Atomistic Kinetic Monte Carlo Research Articles

Cu precipitation in electron-irradiated iron alloys for spent-fuel canisters

In this work, the Cu clustering in Fe under irradiation is investigated using experiments, cluster dynamics and atomistic kinetic Monte Carlo (AKMC) simulations. In experiments, cast iron and model FeCu alloy samples were irradiated with 2 MeV electrons for 143 h at 140 °C. The post-irradiation microstructure was characterized using atom probe tomography. Cluster dynamics and AKMC methods were used to simulate the Cu clustering under the same irradiation conditions. Both simulation methods show satisfactory agreement with experiments, lending strength to the validity of the models. Finally, the Cu clustering in spent-fuel repository conditions for 105 years at 100 °C was simulated using both methods. The results indicate that potential hardening by Cu clustering is insignificant over 105 years.

What can we learn from the exclusion volume of fluctuations and precipitates?

ABSTRACT In the literature, the volume of a cluster (i.e. a fluctuation of the solid solution or a precipitate) is identified as the volume of a sample of the bulk precipitate phase containing the same number of solute atoms (i.e. of the same size). This approximation is suitable for macroscopic precipitates but is debatable for a rigorous description of microscopic clusters, especially during the nucleation and growth stages. To go beyond this simplification, this work focuses on the notion of ‘exclusion volume’, a volume which accounts for real cluster configurations in a given 3D lattice. The exclusion volume normalised by the cluster size is a complex function of the cluster size, which, like the cluster free energy, evolves during precipitation kinetics and converges toward an asymptote in the long-time limit. Furthermore, this asymptote exhibits a bifurcation which clearly separates fluctuations from precipitates, for a cluster size which is found equal to the minimum value of the critical size for nucleation. With the help of Atomistic Kinetic Monte Carlo (AKMC) simulations, it is shown that the evolution of cluster volume during precipitation can be accurately predicted.

Point defect evolution under irradiation: Finite size effects and spatio-temporal correlations

The evolution of point defect concentrations under irradiation is controlled by their diffusion properties, and by their formation and elimination mechanisms. The latter includes the mutual recombination of vacancies and interstitials, and the elimination of point defects at sinks. Two models are traditionally used to predict the defect concentration evolution, the standard rate theory (SRT) and the atomistic kinetic Monte Carlo (AKMC). We show in this work a large discrepancy in the defect concentrations between both methods when the average number of defects in the AKMC simulation box is close or lower than one. The reason is that AKMC naturally captures strong space and time correlations between vacancies and interstitials generated by the finite size of the periodic simulation box. These correlations strongly affect the recombination rate and the point defect concentrations. SRT fails to predict such correlations, and the corresponding solution deviates from the more accurate solution given by the AKMC simulation under similar conditions. These finite size correlation effects are strong when the elimination of point defects occurs by recombination only, but can still be significant in the presence of sinks. In order to account for the spacio-temporal correlation in a continuum framework, we introduce a Correlated Pair Theory (CPT). This theory fully takes into account the correlations between vacancy and interstitial pairs and predicts point defect concentrations in good agreement with AKMC simulations. Inversely, if the goal is for the AKMC to reproduce defect concentrations in bulk using small simulation boxes, the naturally occurring correlations need to be corrected. We show here that the CPT can be used to modify the elimination rates in the AKMC simulations, so as to yield point defect concentrations in agreement with SRT.

Efficient Calculation Methods for the Diffusion Coefficient of Interstitial Solutes in Dilute Alloys.

In the example of oxygen diffusion in dilute ferritic iron alloys it is shown that the calculation of the diffusion coefficient can be separated into a contribution related to the migration in the interaction region between oxygen and the substitutional solute and a part related to diffusion in pure body centered cubic (bcc) Fe. The corresponding diffusion times are determined by analytical expressions using Density-Functional-Theory (DFT) data for the respective binding energies. The diffusion coefficient in the interaction region must be determined by atomistic kinetic Monte Carlo (AKMC) simulations with DFT values for the migration barriers as input data. In contrast to previous calculations, AKMC simulation must only be performed for one concentration of the substitutional solute, and the obtained results can be employed to obtain data for other concentrations in a very efficient manner. This leads to a tremendous decrease of computational efforts. Under certain conditions it is even possible to use analytical expressions where merely DFT data for the binding energies are needed. The limits of applicability of the presented calculation procedures are discussed in detail. The methods presented in this work can be generalized to interstitial diffusion in other host materials with small concentrations of substitutional solutes.

Influence of substitutional atoms on the diffusion of oxygen in dilute iron alloys

A multiscale approach including density functional theory (DFT) and atomistic kinetic Monte Carlo (AKMC) simulations is applied to investigate the diffusion of interstitial oxygen atoms in bcc Fe under the influence of substitutional foreign atoms (Al, Si, P, S, Ti, Cr, Mn, Ni, Y, Mo, and W). The substitutional atoms can be assumed to be immobile since their diffusion coefficient is much smaller than that of oxygen. First, jumps of oxygen in pure bcc Fe between first-, second-, and third-neighbor octahedral interstitial sites are investigated. It is found that the first-neighbor jump is most relevant, with the tetrahedral site as the saddle point. The second-neighbor jump consists of two consecutive first-neighbor jumps. The barrier for a direct third-neighbor jump is too high to be significant for the diffusion process. In the presence of substitutional atoms the most important migration paths are first-neighbor jumps between modified octahedral sites with modified tetrahedral sites as saddle points. Calculations show that Si, P, Ni, Mo, and W cause some modifications of the migration barriers of oxygen and their interaction with O is mainly repulsive. Al, Cr, and Mn have a significant influence on the barriers and they exhibit strong attractive interactions with O. The most important modification of the barriers is found for S, Ti, and Y where deep attractive states exist. Based on the migration energies obtained by DFT, AKMC simulations on a rigid lattice are employed to determine the diffusion coefficient of oxygen in a dilute iron alloy containing the different substitutional atoms. It is found that Si, P, Ni, Mo, and W have almost no influence on the diffusivity of O. The presence of Al, Cr, Mn, S, Ti, and Y causes a significant reduction of the mobility of oxygen. In these cases the temperature dependence of the oxygen diffusion coefficient shows considerable deviations from an Arrhenius law. These phenomena are discussed in detail by considering the occupation time for the different states. The present findings on the strong dependence on the kind of substitutional atoms change the picture of oxygen diffusion in dilute iron alloys substantially.

Interaction between interstitial carbon atoms and a ½ 〈1 1 1〉 self-interstitial atoms loop in an iron matrix: a combined DFT, off lattice KMC and MD study

A static and kinetic study of the interaction between a 19 ½ 〈1 1 1〉 self-interstitial atoms loop and C atoms in body-centred cubic iron is presented in this work. An empirical potential matching the density functional theory calculations is used to study the static properties of the system. The usual kinetic Monte-Carlo (KMC) on-lattice restriction is not valid when the material is highly distorted, especially in the presence of a dislocation loop. Therefore, the dynamics of the system are investigated using both molecular dynamics simulations and k-ART, a self-learning/off-lattice atomic kinetic monte-carlo. The presented work is thus a full study of the C-loop and the C2-loop systems. A good agreement is observed between the statics and the kinetics (e.g. the discovery of a zone of stability of the C atom around the Fe cluster where the C can almost freely move), even though the kinetics show some unexpected behaviours of the studied systems. The pinning time of the loop induced by the C atoms is also estimated.

Precipitation in a copper matrix modeled byab initiocalculations and atomistic kinetic Monte Carlo simulations

A kinetic Monte Carlo approach is used to study the influence of Cr, Fe, Al, or Mg addition on the precipitation in a Cu–Ni–Si alloy. The simulation method is based on a vacancy diffusion model. The crucial parameters of this method are the pairwise mixing energies of all contained elements which are determined by ab initio calculations. The number of dissolved atoms in equilibrium state is used to estimate the influence of Cr, Fe, Al, or Mg on the electrical conductivity. In order to estimate the influence of the alloying elements on strength ab initio calculations of misfit strain at the Cu/Ni3Si interface are performed. The atomistic kinetic Monte Carlo (AKMC) simulations reveal that Cr, Fe, Al, and Mg atoms are located at different positions: Mg and Al atoms are preferably located at the interface of Ni–Si precipitates and Cu matrix, Cr atoms diffuse into Ni–Si precipitates, and Fe atoms form Fe clusters surrounded by Ni–Si shells. Cr addition leads to a significantly reduced fraction of atoms dissolved in the matrix which indicates an increase of electrical conductivity, whereas Mg addition results in a high misfit strain at the Cu/Ni3Si interface which can contribute to increased strength.

Coarse-grained kinetic Monte Carlo simulation of diffusion in alloys

We present a coarse-grained Monte Carlo method designed for the simulation of vacancy-mediated diffusion phenomena. A coarse-grained master equation is derived from the atomic scale master equation using an assumption of local equilibrium within each coarse-grained cell. Atomic kinetic Monte Carlo (AKMC) simulations are used to perform both the thermodynamic and the kinetic parametrization of the coarse-grained simulations. Quantitative reproduction of flux couplings is achieved in the coarse-grained simulation. The ability of the CKMC method to simulate kinetics controlled by diffusion such as a precipitate dissolution and the decay of sinusoidal modulations of the concentration field is illustrated on body-centered-cubic (bcc) model alloys with a clustering tendency. It is shown on the case of a model of the Fe-Cu alloy previously developed that the use of this method can reduce the computational cost of a kinetic simulation by several orders of magnitude.

Correlated recombination and annealing of point defects in dilute and concentrated Fe–Cr alloys

In this work, we present a comprehensive combined modelling approach to study the annealing of lattice defects in dilute and concentrated metallic alloys. The developed approach consists in the combination of molecular dynamics, atomistic kinetic Monte Carlo (AKMC) and mean field rate theory methods, linked at appropriate time and space scales. For the first time, the AKMC tool has been designed to model the evolution of point defects (both vacancies and self-interstitial atoms) in random concentrated alloys, taking into account the influence of lattice distortion on the local migration energy barrier due to the mutual interaction of point defects and solutes. Good accuracy and outstanding speed of calculations has been achieved by introducing the artificial neural network regression as an engine of the AKMC applied to calculate migration barriers for mobile defects. The developed method was applied to study correlated recombination in bcc Fe and random Fe–Cr alloys, aiming at the reproduction of a set of experimental studies after electron irradiation. The obtained results agree well with the available experimental data, implying that the developed modelling procedure correctly captures the undergoing physical process.

Modeling the first stages of Cu precipitation in α-Fe using a hybrid atomistic kinetic Monte Carlo approach

We simulate the coherent stage of Cu precipitation in α-Fe with an atomistic kinetic Monte Carlo (AKMC) model. The vacancy migration energy as a function of the local chemical environment is provided on-the-fly by a neural network, trained with high precision on values calculated with the nudged elastic band method, using a suitable interatomic potential. To speed up the simulation, however, we modify the standard AKMC algorithm by treating large Cu clusters as objects, similarly to object kinetic Monte Carlo approaches. Seamless matching between the fully atomistic and the coarse-grained approach is achieved again by using a neural network, that provides all stability and mobility parameters for large Cu clusters, after training on atomistically informed results. The resulting hybrid algorithm allows long thermal annealing experiments to be simulated, within a reasonable CPU time. The results obtained are in very good agreement with several series of experimental data available from the literature, spanning over different conditions of temperature and alloy composition. We deduce from these results and relevant parametric studies that the mobility of Cu clusters containing one vacancy plays a central role in the precipitation mechanism.

Simulations of Decomposition Kinetics of Fe-Cr Solid Solutions during Thermal Aging

The decomposition of Fe-Cr solid solutions during thermal aging is modeled by Atomistic Kinetic Monte Carlo (AKMC) simulations, using a rigid lattice approximation with composition dependant pair interactions that can reproduce the change of sign of the mixing energy with the alloy composition. The interactions are fitted on ab initio mixing energies and on the experimental phase diagram, as well as on the migration barriers in iron and chromium rich phases. Simulated kinetics is compared with 3D atom probe and neutron scattering experiments.

Physical modeling and implementation scheme of native defect diffusion and interdiffusion in SiGe heterostructures for atomistic process simulation

In order to simulate the diffusion kinetics during thermal treatments in SiGe heterostructures, a physically-based atomistic model including chemical and strain effects has been developed and implemented into a nonlattice atomistic kinetic monte carlo (KMC) framework. This model is based on the description of transport capacities of native point defects (interstitials and vacancies) with different charge states in SiGe alloys in the whole composition range. Lattice atom diffusivities have been formulated in terms of point defect transport, taking into account the different probability to move Si and Ge atoms. Strain effects have been assessed for biaxial geometries including strain-induced anisotropic diffusion, as well as charge effects due to strain-induced modifications of the electronic properties. Si-Ge interdiffusion in heterostructures has been analyzed from an atomistic perspective. A limited set of physical parameters have been defined, being consistent with previously reported ab initio calculations and experiments. The model has been implemented into a nonlattice KMC simulator and the relevant implementation details and algorithms are described. In particular, an efficient point defect mediated Si-Ge exchange algorithm for interdiffusion is reported. A representative set of simulated interdiffusion profiles are shown, exhibiting good agreement with experiments.

Kinetic Monte Carlo Simulations of Anisotropic Lithium Intercalation into LixFePO4 Electrode Nanocrystals

AbstractThe kinetic anisotropy of lithium ion adsorption and lithium absorption for LixFePO4 olivine nanocrystals is simulated and reported. The kinetics depend on the orientation of the electrolyte/LixFePO4 interface with respect to the far‐field ionic flux. As a consequence of these kinetics and a Li miscibility gap in LixFePO4, the particle geometry and orientation also have an effect on the morphology of the two‐phase evolution. These processes accompany the charge and discharge behavior in battery microstructures and a direct influence on battery behavior is suggested. A kinetic Monte Carlo (KMC) algorithm based on a cathode particle rigid lattice is used to simulate the kinetics in this system. In these simulations the adsorption kinetics of the electrolyte/electrode interface are treated by coupling the normal flux outside the particle from a continuum numerical simulation of Li‐ion diffusion in the electrolyte to the atomistic KMC model within the particle. The interfacial reaction depends on local concentration and the potential drop at the interface via the Butler–Volmer (B–V) relation. The atomic potentials for the KMC simulation are derived from empirical solubility limits (as determined by OCV measurements). The main results show that the galvanostatic lithium‐uptake/cell‐voltage has three regimes: 1) a decreasing cell potential for Li‐insertion into a Li‐poor phase; 2) a nearly constant potential after the nucleation of a Li‐rich phase Li(1‐β)FePO4; 3) a decreasing cell potential after the Li‐poor phase has been evolved into a Li‐rich phase. The behavior in the second regime is sensitive to crystallographic orientation.

Stability and mobility of Cu–vacancy clusters in Fe–Cu alloys: A computational study based on the use of artificial neural networks for energy barrier calculations

An atomistic kinetic Monte Carlo (AKMC) method has been applied to study the stability and mobility of copper–vacancy clusters in Fe. This information, which cannot be obtained directly from experimental measurements, is needed to parameterise models describing the nanostructure evolution under irradiation of Fe alloys (e.g. model alloys for reactor pressure vessel steels). The physical reliability of the AKMC method has been improved by employing artificial intelligence techniques for the regression of the activation energies required by the model as input. These energies are calculated allowing for the effects of local chemistry and relaxation, using an interatomic potential fitted to reproduce them as accurately as possible and the nudged-elastic-band method. The model validation was based on comparison with available ab initio calculations for verification of the used cohesive model, as well as with other models and theories.

Kinetic study of phase transformation in a highly concentrated Fe–Cr alloy: Monte Carlo simulation versus experiments

An atomic scale analysis of phase separation in a thermally aged Fe–25 at.% Cr alloy at 500 °C using 3-D atom probe (3DAP) and atomistic kinetic Monte Carlo (AKMC) simulation is presented. Treatment of the simulation data with the Lifshitz–Slyozov–Wagner and Huse laws shows that, whereas diffusion along the interfaces and through the bulk both occur at early stages, diffusion through the matrix quickly controls the growth of domains whereas the structure is still interconnected. Comparison of AKMC results with experimental ones showed that AKMC simulation in the two-band model approximation on a rigid lattice is able to reproduce the behaviour of the concentration field and of the width of the domains observed with 3DAP. This comparison also strongly support that, in real Fe–Cr alloys, as well as in simulated systems, diffusion is predominantly through the bulk and controls the growth of domains, while the structure is still interconnected.

Modelling phase separation in Fe–Cr system using different atomistic kinetic Monte Carlo techniques

Atomistic kinetic Monte Carlo (AKMC) simulations were performed to study α–α′ phase separation in Fe–Cr alloys. Two different energy models and two approaches to estimate the local vacancy migration barriers were used. The energy models considered are a two-band model Fe–Cr potential and a cluster expansion, both fitted to ab initio data. The classical Kang–Weinberg decomposition, based on the total energy change of the system, and an Artificial Neural Network (ANN), employed as a regression tool were used to predict the local vacancy migration barriers ‘on the fly’. The results are compared with experimental thermal annealing data and differences between the applied AKMC approaches are discussed. The ability of the ANN regression method to accurately predict migration barriers not present in the training list is also addressed by performing cross-check calculations using the nudged elastic band method.

Further development of large-scale atomistic modelling techniques for Fe–Cr alloys

In this paper we review the current status of our efforts to model the Fe–Cr system, which is a model alloy for high-Cr ferritic–martensitic steels, using large-scale atomistic methods. The core of such methods are semi-empirical interatomic potentials. Here we discuss their performance with respect to the features that are important for an accurate description of radiation effects in Fe–Cr alloys. We describe their most recent improvements regarding macroscopic thermodynamic properties as well as microscopic point-defect properties. Furthermore we describe a new type of large-scale atomistic kinetic Monte Carlo (AKMC) approach driven by an artificial neural network (ANN) regression method to generate the local migration barrier for a defect accounting for the local chemistry around it. The results of the thermal annealing of the Fe–20Cr alloy modelled using this AKMC approach, parameterized by our newly developed potential, were found to be in very good agreement with experimental data. Furthermore the interaction of a 1/2 〈1 1 1〉 screw dislocation with Cr precipitates as obtained from the AKMC simulations was studied using the same potential. In summary, we critically discuss our current achievements, findings and outline issues to be addressed in the near future development.

Effect of the stress field of an edge dislocation on carbon diffusion inα-iron: Coupling molecular statics and atomistic kinetic Monte Carlo

Carbon diffusion near the core of a $[111](\overline{1}01)$ edge dislocation in $\ensuremath{\alpha}$-iron has been investigated by means of an atomistic model that brings together molecular statics and atomistic kinetic Monte Carlo (AKMC). Molecular statics simulations with a recently developed embedded atom method potential have been carried out in order to obtain atomic configurations, carbon-dislocation binding energies, and the activation energies required for carbon hops in the neighborhood of the line defect. Using information gathered from molecular statics, on-lattice AKMC simulations have been performed for temperatures in the 300--600 K range, so as to study the behavior of a carbon atom as it interacts with the edge dislocation stress field. This model can be seen as a very first step toward the modeling of the kinetics of carbon Cottrell atmosphere formation in iron during the static aging process.

Atomistic Kinetic Monte Carlo studies of microchemical evolutions driven by diffusion processes under irradiation

Atomistic Kinetic Monte Carlo (AKMC) simulations are a powerful tool to study the microstructural and microchemical evolution of alloys controlled by diffusion processes, under irradiation and during thermal ageing. In the framework of the FP6 Perfect program, two main approaches have been applied to binary and multicomponent iron based alloys. The first one is based on a diffusion model which takes into account vacancy and self-interstitial jumps, using simple rigid lattice approximation and broken-bond models to compute the point-defect jump frequencies. The corresponding parameters are fitted on ab initio calculations of a few typical configurations and migration barriers. The second method uses empirical potentials to compute a much larger number of migration barriers, including atomic relaxations, and Artificial Intelligence regression methods to predict the other ones. It is somewhat less rapid than the first one, but significantly more than simulations using “on-the-fly” calculations of all the barriers. We review here the recent advances and perspectives concerning these techniques.

Influence of cluster mobility on Cu precipitation in α-Fe: A cluster dynamics modeling

A cluster dynamics model has been parametrized to quantitatively reproduce results obtained by atomistic kinetic Monte Carlo (AKMC) modeling on the precipitation of Cu in α-Fe under thermal aging. The cluster mobility, highlighted by AKMC, is shown to have a significant effect on the precipitation kinetics and can reconcile the experimentally observed fast kinetics with the relatively low diffusivity of Cu monomers.