Modeling Of Precipitation Kinetics Research Articles

The influence of carbon content gradient and carbide precipitation on the microstructure evolution during carburizing-quenching-tempering of 20MnCr5 bevel gear

Due to the mechanism of microstructural transition, high-strength low-carbon alloy steel gears exhibit high strength and surface hardness after Carburizing-Quenching-Tempering (CQT) combined heat treatment, while maintaining good toughness in the core. The carbon potential gradient and the precipitation of carbides are the main factors affecting the final microstructure, yet there is a shortage of related research. To address this gap, this study developed a multi-physics coupled model considering the impact of carbon content to simulate microstructural transformation. Using finite element analysis, we simulated the microstructural evolution during the CQT heat treatment of 20MnCr5 bevel gear. By comparing the simulation results with the calculations from the commercial heat treatment software DANTE®, the accuracy of this model in predicting microstructural distribution and hardness has been validated. Furthermore, the study established a carbide precipitation kinetics model that takes into account the influence of grain size, to investigate the precipitation behavior of carbides within the matrix during the tempering stage. The cellular automaton method was applied to demonstrate the evolution process of the quenched microstructure during tempering. The study shows that during tempering, carbides primarily precipitate in areas with higher carbon content, preferentially at the boundaries of fine grains with a high density of lattice defects. As the carbon content increases, the number density of precipitates rises, while their average radius decreases. The reduction in matrix supersaturation due to carbide precipitation is one of the causes for the reduction of tooth surface hardness during tempering. The model constructed in this study provides a new perspective for understanding the laws governing material microstructural evolution and offers a solid theoretical foundation for the study of stress and deformation during the heat treatment process.

Carbonitride Precipitation Kinetics Model During Continuous Casting of Ti Microalloyed Steel

Carbonitride Precipitation Kinetics Model During Continuous Casting of Ti Microalloyed Steel

Mean-field modelling of precipitation kinetics with a Fokker-Planck equation in the Lifshitz–Slyozov–Wagner space

Mean-field theory has been widely applied in modelling the kinetics of microstructure evolution. Considering a free energy contribution by microstructure entropy, a Fokker-Planck equation for precipitation kinetics is variationally derived from thermodynamic principles. Using dimensionless transformation into the Lifshitz-Slyozov-Wagner space, the Fokker-Planck equation is effectively solved within a fixed radius interval and an explicit Euler scheme, which is computationally advantageous. Through comparisons with the Ni-12.5 Al at. % alloy, it is shown that the Fokker-Planck equation can represent the typical stages of precipitation kinetics, i.e., nucleation, growth, and coarsening.

Unified theory of nucleation and coarsening in the solid state

ABSTRACT Despite numerous contributions made to the modelling of precipitation kinetics, no theory seems to have unified all the concepts required to simulate the nucleation, growth, and ripening processes as one continuous process. In the present work, a novel approach was used to reshape the master equation calculating the evolution of the size distribution, so that the zero-flux condition can represent a steady-state evolution. This modification was made by introducing explicitly the probability for clusters to grow or to dissolve in each size class. The expression found to calculate this probability allowed the calculation of the entire evolution of the size distribution from the first step of nucleation to the final stages of the ripening process. The winners-losers concept also allowed for a better definition of the average condensation and evaporation rates, which were found by applying the thermodynamic extremum principle and assuming that a population of clusters of a given size behaves on the average as one cluster of the same size and submitted to the same driving forces. Furthermore, the thermodynamic extremum principle allowed us to show that the subcritical growth regime must be interface controlled to maximise the dissipation rate of energy. The computational scheme based on the new theory was applied to calculate the size evolution of precipitates in the Al-Cu system. The influence of aging temperature on the evolution of the number density of precipitates during homogeneous nucleation was found to agree with experimental estimations of number densities made on uniformly distributed θ′-Al2Cu precipitates.

Fully coupled segregation and precipitation kinetics model with ab initio input for the Fe-Au system

For designing new and improved materials, predictive modeling of precipitation and segregation kinetics is required. So far, no modeling approach is available that couples precipitation with ab initio segregation data. We derive the mathematical basis for a model to describe segregation and precipitation kinetics with grain boundary segregation energies obtained from ab initio simulations. The model is implemented rigorously and validated with experimental data on a Fe-Au system from literature.

A combined experimental-analytical modeling study of the artificial aging response of Al–Mg–Si alloys

The isothermal aging response at 170 °C of three extruded Al–Mg–Si alloys that had been solution-treated and quenched was experimentally tracked to understand the influence of Mg + Si content and the Mg/Si ratio on microstructure and mechanical behavior. Transmission electron microscopy (TEM) was employed to quantitatively characterize the needle-shaped β'' precipitates in the underaged, peak-aged and overaged conditions; while the alloy with a high Mg + Si content raised the number density of β'' precipitates, a higher Mg/Si ratio increased the number density but decreased the size of β'' precipitates in the peak-aged condition. In the overaged condition, coarsening of β'' precipitates occurred primarily along its length. Single crystal micropillar compression tests were conducted in the naturally-aged, under-aged, peak-aged and over-aged conditions to correlate the microstructure to the critical resolved shear stress (CRSS). In the peak-aged condition, the alloy with the high Mg + Si content and a Mg/Si ratio corresponding to the β'' stoichiometry (Mg5Si6) exhibited the highest CRSS. A precipitation kinetics model that assumed a cylindrical morphology for the β'' phase with a fixed aspect ratio was developed to describe β'' phase microstructure evolution as a function of alloy chemistry, solution treatment temperature and artificial aging parameters. The results from this model served as input to a dislocation-precipitate interaction model to calculate single crystal yield strength. The predictions from these analytical models were validated with experimental results from this study and from the literature and fundamental insights were obtained on the role of microstructural parameters in affecting yield strength for non-spherical precipitates.

Modelling precipitation hardening in an A356+0.5 wt%Cu cast aluminum alloy

The behavior of a A356+0.5 wt%Cu alloy used to manufacture cylinder heads was studied. Samples were solutionized, quenched and aged at 200 °C for 0.1, 1, 10 and 100 h. TEM characterization showed that for the short aging durations (up to 10 h), the dominating hardening precipitates were β″ rods, while for the long aging duration (100 h), the dominance shifted to the Q-phase (Q′, Q″ precipitates). The length and diameter of the β″ rods were measured to produce size distributions which were later used to calibrate and validate the precipitation model. The physics-based precipitation kinetics model relies on classical nucleation/growth/coarsening equations adapted for the precipitation of Mg–Si precipitates in the aluminum matrix. Indirect coupling to Thermo-Calc software was used in order to determine the essential thermodynamic variables such as the driving force for precipitation and the solubility product for the model. Recent developments regarding the correction of the growth rate equations and the curvature effect were used to take into account the elongated morphology of precipitates. A Kampmann-Wagner Numerical (KWN) based model was used to track the evolution of the size distributions during nucleation, growth and coarsening of the β precipitates. The yield strength of the alloy was modelled using the Pythagorean sum of the contributions of intrinsic strength, solid solution strengthening and precipitation hardening. Both models showed good accuracy when compared to experimental results.

Influence of Intrinsic Heat Treatment (IHT) on Precipitation Kinetics during Laser Beam Powder Bed Fusion of Al-Mg-Sc Alloy

Laser Beam Powder Bed Fusion (LBPBF) process has a unique feature termed as IntrinsicHeat Treatment (IHT), where solidified layers undergo series of heating and cooling (during thesubsequent building of a part). Thus, the LBPBF process offers the opportunity for the formation of microstructuralfeatures, which can have the potential to transform the mechanical properties of the part.In the case of AlMgSc alloy, L12 phase Al3Sc precipitates are thermodynamically favored to nucleatein the Al matrix due to coherency. After post-process analysis, it is evident that Al3Sc precipitatesformed during the LBPBF process, but it is unlikely to monitor (in-situ) the kinetics of precipitation.Therefore, based on inputs from the thermal model, the simulation of precipitation kinetics during theLBPBF process (IHT) is performed. The rapid heating and cooling cause the formation of new vacancies,where Al3Sc precipitates can nucleate and grow. The KWN model based on solid-state phasetransformation is used for modeling of precipitation kinetics. The thermal data at two locations in apart is collected and used to determine the average radius, number density, and volume fraction ofprecipitates. It is found that the IHT does not influence precipitation kinetics, and has no potential toalter the spatial properties of the part.

Role of elastic strain energy in spheroidal precipitates revisited

Improving the treatment of thermodynamic modelling of precipitation kinetics makes it necessary to account for the stored elastic and interface energies of non-spherical strengthening particles. The existing kinetic concepts available for spherical precipitates can be directly adapted to spheroidal precipitates by introducing shape factors, thus approximating shapes differing from spheres. This paper provides shape factors for the stored elastic and interface energies over a wide range of aspect ratios as well as of elastic material properties.

Langer\u2013Schwartz\u2013Kampmann\u2013Wagner precipitation simulations: assessment of models and materials design application for Cu precipitation in PH stainless steels

Quantitative modelling of precipitation kinetics can play an important role in a computational material design framework where, for example, optimization of alloying can become more efficient if it is computationally driven. Precipitation hardening (PH) stainless steels is one example where precipitation strengthening is vital to achieve optimum properties. The Langer–Schwartz–Kampmann–Wagner (LSKW) approach for modelling of precipitation has shown good results for different alloy systems, but the specific models and assumptions applied are critical. In the present work, we thus apply two state-of-the-art LSKW tools to evaluate the different treatments of nucleation and growth. The precipitation modelling is assessed with respect to experimental results for Cu precipitation in PH stainless steels. The LSKW modelling is able to predict the precipitation during ageing in good quantitative agreement with experimental results if the nucleation model allows for nucleation of precipitates with a composition far from the equilibrium and if a composition-dependent interfacial energy is considered. The modelling can also accurately predict trends with respect to alloy composition and ageing temperature found in the experimental data. For materials design purposes, it is though proposed that the modelling is calibrated by measurements of precipitate composition and fraction in key experiments prior to application.Graphic abstract

On the modelling of precipitation kinetics in a turbine disc nickel based superalloy

The precipitation kinetics of gamma prime in the nickel based superalloy RR1000 has been characterised after solid-solution heat treatments and isothermal aging conditions relevant to service conditions. Multi-modal precipitate dispersions are formed within the alloy. Numerical methods are presented for determining the three dimensional size of the particle populations combining information obtained from Scanning Electron microscopy and Transmission Electron microscopy. This information has been used to develop a multi-component mean-field model descriptive of precipitation kinetics. The smallest particle population increases in mean size during isothermal aging at 700 ∘C where classical mean-field models of coarsening kinetics suggest that these particles should dissolve. A phenomenological model has been proposed to capture this behaviour within a statistical formulation that is applicable to both processing and service conditions.

Effect of annealing process on the abnormal grain growth behavior during carburizing of 20CrMnTi steel

Abnormal grain growth (AGG) as an important physical metallurgical behavior is to be avoided during carburizing due to its adverse effect on properties such as fatigue, impact toughness and the quenching distortion. In this work, two types of annealing process which produce different morphology of precipitated particles were conducted to investigate its effect on the subsequent austenite grain growth behavior during carburizing of 20CrMnTi steel. Calculation from Humphreys’ model exhibited a good agreement with experimental observations, which indicates the large sized particle from high temperature annealing led to the locally insufficient pinning force for the blockage of grain boundary migration and is therefore responsible for the AGG. The isothermal precipitation kinetics for TiC particles is able to be described by the diffusional growth and LSW coarsening models with reasonable accuracy provided with the transition time from growth to coarsening stage. Design of microalloying and pre-treatment processing to suppress AGG can be assisted with the combinatory application of Humphrey’s model and precipitation kinetics model.

Dislocation density based model for Al-Cu-Mg alloy during quenching with considering the quench-induced precipitates

The accurate prediction of residual stress for as-quenched Al-Cu-Mg alloy requires profound knowledge on the materials' constitutive behavior. In present work, the flow stress of as-quenched Al-Cu-Mg alloy is modelled using a dislocation density based model which considers the influences of precipitation, solid solution and forest dislocation. The radii and volume fractions of precipitates at different cooling rates and temperatures are predicted by a multi-class precipitation kinetics model. Parameters of the constitutive model are obtained by calibration using isothermal tensile tests. The model predictions are in good agreement with experimental data in terms of the flow stress and volume fraction evolution. The results show that volume fraction of precipitates decreases with the increasing temperature and cooling rate, in contrast to the increasing trend for radius of precipitates. The quench-induced precipitates have an influence only on the magnitudes of flow stress at low temperatures. However, for the large precipitates, the dislocation loops, i.e. geometrically necessary dislocations (GND's), can influence both the strain hardening rates and magnitudes of flow stress.

Rule-based controlling of a multiscale model of precipitation kinetics

One of the most important obstacles of widening of multiscale modelling is its high computational demand. It is caused by the fact, that each of numerous fine scale models has comparable computational requirements to a coarse scale one. There are several ways of decreasing of computational time of multiscale models. Optimization of a structure of a model is one of the most promising. In this paper the Adaptive Multiscale Modelling Methodology is described, including Knowledge-Based adaptation of the multiscale model of precipitation kinetics during heat treatment. Core features of the methodology are introduced. The numerical model of heat treatment of an aluminium alloy based on the methodology and the dedicated framework is presented. Besides modelling of macroscopic heat transfer, models of precipitation kinetics based on thermodynamic calculations are included. To decrease computational requirements arising from coupling of the macroscale model and the thermodynamic models, metamodeling and similarity approaches are applied. Computations with several configuration of rules are described, as well as their results. Reliability and time consumption of computations are discussed. Future perspectives of combining of modelling and metamodeling in one, integrated model are discussed.

High-throughput in-situ characterization and modeling of precipitation kinetics in compositionally graded alloys

The development of new engineering alloy chemistries is a time consuming and iterative process. A necessary step is characterization of the nano/microstructure to provide a link between the processing and properties of each alloy chemistry considered. One approach to accelerate the identification of optimal chemistries is to use samples containing a gradient in composition, ie. combinatorial samples, and to investigate many different chemistries at the same time. However, for engineering alloys, the final properties depend not only on chemistry but also on the path of microstructure development which necessitates characterization of microstructure evolution for each chemistry. In this contribution we demonstrate an approach that allows for the in-situ, nanoscale characterization of the precipitate structures in alloys, as a function of aging time, in combinatorial samples containing a composition gradient. The approach uses small angle X-ray scattering (SAXS) at a synchrotron beamline. The Cu–Co system is used for the proof-of-concept and the combinatorial samples prepared contain a gradient in Co from 0% to 2%. These samples are aged at temperatures between 450°C and 550°C and the precipitate structures (precipitate size, volume fraction and number density) all along the composition gradient are simultaneously monitored as a function of time. This large dataset is used to test the applicability and robustness of a conventional class model for precipitation that considers concurrent nucleation, growth and coarsening and the ability of the model to describe such a large dataset.

Modeling of precipitation kinetics in multicomponent systems: Application to model superalloys

Abstract A new general model dealing with nucleation, growth and coarsening simultaneously has been developed for the simulation of precipitation in non-dilute multicomponent alloys. Nucleation is implemented using the Zeldovich theory that includes regression effects. Growth and coarsening are modeled using the recently developed growth law in multicomponent alloys that accounts for capillarity, mass balance at the interface matrix-precipitate and diffusion-flux couplings. Numerical results are confronted to atom probe tomography (APT) experiments on model NiCrAl superalloys and to rigid lattice kinetics Monte Carlo (LKMC) simulations and are found in very good agreement with both APT experiments and LKMC simulations. We emphasize this work on the evolution of phase concentrations. The temporal evolution of the mean precipitate composition is found to be non-monotonic during the phase transformation, and phase composition does not follow the tie line due to a complex interplay between capillarity and diffusion process. The widespread availability of both thermodynamic and mobility databases makes this new model very suitable for material design.

Experiment and modeling of ultrafast precipitation in an ultrafine-grained Al–Cu–Sc alloy

Experimental results revealed that the aging precipitation behaviors were significantly enhanced in an ultrafine grained (UFG) Al–Cu–Sc alloy when compared with its coarse grained (CG) counterpart. In the UFG Al–Cu–Sc alloy aged at 398K for 20h, a large number of θ′-Al2Cu particles with an average radius of about 38nm were precipitated within the grain interior. While in the CG alloy aged at the same condition, no precipitations were found. The ultrafast precipitation behaviors observed in the UFG alloy is rationalized by developing a precipitation kinetics model. This model is based on the classical N-model framework of Kampmann and Wagner (KWN), but is modified to capture some precipitation features in the UFG regime, such as highly enhanced diffusion and greatly reduced nucleation energy barriers. The modified model yields predictions in good agreement with experimental data, which are several orders of magnitude faster than those predicted by the classical model. This work indicates that the classical N-model, after some modifications, is still applicable for the quantitative description of precipitation behaviors in UFG Al alloys.

PRECIPITATION KINETICS AND YIELD STRENGTH MODEL FOR NZ30K-Mg ALLOY

Age-hardening effect is considerably strong in magnesium alloys containing Nd, making it possible to develop magnesium alloys with low cost and high strength. Although there have been massive researches about the precipitation product sequence and strengthening models in magnesium, aluminum and other light alloys during their ageing processes, those of NZ30K-Mg alloy, a newly-developed magnesium alloy, has not been carefully investigated. The present work mainly focuses on the model of precipitation kinetics and strengthening of NZ30K-Mg alloy. The precipitation kinetics has been investigated using electrical resistivity testing during continuous heating with different heating rates and formulated based on the isoconversional method. Two related model parameters, modified pre-exponential factor A'αrand activation energy Eαrwere respectively determined. The precipitation behavior of NZ30K-Mg alloy during ageing processes can also be intrinsically explained from the variations of Eαrand ln A'αr with the precipitation fraction. This kinetics model with two above-mentioned parameters can accurately describe the precipitation of strengthening phase during different ageing processes. The yield strength of under-aged and peak-aged NZ30K-Mg alloy have been tested and the results show that the testing samples isothermally aged at different temperature from 180 to 250 ℃ have almost the same peak yield strength of about 150 MPa, indicating that the strengthening effect of under-aged and peak-aged NZ30K-Mg alloy is only determined by the precipitation fraction within a certain range of temperatures. The precipitation strengthening model of NZ30KMg alloy has been carefully derived, and the parameter C in the model has then been determined by least squares method based on the tested yield strength data. The value of C is about 93 MPa. The prediction of yield strength of under-aged and peak-aged NZ30K-Mg alloy has been performed and fit well with the tested ones, demonstrating the effectiveness of precipitation strengthening model and its engineering application prospects.

Frühe Stadien der Ausscheidungsbildung – Experimente und Modellierung

Precipitation hardening of structural high-performance materials is considered as one of the most important strengthening mechanisms. In alloys containing precipitate forming elements, a fine dispersion of nanometer-sized particles precipitates from a supersaturated solid-solution. The size, volume fraction, and number density of these precipitates can be controlled by elaborate thermo-mechanical processing or by thermal treatments. The Christian Doppler Laboratory “Early Stages of Precipitation” aims at establishing a deeper understanding of such precipitation processes in steels, nickel-base alloys, and refractory metals. The experimental part of the lab in Leoben focuses on the characterization of small precipitates by state-of-the art high-resolution methods such as atom probe tomography and transmission electron microscopy, by scattering techniques, and by thermal analysis. The modelling part in Vienna concentrates on the modelling of precipitation kinetics mostly within the framework of the scientific software “MatCalc”. The main objective of both parts is the comprehensive study of the microstructure-property relationship in order to develop particle strengthened alloys with improved mechanical properties.

An Atomistic-Based Hierarchical Multiscale Examination of Age Hardening in an Al-Cu Alloy

A large class of modern structural alloys derives its strength from precipitation hardening. Precipitates obstruct the motion of dislocations and thereby increase alloy strength. This paper examines the process using an atomistic-based hierarchical multiscale modeling framework. Atomistic modeling is employed to (1) compute solute-dislocation interaction energies for input into a semi-analytic solute hardening model and (2) evaluate precipitate strengths for use in dislocation line tension simulations. The precipitate microstructure in the dislocation line tension simulations is obtained from simple analytic precipitation kinetics relations. Fitting only the rate constants in the precipitation kinetics model, the macroscopic strength predictions of the hierarchical multiscale model are found to correspond reasonably well with experiments. By analyzing the potential sources of discrepancy between the model’s macroscopic predictions and experiments, this work illuminates the importance of specific atomic-scale processes and highlights important challenges that remain before truly predictive mechanism-based plasticity modeling can be realized.