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Vol. 9. Issue 1.
Pages 89-103 (January - February 2020)
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Vol. 9. Issue 1.
Pages 89-103 (January - February 2020)
Original Article
DOI: 10.1016/j.jmrt.2019.10.033
Open Access
Thermodynamic study on the solute partition coefficients on L/δ and L/δ+γ phase interfaces for 1215 high-sulfur steel solidification by orthogonal design
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Lintao Guia,b, Mujun Longa,b,
Corresponding author
longmujun@cqu.edu.cn

Corresponding authors at: College of Materials Science and Engineering, Chongqing University, No.174 Shazheng Street, Shapingba District, Chongqing, 400044, China.
, Dengfu Chena,b,
Corresponding author
chendfu@cqu.edu.cn

Corresponding authors at: College of Materials Science and Engineering, Chongqing University, No.174 Shazheng Street, Shapingba District, Chongqing, 400044, China.
, Jingjun Zhaoa,b, Qinzheng Wanga,b, Huamei Duana,b
a State Key Laboratory of Coal Mine Disaster Dynamics and Control, College of Materials Science and Engineering, Chongqing University, Chongqing, 400044, China
b Chongqing Key Laboratory of Vanadium-Titanium Metallurgy and New Materials, Chongqing University, Chongqing, 400044, China
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Tables (2)
Table 1. The 6 factors and 5 levels of orthogonal design to determine the sample composition in 1215 steel (in wt%).
Table 2. The composition in 1215 steel of the calculating samples for calculating the solute partition coefficients (in wt%).
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Abstract

The solute partition coefficient (ki), characterizing the solute redistribution on the solid-liquid interface, is an important parameter for the study of segregation. Based on thermodynamic equilibrium, the effects of composition fluctuation, phase transition, MnS precipitation, and temperature on the ki for the solidification of 1215 high-sulfur steel were investigated by orthogonal design. The results show that there is only L→δ phase transition for 0.05–0.07wt% C steel solidification, while there are L→δ and L+δ→γ (peritectic reaction) phase transition in turn for the solidification of steel with 0.07–0.09 wt% C. On the L/δ interface, as the temperature decreases, the kSi and kP increase, the kMn decreases, while the kC and kS both decrease first and then increase because of MnS precipitation. In the L+δ+γ coexistent phase, the kiL/δ+γ is comprehensively determined by kiL/δ and kiL/γ. Moreover, the kiL/δ+γ for each solute is closer to the kiL/δ comparing to kiL/γ, because the mass fraction of γ phase (mγ) in the solidified phase (consisting of δ+γ) is relatively small. The peritectic reaction has small effect on the kiL/δ and kiL/γ, but has significant effect on the mγ, leading to the discrepancy in kiL/δ+γ for the steels with different C contents. The orthogonal analysis indicates that the composition fluctuation in 1215 steel has small effect on the ki. The Mn and S contents have the greatest effect, followed by C content. Si and P contents have the least effect. The quantitative ki in terms of temperature, phase composition, and solute content were fitted.

Keywords:
Solute partition coefficient
Thermodynamic
Orthogonal design
High-sulfur steel
Solidification
Segregation
Full Text
1Introduction

Segregation is one of the common defects in an alloy, which results from the solute redistribution on the solid-liquid interface during an alloy solidification [1]. Severe segregation is harmful to the properties of the steel products [2]. The solute partition coefficient (ki), which is defined as the ratio of the solute equilibrium concentrations in the solid and liquid phases at the solidification front, is an important physical parameter for an alloy solidification. The ki characterizes the redistribution trend of the solute element on the solid-liquid interface and fundamentally determines the degree of solute segregation [3–6]. In theory, if the ki is equal to 1, there will be no redistribution of the solute i between the solid and liquid during alloy solidification. If ki < 1, the solute i will diffuse from the solid into the liquid during solidification, and the solute i is called a positive segregation element. But, if ki > 1, the solute i will diffuse from the liquid into the solid, and the solute i is regarded as a negative segregation element [7]. The greater the ki deviates from 1, the stronger the tendency of positive or negative segregation for solute i. In the study of segregation, as a basic parameter, the ki has a significant effect on predicting results [5,8,9]. Therefore, the accuracy of ki is very important.

The ki for an alloy solidification can be estimated by experiments [10–12] or theoretical calculation [13,14]. Although the experimental method is more intuitive and reliable, there are still some limitations. Firstly, the solidification of an alloy is difficult to be in equilibrium [7] to measure the solute equilibrium concentrations in the solid and liquid, and the contingency exists in the measuring of solute concentrations on the solid-liquid interface [15]. These factors leading to the complex operation of the experiment and the error of measured results. Secondly, the ki generally varies with the temperature and solidified phase (delta or gamma) during a specific alloy solidification [16]. But, a single experiment can only estimate the ki at a certain temperature, resulting in a heavy workload to obtain the quantitative ki with temperature and phase transition for an alloy solidification [10,11]. The theoretical calculation is generally based on the thermodynamic equilibrium of alloy solidification. According to the equilibrium of the chemical potentials (or Gibbs free energy) between the solid and liquid phases, the ki can be determined under the corresponding condition. This method is more simply and efficient. Based on thermodynamic theory, the quantitative ki for various Fe-based alloys solidification have been calculated and verified by Battle et al. [13]. Kagawa et al. [14,15] has also confirmed the feasibility of thermodynamics theory on calculating ki.

During an alloy solidification, the ki is affected not only by the alloy composition [16–18] but also by the temperature and solidified phase [13,19,20]. That is, the ki is changed. Studies suggest that the changed ki caused by composition [21], temperature [22], and solidified phase [23,24] has a significant effect on segregation. However, in previous studies on segregation [1,25,26], the ki was generally regarded as a fixed value. This is inconsistent with the facts and may lead to errors. Therefore, to predict the segregation more accurately, it is necessary to obtain the quantitative ki in terms of temperature and solidified phase, both according to the specific composition in an alloy.

The S content in the high-sulfur steel is higher than that in the conventional steel [27], which may cause a difference in ki. In addition, a large amount of MnS inclusion will precipitate in the high-sulfur steel during solidification [27,28]. The precipitation of MnS inclusion can consume the solutes Mn and S, and affect the solute equilibrium concentrations at the solidification front [5,29–31], which may eventually influence the ki. Besides, the ki is also affected by the composition fluctuation in the steel. However, the existing ki is mainly applicable to the conventional steels with a low S content [1,5,13]. There is a lack of study on the ki for high-sulfur steel. The relations between the ki and the composition, temperature, and solidified phase are not clear. Therefore, in the present study, the ki (i=C, Si, Mn, P, and S) for the 1215 steel, which is a kind of environment-friendly high-sulfur free-cutting steels and is being wildly used in industries [27], were investigated by thermodynamic calculating. The influence of phase transition, MnS precipitation, and temperature on the ki were discussed. Moreover, the quantitative effects of composition fluctuation in 1215 steel on the phase transition (or solidified phase) and ki were analyzed by the orthogonal design method. Finally, the fitting formulas of ki that can be applied for the study of segregation in 1215 steel were obtained.

2Model of solute partition coefficients2.1Model description

If an alloy is in a thermodynamic equilibrium state during solidification, the ratio of the solute concentrations in the solid and liquid is called the solute equilibrium partition coefficient [10].

where ciS and ciL are the equilibrium concentrations of solute i in the solid and liquid, respectively. In steels, the partition coefficients of solutes C, Si, Mn, P, and S are generally less than 1 [30,32]. That is, these solute elements will diffuse from the solid into the liquid during solidification.

Based on the thermodynamic theory, for an alloy solidification, the equilibrium between the solid and liquid is evaluated by equating the chemical potential, which is described by the Gibbs free energy employed for each phase [33,34]. This is also the theory that is applied to determine the equilibrium phase diagram [35].

where μi,0L and μi,0S are the standard chemical potentials of solute i in the liquid and solid, respectively. αiL and αiS are the solute activity in the liquid and solid, respectively. R is the gas constant, and T is the equilibrium temperature.

The definition of the solute activity is as follows.

where γiL and γiS are the solute activity coefficients in the liquid and solid, respectively, which can reflect the interaction between the solute elements in an alloy. The γi is defined as follows.
where γi,o is the activity coefficient of solute i in the infinite diluent solution, and εii,εij,..,εin are the activity interaction coefficients.

Combining Eqs. (1)–(4), the ki is as follows.

It can be found from Eq. (5) that the ki is primarily affected by two factors, i.e.,

The factor of X1 is determined by the characteristic of the element i itself. The factor of X2 represents the interaction between the solute elements in the alloy, which is mainly affected by the solute compositions. Kagawa et al. [14,15] has investigated the quantitative effects of X1 and X2 on the ki for various Fe-C-based and Ni-based alloys solidification, and results showed that the ki in steels was primarily determined by the factor of X1.

It should be pointed out that the molten steel (L) can solidify into δ, γ or δ+γ phase, which is determined by the phase transition at different stages of solidification. Usually, the ki between the L/δ is different from that between the L/γ. Thus, Eqs. (8) and (9) are applied to estimate the ki between the solid/liquid phase during solidification [22,32].

where kiL/S, kiL/δ, and kiL/γ are the ki between the solid/liquid (S/L), δ/L, and γ/L, respectively. mδ and mγ (0 ≤ mδ ≤ 1 and 0 ≤ mγ ≤ 1) are the mass fractions of the δ and γ phases in the solidified phase at the equilibrium temperature, respectively. For steel solidification, if the phase transition is L→δ, that is, the liquid-solid coexistent phase is L+δ (mδ = 1 and mγ = 0). Thus, the solute will only redistribute between the δ/L. So, kiL/S=kiL/δ.  On the contrary, if the phase transition is L→γ, thus, mδ = 0 and mγ = 1. So, kiL/S=kiL/γ. Nevertheless, more importantly, if the phase transition is a peritectic reaction (L+δ→γ), the liquid-solid coexistent phase will consist of L, δ, and γ phases. The solute will redistribute both in the δ/L and γ/L. Thus, the kiL/S(=kiL/δ+γ) is comprehensively determined by the kiL/δ and kiL/γ, as shown in Eq. (8).

In the present study, the ki was calculated based on the equilibrium of the chemical potentials (or Gibbs free energy) between the solid and liquid phases, which depended on the solute concentrations at the solidification interface. The solutions of the solute concentrations were carried out by the ‘Equilib’ module in the thermodynamic software of FactSage 6.3, which is based on the minimization of Gibbs free energy. The database of ‘FSstel’ was adopted. That is, the solute equilibrium concentrations in the liquid and solid phases were firstly estimated by the ‘Equilib’ module if a specific solidification system was in equilibrium at a certain temperature. Then, the ki was determined by Eqs. (1), (8) and (9). Finally, the quantitative ki in terms of solidified phase (δ, γ or δ+γ) and temperature for specific steel could be evaluated. The theory and method in calculating the ki were described in detail in the previous works [30,36].

2.2Model verification

To verify the feasibility of the present solute partition coefficient model, the ki for various Fe-C based alloys solidification were calculated, and the calculated results were compared with the experimental and calculated results in the literature, as shown in Figs. 1 and 2. Fig. 1 illustrates that the calculated kC, kMn, and kSi are in good agreement with the experimental results [14,16]. The deviations between the calculated and experimental kC for the Fe-3.01C-2.20Si ternary and Fe-1.20C-2.26Si-1.06Mn quaternary alloys (in wt%) are −3.1 and 0.7%, respectively. The deviations between the calculated and experimental kMn for the Fe-2.29C-2.07Si-0.90Mn and Fe-0.72C-1.83Si-1.06Mn quaternary alloys (in wt%) are also small, which are 3.2 and 2.6%, respectively. Besides, the deviation of the calculated kSi for the Fe-0.76C-2.82Si alloy (in wt%) is only −1.8%. It should be noted that the above Fe-C based alloys (in Fig. 1(a)) would all solidify into the γ phase during solidification because of the high C content. But, for Fe-0.29C-3.10Si-0.62Mn (in wt%) alloy, it is hyper-peritectic steel, which will solidify into δ phase first and then γ phase. The calculated kC and kSi for it are both in qualitative agreement with the measured results [14], although the measurements were carried out between the L/γ (at T=1673K), as shown in Fig. 1(b).

Fig. 1.

Verification of the calculated solute partition coefficients with the measured results by (a) Kagawa et al. [14] and Ocansey et al. [16], and (b) Kagawa et al. [14].

(0.39MB).
Fig. 2.

Verification of the calculated solute partition coefficients with (a) the calculated results by You et al. [5], and (b) the values of ki that are usually applied in the study of segregation.

(0.33MB).

Fig. 2(a) shows that both in the L/δ and L/γ, the predicted kC, kSi, kMn, and kS for the steel of Fe-0.32C-0.37Si-1.54Mn-0.011P-0.015S (in wt%) are all fully in accordance with the calculated results by You et al. [5]. Moreover, compared to the fixed values of the ki that are usually applied in macro- and micro-segregation prediction for various hypo- and hyper-peritectic steels (the molten steel will solidify into δ phase first) [1,23,26,37–42], the average values of the ki calculated by the present model are in a reasonable range, as shown in Fig. 2(b). The comparison results of Figs. 1 and 2 indicate that the ki estimated by the present model matches well with the experimental and calculated results for various steels, whether on the L/δ or on the L/γ phase interface. So, it can be concluded that the present model is accurate and reliable to investigate the ki for the Fe-C based alloy solidification.

3Orthogonal design of chemical compositions in 1215 steel

For steel solidification, the ki is affected not only by temperature but also by chemical composition [17,18]. In actual production, the composition in steel usually fluctuates within a certain range [43]. In 1215 high-sulfur steel, the composition range are as follows [27,44]: 0.05–0.09% C, <0.1% Si, 1.0–1.4% Mn, 0.05–0.09% P, and 0.25–0.45% S (in wt%). The fluctuation of composition in 1215 steel will cause a change in ki and increase the complexity to determine the ki. Therefore, it is necessary to establish approximate and unified ki for the whole composition range of 1215 steel. This is meaningful for the study of segregation in 1215 steel. Thus, to ensure the applicability and scientificity of the unified ki for 1215 steel, the orthogonal design method was applied to determine the composition of calculating sample within the composition range of 1215 steel.

The common solute elements of C, Si, Mn, P, S, and Fe were considered in the 1215 steel. So, the orthogonal design of 6 factors and 5 levels was adopted, as shown in Table 1. The detailed composition for each calculating sample is listed in Table 2, a total of 25 calculating samples. Then, the composition of each calculating sample was considered as the composition in 1215 steel to calculate the ki. Ultimately, based on the calculated ki of all the 25 calculating samples, the unified and quantitative ki for 1215 steel solidification were fitted. Besides, the effect of composition fluctuation on the ki was discussed.

Table 1.

The 6 factors and 5 levels of orthogonal design to determine the sample composition in 1215 steel (in wt%).

Levels numbersFactors
Si  Mn  Fe 
0.05  0.01  1.0  0.05  0.25  – 
0.06  0.03  1.1  0.06  0.30  – 
0.07  0.05  1.2  0.07  0.35  – 
0.08  0.07  1.3  0.08  0.40  – 
0.09  0.09  1.4  0.09  0.45  – 
Table 2.

The composition in 1215 steel of the calculating samples for calculating the solute partition coefficients (in wt%).

Sample numbersSolute elementsSample numbersSolute elements
Si  Mn  Fe  Si  Mn  Fe 
1  0.05  0.01  1.0  0.05  0.25  –  14  0.07  0.07  1.0  0.07  0.45  – 
2  0.05  0.03  1.1  0.06  0.30  –  15  0.07  0.09  1.1  0.08  0.25  – 
3  0.05  0.05  1.2  0.07  0.35  –  16  0.08  0.01  1.3  0.06  0.45  – 
4  0.05  0.07  1.3  0.08  0.40  –  17  0.08  0.03  1.4  0.07  0.25  – 
5  0.05  0.09  1.4  0.09  0.45  –  18  0.08  0.05  1.0  0.08  0.3  – 
6  0.06  0.01  1.1  0.07  0.40  –  19  0.08  0.07  1.1  0.09  0.35  – 
7  0.06  0.03  1.2  0.08  0.45  –  20  0.08  0.09  1.2  0.05  0.4  – 
8  0.06  0.05  1.3  0.09  0.25  –  21  0.09  0.01  1.4  0.08  0.35  – 
9  0.06  0.07  1.4  0.05  0.30  –  22  0.09  0.03  1.0  0.09  0.4  – 
10  0.06  0.09  1.0  0.06  0.35  –  23  0.09  0.05  1.1  0.05  0.45  – 
11  0.07  0.01  1.2  0.09  0.30  –  24  0.09  0.07  1.2  0.06  0.25  – 
12  0.07  0.03  1.3  0.05  0.35  –  25  0.09  0.09  1.3  0.07  0.3  – 
13  0.07  0.05  1.4  0.06  0.40  –               
4Result and discussion4.1Solidification path and phase transition of 1215 steel

The ki is influenced by the phase composition during steel solidification. That is, the ki between the L/δ is usually different from that between the L/γ [5,22]. For steel solidification, the phase transition is determined by the solute composition. So, the composition fluctuation in steel will lead to the difference in the solidified phase, as well as the ki. Fig. 3 shows the phase diagram of 1215 high-sulfur steel solidification. Within the range of the C content in 1215 steel (corresponding to 0.05–0.09wt% C), there are two solidification paths. Firstly, within the interval of 0.05–0.07wt% C, there is only L→δ phase transition during solidification. That is, the solidified phase is only the δ phase and the mushy zone (solid-liquid coexistent phase) is consists of L and δ phases. So, the solute will only redistribute between the L/δ. Secondly, when 0.07 ≤ [C] < 0.09wt%, the phase transitions of L→δ and L+δ→γ occur successively during solidification, and the mushy zone is successively dominated by the coexistent phases of L+δ and L+δ+γ. This means that after the peritectic reaction (L+δ→γ), the ki is comprehensively determined by the solute redistribution in the L/δ and L/γ, as shown in Eqs. (8) and (9). But, the peritectic reaction occurs only at the last stage of solidification, and the existed temperature interval of the L+δ+γ coexistent phase is small. This indicates that for 1215 steel solidification, the mushy zone is mainly dominated by the L+δ coexistent phase. In other words, the solutes primarily redistribute between the L/δ. Besides, there is no L+γ coexistent phase during solidification for the whole composition range of 1215 steel. Therefore, in the present study, the ki on the L/δ and L/δ+γ phase interfaces were classified and investigated for 1215 steel solidification.

Fig. 3.

The phase diagram for the solidification of 1215 high-sulfur steel.

(0.17MB).
4.2Effects of phase composition and temperature on ki4.2.1ki on L/δ phase interface

For the whole composition range of 1215 steel, the solute mainly redistributed between the L/δ, although the solute would redistribute between the L/δ+γ after the peritectic reaction for the steel with 0.07–0.09wt% C. According to the composition of the calculating samples in Table 2, the ki (i = C, Si, Mn, P, and S) on the L/δ phase interface for the 1215 steel with 0.05–0.09wt% C were explored, as shown in Figs. 4 and 5. Figs. 4(a–c) indicate that for all the 25 calculating samples, with the decreasing temperature, the kSi and kP increase monotonously while the kMn decreases monotonously. From the beginning to the ending of the L→δ phase transition (corresponding to 1790–1730K), the kSi, kMn, and kP are in the range of 0.58–0.76, 0.73–0.65, and 0.32–0.34, respectively. The change percentage for the kSi, kMn, and kP are 31.5, −11.0, and 7.1%, respectively. The temperature has a greater effect on the kSi. Besides, the fluctuation of composition in 1215 steel can affect the temperature range of the L→δ phase transition, thus leading to the difference in the range of the ki for different calculating samples. Such effect of composition fluctuation is more significant at the ending of the L→δ phase transition, especially for kSi.

Fig. 4.

The partition coefficients of solutes Si, Mn, and P on the L/δ phase interface for 1215 steel solidification. (a), (b), and (c) are the calculation results of the kSi, kMn, and kP, and (d), (e), and (f) are the fitting results of the kSi, kMn, and kP for all the 25 calculating samples, respectively.

(0.5MB).
Fig. 5.

The partition coefficients of solutes C and S on the L/δ phase interface for 1215 steel solidification. (a) and (b) are the calculation results of the kC and kS, and (c) and (d) are the fitting results of the kC and kS for all the 25 calculating samples, respectively.

(0.65MB).

Based on the calculated ki of all the 25 calculating samples, the relations between the ki (i=Si, Mn and P) and temperature for 1215 steel solidification were fitted, as shown in Figs. 4(d–f). The fitting formulas are as follows.

Fig. 5 presents the change rules of the kC and kS with temperature. Overall, the effect of temperature on the kC is very small, and the kC remains approximately unchanged in the initial stage of solidification (1790–1770K). But, the relation between the kC and temperature changes suddenly at a certain temperature (around 1770–1765K). Then, the kC gradually increases with the decreasing temperature. In fact, there is a turning point for the kC during solidification, as shown in Fig. 5(a). As with kC, there is also a turning point for the kS. As the temperature decreases, the kS decreases first and then increases, as shown in Fig. 5(b). On the whole, the kC and kS are in the range of 0.147–0.153 and 0.038–0.035, respectively. The kS is very small relative to 1, indicating that the solute S is an easily segregated alloy element. From the beginning to the ending of the L→δ phase transition, the change percentage for the kC and kS are only 2.76 and −3.70%, respectively. This means that on the L/δ phase interface, the changes of the kC and kS with temperature are small. The fitted relations between the ki (i=C and S) and temperature for 1215 steel solidification are as follows.

In brief, on the L/δ phase interface, with the decreasing temperature, the kSi and kP increase monotonously, the kMn decreases monotonously, while the kC and kS first decrease and then increase. The change percentage of the ki are ranked as follows: kSi > kMn > kP > kS > kC. Furthermore, the kC and kS both have a turning point. This is mainly due to the precipitation of MnS inclusion during solidification. In 1215 steel, the S content is higher than that in the conventional steel, which is conducive to the precipitation of MnS inclusion [27]. On the other hand, the kS is small so that the solute S tends to segregate heavily into the liquid during solidification [21,22,26], which will increase the S concentration and promote the precipitation of MnS inclusion in the liquid. But, in turn, the precipitation of MnS inclusion can consume and reduce the concentrations of solutes Mn and S. That is, the concentrations of solutes Mn and S are comprehensively determined by the segregation and MnS precipitation. As a result, the kC and kS are influenced and appear a turning point once the MnS inclusion starts to precipitate during solidification, because the ki can be affected by the solute concentrations on the L/δ interface [14,30].

In the present study, the precipitation of MnS inclusion ([Mn]+[S]=(MnS)), which was based on the thermodynamic theory, was taken into account when calculating the equilibrium of 1215 steel solidification by the ‘Equilib’ module in FactSage 6.3. Fig. 6 shows the variations of the Mn and S concentrations in the liquid (cMnL and cSL), and the precipitation amount of MnS inclusion (cMnS) during the solidification of the 1215 steel with 0.25–0.45wt% S and 1.30wt% Mn (corresponding to the calculating samples of 4, 8, 12, 16 and 25 in Table 2). It can be found that, in the early stage of solidification, the cMnL and cSL both increase with the decreasing temperature due to the segregation of solutes Mn and S in the liquid phase, and there is no MnS precipitate. Then, the MnS inclusion begins to precipitate at a certain temperature (around 1766–1772K) as the cMnL and cSL increase to a certain extent, as shown in Fig. 6(a). But, after that, as the temperature decreases, the cMnL turns to decrease significantly for the steel with 0.25–0.45wt% S, while the cSL turns to increase slightly for the steel with 0.40–0.45wt% S and decrease slowly for the steel with 0.25–0.35wt% S (this is because the increasing effect on the S concentration by the S segregation fails to offset the reducing effect on the S concentration by the MnS precipitation for the steel with a lower S content). In other words, there is a turning point for the cMnL and cSL once the MnS inclusion starts to precipitate. The segregation has an increasing effect while the precipitation of MnS inclusion has a decreasing effect on the cMnL and cSL. So, before the MnS precipitation, the cMnL and cSL are only influenced by the segregation, resulting in both the cMnL and cSL increase monotonously. But, after the MnS precipitation, the cMnL and cSL are comprehensively determined by the segregation and MnS precipitation, leading to a turning point for the cMnL and cSL. It should be pointed out that in the 1215 steel, the S content is excessive relative to the Mn content for the MnS precipitation. This is why the influence of the precipitation of MnS inclusion on the cMnL and cSL is different.

Fig. 6.

The predicted Mn and S equilibrium concentrations in the liquid phase, as well as the MnS precipitation for 1215 steel solidification by phase diagram. (a) MnS, (b) Mn, and (c) S.

(0.33MB).

Fig. 6 proves that the segregation of solutes Mn and S promote the precipitation of MnS inclusion during solidification. But, in turn, the MnS precipitation will consume and reduce the Mn and S concentrations. Finally, the change trends of the concentrations of solutes Mn and S show a turning point. More importantly, the ki is influenced by the solute concentrations [13–15]. Hence, the turning of the Mn and S concentrations, which are caused by the MnS precipitation, will eventually have some effect on the ki. This is the reason why there is a turning point for the kC and kS during solidification. Moreover, Fig. 7 presents that the precipitation starting temperature of MnS inclusion, and the turning point temperature of the cMnL, cSL, kC and kS for each calculating sample are equal. It can be explained that once the MnS inclusion starts to precipitate during solidification, the concentrations of solutes Mn and S are influenced directly, leading to the change in kC and kS. Therefore, it is concluded that the turning of the kC and kS are fundamentally caused by the MnS precipitation. The influence mechanism has been discussed and analyzed in detail in the previous works [30]. It must be noted that the influencing degree of the solute concentration on each ki is different [14,15]. So, only the kC and kS present a pronounced turning.

Fig. 7.

The precipitation starting temperature of MnS inclusion, and the turning point temperature of cMnL, cSL, kC, and kS for all the 25 calculating samples.

(0.12MB).

As mentioned above, the calculating of the ki and MnS precipitation were based on the equilibrium of the chemical potential (or Gibbs free energy) between the solid and liquid for steel solidification. This is also the theory that is usually applied to determine the phase diagram. But, as known to all, the phase diagram (or lever rule) usually fails to properly predict the segregation and inclusion precipitation. In view of this, the precipitation of MnS inclusion in 1215 steel was also investigated by a coupling model of microsegregation and inclusion precipitation. The coupling model was based on the V–B model [1], wherein the changed ki in Figs. 4 and 5 were applied. More details about the coupling model are available in the publication [22]. The segregation and MnS precipitation predicted by the coupling model are shown in Fig. 8. It can be found that before the precipitation of MnS inclusion, the Mn and S concentrations increase gradually because of segregation. Then, the MnS inclusion starts to precipitate when the solid fraction (fs) are 0.52, 0.44, 0.38, 0.31, and 0.24 (corresponding to 1773, 1770, 1775, 1776, and 1775K) for the steels with 0.25, 0.30, 0.35, 0.40, and 0.45wt% S (all with 1.30wt% Mn), respectively. The higher the S content, the earlier the MnS inclusion starts to precipitate. In the 1215 steel, the MnS inclusion begins to precipitate at the middle or early stage of solidification because of the high S content. It is different from that for the steel with common content of S (the MnS inclusion usually precipitates near the complete solidification) [21,26]. After the precipitation of MnS inclusion, the Mn concentration turns to decrease gradually, while the S concentration keeps increasing but with a weaker amplitude (the S content is excessive for MnS precipitation). In other words, the change trends of the Mn and S concentrations both exhibit a turning point once the MnS inclusion starts to precipitate, which are the same as the results in Fig. 6. Besides, the Mn and S concentrations predicted by the coupling model (in Fig. 8) are higher than that predicted by the phase diagram (in Fig. 6), respectively. The predicted precipitation starting temperature of MnS inclusion in Fig. 6 are 1769, 1766, 1771, 1772, and 1771K for the steels with 0.25, 0.30, 0.35, 0.40, and 0.45wt% S, which are all about 4K lower than that predicted by the coupling model (in Fig. 8), respectively. This is because the phase diagram (or lever rule) assumes that the solute is complete diffusion while the coupling model (based on V–B model) supposes that the solute is finite diffusion in the solid during solidification. So, in theory, the solute at the solidification front will segregate more pronouncedly from the solid into the liquid for the coupling model than that for the phase diagram. As a result, the predicted solute concentrations in the liquid are higher and the MnS inclusion precipitates at a higher temperature for the coupling model. But on the whole, both the coupling model and phase diagram can approximately estimate the segregation and MnS precipitation for the 1215 high-sulfur steel solidification, although there are differences between the predicted results of the two methods.

Fig. 8.

The predicted Mn and S concentrations, as well as the MnS precipitation for 1215 steel solidification by the coupling model of microsegregation and inclusion precipitation. (a) MnS, (b) Mn, and (c) S.

(0.44MB).
4.2.2ki on L/δ+γ phase interface

There is no peritectic reaction for the solidification of the 1215 steel with less than 0.07wt% C, as shown in Fig. 3. So, only for the 1215 steel with 0.07–0.09wt% C, the solute redistribution between the L/δ+γ should be considered. Figs. 9 and 10, present the ki for the 1215 steel with 0.07 and 0.09wt% C (corresponding to the calculating samples of ‘14’ and ‘22’ in Table 2), respectively. For the solidification of the 1215 steel with 0.07wt% C, as shown in Fig. 9, the temperature of liquidus, solidus, and peritectic reaction (TL, TS, and TP) are 1782, 1726, and 1729K, respectively. In the L+δ coexistent phase, that is, before the peritectic reaction (corresponding to 1782-1729K), the solutes only redistribute between the L/δ (kiL/S=kiL/δ). With the decreasing temperature, the kSiL/δ, kPL/δ, and kCL/δ increase from 0.619 to 0.795, from 0.325 to 0.343, and from 0.150 to 0.154, while the kMnL/δ and kSL/δ decrease from 0.710 to 0.634 and from 0.037 to 0.035, respectively. The change of the kiL/δ with temperature is more significant for solutes Si and Mn. Then, the peritectic reaction occurs at 1729K, and the mushy zone transfers from the coexistent phases of L+δ into L+δ+γ. But, the temperature interval of the ‘TP - TS’ is only 3K, which is very small relative to the ‘TL - TS=56K’. It means that the solute redistribution between the L/δ+γ lasts for only a short time. In the L+δ+γ coexistent phase, the kCL/γ, kMnL/γ, kSiL/γ,kPL/γ, and kSL/γ are around 0.317, 0.678, 0.763, 0.174, and 0.016, while the kCL/δ, kMnL/δ, kSiL/δ, kPL/δ, and kSL/δ are around 0.154, 0.631, 0.807, 0.344, and 0.035, respectively. The ki between the L/γ is larger than the ki between the L/δ for solutes C and Mn, while the opposite results are found for solutes Si, P, and S. It suggests that the existence of γ phase is conducive to the increase of the kCL/S and kMnL/S, but has a decreasing effect on the kSiL/S, kPL/S, and kSL/S. Furthermore, the discrepancy between the average values of the kiL/γ and kiL/δ (in the L+δ+γ coexistent phase) is more significant for solutes C and P, as shown in Fig. 11, indicating that the phase composition has a greater effect on the ki of solutes C and P. On the L/δ+γ phase interface, the ki is comprehensively determined by the kiL/δ and kiL/γ, as shown in Eqs. (8) and (9). From TP to TS (corresponding to 1729–1726K), the kCL/δ+γ, kSiL/δ+γ, and kMnL/δ+γ increase from 0.154 to 0.178, from 0.795 to 0.809, and from 0.633 to 0.634, while the kPL/δ+γ and kSL/δ+γ decrease from 0.343 to 0.320 and from 0.035 to 0.032, respectively. The change of the kiL/δ+γ with temperature is small, especially for solutes Mn and S. Besides, the kiL/δ+γ for each solute is closer to the kiL/δ comparing to the kiL/γ, which illustrates that the effect of kiL/γ on the kiL/δ+γ is weak. This is because the mass fraction of γ phase (mγ) in the solidified phase (consisting of δ+γ) is only 0.14 even at the end of solidification. In other words, the solidified phase is still primarily dominated by the δ phase after the peritectic reaction. So, the kiL/δ+γ is mainly determined by the kiL/δ.

Fig. 9.

The variation of the ki for the solidification of the steel with 0.07wt% C (corresponding to the calculating sample ‘14’ in Table 2). The mushy zone are dominated successively by the coexistent phases of L+δ (1782 > T > 1729K) and L+δ+γ (1729 ≥ T > 1726K). (a) C and Mn, and (b) Si, P, and S.

(0.4MB).
Fig. 10.

The variation of the ki for the solidification of the steel with 0.09wt% C (corresponding to the calculating sample ‘22’ in Table 2). The mushy zone are dominated successively by the coexistent phases of L+δ (1782 > T > 1733K) and L+δ+γ (1733 ≥ T > 1727K). (a) C and Mn, and (b) Si, P, and S.

(0.41MB).
Fig. 11.

The average values of the kiL/γ and kiL/δ in the L+δ+γ coexistent phase that after the peritectic reaction for the solidification of the steel with 0.07 and 0.09wt% C.

(0.07MB).

For the solidification of the steel with 0.09wt% C, as shown in Fig. 10, the TL, TS, and TP are 1782, 1727, and 1733K, respectively. The ‘TP - TS=6K’ is larger than that for the steel with 0.07wt% C, but is still small relative to the ‘TL - TS=55K’. In the L+δ coexistent phase, the kiL/δ (i = C, Si, Mn, P, and S) are almost the same as that in the steel with 0.07wt% C, respectively. In the L+δ+γ coexistent phase (corresponding to 1733–1727K), as the temperature decreases, the kCL/δ+γ, kSiL/δ+γ, and kMnL/δ+γ increase from 0.153 to 0.210, from 0.764 to 0.784, and from 0.650 to 0.659, while the kPL/δ+γ and kSL/δ+γ decrease from 0.324 to 0.285 and from 0.036 to 0.029, respectively. The discrepancy in the kiL/δ+γ for the steels with 0.09 and 0.07wt% C is not significant, especially for solutes Si, Mn, and S. This is because the maximum mγ for the solidification of the 1215 steel with 0.09wt% C is 0.34. So, the kiL/δ+γ for the 1215 steel with 0.09wt% C is primarily determined by the kiL/δ, which is the same as that for the 1215 steel with 0.07wt% C. Besides, in the L+δ+γ coexistent phase, both the average values of the kiL/δ and kiL/γ (i = C, Si, Mn, P, and S) for the steel with 0.09wt% C are almost the same as that for the steel with 0.07wt% C, respectively, as shown in Fig. 11. This indicates that the peritectic reaction has small effect on the kiL/δ and kiL/γ. But, the mγ and mδ are affected by the peritectic reaction, leading to the discrepancy in kiL/δ+γ for the steels with different C contents.

In the composition range of 1215 steel, the L+δ+γ coexistent phase mainly exists between 1745–1725K for the solidification of the steel with 0.07–0.09wt% C. The calculating and fitting results of the kiL/δ+γ (i = C, Si, Mn, P, and S) are shown in Fig. 12. The fitted relations between the kiL/δ+γ and temperature are as follows.

Fig. 12.

The calculated and fitted ki on the L/δ+γ phase interface for the 1215 steel with 0.07–0.09wt% C.

(0.19MB).

Therefore, for 1215 steel solidification, the Eqs. (10)–(12) can be used to estimate the kiL/δ (on the L/δ phase interface) for the whole composition range of 1215 steel. Moreover, if the C content is in the range of 0.07–0.09wt%, the Eq. (13) should be applied to determine the kiL/δ+γ (on the L/δ+γ phase interface) after the peritectic reaction.

4.3Effect of composition fluctuation on ki

Fig. 3 presents that the temperature interval of the L+δ+γ coexistent phase (‘TP - TS’) for the solidification of the 1215 steel with 0.07–0.09wt% C is less than 7K, which is small relative to the ‘TL - TS’ (around 50K). It is insignificant to discuss the effect of composition fluctuation on the kiL/δ+γ. Therefore, the emphasis was placed on the ki between the L/δ (kiL/δ) for orthogonal analysis. According to the result in Figs. 4 and 5, the solute average partition coefficient (kiAve, the average value of the ki acrosses the whole stage of L→δ phase transition) for each calculating sample was calculated, as shown in Fig. 13. It is found that the kCAve, kPAve, and kSAve are almost unchanged, which are around 0.150, 0.330, and 0.037, respectively. The kSAve and kCAve are too small relative to 1, indicating that the solutes S and C tend to severely segregate from the δ phase into the liquid during solidification. There is a certain difference in the kSiAve and kMnAve for different calculating samples, and the kSiAve and kMnAve mainly fluctuate in the range of 0.63–0.70 and 0.67–0.70, respectively. Overall, the kiAve can be ranked in order as follows: Mn>Si>P>C>S. This order and the range of the ki are consistent with the previous works [1,5,45]. Besides, the minimum, maximum, and the percentage difference between the minimum and maximum (denoted by ‘PD’) of the kiAve in the 25 calculating samples are presented in Fig. 13(f). The results show that the PD for the kPAve, kCAve,  kMnAve, and kSAve are all less than 5%, while the PD for the kSiAve is 10.5%. Furthermore, the difference between the maximum and minimum of the kiAve is less than 0.075 for all the solutes Si, Mn, P, C, and S, showing that the kiAve can only change within a small range due to the composition fluctuation in 1215 steel.

Fig. 13.

The average partition coefficients of solutes (kiAve) for 1215 steel solidification that under the composition of the 25 calculating samples. (a) kMnAve, (b) kSiAve, (c) kPAve, (d) kPAve, (e) kSAve, and (f) the maximum, minimum, and the percentage difference between the minimum and maximum of the kiAve.

(0.49MB).

Theoretically, the ki is mainly determined by two factors (X1 and X2), as shown in Eqs. (6) and (7). Kagawa et al. [14] pointed out that it is impossible to reduce the segregation in Fe-C-based system by the addition of suitable solute elements, because the changes on the ki are small at low levels of these elements. So, the factor X2 has small effect on the ki, and the ki is primarily determined by the factor X1. In other words, the ki mainly depends on the thermodynamic properties of the solute i itself. In general, the physicochemical properties of the solute elements in the steel are difficult to be changed due to the low content of these elements, leading to the small change in the ki. This is why the composition fluctuation in 1215 steel has small effect on the ki, and the order of the ki is as follows: Mn>Si>P>C>S.

The effects of solute contents in 1215 steel on the kiAve were investigated by orthogonal analysis method, as shown in Figs. 14 and 15. Fig. 14 presents that the effects of solutes C, Si, Mn, P, and S on the kiAve are different. Comparatively speaking, the solutes Mn and S have a greater effect on all the kiAve (i = C, Si, Mn, P, and S), which are almost linear relationships. As the Mn content increases, the kMnAve and kSAve increase monotonously while the kCAve, kSiAve, and kPAve decrease monotonously. But, with the increasing S content, an opposite effect is found for all the kiAve of solutes C, Si, Mn, P, and S. Besides, the C content has some effect on the kSAve, kPAve, kMnAve, and kSiAve, but has small effect on the kCAve. With the increasing Si content, the kMnAve decreases monotonously while the other kiAve are almost unchanged. The P content has small effect on all the kiAve. The fitting relations between the kiAve and solute contents, which can be used for the prediction of segregation in 1215 steel, are as follows.

Fig. 14.

The effects of composition fluctuation on the average partition coefficients of solutes Mn, Si, P, C, and S for 1215 steel solidification.

(0.49MB).
Fig. 15.

The range analysis and variance analysis of the average partition coefficients of solutes Mn, Si, P, C, and S for 1215 steel solidification. (a) Range analysis, and (b) variance analysis (F-test).

(0.29MB).

The range analysis and variance analysis of orthogonal design were applied to investigate the effects of solute contents on the kiAve. According to the range (R value) of the kiAve, as shown in Fig. 15(a), the fluctuation of the Mn and S contents in 1215 steel has the largest effect on all the kiAve, and next comes the C content, followed by the Si and P contents. This is consistent with the results in Fig. 14. Besides, the range (R value) for the kSiAve, kMnAve, kPAve, kCAve, and kSAve are all less than 0.04, illustrating that the influence of composition fluctuation in 1215 steel on the kiAve is small, as discussed in Fig. 13. As with the range analysis, the variance analysis for the kiAve (in Fig. 15(b)) also illustrates that the effects of solute contents on the kiAve are ranked as follows: Mn>S>C>Si>P. For 1215 high-sulfur steel solidification, the Mn and S concentrations in the liquid or solid are determined not only by the segregation but also by the precipitation of MnS inclusion. This is different from the concentrations of solutes C, Si, and P, which are intuitively only influenced by the segregation. As a result, the kiAve is comparatively more sensitive to the fluctuation of Mn and S contents in the 1215 steel, because the kiAve is fundamentally affected by the solute concentrations in the liquid and solid during solidification. So, more attention should be paid on the influence of Mn and S contents on the ki for 1215 steel solidification.

5Conclusions

Based on the thermodynamic equilibrium of steel solidification, the quantitative ki for 1215 high-sulfur steel solidification, considering the influences of composition fluctuation, temperature, phase transition, and MnS precipitation, were investigated by orthogonal design method. Conclusions are obtained as follows.

  • (1)

    There were two solidification paths for 1215 steel. Within the interval of 0.05–0.07wt% C, there was only L→δ phase transition during solidification. When 0.07 ≤ [C] < 0.09wt%, the L→δ phase transition occurred first, followed by the peritectic reaction (L+δ→γ) at the last stage of solidification. Therefore, the ki on the L/δ and L/δ+γ phase interfaces were divided and investigated.

  • (2)

    With the decreasing temperature, the kSi and kP on the L/δ phase interface increased monotonously, the kMn decreased monotonously, while the kC and kS decreased first before increasing because of MnS precipitation. The kSi, kMn, kP, kS, and kC were in the range of 0.58–0.76, 0.73–0.65, 0.32–0.34, 0.038–0.035, and 0.147–0.153, respectively. The effect of temperature on the kSi and kMn was more significant, while on the kP, kC, and kS was small.

  • (3)

    In the L+δ+γ coexistent phase (corresponding to 0.07–0.09wt% C), the kiL/γ was greater than the kiL/δ for solutes C and Mn, while the opposite results were found for solutes Si, P, and S. The discrepancy between the kiL/γ and kiL/δ was more significant for solutes C and P. On the L/δ+γ phase interface, the kiL/δ+γ was more closer to the kiL/δ than the kiL/γ for each solute, because the mass fraction of γ phase (mγ) in the solidified phase (consisting of δ+γ) was relatively small. The effects of peritectic reaction on the kiL/δ and kiL/γ were both insignificant. But, the mγ and mδ were affected by the peritectic reaction, leading to the discrepancy in kiL/δ+γ for the steels with different C contents.

  • (4)

    The composition fluctuation in 1215 steel has small effect on the kiAve (i=C, Si, Mn, P, and S). The Mn and S contents have the largest effect on all the kiAve, followed by C content. Si and P contents have the least effect. More attention should be paid on the fluctuation of Mn and S contents for 1215 steel solidification.

  • (5)

    The formulas for the ki, which can be applied for the prediction of segregation in 1215 steel, were fitted in terms of temperature, phase composition, and solute content.

Conflict of interest

The authors declared that they have no conflicts of interest to the submitted manuscript entitled “Thermodynamic study on the solute partition coefficients on L/δ and L/δ+γ phase interfaces for 1215 high-sulfur steel solidification by orthogonal design”.

Acknowledgements

The work is financially supported by the National Natural Science Foundation of China (NSFC, Project No. 51874059, U1960113 and 51874060). The authors would like to thank the support by the Natural Science Foundation of Chongqing (Project No. cstc2018jcyjAX0647, cstc2018jszx-cyzdX0076), and the support by the Fundamental Research Funds for the Central Universities of China (Project No. 2019CDXYCL0031).

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Journal of Materials Research and Technology

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