Johnson Mehl Avrami Kolmogorov Research Articles

Fokker-Planck equation for the crystal-size probability density in progressive nucleation and growth with application to lognormal, Gaussian and gamma distributions

The Fokker Planck (FP) equation for the probability density function (PDF) of crystal size in phase transformations ruled by progressive nucleation and growth, has been derived. Crystals are grouped in sub-sets, we refer to as τ-crystals, where τ is the birth time of the set. It is shown that the size PDF is the superposition of the PDF of the crystal sub-sets (τ-PDFs), with weight given by the nucleation rate. The growth and diffusion coefficients entering the FP equations are estimated as a function of both τ-PDFs and nucleation rate. The functional form of these coefficients is studied for solutions of the FP equation for τ-crystals given by the lognormal, Gaussian and gamma distributions. For the first two distributions, the effect of fluctuations, nucleation rate and growth rate, on the shape of the distribution has been investigated. It is shown that for an exponential decay of the fluctuation term, the shape of the PDF is mainly governed by both the time constant for nucleation and the strength of the fluctuation. It is found that τ-PDFs given by the one-parameter gamma distributions are suitable to deal with KJMA (Kolmogorov Johnson Mehl Avrami) compliant phase transformations, where the fluctuation term is proportional to crystal size. The connection between the FP equation for the size PDF and the evolution equation for the density of crystal populations is also discussed.

Investigating the evolution of U-10Mo fuel foil microstructures during multi-stage hot rolling using coupled potts model-finite element method simulations

In this work, we study the microstructural evolution of U-10Mo foils over multiple stages of hot-rolling and reheating using our previously validated method coupling the Kinetic Monte Carlo Potts Model with the finite element method. Hot rolling and reheating refine the U-10Mo foil microstructure, but the relationships between the foil microstructure and recrystallization behavior over multiple successive reductions have complex relationships with the rolling schedule that have not yet been well quantified. Simulations were employed to forecast the impact of hot rolling reduction per pass, grain size variations, and uranium carbide (UC) distribution on the Johnson Mehl Avrami Kolmogorov (JMAK) recrystallization kinetics of the U-10Mo alloy, as well as it's post-rolling grain growth kinetics. Initial homogenized grain sizes varying from 100 µm to 1 mm and UC volume fractions ranging from 0 to 2 vol% were parametrically evaluated as a part of this study. While some of our simulation results support the findings of our previous analysis for conditions of single-pass rolling and annealing, our extended analysis shows that the grain size within the as-cast and homogenized foil can lead to significant changes in the in the distribution of strain within the final microstructure, which can slow grain coarsening over the final anneal. The magnitude of the hot rolling reduction per pass had a similarly strong impact on the distribution of strain within the foil microstructure and its subsequent grain growth behavior. The implications of these results on U-10Mo fuel foil fabrication procedures are discussed.

Effect mechanism of Nb addition on grain refinement and inhibition of discontinuous precipitation of Cu–15Ni–8Sn alloy

Cu–15Ni–8Sn-xNb alloys (x = 0, 0.2, 0.6 wt%) were prepared using a medium-frequency induction melting furnace. The effect of Nb addition on grain refinement and inhibition of discontinuous precipitation (DP) in Cu–15Ni–8Sn alloys was systematically studied to elucidate the mechanism by which microstructural characteristics contribute to strength improvement. The results indicate that the increase the Nb content from 0 to 0.6 wt% reduces the average grain size of the as-cast alloy from approximately 524.8 μm to approximately 81.3 μm, and significantly decreases the lamellar transition structure region (α+γ). During solution treatment, dispersed needle-like NbNi3 phases were observed in the Cu–15Ni–8Sn-0.2Nb alloy. After aging, the DP growth rate in the Cu–15Ni–8Sn-0.2Nb alloy was notably slower than those of the other alloys. This was attributed to NbNi3 phases at the grain boundaries hindering DP nucleation, with intragranular NbNi3 phases inhibiting DP growth. The phase transformation order of the solid solution Cu–15Ni–8Sn-0.2Nb at 673 K aging is: spinodal structure → D022 ordered phase → L12 ordered phase → DP. The hardness and tensile strength of the Cu–15Ni–8Sn-0.2Nb alloy peaked at 338.3 HV and 725.42 MPa, respectively, after aging for 120 min. Using the Johnson–Mehl–Avrami–Kolmogorov (JMAK) and Arrhenius equations, the activation energies of DP in Cu–15Ni–8Sn-xNb (x = 0, 0.2, 0.6 wt%) alloys were calculated to be 93.19, 148.64, and 98.33 kJ/mol, respectively. These values suggest that the diffusion of DP atoms in the Nb-containing alloys is hindered, which effectively inhibits DP formation.

Grain boundary character and precipitates in 15Cr-30Ni-2Ti-3Cu valve alloy after solution treatment

The microstructure, precipitates and grain boundary character of 15Cr-30Ni-2Ti-3Cu valve alloy after different solution treatments were investigated. The results show that the precipitates in the forged alloy are Laves phase, (Ti,Nb)C carbides and Sigma phase. The grain sizes of the alloy are 35.21 ± 12.17 μm (980 ℃/1 hour) and 53.14 ± 15.88 μm (1015 ℃/1 hour), respectively, both of which exhibit a single peak distribution, indicating that the austenite grains show normal growth mode. A two-stage grain growth model was established to predict austenite grain growth during solution. The average absolute relative error of the predicted values is 4.93 % and the correlation coefficient of linear fitting is 0.983. According to grain boundary engineering analysis, the ratio of special grain boundaries in the alloy after the solution treatment at 980 ℃ for 3 hours reaches the largest (63.1 %). The activation energy for (Ti,Nb)C carbide dissolution is 223.6 kJ/mol based on the modified Johnson-Mehl-Avrami-Kolmogorov (JMAK) model calculation. The microhardness of the alloy is the highest in the forged state. The microhardness decreases first and then increases with the extension of the soaking time, and reaches the maximum value of 238.7 ± 8.4 HV after the solution treatment at 1050 ℃ for 3 hours.

Tracking microstructural evolution and hardening in Al–Zn–Mg–Cu alloys aged artificially via electrical conductivity measurements

Precipitation hardening, a crucial mechanism for strengthening aluminum alloys, involves stages like Guinier–Preston (GP) zone formation, precipitation, peak aging, and precipitate coarsening. This study focuses on the aluminum 7050 alloy, proposing a method to gauge artificial aging through electrical conductivity measurement. The evolving microstructure and time to peak hardness during aging are vital for creating high-strength alloys. The electrical conductivity variation over time is utilized to analyze the diffusion process governing the clustering and growth of specific phases (η′, η, and S) during artificial aging. The paper demonstrates the impact of GP zones, precipitate formation, and grain growth on electrical conductivity, correlating these factors with hardness, microstructure, and tensile strength to determine the hardening stage. Differential electrical conductivity plots, highlighting aging stages, assist in identifying the hardening phase. Tensile strength and hardness plots differentiate the precipitation phases. The Johnson–Mehl–Avrami–Kolmogorov equation models particle growth kinetics, determining growth rates for AA 7050 alloy. The overall activation energy for precipitate growth is 40.77 kJ/mol, with a growth constant ( m) of ∼4, indicating S phase nucleation during η′ and η growth.

Preform design to reduce forging load and grain size simultaneously in disk forging of IN718

In this study, we propose a new methodology for designing preform shapes for turbofan blisk and disk forgings of IN718 superalloy using a CNN (Convolutional Neural Network). The objective function of the preform design model is set to minimize grain size and reduce forging load. Various preform shape candidates were generated using NURBS (Non-Uniform Rational B-Splines) curves, defined by points on the target shape outline, and the grain size was calculated through FE (Finite Element) analysis based on the JMAK (Johnson-Mehl-Avrami-Kolmogorov) model to train the CNN model. The preform shapes derived from the proposed load and grain size-aware CNN model were compared and validated against those from previous studies regarding grain size and load reduction. An objective function f, which is capable of quantitatively expressing the load and grain size improvement, was developed for comparison and validation. The results showed an improvement of 19.64% for the blisk and 18.96% for the disk compared to previous studies. This research is expected to significantly contribute to the forging of aerospace components.

Investigation of effects of aluminum additions on microstructural evolution and mechanical properties of ultra-high strength and vanadium bearing dual-phase steels

The effects of aluminum (Al) additions (0.04, 0.4, and 0.8 wt%) on the microstructural evolution, tensile properties, and sheared-edge ductility of dual phase (DP) steels were investigated. The DP steels were processed with distinct coiling temperatures after finish rolling, cold rolled, and subjected to two continuous galvanizing line (CGL) simulations: standard galvanizing (GI) and supercooling process (SC). Al expanded both the ɑ-ferrite and (ɑ+γ) phase fields at the expense of the γ-austenite region. Continuous cooling transformation (CCT) diagrams showed that Al increased ferrite start temperatures, broadened the ferrite transformation zone, and reduced austenite stability. Austenite growth kinetics during intercritical annealing, quantified using a modified Johnson-Mehl-Avrami-Kolmogorov (JMAK) model, revealed Avrami exponents ranging from 0.72 to 0.81 and apparent activation energies increasing from 422.9 to 496.7 kJ mol−1 with Al additions. Microstructural characterization revealed larger ferrite grain sizes and higher ferrite volume fractions with higher Al levels. Ultimate tensile strength (UTS) was influenced by Al contents, coiling temperatures, and CGL simulation methods. Global ductility (total elongation, TE) was highest for the 0.8Al Steel with a high coiling temperature and GI annealing, while local ductility (hole expansion ratio, HER) was maximized for the 0.04 Al Steel with a low coiling temperature and SC annealing. The majority of the CGL simulated DP steels exhibited good strength-ductility combinations, with the 0.8Al Steels demonstrating superior properties compared to current automotive advanced high-strength steel (AHSS) grades.

Evolution of the σ phase morphology during its isothermal precipitation in the 2205 duplex stainless steel

The present work aims at addressing the change in the σ phase morphology during its isothermal precipitation at 850 °C in a 2205 Duplex Stainless Steel (DSS) and correlate this morphology change with the evolution of the Avrami exponent n obtained using a modified Kolmogorov-Johnson-Mehl-Avrami (KJMA) equation, which account for the time-dependence of n. The microstructural analysis depicted that the σ phase morphology evolves from a blocky-like shape at the beginning of the transformation (t < 1h50) to a choral-like shape at the end of the transformation. The modified KJMA equation, combining the hard impingement introduced using an impingement exponent c and the time-dependent Avrami exponent n, has been used to assess the observed change in the σ phase morphology. N increases to reach a first plateau at n = 1.5 approximately. Then, it increases again and reaches 2–2.3 at the end of the transformation. Very good agreement has been obtained between the experimental and the calculated transformed fraction with a coefficient of determination R2 = 0.999. The evolution of the Avrami exponent n and values it reaches correlate well with the morphological changes observed in the σ phase.

Kinetics of Martensite/Austenite Decomposition during Tempering of Ultrafine Nano-Bainitic Steels.

In this study, the decomposition of a martensite/austenite (M/A) microconstituent in bainitic steels was analyzed using differential scanning calorimetry (DSC) data in conjunction with Kissinger's and Johnson-Mehl-Avrami-Kolmogorov (JMAK)'s formulas. In bainitic steel subjected to austempering heat treatment, the presence of an M/A microstructure adversely affects the mechanical properties. According to the kinetic equations derived, it is observed that after tempering the sample at 600 °C for 4000 s, the generation of each phase reaches its maximum. The SEM images taken before and after tempering reveal extensive decomposition of the M/A constituent in the microstructure. The proportion of the M/A microstructure decreased significantly from about 10% before tempering to less than 1% after. Additionally, the content of residual austenite also reduced nearly to zero. These observations are consistent with the predictions of the kinetic equations.

Microstructure Prediction of 80MnSi8-6 Steel After Hot Deformation Based on Dynamic Recrystallization Kinetics and FEM Simulation

The microstructure evolution during hot deformation of 80MnSi8-6 nanobainitic steel was investigated through hot compression tests at deformation temperatures of 900–1250°C and strain rates of 0.1–20 s−1. The flow curves revealed strain-hardening behavior at the beginning of deformation followed by softening effects caused by microstructure evolution. A Johnson–Mehl–Avrami–Kolmogorov (JMAK) model for grain growth and dynamic recrystallization was developed, and the kinetics were determined. Critical and peak strains were identified, and coefficients for the microstructure evolution models were determined using linear regression. The analysis of S-curves revealed that decreasing the temperature delays the onset of recrystallization and that the strain rate significantly effects the recrystallization rate at lower temperatures. Constitutive modeling and determination of the Zener–Hollomon parameter allowed the determination of the influence of hot processing conditions on material behavior during deformation. Microstructure analysis showed that, at higher deformation temperatures, grain growth occurs simultaneously with grain refinement. Coefficients for the JMAK model were implemented in QForm software. Simulation results were compared with experimental measurements exhibited good arrangement, which confirms the accuracy of the JMAK model in predicting the microstructure evolution. This study demonstrated how microstructure evolution modeling and FEM simulations combined can be used to predict the grain size of 80MnSi8-6 steel after hot deformation.

Microstructure Evolution and Homogenization Kinetics of 15Cr–30Ni–Fe Heat‐Resistant Alloy

The as‐cast microstructure, microsegregation of alloying elements, and microstructure evolution during homogenization of the newly designed 15Cr–30Ni–Fe heat‐resistant alloy are studied. Ti, Nb, Si, and Mo segregate at the interdendritic region. The brittle eutectic Laves phase contains Fe, Ti, Ni, Nb, Si, and Mo, which is identified as the Fe2Ti Laves phase. The activation energy for the Fe2Ti Laves phase dissolution is close to the activation energy for Ti diffusion in the γ matrix based on the Johnson–Mehl–Avrami–Kolmogorov analysis. The back‐diffusion of Ti to the γ matrix is the limiting step for the dissolution of Fe2Ti Laves phase. The kinetic equation of homogenization considering lattice parameters and pre‐exponential factor correction is established. After the double‐stage homogenization at 980 °C/10 h + 1140 °C/4 h, the volume fraction of the Laves phase decreases from 5.11% to 0.26%, and the large eutectic Laves phase is completely dissolved. The reduction in the standard deviation of microhardness at different soaking times effectively reflects the gradual decrease in the degree of microsegregation of alloying elements. The average grain size of the alloy after homogenization is 410 ± 63.1 μm, which is beneficial to control the further refinement of grain during the subsequent hot working.

Detailed preform design procedure considering the effect of heat treatment in IN718 disk forging

We investigated the impact of heat treatment on preform deduction during the hot forging of In718 turbine disks. Proposed preform design method leverages a Convolutional Neural Network (CNN) to aim for minimized damage and uniform grain size in the final forged products. For damage calculation, we employ the Lemaitre damage model, while the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model is utilized for grain size calculation. The CNN, trained on datasets of NURBS-generated preform shapes and final forged disks from them, showed superior performance in terms of damage and grain size of forged disk compared with other existing approaches. Moreover, the results indicate that preforms deduced with heat treatment considerations can lead to significant improvements in forging results, including up to a 17% reduction in average grain size and a 16% decrease in standard deviation compared to preforms without heat treatment consideration. These results underscore the importance of considering heat treatment in preform design, offering valuable insights for industries where the integrity of forged products, like aerospace, is critical.

Weibull function to describe the cumulative size distribution of clumps formed by two-dimensional grains randomly arranged on a plane.

Many manifestations of natural processes give rise to interesting morphologies; it is all too easy to cite the corrugation of the Earth's surface or of planets in general. However, limiting ourselves to 2D cases, the morphology to which crystal growth gives rise is also intriguing. In particular, it is interesting to study some characteristics of the cluster projection in 2D, namely the study of the shapes of the speckles (fractal dimension of their rims) or the distribution of their areas. Recently, for instance, it has been shown that the size cumulative distribution function (cdf) of "voids" in a corrole film on Au(111) is well described by the well known Weibull distribution. The present article focuses on the cdf of cluster areas generated by numerical simulations: the clumps (clusters) are generated by overlapping grains (disks) whose germs (disk centers) are chosen randomly in a 2000×2000 square lattice. The obtained cdf of their areas is excellently fitted to the Weibull function in a given range of surface coverage. The same type of analysis is also performed for a fixed-time clump distribution in the case of Kolmogorov-Johnson-Mehl-Avrami (KJMA) kinetics. Again, a very good agreement with the Weibull function is obtained.

The recrystallization behavior of cryo- and cold-rolled AlCoCrFeNiTi high entropy alloy

The present work was conducted on the non-equiatomic Al0·2Co1·5CrFeNi1·5Ti0.3 high entropy alloy (HEA), which was separately subjected to 50% cold- and cryo-rolling. Subsequent annealing was performed within the temperature range of 800–1000 °C to investigate their recrystallization behavior. A partial recrystallization was achieved up to 900 °C for the cold-rolled material, while the cryo-rolled one had shown nearly fully recrystallized microstructure after 800 °C. A significant reduction in grain size from the solution-treated state (∼784 μm) to fully recrystallized state (∼71 μm for cold-rolled and ∼6 μm for cryo-rolled) was observed. A combined XRD and TEM analysis confirms that the cryo-rolled sample exhibits higher dislocation density than the cold-rolled sample, which resulted in faster recrystallization kinetics observed in the former than in the latter. Additionally, the Johnson–Mehl–Avrami–Kolmogorov model was employed to understand the recrystallization behavior.

Long term hydrogen storage properties of ZK60 Mg-alloy as processed by different methods of SPD

Hydrogen storage characteristics is studied in the Mg-alloy ZK60 after processing by different SPD (Severe Plastic Deformation) methods such as High Pressure Torsion (HPT) and Friction Stir Processing (FSP), applying various deformation extents and rates. The capacity and kinetics of hydrogen storage was investigated and analysed, up to 100 storage cycles. While the degree of SPD deformation is less important for the storage capacity, the SPD processing method itself matters, yielding about ~ 30% more capacity in FSP than in HPT. As shown by DSC and XRD analyses, it is the density of SPD-induced vacancy agglomerates which is significantly higher in FSP than in HPT (~ 10–3 instead of ~ 10–4) because of the enhanced dislocation slip activity. Thanks to their stabilization through Mg(Zn,Zr) precipitates, the vacancy agglomerates survive numerous cycles of hydrogen storage in spite of the high storage temperature of 350 °C, and can act as thermally stable heterogeneous nuclei for the hydrogenation. This latter mechanism was found in all SPD methods applied irrespective of the deformation extent, on the basis of Johnson–Mehl–Avrami-Kolmogorov analysis providing the Avrami exponent n = 1, already from the second up to the highest hydrogen storage cycles.

Comparative study of microstructural evolution in W-3Re alloy under high-temperature conditions: High heat flux loading versus furnace heating

In tokamak devices, tungsten (W), as the main candidate for plasma facing material, will suffer from the simultaneous challenges of high heat flux (HHF) and neutron irradiation. This study involved annealing experiments on W-3Re alloy using two distinct heating methods: furnace heating and electron beam heating. The evolution of hardness and microstrure of tungsten-rhenium (W-3Re) alloy after isochronous annealing and isothermal annealing has been analyzed and compared with pure tungsten (PW). The recrystallization kinetics of W-3Re alloy have been obtained employing the Johnson Mehl-Avrami Kolmogorov (JMAK) equation. Experimental results showed that recrystallization kinetics during electron beam annealing were faster than those in furnace annealing. This phenomenon can be attributed to the rapid heating process of electron beam annealing, which preserved more deformation energy within the material, thus providing more driving force for recrystallization at high temperatures. Additionally, Re appeared to inhibit grain boundary migration in W-3Re alloy, causing the recrystallization temperature to be approximately 100 °C higher than in pure tungsten, and also resulted in a smaller average grain size compared to pure tungsten. This research provides experimental data and reference for predicting the microstructural evolution of tungsten-rhenium alloy in future fusion reactor environments.

Microstructure evolution of the rolled tungsten during the current-assisted annealing treatment

Current-assisted annealing treatment (CAAT) is a promising method for adjusting the microstructure of metals. It was employed to heat treat the tungsten (W) with a rolling deformation of 90% to clarify its microstructure evolution under the action of current. The grain morphologies before and after heat treatment were observed by metallographic microscopy. Compared to traditional heat treatment, the CAATed W is easier to achieve full recrystallization with finer and more uniform grains. This indicates that the current increases the nucleation rate of recrystallization. The Johnson-Mehl-Avrami-Kolmogorov (JMAK) model was applied to describe the increase rate of recrystallization fraction. It can be calculated that the apparent activation energy of recrystallization under current is only 352.9 kJ/mol. Combined with electron backscatter diffraction (EBSD) characterization, the rolled W prefers to align along the 〈100〉 // ND orientation with the prolonging of current action time. In essence, the acceleration of recrystallization and the tendency of grain orientation are both related to the atomic migration enhanced by current. The electric current can generate an additional driving force for atomic migration, especially in regions with high electrical resistivity, where this driving force becomes more pronounced, leading to intensified atomic migration.

Effect of the roughness on the photoinduced growth of crystalline tellurium on MoTe2 matrix

In this work, a series of 5 nm thick films of molybdenum oxide (MoO3) were synthesized on SiO2/Si substrates by magnetron sputtering of molybdenum in low vacuum conditions. We used closed-space sublimation (CSS) to tellurize the Mo oxide and to produce pure-phase 1 T′-MoTe2 films by controlling the temperature inside the CSS reactor. Surface analyses showed that tellurium spreads over the surface of the as-grown films, forming localized crystalline precipitates whose distribution depends on the tellularization time. From Raman spectroscopy investigations, we observed that the exposure of these films to visible light leads to a growth of a highly-ordered trigonal phase of tellurium. The intensity evolution of the vibrational modes related to the crystalline phase of tellurium allows us to perform a quantitative analysis of the crystallization. Our results indicate that this process is consistent with the Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory for the overall crystallization process. By the characteristic growth exponent, we introduce a more specific model which takes into account the diffusion and the dissociation of tellurium chains, which is used to probe how the effective growth rate is affected by the roughness of the samples.

Effect of Austenite Grain Size on Phase Transformation Structure of Low‐Carbon Microalloyed Steel

The effect of austenite grain on the phase transformation of low‐carbon microalloyed steel during continuous cooling is investigated by using Gleeble thermal simulator. The austenite grain size is changed by altering the holding period at the peak temperature. The findings demonstrate that for a given cooling rate, the longer the holding time, the microstructure becomes finer, and bainite begins to show up in the microstructure. The critical temperature of phase transformation and the diagram of phase transformation reaction rate during continuous cooling are determined by dilatometry. The Johnson–Mehl–Avrami–Kolmogorov equation is used to fit the transformation volume fraction, and the variation of kinetic parameters k and n with prior austenite grain size under nonisothermal condition is determined. The n value is trending downward as the prior austenite grain size increases, while the kinetic parameter k barely changes at all. The activation energy of phase transition is calculated by Kissinger method. The activation energy of fine austenite grain increases from 248.6 to 286.3 kJ mol−1 with the increase of austenite grain size, while the activation energy of coarse austenite grain decreases to 176.5 kJ mol−1 due to the competitive nucleation mechanism.

Effect of initial grain size on the recrystallization behavior and recrystallization texture of a Mg–3Gd alloy

The effect of initial grain size on the recrystallization and recrystallization texture of a rolled Mg–3Gd (wt.%) alloy is studied in detail. The results show that the deformation microstructure of an initially coarse-grained (CG) sample has a larger twinned area and a higher density of twin boundaries than a fine-grained (FG) sample. After annealing, the CG sample recrystallizes preferentially in the twinned area, whereas the FG sample adopts the higher density grain boundaries as the nucleation sites. Furthermore, weak recrystallization texture components appear from the grain nucleation stage, regardless of the initial grain size, and are preserved after complete recrystallization due to uniform grain growth. The majority of recrystallization texture is deviated 20°–45° away from normal direction (ND), accounting for more than 50 %. Especially, the recrystallization texture of the FG sample is a “Rare Earth texture”, in contrast to the widely reported texture modification unrelated to grain boundary nucleation. Only a scattered basal texture is observed in the CG sample, which also differs from the reported “Rare Earth texture” originating from shear band nucleation in dilute Mg–Gd alloys. Finally, based on the Johnson–Mehl–Avrami–Kolmogorov (JMAK) model, the recrystallization kinetics are calculated, and it is found that the initial grain size mainly affects the nucleation rate, and has limited effect on the grain growth rate.