Model for Inclusion Precipitation Kinetics During Solidification of Steel Applications in MnS and TiN Inclusions

During the solidification of steel, sulfur is often heavily segregated. The reaction for precipitation of manganese sulfide during solidification of steel is often considered:The solubility product K_MnS is given by[23,37]

Precipitation of titanium nitrides occurs during solidification of Ti-stabilized stainless steel or other steel grades containing Ti. The reaction can be represented byThe solubility product K_TiN is given by[38]

The nucleation of inclusion could follow the way of homogenous or heterogeneous nucleation. In this work, for the sake of simplification, only homogenous nucleation is taken into consideration for precipitation of inclusions during cooling. Thus, the steady-state nucleation rate I can be expressed as[39]where I_A is a pre-exponential factor, k_b is the Boltzmann constant, and \( \Delta G_{ \hom }^{*} \) is the free energy barrier for homogeneous nucleation and can be calculated aswhere σ is the interfacial tension between inclusion and residue steel liquid and\( \Delta G_{V} \) is Gibbs free energy change per molar volume for inclusion formation.

The crystals would undergo growth or dissolution depending on whether the interfacial concentration is larger than the matrix concentration. The rate for growth or dissolution can be calculated as follows:[40]where \( \bar{C} \) is the concentration in residue liquid steel, C_p is the concentration in inclusion, C_i is the concentration at the interface, D is the diffusion coefficient of the element in liquid steel, and r is the radius of the inclusion.

The concentration at the inclusion-steel interface could be calculated from equilibrium concentration C_e by the Gibbs–Thomson equation:where σ is the interfacial tension between inclusion and residue steel liquid and V_m is the molar volume of inclusion.

Through Eqs. [12] and [13], both growth and coarsening of inclusion particles can be modeled. The critical size r* for the transition from dissolution to growth can be calculated by combining Eqs. [12] and [13]:

Myhr and Grong[32] proposed a control volume approach to calculate the particle size distribution using the Kampmann–Wagner theory. Particle size distribution was discretized into a series of small radius elements, which could be treated as control volumes according to the definition and terminology used by Patankar.[41] Following the method by Myhr and Grong,[32] in this work, the total size distribution was discrete into a series of radius elements r_i with width of ∆r_i. The radius for each element increases by a constant factor of R_r compared with the previous neighbor element. According to the classic grid configuration for the control volume method in CFD, the grids employed in the present study are shown in Figure 1. If we use the up-wind scheme by Patankar,[41] the following discrete equations can be obtained for calculation of number density at grid P from the number densities at grids P, W (west), and E (east) in the last iteration:v_W > 0:v_W < 0:v_E > 0:v_E < 0:where \( N_{\text{P}} \) denotes the present number density at the P grid; \( N_{\text{P}}^{0} \), \( N_{\text{W}}^{0} \), and \( N_{\text{E}}^{0} \) denote the number densities at P, W, and E grids before iteration, respectively; \( \Delta t \) is the time-step; and \( \Delta r_{\text{W}} \), \( \Delta r_{\text{P}} \), and \( \Delta r_{\text{E}} \)are the radius width for W, P, and E grids, respectively. \( v_{\text{W}} \) and \( v_{\text{E}} \) are the rates of growth and dissolution for grids W and E.

The mass conservation law enables us to calculate the matrix solute concentration in interdendritic liquid steel after the inclusion precipitation:where C_p represents the solute concentration in inclusions; \( C_{0} \) is the initial solute concentration in interdendritic liquid steel; \( \rho_{\text{st}} \) and \( \rho_{\text{P}} \) represent the density of liquid steel and inclusion, respectively; and \( V_{\text{p}} \) is the total volume of inclusions.

In the residual liquid during solidification, there also could be some agglomeration of inclusions due to collisions. In the ladle treatment and tundish, different collisions, including Brownian, stokes, and turbulent collisions, were considered in modeling the behavior of inclusions.[33,34,42] However, in intradendritic liquid during casting, the velocity of liquid cannot be easily determined, but it can be assumed that the turbulence of liquid is weak due to the lack of strong stirring. Especially for some types of inclusions (e.g., MnS), which are only precipitated at very high solidification fraction, the motion of inclusions is rather limited. Accordingly, only agglomeration due to Brownian collision was accounted for in the present work.

The well-known PBE was employed to describe the agglomeration of inclusions in residual liquid[43]:where the Brownian collision frequency function \( \beta_{ij}^{B} \) can be represented byThe PSG method[42] is an efficient method to solve the PBE. According to this method, the particles can be divided into M groups. The size of particles in each group is increasing by R_r times compared with the size of particles in the previous group. The following equation could be obtained to replace the PBE:where n_i denotes the number density of group i and \( i_{c,k} \) denotes the critical size number, which was determined by the R_r ratio. When R_r = 2^1/3, \( i_{c,k} = k - 1 \).\( \xi_{i,k - 1} \) and \( \xi_{i,k} \) are the correction factors and can be calculated as follows:where v_i denotes the volume of particle in group i.

According to the assessment performed by Bester and Lange,[50] the diffusion coefficients of sulfur in liquid iron can be calculated according to the following equation:where D_S is the diffusion coefficient of sulfur in liquid iron, R is the universal gas constant, and T is the temperature in Kelvin. The temperature fitting range for Eq. [26] is from the melting point of iron to 1700 °C. The diffusion coefficient of nitrogen in steel can be obtained by extrapolating Eq. [26] to the temperature below the melting point.

The value of interfacial tension between steel and MnS has not been reported in the literature. However, Oikawa et al.[51] estimated this value to be in the region of 0.2 N/m by extrapolating the data on the interfacial tension between steel and MnO-SiO₂-CaO-MnS slag. In a simulation by You et al.,[23] 0.2 N/m was also employed for interfacial tension between steel and MnS; we adopted this value.

The equation for calculating the density of liquid steel ρ_st at various temperatures was obtained from a handbook[52] compiled by The Iron and Steel Institute of Japan as the following equation for Fe-0.29 mass pct C:where T is the temperature in Kelvin.

The model parameters for the simulation of MnS precipitation during solidification are shown in Table II.

You et al.[23] simulated the solidification process of steel using the submerged split chill tensile (SSCT) experiment. Three steels with different sulfur contents (60, 50, and 21 ppm) were melted and solidified using the SSCT experiment. The chemical compositions of steel investigated are shown in Table III. The inclusions in the sample solidified were measured by using automated energy-dispersive X-ray spectroscopy of a scanning electron microscope. The size distribution and number density of inclusions in a defined area were determined. The effects of the cooling rate and the sulfur concentration on the precipitation of MnS inclusions were investigated by sampling different horizontal positions in the solid steel shell and varying sulfur concentrations in the sample, respectively.

Figure 2 shows the comparison between the measured size distribution and calculated size distribution of MnS in cooled steel samples S1 at a cooling rate of 25.4 °C/s. It can be seen that the calculated size distribution fits well with the experimental size distribution, which indicates that the present model could give a good prediction of the size distribution of MnS during solidification.

The cooling rate during steel solidification has an impact on the MnS precipitation. It was reported in a high carbon steel that the number of oxides per unit area decreased and the size of oxides increased with increasing distance from the surface of the bloom, which had solidified at the highest cooling rate.[54] You et al.[23] measured the size distribution of MnS inclusions at the different positions in the test body, which correspond to different cooling rates (42.3, 25.4, and 13.5 °C/s). The calculated size distributions of MnS in steel sample S1 by the present model at cooling rates of 42.3, 25.4, and 13.5 °C/s are shown in Figure 3. As seen in Figure 3, the radius corresponding to the maximum number density (R_p) increases with decreasing cooling rates. It can be seen also that the maximum number density decreases at lower cooling rates. The number density of MnS decreases with decreasing cooling rates for particles with a radius less than R_p, while the number density increases with decreasing cooling rates for particles with a radius larger than R_p. This indicates a more developed coarsening of MnS inclusions with lower cooling rates. The present results are consistent with the results by You et al. regarding the changing trend of mean particle size with decreasing cooling rates. The tendency for variation of the number density and mean size with cooling rate is also consistent with the investigation on high-carbon steel bloom by Faraji et al.[54]

The sulfur concentration in steel could also affect the precipitation of MnS in steel by changing the supersaturation for nucleation. The size distributions of MnS for three steels with different sulfur concentrations (60, 50, and 40 ppm) were calculated by the present model and are shown in Figure 4. It can be seen that R_p decreases for steels with lower sulfur concentration, as does the maximum number density for these steels. This is consistent with the experimental data by You et al.[23] with respect to the changing tendency of R_p and maximum number density. However, the experimental result for steel with 20 ppm cannot be reproduced since very few MnS could be precipitated according to the model calculation results. This inconsistency between model calculations and experiments could be due to the possible heterogeneous nucleation in steel samples with low sulfur concentration. There is no micrograph for inclusions reported in the work by You et al.,[23] and the possibility of heterogeneous nucleation cannot be excluded. Heterogeneous nucleation would promote the precipitation of MnS in samples with low sulfur concentration. However, heterogeneous nucleation was not accounted for in the present model, leading to the inconsistency of samples with low sulfur concentration. The reduced number density with lower sulfur concentrations should be attributed to the lower nucleation rate due to reduced supersaturation. According to Eq. [13], as the concentration of sulfur decreases, the growth rate of crystals also decreases, which would lead to the movement of size distribution toward a smaller radius.

Bester and Lange[50] critically assessed the diffusion coefficients of nitrogen in liquid iron and obtained the following equation:where D_N is the diffusion coefficient of nitrogen in liquid iron and T is the temperature in Kelvin. The temperature fitting range for Eq. [28] is from the melting point of iron to 1700 °C. An extrapolating procedure can be made to calculate the diffusion coefficient of nitrogen in steel below the melting point.

There is no experimental data reported for interfacial tension between TiN and steel. Nishizawa et al.[64] estimated the interfacial tension between TiN and liquid iron to be 0.3 N/m by employing an improved version of the Becker’s model for interfacial energy calculations at liquid metal–liquid metal interfaces to calculate the total energy of interatomic bonds across an interface. Considering that there are some surface-active elements, such as oxygen and sulfur, in liquid steel, we employed an interfacial tension value of 0.28 N/m between TiN and liquid steel for samples with low sulfur concentration. The high sulfur concentration of steel could lead to a further decrease of interfacial tension between TiN and liquid steel. The influence of interfacial tension between TiN and liquid steel on the inclusion distribution is discussed in Section II–C.

The model parameters for precipitation of TiN during solidification of steel are listed in Table IV.

Rocabois et al.[20] investigated the precipitation of TiN inclusions in quenched directionally solidified steel rods (5-mm diameter and 300-mm long) with a conventional vertical apparatus with a static induction coil under controlled thermal conditions. The cooling rate employed in their experiments was 0.37 °C/s. Longitudinal and transverse cross sections containing the quenched mushy zone were analyzed using the metallographic method. The observation of the position of inclusion toward dendrites showed that TiN inclusions were pushed out of the solid-liquid front in the interdendritic regions. Two compositions of steels with low (25 ppm) and high (440 ppm) sulfur concentrations were investigated. The chemical compositions of these two steels are shown in Table V. The sizes of the TiN inclusions at different stages of the solidification for low sulfur steel were measured.

The precipitation of TiN during solidification was simulated by the present model. The model parameters in Table IV were employed for the calculation of low sulfur (25 ppm) steel. It is assumed that all inclusions were pushed out of solid steel during solidification.

The calculated mean sizes of TiN inclusions are shown in Figure 5. For comparison, the experimental mean sizes of TiN from Rocabois et al.[20] are also shown in Figure 5. It can be seen from the figure that the growth is predicted to start already at higher temperatures and continue above the plateau level measured by Rocabois et al.[20] Generally, the calculated mean sizes of TiN inclusions at different temperatures were in reasonable agreement with the experimental data from Rocabois et al.[20] This indicates that the present model could give a good prediction of precipitation characteristics of TiN inclusion during solidification.

The size distributions of inclusion at different temperatures were calculated and are shown in Figure 6. It can be seen that the total distribution shifts toward the larger radius as the solidification proceeds. The peak-value radius of TiN inclusion increases with decreasing temperature, while the maximum and total number density only change slightly. All size distributions can be found to obey lognormal distribution, which is a common characteristic for distribution after nucleation and growth.[2,4,29‐32] These characteristics of size distribution can be explained with competitive nucleation and growth of TiN inclusions. The growth and coarsening of inclusion particles would lead to larger particles as the solidification proceeds. The lowering of number density due to the coarsening is compensated by the continuous nucleation of inclusion particles, so the maximum and total number density remains nearly invariant. Rocabois et al.[20] also found that the number density N_V remains nearly constant during the solidification with a value of around 3.3 × 10¹²/m³. A comparison could be made between the calculated and experimental results on number density. The calculated maximum number density in the present work is also almost constant during solidification. The calculated maximum number density (10¹³ m⁻³) is somewhat higher than the experimental data (3.3 × 10¹² m⁻³). But, considering that the number density values are from scanning electron microscope examinations on sections, the difference between calculated and experimental data is still reasonable.

Oxygen and sulfur are well-known surface-active elements in liquid steel. The surface tension of liquid steel can be decreased with increasing concentrations of oxygen and sulfur. In experiments by Rocabois et al.,[20] two steel compositions with low and high sulfur concentration were investigated. The interfacial tensions between TiN and two liquid steel compositions should differ from one another.

Since the information on the interfacial tension between TiN and liquid steels with different sulfur concentrations is scarce, a sensitivity analysis on the effect of the interfacial tension and sulfur concentration on the precipitation of TiN was conducted.

The final size distributions of TiN for steels were calculated with varying interfacial tensions between TiN and liquid steels (γ = 0.28, 0.275, 0.27, 0.26, and 0.25 N/m). The other parameters in Table IV were employed for calculation. The size distributions of TiN with varying interfacial tension are shown in Figure 7. It can be seen that the maximum number density and peak-value radius of TiN inclusions varied with decreasing interfacial tension from 0.28 to 0.25 N/m. As the interfacial tension decreases from 0.28 to 0.27 N/m, the peak-value radius of TiN decreases, while the maximum number density increases. The further decrease of interfacial tension from 0.27 to 0.25 N/m only produces a slight change of peak-value radius and maximum number density. Rocabois et al.[20] also observed in their experiments that the number density of samples with high sulfur concentration is higher than that of samples with low sulfur concentration. The mean sizes of TiN in experimental samples showed no large difference, but the mean sizes calculated by their model showed a decrease with increasing sulfur concentration. The increase of the maximum number density in size distribution with decreasing interfacial tension could be attributed to the enhanced nucleation due to the lower energy barrier for homogeneous nucleation, as shown in Eq. [12]. The decrease of the peak-value radius of TiN inclusions with decreasing interfacial tension should be due to the suppression of coarsening. The coarsening (Ostwald ripening) of particles is often interpreted by employing Lifshitz–Sloyzov–Wagner theory.[66,67] If it is assumed that the ripening during solidification is controlled by diffusion, we can obtainwhere \( \bar{d} \) and \( \bar{d}_{0} \) are the mean crystal size at time t and at the beginning of coarsening, respectively; D is the effective diffusion coefficient; \( \sigma_{\text{SL}} \) is the interfacial tension between solid and liquid; V_S is the molar volume of crystal; and c₀ is the mass concentration of mobile species in liquid equilibrated with a crystal with infinite large size.

According to Eq. [29], the coarsening rate \( k \) is directly proportional to the interfacial tension \( \sigma_{\text{SL}} \). Consequently, a decrease in \( \sigma_{\text{SL}} \) leads to a corresponding relative decrease in the value of k. The decrease of peak-value radius with decreasing interfacial tension should be attributed to the suppressed coarsening.

As seen in Figure 7, the size distribution of TiN becomes wider also with decreasing interfacial tension. All these findings indicate that the size distribution of inclusions is very sensitive to interfacial tension between inclusion and steel. Consequently, in order to make more detailed calculations in the future, it might be relevant to direct more research using experimental techniques toward clarifying that aspect.

The cooling condition has an impact on the precipitation of TiN particles from steel during solidification.[7] Sage and Cochrane[59] established an inverse linear relationship between the size of TiN particles and the logarithmic cooling rate during and after solidification. Stock et al.[60] confirmed this relationship by using their data for higher cooling rates. Although Rocabois et al.[20] only investigated the TiN precipitation at one cooling rate (0.37 °C/s), it will be interesting to investigate effect of different cooling rates by the model simulation.

The TiN precipitations at three cooling rates (0.37, 3.7, and 37 °C/s) for a low S steel sample whose composition is shown in Table V during solidification were simulated by using the present model. The size distributions of TiN inclusions precipitated from samples cooled at different rates are shown in Figure 8. As seen in Figure 8, the size distributions at cooling rates of 3.7 and 37 °C/s have distinct characteristics that differ from those at a cooling rate of 0.37 °C/s. The distribution at 0.37 °C/s basically follows a lognormal distribution, while the distributions at higher cooling rates exhibit bimodal distributions. This kind of bimodal distribution for TiN inclusions was also observed by Ohta and Suito[61] in Fe-1.5 pct Mn-0.05 pct C-0.1 pct Ti alloys and Stock et al.[60] in a low-carbon steel. It can be also seen in Figure 8 that the size distribution curves move toward lower radius positions, indicating that the size of TiN precipitations decreases with increasing cooling rates. The mean radius values at the three cooling rates (0.37, 3.7, and 37 °C/s) are 5.42, 0.107, and 0.0229 μm, respectively. This kind of cooling rate dependence of TiN size has been reported by many researchers.[59,60] Meanwhile, the maximum number density in size distribution curves increases with increasing cooling rates, which is also in line with the size distribution previously reported.

Calculating the microsegregation of titanium and nitrogen at different cooling rates should be useful for interpreting the results of the calculated size distributions. Figure 9 shows the calculated Ti and N concentration using the Ohnaka model[17] in the interdendritic liquid steel during solidification of the low S steel sample shown in Table V. It can be seen that there is a rapid increase of Ti concentration near the solidus temperature. This increase becomes more significant as the cooling rate increases. It is presumed that the nucleation rate at the final stage of solidification is so high that many smaller particles form, which leads to the second logarithm normal peak. The calculated size distributions of TiN at different solidification fractions (f_s) with a cooling rate of 37 °C/min are shown in Figure 10. It can be seen that the bimodal distributions only appear after solidification proceeds at f_s = 0.95. This supports our presumption that the bimodal distribution should be due to the high nucleation rate at high solidification fraction. The decreased size of precipitated TiN particles at a higher cooling rate should be due to the lack of sufficient coarsening. The increased maximum number density of TiN particles observed in our calculation at high cooling rates also supports the notion that the coarsening is weaker than that at lower cooling rates.