Atomistic Kinetic Monte Carlo Simulations Research Articles

Self-Evolving Atomistic Kinetic Monte Carlo simulations of defects in materials

The recent development of on-the-fly atomistic kinetic Monte Carlo methods has led to an increased amount attention on the methods and their corresponding capabilities and applications. In this review, the framework and current status of Self-Evolving Atomistic Kinetic Monte Carlo (SEAKMC) are discussed. SEAKMC particularly focuses on defect interaction and evolution with atomistic details without assuming potential defect migration/interaction mechanisms and energies. The strength and limitation of using an active volume, the key concept introduced in SEAKMC, are discussed. Potential criteria for characterizing an active volume are discussed and the influence of active volume size on saddle point energies is illustrated. A procedure starting with a small active volume followed by larger active volumes was found to possess higher efficiency. Applications of SEAKMC, ranging from point defect diffusion, to complex interstitial cluster evolution, to helium interaction with tungsten surfaces, are summarized. A comparison of SEAKMC with molecular dynamics and conventional object kinetic Monte Carlo is demonstrated. Overall, SEAKMC is found to be complimentary to conventional molecular dynamics, especially when the harmonic approximation of transition state theory is accurate. However it is capable of reaching longer time scales than molecular dynamics and it can be used to systematically increase the accuracy of other methods such as object kinetic Monte Carlo. The challenges and potential development directions are also outlined.

Solute-defect interactions in Al-Mg alloys from diffusive variational Gaussian calculations

Resolving atomic-scale defect topologies and energetics with accurate atomistic interaction models provides access to the nonlinear phenomena inherent at atomic length and time scales. Coarse graining the dynamics of such simulations to look at the migration of, e.g., solute atoms, while retaining the rich atomic-scale detail required to properly describe defects, is a particular challenge. In this paper, we present an adaptation of the recently developed ``diffusive molecular dynamics'' model to describe the energetics and kinetics of binary alloys on diffusive time scales. The potential of the technique is illustrated by applying it to the classic problems of solute segregation to a planar boundary (stacking fault) and edge dislocation in the Al-Mg system. Our approach provides fully dynamical solutions in situations with an evolving energy landscape in a computationally efficient way, where atomistic kinetic Monte Carlo simulations are difficult or impractical to perform.

Confined phase separation in SiOX nanometric thin layers

Phase separation in silicon-rich silica/silica multilayers was investigated using Atom Probe Tomography and Atomistic Kinetic Monte Carlo simulation. It is shown that the thickness of silicon-rich silicon oxide sublayers plays an important role during phase transformation. It determines the morphology of Si-rich phase formed after subsequent annealing, which is of prime interest for microelectronic and optoelectronic applications. Monte Carlo simulation reveals that the formation of isolated Si clusters can be achieved even in the case of spinodal decomposition and is directly related to the ratio between the spinodal wavelength and the sublayer thickness.

On the mobility of vacancy clusters in reduced activation steels: an atomistic study in the Fe–Cr–W model alloy

Reduced activation steels are considered as structural materials for future fusion reactors. Besides iron and the main alloying element chromium, these steels contain other minor alloying elements, typically tungsten, vanadium and tantalum. In this work we study the impact of chromium and tungsten, being major alloying elements of ferritic Fe–Cr–W-based steels, on the stability and mobility of vacancy defects, typically formed under irradiation in collision cascades. For this purpose, we perform ab initio calculations, develop a many-body interatomic potential (EAM formalism) for large-scale calculations, validate the potential and apply it using an atomistic kinetic Monte Carlo method to characterize the lifetime and diffusivity of vacancy clusters. To distinguish the role of Cr and W we perform atomistic kinetic Monte Carlo simulations in Fe–Cr, Fe–W and Fe–Cr–W alloys. Within the limitation of transferability of the potentials it is found that both Cr and W enhance the diffusivity of vacancy clusters, while only W strongly reduces their lifetime. The cluster lifetime reduction increases with W concentration and saturates at about 1–2 at.%. The obtained results imply that W acts as an efficient ‘breaker’ of small migrating vacancy clusters and therefore the short-term annealing process of cascade debris is modified by the presence of W, even in small concentrations.

First principle-based AKMC modelling of the formation and medium-term evolution of point defect and solute-rich clusters in a neutron irradiated complex Fe–CuMnNiSiP alloy representative of reactor pressure vessel steels

The formation and medium-term evolution of point defect and solute-rich clusters under neutron irradiation have been modelled in a complex Fe–CuMnNiSiP alloy representative of RPV steels, by means of first principle-based atomistic kinetic Monte Carlo simulations. The results obtained reproduce most features observed in available experimental studies, highlighting the very good agreement between both series. According to simulation, solute-rich clusters form and develop via an induced segregation mechanism on either the vacancy or interstitial clusters, and these point defect clusters are efficiently generated only in cascade debris and not Frenkel pair flux. The results have revealed the existence of two distinct populations of clusters with different characteristic features. Solute-rich clusters in the first group are bound essentially to interstitial clusters and they are enriched in Mn mostly, but also Ni to a lesser extent. Over the low dose regime, their density increases in the alloy as a result of the accumulation of highly stable interstitial clusters. In the second group, the solute-rich clusters are merged with vacancy clusters, and they contain mostly Cu and Si, but also substantial amount of Mn and Ni. The formation of a sub-population of pure solute clusters has been observed, which results from annihilation of the low stable vacancy clusters on sinks. The results indicate finally that the Mn content in clusters is up to 50%, Cu, Si, and Ni sharing the other half in more or less equivalent amounts. This composition has not demonstrated any noticeable modification with increasing dose over irradiation.

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 simulations, using a rigid lattice approximation with pair interactions that depend on the local composition and temperature. The pair interactions are fitted on ab initio calculations of mixing energies and vacancy migration barriers at 0 K. The entropic contributions to the mixing of Fe-Cr alloys and to the vacancy formation and migration free energies are taken into account. The model reproduces the change in sign of the mixing energy with the alloy composition and gives realistic thermodynamic and kinetic properties, including an asymmetrical miscibility gap at low temperature and diffusion coefficients in good agreement with available experimental data. Simulations of short-range ordering and $\ensuremath{\alpha}\ensuremath{-}\ensuremath{\alpha}$${}^{\ensuremath{'}}$ decomposition are performed at 773 and 813 K for Cr concentrations between 10$%$ and 50$%$. They are compared with experimental kinetics based on three-dimensional atom probe and neutron scattering measurements. The possible effect of magnetic properties on diffusion in the $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}$ phases, and therefore on the decomposition kinetics, is emphasized.

Theoretical investigation of the influence of reaction and diffusion kinetics upon thin-film reactive diffusion

Atomistic kinetic Monte Carlo simulations, considering the thermodynamic model of the nonregular solid solution on a rigid fcc lattice and using the first-neighbor atomic pair interaction Ising model, were performed to investigate the influence of reaction and diffusion kinetics on thin-film reactive diffusion with a semi-infinite substrate. The simulations show that compound bulk energies have no effect on the phase sequence of appearance during sequential phase formation. However, a diffusion asymmetry between the elements composing the solid solution has a strong effect on sequential phase formation. Diffusion asymmetry increases the critical thickness of the simultaneous-to-sequential phase formation transition, modifies the phase formation sequence, and controls which phase is the first to form. Asymmetrical diffusion can promote transient phase formation and can prevent some phases of the bulk phase diagram from forming. In addition, it influences the thicknesses of phases and their lifetimes during thin-film reactive diffusion. Consequently, the control of atom diffusion during reactive diffusion, for example, thanks to the controlled introduction of impurities, should allow the phase formation sequence of a given binary system to be modified.

Diffusion and Cluster Growth of Binary Alloys on Surfaces

Abstract Metastable nanoclusters grown on surfaces by vapour deposition or molecular beam epitaxy techniques have become an active topic in surface science because of their potential to display new physical properties useful for applications. Atomistic modelling and Kinetic Monte Carlo (KMC) simulations of these processes are reviewed with emphasis on two-component adatom systems. The situation we consider is that two types of atoms are co-deposited to the substrate. In this field of binary growth, systematic theoretical investigation is only at its beginning. Surface diffusion and nucleation leads to the formation of two-dimensional islands, before two-dimensional cluster growth sets in. In distinction to well-known growth scenarios for one species of adatoms, a wealth of new aspects arises in binary systems. Already in the regime of submonolayer growth, differences in adatom diffusion coefficients and binding energies make it necessary to generalise traditional scaling relations for island densities. Surface segregation and compositional fluctuations in growing 3-D clusters are issues requiring renewed examination under the point of view that the atomic short-range order, which is frozen in the bulk, was generated through the surface kinetics during previous stages of growth. We discuss the current theoretical understanding and experimental implications of these problems, including a description of perpendicular magnetic anisotropy (PMA) in metastable CoPt3 nanoalloys, as detected recently.

“Order–Order” Kinetics In Ni50.5Al49.5 single crystal by quasi-residual resistometry

Kinetics of “order–order” reactions in B2-ordered Ni50.5Al49.5 intermetallic was examined by means of quasi-residual resistometry. Isothermal relaxations of quasi-residual electrical resistance of the sample, measured in liquid nitrogen bath, were associated with corresponding relaxations of the concentration of antisite defects, proportional to the long-range order parameter.Similarly as in the case of previous “in situ” experiments, the observed relaxations were very sluggish with relatively low activation energies: EA(ord) = (2.37 ± 0.10) eV for ordering and EA(dis) = (1.93 ± 0.19) eV for disordering process. These energies are lower than the activation energy for Ni self-diffusion in NiAl (3.0 eV), but somewhat higher than the one previously determined by means of “in situ” resistometry for “order–order” processes in Ni50.5Al49.5. The origin of low relaxation rates observed in the experiment has recently been elucidated by atomistic Kinetic Monte Carlo (KMC) simulations in terms of low disordering efficiency of atomic jumps in triple-defect B2-ordering binary systems.

Comparison of atomistic and elasticity approaches for carbon diffusion near line defects in α-iron

Energy barriers for carbon migration in the neighborhood of line defects in body-centered cubic iron have been obtained by atomistic simulations. For this purpose, molecular statics with an Fe–C interatomic potential, based on the embedded atom method, has been employed. Results of these simulations have been compared to the predictions of anisotropic elasticity theory. The agreement is better for a carbon atom sitting on an octahedral site (energy minimum) than one on a tetrahedral site (saddle point). Absolute differences in the energy barriers obtained by the two methods are usually below 5 meV at distances larger than 1.5 nm from a screw dislocation and 2 nm (up to 4 nm in the glide plane) from the edge dislocation. Atomistic kinetic Monte Carlo simulations performed at T = 300 K and additional analysis based on the activation energies obtained by both methods show that they are in good qualitative agreement, despite some important quantitative discrepancies due to the large absolute errors found near the dislocation cores.

Theoretical and Experimental Evidences of Sequential Phase Formation during Sub-Nanometric-Thick Film Reactive Diffusion

Silicide sequential phase formation during tens-of-nanometer-thick metallic film reaction on Si substrate has been extensively studied. Nevertheless, the reasons of sequential phase formation are still under debate, and have been poorly studied at the atomic scale. Using atomistic kinetic Monte Carlo simulations, we show that considering a binary fcc non-regular solid solution, without diffusion asymmetries, the diffusive reaction of a sub-nanometer-thick film (~5 atomic monolayers) on a semi-infinite substrate leads to the sequential formation of all the phases present in the binary phase diagram, starting with the film atom richest phase. These predictions are supported by experimental observations: the dissolution of a 4 monolayer-thick Si film on a Ni(111) substrate, during in-situ ultra high vacuum Auger electron spectroscopy, shows delays and kinetic changes in the dissolution process that may correspond to the sequential formation of the Ni-Si compounds, i.e. NiSi2, NiSi, Ni3Si2, Ni2Si, Ni31Si12 and Ni3Si.

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.

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.

Electrical signature of nanoscale coalescence in a percolating Bi nanocluster film

We explore changes in the electrical conductance of a percolating Bi nanocluster film due to coalescence. A power law increase in conductance is observed immediately after deposition and we show this corresponds to power law changes in the radius of the necks between clusters. The power-law exponent $(\ensuremath{\lesssim}0.04)$ is much smaller than expected from classical models of microparticle coalescence. Atomistic kinetic Monte Carlo simulations reveal similar behavior during a late stage of coalescence where faceting near the necks slows the effects of surface diffusion.

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.

Isochronal annealing of electron-irradiated dilute Fe alloys modelled by an ab initio based AKMC method: Influence of solute–interstitial cluster properties

The evolution of the microstructure of dilute Fe alloys under irradiation has been modelled using a multiscale approach based on ab initio and atomistic kinetic Monte Carlo simulations. In these simulations, both self interstitials and vacancies, isolated or in clusters, are considered. Isochronal annealing after electron irradiation experiments have been simulated in pure Fe, Fe–Cu and Fe–Mn dilute alloys, focusing on recovery stages I and II. The parameters regarding the self interstitial – solute atom interactions are based on ab initio predictions and some of these interactions have been slightly adjusted, without modifying the interaction character, on isochronal annealing experimental data. The different recovery peaks are globally well reproduced. These simulations allow interpreting the different recovery peaks as well as the effect of varying solute concentration. For some peaks, these simulations have allowed to revisit and re-interpret the experimental data. In Fe–Cu, the trapping of self interstitials by Cu atoms allows experimental results to be reproduced, although no mixed dumbbells are formed, contrary to the former interpretations. Whereas, in Fe–Mn, the favorable formation of mixed dumbbell plays an important role in the Mn effect.

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.

Calculation of proper energy barriers for atomistic kinetic Monte Carlo simulations on rigid lattice with chemical and strain field long-range effects using artificial neural networks

In this paper we take a few steps further in the development of an approach based on the use of an artificial neural network (ANN) to introduce long-range chemical effects and zero temperature relaxation (elastic strain) effects in a rigid lattice atomistic kinetic Monte Carlo (AKMC) model. The ANN is trained to predict the vacancy migration energies as calculated given an interatomic potential with the nudged elastic band method, as functions of the local atomic environment. The kinetics of a single-vacancy migration is thus predicted as accurately as possible, within the limits of the given interatomic potential. The detailed procedure to apply this method is described and analyzed in detail. A novel ANN training algorithm is proposed to deal with the necessarily large number of input variables to be taken into account in the mathematical regression of the migration energies. The application of the ANN-based AKMC method to the simulation of a thermal annealing experiment in Fe-20%Cr alloy is reported. The results obtained are found to be in better agreement with experiments, as compared to already published simulations, where no atomic relaxation was taken into account and chemical effects were only heuristically allowed for.