Cross-Scale Modeling of MnS Precipitation for Steel Solidiﬁcation

: One of the advantages of numerical simulations over traditional experimental methodologies is that they can synchronize nucleation, growth and coarsening during solidiﬁcation from the point of view of microstructural analysis. However, the computational cost and accuracy are bottlenecks restricting simulation approaches. Here, two cellular automaton (CA) modules with different grid dimensions are coupled to form a cross-scale model in order to simulate MnS precipitation, accompanied by the matrix growth of dendrites during the solidiﬁcation of a Fe-C-Mn-S steel, where the matrix growth is computed through the CA module with large grids based on the solute conservation and the undercooling of thermal, constitutional, and curvature, and increments of solid fraction of MnS are solved in combination with the transient thermodynamic equilibrium on the locally re-meshed grids once the MnS precipitation is formed. We utilize the cross-scale mode to illustrate MnS evolution in a solidifying matrix and explain the reason why it coexists in three shapes. Further, we study the effects of the content of elements Mn and S on MnS precipitation based on two continuously cast steel objects, with the factor of concentration product ﬁxed as a constant. A of MnS is observed during the solidiﬁcation of a system with a high content of Mn and low content of S. Simultaneous computation using cross-scale modeling can effectively save on computational resources, and the simulation results agree well with the experimental cases, which conﬁrm its reliable accuracy.


Introduction
Solute enrichment, caused by the segregation of the elements Mn and S, occurring at the solid-liquid interface, leads to a reduction of Gibbs free energy, which is bound to induce MnS precipitation in solidifying steel. MnS precipitation plays an entirely different role in the performance of steel. It is seriously detrimental to intergranular cracking due to the precipitation reaction within solidifying metal acting as a strong sink of free sulfur segregation to the grain boundaries [1], thus decreasing corrosion resistance by accelerating hydrogen absorption into the matrix [2] and weakening the hot ductility because boundary sliding is enhanced at austenite grains [3]. The lower modulus and hardness of MnS inclusions compared to those of a matrix give rise to a drop in the fatigue strength of steel [4], while the machinability is amended simultaneously [5]. MnS can also refine the matrix grains by suppressing austenite growth with the pinning effect of MnS-rich precipitates [6] and facilitate acicular ferrite nucleation in the coarse-grained zone so as to promote fractions of the high-toughness phase [7,8]. Many works have been conducted from a variety of perspectives, such as thermodynamics, kinetics, solute segregation and nucleation, in order to determine MnS precipitation to better control its impact on steel performance. The formation and evolution of precipitation, as well as the morphology of MnS and the matrix on which it is growth shares a common thermal-solutal environment, in which the liquid phase is transformed into the matrix and MnS simultaneously and the two transformations interact with each other. Hence, a model that covers the entire evolution with precipitation and growth of solidification in the same computational domain is needed.
Our study achieved the cross-scale simultaneous computation for solidification of Fe-C-Mn-S steel by applying two CA modules with different grid dimensions to a solidifying matrix and a precipitated MnS, respectively. One CA module with large grids was used to compute the matrix growth of dendrites by acquiring the growth velocity, based on the solute conservation, and the undercooling of thermal, constitutional, and curvature, where the thermal-solutal diffusion in grids was determined using finite difference method. Then, local domains were re-meshed immediately to a mesh of a finer size once MnS was found, and the thermal-solutal diffusion of the large grids was assigned to re-meshed grids using linear interpolation to present MnS precipitation using another CA module with small grids where the increment of solid fraction of MnS was solved by combining it with the transient thermodynamic equilibrium. Here, two CA modules with different grid dimensions were coupled to be a cross-scale model to simulate the evolution and morphology of MnS precipitation, accompanied by the matrix growth of dendrites. Cross-scale modeling broke through the scale gap between the matrix and the precipitates and saved on computational costs. Its reliability was confirmed via a comparison with the matrix growth from a modified Lipton-Glicksman-Kurz (LGK) model, and experimental analysis for MnS inclusions, respectively, because the LGK dendritic tip analytical model can accurately solve the growth characteristics of dendritic tips.

Heat and Mass Transfer
Thermal-solutal diffusion, shared by solidifying matrix and precipitated MnS, was formulated using Equations (1)-(3) for each grid cell in the computational domain. Equation (1) is the uniform governing formula for heat transfer in liquid and solid phases. However, diffusivity of solutes in different phases needed to be distinguished because mass transfer in the solid was an order of magnitude less than that of the liquid. Equation (2) is the governing formula for solutal diffusion in the liquid phase in considering interactions between solutes. Equation (3) is for the solid phase of neglecting interaction.
where ρ, c p and f S are density (kg·m −3 ), specific heat (J·kg −1 ·K −1 ) and the solid fraction of matrix, λ was thermal diffusivity (kW·m −1 ·K −1 ), L is latent heat (J·kg −1 ), q w is heat flux on heat dissipation wall (W·m −2 ), c L,i and c S,i were concentration of solute element i (1, 2, . . . , n − 1) in the liquid and solid phases (wt. %), element n denotes solvent, D S,i is diffusivity of element i in solid phase (m −2 ·s −1 ), and D L,ij is the Darken diffusivity of the matrix in the liquid phase, expressed as [27,28]: where a k , x k and µ k are activity, mole fraction and chemical potential (J·mol −1 ) of element k (1, 2, . . . , n), δ ki is the Kronecker delta (δ ki = 1 if i = k, otherwise δ ki = 0), R is the gas constant (J·K −1 ·mol −1 ), M k is the atomic mobility (J·m 2 ·mol −1 ·s −1 ) determined using the Einstein relation and D k * = RTM k , D k * is the diffusivity of traced element k in the liquid phase (m −2 ·s −1 ).

Matrix Growth of Dendrites
The growth velocity of dendrites at solid-liquid interface I was acquired by solving the simultaneous equations of solute conservation for elements C, Mn, and S, which follow the law of solute partition under a local thermodynamic equilibrium.
where υ n is growth velocity (m·s −1 ), c S,i* and c L,i* are equilibrium concentrations (wt. %) of element i in solid-liquid interface I, c S,i* = k i c L,i* , k i is partition coefficient, and n denotes the growth orientation of a normal vector, from solid to liquid: The increment of the solid fraction for all grid cells at the solid-liquid interface was expressed as: L φ = ∆l max(|sin θ|, |cos θ|) (10) where θ is the angle (rad) between the growth orientation and x axis, ∆l is the grid dimension (m), Lφ is the length of the line segment (m) along the normal direction n going through the cell center in Figure 1. where ak, xk and μk are activity, mole fraction and chemical potential (J·mol −1 ) of element k (1, 2, …, n), δki is the Kronecker delta (δki = 1 if i = k, otherwise δki = 0), R is the gas constant (J·K −1 ·mol −1 ), Mk is the atomic mobility (J·m 2 ·mol −1 ·s −1 ) determined using the Einstein relation and Dk * = RTMk, Dk * is the diffusivity of traced element k in the liquid phase (m −2 ·s −1 ).

Matrix Growth of Dendrites
The growth velocity of dendrites at solid-liquid interface I was acquired by solving the simultaneous equations of solute conservation for elements C, Mn, and S, which follow the law of solute partition under a local thermodynamic equilibrium.
where υn is growth velocity (m·s −1 ), cS,i* and cL,i* are equilibrium concentrations (wt. %) of element i in solid-liquid interface I, cS,i* = ki cL,i*, ki is partition coefficient, and n ⃑ denotes the growth orientation of a normal vector, from solid to liquid: The increment of the solid fraction for all grid cells at the solid-liquid interface was expressed as: L φ = ∆l max(|sin θ|, |cos θ|) (10) where θ is the angle (rad) between the growth orientation and x axis, Δl is the grid dimension (m), Lφ is the length of the line segment (m) along the normal direction n ⃑ going through the cell center in Figure 1. Here, the matrix growth of the dendrites at the solid-liquid interface was driven by the undercooling of thermal, constitutional, and curvature: Here, the matrix growth of the dendrites at the solid-liquid interface was driven by the undercooling of thermal, constitutional, and curvature: where T l is the liquidus temperature (K), T cell is the temperature of the liquid cell (K), T l − T cell is the thermal undercooling, ∆T c is the constitutional undercooling (K), c L,io and m L,i are the initial concentration (wt. %) and liquidus line slope (K·wt. %) of element i, and ∆T r is the curvature undercooling (K) expressed as [22,26]: where Γ is the Gibbs-Thomson coefficient (m·K), K ave is the average curvature, f (θ, Φ) is the anisotropy of surface tension, is the parameter for anisotropy of surface tension, Φ is the angle (rad) of preferential growth direction with respect to the x axis, N = 8 is the number of neighbor cells used, and f S,i is the solid fraction of neighbor cells.

MnS Precipitation
The formation reaction of MnS in Fe-C-Mn-S Steel is expressed as: [Mn] + [S] = (MnS) ∆G 0 = RT ln K (18) log K = 8817 T − 5.16 (19) where K is the equilibrium constant of the reaction, a MnS = 1 is the activity for pure MnS, a [Mn] and a [S] are the Henry activities with respect to 1 mass % standard state and were determined, as follows, using Mn as an example: where [%i] is the concentration (wt. %) of element i, f [Mn] is the activity coefficient of Mn with respect to 1 mass % standard state, e i Mn is the interaction coefficient of element i to Mn and were obtained through a thermodynamic database. The thermodynamic condition for the precipitation reaction needs to be satisfied:  [29,30]: where I v is the nucleation rate, A = 10 33 m −3 ·s −1 is the frequency factor, ∆G n is the activation energy (J·mol −1 ), V m is the molar volume of MnS (L·mol −1 ), σ MnS is the interfacial energy (N·m −1 ) between MnS and matrix, and k 0 is the Boltzmann constant (J·K). The kinetic condition for MnS nucleation in matrix was defined as nucleation rate I v and was larger than a random number between 0 and 1.
Since the size of the precipitated MnS was too small to present its morphology, if only the meshed grids for solidifying the matrix were employed during the simulation, we re-meshed local domains into 10 × 10 liquid precipitation cells (PCs) once thermodynamic and kinetic conditions were achieved and randomly allocated MnS seeds into one PC of every meshed matrix grid to become a new interface cell of MnS. Here, chemistry equilibrium required for the growth of MnS could be attained instantaneously due to the high temperature, expressed as: Here, chemistry equilibrium required for the growth of MnS could be attained instantaneously due to the high temperature, expressed as: where HMn and HS are the relative atomic mass of Mn and S, HMnS is the relative molecular mass of

Model Verification
A CA model was applied to undercooled molten steel, with components of Fe-0.6% C-1.0% Mn-0.3% S in an adiabatic domain of 0.3 × 0.3 mm 2 , in order to compute the matrix growth of equiaxed dendrites with an initial temperature 1800 K. The computational domain was meshed into 300 × 300 grids and a dendrite nucleus was placed in the centers. Figure 3 showed solutal concentrations of trace elements C, Mn and S, and the morphology of growing dendrites. The diffusivities of the solute elements conformed to the relationship DC * > DS * > DMn * , resulting in a rapid rejection of element C from dendrites into the liquid phase and the formation of a large diffusion zone. The zone areas of elements S and Mn declined in turn. Solute segregation led to an enrichment of elements at the frontier of dendrites, thus forming a radical concentration gradient centered on the dendrite nucleus due to concentration being superior to the liquid phase. The tips of the dendrites grew preferentially compared to their arms because of the axial and radial diffusion of solute

Model Verification
A CA model was applied to undercooled molten steel, with components of Fe-0.6% C-1.0% Mn-0.3% S in an adiabatic domain of 0.3 × 0.3 mm 2 , in order to compute the matrix growth of equiaxed dendrites with an initial temperature 1800 K. The computational domain was meshed into 300 × 300 grids and a dendrite nucleus was placed in the centers. Computational data used in the  Tables A1 and A2 (Appendix A). Figure 3 showed solutal concentrations of trace elements C, Mn and S, and the morphology of growing dendrites. The diffusivities of the solute elements conformed to the relationship D C * > D S * > D Mn *, resulting in a rapid rejection of element C from dendrites into the liquid phase and the formation of a large diffusion zone. The zone areas of elements S and Mn declined in turn. Solute segregation led to an enrichment of elements at the frontier of dendrites, thus forming a radical concentration gradient centered on the dendrite nucleus due to concentration being superior to the liquid phase. The tips of the dendrites grew preferentially compared to their arms because of the axial and radial diffusion of solute elements, rather than only the radial for the arm, causing a simultaneous enhancement in concentration gradient and growth velocity. Solutal diffusion at the roots of the dendrites was hindered by the surrounding arms so that the radial coarsening growth was slowed and caused root necking and solute enrichment. The LGK model is an analytic method for accurately computing the axial growth velocity of dendrites because the shape of tip is solved in strict accordance with thermal and mass transfers, as well as the critical stability of the solid-liquid interface. Thus, the reliability of the CA model can be verified by comparing the growth velocity of the tips of two models. Figure 4a shows that the velocity predicted by the LGK model was constant at a fixed undercooling of 7.0 K due to the fact that the Péclet number, depending on thermal diffusivity and tip radius, was not changed [31]. However, a maximum velocity was obtained by CA model at the beginning of nucleation because a large instantaneous concentration gradient was induced in the liquid phase by a high concentration of solidifying frontier around the dendrite nucleus. This concentration gradient was decreased with the diffusion of rejected solutes into the liquid phase until the growth velocity tended to be stable (when solutal diffusion and tip growth reached equilibrium). The relative tolerance of velocity of the two models was limited within 0.6 × 10 −2 to 0.2 from solidification times ranging from 0.15 s to 0.25 s. Figure 4b shows that the velocity predicted by the CA model at a low undercooling was higher than that of by the LGK model after solidification for 0.2 s because the LGK model can only be used to solve the shape of the tip without coarsening of the dendrite arm, while the CA model contained sufficient diffusion of solutes around the dendrite arm, so that a rapid growth of the dendrite, caused by low undercooling, was included. This result can be explained in detail by formula as: Radial arm coarsening increased the solid fraction fS,i near the tip and decreased the average curvature Kave in Equation (14); then, the curvature undercooling ΔTr was dropped accordingly in Equation (13), causing the constitutional undercooling ΔTc to be promoted in Equation (11), which led to an enhancement of growth velocity υn due to a larger concentration gradient ∇cL,j in Equation (7). Further, arm coarsening was gradually weakened as undercooling rose, resulting in the growth velocity from the CA model being lower than that of the LGK model. Comparison of the two models showed that the predicted results were consistent for both fixed and different undercoolings. The reliability of the CA model was verified using an analytical LGK model. The LGK model is an analytic method for accurately computing the axial growth velocity of dendrites because the shape of tip is solved in strict accordance with thermal and mass transfers, as well as the critical stability of the solid-liquid interface. Thus, the reliability of the CA model can be verified by comparing the growth velocity of the tips of two models. Figure 4a shows that the velocity predicted by the LGK model was constant at a fixed undercooling of 7.0 K due to the fact that the Péclet number, depending on thermal diffusivity and tip radius, was not changed [31]. However, a maximum velocity was obtained by CA model at the beginning of nucleation because a large instantaneous concentration gradient was induced in the liquid phase by a high concentration of solidifying frontier around the dendrite nucleus. This concentration gradient was decreased with the diffusion of rejected solutes into the liquid phase until the growth velocity tended to be stable (when solutal diffusion and tip growth reached equilibrium). The relative tolerance of velocity of the two models was limited within 0.6 × 10 −2 to 0.2 from solidification times ranging from 0.15 s to 0.25 s. Figure 4b shows that the velocity predicted by the CA model at a low undercooling was higher than that of by the LGK model after solidification for 0.2 s because the LGK model can only be used to solve the shape of the tip without coarsening of the dendrite arm, while the CA model contained sufficient diffusion of solutes around the dendrite arm, so that a rapid growth of the dendrite, caused by low undercooling, was included. This result can be explained in detail by formula as: Radial arm coarsening increased the solid fraction f S,i near the tip and decreased the average curvature K ave in Equation (14); then, the curvature undercooling ∆T r was dropped accordingly in Equation (13), causing the constitutional undercooling ∆T c to be promoted in Equation (11), which led to an enhancement of growth velocity υ n due to a larger concentration gradient ∇c L,j in Equation (7). Further, arm coarsening was gradually weakened as undercooling rose, resulting in the growth velocity from the CA model being lower than that of the LGK model. Comparison of the two models showed that the predicted results were consistent for both fixed and different undercoolings. The reliability of the CA model was verified using an analytical LGK model. Equation (13), causing the constitutional undercooling ΔTc to be promoted in Equation (11), which led to an enhancement of growth velocity υn due to a larger concentration gradient ∇cL,j in Equation (7). Further, arm coarsening was gradually weakened as undercooling rose, resulting in the growth velocity from the CA model being lower than that of the LGK model. Comparison of the two models showed that the predicted results were consistent for both fixed and different undercoolings. The reliability of the CA model was verified using an analytical LGK model.

MnS Evolution in Solidifying Matrix
Cross-scale modelling consisting of two CA modules was used to simulate MnS precipitation, accompanied by the matrix growth of dendrites during the solidification in the system of Fe-0.6%C-1.0%Mn-0.3%S within a computational domain of 300 × 300 µm, as shown in Figure 5. Undercooling was induced by the boundary condition of heat flux q w = 0.4 W·m −2 on two face-to-face borders and the other two sides were adiabatic, considering a balance of heat exchange. Nuclei were allocated into the domain using the Gaussian formula [25], and those at the borders began to grow into initial columnar dendrites. Mutual competition of initial dendrites enabled some of them to grow preferentially as superior ones, and rapidly reject solutes, which caused local enrichment for other columnar dendrites, resulting in the growth of these inferior dendrites to be further restrained because of a decrease in the concentration gradient. Meanwhile, secondary dendrites were formed as surface protrusions of the growing dendrite arm, and those of adjacent arms bridged each other to solidify the matrix in Zone I (inward about 80 µm from the boundary). Superior columnar dendrites that continue to extend were coarsened due to the concentration gradient rising in the liquid phase between them, also causing accompanying secondary dendrites to develop well in Zone II of the domain. MnS precipitation depended on local undercooling, solutal concentration and dendritic morphology.

MnS Evolution in Solidifying Matrix
Cross-scale modelling consisting of two CA modules was used to simulate MnS precipitation, accompanied by the matrix growth of dendrites during the solidification in the system of Fe-0.6%C-1.0%Mn-0.3%S within a computational domain of 300 × 300 μm, as shown in Figure 5. Undercooling was induced by the boundary condition of heat flux qw = 0.4 W·m −2 on two face-to-face borders and the other two sides were adiabatic, considering a balance of heat exchange. Nuclei were allocated into the domain using the Gaussian formula [25], and those at the borders began to grow into initial columnar dendrites. Mutual competition of initial dendrites enabled some of them to grow preferentially as superior ones, and rapidly reject solutes, which caused local enrichment for other columnar dendrites, resulting in the growth of these inferior dendrites to be further restrained because of a decrease in the concentration gradient. Meanwhile, secondary dendrites were formed as surface protrusions of the growing dendrite arm, and those of adjacent arms bridged each other to solidify the matrix in Zone I (inward about 80 μm from the boundary). Superior columnar dendrites that continue to extend were coarsened due to the concentration gradient rising in the liquid phase between them, also causing accompanying secondary dendrites to develop well in Zone II of the domain. MnS precipitation depended on local undercooling, solutal concentration and dendritic morphology. For example, MnS began to precipitate at a solid fraction of fS = 0.69 in Zone I, while it needed to be fS = 0.73 in Zone II. Details are shown in Figures 6 and 7 by taking Parts A and B as cases, respectively.     Preferential precipitation of MnS at a solid fraction of 0.69 in Zone I was caused by the reduction of Gibbs freedom energy due to the low temperature and high solutal concentration, as shown in Figure 6a. The residual liquid phase between solidifying dendrites was difficult to be transported into the liquid phase and formed isolated liquid holes by the bridging of secondary dendrites. The high concentrations in the holes facilitated the nucleation of MnS but the size was confined by the limited space and solute mass. Thus, the amount of MnS increased gradually from a solid fraction of 0.73 to 0.77, and the volume was hardly changed, as shown in Figure 6b,c. Precipitated MnS tended to be saturated at solid fraction fS = 0.87 until it was completely solidified, as shown in Figure 6d,e, where the content of element Mn decreased significantly and only increased in the corresponding position of holes because it failed in the formation reaction of MnS. The predicted morphology of MnS was compared with those observed using an optical microscope and scanning electron micrograph [32] for similar compositional systems of Fe-1.0%Mn-0.3%S-1.0%Ti and Fe-1.0%Mn-0.3%S, as shown in Figure 6f,g. The simulation results agreed well with the experimental characterizations in the geometric dimensions, and the inconsistency in shape was due to the fact that our meshed PCs still needed to be refined.
Solutal transport was active in Zone II and almost no liquid holes formed because the large space between the columnar dendrites caused second dendrites to no longer play an important role in the liquid phase transfer. Here, precipitated MnS usually existed in three shapes. For example, the seeds α and β of MnS in Part B at solid fraction fS = 0.73 were immersed freely in the liquid phase or attached to a solidified matrix containing columnar and secondary arms, respectively, as shown in Preferential precipitation of MnS at a solid fraction of 0.69 in Zone I was caused by the reduction of Gibbs freedom energy due to the low temperature and high solutal concentration, as shown in Figure 6a. The residual liquid phase between solidifying dendrites was difficult to be transported into the liquid phase and formed isolated liquid holes by the bridging of secondary dendrites. The high concentrations in the holes facilitated the nucleation of MnS but the size was confined by the limited space and solute mass. Thus, the amount of MnS increased gradually from a solid fraction of 0.73 to 0.77, and the volume was hardly changed, as shown in Figure 6b,c. Precipitated MnS tended to be saturated at solid fraction f S = 0.87 until it was completely solidified, as shown in Figure 6d,e, where the content of element Mn decreased significantly and only increased in the corresponding position of holes because it failed in the formation reaction of MnS. The predicted morphology of MnS was compared with those observed using an optical microscope and scanning electron micrograph [32] for similar compositional systems of Fe-1.0%Mn-0.3%S-1.0%Ti and Fe-1.0%Mn-0.3%S, as shown in Figure 6f,g. The simulation results agreed well with the experimental characterizations in the geometric dimensions, and the inconsistency in shape was due to the fact that our meshed PCs still needed to be refined.
Solutal transport was active in Zone II and almost no liquid holes formed because the large space between the columnar dendrites caused second dendrites to no longer play an important role in the liquid phase transfer. Here, precipitated MnS usually existed in three shapes. For example, the seeds α and β of MnS in Part B at solid fraction f S = 0.73 were immersed freely in the liquid phase or attached to a solidified matrix containing columnar and secondary arms, respectively, as shown in Figure 7a. Seed α was fully in contact with the liquid phase and was likely to be fostered into an isotropic two-dimensional dendrite shape. Its arms that were close to the matrix grew slowly due to local insufficiency of solutes. However, MnS grown from seed β was restricted by the surrounding matrix and could only develop into an irregular strip-and cluster-shape. The three MnS shapes are shown in Figure 7b at solid fraction f S = 0.77, and they continued to grow, as shown in Figure 7c,d, until they turned into solids (Figure 7e). Predicted dendrite-, strip-and cluster-shaped MnS were compared with those of captured microscopic images [32] (Figure 7f,g). Simulated and observed results agree well with shape and size. Our work confirmed that MnS (three shapes) had a high possibility of coexistence, and also explained why this occurs.
We found that the changing trend in the content of elements Mn and S was opposite during solidification. Element Mn was instantaneously consumed once MnS was precipitated, in Figure 6a, while element S was still abundant, as shown in Figure 7e. This was because the partition coefficient of element S k S = 0.035 was far less than that of element Mn k Mn = 0.785 at the solid-liquid interface, resulting in an amount of rejected element S that was higher than the level of consumption for MnS formation. Further, the solubility product of MnS, E MnS , was fixed at a constant temperature so that the content of element Mn was reduced; see Equation (22).

Effect of Content of Mn and S
Cross-scale modeling was used to predict the MnS precipitations for two continuously-cast steel objects with systems of Fe-0.6%C-0.2%Mn-0.02%S and Fe-0.6%C-0.8%Mn-0.005%S in order to clarify the effects of the content of elements Mn and S. The initial contents of Mn and S were distinctly differentiated in the two systems-high content of S for former (High-S) and high content of Mn for latter (High-Mn)-but the concentration products were fixed as a constant for comparison. MnS was precipitated at solid fractions f S = 0.91 and 0.93 in two systems, respectively. They were significantly higher than that of 0.69 in the system of Fe-0.6%C-1.0%Mn-0.3%S, discussed previously. This was because the low content of Mn and S enabled them to be sufficiently enriched in the later stage of solidification in order to achieve the Gibbs free energy required for the nucleation of MnS in the residual liquid phase. Hence, MnS can only be presented as a cluster shape and fails to be formed in the dendrite and strip shapes due to the narrow space and insufficient solutes, as shown in Figure 8a,b. Further, element S played a dominant role in the concentration product of solutes because of its low partition coefficient.
Because of this, it was necessary that the Gibbs free energy be reduced and the nucleation rate of MnS was enhanced in the High-S system, making the solid fraction of MnS, when it was beginning to be precipitated, slightly lower than that of the High-Mn system. Moreover, the content of Mn was a restrictive factor in deciding the amount of precipitated MnS because it was a supplement to the rejected S during MnS growth, attributed to its high partition coefficient. We counted the precipitated MnS of all PCs in the computational domain and determined that the volume ration of MnS for the two systems were 9.73 × 10 −6 and 3.76 × 10 −5 , respectively, showing a relationship of 4 times that was coincident with the initial concentration of Mn. It is noteworthy that a re-precipitation of MnS was observed during solidification in the High-Mn system. Precipitated MnS was immobilized by the solidified matrix and the residual liquid phase was also partitioned into small pools, as shown in Figure 8c. Excessive Mn had great potential to react with residual S, which was able to form a new seed γ in these pools when the required reaction conditions for MnS formation could be satisfied again, as shown in Figure 8d. However, re-precipitation of MnS was observed in the High-S system because rejected Mn was rapidly consumed by the high content of S, and it was difficult to reach a sufficient concentration to reduce the Gibbs free energy in the pools. We confirmed that it was likely to have a re-precipitation of MnS for solidification of Fe-C-Mn-S steel with a high content of Mn and low content of S. The predicted morphology of MnS was compared with those of in-cast billet samples that were observed using a scanning electron micrograph and energy dispersive spectrometer [13] (Figure 8e,f). They were in good agreement in terms of shape and size, as well as site. Our model was able to simulate the precipitation and re-precipitation of MnS simultaneously. This was difficult to conduct through experimentation.
counted the precipitated MnS of all PCs in the computational domain and determined that the volume ration of MnS for the two systems were 9.73 × 10 −6 and 3.76 × 10 −5 , respectively, showing a relationship of 4 times that was coincident with the initial concentration of Mn. It is noteworthy that a re-precipitation of MnS was observed during solidification in the High-Mn system. Precipitated MnS was immobilized by the solidified matrix and the residual liquid phase was also partitioned into small pools, as shown in Figure 8c. Excessive Mn had great potential to react with residual S, which was able to form a new seed γ in these pools when the required reaction conditions for MnS formation could be satisfied again, as shown in Figure 8d. However, re-precipitation of MnS was observed in the High-S system because rejected Mn was rapidly consumed by the high content of S, and it was difficult to reach a sufficient concentration to reduce the Gibbs free energy in the pools. We confirmed that it was likely to have a re-precipitation of MnS for solidification of Fe-C-Mn-S steel with a high content of Mn and low content of S. The predicted morphology of MnS was compared with those of in-cast billet samples that were observed using a scanning electron micrograph and energy dispersive spectrometer [13] (Figure 8e,f). They were in good agreement in terms of shape and size, as well as site. Our model was able to simulate the precipitation and re-precipitation of MnS simultaneously. This was difficult to conduct through experimentation.

Conclusions
We developed a cross-scale model consisting of two CA modules with different grid dimensions in order to conduct a simultaneous computation for MnS precipitation, accompanied by the matrix growth of dendrites during the solidification of Fe-C-Mn-S steel. The reliability of a single CA model was verified by comparison with an analytical LGK model and the predicted results from the cross-scale model agreed well with those observed in the experimental cases.
(1) By virtue of cross-model, we illustrated MnS evolution, accompanied by the matrix growth of dendrites during solidification, and indicated that MnS precipitation depends on local undercooling, solutal concentration, and dendritic morphology, thus coexisting in three different shapes. (2) We found that element S played a dominant role in the concentration product and caused MnS to be precipitated at a low solid fraction and the content of Mn was a restrictive factor in deciding the amount of precipitated MnS, due to its high partition coefficient. (3) A re-precipitation of MnS was observed in a system with high content of Mn and low content of S because excessive Mn had great potential to react with residual S, which was able to form new seeds when the required reaction conditions for MnS formation could be satisfied again.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
Computational data used in the present study are listed in Tables A1 and A2, respectively.