Al has been constructed on the foundation ofthe Zn–Fe binary model [14]. The present model is consideredvalid from 420 to 500 ◦C, the temperature range of mostrelevance to galvanizing practice. Good agreement has been

Fig. 12. A zinc-rich corner at 480 ◦C with experimental data.

demonstrated between the predictions of the model and thebest available experimental data, thereby helping to confirmthe extensibility of the crystallographically-consistent binarymodel to higher order systems.

Acknowledgements

The authors are grateful to Professor Joseph R. McDermidfor providing his unpublished data on liquidus used in theoptimization, and to the referees for their constructive criticismof an earlier version of this manuscript.

Financial support from the McMaster Steel Research Centreis gratefully acknowledged.


Fig. 13. Liquidus projections and primary solidification fields of Zn–Fe–Al (Γ2is suspended).

Fig. 14. A Zn-rich portion of the Zn–Fe–Al phase diagram at full equilibrium(450 ◦C).

Appendix. Experimental values of activity of Al withrespect to FCC

Electromotive forces presented in this appendix weremeasured by using the following concentration cell:

Pure solidAl

Saturated solution ofNaCl in molten AlCl3

Two-phase orthree-phase mixture

A portion of experimental data was reported in [8,23]. Amethod used for estimating the concentration of Al in the liquidphase is explained elsewhere [8]. In Table A.1, experimentalquantities are characterized by the following estimations oferrors: 1T = 1 ◦C , 1WAl = 0.06%, 1EMF = 5 mV. Anerror associated with the calculated activities of aluminum doesnot exceed 0.005.

Table A.1

Temperature Wt% of Al in the liquid phase EMF aAl

ζ + L

423.2 0.080 88.01 0.0123427.7 0.074 90.62 0.0110428.1 0.074 90.72 0.0111432.9 0.080 90.26 0.0117434.9 0.074 92.31 0.0106442.8 0.073 94.45 0.0101443.2 0.032 111.52 0.0044448.1 0.053 102.33 0.0072458.0 0.031 116.02 0.0040458.0 0.052 105.15 0.0066

δ + L

487.1 0.023 130.10 0.0026492.5 0.023 131.55 0.0025475.4 0.039 115.60 0.0047487.3 0.040 118.06 0.0045497.0 0.040 120.51 0.0043471.8 0.049 109.80 0.0059472.3 0.048 110.37 0.0058479.6 0.048 112.17 0.0055482.2 0.048 112.81 0.0055492.1 0.048 115.26 0.0053506.7 0.050 117.95 0.0051464.7 0.058 104.47 0.0072467.5 0.059 104.79 0.0072472.9 0.059 106.09 0.0071477.8 0.060 106.91 0.0070482.5 0.059 108.40 0.0068492.4 0.059 110.79 0.0065502.3 0.058 113.56 0.0061462.6 0.066 101.23 0.0082464.7 0.067 101.41 0.0084467.7 0.067 102.13 0.0083469.9 0.066 102.97 0.0080473.2 0.067 103.43 0.0080477.0 0.067 104.34 0.0079482.2 0.068 105.25 0.0078502.0 0.068 109.94 0.0071451.7 0.079 94.89 0.0104454.5 0.078 95.81 0.0103457.4 0.078 96.49 0.0101467.7 0.077 99.16 0.0095477.4 0.078 101.15 0.0092487.5 0.079 103.23 0.0089447.5 0.088 91.68 0.0119449.7 0.087 92.42 0.0117462.9 0.088 95.22 0.0111482.2 0.089 99.41 0.0103501.6 0.093 102.88 0.0098444.9 0.101 88.24 0.0139444.5 0.101 88.15 0.0138449.7 0.104 88.71 0.0139459.2 0.103 91.05 0.0132468.7 0.104 92.99 0.0128479.8 0.108 94.67 0.0125498.0 0.110 98.33 0.0118

Γ2 + L

427.7 0.108 83.01 0.0162430.3 0.107 83.78 0.0158432.1 0.103 84.95 0.0150435.2 0.103 85.65 0.0149

(continued on next page)


Table A.1 (continued)


δ + Fe2Al5 + L

459.5 0.155 82.51 0.0198459.5 0.148 83.49 0.0189459.5 0.138 84.96 0.0176460.2 0.143 84.36 0.0182462.1 0.151 83.62 0.0191465.1 0.144 85.27 0.0179465.4 0.139 86.09 0.0173469.8 0.139 87.04 0.0169469.8 0.139 87.04 0.0169470.0 0.143 86.48 0.0174470.1 0.143 86.50 0.0174470.8 0.149 85.77 0.0180471.9 0.144 86.74 0.0174479.4 0.150 87.47 0.0175479.6 0.142 88.70 0.0165479.7 0.149 87.68 0.0173480.4 0.142 88.87 0.0165480.5 0.147 88.14 0.0170482.0 0.148 88.32 0.0170482.0 0.154 87.46 0.0177489.4 0.151 89.47 0.0168489.4 0.139 91.28 0.0155489.7 0.150 89.68 0.0167489.9 0.143 90.77 0.0159490.2 0.143 90.83 0.0159490.6 0.146 90.46 0.0162491.6 0.151 89.94 0.0166499.5 0.138 93.63 0.0147499.5 0.150 91.78 0.0160500.0 0.146 92.49 0.0155500.0 0.153 91.45 0.0163500.2 0.154 91.34 0.0163500.5 0.156 91.12 0.0165500.7 0.141 93.41 0.0149501.7 0.150 92.25 0.0158511.1 0.150 94.27 0.0152

δ + Γ2 + L

440.0 0.110 85.38 0.0155440.8 0.104 86.71 0.0146442.8 0.111 85.82 0.0154443.2 0.107 86.67 0.0148443.6 0.112 85.82 0.0154444.1 0.081 92.61 0.0111444.1 0.111 86.11 0.0153445.0 0.112 86.13 0.0153445.4 0.116 85.49 0.0158445.8 0.111 86.49 0.0151446.2 0.113 86.21 0.0154449.3 0.111 87.27 0.0149449.7 0.084 93.15 0.0112450.0 0.119 85.98 0.0159450.1 0.117 86.36 0.0157450.6 0.115 86.83 0.0153450.9 0.122 85.66 0.0162451.0 0.124 85.35 0.0165451.9 0.116 86.93 0.0154452.5 0.115 87.25 0.0152453.0 0.118 86.82 0.0156455.2 0.126 85.93 0.0164455.9 0.130 85.43 0.0169457.2 0.129 85.88 0.0166



459.1 0.138 84.87 0.0176459.9 0.138 85.05 0.0176460.3 0.143 84.38 0.0182462.7 0.095 93.55 0.0120

Fe4Al13 + Fe2Al5 + L

421.6 1.297 32.00 0.2011430.5 1.280 33.62 0.1894431.4 1.222 34.70 0.1800441.0 1.276 35.29 0.1789450.3 1.274 36.74 0.1706460.1 1.170 40.03 0.1493460.8 1.270 38.41 0.1616459.2 1.281 37.99 0.1642460.1 1.341 37.16 0.1712469.7 1.276 39.67 0.1557480.2 1.300 40.87 0.1511480.2 1.251 41.71 0.1454480.2 1.227 42.13 0.1426489.3 1.259 42.96 0.1405499.8 1.258 44.59 0.1341499.8 1.234 45.02 0.1316

Γ2 + Fe2Al5 + L

425.0 0.137 77.63 0.0208425.7 0.137 77.78 0.0208430.7 0.139 78.57 0.0205435.7 0.141 79.36 0.0202439.3 0.142 80.00 0.0200439.5 0.142 80.04 0.0200439.7 0.143 79.94 0.0201440.1 0.140 80.46 0.0197445.6 0.142 81.36 0.0194449.1 0.141 82.26 0.0189450.0 0.141 82.45 0.0188454.2 0.140 83.51 0.0183455.2 0.141 83.58 0.0184

ζ + δ + L

438.5 0.088 89.61 0.0125440.0 0.105 86.34 0.0148440.0 0.104 86.53 0.0146441.2 0.086 90.71 0.0120442.0 0.102 87.38 0.0142443.0 0.104 87.21 0.0144444.0 0.098 88.66 0.0135444.4 0.081 92.68 0.0111445.0 0.103 87.86 0.0141446.1 0.094 90.00 0.0128447.7 0.073 95.60 0.0099449.8 0.070 96.96 0.0094449.8 0.072 96.38 0.0096450.0 0.107 88.19 0.0143450.0 0.099 89.81 0.0133450.0 0.099 89.81 0.0133450.4 0.085 93.07 0.0113453.0 0.067 98.64 0.0088453.2 0.071 97.47 0.0093455.0 0.103 90.11 0.0134455.1 0.081 95.16 0.0105455.1 0.067 99.13 0.0087455.1 0.069 98.52 0.0090455.7 0.065 99.91 0.0085




455.9 0.065 99.96 0.0085456.3 0.069 98.80 0.0089457.7 0.077 96.83 0.0099457.8 0.062 101.41 0.0080458.9 0.063 101.33 0.0080460.3 0.074 98.27 0.0094460.4 0.098 92.37 0.0124460.4 0.068 100.08 0.0086460.4 0.078 97.19 0.0099460.5 0.102 91.55 0.0130460.8 0.058 103.53 0.0073461.8 0.058 103.77 0.0073462.9 0.055 105.16 0.0069463.0 0.072 99.49 0.0090463.2 0.073 99.24 0.0091465.4 0.069 100.96 0.0085465.4 0.053 106.56 0.0066466.4 0.069 101.19 0.0085467.3 0.047 109.58 0.0058467.9 0.064 103.15 0.0078469.6 0.041 113.06 0.0050470.5 0.060 105.15 0.0073471.3 0.055 107.21 0.0066473.0 0.058 106.48 0.0070473.6 0.055 107.76 0.0066475.8 0.052 109.51 0.0061477.2 0.046 112.49 0.0054477.6 0.031 121.11 0.0036482.2 0.024 127.86 0.0028485.0 0.034 121.02 0.0039490.1 0.027 127.39 0.0030

ζ + Γ2 + L

422.1 0.094 84.53 0.0145422.3 0.093 84.79 0.0143425.0 0.095 84.98 0.0144425.6 0.095 85.11 0.0144426.1 0.097 84.81 0.0146428.0 0.096 85.45 0.0143429.6 0.100 84.99 0.0148430.0 0.102 84.68 0.0151430.0 0.096 85.90 0.0142430.0 0.100 85.08 0.0148430.2 0.102 84.72 0.0150430.3 0.097 85.76 0.0143432.2 0.096 86.40 0.0141434.1 0.101 85.80 0.0146439.2 0.106 85.96 0.0149

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zn-fe-al phase diagram low temp

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