A Kinetic Model for Oxide–Carbonitride Inclusion Heterogeneous Nucleation and Precipitation during Superalloy Solidification

Abstract

Complex oxide–carbonitrides (MgO-Ti(CN), Al₂O₃-Ti(CN), and MgO·Al₂O₃-Ti(CN)) are the most common non-metallic inclusions presented in cast and wrought superalloys. In this work, a coupled kinetics model was proposed to predict the complex oxide–carbonitride inclusion’s precipitation behavior during the solidification of superalloys. This model takes into account thermodynamics, micro-segregation, heterogeneous nucleation in the inter-dendritic liquid, and growth controlled by the diffusion of solute elements and kinetics of interfacial reaction. The results demonstrated that both the cooling rate and nitrogen content take significant effects on the final size of complex oxide–carbonitride inclusions, as the former controls the total growth time and the latter determines the initial precipitation temperature. In comparison, the particle size of primary oxides shows a negligible impact on the final size of complex inclusions. The practice of an industrial vacuum arc remelting confirmed that the inclusion size variation predicted by the present model is reasonably consistent with the experimental results.

1. Introduction

Nickel-based superalloys exhibit an exceptional combination of high-temperature mechanical properties, fatigue resistance, and corrosion resistance, widely applied in aero-engine turbine components [1,2,3]. The production of components with required mechanical properties and free from metallurgical defects, requires advanced manufacturing processes. Vacuum arc remelting (VAR) is considered one of the most suitable melting processes and is typically used as the final step in the triple-melting process of Ni-based superalloys [4]. Due to the excellent cooling condition and vacuum environment, the VAR ingot generally exhibits a dense microstructure, little segregation, and exceptional cleanliness. Nevertheless, non-metallic inclusion contamination seems inevitable, particularly oxides, carbonitrides and oxide–carbonitride complex inclusions precipitated during solidification and may change the metallurgical properties of various superalloy grades. A typical risk is that these inclusions can cause stress concentrations under cyclic loading, thereby promoting crack initiation and leading to low cycle fatigue failure [5].

Similar to the goal of clean steel production, superalloy manufacturers also strive to achieve the harmlessness of non-metallic inclusions. Hence, extensive studies were conducted on inclusions in superalloys, including thermodynamic analysis [6], metallographic characterization [7], quantitative statistics [8], evolutionary behavior [9], and their detrimental effects on turbine disk serviceability [10]. From the reports in the literature, oxide-nitride complex inclusions, like MgO-Ti(CN), Al₂O₃-Ti(CN), MgO·Al₂O₃-Ti(CN), appear to be the most common class of inclusions in all cast and wrought superalloys. MgO, Al₂O₃, and MgO·Al₂O₃ are the primary oxide inclusions and can act as nucleation cores for nitrides, carbides, and carbonitrides to form inclusions with a multi-layered structure [11]. Some studies [12,13,14] have pointed out that carbonitride inclusions consist of TiN and TiC with a face-centered cubic structure, which share the same crystal structure and similar lattice parameters, allowing carbon and nitrogen atoms to be exchanged between them without a distinct boundary separating the two phases. Although these studies have helped us to understand the formation mechanisms of non-metallic inclusions in superalloy and have pointed the way to processes for producing “clean” superalloys, such as reducing the impurity content and increasing the cooling rate [15], information on quantitative studies is not widely available. Considering that nickel-based superalloys are mainly prepared by a combination of vacuum induction melting, electroslag remelting, and vacuum arc remelting, to accurately predict and control the complex inclusion formation, it is necessary to define the allowable impurity content and oxide inclusion size for the melting process, as well as the optimal cooling rate for the solidification process.

It is practical to study the process of inclusion precipitation and growth through theoretical calculations, which provides convenience for predicting and controlling the final size of inclusions. Recently, some thermodynamic and kinetic models have been proposed to simulate the precipitation and growth of oxide–carbonitride complex inclusions during the solidification of steel [16,17,18,19,20,21]. In the present work, a coupled kinetic model was proposed to predict the complex oxide–carbonitride inclusion’s precipitation behavior during superalloy solidification. This model takes into account thermodynamics, micro-segregation, heterogeneous nucleation in the inter-dendritic liquid, and growth controlled by a diffusion of solute elements and kinetics of interfacial reaction. Calculated and experimental results obtained by an industrial vacuum arc remelting of a GH4742 superalloy are compared. The influence of various factors, e.g., nitrogen concentration, cooling rate and particle size of primary oxides were investigated in the model calculations, providing references for the control of complex inclusions during the remelting and solidification of the superalloy.

2. Model Description and Assumption

2.1. Thermodynamics of Carbonitride Precipitation

Since oxides exhibit a high melting point and stability, it is supposed to be a solid particle that exists in the liquid superalloy. TiN and TiC inclusions are preferred to nucleate on these oxides during the solidification. The precipitation reaction and their standard Gibbs free energy of TiN and TiC inclusions can be represented by [22]:

$$\left[\mathrm{Ti}\right]+\left[\mathrm{N}\right]=\mathrm{TiN}\left(\mathrm{s}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathsf{\Delta }{G}_{\mathrm{TiN}}^{\theta }=-291,000+107.91T \mathrm{J/mol}$$

(1)

$$\left[\mathrm{Ti}\right]+\left[\mathrm{C}\right]=\mathrm{TiC}\left(\mathrm{s}\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathsf{\Delta }{G}_{\mathrm{TiC}}^{\theta }=-157,524+88.362T \mathrm{J/mol}$$

(2)

$$\mathsf{\Delta }{G}_{\mathrm{TiN}}=\mathsf{\Delta }{G}_{\mathrm{TiN}}^{\theta }+\mathrm{R}T\ln {\displaystyle \frac{a_{\mathrm{TiN}}}{a_{[\mathrm{Ti}]}\cdot a_{[\mathrm{N}]}}}=\mathsf{\Delta }{G}_{\mathrm{TiN}}^{\theta }+\mathrm{R}T\ln {\displaystyle \frac{1}{f_{\left[\mathrm{Ti}\right]}{w}_{[\mathrm{Ti}]}\cdot f_{\left[\mathrm{N}\right]}{w}_{[\mathrm{N}]}}}$$

(3)

$$\mathsf{\Delta }{G}_{\mathrm{TiC}}=\mathsf{\Delta }{G}_{\mathrm{TiC}}^{\theta }+\mathrm{R}T\ln {\displaystyle \frac{a_{\mathrm{TiC}}}{a_{[\mathrm{Ti}]}\cdot a_{[\mathrm{C}]}}}=\mathsf{\Delta }{G}_{\mathrm{TiC}}^{\theta }+\mathrm{R}T\ln {\displaystyle \frac{1}{f_{\left[\mathrm{Ti}\right]}{w}_{[\mathrm{Ti}]}\cdot f_{\left[\mathrm{C}\right]}{w}_{[\mathrm{C}]}}}$$

(4)

where $\mathsf{\Delta }{G}_{\mathrm{TiN}}^{\theta }$ and $\mathsf{\Delta }{G}_{\mathrm{TiC}}^{\theta }$ are standard Gibbs free energy of TiN and TiC inclusion, J/mol; R is the ideal gas constant, 8.314 J/(mol·K); T is temperature, K; $a_{\mathrm{TiN}}$, $a_{\mathrm{TiC}}$, $a_{\left[\mathrm{Ti}\right]}$, $a_{\left[\mathrm{N}\right]}$, and $a_{\left[\mathrm{C}\right]}$ are activities of TiN, TiC, [Ti] and [N] and [C] in the liquid metal; $f_i$ is the mass-based activity coefficient of i; and $w_{[i]}$ is the mass fraction of i, wt%. The activity coefficients of [Ti], [C], and [N] in liquid superalloy are calculated by the Wagner model, and all equations are considered as functions of temperature, as shown in Equations (5)–(8). The first-order interaction coefficients $e_i^j$ are used to calculate $f_i$, where j is the solute element in liquid metal.

Table 1. Interaction coefficient of elements in nickel-based superalloy at 1600 °C [23,24,25,26,27].

j	Al	Co	Cr	Mo	Nb	Ti	C	N
$e_{\mathrm{Ti}}^j$	0.037	-	0.015	0.016	-	0.013	−0.165	−0.1982
$e_{\mathrm{N}}^j$	0	−0.0054	−0.101	−0.04	−0.075	−0.20	-	-
$e_{\mathrm{C}}^j$	0.027	−0.005	−0.011	−0.001	−0.014	−0.022	0.42	-

lists the interaction coefficients reported by the literature:

$$\lg f_i={\displaystyle \sum e_i^j\cdot w_{[j]}}$$

(5)

$$\lg f_{[\mathrm{Ti}]}=\lg f_{[\mathrm{Ti}],1600℃}\cdot \left({\displaystyle \frac{2557}{T}}-0.365\right)$$

(6)

$$\lg f_{[\mathrm{N}]}=\lg f_{[\mathrm{N}],1600℃}\cdot \left({\displaystyle \frac{3280}{T}}-0.75\right)$$

(7)

$$\lg f_{[\mathrm{C}]}=\lg f_{[\mathrm{C}],1600℃}\cdot \left({\displaystyle \frac{2538}{T}}-0.355\right)$$

(8)

The solubility product of TiN and TiC in liquid superalloy can be calculated, as shown in our previous work [13]. According to the [Ti]-[N] and [Ti]-[C] equilibrium diagrams, a TiN inclusion is generally precipitated in the liquids, while a TiC inclusion is precipitated in the mushy zone during the solidification, indicating that TiN inclusions will precipitate earlier than TiC inclusions, which are consistent with the experiment results studied by Alec Mitchell [28]. However, considering the microsegregation and heterogeneous nucleation during solidification, TiN and TiC would inevitably precipitate at even higher temperatures when the actual concentration product is higher than the equilibrium solubility product.

2.2. Microsegregation Model

Some microsegregation models with different assumptions and simplifications were proposed to describe solute concentrations in the residual liquid as a function of a solid fraction. Here, the C-K model proposed by T.W. Clyne and W. Kurz [29] was used to calculate the microsegregation of solute in residual liquid, which considers the solute diffusion both in solid and liquid metal:

$$C_{\mathrm{l}}=C_0{[1-(1-2\mathsf{\Omega }\left({\alpha }_f\right)k_0)f_s]}^{{\displaystyle \frac{k_0-1}{1-2\mathsf{\Omega }\left({\alpha }_f\right)k_0}}}$$

(9)

$$\mathsf{\Omega }\left({\alpha }_f\right)={\alpha }_f\left[1-\exp \left(-{\displaystyle \frac{1}{{\alpha }_f}}\right)\right]-{\displaystyle \frac{1}{2}}\exp \left(-{\displaystyle \frac{1}{2{\alpha }_f}}\right)$$

(10)

$${\alpha }_f={\displaystyle \frac{D_{\mathrm{s}}{t}_f}{{\left(0.5{\lambda }_2\right)}^2}}$$

(11)

$$t_f={\displaystyle \frac{T_{\mathrm{l}}-T_{\mathrm{s}}}{C_R}}$$

(12)

where $C_{\mathrm{l}}$ and $C_0$ are the solute concentration in the liquid and the initial solute concentration, mol/L; ${\alpha }_f$ is the Fourier constant; $f_s$ is the solidification fraction; $k_0$ is the equilibrium partition coefficient of the solute element; $D_{\mathrm{s}}$ is the diffusion coefficient of solute in solid metal, m²/s; ${\lambda }_2$ is the secondary dendrite arm spacing, μm; $t_f$ is the local solidification time, s; $T_{\mathrm{l}}$ and $T_{\mathrm{s}}$ are the liquid and solid temperature, °C; and $C_R$ is the cooling rate, °C/s.

In the process of solidification, solute elements are inevitably enriched in the residual liquid phase and affected by back diffusion, which continuously changes in the solid and liquid phases. $k_0$ in Equation (9) is the equilibrium partition coefficient of the solute element, which is defined as the ratio of the solid composition to the liquid composition in equilibrium. In the previous work, it is invariably used to estimate the concentration profiles in the model as a constant value, regardless of the back diffusion in solid phase at different cooling rates. However, according to our previous study [30] and similar work reported in [31], solute elements did show strong segregation tendencies, and the segregation degree first increased and then decreased with increasing cooling rate, which suggests that it should not be a constant value, but a variation value. Accordingly, applying the variable effective partition coefficient in each cooling rate instead of the constant is more accurate in predicting the effects of segregation and back diffusion under different cooling conditions. So the $k_0$ used in this study is the actual value obtained from our previous experiments, as shown in

Table 2. Equilibrium partition coefficient $k_0$ of solute elements at different cooling rates [30].

Cooling Rate °C/s	Co	Cr	Mo	Nb	Ti	C	N
0.019	1.04592	1.0754	0.91959	0.4964	0.74594	1.0587	0.89434
0.038	1.05101	1.08196	0.93337	0.4837	0.70492	0.99996	0.98009
0.076	1.09037	1.07426	0.89592	0.36363	0.60161	0.95269	0.90398
0.152	1.1154	1.09431	0.8605	0.3622	0.55801	0.90482	0.89371
0.304	1.06625	1.07222	0.90944	0.41758	0.62936	1.09711	0.90559

.

2.3. Kinetic Model of Inclusion Heterogeneous Nucleation and Precipitation

It was introduced that oxide–carbonitride complex inclusions are generally treated as the competitive result of heterogeneous nucleation during the solidification of nickel-based superalloy. Although many studies in the literature report that the complex inclusions in superalloys are a combination of oxides and carbonitrides, we must admit that we cannot accurately indicate how carbon and nitrogen atoms simultaneously bind with titanium atoms. So, we simplified the heterogeneous nucleation and precipitation kinetic model by simplifying carbonitride into nitride. Since the concentration of main solute elements is much higher than that of nitrogen, the [N] diffusion is considered as the limiting factor of complex inclusion growth. For the sake of modeling the oxide–carbonitride complex inclusion, heterogeneous nucleation, and precipitation, it is assumed that the diffusion of [N] from liquid melts to the surface of inclusion and can be divided into three processes, as shown in Figure 1a: (i) diffusion from liquid metal to the inclusion boundary layer; (ii) diffusion from the inclusion boundary layer to the inclusion interface; and (iii) inclusion, precipitation, and growth at the interface. Because the reaction at the interface of inclusions always occurs instantaneously, the precipitation reaction at the interface of inclusions is not considered as the crucial controlling step. Hence, the precipitation and growth of oxide–carbonitride complex inclusions could be described as the diffusion process of [N] from liquid metal to the inclusion boundary:

$$J_{\mathrm{N}}=-{\displaystyle \frac{1}{4\pi r^2}}{\displaystyle \frac{d{n}_{\mathrm{N}}}{dt}}=k_{\mathrm{m},\mathrm{N}}\left(C_{j-\mathrm{N}}-C_{\mathrm{s}-\mathrm{N}}\right)$$

(13)

$$k_{\mathrm{m},\mathrm{N}}={\displaystyle \frac{D_{\mathrm{N}}}{r}}$$

(14)

$${\displaystyle \frac{d{n}_{\mathrm{s}}}{dt}}={\displaystyle \frac{4\pi r^2{\rho }_{\mathrm{s}}}{M_{\mathrm{s}}}}{\displaystyle \frac{dr}{dt}}$$

(15)

where $J_{\mathrm{N}}$ is the diffusion flux of [N]; $r$ is the radial of inclusion, m; $k_{\mathrm{m},\mathrm{N}}$ is the mass transfer coefficient of [N] at the interface of inclusion and liquid alloy, m/s; $D_{\mathrm{N}}$ is the diffusion coefficient of [N] in the liquid alloy, m²/s; $C_{j-\mathrm{N}}$ is the concentration of [N], where $C_{\mathrm{l}-\mathrm{N}}$ is the equilibrium concentration of [N] in liquid metal; $C_{f-\mathrm{N}}$ is the equilibrium concentration of [N] in inclusion boundary layer; $C_{s-\mathrm{N}}$ is the equilibrium concentration of [N] in inclusion; $n_{\mathrm{s}}$ is the mole of inclusion; and ${\rho }_{\mathrm{s}}$ and $M_{\mathrm{s}}$ are the density and molecular mass, respectively.

According to mass balance law, the diffusion flux of [N] towards particles can be written as Equations (16) and (17):

$$-{\displaystyle \frac{d{n}_{\mathrm{N}}}{dt}}={\displaystyle \frac{d{n}_{\mathrm{s}}}{dt}}$$

(16)

$$r{\displaystyle \frac{dr}{dt}}={\displaystyle \frac{M_{\mathrm{s}}}{{\rho }_{\mathrm{s}}}}{D}_{\mathrm{N}}\left(C_{j-\mathrm{N}}-C_{\mathrm{s}-\mathrm{N}}\right)$$

(17)

when the [N] diffusion from liquid metal to the inclusion boundary layer is the controlling step (where $C_{\mathrm{s}-\mathrm{N}}=C_{\mathrm{f}-\mathrm{N}}$), the growth model can be written as:

$$r_t=\sqrt{{\displaystyle \frac{2M_{\mathrm{s}}}{{\rho }_{\mathrm{s}}}}{D}_{\mathrm{N}}\left(C_{\mathrm{l}-\mathrm{N}}-C_{\mathrm{s}-\mathrm{N}}\right)t+r_0^2}$$

(18)

Given that the [N] content in the liquid melt are always in thermodynamic equilibrium during the precipitation process of inclusion, the $C_{\mathrm{l}-\mathrm{N}}-C_{\mathrm{s}-\mathrm{N}}=0$, which indicates that the model is no longer applicable. Thus, the [N] diffusion from inclusion boundary layer to inclusion interface is the main controlling step (where $C_{\mathrm{f}-\mathrm{N}}=C_{\mathrm{l}-\mathrm{N}}$), as shown in Figure 1b, and the kinetic model of TiN inclusion growth can be written as:

$$r_t=\sqrt{{\displaystyle \frac{2M_{\mathrm{s}}}{{\rho }_{\mathrm{s}}}}{D}_{\mathrm{N}}\left(C_{\mathrm{f}-\mathrm{N}}-C_{\mathrm{s}-\mathrm{N}}\right)t}, r_0=0$$

(19)

$$r_t=\sqrt{{\displaystyle \frac{2M_{\mathrm{s}}}{{\rho }_{\mathrm{s}}}}{D}_{\mathrm{N}}\left(C_{\mathrm{f}-\mathrm{N}}-C_{\mathrm{s}-\mathrm{N}}\right)t+{r_{\mathrm{t}-1}}^2}, r_0=r_{t-1}\ne 0$$

(20)

According to a study conducted by Shu et al. [18,19], the concentration of [N] element at the boundary layer of the inclusion–liquid interface satisfies the Gibbs–Thomson equation:

$$C_{\mathrm{f}-\mathrm{N}}=C_{\mathrm{s}-\mathrm{N}}\exp \left({\displaystyle \frac{2{\sigma }_{\mathrm{sl}}{V}_{\mathrm{m},\mathrm{s}}}{r_{t}R{T}_t}}\right)$$

(21)

where ${\sigma }_{\mathrm{sl}}$ is the interface tension between inclusions and liquid melt, N/m; $V_{\mathrm{m},\mathrm{s}}$ is the molar volume of inclusions, m³/mol; and $T_t$ is the thermodynamic temperature at time t, K.

The concentration of [N] in the inclusion boundary layer satisfied mass conservation during TiN precipitation:

$$C_{j-\mathrm{N}}={\displaystyle \frac{m_{\mathrm{N}}}{M_{\mathrm{N}}\frac{m_{\mathrm{alloy}}}{{\rho }_{\mathrm{alloy}}}}}={\displaystyle \frac{{\rho }_{\mathrm{alloy}}{w}_{j-N}}{M_{\mathrm{N}}\cdot 100}}$$

(22)

Combining Equations (20)–(22), the heterogeneous nucleation and growth of oxide–carbonitride complex inclusions are expressed by Equation [23]:

$$r_t=\sqrt{{\displaystyle \frac{M_{\mathrm{s}}{\rho }_{\mathrm{alloy}}}{50M_{\mathrm{N}}{\rho }_{\mathrm{s}}}}{D}_{\mathrm{N}}\left(w_{\mathrm{f}-\mathrm{N}}-w_{\mathrm{s}-\mathrm{N}}\right)t+r_{t-1}^2}$$

(23)

2.4. Calculation Procedure and Model Validation

The present model consists of a series of sub-models accounting for the thermodynamic model, micro-segregation model, and kinetic model of heterogeneous nucleation and growth. The calculation procedure can be described by the flow chart, as shown in Figure 2. First, parameters such as the initial alloy composition, initial size of primary oxide inclusions, solidus and liquidus temperature of the alloy, cooling rate, and interaction coefficient of each element are input to calculate the thermodynamic model of Ti(CN) inclusion and obtain the precipitation temperature. Then, once the melt temperature has been reduced to the temperature at which complex inclusions begin to precipitate, the growth model begins calculations, and the concentration of solutes are updated according to the microsegregation model and precipitation model described in Section 2.3 and Section 2.4. The parameters used in the model calculation are shown in

Table 3. The parameters used in the model calculation [18,32,33].

Parameter	${\mathit{M}}_{\mathbf{s}}$ (g/mol)	${\mathit{M}}_{\mathbf{N}}$ (g/mol)	${\mathit{\rho }}_{\mathbf{alloy}}$ (kg/m3)	${\mathit{\rho }}_{\mathbf{s}}$ (kg/m3)
Value	61.87	14	7290	5430
Parameter	$V_{\mathrm{m},\mathrm{s}}$ (m3/mol)	$D_{\mathrm{N}}$ (m2/s)	$D_{\mathrm{s},\mathrm{Ti}}$ (m2/s)	${\sigma }_{\mathrm{sl}}$ (N/m)
Value	1.18 × 10−5	$3.25\times {10}^{-7}{e}^{{\displaystyle \frac{-11500}{\mathrm{R}T}}}$	$4.1\times {10}^{-7}{e}^{{\displaystyle \frac{-27500}{\mathrm{R}T}}}$	0.7

. To validate the accuracy of the proposed model, samples of the GH4742 superalloy extracted from industrial VAR ingot were characterized. The average size of complex inclusions on the surface, 1/2R, and center of the ingot are compared with the calculation results.

3. Results and Discussion

3.1. Effect of Cooling Rate on the Final Particle Size of Oxide–Carbonitride Inclusions

Numerous studies confirmed that the cooling rate takes significant effects on the final particle size of inclusions. However, most of this work focuses on the precipitation of pure TiN inclusions, with few investigations into the effect of cooling rate on the precipitation of complex inclusions. From our previous work, it can be determined that as the cooling rate increases, the average size of inclusions decreases obviously. On the one hand, it limits the total precipitation time for inclusion growth; and on the other hand, the cooling rate affects the element segregation in the interdendritic region, which in turn influences the concentration difference at the inclusion interface. Particularly, Ti is one of the most easily segregated elements in nickel-based superalloys. Figure 3a shows the variation of [Ti] content during the solidification process calculated using the above microsegregation model. The results presented that the concentration of the [Ti] element in the residual liquids shows a significant enrichment, and as the cooling rate increases from 0.019 °C/s to 0.304 °C/s, the concentration of the Ti element in the residual liquid shows a trend in first increasing and then decreasing. This trend could be attributed to the back diffusion of the solute element. At a lower cooling rate, the back-diffusion effect in the solid phase is significant owing to the longer solidification time, thus the enrichment of solute in the residual liquid is smaller. Meanwhile, with the increase in cooling rate, the effect of the degree of primary solid diffusion on segregation is gradually weakening, and the enrichment of solute in the residual liquid becomes dominant. When the cooling rate further increases, the effect of the degree of primary solid diffusion on segregation is negligible and the liquid-phase diffusion of elements becomes a limiting factor. At this time, the larger the cooling rate, the smaller the degree of segregation.

Figure 3b shows the variation of [N] concentration deviation ($w_{\mathrm{f}-\mathrm{N}}-w_{\mathrm{s}-\mathrm{N}}$) at the interface of primary oxide with an initial size of 1.8 μm. It can be found that the concentration deviation of [N] at the interface of inclusions rapidly decreases with decreasing temperature. When the temperature drops below the liquidus line, the concentration difference at the interface gradually stabilizes. Additionally, the smaller the cooling rate, the faster the interfacial concentration difference decreases, which means the better the solute diffusion.

The particle size of oxide–carbonitride complex inclusions at different cooling rates was calculated using the present model, and the results are shown in Figure 4. It is interesting to note that the complex inclusions rapidly grow and then tend to stabilize during the cooling and solidification process, and the radius corresponding to the maximum size of inclusions decreases with increasing cooling rates. This trend indicates that the nucleation and growth of inclusions mainly take place above the liquidus temperature (1340 °C), and the effect of microsegregation during solidification has relatively little effects. Combined with the calculation results in Figure 3, the earlier precipitation of TiN at a higher temperature reduces the concentration difference in the element N, which is the main driving force for the TiN inclusion growth process.

3.2. Effect of Initial N Content on the Final Particle Size of Oxide–Carbonitride Inclusions

The variation of initial [N] content in the liquid metal will change the concentration difference in [N] that affects the precipitation behavior as well as the final size distribution of TiN inclusions. The calculated concentration difference in [N] at the inclusion interface for initial [N] content of 0.0016%, 0.0024%, 0.0032%, 0.0040%, and 0.0048% are shown in Figure 5. It should be mentioned that the calculation was carried out assuming a cooling rate of 0.038 °C/s and a primary oxide core size of 1.8 μm. As seen in Figure 5, as the initial [N] content rises from 0.0016% to 0.0048%, the maximum concentration difference in [N] at the inclusion interface increases, and with the decrease in temperature during the cooling process, the concentration difference rapidly decreases.

Subsequent calculations were performed to assess the effect of the initial [N] content on the precipitation and growth behavior of TiN, based on the current model. As shown in Figure 6a, with the increase in initial nitrogen content, oxide–carbonitride complex inclusions precipitate earlier in liquid metal, with a higher precipitation temperature and ultimately larger size. It should be noted that the higher the initial N content, the faster the growth rate of inclusions in the liquid phase. In addition, regardless of the initial N content, the growth rate of inclusions decreases and tends to stabilize with the progress of cooling and solidification. The principal reason is that the precipitation of TiN at a higher temperature result in a reduction in the concentration difference in the element N, which represents the main driving force for the TiN inclusion growth process. Figure 6b shows the variation of inclusion precipitation temperature and the final size with the initial N content in the liquid metal. It can be seen that the precipitation temperature and final size of inclusions exhibit an approximately linear relationship with the initial N content. When increasing the initial N content from 0.0016% to 0.0048%, the precipitation temperature of Ti(CN) increased from 1358 °C to 1496 °C, and the corresponding final particle size of oxide–carbonitride complex inclusions increased from 6.17 μm to 10.2 μm. This increase can be attributed to the elevated initial N content, which elevates the precipitation temperature of inclusions, prolongs the growth time of inclusions and, in turn, increases inclusion size. It can therefore be concluded that the initial nitrogen content exerts a controlling influence on the initial precipitation temperature of inclusions, in comparison to the cooling rate.

3.3. Effect of Primary Oxide Size on the Final Particle Size of Oxide–Carbonitride Inclusions

Given that oxides are the core of heterogeneous nucleation in oxide–carbonitride complex inclusions, it is also worth considering whether the initial size of oxide may exert a modifying influence on the ultimate size of the inclusions. Figure 7 and Figure 8 present the concentration difference in [N] at the inclusion interface with varying sizes and the evolution of the final size of inclusions, respectively. The calculated cooling rate is 0.038 °C/s, and the initial [N] content is 0.0032%. Interestingly, the calculations show that the size of the oxide core has little effect on the final size of the inclusions. The oxide core size increases from 1 μm to 3 μm, and the final size of the inclusions only increases from 8.41 μm to 8.51 μm. The present results are consistent with the results by Shu et al. [18] regarding the changing trend in mean particle size with the varying size of primary inclusions. This changing trend could be due to the fact that the inclusions with smaller sizes have a larger growth rate, resulting in a very narrow final inclusion size distribution.

3.4. Validation and Application of the Model in Vacuum Arc Remelting Process

From the previous calculations, it can be concluded that the size of oxide–carbonitride complex inclusions is mainly determined by the cooling rate and the initial [N] content for a certain nickel-based alloy with a definite composition. Among them, the cooling rate mainly controls the total growth time, and the initial [N] content determines the starting precipitation temperature. The size of the oxide has a negligible effect on the final size of the complex inclusions.

The proposed model was further validated and applied to the vacuum arc remelting process to predict the size distribution of complex inclusions in an industrial GH4742 nickel-based superalloy ingot. The initial [N] content used for calculation was taken as the average nitrogen content of the ingot obtained from the industrial experiment (0.0020%), and the size of the oxide core was taken as 1.2 μm. Considering that the cooling rate inside industrial ingots is difficult to monitor experimentally, we calculated it here with the help of commercial software Meltflow-VAR [4,30]. As shown in Figure 9, the cooling rate of the top, middle, and bottom of the ingot gradually decreases radially from the center to the edge. The cooling rate in the middle of the ingot center is 0.018 °C/s, the cooling rate at 1/2R is 0.021 °C/s, and the edge rate is 0.144 °C/s. Figure 10 and Figure 11 present the calculated and experimental results obtained by the industrial vacuum arc remelting ingot. It can be seen that the size of complex inclusions decreases from the center to the edge, and the calculated results are reasonablly consistent with the experimental results.

4. Conclusions

A coupled kinetic model that considers thermodynamics, micro-segregation, heterogeneous nucleation in the inter-dendritic liquid, and the growth controlled by the diffusion of solute elements and the kinetics of interfacial reaction was proposed to predict the complex oxide–carbonitride inclusion’s precipitation behavior during superalloy solidification. The effect of cooling rates, initial nitrogen content, and size distribution of primary oxides on the final size of complex inclusions were investigated by the model calculation. The main conclusions are as follows:

(1)

The growth of complex inclusions starts from the thermodynamic precipitation temperature of TiN, and the diffusion of the N element from the inclusion boundary layer to the inclusion interface is the limiting step.

(2)

Both the cooling rate and N content take significant effects on the final size of complex inclusions, as the former controls the total growth time and the latter determines the initial precipitation temperature. The initial particle size of primary oxides has only a slight effect on the final size of complex inclusions.

(3)

Further validation and application of the present model in precipitation of oxide–carbonitride was carried out in an industrial vacuum arc remelting experiment. The calculated particle sizes of precipitated complex inclusions are in good agreement with the experimental data.