Effect of Different Cooling Media After Solid Solution on the Microstructure and Yield Strength in a Ni-Al Alloy During Aging: Experimental Measurement and Computational Modeling

Abstract

In this paper, the effect of different cooling media, i.e., water quenching, air cooling and furnace cooling, after solid solution treatment on the microstructure and yield strength of Ni-15.9Al at. pct alloy during aging at 800 °C was first experimentally investigated. It was found that the morphologies and the particle sizes of γ′ precipitates as well as the yield strength of the target alloys during aging were strongly affected by the cooling media after solid solution. The yield strengths of the target alloys after aging with water quenching and air cooling after solid solution are similar, and higher than that with furnace cooling. By further considering the cost and environment factors, the air cooling after solid solution treatment was thus proposed for industry alloys. Meanwhile, a quantitative simulation of the microstructure evolution in the target alloy during aging was realized by means of phase-field modeling coupling with CALPHAD thermodynamic and atomic mobility descriptions. Moreover, the extracted experimental microstructure of the Ni-15.9Al at. pct alloy with air cooling after solid solution was inputted as the initial microstructure of phase-field simulation. Subsequently, the microstructural features obtained from both phase-field simulations and experiments were imported into the strengthening models to predict the evolution of the total yield strengths during aging. The model-predicted total yield strengths in the Ni-15.9Al at. pct alloy were found to be in the excellent agreement with the experimental results from the tensile tests.

Introduction

Ni-based superalloys possess exceptional combinations of high-temperature strength, toughness, inherent oxidation and hot corrosion resistance and are thus widely used in aircraft, power-generation turbines and rocket engines.[1,2,3] During the manufacture of Ni-based superalloys components, different heat treatments are usually used to obtain optimal mechanical properties suitable for the application by tailoring the size, distribution and morphology of the microstructures in Ni-based superalloy.[4,5] The common heat treatments include solid solution, aging and so on. Up to now, numerous experimental and theoretical investigations focusing on the effect of solid solution and aging temperature/time on different properties of Ni-based superalloys have been reported.[6,7,8] In fact, either in laboratory experiments or in industrial manufactures, one cooling medium like water quenching (WQ), air cooling (AC) or furnace cooling (FC) should be chosen after solid solution. Such cooing medium has a genetic effect on the evolution of the microstructure during the subsequent aging process and then the properties of the final Ni-based superalloys.[9] However, there is little research work on the effect of different cooling media in the literature. Therefore, it is very meaningful to remedy this situation.

Systematic experimental investigations are typical labor- and money-consuming. As the rapid development of computational material science, the underlying computation modeling/simulation may serve as an effective and efficient way to investigate microstructural evolution as well as predict the correlative properties in materials during the preparation and service processes.[10,11,12] For the past decades, the phase-field simulation coupling with the accurate CALPHAD thermodynamic databases[13] has become one effective approach to realize the quantitative simulation of microstructure evolution, i.e., in Ni-based superalloys during different preparation processes.[14,15,16,17,18] Meanwhile, the strengthening models[19,20,21] can predict the mechanical properties of the materials with the microstructure information, including grain size, its distribution, phase fraction and compositions, which can be also provided by the quantitative phase-field simulations. Thus, it is very promising to effectively establish the “composition/process—microstructure—property” relation for the target materials by combining the quantitative phase-field simulation with the strengthening model.

Consequently, a binary Ni-Al alloy, Ni-15.9 at. pct Al is chosen as the target in the present work, and the major aims are (i) to perform the experimental investigation of the effect of different cooling media, including WQ, AC and FC, after solid solution on the microstructure and yield strength in the target binary alloy during aging process; (ii) to conduct the phase-field simulation of the microstructure evolution of in Ni-15.9Al at. pct alloy during aging process at 800 °C by using the phase-field simulation coupling with the CALPHAD-type thermodynamic and atomic mobility descriptions. Moreover, the real experimental structures of the target Ni-Al alloy after solution treatment with air cooling were read into MICRESS code as the initial microstructures for the subsequent phase-field simulation, and (iii) to predict the yield strength of the target Ni-Al alloy after solid solution with air cooling during aging process at 800 °C by linking the experimental and phase-field simulated microstructure information to the strengthening models.

Experimental Procedure

The nominal Ni-15.9 at. pct Al alloy was melted from high-purity Al (purity: 99.95 pct) and Ni (purity: 99.99 pct) in a vacuum medium frequency induction melting furnace (CX2G-0.5, Shanghai Chen Xin Electric Furnace Co., Ltd, Shanghai, China) using a graphite crucible, and cast in a steel mold. The sheet samples were cut from the castings and encapsulated in vacuum quartz. Then, the samples were annealed at 1100 °C for 48 hours and 1160 °C for 16 hours in a high-temperature furnace (GSL-1700X, Hefei Kejing Materials Technology Co., Ltd, Hefei, China), to improve homogenization, followed by air cooling. Afterwards, the samples were divided into three parts and solid solution treated in the GSL-1700X furnace at 1280 °C for 2 hours, followed by WQ, AC and FC, respectively. Subsequently, the samples were subject to aging in the GSL-1700X furnace at 800 °C for 4 to 16 hours, followed by WQ. The samples for microstructural study were polished by standard metallographic procedures and etched in a solution of 4g CuSO4 + 20 mL HCl + 20 mL H2O ethanol. The experimental microstructures were characterized by scanning electron microscope (SEM) on a Nova Nano SEM 230 (FEI Electron Optics B.V, Czech). The microstructural features were quantitatively acquired from the image analyzer software (Image J). Moreover, the samples were machined into the plate-type tensile specimens shown in Figure 1, and the room-temperature tensile tests were conducted on the specimens by a universal testing machine (Instron3369) with a loading strain rate of 10−4 s−1. Three identical tensile specimens were tested under each condition, and the tensile failure for all the specimens occurs within the gage length.

Fig. 1
figure1

Schematic geometry of the tensile sample (unit: mm)

Model Description

Phase-Field Model

In this work, the multi-phase-field (MPF) model incorporated in the MICRESS (MICRostrcture Evolution Simulation Software)[22] was employed to perform the microstructure evolution in the target Ni-Al alloys during aging process. In the framework of MPF model, the free energy functional for a binary two-phase system during aging process can be constructed as[23]

$$ F = \int {f{}^{\text{int \,f}} + f{}^{\text{chem}} + f{}^{\text{elast}}} $$
(1)
$$ f{}^{\text{int \,f}} = \frac{{4\sigma_{\alpha \beta } }}{{\eta_{\alpha \beta } }}\left\{ { - \frac{{\eta {}^{{{}^{2}}}_{\alpha \beta } }}{{\pi {}^{2}}}\nabla \phi_{\alpha } \cdot \nabla \phi_{\beta } + \phi_{\alpha } \phi_{\beta } } \right\} $$
(2)
$$ f^{\text{chem}} = \sum\limits_{\alpha = 1,2} {h\left( {\phi_{\alpha } } \right)} f_{\alpha } \left( {c_{\alpha }^{i} } \right) + \tilde{\mu }^{i} \left( {c - \sum\limits_{\alpha = 1,2}^{N} {\phi_{\alpha } c_{\alpha }^{i} } } \right) $$
(3)
$$ f^{\text{elast}} = \frac{1}{2}\left\{ {\sum\limits_{\alpha = 1,2} {h\left( {\phi_{\alpha } } \right)\left( {\bar{\varepsilon }_{\alpha } - \bar{\varepsilon }^{ * }_{\alpha } - c_{\alpha }^{i} \bar{\varepsilon }_{\alpha }^{i} } \right){{\overline{\overline {C}}}_{\alpha}} \left( {\bar{\varepsilon }_{\alpha } - \bar{\varepsilon }_{\alpha }^{*} - c_{\alpha }^{j} \bar{\varepsilon }_{\alpha }^{j} } \right)} } \right\} $$
(4)

Based on the above free energy functional, the following governing equations can be derived:

$$ \dot{\phi }_{\alpha } = \mu_{\alpha \beta } \left\{ {\sigma_{\alpha \beta } \left[ {\phi_{\beta } \nabla^{2} \phi_{\alpha } - \phi_{\alpha } \nabla^{2} \phi_{\beta } + \frac{{\pi^{2} }}{{2\eta_{\alpha \beta }^{2} }}\left( {\phi_{\alpha } - \phi_{\beta } } \right)} \right] + \frac{\pi }{{\eta_{\alpha \beta } }}\sqrt {\phi_{\alpha } \phi_{\beta } } \Delta G_{\alpha \beta } } \right\} $$
(5)
$$ \dot{c}^{\text{i}} = \nabla \sum\limits_{\alpha = 1,2} {\phi_{\alpha } } M_{\alpha } \nabla \tilde{\mu }_{\alpha }^{\text{i}} $$
(6)
$$ 0 = \nabla \frac{\delta F}{\delta \varepsilon }\nabla \sum\limits_{\alpha = 1,2} {\phi_{\alpha } } {{\overline{\overline {C}}}_{\alpha}} \left( {\bar{\varepsilon }_{\alpha } - \bar{\varepsilon }_{\alpha }^{ * } - c_{\alpha }^{\text{i}} \bar{\varepsilon }_{\alpha }^{\text{i}} } \right) $$
(7)

where \( f^{\text{int f}} \) is the interfacial energy density, \( f{}^{\text{chem}} \) the chemical free energy density and \( f{}^{\text{elast}} \) is the elastic energy density. \( \phi_{\alpha } \) is the phase field value of α phase/grain and one can have the sum constraint \( \sum\nolimits_{\alpha = 1, \ldots ,N} {\phi_{\alpha } } = 1 \). \( h\left( \phi \right) \) is a monotonous coupling function. \( f_{\alpha } \left( {c_{\alpha } {}^{\text{i}}} \right) \) is the bulk free energy density of the individual phases depended on the phase concentrations \( c_{\alpha } {}^{\text{i}} \). \( \sigma_{\alpha \beta } \) is the interfacial energy between phase/grain α and β. \( \eta_{\alpha \beta } \) is the interface width and \( \mu_{\alpha \beta } \) is the interfacial mobility. \( \Delta G_{\alpha \beta } \) is the local deviation from thermodynamic equilibrium. \( \tilde{\mu }{}^{\text{i}} \) is the diffusion potential of component i, which is introduced as a Lagrange multiplier to conserve the mass balance between the phases \( c{}^{\text{i}} = \sum\nolimits_{\alpha = 1, \ldots ,N} {\phi_{\alpha } c_{\alpha } {}^{\text{i}}} \). \( \bar{\varepsilon }_{\alpha } \) is the total strain tensor in phase α, \( \bar{\varepsilon }_{\alpha }^{ * } \) is the Eigenstrain of transformation, \( \bar{\varepsilon }_{\alpha }^{\text{i}} \) is the chemical expansion of component i in Vegard approximation, \( {{\overline{\overline {C}}}_{\alpha}} \) is the elasticity or Hook's matrix.

In this study, we performed the phase-filed simulations in a two-dimensional (2-D) domain of 400 × 400 grids with the grid spacing of 10 nm. The periodic boundary condition was utilized. The simulation temperature was 800 °C. The lattice misfit between the γ and γ′ phases was 0.00377.[16] The elastic constants[16] for γ and γ′ phases were C11 = 207.5 GPa, C12 = 148.5 GPa, C44 = 93.5 GPa for γ phase, C11 = 197.8 GPa, C12 = 138 GPa, C44 = 99.2 GPa for γ′ phase. The interfacial energy of γ/γ′ was set to be 80 mJ/m2, while the interfacial energy of γ′/γ′ was set to be three times higher than that of γ/γ′ interface in order to prevent the coalescence of the γ′ precipitates.[24] The interfacial mobility was evaluated as 2.66 × 10-9 cm4 /J s according to Eq. [35] proposed by Steinbach[23] to guarantee the diffusion-controlled transformation.

The thermodynamic[25] and atomic mobility[26] descriptions of γ and γ′ phases in the Ni-Al system were used in the simulation. The linking to the CALPHAD thermodynamic and atomic mobility databases is attained via the TQ interface incorporated in MICRESS code.

It should be noted that the major difference between the present work and previous reports like Reference 16 lies in that the real structures in binary Ni-Al alloys after solution treatment were digitized and read into MICRESS code as the initial microstructure for phase-field simulation of the aging process as performed in Reference 27, instead of a single-crystal γ matrix in literature reports.[16,28]

Strengthening Model

The strengthening models were then utilized to predict the variation of mechanical properties corresponding to the microstructure evolution due to the phase-field simulation as well as the experimental measurement. In the present work, the yield strength of Ni-Al alloys was chosen for a representative demonstration. According to the classic strengthening model,[19,20,21] the yield strength σy in the annealed polycrystal materials is constituted by grain boundary strengthening σg, solid solution strengthening σss and precipitation strengthening σp (σp=MΔτ0, where M is the Taylor orientation factor (i.e., equals to 3 here), and Δτ0 is the critical resolved shear stress increment),[29]

$$ \sigma_{\text{y}} = \sigma_{\text{g}} + \sigma {}_{\text{ss}} + \sigma_{\text{p}} $$
(8)

where σg can be represented by Hall–Petch relationship,[30,31]

$$ \sigma_{\text{g}} = \sigma_{\text{i}} + K_{\text{Y}} /D^{1/2} $$
(9)

σi is the friction stress of the crystal lattice to dislocation movement and assumed to be 200 MPa.[8] D is the average grain size of γ matrix. KY is the parameter of grain boundary hardening which varies with the average radius (r), volume fraction (f) of γ′ precipitates in Ni-based superalloys.[32] The solid solution strength σss can be given by Butt model,[33]

$$ \sigma_{\text{ss}} = K_{\text{s,i}} C_{\text{i}}^{p} $$
(10)

Ks,i is a strengthening constant for solute i, Ci is the concentration of solute i and p is a constant. In this work, Ks,i is 32.75 MPa/wt pct[34] and p is 2/3. The γ′ phase with the L12-type crystal structure is a chemically ordered precipitate in the disordered γ matrix with the fcc-type crystal structure. The order strengthening should be the predominant precipitation strengthening mechanism in nickel base alloys.[19] As the precipitate diameter increases, the strengthening process changes from the order shearing process to the Orowan looping process. Thus, for the precipitation strengthening increment due to the γ′ precipitates in Ni-Al alloys, it depends on the mechanism of order shearing process and Orowan looping process. The corresponding equations are respectively given by[19,20,21,35,36]

$$ \Delta \tau_{WPC} = \left( {\frac{1}{2}\left( {\frac{{\gamma_{\text{APB}} }}{b}} \right)^{3/2} \left( {\frac{{b{\text{d}}f}}{T}} \right)^{1/2} 0.72 - \frac{1}{2}\left( {\frac{{\gamma_{\text{APB}} }}{b}} \right)f} \right) \times 0.95 \times \left( {1 + C\eta_{0} } \right) $$
(11)
$$ \Delta \tau_{\text{SPC}} = \left( {\frac{1}{2}} \right)1.72 \times \frac{{Tf^{1/2} w}}{bd}\left( {1.28\frac{{{\text{d}}\gamma_{\text{APB}} }}{wT} - 1} \right)^{1/2} \times\, 0.95 \times \left( {1 + C\eta_{0} } \right) $$
(12)
$$ \Delta \tau_{\text{Orowan}} = \frac{2T}{b\lambda },\lambda = \sqrt {2/3} d\left( {\sqrt {\frac{\pi }{4f}} - 1} \right) $$
(13)

Here, d is the average particle diameter and f is the volume fraction of γ′ precipitates. b is the Burgers vector (i.e., 0.258 nm), G is shear modulus of matrix (i.e., 76 Gpa), and γAPB is the anti-phase boundary (APB) energy (i.e., 0.185 J/m2). T is the line tension of dislocation. Here, we roughly assumed TWPC = TSPC ≈ 1/2Gb2 and TOrowan ≈ 3/2 (Gb2/2).[20] w is a parameter introduced for elastic repulsion between the strongly paired dislocations.[37] The value of C can be estimated by plotting the normalized critical stress in term of the reduced particle depth. η0 is proportional to the volume fraction f.[20] λ is the average particle spacing.

Results and Discussion

Experimental Results

The experimental microstructures of Ni-15.9 at. pct Al alloy before and after the solid solution at 1280 °C for 2 hours are demonstrated in Figures 2(a) through (d), in which the dark-gray particles are γ′ precipitates while the gray area is γ matrix. As shown in Figure 2, the γ′ precipitates with irregular morphology before solid solution turn into regular fine cube particles after solid solution treatment with WQ or AC but relatively larger block particles with FC. As can be seen in Figure 2, the morphology of γ′ precipitates changes greatly after the solid solution treatment.

Fig. 2
figure2

Experimental microstructures for Ni-15.9 at. pct Al alloy before (a) and after the solid solution at 1280 °C for 2 h with different cooling media: (b) WQ; (c) AC; (d) FC

The samples were further etched in order to better observe the microstructures after the solid solution at 1280 °C with different cooling media. The morphologies of γ′ precipitates in etched γ matrix after solid solution with WQ, AC and FC are shown in Figures 3(a) through (c), respectively. As can be seen in Figure 3, the bimodal size distribution of γ′ precipitates exists in all the three samples. Moreover, as the decrease in cooling rate from WQ, AC to FC, the quantities of γ′ precipitates increase significantly, and the shape of γ′ precipitates changes from the cube to large block. The average diameters of γ′ precipitates after different cooling media are measured and demonstrated in Figure 4.

Fig. 3
figure3

Experimental microstructures (etched) for Ni-15.9 at. pct Al alloy after the solid solution at 1280 °C for 2 h but with different cooling media: (a) WQ; (b) AC; (c) FC

Fig. 4
figure4

Average diameters of secondary (d1) or tertiary (d2) γ′ precipitates in experimental microstructures together with yield strengths of Ni-15.9 at. pct Al alloy after the solid solution at 1280 °C for 2 h but with different cooling media, including WQ, AC, and FC

The average diameter of the secondary or tertiary γ′ precipitates after WQ is the smallest, while that after FC is the largest. Moreover, the measured yield strength values of the three alloys are displayed in Figure 4. The yield strength of the alloy after solution with WQ ranks the highest, followed by AC, and then FC. Thus, it is obvious that the cooling media after solid solution treatment show very strong effect on the morphology and size of γ′ precipitates in γ matrix in the target Ni-Al alloy, and then on the yield strength.

The microstructures of the Ni-Al alloy after solid solution with different cooling media during aging at 800 °C are displayed in Figure 5. As can be seen in Figure 5, the morphological evolution of γ′ precipitates in γ matrix for the alloys with WQ and AC during aging is quite similar. Three noticeable changes for the precipitates can be clearly observed. Firstly, the size of γ′ precipitates increases with the aging time. Secondly, the morphology of γ′ precipitates changes from cubic to nearly spherical shape or irregular shape as aging time increases. Thirdly, the number density of precipitates decreases during aging process. While the morphological evolution of γ′ precipitates during aging in the alloy after solid solution treatment with FC is completely different from the above two. The large block γ′ precipitates gradually dissolve as the aging time increases, but still exist.

Fig. 5
figure5

Experimental microstructures for Ni-15.9 at. pct Al alloy aging at 800 °C for 4, 8, and 16 h after solid solution with different cooling media: (a) to (c) WQ; (d) to (f) AC; (g) to (i) FC

Figure 6 shows the experimental average diameters of γ′ precipitates during aging in the target alloys with different cooling media. Again, the average diameters of γ′ precipitates in the alloys with WQ and AC are quite similar, as demonstrated in Figures 6(a) through (b). While the average diameters of γ′ precipitates in the alloys with FC are much larger than those with WQ and AC. Tensile stress-strain curves of Ni-15.9 at. pct Al alloy aging at 800 °C with different cooling media after solid solution are shown in Figure 7(a) through (c), and the average yield strengths of these alloys are presented in Figure 7(d). In general, the yield strengths of all the alloys decrease as the aging time increases. In consistency with the average diameters of γ′ precipitates, the yield strengths of the alloys after solid solution with FC are still the lowest, and the values of the alloys in solid solution cooling by WQ and AC after 16 hours aging are similar. Hence, it can be clearly concluded that the cooling medium after solid solution still plays a very important role in the microstructure and mechanical property evolution of the target alloys during aging process. Moreover, it can be seen that the microstructure and yield strength of the target alloys with WQ and AC after solution are similar, and their yield strengths are much larger than that with FC. Therefore, from the perspective of cost saving as well as environmental protection, the AC approach after solid solution treatment is proposed for the industrial Ni-based superalloys, which is consistent with the common treatment in the industrial manufacture.

Fig. 6
figure6

Average diameters of secondary (d1) and tertiary (d2) γ′ precipitates for Ni-15.9 at. pct Al alloy aging at 800 °C for 4, 8, and 16 h: after solid solution with different cooling media (a) WQ; (b) AC; (c) FC

Fig. 7
figure7

Stress–strain curves due to the room-temperature tensile test of Ni-15.9 at. pct Al alloy aging at 800 °C for 0, 4, 8, and 16 h with different cooling media after solid solution: (a) WQ; (b) AC; (c) FC; and (d) the measured yield strengths for the corresponding samples

Phase-Field Simulation Results

The phase-field simulation of microstructure evolution during aging in the Ni-15.9 at. pct Al alloy after solid solution with AC was then conducted for a demonstration here. For the specimens subjected to solid solution and subsequently cooled in air, the average grain size of γ matrix (D) roughly equals to 196 μm according to the experimental measurement. The grain size of γ matrix changes slightly during aging process, and thus, the variation of grain size of γ matrix can be neglected here. In fact, this work focuses on the evolution process of γ′ precipitates during aging.

In order to reproduce the microstructure evolution of γ′ precipitates in Ni-15.9 at. pct Al alloy during aging at 800 °C, a two-dimensional square domain of 400 × 400 grid points with the grid spacing of 10 nm, i.e., 4 × 4 μm2 area, was chosen from the specimen after solution heat treatment as the initial state of aging in the subsequent phase-field simulation. Figures 8(e) through (h) shows the simulated microstructure evolution in Ni-15.9 at. pct Al alloy during aging at 800 °C. As shown in Figures 8(e) through (h), the experimental microstructure characteristics were reproduced by the phase-field simulation. For instance, the simulation results indicate the existence of bimodal size distribution of γ′ precipitates in γ matrix, similar to the experimental microstructures shown in Figures 8(a) through (d). On one hand, the morphology of secondary γ′ precipitates changes from the cubic to nearly spherical shape or irregular shape as the aging time increases. On the other hand, tertiary γ′ precipitates grow up at the beginning and then slowly disappear. Moreover, the existence of the interconnected γ′ precipitates can be also observed. Figure 9 displays the simulated average diameter, volume fraction and area fraction of γ′ precipitates during aging, compared with the experimental values. As can be clearly seen, the simulation results are in good agreement with the experimental data, indicating that a quantitative phase-field simulation of the microstructure evolution in Ni-15.9 at. pct Al alloy was achieved.

Fig. 8
figure8

Microstructure slides for Ni-15.9 at. pct Al alloy aging at 800 °C for 0, 4, 8, and 16 h after solid solution with air cooling: (a) through (d) the experimental microstructure, while (e) through (h) the phase-field simulation results

Fig. 9
figure9

Comparison of (a) average diameter, (b) volume fraction and (c) area fraction of γ′ precipitates in Ni-15.9 at. pct Al alloy aging at 800 °C after solid solution with AC obtained by phase-filed simulations and experimental measurements. Here, Sim: simulation results; Exp: experiment results; d1, f1 and A1 are average diameter, volume fraction, and area fraction of secondary γ′ precipitates, respectively; d2, f2 and A2 are average diameter, volume fraction, and area fraction of tertiary γ′ precipitates, respectively

Evaluation of Yield Strength

During the aging process after solid solution with AC, there are secondary γ′ and tertiary γ′ dispersing in the γ matrix, as can be seen in Figure 8. Based on Eqs. [11] through [13], the calculated critical shear stress increments in the weak/strong order shear strengthening mechanism as well as Orowan mechanism for secondary and tertiary γ′ precipitates are presented in Figure 10. One can easily observe from the results that the operative strengthening mechanism for secondary γ′ precipitates is Orowan mechanism, while that for tertiary γ′ precipitates is strong order shear mechanism, as respectively shown in Figures 10(a) and (b).

Fig. 10
figure10

Analysis of the critical resolved shear stress increments of the bimodal size distribution of γ′ precipitates in γ matrix: (a) secondary γ′ precipitates; (b) tertiary γ′ precipitates. The red solid circles denote the predicted values from the simulated microstructure features, while the black solid squares represent the predicted values from the experimental microstructure features. It should be noted that the lowest strengthening value is assumed to define the operative strengthening mechanism

We have so far considered the strength imparted by random arrays of obstacles of different classes of γ′ precipitates. The critical resolved shear stress increment Δτ0 of the bimodal γ′ precipitates may be described by a mixing rule,[38] Δτ0 = A1Δτ01 + A2Δτ02, in which the weight factors A1 and A2 are the area fractions of the secondary and tertiary γ′ precipitates, respectively, as displayed in Figure 9(c). The critical resolved shear stress of γ′ precipitates can be calculated by the mixing rule model with the microstructure information from either simulation or experiment.

Based on the measured and simulated microstructure characteristic parameters, the evolution of yield strength, σy, of Ni-15.9 at. pct Al alloy during the aging process after solid solution cooling by AC can be evaluated. As stated above, the contributions of each strengthening factor to the yield strength, σy, of Ni-15.9 at. pct Al alloy can be divided into three parts: σg, σss and σp. In the present Ni-15.9 at. pct Al alloy, the γ′ precipitates form in γ grain, and thus the parameter for grain boundary hardening, KY, can be calculated based on the average radius (r) and volume fraction (f) of γ′ precipitates from both experimental and phase-field simulated microstructures, which vary during the aging process. Based on Eq. [9], σg also varies in the range of 214 to 224 MPa, as shown in Figure 11. While the solid solution strengthening increment σss due to the addition of aluminum element is 129.6 MPa according to Eq. [10]. As for the precipitation strengthening increment, σp varies over the range of 280 to 335 MPa depending on the aging time. Figure 12 summarizes the computed total yield strength evolution, σy, that includes the contribution of individual strengthening of Ni-Al alloy during the aging process after solid solution cooling by AC at 800 °C, compared with the experimental results from tensile tests. As can be seen in Figure 12, all the computed yield strengths are within the errors of the experimental values, except for the computed yield strengths after aging for 4 hours, which are slightly lower than the test value. That is to say, it is successful to combine the phase-field simulation with the strengthening models for prediction of the yield strength for the present Ni-Al alloy during aging. Though only the yield strength was demonstrated here, it is anticipated that different mechanical properties, e.g., the ultimate tensile strength, elongation and even creep resistance, can be also predicted if the corresponding strengthening modeling are available.

Fig. 11
figure11

Evolution of the parameter for grain boundary hardening, KY, and the grain boundary strengthening σg during aging time due to the experimental/phase-field simulated microstructure information

Fig. 12
figure12

Comparison among the total yield strengths predicted from the strengthening models based on the microstructure information from phase-field simulation and experimental measurement, and the values from the experimental tensile tests. The three parts of contributions to total yield strength, σg, σss and σp, are also superimposed

Further Discussion

With the input of truly experimental microstructure after solution treatment with AC as the initial microstructure for the later simulation of aging process as well as the reasonable thermophysical parameters, the quantitative phase-field simulation of microstructure evolution in Ni-Al alloy during aging can be realized. The simulated results are in good agreement with the experimental data, indicating that the phase-field simulation can reproduce well the microstructure evolution of γ′ precipitates in the alloy during aging at 800 °C. Moreover, the phase-field simulation can provide the continuous microstructure information (including size, volume fraction of γ′ precipitates, and so on) for replacing the scattered experimental data as the input for the strengthening model. Then, the mechanical properties predicted from the strengthening model should be continuous over the entire aging process. Thus, the present approach by combing the quantitative phase-field simulation with the strengthening models to predict yield strength can be used to establish the quantitative “composition/process—microstructure—mechanical property” relation in the target alloys, from which the optimal composition and aging treatment process can be effectively chosen.

Conclusions

  1. 1.

    The effect of different cooling media (i.e., WQ, AC and FC) after solid solution on the evolution of microstructure and yield strength in Ni-15.9 at. pct Al alloy during aging at 800 °C was experimentally investigated in detail. The results indicate that the cooling route after sold solution plays a very important role in the microstructure and yield strength evolution during the subsequent aging process.

  2. 2.

    The evolution of microstructure in Ni-15.9Al at. pct alloy with AC during aging process at 800 °C was simulated by using the phase-field simulation coupled with the CALPHAD databases and was also verified with the experimental observations. Moreover, the truly experimental microstructure after solid solution with AC was read in the MICRESS code as the initial microstructure for the phase-field simulation.

  3. 3.

    The microstructural features acquired from the quantitative phase-field simulations as well as the experimental measurements were imported into the strengthening modeling, resulting in the evolution of total yield strength along the aging time. It was finally found that the excellent agreement between the predicted yield strengths and the tensile test results exists.

  4. 4.

    The successful application in the present Ni-Al alloy proves that the combination of the quantitative phase-field simulation with the strengthening models provides a feasible way to establish the quantitative “composition/process—microstructure—mechanical property” relation in different alloys, which serves as the basis for the later alloys design.

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Acknowledgments

The work was supported by the Youth Talent Project of Innovation-driven Plan at Central South University (Grant No. 2019CX027), and the Hunan Provincial Science and Technology Program of China (Grant No. 2017RS3002)—Huxiang Youth Talent Plan. Ming Wei acknowledges the financial support from the program of China Scholarship Council (No. 201706370128).

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Lin, Y., Li, G., Wei, M. et al. Effect of Different Cooling Media After Solid Solution on the Microstructure and Yield Strength in a Ni-Al Alloy During Aging: Experimental Measurement and Computational Modeling. Metall Mater Trans A 50, 4920–4930 (2019). https://doi.org/10.1007/s11661-019-05400-z

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