*3.1. Segregation Models*

The above results shown in Figure 1 are based on the uniform distributions for elements N and Ti in molten steel without considering micro-segregation. However, due to the decrease in solubility of solute elements N and Ti during the solidification process, micro-segregation will occur inevitability, which will further lead to the precipitation of TiN. With respect to this issue, several micro-segregation models were proposed to describe the concentration changes of solute elements as a function of the solid fraction, as listed in Table 4 [14].


**Table 4.** Micro segregation models for solute elements during the solidification process [14].

where *w*[*i*] and *w*<sup>0</sup> [*i*] denote the instantaneous and initial concentration of solute elements (N and Ti) in the liquid phase zone during the solidification process, respectively; *ki* is the equilibrium distribution coefficient between liquid and γ-Fe phase, herein *k*<sup>C</sup> = 0.34, *k*<sup>N</sup> = 0.48, and *k*Ti = 0.30 [18–20]; *g* represents the solid fraction; φ (in the range of 0–1) denotes the inverse diffusion coefficient and α is the Fourier parameter.

#### *3.2. Usage of the LRSM Model*

As shown in Table 4, it can be seen that the Lever-rule model is obtained based on the assumption that solute elements are completely diffused in both liquid and γ-Fe phases; however, the Scheil model neglects such diffusion in the γ-Fe phase, which means the solute elements are completely diffused in liquid and have no diffusion in the γ-Fe phase. Due to the fact that the diffusion coefficient of N is much larger than that of Ti in the γ-Fe phase, as comparison of Equations (18) and (19) indicates [18,21], and more obviously supported by Figure 2, so, it is reasonable to assume that solute element N is completely diffused in the γ-Fe phase and the diffusion in the γ-Fe phase for Ti is neglected. That is to say, the Lever-rule model is applied for the N and Scheil model for Ti. In the current paper, this model combination was named as the LRSM model. Then the corresponding concentration expressions of solute elements N and Ti can be described by Equations (20) and (21), respectively.

$$D\_{\rm N}^{\prime\prime} = 0.91 \exp(-168600/RT) \tag{18}$$

$$D\_{\rm Ti}^{\gamma} = 0.15 \exp(-250000/RT) \tag{19}$$

$$w\_{\rm [N]}^{\rm act} = \frac{w\_{\rm [N]}^0}{1 - (1 - k\_{\rm N})g} \tag{20}$$

$$w\_{\rm [Ti]}^{\rm act} = w\_{\rm [Ti]}^0 (1 - \lg)^{k\_{\rm Ti} - 1} \tag{21}$$

**Figure 2.** Diffusion coefficients of solute elements N and Ti in the γ-Fe phase at different temperatures.

Prior to analyzing the solidification process of molten steel, the actual solubility products of N and Ti should be calculated, which can be considered as *Q*act <sup>3</sup> (*Q*act <sup>3</sup> <sup>=</sup> *<sup>w</sup>*act [N] · *<sup>w</sup>*act [Ti] ), as shown by Equation (22),

$$Q\_3^{\rm act} = \frac{(1 - g)^{k\_{\rm Ti} - 1}}{1 - (1 - k\_{\rm N})g} \cdot w\_{[\rm N]}^0 \cdot w\_{[\rm Ti]}^0 \tag{22}$$

In addition, the relationship between solidification front temperature (*T*L-S) and solid fraction (*g*) can be expressed by Equation (23) [22],

$$T\_{\rm L-S} = T\_{\rm Fe} - \frac{T\_{\rm Fe} - T\_{\rm L}}{1 - g\frac{T\_{\rm L} - T\_{\rm S}}{T\_{\rm Fe} - T\_{\rm S}}} \tag{23}$$

At the same time, by substituting Equation (23) into Equation (17), the relationship between lg*K*equ <sup>3</sup> and solid fraction (*g*) can also be obtained. As is well known, if the actual solubility product reaches the equilibrium value (or lg*Q*act <sup>3</sup> <sup>≥</sup> lg*K*equ <sup>3</sup> ), TiN will precipitate. The values of lg*K*equ <sup>3</sup> and lg*Q*act 3 (calculated by LRSM model) are depicted in Figure 3, from which it can be easily seen that TiN will only precipitate at the very late stage of the solidification process, with a solid fraction bigger than 0.9966. When substituting 0.9966 into Equation (23), the solidification front temperature (*T*L-S = 1637 K) can be easily deduced, which is almost the same as the theoretical solidus temperature (*T*<sup>S</sup> = 1636 K) of the studied tire cord steel. This result suggests that TiN will not precipitate in the mushy zone until nearly close to complete solidification.

#### *3.3. Usage of Ohnaka Model on Considering the E*ff*ect of Carbon on SDAS L*

The above analytical results (Figure 3) were obtained based on the assumption that N is completely diffused and Ti has no diffusion in the γ-Fe phase. In fact, both N and Ti would diffuse to some extent in the γ-Fe phase, as shown in Equations (18) and (19), respectively.

**Figure 3.** Comparison of equilibrium solubility product with the calculated value obtained by the LRSM (Lever-rule model was applied for the N and Scheil model for Ti) model.

In order to achieve a more realistic representation on the precipitation of TiN in SWRH 92A tire cord steel, the finite diffusion of the solute elements in the γ-Fe phase are now considered. The basic concentration expression is shown in Equation (24)). Herein, it can be easily found that if φ equals to zero, Equation (24) will change into the Scheil model; while if φ equals to one, Equation (24) will change into the Lever-rule model, instead. Furthermore, if the Brody–Fleming model [23] is adopted (as seen in Equation (25)), φ will be not physically reasonable when the Fourier parameter α is bigger than 0.5; the Clyne–Kurz model [24] lacks the actual physical meaning for φ if Equation (26) is used. In order to solve those problems, Ohnaka [25] presented a simple modification of φ based on comparison with the approximate solution of the diffusion equation, as shown in Equation (27) [25], which showed better agreement with the experimental data of Matsumiya et al. [26] than did predictions using Equation (25). Therefore, the Ohnaka model was used in this paper. The Fourier parameter α involves the diffusion coefficient *D*<sup>γ</sup> *<sup>i</sup>* (cm2/s), SDAS *<sup>L</sup>* [cm, the unit of *<sup>L</sup>* was converted from <sup>μ</sup>m (calculated by Equations (29) and (31)) to cm for the calculation in Equation (28), corresponding to the unit of *D*<sup>γ</sup> *<sup>i</sup>* ], and the local solidification time τ (s), as seen in Equation (28) [27],

$$w\_{[i]} / w\_{[i]}^0 = [1 - (1 - \phi k\_i)g]^{(k\_i - 1) / (1 - \phi k\_i)} \tag{24}$$

$$
\phi = 2\alpha \tag{25}
$$

$$
\phi = 2\alpha (1 - e^{-\frac{1}{a}}) - e^{-\frac{1}{2\pi}} \tag{26}
$$

$$
\phi = 4\alpha/(1+4\alpha)\tag{27}
$$

$$a = \frac{4D\_i^\circ \pi}{L^2} \tag{28}$$

The expressions of *D*<sup>γ</sup> *<sup>i</sup>* for solute elements N and Ti are shown in Equations (18) and (19), respectively; SDAS *L* (herein, the unit of *L* calculated by Equation (29) was μm) is related with the cooling rate *R*<sup>C</sup> (K/s) and the carbon concentration *w*[C], as expressed by Equation (29) [28]; the local solidification time τ (s) was calculated by Equation (30) [21,29].

$$L = 143.9 \cdot R\_{\mathbb{C}}^{-0.3616} \cdot w\_{[\mathbb{C}]}^{\left(0.5501 - 1.996 \cdot w\_{[\mathbb{C}]}\right)} \left(w\_{[\mathbb{C}]}^{0} > 0.15\right) \tag{29}$$

$$
\pi = \frac{T\_L - T\_S}{R\_\odot} \tag{30}
$$

By substituting Equations (27)–(30) and the equilibrium distribution coefficients and diffusion coefficients of different solute elements into Equation (24), the corresponding segregation ratio of different solute elements during the solidification process can be calculated, as shown in Figure 4. From Figure 4 it can be seen that the segregation ratio of both N and Ti increased with the increasing solid fraction, and the final segregation ratios almost equaled each other although the cooling rates were different, which suggests that the effect of the cooling rate can be ignored. However, it can also be seen that the cooling rate has a certain effect on the intermediate segregation process for Ti, the faster the cooling rate, the larger the segregation ratio would be. As for N, however, the effect can be ignored.

**Figure 4.** Effects of cooling rates on the segregation ratios of solute elements N and Ti during the solidification process (the results obtained by the Lever-rule model are given as a reference).

φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ What is more, when comparing with the results obtained by the Lever-rule model, it is surprising to find that the final segregation ratio almost kept the same for both N and Ti. The possible reasons may be explained as follows. When the segregation of solute element carbon was considered, the concentration of carbon would increase during the solidification process (the plot was not given in this paper), then the value of SDAS *L* would decrease (according to Equation (29) in the current condition), which may accelerate the diffusion velocity of solute elements between the liquid and the γ-Fe phases and even arrive complete. Besides, when the cooling rate became slower, the diffusion of solute element between liquid and γ-Fe phases would be more complete due to the adequate time, as shown in Figure 5, which makes the inverse diffusion coefficient closer to one and the results are similar to the case obtained by the Lever-rule model. In addition, it can also be seen from Figure 5 that the inverse diffusion coefficients of N almost equaled one during the total solidification process at the different cooling rates, so the results were almost the same as those obtained by the Lever-rule model. As for Ti, the inverse diffusion coefficients gradually increased with increasing solid fraction. The slower the cooling rate, the larger the inverse diffusion coefficients would be. However, the final values almost equaled one for the four different cooling rates (0.1, 1, 10, and 100 K/s). That is to say, the Ohnaka model applied will change into the Lever-rule model at the end of the solidification process.

In addition, when the current Ohnaka model (considering the effect of carbon on SDAS *L*) was applied, precipitation of TiN during the solidification process would not happen because the value of the actual solubility product lg*Q*act <sup>3</sup> was much smaller than that of the equilibrium value lg*K*equ <sup>3</sup> , as can be seen in Figure 6. The current results clearly indicate that TiN cannot precipitate in the solid–liquid two-phase region (mushy zone).

**Figure 5.** Inverse diffusion coefficients of solute elements N and Ti with different cooling rates.

**Figure 6.** Comparisons of equilibrium solubility product with the calculated value obtained by the Ohnaka model (considering the effect of carbon on SDAS *L*).
