Kinetics on Formation, Growth, and Removal of Alumina Inclusions in Molten Steel

Abstract

A series of Al deoxidation mechanisms from the nucleation and growth of Al₂O₃ nuclei immediately after the addition of Al to the growth, agglomeration, and removal of Al₂O₃ inclusions after the deoxidation equilibrium has been analyzed in light of the kinetics taking into consideration the influences of the interfacial properties on the basis of Al deoxidation experiments of molten steel. The nucleation number density of Al₂O₃ is (0.72 to 1.62) × 10¹⁴ m⁻³ and increases as the degree of supersaturation increases and the interfacial tension between the nuclei and molten steel decreases. These tendencies can be explained by the homogeneous nucleation theory, and the average interfacial tension, frequency factor, nucleation time, and average nucleation rate are respectively estimated to be 1.43 N·m⁻¹, 4.27 × 10³⁵ m⁻³·s⁻¹, 0.01 s, and 1.96 × 10¹⁶ m⁻³·s⁻¹ for the nucleation of Al₂O₃. Al₂O₃ nuclei rapidly grow to Al₂O₃ single inclusions having diameters of 2.0 to 2.6 µm through diffusion growth of supersaturated O in molten steel within 2.2 to 3.7 s after the addition of Al, and the molten steel reaches the deoxidation equilibrium. In the subsequent deoxidation equilibrium, the growth rate of Al₂O₃ single inclusions increases as the O concentration in molten steel increases, and their growth mechanism can be explained by Ostwald ripening. Meanwhile, Al₂O₃ cluster inclusions grow with the increase in the agglomeration force while agglomerating not only with single inclusions dispersed in molten steel but also with other cluster inclusions existing in the floating paths.

1. Introduction

The deoxidation control implemented in the final process of steelmaking plays an extremely important role in determining the quality and properties of steel products. Thus, there have been many studies to reveal the principle of deoxidation, and these research outcomes were reviewed in detail by, for example, Sakao et al.¹⁾ An overview of these studies shows that the thermodynamic data on the deoxidation equilibrium of almost all types of industrially available deoxidizing agents have been measured and evaluated, and have been effectively used for operation analysis in steelmaking based on the equilibrium theory. In contrast, according to research on kinetics, the agglomeration and floating separation of inclusions are considered to be the rate-determining step for deoxidation rates. However, because of the difficulty of conducting dissolution experiments for each elementary step and uncertainty about the influences of the interfacial properties, it seems that there is no common understanding of the mechanisms of the formation and growth of inclusion nuclei immediately after deoxidation as well as the mechanisms of the growth and agglomeration of inclusions after the deoxidation equilibrium in each elementary process.

The author has conducted fundamental studies on the agglomeration mechanism of Al₂O₃ inclusions in molten steel from the viewpoint of interface chemistry. In the course of these studies, the agglomeration force acting between Al₂O₃ particles in molten steel was successfully measured and it was verified that Al₂O₃ inclusions are subject to a strong mutual attracting force generated by the cavity bridge due to their low wettability with molten steel.^2,3,4) Also, through Al deoxidation experiments on molten steel, the author quantitatively showed that the particle diameters of Al₂O₃ cluster inclusions increased in proportion to the agglomeration force generated by the cavity bridge, and that Al₂O₃ cluster inclusion removal rates also increased as particle diameters increased.⁵⁾ Based on these fundamental studies, it can be concluded that the Al₂O₃ inclusions produced in molten steel through Al deoxidation agglomerate and coalesce by the agglomeration force due to the cavity bridge force and are removed as Al₂O₃ cluster inclusions. However, in order to improve the quality and properties of steel products by controlling the diameters and quantity of Al₂O₃ inclusions in molten steel, it is still necessary to kinetically ascertain the overall mechanisms of Al deoxidation, including the formation, growth, agglomeration, and removal of Al₂O₃ inclusions.

In the present study, the formation and growth of Al₂O₃ nuclei immediately after the addition of Al and the growth, agglomeration, and removal of Al₂O₃ inclusions in the subsequent deoxidation equilibrium were analyzed through mathematical models derived by assuming appropriate mechanisms on the basis of Al deoxidation experiments that controlled the concentrations of oxygen and sulfur in molten steel, which are interfacial active elements. Through these examinations, the series of Al deoxidation mechanisms were clarified from a perspective of the kinetics in consideration of the influences of the interfacial properties.

2. Experimental Method

2.1. Al Deoxidation Experiments of Molten Steel

The Al deoxidation experiments were conducted using a graphite resistance heating furnace with a capacity of 30 kW.^2,3,4,5) In the experiments, 500 g of electrolytic iron (with C concentration of 0.001 mass%, S concentration of 0.0001 mass%, O concentration of 0.005 mass%, and Mn concentration of 0.0001 mass%) was melted in an alumina crucible with an inner diameter of 40 mm and a height of 150 mm under an Ar gas atmosphere. The molten steel temperature was constantly set at 1600°C. After adjusting the O concentration to 0.018 to 0.044 mass% by adding Fe₂O₃, Al of 0.02 to 0.06 mass% was added to the molten steel based on the stoichiometric proportion of Eq. (1) (reaction between O of 0.018 mass% and Al of 0.02 mass%) so as to adjust the oxygen concentration of total Al₂O₃ inclusions immediately after the addition of Al to approximately 0.017 mass%, and so as to vary the O concentrations after the deoxidation equilibrium in the range of 0.0006 to 0.0261 mass%. In this study, X represents the dissolved state of the element X.   

$$2\underset{\_}{\mathrm{A}\mathrm{l}}+3\underset{\_}{\mathrm{O}}=\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$$(1)

The experiments also included cases of Al deoxidation with the O concentration and S concentration in molten steel adjusted to 0.0009 mass% or less and 0.018 to 0.073 mass%, respectively. These experiments were started 60 s after the addition of Al and continued for 600 s. The samples of molten steel were taken with transparent quartz tubes with an inner diameter of 6 mm immediately before the addition of Al, after the start of the experiments, and during the experiments; the total oxygen, Al, and S concentrations of the molten steel were analyzed using the samples.

The O concentrations during the experiments were obtained from the analytical values of the Al concentrations using the thermodynamic reevaluation value for the Al deoxidation equilibrium proposed by Itoh et al.⁶⁾ except for the case of high O concentrations, with Al concentrations less than the analytical limit, in which the O concentrations were calculated using the thermodynamic reevaluation value for the Al deoxidation equilibrium and mass balances based on the O concentrations before the addition of Al and the additive amounts of Al. The oxygen concentrations of total Al₂O₃ inclusions [I.O]_T were obtained by subtracting the O concentrations during the experiments from the analytical values of the total oxygen concentrations at the respective sampling times.

2.2. Observation of Inclusions by the Shapes with Optical Microscope

10 mm long specimens subject to microscopic observation of inclusions were taken from the central portions of rod-like molten steel samples, and the circular cut faces of the specimens were mirror polished. Using an optical microscope, the particle size distribution was obtained for the following: the Al₂O₃ cluster inclusions having diameters of 10 μm or more existing in the cross-sectional areas with 5 mm diameter excluding the outermost peripheries at 100x magnification; and the Al₂O₃ single inclusions having diameters of 0.5 μm or more existing in the areas of 1 to 4 mm² at 1000x magnification. Based on the particle size distribution of the Al₂O₃ single and cluster inclusions, the average particle diameters and the volume number density of the inclusions in the molten steel samples were calculated using DeHoff’s equation.⁷⁾ Also, using an EPMA (Electron Probe Micro Analyzer), composition analyses were conducted for typical molten steel samples at each experimental level, and it was confirmed that all inclusions were Al₂O₃.

2.3. Evaluation of the Formation Number Density of Total Al₂O₃ Single Inclusions

Because the Al₂O₃ cluster inclusions are the aggregates of Al₂O₃ single inclusions, the number density of total Al₂O₃ single inclusions is considered to be the sum of the number density of the single inclusions suspended in molten steel and that of the single inclusions that form clusters. Because the number density of total Al₂O₃ single inclusions N_V,N (m⁻³) formed immediately after the addition of Al could not be obtained directly by experiment, it was evaluated by Eq. (2) which corrects the number density of total Al₂O₃ single inclusions at the start time of the experiments (60 s after the addition of Al) with the ratios of the oxygen concentrations of total Al₂O₃ inclusions immediately after the addition of Al to those at the start time of the experiments.   

$$\begin{array}{l}{\mathrm{N}}_{\mathrm{V},\mathrm{N}}=\\ \left[{\mathrm{N}}_{\mathrm{V},\mathrm{S}(60)}+{({\mathrm{d}}_{\mathrm{C}\mathrm{I}(60)}/{\mathrm{d}}_{\mathrm{S}\mathrm{I}(60)})}^3\cdot \epsilon \cdot {\mathrm{N}}_{\mathrm{V},\mathrm{C}(60)}\right]\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}(0)}/{[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}(60)}\end{array}$$(2)

where N_V,C is the volume number density of Al₂O₃ cluster inclusions (m⁻³), N_V,S is the volume number density of Al₂O₃ single inclusions (m⁻³), ε is the filling rate of Al₂O₃ in Al₂O₃ cluster inclusions with a value of 0.244,⁵⁾ d_CI is the average particle diameter of Al₂O₃ cluster inclusions (m), d_SI is the average particle diameter of Al₂O₃ single inclusions (m), and the subscripts of (0) and (60) represent the time immediately after the addition of Al and the start of the experiment 60 s after that, respectively.

3. Experimental Results

3.1. Volume Number Density of Al₂O₃ Nuclei

The degree of supersaturation S_S of the Al deoxidation reaction in Eq. (1) is defined by Eq. (3).   

$${\mathrm{S}}_{\mathrm{S}}={\mathrm{K}}_{\mathrm{S}}/{\mathrm{K}}_{\mathrm{E}}$$(3)

where K_S is the solubility product in Eq. (1) expressed by [Al]²·[O]³ and K_E is the equilibrium constant of Eq. (1) which uses the reevaluation value of the Al deoxidation equilibrium proposed by Itoh et al.⁶⁾ [X] represents the X concentration. Figure 1 shows the relation between N_V,N and S_S. As can be seen, N_V,N increased as S_S increased. Figure 2 shows the interfacial tension between molten steel and Al₂O₃, ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ (N·m⁻¹), obtained by substituting into Young’s Eq. (4) the surface tension of molten steel, σ_Fe (N·m⁻¹), and the contact angle between molten steel and Al₂O₃, ${\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ (°), noted in previous studies.^4,5)   

$${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}={\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}-{\sigma }_{\mathrm{F}\mathrm{e}}\cdot \mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$$(4)

where ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$ is the surface tension of Al₂O₃ with a value of 0.75 N·m⁻¹.⁸⁾ Then, ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ after the deoxidation equilibrium in each experiment was estimated from Fig. 2, and the relation between ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ and N_V,N was obtained as shown in Fig. 3. N_V,N demonstrated an increasing trend as ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ decreased, but the influence of ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ was smaller than that of S_S in Fig. 1. According to the homogeneous nucleation theory, N_V,N is predicted to increase when nucleation is enhanced since the critical nucleus radii become smaller as the supersaturation degree increases and the interfacial tension decreases. Thus, the relation of Figs. 1 and 3 can be explained qualitatively.

Fig. 1.

Relation between degree of supersaturation S_S and formation number density N_V,N of total Al₂O₃ single inclusions in Al deoxidation reaction.

Fig. 2.

Influences of the concentrations of O and S in molten steel on interfacial tension ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ between molten steel and Al₂O₃.

Fig. 3.

Relation between molten steel-Al₂O₃ interfacial tension ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ and formation number density N_V,N of total Al₂O₃ single inclusions.

When critical nuclei Al₂O₃ uniformly distributed in molten steel with a volume number density of N_V,N independently grow by promptly consuming supersaturated Al and O, Al₂O₃ single inclusions with a particle diameter d are produced according to Eq. (5).   

$$\mathrm{d}=2{\left[3({\mathrm{C}}_{\mathrm{B}(0)}-{\mathrm{C}}_{\mathrm{E}\mathrm{q}})/(4\pi \cdot {\mathrm{N}}_{\mathrm{V},\mathrm{N}}\cdot {\mathrm{C}}_{\mathrm{P}})\right]}^{1/3}$$(5)

where C_B is the O molar concentration in molten steel (mol·m⁻³), C_Eq is the equilibrium O molar concentration in molten steel (mol·m⁻³), and C_P is the O molar concentration in Al₂O₃ calculated by $3{\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$/${\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$ = 1.17 × 10⁵ mol·m⁻³. In the Al deoxidation experiments, because the production amount of Al₂O₃ single inclusions corresponds to an oxygen mass percentage of 0.017 mass%O, C_B(0) − C_Eq can be calculated by 0.017/100·ρ_Fe/M_O = 74.4 mol·m⁻³, where ρ_Fe is the density of molten steel with a value of 7000 kg·m⁻³, ${\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$ is the density of Al₂O₃ with a value of 3970 kg·m⁻³, M_O is the atomic mass of O (kg·mol⁻¹), and ${\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$ is the molecular mass of Al₂O₃ (kg·mol⁻¹). Figure 4 shows a comparison of the relation between d_SI and N_V,N with the calculation results of Eq. (5). At a relatively early stage 60 s after the addition of Al, d_SI gradually decreased with the increase in N_V,N and roughly corresponded to the particle diameter predicted by Eq. (5). Thus, N_V,N can be regarded to be the volume number density of critical nuclei Al₂O₃. Also, the decrease in d_SI with the increase in N_V,N was barely observed after 60 s elapsed since the addition of Al; contrarily, d_SI showed an increase from the values at 60 s after the addition of Al with the increase in N_V,N. Accordingly, it was considered that the particle diameters of Al₂O₃ single inclusions were dominated by nucleation until approximately 60 s after the addition of Al, but the influence of the nucleation became diluted as time passed thereafter and another particle growth mechanism contributed to the increase in the particle diameters of the Al₂O₃ single inclusions.

Fig. 4.

Relation between formation number density N_V,N of total Al₂O₃ single inclusions and average particle diameter d_SI of Al₂O₃ single inclusions.

3.2. Particle Diameter Changes and Decrease Rates of Al₂O₃ Inclusions in Molten Steel

Figure 5 shows the changes in d_CI and d_SI with time. d_CI gradually increased as time advanced before decreasing after peaking at 240 s. The maximum value of d_CI was reduced with increases in the concentrations of O and S in the molten steel. This is because the agglomeration force acting among Al₂O₃ inclusions in the molten steel is reduced with increases in the concentrations of O and S. The details of this matter are discussed later in 4.2.4. Also, d_SI showed an increasing tendency as the O concentration increased. This growth mechanism of Al₂O₃ single inclusions also observed in Fig. 4 is discussed in detail in 4.2.3. The time change in the logarithms of [I.O]_T shown in Fig. 6 was a linear decreasing with time according to the first-order kinetics expressed by Eq. (6).   

$$-d{[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}/dt={\mathrm{k}}_{\mathrm{T}}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}$$(6)

where t is time (s) and k_T is the decrease rate constant for the oxygen concentration of total Al₂O₃ inclusions (s⁻¹). The decrease rate of [I.O]_T was reduced with the increases in the concentrations of O and S in the molten steel. This is because d_CI decreased as the concentrations of O and S increased as shown in Fig. 5, thereby reducing the floating separation rates of the Al₂O₃ cluster inclusions.

Fig. 5.

Time changes in average particle diameter d_CI of Al₂O₃ cluster inclusions and average particle diameter d_SI of Al₂O₃ single inclusions.

Fig. 6.

Time change in oxygen concentration [I.O]_T of total Al₂O₃ inclusions.

4. Discussion

Because when a deoxidizing agent is added to quiescent molten steel the dissolution of the deoxidation agent and its chemical reaction with oxygen are quite fast,⁹⁾ it is considered that nucleation and the growth of nuclei occur immediately with a rapid reduction in dissolved oxygen to an equilibrium state, and then deoxidation products grow relatively gradually while floating and separating from molten steel.^10,11) Also, according to previous Al deoxidation experiments, it is known that when the O concentration increases after reaching the deoxidation equilibrium, Al₂O₃ single inclusions gradually grow concurrently with the agglomeration growth of Al₂O₃ cluster inclusions. Thus, in what follows, a series of Al deoxidation mechanisms are examined in a manner that classifies the Al deoxidation process into the following elementary steps: ① nucleation and ② growth of Al₂O₃ nuclei immediately after the addition of Al, and ③ growth of Al₂O₃ single inclusions and ④ agglomeration growth and removal of Al₂O₃ cluster inclusions during floating after reaching the deoxidation equilibrium; and derives rate equations capable of appropriately expressing the time changes in particle number density and particle diameters in each elementary step.

4.1. Al₂O₃ Nucleation Mechanism ①

4.1.1. Al₂O₃ Nucleation Theory

Because even a slight decline in supersaturation significantly reduces nucleation rates, Al₂O₃ nucleation is analyzed by applying the assumption that Bogdandy¹²⁾ used to calculate the nucleation of ferric oxide vapor to Al deoxidation. That is, Eq. (7) holds when assuming that the nucleation of Al₂O₃ occurs at a constant nucleation rate I₀ (nucleus·m⁻³·s⁻¹) during a period t_k (s) when the initial number density of molecules of supersaturated Al₂O₃ N₀ (molecule·m⁻³) is reduced to 0.9N₀ due to nucleation and, after that, no additional nucleation occurs.   

$$0.1{\mathrm{N}}_0=1/2{\mathrm{I}}_0\cdot {\int }_0^{{\text{t}}_{\text{k}}}{\int }_{\tau }^{{\text{t}}_{\text{k}}}\text{w}dtd\tau$$(7)

where τ is time (s), w is the growth rate of Al₂O₃ nuclei (molecule·s⁻¹·nucleus⁻¹), and I₀ is the nucleation rate when the number density of molecules of dissolved Al₂O₃ in supersaturation becomes 0.95N₀ (with the degree of supersaturation S_S0) that can be obtained by Eqs. (8), (9), (10), (11), (12) according to the homogeneous nucleation theory.^13,14)   

$${\mathrm{I}}_0={\mathrm{K}}_{\mathrm{V}}\cdot \mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}/({\mathrm{k}}_{\mathrm{B}}\cdot \mathrm{T}))$$(8)

  

$$\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}=16/3\pi \cdot {\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}{}^3\cdot {\left[{\mathrm{V}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/(\mathrm{R}\cdot \mathrm{T}\cdot \mathrm{l}\mathrm{n}({\mathrm{S}}_{\mathrm{S}0}))\right]}^2$$(9)

  

$$\begin{array}{l}{\mathrm{K}}_{\mathrm{V}}={\text{n}}_{{\text{Al}}_2{\text{O}}_3}^{*}\cdot {\left[{\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}/({\mathrm{k}}_{\mathrm{B}}\cdot \mathrm{T})\right]}^{1/2}\cdot {\left[2{\mathrm{v}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/(9\pi )\right]}^{1/3}\cdot \\ \hspace{1em}\hspace{1em}\hspace{1em}{\text{n}}_{{\text{Al}}_2{\text{O}}_3}\cdot ({\mathrm{k}}_{\mathrm{B}}\cdot \mathrm{T}/\mathrm{h})\cdot \mathrm{e}\mathrm{x}\mathrm{p}(-\mathrm{\Delta }{\mathrm{G}}_{\mathrm{A}}/({\mathrm{k}}_{\mathrm{B}}\cdot \mathrm{T}))\end{array}$$(10)

  

$${\text{n}}_{{\text{Al}}_2{\text{O}}_3}^{*}=4\pi \cdot {\mathrm{r}}^{*2}/{\left\{2{\left[3{\mathrm{v}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/(4\pi )\right]}^{1/3}\right\}}^2$$(11)

  

$${\mathrm{r}}^{*}=2{\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}\cdot {\mathrm{V}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/(\mathrm{R}\cdot \mathrm{T}\cdot \mathrm{l}\mathrm{n}({\mathrm{S}}_{\mathrm{S}0}))$$(12)

where K_V is the frequency factor (m⁻³·s⁻¹), k_B is the Bolzmann constant (J·K⁻¹), $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$ is the change in the free energy of formation of Al₂O₃ critical nuclei at the degree of supersaturation S_S0 (J), T is the absolute temperature of 1873 K, ${\mathrm{V}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$ is the molar volume of Al₂O₃ (m³·mol⁻¹), R is the gas constant (J·K⁻¹·mol⁻¹), ${\mathrm{v}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$ is the volume of a single molecule of Al₂O₃ (m³·molecule⁻¹), h is the Planck constant (J·s), ΔG_A is the diffusion activation energy with a value of 1.36 × 10⁻¹⁹ J,¹⁵⁾ r^* is the radius of the critical nucleus of Al₂O₃ (m), ${\text{n}}_{{\text{Al}}_2{\text{O}}_3}^{*}$ is the number of molecules on the surface of the critical nucleus of Al₂O₃ approximated by Eq. (11), and ${\mathrm{n}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$ is the number of molecules of Al₂O₃ in the unit volume of molten steel with a value of 1% of N₀ ([I.O]_T = 0.017 mass%) on the assumption that almost all supersaturated Al₂O₃ was in a dissolved state at the time of nucleation (with the degree of supersaturation S_S0). Given that Al₂O₃ nuclei grow through steady diffusion of O as with the analysis of Si deoxidation by Sano et al.,¹¹⁾ w in Eq. (7) can be expressed by Eqs. (13) and (14).   

$$\mathrm{w}=4\pi /(3{\mathrm{v}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3})\cdot d{({\mathrm{r}}^{*2}+2{\mathrm{k}}_{\mathrm{D}}\cdot \mathrm{t})}^{3/2}/dt$$(13)

  

$${\mathrm{k}}_{\mathrm{D}}={\mathrm{D}}_{\mathrm{O}}\cdot ({\mathrm{C}}_{\mathrm{B}0}-{\mathrm{C}}_{\mathrm{I}})\cdot {\mathrm{V}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/3$$(14)

where D_O is the diffusion coefficient of O in molten steel with a value of 2.3 × 10⁻⁹ m²·s⁻¹,¹⁶⁾ C_B0 is the molar concentration of O in molten steel at the degree of supersaturation S_S0 (mol·m⁻³), and C_I is the molar concentration of O on the interface between molten steel and Al₂O₃ at the molten steel side (mol·m⁻³), which can be considered equal to C_Eq on the basis of the deoxidation equilibrium. Equation (15) can be obtained by substituting Eq. (13) into Eq. (7) and integrating it. With Eq. (15) nucleation time t_k can be calculated.   

$$\begin{array}{l}0.1{\mathrm{N}}_0=2\pi \cdot {\mathrm{I}}_0\cdot \\ \left[{({\mathrm{r}}^{*2}+2{\mathrm{k}}_{\mathrm{D}}\cdot {\mathrm{t}}_{\mathrm{k}})}^{3/2}\cdot {\mathrm{t}}_{\mathrm{k}}-{({\mathrm{r}}^{*2}+2{\mathrm{k}}_{\mathrm{D}}\cdot {\mathrm{t}}_{\mathrm{k}})}^{5/2}/(5{\mathrm{k}}_{\mathrm{D}})+{\mathrm{r}}^{*5}/(5{\mathrm{k}}_{\mathrm{D}})\right]/\\ (3{\mathrm{v}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3})\end{array}$$(15)

4.1.2. Al₂O₃ Nucleation Rates and Nucleation Time

The volume number density of Al₂O₃ nuclei can be expressed by the equality N_V,N = I₀·t_k. Equation (16) can be obtained by substituting Eq. (8) into I₀ in the equality and taking the logarithm of both sides.   

$$\mathrm{l}\mathrm{n}({\mathrm{N}}_{\mathrm{V},\mathrm{N}})=\mathrm{l}\mathrm{n}({\mathrm{K}}_{\mathrm{V}}\cdot {\mathrm{t}}_{\mathrm{k}})-\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}/({\mathrm{k}}_{\mathrm{B}}\cdot \mathrm{T})$$(16)

Assuming that K_V and t_k are constant under Al deoxidation, a linear relation with an inclination of −1 can be expected between ln(N_V,N) and $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$/(k_B·T).¹⁰⁾ Figure 7 shows the relation between ln(N_V,N) and $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$/(k_B·T) obtained for several cases. In the figure, ☐ represents ln(N_V,N) corresponding to $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$/(k_B·T) calculated by Eq. (9) using ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ of each experiment obtained from Fig. 2 in accordance with the concentrations of O and S in molten steel after deoxidation, and ○ represents ln(N_V,N) corresponding to $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$/(k_B·T) calculated by Eq. (9) using ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ = 2.40 N·m⁻¹ that Turpin and Elliott¹⁷⁾ applied to the nucleation of Al₂O₃. The dispersion of ○ with a constant value of ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ was smaller than ☐, but no linear relation could be found between ln(N_V,N) and $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$/(k_B·T). In contrast, ■, ●, and ▲ in the figure represent ln(N_V,N) corresponding to $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$/(k_B·T) calculated by Eq. (9) using the average of ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ obtained from Fig. 2 for each experiment assuming that the deoxidation ratio f_DI on the interfaces between molten steel and Al₂O₃ nuclei were 1.0, 0.95, and 0.9, respectively. Table 1 shows the average ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ for each f_DI used in the calculation. f_DI is the deoxidation ratio on the interfaces of Al₂O₃ nuclei defined in the same way as in Eq. (19), and an f_DI value of 1 indicates the deoxidation equilibrium. The linear relations with an inclination of −1 with respect to each f_DI were calculated through the least square method and drawn as solid lines in the figure. Every case of using the average ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ for each f_DI in the calculation shows a linear relation with an inclination of −1 between ln(N_V,N) and $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$/(k_B·T) with less dispersion than the cases of □ and ○. It is difficult to accurately determine the interfacial tension between molten steel and Al₂O₃ nuclei while taking into consideration the influences of the interfacial active elements in the nucleation process, where ultrafine Al₂O₃ nuclei form in an extremely short period of time. Using different ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ for each experiment causes the dispersion of the experimental results to increase because, as can be seen in Eq. (9), $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$/(k_B·T) varies as the cube of ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$. Because the influence of ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ on N_V,N is small according to the experimental results shown in Fig. 3, in what follows, a discussion on nucleation is carried out using a constant value for ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$, namely the average, so as to easily obtain the linear relation and alleviate the influence of the differences in ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$.

Fig. 7.

Relation between volume number density N_V,N of Al₂O₃ nuclei and changes in nucleation free energy $\mathrm{\Delta }{\text{G}}_{\text{N}0}^{*}$.

Table 1. Average interfacial tension ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$, average critical nucleus radius r^*, frequency factor K_V, and average nucleation rate I₀ with respect to interfacial deoxidation ratio f_DI.

Interfacial deoxidation ratio fDI	Average interfacial tension ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ (N·m−1)	Average nucleus radius r* (m)	Frequency factor KV (m−3·s−1)	Average nucleation rate I0 (m−3·s−1)
1	1.59	4.87×10−10	5.56×1035	1.46×109
0.95	1.43	4.38×10−10	4.27×1035	1.96×1016
0.9	1.26	3.86×10−10	3.11×1035	1.84×1022
0.85	1.18	3.62×10−10	2.64×1035	3.60×1024
0.8	1.11	3.40×10−10	2.26×1035	2.05×1026

The interceptions of the solid lines in Fig. 7 are equal to ln(K_V·t_k) as is evident in Eq. (16). Table 1 shows the average r^*, K_V and the average I₀ calculated by Eqs. (8), (9), (10), (11), (12) using the average ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ and the average S_S0 of 47000 corresponding to each f_DI. The values of t_k calculated on the basis of ln(K_V·t_k) read from Fig. 7 and K_V in Table 1 for each f_DI are indicated by ○ marks in Fig. 8. Also, the dashed-dotted line in Fig. 8 represents the t_k theoretically predicted by Eq. (15) using the averages of r^* and I₀ shown in Table 1. When f_DI was 0.95, the experimental and theoretical values of t_k was approximately 0.01 s and both values was coincident. Therefore, the nucleation time of Al₂O₃ can be considered to be on the order of 0.01 s. On the interfaces between Al₂O₃ nuclei and molten steel, the O concentration was higher than the equilibrium value by approximately 5% (f_DI = 0.95), and ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ was 1.43 N·m⁻¹. Also, as can be seen in Table 1, I₀ can be expressed by Eq. (8) when K_V is 4.27 × 10³⁵ m⁻³·s⁻¹, and the average I₀ in these experiments was 1.96 × 10¹⁶ m⁻³·s⁻¹. Wakou and Sano¹⁸⁾ claimed an I₀ of 8.3 × 10¹⁵ m⁻³·s⁻¹ on the basis of the evaluated values of the inclusion number density 1 s after Al deoxidation. This value is close to the average I₀ obtained in this study. Based on the above explanation, the discussion about the nucleation of Al₂O₃ is considered to be appropriate, and also the quantitative results for the interfacial tension, frequency factor, nucleation time, and nucleation rate are considered to be valid.

Fig. 8.

Relation between nucleation time t_k and interfacial deoxidation ratio f_DI.

4.2. Mechanisms for Growth and Removal of Al₂O₃ Nuclei and Al₂O₃ Inclusions

4.2.1. Mathematical Models for Growth and Removal of Al₂O₃ Nuclei and Al₂O₃ Inclusions

The following mechanisms are considered to contribute to the growth and removal of Al₂O₃ inclusion particles in elementary steps ② to ④: (1) growth through diffusion, (2) diffusion growth based on the difference in solubility due to the sizes of inclusion particles (Ostwald ripening), (3) collision agglomeration due to Brownian motion, and (4) collision agglomeration due to the difference in floating rates and floating separation. Below, rate equations based on the respective mechanisms are derived in order to discuss the mechanisms that contribute to the growth and removal of Al₂O₃ inclusion particles.

(1) Growth through diffusion

The mechanism by which Al₂O₃ inclusion particles individually grow by consuming supersaturated Al and O with Al₂O₃ inclusion particles uniformly dispersed in molten steel is discussed in the following. When dividing molten steel into spherical elements with Al₂O₃ inclusion particles individually positioned at the center of the respective spherical elements, the radii R_N (m) of the spherical elements can be expressed by Eq. (17) using the volume number density N_V (m⁻³) of the Al₂O₃ inclusion particles.   

$${\mathrm{R}}_{\mathrm{N}}={\left[3/(4\pi \cdot {\mathrm{N}}_{\mathrm{V}})\right]}^{1/3}$$(17)

Assuming that deoxidation equilibrium is reached on the interface between Al₂O₃ and molten steel while the Al₂O₃ inclusion particles grow through the diffusion of O in the molten steel, the growth of Al₂O₃ inclusion particles can be calculated by Eqs. (18), (19), (20) derived by Turkdogan¹⁹⁾ while taking into consideration the reduction in solute concentrations associated with the growth of deoxidation products. Here, because C_P of the Al₂O₃ inclusion particles is significantly larger than C_Eq, (C_P − C_Eq) in Eq. (18) is approximated by C_P.   

$$\begin{array}{l}{\mathrm{D}}_{\mathrm{O}}\cdot \mathrm{t}/{\mathrm{R}}_{\mathrm{N}}{}^2\cdot {\left[({\mathrm{C}}_{\mathrm{B}(\mathrm{S})}-{\mathrm{C}}_{\mathrm{E}\mathrm{q}})/({\mathrm{C}}_{\mathrm{P}}-{\mathrm{C}}_{\mathrm{E}\mathrm{q}})\right]}^{1/3}=\\ 1/6\mathrm{l}\mathrm{n}\left[({\mathrm{u}}^2+\mathrm{u}+1)/{(\mathrm{u}-1)}^2\right]-1/\sqrt{3}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left((2\mathrm{u}+1)/\sqrt{3}\right)\\ +1/\sqrt{3}\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}\left(1/\sqrt{3}\right)\end{array}$$(18)

  

$${\mathrm{u}}^3={\mathrm{f}}_{\mathrm{D}\mathrm{B}}=({\mathrm{C}}_{\mathrm{B}(\mathrm{S})}-{\mathrm{C}}_{\mathrm{B}})/({\mathrm{C}}_{\mathrm{B}(\mathrm{S})}-{\mathrm{C}}_{\mathrm{E}\mathrm{q}})$$(19)

  

$$\mathrm{d}=2{\mathrm{R}}_{\mathrm{N}}\cdot {\left[({\mathrm{C}}_{\mathrm{B}(\mathrm{S})}-{\mathrm{C}}_{\mathrm{B}})/{\mathrm{C}}_{\mathrm{P}}\right]}^{1/3}$$(20)

where the subscript of (S) represents the origin of time, and u³ and f_DB are the deoxidation ratios of molten steel defined by Eq. (19) along with the deoxidation ratios of interfaces. In these equations, the diameters of the Al₂O₃ inclusion particles at early stages are considered to be negligibly small.

(2) Diffusion growth based on the difference in solubility due to the sizes of inclusion particles (Ostwald ripening)

Regarding Ostwald ripening of precipitated compounds in steel, Hasegawa et al.²⁰⁾ have already analytically derived the growth model of MnS. Using this growth model, the growth rate equations of Al₂O₃ inclusion particles based on O diffusion rate determination can be obtained as shown in Eqs. (21) and (22).   

$$d\tilde{\mathrm{r}}/dt={\mathrm{K}}_{\mathrm{O}\mathrm{W}}\cdot (1/\tilde{\mathrm{r}})\cdot (1/{\mathrm{r}}_{\mathrm{C}\mathrm{r}}-1/\tilde{\mathrm{r}})$$(21)

  

$${\mathrm{K}}_{\mathrm{O}\mathrm{W}}=2{\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}\cdot {\mathrm{D}}_{\mathrm{O}}\cdot {\mathrm{V}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/(\eta \cdot \mathrm{R}\cdot \mathrm{T})\cdot \left[{\mathrm{C}}_{\mathrm{B}}/({\mathrm{C}}_{\mathrm{P}}-{\mathrm{C}}_{\mathrm{B}})\right]$$(22)

where $\tilde{\mathrm{r}}$ is the radius of Al₂O₃ inclusion particles (m), η is the stoichiometric factor of compounds with a value of 3/5 in the case of Al₂O₃, and r_Cr is the critical radius at which a particle neither grows nor dissolves and corresponds to the average particle radius. Solving Eq. (21) by applying the Lifshitz and Slyozov theory²¹⁾ considering the particle diameter distribution gives Eq. (23), which expresses the time changes in average particle diameters due to Ostwald ripening.   

$${(\mathrm{d}/2)}^3-{({\mathrm{d}}_{(\mathrm{S})}/2)}^3=4/9{\mathrm{K}}_{\mathrm{O}\mathrm{W}}\cdot \mathrm{t}$$(23)

When η is equal to 1, Eq. (23) is consistent with the equation, capable of being applied to the growth of single precipitates, suggested by Lindborg and Torssell.²²⁾

(3) Collision agglomeration due to Brownian motion

The mechanism assumed in this section is that fine Al₂O₃ inclusion particles (primary particles) that have initially dispersed uniformly in molten steel agglomerate through two particles collision with each other due to Brownian motion. By applying the agglomeration rate of two particles, which is obtained on the assumption that the Brownian motion is equivalent to the diffusion process of particles, to the balance equation of particle number density, and solving this equation under the initial agglomeration conditions (i.e., the conditions in which the majority of particles are primary particles with nearly identical particle diameters), Eq. (24) is obtained as the rate equation of the number density of Al₂O₃ inclusion particles.²³⁾   

$$d{\mathrm{N}}_{\mathrm{V}}/dt=-{\mathrm{K}}_{\mathrm{B}\mathrm{R}}\cdot {\mathrm{N}}_{\mathrm{V}}{}^2$$(24)

where K_BR is the rate constant of Brownian agglomeration (m³·s⁻¹) and expressed by Eq. (25) according to Smoluchowski.²⁴⁾   

$${\mathrm{K}}_{\mathrm{B}\mathrm{R}}=4{\mathrm{k}}_{\mathrm{B}}\cdot \mathrm{T}/(3\mu )$$(25)

where μ is the coefficient of viscosity of molten steel with a value of 0.005 Pa·s. Assuming that N_V(S) is the initial number density of Al₂O₃ inclusion particles, Eq. (24) can be solved to obtain Eq. (26).   

$${\mathrm{N}}_{\mathrm{V}}={\mathrm{N}}_{\mathrm{V}(\mathrm{S})}/(1+{\mathrm{K}}_{\mathrm{B}\mathrm{R}}\cdot {\mathrm{N}}_{\mathrm{V}(\mathrm{S})}\cdot \mathrm{t})$$(26)

Because the total volume of fine particles is constant as they are not removed by flotation, Eq. (26) can be used to obtain Eq. (27), which describes the time changes in particle diameters due to Brownian agglomeration.   

$${(\mathrm{d}/2)}^3-{({\mathrm{d}}_{(\mathrm{S})}/2)}^3={\mathrm{K}}_{\mathrm{B}\mathrm{R}}\cdot {\mathrm{N}}_{\mathrm{V}(\mathrm{S})}\cdot {({\mathrm{d}}_{(\mathrm{S})}/2)}^3\cdot \mathrm{t}$$(27)

(4) Collision agglomeration due to the difference in floating rates and floating separation

i) Collision agglomeration of floating Al₂O₃ cluster inclusions with Al₂O₃ single inclusions

The mechanism assumed in this section is that Al₂O₃ cluster inclusions floating in molten steel grow mainly through agglomeration with fine Al₂O₃ single inclusions suspended in the molten steel. Given molten steel having a spherical element of volume V_E (m³) with one large Al₂O₃ cluster inclusion located at the center and fine Al₂O₃ single inclusions suspended around it, because the agglomeration rate of Al₂O₃ single inclusions onto the Al₂O₃ cluster inclusion is equal to the decrease rate of Al₂O₃ single inclusions, the time changes in the oxygen concentration of an Al₂O₃ single inclusion can be expressed by Eqs. (28) and (29).   

$$d{\mathrm{C}}_{\mathrm{S}\mathrm{I}}/dt=-{\mathrm{k}}_{\mathrm{S}}\cdot {\mathrm{C}}_{\mathrm{S}\mathrm{I}}$$(28)

  

$${\mathrm{k}}_{\mathrm{S}}=({\mathrm{A}}_{\mathrm{C}}/{\mathrm{V}}_{\mathrm{E}})\cdot {\mathrm{k}}_{\mathrm{m}}$$(29)

where C_SI is the molar concentration of O included in Al₂O₃ single inclusions in the molten steel (mol·m⁻³), A_C is the surface area of Al₂O₃ cluster inclusions (m²), k_m is the mass transfer coefficient of Al₂O₃ single inclusions to Al₂O₃ cluster inclusions at the molten steel side (m·s⁻¹), and k_S is the agglomeration rate constant of Al₂O₃ single inclusions onto floating Al₂O₃ cluster inclusions and, at the same time, the decrease rate constant of Al₂O₃ single inclusions (s⁻¹).⁵⁾ In addition, Eq. (30) holds considering that the growth rate of an Al₂O₃ cluster inclusion is equal to the agglomeration rate of Al₂O₃ single inclusions onto the cluster inclusion.   

$$d({\mathrm{d}}_{\mathrm{C}\mathrm{I}})/dt=2{\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot {\mathrm{k}}_{\mathrm{S}}\cdot {\mathrm{C}}_{\mathrm{S}\mathrm{I}}/\left[3\epsilon \cdot {\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot ({\mathrm{A}}_{\mathrm{C}}/{\mathrm{V}}_{\mathrm{E}})\right]$$(30)

By integrating Eq. (30) after substituting C_SI obtained from Eq. (28) into it on the grounds that A_C/V_E during the Al deoxidation experiments are nearly constant,⁵⁾ Eq. (31) is obtained as an expression of the diameter of Al₂O₃ cluster inclusion.   

$$\begin{array}{l}{\mathrm{d}}_{\mathrm{C}\mathrm{I}}={\mathrm{d}}_{\mathrm{C}\mathrm{I}(\mathrm{S})}+2{\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot {\mathrm{C}}_{\mathrm{S}\mathrm{I}(\mathrm{S})}\cdot \\ \hspace{1em}\hspace{1em}\left[1-\mathrm{e}\mathrm{x}\mathrm{p}(-{\mathrm{k}}_{\mathrm{S}}\cdot \mathrm{t})\right]/\left[3({\mathrm{A}}_{\mathrm{C}}/{\mathrm{V}}_{\mathrm{E}})\cdot \epsilon \cdot {\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\right]\end{array}$$(31)

ii) Collision agglomeration of floating Al₂O₃ cluster inclusions with Al₂O₃ single inclusions and other Al₂O₃ cluster inclusions

The mechanism assumed in this section is that Al₂O₃ cluster inclusions in molten steel grow through agglomeration with other Al₂O₃ cluster inclusions during floating in addition to the agglomeration with Al₂O₃ single inclusions. Given that the floating rate v_C of Al₂O₃ cluster inclusions in molten steel (m·s⁻¹) follows Stokes’ law expressed by Eq. (32), the growth rate of Al₂O₃ cluster inclusions through agglomeration with other Al₂O₃ cluster inclusions in the volume of molten steel in the floating path is given by Eq. (33).¹¹⁾   

$${\mathrm{v}}_{\mathrm{C}}=2\mathrm{g}\cdot ({\rho }_{\mathrm{F}\mathrm{e}}-{\rho }_{\mathrm{C}})\cdot {({\mathrm{d}}_{\mathrm{C}\mathrm{I}}/2)}^2/(9\mu )$$(32)

  

$$d({\mathrm{d}}_{\mathrm{C}\mathrm{I}})/dt={\alpha }_{\mathrm{C}}\cdot {\mathrm{v}}_{\mathrm{C}}/2={\alpha }_{\mathrm{C}}\cdot \mathrm{g}\cdot ({\rho }_{\mathrm{F}\mathrm{e}}-{\rho }_{\mathrm{C}})\cdot {\mathrm{d}}_{\mathrm{C}\mathrm{I}}{}^2/(36\mu )$$(33)

where g is gravitational acceleration (m·s⁻²) and ρ_C is the density of Al₂O₃ cluster inclusions (kg·m⁻³), which can be expressed by Eq. (34) using the filling rate of Al₂O₃ cluster inclusions.   

$${\rho }_{\mathrm{C}}=\epsilon \cdot {\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}+(1-\epsilon )\cdot {\rho }_{\mathrm{F}\mathrm{e}}$$(34)

where α_C is the volume of Al₂O₃ cluster inclusions in the unit volume of molten steel that can be expressed by Eq. (35) using the initial value of the oxygen concentration [I.O]_C in Al₂O₃ cluster inclusions in the molten steel.   

$${\alpha }_{\mathrm{C}}={\rho }_{\mathrm{F}\mathrm{e}}\cdot {\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot {[\mathrm{I}.\mathrm{O}]}_{\mathrm{C}(\mathrm{S})}/(300{\mathrm{M}}_{\mathrm{O}}\cdot \epsilon \cdot {\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3})$$(35)

Because the growth of Al₂O₃ cluster inclusions occurs by the mechanisms of both agglomeration with Al₂O₃ single inclusions (Eq. (30)) and agglomeration with other Al₂O₃ cluster inclusions (Eq. (33)), Eq. (36) holds.   

$$\begin{array}{l}d({\mathrm{d}}_{\mathrm{C}\mathrm{I}})/dt=\left[{\alpha }_{\mathrm{C}}\cdot \mathrm{g}\cdot ({\rho }_{\mathrm{F}\mathrm{e}}-{\rho }_{\mathrm{C}})/(36\mu )\right]\cdot {\mathrm{d}}_{\mathrm{C}\mathrm{I}}{}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}+2{\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot {\mathrm{k}}_{\mathrm{S}}\cdot {\mathrm{C}}_{\mathrm{S}\mathrm{I}(\mathrm{S})}/\left[3\epsilon \cdot {\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot ({\mathrm{A}}_{\mathrm{C}}/{\mathrm{V}}_{\mathrm{E}})\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\mathrm{A}}_4\cdot {\mathrm{d}}_{\mathrm{C}\mathrm{I}}{}^2+{\mathrm{B}}_4\end{array}$$(36)

Here, C_SI in Eq. (30) is approximated by C_SI(S) for simplification. Integrating Eq. (36) obtains Eq. (37) as an expression of the diameter of Al₂O₃ cluster inclusion.   

$$\begin{array}{l}{\mathrm{d}}_{\mathrm{C}\mathrm{I}}=\\ \mathrm{t}\mathrm{a}\mathrm{n}\left({({\mathrm{A}}_4\cdot {\mathrm{B}}_4)}^{0.5}\cdot \mathrm{t}+\mathrm{t}\mathrm{a}{\mathrm{n}}^{-1}({({\mathrm{A}}_4/{\mathrm{B}}_4)}^{0.5}\cdot {\mathrm{d}}_{\mathrm{C}\mathrm{I}(\mathrm{S})})\right)/{({\mathrm{A}}_4/{\mathrm{B}}_4)}^{0.5}\end{array}$$(37)

iii) Floating separation rates of total Al₂O₃ single inclusions

In Al deoxidation molten steel, Al₂O₃ single inclusions having diameters of a few μm coexist with coarse Al₂O₃ cluster inclusions having diameters of a few dozen μm formed through the agglomeration of Al₂O₃ single inclusions. Thus, [I.O]_T can be expressed by Eq. (38) using the volume number density N_V,T (m⁻³) of total Al₂O₃ single inclusions combining the single and cluster inclusions.   

$$\begin{array}{l}{[\mathrm{I}.\mathrm{O}]}_{\mathrm{T}}=100\cdot (4\pi /3)\cdot {({\mathrm{d}}_{\mathrm{S}\mathrm{I}}/2)}^3\cdot {\mathrm{N}}_{\mathrm{V},\mathrm{T}}\cdot 3{\mathrm{M}}_{\mathrm{O}}\cdot \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}/({\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}\cdot {\rho }_{\mathrm{F}\mathrm{e}})\end{array}$$(38)

By substituting Eq. (38) into Eq. (6), the removal rate of total Al₂O₃ single inclusions is given by Eq. (39).   

$$-d{\mathrm{N}}_{\mathrm{V},\mathrm{T}}/dt={\mathrm{k}}_{\mathrm{T}}\cdot {\mathrm{N}}_{\mathrm{V},\mathrm{T}}$$(39)

4.2.2. Growth Mechanism of Al₂O₃ Nuclei ②

Inclusion nuclei grow very quickly. It is reported that the diameters of inclusion nuclei increase to 0.3 to 0.8 μm in about 1 second after addition of a deoxidizing agent in the case of Al deoxidation.¹⁸⁾ In the case of Si deoxidation, the diameters reportedly reach 1.4 μm¹⁰⁾ concurrently with a rapid decrease in dissolved oxygen to equilibrium values.^9,10) As is clear from the derivation of the growth rate equations explained above, the process of rapid growth of Al₂O₃ nuclei while consuming supersaturated Al and O can be explained by the diffusion growth model expressed by Eqs. (17), (18), (19), (20). Figure 9 shows the time changes in f_DB and d_SI calculated by the diffusion growth model. The solid and dotted lines in the figure respectively represent the results calculated using the maximum Al₂O₃ nucleus volume number density (N_V,N = 1.62 × 10¹⁴ m⁻³) and the minimum Al₂O₃ nucleus volume number density (N_V,N = 7.15 × 10¹³ m⁻³) obtained from experiments in terms of N_V. Also, these time changes are based on C_P − C_Eq ≈ C_P = 1.17 × 10⁵ mol·m⁻³ and C_B(S) − C_Eq = 74.4 mol·m⁻³. As can be seen in the figure, d_SI increased along with the rapid decrease in supersaturated O immediately after the addition of Al, and grew to 2.0 to 2.6 μm after 2.2 to 3.7 s had elapsed (f_DB = 0.9999) since the addition of Al depending on the number density of Al₂O₃ nuclei. These particle diameters roughly correspond to the experimental values 60 s after the addition of Al. Accordingly, the Al₂O₃ nuclei that formed immediately after the addition of Al are considered to have grown rapidly due to the diffusion of O in molten steel.

Fig. 9.

Changes in deoxidation ratio f_DB and particle diameters d_SI of Al₂O₃ single inclusions associated with diffusion growth of Al₂O₃ nuclei.

4.2.3. Growth Mechanism of Al₂O₃ Single Inclusions ③

As can be seen in Fig. 5, Al₂O₃ single inclusions grew as the O concentration in the molten steel increased. Considering that such growth occurred after reaching the deoxidation equilibrium 60 s after the addition of Al, the growth mechanism cannot be (1) growth due to diffusion. Also, because Al₂O₃ single inclusions have particle diameters of up to approximately 3 μm and are scarcely able to float according to Stokes’ law, (4) collision agglomeration due to the differences in floating rates can be excluded from the possible growth mechanisms of Al₂O₃ single inclusions. So, either (2) Ostwald ripening or (3) collision agglomeration due to Brownian motion can be considered to be the growth mechanism of Al₂O₃ single inclusions. Figure 10 shows the particle diameters of Al₂O₃ single inclusions based on the growth rate equations obtained as Eqs. (23) and (27). Despite some dispersion, linear relations can be found between (d_SI/2)³ − (d_SI(S)/2)³ and time as indicated by the solid lines in the figure. Figure 11 shows the influences of the concentrations of O and S on the inclinations of the lines (= [(d_SI/2)³ − (d_SI(S)/2)³]·t⁻¹) obtained from Fig. 10. The dotted line represents the inclination (=K_BR·N_V,S(S)·(d_SI(S)/2)³) of the Brownian agglomeration obtained on the basis of Eqs. (25) and (27) using the average volume number density of 6.42 × 10¹³ m⁻³ and the average particle diameter of 2.10 μm at the start of the experiments. The solid lines are the inclinations of Ostwald ripening (=4/9K_OW) calculated from Eqs. (22) and (23) using ${\sigma }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3-\mathrm{F}\mathrm{e}}$ of each experiment in Fig. 2 in accordance with the concentrations of O and S in the molten steel. The inclination of Brownian agglomeration was close to the experimental values for the cases with an O concentration of 0.0009 mass% or less, but smaller than the other experimental values for the cases with a higher O concentration because Brownian agglomeration does not depend on O concentration. Thus, although it cannot be completely excluded from the possible growth mechanisms of Al₂O₃ single inclusions, (3) the collision agglomeration due to Brownian motion cannot be considered to be the primary one. When including the dependency on the concentrations of O and S, the inclination of Ostwald ripening roughly corresponds to the experimental results. Considering that the influences of (3) the collision agglomeration due to Brownian motion on the inclination are small, it is considered that the growth mechanism of Al₂O₃ single inclusions can be explained mainly by (2) diffusion growth based on the difference in solubility due to the sizes of inclusion particles.

Fig. 10.

Relation between (d_SI/2)³ − (d_SI(S)/2)³ and time.

Fig. 11.

Influences of concentrations of O and S on [(d_SI/2)³ − (d_SI(S)/2)³]·t⁻¹.

4.2.4. Agglomeration Growth and Removal Mechanisms of Al₂O₃ Cluster Inclusions ④

The author has experimentally and theoretically verified that Al₂O₃ inclusions in molten steel are subject to the action of the agglomeration force due to the cavity bridge force, which is far stronger than the van der Waals force and the capillary force on the surfaces of molten steel,^2,3,4) and that agglomeration rates increase as the agglomeration force increases, thereby increasing the particle diameters of Al₂O₃ cluster inclusions.⁵⁾ Figure 12 shows the relation between the maximum particle diameters d_CI,Max (corresponding to d_CI after 240 s in Fig. 5) of Al₂O₃ cluster inclusions of each experiment and the agglomeration force F_A,S (acting between spherical Al₂O₃ inclusions with d_SI = 2 μm) due to the cavity bridge force reported in the previous paper.⁵⁾ Also, for comparison purposes, the particle diameters of Al₂O₃ cluster inclusions calculated by Eqs. (31) and (37) are plotted in Fig. 12 as □ and ◇, respectively. In the calculation of the agglomeration model, the ratio of the oxygen concentration in Al₂O₃ cluster inclusions to the oxygen concentration of total Al₂O₃ inclusions ([I.O]_T = [I.O]_C + [I.O]_S, [I.O]_S: oxygen concentration in Al₂O₃ single inclusions in molten steel) is 0.533,⁵⁾ and the average [I.O]_T(S) is 0.0152 mass%. Therefore, C_SI(S) is 0.0152·(1 – 0.533)/100·ρ_Fe/M_O = 31.1 mol·m⁻³, and α_C is 0.0152·0.533·ρ_Fe·${\mathrm{M}}_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$/(300M_O·ε·${\rho }_{\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3}$) = 1.24 × 10⁻³ according to Eq. (35). In the previous paper,⁵⁾ the experimental values of k_S were obtained by applying the first-order kinetics, Eq. (28), to the changes in the oxygen concentration of Al₂O₃ single inclusions in molten steel, and the appropriateness of the experimental values has been verified with a mass transfer model using the penetration theory. Also, A_C/V_E during experiments can be considered to have a constant value of approximately 195 m^{−1 5)} due to mutual compensation of the influences of particle coarsening and floating separation, and d_CI(S) was determined to be 24.9 μm, which was the average of the actual measurements. According to the favorable linear relation between d_CI,Max and F_A,S, it is obvious that the agglomeration growth of Al₂O₃ cluster inclusions is strongly affected by the agglomeration force due to the cavity bridge force acting between Al₂O₃ inclusions. Also, d_CI,Max is consistent with the agglomeration model involving single inclusions only when F_A,S is relatively small. However, when F_A,S becomes larger, d_CI,Max increases and is more consistent with the agglomeration model that involves both cluster and single inclusions than the model that only involves single inclusions. Accordingly, with the increase in the agglomeration force between inclusions, it is considered that Al₂O₃ cluster inclusions were removed by flotation while growing such that they not only agglomerated with suspended fine single inclusions but also strongly attracted and agglomerated with other cluster inclusions existing in the floating paths.

Fig. 12.

Comparison of maximum Al₂O₃ cluster inclusion diameters d_CI,Max and calculated Al₂O₃ cluster inclusion diameters using the agglomeration model.

4.3. Influences of Deoxidation Mechanisms in Each Elementary Step on the Formation, Growth, and Removal of Al₂O₃ Inclusion Particles

In 4.1 and 4.2, the Al deoxidation process in molten steel was divided into several elementary steps, and the mechanisms of formation, growth, and removal of Al₂O₃ inclusion particles in each elementary step were clarified through proposals of mathematical models based on said mechanisms. In this section, to coherently verify a series of mechanisms, the changes in the diameters and volume number density of Al₂O₃ inclusion particles are calculated from immediately after addition of Al to after the deoxidation equilibrium by using the aforementioned mathematical models while taking into consideration the continuity of the elementary steps. Figures 13 and 14 show comparisons of the calculation results and experimental values of the diameters and volume number density of Al₂O₃ inclusion particles, respectively. Because the growth, agglomeration, and removal of Al₂O₃ inclusion particles after reaching deoxidation equilibrium change in accordance with the O concentration in the molten steel, calculated values with a low O concentration (0.0006 mass%) and high O concentration (0.0261 mass%) are shown as solid and dashed-dotted lines in Figs. 13 and 14, respectively. In these calculations, it was assumed in the calculation that the rapid agglomeration and removal of Al₂O₃ inclusions due to the stirring of the molten steel by the addition of Al occurred instantaneously immediately after reaching the Al deoxidation equilibrium, and almost no further changes occurred until 60 s elapsed thereafter. Thus, immediately after reaching the deoxidation equilibrium, the diameter of Al₂O₃ cluster inclusions was the average diameter at the start of the experiments, and the number density of total Al₂O₃ single inclusions was 89% of that immediately after the addition of Al (= [I.O]_T(60)/[I.O]_T(0) = 0.0152/0.017). Also, the experimental values for the number density of total Al₂O₃ single inclusions were obtained by converting the actually measured [I.O]_T with Eq. (38), which uses the average particle diameter of Al₂O₃ single inclusions of 2.10 μm as d_SI instead of using microscopic observation results with large dispersion.

Fig. 13.

Time changes in diameters of Al₂O₃ inclusion particles.

Fig. 14.

Time changes in total volume number density of Al₂O₃ inclusion particles.

During the period from immediately after the addition of Al until t_k = 0.01 s elapsed once nucleation was completed with Al₂O₃ nuclei having a diameter of 8.8 × 10⁻⁴ μm formed at I₀ = 2.0 × 10¹⁶ m⁻³·s⁻¹, the volume number density of Al₂O₃ nuclei reached 2.0 × 10¹⁴ m⁻³, which is close to the experimental value range of N_V,N = 0.72 to 1.62 × 10¹⁴ m⁻³. The formed Al₂O₃ nuclei then consumed all supersaturated O and, after 1.9 s, equilibrium was reached through the diffusion growth of the Al₂O₃ nuclei into Al₂O₃ single inclusions having a diameter of 1.83 μm. During this period, considering the very slow floating rates of fine Al₂O₃ single inclusions, it was assumed that Al₂O₃ single inclusions did not undergo any change in volume number density. In the equilibrium state after 1.9 s from the addition of Al, Al₂O₃ single inclusions and Al₂O₃ cluster inclusions were removed while concurrently growing through different mechanisms. The Al₂O₃ single inclusions with a diameter of 1.83 μm grew through Ostwald ripening with a growth rate determined by O diffusion, and the particle diameter increased as the O concentration increased, which corresponded well with the experimental values. Meanwhile, the Al₂O₃ cluster inclusions with a diameter of 24.9 μm which rapidly agglomerated due to stirring of the molten steel in association with the addition of Al were removed by flotation while agglomerating single inclusions when the O concentration was high, and both single inclusions and other cluster inclusions when the O concentration was low. Thus, the diameters of Al₂O₃ cluster inclusions increased as the O concentration decreased and with the acceleration of the decrease in the number density of total Al₂O₃ single inclusions accordingly. All these calculation results roughly reproduced the experimental values.

As explained above, the Al deoxidation mechanisms proposed in this research were verified to be appropriate because the mathematical models of the formation, growth, and removal of Al₂O₃ inclusion particles derived by appropriately assuming mechanisms for each elementary step could coherently reproduce the Al deoxidation behavior from immediately after the addition of Al to after the deoxidation equilibrium. In addition, by the Al deoxidation mechanisms elucidated in this study, it is possible to optimize the rate controlling factors such as the degree of supersaturation and O, which is the interfacial active element, as a whole in consideration of the effects of the factors on each elementary step. As a result, the diameter and quantity of Al₂O₃ inclusion can be controlled at a high lebel, and it is considered that this can greatly contribute to the improvement of the quality and properties of steel products.

5. Conclusions

The following conclusions were obtained as a result of analyses of a series of Al deoxidation behavior from nucleation and growth of Al₂O₃ nuclei immediately after the addition of Al to the growth, agglomeration, and removal of Al₂O₃ inclusions after reaching deoxidation equilibrium in light of the kinetics on the basis of Al deoxidation experiments in which the interfacial tension between molten steel and Al₂O₃ inclusions was controlled by changing the concentrations of O and S in the molten steel.

(1) The nucleation number density of Al₂O₃ was (0.72 to 1.62) × 10¹⁴ m⁻³ and increased as the degree of supersaturation increased and the interfacial tension between the nuclei and molten steel decreased. However, the influence of the interfacial tension was relatively small. These tendencies can be explained by the homogeneous nucleation theory and the interfacial tension, frequency factor, nucleation time, and average nucleation rate in the Al deoxidation experiments were estimated to be 1.43 N·m⁻¹, 4.27 × 10³⁵ m⁻³·s⁻¹, 0.01 s, and 1.96 × 10¹⁶ m⁻³·s⁻¹, respectively.

(2) The growth mechanism of Al₂O₃ nuclei can be explained by the diffusion growth of supersaturated O in molten steel, and the critical nuclei rapidly grew to Al₂O₃ single inclusions having diameters of 2.0 to 2.6 μm within 2.2 to 3.7 s after addition of Al in accordance with the number density of Al₂O₃ nuclei and the deoxidation equilibrium was reached.

(3) In the deoxidation equilibrium, Al₂O₃ single inclusions and Al₂O₃ cluster inclusions concurrently grew and were removed through different growth mechanisms.

(4) The growth rate of Al₂O₃ single inclusions increased as the O concentration in the molten steel increased, and their growth mechanism can be explained by diffusion growth (Ostwald ripening) based on the difference in solubility due to the sizes of the inclusion particles.

(5) Al₂O₃ cluster inclusions floated and separated while growing along with the increase in the agglomeration force originating from the cavity bridge force such that they not only agglomerated with fine single inclusions dispersed in the molten steel but also attracted and agglomerated with other cluster inclusions existing in the floating paths.

(6) The Al deoxidation mechanisms proposed in this research were verified to be appropriate because the mathematical models of the formation, growth, agglomeration, and removal of Al₂O₃ inclusions derived by appropriately assuming mechanisms for each elementary step could coherently reproduce the Al deoxidation behavior from immediately after the addition of Al to after reaching the deoxidation equilibrium.

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