Permeability Coefficient of Oxygen in Molten Slags and Its Mechanisms

Abstract

Molten silicate slags with transition metal oxides roles the accelerated oxidation and decarburization of steel. In 1978 to 1990, Sasabe et al. measured the flow rate of oxygen permeating through molten slags and realized the marvelous increase of the flow rate by 10¹⁰ times by the addition of only 0.2 mass% iron oxide to molten silicate slags. The permeability coefficient of oxygen of molten slags as a physical property is calculated from the flow rate of oxygen. The marvelous increase is interpreted as oxygen transfer through slag like a ball thrust, that is caused by the redox reaction of oxygen at both surfaces and the move of all ions in slag under electric field generated by oxygen potential gradient.

1. Introduction

A Japanese sword is made of special steel, so-called “Tamahagane” produced by Tatara process that is a traditional ironmaking process in Japan. “Tamahagane” is made tough by “Tanren” that is forge-and-welding operation in several or more than ten times. It can be welded steel and steel only by heating in burning charcoal. This is because the surface of steel is heated by the oxidation heat up to a temperature close to 1500°C and melted to weld each other. At the time of melting, white sparks in flame, so-called “Wakibana”, occur violently. The surface is covered with molten FeO and microbubbles of CO gas are suddenly generated at the interface between steel and molten FeO by the reaction of carbon in steel and FeO. The fine particles of steel are trapped by bubbles with strong sweeping force and oxidized in air to generate white sparks.¹⁾ The oxidation of steel is very fast, even though oxygen gas in air is blocked by the layer of molten FeO. How is oxygen supplied to steel surface from air?

The decarburization of molten steel in the converter of steelmaking process occurs at the fire spot of blowing oxygen gas. Droplets of molten FeO generated at the fire spot are dispersed in molten steel by convection and react with carbon in molten steel to generate CO gas bubbles. The oxidation of carbon occurs at the interface between molten steel and FeO droplet.

In the open hearth, although not already used in recent years, oxygen was supplied through slag layer to decarburize molten steel. Further back in the history of pre-modern steelmaking processes, the paddling process was to produce decarburized wrought iron by stirring molten iron with FeO-containing slag. In the refining furnace (Finery) to produce wrought iron, pig iron was decarburized with molten FeO that was produced by oxidizing iron with blowing air. In the Tatara process, wrought iron, so-called “Waritetsu” or “Hochoutetsu”, was produced as the almost same process.²⁾ In the Tatara process, slag containing iron oxide, so-called “Noro”, blocked oxygen gas to prevent reduced iron from re-oxidizing.³⁾

In the ladle for adjusting the concentration of alloy elements and the tundish and mold of continuous casting, molten steel is covered with a slag with less FeO in order to prevent molten steel from oxidation by air.

The redox reaction rate of elements between molten steel and slag is limited by the diffusion rate of elements through the boundary layer generated in the slag side. Exceptionally, the redox reaction of silicon is said to be limited by the chemical reaction rate.

The diffusion coefficient of oxygen in slag of 40CaO-40SiO₂-20Al₂O₃ (mass%) is greater than the other elements,⁴⁾ though oxygen forms the network of polymer structure with silicon by covalent bond. Why is the diffusion coefficient of oxygen larger than the other elements?

Sasabe et al.^5,6,7,8) measured the flow rate of oxygen permeating through molten silicate slag layer. In this study, the permeability coefficient of oxygen is calculated from the flow rate of oxygen.

They realized that the flow rate of oxygen in molten silicate slag with only 0.2 mass% iron oxide increased by 10¹⁰ times of that of slag without iron oxide.⁶⁾ Also, the addition of CaF₂⁷⁾ and ZnO⁸⁾ to molten slag showed the increase of the flow rate of oxygen. However, they did not explain the cause of the increase. In the present study the mechanisms of flowing oxygen in molten slag is revealed.

2. Experimental

The measurement method of flow rate of oxygen through molten slag was reported by Sasabe et al.⁶⁾ Here is a brief introduction.

An alumina tube with inner diameter of 6 mm was closed with a slag layer. The slag layer was melted by heating up to 1600°C. In the inside of tube, oxygen gas was flowed with 100 cc/min under 1 atm. In the outside, Ar gas was switched from air with 700 cc/min under 1 atm. Oxygen partial pressure in Ar was about 1×10⁻¹⁶ atm. Then, the change of oxygen partial pressure in Ar gas was measured by an oxygen concentration cell using a one-end-closed tube of stabilized zirconia (ZrO₂-15 mol% CaO) as a solid electrolyte. The mixture of Ni and NiO powder was stuffed in the tube as the standard electrode. The temperature of the cell was 800°C. Oxygen partial pressure in Ar, p_O2, is given by the following equation.   

$$P_{{\text{O}}_2}\left(\mathrm{a}\mathrm{t}\mathrm{m}\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left\{\frac{2\mathrm{\Delta }{G}_{f\mathrm{\hspace{0.17em} }NiO}^0-4EF}{RT}\right\}$$(1)

where E is the electromotive force of the cell, T is the absolute temperature of 1073 K of the cell, R is the Gas constant, F is the Faraday constant and ΔG_f⁰_NiO is the Gibbs energy of formation of NiO⁹⁾ at 1073 K.

When the electromotive force was stabilized after 40 minutes, the oxygen partial pressure in Ar gas was calculated as a value of the steady state. It has reached about 98% of the steady state from the calculation of the concentration distribution of oxygen in the slag layer using the diffusion coefficient of oxygen.

The amount of permeating oxygen gas, Q (mol/s), was calculated using the state equation of ideal gas.   

$$\text{Q}=\frac{0.1013P_{O_2}{V}_{Ar}}{R{T}_r}$$(2)

where V_Ar is the flow rate of Ar gas (cc/s) and T_r is the room temperature of 298 K.

The shape of solidified slag after cooling was a plane at the end of the alumina tube and a parabolic surface in the inner side of the tube. This was also confirmed in molten state by X-ray transmission observation.

3. Permeability Coefficient of Oxygen

The amount of permeating oxygen gas is proportional to the cross-sectional area of slag layer, S, and inversely proportional to the length, L. Sasabe et al. defined the flow rate of oxygen, P, and measured it.   

$$\text{P}=\frac{QL}{S}$$(3)

They calculated L to be about 5 mm by the area average.

The flow rate of oxygen in unit cross-sectional area is proportional to the chemical potential gradient of oxygen.   

$$\frac{Q}{S}=\frac{\text{P}}{L}=-\text{B}\frac{d{\mu }_{O_2}}{dx}$$(4)

μ_O2 is the chemical potential of oxygen. B is the permeability coefficient of oxygen that is represented by the function of only temperature and composition of slag.

Under the steady state, the chemical potential gradient of oxygen is described as   

$$\frac{d{\mu }_{O_2}}{dx}=\frac{\mathrm{\Delta }{G}_{O_2}}{L}$$(5)

$\mathrm{\Delta }{G}_{O_2}$ is the oxygen potential difference between both sides of slag layer.   

$$\mathrm{\Delta }{G}_{O_2}={\mu }_{O_2}\left(out\right)-{\mu }_{O_2}\left(in\right)=RTln\frac{p_{O_2}\left(out\right)}{p_{O_2}\left(in\right)}$$(6)

$p_{O_2}(out)$ is obtained by Eq. (1) and $p_{O_2}(in)$ is 1 atm in an alumina tube.

Eliminating $p_{O_2}$ using the Eqs. of (2), (3), (4), (5), (6), the relation of P and B is obtained.   

$$\frac{\text{P}}{\text{B}}=-1\mathrm{\hspace{0.17em} }073R\left[ln\text{P}+ln\left\{\left(\frac{S}{L}\right)\left(\frac{298R}{0.1013V_{Ar}}\right)\right\}\right]$$(7)

The value of B calculated by the Eq. (7) from P is shown in Table 1 and Fig. 1. B is 5 orders smaller than P. Further, the permeability coefficient of oxygen increases by about 11 orders of magnitude larger by the addition of iron oxide.

Table 1. Apparent activation energy (E) and parameter (B₀) in B=B₀exp(-E/RT) of the permeability coefficient of oxygen for molten slags.

Slag			B0 (mol2/J·m·s)	E (kJ/mol·K)	Temp. range (°C)
Solvent (mass%): x	Solute y: mass%
25CaO-65SiO2-10Al2O3			6.23×10−5	117	1450–1500
40CaO-40SiO2-20Al2O3			2.18×10−7	82	1350–1450
45CaO-40SiO2-15Al2O3			7.96×10−8	69
40CaO-40SiO2-20Al2O3	Fe2O3	0.2	1.33×10	187
		1	2.46×10	189
		3	5.27×10	192
		5	1.22×10	195
		10	1.65×102	198
	CaF2	3.5	7.74×10−4	101
		6.9	5.28×10−3	103
		13.4	5.74×10−2	106
		19.7	8.43×10−2	107
	10 Fe2O3-6CaF2		1.63×10−1	109
33CaO-67SiO2	Fe2O3	10	1.72×10−1	112	1400–1500
		15	4.29×10−1	113
		20	5.05×10−1	114
		30	3.98×10−1	113
		40	3.67×10−1	113
50CaO-50SiO2	Fe2O3	5	2.63×10−1	111
		10	4.51×10−1	113
		15	6.33×10−1	115
		20	7.59×10−1	116
		30	5.59×10−1	114
		40	4.86×10−1	113
40CaO-60SiO2	Fe2O3	15	6.78×10−1	115	1400–1600
		20	8.35×10−1	116	1400–1500
		30	5.90×10−1	115
		40	5.01×10−1	114
50CaO-50SiO2	ZnO	10	1.09×10−2	93	1450–1550
	NiO	7.5	1.14×10−2	93
FeO			8×10−7		1400

Note: x+y=100 (mass%)

Fig. 1.

Permeability coefficients of oxygen in molten slags.

4. State of Permeating Oxygen

4.1. Silicate Slags without Transition Metal Ions

The crystal structure of silica is that a silicon atom is arranged in the gap of the center of 4 coordination of oxygen atoms in the closed-packed structure. Then one oxygen atom is coordinated to two silicon atoms. Oxygen atoms are covalently bonded to silicon atoms and strongly attracted to them. Therefore, the local structure is a tetrahedron of SiO₄. The oxygen atoms at the apex of SiO₄ are shared with the other SiO₄ to create a three-dimensional periodic structure (–Si–O–Si–). In the glass state, the structure has no regularity and is called the irregular network structure model.

The addition of alkali metal oxides and alkaline earth oxides to silica makes a more gap-like structure in which the bond of silicon and oxygen is cut by supplying oxygen atom to the apex of SiO₄.⁴⁾ Two oxygen atoms at both ends of SiO₄ structures near the gap are called non-bridging oxygen and electrically charged to −1 value, respectively. Ca²⁺ ion is in the vicinity of two non-bridging oxygen ions of O⁻ and attracts them.

Na⁺ ions are in the vicinity of each non-bridging oxygen ion and create wider gaps.¹⁰⁾ When CaF₂ is dissolved in silicate slag containing CaO, wider gaps are made by coordinating CaF⁺ ions in the vicinity of each non-bridging oxygen.

Since Na₂O–SiO₂ silicate slag exhibits the conductivity by Na⁺ ions, the correlation coefficient representing the diffusion path of ions obtained from the equation of Nernst-Einstein is between 0.2 and 0.4. This value coincides the value of the interstitial diffusion mechanism calculated from the theory of random walk. It is realized that sodium ions diffuse through the gap of silicate network.⁴⁾

In the molten silicate slag of 40CaO-40SiO₂-20Al₂O₃ (mass%) the tracer diffusion coefficients of elements are reduced in the order of oxygen, calcium, aluminum and silicon, though the transference number of calcium ion is close to unity. Calcium ions are most likely to move under the electric field. Thus, oxygen is most likely to move as an electrically neutral molecule. Sasabe et al.⁵⁾ showed that the permeability of oxygen is proportional to oxygen partial pressure, indicating that oxygen is physically dissolved in slag as an oxygen molecule near the gap of silicate network.

4.2. Silicate Slags Containing Transition Metal Ions

In the case of iron oxide-containing silicate slag, the redox reaction of iron ions between Fe²⁺ and Fe³⁺ occurs. The atomic defect structure of FeO and NiO crystals is called Schottky type that oxygen atom is excessive. Oxygen gas is decomposed to produce oxygen atoms, O_O, in the oxygen sublattice and the atomic defects, V_M²⁻, in the metal sublattice and hole, h.   

$$1/2{\mathrm{O}}_2={\mathrm{O}}_{\mathrm{O}}+{\mathrm{V}}_{\mathrm{M}}{}^{2-}+2\mathrm{h}$$(8)

In addition, some of Fe²⁺ ion becomes Fe³⁺ ion.   

$$\mathrm{F}{\mathrm{e}}^{2+}+\mathrm{h}=\mathrm{F}{\mathrm{e}}^{3+}$$(9)

Since the concentration of V_M²⁻ is equal to twice concentration of hall from the electrical neutral condition, the concentration of V_M²⁻ is proportional to the 1/6 power of oxygen partial pressure from the law of mass action. Since iron ions move through the atomic defects, the diffusion coefficient of iron ion is also proportional to the 1/6 power of the oxygen partial pressure.¹¹⁾

In the case of molten silicate slag containing transition metal oxide, it is supposed that the Eqs. of (8) and (9) are available. When alkaline metal oxide or alkaline earth metal oxide is dissolved in molten silicate, many gaps are made at the cutting point of the silicate network. Then, the atomic defect concentration is almost constant. Therefore, the concentration ratio of Fe³⁺ to Fe²⁺ is proportional to the 1/4 power of oxygen partial pressure from the Eqs. (8) and (9). Sasabe et al.⁶⁾ showed that for molten silicate slag containing iron oxide the flow rate of oxygen was proportional to the 1/4 power of oxygen partial pressure. Then, oxygen gas is dissolved as oxygen atoms at the apex of SiO₄ with the redox reaction of iron ions.

4.3. Zinc Oxide-containing Silicate Slags

ZnO crystal takes the Frenkel-type defect structure in which there are Zn⁺ ions between the lattices and electrons, e, by the reaction with zinc gas.¹¹⁾   

$$\mathrm{Z}\mathrm{n}\left(\mathrm{g}\mathrm{a}\mathrm{s}\right)=\mathrm{Z}{\mathrm{n}}^{+}+\mathrm{e}$$(10)

  

$$C_{{\text{Zn}}^{+}}\cdot C_{\text{e}}=K_{10}{p}_{\text{Zn}}$$(11)

K₁₀ is the equilibrium constant of the Eq. (10) and p_Zn is the partial pressure of zinc. The concentrations of Zn⁺ ion and electron are equal from the electrical neutrality.   

$$C_{{\text{Zn}}^{+}}=C_{\text{e}}$$(12)

When ZnO dissolves in molten silicate slag, oxygen ion cuts the binding of the apex of SiO₄ to create a gap.   

$$\mathrm{Z}\mathrm{n}\mathrm{O}+\left(-\mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{S}\mathrm{i}-\right)=\mathrm{Z}{\mathrm{n}}^{2+}+2\left(-\mathrm{S}\mathrm{i}-{\mathrm{O}}^{-}\right)$$(13)

Further zinc gas reacts with oxygen gas to produce ZnO.   

$$\mathrm{Z}\mathrm{n}\left(\mathrm{g}\mathrm{a}\mathrm{s}\right)+\left(1/2\right){\mathrm{O}}_2=\mathrm{Z}\mathrm{n}\mathrm{O}$$(14)

  

$$K_{14}=\frac{a_{\text{ZnO}}}{p_{\text{Zn}}{p}_{{\text{O}}_2}^{1/2}}$$(15)

The equilibrium constant K₁₄ is a function of temperature. activity of ZnO, a_ZnO, is a function of temperature and composition. From the Eqs. of (11), (12) and (15) the concentrations of Zn⁺ ion and electron are proportional to −1/4 power of oxygen partial pressure.

For ZnO-containing slag, Zn²⁺ ion reacts with electron to produce Zn⁺ ion.   

$$\mathrm{Z}{\mathrm{n}}^{2+}+\mathrm{e}=\mathrm{Z}{\mathrm{n}}^{+}$$(16)

From the Eqs. (10), (13), (14) and (16), oxygen gas and zinc ions react with silicate network in molten slag as the following reaction.   

$$\left(1/2\right){\mathrm{O}}_2+2\mathrm{Z}{\mathrm{n}}^{+}+\left(-\mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{S}\mathrm{i}-\right)=2\left(-\mathrm{S}\mathrm{i}-{\mathrm{O}}^{-}\right)+2\mathrm{Z}{\mathrm{n}}^{2+}$$(17)

Then, oxygen gas is also dissolved as oxygen atoms at the apex of SiO₄ like the redox reaction of iron ions.

5. Permeation Mechanisms of Oxygen in Molten Slags

Sasabe et al.¹²⁾ showed that the transfer rate of oxygen through molten slag from gas phase was limited by not chemical reaction at the surface but diffusion in slag.

5.1. Slag without Iron Oxide

When the oxygen in the slag diffuses as O₂, from the Fick’s first law the flow rate of O₂ is represented by   

$$\frac{Q}{S}=\frac{\text{P}}{L}=-D_{{\text{O}}_2}\frac{\mathrm{d}{C}_{{\text{O}}_2}}{\mathrm{d}\mathrm{x}}$$(18)

where $D_{{\text{O}}_2}$ and $C_{{\text{O}}_2}$ are the diffusion coefficient and the concentration of O₂, respectively. Sasabe et al.⁵⁾ measured the diffusion coefficient of oxygen in CaO–SiO₂–Al₂O₃ molten slags in the atmosphere near air.

Under the steady state, $\frac{\mathrm{d}{C}_{{\text{O}}_2}}{\mathrm{d}\mathrm{x}}$ can be represented as $-\frac{\mathrm{\Delta }{C}_{{\text{O}}_2}}{L}$. Then, the Eq. (18) is written as $\text{P}=D_{{\text{O}}_2}\mathrm{\Delta }{C}_{{\text{O}}_2}$. When the concentration of O₂ in the side with low oxygen partial pressure is negligibly small, the following equation is obtained,   

$$\text{P}=D_{{\text{O}}_2}{C}_{{\text{O}}_2}$$(19)

Sasabe et al. calculated $C_{{\text{O}}_2}$ in the slags from P and $D_{{\text{O}}_2}$. The oxygen solubility of the slags is 1×10⁻⁹ to 3×10⁻⁸mol/m³ at 1350 to 1500°C under the atmosphere near air. The oxygen solubility increases with increasing the molar volume of slag and decreasing the viscosity. From these results, they concluded that oxygen gas molecules pass through the cleavage points of silicate network.

5.2. Slag Containing Iron Oxide

Sasabe et al.⁶⁾ reported that the flow rate of oxygen increases by about 10¹⁰ times when 0.2 mass% of Fe₂O₃ is added to molten silicate slags of CaO–SiO₂–Al₂O₃. Also, they showed that for molten silicate slag containing iron oxide the flow rate of oxygen was proportional to the 1/4 power of oxygen partial pressure. It is known that the electrical conductivity of molten slag increases more than 10 times by the addition of 0.14 mass% of FeO.^13,14,15) However, the electrical conductivity of the slags mainly by Ca²⁺ ions hardly changes. These facts indicate that the redox reaction of Fe²⁺ and Fe³⁺ relates to the electric conduction.

As mentioned in 4.2, in the side with higher oxygen partial pressure, oxygen gas is dissolved in the slags as oxygen ions that take electrons from Fe²⁺ to make Fe³⁺. On the other hand, the reverse reaction occurs in the side with lower oxygen partial pressure.   

$$2\mathrm{F}{\mathrm{e}}^{2+}+1/2{\mathrm{O}}_2+\left(-\mathrm{S}\mathrm{i}-\mathrm{O}-\mathrm{S}\mathrm{i}-\right)\rightleftarrows 2\mathrm{F}{\mathrm{e}}^{3+}+2\left(-\mathrm{S}\mathrm{i}-{\mathrm{O}}^{-}\right)$$(20)

Thus, oxygen gas is apparently transported through slag and all of ions including Fe³⁺ and Fe²⁺ mutually diffuse.

The average flow rate of each atom in slag can be represented as that of individual element ion.¹⁶⁾ The average flow rates of ions, J_i, are determined on a coordinate. From no net electric current in a local area of slag, the average flow rate of oxygen is represented by those of the other elements.   

$$J_{O^{-}}=3J_{F{e}^{3+}}+2J_{F{e}^{2+}}+\sum z_i{J}_i\hspace{1em}\left(\mathrm{i}\ne {\mathrm{O}}^{2-},\mathrm{\hspace{0.17em} } \mathrm{F}{\mathrm{e}}^{3+},\mathrm{\hspace{0.17em} } \mathrm{F}{\mathrm{e}}^{2+}\right)$$

z_i is the electric charge of i ion.

The flow rates of ions are described as J_i=C_iv_i, where C_i and v_i are the concentration and velocity of i ion, respectively. The average velocity of ions on their concentrations, v_o, is represented by   

$$v_o=\sum C_{\mathrm{i}}{\mathrm{v}}_{\mathrm{i}}/\sum C_{\mathrm{i}}$$(21)

The flow rates of ions on the coordinate placed on v_o are determined as j_i=C_i (v_i−v_o). The sum of the flow rate of ions, Σj_i, is zero. This means that the coordination placed on v_o is equal to that placed on the alumina tube because slag layer does not move relative to the alumina tube. In the case that the coordination of Ji is placed on a marker in diffusion couple that moves relative to the diffusion holder, this phenomenon is called the Kirkendall effect.

As net electric current in a local area is zero, Σz_ij_i=0. Thus,   

$$j_{O^{-}}=3j_{F{e}^{3+}}+2j_{F{e}^{2+}}+\sum z_i{j}_i\hspace{1em}\left(\mathrm{i}\ne {\mathrm{O}}^{2-},\mathrm{\hspace{0.17em} } \mathrm{F}{\mathrm{e}}^{3+},\mathrm{\hspace{0.17em} } \mathrm{F}{\mathrm{e}}^{2+}\right)$$(22)

When the reaction described as the Eq. (20) takes place, all ions move under electric field that is generated by oxygen potential gradient.¹⁷⁾ As slag is fixed on an alumina tube, only the flow rate of oxygen element is apparently detected though the other elements move in slag. This phenomenon seems to be pushed out oxygen gas like a ball thrust.

When O₂ chemically absorbed in slag as O⁻ ions of non-bridging oxygen, Fe³⁺ ion mutually moves with Fe²⁺ ion near slag surface. Then, the apparent flow rate of O₂ can be represented as   

$$\frac{Q}{S}=j_{O_2}=4j_{F{e}^{3+}}$$(23)

  

$$j_{F{e}^{3+}}=-{\text{B}}_{\text{Fe}}\frac{\mathrm{d}{\eta }_{F{e}^{3+}}}{\mathrm{d}\mathrm{x}}$$(24)

where B_Fe is the mobility of iron ion that is equivalent to 4 times of the permeability coefficient of oxygen, B. ${\eta }_{F{e}^{3+}}$ is the electrochemical potential of Fe³⁺ ion. As Fe³⁺ ions make Fe₂O₃ with oxygen ions under the assumption of local equilibrium.¹⁷⁾ The following equation is obtained for the small concentration of Fe₂O₃.   

$$\frac{\mathrm{d}{\eta }_{{\text{Fe}}^{3+}}}{\mathrm{d}\mathrm{x}}=\frac{1}{2}\frac{\mathrm{d}{\mu }_{{\text{Fe}}_2{\text{O}}_3}}{\mathrm{d}\mathrm{x}}=\frac{1}{2}\frac{RT}{C_{{\text{Fe}}_2{\text{O}}_3}}\frac{\mathrm{d}{C}_{{\text{Fe}}_2{\text{O}}_3}}{\mathrm{d}\mathrm{x}}$$

From the Fick’s first law,   

$$j_{F{e}^{3+}}=-\frac{1}{2}{D}_{{\text{Fe}}^{3+}}\frac{\mathrm{d}{C}_{{\text{Fe}}_2{\text{O}}_3}}{\mathrm{d}\mathrm{x}}$$

the following equation is obtained.   

$$D_{{\text{Fe}}^{3+}}=\frac{RT{B}_{Fe}}{C_{{\text{Fe}}_2{\text{O}}_3}}$$(25)

where $D_{{\text{Fe}}^{3+}}$ is the diffusion coefficient of Fe³⁺ and is reported¹⁸⁾ as   

$$D_{{\text{Fe}}^{3+}}=0.708\times {10}^{-8}\text{exp}\frac{-279\left(\text{kJ}\right)}{RT}\hspace{1em}\left({\mathrm{m}}^2/\mathrm{s}\right)$$(26)

At 1400°C, $D_{{\text{Fe}}^{3+}}$ is 1.39×10⁻⁹m²/s. The concentration of Fe₂O₃ of 0.2 mass% is 45 mol/m³. B_Fe at 1400°C is calculated to be 6.2×10⁻¹⁰ mol²/J·m·s. Then, B is 1.6 × 10⁻¹⁰ mol²/J·m·s. This value is 10¹¹ times larger than slags without iron oxide.

Sasabe et al. showed that P appeared to be maximum in the vicinity of 30 mass% of Fe₂O₃ in 50CaO-50SiO₂ molten slag where the ratio of $\frac{C_{{\text{Fe}}^{3+}}}{C_{{\text{Fe}}^{3+}}+C_{{\text{Fe}}^{2+}}}$ was maximum. The permeability coefficient of oxygen, B, also appears a maximum value in the same manner. Therefore, the permeability coefficient of oxygen depends on the concentration of Fe³⁺ ion.

The electrical conductivity of the 50CaO-50SiO₂ molten slag with the low concentration range of Fe_xO is about 30 Ω⁻¹m⁻¹ at 1400°C, but increases a little by increasing the concentration of Fe_xO. It rapidly increases from 100 Ω⁻¹m⁻¹ over 50 mass% Fe_xO and 30000Ω⁻¹m⁻¹ in pure Fe_xO.¹⁹⁾ The cause is that the conduction mechanisms changes from ionic conduction to hole conduction more than around 50 mass%.

On the slag side with higher oxygen partial pressure, Fe²⁺ reacts with hole to produce Fe³⁺ and oxygen molecules become oxygen ions to produce holes.   

$${\mathrm{O}}_2\to 2{\mathrm{O}}^{2-}+4\mathrm{p}$$(27)

Holes move via Fe²⁺ ions and Fe³⁺ ions and the reverse reaction occurs on the other side with lower oxygen partial pressure. All ions and hole move, and oxygen gas is apparently pushed out from the slag like a ball thrust.

The mobility B_e of hole is represented as   

$$\kappa =F^2{\mathrm{B}}_{\mathrm{e}}$$(28)

where κ is the electrical conductivity. When the electrical conductivity of Fe_xO at 1400°C is 30000 Ω⁻¹m⁻¹, B_e is 3.2×10⁻⁶ mol²/J·.m·s. Since the oxygen gas is generated two oxygen ions and four holes, the permeability coefficient of oxygen, B, is obtained by the division of B_e by 4 and is 8×10⁻⁷ mol²/J·m·s. The temperature dependence of the permeability coefficient is small. This value is 10¹⁵ times larger than 40CaO-40SiO₂-20Al₂O₃ (mass%) slag. It shows that Fe_xO molten slag plays the acceleration of oxidation and decarburization of steel.

5.3. ZnO-containing Slag

As shown in the Eq. (17), oxygen gas reacts with Zn⁺ to generate non-bridging oxygen ion and Zn²⁺ on the slag side with higher oxygen partial pressure. The reverse reaction proceeds on the other side. Then, Zn⁺ and Zn²⁺ ions mutually diffuse and the other ions also move. Apparently, oxygen gas seems to transfer through molten slag like a ball thrust.   

$$j_{{\text{O}}_2}=4j_{{\text{Zn}}^{2+}}$$(31)

  

$$D_{Z{n}^{2+}}=\frac{RT{\text{B}}_{\text{Zn}}}{C_{ZnO}}$$(32)

The diffusion coefficient of Zn²⁺ in 25CaO-25MgO-50SiO₂ (mol%) molten slag²⁰⁾ is reported as   

$$D_{Zn}=3.3\times {10}^{-5}\text{exp}\frac{-169\left(\text{kJ}\right)}{RT}\hspace{1em}\left({\mathrm{m}}^2/\mathrm{s}\right)$$(33)

and is 2.5×10⁻¹⁰ m²/s at 1450°C. Assuming that the diffusion coefficient of Zn in the molten slag of 10ZnO-(50CaO-50SiO₂) (mass%) is close to this value and using the concentration of zinc oxide of 3.9×10³ mol/m³, the value of B_Zn is 6.0×10⁻¹² mol²/J·m·s. Then, B is calculated to be 1.5 × 10⁻¹² mol²/J·m·s from B_Zn/4. This value is near the measured one of 9.7×10⁻¹² mol²/J·m·s.

As described in Section 4.3, the concentrations of zinc ion and electron are proportional to −1/4 power of oxygen partial pressure. The permeability coefficient of B could take the same tendency. Sasabe et al.⁸⁾ showed that the flow rate of oxygen through 10ZnO-(50CaO-50SiO₂) molten slag was proportional to −1/4.2 power of oxygen partial pressure.

However, the flow rates of oxygen in the molten slags with 20 mass% and 30 mass%ZnO is proportional to 1/5.7 power and 1/1.6 power of oxygen partial pressure, respectively. This indicates the behavior of the p-type semiconductor and is difficult to explain for the slag containing zinc oxide.

50CaO-50SiO₂ silicate slag containing zinc oxide has the compound of Ca₂ZnSi₂O₇ with the melting point of 1425°C and the solubility of the solid ZnO in this slag is 50 mass% at 1500°C.²¹⁾ This means that the activity of ZnO relative to solid state positively deviates from the Raoult’s law. The vapor pressure of zinc equilibrated with solid ZnO at 1500°C is 6.31×10⁻⁴ atm under the oxygen partial pressure of 1 atm and 1 atm under that of 3.98×10⁻⁷ atm. Sasabe et al.⁸⁾ performed the experiments in the concentration range of 20 to 30 mass% ZnO from 1550 to1450°C in the oxygen partial pressure of about 2.5×10⁻⁴ atm. In this condition, the partial pressure of zinc becomes about 0.04 atm and Zn seemed to evaporate. In the concentration of zinc oxide over 20 mass% under the low oxygen partial pressure, the composition of ZnO in slag could change and then the flow rate of oxygen decreased.

6. Conclusion

The permeability coefficient of oxygen of molten silicate slags is the order of 10⁻²² to 10⁻⁸ mol²/J·m·s at 1350 to 1550°C. The value increases by about 11 orders of magnitude larger by the addition of iron oxide of only 0.2 mass% to molten silicate slags. In molten Fe_xO, it is the order of 10⁻⁶ mol²/J·m·s. The state of oxygen permeating is oxygen molecules in molten silicate without transition metal oxide. In molten slag with transition metal oxide, oxygen gas makes redox reaction with transition metal ions to produce non-bridging oxygen ions in silicate network. Under the electric field generated by oxygen potential gradient, all of ions in slag move, and oxygen gas seems to permeate through slag like a ball thrust.

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