Effect of Nb and B on the Precipitation Behaviors in Al-Ti-Nb Balanced-Ratio Ni-Based Superalloy: A Phase-Field Study

Abstract

In this paper, quantitative two-dimensional (2-D) phase-field simulations were performed to gain insight into the effects of B and Nb for Al-Ti-Nb balanced-ratio GH4742 alloys. The microstructure evolution during the precipitation process was simulated using the MICRESS (MICRostructure Evolution Simulation Software) package developed in the formalism of the multi-phase field model. The coupling to CALPHAD (CALculation of PHAse Diagram) thermodynamic databases was realized via the TQ interface. The morphological evolution, concentration distribution, and thermodynamic properties were extensively analyzed. It is indicated that a higher Nb content contributes to a faster precipitation rate and higher amounts and the smaller precipitate size of the γ′ phase, contributing to better mechanical properties. The segregation of the W element in γ′ precipitate due to its sluggish diffusion effect has also been observed. Higher temperatures and lower B contents accelerate the dissolution of boride and reduce the precipitation of borides. With the increased addition of B, the formation of borides may have a pinning effect on the grain boundary to hinder the kinetic process. In addition, borides are prone to precipitate around the interface rather than in the bulk phase. Once the M₃B₂ borides nucleate, they grow in the consumption of γ′ phases.

1. Introduction

As a typical Al, Ti, Nb balanced-ratio, nickel-based superalloy, the GH4742 alloy is widely used for the turbine blade. The total mass fraction of the γ′ phase formation elements Al, Ti, and Nb in the GH4742 alloy exceeds 7%, and the equilibrium γ′ phase fraction is close to 40%. Thus, GH4742 is a typical high-alloyed and hard-to-deform alloy [1]. As a precipitation-enhanced nickel-based alloy with high performance, the GH4742 alloy is suitable for preparing key hot, rotating parts, such as heavy gas turbines, with service temperatures below 750 °C under extreme conditions of complex stress for long periods of service [1,2]. Therefore, while boasting a high temperature holding capacity and excellent mechanical properties, it also has the problems of difficult thermoplastic processing and a microstructure highly sensitive to the preparation process. The content of the main elements in the alloy has a great influence on its structural stability and performance.

Nb, the formation element of the γ′ precipitation phase, can take the place of Al and Ti, thus reducing stacking fault energy, improving the high-temperature creep resistance of the nickel-based alloy by enhancing the γ′ precipitation strengthening and the γ matrix solid solution strengthening effect, and refining the grains so that M₂₃C₆ is distributed in a chain shape along the grain boundary [3]. In Ni-based alloys with a high content of niobium, the formation of the γ′-Ni₃Nb phase also greatly improves the yield strength of the alloy [4]. When the Nb contents are larger than 2 wt.%, the ductility and toughness of the alloy sharply decrease. Furthermore, Nb can increase the activity of Al, reducing the solubility of O in the alloy and forming a composite oxide layer with Nb and Al to hinder element diffusion [5,6]. While Nb is also a strong carbide-forming element, its carbides are very stable at high temperatures, which may be disadvantageous to the high-temperature strength of the alloy. Moreover, the content of Nb also has a significant impact on the microstructure of the alloy, which can change the type, morphology distribution, and quantity of grain boundary precipitates and strengthen or weaken the grain boundary [7,8,9].

Another component, B, is also the most widely used microalloying element in high-temperature alloys. B has the most significant impact on the durability and creep properties of high-temperature alloys, usually with an optimal content range [10,11,12]. When the B content is around 0.006 wt.%, the endurance time reaches its peak, which is 3–6 times higher than the original value. The endurance plasticity is also good. However, when the B content is either too high or too low, the endurance time of the alloy decreases [13]. Furthermore, the segregation of B in grain boundaries, such as in vacancy defects of the Ni₃Al boundary, leads to local alloying. Equilibrium segregation is generally limited to a few atomic layers, significantly changing the bonding state of the grain boundary and reducing the diffusion kinetics of elements around the grain boundary, thus strengthening the grain boundary and increasing the bonding force [14]. In addition, the distribution state of B within the crystal can also affect the stability of the γ′ phase and slow down the growth rate of the γ′ phase. B can also affect the precipitation of carbides or some intermetallic compounds in the alloy, improving the dense and uneven state of carbides at grain boundaries, which is beneficial for thermal strength [15]. Meanwhile, excessive B can form low-melting-point eutectic compounds, which have an adverse effect as impurities form. During the aging process, the addition of B exists in the forms of M₃B₂, M₂B, and MB₂, with coarse block, extremely fine block, and needle-like shapes as grain boundary precipitation [16,17,18]. The primary borides rich in Cr-Mo-Nb are relatively stable during heat treatment below 1100 °C and exhibit significant dissolution above 1150 °C [18]. Therefore, when the aging temperature is around 1150 °C, the dissolution of borides is not complete, and there will be residual borides that will precipitate again during the subsequent cooling process [19].

The typical precipitation-strengthening structure of the GH4742 alloy consists of the γ phase and the γ′ phase, together with some MC-type carbides, M₆C-type carbides, TiN inclusions, etc. The γ′ precipitates and the stress field will hinder the movement of dislocations, thus having a strengthening effect. The lattice misfits between the γ′ and γ phases can be significantly changed via even small changes in the composition of the two phases, thereby weakening the strengthening effect, contributing to the instability of the γ′ phase itself, and ultimately leading to a degradation in the alloy performance [3,4]. Therefore, to understand the distribution characteristics of elements at different temperatures, the aging time in alloys is of great significance for further designing the optimal microstructure, heat treatment process, and high-temperature mechanical properties of the alloys. At present, research on the GH4742 superalloy mainly focuses on the smelting process, the law of dendritic segregation, the characteristics of element distribution, and its microstructure after homogenization treatment. Long et al. [1] studied the influence of the heat treatment mechanism on the microstructure and the mechanical properties of the GH4742 superalloy, showing that a lower aging temperature and longer aging time contribute to smaller γ′ precipitates and excellent mechanical properties. Zhang et al. [3] studied the hot deformation behavior and microstructure evolution in the GH4742 superalloy and found that the γ′ phase has a high apparent activation energy during deformation in the γ/γ′ two-phase region, with a decrease in deformation temperature and an increase in strain rate. Lu et al. [2] believed that the poor hot workability of GH4742 was due to the interaction between the γ′ and γ matrix, as well as to the solution strengthening of various alloy elements. Generally speaking, there are two methods for γ′ coarsening: one is to cool it slowly in the air in the two-phase zone and the other is to keep it at a suitable temperature for a long time in a γ + γ′ two-phase zone. The morphology of the microstructure changes greatly after slow cooling, mainly forming a rough, spherical, and irregular petal-shaped γ′ phase. It is observed that the slower the cooling rate is, the higher the concentrations of the phase-forming elements Al, Ti, and Nb are and the easier it is to form rough and irregular shapes. The present work aims to investigate the influence of Nb and B contents on the microstructure characteristics of γ′ precipitates. However, due to the large number of heat treatment process parameters, it is necessary to study the precipitation process by means of phase-field simulation in order to assist the alloy design, and so as to further understand the strengthening mechanism and provide theoretical guidance for improving the properties at high temperatures.

After rapid development over the past decades, phase-field modeling has become a convenient tool for investigating the microstructure evolution process in alloy systems [20,21,22,23]. Even for Ni-based alloys, there also exists a large quantity of phase-field simulations in the literature. Warnken [24] employed the multi-phase-field (MPF) model to study the solidification and dissolution process. The major contribution is from Wang’s research group, which has made a series of phase-field simulations of the solid-state phase transformation, including precipitation, coarsening, etc. [25,26,27,28]. Wu et al. carried out some phase-field simulations of the inter-diffusion process [29]. Moreover, a large-scale 3-D phase-field modeling of γ′-rafting and creep behavior was performed by Zhou et al. [30]. The above excellent work indicates the important role of the phase-field method in promoting the design and research of high-temperature alloys. Nevertheless, very little phase-field simulation research has focused on the precipitation process in high-alloyed, industrial, Ni-based superalloys, such as GH4742, in which the interaction of the complicated alloy composition needs careful investigation and understanding.

Consequently, the aging process of the GH4742 alloy will be quantitatively simulated by an MPF (multi-phase-field) model, with the aid of MICRESS (Microstructure Evolution Simulation Software) coupling to CALPHAD (Calculation of Phase Diagram) databases. The phase-field modeling method used in the present work is described in Section 2, and the database coupling details and other parameters necessary for the simulation are demonstrated in Section 3 and Section 4. Finally, the phase-field simulation of the precipitation process of the γ′ phase in the γ matrix in GH4742 alloys with different Nb compositions (2.1 wt.%, 2.4 wt.%, 2.9 wt.%) is presented in Section 5 to investigate the influence of the alloy compositions of Nb and B. The results are compared with the experimental results.

2. MPF Model

A reliable model is necessary to quantitatively describe the microstructural evolution. Here, the MP model is used for the present simulation. For the present description of the precipitation process, the functional energy consists of the following three parts [31,32,33,34],

$$F={\displaystyle {\int }_{\Omega }{f}^{\mathrm{intf}}+f^{\mathrm{chem}}+f^{\mathrm{elast}}}$$

(1)

where the specific formula of interfacial energy density $f^{\mathrm{intf}}$, chemical free-energy density $f^{\mathrm{chem}}$, and elastic free-energy density $f^{\mathrm{elast}}$ are presented as follows,

$$f^{\mathrm{intf}}={\displaystyle \sum _{\alpha ,\beta =1,\dots ,N,\alpha \ne \beta }\frac{4{\sigma }_{\alpha \beta }}{{\eta }_{\alpha \beta }}\left\{-\frac{{\eta }_{\alpha \beta }^2}{{\pi }^2}\nabla {\varphi }_{\alpha }\cdot \nabla {\varphi }_{\beta }+{\varphi }_{\alpha }{\varphi }_{\beta }\right\}}$$

(2)

$$f^{\mathrm{chem}}={\displaystyle \sum _{\alpha =1,\dots ,N}h({\varphi }_{\alpha })f_{\alpha }(c_{\alpha }^i)+{\mu }^i\left(c^i-{\displaystyle \sum _{\alpha =1}^N{\varphi }_{\alpha }{c}_{\alpha }^i}\right)}$$

(3)

$$f^{\mathrm{elast}}=\frac{1}{2}\left\{{\displaystyle \sum _{\alpha =1,\dots ,N}h({\varphi }_{\alpha })({\overline{\epsilon }}_{\alpha }-{\overline{\epsilon }}_{\alpha }^{*}){\overline{C}}_{\alpha }({\overline{\epsilon }}_{\alpha }-{\overline{\epsilon }}_{\alpha }^{*})}\right\}$$

(4)

where ${\varphi }_{\alpha }$ denotes the phase field of α phase. The total number of phases in the system is indicated by N, and the sum of the phase fields always satisfies the constraints ${\displaystyle \sum _{\alpha =1,\dots ,N}{\varphi }_{\alpha }}=1$. The interfacial energy ${\sigma }_{\alpha \beta }$ and interfacial width ${\eta }_{\alpha \beta }$ are the input parameters to describe the interfacial properties of α/β phases. As for the chemical free-energy part, one monotonic coupling function $h({\varphi }_{\alpha })$ is included to make up the concentration $c_{\alpha }^i$ -related bulk chemical free-energy density $f_{\alpha }(c_{\alpha }^i)$ together. Where the diffusion potential of component i is derived as a Lagrange multiplier to make sure the mass balance condition is fulfilled, $c^i={\displaystyle \sum _{\alpha =1,\dots ,N}{\varphi }_{\alpha }{c}_{\alpha }^i}$. Considering the present case for the precipitation process, elastic stress field coupling is necessary, in which the total elastic energy is split into the summation of the elastic energy for individual phases based on Hooke’s law, where ${\overline{\epsilon }}_{\alpha }$ denotes the total strain tensor in phase α, ${\overline{\epsilon }}_{\alpha }^{*}$ denotes the eigenstrain of α phase, and ${\overline{C}}_{\alpha }$ denotes the elasticity stiffness coefficient matrix. The kinetic evolution equations for the principal variables, phase field ${\varphi }_{\alpha }$, concentration $c^i$, and strain tensor ${\overline{\epsilon }}_{\alpha }$ can then be solved numerically based on the variations in the above functions [31,32,33,34] as follows,

$$\frac{d{\varphi }_{\alpha }}{dt}={\displaystyle \sum _{\beta =1,\dots ,N}{\mu }_{\alpha \beta }\left\{{\sigma }_{\alpha \beta }\left[{\varphi }_{\beta }{\nabla }^2{\varphi }_{\alpha }-{\varphi }_{\alpha }{\nabla }^2{\varphi }_{\beta }+\frac{{\pi }^2}{2{\eta }_{\alpha \beta }^2}\left({\varphi }_{\alpha }-{\varphi }_{\beta }\right)\right]+\frac{\pi }{{\eta }_{\alpha \beta }}\sqrt{{\varphi }_{\alpha }{\varphi }_{\beta }}\Delta G_{\alpha \beta }\right\}}$$

(5)

$$\frac{d{c}^i}{dt}=\nabla {\displaystyle \sum _{\alpha =1,\dots ,N}{\varphi }_{\alpha }{M}_{\alpha }\nabla {\mu }_{\alpha }^i}$$

(6)

$$0=\nabla {\displaystyle \sum _{\alpha =1,\dots ,N}{\varphi }_{\alpha }{\overline{C}}_{\alpha }({\overline{\epsilon }}_{\alpha }-{\overline{\epsilon }}_{\alpha }^{*})}$$

(7)

in which ${\mu }_{\alpha \beta }$ denotes the interfacial mobility and $M_{\alpha }$ denotes the atomic mobility in the α phase, which can then be linked to the kinetic mobility databases. ${\mu }_{\alpha }^i$ denotes the diffusion potential of component i in phase α.$\Delta G_{\alpha \beta }$ denotes the total driving force of the phase transformation, including the chemical driving force $\Delta G_{\alpha \beta }^{\mathrm{chem}}$ and elastic driving force $\Delta G_{\alpha \beta }^{\mathrm{elast}}$:

$$\Delta G_{\alpha \beta }^{\mathrm{chem}}=-f_{\alpha }(c_{\alpha }^i)+f_{\beta }(c_{\beta }^i)+{\tilde{\mu }}^i\left(c_{\alpha }^i-c_{\beta }^i\right)$$

(8)

$$\Delta G_{\alpha \beta }^{\mathrm{elast}}=({\overline{\epsilon }}_{\alpha }^{*}-{\overline{\epsilon }}_{\beta }^{*}){\overline{s}}_{\alpha }$$

(9)

The chemical driving force $\Delta G_{\alpha \beta }^{\mathrm{chem}}$ can be obtained by coupling to the CALPHAD thermodynamic databases. The formula ${\overline{s}}_{\alpha }={\overline{C}}_{\alpha }({\overline{\epsilon }}_{\alpha }-{\overline{\epsilon }}_{\alpha }^{*})$ indicates the stress tensor.

3. Coupling to CALPHAD Databases

By coupling to real CALPHAD databases, a quantitative description of the microstructural evolution process can be realized. The phase-field simulation can be performed with temperature- and composition-dependent chemical free-energy and atomic mobility or kinetic diffusivities. In the current case, the coupling to the CALPHAD thermodynamic database is realized via the TQ interface built into the MICRESS code.

The thermodynamic database from TCNI9 [35] is employed in the present work to provide the chemical driving force and diffusion potentials for the Ni-based superalloy GH4742 during the present phase-field simulation. All the initial alloy compositions are also appended in

Table A1. Alloy composition for the present simulated GH4742 alloy.

Component	Weight Percent wt. %
Cr	14.000	Nb	2.100	Mo	5.050
Co	10.000	B	0.008	Si	0.100
Al	2.750	C	0.040	W	0.200
Ti	2.700	Fe	0.020	-	-

in Appendix A, which has been simplified as compared with the industrial alloy composition due to the thermodynamic database but includes most of the principal elements. In this appended table, the content of Mo is approximately 5%, which could improve the high-temperature strength and creep resistance of alloys due to its excellent solid solution-strengthening effect and good corrosion resistance. As an Al-Ti-Nb balanced-ratio alloy, the similar content of Ti to Nb contained in the alloy, which is also a kind of γ′ precipitation-strengthening element, contributes to the creep rupture strength. As for the diffusivities of different components in the γ and γ′ phases, they are taken directly from the averaged experimental data over the investigated composition range and are also calculated based on the atomic mobility database [36,37]. Moreover, these diffusivities are simply assumed to be independent of composition.

Considering that Nb is one of the most important elements in the GH4742 alloy, especially for the precipitation of the γ′ phases, then the contents of the Nb element are adjusted in the ranges of 2.1 wt.%, 2.4 wt.%, and 2.9 wt.% around their reasonable limits to investigate its influence on the microstructural evolution of the γ/γ′ phases through a series of phase-field simulations at 1053 K.

4. Other Thermophysical and Numerical Parameters

4.1. Interfacial Energy and Interfacial Mobility

In the present work, the target γ and γ′ phases are assumed to be isotropic in the GH4742 alloy. Considering that the temperature is constant in the present simulation, the interface energy between the γ and γ′ phases was set at 5 × 10⁻⁶ J/cm² according to our previous work [38]; meanwhile, the interface mobility was set at 2.703 × 10⁻¹¹ cm⁴/Js to guarantee the diffusion-controlled transformation [33].

4.2. Lattice Misfit and Elastic Constant

The input eigenstrain parameter, ${\overline{\epsilon }}_{\alpha }^{*}$, used in Equations (7) and (9)—namely the lattice mismatch between γ′ phase and γ phase—can be calculated as a function of lattice constants,

$$\delta =\left(a_{\gamma \prime }-a_{\gamma }\right)/a_{\gamma }$$

(10)

where $a_{\gamma \prime }$ and $a_{\gamma }$ are the lattice constants of the γ′ phase and the γ phase, respectively, which are generally composition- and temperature-dependent parameters. The experimental lattice parameters of the γ and γ′ phases of the GH4742 alloy in the literature are rather limited. In the present simulation, the lattice parameters are considered as constant values. According to the research results of Long et al. [39], the lattice parameters at 780 °C (1053 K) used in the present simulation are calculated as a_γ = 0.36143 nm and a_γ′ = 0.36210 nm.

$$\delta =\frac{a_{{\gamma }^{\prime }}-a_{\gamma }}{a_{\gamma }}=\frac{0.36210-0.36143}{0.36143}=0.0018537$$

(11)

The elastic coefficients of the γ and γ′ phases were set based on either experimental data or first-principle calculated data from Fahrmann et al. [40] and Meher et al. [41]. The elastic coefficients of the γ phase and γ′ phase used in the present simulation at 1053 K (GPa) are $C_{11 }^{\gamma }$= 154.000, $C_{12}^{\gamma }$ = 103.000,$C_{44}^{\gamma }$ = 81.000, $C_{11}^{\mathsf{\gamma }\prime }$= 173.468, $C_{12}^{\mathsf{\gamma }\prime }$ = 120.393, and$C_{44}^{\mathsf{\gamma }\prime }$ = 90.168.

4.3. Numerical Parameters and Initial Conditions

All the modeling conditions and parameters were chosen to be consistent with practical heat treatment conditions. A two-dimensional (2-D) region with 200 × 200 grids (grid spacing: 0.1 μm (1 × 10⁻⁷ m)) was set up as the modeling domain, i.e., the actual simulation area is 20 × 20 μm. Then, the reasonable interface width with four grids was 0.4 μm (4 × 10⁻⁷ m). The periodic boundary condition was set for both the phase field and the concentration field. Moreover, the normal expansion boundary condition was set for elastic stress calculation. The other numerical simulation parameters are listed in

Table 1. List of the numerical and materials parameters for the present phase-field simulations.

Parameters	Symbol	Value
Interface thickness	η	0.4 μm
Grid spacing	dx	0.1 μm
Interfacial energy	σ	5 × 10−6 J/m2
Interfacial mobility	μ	2.703 × 10−11 m4/Js
Eigenstrain	ɛ*	0.0018537
Elastic coefficient	Cij	[40,41]
Diffusivities in γ phase	Dγ	[36]
Driving force	ΔG	TCNI9 database [35]

.

5. Results and Discussion

5.1. Microstructure Evolution of GH4742 Alloy during Aging

The precipitation process in the GH4742 alloy with three different Nb compositions (2.1 wt.%, 2.4 wt.%, 2.9 wt.%) at 1053K was investigated using the phase-field simulations. While Nb is varied, the matrix alloy element Ni makes up the balance as the content of Nb is adjusted slightly around the usual nominal composition of the industrial alloy, which could be reasonable based on the practical experimental conditions. The initial state started with the supersaturated γ matrix phase, and then the γ′ phase precipitated during the aging process; the so-called ‘seed density model’ developed by Böttger et al. [42] was employed for simulating the nucleation of γ′ from the γ matrix in the present work. The microscopic morphology evolution results are shown in Figure 1. The γ′ phase precipitated from the supersaturated γ matrix gradually and then transformed into a cuboidal shape due to the coherent strain between the γ and γ′ phases, as while the lattice mismatch is not so large in the multicomponent alloy, the cubic morphology is not so obvious.

As shown in Figure 2, the simulated results aging at 780 °C with three different Nb contents are compared with the experimental results aging at 700 °C for 500 h (Figure 2d). The morphology of the precipitation phase is not regular; it is nearly square with the effect of the elastic strain field and the precipitate size is random, which can be considered to better reflect the morphology characteristics of precipitates during the practical aging process. In addition, the simulation results with three different Nb contents are compared and analyzed. The morphology shows negligible differences. It is speculated that the influence on the morphology is not obvious when the Nb content changes in a small range. Under the condition of a certain phase fraction calculated via thermodynamic equilibrium, the simulated precipitate sizes are highly sensitive to the initial particle number density of the precipitate phase. Consequently, the larger nucleation density is close to the experimental condition, which contributes to the approached precipitate size as shown in the experiments. Considering that the statistical law of the average precipitate size is not the focus of the present work, the initial particle number density was, therefore, set lower in order for higher computational efficiency, which led to a relatively large grain size compared to the experimental condition.

Besides the above Nb concentration field, the simulated concentration evolution of the other two balanced-ratio components in the alloy, Al and Ti, aged at 1053 K (see Figure 3). The concentration field evolution is presented in Appendix A Figure A1. With the increase in simulation time, the composition of Al and Ti in the precipitation phase keeps decreasing but is always higher than the concentration in the matrix phase. The reason is that Nb, Al, and Ti are the formation elements of the γ′ phase. As important constituents of the phase, their diffusion rate is slow. The solute diffusion shows the competitive effects due to the surrounding multiple precipitates. The concentration evolution of the doped elements Fe and Si are also presented in Figure 4, whose concentration field evolution is also available in Appendix A Figure A2. Fe atoms are segregated at the γ/γ′ interface and the concentration of Si near the precipitate appears to exceed the mean concentration, while the overall concentration is conserved.

As shown in Figure 5, the concentration of component W evolves with the aging time. The variation range of the concentration increase is very limited, but its concentration distribution has a sudden change around the phase interface. The concentration of W in the matrix phase remains unchanged, while the distribution inside the precipitated phase has shown a complicated variation characteristic. There are two inflection points in the concentration field of the precipitated phase. With the increase in simulation time, the concentration gradient at the interface becomes larger and larger. The concentration distribution inside the precipitation phase is not uniform, but there exist two concentration peaks. It is speculated that component W has shown an interesting segregation or solute atom aggregation behavior inside the precipitation phase, which is mainly due to the slow diffusion rate of the W component.

According to the analysis in Figure 5, as an element with less content in the GH4742 alloy, B is mainly distributed in the matrix phase, showing a non-significant variation, indicating that the concentration field of B has been less affected by the competitive diffusion among the multiple precipitates. On the one hand, the content is less; on the other hand, B is basically absent in the precipitation phase and evenly distributed in the matrix phase.

The thermodynamically calculated precipitation volume fraction curve versus temperature in the GH4742 alloy based on [35] has been shown in Figure 6a. To ensure the reliability of the present simulation results, it is also necessary to make a comparison between the phase-field simulations and the thermodynamic calculations via the database. The phase-field-simulated volume fractions of the γ and γ′ phases at 1053 K with 2.1 wt.% Nb contents are presented in Figure 6b.

Comparing the thermodynamic calculation and the phase-field simulation results in Figure 6, the former calculated volume fraction of the γ′ phase is 34.9%, while the simulated one is 47%. Such a large difference can be attributed to two aspects: firstly, the thermodynamic calculated result is a volume fraction. Secondly, the thermodynamic calculations based on the database correspond to equilibrium conditions without considering the interfacial energy and elastic effects, while the experimental measurement or phase-field simulation corresponds to the non-equilibrium state. The latter is a dynamic change process, but after a long enough time, it will tend toward the result of chemical thermodynamic equilibrium.

As shown in Figure 7, it can be clearly seen that with the increase in simulation time, the internal stress of the precipitated phase is uniformly distributed. However, the stress gradually decreases, while the stress distribution in the matrix phase far from the precipitated phase gradually increases. This could be explained by the elastic energy being proportionate to the volume of the precipitate. Moreover, the elastic energy is anisotropic, and in this specific case, the elastically soft directions are <100>, <010>, and <001>. After comparing the stress distribution of multiple grains with that of single crystal grains, it is found that the stress field distribution of multiple grains is closer to the sine function curve. The reason is that the stress field of multiple precipitates may form a stress field similar to a sine function after superposition. The stress in the matrix phase increases with the increase in simulation time. It is concluded that the stress distribution around the precipitate changes due to the continuous growth of the precipitate, and the interaction with the stress field of other precipitates causes a certain stress in the matrix.

5.2. Effect of Different Nb Compositions on Precipitation Rate

In general, the precipitation rate of the precipitated phase can be analyzed according to the growth rate of particle diameter. However, because there are many particles in the simulation and the variation range is too small, the simulation with different Nb contents has a non-significant difference and, thus, the change of particle diameter cannot be accurately measured and distinguished. Therefore, to ensure the accurate reflection of the precipitation rate of the γ′ phase, the change in the total volume fraction over time is used as the standard to indicate the precipitation rate. Choosing t = 20,000 s as the initial time step, the total volume fraction change in the precipitated phase was counted within t = 125,000 s, and then the volume fraction increment of the γ′ phase was obtained every 20,000 s, as shown in Figure 8a. Considering that the phase fraction variations at a higher time resolution lower than 20,000 s nearly present similar trends, three representative time steps were chosen to illustrate the effects of Nb. At the initial stage, the phase fraction kept growing linearly and the precipitation rate kept accelerating. The higher the Nb content is, the faster the precipitation rate is. The reason for this is that the higher the content of the Nb element is, the greater the concentration gradient formed between the matrix and the precipitation phase is; the greater the driving force is, the greater the acceleration of the precipitation rate of the γ′ phase is. However, with the increase in simulation time, the precipitation rate decreases gradually, which is because as the phase volume fraction gradually reaches the equilibrium value, the precipitation becomes slower. According to the previous morphology analysis, the abrupt changes in Figure 8 at ~80,000 s may be caused by the beginning of the ripening process; the precipitation growth process has been slowed down and the elastic field around the γ′ precipitate becomes stronger since the elastic energy is proportionate to the volume of the precipitate. The Nb contents definitely indicate some influence on the kinetic process. The precipitation rate change law reflected in the above figure can be explained well by the principle of phase transformation dynamics, which also proves the reliability of the present field simulation results.

In order to study the influence of different Nb contents on the volume fraction of γ′ precipitates, phase-field simulation results with three different Nb contents (2.1 wt.%, 2.4 wt.%, 2.9 wt.%) have been compared together. Figure 8b shows the variation law of the phase fraction for three different Nb components vs. time. It can be seen that with the increasing simulation time, the Nb contents are higher and the phase fraction is higher. According to the research literature, it is known that Nb is an important formation element of the γ′ phase. This means that Nb strongly promotes the precipitation of the γ′ phase. Therefore, the higher Nb content is beneficial to γ′ precipitation; consequently, the volume fraction of the γ′ precipitated phase is larger.

5.3. Effect of Different Nb Compositions on the Morphology of the Precipitates

The statistical distribution of the grain size has been analyzed based on the simulation results with three different Nb contents using Image-Pro software. Moreover, the influence of Nb contents on grain size has been further studied according to these results. As shown in Figure 9, for Nb contents of 2.1 wt.%, the grain sizes are concentrated in the range of 1.0–1.5 μm. In addition, some precipitates are still observed with smaller grain sizes. By comparing the simulation results between 2.4 wt% Nb and 2.9 wt% Nb, it is found that the grain size distributions are basically similar to each other. According to the above analysis, the grain size is affected by the Nb content. Within the present investigated range, the larger Nb content leads to a larger grain size. However, the comprehensive effects are not so significant. From the simulation results of 2.4 wt.% and 2.9 wt.%, it can be seen that the influence effect has been slightly reduced.

5.4. Phase-Field Simulations of the Boride Precipitation

According to the thermodynamic calculation results in Figure 10, there exist three types of borides—M₃B₂, MB₂_C32, and M₂B_{_}TETR—within the present temperature range. Only M₃B₂ precipitated at 1053 K, while the two other types of borides, i.e., M₃B₂ and MB₂_C32, precipitated at 1273 K. According to the thermodynamic calculations, the equilibrium volume fraction of M₃B₂ is 0.915% at 1053 K. Above 1300 K, as shown in the figure, only MB₂ precipitated.

Figure 11 further presents the effect of B contents on the main diffusion coefficients of the major γ′ formation components in the γ and γ′ phases based on thermodynamic calculations [35]. It is indicated that the diffusivities of Al in the γ′ phase and Ti in both phases decrease with the increase in the B contents and that the diffusion of Al then accelerated with the addition of B after 7 wt.% B. Meanwhile, the diffusion rate of Nb is basically unaffected by B contents. The diffusion coefficient of Al in the γ phase presents an upward trend with an increase in B contents. Therefore, it is hard to determine whether the B component accelerates or slows down the diffusion rate of the components in terms of chemical energy. Correspondingly, with the increased addition of B, the formation of borides may have a pinning effect on the grain boundary and hinder the kinetic process.

Starting from the thermodynamic calculations, the 2-D phase-field simulation of the precipitation process at 1053 K was performed, during which only M₃B₂ borides were observed to precipitate around the interface between the γ′ and γ phases. Boron is the main grain boundary-strengthening solute in superalloys, which usually tends to aggregate at grain boundaries and react with refractory elements to form borides, increasing the binding force in these areas and reducing the diffusion kinetics of the elements around the grain boundaries. This functions to prevent grain boundary slip and inhibit the connection and expansion of grain boundary pores, thereby improving the stability of the alloy structure. The concentration distribution and phase information at t = 75,000 s have been shown in Figure 12. As shown in the figure, the M₃B₂ boride is rich in Cr, B, C, Fe, Mo, and Si. Moreover, the components Cr, Fe, Mo, and W present a significant diffusion behavior of grain boundary segregation, while the diffusion of carbon shows an opposite tendency. The simulated phase fraction of M₃B₂ boride is about 0.292% at the present time step, compared to the previous equilibrium value in Figure 10, where the aging process was not at a final equilibrium state. According to the simulation results, the M₃B₂ borides are prone to precipitate around the interface rather than in the bulk phase, and the precipitation of borides considerably slows down the precipitation rate of the γ′ phases, which leads to finer precipitate distribution, further confirming the previous inference that the precipitation of borides may have a pinning effect that hinders the kinetic process. Once the M₃B₂ borides nucleate, they may grow in the consumption of the γ′ phases.

6. Conclusions

In the present work, a series of phase-field simulations for the precipitation process of the γ′ phase in the GH4742 alloy were performed by coupling with the CALPHAD database. The morphology evolution, concentration distribution, and thermodynamic properties were analyzed, which hopefully accelerate the experimental design of Ni-based superalloys, taken as potential microstructural data storage for machine learning. Moreover, the effects of B and Nb behavior on the precipitation strengthening in the Al-Ti-Nb balanced-ratio alloy have been discussed, and the major conclusions are summarized as follows.

First, higher Nb contributes to the faster precipitation rate of the γ′ phase. Furthermore, the phase fraction of precipitates increases along with the increase in Nb contents. The larger γ′ phase amount results in a smaller grain size and better mechanical properties. The segregation of the W element in the γ′ precipitate due to its sluggish diffusion effect has been observed, which may lead to segregation or grain boundary defects.

Second, higher temperatures and lower B contents accelerate the dissolution of boride and reduce the precipitation of borides. With the increased addition of B, the formation of borides may have a pinning effect on the grain boundary, which hinders the kinetic process. Moreover, borides are prone to precipitate around the interface rather than in the bulk phase, as once the M₃B₂ borides nucleated, they would grow in the consumption of the γ′ phases.