Simulation of δ-phase precipitation behavior in hot compression deformation of inconel 718 superalloy

A model for δ-phase precipitation in Inconel 718 superalloy during hot working, grounded in experimental data, was developed. A Cellular Automata (CA) simulation platform was designed to simulate the precipitation of second-phase particles in the alloy. The δ-phase precipitation behavior of Inconel 718 alloy during hot aging and hot compression was simulated. The results revealed that during hot aging, δ-phase initially precipitated on grain boundaries as particles and short rods, followed by the formation of long needle-like δ-phase with similar growth direction within grains. During hot compression deformation, flat needle-like δ-phase gradually dissolved and transformed into short rods and granular forms dispersed around grain boundaries as thermal deformation temperature increased. The simulation results for δ-phase content, morphology, and distribution were in good agreement with experimental results, demonstrating the model’s strong predictive potential for second-phase evolution.


Introduction
Inconel 718 is a precipitation-strengthened Ni-based alloy with high strength, good oxidation resistance, and excellent fatigue resistance [1].It is suitable for manufacturing high-temperature and high-strength components such as aircraft engines, gas turbines, and solid rocket engines [2].The alloy's high-temperature performance depends largely on hot working technology and microstructure control [3].
γ″-phase is the main reinforcing phase in nickel-based alloys, the crystal structure is BCT(DO 22 ) type, the shape is disk-like, causing lattice distortion in the precipitation process, forming coherent strain strengthening.γ′-phase is the auxiliary strengthening phase, the crystal structure is FCC(L 12 ) type, spherical particles, the size of which is smaller than that of the γ″-phase.δ-phase is a stable common lattice phase with orthorhombic crystal system structure, and its chemical formula is Ni 3 ( Nb 0.8 Ti 0.2 ).The δ-phase, which has an appropriate amount, form, and distribution, can refine the grain of Inconel 718 forging, improve impact toughness and plasticity, reduce notch sensitivity, and prevent crack initiation and propagation [4,5].The precipitation behavior of Inconel 718 during hot aging has been widely studied.Generally, when aged above 900 °C, δ-phase precipitates directly from the austenitic base as short rods.When aged below 900 °C, δ-phase precipitates after γ′ and γ″ phases.It first precipitates on grain and twin boundaries before gradually appearing within crystals as aging time increases.The precipitate morphology is mainly needle-like [6][7][8].
The microstructure evolution of materials is complex and diverse, and its quantitative research and prediction by theoretical and experimental methods is challenging.Computer simulation technology, which has developed with computational materials science, is increasingly used to study microstructure evolution [9].The simulation studies on the precipitation behavior of second-phase particles in alloys reveal their mechanisms and processes [10].This provides a basis for effectively controlling the particle size, shape, and distribution, and developing reasonable hot working technology to improve the microstructure and performance of the alloys [11][12][13].
The cellular automaton (CA) model has been widely used to simulate microstructure evolution during various processes such as solidification, grain growth, static recrystallization, and dynamic recrystallization [14][15][16][17].The model was first used by Hesselbarth et al [18] to study the dynamics and growth of recrystallization nucleation.The model was further developed by Goetz et al [19] to simulate static and dynamic recrystallization.Several studies have been conducted on the recrystallization behavior of Ni-based alloys [20][21][22][23].Reyes et al [21] successfully predicted the grain size evolution in Ni-based superalloys during dynamic recrystallization using the CA method.Liu et al [24] applied a 2D CA model to simulate the thermal deformation behavior under different thermomechanical conditions.Azarbarmas et al [25] used experimental data and an improved cellular automaton model to simulate dynamic recrystallization and texture in Inconel 718.Chen et al [26] coupled the basic metallurgical principles of dynamic recrystallization with a 2D CA model to study the microstructure evolution and plastic flow characteristics in Ni-based superalloys.The CA method has unique advantages in displaying particle morphology evolution and can be applied to second-phase particle precipitation simulation by providing reasonable transformation rules.However, few studies have used CA model for second-phase particle precipitation simulation in Ni-based alloys [10].
In this study, a CA model was applied to simulate the δ-phase precipitation behavior in Inconel 718 superalloy during hot aging and compression.A negative exponential stochastic nucleation function, based on experimental data, was embedded in the CA method to simulate the δ-phase precipitation behavior.The δ-phase content in Inconel 718 during hot compression deformation was measured to improve the CA simulation platform.The δ-phase content, morphology, and distribution during hot compression deformation were simulated by the CA model.
To ensure that the material structure was close to the actual pre-processing state, the specimens were heated in a chamber electric furnace.The specimens were heated in a chamber electric furnace at a rate of 10 °C s −1 until the furnace reached 975 °C, which was maintained for 30 min to achieve temperature uniformity.The Inconel 718 specimens were then rapidly inserted into the furnace and held for one hour before being air-cooled [27].After solution treatment, most of the δ-phase in the specimens dissolved, leaving only a small amount of granular δ-phase distributed along the grain boundary.The grain size remained unchanged, with an average value of about 10.8 μm.
The compression test was carried out on a Gleeble thermal-mechanical simulator to determine the mechanical properties of Inconel 718 under compression loading conditions.To reduce bulging and improve the accuracy of true stress-true strain experimental data collected by the computer, graphite powders were uniformly filled as a lubricant on the contact surface between the end of specimen and the moving head of the simulator.High-purity nitrogen gas was filled around the specimens to prevent oxidation of the surface in a high-temperature environment.
The specimens underwent heating to the test temperature at a rate of 10 °C s −1 and were subsequently held for 3 min in the Gleeble simulator.Compression tests were then conducted at five temperatures: 930 °C, 980 °C, 1020 °C, 1040 °C, and 1080 °C, respectively, with a deformation rate of 50 mm s −1 and a deformation degree of 0.5.The specimens were immediately water-quenched following the completion of the hot compression test.
After the compression experiments, the specimens with different deformation were cut from the middle, and mechanically lapped and polished at the sections.The polished specimens were rinsed with water and dehydrated with ethyl alcohol absolute.Then, the polished surfaces of specimens were etched by a mixture of 50% HCl and 50% H 2 O 2 .The specimens were then rinsed with water, dehydrated in alcohol, and dried by air heater.Finally, a metalloscope was used to observe the microstructure of the specimens after hot deformation.

Models for δ-phase precipitation during hot aging
The content, morphology and distribution of δ-phase of Inconel 718 alloy vary with time during the hot aging process [28].The δ-phases initially form along the grain boundary and then precipitate inside the grain.The δphase undergoes a transition from granular to short rod-like and long needle-like shapes.The granular and short rod-shaped δ-phases are primarily located on the grain boundary.
In addition, most of the needle-like δ-phase in the grain stop at the grain boundary without crossing it in the original direction.This may be due to the different lattice orientations among different grains, which block the precipitation of δ-phase along the crystal plane at the distorted lattice region at the grain boundary [8], In general, the amount of precipitated δ-phase increases with aging time, which follows approximately the Avrami equation (equation ( 1)) [29] Where W s is the precipitation equilibrium content of the δ-phase at a given aging temperature in wt%, a is the precipitation speed of the δ-phase, n is the time index.The δ-phase precipitation data of the solid solution Inconel 718 alloy measured with different aging treatments in the literature [4], which is shown in figure 1.

Models for δ-phase precipitation during hot compression deformation
Figure 2 presents the microstructure of the specimens after hot compression deformation at a rate of 50 mm s −1 and 0.5 deformation.Predominantly, the δ-phases precipitate on the grain boundary in granular and short rods forms, with a minor quantity of δ-phases precipitating within the grain as short rods.As thermal deformation temperature rises, there is an intensification in fracture and dissolution of δ-phase, causing flat needle-like δ-phases  to become irregular and gradually evolve into short rod-like and granular shapes dispersed around the grain boundary.A detailed examination of the microstructure reveals a distinct transition region between the dissolved δ-phase and the matrix phase structure.This observation suggests an interfacial reaction occurrence.The interfacial reaction propagates from the edge of the δ-phase towards its interior, consuming external δ-phase and resulting in an irregular shape of the δ-phase edge.Furthermore, high temperatures amplify atomic diffusion and stimulate interfacial reaction.
To assess the variation in the δ-phase content of Inconel 718 alloy post high-temperature deformation, the volume fraction of δ-phase in the microstructure image was determined by the 'linear intercept method'.Measurements of the δ-phase content were taken at varying deformation temperatures, maintaining a deformation rate of 50 mm s −1 and a reduction of 0.5.These measurements are depicted in figure 3, which illustrates a linear decrease in δ-phase content with increasing deformation temperature.
A linear regression analysis was performed on the data from figure 3, correlating δ-phase content with deformation temperatures, and a linear function was obtained, which provided the core mathematical model for constructing a CA method to simulate the precipitation behavior of δ-phase during hot compression deformation of Inconel 718 alloy.

Rules for grain nucleation and growth
The CA method discretizes the space into finite cells.A representative area of 50 μm × 50 μm is discretized into an array of same square cells of 200 × 200, each with a side length of 0.25 μm.For the proposed simulation program of Inconel 718 alloy δ-phase precipitation, each cell is characterized by 7 state variables, and the number of grain nucleation can be obtained as.
Where d 0 = 10.8μm, is the average grain diameter of the Inconel 718 alloy after solution treatment.
The simulation program for δ-phase precipitation behavior was developed based on the CA simulation program for matrix phase structure and experimental results regarding δ-phase precipitation content, morphology, and distribution in Inconel 718 alloy post hot aging.To simplify the δ-phase precipitation mechanism, the following assumptions are considered.
(1) The presences of twin crystals, stacking faults, vacancies, interstitial atoms, carbides and nitrides in the crystal are ignored, and there are only three kinds of microstructure: grains, grain boundaries and δ-phase particles.
(2) The precipitation content of the δ-phase is only related to the aging time and it is not affected by other external factors.
(3) The precipitation location of the δ-phase is random on grain boundary and inside grain.The second phase particle first undergoes random nucleation.The δ-phase precipitates both on the grain boundary and within the grain, exhibiting different morphologies and contents.Consequently, we separately constructed the nucleation and growth algorithms for the grain boundary and intragranular.
The random nucleation function of second phase particles directly influences the variation trend of δ-phase precipitation content with time step, which is the key to the successful simulation of δ-phase precipitation.After data comparison, a negative exponential function for the random nucleation function was found to agree with the practical situation [3].N 1 and N 2 are considered as the negative exponential functions.
Where b 1 , b 2 , c 1 and c 2 are positive constants, k is the simulation time step.Upon completion of the random nucleation of second phase particles, the growth of these particles commences.The entire cell space is initially traversed, and cells at nucleation locations where growth has not yet occurred are identified.The slope of needle-like second phase particles in these cells is then randomly assigned.Grain boundary identification is conducted on cells that have not grown.If identified as a grain boundary cell, the δ-phase precipitates in short rod-like or granular forms.
Once the growth of second phase particles is complete, the growth state number of cells at corresponding nucleation locations is set to 1, and the second phase state number of newly formed cells into second phase particles is also set to 1. Subsequently, the current time step concludes and the program advances to the next time step to continue the loop.
In constructing the CA model for simulating δ-phase precipitation behavior during hot compression deformation, the temperature variable is treated as equivalent to the simulated time step variable.Emphasis is placed on adjusting the location probability of second phase particles in the random nucleation across the entire matrix phase structure, as well as fine-tuning the specific form of the random nucleation function.Given the approximate negative linear relationship between δ-phase content and deformation temperature depicted in figure 3, the random nucleation function can be determined as.
Where u, v are positive numbers.
To achieve the 'dissolution' of the second phase particles as simulation time steps increase, a specified amount of the second phase particles are precipitated in the initial time step.Consequently, in subsequent time steps, the content of the second phase particles changes according to the random nucleation function model integrated into the program.By adjusting the morphological judgment probability during secondary phase particle growth, the morphological tendency of the second phase particle growth in the CA simulation can be changed.
At the end of the kth time step of the simulation program, the δ-phase content is calculated as.

=
´( ) Where N k is the total number of second phase particle cells at the end of the kth time step, N is the total number of cells at the end of the kth time step.

Results and discussion
The δ-phase precipitation behavior at 960 °C was simulated by the δ-phase precipitation program developed for Inconel 718 alloy during hot aging.The simulation results are presented in figure 5. Figure 5 reveals that the δ-phase initially precipitates in small quantities as particles and short rods on the grain boundary.Subsequently, long needle-like δ-phase forms within the grains and increases with time steps.Most long needle-like δ-phases within the same grain exhibit relatively similar growth directions, with a few growing at irregular angles.The long needle-like δ-phase that intersects with the grain boundary is halted by the grain boundary.The δ-phase precipitation content rapidly increases in the initial stage, but as the simulation time step increases, the growth rate decreases, and the precipitation content value gradually approaches a certain equilibrium content.
Figure 6 compares the simulated and experimental results of Inconel 718 δ-phase precipitation during hot aging.The changes in morphology, distribution, and content of δ-phase are similar to experimental results, validating the feasibility and effectiveness of using the cellular automaton method to simulate δ-phase precipitation behavior during hot aging.
Figure 7 visually depicts the dissolution course of δ-phase changing with time steps in CA simulation.From the third time step to the 18th time step (equivalent to deformation temperatures from 930 °C to 1080 °C), the content of δ-phase on the grain boundary and within the crystal gradually decreases due to dissolution as thermal deformation temperature increases.Figures 7(a)-(e) show that most δ-phase is distributed in grain boundaries as particles and short rods, while a small amount of acicular δ-phase precipitates inside the crystal grain.This phenomenon aligns with metallographic microstructure test results shown in figure 2, further validating the validity and accuracy of CA simulation results.
Figure 8 presents simulated δ-phase content during hot compression deformation with respect to simulation time step.As time step increases, δ-phase content decreases approximately linearly, At 930 °C, the proportion of the δ-phase is 9.7%, which decreases as the temperature increases.At 1080 °C, the proportion of the δ-phase is 3%, indicating a 69% reduction in phase content.The δphase is usually directly precipitated from austenite at temperatures above 900 °C.As the temperature increases, the quantity gradually decreases and transforms from short rod-shaped to granular, ultimately leading to complete dissolution.The trend of phase content variation with hot compression deformation temperature is very similar to the ones shown in figure 3.
Figure 9 presents a comparison of simulated and experimental results of δ-phase precipitation content.The calculation equation for δ-phase precipitation content in the CA method, as described in equation (5), is the ratio of the total number of second phase particle cells to the total number of cells in the entire system.The 'linear intercept method' is used to obtain δ-phase precipitation content in experimental measurements.As both methods share the same physical meaning, their values can be directly compared.Figure 9 reveals that the maximum difference between simulated and experimental measurements of δ-phase precipitation content is 0.8%, the minimum difference is 0.1%, and the average difference is 0.4%.These findings indicate that simulation results align well with measured results, validating the accuracy of simulation results and demonstrating the feasibility of the simulation method constructed in this study.Despite its effectiveness, the simulation method for δ-phase precipitation in hot compression deformation of Inconel 718 alloy established in this study has some limitations, such as the absence of a theoretical model to describe the evolution mechanism of δ-phase from a physical metallurgy perspective.However, this study's simulation results clearly demonstrate that the cellular automata method, which emphasizes random probability, effectively simulates secondary phase precipitation and dissolution evolution behavior.

Conclusion
A CA simulation platform was designed to simulate the precipitation behavior of the second phase particles of Inconel 718 alloy.The δ-phase precipitation behavior of hot aging and hot compression of Inconel 718 alloy were simulated respectively.The following conclusions can be drawn: (1) The simulated precipitation morphology, distribution, and content align well with experimental results, verifying the accuracy of the CA model and demonstrating its predictive potential for second phase evolution.
(2) When the random nucleation function of second phase particles is a negative exponential function, the simulated δ-phase precipitation content during hot aging most closely resembles actual conditions.The δphase precipitation content increases rapidly initially before growth rate decreases and precipitation content value gradually approaches equilibrium.
(3) Under deformation at a rate of 50 mm s −1 and 0.5 deformation, δ-phase content decreases linearly with increasing thermal deformation temperature.Needle-like δ-phase gradually fractures and evolves into short rods and granules dispersed around grain boundaries.

Figure 1 .
Figure 1.Relationship between amount of δ-phase and aging time at different temperatures.

Figure 2 .
Figure 2. Microstructures of alloy Inconel 718 after hot deformation at different temperatures.

Figure 4
Figure 4 shows the algorithm flowchart of the simulation program at the step k.The main program performs cyclic operations according to time steps.Each time step comprises two stages: nucleation and growth of second phase particles.The second phase particle first undergoes random nucleation.The δ-phase precipitates both on the grain boundary and within the grain, exhibiting different morphologies and contents.Consequently, we separately constructed the nucleation and growth algorithms for the grain boundary and intragranular.The random nucleation function of second phase particles directly influences the variation trend of δ-phase precipitation content with time step, which is the key to the successful simulation of δ-phase precipitation.After data comparison, a negative exponential function for the random nucleation function was found to agree with the practical situation[3].N 1 and N 2 are considered as the negative exponential functions.= -= -( ) ( ) ( ) N b c k N b c k exp , exp 3

Figure 4 .
Figure 4. Process chart of δ-phase precipitation simulation with CA method.

Figure 5 .
Figure 5. Results of δ-phase precipitation simulated with CA method.

Figure 6 .
Figure 6.Comparison of simulated and experimental results of Inconel 718 δ-phase precipitation course during hot aging.

Figure 7 .
Figure 7. Dissolution course of δ-phase changing with time steps in CA simulation.

Figure 8 .
Figure 8. Curve for amount of δ-phase changing with time step.