Composition–Oxygen Partial Pressure Diagram of the Cr–B–O Ternary System Based on the Standard Gibbs Energies of Formation of CrB₄, CrB₂, Cr₃B₄, Cr₅B₃ and CrBO₃ Determined by Solid Electrolyte

Abstract

The standard Gibbs energies of formation, Δ_fG^o, of CrB₄, CrB₂, Cr₃B₄, Cr₅B₃, and CrBO₃ existing along the oxidation path of Cr–B binary alloy were determined using the electromotive forces of galvanic cell composed of the ZrO₂–Y₂O₃ solid electrolyte. The electromotive force showed plateaus and decreases in the process that CrB₄–CrB₂ two-phase alloy used for the cell materials were oxidized via two- and three-phase regions along its oxidation path. From the plateaus of electromotive forces corresponding to the cell materials in the three-phase regions, the values of Δ_fG^o(CrB₄), Δ_fG^o(CrB₂), Δ_fG^o(Cr₃B₄), Δ_fG^o(Cr₅B₃), and Δ_fG^o(CrBO₃) in the temperature range from 1273 to 1346 K were determined as follows:

Δ_fG^o(CrB₄)/J (mol of compd.)⁻¹ = 167500 − 261.2 T ± 7200

Δ_fG^o(CrB₂)/J (mol of compd.)⁻¹ = −21020 − 79.22 T ± 2100

Δ_fG^o(Cr₃B₄)/J (mol of compd.)⁻¹ = −173400 − 95.47 T ± 2500

Δ_fG^o(Cr₅B₃)/J (mol of compd.)⁻¹ = −318900 + 20.64 T ± 5800

Δ_fG^o(CrBO₃)/J (mol of compd.)⁻¹ = −958800 + 59.24 T ± 4100

The Δ_fG^o values determined in the present study satisfied the phase equilibria in the Cr–B binary system. Using the determined Δ_fG^o values, the composition-oxygen partial pressure diagram of the Cr–B–O system was constructed under the conditions at 1300 K and a total pressure of 1 bar (100 kPa). It is useful to understand the oxidation property of the Cr–B binary alloys.

Fig. 8 Composition-oxygen partial pressure diagram of the Cr–B–O ternary system at 1300 K. The horizontal lines show the following three-phase equilibria: (1) CrB₂–CrB₄–B₂O₃; (2) Cr₃B₄–CrB₂–B₂O₃; (3) CrB–Cr₃B₄–B₂O₃; (4) Cr₅B₃–CrB–B₂O₃; (5) Cr₅B₃–CrBO₃–B₂O₃; (6) Cr₂B–Cr₅B₃–CrBO₃; (7) Cr₂B–Cr₂O₃–CrBO₃; (8) Cr–Cr₂B–Cr₂O₃; (9) CrB₄–B–B₂O₃.

1. Introduction

The Cr–B binary system is a fundamental system of the Ni-based corrosion and heat resistance alloys composed of Ni, Cr, Fe, Mo, W, Si, B, and so on.^1–5) To evaluate the structure of the high-temperature oxidation films over the Cr–B binary alloy from the viewpoint of chemical thermodynamics, it is useful to construct the composition-equilibrium partial oxygen pressure, $p_{\text{O}_{2}}$, diagram which visualizes the relationships between the phases existed on the oxidation path and their $p_{\text{O}_{2}}$ values.^6,7) In our previous studies, the composition-$p_{\text{O}_{2}}$ diagrams of the Ni–B–O,⁸⁾ Mo–B–O,⁹⁾ and W–B–O¹⁰⁾ ternary systems were constructed on the basis of the standard Gibbs energies of formation, Δ_fG^o, determined by the electromotive force (EMF) measurements using the ZrO₂–Y₂O₃ solid electrolyte. In the present study, the Δ_fG^o values of the intermediate compounds in the Cr–B–O ternary system were determined to construct the composition-$p_{\text{O}_{2}}$ diagram of the Cr–B–O ternary system.

There exist six stoichiometric compounds, namely, CrB₄, CrB₂, Cr₃B₄, CrB, Cr₅B₃, and Cr₂B in the Cr–B binary system,¹¹⁾ Cr₂O₃ in the Cr–O binary system,¹²⁾ B₂O₃ in the B–O binary system,¹³⁾ and CrBO₃ in the Cr–B–O ternary system^14,15) at the experimental conditions in the present study. The thermodynamic properties of Cr₂O₃^16,17) and B₂O₃^18,19) are listed on the thermodynamic tables, however, those for the Cr–B binary system are limited. Barin²⁰⁾ listed the heat capacities, the third law enthalpies, and the Δ_fG^o values of CrB and CrB₂ in the thermodynamic tables, however, the information regarding how and why the values were selected was not provided. Topor and Kleppa²¹⁾ used high-temperature drop solution calorimetry to determine the standard enthalpies of formation of CrB₂ at 298 K, Δ_fH^o₂₉₈(CrB₂), by allowing the borides to react with Pt, Pd, or Ni to form liquid alloys: Δ_fH^o₂₉₈(CrB₂) = −119.4 ± 3.4 kJ (mol of compd.)⁻¹. Liao and Spear²²⁾ assessed the phase diagram of the Cr–B binary system based on the invariant points, crystal structure, and lattice parameter data of Andersson and Lundstrom,²³⁾ Portnoi et al.,^24,25) and Guy and Uraz,²⁶⁾ and calculated the phase diagram by using the optimized Δ_fG^o values of chromium borides and the realistic empirical expression for the excess Gibbs energy of the liquid phase, not describing which data were used to optimize. The optimized value of Δ_fH^o(CrB₂) = −124.2 kJ (mol of compd.)⁻¹ by Liao et al. was in good agreement with the experimental value by Topor and Kleppa²¹⁾ mentioned above and the estimated value by Brewer and Haraldsen:²⁷⁾ Δ_fH^o(CrB₂) = −126 kJ (mol of compd.)⁻¹. There is little experimental information for the Δ_fG^o values of chromium borides.

In our recent study,¹⁵⁾ the phase relationships for the Cr–B–O ternary system were clarified experimentally to conduct the electromotive force (EMF) measurement using a ZrO₂–Y₂O₃ solid electrolyte and the Δ_fG^o values of Cr₃B₄, CrB₂, and CrB₄ in the boron-rich side of the Cr–B binary system were determined. According to the phase relationships, the Cr–B binary alloy is oxidized via two- and three-phase regions alternately along the oxidation path. In the EMF method used in our previous studies,^{9,10,15,28,29)} the EMFs derived from the $p_{\text{O}_{2}}$ values of the cell materials in the three-phase region involving the target substances were measured under the well-deoxidized argon atmosphere. The EMF measurement for determining the Δ_fG^o value of target substance is reasonable as long as the cell materials remain within the initial three-phase region. In other words, the life time of the cell is until the cell materials are oxidized to the subsequent two-phase region. When the $p_{\text{O}_{2}}$ is low or the activation energy of oxidation of the cell materials is low, it makes the oxidation rate of cell materials high and shortens the life time of the cell. However, it may be possible to obtain the EMF information on materials with high oxidation rates by measuring the change in EMF corresponding to the oxidation process of the cell materials. In the present EMF measurement, the oxidation process of the Cr–B binary alloy used in the cell materials was measured as the change in EMF and $p_{\text{O}_{2}}$ with time. In the present study, the Δ_fG^o values of CrB₄, CrB₂, Cr₃B₄, Cr₅B₃, and CrBO₃ in the Cr–B–O ternary system were determined on the basis of the change in EMF with time for the oxidation process of the Cr–B binary alloy, and the composition-$p_{\text{O}_{2}}$ diagrams of the Cr–B–O ternary system were constructed.

2. Experimental Procedure

2.1 Preparation of cell materials

Figure 1 shows the phase relationships for the Cr–B–O ternary system at 1273 K determined in our previous study.¹⁵⁾ There are following 11 three-phase equilibria in the Cr–B–O ternary system: CrB₂–CrB₄–B₂O₃; Cr₃B₄–CrB₂–B₂O₃; CrB–Cr₃B₄–B₂O₃; Cr₅B₃–CrB–B₂O₃; Cr₅B₃–CrBO₃–B₂O₃; Cr₂B–Cr₅B₃–CrBO₃; Cr₂B–Cr₂O₃–CrBO₃; Cr–Cr₂B–Cr₂O₃; CrB₄–B–B₂O₃; CrBO₃–B₂O₃–O₂; CrBO₃–Cr₂O₃–O₂. The oxidation path of the 25.0 mol%Cr–75.0 mol%B binary alloy (CrB₂–CrB₄ two-phase alloy) are shown as a dotted line in Fig. 1. The CrB₂–CrB₄ two-phase alloy is oxidized to CrBO₃ and B₂O₃ via two- and three-phase regions in the following order: (1) CrB₂–CrB₄–B₂O₃, CrB₂–B₂O₃, (2) Cr₃B₄–CrB₂–B₂O₃, Cr₃B₄–B₂O₃, (3) CrB–Cr₃B₄–B₂O₃, CrB–B₂O₃, (4) Cr₅B₃–CrB–B₂O₃, Cr₅B₃–B₂O₃, (5) Cr₅B₃–CrBO₃–B₂O₃. To measure the EMF corresponding to the oxidation of CrB₂–CrB₄ two-phase alloy, the following galvanic cell was constructed:   

\begin{equation} \text{Cell: W $|$ CrB$_{2}$, CrB$_{4}$, B$_{2}$O$_{3}{}|$ YSZ(8) $|$ O$_{\text{2 in air}}{}|$ Pt}, \end{equation} (1)

where YSZ(8) is one end-closed tube composed of ZrO₂ solid electrolyte stabilized by 8 mol% Y₂O₃ (outer diameter: 13 mm, inner diameter: 9 mm, length: 300 mm, ZR–8Y, Nikkato Corp., Sakai-shi Osaka Japan). The CrB₂–CrB₄ two-phase alloy was prepared by arc melting using a tungsten electrode and a water cooled copper hearth in an argon atmosphere for the compressed compact of the mixtures of Cr (purity: 99.9 mass%, High Purity Chemical Institute Co., Ltd., Sakado-shi Saitama Japan) and B (purity: 99.5 mass%, High Purity Chemical Institute Co., Ltd., Sakado-shi Saitama Japan) powders in the composition of 25.0 mol%Cr–75.0 mol%B. The two-phase alloy was heat-treated at 1373 K for 1.2 Ms under vacuum and were crushed to fine powders in a mortar with a pestle made of tungsten carbide. The formation of CrB₂–CrB₄ two-phase alloy was confirmed by X-ray diffractometer (Ultima IV, Rigaku Co.). The alloy powders were mixed with 2.5 mol% B₂O₃ (purity: 99.99 mass%, High Purity Chemical Institute Co., Ltd., Sakado-shi Saitama Japan), and used as the cell materials.

Fig. 1

Phase relationships for the Cr–B–O ternary system at 1273 K. A dotted line shows the oxidation path of 25.0 mol%Cr–75.0 mol%B binary alloy.

2.2 Measurement of electromotive force

Figure 2 shows a schematic diagram of the cell used for the EMF measurement. The total weight of 1.4 g of the cell materials, which was equivalent to the amount that the height of the cell materials in the YSZ(8) tube was approximately 10 mm, was used for the measurement. A tungsten wire (purity: 99.95 mass%, diameter: 0.25 mm, The Nilaco Co., Tokyo Japan) was inserted to the cell materials so that its tip was located in the center of cell materials, and the YSZ(8) tube was sealed with a silicon rubber stopper. The boundary face between the tungsten wire and the cell materials were sealed with the heat-resistant inorganic adhesive (TOAGOSEI Co., Ltd., Tokyo Japan) except for the tip of wire. A platinum wire (purity: 99.98 mass%, diameter: 0.25 mm, The Nilaco Co., Tokyo Japan) was attached to the outer bottom of the YSZ(8) tube using platinum paste (No. 8105, TOKURIKI HONTEN Co., Ltd., Tokyo Japan), and the contact point was placed between the YSZ (8) tube and the porcelain tube and fixed with springs. The cell was vacuumed at 473 K for 60 minutes and heated up to experimental temperature, and filled with argon gas at a flow rate of 30 mL min⁻¹ (ntp) deoxidized and dehydrated by concentrated sulfuric acid, silica gel, and molecular sieve (pore diameter: 0.3 nm, Merck KGaA, Darmstadt Germany). The argon gas replacement was conducted once in the heating process of the cell without careful deoxidation by using titanium mentioned in our previous studies.^9,10) The EMF, which was derived from the difference in the oxygen partial pressures between the cell materials and the reference (O₂ gas in air), was measured every 5 minutes by a digital multimeter (R6551, ADVANTEST Co., Tokyo Japan) using the tungsten wire and the platinum wire as lead wires, and recorded on a PC connected by the GP-IB interface. The measured EMF values were corrected by subtracting the thermoelectric force generated between tungsten and platinum wires, measured before experiments. The temperature was measured with a platinum/platinum-rhodium thermocouple located at the outer bottom of the YSZ(8) tube. The thermocouple was corrected by measuring the melting points of pure gold and pure aluminum.

Fig. 2

Schematic diagram of the cell. 1: thermocouple for controlling the furnace, 2: thermocouple for measuring temperature, 3: electric furnace, 4: porcelain tube, 5: cell materials, 6: YSZ(8) tube, 7: spring, 8: silicon rubber stopper, 9: Pt wire, 10: W wire, 11: Pt paste, 12: sulfuric acid, 13: silica gel, 14: molecular sieve.

2.3 Partial oxygen pressure of cell materials

According to Schmalzried,³⁰⁾ the value of EMF, E, of the cell using solid electrolyte is expressed as follows:   

\begin{equation} E = (RT/4F) \ln[p_{\text{O${_{2}}$ in air}}/p_{\text{O${_{2}}$}}] \end{equation} (2)

where R, T, F, and $p_{\text{O}_{2}\ \text{in air}}$ are the gas constant, the absolute temperature, the Faraday constant, and the partial oxygen pressure in air, 0.21226 bar (21226 Pa),³¹⁾ respectively. Therefore, the values of $p_{\text{O}_{2}}$ can be known by measuring E of the cell at the constant temperature.

The cell materials in the cell shown in eq. (1) are oxidized via the following three-phase equilibria in this order along the oxidation path: (1) CrB₂–CrB₄–B₂O₃, (2) Cr₃B₄–CrB₂–B₂O₃, (3) CrB–Cr₃B₄–B₂O₃, (4) Cr₅B₃–CrB–B₂O₃, (5) Cr₅B₃–CrBO₃–B₂O₃. These three-condensed phases are in equilibrium with oxygen gas as known from the total cell reactions eqs. (3) through (7) described later in the section “2.4 Calculation Principle of Gibbs Energy of Formation”. According to the phase rule,³²⁾ the activity of oxygen, that is, the partial oxygen pressure of the cell materials is invariant at the constant temperature. Therefore, the value of E is constant when the cell materials are in the three-condensed phases.

2.4 Calculation principle of Gibbs energy of formation

When the cell materials are in (1) CrB₂–CrB₄–B₂O₃, (2) Cr₃B₄–CrB₂–B₂O₃, (3) CrB–Cr₃B₄–B₂O₃, (4) Cr₅B₃–CrB–B₂O₃, and (5) Cr₅B₃–CrBO₃–B₂O₃ three-phase regions, the total cell reactions of the cell are shown as follows:   

\begin{equation} \text{(2/3) CrB$_{4}$} + \text{O$_{\text{2 in air}}$} = \text{(2/3) CrB$_{2}$} + \text{(2/3) B$_{2}$O$_{3}$}, \end{equation} (3)

  

\begin{equation} \text{2 CrB$_{2}$} + \text{O$_{\text{2 in air}}$} = \text{(2/3) Cr$_{3}$B$_{4}$} + \text{(2/3) B$_{2}$O$_{3}$}, \end{equation} (4)

  

\begin{equation} \text{(4/3) Cr$_{3}$B$_{4}$} + \text{O$_{\text{2 in air}}$} = \text{4 CrB} + \text{(2/3) B$_{2}$O$_{3}$}, \end{equation} (5)

  

\begin{equation} \text{(10/3) CrB} + \text{O$_{\text{2 in air}}$} = \text{(2/3) Cr$_{5}$B$_{3}$} + \text{(2/3) B$_{2}$O$_{3}$}, \end{equation} (6)

  

\begin{equation} \text{(1/6) Cr$_{5}$B$_{3}$} + \text{(1/6) B$_{2}$O$_{3}$} + \text{O$_{\text{2 in air}}$} = \text{(5/6) CrBO$_{3}$}. \end{equation} (7)

On the assumption that oxygen gas can be treated as an ideal gas and the solubility of the solid phase in the liquid B₂O₃ phase is negligibly small, the Gibbs energies of cell reactions, Δ_rG, are written as follows:   

\begin{align} \Delta_{\text{r}}G(1) & = \text{(2/3) $\Delta_{\text{f}}G^{\text{o}}$(CrB$_{2}$)} + \text{(2/3) $\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)}\\ &\quad - \text{(2/3) $\Delta_{\text{f}}G^{\text{o}}$(CrB$_{4}$)} - RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (8)

  

\begin{align} \Delta_{\text{r}}G(2) & = \text{(2/3) $\Delta_{\text{f}}G^{\text{o}}$(Cr$_{3}$B$_{4}$)} + \text{(2/3) $\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)}\\ &\quad - \text{2 $\Delta_{\text{f}}G^{\text{o}}$(CrB$_{2}$)} - RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (9)

  

\begin{align} \Delta_{\text{r}}G(3) & = \text{4 $\Delta_{\text{f}}G^{\text{o}}$(CrB)} + \text{(2/3) $\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)}\\ &\quad - \text{(4/3) $\Delta_{\text{f}}G^{\text{o}}$(Cr$_{3}$B$_{4}$)} - RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (10)

  

\begin{align} \Delta_{\text{r}}G(4) & = \text{(2/3) $\Delta_{\text{f}}G^{\text{o}}$(Cr$_{5}$B$_{3}$)} + \text{(2/3) $\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)}\\ &\quad - \text{(10/3) $\Delta_{\text{f}}G^{\text{o}}$(CrB)} - RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (11)

  

\begin{align} \Delta_{\text{r}}G(5) & = \text{(5/6) $\Delta_{\text{f}}G^{\text{o}}$(CrBO$_{3}$)} - \text{(1/6) $\Delta_{\text{f}}G^{\text{o}}$(Cr$_{5}$B$_{3}$)}\\ &\quad - \text{(1/6) $\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)} - RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (12)

where p^o express the standard pressure (1 bar (10⁵ Pa)). Under the condition that the transport number of the oxide ion in the solid electrolyte is regarded as unity, Δ_rG = −4EF for eqs. (8) through (12). By determining the values E(1) through E(5) for the cell materials in (1) through (5) three-phase regions, the values of Δ_fG^o(CrB₄), Δ_fG^o(CrB₂), Δ_fG^o(Cr₃B₄), Δ_fG^o(Cr₅B₃), and Δ_fG^o(CrBO₃) are given by the following equations:   

\begin{align} \text{$\Delta_{\text{f}}G^{\text{o}}$(CrB$_{4}$)} & = 6\ E(1)F + \text{$\Delta_{\text{f}}G^{\text{o}}$(CrB$_{2}$)}+ \text{$\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)}\\ &\quad - (3/2)\ RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (13)

  

\begin{align} \text{$\Delta_{\text{f}}G^{\text{o}}$(CrB$_{2}$)} & = 2\ E(2)F + \text{(1/3) $\Delta_{\text{f}}G^{\text{o}}$(Cr$_{3}$B$_{4}$)}\\ &\quad + \text{(1/3) $\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)} \\ &\quad - (1/2)\ RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (14)

  

\begin{align} \text{$\Delta_{\text{f}}G^{\text{o}}$(Cr$_{3}$B$_{4}$)} & = 3\ E(3)F + \text{3 $\Delta_{\text{f}}G^{\text{o}}$(CrB)}\\ &\quad + \text{(1/2) $\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)}\\ &\quad - (3/4)\ RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (15)

  

\begin{align} \text{$\Delta_{\text{f}}G^{\text{o}}$(Cr$_{5}$B$_{3}$)} & = - 6\ E(4)F + \text{5 $\Delta_{\text{f}}G^{\text{o}}$(CrB)} - \text{$\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)} \\ &\quad+ (3/2)\ RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}), \end{align} (16)

  

\begin{align} \text{$\Delta_{\text{f}}G^{\text{o}}$(CrBO$_{3}$)} & = - (24/5)\ E(5)F + \text{(1/5) $\Delta_{\text{f}}G^{\text{o}}$(Cr$_{5}$B$_{3}$)}\\ &\quad + \text{(1/5) $\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)} \\ &\quad + (6/5)\ RT\ln(p_{\text{O${_{2}}$ in air}}/p^{\text{o}}). \end{align} (17)

The values of Δ_fG^o(B₂O₃)¹⁸⁾ in the temperature range from 1000 to 1500 K and Δ_fG^o(CrB)²⁰⁾ in the temperature range from 600 to 1200 K were taken from the thermochemical tables and are expressed as linear functions as follows:   

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(B$_{2}$O$_{3}$)/J (mol of compd.)$^{-1}$} \\ &\quad = - 1231000 + 213.2\ T \pm 600, \end{align} (18)

  

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(CrB)/J (mol of compd.)$^{-1}$} \\ &\quad= - 74950 - 6.551\ T \pm 10. \end{align} (19)

The above errors were estimated as twice the standard deviations at the 0.95 level of confidence for the linear least-squares approximations. Since the approximation shown in eq. (19) shows a strong correlation between temperature and Δ_fG^o(CrB) (coefficient of determination, R² = 0.99998) and there is no phase transition below 1346 K,¹¹⁾ the approximation is also valid at the experimental temperature between 1273 and 1346 K.

2.5 Activities of Cr and B in the Cr–B binary system

When the elemental bcc Cr and rhombohedral B are adopted as the standard state, the chemical potential of Cr, Δμ_Cr, and the chemical potential of B, Δμ_B, are expressed as   

\begin{equation} \Delta\mu_{\text{Cr}} = \mu_{\text{Cr}} - \mu^{\text{o}}{}_{\text{Cr}} = RT\ln a_{\text{Cr}} \end{equation} (20)

  

\begin{equation} \Delta\mu_{\text{B}} = \mu_{\text{B}} - \mu^{\text{o}}{}_{\text{B}} = RT\ln a_{\text{B}} \end{equation} (21)

where a_Cr and a_B are the activities of Cr and B, respectively. The values of a_Cr and a_B are given by   

\begin{equation} a_{\text{Cr}} = \exp(\Delta\mu_{\text{Cr}}/RT) \end{equation} (22)

  

\begin{equation} a_{\text{B}} = \exp(\Delta\mu_{\text{B}}/RT) \end{equation} (23)

According to the equilibrium phase diagram of the Cr–B binary system,¹¹⁾ all of the intermediate compounds are stoichiometric, and the following two-phase equilibria exist: Cr–Cr₅B₃, Cr₅B₃–CrB, CrB–Cr₃B₄, Cr₃B₄–CrB₂, CrB₂–CrB₄, and CrB₄–B. The values of Δμ_Cr and Δμ_B as well as a_Cr and a_B in the two-phase equilibria can be calculated on the basis of the two-phase equilibria. Note that Δμ_Cr is zero in the Cr–Cr₅B₃ two-phase region where a_Cr is unity and Δμ_B is zero in the CrB₄–B two-phase region where a_B is unity.

3. Results and Discussions

3.1 Results of electromotive force measurements

Figure 3 shows the change in the EMF of the cell and the $p_{\text{O}_{2}}$ value of the cell materials at 1319 K. The $p_{\text{O}_{2}}$ of the cell materials was converted from the EMF by using eq. (2) and shown on the right-hand axis of ordinate. In the beginning of measurement, the EMF showed approximately 1.51 V after steep increases and decreases for 14 ks. It showed plateaus and decreases, and reached to 1.31 V. In this process, the five plateaus of EMF indicated as (1) through (5) in Fig. (3) were observed at around 30, 80, 140, 185, and 246 ks from the beginning of the measurement. With this change in EMF, the $p_{\text{O}_{2}}$ of the cell materials showed plateaus and increases, that is, they were oxidized in this process. As mentioned in the sections “2.1 Preparation of Cell Materials” and “2.2 Partial Oxygen Pressure of Cell Materials”, the CrB₂–CrB₄ two-phase alloy used for the cell materials are oxidized to CrBO₃ and B₂O₃ via two- and three-phase regions along its oxidation path, and the value of E is constant when the cell materials are in the three-phases regions. In our previous study,¹⁵⁾ the EMF values of the cells having CrB₂–CrB₄–B₂O₃, Cr₃B₄–CrB₂–B₂O₃, and CrB–Cr₃B₄–B₂O₃ as cell materials were measured. Since the EMF values in the plateaus of (1), (2), and (3) shown in Fig. 3 were in good agreement with those of our previous study, it is considered that the cell materials in the plateau regions of (1), (2), and (3) correspond to the three-phase regions of (1) CrB₂–CrB₄–B₂O₃, (2) Cr₃B₄–CrB₂–B₂O₃, and (3) CrB–Cr₃B₄–B₂O₃, respectively. Moreover, the cell materials corresponding to plateau regions were analyzed by XRD. When the EMF of the cell reached the value of each plateau region, the measurement was stopped, the cell was cooled, and then the cell materials were analyzed by XRD. As a result, CrB₂–CrB₄–B₂O₃, Cr₃B₄–CrB₂–B₂O₃, and Cr₅B₃–CrB–B₂O₃ phases were detected in the cell materials obtained in the plateau regions (1), (2), and (4), respectively. However, CrB–Cr₃B₄–B₂O₃ and Cr₅B₃–CrBO₃–B₂O₃ were not detected in the cell materials obtained in the plateau regions (3) and (5), because their plateau regions were short and the oxidation proceeded during cooling. In addition, the final oxides, CrBO₃ and B₂O₃, were detected in the cell materials after passing through the plateau region (5). On the basis of the oxidation path shown as the dashed line in Fig. 1, the phases for plateau region (5) is considered to be (5) Cr₅B₃–CrBO₃–B₂O₃, which is the only three-phase region that exists between the plateau region (4) and the final oxides. Therefore, the measured five plateau values correspond to the values of E for the cell materials in the three-phase regions. From the measured five plateaus, the values of E(1) through E(5) were determined.

Fig. 3

Change in the electromotive force of the cell and the partial oxygen pressure of the cell materials at 1319 K. The five plateaus (1) through (5) correspond to the cell materials in the following three-phase regions: (1) CrB₂–CrB₄–B₂O₃, (2) Cr₃B₄–CrB₂–B₂O₃, (3) CrB–Cr₃B₄–B₂O₃, (4) Cr₅B₃–CrB–B₂O₃, (5) Cr₅B₃–CrBO₃–B₂O₃.

Table 1 summarizes the two- and three-phase regions in the EMF measurement, time duration, the (ΔE/Δt) value, and the average values of E for three-phase regions. The two- and three-phase regions for the cell materials in the EMF measurement were distinguished by using the (ΔE/Δt) value, which were calculated by the division of the change in EMF, ΔE, by its time duration, Δt. The (ΔE/Δt) values for the three-phase regions were more than 2 times smaller than those for two-phase regions.

Table 1 Summary of the two- and three-phase regions, time duration, the (ΔE/Δt) value, and the average electromotive forces for three-phase regions in the electromotive force measurement at 1319 K.

In the same way, the E values in the oxidation path of the cell materials were measured at 1273, 1301, and 1346 K.

Figure 4 shows the values of E corresponding to the cell materials in the (1) CrB₂–CrB₄–B₂O₃, (2) Cr₃B₄–CrB₂–B₂O₃, (3) CrB–Cr₃B₄–B₂O₃, (4) Cr₅B₃–CrB–B₂O₃, and (5) Cr₅B₃–CrBO₃–B₂O₃ three-phase regions as a function of temperature. The values of E decreased with temperature, and they were expressed as a function of temperature in the temperature range from 1273 to 1346 K by using the linear least-squares method as follows:   

\begin{equation} E(1)/\text{V} = 2.452 - 7.161\times 10^{-4}\ T \pm 1.2\times 10^{-2}, \end{equation} (24)

  

\begin{equation} E(2)/\text{V} = 2.317 - 6.473\times 10^{-4}\ T \pm 1.0\times 10^{-2}, \end{equation} (25)

  

\begin{equation} E(3)/\text{V} = 2.304 - 6.636\times 10^{-4}\ T \pm 8.6\times 10^{-3}, \end{equation} (26)

  

\begin{equation} E(4)/\text{V} = 2.030 - 4.939\times 10^{-4}\ T \pm 1.0\times 10^{-2}, \end{equation} (27)

  

\begin{equation} E(5)/\text{V} = 1.401 - 6.032\times 10^{-5}\ T \pm 8.5\times 10^{-3}. \end{equation} (28)

The errors in eqs. (24) through (28) were estimated as twice the standard deviations at the 0.95 level of confidence for the linear least-squares approximations.

Fig. 4

Electromotive forces corresponding to the cell materials in the (1) CrB₂–CrB₄–B₂O₃, (2) Cr₃B₄–CrB₂–B₂O₃, (3) CrB–Cr₃B₄–B₂O₃, (4) Cr₅B₃–CrB–B₂O₃, and (5) Cr₅B₃–CrBO₃–B₂O₃ three-phase regions as a function of temperature.

3.2 Δ_fG^o values for the Cr–B binary and Cr–B–O ternary compound

Using the measured values of E(1) through E(5), Δ_fG^o(CrB₄), Δ_fG^o(CrB₂), Δ_fG^o(Cr₃B₄), Δ_fG^o(Cr₅B₃), and Δ_fG^o(CrBO₃) in the temperature range from 1273 to 1346 K were determined from eqs. (13) through (17).   

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(CrB$_{4}$)/J (mol of compd.)$^{-1}$}\\ &\quad = 167500 - 261.2\ T \pm 7200 \end{align} (29)

  

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(CrB$_{2}$)/J (mol of compd.)$^{-1}$} \\ &\quad= - 21020 - 79.22\ T \pm 2100 \end{align} (30)

  

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(Cr$_{3}$B$_{4}$)/J (mol of compd.)$^{-1}$} \\ &\quad= - 173400 - 95.47\ T \pm 2500 \end{align} (31)

  

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(Cr$_{5}$B$_{3}$)/J (mol of compd.)$^{-1}$} \\ &\quad= - 318900 + 20.64\ T \pm 5800 \end{align} (32)

  

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(CrBO$_{3}$)/J (mol of compd.)$^{-1}$} \\ &\quad= - 958800 + 59.24\ T \pm 4100 \end{align} (33)

The Δ_fG^o value for one mole of atoms is convenient for comparison of the molar thermodynamic values in the Cr–B binary system. The Δ_fG^o values for one mole of atoms of CrB₄, CrB₂, Cr₃B₄, and Cr₅B₃ were calculated from eqs. (29) through (32) as follows:   

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(CrB$_{4}$)/J (mol of atoms)$^{-1}$} \\ &\quad= 33500 - 52.24\ T \pm 1400, \end{align} (34)

  

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(CrB$_{2}$)/J (mol of atoms)$^{-1}$} \\ &\quad= - 7006 - 26.41\ T \pm 700, \end{align} (35)

  

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(Cr$_{3}$B$_{4}$)/J (mol of atoms)$^{-1}$} \\ &\quad= - 24770 - 13.64\ T \pm 360, \end{align} (36)

  

\begin{align} &\text{$\Delta_{\text{f}}G^{\text{o}}$(Cr$_{5}$B$_{3}$)/J (mol of atoms)$^{-1}$} \\ &\quad= - 39860 + 2.580\ T \pm 730. \end{align} (37)

These values were calculated on the assumption that the solubility of the solid phase in the liquid B₂O₃ phase is negligibly small. When the solid phase dissolves into the liquid B₂O₃ phase, the terms related to the activity of each component, that is, the coefficients of $RT\ln (p_{\text{O}_{2}\ \text{in air}}/p^{\text{o}})$ terms in eqs. (13)∼(17) are affected, and the temperature dependencies of the determined Δ_fG^o values may have some errors.

To determine the Δ_fG^o value using a solid oxide electrolyte, the EMF measurement must be carried out in the ionic conduction region in which the transport number of oxygen ion, t_ion, can be regarded as unity. The t_ion value is given by³⁰⁾   

\begin{equation} t_{\text{ion}} = 1/[1 + (p_{\theta}/p_{\text{O${_{2}}$}})^{1/4}], \end{equation} (38)

where p_θ is the partial pressure where t_ion becomes 0.5, owing to the formation of excess electrons, and is given by the following equation for YSZ(8) used in the present study.³³⁾   

\begin{equation} \log(p_{\theta}/\text{Pa}) - \log(101325) = 18.08 - 6.24 \times 10^{4}\ T^{-1} \end{equation} (39)

The value of t_ion decreases with temperature. Thus, the values of t_ion at the temperature of 1346 K, where the highest temperature in the present study, were calculated from eqs. (38) and (39) by using the $p_{\text{O}_{2}}$ values converted from E values shown in eqs. (24) through (28).

Table 2 summarizes the values of $p_{\text{O}_{2}}$ of the cell materials and t_ion of YSZ(8) at 1346 K. The values of t_ion corresponding to the three-phase regions of cell materials were at least 0.956 and were regarded as 1.0. As described in our previous paper,⁹⁾ the EMF measurements with t_ion less than 0.95 were unstable due to electronic conduction. Therefore, all the EMF measurements have been carried out in the ionic conduction region of the solid electrolyte.

Table 2 Summary of the values of $p_{\text{O}_{2}}$ of cell materials and tion of YSZ(8) at 1346 K.

3.3 Validity of determined Δ_fG^o value

Figure 5 shows the comparison of Δ_fG^o(CrB₄), Δ_fG^o(CrB₂), and Δ_fG^o(Cr₃B₄) values for one mole of atoms determined by the present EMF method with those determined by the EMF method in the previous study. In the previous study,¹⁵⁾ three types of cells having different cell materials such as CrB₂–CrB₄–B₂O₃, Cr₃B₄–CrB₂–B₂O₃, and CrB–Cr₃B₄–B₂O₃ were prepared and measured their EMFs. The experimental temperature was varied and the equilibrium EMF value was measured at each temperature. On the other hand, in the present study, the cell having CrB₂–CrB₄–B₂O₃ as cell materials was prepared, and the oxidation process was measured as the change in EMF while the temperature was fixed. Since the same solid electrolyte was used, the differences in the values shown in Fig. 5 were due to the difference in the way of EMF measurement, that is, the difference in the measurement of equilibrium EMF and the measurement of EMF during the oxidation process. The temperature gradients of Δ_fG^o values determined by the present method were slightly higher than those determined in the previous study. However, the Δ_fG^o values determined by both methods were in good agreement within the limit of experimental temperature.

Fig. 5

Comparison of Δ_fG° values for one mole of compounds of CrB₄, CrB₂, and Cr₃B₄ determined by the present EMF method with those determined by the EMF method in previous study.¹⁵⁾

Figure 6 shows the Δ_fG^o values for one mole of atoms of CrB₄, CrB₂, Cr₃B₄, and Cr₅B₃ at 1300 K in the Cr–B binary system determined in the present study compared with those in the thermodynamic tables²⁰⁾ and the optimized values for phase diagram calculation.²²⁾ The present Δ_fG^o values of Cr₅B₃, Cr₃B₄, and CrB₂ exhibited good agreement with those in Ref. 22), whereas the present Δ_fG^o value of CrB₂ was approximately 8 kJ (mol of atoms)⁻¹ smaller than that in Ref. 20) and the present Δ_fG^o value of CrB₄ was approximately 9 kJ (mol of atoms)⁻¹ smaller than that in Ref. 22). Since the methods of Δ_fG^o determination are different, the reasons for such discrepancies are unknown.

Fig. 6

The Δ_fG° values for one mole of atoms of CrB₄, CrB₂, Cr₃B₄, and Cr₅B₃ at 1300 K in the Cr–B binary system determined in the present study compared with those in the Refs. 20, 22).

The value of Δ_fG^o(Cr₂B) could not be obtained experimentally in the present study. However, it can be estimated by the phase equilibria in the Cr–B binary system at 1300 K. The mole fraction of boron, X_B, of Cr₂B is 0.333 and it must be more stable than the hypothetical two-phase equilibrium of Cr (X_B = 0) and Cr₅B₃ (X_B = 0.375). That is, the Δ_fG^o(Cr₂B) value must be smaller than the value at X_B = 0.333 on the line segment between the Δ_fG^o values of Cr and Cr₅B₃. Therefore, the upper limit of Δ_fG^o(Cr₂B) can be estimated to be −32.41 kJ (mol of atoms)⁻¹. In the same manner, the Δ_fG^o(Cr₅B₃) value must be smaller than the value at X_B = 0.375 on the line segment between the Δ_fG^o(Cr₂B) and the Δ_fG^o(CrB)²⁰⁾ values for the stability of the Cr₅B₃ phase. In other words, the Δ_fG^o(Cr₂B) value must be higher than the value at X_B = 0.333 on the straight line passing through the Δ_fG^o(Cr₅B₃) and the Δ_fG^o(CrB) values. The lower limit of Δ_fG^o(Cr₂B) can be estimated to be −34.74 kJ (mol of atoms)⁻¹. Therefore, the Δ_fG^o(Cr₂B) value at 1300 K can be estimated to be −34.74 < Δ_fG^o(Cr₂B)/J (mol of atoms)⁻¹ < −32.41.

Figure 7 shows a_Cr and a_B in the Cr–B binary system at 1300 K calculated on the basis of the two-phase equilibria. The activities are shown in the mole fraction of boron, X_B, from 0.3 to 1.0, related to the Δ_fG^o values of Cr₅B₃, Cr₃B₄, CrB₂, and CrB₄ determined experimentally in the present study. The activities exhibit monotonic and stepwise change, that is, the Δ_fG^o values determined in the present study satisfy the phase equilibria for the Cr–B binary system.

Fig. 7

Activity of Cr, a_Cr, and B, a_B, in the Cr–B binary system at 1300 K.

3.4 Composition-oxygen partial pressure diagram

Figure 8 shows the composition-$p_{\text{O}_{2}}$ diagram of the Cr–B–O ternary system at 1300 K under the total pressure of 1 bar (100 kPa). This figure was constructed by considering the relationships between the oxidation paths of Cr–B binary alloys as understood from Fig. 1 and the $p_{\text{O}_{2}}$ values of three-phase equilibria on the oxidation path. The $p_{\text{O}_{2}}$ values of (1) CrB₂–CrB₄–B₂O₃, (2) Cr₃B₄–CrB₂–B₂O₃, (3) CrB–Cr₃B₄–B₂O₃, (4) Cr₅B₃–CrB–B₂O₃, (5) Cr₅B₃–CrBO₃–B₂O₃, (8) Cr–Cr₂B–Cr₂O₃, and (9) CrB₄–B–B₂O₃ three-phase equilibria were calculated by using the present Δ_fG^o values of CrB₄, CrB₂, Cr₃B₄, Cr₅B₃, and CrBO₃, and the literature values of CrB,²⁰⁾ B₂O₃,¹⁸⁾ and Cr₂O₃.¹⁶⁾ The $p_{\text{O}_{2}}$ values of (6) Cr₂B–Cr₅B₃–CrBO₃ and (7) Cr₂B–Cr₂O₃–CrBO₃ three-phase equilibria shown as dashed lines in Fig. 8 were calculated by using a estimated Δ_fG^o(Cr₂B) value at 1300 K of −33.00 kJ (mol of atoms)⁻¹. This diagram is useful for understanding the relationship between the composition of Cr–B binary alloy and the oxidation product at any specific oxygen partial pressure.

Fig. 8

Composition-oxygen partial pressure diagram of the Cr–B–O ternary system at 1300 K. The horizontal lines show the following three-phase equilibria: (1) CrB₂–CrB₄–B₂O₃; (2) Cr₃B₄–CrB₂–B₂O₃; (3) CrB–Cr₃B₄–B₂O₃; (4) Cr₅B₃–CrB–B₂O₃; (5) Cr₅B₃–CrBO₃–B₂O₃; (6) Cr₂B–Cr₅B₃–CrBO₃; (7) Cr₂B–Cr₂O₃–CrBO₃; (8) Cr–Cr₂B–Cr₂O₃; (9) CrB₄–B–B₂O₃.

4. Conclusion

The electromotive force measurement using the solid electrolyte was applied along the oxidation path of the Cr–B binary alloy to measure the partial oxygen pressure, $p_{\text{O}_{2}}$. From the plateaus of electromotive forces corresponding to the three-phase regions along the oxidation path, the Δ_fG^o values of CrB₄, CrB₂, Cr₃B₄, Cr₅B₃, and CrBO₃ in the Cr–B–O ternary system were determined. The obtained Δ_fG^o values of CrB₄, CrB₂, Cr₃B₄, and Cr₅B₃ satisfied the phase equilibria in the Cr–B binary system. Using the Δ_fG^o values determined in the present study, the composition-oxygen partial pressure diagram of the oxidation path was constructed.

REFERENCES

• 1)   K.  Gurumoorthy,  M.  Kamaraj,  K.  Prasad Rao,  A.  Sambasiva Rao and  S.  Venugopal: Mater. Sci. Eng. A 456 (2007) 11–19.
• 2)   E.S.  Puchi Cabrera,  J.A.  Berríos-Ortiz,  J.  Da-Silva and  J.  Nunes: Surf. Coat. Tech. 172 (2003) 128–138.
• 3)   G.  Zhang,  M.  Morishita and  K.  Koyama: J. Japan Inst. Metals 64 (2000) 1252–1256.
• 4)   A.  Thomas,  M.  El-Wahabi,  J.M.  Cabrera and  J.M.  Prado: J. Mater. Process. Technol. 177 (2006) 469–472.
• 5)   F.-Y.  Ouyang,  C.-H.  Chang and  J.-J.  Kai: J. Nucl. Mater. 446 (2014) 81–89.
• 6)   K.T.  Jacob: J. Mater. Sci. 12 (1977) 1647–1652.
• 7)   K.T.  Jacob,  G.M.  Kale and  G.N.K.  Iyengar: J. Mater. Sci. 22 (1987) 4274–4280.
• 8)   H.  Yamamoto and  M.  Morishita: J. Alloy. Compd. 438 (2007) L1–L3.
• 9)   H.  Yamamoto,  M.  Morishita,  T.  Yamamoto and  K.  Furukawa: Metall. Mater. Trans. B 42 (2011) 114–120.
• 10)   H.  Yamamoto,  M.  Morishita,  Y.  Miyake and  S.  Hiramatsu: Metall. Mater. Trans. B 48 (2017) 1703–1714.
• 11)  T.B. Massalski, H. Okamoto, P.R. Subramanian and L. Kacprzak: Binary Alloy Phase Diagrams, 2nd ed., (ASM International, Ohio, 1990) pp. 471–474.
• 12)   H.  Okamoto: J. Phase Equilib. 18 (1997) 402.
• 13)  T.B. Massalski, H. Okamoto, P.R. Subramanian and L. Kacprzak: Binary Alloy Phase Diagrams, 2nd ed., (ASM International, Ohio, 1990) pp. 510–512.
• 14)   N.C.  Tombs,  W.J.  Croft and  H.C.  Mattraw: Inorg. Chem. 2 (1963) 872–873.
• 15)   H.  Yamamoto,  Y.  Wada,  K.  Nishiyama,  Y.  Taniguchi,  A.  Nozaki and  M.  Morishita: Mater. Trans. 61 (2020) 2357–2362.
• 16)  M.W. Chase, Jr.: NIST-JANAF Thermochemical Tables, 4th Ed., (American Chemical Society and American Institute of Physics for National Institute of Standards and Technology, New York, 1998) p. 972.
• 17)  I. Barin: Thermochemical Data of Pure Substances, (VCH Verlagsgesellschaft, Weinheim, 1989) p. 573.
• 18)  M.W. Chase, Jr.: NIST-JANAF Thermochemical Tables, 4th Ed., (American Chemical Society and American Institute of Physics for National Institute of Standards and Technology, New York, 1998) p. 272.
• 19)  I. Barin: Thermochemical Data of Pure Substances, (VCH Verlagsgesellschaft, Weinheim, 1989) p. 122.
• 20)  I. Barin: Thermochemical Data of Pure Substances, (VCH Verlagsgesellschaft, Weinheim, 1989) p. 558.
• 21)   L.  Topor and  O.J.  Kleppa: J. Chem. Thermodyn. 17 (1985) 109–116.
• 22)   P.K.  Liao and  K.E.  Spear: Bull. Alloy Phase Diagrams 7 (1986) 232–237.
• 23)   S.  Andersson and  T.  Lundstrom: Acta Chem. Scand. 22 (1968) 3103–3110.
• 24)   K.I.  Portnoi,  V.M.  Romashov and  I.V.  Romanovich: Poroshk. Metall. 4 (1969) 51–57.
• 25)   K.I.  Portnoi and  V.M.  Romashov: Poroshk. Metall. 5 (1972) 48–56.
• 26)   C.N.  Guy and  A.A.  Uraz: J. Less-Common Met. 48 (1976) 199–203.
• 27)   L.  Brewer and  H.  Haraldsen: J. Electrochem. Soc. 102 (1955) 399–406.
• 28)   H.  Yamamoto,  M.  Masao,  A.  Miyata and  K.  Koyama: Metall. Mater. Trans. B 37 (2006) 607–613.
• 29)   H.  Yamamoto,  A.  Miyata and  K.  Koyama: J. Japan Inst. Metals 70 (2006) 233–237.
• 30)   H.  Schmalzried: Z. Phys. Chem. 38 (1963) 87–102.
• 31)  The Chemical Society of Japan: Handbook of Chemistry, Basic Part I, 4th ed., (Maruzen, Tokyo, 1993) p. I-52.
• 32)  P. Atkins, J. Paula and J. Keeler: Atkins’ Physical chemistry, 11th ed., (Oxford University Press, Oxford, 2018) pp. 123–125.
• 33)   M.  Iwase,  E.  Ichise,  M.  Takeuchi and  T.  Yamasaki: Trans. JIM 25 (1984) 43–52.