Simulations of Non-Equilibrium and Equilibrium Segregation in Nickel-Based Superalloy Using Modified Scheil-Gulliver and Phase-Field Methods

Abstract

Solute segregation significantly affects material properties and is an essential issue in the additive manufacturing (AM) process. In the present study, we have investigated (i) non-equilibrium segregation in solidification, and (ii) equilibrium segregation at grain boundary in Ni-based Hastelloy-X (HX) superalloy using the modified Scheil-Gulliver model (i.e., Scheil-Gulliver model with back diffusion) and a phase-field model. We have found that the concentrations of all solute elements on grain boundary differ from those in face-centered cubic (FCC) phase matrix even in equilibrium state. In the non-equilibrium segregation, the segregations of Mo, Cr, and Mn and the depletion of Fe become more remarkable than the equilibrium segregation. Moreover, we have investigated the segregation in HX-based alloys with different Fe concentrations to propose a guide for tailoring the chemical composition of HX via the control of the segregation behaviors. The equilibrium-segregation simulation revealed that the Cr segregation in the grain boundary phase increased with the increase of Fe concentration. This result suggests that by controlling the Fe concentration, the Cr concentrations on grain boundaries can be controlled without changing directly the Cr concentrations. This finding opens new way of controlling materials properties which are dominated by the nature of grain boundaries such as corrosion resistance, crack sensitivity, high temperature strength.

Cr concentration profiles across a grain boundary in the one-dimensional phase field models of Hastelloy-X (HX) based alloys modified with different Fe concentrations.

1. Introduction

Additive manufacturing (AM) technologies have attracted much attention because it enables us to build complicated geometry 3-D parts easily. Among the wide family of AM techniques available, Selective Laser Melting (SLM) has become a preferred technique for the metal AM.^1–3) SLM is one of the powder bed fusion (PBF) type AM technologies, in which the metal products are built in layer-by-layer manner by scanning laser to fuse metal powder particles to form objects. Moreover, the PBF process can control crystallographic textures via the control of solidification conditions such as power, diameter, and scanning speed of the laser beam.^3–5)

In particular, additively manufactured Ni-based superalloys are desired to be applied to aerospace industries.^5–7) However, the superalloys contain high concentrations of solute elements such as Al, Cr, Mo, Ta, and W, and due to its rapid solidification in the AM process, segregations of these solute elements become remarkable and cause liquation cracking during AM fabrication process.^5–7) Segregation of solute elements also occurs during homogenizing heat-treatment in which the components are subjected to a high-temperature environment.⁸⁾ This segregation is called “equilibrium segregation” and greatly affects grain-boundary properties. These non-equilibrium and equilibrium segregation in AM processes need to be understood to ensure process reliability and quality of AM components. However, because of enormous combinations of elements and their concentrations in multi-component Ni-based alloys, it takes a lot of effort to measure various kinds of segregation. If segregation can be predicted using simulation methods, material development can be accelerated.

In this study, we have investigated non-equilibrium and equilibrium segregation behavior of Ni-based superalloy Hastelloy-X (HX), to which is expected to be applied to AM.⁹⁾ The non-equilibrium segregation during solidification was simulated using the modified Scheil-Gulliver model. The original Scheil-Gulliver model was developed by Scheil¹⁰⁾ and Gulliver,¹¹⁾ and various improved models were proposed.^12–15) The differences between these models have been reviewed by Nakajima et al.¹⁶⁾ However, these analytical models cannot be applied to multi-component systems because of their assumptions of diffusivities and a partition coefficient.^17,18) Therefore, we have used a modified Scheil-Gulliver model for multi-component alloys within the Thermo-Calc software¹⁹⁾ in which thermal stabilities and back diffusions of all solute elements are calculated by thermodynamic and mobility databases. On the other hand, equilibrium segregation was investigated by using the “multi-component phase-field model”, which was developed based on the phase-field model of Koizumi et al.²⁰⁾ The phase-field model can predict spatial distributions of solute concentration across a grain boundary, and is consistent with the parallel tangent rule of Hillert,²¹⁾ which is generally used to predict equilibrium segregation. Based on the simulation results, differences between the non-equilibrium and the equilibrium segregations are discussed. In addition, the effects of Fe concentration on segregation behavior were also investigated to propose tailoring the chemical composition of HX.

2. Method

2.1 Non-equilibrium segregation simulation

The solidification segregation of HX was simulated by the modified Scheil-Gulliver model using the TCNI9 thermodynamic database²²⁾ and the MOBNi5 mobility database²³⁾ within the Thermo-Calc software.¹⁹⁾ Table 1 shows the chemical composition of HX, and the corresponding property diagram (a type of phase diagram showing the volume fraction of constituent phases as a function of temperature) calculated by Thermo-Calc is shown in Fig. 1. The property diagram indicates the face-centered cubic (FCC) and σ phases appear as the equilibrium solid phases. However, according to the time–temperature-transformation (TTT) diagram,²⁴⁾ the σ phase are formed after tens of hours at temperature in the range of 1073–1173 K (800–900°C). Therefore, only liquid and FCC phases were assumed to exist in the modified Scheil-Gulliver simulation. In the simulation, the models of HX was cooled from 2500 K until the solid fraction reach 0.99 with various cooling rates of 10⁰–10⁶ K s⁻¹, and the concentration profiles were compared. Moreover, the modified Scheil-Gulliver simulation were performed for HX-based model alloys with different Fe concentration of 14.08 at% (−5 at% Fe), 24.08 at% (+5 at% Fe), and 29.08 at% (+10 at% Fe) with a cooling rate of 10⁶ K s⁻¹ as well in order to investigate the effects of Fe concentration on non-equilibrium segregation behavior. These alloys with different Fe concentration are hereafter referred to as HX−5Fe, HX+5Fe, and HX+10Fe, respectively.

Table 1 Chemical compositions of Hastelloy-X (HX) models (at%).

Fig. 1

Equilibrium volume fraction of face-centered cubic (FCC), sigma, and liquid phases of Hastelloy-X (HX) model.

2.2 Equilibrium segregation simulation

Equilibrium segregation simulation was performed using a phase-field model developed by Koizumi et al.²⁰⁾ We assumed that the Gibbs free energy of liquid phase at temperatures below melting point can be an approximation of that grain boundary, and equilibrium grain boundary segregation can be simulated by the phase-field model consisting of FCC and liquid phases. Phase-field order parameter, ϕ, set to −1 or 1 for the FCC phase and 0 for grain boundary.

The local free energy density is assumed to be the sum of chemical-free energy density, f_chem, double-well potential, f_doub, and gradient energy density, f_grad:   

\begin{equation} f = f_{\text{chem}} + f_{\text{doub}} + f_{\text{grad}}. \end{equation} (1)

The chemical-free energy density f_chem can be expressed as   

\begin{equation} f_{\text{chem}} = h(\phi)f_{\text{FCC}} + \{1 - h(\phi)\}f_{\text{liq}} \end{equation} (2)

  

\begin{equation} h(\phi) = \begin{cases} \phi^{3}(10 - 15\phi + 6\phi^{2}), & \text{when $\phi \geq 0$}\\ -\phi^{3}(10 + 15\phi + 6\phi^{2}), & \text{when $\phi < 0$} \end{cases} \end{equation} (3)

Here, f_FCC and f_liq are the chemical free-energy densities of the FCC phase and the liquid phase, respectively. The double-well potential f_doub = Wg(ϕ) was introduced to reproduce a saddle point between equilibrium and semi-equilibrium states. Where g(ϕ) is the double-well function defined by   

\begin{equation} g(\phi) = \begin{cases} \phi^{2}(1 - \phi)^{2}, & \text{when $\phi \geq 0$}\\ \phi^{2}(1 + \phi)^{2}, & \text{when $\phi < 0$} \end{cases} \end{equation} (4)

and W is a coefficient that determines the magnitude of the double-well potential. The gradient energy f_grad is in terms of gradient coefficient κ_ϕ   

\begin{equation} f_{\text{grad}} = \kappa_{\phi}(\nabla\phi)^{2}. \end{equation} (5)

In the phase-field simulation, the time evolution of the solute concentration c_i and the phase-field order parameter, ϕ, are calculated for minimizing the total free energy of the system F, which is given by integrating the local Gibbs free energy throughout the system. The time evolution of ϕ is the Allen-Cahn equation:²⁵⁾   

\begin{equation} \frac{\partial\phi}{\partial t} = -M_{\phi}\frac{\delta F}{\delta \phi} \end{equation} (6)

where M_ϕ is a phase-field interface mobility. On the other hand, the time evolution equation of c_i was derived from the interdiffusion flow of multi-component alloys.²⁶⁾ When the phase to be calculated is a liquid phase or a substitutional solid solution, interdiffusion flow $\skew4\tilde{J}$ of n + 1 elements can be expressed   

\begin{equation} \skew4\tilde{J}_{i} = -\sum\nolimits_{j = 1}^{n}L_{ij}\nabla(\tilde{\mu}_{j} - \tilde{\mu}_{0}),\quad (i = 1, \cdots n) \end{equation} (7)

where L_ij is the Onsager coefficient, and $\tilde{\mu }_{j}$ and $\tilde{\mu }_{0}$ are the chemical potentials of solute and solvent elements. The Onsager coefficient L_ij is expressed as   

\begin{equation} L_{ij} = \left\{(\delta_{ij} - c_{i})M_{j} - c_{i}M_{i} + c_{i}\sum\nolimits_{k = 0}^{n}c_{k}M_{k}\right\}c_{j} \end{equation} (8)

Cogswell²⁷⁾ generalized diffusion potential, $\tilde{\mu }_{i}$, and derived the relation   

\begin{equation} \tilde{\mu}_{i} - \tilde{\mu}_{0} = \frac{\delta F}{\delta c_{i}}. \end{equation} (9)

Assuming that the atomic mobility M_i is a constant M_c for all element, eq. (7) can be transformed into   

\begin{equation} \frac{\partial c_{i}}{\partial t} = \nabla\left[M_{c}c_{i}\left\{(1 - c_{i})\nabla \frac{\delta F}{\delta c_{i}} - \sum\nolimits_{j = 1,j \neq i}^{n} c_{j}\nabla\frac{\delta F}{\delta c_{j}}\right\}\right]. \end{equation} (10)

Note that the phase-field simulations were performed to study equilibrium segregation, and this assumption does not affect the behavior.

Equilibrium segregations in Ni solvent were simulated using this phase-field model. The chemical-free energy density, f_chem, was calculated from the TCNI9 thermodynamic database²²⁾ as in the case for the non-equilibrium segregation. The energy density at the FCC/liquid interface was assumed to be same as that of the pure Ni interface energy density, σ [J m⁻²], as below.²⁸⁾   

\begin{align} \sigma &= 0.354\ [\text{J$\,$m$^{-2}$}] + 5.6 \\ &\quad \times 10^{-5}\ [\text{J$\,$m$^{-2}{}\,$K$^{-1}$}] \cdot (T-1300\ [\text{K}]) \end{align} (11)

According to Warren and Boettinger,²⁹⁾ diffusion coefficients of solute elements, M_c, and the phase-field interface mobility, M_ϕ, were 1.0 m² s⁻¹ and 1.3 m⁴ J⁻¹ s⁻¹, respectively. For the reason mentioned above, these kinetic values do not affect the equilibrium-segregation behavior.

A one-dimensional model with sides of 6 nm composed of 128 grids was used for the equilibrium simulation, and a grain boundary phase (ϕ = 0) was introduced by reversing ϕ from −1 to +1 at the center of the grid. The grain boundary width was 1 nm. The solute concentrations fields in the simulation domains were initially set to be uniformly same as those for the chemical composition of HX alloy as shown in Table 1 and the model alloys with different Fe concentration ranging from 14.08 at% Fe (−5 at% Fe) to 29.08 at% Fe (+10 at% Fe). Periodic boundary condition was employed, and the concentrations at both ends of the grid were kept constant throughout the simulation. This is for avoiding the change of the composition in the matrix during the simulation. This condition corresponds to the situation where the system is embedded in a bath in which the chemical potentials of all the solute element are same as those of the alloy with the initial composition. The system was equilibrated at 1073–1643 K for 150,000 timesteps and the concentration profiles of the equilibrium models were calculated.

3. Results and Discussion

3.1 Segregation in Hastelloy-X alloy

Figure 2(a) shows the solidification paths of the HX and model alloys obtained by the modified Scheil-Gulliver simulation with various cooling rates of 10⁰–10⁶ K s⁻¹. When the HX alloy liquid was cooled from 2500 K, the FCC phase formed below 1645 K, which is corresponding to the equilibrium liquidus temperature. The cooling rate significantly affects the solidification paths in the low-temperature region below 1620 K in which the solid fraction has over 0.8, while it does not in the high-temperature region near the melting point. The higher the cooling rate is, the lower the temperature at which the solidification completed is, that is, the liquid phase remained to a lower temperature as the cooling rate increased. The temperatures at which the solid fraction reached 0.99, T_0.99, are plotted as a function of cooling rate in Fig. 2(b). These temperatures are lower than the equilibrium solidus temperature of 1628 K shown in the property diagram (Fig. 1). The T_0.99 decreased as the cooling rate increased, and there is a logarithmic relationship in the range from 10⁰ to 10⁶ K s⁻¹:   

\begin{equation} T_{0.99} = -10.3\cdot \log_{10}{}\cdot |\text{Cooling rate [K$\,$s$^{-1}$]}| + 1581.2. \end{equation} (12)

The coefficient of determination, R², was 0.9994. The concentrations of the solute elements in the liquid phase of the HX alloy model cooling in the modified Scheil-Gulliver simulation is shown in Fig. 3(a). The cooling rates were 10⁰ K s⁻¹ (dotted line) and 10⁶ K s⁻¹ (solid line). When the FCC phase started to appear, Mo, Si, Mn, and W were segregated to the liquid phase, while Ni, Fe, and Co were depleted at all temperatures. When the solid fraction has over 0.8, the effect of cooling rates on the segregation behavior was remarkable: concentrations of Mo and Si in the last liquid phase increase and that of Fe decrease. Cooling rates affect both solidification paths and concentrations of solute elements significantly when the solid fraction has over 0.8. Therefore, it is suggested that the segregation of solute elements lowers solidification temperature. Figure 3(b) shows the concentration differences of solute elements between the last liquid phase and chemical composition (Table 1) with different cooling rates. As the cooling rate increased, the segregation of Mo and Si increased, while that of Fe decrease. The concentrations of other elements (Cr, Co, Mn, and W) are almost independent of the cooling rate.

Fig. 2

(a) Solidification paths of HX alloy model in modified Scheil-Gulliver simulations with various cooling rates. (b) Completely solidified temperatures when the models have solid phase fraction of 0.99, T_0.99.

Fig. 3

(a) Concentrations of solute elements in liquid phases of HX model in modified Scheil-Gulliver simulation. (b) Concentration differences of constitute elements between the finally residual liquid phases and the chemical composition.

The equilibrium segregation behavior of the HX alloy was investigated using the phase-field method. Figure 4 shows the phase-field order parameter, ϕ, and the concentration profiles of the HX alloy model equilibrated at 1073 K (800°C). The order parameter, ϕ, shown in Fig. 4(a) changes smoothly from 1 to −1 around the center grid through the grain boundary phase of ϕ = 0. In equilibrium, the concentrations of all solute elements in the grain boundary phase were different from those in the FCC matrix phase, as shown in Fig. 4(b) in which Cr, Fe, Mo, Si, and Mn are segregated to the grain boundary, while Ni, Co, and W depleted. Figure 5 shows the difference between the concentration in FCC solid-phase and that at the center of the grain boundary in the model equilibrated at 1073–1693 K. Below the melting point of 1643 K, the concentrations of all the solute elements on grain boundary are different from those in FCC matrix phase at all the temperatures simulated. The solutes of Cr, Mo, Si, and Mn were segregated to the grain boundary, while Ni, Co, and W were depleted. On the other hand, the segregation of Fe depends on the temperature. It is negative at temperatures above 1213 K and positive below 1203 K. We note that the Fe concentration profile of Fig. 4 has a minimum at the center and, as shown in Fig. 6, this double peak become remarkable around the temperature at which the sign of segregation changes from negative to positive.

Fig. 4

(a) Phase-field order parameter, ϕ, and (b) concentration profiles of the Hastelloy-X (HX) phase-field model equilibrated at 1073 K.

Fig. 5

Concentration differences between liquid and solid phases of HX model in Phase-field simulation.

Fig. 6

Fe concentration profiles of the HX phase-field model equilibrated at 1073, 1173, 1273, 1373, and 1473 K.

In order to compare the non-equilibrium segregation with the equilibrium segregation, the concentration difference of the last liquid phase in the modified Scheil-Gulliver simulation with the cooling rate of 10⁶ K s⁻¹ and the that of grain boundary in the phase-field simulation at the lowest simulation temperature of 1073 K (800°C) were shown in Table 2. In the non-equilibrium segregation, the segregations of Mo, Cr, and Mn and the depletion of Fe become more remarkable than the equilibrium segregation. In equilibrium condition, segregation behavior would be determined by partition coefficients of solute atoms between a grain boundary and a solid phase.³⁰⁾ On the other hand, solute-elements diffusivities also affect segregation behavior during the solidification.³¹⁾ It is suggested that the kinetic effect caused the difference in segregation behavior between non-equilibrium and equilibrium segregations. As mentioned in the introduction, the properties of AM-processed Ni-based superalloy are greatly affected by both non-equilibrium segregation and equilibrium segregations. The simulations of this study revealed that the non-equilibrium and equilibrium segregation had different segregation behavior to the grain boundaries, and it is suggested that different alloy design guidelines should be established for each segregation.

Table 2 Concentration differences of solute elements between liquid and solid phases of HX models in modified Scheil-Gulliver (MSG) and phase-field (PF) simulations (at%).

3.2 Effects of Fe concentration on segregation behavior

To propose a guide for tailoring the chemical composition of Hastelloy-X, the simulations of non-equilibrium segregation and equilibrium segregation in HX alloy and HX-based alloy models with different Fe concentrations were performed. Figure 7(a) shows solidification paths of HX−5Fe, HX, HX+5Fe, and HX+10Fe models with a cooling rate of 10⁶ K s⁻¹ and the completely solidified temperatures, T_0.99, are plotted in Fig. 7(b). The T_0.99 decreased as the Fe concentration increased, and there is a linear relationship between T_0.9 and Fe-concentration in the range from 14.08 to 29.08 at% Fe:   

\begin{equation} T_{0.99} = -3.2\cdot |\text{Fe concentration [at%]}| + 1580.3. \end{equation} (13)

The coefficient of determination, R², was 0.998378. Concentration differences between the last liquid phase and chemical compositions are shown in Fig. 8. The Fe segregation in the liquid phase decreased as the Fe concentration increased. There is no significant difference in the segregation behavior of the other solute elements.

Fig. 7

(a) Solidification paths of HX-based alloy models with different Fe concentrations in modified Scheil-Gulliver simulations. (b) Solidification-completed temperatures at which solid phase fraction reached 0.99, T_0.99.

Fig. 8

Concentration differences between the last liquid phases and the chemical compositions of HX-based alloy models with different Fe concentrations in modified Scheil-Gulliver simulation.

Effect of Fe concentration on equilibrium solidification behavior was investigated using HX phase-field models with different Fe concentration equilibrated at 1073 K. Figure 9 shows the concentration difference between the FCC-matrix phase and the center of the grain boundary. The segregation behavior of Mo and Cr significantly depends on the Fe concentration: as the Fe concentration increase, the segregation of Mo decrease and that of Cr increase.

Fig. 9

Concentration differences between liquid and solid phases of HX-based alloy models with different Fe concentrations in phase-field simulations.

Based on these simulation results, let us consider tailoring the chemical composition of HX. The equilibrium-segregation simulation revealed that the Cr segregation in the grain boundary phase increased with the increase of Fe concentration. This result suggests that by controlling the Fe concentration, the Cr concentrations at grain boundaries in equilibrium can be controlled without changing directly Cr concentration of the alloy and that in the FCC phase. HX alloy has been mainly used for gas turbine combustors where high-temperature corrosive environment causes oxidation and sulfidation. The corrosion under the oxidating and/or sulfurizing condition can be suppressed by increasing the Cr concentration at grain boundaries.³²⁾ Therefore, the corrosion resistance can be improved by designing an alloy with a high Fe content. On the other hand, Ni-based superalloys have been proposed to be applied to next-generation nuclear reactors, in which molten fluoride will be a candidate for a coolant.³³⁾ In a molten fluoride environment, Cr in grain boundary can be a corrosion site, and the lower the Cr concentration can improve corrosion resistance.³⁴⁾ When used in next-generation nuclear reactors, intergranular corrosion resistance can be improved by alloy design with a low Fe content.

4. Conclusion

In this study, solidification segregation and equilibrium segregation behavior of Ni-based superalloy Hastelloy-X have been investigated using modified Scheil-Gulliver, i.e., Scheil-Gulliver simulation with back diffusion, and phase-field simulation methods. To propose tailoring the chemical composition of Hastelloy-X, segregation simulations of HX-based alloy models with different Fe concentrations were also performed. The results and conclusions are summarized as follows:

In the phase-field simulations of the HX alloy model, concentrations of all solute elements in the grain boundary phase were different from the FCC phase although the simulation model was an equilibrium state: Cr, Fe, Mo, Si, and Mn were segregated to the grain boundary, while Ni, Co, and W were depleted. In the non-equilibrium segregation, the segregations of Mo, Cr, and Mn and the depletion of Fe become more remarkable than the equilibrium segregation. The non-equilibrium and equilibrium segregation had different segregation behavior to the grain boundaries, and it is suggested that different alloy design guidelines should be established for each segregation.

The equilibrium-segregation simulation of HX-based alloy models with different Fe concentrations revealed that the Cr segregated in the grain boundary phase increased with the increase of Fe concentration. This result suggests that by controlling the Fe concentration, the Cr and Mo concentrations in grain boundaries can be controlled without changing these concentrations in the FCC phase.

Acknowledgement

This work was supported by Cabinet Office, Government of Japan, Cross-ministerial Strategic Innovation Promotion Program (SIP), “Materials Integration for Revolutionary Design System of Structural Materials”, (funding agency: The Japan Science and Technology Agency).

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