Phase equilibria of Fe₃O₄–Cr₂O₃–CaO system: experimental measurements and thermodynamic calculations

Abstract

Understanding the phase equilibria of the Fe₃O₄–Cr₂O₃–CaO system is essential for the efficient recycling of stainless steel pickling sludge. The isothermal section of this system at 1473 K under oxygen partial pressure of 0.15 Pa was investigated. Key experiments on the relevant binary systems were conducted using a combination of equilibrium-quenching techniques, X-ray diffraction, high-resolution transmission electron microscope, and electron probe microanalysis. These systems were rigorously assessed using the CALPHAD (CALculation of Phase Diagram) method, incorporating the present experimental data. The liquid phase was modeled using the ionic two-sublattice model, represented as (Ca²⁺, Cr³⁺, Fe²⁺)_P(O²⁻, Va, FeO_1.5)_Q, where Va represents vacancy, and P and Q denote the number of sites on the cation and anion sublattices, respectively. To ensure electroneutrality, the values of P and Q adjust according to the composition of the mixture. From this, the isothermal section of the Fe₃O₄–Cr₂O₃–CaO system at 1473 K under the specified oxygen partial pressure was obtained based on the thermodynamic parameters of the binary systems. The present experimental data and calculation results hold significant implications for the comprehensive recycling of stainless steel pickling sludge.

1 Introduction

Stainless steel, a critical material in the modern steel industry, is widely used due to its extensive application potential. During its production, acids like H₂SO₄, HNO₃, and HF are employed for surface pickling, enhancing its corrosion resistance and appearance [1, 2]. This process generates pickling wastewater, typically treated through neutralization and sedimentation, leading to the generation of stainless steel pickling sludge (SSPS). SSPS, containing valuable metals like iron and chromium, is categorized as hazardous solid waste due to its high concentration of heavy metals (HMs) and fluorine [3, 4]. Chromium, a costly and vital raw material in metallurgical and chemical industries [5], is a key focus in SSPS recovery research. The SSPS mainly contains FeO_n, Cr₂O₃, and calcium containing compounds. If FeO_n and Cr₂O₃ can be reorganized into spinel phase and separated, high-value chromite can be obtained from the sludge. Understanding the phase equilibrium relationship of the FeO_n–Cr₂O₃–CaO system and identifying the chromite phase region are crucial for efficient utilization of SSPS resources. To enhance the efficiency of obtaining phase equilibrium information, thermodynamic calculations using appropriate databases and software are recommended. Using CALPHAD (CALculation of PHAse Diagram) method, in particular, can provide a self-consistent database. When combined with suitable software, this method can significantly reduce costs and time in material development, offering effective guidance for material design [6].

Currently, the ternary phase diagram of the FeO_n–Cr₂O₃–CaO system remains undocumented, highlighting the importance of investigating its phase equilibrium. Previous studies have reported phase diagrams [7,8,9,10,11,12,13,14] and thermodynamic evaluations [15,16,17,18] for related binary systems. Taylor and Dinsdale [15] focused on the Fe–Cr–O system, adopting the ionic two-sublattice model to represent the liquid phase as (Fe²⁺, Fe³⁺, Cr³⁺)_P(O²⁻, Va)_Q, where Va represents vacancy, and P and Q denote the number of sites on the cation and anion sublattices, respectively. To ensure electroneutrality, the values of P and Q adjust according to the composition of the mixture. In contrast, Kjellqvist et al. [16] deviated from this approach in their evaluation of liquid Fe₂O₃. They introduced FeO_1.5 as a neutral molecule on the anion sublattice, rather than using separate cation and anion sublattices. This modification arose due to complications in more complex systems. Selleby and Sundman [17] evaluated the Fe–Ca–O system, also using the ionic two-sublattice model for the liquid phase. Meanwhile, Degterov and Pelton [18] focused on the CaO–CrO_n system, employing a modified quasi-chemical model for the liquid phase. However, their study was confined to conditions of low oxygen partial pressure. The ionic two-sublattice model, which has been used to describe both oxide and metallic liquids in various studies [19, 20], is adopted in the current work. This model offers a versatile framework for parameterizing these complex systems, facilitating a deeper understanding of the FeO_n–Cr₂O₃–CaO system.

The primary objective of this study is to perform crucial experiments on the binary systems within the Fe₃O₄–Cr₂O₃–CaO system. Following these experiments, the isothermal section of the Fe₃O₄–Cr₂O₃–CaO system at 1473 K under oxygen partial pressure of 0.15 Pa will be determined using the CALPHAD method.

2 Experimental

2.1 Sample preparation

This study aims to investigate the phase equilibrium relationship of the Fe₃O₄–Cr₂O₃–CaO system at 1473 K. To achieve this, we conducted experiments on the binary systems of Fe₃O₄–Cr₂O₃, Fe₃O₄–CaO and CaO–Cr₂O₃. The starting materials were Fe₃O₄ powder (99.99 wt.% purity, Sigma-Aldrich, USA), Cr₂O₃ powder (99.97 wt.% purity, Alfa Aesar, USA) and CaO powder (99.99 wt.% purity, Sinopharm Chemical Reagent Co., Ltd., China). These powders were proportionally mixed and ground using an agate mortar for 30 min to ensure homogeneity. The composition of the samples is detailed in Table 1. Approximately 1 g of this mixture was then compressed into a cylinder with diameter of 10 mm at a uniaxial pressure of 10 MPa and placed in a nickel crucible. During the heating process, the nickel crucible with the sample, secured with nickel wire, was suspended in a tube furnace (Henan Nobody Materials Science and Technology Co., Ltd., China) and heated by a MoSi₂ heating element. Temperature control was achieved using an S-type thermocouple, maintaining accuracy within ± 5 K. The samples were equilibrated at the target temperature for 48 h under a 99.999% pure argon atmosphere, which maintained the oxygen partial pressure at 0.15 Pa. After equilibration, the samples, still in their nickel crucibles, were rapidly quenched in quenching oil. For further analysis, these quenched samples were embedded in epoxy resin and polished.

Table 1 Initial composition of samples (wt.%)

2.2 Sample characterization

Phase identification in this study was conducted using an X-ray diffractometer (XRD, D8-Advanced, Bruker, Germany) equipped with Cu Kα radiation at 40 kV and 40 mA. Scans spanned a 2θ range from 10° to 80°, with a step size of 0.02°. Additionally, the microstructures of the samples in the Fe₃O₄–Cr₂O₃ system were examined using a high-resolution transmission electron microscope (HRTEM, Tecnai G2 F20 S-Twin, FEI, America). The solid solubility of the system was determined using an energy-dispersive X-ray spectrometer (EDS). To identify the eutectic reaction temperature in the Fe₃O₄–CaO system, a simultaneous thermal analyzer (STA449F3, Netzsch, Germany) was employed, with a heating rate of 5 K/min and argon as the carrier gas. Further characterization of the microstructures in the Fe₃O₄–CaO system was performed using a scanning electron microscope (SEM), complemented by electron probe microanalysis (EPMA, EPMA–1720T, Shimadzu, Japan). EPMA analyses were conducted at 15 kV and 2 × 10⁻⁸ A. For precise phase composition and elemental measurement, pure Fe₂O₃ (99.99 wt.%) and CaCO₃ (99.99 wt.%) were used as standard materials in the wavelength-dispersive X-ray spectrometer (WDS).

3 Thermodynamic models

In this study, the CALPHAD method [21] was employed to compute the phase diagram of the Fe₃O₄–Cr₂O₃–CaO system. For the thermodynamic assessment, the Thermo-Calc software package [22] was utilized. The thermodynamic models for all stable phases present in the Fe₃O₄–Cr₂O₃–CaO system are comprehensively outlined in Table 2.

Table 2 Thermodynamic models of all stable phases in Fe₃O₄–Cr₂O₃–CaO system in present work

3.1 Liquid phase

The liquid phase in this study is characterized using the ionic two-sublattice model, represented by the formula (Ca²⁺, Cr³⁺, Fe²⁺)_P(O²⁻, Va, FeO_1.5)_Q. The Gibbs energy of the liquid phase ${{G}}_{\text{m}}^{{\text{Liq}}{\text{uid}}}$ is expressed as follows:

$$\begin{aligned} {{G}}_{{\text{m}}}^{{{\text{Liquid}}}} & = {{y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{^0}{G}}_{{{\text{Ca}}^{{2 + }} {\text{:O}}^{{2 - }} }}^{{{\text{Liquid}}}} +{{y}}_{{{\text{Cr}}^{{3 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{^0}{G}}_{{{\text{Cr}}^{{3 + }} {\text{:O}}^{{2 - }} }}^{{{\text{Liquid}}}} + {{y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{^0}{G}}_{{{\text{Fe}}^{{2 + }} {\text{:O}}^{{2 - }} }}^{{{\text{Liquid}}}} +{{Q}}\left( {{{y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{Va}}}} {{^0}{G}}_{{{\text{Ca}}}}^{{{\text{Liquid}}}} {\text{ + y}}_{{{\text{Cr}}^{{3 + }} }} {{y}}_{{{\text{Va}}}} {{^0}{G}}_{{{\text{Cr}}}}^{{{\text{Liquid}}}} +{{y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{Va}}}} {{^0}{G}}_{{{\text{Fe}}}}^{{{\text{Liquid}}}}+ {{y}}_{{{\text{FeO}}_{{1.5}} }} {{^0}{G}}_{{{\text{FeO}}_{{1.5}} }}^{{{\text{Liquid}}}} } \right) + {{PRT}}\left( {{{y}}_{{{\text{Ca}}^{{2 + }} }} {\text{ln}y}_{{{\text{Ca}}^{{2 + }} }} + {{y}}_{{{\text{Cr}}^{{3 + }} }} {\text{ln}y}_{{{\text{Cr}}^{{3 + }} }} + {{y}}_{{{\text{Fe}}^{{2 + }} }} {\text{ln}}y_{{{\text{Fe}}^{{2 + }} }} } \right) +{{QRT}}\left( {{{y}}_{{{\text{O}}^{{2 - }} }} {{\text{ln}}y}_{{{\text{O}}^{{2 - }} }}+ {{ y}}_{{{\text{Va}}}} {\text{ln}y}_{{{\text{Va}}}}+ {{y}}_{{{\text{FeO}}_{{1.5}} }} {\text{ln}y}_{{{\text{FeO}}_{{1.5}} }} } \right){{ + }}^{{\text{E}}} {{G}}_{{\text{m}}}^{{{\text{Liquid}}}} \\ \end{aligned}$$

(1)

where y represents the site fraction; R is the universal gas constant; T is temperature, K; and ${}^{0}{{{G}}}_{i}^{{\text{Liq}}{\text{uid}}}$ is the molar Gibbs energy of the end member in the model. The expression for ${}^{\text{E}}{{{G}}}_{\text{m}}^{{\text{Liq}}{\text{uid}}}$ is given by the following equation, including only the parameters that have been used.

$$\begin{aligned} ^{{\text{E}}} {{G}}_{{\text{m}}}^{{{\text{Liquid}}}} & = {{ y}}_{{{\text{Cr}}^{{3 + }} }} {{y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{L}}_{{{\text{Cr}}^{{3 + }} {\text{,Fe}}^{{2 + }} {\text{:O}}^{{2 - }} }}^{{{\text{Liquid}}}} +{{y}}_{{{\text{Cr}}^{{3 + }} }} {{y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{Va}}}} {{L}}_{{{\text{Cr}}^{{3 + }} {\text{,Fe}}^{{2 + }} {\text{:Va}}}}^{{{\text{Liquid}}}} \\ & \quad + {{y}}_{{{\text{Cr}}^{{3 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{y}}_{{{\text{Va}}}} {{L}}_{{{\text{Cr}}^{{3 + }} {\text{:O}}^{{2 - }} {\text{,Va}}}}^{{{\text{Liquid}}}} + {{ y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{y}}_{{{\text{Va}}}} {{L}}_{{{\text{Fe}}^{{2 + }} {\text{:O}}^{{2 - }} {\text{,Va}}}}^{{{\text{Liquid}}}} \\ & \quad + {{ y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{y}}_{{{\text{FeO}}_{{1.5}} }} {{L}}_{{{\text{Fe}}^{{2 + }} {\text{:O}}^{{2 - }} {\text{,FeO}}_{{1.5}} }}^{{{\text{Liquid}}}} +{{ y}}_{{{\text{Cr}}^{{3 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{y}}_{{{\text{FeO}}_{{1.5}} }} {{L}}_{{{\text{Cr}}^{{3 + }} {\text{:O}}^{{2 - }} {\text{,FeO}}_{{1.5}} }}^{{{\text{Liquid}}}} \\ & \quad + {{ y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{FeO}}_{{1.5}} }} {{y}}_{{{\text{Va}}}} {{L}}_{{{\text{Fe}}^{{2 + }} {\text{:FeO}}_{{1.5}} {\text{,Va}}}}^{{{\text{Liquid}}}} +{{ y}}_{{{\text{Cr}}^{{3 + }} }} {{y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{y}}_{{{\text{Va}}}} {{L}}_{{{\text{Cr}}^{{3 + }} {\text{,Fe}}^{{2 + }} {\text{:O}}^{{2 - }} {\text{,Va}}}}^{{{\text{Liquid}}}} \\ & \quad +{{ y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{L}}_{{{\text{Ca}}^{{2 + }} {\text{,Fe}}^{{2 + }} {\text{:O}}^{{2 - }} }}^{{{\text{Liquid}}}} +{{ y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{y}}_{{{\text{FeO}}_{{1.5}} }} {{L}}_{{{\text{Ca}}^{{2 + }} {\text{,Fe}}^{{2 + }} {\text{:O}}^{{2 - }} {\text{,FeO}}_{{1.5}} }}^{{{\text{Liquid}}}} \\ & \quad +{{ y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{y}}_{{{\text{FeO}}_{{1.5}} }} {{L}}_{{{\text{Ca}}^{{2 + }} {\text{:O}}^{{2 - }} {\text{,FeO}}_{{1.5}} }}^{{{\text{Liquid}}}} +{{ y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{y}}_{{{\text{Va}}}} {{L}}_{{{\text{Ca}}^{{2 + }} {\text{:O}}^{{2 - }} {\text{,Va}}}}^{{{\text{Liquid}}}} \\ & \quad +{{ y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{Fe}}^{{2 + }} }} {{y}}_{{{\text{Va}}}} {{L}}_{{{\text{Ca}}^{{2 + }} {\text{,Fe}}^{{2 + }} {\text{:Va}}}}^{{{\text{Liquid}}}} +{{ y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{Va}}}} {{y}}_{{{\text{FeO}}_{{1.5}} }} {{L}}_{{{\text{Ca}}^{{2 + }} {\text{:Va,FeO}}_{{1.5}} }}^{{{\text{Liquid}}}} \\ & \quad +{{ y}}_{{{\text{Ca}}^{{2 + }} }} {{y}}_{{{\text{Cr}}^{{3 + }} }} {{y}}_{{{\text{O}}^{{2 - }} }} {{L}}_{{{\text{Ca}}^{{2 + }} {\text{,Cr}}^{{3 + }} {\text{:O}}^{{2 - }} }}^{{{\text{Liquid}}}} \\ \end{aligned}$$

(2)

where L signifies the interaction parameter, which is designed to vary linearly with temperature. The initial fourteen interaction parameters are derived from binary subsystems, as referenced in previous studies [16, 17, 23]. The final interaction parameter has been specifically evaluated in this work.

3.2 CF and C₂F phases

The molar Gibbs energy for the CF and C₂F phases is characterized using the following descriptions:

$$^{0} G - H_{i}^{{{\text{SER}}}} = a + bT + cT{\text{ln}}T + {{d}}T^{{ - 1}} + eT^{2}$$

(3)

where ${}^{0}{{G}}$ represents the molar Gibbs energy of CF or C₂F; ${{H}}_{{i}}^{\text{SER}}$ represents the enthalpy of element i at the standard state of 298.15 K and 10⁵ Pa, as well as the reference state relative to the stable element reference (SER); and a, b, c, d, and e are coefficients. The thermodynamic parameters of CF and C₂F previously reported in Ref. [17] are deemed reliable and have thus been adopted in the present study.

3.3 Halite, spinel and corundum phases

The halite, spinel and corundum phases in this study are modeled using the compound energy formalism (CEF) [23]. Specifically, in the halite phase model, it is assumed that two Fe³⁺ ions can replace three Fe²⁺ ions on the cation sublattice. This substitution results in the formation of neutral vacancies on the cation sublattice to maintain a balance of sites. Additionally, it is presumed that Ca²⁺ and Cr³⁺ can substitute the cation sublattice as well. Therefore, the halite phase is formulated as (Ca²⁺, Cr³⁺, Fe²⁺, Fe³⁺, Va)₁(O²⁻)₁.

Based on the evaluation of the Fe–O system [24], the spinel phase is modeled using a four-sublattice approach. This model designates two sublattices for typical tetrahedral and octahedral cation sites, one for oxygen anions and the other to accommodate Fe²⁺ ions at the normally cation-free octahedral sites. In this framework, Ca is assumed to only enter the second sublattice, while Cr can enter both the first and second sublattices. The spinel phase is thus described by the CEF as (Cr²⁺, Cr³⁺, Fe²⁺, Fe³⁺)₁(Ca²⁺, Cr³⁺, Fe²⁺, Fe³⁺, Va)₂(Fe²⁺, Va)₂(O²⁻)₄.

The corundum phase in this study is modeled as a solid solution, representing a complete miscibility between stoichiometric Fe₂O₃ and Cr₂O₃ across all temperatures. Accordingly, the corundum phase is formulated as (Cr²⁺, Cr³⁺, Fe³⁺)₂(Cr³⁺, Va)₁(O²⁻)₃. This model reflects the interchangeable nature of the oxides within the corundum structure.

3.4 α-CaCr₂O₄, β-CaCr₂O₄ and CF₂ phases

The low-temperature form of β-CaCr₂O₄ exhibits a spinel structure [25], while the high-temperature counterpart, α-CaCr₂O₄, shares the same structural characteristics as SrCr₂O₄ [26]. Both α-CaCr₂O₄ and β-CaCr₂O₄, along with the CF₂ phase, are stoichiometric compounds. Their Gibbs energies can be described in terms of the corresponding pure oxide and its deviations, under the assumption that the Neumann–Kopp rule is applicable to the heat capacity. The expression for their Gibbs energies is formulated as follows:

$${}^{0}{{{G}}}_{\text{m}}^{{\text{Ca}}{\text{Cr}}_{2}{{\text{O}}}_{4}}={}^{0}{{{G}}}_{\text{CaO}}+{}^{0}{{G}}_{{\text{Cr}}_{2}{{\text{O}}}_{3}}+{{a}}+{{bT}}$$

(4)

$${}^{0}{{{G}}}_{\text{m}}^{\text{{C}}{{F}}_{2}}={}^{0}{{{G}}}_{\text{CaO}}+ {2} {}^{0}{{G}}_{{\text{Fe}}_{2}{{\text{O}}}_{3}}+{{a}}+{{bT}}$$

(5)

4 Results

4.1 Phase equilibria of Fe₃O₄–Cr₂O₃ system under oxygen partial pressure of 0.15 Pa at 1473 K

The phase equilibrium relations of five distinct component samples in the Fe₃O₄–Cr₂O₃ system were examined. Initial characterization of the crystal phases in the quenched samples was conducted through XRD measurements. To provide deeper insights into the microstructure and composition of each crystal phase, HRTEM and EDS were employed. Figure 1 displays the XRD patterns of five quenched samples at 1473 K in the Fe₃O₄–Cr₂O₃ system. Analysis of these patterns reveals the presence of two crystal phases at 1473 K: Cr₂O₃ (ICDD No. 00-038-1479) and FeCr₂O₄ (ICDD No. 01–089-2618). In quenched Samples 1 and 2, both Cr₂O₃ and FeCr₂O₄ phases were identified, suggesting that the compositions of these samples fall within the two-phase region of Cr₂O₃ and FeCr₂O₄. In contrast, only the FeCr₂O₄ phase was observed in quenched Samples 3–5, indicating complete dissolution of Cr₂O₃ into Fe₃O₄, resulting in the formation of FeCr₂O₄.

Fig. 1

XRD patterns of five quenched samples at 1473 K in Fe₃O₄–Cr₂O₃ system

HRTEM and EDS were utilized to delve into the microstructure and composition of two crystal phases. Figure 2 presents the transmission electron microscopy (TEM) images of quenched Sample 2. Specifically, Fig. 2a displays the TEM image of FeCr₂O₄, while Fig. 2b reveals the HRTEM image taken from Fig. 2a. In this image, an interplanar spacing d of 0.259 nm is evident, aligning with the (3$\bar{1}\bar{1}$) crystal plane of FeCr₂O₄. FeCr₂O₄ is characterized by a spinel structure within the cubic crystal system and is classified under the space group Fd$\bar{3}$m. The selected area electron diffraction (SAED) pattern in Fig. 2c showcases the diffraction spots of the (11$\bar{1}$), (3$\bar{1}\bar{1}$) and (1$\bar{3}$1) planes along the [112] direction. Turning to Cr₂O₃, Fig. 2d exhibits the TEM image, with an interplanar spacing of 0.281 nm highlighted in Fig. 2e, corresponding to the (0$\bar{1}$4) crystal plane of Cr₂O₃. Cr₂O₃ possesses a corundum structure and falls within the rhombohedral crystal system, under the space group R$\bar{3}$c. The SAED pattern in Fig. 2f, related to the image in Fig. 2e, displays the diffraction spots of the ($\bar{2}$1$\bar{3}$), ($\bar{2}$01) and (0$\bar{1}$4) planes along the [182] direction. A comparison with the standard PDF card (ICDD No. 00-038-1479) reveals a slight increase in the interplanar spacing of the (0$\bar{1}$4) crystal plane of Cr₂O₃, potentially attributed to Fe³⁺ incorporation into Cr₂O₃. The compositional details of these phases are provided in Table 3. In conclusion, at 1473 K, the Fe₃O₄–Cr₂O₃ system exhibits three-phase equilibrium relationships: a two-phase equilibrium between FeCr₂O₄ and Cr₂O₃, a singular FeCr₂O₄ phase, and a singular Cr₂O₃ phase.

Fig. 2

TEM pattern of quenched Sample 2. a TEM image of FeCr₂O₄; b HRTEM image of FeCr₂O₄; c SAED pattern corresponding to b; d TEM image of Cr₂O₃; e HRTEM image of Cr₂O₃; f SAED pattern corresponding to e

Table 3 Composition of corresponding phases in Fe₃O₄–Cr₂O₃ system (wt.%)

4.2 Eutectic reaction of Fe₃O₄–CaO system under oxygen partial pressure of 0.15 Pa

Based on the thermodynamic parameters assessed by Selleby and Sundman [17], a eutectic reaction in the Fe₃O₄–CaO system was identified under oxygen partial pressure of 0.15 Pa, with the calculated eutectic reaction temperature being 1475 K. To validate these calculated results, experimental studies on the eutectic reaction were conducted. The eutectic temperature was initially determined using differential scanning calorimetry (DSC) in a simultaneous thermal analyzer. The DSC curve of Sample 8 is depicted in Fig. 3. This curve, which extends from 800 to 1600 K, is shown as a solid line, with a dashed line indicating the enlarged section of the curve. Generally, the DSC curve tends downward, indicating that the specific heat of the sample does not vary significantly with temperature. Notably, an endothermic peak is observed on the DSC curve, which corresponds to the eutectic reaction of Liquid → Ca₂Fe₂O₅ + Fe₃O₄. The onset temperature of the eutectic reaction, based on the DSC curve, is determined to be 1485 K. However, considering the potential lag caused by the heating rate during detection, the calculated eutectic temperature of 1475 K is deemed reasonable. This consideration helps align the experimental observations with the thermodynamic calculations.

Fig. 3

DSC curve of Sample 8 in heat flow–temperature graph. Solid line means DSC curve from 800 to 1600 K and dashed line means locally enlarged area

Quenched Sample 6 was examined to study the liquid phase composition. Figure 4 illustrates the XRD patterns of the quenched Sample 6 at 1485 K, along with the backscattered electron (BSE) micrograph. In Fig. 4a, the XRD patterns indicate the presence of Ca₂Fe₂O₅. As depicted in Fig. 4b, two distinct phases are observed in quenched Sample 6. The brighter phase is identified as the liquid phase, while the darker phase is recognized as Ca₂Fe₂O₅. Additionally, black and gray areas in the image represent pores and carbon, respectively. The liquid phase composition measured using the WDS was 81.9 wt.% Fe₃O₄. Quenched Sample 7 was used to investigate the maximum solid solubility of CaO in Fe₃O₄. The XRD patterns and BSE micrograph for quenched Sample 7 are presented in Fig. 5. The XRD patterns reveal the presence of a spinel phase in Fig. 5a. Based on Fig. 5b, both spinel and liquid phases are identified in quenched Sample 7. The maximum solid solubility of CaO in Fe₃O₄ measured using the WDS was 2.9 wt.%.

Fig. 4

XRD patterns (a) and backscattered electron micrograph (b) of quenched Sample 6

Fig. 5

XRD patterns (a) and backscattered electron micrograph (b) of quenched Sample 7

4.3 Phase equilibria of β-CaCr₂O₄ under oxygen partial pressure of 0.15 Pa at 1473 K

The phase diagram of the CaO–Cr₂O₃ system is significantly influenced by the oxygen partial pressure. Under reducing conditions, only CaCr₂O₄, which contains chromium in the (+ 3) valence state, forms in the CaO–Cr₂O₃ system. CaCr₂O₄ crystals are classified into high-temperature (α), low-temperature (β), and metastable (δ) forms [11, 27, 28]. Kaiser et al. [11] reported the transformation of δ-CaCr₂O₄ to β-CaCr₂O₄ at 1516 K under reducing conditions. Quenched Samples 9 and 10 were analyzed to investigate the phase equilibrium relationship of the CaO–Cr₂O₃ system at 1473 K. The XRD patterns of these samples, presented in Fig. 6, show Cr₂O₃ and β-CaCr₂O₄ in quenched Sample 9 and CaO and β-CaCr₂O₄ in quenched Sample 10. The experimental results suggest that the crystal form of CaCr₂O₄ at 1473 K is the low-temperature (β) form, identified using the ICDD No. 01-070-2387. The solid solubility of Cr₂O₃ in CaO decreases rapidly with temperature and is considered very small at 1473 K, making accurate measurement challenging. Consequently, the solid solubility of Cr₂O₃ in CaO at 1473 K has not been comprehensively explored in this work. Parameters reflecting this solid solubility will be addressed in the subsequent optimization process.

Fig. 6

XRD patterns of quenched samples at 1473 K in CaO–Cr₂O₃ system

4.4 Thermodynamic calculations

Utilizing the CALPHAD method, thermodynamic evaluations of the Fe₃O₄–Cr₂O₃, Fe₃O₄–CaO, and Cr₂O₃–CaO systems were conducted using the PARROT module in Thermo-Calc software. This approach operates by minimizing the sum of the squared differences between experimental data and calculated results. The optimized parameters assessed in the present work are presented in Table 4. GCAOSOL and GCR2O3 are Gibbs energy expression abbreviations for solid phase CaO and Cr₂O₃, respectively, as detailed in supplementary materials. In supplementary materials, all parameters for the system are listed together with the models used.

Table 4 Thermodynamic parameters assessed in present work

The phase diagram of the Fe₃O₄–Cr₂O₃ system was calculated using the thermodynamic parameters provided by Kjellqvist et al. [16], with the oxygen partial pressure set at 0.15 Pa. According to the calculations of Kjellqvist et al. [16], the solid solubility of Cr₂O₃ in Fe₃O₄ is 62.3 wt.% at 1473 K, aligning with the experimental data of this study. However, there is a discrepancy in the solid solubility of Fe₃O₄ in Cr₂O₃ at 1473 K; the experimental results indicate a solubility of 3.2 wt.%, which does not match the value of 9.2 wt.% calculated by Kjellqvist et al. [16]. To reconcile the experimental data with the calculated results, optimization of the parameter ${{}^{0}{{L}}}_{{\text{Cr}}^{3+}\text{,}{\text{Fe}}^{3+}\text{:Va:}{\text{O}}^{2-}}^{{\text{Corundum}}}$ was undertaken. The optimized phase diagram of the Fe₃O₄–Cr₂O₃ system is illustrated in Fig. 7. And at high temperature, there is only one liquid phase and no miscibility gap.

Fig. 7

Calculated phase diagram of Fe₃O₄–Cr₂O₃ system under oxygen partial pressure of 0.15 Pa with experimental data

The phase diagram of the Fe₃O₄–CaO system, calculated using the thermodynamic parameters provided by Selleby and Sundman [17] at oxygen partial pressure of 0.15 Pa, is depicted in Fig. 8. The experimentally measured solid solubility of CaO in Fe₃O₄ is 2.9 wt.%, aligning with the calculated results. Additionally, the calculated eutectic reaction temperature for the Liquid → Ca₂Fe₂O₅ + Fe₃O₄ is 1475 K, matching the temperature measured by DSC in this work. However, there is a slight discrepancy in the liquid composition. The calculated value is 77.6 wt.% Fe₃O₄, whereas the experimentally measured composition is 81.9 wt.% Fe₃O₄. The Fe₃O₄–CaO binary system is actually a projection of the FeO–Fe₂O₃–CaO ternary system along the Fe₃O₄–CaO direction. In the actual process, the starting position of Fe₃O₄ may deviate slightly from the calculated position, because in the FeO–Fe₂O₃–CaO ternary system, the position of Fe₃O₄ is a region rather than a point at high temperature. Therefore, this set of thermodynamic parameters has been selected for extrapolating the isothermal section of the Fe₃O₄–Cr₂O₃–CaO system at 1473 K.

Fig. 8

Calculated phase diagram of Fe₃O₄–CaO system under oxygen partial pressure of 0.15 Pa with experimental data

The calculated phase diagram of the CaO–Cr₂O₃ system under oxygen partial pressure of 0.15 Pa is illustrated in Fig. 9. Within this system, only CaCr₂O₄ is taken into consideration for optimization. The low-temperature β form of CaCr₂O₄ exhibits a spinel structure, whereas the high-temperature α form parallels the structure of SrCr₂O₄. The transition temperature between α-CaCr₂O₄ and β-CaCr₂O₄ is calculated to be 1918 K, aligning with the experimental findings of Panek and Kanclir [13]. Additionally, CaCr₂O₄ also possesses a metastable δ form. The CALPHAD method focuses on determining the variation of Gibbs free energy in relation to the system's composition at each temperature, thereby identifying the phase of equilibrium coexistence when the system's total Gibbs energy is minimized. Since a metastable phase is not thermodynamically stable and only temporarily stable due to thermodynamic energy barriers or dynamic factors, it typically has a higher Gibbs energy compared to the stable phase. Therefore, the δ-CaCr₂O₄ form is not included in the optimization process. The calculated heat capacity of β-CaCr₂O₄, compared with experimental data [29], is shown in Fig. 10. The heat capacity of CaCr₂O₄ was estimated using the Neumann–Kopp rule based on heat capacity data of CaO and Cr₂O₃, yielding results broadly consistent with the experimental data reported by Lee and Nassaralla [29]. The calculated eutectic temperature for Liquid → CaO + CaCr₂O₄ is 2238 K, in agreement with the experiments conducted by Panek [12]. Ol’shanskii et al. [14] reported a eutectic reaction involving Liquid → Cr₂O₃ + CaCr₂O₄. The eutectic temperature for this reaction is higher than that of Liquid → CaO + CaCr₂O₄, though it has not been precisely measured. In this study, a eutectic temperature of 2314 K is adopted. The maximum solid solubility of Cr₂O₃ in the CaO phase is calculated to be 4.7 wt.%.

Fig. 9

Calculated phase diagram of CaO–Cr₂O₃ system under oxygen partial pressure of 0.15 Pa with experimental data [12, 13]

Fig. 10

Calculated heat capacity of β-CaCr₂O₄ with the experimental data [29]

5 Discussion

The ternary phase diagram can be inferred from the thermodynamic data of binary systems. This is accomplished by multiplying the thermodynamic parameters of the binary systems by weight factors and summing them up to represent the thermodynamic properties of the ternary system. By integrating the phase diagrams of the three binary systems of Fe₃O₄–Cr₂O₃, Fe₃O₄–CaO and CaO–Cr₂O₃, the ternary phase diagram of the Fe₃O₄–Cr₂O₃–CaO system can be extrapolated. However, it is important to note that such direct extrapolation does not account for the solid solubility of compounds or the potential formation of new ternary compounds. Therefore, some discrepancies may exist between the extrapolated phase diagram and the actual experimental observations. The extrapolated isothermal section of the Fe₃O₄–Cr₂O₃–CaO system at 1473 K under oxygen partial pressure of 0.15 Pa is presented in Fig. 11.

Fig. 11

Extrapolated isothermal section of Fe₃O₄–Cr₂O₃–CaO system at 1473 K under oxygen partial pressure of 0.15 Pa

As depicted in Fig. 11, the Fe₃O₄–Cr₂O₃–CaO system at 1473 K exhibits twelve major types of phase equilibrium relationships. These include three-phase equilibrium among β-CaCr₂O₄, Ca₂Fe₂O₅, and halite, three-phase equilibrium involving the liquid phase, Ca₂Fe₂O₅, and spinel, three-phase equilibrium among β-CaCr₂O₄, Ca₂Fe₂O₅, and spinel, three-phase equilibrium among β-CaCr₂O₄, corundum, and spinel, two-phase equilibrium between the liquid phase and Ca₂Fe₂O₅, two-phase equilibrium between the liquid phase and spinel, two-phase equilibrium between Ca₂Fe₂O₅ and spinel, two-phase equilibrium between β-CaCr₂O₄ and spinel, two-phase equilibrium between corundum and spinel, two-phase equilibrium between β-CaCr₂O₄ and corundum, a single spinel phase, and a single liquid phase. Interestingly, despite the absence of a liquid phase in the corresponding binary systems at 1473 K, the isothermal section extrapolated from thermodynamic parameters of the binary systems reveals a narrow liquid phase region. This observation underscores the complexities and interactions within the ternary system that may not be immediately apparent from the binary phase diagrams alone.

In the Fe₃O₄–Cr₂O₃–CaO system at 1473 K, Cr³⁺ and Ca²⁺ ions have the potential to substitute Fe³⁺ and Fe²⁺ ions in Fe₃O₄, respectively, leading to the formation of a (Fe, Ca)(Fe, Cr)₂O₄ solid solution. This substitution results in a spinel single-phase region within the isothermal section of the system. However, the solid solubility limits of CaO and Cr₂O₃ in Fe₃O₄ are 2.9 and 62.8 wt.%, respectively, suggesting that the spinel phase is predominantly chromite (FeCr₂O₄). Identifying the phase region of chromite is critical for the recovery of valuable chromium from stainless steel pickling sludge. β-CaCr₂O₄ and CaFe₂O₄ share similar structural characteristics, and both are part of the orthorhombic system with the space group Pbnm. Consequently, a solid solution forms between β-CaCr₂O₄ and CaFe₂O₄. Amberúz et al. [30] have previously studied the phase equilibrium of the CaCr₂O₄–CaFe₂O₄ system and reported the formation of a Ca(Cr, Fe)₂O₄ solid solution. However, the extrapolation results do not account for the solid solubility of the third component in the binary compound. Thus, future work should focus on exploring the Ca(Cr, Fe)₂O₄ solid solution and completing the isothermal section of the system at 1473 K under oxygen partial pressure of 0.15 Pa, to provide a more comprehensive understanding of the system's phase behavior.

6 Conclusions

• 1. The solid solubility of Cr₂O₃ in Fe₃O₄ at 1473 K was clarified. Additionally, the eutectic reaction involving Liquid → Ca₂Fe₂O₅ + Fe₃O₄ was investigated. Under the specified oxygen partial pressure at 1473 K, the CaO–Cr₂O₃ system was found to form β-CaCr₂O₄.

• 2. Utilizing the ionic two-sublattice model to describe the liquid phase in the CaO–Cr₂O₃ binary system, a self-consistent set of thermodynamic parameters for this system was established. This was achieved using reliable experimental data from both the literature and the current study. Furthermore, the solid solubility of Fe₃O₄ in Cr₂O₃ at 1473 K was optimized based on experimental findings from this work.

• 3. The calculated isothermal section of the Fe₃O₄–Cr₂O₃–CaO system at 1473 K under oxygen partial pressure of 0.15 Pa was determined using the thermodynamic parameters of the binary systems. This enabled the identification of the phase region of chromite at 1473 K, providing a theoretical foundation for the recycling of valuable metal chromium from stainless steel pickling sludge.