Thermodynamic investigation of the galvanizing systems, I: Refinement of the thermodynamic description for the Fe–Zn system

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 433–440

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Thermodynamic investigation of the galvanizing systems, I: Refinement of the thermodynamic description for the Fe–Zn system Wei Xiong a,b , Yi Kong b , Yong Du b,∗ , Zi-Kui Liu c , Malin Selleby a , Wei-Hua Sun b a

Department of Materials Science and Engineering, Royal Institute of Technology, SE-100 44 Stockholm, Sweden

b

State Key Laboratory of Powder Metallurgy, Central South University, 410083, Changsha, PR China

c

Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802, USA

article

info

Article history: Received 16 December 2008 Received in revised form 2 January 2009 Accepted 5 January 2009 Available online 31 January 2009 Keywords: Fe–Zn Galvanizing Thermodynamic description Phase diagram Ab initio calculations

a b s t r a c t The thermodynamic description for the Fe–Zn system was updated using CALPHAD approach. A set of selfconsistent thermodynamic model parameters for this system was obtained by considering the available experimental data. Compared with the previous thermodynamic modeling, the present assessment with fewer parameters shows not only a better agreement with the experiments but also sounder physical meaning. The present CALPHAD modeling coupled with the ab initio calculations were used to predict the enthalpies of formation of the solid phases in the Fe–Zn system. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, there has been great interest in the ternary system Al–Fe–Zn owing to its substantial applications on the galvanizing processes, in which the steel is coated with Zn and Al for corrosion protection. So far, the binary Fe–Zn system has been thermodynamically modeled for seven times [1–7] based on the experimental thermodynamic and crystallographic data. Most recently, Nakano et al. [7] reported their modeling of the Fe–Zn system with the well-established thermodynamic models for the binary compounds. However, their modeling has four shortcomings as discussed below. (i) An inverse miscibility gap exists with a minimumtemperature critical point at 1529 ◦ C as shown in Fig. 1a, similar to the case in the work by Su et al. [6] as shown in Fig. 1b; (ii) the reference Gibbs energy for four of the eight end-members of the compound Γ have not been given any values which means that they are set to zero in an unrecommended way within the compound energy formalism [8]; (iii) the Γ and δ phases are predicted to decompose at the temperatures 247.9 and 77.3 ◦ C, respectively, which is not supported by the experimental data; and (iv) concerning the solidus of the (α Fe) solution above 780 ◦ C based on the

∗

Corresponding author. Tel.: +86 731 8836213; fax: +86 731 8710855. E-mail addresses: [email protected], [email protected] (Y. Du). URL: http://www.imdpm.net (Y. Du).

0364-5916/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2009.01.002

evaluation by Burton and Perrot [9], there are visible discrepancies between the evaluation and experiments, which were mentioned in Ref. [6]. The assessed results [9] rather than the original experimental data [10] were adopted for the optimization by Nakano et al. [7]. In the present work, the thermodynamic models of the compounds proposed by Nakano et al. [7] were adopted but the model parameters are refined to overcome the above shortcomings with the aid of first principles calculations. 2. Literature review Since the literature data on the Fe–Zn system have been reviewed several times [5–7], this part is brief in the present paper, and a summary of the literature is given in Table 1 to facilitate reading. 2.1. Phase diagram data There are eight phases in the Fe–Zn system: the liquid phase, four binary compounds (Γ , Γ1 , δ and ζ ) and three terminal solid solutions ((α Fe), (γ Fe) and (Zn)). The γ -loop was investigated by Budurov et al. [10] and Kirchner et al. [11]. The experiments carried out by Budurov et al. [10] show considerable scatter and unexpected high Zn contents at the highand low-temperature limits. In contrast, the subsequent work by

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W. Xiong et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 433–440 Table 1 Summary of the experimental phase diagram and thermodynamic data in the Fe–Zn system.

Fig. 1a. Calculated phase diagram in the work of Nakano et al. [7].

Property

Ref.

Experimental techniquea

QMb

Zinc solubility in (α Fe)

[12] [13] [50]

Microscopy and XRD Microscopy and HTXRD Microscopy and XRD

+ + +

Ferromagnetic transition temperature

[22]

MA

+

[23] [14]

MA XRD and EPMA

+ +

γ -loop in the Fe–Zn system

[10] [11]

X-ray microanalysis Microprobe analysis

− +

Invariant reactions, phase boundary

[15]

Micrography, XRD, TA and MA

+

Phase boundary

[10] [17,18] [18] [28]

CA XRD and EPMA DTA, EDX and XRD Evaluationc

+ + + −

Invariant reactions

[31]

DSC

+

Gibbs energy of formation

[25]

KEVPM

+

Activity of Zn

[28]

Transportation technique

+

[25] [24] [26]

KEVPM KEVPM CA, XRD, KEVPM

+ + +

[27]

Isopiestic method

+

700, 750 and 800 ◦ C 344 ◦ C, 344 ◦ C 350, 375, 400 and 425 ◦ C 703.5, 757, 793 and 900 ◦ C 1585 ◦ C

Enthalpy of formation

[29,30]

Vapor pressure method

−

[31]

Calorimetry

−

a XRD = X-ray diffraction; HTXRD = High-temperature X-ray diffraction; MA = Magnetic analysis; EPMA = Electron probe microanalysis; TA = Thermal analysis; DTA = Differential thermal analysis; EDX = Energy dispersive X-ray spectroscopy; CA = Chemical analysis; KEVPM = Knudsen effusion vapor pressure measurement. b QM means Quoted Mode, which indicates whether the data are used in the parameter optimization: + used; − not used. c Evaluated from the experimental activity–composition diagrams.

Fig. 1b. Calculated phase diagram in the work of Su et al. [6].

The solidus of the (α Fe) solution above 600 ◦ C was determined by Budurov et al. [10]. The solidus in a phase diagram proposed by Burton et al. [9] differs considerably from that information [10]. The experimental results of the solidus of the (α Fe) solution in equilibrium with the Γ phase were reported in Refs. [10–14], showing consistency among different measurements. Consequently, these experimental data were used in the present work. The liquidus in the Fe–Zn binary was determined by two groups of authors [10,15]. The phase boundaries of the four binary compounds were measured systemically and reported in Refs. [16– 18]. The above-mentioned experiments were considered as the reliable data and were adopted consequently for evaluating model parameters. The solubility of Fe in liquid Zn is 0.029 wt% [19] or 0.030 wt% at 450.8 ◦ C [20], and 0.038 wt% at 460.8 ◦ C [21]. The investigation on the ferromagnetic transition temperature of the (α Fe) phase was performed by Takayama et al. [14], Fallot [22], and Wriedt and Arajs [23]. Their experimental results agree with each other, and were used in the present work. 2.2. Thermodynamic data

Fig. 2. Calculated γ -loop of the Fe–Zn phase diagram.

Kirchner et al. [11] indicated a more reasonable shape of the γ loop as shown in Fig. 2, due to the improved technique of materials characterization.

The activities of Zn below 400 ◦ C were measured by three groups of authors [24–26]. In addition, the Gibbs energy of formation for the phases at 344 ◦ C was measured by Cigan [24] and Gellings et al. [25]. According to the previous evaluation by Su et al. [6], both Gibbs energy data and activities of zinc at 344 ◦ C

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preformed for the enthalpy of formation of the Γ phase as discussed in the next section. Comparisons of the enthalpies of formation for the solid phases among experiment, ab initio calculations and CALPHAD modeling will be discussed further in Section 6. 3. Ab initio calculations

Fig. 3. Calculated Gibbs energy of formation at 344 ◦ C with the experimental data [24,25].

Ab initio calculations within the generalized gradient approximation (GGA) [32], as implemented in the highly efficient Vienna ab initio simulation package (VASP) [33], were carried out to evaluate the enthalpies of formation at 0 K for the Γ phase. The Perdew et al. (GGA-PW91) [34] GGA for the exchange–correlation potential was used for all the calculations, and the valence electrons were explicitly treated by projector augmented plane-wave (PAW) potentials [35]. The atoms were relaxed toward equilibrium until the Hellmann–Feynman forces were less than 10−2 eV Å−1 . A planewave cutoff energy of 400 eV and an energy convergence criterion of 10−5 eV for electronic structure self-consistency were used in the calculations. Brillouin zone integrations were performed using the Monkhorst–Pack [36] k-point meshes scheme, and the total energy differences were converged to within 0.1 kJ (mol of atoms)−1 . Both the unit cell sizes and the ionic coordinates were fully relaxed to find the stable state. The enthalpy of formation, 1Hf (Fex Zny ), is given by the energy of Fex Zny relative to the composition-weighted average of the energies of the pure constituents in their equilibrium crystal structures:

1Hf (Fex Zny ) = E (Fex Zny ) − [xFe E (Fe) + xZn E (Zn)]

(1)

where E (Fex Zny ), E (Fe), and E (Zn) are the energies (per mol of atoms) of Fex Zny and components (Fe and Zn), respectively. Each quantity is relaxed to its equilibrium geometry at zero-pressure. xi (i = Fe or Zn) is the atomic fraction of the component. In addition, the reference states are Bcc_A2 Fe (ferromagnetic) and Hcp Zn (nonmagnetic). 4. Thermodynamic models

Fig. 4. Calculated activity of Zn with the experimental data [24–26].

measured by Gellings et al. [25] are too low, as shown in Figs. 3 and 4, respectively. Besides, the claimed error for the measurement by Gellings et al. [25] is ±2 kJ mol−1 , which is too large to be accepted for assessment. As a consequence, the experimental data reported by Gellings et al. [25] were given relatively low weight in the present assessment. Wriedt et al. [27] and Tomita et al. [28] reported the activities of Zn between 700 and 900 ◦ C. Wriedt et al. [27] focused on the composition range between 0 and 40 at.% Zn, and Tomita et al. [28] paid more attention to the composition range above 40 at.% Zn. Both the experimental values were used in the present work. The measurements of Zn activities at 1585 ◦ C with very low Zn contents were carried out by Stell [29] and Dimov et al. [30]. Since the measurements of the molten Fe–Zn alloys were only up to 0.005 at.% Zn, both the experimental data were used merely for the comparison with the present calculated results. Due to the high vapor pressure of Zn, measurements on enthalpies of formation for solid phases in Zn-containing systems are very difficult. The only enthalpy of formation for solid phases in the Fe–Zn system is −2178 ± 772 J (mol of atoms)−1 at 93.0 at.% Zn for the ζ phase from Feutelais et al. [31]. In the present work, this datum was used to verify the present description but was not used in the optimization. In view of the difficulties of the calorimetry measurement for Zn alloys, Density Functional Theory (DFT) calculations were

The reported crystallographic information for the binary compounds is given in Table 2 [37–39]. Among the previous thermodynamic descriptions [1–7], thermodynamic models of the intermetallics were well established by Nakano et al. [7] and adopted in the present work. 4.1. Unary phases ϕ

ϕ

The Gibbs energy function 0 Gi (T ) = Gi (T ) − HiSER for element i (i = Fe or Zn) in the phase φ (φ = (α Fe), (γ Fe), (Zn) or liquid) is described by an equation of the form 0

ϕ

Gi (T ) = a + b · T + c · T · ln(T ) + d · T 2 + e · T −1

+ f · T 3 + g · T 7 + h · T −9

(2)

HiSER

in which is the molar enthalpy of the element i at 298.15 K and 1 bar in its Stable element reference (SER) state, and T is the absolute temperature. The last two terms in Eq. (2) are used only outside the ranges of stability [40], the term g · T 7 for a liquid below the melting point, and h · T −9 for solid phases above the melting point. In the present modeling, the Gibbs energies as functions of temperature are taken from the SGTE compilation by Dinsdale [41]. 4.2. Solution phases The liquid phase and terminal solid solutions are described by a substitutional solution model:

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Table 2 Crystallographic data of the binary compounds in the Fe–Zn system. Phase

Pearson symbol

Γ Γ1 δ ζ

Space group

Structure type

¯ I 43m ¯ F 43m P63 /mmc C2/m

cI52 cF408 hP556 mC28

ϕ

Cubic Cubic Hexagonal Monoclinic

ϕ

Gϕm = xFe · 0 GFe + xZn · 0 GZn + RT · (xFe ln xFe + xZn ln xZn ) ex

ϕ

+ Gm +

mag

ϕ

Gm

(3) ϕ

where xi is the mole fraction of element i (i = Fe or Zn) and 0 Gi the Gibbs energy of element i with the state of ϕ . The third term in Eq. (3) corresponds to the contribution to Gibbs energy from the ideal entropy of mixing, the fourth term to the excess Gibbs energy, and the last term to the magnetic contribution to the Gibbs energy. The excess Gibbs energy is described by the Redlich–Kister polynomial [42]: ex

Gϕm = xFe · xZn



ϕ 0

aFe,Zn + ϕ b0Fe,Zn · T

+ (xFe − xZn )




ϕ 1

aFe,Zn + ϕ b1Fe,Zn · T




.

(4)

The parameters denoted as ϕ Lij = ϕ ai,j + ϕ bi,j · T (i, j = Fe or Zn) are the interaction parameters for the binary system. mag ϕ Gm is equal to zero in the description of liquid, (γ Fe) and (Zn). For (α Fe), there is a magnetic contribution to Gibbs energy. This magnetic contribution is given by the following equation: mag

Gϕm = RT · ln (β ϕ + 1) g (τ ϕ )

(5)

where τ is T /T ∗ , T ∗ being the critical temperature for magnetic ordering (Curie temperature Tc for ferromagnetic materials or Néel Temperature TN for antiferromagnetic materials), and β is the average magnetic moment per atom expressed in Bohr magnetrons. The polynomial g (τ ) has been defined by Hillert and Jarl [43]. Being consistent with the description by Nakano et al. [7], in the present modeling, the magnetic transition temperature TC for the phase (α Fe) is considered as the function of the Zn content as follows: TC = T C Fe (1 − xZn ) + ex TC Zn,Fe (1 − xZn ) xZn .

(6)

The magnetic moment β can be expressed in the same way. The excess term in Eq. (6) was determined by using the experimental results from the work of Wriedt and Arajs [23].

The thermodynamic models of the intermetallic compounds are the same as the ones proposed in the work due to Nakano et al. [7] as shown in Table 3. All the four binary compounds are described by using the sublattice model [8,44]. Taking the ζ phase for example, its Gibbs energy per mol formula unit is expressed as: SER SER Gζm − 0.072 · HFe − 0.928 · HZn

ζ

ζ

0 0 000 0 = y0Fe y000 Zn GFe:Zn:Zn + yFe yVa GFe:Zn:Va

ζ

Ref. b/β

c /γ

8.994/90◦ 17.963/90◦ 12.787/120◦ 13.410/90◦

– – – 7.606/127◦ 180

– – 57.222/90◦ 5.076/90◦

[37] [38] [37] [39]

Table 3 Summary of the evaluated thermodynamic parameters of the Fe–Zn system* . Liquid: Model (Fe, Zn)1 0 Liq LFe,Zn = 20696.5073 1 Liq LFe,Zn 2 Liq LFe,Zn

= 14782.0192 − 8.9768 · T = −11266.6992 + 7.3942 · T

(γ Fe): Model (Fe, Zn)1 (Va)1 = 11774.2521 + 0.8668 · T

0 fcc LFe,Zn:Va

(αFe): Model (Fe, Zn)1 (Va)3 = −1684.7191 + 9.9211 · T = 8461.5520 − 5.6064 · T = 504.3

0 bcc LFe,Zn:Va 1 bcc LFe,Zn:Va bcc ex TcFe ,Zn:Va

(Zn): Model (Fe, Zn)1 (V a)0.5 0 hcp LFe,Zn:Va = 23601.6208

Γ : Model(Fe, Zn)0.154 (Fe, Zn)0.154 (Fe, Zn)0.231 (Zn)0.461

0 GΓFe:Fe:Fe:Zn − 0.539 · 0 Gfcc Fe − 0.461 · GZn = 0 0 hcp GΓZn:Fe:Fe:Zn − 0.385 · 0 Gfcc − 0 . 615 · GZn = 0 Fe 0 Γ 0 fcc 0 hcp GFe:Zn:Fe:Zn − 0.385 · GFe − 0.615 · GZn = −5883.5577 + 3.0932 · T 0 Γ 0 hcp GZn:Zn:Fe:Zn − 0.231 · 0 Gfcc Fe − 0.769 · GZn = 4689.6066 0 Γ 0 hcp GFe:Fe:Zn:Zn − 0.308 · 0 Gfcc − 0 . 692 · GZn = 0 Fe 0 Γ 0 hcp GZn:Fe:Zn:Zn − 0.154 · 0 Gfcc − 0 . 846 · GZn = 0 Fe 0 Γ 0 hcp GFe:Zn:Zn:Zn − 0.154 · 0 Gfcc Fe − 0.846 · GZn = −2661.6129 hcp 0 Γ GZn:Zn:Zn:Zn − 0 GZn = 4830.0580 − 6.9361 · T 0 Γ LFe:Zn:Fe,Zn:Zn = −8619.0736 + 8.7279 · T hcp

0 0

Γ 1 : Model Fe0.137 (Fe, Zn)0.118 Zn0.745 0

hcp

0 GFe1:Fe:Zn − 0.255 · 0 Gfcc Fe − 0.745 · GZn = −6804.9287 + 3.2482 · T 0 0 hcp GFe1:Zn:Zn − 0.137 · 0 Gfcc − 0 . 863 · GZn = −3507.8523 Fe 0 01 LFe:Fe,Zn:Zn = −2448.0226 + 4.3133 · T 0

0

δ: Model Fe0.058 (Fe, Zn)0.18 Zn0.525 Zn0.237 0 δ 0 hcp GFe:Fe:Zn:Zn − 0.238 · 0 Gfcc Fe − 0.762 · GZn = −2658.8743 0 δ 0 hcp GFe:Zn:Zn:Zn − 0.058 · 0 Gfcc − 0 . 942 · GZn = −2098.9962 Fe 0 δ LFe:Fe,Zn:Zn:Zn = −6566.3394 + 5.1381 · T ζ : Model (Fe, Va)0.072 Zn0.856 (Zn, Va)0.072 0 ζ 0 hcp GFe:Zn:Va − 0.072 · 0 Gfcc Fe − 0.856 · GZn = 980.4046 0 ζ 0 hcp GVa:Zn:Va − 0.856 · GZn = 81.0806 0 ζ 0 hcp GFe:Zn:Zn − 0.072 · 0 Gfcc Fe − 0.928 · GZn = −2722.7359 0 ζ 0 hcp GVa:Zn:Zn − 0.928 · GZn = 763.0428 * In J (mol of atoms)−1 ; temperatures (T ) in kelvin. The Gibbs energies for the pure elements are from the SGTE compilation [40].

4.3. Intermetallic compounds

0

Lattice parameters (Å) a/α

ζ

0 0 000 0 + y0Va y000 Zn GVa:Zn:Zn + yVa yVa GVa:Zn:Va

+ 0.072 · RT · y0Fe ln y0Fe + y0Va ln y0Va




000 000 000 + 0.928 · RT · y000 Zn ln yZn + yVa ln yVa




0 + y0Fe y0Va y000 Zn · LFe,Va:Zn:Zn 0 0 000 000 0 + y0Fe y0Va y000 Va · LFe,Va:Zn:Va + yFe yZn yVa · LFe:Zn:Zn,Va

000 0 + y0Va y000 Zn yVa · LVa:Zn:Zn,Va + · · ·

(7)

in which y0Fe and y0Va are the site factions of Fe and vacancy on the 000 first sublattice, and y000 Zn and yVa on the third sublattice. The four ζ

parameters denoted 0 G∗:∗:∗ (* means the species, Fe, Zn or vacancy), which represent the Gibbs energies of the end-members of the ζ phase, are expressed relative to the Gibbs energy of (γ Fe) and (Zn) at the same temperature. The L0 parameters represent the interactions primarily within each sublattice. It should be noted that, the numerical values of the Gibbs energy for the end-member compounds can only be given relative to some standard states of the components [8]. As an example, the Thermo-Calc software interprets all numerical values of Gibbs energy as G − H SER . Nakano et al. [7] modeled the Γ phase as (Fe, Zn)0.154 (Fe, Zn)0.154 (Fe, Zn)0.231 (Zn)0.461 and four end-members, 0 GΓFe:Fe:Fe:Zn , 0 GΓZn:Fe:Fe:Zn , 0 GΓFe:Fe:Zn:Zn , and 0 GΓZn:Fe:Zn:Zn , that should have only small effects on the solubility of the phase, were given the value Zero. However, Thermo-Calc then interpreted

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Fig. 5. Calculated phase diagram compared with various experimental data [10–18,22,23,28,50].

this as G = H SER , which is certainly not what the authors intended. We have instead applied the values GFex Zny − H SER = 0 + x · 0




SER Gfcc +y· Fe − HFe



0

Hcp



SER GZn − HZn . Nakano et al. [7] treated the

Table 4 The number of the assessed parameters in this work and the assessment by Nakano et al. [7]. Phase

Γ and ζ phases in the same way in another assessment of the Al–Fe–Zn system [45], and should be reassessed. 5. Optimization strategy The PARROT module [46] in Thermo-Calc was employed for the current optimization. At the first stage of the optimization, only the experimental data on the phase diagram and the Gibbs energy of formation of the solid phases were considered. Since the description of the invariant reaction, L + (α Fe) = Γ , has a significant effect on the description of the phase diagram, this reaction was assessed at the beginning of the first stage. In contrast, the γ -loop was assessed in the final step of the first stage, since it can be easily adjusted due to its limited temperature and composition ranges. In view of the experimental uncertainty of the alloy preparation and thermal analysis, the discrepancies of the phase transition temperatures between the current optimization and the adopted experimental data were kept in 5 K for the assessment. After achieving an overall fit to the phase diagram and the Gibbs energies of formation for the solid phases, the measurements of the activities were then considered to perform a refinement of the preliminary optimization as the second stage. It is noteworthy that the non-existence of the inverse miscibility gap in the liquid phase has been checked at each stage of the optimization. At the final stage of the optimization, all of the selected experiments were used in order to get the self-consistent set of thermodynamic parameters. 6. Results and discussion Table 3 lists the evaluated parameters according to the present work. Compared with the previous work by Nakano et al. [7], there are eight parameters less in the present work, see Table 4. The liquid phase parameters were carefully checked during the optimization in order to avoid the occurrence of the inverse miscibility gap. Fig. 5 shows the calculated phase diagram, compared with the experimental data. It is demonstrated that the present set of parameters can well describe all the accepted experiments of the phase diagram. Because the solidus by Burton et al. [9] is not from experiments, it is not included in Fig. 5.

Liquid (γ Fe) (α Fe) (Zn)

Γ Γ1 δ ζ In total

Number of the assessed parameters Nakano et al. [7]

This work

5 3 5 1 10 4 6 8 42

5 2 5 1 8 5 4 4 34

A comparison between Figs. 1a, 1b and 5 shows some significant improvements of the present modeling, in comparison with the previous modeling. Firstly, the inverse miscibility gap is eliminated in the revised modeling. Secondly, the experimental phase boundary between the (α Fe) and liquid phases is better described by using the present parameters. As recommended [47], during the thermodynamic assessment, the original experimental data rather than the evaluated data should always be given the highest priority, especially when, as in the present case, distinct discrepancies exist between evaluation and experiments. Thirdly, the Γ and δ phases are stable down to 0 ◦ C, which is more reasonable. The calculated γ -loop in the Fe–Zn system is enlarged in Fig. 2. The current assessment can well describe the experimentally determined shape by Kirchner et al. [11], who constructed a more reasonable shape of the γ -loop than the one by Budurov, et al. [10]. Fig. 6 shows the calculated Zn-rich corner of the Fe–Zn phase diagram with the experimental data superimposed. The calculated phase diagram agrees well with the experimental data [19–21]. The comparison of the invariant reactions among calculations [6, 7], experiments [49] and evaluations [9,48] are listed in Table 5. The present assessment can well describe all of the invariant reactions in the Fe–Zn binary system. The calculated Gibbs energies of formation for the solid phases together with the measured results by Cigan [24] and Gellings et al. [25] are presented in Fig. 3, showing good agreement with the experimental data by Cigan [24] as well. The calculated activities of Zn at 344, 400, 700, 750, 800, 900 and 1585 ◦ C, are shown in Figs. 4 and 7–12 together with the experimental results. An overall satisfactory agreement was obtained between the present calculation and experiments. Additionally, the comparisons of two sets of the activity data (344

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Table 5 Comparison of the invariant reactions among the calculations, experiments and evaluations. T (◦ C)

Reaction

Type

Reference

Type

at.% Zn

(α Fe) + L = Γ

Peritectic

This work [7] [6] [18] [48] [9] This work [7] [6] [15] [18] [48] [9] This work [7] [6] [18] [48] [9] This work [7] [6] [15] [18] [48] This work [7] [6] [49] [15] [18] [48]

H H H N

42.3 41.8 41.5

92.0 91.8 91.2

72.6 72.1 73.2



42 42 82.7 82.5 80.7

92 92 97.3 96.4 96.4

72 72 85.6 86.6 86.9

82.5 82.5 76.8 76.6 77.8

97.5 96.5 85.6 86.5 86.9

86.5 86.5 81.0 80.9 80.1

76.5 76.5 91.4 91.4 91.4

86.5 86.5 99.85 99.6 99.6

81 81 93.0 92.8 92.9

92 99.9981 99.95 99.9878 99.979

99.5 93.9 94.4 92.9

92.7 99.9886 99.9899 ∼ 100 99.997

99.989

94

∼ 100

Γ +L=δ

Γ + δ = Γ1

Peritectic

Peritectoid

δ+L=ζ

Peritectic

L = ζ + (Zn)

Eutectic (Degenerated invariant equilibrium)

H H H N N

H H H N

H H H N N

H H H N N N

782 782 782 782 782 782 668 672 672 672 672 665 672 550 550 550 550 550 550 526 530 530 530 530 530 419.48 419.50 419.46 419.40 419.40 419.40 419.53

H stands for calculation, N stands for experiment, stands for evaluation.

Fig. 6. Calculated Zn-rich corner of the Fe–Zn phase diagram with the experimental data [19–21,49].

Fig. 7. Calculated activity of Zn at 400 ◦ C with the experimental data [26].

and 400 ◦ C) between the present modeling and experiments need to be interpreted in detail. The reported experimental data of the activities of Zn at 344 ◦ C [24–26] are very scattered. Besides there is no detailed information on the experimental procedure in Ref. [25]. Thus the reliability of this experiment cannot be judged properly. During the optimization, we found that any attempt to fit the data [26] at both 344 and 400 ◦ C will cause significant upward deviation from the measured Gibbs energy of formation of the solid phases at 344 ◦ C [24,25]. Although the experiments on the determination of the activities of zinc at 1585 ◦ C by Stell [29] and Dimov et al. [30] were not used for the optimization, the present calculation shows a good fit to the experiments by Stell [29] as shown in Fig. 12.

Using the present thermodynamic parameters, the enthalpies of formation of the solid phases in the Fe–Zn system are calculated and presented in Fig. 13. The enthalpy of formation of the Γ phase was predicted by using the VASP code based on the available crystallographical information [37]. The obtained enthalpy of formation of the stoichiometric Γ phase is −3540 J (mol of atoms)−1 . The present CALPHAD modeling gives a minimum value of −4480 J (mol of atoms)−1 at 75 at.% Zn. The discrepancy could be due to the disordering of the Γ phase at finite temperatures. Since the present ab initio calculation is based on perfect crystalline structure and at zero kelvin, while the CALPHAD approach is based on sublattice model and at finite temperature.

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Fig. 8. Calculated activity of Zn at 700 ◦ C with the experimental data [23,28].

Fig. 11. Calculated activity of Zn at 900 ◦ C with the experimental data [23].

Fig. 9. Calculated activity of Zn at 750 ◦ C with the experimental data [23,28].

Fig. 12. Calculated activity of Zn at 1585 ◦ C with the experimental data [29,30].

Fig. 10. Calculated activity of Zn at 800 ◦ C with the experimental data [23,28].

Such a kind of discrepancy leads to the deviation of the results from ab initio calculation to CALPHAD approach. The CALPHAD model predicted enthalpy of formation of the ζ phase at 93 at.% Zn is −1966 J (mol of atoms)−1 , which is close to the measured value of −2178 ± 772 J (mol ofatoms)−1 by Feutelais

Fig. 13. Calculated enthalpy of formation for the solid phases of the Fe–Zn system at 25 ◦ C.

et al. [31] and the ab initio results (−2454 J (mol of atoms)−1 ) in this work. It is observed that in Fig. 13, the enthalpy of formation by Nakano et al. [7] is significantly more negative.

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W. Xiong et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 433–440

Schmid-Fetzer (TU Clausthal, Germany) and Dr. Xing-Qiu Chen (Oak Ridge National Laboratory, USA). Authors are grateful to Professor Mats Hillert (Royal Institute of Technology, Sweden) for a critical review of this paper. References

Fig. 14. Calculated excess entropy of mixing of the liquid phase at 1600 ◦ C.

The comparison of the predicted excess entropy of mixing of the liquid phase at 1600 ◦ C from several calculations [6,7] is shown in Fig. 14. Those calculations gave strongly positive values on the Fe-rich side and a very strong curvature close to the Zn-rich side. Those results were not supported by any experimental information and they were not based on any physical effect included in the mathematical model. They are simply the results of extrapolation from the fitting to information at lower temperatures. Such results are not uncommon in assessment and are often unnoticed or have been accepted as harmless because they concern ranges of the phase diagram that are without practical interest. However, the present assessment has given very low values and demonstrates that such effects of extrapolation can be easily avoided. 7. Conclusions – The Fe–Zn phase diagram was remodeled by using fewer thermodynamic parameters compared with previous works. The undesirable inverse miscibility gap was avoided in the revised thermodynamic description. – The original experimental results were carefully considered and utilized for the current modeling, which generate a better description for the Fe–Zn system compared with the previous calculations. – Both ab initio calculated results and the experimental data show that the thermodynamic properties can be well predicted by using the present thermodynamic parameters. Acknowledgments This research work is supported by Creative Research Group of National Natural Science Foundation of China (Grant no. 50721003) and National Natural Science Foundation of China (Grant no. 50831007). Wei Xiong acknowledges the financial support from Chinese Scholarship Council (Grant no. 2008637018) and Royal Institute of Technology in Sweden. Zi-Kui Liu would like to acknowledge the partial support from the National Science Foundation through Grant DMR-0510180. Yong Du and Zi-Kui Liu also acknowledge Cheung Kong Chair Professorships by Ministry of Education of China for financial support. In addition, Wei Xiong is grateful to the helpful discussion with Prof. Dr. Rainer

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