Polycrystalline Nickel-based Superalloy Research Articles

Characterisation of the creep cavitation process on grain boundaries in a polycrystalline nickel-base alloy 247

ABSTRACT Hot gas components such as gas turbine blades are loaded with constant centrifugal stress at high temperatures and thus the formation of creep cavitation on grain boundaries is expedited. Conventionally cast polycrystalline Nickel-base superalloys used as blade materials are prone to creep cavitation as unfavourably orientated grain boundaries exist. This research presents a diffusion-based probabilistic creep model which describes the creep cavitation process on grain boundaries. It includes the three mechanisms: pore nucleation, growth, and coalescence. The calibration of the model has been carried out by analysing Alloy 247 as-received specimens and specimens with pre-strain. For the evaluation of the pore numbers and sizes, a deep learning model for pore detection was trained on light microscopic and scanning electron microscopy images. Electron backscatter diffraction images are analysed for further investigations regarding grain orientations and grain boundary angles to the loading direction. The calibrated model allows predictions of pore size distributions over time.

Micro-strain and cyclic slip accumulation in a polycrystalline nickel-based superalloy

This work provides a comprehensive characterization and analysis of deformation and fatigue damage mechanisms in a nickel-based superalloy during ambient temperature fatigue and points to a fundamental deformation mechanism that results in the onset of crack nucleation. Strain and slip irreversibility are investigated at the nanometer scale using high-resolution digital image correlation and high-resolution electron backscatter diffraction, highlighting distinct deformation mechanisms contributing to crack nucleation. It is observed during early fatigue cycling at relatively low applied stress, the formation of intense slip events that induce grain boundary shearing. This results in intense micro-scale strain in the neighboring grains, producing localized plasticity and stresses. Such stresses facilitate fatigue extrusion–intrusion mechanisms during subsequent cycling, resulting in preferred crack nucleation. Finally, the configurations within the microstructure that promote such deformation and damage mechanisms sequence are highlighted.

Delayed parabolic oxidation via transient thermal exposures on a polycrystalline nickel-based superalloy

Multiple oxides were observed during the transient oxidation stage of the polycrystalline Ni-based superalloy, RR1000, before a protective Cr2O3 scale formed. Thermogravimetric analysis, synchrotron grazing incidence X-ray diffraction, and electron microscopy were performed on samples subjected to isothermal exposures at 800 °C for up to 100 h. Transient effects governed the early stages up to 40 h. NiO, spinels (NiCr2O4, (Ni,Co)(Cr,Co)2O4), Cr2O3/(Cr0.88Ti0.12)2O3, NiTiO3, and CrTaO4 formed during the initial stage with pseudo-linear kinetics. At the onset of parabolic kinetics, extensive Cr2O3 and TiO2 growth dominated scale formation with the former emerging as the major passivating oxide.

Discontinuous dynamic recrystallization and nucleation mechanisms associated with 2-, 3- and 4-grain junctions of polycrystalline nickel-based superalloys

Despite the important role of grain junctions to the dynamic recrystallization (DRX) of nickel-based superalloys, the nucleation mechanisms operating there and the influence factors besides deformation condition on nucleation priority are still unclear. The microstructure evolution considering grain junction effects has not been explored in depth yet. In this paper, DRX nucleation at the 2-, 3- and 4-grain junctions of a nickel-based superalloy was investigated. A proposed passive grain boundary bulging (PGBB) mechanism operated well at 2-grain junctions, and accounted for the continuity of necklace structure during the early stage of hot deformation. For the 3-grain junctions, a two-step strain-induced bulging of grain boundary fragments, which closely adjoined the junctions, was found to dominate the DRX nucleation. In addition, an increasing nucleation priority from 2-, 3- to 4-grain junctions was confirmed by the established thermodynamic model, while the nucleation priority differences of the same kind grain junctions were quantitatively analyzed by using the Taylor factors of their component grains. Finally, the DRX microstructure evolution of a 4-grain stacking unit during hot deformation was described. The understanding of DRX associated with 2-, 3- to 4-grain junctions made it more effective to tailor the microstructure of nickel-based superalloy forgings.

Supersolvus Recrystallization and Grain Growth Kinetics for the Fine Tuning of Grain Size in VDM Alloy 780 Forgings

VDM Alloy 780 is a new polycrystalline nickel-based superalloy developed for aeronautical applications. In most of the targeted applications, grain size after forging must be precisely controlled to meet the targeted mechanical properties and in-service life requirements. Grain size in forgings is the direct consequence of the recrystallization and grain growth kinetics which are addressed in this paper at high temperatures, above the solvus temperature of γ′ and η/δ phases. The dynamic and post-dynamic recrystallization kinetics as well as the grain growth kinetics of VDM Alloy 780 are detailed over a range of thermomechanical conditions. Dynamic recrystallization appears to be limited, with only 30 pct recrystallized at quite high strain of 1.7 applied at 1050 °C and 0.01 s−1 for instance, but this is compensated by fast post-dynamic evolution. Within the investigated thermomechanical range, recrystallization is completed with 5 minutes of post-deformation hold in VDM Alloy 780 independent of the prior strain, strain rate and dynamic recrystallization fraction. For a strain as low as 0.08, an isothermal annealing of 30 minutes at 1050 °C generates a homogenous and fully recrystallized microstructure. Capillarity driven grain growth following recrystallization is also relatively slow, for instance an exposure at 1050 °C (50 °C above the solvus temperature) for 2 hours results in an increase in average grain size from 20 to 70 μm. This opens the possibility to fine tune the grain sizes by subsequent heat treatments within a time scale that is compatible with industrial conditions. The high cobalt content (25 pct) is suspected to play a role in the control of microstructure evolution kinetics. It is noteworthy that VDM Alloy 780 is shown here to not undergo the heterogeneous grain growth phenomenon reported in low strain regions for other nickel-based superalloys, which is also an asset for applications requiring strict control of grain sizes and grain size distributions.

Dynamic Precipitation During Forging of a γ–γ′ Nickel-Based Superalloy

Hot compression and creep tests under different conditions were performed on a typical γ–γ′ polycrystalline nickel base superalloy below the γ′-phase solvus temperature, to shed light on the mechanisms of γ′ precipitation during deformation and the driving forces behind these mechanisms. Scanning Electron Microscopy (SEM) and image analysis were employed to characterize the different populations of γ′ precipitates and their evolutions. The γ′-phase area fractions observed after hot compression are systematically higher than the fraction prior to deformation within the investigated range of thermomechanical conditions. Two new populations of precipitates formed during deformation are identified and characterized. These are interpreted to result from a dynamic precipitation process, the hypothesis of their formation during the quenching step being very unlikely with regards to their size and amount. The fraction of γ′-phase produced during deformation increases with increasing deformation magnitude, decreasing deformation temperature and for slower strain rates. Finally, a scenario for dynamic γ′ precipitate nucleation and growth is proposed to explain the experimental observations and the influence of dislocation density.

Quantification of Slip Band Distribution in Polycrystals: An Automated Fast Fourier Transform Decomposition Approach.

Plastic deformation is accumulated in slip bands in a wide variety of engineering alloys. Multiple material and loading conditions impact their distribution and degree of slip localization, but these effects are rarely quantified. To tackle this, the current work introduces a fast Fourier transform (FFT) decomposition method and applies it to a tensile-loaded polycrystalline nickel-based superalloy imaged via high-resolution digital image correlation and electron backscatter diffraction. This approach identifies active slip planes over the FFT images of individual grains and performs inverse transforms such that slip band traces with shared orientations are isolated. This technique enabled the largest quantification of slip band spacings and in-plane strains to date, with a total of 6,557 slip bands detected. The results show that the slip band spacings increase with grain size, with no evident dependence on grain orientation and Schmid factor. Slip bands are found to develop similar spacings along different octahedral planes and continue to spread over larger regions of the grain as the resolved shear stress of the active slip system increases. The FFT decomposition technique, which could be employed with multiple microscopy techniques, will allow for much-needed large-scale quantitative studies of slip localization.

Short-Time Creep Deformation of the Coarse-Grained Nickel-Base Alloy 247 and Its Implications on the High-Cycle Fatigue Behavior

Abstract Cast polycrystalline nickel-base superalloys are typically used for critical high temperature aerospace and automotive components, such as turbine blades or turbocharger wheels. These high temperature components can undergo creep damage caused by, e.g., centrifugal forces on the turbine rotor blades, as well as high-cycle fatigue (HCF) from, for instance, vibrations of the rotor blade. Therefore, both, creep resistance and fatigue strength are important mechanical properties of these materials. The study presented here addresses the introduction of creep damage and its influence on the high-cycle fatigue behavior. For this purpose, fatigue specimens of coarse-grained Alloy 247 LC CC blade root material were prestressed on a pneumatic creep test rig to achieve creep-induced microstructural damage. The isothermal 900 °C short-time creep investigations below 1000 h test duration were performed at different tensile stresses which result in different strain rates. After the creep tests, scanning electron- and optical microscopy investigations of metallographic cross section of selected specimens were performed to determine the creep-induced grain boundary damage such as pore count, pore size, and coarsening of the γ' structure. This was followed by uniaxial, stress-controlled isothermal 850 °C high cycle fatigue tests on undamaged and predamaged specimens at a frequency of 10 Hz and a load ratio of R = −1. Subsequently, fractographic and cross section investigations, which provide information on the fatigue cracking, the failure initiating defects, and the pore morphology, were performed. The results of this study show that the degree of creep induced porosity is a dominant structural parameter for the HCF behavior of the investigated nickel-base superalloy.

High Temperature Fatigue Properties of Boron-doped Superalloy B1914

The B1914 superalloy is engaged in the power and aerospace industry as a material for critical hot section of gas turbines. Typical findings in these critical components with a high number of operations are damages by low cycle fatigue where high stresses comparable to the material yield strength are developed due to mechanical loading or thermal gradients causing large plastic deformation. The present contribution refers to the high-temperature low cycle fatigue behaviour of polycrystalline nickel-based superalloy B1914. Cylindrical specimens were fatigued under strain control with constant total strain amplitude in symmetrical cycling at 800°C and 900°C in air. The microstructure is based on a γ matrix (face-centered cubic lattice) with coherent L12 γ´ precipitates and γ/γ´ eutectic. The higher content of grain boundary stabilizing boron allows for lower carbon contents and thus reduces the carbide content that can facilitate fatigue crack initiation. Cyclic hardening/softening curves and fatigue life diagrams were determined. An increase in testing temperature accelerated fatigue crack initiation at grain boundaries associated with carbides. The SEM and TEM analysis of fatigued material further the discussion of fatigue behaviour of the B1914 superalloy.

High-cycle fatigue behavior of IN 738LC superalloy at high temperatures

The presented study is focused on experimental results of high-cycle fatigue life of a cast polycrystalline nickel-based superalloy IN 738LC at temperatures of 800, 900, and 950°C. The absence of hot isostatic pressing (HIP) led to the occurrence of casting defects in the microstructure. The fatigue tests were performed in symmetrical loading in laboratory air. The fracture surfaces of specimens were studied by scanning electron microscopy to describe the influence of temperature on the fatigue damage mechanism and fatigue crack initiation. Change in the primary crack propagation mechanism from crystalline (stage I regime) to non-crystalline (stage II regime) mechanism was observed with increasing temperature. However, the effect of changing mechanism caused by temperature dependence was unfavorably influenced by casting defects.

Weldability of Nickel/Aluminum Dissimilar Materials Laser Welding Part I: Microstructure Development, Solidification Behavior and Mechanical Properties of Polycrystalline Nickel-based Superalloy

Weldability-related issues, including asymmetrical weld pool shape, γ phase multicomponent microstructure development, solidification behavior and mechanical properties, of γʹʹ precipitation-strengthened polycrystalline nickel-based superalloy are experimentally analyzed during keyhole laser welding repair. The crucial relationship between them provides weldability improvement opportunities and defect-free high quality weld for feasible aerospace materials laser processing. Attractive neck transition region of weld is more vulnerable to coarser dendrite than bottom part in the vicinity of fusion boundary, and this location is also liable to abundant Laves/γ eutectic phase formation because of heat and solute accumulation. The dominant chemical inhomogeneity brings about microstructure anomalies and irregular morphology to deteriorate hot cracking resistance. Because of thermometallurgical factors of selective alloying partition in the molten pool, partition-resistant Cr is chemically accumulated in dendrite core, while partition-susceptible Ti, Al, Mo and Nb are enriched in high-supersaturation interdendrite region, which consequently result in serious segregation and extensive Nb-rich intermetallic phase formation across solidification interface. Dendrite refinement and aggressive Laves/γ eutectic phase formation suppression are not simultaneously satisfied by low heat input under nonequilibrium solidification conditions. Therefore, mechanical properties are only partially improved by diffusion-limited microstructure refinement as result of phase instability. In other words, reductions of tensile strength and ductility are attributed to brittle Laves/γ eutectic phase formation during terminal stage of solidification. The severity of eutectic reaction thermodynamically depends on nail-shaped keyhole profile, solidification conditions and solute redistribution ahead of solid/liquid interface. Additionally, viable control of weld metallurgical and mechanical properties progressively encourages optimal combination of laser power and welding speed to interdendritically minimize disadvantages of eutectic phase formation by a variety of welding conditions optimization.

Thermal, Metallurgical and Mechanical Determinants of Laminar Nickel/Aluminum Dissimilar Alloys during Laser-material Interaction Part I: Nickel-based Superalloy

To develop in-depth understanding of metallurgical phenomena for completeness, mechanical heterogeneity is kinetically and thermodynamically explained by thorough microstructure characterization of engineering materials, e.g. polycrystalline nickel-based superalloy, for alleviation of weldability-related problems during laser-induced keyhole fabrication. Because of nonequilibrium solidification behavior inside weld pool, finer γ phase dendrite substructure is confined to partition-resistant keyhole bottom part, coarser dendrite is crystallography-independently circumscribed at partition-vulnerable neck transition region of full penetration weld, and thus microstructure is lack of homogeneity. Abundance of segregation-driven eutectic phase or intermetallic phase in the interdendrite area throughout weld is essentially attributable to nonequilibrium solidification conditions, and morphologically increases susceptibility to mechanical properties deterioration. As a result of γ phase instability, Niobium-aided Laves/γ eutectic reaction in the vicinity of dendrite interstices at terminal stage of solidification contributes to severe dendrite boundaries brittleness, impairs mechanical properties, which is consistent with metallography and fractography results, and is more deleterious to weldability, since solute redistribution and supersaturation adversely exacerbate segregation behavior in the residual interdendrite liquid, especially asymmetric weld pool shape. There is inverse parabolic relationship between secondary dendrite arm spacing and solute partition coefficient, when location varies from nail-shaped weld upper site to bottom site. Chemical, microstructural and mechanical heterogeneities are more geometrically favorable in the curvature-related neck transition region. In addition, the mechanism of thermal, metallurgical and mechanical inhomogeneities, which are attributed to asymmetric weld pool shape, is consequently proposed. Untoward metallurgical phenomena, such as microstructure heterogeneity and brittle Niobium-rich Laves/γ eutectic phases mitigate strength, ductility and toughness of weld. In order to macroscopically and microscopically satisfy superior mechanical properties requirement, chemistry and microstructure of high quality weld are metallurgically controlled. Fruitful metallurgical information and mechanical data further support the reasonable explanations. It is imperative to progressively advance welding metallurgy, weldability and fabricability of intricate shape for welding defects minimization, suppress segregation and further develop mechanical properties through viable design and control of laser processing, simultaneously.

Diffusion-limited Nonequilibrium Phase Transformation of Nickel/Aluminum Dissimilar Materials during Laser Welding Part I: Precipitation-strengthened Polycrystalline Nickel-based Superalloy

Polycrystalline heterogeneities of grain growth, phase transformation and tensile properties are discussed by metallography, spectroscopy and fractography when choosing laser welding with butt-joint configuration to satisfactorily minimize effect of inappropriate weld pool shape on microstructure and mechanical properties of γʹʹ precipitation hardable nickel-based superalloy in aerospace industry. There is parabolic relationship between weld penetration and diffusion-limited nucleation, growth and coarseness of secondary dendrite arm spacing alongside fusion boundary through nonequilibrium solidification process, which is dendritically susceptible to grain growth variation and metallurgical discontinuities. The amount, size, morphology and distribution of nonequilibrium intermetallic phase near the dendrite boundaries are kinetically and thermodynamically rely on weld pool shape whose crystal structure is incoherent with γ solid solution austenite phase and increases preference for crack initiation and propagation of brittle intergranular and ductile dimple fracture failures to enormously contribute to losses of strength and ductility. Unsymmetrical keyhole weld is thermometallurgically inconvenient for amelioration of microstructure and mechanical properties on either side, and adversely mitigates resistance to hypereutectic-aided dendrite embrittlement. Beneficial grain refinement and suppression of detrimental nonequilibrium intermetallic phase at the same time are challenging. This problem is an integral part of inhomogeneous microstructure and mechanical properties. The finer grain size occurs, the larger grain boundary area is available for Laves/γ eutectic-type reaction and vice versa. Contributions of coarse grain kinetics and metallurgical reaction thermodynamics to weld disintegration and fracture failure mechanism of tensile properties are explained by microstructure characterization of multicomponent and multiphase weld. Finally, it is imperative to dendritically balance these important factors to minimize inevitable interface solidification products and anomalous substructure growth, and reasonably advance superior mechanical properties of reliable weld.

Slip localization and Cr23C6 carbide formation during low cycle fatigue of a polycrystalline nickel-base superalloy

This paper reports the formation mechanism of Cr23C6 carbide and its role in the localization of plasticity and fatigue deformation. The strong elastic interaction between carbon and dislocations at the matrix/MC carbide interface promotes carbon segregation and Cr23C6 carbide precipitation. Significant dislocation density is noticed around the Cr23C6 carbide and is one of the contributing factors to the hardening behaviour observed at 750 °C and strain amplitude of 0.6%.

An innovative approach towards forming the serrated grain boundaries and refining the γ′ precipitates in nickel-based superalloys

The serrated grain boundaries have been proven to improve the creep performance and reduce crack growth rate of polycrystalline nickel-based superalloys. However, a slow cooling rate to form the serrated grain boundaries often results in the coarsening of γ′ precipitates, thereby deteriorating the high temperature strength of the nickel-based superalloys. Hereof, a new heat treatment processing, high temperature pre-precipitation treatment (HTPT), is developed to simultaneously achieve the serrated grain boundaries and fine γ′ precipitates. The effects of HTPT conditions on the microstructure of the powder metallurgy nickel-based superalloys are systematically characterized. Surprisingly, the HTPTed samples possess the serrated grain boundaries and obtain even finer γ′ precipitates than those in air cooled samples. With the increasing of holding time from 10 min to 30 min and 60 min at 1140 ℃ for pre-precipitation, the average size of tertiary γ′ precipitates decreases from 43.9 nm to 34.7 nm and 30.4 nm, and the amplitude of the serrated grain boundaries increases from 0.84 µm to 1.71 µm and 1.87 µm, respectively. On the other hand, the excessively long holding time causes the abnormal growth of the secondary γ′ precipitates and the suitable holding time is considered to be 30 min. The formation of the serrated grain boundary during HTPT is related to the growth of the pre-precipitated γ′ precipitates and the hindering effect on the migration of grain boundaries. Our work demonstrates a new way to control the microstructure of nickel-based superalloys.

Ultra-fast quantification of polycrystalline texture via single shot synchrotron X-ray or neutron diffraction

Tracking texture evolution during in situ loading is critical to understand and simulate the dynamic behaviour of microstructure in polycrystalline materials, yet conventional texture quantification methods are sometimes restricted due to various factors, such as acquisition time, sample environment and complex setup. To address this, a novel approach to extract texture information from single shot Time-Of-Flight neutron diffraction pattern has been developed. Another texture analysis approach based on single shot synchrotron X-ray diffraction has also been demonstrated. The effectiveness of two methods is assessed for polycrystalline Nickel-based superalloy polycrystalline samples possessing different textures. Both methods feature a moderate acquisition time of ~10 min and 30 s respectively, as well as a simplified setup which allows adding complex sample environments and the use of additional equipment. Comparison with the referential EBSD texture suggests that both approaches achieve a satisfactory match, though some details of the complex contour profiles in inverse pole figures may be missing. Besides that, a novel metric has been proposed to quantify the matching quality of pole figures. By employing the EPSC modelling approach, it is shown that the texture deviation due to the technique chosen for its evaluation exerts a subtle influence on th macro- and mesoscale simulation results, highlighting the significance of this approach for underpinning robust computational modelling. • Two novel approaches presented for single shot texture analysis by synchrotron X-ray and Time-Of-Flight neutron diffraction. • New methods are validated by large-area EBSD analysis of a new Nickel-based superalloy. • A novel metric has been proposed to quantify the error of texture determination. • EPSC modelling approach confirms negligible influence of texture errors on computational simulation.

Numerical Investigation into the Influence of Grain Orientation Distribution on the Local and Global Elastic-Plastic Behaviour of Polycrystalline Nickel-Based Superalloy INC-738 LC

Polycrystalline nickel-based superalloys tend to have large grains within component areas where high loads are dominant during operation. Due to these large grains, caused by the manufacturing and cooling process, the orientation of each grain becomes highly important, since it influences the elastic and plastic behaviour of the material. With the usage of the open source codes NEPER and FEPX, polycrystalline models of Inconel 738 LC were generated and their elastic and crystal plasticity behaviour simulated in dependence of different orientation distributions under uniaxial loading. Orientation distributions close to the [100] direction showed the lowest Young’s moduli as well as the highest elastic strains before yielding, as expected. Orientations close to the [5¯89] direction, showed the lowest elastic strains and therefore first plastic deformation under strain loading due to the highest shear stress in the slip systems caused by the interaction of Young’s modulus and the Schmid factor.

Abnormal feather-like grains induced by the thin-wall effect in a polycrystalline nickel-based superalloy

The effect of wall thickness on the casting microstructures of polycrystalline Ni-based superalloy K465 was investigated. Abnormal feather-like grains with high-angle grain boundaries (HAGBs) were detected. The quantity of the feather-like grains increased with the decrease of the specimen thickness. Experimental investigation and numerical simulation results showed the cooling rate and the residual stress increased with the decrease of the specimen thickness. The increased residual stress and the heat treatment resulted from the sand box with elevated temperature induced the formation of the abnormal feather-like recrystallized grains. This work provides a guidance for producing turbine blades made of cast superalloys.

Revealing the true partitioning character of zirconium in additively manufactured polycrystalline superalloys

• Experimental evidence of the migration of zirconium away from grain boundaries during heat-treatment of nickel-based superalloys fabricated by laser powder bed fusion. This contradicts the classic literature on the subject. • Experimental evidence of the reduction of alumina oxides by zirconium during heat-treatment. • Experimental evidence of zirconium dissolved in the γʹ precipitates and the γ matrix after heat-treatment.. • Quantification of the zirconium partitioning after heat-treatment. Minor addition of zirconium is common in polycrystalline nickel-based superalloys, where it is believed that it segregates at grain boundaries and contributes to increase the creep resistance. However, in superalloys produced by additive manufacturing, zirconium may become detrimental as it promotes hot-cracking during the fabrication stage. Here, we clarify the controversial role of this element by studying its distribution at near atomic scale in the as-built and heat-treated microstructures. In the as-built microstructure, zirconium is almost exclusively found at grain boundaries. However, after heat-treatment, zirconium is no longer found at grain boundaries. Instead, it partitions in γʹ precipitates and zirconium oxides particles. The formation of zirconia is shown to originate from the reduction of nano-particles of alumina by zirconium during heat-treatment. The absence of zirconium at grain boundaries in this state challenges the classic view often reported in the literature for superalloys.

Dissolution of the Primary γ′ Precipitates and Grain Growth during Solution Treatment of Three Nickel Base Superalloys

Second phase particles (SPP) play an essential role in controlling grain size and properties of polycrystalline nickel base superalloys. The understanding of the behavior of these precipitates is of prime importance in predicting microstructure evolutions. The dissolution kinetics of the primary γ′ precipitates during subsolvus solution treatments were investigated for three nickel base superalloys (René 65, AD730 and N19). A temperature-time codependency equation was established to describe the evolution of primary γ′ precipitates of each material using experimental data, the Thermo-Calc software and the Johnson–Mehl–Avrami–Kolmogorov (JMAK) model. The dissolution kinetics of precipitates was also simulated using the level-set (LS) method and the former phenomenological model. The precipitates are represented using an additional LS function and a numerical treatment around grain boundaries in the vicinity of the precipitates is applied to reproduce their pinning pressure correctly. Thus, considering the actual precipitate dissolution, these simulations aim to predict grain size evolution in the transient and stable states. Furthermore, it is illustrated how a population of Prior Particle Boundaries (PPB) particles can be considered in the numerical framework in order to reproduce the grain size evolution in the powder metallurgy N19 superalloy. The proposed full-field strategy is validated and the obtained results are in good agreement with experimental data regarding the precipitates and grain size.