Nucleation and Growth Kinetics of MgO in Molten Steel

Transcription

1 J. Mater. Sci. Technol., 202, 28(7), Nucleation and Growth Kinetics of MgO in Molten Steel Hong Lei and Jicheng He Key Lab of Electromagnetic Processing of Materials, Ministry of Education, Northeastern University, Shenyang 89, China [Manuscript received May 6, 20, in revised form August 6, 20] The size, number, morphology and type of inclusion particles are the key factors to estimate the quality of steel product. Although considerable efforts have been made in the mathematical modeling of inclusion growth, few papers were involved in inclusion s nucleation and collision-growth, and all the existing researches about the behaviors of magnesia inclusion were based on the experiments. Thus, a mathematical model was developed to investigate the nucleation, Ostwald ripening and collision-growth of magnesia inclusion in the molten steel. Numerical results showed that the predicted particle size distributions are consistent with the previous experimental data. For the magnesia inclusions smaller than 0 nm, Brownian collision is the main collision modes. For the inclusions ranging from to µm, Brownian collision and turbulent collision are the main collision modes. For the inclusions ranging from to 0 µm, turbulent collision and Stokes collision are the main collision modes. Thus, the strong turbulent flow can decrease the peak-value diameter of the magnesia inclusion effectively. KEY WORDS: Magnesia; Homogeneous nucleation; Ostwald ripening; Brownian collision; Stokes collision; Turbulent collision; Particle size distribution. Introduction The inclusions in the steel were generally considered to be harmful to the properties of steel such as workability, surface quality, and fatigue strength, so special attention was paid to the inclusion removal in the metallurgical reactors [ 3]. Recently, some experiments showed that the fine inclusions with special composition could control the microstructure, so many researchers paid more attention to the inclusion engineering [4 7]. In both cases, the key technology is to control the number, composition, morphology, size and spatial distribution of the inclusions in the steel. The super-saturation degree, the temperature and the composition of steel are important factors to affect the formation and growth of the inclusions, which yield the distinctive inclusions morphology and size. Thus, it is necessary to have a deep insight into the inclusions formation and growth in order to control Corresponding author. Ph.D.; Tel.: ; Fax: ; address: cn leihong@yahoo.com (H. Lei). their composition and the related size distribution during the deoxidation process. Magnesium has been used as a deoxidizer because of its strong affinity with oxygen. There were many research papers about magnesium deoxidation equilibrium in molten iron [6,7]. But few papers referred to the nucleation and growth of magnesia inclusion. The observation of the inclusion soon after nucleation is a great challenge for experimental research, but numerical simulation method can predict inclusion size distribution during its formation and growth. Thus, a mathematical model was developed in this work to predict the size distribution of magnesia inclusion and to investigate the effects of key parameters on the growth of inclusion particle. 2. Assumptions The current mathematical model of nucleationgrowth of the magnesia inclusion was based on the following assumptions. () The spatial distribution of steel chemical com-

2 H. Lei et al.: J. Mater. Sci. Technol., 202, 28(7), position and deoxidation product is uniform in the Fe Mg O system. (2) The Gibbs Thomson equation holds for inclusion particles of all sizes. (3) The chemical reaction and nucleation-growth of inclusions occur in the isothermal system. (4) The basic unit is the inclusion molecule. The embryos or nuclei form and grow only by addition of one molecule. These embryos are unstable until they are assembled into groups large enough to reach the critical size. (5) Embryos and inclusion particles are spherical. (6) The interfacial energy between embryos (or inclusion particles) and molten steel is a constant. 3. Homogeneous Nucleation During steelmaking process, the deoxidization of magnesium is a basic chemical reaction. [Mg] + [O] = (MgO) () The formation of magnesia particles should follow two steps. The first step is to produce plenty of dissolved magnesia molecules. [Mg] + [O] = [MgO] (2) In this case, the number density of one-molecule magnesia inclusion n follows the Kampmann s theory [8] with and n = css [MgO] c eq [MgO] n,eq = n,eq [ exp( Bt )] (3) t = β n,eq t (4) ( ic k= n k i k ) eq (5) Here, the superscripts ss and eq mean a supersaturation state and an equilibrium state, respectively. The molar concentration of magnesia c [MgO] depends on the number density of soluble oxygen and soluble magnesium: c [MgO] = min(ρ Fe [%Mg]/M Mg, ρ Fe [%O]/M O ) (6) where M Mg and M O are the molar weight of Mg and O, respectively and ρ Fe is the density of liquid steel. [%Mg] and [%O] are the mass fraction of soluble magnesium and soluble oxygen, respectively. The second step is that these magnesia molecules are assembled into molecular group. If the radius of the molecular group is greater than the critical radius r C, the magnesia particle appears as the second phase. Such a growth mechanism is processed by the Ostwald ripening [9,] dn k dt = β,k n n k + α k+ A k+ n k+ with and β,k n n k α k A k n k (k 2) (7) β,k = 4πD r k (8.) α k A k = β,k n (8.2) where D is the diffusion coefficient of a molecule, and t is the time. Literature [9,] assumed that the diffusion coefficient of an alumina molecule in the molten steel is equal to that of an oxygen atom (D =3.0 9 m 2 /s) in the molten steel. But the size of the alumina molecule is greater than that of oxygen atom, so the diffusion coefficient of an inclusion molecule should be less than that of an oxygen atom in the molten steel. In this case, Stokes Einstein relation [] is applied to obtain the diffusion coefficient of MgO molecule D = K 0T 6πµr (9) where K 0 is the Boltzmann s constant, µ is the viscosity of molten steel, T is the temperature. Here, the diffusion coefficient of MgO molecule is equal to.8 9 m 2 /s. The first and second terms on the right hand side of Eq. (7) are the generation rates of k-molecules group, and the third and fourth terms are the depletion rates of k-molecules group. The first term on the right hand side of Eq. (7) means that (k )-molecules group catches -molecule and then forms k-molecules group. The second one means that (k+)-molecules group loses -molecule and then forms k-molecules group. The third one means that k-molecules group catches -molecule and then forms (k+)-molecules group. And the fourth one means that k-molecules group loses -molecule and then forms (k-)-molecules group. The homogeneous nucleation theory shows that the critical radius r C of the magnesia nuclei can be expressed as the function of interfacial energy σ between MgO nuclei and molten steel, the molar volume V m, the temperaturet, the super-saturation degree S and the gas constant R. r C = 2σV m RT ln S () Here, the super-saturation degree S [2,3,4] can be expressed as follows: S = (an [Mg] am [O] )ss (a n [Mg] am [O] )eq () where a [Mg] and a [O] are the activity of [Mg] and [O], respectively. The critical number of inclusion molecules to form a nucleus can be formulated by i C = 4 3 πr3 C 4 = 4πN ArC 3 (2) 3 πr3 3V m

3 644 H. Lei et al.: J. Mater. Sci. Technol., 202, 28(7), where N A is the Avogadro s number. Here, the inclusions are divided into two types according to the inclusion s size. If the inclusion radius is less than the critical radius r C, it is called embryo or quasi nuclei. These embryos ( k<i C ) exist in the solvent state. If the inclusion radius is greater than the critical radius r C, it is called inclusion particle. These inclusion particles (k i C ) exist as the second phase in the molten steel. 4. Particle Growth Kinetics Inclusion particle s growth depends on not only the Ostwald ripening mechanism but also the collision-aggregation mechanism. dn k dt = (β,k n n k + α k+ A k+ n k+ β,k n n k α k A k n k ) + 2 k i C i=i C,i+j=k (β B ij + β S ij + β T ij)n i n j i=i C (β B ij + β S ij + β T ij)n i n k (3) Here, βij B, βs ij and βt ij are Brownian collision rate, Stokes collision rate and turbulent collision rate, respectively. βij B = 2K ( 0T + ) (r i + r j ) (4.) 3µ r i r j βij S = 2gπ(ρ Fe ρ in ) ri 2 rj 2 (r i + r j ) 2 (4.2) 9µ β T ij =.3α t πρfe ε µ (r i + r j ) 3 (4.3) where g is the gravity acceleration, ε is the turbulent energy dissipation rate and α t is the coagulation coefficient. The first term on the right hand side of Eq. (3) describes the effect of Ostwald ripening on the inclusion s growth. The second one is the generation rate because i-molecules particle catches j-molecule particle and then forms k-molecules particle, and the last one is the depletion rate because k-molecules particle catches i-molecule particle and then forms (k+i)- molecules particle. 5. Numerical Methods The mathematical model for the nucleationgrowth kinetics of inclusion and the related computational method are not complicated, but the size-span is great during the inclusion s nucleation and growth process. For example, the nucleus size is around nm, but the size of the inclusion particles observed in the sample is around µm. In other words, the current challenge is how to solve the mathematical model quickly and accurately. Thus, the particle-sizegrouping method [5,6] and the four-order adaptivestep-size Gill method are introduced to solve these governing equations. Numberdensity of inclusion / 6 mm -3 n max d pv t/s predicted Experimental Inclusion diameter / m Fig. Inclusion size distribution at different moments The following parameters were chosen to describe the formation of magnesia in the molten steel. ρ Fe =6970 kg m 3, µ= Pa s, T =873 K, σ=0.9 N m [7], ρ MgO =3393 kg m 3, M Mg = kg mol, M O =6.0 3 kg mol, S=8.4 [6] and B= [8] 6. Results and Discussion Fig. shows that the predicted size distribution of magnesia particles in the molten steel is consistent with the experimental data [8]. But the predicted peak-value diameter d pv is 66 µm at 60 s, which is 2.97 times of the experimental data. And the predicted maximum number density n max is.52 6 mm 3 at 60 s, which is /3.76 time of the experimental data. Such results are better than the existing numerical results because the predicted peakvalue diameter of alumina at s by Zhang [] is far (about 30 times) greater than the experimental data. Several reasons lead to the differences. () During the formation of magnesia particle, it is impossible for all the magnesium in Ni Mg alloy to be remained in the melt due to high vapor pressure of magnesium. In other words, the remaining rate of magnesium is less than one. (2) The magnesia inclusion has the cluster shape [7], but it is assumed to be spherical in the current mathematical model. (3) In the experiment, the breakup of cluster-shape magnesia particles will increase the number density of smaller inclusions and decrease the peak-value diameter. (4) Particlesize-grouping method is an approximation method to solve Smoluchowski s model. The total number density and the total volume for inclusion particle by particle-size-grouping method is less than the exact value by Smoluchowski s model [5,6]. Fig. 2 is the isometric contours of the collision rate. If the particle is smaller than 0 nm, Brownian collision rate is from 7 to 5 m 3 s, which is at least 6 orders greater than Stokes collision rate and 2 orders greater than turbulent collision rate (ε= m 2 s 3 ). With increasing inclusion size, the role of turbulent collision becomes more important. If the inclusion size is in the range from 0 nm to µm, turbulent collision rate and Brownian collision are in the same order and at least one order greater

4 H. Lei et al.: J. Mater. Sci. Technol., 202, 28(7), Fig. 2 Collision rate for different collision mechanism: (a) Brownian collision, (b) Stokes collision, (c) turbulent collision Number density of inclusion / 6 mm -3 n max d pv Inclusion size / m (m 2 /s 3 ) Fig. 3 Relationship between inclusion size and inclusion number density under different turbulent energy dissipation rate than Stokes collision rate. If the inclusion size is in the range from to 0 µm, Stokes collision rate is in the range from 5 to m 3 s, which is at least two orders greater than Brownian collision rate, and it has the same order as turbulent collision rate. Fig. 3 gives the effect of turbulent flow on the inclusion growth. With increasing turbulent energy dissipation rate, the peak-value diameter d pv decreases and the maximum number density n max also decrease slowly. Fig. 2 shows that the turbulent collision is one of the key factors for the growth of the magnesia inclusion (d>0 nm). In the case of strong stirring, there are more chances for bigger inclusions to collide with each other. In other words, it is easier for the bigger inclusion to grow up and to be removed. In this way, with increasing turbulent energy dissipation rate, smaller inclusions also have more chances to be removed with the help of bigger inclusion particles, so the peak-value diameter d pv also decreases. By comparison with bigger inclusion particles, smaller inclusion particles are not sensitive to the turbulent flow. Thus, the maximum number density n max decreases slowly with increasing turbulent energy dissipation rate. Fig. 4 shows the history for the number density of embryos and particles. After magnesium addition, magnesia molecules appear in the Fe O Al system at once, and then groups of molecules are generated from Number density of embryos / mm Incubation Nucleation Growth Number density of particles / mm Time / s Fig. 4 History for the number density of embryos and particles Ostwald ripening. The features of the incubation periods are that the density of embryos increases and no inclusion particle appears. After t=0.08 µs, nucleation starts because of the appearance of the inclusion particle. In the nucleation period, the number density of embryos continues to increase and then decrease after t=0.27 µs. But after t=0.35 ms, the number density of embryos keeps the constant because the appearance of new-born inclusions and the disappearance of the existing inclusions are at the equilibrium state. Nucleation and growth usually occur at the same time. During the growth period, Ostwald ripening mechanism acts for both embryos and inclusion particle of all size class. 7. Conclusion A mathematical model has been developed to describe the nucleation, Ostwald ripening and collisiongrowth for magnesia inclusion in the molten steel. Such a model consists of the entire process from homogeneous nucleation to Ostwald ripening and various collision-aggregation mechanisms (Brownian collision, Stokes collision and turbulent collision). As an application of the current model, the predicted size distribution of magnesia particles conforms with the existing experimental data. For inclusion particles with different size, three types of collision modes have different contributions to the growth of inclusions. For the magnesia inclusions smaller than 0 nm, Brownian 5 0-5

5 646 H. Lei et al.: J. Mater. Sci. Technol., 202, 28(7), collision is the main factor for the inclusion particles to collide with each other; turbulent collision has the minor effect and Stokes collision can be ignored. For the inclusions ranging from to µm, Brownian collision and turbulent collision are the main collision modes. For the inclusion ranging from to 0 µm, turbulent collision and Stokes collision are the main collision modes. Thus, the strong turbulent flow can effectively decrease the peak-value diameter of magnesia particles. Acknowledgements The authors gratefully acknowledge the financial support of the project from the National High-Tech R&D Program of China (No. 2009AA03Z530), the National Natural Science Foundation of China and Shanghai Baosteel (No ), and the Fundamental Research Funds for the Central Universities (N ). REFERENCES [ ] Y. Liu, Y. Qian, M. Zhang, Z. Chen and C. Wang: Mater. Res. Bull., 996, 3, 29. [2 ] Z. Hussain: J. Mater. Res., 200, 6, [3 ] M.R. Aboutalebi, M. Hasan and R.I.L. Guthrie: Metall. Mater. Trans. B, 995, 26, 73. [4 ] Y. Sahai and T. Emi: ISIJ Int., 996, 36, 66. [5 ] Y. Miki, B.G. Thomas, A. Denissov and Y. Shimada: Iron Steelmaker, 997, 24(8), 3. [6 ] N. Kikuchi, S. Nabeshima, Y. Kishimoto, T. Matsushita and S. Sridhar: ISIJ Int., 2007, 47, 255. [7 ] C. Wang, N.T. Nuhfer and S. Sridhar: Metall. Mater. Trans. B, 20, 4, 84. [8 ] H. Suito and H. Ohta: ISIJ Int., 2006, 46, 33. [9 ] A.V. Karasev and H. Suito: ISIJ Int., 2008, 48, 507. [] L. Kampmann and M. Kahlweit: Ber. Bunsen-Ges. Phys. Chem., 970, 74, 456. [] L. Zhang and W. Pluschkell: Iron. Steel., 2003, 30, 6. [2] J. Zhang and H.G. Lee: ISIJ Int., 2004, 44, 629. [3] H.B. Shi, G.H. Gao and Y.X. Yu: Fluid Phase Equilibria, 2005, 228, 535. [4] M.L. Turpin and J.F. Elliott: Iron Steel Inst. J., 966, 204, 27. [5] G. Forward and J.F. Elliott: J. Metals, 967, 9, 54. [6] J. Tanabe and H. Suito: Metall. Mater. Trans. B, 995, 26, 95. [7] T. Nakaoka, S. Taniguchi, K. Matsumot and S.T. Johansen: ISIJ Int., 200, 4, 3. [8] H. Lei, K. Nakajima and J.C. He: ISIJ Int., 20, 50, 735. [9] K. Nakajima: Tetsu-to-Hagane, 994, 80, 383. [20] K. Nakajima, H. Ohta, H. Suito and P. Jonsson: ISIJ Int., 2006, 46, 807. Nomenclatures a i Activity A k Surface area of molecules (k-molecules) (m 2 ) B Constant c Molar concentration (mol m 3 ) D Diffusion coefficient of a molecule in molten steel (m 2 s ) g Gravitational acceleration (m s 2 ) [%i] Mass concentration i C Critical number of molecules to form a nuclei with critical radius r C i k Number of molecules (k-molecules) K 0 Boltzmann s constant (J K ) M Molar weight (kg mol ) N A Avogadro s number (mol ) n Number density of a molecule (m 3 ) n,eq Equilibrium number density of a molecule (m 3 ) n k Number density of k-molecules (m 3 ) r i Radius of i-molecules (m) r C Critical radius of inclusion (m) R Gas law constant (J K mol ) S Super-saturation degree t Time (s) t Dimensionless time T Absolute temperature (K) V m Molar volume (m 3 mol ) α k Number of molecules dissociating from unit area of k-molecules in unit time (m 2 s ) α t Coagulation coefficient for turbulent collision β ij Rate constant of the reaction (i)+(j) (i + j) (m 3 s ) βij B Brownian collision rate (m 3 s ) βij S Stokes collision rate (m 3 s ) βij T Turbulent collision rate (m 3 s ) ε Turbulent energy dissipation rate (m 2 s 3 ) µ Molecular viscosity of molten steel (kg m s ) ρ Density (kg m 3 ) σ Interfacial free energy between inclusion and molten steel (N m )