Modeling of manganese sulfide formation during the solidification of steel
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Abstract
A comprehensive model was developed to simulate manganese sulfide formation during the solidification of steel. This model coupled the formation kinetics of manganese sulfide with a microsegregation model linked to thermodynamic databases. Classical nucleation theory and a diffusioncontrolled growth model were applied to describe the formation process. Particle size distribution (PSD) and particlesizegrouping (PSG) methods were used to model the size evolution. An adjustable parameter was introduced to consider collisions and was calibrated using the experimental results. With the determined parameters, the influences of the sulfur content and cooling rate on manganese sulfide formation were well predicted and in line with the experimental results. Combining the calculated and experimental results, it was found that with a decreasing cooling rate, the size distribution shifted entirely to larger values and the total inclusion number clearly decreased; however, with increasing sulfur content, the inclusion size increased, while the total inclusion number remained relatively constant.
Keywords
Cool Rate Sulfur Content Solid Fraction Liquid Steel Acicular FerriteIntroduction
Nonmetallic inclusions formed during solidification processes can essentially influence the final product quality. On the one hand, their presence can negatively affect steel properties [1, 2, 3]. On the other hand, they can contribute to a beneficial microstructure by acting as heterogeneous nucleation sites. To combine a preferably high steel cleanness with the creation of specific inclusion types and sizes for microstructure evolution, comprehensive knowledge of the inclusion formation is needed.
A typical inclusion type that is formed in nearly every steel grade is manganese sulfide (MnS). The latter can lead to anisotropy of the steel matrix and act as a possible starting point for crack formation or corrosion [2, 3]. Apart from these negative effects, in the field of ‘Oxide Metallurgy’ [4, 5], MnS, whether as singlephase inclusion or together with titanium oxides, is known to act as a potential nucleation agent for the formation of acicular ferrite [6, 7, 8]. In addition, the formation of MnS prevents internal cracks resulting from the appearance of FeS and reduces hot tearing segregation [9]. Two factors have a significant impact on number density, size distribution, and total amount of formed MnS: the cooling rate and the sulfur content. Both parameters play an important role in process control and optimization, especially during casting, and can therefore directly affect the final product quality. Thus, it is not surprising that MnS formation has been extensively studied over the last several decades.
Mathematical modeling provides a useful tool to investigate the formation of inclusions during the solidification of steel. Different researchers [10, 11, 12, 13] developed several models describing MnS formation. MnS is normally generated from the enrichment of Mn and S in the residual liquid during the solidification process. Thus, it is important to consider the microsegregation of solutes when simulating MnS formation. Ueshima et al. [10] thermodynamically evaluated MnS formation based on an analysis of the interdendritic segregation. Imagumbai [11] applied a SolidificationUnitCell method to calculate the mean diameter of MnS, which depends on the cell volume, temperature gradient, and solidification speed. Valdez et al. [12] coupled Scheil’s model [14] and MnS growth to predict the size evolution. In their mean size prediction, Diederichs and Bleck [13] modified the empirical equation from Schwerdtfeger [15] into a function of manganese and sulfur contents, cooling rate, and secondary dendrite arm spacing. In this model, the concentrations of manganese and sulfur were calculated using the model of Clyne–Kurz [16]. In total, an enhanced model covering microsegregation, thermodynamics, and kinetics to describe the MnS size distribution has not been published thus far.
The present paper proposes a comprehensive model of MnS formation during the solidification of steel. A deeper understanding of the nucleation and growth of manganese sulfide during the solidification of steels is desirable to reduce, control, and even benefit from the formation of MnS. For that purpose, the development of a comprehensive modeling approach for inclusion formation is continued. As a first step, a microsegregation model linked to thermodynamic databases has been developed [17, 18]. Second, coupled with the proposed microsegregation model, the thermodynamics of inclusion formation during the solidification process has been simulated [19]. In the present case, the modeling of inclusion formation is conducted by simultaneously considering the kinetics, microsegregation, and thermodynamics.
Microsegregation is estimated using Ohnaka’s model [20]. The thermodynamic equilibrium is calculated with ChemApp [21] to determine the liquidus temperature and solute partition coefficients at the solidification interface based on commercial databases. MnS trapping at the solidification interface in the residual liquid is assumed to be proportional to the step value of the solid fraction. The kinetics of MnS formation are described using classical nucleation theory [22, 23] and a diffusioncontrolled growth model. Particle size distribution (PSD) [24] and particlesizegrouping (PSG) [25] methods are applied to model the size evolution process. An adjustable parameter is introduced to consider collisions and is calibrated using the experimental results. Steels with different cooling rates and sulfur contents are calculated. The size distribution and evolution, as well as the amount of manganese sulfide, are obtained and compared with the experimental results. The influences of the cooling rate and sulfur content on MnS formation are summarized.
Modeling
Model background

Only MnS formed in the residual liquid is considered in this work. The particles trapped in the solid are assumed to be inert. The particles are distributed homogeneously in liquid steel.

MnS particles are independent. The formation behaviors of nucleation, growth, and collision occur independently for the particles.

The morphology of the particles is spherical.

Diffusioncontrolled growth is assumed. A local equilibrium exists at the interface of the inclusions and liquid steel.
Microsegregation
Thermodynamics
Nucleation
Growth
Collisions
Class model

The size of the particles is classified into several groups (G _{ i }) according to the boundary values (R _{ i }) in both solid and liquid steel. The groups are characterized by the average radius (r _{ i }) and related number density (n _{ i }). The superscripts S1 and L1 indicate that it is in the solid state and liquid state at the current (‘1’) calculation step, respectively. S2 and L2 are for the subsequent (‘2’) step after a series of activities, such as trapping, nucleation, and growth.
 Nucleation: The particles created by nucleation with the radius and number density (r _{0}, n _{0}) are classified into the first group (G _{1}). As given by Eq. (19), the number density (\( n_{1}^{{{\text{L}}2}} \)) of G _{1} at the second step is the sum of n _{0} and existing number density (\( n_{1}^{{{\text{L}}1}} \)). The average radius changes to \( r_{1}^{{{\text{L}}2}} \) based on the calculation with total volume and number [Eq. (20)].$$ n_{1}^{{{\text{L}}2}} = n_{1}^{{{\text{L}}1}} + n_{0} $$(19)$$ r_{1}^{{{\text{L}}2}} = \sqrt[3]{{\frac{{(r_{1}^{{{\text{L}}1}} )^{3} \cdot n_{1}^{{{\text{L}}1}} + r_{0}^{3} \cdot n_{0} }}{{n_{1}^{{{\text{L}}1}} + n_{0} }}}} $$(20)
 Growth: The particles after growing from (\( r_{i  1}^{{{\text{L}}1}} \), \( n_{i  1}^{{{\text{L}}1}} \)) to (r _{ g }, n _{ g }) can be grouped into G _{ i−1} or G _{ i }. If R _{ i−2} < r _{ g } ≤ R _{ i−1} they belong to G _{ i−1} (the same group before growing); the size of this group is r _{ g }(\( r_{i  1}^{{{\text{L}}2}} \) = r _{ g }) and the number is n _{ g } (\( n_{i  1}^{{{\text{L}}2}} \) = n _{ g }). If R _{ i−1} < r _{ g } ≤ R _{ i }, they upgrade to the larger group G _{ i }; the number of the group (\( n_{i}^{{{\text{L}}2}} \)) becomes the sum of \( n_{i}^{{{\text{L}}1}} \) and n _{ g }(\( n_{i}^{{{\text{L}}2}} = n_{i}^{{{\text{L}}1}} + n_{g} \)); and the radius of this group renews to \( r_{i}^{{{\text{L}}2}} \) as calculated in Eq. (21).$$ r_{i}^{{{\text{L}}2}} = \sqrt[3]{{\frac{{(r_{i}^{{{\text{L}}1}} )^{3} \cdot n_{i}^{{{\text{L}}1}} + r_{g}^{3} \cdot n_{g} }}{{n_{i}^{{{\text{L}}1}} + n_{g} }}}} $$(21)
 Collision: The new size class (G _{ i+2}) is easier to create due to collision compared with diffusioncontrolled growth. The calculation of the radius and number are similar to calculations described in the nucleation and growth processes. The number of particles contributing to the collisions is reduced.

The inclusions in solid steel are trapped particles and inert in the following solidification process. Therefore, at each calculation step, the number densities of the particles in different classes increase according to the trapped number (dark volumes in Fig. 3) in the corresponding classes. The trapped number or amount of each group in the liquid is proportional to the step value of the solid fraction as given in Eq. (22) [35]. The average radius of each class is obtained based on the total volume and number of particles in the group.
$$ {\text{Amount}}_{\text{trapped}} = {\text{Amount}}_{{{\text{in}}\;{\text{liquid}}}} \times \Delta f_{\text{s}} /(1  f_{\text{s}} ) $$(22) 
 At one solidification step, the radius and number of particles in different size groups, as well the size classes, are refreshed once after the inclusions experience all of the activities (trapped, nucleation, growth, and collision). Note that the boundary values of the size group (R _{ i }) are settled during the calculation. Hence, the particles can be classified into the appropriate group according to their own radius (r _{ i }) and the boundary values.
At each solidification step, after the nucleation and growth of inclusions, the increase of inclusion amount is recorded. This further causes decrease of the amount of Mn and S in the residual liquid. The changes of solute concentration are accounted for, and the new concentrations of solutes in the residual liquid are used for the next calculation.
Experiments
As shown in Fig. 4a, liquid steel is premelted in an induction furnace (25 kg). A cylindrical chill body is submerged into liquid steel. A steel shell starts to solidify on the cylindrical body with the Zroxide coating surface. The crystallographic growth of the shell mainly originates perpendicular to the cylinder. After approximately 30 s, the sample is lifted out of the liquid melt. The temperature changes during shell solidification are measured by thermocouples inside the test body. The measured temperatures serve as input data for thermal analysis and heat flux calculation. Furthermore, shell growth, cooling rates, solid fractions, and temperature distributions are obtained using an inhouse developed solidification model. The detailed descriptions of SSCT and the interpretation of the results can be found elsewhere [37, 38]. Figure 4b displays the sample preparation procedure. The solidified shell is cut into 16 pieces at room temperature. The piece with a relatively even shell thickness is selected. Then, the sample is metallographically prepared for observation.
Chemical compositions of analyzed steels (wt%)
Samples  C  Si  Mn  S  P 

S1  0.22  0.03  1.40  0.0060  0.0055 
S2  0.22  0.03  1.46  0.0050  0.0048 
S3  0.21  0.04  1.50  0.0021  0.0036 
In the SEM measurements inclusions are detected due to material contrast differences in the backscattered electron (BSE) image. Usually, nonmetallic inclusions are displayed as darker compared to the steel matrix. This method enables the definition of a measurement field on the specimen which is automatically scanned for inclusions. The output consists of the position and the morphological data of every detected particle as well as its chemical composition. Thus, in contrast to manual SEM/EDS analysis, a huge amount of data is obtained that enables statistical evaluation. With this method, the size distribution and number density of inclusions on a defined area can be determined.
Parameter fitting
Symbol (unit)  Name  Values  Symbol (unit)  Name  Values 

V _{m} (m^{3} mol^{−1})  Molar volume of manganese sulfide  2.2 × 10^{−5}  ρ _{in} (kg m^{−3})  Density of manganese sulfide  4.0 × 10^{3} 
R (J K^{−1} mol^{−1})  Gas constant  8.314  μ (kg m^{−1} s^{−1})  Dynamic viscosity of liquid steel  6.2 × 10^{−3} 
k _{b} (J K^{−1})  Boltzmann constant  1.38 × 10^{−23}  ρ _{Fe} (kg m^{−3})  Density of liquid steel  7.9 × 10^{3} 
\( D_{\text{l}}^{\text{Mn}} \) (m^{2} s^{−1})  Manganese(Mn) diffusion coefficient in liquid  1.3 × 10^{−9}  \( D_{\text{l}}^{\text{S}} \) (m^{2} s^{−1})  Sulfur(S) diffusion coefficient in liquid  2.1 × 10^{−9} 
M _{Mn} (g mol^{−1})  Manganese(Mn) molar mass  55.0  M _{S} (g mol^{−1})  Sulfur(S) molar mass  32.0 
Δf _{s}  Solidification step (f _{s} < 0.96)  5.0 × 10^{−3}  Δf _{s}  Solidification step (f _{s} ≥ 0.96)  2.5 × 10^{−5} 
I _{A} (m^{−3} s^{−1})  Preexponent  10^{42}  σ (J m^{−2})  Interfacial energy  0.2 
π (–)  Circumference ratio  3.14  g (m^{2} s^{−1})  Gravitational acceleration  9.8 
Comparisons of the measured and calculated mean diameter and the number density of MnS
Sources  Collision factor (f)  Mean diameter (µm)  Number (mm^{−3}) 

Calculations  1  0.48  2.12 × 10^{6} 
10  0.49  1.83 × 10^{6}  
100  0.60  9.67 × 10^{5}  
200  0.67  6.78 × 10^{5}  
300  0.71  5.34 × 10^{5}  
Experiment  0.54–0.65  5.05–6.22 × 10^{5} 
As a whole, a collision factor equal to 200 is regarded to be effective for simulating MnS formation using the present model under the SSCT experimental conditions. Therefore, in the following calculations, a collision factor of 200 is applied to study the influence of cooling rate and sulfur content.
Influence of the cooling rate and sulfur content
After fitting the parameters to the experimental results, the present model gives highquality predictions for MnS formation during the solidification process. Furthermore, the model is utilized to investigate the influences of two important process parameters on MnS formation: the cooling rate and sulfur content. For reasonable comparisons of the calculated and experimental results, the measured 2D size distributions from the SSCT samples are also converted into 3D ones using the described CSD Corrections v. 1.5 with the lognormal bin size setting; size classes smaller than 0.1 µm are not considered to avoid differences caused by the measurement limitations and precipitation in the solid steel.
Influence of cooling rate
From Fig. 9a, it is found that the size of the inclusions has an increasing trend with a decreasing cooling rate. When the cooling rate slows from 42.3 to 13.5 K s^{−1}, the maximum particle size class increases from approximately 1.3 to 2.5 µm; the diameter of particles with peak frequencies enlarges from 0.42 to 0.76 µm; the frequency of particles with diameters of approximately 0.25 µm (the smallest size class) decreases from 20 to 5 %. In Fig. 9b, the experimental size distributions display similar trends with the predicted results with decreasing cooling rates. The calculated and predicted size ranges and the diameter of the particles with the largest proportion are the same.
Influence of sulfur content
As shown in Fig. 11a, the size significantly increases with the increase of sulfur content from 20 ppm to 50 and 60 ppm. In the sample containing 20 ppm sulfur (S3), a diameter of approximately 0.25 µm, that is, the smallest size class, has a peak frequency of 80 %, which is only approximately 15 % in the other two samples with higher sulfur contents; the maximum size of the particles is approximately 0.76 µm, while it is 2.5 µm in the higher sulfur samples. The size distributions of the samples with 50 and 60 ppm sulfur are close to each other. Comparing the predicted size distributions with experimental ones (Fig. 11b), the agreement can be considered satisfactory when bearing in mind the complexity of the phenomenon and the uncertainty of the physical properties and measurements.
Based on the above discussion in this section, the cooling rate and sulfur content are important process parameters on deciding the MnS formation, final size distribution, and number density. Finely dispersed MnS is desirable for both controlling and utilizing the inclusions. For instance, the small size can effectively relieve the steel anisotropy due to MnS elongation after rolling; meanwhile, it can retain the austenite growth through pinging grain boundaries and further improve the steel properties. Faster cooling and reducing the sulfur content are two approaches for obtaining finer particles. At the same time, the other effects of these two methods should be considered, such as the increasingly serious manganese microsegregation, as shown in Fig. 10b and Fig. 12b, which is detrimental to steel properties. Additionally, there is a favorable diameter, ranging from approximately 0.3 to 0.9 µm when utilizing inclusions such as heterogeneous nucleation sites for acicular ferrite [51, 52]. In industrial solidification processes, the local cooling rate varies between several hundred degrees per second for welding and strip casting, whereas for continuous casting or ingot casting the local cooling rate might decrease to less than 0.1 degree per second close to the center of the cast part. The presented model allows the precise evaluation of solidification processes with respect to the formation of sulfides with prescribed diameter or—vice versa—the adjustment of the manganese and sulfur content for the cooling conditions of a certain solidification process. Thus, the model provides a valuable tool for further activities in the field of inclusion metallurgy.
Summary
MnS formation in the solidification of steels influences both the final product quality and the casting process. In addition to the negative effect on mechanical steel properties, MnS is also known to improve the machinability of free cutting steel, enhance the hot ductility during continuous casting, and promote the acicular ferrite formation. For detailed studies of these aspects, apart from thermodynamics and the mean diameters, considering the formation kinetics of MnS including the number density, the size evolution, and distribution, as well as the amount and the resultant concentrations of the reactants is necessary.

The suggested comprehensive model can be applied to simulate the formation of manganese sulfide during solidification. The calculated size distribution of manganese sulfide fits well with the experimental results. The influences of the sulfur content and cooling rate on manganese sulfide formation were well predicted and in line with the experimental results.

With the decreasing cooling rate, the size distribution shifted entirely to a larger size direction and the total number clearly decreased. The content of manganese sulfides decreased slightly due to the lesser enrichment of the solutes.

With the increasing sulfur content, the MnS size increased, while the total number was similar. The overall inclusion amount significantly increased. Finer manganese sulfides can be achieved via faster cooling or reducing the sulfur content.
It is common to find heterogeneous nucleation inclusions in an alloyed steel. In the meantime, compound inclusions play indispensable roles in inclusion metallurgy. So in the future, based on the current work, heterogeneous nucleation of inclusions, such as sulfides and nitrides, on oxides will be the primary dedicated object. Additionally, the competitive formation thermodynamics and kinetics of multioxides during cooling and solidification will be simulated. As a whole, a comprehensive model of the both homogeneous and heterogeneous types of inclusions accounting for microsegregation, thermodynamics, and kinetics is the final target. A further ambitious idea is connecting the inclusion formation to the microstructure evolutions in the following metallurgical process.
Notes
Acknowledgements
Open access funding provided by Montanuniversität Leoben. The authors are grateful for financial support from the Federal Ministry for Transport, Innovation and Technology (bmvit) and the Austrian Science Fund (FWF) [TRP 266N19]. The authors also sincerely acknowledge the laboratories at voestalpine Stahl GmbH in Linz for assistance in the analysis of samples. Michael Higgins is thanked for offering the program of CSD Corrections v. 1.5 and answering questions during usage.
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