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Thermodynamic calculation of phase equilibria in the Al–Fe–Zn–O system

  • Naoki Matsumoto and Tatsuya Tokunaga EMAIL logo
Published/Copyright: December 6, 2022
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High Temperature Materials and Processes
From the journal High Temperature Materials and Processes Volume 41 Issue 1

Abstract

The thermodynamics of the phase equilibria in the Al–Fe–Zn–O quaternary system was studied using the calculation of phase diagrams method to understand the oxidation behavior of the Zn bath surface during galvanizing process. The thermodynamic parameters for the Gibbs energies of the different constituent phases in the binary and ternary systems relevant to this quaternary system were taken mainly from previous studies. In this study, the thermodynamic assessment of the Al2O3–ZnO system was carried out based on the available experimental data, and some modifications to the thermodynamic model and/or parameters for the Fe–Zn–O ternary system were made to maintain consistency with the thermodynamic descriptions of other binary and ternary systems, making up the Al–Fe–Zn–O quaternary system adopted in this study. The calculated results on the ternary and quaternary systems generally agreed with the available experimental results on phase equilibria. The set of thermodynamic parameters enabled us to calculate the phase equilibria in the Al–Fe–Zn–O quaternary system over the entire composition and temperature ranges.

Keywords: calculation of phase diagrams; hot-dip galvannealed steel; phase equilibria; Al–Fe–Zn–O system; oxide-type dross

1 Introduction

Hot-dip galvannealed (GA) steel sheets with a Zn–Fe coating from hot-dip galvanizing and subsequent annealing have been widely applied for automobile body panels because of their high corrosion resistance, formability, weldability, and paintability [1]. A galvanizing bath is composed of mainly Zn and contains a small amount of Al ranging from 0.1 to 0.3 mass% to suppress the formation of brittle Zn–Fe-based intermetallic compounds formed between the Zn coating and the steel substrate during the galvanizing process [1]. In addition, undesired intermetallic compounds called dross are formed in the bath during galvanizing [2]. To maintain the mechanical properties and formability of GA steel sheets properly, it is important to control the composition and temperature in the galvanizing zinc bath. Information on the phase equilibria in the bath can be obtained from an experimental Zn–Al–Fe ternary phase diagram, which has been applied practically to the galvanizing process [3]. The calculation of phase diagrams (CALPHAD) approach [4,5] is recognized as a powerful tool for material and process development because this approach provides information on various thermodynamic properties, such as the activities of the constituents and the driving force for phase formation in addition to stable and metastable phase equilibria. Therefore, thermodynamic assessments of the Zn–Al–Fe ternary system including the Zn–Fe binary system were conducted previously [6,7]. However, the bath is exposed to atmospheres with different oxygen partial pressures p O 2 ; therefore, information on the formation of oxide-type dross on the bath surface in addition to the formation of intermetallic compound-type dross in the bath may be necessary. However, information on the phase equilibria concerning the formation of oxides is not sufficient.

In this study, we conducted a thermodynamic assessment of phase equilibria in the Al–Fe– Zn–O quaternary system using the CALPHAD approach and developed a thermodynamic database for calculations of the phase equilibria in the Al–Fe–Zn–O quaternary system over the entire composition and temperature ranges.

2 Calculation procedure

The thermodynamic analysis of the Al–Fe–Zn–O quaternary system was performed using the CALPHAD approach [4,5]. The thermodynamic data for the pure elements were taken from the Scientific Group Thermodata Europe (SGTE) database [8]. The thermodynamic parameters for each phase were mainly taken from previous analyses for Al–O [9,10,11], Fe–O [12,13,14,15], Al–Fe–Zn [7], Al–Fe–O [16], and Fe–Zn–O [17] systems. The adopted parameters are listed in Table 1, together with the parameters assessed in this study. The calculations of phase equilibria in the Al–Fe–Zn–O quaternary system were performed using CaTCalc, a commercial thermodynamic analysis software [18].

Table 1

Thermodynamic parameters for the Al–Fe–Zn–O quaternary system (in units: J, mole, and K)

Liquid: (Al3+, Fe2+, Zn2+) p (O2–, Va, AlO1.5, FeO1.5) Q Ref.
G Al 3 + : O 2 − Liquid 0 − 2 H Al SER 0 − 3 H O SER 0 GLAL2O3 + 1,000,000 [10]
G Fe 2 + : O 2 − Liquid 0 − 2 H Fe SER 0 − 2 H O SER 0 4GFEOLIQ [12]
G Zn 2 + : O 2 − Liquid 0 − 2 H O SER 0 − 2 H Zn SER 0 2GZINCITE + 140,000 − 62.2222222T [19]
G Al 3 + : Va Liquid 0 − H Al SER 0 GALLIQ [8]
G Fe 2 + : Va Liquid 0 − H Fe SER 0 GFELIQ [8]
G Zn 2 + : Va Liquid 0 − H Zn SER 0 GZNLIQ [8]
G Al O 1 . 5 Liquid 0 − H Al SER 0 − 1 . 5 H O SER 0 0.5GLAL2O3 [9]
G Fe O 1 . 5 Liquid 0 − H Fe SER 0 − 1 . 5 H O SER 0 −89,819 + 39.962T + 2.5GFEOLIQ [15]
L Al 3 + , Fe 2 + : Va Liquid 0 −91976.5 + 22.1314T [30]
L Al 3 + , Fe 2 + : Va Liquid 1 −5672.58 + 4.8728T [30]
L Al 3 + , Fe 2 + : Va Liquid 2 121.9 [30]
L Al 3 + , Zn 2 + : Va Liquid 0 10465.55 − 3.39259T [31]
L Fe 2 + , Zn 2 + : Va Liquid 0 58,088 − 23.665T [7]
L Fe 2 + , Zn 2 + : Va Liquid 1 92,219 − 55.584T [7]
L Fe + 2 , Zn + 2 : Va Liquid 2 13,570 [7]
L Al 3 + : O 2 − , Va Liquid 0 −829,000 + 106T [10]
L Fe 2 + : O 2 − , Va Liquid 0 176,681 − 16.368T [14]
L Fe 2 + : O 2 − , Va Liquid 1 −65,655 + 30.869T [14]
L Fe 2 + : O 2 − , Fe O 1 . 5 Liquid 0 −26,362 [14]
L Fe 2 + : O 2 − , Fe O 1 . 5 Liquid 1 13,353 [14]
L Fe 2 + : O 2 − , Al O 1 . 5 Liquid 0 −40,000 + 25T [16]
L Zn 2 + : Al O 1 . 5 , O 2 − Liquid 0 416911.37 − 230.042T This study
L Zn 2 + : Fe O 1 . 5 , O 2 − Liquid 0 −33,360 This study
L Al 3 + : Va , Al O 1 . 5 Liquid 0 110,000 + 46T [10]
L Al 3 + : Va , Fe O 1 . 5 Liquid 0 110,000 [16]
L Fe 2 + : Va , Al O 1 . 5 Liquid 0 178,992 [16]
L Fe 2 + : Va , Fe O 1 . 5 Liquid 0 110,000 [14]
L Al 3 + , Fe 2 + : O 2 − , Va Liquid 0 −740,767 [16]
Spinel: (Al3+, Fe2+, Fe3+, Zn2+)1(Al3+, Fe2+, Fe3+, Va, Zn2+)2(Fe2+, Va)2(O2–)4
G Al 3 + : Al 3 + : Va : O 2 − Spinel 0 − 3 H Al SER 0 − 4 H O SER 0 GPP [16]
G Al 3 + : Fe 2 + : Va : O 2 − Spinel 0 − H Al SER 0 − 2 H Fe SER 0 − 4 H O SER 0 GP2 [16]
G Al 3 + : Fe 3 + : Va : O 2 − Spinel 0 − H Al SER 0 − 2 H Fe SER 0 − 4 H O SER 0 GP3 [16]
G Al 3 + : Va : Va : O 2 − Spinel 0 − H Al SER 0 − 4 H O SER 0 GPV [16]
G Al 3 + : Zn 2 + : Va : O 2 − Spinel 0 − H Al SER 0 − 4 H O SER 0 − 2 H Zn SER 0 GALZN2O4 [43]
G Fe 2 + : Al 3 + : Va : O 2 − Spinel 0 − 2 H Al SER 0 − H Fe SER 0 − 4 H O SER 0 GFEAL2O4 [16]
G Fe 2 + : Fe 2 + : Va : O 2 − Spinel 0 − 3 H Fe SER 0 − 4 H O SER 0 7GFE3O4 + BFE3O4 [12]
G Fe 2 + : Fe 3 + : Va : O 2 − Spinel 0 − 3 H Fe SER 0 − 4 H O SER 0 7GFE3O4 [12]
G Fe 2 + : Va : Va : O − 2 Spinel 0 − H Fe SER 0 − 4 H O SER 0 5GFE3O4 + CFE3O4 [12]
G Fe 2 + : Zn 2 + : Va : O 2 − Spinel 0 − H Fe SER 0 − 4 H O SER 0 − 2 H Zn SER 0 −7GFE3O4 + 2GZNFE3−2IF2F3 + IZNF3 + DF2F3 [17]
G Fe 3 + : Al 3 + : Va : O 2 − Spinel 0 − 2 H Al SER 0 − H Fe SER 0 − 4 H O SER 0 G3P [16]
G Fe 3 + : Fe 2 + : Va : O 2 − Spinel 0 − 3 H Fe SER 0 − 4 H O SER 0 7GFE3O4 [12]
G Fe 3 + : Fe 3 + : Va : O 2 − Spinel 0 − 3 H Fe SER 0 − 4 H O SER 0 7GFE3O4 − BFE3O4 [12]
G Fe 3 + : Va : Va : O 2 − Spinel 0 − H Fe SER 0 − 4 H O SER 0 5GFE3O4 − BFE3O4 + CFE3O4 [12]
G Fe 3 + : Zn 2 + : Va : O 2 − Spinel 0 − H Fe SER 0 − 4 H O SER 0 − 2 H Zn SER 0 −7GFE3O4 + 2GZNFE3 − IF2F3 + IZNF3 [17]
G Zn 2 + : Al 3 + : Va : O 2 − Spinel 0 − 2 H Al SER 0 − 4 H O SER 0 − H Zn SER 0 GZNAL2O4 [19]
G Zn 2 + : Fe 2 + : Va : O 2 − Spinel 0 − 2 H Fe SER 0 − 4 H O SER 0 − H Zn SER 0 GZNFE3 − IF2F3 + DZNF3 [17]
G Zn 2 + : Fe 3 + : Va : O 2 − Spinel 0 − 2 H Fe SER 0 − 4 H O SER 0 − H Zn SER 0 GZNFE3 [17]
G Zn 2 + : Va : Va : O 2 − Spinel 0 − 4 H O SER 0 − H Zn SER 0 −2GFE3O4 + GZNFE3 + VF3 − IF2F3 + DZNF3 − DF3ZNV [17]
G Zn 2 + : Zn 2 + : Va : O 2 − Spinel 0 − 4 H O SER 0 − 3 H Zn SER 0 −14GFE3O4 + 3GZNFE3 − 2IF2F3 + IZNF3 + DZNF3 [17]
G Al 3 + : Al 3 + : Fe 2 + : O 2 − Spinel 0 − 3 H Al SER 0 − 2 H Fe SER 0 − 4 H O SER 0 GPP + 2GFE3O4 − BFE3O4 + DFE3O4 [16]
G Al 3 + : Fe 2 + : Fe 2 + : O 2 − Spinel 0 − H Al SER 0 − 4 H Fe SER 0 − 4 H O SER 0 GP2 + 2GFE3O4 − BFE3O4 + DFE3O4 [16]
G Al 3 + : Fe 3 + : Fe 2 + : O 2 − Spinel 0 − H Al SER 0 − 4 H Fe SER 0 − 4 H O SER 0 GP3 + 2GFE3O4 − BFE3O4 + DFE3O4 [16]
G Al 3 + : Va : Fe 2 + : O 2 − Spinel 0 − H Al SER 0 − 2 H Fe SER 0 − 4 H O SER 0 GPV + 2GFE3O4 − BFE3O4 + DFE3O4 [16]
G Al 3 + : Zn 2 + : Fe 2 + : O 2 − Spinel 0 − H Al SER 0 − 2 H Fe SER 0 − 4 H O SER 0 − 2 H Zn SER 0 GALZN2O4 + 2GFE3O4 − BFE3O4 + DFE3O4 This study
G Fe 2 + : Al 3 + : Fe 2 + : O 2 − Spinel 0 − 2 H Al SER 0 − 3 H Fe SER 0 − 4 H O SER 0 GFEAL2O4 + 2GFE3O4 − BFE3O4 + DFE3O4 [16]
G Fe 2 + : Fe 2 + : Fe 2 + : O 2 − Spinel 0 − 5 H Fe SER 0 − 4 H O SER 0 9GFE3O4 + DFE3O4 [12]
G Fe 2 + : Fe 3 + : Fe 2 + : O 2 − Spinel 0 − 5 H Fe SER 0 − 4 H O SER 0 9GFE3O4 − BFE3O4 + DFE3O4 [12]
G Fe 2 + : Va : Fe 2 + : O 2 − Spinel 0 − 3 H Fe SER 0 − 4 H O SER 0 7GFE3O4 + CFE3O4 − BFE3O4 + DFE3O4 [12]
G Fe 2 + : Zn 2 + : Fe 2 + : O 2 − Spinel 0 − 3 H Fe SER 0 − 4 H O SER 0 − 2 H Zn SER 0 −5GFE3O4 + 2GZNFE3 − 2IF2F3 + IZNF3 + DF2F3 − BFE3O4 + DFE3O4 This study
G Fe 3 + : Al 3 + : Fe 2 + : O 2 − Spinel 0 − 2 H Al SER 0 − 3 H Fe SER 0 − 4 H O SER 0 G3P + 2GFE3O4 − BFE3O4 + DFE3O4 [16]
G Fe 3 + : Fe 2 + : Fe 2 + : O 2 − Spinel 0 − 5 H Fe SER 0 − 4 H O SER 0 9GFE3O4 − BFE3O4 + DFE3O4 [12]
G Fe 3 + : Fe 3 + : Fe 2 + : O 2 − Spinel 0 − 5 H Fe SER 0 − 4 H O SER 0 9GFE3O4 − 2BFE3O4 + DFE3O4 [12]
G Fe 3 + : Va : Fe 2 + : O 2 − Spinel 0 − 3 H Fe SER 0 − 4 H O SER 0 7GFE3O4 + CFE3O4 − 2BFE3O4 + DFE3O4 [12]
G Fe 3 + : Zn 2 + : Fe 2 + : O 2 − Spinel 0 − 3 H Fe SER 0 − 4 H O SER 0 − 2 H Zn SER 0 −5GFE3O4 + 2GZNFE3 − IF2F3 + IZNF3 − BFE3O4 + DFE3O4 This study
G Zn 2 + : Al 3 + : Fe 2 + : O 2 − Spinel 0 − 2 H Al SER 0 − 2 H Fe SER 0 − 4 H O SER 0 − H Zn SER 0 GZNAL2O4 + 2GFE3O4 − BFE3O4 + DFE3O4 This study
G Zn 2 + : Fe 2 + : Fe 2 + : O 2 − Spinel 0 − 4 H Fe SER 0 − 4 H O SER 0 − H Zn SER 0 GZNFE3 − IF2F3 + DZNF3 + 2GFE3O4 − BFE3O4 + DFE3O4 This study
G Zn 2 + : Fe 3 + : Fe 2 + : O − 2 Spinel 0 − 4 H Fe SER 0 − 4 H O SER 0 − H Zn SER 0 GZNFE3 + 2GFE3O4 − BFE3O4 + DFE3O4 This study
G Zn 2 + : Va : Fe 2 + : O 2 − Spinel 0 − 2 H Fe SER 0 − 4 H O SER 0 − H Zn SER 0 GZNFE3 + VF3 − IF2F3 + DZNF3 − DF3ZNV − BFE3O4 + DFE3O4 This study
G Zn 2 + : Zn 2 + : Fe 2 + : O 2 − Spinel 0 − 2 H Fe SER 0 − 4 H O SER 0 − 3 H Zn SER 0 − 12GFE3O4 + 3GZNFE3 − 2IF2F3 + IZNF3 + DZNF3 − BFE3O4 + DFE3O4 This study
L Al 3 + , Zn 2 + : Al 3 + : Va : O 2 − Spinel 0 270122.5 − 157.5T This study
L Fe 2 + : Al 3 + , Fe 3 + : Va : O 2 − Spinel 0 16,427 − 6.4653T [16]
L Fe 3 + : Al 3 + , Fe 3 + : Va : O 2 − Spinel 0 −132,425 + 39.326T [16]
L Fe 3 + : Al 3 + , Fe 3 + : Va : O 2 − Spinel 1 −91,226 + 80.135T [16]
L Fe 3 + : Al 3 + , Fe 3 + : Va : O 2 − Spinel 2 −91.20798T [16]
L Fe 3 + , Zn 2 + : Fe 2 + : Va : O 2 − Spinel 0 −77,500 This study
T i : j : ∗ : O 2 − Spinel 0 (i, j ≠ Al 3 + , Zn 2 + ) 848 [12]
β i : j : ∗ : O 2 − Spinel 0 (i, j ≠ Al 3 + , Zn 2 + ) 44.54 [12]
T Zn 2 + : Fe 3 + : Va : O 2 − Spinel 0 −3 × 9.5 [17]
β Zn 2 + : Fe 3 + : Va : O 2 − Spinel 0 −3 × 3.987 [17]
Corundum: (Al3+, Fe2+, Fe3+)2(Fe3+, Va)1(O2−)3
G Al 3 + : Fe 3 + : O 2 − Corundum 0 − 2 H Al SER 0 − H Fe SER 0 − 3 H O SER 0 GAL2O3 + 85,000 [16]
G Al 3 + : Va : O 2 − Corundum 0 − 2 H Al SER 0 − 3 H O SER 0 GAL2O3 [9]
G Fe 2 + : Fe 3 + : O 2 − Corundum 0 − 3 H Fe SER 0 − 3 H O SER 0 GFE2O3 + 85,000 [15]
G Fe 2 + : Va : O 2 − Corundum 0 − 2 H Fe SER 0 − 3 H O SER 0 GFE2O3 [15]
G Fe 3 + : Fe 3 + : O 2 − Corundum 0 − 3 H Fe SER 0 − 3 H O SER 0 GFE2O3 + 85,000 [15]
G Fe 3 + : Va : O 2 − Corundum 0 − 2 H Fe SER 0 − 3 H O SER 0 GFE2O3 [15]
L Al 3 + , Fe 3 + : Va : O 2 − Corundum 0 110,010−31.781T [16]
L Al 3 + , Fe 3 + : Va : O 2 − Corundum 1 25,408 [16]
L Al 3 + , Fe 3 + : Va : O 2 − Corundum 2 −65,489 [16]
T Fe 3 + : ∗ : O 2 − Corundum 0 −2,867 [12,15]
β Fe 3 + : ∗ : O 2 − Corundum 0 −25.1 [12,15]
Halite: (Al3+, Fe2+, Fe3+, Va, Zn2+)1(O2–)1
G Al 3 + : O 2 − Halite 0 − H Al SER 0 − H O SER 0 50,000 + 0.5GAL2O3 [11,16]
G Fe 2 + : O 2 − Halite 0 − H Fe SER 0 − H O SER 0 GWUSTITE [12]
G Fe 3 + : O 2 − Halite 0 − H Fe SER 0 − H O SER 0 1.25GWUSTITE + 1.25AWUSTITE [12]
G Va : O 2 − Halite 0 − H O SER 0 0 [12]
G Zn 2 + : O 2 − Halite 0 − H O SER 0 − H Zn SER 0 GZINCITE + 12,552 [17]
L Fe 2 + , Fe 3 + : O 2 − Halite 0 −12324.4 [12]
L Fe 2 + , Fe 3 + : O 2 − Halite 1 20,070 [12]
L Fe 2 + , Zn 2 + : O 2 − Halite 0 8527.095−7.015T This study
L Fe 2 + , Zn 2 + : O 2 − Halite 1 −11,000 This study
L Fe 3 + , Zn 2 + : O 2 − Halite 0 −104400.665 + 81.075T This study
L Fe 3 + , Zn 2 + : O 2 − Halite 1 −80,000 This study
AlFeO3 : (Al3+)1(Fe3+)1(O2−)3
G Al 3 + : Fe 2 + : O 2 − Al Fe O 3 0 − H Al SER 0 − H Fe SER 0 − 3 H O SER 0 GALFEO3 [16]
Zincite : ( FeO , FeO 1 . 5 , ZnO )
G FeO Zincite 0 − H O SER 0 GWUSTITE + 16,736 [17]
G FeO 1.5 Zincite 0 − H Fe SER 0 − 1.5 H O SER 0 0.5GFE2O3 + 3,766 [17]
G ZnO Zincite 0 − H O SER 0 − H Zn SER 0 GZINCITE [19]
L FeO , ZnO Zincite 0 −6,276 [17]
L FeO 1 . 5 ,  ZnO Zincite 0 38012.17 − 20.23T This study
L FeO 1 . 5 ,  ZnO Zincite 1 −16,071 This study
BCC_A2: (Al, Fe, Zn)1(O, Va)3
G Al : O BCC_A 2 0 − H Al SER 0 − 3 H O SER 0 GHSERAL + 1.5GO2GAS + 195T [13,16]
G Fe : O BCC _ A 2 0 − H Fe SER 0 − 3 H O SER 0 GHSERFE + 1.5GO2GAS + 195T [13,15]
G Zn : O BCC _ A 2 0 − H Zn SER 0 − 3 H O SER 0 GZNBCC + 1.5GO2GAS + 195T This study
G Al : Va BCC _ A 2 0 − H Al SER 0 GALBCC [8]
G Fe : Va BCC _ A 2 0 − H Fe SER 0 GHSERFE [8]
G Zn : Va BCC _ A 2 0 − H Zn SER 0 GZNBCC [8]
L Al , Fe : O BCC _ A 2 0 −122,960 + 7.9972T [16,30]
L Al , Fe : O BCC _ A 2 1 2945.2 [16,30]
L Al , Fe : Va BCC _ A 2 0 −122,960 + 7.9972T [30]
L Al , Fe : Va BCC _ A 2 1 2945.2 [30]
L Fe , Zn : Va BCC _ A 2 0 −10,494 + 18.299T [6]
L Fe , Zn : Va BCC _ A 2 1 15,513 − 12.608T [6]
L Fe : O , Va BCC _ A 2 0 −517,549 + 71.83T [13]
T Fe : Va BCC _ A 2 0 1043.85 [8]
β Fe : Va BCC _ A 2 0 2.22 [8]
T Zn : Va BCC _ A 2 0 0 [6]
T Al , Fe : O BCC _ A 2 0 0 [16,30]
T Al , Fe : O BCC _ A 2 1 504 [16,30]
T Al , Fe : Va BCC _ A 2 0 0 [30]
T Al , Fe : Va BCC _ A 2 1 504 [30]
T Fe , Zn : Va BCC _ A 2 0 504.3 [6]
FCC_A1: (Al, Fe, Zn)1(O, Va)1
G Al : O FCC _ A 1 0 − H Al SER 0 − H O SER 0 GHSERAL + 0.5GO2GAS − 236446.62 [11,16]
G Fe : O FCC _ A 1 0 − H Fe SER 0 − H O SER 0 GFEFCC + 0.5GO2GAS + 65T [13,15]
G Zn : O FCC _ A 1 0 − H Zn SER 0 − H O SER 0 GZNFCC + 0.5GO2GAS + 65T This study
G Al : Va FCC _ A 1 0 − H Al SER 0 GHSERAL [8]
G Fe : Va FCC _ A 1 0 − H Fe SER 0 GFEFCC [8]
G Zn : Va FCC _ A 1 0 − H Zn SER 0 GZNFCC [8]
L Al , Fe : O FCC _ A 1 0 −76066.1 + 18.6758T [16,30]
L Al , Fe : O FCC _ A 1 1 21167.4 + 1.3398T [16,30]
L Al , Fe : Va FCC _ A 1 0 −76066.1 + 18.6758T [30]
L Al , Fe : Va FCC _ A 1 1 21167.4 + 1.3398T [30]
L Al , Zn : Va FCC _ A 1 0 7297.48 + 0.47512T [31]
L Al , Zn : Va FCC _ A 1 1 6612.88 − 4.5911T [31]
L Al , Zn : Va FCC _ A 1 2 −3097.19 + 3.30635T [31]
L Fe , Zn : Va FCC _ A 1 0 6934.7 + 4.212T [6]
L Fe , Zn : Va FCC _ A 1 1 691 [6]
L Al : O , Va FCC _ A 1 0 −90252.23 [11,16]
L Fe : O , Va FCC _ A 1 0 −168,758 + 19.17T [13]
T Fe : Va FCC _ A 1 0 67 [8]
β Fe : Va FCC _ A 1 0 0.7 [8]
HCP_A3: (Al, Fe, Zn)1(Va)0.5
G Al : Va HCP − A 3 0 − H Al SER 0 GALHCP [8]
G Fe : Va HCP − A 3 0 − H Fe SER 0 GFEHCP [8]
G Zn : Va HCP − A 3 0 − H Zn SER 0 GHSERZN [8]
L Al , Fe : Va HCP − A 3 0 −106,903 + 20T [30]
L Al , Zn : Va HCP − A 3 0 18820.95 − 8.95255T [31]
L Al , Zn : Va HCP − A 3 1 0 [31]
L Al , Zn : Va HCP − A 3 2 0 [31]
L Al , Zn : Va HCP − A 3 3 −702.79 [31]
L Fe , Zn : Va HCP − A 3 0 12,786 [6]
Γ: (Fe, Zn)0.154(Fe, Zn)0.154(Al, Fe, Zn)0.231(Zn)0.461
G Fe : Fe : Al : Zn Γ 0 − 0.231 H Al SER 0 − 0.308 H Fe SER 0 − 0.461 H Zn SER 0 0.231GHSERAL + 0.308GHSERFE + 0.461GHSERZN [7]
G Fe : Zn : Al : Zn Γ 0 − 0.231 H Al SER 0 − 0.154 H Fe SER 0 − 0.615 H Zn SER 0 −2,000 − 0.5T + 0.231GHSERAL + 0.154GHSERFE + 0.615GHSERZN [7]
G Fe : Zn : Fe : Zn Γ 0 − 0.385 H Fe SER 0 − 0.615 H Zn SER 0 −5900.8 + 2.406T + 0.385GHSERFE + 0.615GHSERZN [6]
G Fe : Zn : Zn : Zn Γ 0 − 0.154 H Fe SER 0 − 0.846 H Zn SER 0 −2959.6 − 0.448T + 0.154GHSERFE + 0.846GHSERZN [7]
G Zn : Fe : Al : Zn Γ 0 − 0.231 H Al SER 0 − 0.154 H Fe SER 0 − 0.615 H Zn SER 0 0.231GHSERAL + 0.154GHSERFE + 0.615GHSERZN [7]
G Zn : Zn : Al : Zn Γ 0 − 0.231 H Al SER 0 − 0.769 H Zn SER 0 0.231GHSERAL + 0.769GHSERZN [7]
G Zn : Zn : Fe : Zn Γ 0 − 0.231 H Fe SER 0 − 0.769 H Zn SER 0 793 + 4.782T + 0.231GHSERFE + 0.769GHSERZN [6]
G Zn : Zn : Zn : Zn Γ 0 − H Zn SER 0 6602.65 − 8.157T + GHSERZN [6]
L Fe : Zn : Al , Fe : Zn Γ 0 −15000 − 4T [7]
L Fe : Zn : Al , Zn : Zn Γ 0 −100 – 12T [7]
L Fe : Zn : Fe , Zn : Zn Γ 0 −10394.77 + 12.1876T [7]
Γ1: (Fe)0.137(Al, Fe, Zn)0.118 (Zn)0.745
G Fe : Al : Zn Γ 1 0 − 0.118 H Al SER 0 − 0.137 H Fe SER 0 − 0.745 H Zn SER 0 −6,900 + 0.137GHSERFE + 0.118GHSERAL + 0.745GHSERZN [7]
G Fe : Fe : Zn Γ 1 0 − 0.255 H Fe SER 0 − 0.745 H Zn SER 0 −8609.4 + 5.4T + 0.255GHSERFE + 0.745GHSERZN [6]
G Fe : Zn : Zn Γ 1 0 − 0.137 H Fe SER 0 − 0.863 H Zn SER 0 −5089.67 + 1.898T + 0.137GHSERFE + 0.863GHSERZN [6]
L Fe : Al , Fe : Zn Γ 1 0 −5,300 [7]
L Fe : Al , Zn : Zn Γ 1 0 −2,500 [7]
Γ2: (Al, Fe, Zn)0.255 (Zn)0.745
G Al : Zn Γ 2 0 − 0.255 H Al SER 0 − 0.745 H Zn SER 0 7370.454 + 0.255GHSERAL + 0.745GHSERZN [7]
G Fe : Zn Γ 2 0 − 0.255 H Fe SER 0 − 0.745 H Zn SER 0 0.255GHSERFE + 0.745GHSERZN [7]
G Zn : Zn Γ 2 0 − H Zn SER 0 2665.3728 + GHSERZN [7]
L Al , Fe : Zn Γ 2 0 −32689.0355 [7]
L Al , Zn : Zn Γ 2 0 −16069.1316 + 0.781T [7]
L Fe , Zn : Zn Γ 2 0 −12561.3751 [7]
L Al , Fe , Zn : Zn Γ 2 0 −15588.2333 [7]
δ: (Fe)0.058(Al, Fe, Zn)0.180 (Zn)0.525(Zn)0.237
G Fe : Al : Zn : Zn δ 0 − 0.18 H Al SER 0 − 0.058 H Fe SER 0 − 0.762 H Zn SER 0 10,919−10.5T + 0.18GHSERAL + 0.058GHSERFE + 0.762GHSERZN [7]
G Fe : Fe : Zn : Zn δ 0 − 0.238 H Fe SER 0 − 0.762 H Zn SER 0 −3,886 + 1.365T + 0.238GHSERFE + 0.762GHSERZN [6]
G Fe : Zn : Zn : Zn δ 0 − 0.058 H Fe SER 0 − 0.942 H Zn SER 0 −3,072 + 0.893T + 0.058GHSERFE + 0.942GHSERZN [6]
L Fe : Al , Fe : Zn : Zn δ 0 −23,514 [7]
L Fe : Al , Zn : Zn : Zn δ 0 −12,317 [7]
L Fe : Al , Zn : Zn : Zn δ 1 −4,318 [7]
L Fe : Fe , Zn : Zn : Zn δ 0 −5742.666 + 3.755T [6]
ζ: (Fe, Va)0.072(Al, Va, Zn)0.072 (Al, Zn)0.856
G Fe : Al : Zn ζ 0 − 0.072 H Al SER 0 − 0.072 H Fe SER 0 − 0.856 H Zn SER 0 1,023 − 7.6T + 0.072GHSERAL + 0.072GHSERFE + 0.856GHSERZN [7]
G Fe : Va : Zn ζ 0 − 0.072 H Fe SER 0 − 0.856 H Zn SER 0 700.31 − 2.562T + 0.072GHSERFE + 0.856GHSERZN [7]
G Fe : Zn : Zn ζ 0 − 0.072 H Fe SER 0 − 0.928 H Zn SER 0 −3861.9 + 1.152T + 0.072GHSERFE + 0.928GHSERZN [7]
G Va : Al : Zn ζ 0 − 0.072 H Al SER 0 − 0.856 H Zn SER 0 100 + 0.072GHSERAL + 0.856GHSERZN [7]
G Va : Va : Zn ζ 0 − 0.856 H Zn SER 0 14T + 0.856GHSERZN [7]
G Va : Zn : Al ζ 0 − 0.072 H Fe SER 0 − 0.928 H Zn SER 0 2,000 [7]
G Va : Zn : Zn ζ 0 − 0.928 H Zn SER 0 808.7 − 0.102T + 0.928GHSERZN [7]
Al13Fe4: (Al)0.6275(Fe, Zn)0.235 (Al, Va, Zn)0.1375
G Al : Fe : Zn Al 13 Fe 14 0 − 0.6275 H Al SER 0 − 0.235 H Fe SER 0 − 0.1375 H Zn SER 0 −23,500 + 0.6275GHSERAL + 0.235GHSERFE + 0.1375GHSERZN [7]
G Al : Zn : Al Al 13 Fe 14 0 − 0.765 H Al SER 0 − 0.235 H Zn SER 0 3,147 + 0.765GHSERAL + 0.235GHSERZN [7]
G Al : Zn : Va Al 13 Fe 14 0 − 0.6275 H Al SER 0 − 0.235 H Zn SER 0 2,000 + 0.6275GHSERAL + 0.235GHSERZN [7]
G Al : Zn : Zn Al 13 Fe 14 0 − 0.6275 H Al SER 0 − 0.3725 H Zn SER 0 0.6275GHSERAL + 0.3725GHSERZN [7]
G Al : Fe : Al Al 13 Fe 14 0 − 0.765 H Al SER 0 − 0.235 H Fe SER 0 −30714.3 + 7.44T + 0.765GHSERAL + 0.235GHSERFE [30]
G Al : Fe : Va Al 13 Fe 14 0 − 0.6275 H Al SER 0 − 0.235 H Fe SER 0 −27781.3 + 7.2566T + 0.6275GHSERAL + 0.235GHSERFE [30]
Al5Fe2: (Fe)2(Al)5(Va, Zn)3
G Fe : Al : Va Al 5 Fe 2 0 − 5 H Al SER 0 − 2 H Fe SER 0 −228,576 + 48.99503T + 5GHSERAL + 2GHSERFE [30]
G Fe : Al : Zn Al 5 Fe 2 0 − 5 H Al SER 0 − 2 H Fe SER 0 − 3 H Zn SER 0 −277,947 + 121.95T + 5GHSERAL + 2GHSERFE + 3GHSERZN [7]
Al2Fe: (Fe)1(Al)2(Va, Zn)0.035
G Fe : Al : Va Al 2 Fe 0 − 2 H Al SER 0 − H Fe SER 0 −98097.0 + 18.7503T + 2GHSERAL + GHSERFE [30]
G Fe : Al : Zn Al 2 Fe 0 − 2 H Al SER 0 − H Fe SER 0 − 0.035 H Zn SER 0 −96,068 + 16T + 2GHSERAL + GHSERFE + 0.035GHSERZN [7]
Table 1: Continued
Al5Fe4: (Al, Fe)1
G Al Al 5 Fe 4 0 − H Al SER 0 12178.9 − 4.813T + GHSERAL [30]
G Fe Al 5 Fe 4 0 − H Fe SER 0 5009.03 + GHSERFE [30]
L Al , Fe Al 5 Fe 4 0 −131,649 + 29.4833T [30]
L Al , Fe Al 5 Fe 4 1 −18619.5 [30]
Function
GHSERAL 298.15 < T < 700 −7976.15 + 137.093038T − 24.3671976Tln(T) − 0.001884662T 2 − 8.77664 × 10−7 T 3 + 74092T −1 [8]
700 < T < 933.47 −11276.24 + 223.048446T − 38.5844296Tln(T) + 0.018531982T 2 − 5.764227 × 10−6 T 3 + 74092T −1
933.47 < T < 4,000 −11278.378 + 188.684153T − 31.748192Tln(T) − 1.230524 × 10²⁸T −9
GHSERFE 298.15 < T < 1,811 1225.7 + 124.134T − 23.5143Tln(T) − 0.00439752T 2 − 5.89269 × 10−8 T 3 + 77358.5T −1 [8]
1,811 < T < 6,000 −25383.581 + 299.31255T − 46Tln(T) + 2.2960305 × 1031 T −9
GHSERZN 298.15 < T < 692.68 −7285.787 + 118.470069T − 23.701314Tln(T) − 0.001712034T 2 − 1.264963 × 10−6 T 3 [8]
692.68 < T < 4,700 −11070.559 + 172.34566T − 31.38Tln(T) + 4.70514 × 10²⁶T −9
GALBCC 298.15 < T < 4,000 10,083 − 4.813T + GHSERAL [8]
GZNBCC 298.15 < T < 1,700 2886.96 − 2.5104T + GHSERZN [8]
GFEFCC 298.15 < T < 1,811 −1462.4 + 8.282T − 1.15Tln(T) + 6.4 × 10−⁴T 2 + GHSERFE [8]
1,811 < T < 4,000 −27098.266 + 300.25266T − 46Tln(T) + 2.78854 × 1031 T −9
GZNFCC 298.15 < T < 1,700 2969.82 − 1.56968T + GHSERZN [8]
GALHCP 298.15 < T < 2,900 5481 − 1.8T + GHSERAL [8]
GFEHCP 298.15 < T < 1,811 −3705.78 + 12.591T − 1.15Tln(T) + 6.4 × 10−4 T 2 + GHSERFE [8]
1,811 < T < 4,000 −3957.199 + 5.24951T + 4.9251 × 10−30 T −9 + GHSERFE
GALLIQ 298.15 < T < 700 11005.029 − 11.841867T + 7.934 × 10−20 T ⁷ + GHSERAL [8]
700 < T < 933.47 11005.03 − 11.841867T + 7.9337 × 10−20 T⁷ + GHSERAL
933.47 < T < 4,000 10482.382 − 11.253975T + 1.230524 × 10²⁸T −9 + GHSERAL
GFELIQ 298.15 < T < 1,811 12040.17 − 6.55843T − 3.67516 × 10−²¹T⁷ + GHSERFE [8]
1,811 < T < 4,000 14544.8 − 8.01055T − 2.29603 × 10³¹T −9 + GHSERFE
GZNLIQ 298.15 < T < 692.68 7157.213 − 10.29299T − 3.58958 × 10−19 T⁷ + GHSERZN [8]
692.68 < T < 4,700 7450.168 − 10.737066T − 4.70514 × 10²⁶T −9 + GHSERZN
GLAL2O3 298.15 < T < 600 −1607850.8 + 405.559491T − 67.4804Tln(T) − 0.06747T 2 + 1.4205433 × 10−⁵T 3 + 938780T −1 [9]
600 < T < 1,500 −1625385.57 + 712.394972T − 116.258Tln(T) − 0.0072257T 2 + 2.78532 × 10−⁷T 3 + 2120700T −1
1,500 < T < 1,912 −1672662.69 + 1010.9932T − 156.058Tln(T) + 0.00709105T 2 − 6.29402 × 10−⁷T 3 + 12366650T −1
1,912 < T < 2,327 29178041.6 − 168360.926T + 21987.1791Tln(T) − 6.99552951T 2 + 4.10226192 × 10−⁴T 3 − 7.98843618 × 10⁹T −1
2,327 < T < 4,000 −1757702.05 + 1344.84833T − 192.464Tln(T)
GFEOLIQ 298.15 < T < 3,000 −137252 + 224.641T − 37.1815Tln(T) [12]
GZINCITE 298.15 < T < 2,250 −367388.671 + 280.979572T − 47.584Tln(T) − 0.00195155T 2 − 2.1322 × 10−7 T 3 + 375180T −1 [19]
2,250 < T < 4,000 −396004.038 + 438.167206T − 67Tln(T)
GPP 298.15 < T < 6,000 1.5GFEAL2O4 − 3.5GFE3O4 − 0.5BFE3O4 + J [16]
GP2 298.15 < T < 6,000 J + 0.5GFEAL2O4 + 3.5GFE3O4 + 0.5BFE3O4 [16]
GP3 298.15 < T < 6,000 7GFE3O4 + 0.5GFEAL2O4 + J − 3.5GFE3O4 − 0.5BFE3O4 [16]
GALZN2O4 298.15 < T < 6,000 2GZNAL2O4 − GPP + IZNAL [43]
GPV 298.15 < T < 6,000 8GGAMMA − 7.5GFEAL2O4 + 17.5GFE3O4 + 2.5BFE3O4 − 5J + 44.9543481T [16]
GFEAL2O4 298.15 < T < 6,000 −2038654.7 + 958.16741T − 155.3938Tln(T) − 0.0097141T 2 + 1566908T −1 [16]
GFE3O4 298.15 < T < 3,000 −161,731 + 144.873T − 24.9879Tln(T) − 0.0011952256T 2 + 206520T −1 [12]
BFE3O4 298.15 < T < 3,000 46,826 − 27.266T [12]
CFE3O4 298.15 < T < 3,000 120,730 − 20.102T [12]
DFE3O4 298.15 < T < 3,000 402,520 − 30.529T [12]
GZNFE3 298.15 < T < 4,000 −1,175,354 + 1615.28T − 5186.49T 0.5 − 7359.34ln(T) − 237.559Tln(T) [17]
IF2F3 298.15 < T < 6,000 −31,229 + 22.063T [17]
IZNF3 298.15 < T < 6,000 51,800 [17]
DF2F3 298.15 < T < 6,000 15,781 [17]
G3P 298.15 < T < 6,000 −7GFE3O4 + 7GFE3O4 − BFE3O4 + GFEAL2O4 + DG3P3P [16]
GZNAL2O4 298.15 < T < 6,000 −2137061.09 + 1058.77749T − 166.523Tln(T) − 0.0077405T 2 + 2301000T −1 [19]
DZNF3 298.15 < T < 6,000 40,000 [17]
VF3 298.15 < T < 6,000 29,932 + 28.547T [17]
DF3ZNV 298.15 < T < 6,000 −36,000 [17]
GAL2O3 298.15 < T < 600 −1707351.3 + 448.021092T − 67.4804Tln(T) − 0.06747T 2 + 1.4205433 × 10−⁵T 3 + 938780T −1 [9]
600 < T < 1,500 −1724886.06 + 754.856573T − 116.258Tln(T) − 0.0072257T 2 + 2.78532 × 10−⁷T 3 + 2120700T −1
1500 < T < 3,000 −1772163.19 + 1053.4548T − 156.058Tln(T) + 0.00709105T 2 − 6.29402 × 10−⁷T 3 + 12366650T −1
GFE2O3 298.15 < T < 6,000 −858,683 + 827.946T − 137.0089Tln(T) + 1,453,810T −1 [12]
GWUSTITE 298.15 < T < 3,000 −279,318 + 252.848T − 46.12826Tln(T) − 0.0057402984T 2 [12]
AWUSTITE 298.15 < T < 3,000 −55,384 + 27.888T [12]
GALFEO3 298.15 < T < 6,000 −1284485.36 + 795.2115T − 126.28Tln(T) − 0.013681T 2 + 1.1664 × 10−⁷T 3 + 1512700T −1 [16]
GGAMMA 298.15 < T < 600 −1689977.34 + 469.458181T − 70.5452Tln(T) − 0.070794T 2 + 1.491345 × 10−⁵T 3 + 981165T −1 [19]
600 < T < 1,500 −1708389.72 + 791.591946T − 121.754Tln(T) − 7.5467 × 10−³T 2 + 2.89573 × 10−⁷T 3 + 2,222,750T −1
1,500 < T < 3,000 −1758861.74 + 1110.41976T − 164.253Tln(T) + 7.75305 × 10−³T 2 − 6.8247 × 10−⁷T 3 + 13,162,750T −1
GO2GAS 298.15 < T < 1,000 −6961.742 − 51.0061T − 22.271Tln(T) − 0.01019775T 2 + 1.32369 × 10−⁶T 3 − 76729.5T −1 [8]
1,000 < T < 3,300 −13137.527 + 25.31976T − 33.6276Tln(T) − 0.001191595T 2 + 1.35611 × 10−⁸T 3 + 525810T −1
3,300 < T < 6,000 −27973.4908 + 62.5195726T − 37.9072074Tln(T) − 8.50483772 × 10−⁴T 2 + 2.14409777 × 10−⁸T 3 + 8766421.4T −1
J 298.15 < T < 6,000 29163.95 + 7.88T [16]
DG3P3P 298.15 < T < 6,000 133833.21 − 52.276862T [16]
IZNAL 298.15 < T < 6,000 170,000 − 18T [43]

2.1 Gas phase

The gas phase was treated as an ideal mixture of Al, AlO, AlO2, Al2O, Al2O2, Fe, FeO, FeO2, Fe2, O, O2, O3, Zn, and ZnO. The Gibbs energy is represented using the following equation:

(1) G m Gas = ∑ i x i G i Gas 0 + R T ∑ i x i ln x i ,

where x i denotes the mole fraction of species i, R denotes the universal gas constant, and T denotes the temperature in Kelvin. The term G i Gas 0 denotes the molar Gibbs energy of species i in the gaseous state, and the descriptions of these parameters for each species were taken from the Thermo-Calc Software SSUB6 SGTE substances database [19].

2.2 Liquid phase

The Gibbs energy of the liquid phase was described using the ionic two-sublattice liquid model [20,21] with the formula (Al3+, Fe2+, Zn2+) P (O2–, Va, AlO1.5, FeO1.5) Q where P and Q are the number of sites on the cation and anion sublattice, respectively. Electroneutrality is maintained because the number of sites varies. The Gibbs energy of the liquid phase with the formula ( C i v j + ) P ( A j v j − , Va, B k 0 ) Q using a formula unit that contains P + Q(1−y Va) moles of atoms is given by:

(2) G m Liquid = ∑ i ∑ j y C i y A j G C i : A j Liquid 0 + Q y Va ∑ i y C i G C i : Va Liquid 0 + Q ∑ k y B k G B k Liquid 0 + RT P ∑ i y C i ln y C i + Q ∑ j y A j ln y A j + y Va ln y Va + ∑ k y B k ln y B k + G m Liquid E ,

where C represents cations, A represents anions, Va represents hypothetical vacancies, and B represents neutral species. The superscript v i on ions denotes the charge of ions, and the indices i, j, and k denote specific constituents. The colon separates the constituent elements in the sublattice. y denotes the site fractions of a constituent on each sublattice. G C i : A j Liquid 0 represents the Gibbs energy per (v i + v j ) mole of atoms of liquid C i A j . G C i : Va Liquid 0 and G B i Liquid 0 represent the Gibbs energies per mole of atoms of liquid C i ( G C i Liquid 0 ) and B i , respectively, and the parameters G C i Liquid 0 were taken from the SGTE database [8]. G m Liquid E represents the excess Gibbs energy and is expressed by:

(3) G m Liquid E = ∑ i 1 ∑ i 2 ∑ j y C i 1 y C i 2 y A j L C i 1 , C i 2 : A j Liquid + ∑ i 1 ∑ i 2 y C i 1 y C i 2 y Va 2 L C i 1 , C i 2 : Va Liquid + ∑ i ∑ j 1 ∑ j 2 y C i y A j 1 y A j 2 L C i : A j 1 , A j 2 Liquid + ∑ i ∑ j y C i y A j y Va L C i : A j , Va Liquid + ∑ i ∑ j ∑ k y C i y A j y B k L C i : A j , B k Liquid + ∑ i ∑ k y C i y B k y Va L C i : Va , B k Liquid + ∑ k 1 ∑ k 2 y B k 1 y B k 2 L B k 1 , B k 2 Liquid .

Note that only the binary parameters are included in the expression for G m Liquid E in equation (3). L C i 1 , C i 2 : A j Liquid represents the interaction between two cations with a common anion, L C i 1 , C i 2 : Va Liquid represents the interaction between two metallic elements, L C i : A j 1 , A j 2 Liquid represents the interaction between two anions with a common cation, L C i : A j , Va Liquid represents the interaction between a metallic atom and anion, L C i : A j , B k Liquid represents the interaction between an anion and a neutral species, L C i : Va , B k Liquid represents the interaction between a metal and a neutral species, and L B k 1 , B k 2 Liquid represents the interaction between two neutral species, where a comma is used to differentiate between constituents on the same sublattices. The interaction parameter has a compositional dependency using an nth degree Redlich–Kister polynomial [22]. For example, the description for L C i 1 , C i 2 :Va Liquid is expressed as follows:

(4) L C i 1 , C i 2 : A j Liquid = ∑ n = 0 L C i 1 , C i 2 : A j Liquid n ( y C i 1 − y C i 2 ) n .

The parameters L C i 1 , C i 2 : A j Liquid n in equation (4) can be temperature-dependent as follows:

(5) L C i 1 , C i 2 : A j Liquid n = A n + B n T + C n T ln T + D n T 2 + E n T 3 + F n T − 1 .

2.3 Spinel, corundum, halite, FCC_A1, BCC_A2, and HCP_A3 phases

The spinel phase is the cubic AE2O4-type solid solution of the magnetite (Fe3O4), hercynite (FeAl2O4), and gahnite (ZnAl2O4), where A and E are divalent and trivalent cations, respectively, and exhibits the deviation from the ideal stoichiometry. The corundum phase is the solid solution of corundum (Al2O3) and hematite (Fe2O3). Halite is the NaCl-type solid solution of the wüstite (FeO) dissolving of Al and Zn. The FCC_A1, BCC_A2, and HCP_A3 phases are the fcc-type solid solution of Al and Fe dissolving O and Zn, the bcc-type solid solution of Fe dissolving Al, O, and Zn, and the hcp-type solid solution of Zn dissolving Al and Zn, respectively.

The Gibbs energies of spinel, corundum, halite, FCC_A1, BCC_A2, and HCP_A3 phases were described using the sublattice formalism [23] based on the two-sublattice model [24]. For the simple case of a phase with formula (A,B) m (C,D) n , m and n are the numbers of the sites of sublattices 1 and 2, respectively, and the constituents A, B, C, and D can represent atoms, ions, vacancies, and so forth. The Gibbs energy of the ϕ phase per mole of formula unit is expressed by:

(6) G m ϕ = y A ( 1 ) y C ( 2 ) G A : C ϕ 0 + y B ( 1 ) y C ( 2 ) G B : C ϕ 0 + y A ( 1 ) y D ( 2 ) G A : D ϕ 0 + y B ( 1 ) y D ( 2 ) G B : D ϕ 0 + mRT ( y A ( 1 ) ln y A ( 1 ) + y B ( 1 ) ln y B ( 1 ) ) + nRT ( y C ( 2 ) ln y C ( 2 ) + y D ( 2 ) ln y D ( 2 ) ) + G m ϕ E ,

where G i : j ϕ 0 denotes the Gibbs energy of a hypothetical compound (also called a constituent array), i m j n per mole of formula unit, in which all the sites in sublattice 1 are occupied by constituent i, and all the sites in sublattice 2 are occupied by constituent j. In this case, the existence of four end-members, A m C n , A m D n , B m C n , and B m D n , is assumed. It should be noted that G i : Va ϕ 0 represents the Gibbs energy per mole of atoms of pure element i in the ϕ phase and is called the lattice stability. The descriptions of the lattice stability parameters and stable end-member parameters were taken from the SGTE database [8] and the Thermo-Calc software SSUB6 SGTE substance database [19], respectively. The site fraction of the constituent on the sth sublattice is denoted by y i s . The term G m ϕ E is the excess Gibbs energy term containing the interaction energy between unlike constituents and is expressed by the following equation:

(7) G m ϕ E = y A ( 1 ) y B ( 1 ) y C ( 2 ) L A , B : C ϕ + y A ( 1 ) y B ( 1 ) y D ( 2 ) L A , B : D ϕ + y A ( 1 ) y C ( 2 ) y D ( 2 ) L A : C , D ϕ + y B ( 1 ) y C ( 2 ) y D ( 2 ) L B : C , D ϕ ,

where L i , j : k ϕ (or L i : j , k ϕ ) is the interaction parameter between unlike constituents on the same sublattice and is described by equations similar to equations (4) and (5).

In this calculation, the formulas (Al3+, Fe2+, Fe3+, Zn2+)1(Al3+, Fe2+, Fe3+, Va, Zn2+)2(Fe2+, Va)2(O2–)4, (Al3+, Fe2+, Fe3+)2(Fe3+, Va)1(O2–)3, (Al3+, Fe2+, Fe3+, Zn2+, Va)1(O2–)1, (Al, Fe, Zn)1(O, Va)1, (Al, Fe, Zn)1(O, Va)3, and (Al, Fe, Zn)1(Va)0.5 were adopted for the spinel, corundum, halite, FCC_A1, BCC_A2, and HCP_A3 phases, respectively. The magnetic contribution to the Gibbs energy was considered for the spinel, corundum, FCC_A1, and BCC_A2 phases, and the approach proposed by Hillert and Jarl [25,26] was used to describe the magnetic contribution.

2.4 Zincite phase

The zincite phase is the solid solution of zincite (ZnO) dissolving Fe. The solubility of Al in the zincite was not considered in this study because the solid solubility of Al is reported to be very small as reported previously [27].

The Gibbs energy of the zincite phase was represented using the regular solution approximation with the components of FeO, FeO1.5, and ZnO. The Gibbs energy of the zincite phase is described by:

(8) G m Zincite = ∑ i x i G i Zincite 0 + R T ∑ i x i ln x i + G m Zincite E

where x i and G i Zincite 0 denote the mole fraction of component i and the molar Gibbs energy of component i in the zincite state, respectively. The term G i Zincite 0 is the excess Gibbs energy described by an equation similar to equation (7).

2.5 Orthorhombic AlFeO3 phase

Although the orthorhombic AlFeO3 with a FeGaO3-type structure was found to exhibit some homogeneity range [28], this phase was treated as a stoichiometric compound for simplicity, and the Gibbs energy is given by:

(9) G m AlFe O 3 0 − H Al SER 0 − H Fe SER 0 − 3 H O SER 0 = a + bT + cT ln T + d T 2 + e T 3 + f T − 1 .

2.6 Intermetallic compound phases

The detailed thermodynamic models of remaining intermetallic compound phases appearing in the Al–Fe–Zn ternary system were described previously by Nakano et al. [7].

3 Results and discussion

3.1 Al–O, Fe–O, and Zn–O binary systems

The thermodynamic descriptions from previous assessments were adopted for the Al–O [9,10,11] and Fe–O [12,13,14,15] binary systems, which reproduced available experimental data on phase boundaries and thermodynamic properties. Regarding the Zn–O binary system, although a phase diagram was not found in the literature, a diagram of the condensed system was proposed [29]. According to the diagram, this binary system consists of an HCP_A3 solid solution of Zn, ZnO, and ZnO2 for the condensed phase, and the solubility of oxygen in Zn is almost zero. Furthermore, the thermodynamic property and melting point of ZnO2 have not been clarified. Thus, the solubility of oxygen in Zn was not considered in this calculation, and the Gibbs energy functions of the solid (zincite) and liquid ZnO phases were taken from the Thermo-Calc Software SSUB6 SGTE substances database [19].

3.2 Al–Fe–Zn ternary system

Thermodynamic parameters of a previous assessment [7], based on the thermodynamic descriptions of the Al–Fe [30], Al–Zn [31], and Fe–Zn [7], were used in this study. The thermodynamic assessment by Nakano et al. [7] was mainly conducted based on the experimental data for phase boundaries [32,33,34,35] and Al activities in various two- and three-phase regions [7,33,36]. The assessed parameters can be used to calculate the phase equilibria in the Al–Fe–Zn ternary system in the temperature range important for the galvanizing process [7].

3.3 Al–Fe–O ternary system

The thermodynamic assessment of this ternary system was conducted using the CALPHAD method [16], and the calculated results were in good agreement with the available experimental results on the phase equilibria in the Al2O3–Fe2O3 [37], FeAl2O4–Fe3O4 [38], and Al2O3–FeO–Fe2O3 [39] systems and also on thermodynamic properties, such as the activities of Fe3O4 and FeAl2O4 in the Fe3O4–FeAl2O4 system [40,41]. Therefore, the thermodynamic parameters assessed by Lindwall et al. [16] were adopted in this study.

3.4 Al–Zn–O ternary system

In the Al–Zn–O ternary system, the phase diagram in the Al2O3–ZnO system was proposed by Hansson et al. [27] based on their own experimental results and the data from Bunting [42]. The Al2O3–ZnO system is composed of liquid, corundum, spinel, and zincite phases. According to the phase diagram, the solubility of ZnO in the corundum phase is about 2 mol% at temperatures between 1,250 and 1,695°C, and the maximum solubility of Al2O3 in the zincite phase is about 4.7 mol% at 1,695°C, which decreases with decreasing temperature to less than 0.5 mol% below 1,550°C. The spinel phase has a large homogeneity range at higher temperatures on the Al2O3 side; however, the range decreases rapidly in the lower temperature region, whereas the composition is nearly stoichiometric ZnAl2O4 on the ZnO side. Shevchenko and Jak [43] also proposed the phase diagram in the Al2O3–ZnO system based on their own experimental results and then performed thermodynamic analysis. Their proposed phase diagram is almost the same as that proposed by Hansson et al. [27]; however, they reported two-phase separation in the zincite phase on the ZnO-rich side. Considering the phase equilibrium reported by Hansson et al. [44] in the Fe2O3–ZnO system, the two-phase separation of the ZnO phase in the Al2O3–ZnO system is questionable and was not considered in this study. In this study, the thermodynamic assessment of the Al2O3–ZnO system was conducted based on the experimental data on phase boundaries [27,42], where the corundum and zincite phases were treated as pure binary oxides for simplicity. In the assessment, the thermodynamic parameters of end-members, (Al3+)1(Zn2+)2(Va)2(O2–)4 and (Zn2+)1(Al3+)2(Va)2(O2–)4, for the spinel phase were taken from the previous assessment by Shevchenko and Jak [43] and the Thermo-Calc Software SSUB6 SGTE substances database [19], respectively. Furthermore, the parameter, G Zn : O FCC_A 1 0 , was given following the previous assessments [13,15]. Figure 1 shows the calculated phase diagram of the Al2O3–ZnO system, together with the experimental data. The calculated results approximately reproduced the experimental phase boundary data, and the assessed thermodynamic parameters are listed in Table 1.

Figure 1 Calculated phase diagram of the Al2O3–ZnO system.
Figure 1

Calculated phase diagram of the Al2O3–ZnO system.

3.5 Fe–Zn–O ternary system

The Fe–Zn–O system was thermodynamically assessed by Degterov et al. [17] based on their own experimental data as well as available literature data, and the calculated results reproduced the experimental results satisfactorily. In their assessment, however, the modified quasi-chemical model [45,46,47,48] was applied for the description of the Gibbs energy of the liquid phase, whereas the Gibbs energy of the halite phase was represented by the regular solution approximation model with the Kohler-like equation [49,50] for the ternary excess Gibbs energy term. In addition, the thermodynamic model for the spinel phase, (Fe2+, Fe3+, Zn)1(Fe2+, Fe3+, Va, Zn)2(O2–)4, was different from that adopted in this study. Therefore, the thermodynamic parameters assessed by Degterov et al. [17] were modified to reproduce the available experimental data using the thermodynamic models for the liquid, halite, and spinel phases adopted in this study. It should be noted that the parameters of end-members, G ∗ : ∗ : Fe 2 + : O 2 − spinel 0 , for the spinel phase, which correspond to the substitution of Va for Fe2+ on the third sublattice of end-members, (∗)1(∗)2(Va)2(O2–)4, were set to have the same energy difference, G Fe 3 + : Fe 2 + : Fe 2 + : O 2 − spinel 0 − G Fe 3 + : Fe 2 + : Va : O 2 − spinel 0 , as for the Fe3O4 in the Fe–O binary system [12,15]. Furthermore, the parameter, G Zn : O BCC_A 2 0 , was given following the previous assessments [13,15]. The evaluated parameters in this study are listed in Table 1. The calculated isothermal section diagrams of the Fe–Zn–O system at T = 827°C and of the FeO–Fe2O3–ZnO system at T = 900°C are shown in Figure 2, together with the experimental results [17,51]. The calculated vertical section diagram of the FeO–ZnO system is compared with the experimental data [51,52,53] in Figure 3. As shown in Figures 2 and 3, the calculated phase equilibria reproduced the experimental data satisfactorily.

Figure 2 Calculated isothermal section diagrams: (a) Fe–Zn–O system at T = 827°C and (b) FeO–Fe2O3–ZnO system at T = 900°C.
Figure 2

Calculated isothermal section diagrams: (a) Fe–Zn–O system at T = 827°C and (b) FeO–Fe2O3–ZnO system at T = 900°C.

Figure 3 Calculated vertical section diagram of the FeO–ZnO system.
Figure 3

Calculated vertical section diagram of the FeO–ZnO system.

3.6 Al–Fe–Zn–O quaternary system

Regarding the Al–Fe–Zn–O quaternary system, the phase equilibria in the Al2O3–FeO–Fe2O3–ZnO region in air [27] and at intermediate p O 2 [54] were investigated, and some isothermal projections on to the Al–Fe–Zn plane were presented. In this study, phase diagram calculations on the Al–Fe–Zn–O quaternary system were performed based on simple extrapolation of the constituent ternary systems with no additional quaternary thermodynamic parameters. It should be noted that the parameters of end-members, G Al 3 + : Zn 2 + : Fe 2 + : O 2 − spinel 0 and G Zn 2 + : Al 3 + : Fe 2 + : O 2 − spinel 0 , for the spinel phase, which correspond to the substitution of Va for Fe2+ on the third sublattice of end-members, (Al3+)1(Zn2+)2(Va)2(O2–)4 and (Zn2+)1(Al3+)2(Va)2(O2–)4, were set to have the same energy difference, G Fe 3 + : Fe 2 + : Fe 2 + : O 2 − spinel 0 − G Fe 3 + : Fe 2 + : Va : O 2 − spinel 0 , as for the Fe3O4 in the Fe–O binary system [12,15].

Figures 4 and 5 show the calculated isothermal phase equilibria of the Al2O3–FeO–Fe2O3–ZnO region projected onto the Al–Fe–Zn ternary composition triangle in air and p O 2 = 1 × 10–6 atm, respectively, together with the experimental results [27,54]. These figures show that the calculated results approximately agree with the experimental results. As mentioned by Saunders and Miodownik [5] for substitutional solution phases in multicomponent alloys, it seems that interaction parameters of a higher order than the ternary system are unnecessary in the calculation of the present quaternary phase diagrams.

Figure 4 Calculated isothermal phase equilibria of the Al2O3–FeO–Fe2O3–ZnO region projected onto the Al–Fe–Zn ternary composition triangle in air at (a) T = 1,250°C, (b) T = 1,400°C, and (c) T = 1,550°C, together with the experimental results [27].
Figure 4

Calculated isothermal phase equilibria of the Al2O3–FeO–Fe2O3–ZnO region projected onto the Al–Fe–Zn ternary composition triangle in air at (a) T = 1,250°C, (b) T = 1,400°C, and (c) T = 1,550°C, together with the experimental results [27].

Figure 5 Calculated isothermal phase equilibria of the Al2O3–FeO–Fe2O3–ZnO region projected onto the Al–Fe–Zn ternary composition triangle p O 2 {p}_{{\text{O}}_{2}} = 1 × 10−6 atm at (a) T = 1,200°C, (b) T = 1,300°C, and (c) T = 1,400°C, together with the experimental results [54].
Figure 5

Calculated isothermal phase equilibria of the Al2O3–FeO–Fe2O3–ZnO region projected onto the Al–Fe–Zn ternary composition triangle p O 2 = 1 × 10−6 atm at (a) T = 1,200°C, (b) T = 1,300°C, and (c) T = 1,400°C, together with the experimental results [54].

4 Conclusions

The phase equilibria in the Al–Fe–Zn–O quaternary system were studied using the thermodynamic descriptions of four ternary systems based on the CALPHAD approach. In this study, the thermodynamic assessment of the Al2O3–ZnO system was conducted based on the experimental data on phase boundaries, and some thermodynamic parameters of the Fe–Zn–O system were modified to maintain consistency with the thermodynamic descriptions of other binary and ternary systems adopted in this study. The set of thermodynamic parameters enabled us to calculate the phase equilibria in the Al–Fe–Zn–O quaternary system over the entire composition and temperature ranges.

  1. Funding information: The authors state no funding is involved.

  2. Author contributions: Naoki Matsumoto: formal analysis, investigation, methodology, writing – original draft; Tatsuya Tokunaga: conceptualization, resources, supervision, writing – review & editing.

  3. Conflict of interest: The authors state no conflict of interest.

References

[1] Marder, A. R. The metallurgy of zinc-coated steel. Progress in Materials Science, Vol. 45, No. 3, 2000, pp. 191–271.10.1016/S0079-6425(98)00006-1Search in Google Scholar

[2] Kato, C., H. Koumura, K. Mochizuki, and N. Morito. Dross formation and flow phenomena in molten zinc bath. Proceeding of the 3rd International Conference on Zinc and Zinc Alloy Coated Steel Sheet, Iron and Steel Society, Chicago, September 17–21, 1995, pp. 801–806.Search in Google Scholar

[3] Tang, N. Y. Practical Applications of phase diagrams in continuous galvanizing. Journal of Phase Equilibria and Diffusion, Vol. 27, No. 5, 2006, pp. 462–468.10.1361/154770306X131994Search in Google Scholar

[4] Kaufman, L. and H. Bernstein. Computer calculation of phase diagrams, with special reference to refractory metals, Academic Press, New York, NY, 1970.Search in Google Scholar

[5] Saunders, N. and A. P. Miodownik. CALPHAD, calculation of phase diagrams, a comprehensive guide, Elsevier, Science, Oxford, UK, 1998.Search in Google Scholar

[6] Nakano, J., D. V. Malakhov, and G. R. Purdy. A crystallographically consistent optimization of the Zn–Fe system. Calphad, Vol. 29, No. 4, 2005, pp. 276–288.10.1016/j.calphad.2005.08.005Search in Google Scholar

[7] Nakano, J., D.V. Malakhov, S. Yamaguchi, and G. R. Purdy. A full thermodynamic optimization of the Zn–Fe–Al system within the 420–500°C temperature range. Calphad, Vol. 31, No. 1, 2007, pp. 125–140.10.1016/j.calphad.2006.09.003Search in Google Scholar

[8] Dinsdale, A. T. SGTE data for pure elements. Calphad, Vol. 15, No. 4, 1991, pp. 317–425.10.1016/0364-5916(91)90030-NSearch in Google Scholar

[9] Taylor, J. R., A. T. Dinsdale, M. Hilleit, and M. Selleby. A critical assessment of thermodynamic and phase diagram data for the Al–O system. Calphad, Vol. 16, No. 2, 1992, pp. 173–179.10.1016/0364-5916(92)90005-ISearch in Google Scholar

[10] Hallstedt, B. Thermodynamic calculation of some subsystems of the Al–Ca–Mg–Si–O system. Journal of Phase Equilibria, Vol. 14, No. 6, 1993, pp. 662–675.10.1007/BF02667878Search in Google Scholar

[11] Mozaffarie-Jovein, H. Enwincklung und Einsatz von keramischen Materialien als Wärmedammschicht fur Ni-Basis-legierungen für Temperaturen gösser als 1350°C. PhD thesis, Universität Stuttgart, Stuttgart, 2003.Search in Google Scholar

[12] Sundman, B. An assessment of the Fe–O system. Journal of Phase Equilibria, Vol. 12, No. 2, 1991, pp. 127–140.10.1007/BF02645709Search in Google Scholar

[13] Kowalski, M. and P. J. Spencer. Thermodynamic reevaluation of the Cr–O, Fe–O and Ni–O systems: Remodelling of the liquid, BCC and FCC phases. Calphad, Vol. 19, No. 3, 1995, pp. 229–243.10.1016/0364-5916(95)00024-9Search in Google Scholar

[14] Selleby, M. and B. Sundman. A reassessment of the Ca–Fe–O system. Calphad, Vol. 20, No. 3, 1996, pp. 381–392.10.1016/S0364-5916(96)00039-9Search in Google Scholar

[15] Kjellqvist, L., M. Selleby, and B. Sundman. Thermodynamic modelling of the Cr–Fe–Ni–O system. Calphad, Vol. 32, No. 3, 2008, pp. 577–592.10.1016/j.calphad.2008.04.005Search in Google Scholar

[16] Lindwall, G., X. L. Liu, A. Ross, H. Fang, B. C. Zhou, and Z. K. Liu. Thermodynamic modeling of the aluminum–iron–oxygen system. Calphad, Vol. 51, 2015, pp. 178–192.10.1016/j.calphad.2015.09.004Search in Google Scholar

[17] Degterov, S. A., E. Jak, P. C. Hayes, and A. D. Pelton. Experimental study of phase equilibria and thermodynamic optimization of the Fe–Zn–O system. Metallurgical and Materials Transactions B, Vol. 32, No. 4, 2001, pp. 643–657.10.1007/s11663-001-0119-2Search in Google Scholar

[18] Shobu, K. CaTCalc: New thermodynamic equilibrium calculation software. Calphad, Vol. 33, No. 2, 2009, pp. 279–287.10.1016/j.calphad.2008.09.015Search in Google Scholar

[19] Scientific Group Thermodata Europe (SGTE), Thermo-Calc Software SSUB6 SGTE substances database (accessed 9 June 2022).Search in Google Scholar

[20] Hillert, M., B. Jansson, B. Sundman, and J. Ågren. A two-sublattice model for molten solutions with different tendency for ionization. Metallurgical Transactions A, Vol. 16, No. 2, 1985, pp. 261–266.10.1007/BF02816052Search in Google Scholar

[21] Sundman, B. Modification of the two-sublattice model for liquids. Calphad, Vol. 15, No. 2, 1991, pp. 109–119.10.1016/0364-5916(91)90010-HSearch in Google Scholar

[22] Redlich, O. and A. T. Kister. Algebraic representation of thermodynamic properties and the classification of solutions. Industrial & Engineering Chemistry, Vol. 40, No. 2, 1948, pp. 345–348.10.1021/ie50458a036Search in Google Scholar

[23] Sundman, B. and J. Ågren. A regular solution model for phases with several components and sublattices, suitable for computer applications. Journal of Physics and Chemistry of Solids, Vol. 42, No. 4, 1981, pp. 297–301.10.1016/0022-3697(81)90144-XSearch in Google Scholar

[24] Hillert, M. and L. Staffansson. The regular-solution model for stoichiometric phases and ionic melts. Acta Chemica Scandinavica, Vol. 24, 1970, pp. 3618–3626.10.3891/acta.chem.scand.24-3618Search in Google Scholar

[25] Inden, G. Determination of chemical and magnetic interchange energies in BCC alloys. Zeitschrift für Metallkunde, Vol. 66, No. 10, 1975, pp. 577–582.10.1515/ijmr-1975-661003Search in Google Scholar

[26] Hillert, M. and M. Jarl. A model for alloying in ferromagnetic metals. Calphad, Vol. 2, No. 3, 1978, pp. 227–238.10.1016/0364-5916(78)90011-1Search in Google Scholar

[27] Hansson, R., P. C. Hayes, and E. Jak. Experimental study of phase equilibria in the Al-Fe–Zn–O system in air. Metallurgical and Materials Transactions B, Vol. 35, No. 4, 2004, pp. 633–642.10.1007/s11663-004-0004-xSearch in Google Scholar

[28] Rhamdhani, M. A., T. Hidayat, P. C. Hayes, and E. Jak. Subsolidus phase equilibria of Fe–Ni–X–O (X = Mg, Al) systems in air. Metallurgical and Materials Transactions B, Vol. B40, No. 1, 2009, pp. 25–38.10.1007/s11663-008-9213-zSearch in Google Scholar

[29] Wriedt, H. A. The O−Zn (oxygen–zinc) system. Journal of Phase Equilibria, Vol. 8, No. 2, 1987, pp. 166–176.10.1007/BF02873202Search in Google Scholar

[30] Ansara, I., A. T. Dinsdale, and M. H. Rand, Eds., Thermochemical Database for Light Metal Alloys, European Commission, Luxembourg, 2, 1998, pp. 34–39.Search in Google Scholar

[31] Mey, S. Reevaluation of the Al–Zn system. Zeitschrift für Metallkunde, Vol. 84, No. 7, 1993, pp. 451–455.10.1515/ijmr-1993-840704Search in Google Scholar

[32] Perrot, P., J. C. Tissier, and J. Y. Dauphin. Stable and metastable equilibria in the Fe–Zn–Al system at 450°C. Zeitschrift für Metallkunde, Vol. 83, No. 11, 1992, pp. 786–790.10.1515/ijmr-1992-831104Search in Google Scholar

[33] Yamaguchi, S., H. Makino, A. Sakatoku, and Y. Iguchi. Phase stability of dross phases in equilibrium with liquid Zn measured by the Al sensor. Proceeding of the 3rd International Conference on Zinc and Zinc Alloy Coated Steel Sheet, Iron & Steel Society, Chicago, September 17–21, 1995, pp. 787–794.Search in Google Scholar

[34] Tang, N. Y. and X. Su. On the ternary phase in the zinc-rich corner of the Zn–Fe–Al system at temperatures below 450°C. Metallurgical and Materials Transactions A, Vol. 33, No. 5, 2002, pp. 1559–1561.10.1007/s11661-002-0078-5Search in Google Scholar

[35] McDermid, J. R., É. Baril, and W. T. Thompson. Fe solubility in the Zn–Al–Fe system for use in continuous galvanizing and galvannealing. Proceedings of the 6th International Conference on Zinc and Zinc Alloy Coated Steel Sheet, Association for Iron and Steel Technology, Chicago, April 4–7, 2004, pp. 491–499.Search in Google Scholar

[36] Yamaguchi, S. Thermochemical stability and precipitation behavior of dross phases in CGL bath. Proceedings of the 3rd International Conference on Zinc and Zinc Alloy Coated Steel Sheet, Iron and Steel Institute of Japan, Makuhari, Chiba, September 20–23, 1998, pp. 84–89.Search in Google Scholar

[37] Raghavan, V. Al–Fe–O (aluminum–iron–oxygen). Journal of Phase Equilibria and Diffusion, Vol. 31, No. 4, 2010, id. 367.10.1007/s11669-010-9712-xSearch in Google Scholar

[38] Turnock, A. C. and H. P. Eugster. Fe–Al oxides: Phase relationships below 1000°C. Journal of Petrology, Vol. 3, No. 3, 1962, pp. 533–565.10.1093/petrology/3.3.533Search in Google Scholar

[39] Raghavan, V. The Al–Fe–O (aluminum–iron–oxygen) System. Phase Diagrams of Ternary Iron Alloys. Part 5: Ternary Systems Containing Iron and Oxygen, Indian Institute of Metals, Calcutta, 1989, pp. 10–28.Search in Google Scholar

[40] Schmahl, N. G. and H. Dillenburg. Gleichgewichtsuntersuchungen an eisenoxidhaltigen mischphasen innerhalb der dreistoffsysteme Fe–Al–O, Fe–Cr–O Und Fe–V–O. Zeitschrift für Physikalische Chemie, Vol. 65, No. 1_4, 1969, pp. 119–138.10.1524/zpch.1969.65.1_4.119Search in Google Scholar

[41] Petric, A., K. T. Jacob, and C. B. Alcock. Thermodynamic properties of Fe3O4–FeAl2O4 spinel solid solutions. Journal of the American Ceramic Society, Vol. 64, No. 11, 1981, pp. 632–639.10.1111/j.1151-2916.1981.tb15860.xSearch in Google Scholar

[42] Bunting, E. N. Phase equilibria in the system SiO2–ZnO–Al2O3. Bureau of Standards Journal of Research, Vol. 8, 1932, pp. 279–287.10.6028/jres.008.020Search in Google Scholar

[43] Shevchenko, M. and E. Jak. Integrated experimental phase equilibria study and thermodynamic modelling of the binary ZnO–Al2O3, ZnO–SiO2, Al2O3–SiO2 and ternary ZnO–Al2O3–SiO2 systems. Ceramics International, Vol. 47, No. 15, 2021, pp. 20974–20991.10.1016/j.ceramint.2021.04.098Search in Google Scholar

[44] Hansson, R., P. C. Hayes, and E. Jak. Phase equilibria in the ZnO-rich area of the Fe–Zn–O system in air. Scandinavian Journal of Metallurgy, Vol. 33, No. 5, 2004, pp. 294–304.10.1111/j.1600-0692.2004.00696.xSearch in Google Scholar

[45] Pelton, A. D., S. A. Degterov, G. Eriksson, C. Robelin, and Y. Dessureault. The modified quasichemical model I - Binary solutions. Metallurgical and Materials Transactions B, Vol. 31, No. 4, 2000, pp. 651–659.10.1007/s11663-000-0103-2Search in Google Scholar

[46] Pelton, A. D. A general “geometric” thermodynamic model for multicomponent solutions. Calphad, Vol. 25, No. 2, 2001, pp. 319–328.10.1016/S0364-5916(01)00052-9Search in Google Scholar

[47] Pelton, A. D. and P. Chartrand. The modified quasi-chemical model: Part II. Multicomponent solutions. Metallurgical and Materials Transactions A, Vol. 32, No. 6, 2001, pp. 1355–1360.10.1007/s11661-001-0226-3Search in Google Scholar

[48] Pelton, A. D., P. Chartrand, and G. Eriksson. The modified quasi-chemical model: Part IV. Two-sublattice quadruplet approximation. Metallurgical and Materials Transactions A, Vol. 32, No. 6, 2001, pp. 1409–1416.10.1007/s11661-001-0230-7Search in Google Scholar

[49] Kohler, F. Zur berechnung der thermodynamischen daten eines ternären systems aus den zugehörigen binären systemen. Monatshefte für Chemie und verwandte Teile anderer Wissenschaften, Vol. 91, No. 4, 1960, pp. 738–740.10.1007/BF00899814Search in Google Scholar

[50] Pelton, A. D. A database and sublattice model for molten salts. Calphad, Vol. 12, No. 2, 1988, pp. 127–142.10.1016/0364-5916(88)90015-6Search in Google Scholar

[51] Itoh, S. and T. Azakami. Activities of the components and phase relations in Zn–Fe–O and ZnO–FeO–SiO2 systems. Metallurgical Review of MMIJ (Mining and Metallurgical Institute of Japan), Vol. 10, No. 2, 1993, pp. 113–133.Search in Google Scholar

[52] Lykasov, A. A., V. V. D’yachuk, M. S. Pavlovskaya, and T. Popova. Monovariant equilibria in the Fe–Zn–O system. Izvestiya Akademii Nauk SSSR. Neorganicheskie Materialy, Vol. 27, No. 3, 1991, pp. 539–543.Search in Google Scholar

[53] Jak, E., B. Zhao, and P. C. Hayes. Experimental study of phase equilibria in the systems Fe–Zn–O and Fe–Zn–Si–O at metallic iron saturation. Metallurgical and Materials Transactions B, Vol. 31, No. 6, 2000, pp. 1195–1201.10.1007/s11663-000-0006-2Search in Google Scholar

[54] Hansson, R., P. C. Hayes, and E. Jak. Phase equilibria in the system Al–Fe–Zn–O at intermediate conditions between metallic iron saturation and air. Canadian Metallurgical Quarterly, Vol. 44, No. 1, 2005, pp. 111–118.10.1179/cmq.2005.44.1.111Search in Google Scholar
Received: 2022-08-25
Revised: 2022-10-31
Accepted: 2022-11-01
Published Online: 2022-12-06

© 2022 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/htmp-2022-0249
Keywords for this article
calculation of phase diagrams; hot-dip galvannealed steel; phase equilibria; Al–Fe–Zn–O system; oxide-type dross
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  23. Smelting high purity 55SiCr automobile suspension spring steel with different refractories
  24. Investigation into the thermal stability of a novel hot-work die steel 5CrNiMoVNb
  25. Residual stress relaxation considering microstructure evolution in heat treatment of metallic thin-walled part
  26. Experiments of Ti6Al4V manufactured by low-speed wire cut electrical discharge machining and electrical parameters optimization
  27. Effect of chloride ion concentration on stress corrosion cracking and electrochemical corrosion of high manganese steel
  28. Prediction of oxygen-blowing volume in BOF steelmaking process based on BP neural network and incremental learning
  29. Effect of annealing temperature on the structure and properties of FeCoCrNiMo high-entropy alloy
  30. Study on physical properties of Al2O3-based slags used for the self-propagating high-temperature synthesis (SHS) – metallurgy method
  31. Low-temperature corrosion behavior of laser cladding metal-based alloy coatings on EH40 high-strength steel for icebreaker
  32. Study on thermodynamics and dynamics of top slag modification in O5 automobile sheets
  33. Structure optimization of continuous casting tundish with channel-type induction heating using mathematical modeling
  34. Microstructure and mechanical properties of NbC–Ni cermets prepared by microwave sintering
  35. Spider-based FOPID controller design for temperature control in aluminium extrusion process
  36. Prediction model of BOF end-point P and O contents based on PCA–GA–BP neural network
  37. Study on hydrogen-induced stress corrosion of 7N01-T4 aluminum alloy for railway vehicles
  38. Study on the effect of micro-shrinkage porosity on the ultra-low temperature toughness of ferritic ductile iron
  39. Characterization of surface decarburization and oxidation behavior of Cr–Mo cold heading steel
  40. Effect of post-weld heat treatment on the microstructure and mechanical properties of laser-welded joints of SLM-316 L/rolled-316 L
  41. An investigation on as-cast microstructure and homogenization of nickel base superalloy René 65
  42. Effect of multiple laser re-melting on microstructure and properties of Fe-based coating
  43. Experimental study on the preparation of ferrophosphorus alloy using dephosphorization furnace slag by carbothermic reduction
  44. Research on aging behavior and safe storage life prediction of modified double base propellant
  45. Evaluation of the calorific value of exothermic sleeve material by the adiabatic calorimeter
  46. Thermodynamic calculation of phase equilibria in the Al–Fe–Zn–O system
  47. Effect of rare earth Y on microstructure and texture of oriented silicon steel during hot rolling and cold rolling processes
  48. Effect of ambient temperature on the jet characteristics of a swirl oxygen lance with mixed injection of CO2 + O2
  49. Research on the optimisation of the temperature field distribution of a multi microwave source agent system based on group consistency
  50. The dynamic softening identification and constitutive equation establishment of Ti–6.5Al–2Sn–4Zr–4Mo–1W–0.2Si alloy with initial lamellar microstructure
  51. Experimental investigation on microstructural characterization and mechanical properties of plasma arc welded Inconel 617 plates
  52. Numerical simulation and experimental research on cracking mechanism of twin-roll strip casting
  53. A novel method to control stress distribution and machining-induced deformation for thin-walled metallic parts
  54. Review Article
  55. A study on deep reinforcement learning-based crane scheduling model for uncertainty tasks
  56. Topical Issue on Science and Technology of Solar Energy
  57. Synthesis of alkaline-earth Zintl phosphides MZn2P2 (M = Ca, Sr, Ba) from Sn solutions
  58. Dynamics at crystal/melt interface during solidification of multicrystalline silicon
  59. Boron removal from silicon melt by gas blowing technique
  60. Removal of SiC and Si3N4 inclusions in solar cell Si scraps through slag refining
  61. Electrochemical production of silicon
  62. Electrical properties of zinc nitride and zinc tin nitride semiconductor thin films toward photovoltaic applications
  63. Special Issue on The 4th International Conference on Graphene and Novel Nanomaterials (GNN 2022)
  64. Effect of microstructure on tribocorrosion of FH36 low-temperature steels
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