Thermodynamic Modeling of Oxide Phases in the Mn-O System

Abstract

A critical evaluation and thermodynamic modeling for thermodynamic properties of all oxide phases and phase diagrams in the Mn-O system are presented. Optimized Gibbs energy parameters for the thermodynamic models of the oxide phases were obtained which reproduce all available and reliable experimental data within error limits from 298 K (25 °C) to above the liquidus temperature at compositions covering from MnO to MnO₂, and oxygen partial pressure from 10⁻¹⁵ to 10² (bar). The optimized thermodynamic properties and phase diagrams are believed to be the best estimates presently available. Two spinel phases ($\alpha$- and $\beta$-Mn₃O₄) were modeled using Compound Energy Formalism (CEF) with the use of physically meaningful parameters. Valence states of the spinels are interpreted based on the available thermopower measurement, for which Mn⁴⁺ was considered in the cubic spinel ($\beta$-Mn₃O₄). The present Mn₃O₄ spinel solutions can be integrated into a larger spinel solution database, which has been already developed. The database of the model parameters can be used along with a software for Gibbs energy minimization in order to calculate any type of phase diagram sections and thermodynamic properties.

Introduction

Manganese oxides are of great interest in metallurgical fields (iron and steelmaking, ferroalloy production), since the manganese plays an important role as a major alloying element in steel, and manganese silicate is one of the abundant inclusions in steels. Also manganese oxides have been interesting materials for their applications to oxide fuel cell as one of the energy materials, and magnetic storage materials. Mn nodule under pacific ocean also has been interesting as noble metal resources. Therefore, knowledge for the manganese oxide containing system is valuable to the industries. In this regard, a research on the thermodynamic properties and phase equilibria in the Mn-O system is necessary. Extensive reviews of several investigations and comparisons of data obtained in the Mn-O system were given by Wang and Sundman[1] and Grundy et al.[2] by critical evaluations and thermodynamic assessments.

Apart from metallic Mn ($\alpha$-, $\beta$-, $\gamma$-, $\delta$-Mn and L-Mn), the Mn-O system is composed of the following phases: manganosite (Mn$_{1-x}$O), hausmannite ($\alpha$- and $\beta$-Mn₃O₄), bixbyite ($\alpha$- and $\beta$-Mn₂O₃), pyrolusite (MnO₂), and liquid oxide (slag). Mn may exist as divalent, trivalent, and tetravalent ions in those oxide phases depending on phase, temperature, and oxygen partial pressure. Non-stoichiometries of the Mn₂O₃ and MnO₂ are very small, while manganosite deviates considerably from its stoichiometry at high temperature/oxygen partial pressure. Non-stoichiometries of $\alpha$- and $\beta$-Mn₃O₄ were observed.[3]

The earlier thermodynamic assessment by Wang and Sundman[1] used Compound Energy Formalism[4] to model the manganosite and Two Sublattice Ionic Model[5] for the liquid phase which extends from metallic liquid to ionic liquid. Other phases were regarded as stoichiometric compounds. Later, Grundy et al.[2] pointed out that the thermodynamic description of Wang and Sundman[1] had some flaws and they re-assessed the Mn-O system using the same modeling approach but different set of model parameters. In both studies, $\alpha$- and $\beta$-Mn₃O₄ were treated as stoichiometric compounds.

However, when an optimized thermodynamic database of the Mn-O system is to be integrated into an already developed large thermodynamic database, keeping consistency between the databases is indispensible. Since the manganese spinels ($\alpha$- and $\beta$-Mn₃O₄, both) form extensive solid solutions with Fe, Al, Cr, Co, Mg, etc.,[6] $\alpha$-Mn₃O₄ and $\beta$-Mn₃O₄ have to be modeled as a part of larger thermodynamic database of spinel solid solution, for example, developed at the CRCT, École Polytechnique de Montréal.[7] Therefore, in the present study, it has been attempted to model both $\alpha$-Mn₃O₄ and $\beta$-Mn₃O₄ spinels as solutions, not stoichiometric compounds. Recently, Kjellqvist and Selleby updated thermodynamic modeling of the Mn-O system by Grundy et al.,[2] by taking into account the two spinel phases as a part of larger solution composed of Fe-Mn-O,[8] but only limited results were presented in their article.

The present study is a part of a complete database development of the Al₂O₃-CaO-CoO-CrO-Cr₂O₃-Cu₂O-FeO-Fe₂O₃-MgO-MnO-Mn₂O₃-NiO-PbO-SiO₂-TiO₂-Ti₂O₃-ZnO systems for applications in the ferrous, non-ferrous, ceramic, and electric/electronic industries. In particular, in conjunction with the present study of Mn oxide, critical evaluation and thermodynamic optimization for the Mn-Cr-O system were already published,[9] and those of Fe-Mn-O, Mn-Si-O, and Fe-Mn-Si-O systems will be described elsewhere.[10,11] Some of the results obtained in the present study was presented in a conference proceeding.[12] All the thermodynamic calculations in the present study were carried out using FactSage.[13,14]

Thermodynamic Models

Figure 1 shows a phase diagram of the Mn-O system as a function of temperature and oxygen partial pressure, calculated from the thermodynamic models of the present study. The following phases are found in this system: Slag, Mn$_{1-x}$O, $\alpha$-Mn₃O₄, $\beta$-Mn₃O₄, $\alpha$-Mn₂O₃, $\beta$-Mn₂O₃, MnO₂, $\alpha$-Mn, $\beta$-Mn, $\gamma$-Mn, $\delta$-Mn, and liquid Mn (L-Mn).

Fig. 1

Calculated phase diagram of the Mn-O system along with experimental data[20,21,55,67–70]

$\alpha$-Mn₃O₄ and $\beta$-Mn₃O₄ (Spinel)

Thermodynamic models for the spinel solutions were developed using two sublattice model within the framework of the Compound Energy Formalism (CEF).[22] Using the model, a thermodynamic database for a multicomponent spinel composed of Al-Co-Cr-Fe-Mg-Ni-Zn-O has been developed.[7] In the present study, this model was extended to incorporate Mn. Moreover, since the model and the database were previously developed only for the cubic spinel, a new thermodynamic model using the similar approach (CEF) was developed for the tetragonal spinel ($\alpha$-Mn₃O₄).

In the cubic spinel, oxygen ions form a fcc close packing structure. Cations occupy half of octahedral interstices and one-eighth of tetrahedral interstices. The number of the octahedral sites is twice the number of the tetrahedral sites. Therefore, a formula unit of spinel may be written as (i)${^{\text{T}}}$[j]${^{\text{O}}_2}$O₄. For the tetragonal spinel, the same close packing scheme of oxygen in the tetragonal structure may be used. Additional interstitial site to describe a deviation toward Mn side was not considered in the present study.

The Gibbs energy expression in the CEF per formula unit of spinel is

$$\begin{aligned} G = \sum _i\sum _jy_i^{\text{T}}y_j^{\text {O}}G_{ij}-TS^{\text{config}} + G^{\text{excess}} + G^{\text {magnetic}} \end{aligned},$$

(1)

where $y_i^{\text{T}}$ and $y_j^{\text{O}}$ represent the site fractions of species i and j on the tetrahedral and octahedral sites, and $G_{ij}$ is the Gibbs energy of an “end-member” $(i)^{\text{T}}[j]_2^{\text{O}}{\text{O}_4}$ of the solution, in which the first sublattice is occupied only by the species i and the second sublattice only by the species j. $S^{\text{config}}$ is the configurational entropy assuming random mixing on each sublattice:

$$\begin{aligned} S^{\text{config}} = -{\text{R}}\Big (\sum _iy_i^{\text{T}}\ln y_i^{\text{T}} + 2\sum _jy_j^{\text{O}}\ln y_j^{\text{O}}\Big ) \end{aligned}.$$

(2)

$G^{\text{magnetic}}$ is the contribution of magnetic properties on the Gibbs energy, which is explained in Section II–E.

Since the Mn₃O₄ exhibits two different structures (tetragonal and cubic), thermodynamic modeling was carried out separately in the present study.

$\alpha$-Mn₃O₄ (tetragonal spinel)

For the tetragonal Mn₃O₄ spinel solution, the following model structure was considered:

$$\begin{aligned} \left({\text{Mn}}^{2+}, {\text{Mn}}^{3+}\right)^{\text{T}}\left[{\text{Mn}}^{2+}, {\text {Mn}}^{3+}, {\text{Va}}\right]^{\text{O}}_2{\text{O}_4} \end{aligned},$$

(3)

for which the reason is discussed in detail in Section III–B.

In the present study, no excess Gibbs energy terms were required, that is $G^{\text{excess}} = 0$. The Gibbs energy of the tetragonal spinel solution is thus fully defined by the Gibbs energies $G_{ij}$ of the six end-members. These are denoted as $G_{\text{JJ}}$, $G_{\text{JK}}$, $G_{\text{JV}}$, $G_{\text{KJ}}$, $G_{\text{KK}}$, and $G_{\text{KV}}$, where J, K, and V mean Mn²⁺, Mn³⁺, and Va, respectively. These are also connected each other by certain linear combinations of the $G_{ij}$, having physical sense as

$$\begin{aligned} G_{\text{JK}}&\end{aligned},$$

(4)

$$\begin{aligned} G_{\text{KJ}}= \,& {} G_{\text{JK}} \end{aligned},$$

(5)

$$\begin{aligned} G_{\text{KK}}= \,& {} G_{\text{JK}} + I_{\text{JK}} \end{aligned},$$

(6)

$$\begin{aligned} G_{\text{JJ}}= \,& {} G_{\text{JK}} - I_{\text{JK}} + \Delta _{\text{JK}} \end{aligned},$$

(7)

$$\begin{aligned} G_{\text{KV}}= \,& {} 8G^\circ ({\text{Mn}}_2{\text{O}}_3, \gamma ) - 2{\text{R}}T(5\ln 5 - 6\ln 6) - 5G_{\text{JK}} - 5I_{\text{JK}} \end{aligned},$$

(8)

$$\begin{aligned} G_{\text{JV}}= \,& {} \Delta _{\text{JK}} - \Delta _{\text{KJV}} - 6I_{\text{JK}} \nonumber \\&+ 8G^\circ ({\text{Mn}}_2{\text{O}}_3, \gamma ) - 2{\text{R}}T(5\ln 5 - 6\ln 6) - 5G_{\text{JK}} \end{aligned}.$$

(9)

$G_{\text{JK}}$ represents a Gibbs energy of $({\text{Mn}}^{2+})^{\text{T}}[{\text{Mn}}^{3+}]_2^{\text{O}}{\text{O}}_4$ end-member in the tetragonal spinel structure. $G_{\text{KJ}}$ was set to the same to $G_{\text{JK}}$ as the reference state for charged end-members of spinels, as was done for cubic spinel.[22] $I_{\text{JK}}$ was used to set $G_{\text{KK}}$, and represents degree of inversion of cation distribution. $\Delta _{\text{JK}}$ was used to set $G_{\text{JJ}}$. Gibbs energy of end-members containing Va ($G_{\text{KV}}$ and $G_{\text{JV}}$) were described by Eqs. [8] and [9], using Gibbs energy of a hypothetical “$\gamma$-Mn₂O₃” (Mn deficient spinel structure). The Gibbs energy of the “$\gamma$-Mn₂O₃” was optimized in order to model experimentally observed non-stoichiometry of $\alpha$-Mn₃O₄[3] toward O-rich side, assuming an ideal dissolution of a hypothetical “$\gamma$-Mn₂O₃” (Mn deficient spinel structure) in the spinel.

All the parameters are listed in Table I.

Table I Optimized Model Parameters for Tetragonal Spinel Phases

Cubic spinel solution ($\beta$-Mn₃O₄)

For the cubic Mn₃O₄ spinel solution, the following model structure was considered:

$$\begin{aligned} ({\text{Mn}}^{2+})^{\text {T}}[{\text {Mn}}^{2+}, {\text{Mn}}^{3+}, {\text{Mn}}^{4+}, {\text{Va}}]^{\text {O}}_2{\text{O}}_4 \end{aligned},$$

(10)

for which the reason is discussed in detail in Section III–B. As was done for the tetragonal spinel, no excess Gibbs energy terms were required, that is $G^{\text{excess}} = 0$. The Gibbs energy of the cubic spinel solution is thus fully defined by the Gibbs energies $G_{ij}$ of the four end-members. These are denoted as $G_{\text{JJ}}$, $G_{\text{JK}}$, $G_{\text{JL}}$, and $G_{\text{JV}}$, where J, K, L, and V mean Mn²⁺, Mn³⁺, Mn$^{{4+}}$, and Va, respectively. Since the present study is part of a larger effort aimed at developing a complete database for multicomponent spinel, it is necessary to maintain consistency in parameters which are common to several sub-systems. Therefore, although only Mn₃O₄ spinel is considered in this article, the Gibbs energies of end-members $G_{ij}$ in the Mn-O system are linked to the Gibbs energies of end-members in other sub-systems. In the present study, Gibbs energies of end-members in the Fe₃O₄,[22] and (Fe,Mn)₃O₄[10] which was simultaneously optimized in the present study, were used as parts of Gibbs energies of end-members in the $\beta$-Mn₃O₄. Also, certain linear combinations of the $G_{ij}$, which have physical sense, are used as the optimized model parameters. These linear combinations are related each other as

$$\begin{aligned} G_{\text{JK}}&\end{aligned},$$

(11)

$$\begin{aligned} G_{\text{JJ}}= & {} -2G_{\text{AE}} - 2I_{\text{AE}} + 3G_{\text{JE}} + I_{\text{JE}} + \Delta _{\text{JE}} \end{aligned},$$

(12)

$$\begin{aligned} G_{\text{JL}}= \,& {} 2G_{\text{JK}} - G_{\text{JJ}} + \Delta _{\text{JL}}\end{aligned},$$

(13)

$$\begin{aligned} G_{\text{JV}}= \,& {} \frac{5}{7}G_{\text{JK}} + V_x \end{aligned},$$

(14)

where A and E represent Fe²⁺ and Fe³⁺, respectively. $G_{\text{JK}}$ represents Gibbs energy of $({\text{Mn}}^{2+})^{\text{T}}[{\text{Mn}}^{3+}]_2^{\text{O}}{\text{O}}_4$ end-member in the cubic spinel structure. $V_x$ is used to set $G_{\text{JV}}$ in order to reproduce non-stoichiometry of $\beta$-Mn₃O₄ toward O-rich side, followed by the suggestion of Degterov et al.[22] $\Delta _{\text{JL}}$ is used to set $G_{\text{JL}}$ in order to reproduce the degree of inversion. Description of $G_{\text{JJ}}$ is connected to the Gibbs energy of other end-members in the Fe–O system[22] and those in the Fe–Mn-O system, which can be found elsewhere.[10] All the parameters are listed in Table II.

Table II Optimized Model Parameters for Cubic Spinel Phases

Slag (Molten Oxide)

For the slag (molten oxide) phase, the Modified Quasichemical Model in the pair approximation (MQM) was used.[15,16] This model has been recently further developed and summarized.[17,18] Short-range ordering (SRO) is taken into account by considering second-nearest neighbor (SNN) pair exchange reactions. For example, for the MnO-MnO$_{1.5}$ slag, this reaction is

$$\begin{aligned} ({\text{Mn}}^{2+}-{\text{O}}-{\text{Mn}}^{2+}) + ({\text {Mn}}^{3+}-{\text{O}}-{\text{Mn}}^{3+}) = 2({\text{Mn}}^{2+}-{\text{O}}-{\text{Mn}}^{3+})\quad \Delta g_{\text {Mn}^{2+}{\text{Mn}}^{3+}} \end{aligned},$$

(15)

where (i-O-j) represents a SNN pair. The parameter of the model is the Gibbs energy of this pair forming reaction, $\Delta g_{\text {Mn}^{2+}{\text{Mn}}^{3+}}$, which may be expanded as empirical functions of composition. There being no evidence to the contrary in the present optimization, it was assumed that only Mn²⁺ and Mn³⁺ are considered for the molten oxide. “Coordination numbers” of Mn²⁺ and Mn³⁺ required in the MQM were set to be 1.3774 ($Z_{{\text{Mn}}^{2+}{\text{Mn}}^{2+}}$) and 2.0662 ($Z_{{\text{Mn}}^{3+}{\text{Mn}}^{3+}}$), respectively. Gibbs energy of pure liquid MnO was taken from Wu et al.[19] However, a pure liquid “Mn₂O₃” is not stable under normal condition, and its thermodynamic properties has not been known. Therefore, in the present study, the Gibbs energy of hypothetical liquid “Mn₂O₃” was assumed to be 2.5 times of Gibbs energy of the pure liquid MnO, and an additional enthalpic term was used as a model parameter during optimization in order to reproduce measured liquidus temperatures[20,21] (Actual model component is MnO$_{1.5}$, not Mn₂O₃). $\Delta g_{{\text{Mn}}^{2+}-{\text{Mn}}^{3+}}$ was set to zero. The optimized thermodynamic values for the molten oxide are shown in Table III.

Table III Optimized Model Parameters for Slag, Manganosite, and Stoichiometric Compounds

Manganosite Solution (Mn$_{1-x}$O)

The manganosite solution of rock-salt structure was modeled as a simple random mixture of Mn²⁺ and Mn³⁺ ions on cation site using a simple polynomial excess Gibbs energy. It is assumed that cation vacancies associate with the Mn³⁺ cation in order to maintain electrical neutrality and do not contribute to the configurational entropy independently. The Gibbs energy per mole of the solution is given as

$$ G = X_{{\text{Mn}}^{2+}}G_{\text{MnO}} + X_{{\text{Mn}}^{3+}}G_{{\text{MnO}}_{1.5}} + {\text{R}}T (X_{{\text{Mn}}^{2+}}\ln X_{{\text{Mn}}^{2+}} + X_{{\text{Mn}}^{3+}}\ln X_{{\text{Mn}}^{3+}}) + X_{{\text{Mn}}^{2+}}X_{{\text{Mn}}^{3+}}L_{{\text{Mn}}^{2+}{\text{Mn}}^{3+}} $$

(16)

where $L_{{\text{Mn}}^{2+}{\text{Mn}}^{3+}}$ may be expanded as a function of polynomial:

$$\begin{aligned} L_{{\text{Mn}}^{2+}{\text{Mn}}^{3+}} = \sum _{i,j\ge 0}q_{{\text{Mn}}^{2+}{\text{Mn}}^{3+}}^{ij}X_{{\text{Mn}}^{2+}}^iX_{{\text{Mn}}^{3+}}^j \end{aligned}$$

(17)

Metallic Phases

All metallic Mn ($\alpha$-Mn, $\beta$-Mn, $\gamma$-Mn, $\delta$-Mn, and L-Mn) were treated as stoichiometric phases and their Gibbs energies were taken from Dinsdale.[23]

Magnetic Properties

$\alpha$-Mn₃O₄, $\alpha$-Mn₂O₃, and MnO₂ have magnetic transition associated with Néel temperatures. Their magnetic contributions to the Gibbs energies were considered by an empirical relationship suggested by Inden[24] and modified by Hillert and Jarl.[25] Although the suggested Gibbs energy equation for the magnetic property was developed for cubic (FCC, BCC) and hexagonal (HCP) structures,[25] in the present study, the same equation with the same p factor for FCC (p = 0.28) was assumed for $\alpha$-Mn₃O₄ (tetragonal), $\alpha$-Mn₂O₃ (orthorhombic), and MnO₂ (tetragonal) phases.

Critical Evaluation of Experimental Data and Thermodynamic Optimization

Thermodynamic and Magnetic Properties

Gibbs energy of stoichiometric MnO was taken from Wu et al.[19]

Low-temperature heat capacities and heat contents of Mn₃O₄, Mn₂O₃, and MnO₂ were measured by a number of researchers. Millar measured the heat capacities of Mn₃O₄ and MnO₂ calorimetrically down to ~72 K (−201 °C).[26] He did not observe $\lambda$-type anomaly of hear capacity of Mn₃O₄, which was later found by Robie and Hemingway[27] and Chhor et al.[28] at ~43 K (−230 °C). Also, the measured heat capacity of MnO₂ by Millar is systematically higher than later measurements by Robie and Hemingway[27] and Kelley and Moore.[29] Kelley and Moore pointed out that Millar’s calibration on his calorimeter had small error which caused the systematic error. Other measurements of the heat capacities of Mn₂O₃ by King[30] and Robie and Hemingway[27] were in good agreement with each other. Robie and Hemingway also observed a broad change of heat capacity near 307.5 K (34.5 °C), which is associated with the orthorhombic to cubic transition discovered by Geller and Espinosa.[31] All the measured heat capacity data are shown in Figures 2 through 4 with optimized heat capacities in the present study. By fitting the heat capacities considering magnetic transitions, the third law entropies for these phases were calculated as 168.06 J/mol$\cdot$K for the Mn₃O₄, 115.78 J/mol K for the Mn₂O₃, and 53.48 J/mol K for the MnO₂, while the reported values are in the ranges of 154.07 to 167.1 J/mol K for the Mn₃O₄, 110.5 to 113.7 J/mol K for the Mn₂O₃, and 52.75 to 58.32 J/mol K for the MnO₂.[2] All these phases experience phase transitions from the antiferromagnetic to the paramagnetic states and corresponding Néel temperatures are shown in Tables I and III along with optimized magnetic momentum $\beta$. Large heat capacity data of $\alpha$-Mn₃O₄ at the magnetic transition temperature observed by Robie and Hemingway[27] and Chhor et al.[28] could not be accounted for by the current magnetic model of Hillert and Jarl.[25]

Fig. 2

Optimized heat capacity of $\alpha$-Mn₃O₄ along with experimental data[26–28]

Fig. 3

Optimized heat capacity of $\alpha$- and $\beta$-Mn₂O₃ along with experimental data[27,30]

Fig. 4

Optimized heat capacity of MnO₂ along with experimental data[26,27,29]

Heat contents of Mn₃O₄, Mn₂O₃, and MnO₂ were also measured calorimetrically by Fritsch and Navrotsky,[32] Southard and Moore,[33] Orr,[34] and Moore.[35] A discontinuity of heat content of Mn₃O₄ at 1441 K to 1448 K (1168 °C to 1175 °C) associated with $\sim$20 kJ/mol was observed by Southard and Moore.[33] This corresponds to the enthalpy of transformation from tetragonal to cubic structure of the Mn₃O₄ spinel. Optimized heat contents for these phases are in good agreement with the measured data as shown in Figure 5.

Fig. 5

Optimized heat contents of $\alpha$- and $\beta$-Mn₃O₄, $\beta$-Mn₂O₃, and MnO₂ along with experimental data[32–35]

Valence State in Spinel

The valence state of cations in Mn₃O₄ is still controversial. Some investigations suggested “2+/4+” structure (Mn$^{2+}_2$Mn⁴⁺O₄),[36–38] while others are in favor of the “2+/3+” structure (Mn²⁺Mn$^{3+}_2$O₄).[39–41] In spite of this controversy, it is generally accepted that Mn₃O₄ is almost normal spinel due to high site preference energies of Mn³⁺ and Mn⁴⁺ for octahedral sites,[42–44] and Mn³⁺ in octahedral site is in charge of tetragonal distortion of spinel containing Mn.[40]

A later investigation by Dorris and Mason[45] using thermopower measurement on electrical properties and cation valences in Mn₃O₄ suggested that “2+/3+” structure cannot explain the electrical properties of the cubic Mn₃O₄. They observed a sudden increase of electrical conductivity and decrease of thermopower by passing the tetragonal-to-cubic transformation temperature. This implies a significant change of valence status between these two spinels. For the tetragonal Mn₃O₄, they concluded that the “2+/3+” structure is reasonable on the basis of a direct relationship between the concentration of octahedral Mn³⁺ and the tetragonal-to-cubic transformation temperature,[46,47] and the high transformation temperature in Mn₃O₄. On the other hand, in order to see feasibility the “2+/3+” structure for the cubic Mn₃O₄, they considered the following cation distribution reaction in sublattices:

$$\begin{aligned} {\text{Mn}}^{2+}_{\text{T}} + {\text{Mn}}^{3+}_{\text{O}} \Leftrightarrow {\text{Mn}}^{2+}_{\text{O}} + {\text{Mn}}^{3+}_{\text{T}} \qquad \Delta H_{\text{dis}} \end{aligned},$$

(18)

where values for the $\Delta H_{\text{dis}}$ have been estimated in the range of 75 to 80 kJ/mole.[42,43] Assuming that conduction can occur on either tetrahedral or octahedral sites, not simultaneously on both sites, they considered four possible conduction models for the cubic Mn₃O₄ as follows:

• 1. The conduction occurs on octahedral sites by electron holes.

• 2. The conduction occurs on tetrahedral sites by electron holes.

• 3. The conduction occurs on octahedral sites by electrons.

• 4. The conduction occurs on tetrahedral sites by electrons.

Using the above four models along with appropriate site balance equation and the measured thermopower data, they showed that the models (1) and (3) gave site fraction of Mn³⁺ on tetrahedral site to be higher than 1. Also, derived enthalpy of the distribution reaction from the models 2 and 4 is calculated to be only 15 to 45 kJ/mole, as opposed to the reported values of 75 to 80 kJ/mole.[42,43] Moreover, non-insulating characteristic of cubic Mn₃O₄ compared to near insulation of tetragonal Mn₃O₄ implies that the “2+/3+” structure is not appropriate to the cubic Mn₃O₄, since the conduction can occur by electrons or electron holes between Mn²⁺ and Mn³⁺ in spite of high octahedral preference energy of Mn³⁺. Hence, they employed an alternate model in which Mn⁴⁺ was considered in octahedral sites along with Mn²⁺ and Mn³⁺, while tetrahedral sites are assumed to be occupied solely by Mn²⁺ due to high octahedral site preference energies of Mn³⁺ and Mn⁴⁺. The following “disproportionate equilibrium” was considered:

$$\begin{aligned} 2{\text{Mn}}^{3+}_{\text{O}} = {\text{Mn}}^{2+}_{\text{O}} + {\text{Mn}}^{4+}_{\text{O}} \qquad \Delta G_{\text{D}} \end{aligned}.$$

(19)

After careful considerations, they concluded that conduction proceeds by the hopping of electron holes between Mn⁴⁺ and Mn³⁺ on octahedral sites. Other conduction mechanisms (conduction by the hopping of electrons or electron holes between Mn²⁺ and Mn³⁺, or conduction by the hopping of electrons between Mn³⁺ and Mn⁴⁺) seemed unlikely. Therefore, in the present study, the valence status of cubic spinel was followed by the suggestion of Dorris and Mason in which Mn²⁺, Mn³⁺, and Mn⁴⁺ mix each other in octahedral site, while Mn²⁺ is the only constituent in tetrahedral site, as seen in Eq. [10]. This is also confirmed by recent electrical conductivity measurement on manganese chromite (MnCr₂O₄) of Lu et al.[48] that the Mn₂CrO₄ can be thought as $({\text{Mn}}^{2+})^{\text{T}}[{\text{Mn}}^{2+}, {\text{Mn}}^{3+}, {\text{Mn}}^{4+}, {\text{Cr}}^{3+}]^{\text{O}}_2{\text{O}}_4$. (Manganese chromite was already modeled in the framework of the present spinel modeling by one of the present authors,[9] keeping consistency to this observation.)

For the tetragonal Mn₃O₄, the usual “2+/3+” model was used, as seen in Eq. [3]. Moreover, vacancy was also considered on octahedral sites for both Mn₃O₄ in order to model non-stoichiometries, although it is known to be very small.[3] However, it is also required to incorporate this spinel model into large spinel database already developed.[22]

Figure 6 shows the mole number of cations in each site per mole of Mn₃O₄ for tetragonal and cubic Mn₃O₄, derived from appropriate conduction mechanisms along with the measured thermopower data by Dorris and Mason,[45] as symbols. Lines in the Figure 6 are calculated in the present study. In order to reproduce the derived cation valencies in these spinels, $\Delta _{\text{JL}}$ for the cubic spinel and $I_{\text{JK}}$ and $\Delta _{\text{JK}}$ for the tetragonal spinel were optimized. As seen in the figure, Mn³⁺ on octahedral site becomes dominant species during cubic-to-tetragonal transformation at 1445 K (1172 °C) in accord with the high preference energy of octahedral site of Mn³⁺.

Fig. 6

Calculated cation distributions in $\alpha$- and $\beta$-Mn₃O₄. For details, see Section III–B. Experimental data were taken from thermopower measurement[45]

Non-stoichiometry in Manganosite and Spinel

Manganosite (Mn$_{1-x}$O) has a non-stoichiometry toward oxygen due to oxidation of Mn (Mn²⁺ $\rightarrow$ Mn³⁺) under high oxygen partial pressure. Contrary to wüstite (Fe$_{1-x}$O), the manganosite has also a non-stoichiometry toward Mn ($-0.128< x < 0$).[49] However, in the present study, only the non-stoichiometry toward oxygen was considered. Defect mechanism of the non-stoichiometry has generally been known to be cation vacancy.[50–54] However, most of those researches showed that a single defect species does not applicable over wide range of oxygen partial pressure. For example, experimental data of non-stoichiometry by Keller and Dieckmann[53] showed that isotherms of log x (measured by TGA) vs log $p_{{\text{O}}_2}$ do not vary linearly. This implies that more than one defect species play in the manganosite. However, for the sake of simplicity, only neutral cation vacancy associated with Mn³⁺ in cationic site was considered in the present study. Figure 7 shows the measured defect concentration as functions of oxygen partial pressure at different temperatures. Lines are calculated in the present study assuming only the neutral cation vacancy as defect species. Agreement between experiments and calculations seems reasonable.

Fig. 7

Calculated oxygen partial pressures over manganosite at various temperatures along with experimental data[51–53]

Non-stoichiometries of both $\alpha$- and $\beta$-Mn₃O₄ are very small, and those were neglected in previous thermodynamic assessments.[1,2] In fact, thermogravimetrically measured non-stoichiometry of $\beta$-Mn₃O₄ is maximum 0.05% of total cation site.[53] However, in order to keep consistency with already developed spinel model by Degterov et al.[22] and extend thermodynamic database for spinel solid solution, it was tried to model the non-stoichiometries of spinels in the present study. For the $\alpha$-Mn₃O₄, it was assumed that the vacancy in the octahedral site was generated due to the dissolution of hypothetical “$\gamma$-Mn₂O₃.” For the $\beta$-Mn₃O₄, however, it was assumed that no Mn³⁺ ion in tetrahedral site exists. Consequently, the same approach for tetragonal spinel could not be applicable. Therefore, as suggested by Degterov et al.,[22] the parameter $V_x$ was fitted to the experimental data of oxygen non-stoichiometry, directly. Figures 8 and 9 show the calculated non-stoichiometries in $\alpha$- and $\beta$-Mn₃O₄ along with experimental data by Keller and Dieckmann.[3] Non-stoichiometry toward MnO was neglected, and the corresponding experimental data by Keller and Dieckmann[53] are not shown. Hahn and Muan also carried out to measure excess oxygen in Mn₃O₄ using quenching and titration analysis.[55] However, their results of non-stoichiometry of Mn$_{3-x}$O₄ was quiet larger (x up to  0.008) than those of Keller and Dieckmann[3] (x up to  0.0015) and scattered. Therefore, the data of Hahn and Muan[55] was not considered in the present study.

Fig. 8

Calculated oxygen partial pressures over $\alpha$-Mn₃O₄ at various temperatures along with experimental data[3]

Fig. 9

Calculated oxygen partial pressures over $\beta$-Mn₃O₄ at various temperatures along with experimental data[3]

Equilibrium Oxygen Partial Pressures

In the Mn-O system, the following redox reactions can be considered:

$$\begin{aligned} 3{\text{Mn}}_{1-x}{\text{O}} + \frac{1-4x}{2}{\text{O}}_2 = (1-x){\text{Mn}}_3{\text{O}}_4 \end{aligned},$$

(20)

$$\begin{aligned} 2{\text{Mn}}_3{\text{O}}_4 + \frac{1}{2}{\text{O}}_2 = 3{\text{Mn}}_2{\text{O}}_3 \end{aligned},$$

(21)

$$\begin{aligned} {\text{Mn}}_{2}{\text{O}}_3 + \frac{1}{2}{\text{O}}_2 = 2{\text{MnO}}_2 \end{aligned}.$$

(22)

A number of investigations have been carried out in order to measure the equilibrium oxygen partial pressure of the above reactions using EMF or manometer. Extensive review on these investigations are given by Grundy et al.[2] The measured experimental data[56–65] are compared along with the calculations in the present study and shown in Figure 10. Experimental data (equilibrium oxygen partial pressure) for the Reaction [20] are in generally good agreement with those investigations. Scatter of the reported data for the Reaction [21] at low temperature is probably due to hysteresis between redox Reaction [21].[66] In the present study, experimental data which were obtained relatively high temperature by Shenouda and Aziz[66] and Otto[58] were preferred for the Reaction [21]. For the Reaction [22], experimental data of Otto[59] was used due to relatively longer experimental time.

Fig. 10

Calculated oxygen partial pressures in three-phase equilibria (gas phase with Mn$_{1-x}$O-Mn₃O₄, Mn₃O₄-Mn₂O₃, Mn₂O₃-MnO₂), as functions of temperature, along with experimental data[56–59,61–65]

Phase Diagrams

Figure 1 shows the optimized phase diagram of the Mn-O system with the oxygen partial pressure and temperature as axis variables. Several phase-equilibrium data measured mainly using quenching and XRD are shown as symbols.[20,21,55,67–70] It should be noted that the dashed lines are only predicted by the model because there is no experimental data available. Also, in the present study, the molten oxide phase is defined up to $X_{\text{O}}$ = 0.6 which corresponds to the hypothetical pure liquid “Mn₂O₃”. Therefore, the liquidus of MnO₂ is only imaginary and is pending to further experimental verification. Calculated phase boundary of manganosite in equilibrium with $\alpha$- and $\beta$-Mn₃O₄ are shown in Figure 11 along with measured experimental data.[54,64,71] The concentration range of the manganosite becomes wide quickly as temperature increases, due to dissolution of Mn³⁺. In order to reproduce this and the equilibrium oxygen partial pressures over the manganosite, one interaction parameter and Gibbs energy of dissolution of MnO$_{1.5}$ in the manganosite solution were optimized (Table III). Also in order to reproduce liquidus of manganosite and $\beta$-Mn₃O₄,[20,21,69] Gibbs energy of the hypothetical pure liquid “MnO$_{1.5}$” was used as a model parameter (Table III). Since there is no further experimental data for the slag in the Mn-O system, it was decided not to use interaction parameter between Mn²⁺ and Mn³⁺. At the present stage, it has been found that the current description of the Mn-O molten oxide phase results in a good agreement with reported experimental data in multicomponent systems containing up to Fe, Cr, Mn, Si, and O from metal saturation to 1 atm O₂ atmosphere. The details of the thermodynamic optimization for these systems will be shown elsewhere.[9–11]

Fig. 11

Enlarged phase diagram at Mn$_{1-x}$O region along with experimental data[54,64,71]

From the present thermodynamic optimization, the phase diagram of Mn-O system with oxygen iso-bar was drawn as shown in Figure 12(a) from MnO to MnO₂. Gas phase was suspended during the calculation. Again, part of the calculated phase diagram at high oxygen partial pressure (log $p_{{\text{O}}_2} > {\sim}0$) is imaginary and only predicted by thermodynamic models. Nevertheless, since the non-stoichiometries of $\alpha$- and $\beta$-Mn₃O₄ were considered by introducing the vacancies, it could be observed that the spinel solutions deviate from its stoichiometric composition as the oxygen partial pressure increases. A similar phase diagram reported by Grundy et al. is also shown in Figure 12(b) for a comparison.

Fig. 12

Calculated phase diagram of the Mn-O system in the range of composition (a) from $X_{\text{O}} = 0.5$ to $X_{\text{O}} = 0.7$ in the present study (modified from a figure presented in Ref. [12]), and (b) from $X_{\text{O}} = 0.45$ to $X_{\text{O}} = 0.7$, reprinted with permission from Grundy et al.[2] Dashed lines are oxygen iso-bars

Various special points including invariant reactions are calculated using the models of the present study and are shown in Table IV. Other calculated values from previous thermodynamic modeling[1,2] are also compared.

Table IV Special Points Including Invariant Reactions Involving Oxides in Mn-O System

Conclusion

A critical evaluation/optimization of experimental data for the Mn-O system was performed in the present study. The optimized model parameters can reproduce all reliable thermodynamic and structural data as well as the phase diagrams of the Mn-O system within experimental error limits. In order to construct a larger thermodynamic database, $\alpha$- and $\beta$-Mn₃O₄ were modeled as a part of spinel solid solutions. This enables us to integrate the presently developed database into a larger thermodynamic database. The database of model parameters can be used with general thermodynamic software, such as FactSage,[13,14] in order to calculate the thermodynamic properties, the distribution of cations in the spinel solutions which is related to the electrical properties of the spinel, and phase equilibria for temperatures, compositions, and oxygen partial pressures where experimental data are not available. The present thermodynamic modeling has been extended to incorporate Cr,[9] Fe,[10] and Si,[11] respectively.