Production Bias Model Research Articles

Effect of Carbon Impurity Content on Microstructural Evolution in Neutron-Irradiated Alpha Iron: Cluster Dynamics Modeling

ABSTRACTCluster dynamics (CD) modeling has been used to estimate the long-term evolution of point defect (PD) clusters. However, previous studies have often simplified the governing equations by assuming the maximum size of mobile self-interstitial atom (SIA) clusters and by ignoring the one-dimensional (1D) reaction kinetics of SIA loops. They have also conducted parameter fittings, such as the clustered fraction and the maximum size of clusters produced by collision cascade, to reproduce experimental data. In this study, in addition to modeling the 1D motion of SIA loops in the framework of the production bias model (PBM), reaction rates associated with carbon impurity atoms present in alpha iron were formulated to consider the trapping effect of one-dimensionally migrating SIA loops by a vacancy-carbon (V-C) complex that was shown to have strong bindings with SIA loops by previous atomistic simulations. Calculations results for neutron-irradiated alpha iron showed that the developed CD model can successfully reproduce the saturation trend of the number density of immobile SIA loops in contrast to the prediction using a model without the trapping effect.

Radiation Growth of HCP Metals under Cascade Damage Conditions

ABSTRACTModels of radiation growth proposed to date are all based on the assumption that the primary damage is produced by neutron irradiation in the form of single defects. These models do not account for the features of the cascade damage: intra-cascade clustering of self‑interstitial atoms (SIAs) and their one‑dimensional diffusion. During the last twenty years, a ‘Production Bias Model’ has been developed, which shows that the damage accumulation in cubic metals depends crucially on the cascade properties. The cascades in hcp zirconium are similar to those in cubic crystals; hence the model can provide a realistic framework for the hcp metals as well. In this work we present such a model in application to low-temperature (below 300°C) radiation growth.

Simulations of the effects of 1-d interstitial diffusion on void lattice formation during irradiation

This paper describes the use of simulation techniques to examine some of the processes involved in the alignment of voids under the influence of one-dimensional self interstitial atom (1-d SIA) transport. The work follows the paper of Heinisch and Singh [1] on this topic but a different and simpler methodology is used. Besides repeating the scenarios studied by Heinisch and Singh, the effects of re-nucleation and the influence of vacancies have been introduced. One of the important processes that emerged from the results was the barrier to precise void alignment caused by the SIA-induced coalescence of aligned voids. This appears to prevent the formation of stable void lattices by any 1-d SIA transport mechanism, a point supported by the initial void alignment in the mechanism requiring swelling values well above those found experimentally. A full consideration of the void lattice phenomenon shows that the one-dimensional diffusion of self-interstitials central to the production bias model of irradiation damage cannot be the only mode of anisotropic diffusion available under irradiation.

Kinetic Monte Carlo simulations of void lattice formation during irradiation

Over the last decade, molecular dynamics simulations of displacement cascades have revealed that glissile clusters of self-interstitial crowdions are formed directly in cascades and that they migrate one-dimensionally along close-packed directions with extremely low activation energies. Occasionally, under various conditions, a crowdion cluster can change its Burgers vector and glide along a different close-packed direction. The recently developed production bias model (PBM) of microstructure evolution under irradiation has been structured specifically to take into account the unique properties of the vacancy and interstitial clusters produced in the cascades. Atomic-scale kinetic Monte Carlo (KMC) simulations have played a useful role in understanding the defect reaction kinetics of one-dimensionally migrating crowdion clusters as a function of the frequency of direction changes. This has made it possible to incorporate the migration properties of crowdion clusters and changes in reaction kinetics into the PBM. In the present paper we utilize similar KMC simulations to investigate the significant role that crowdion clusters can play in the formation and stability of void lattices. The creation of stable void lattices, starting from a random distribution of voids, is simulated by a KMC model in which vacancies migrate three-dimensionally and self-interstitial atom (SIA) clusters migrate one-dimensionally, interrupted by directional changes. The necessity of both one-dimensional migration and Burgers vectors changes of SIA clusters for the production of stable void lattices is demonstrated, and the effects of the frequency of Burgers vector changes are described.

Modeling of void nucleation under cascade damage conditions

In recent years, significant progress has been made in understanding the void growth and swelling behaviour in metals under cascade damage conditions in terms of differences in the intra-cascade clustering of self-interstitials (SIAs) and vacancies, and the thermal stability and (one-dimensional) mobility of the resulting clusters. The problem of void nucleation was, however, not treated in any detail within this ‘production bias model’. In treating void nucleation, the following experimental findings have to be considered: (1) Both nucleation and growth of voids are much more efficient under cascade damage conditions than under electron irradiation. (2) Void nucleation occurs at low dose, particularly in pure metals, and tends to saturate at higher doses. (3) There are significant differences in the void nucleation behaviour between pure metals of different crystal structure and between pure metals and alloys. In the present contribution, we discuss these findings in terms of a production bias controlled vacancy supersaturation rising in the transient to quasi-steady-state and falling during cascade damage accumulation, and thus inducing a limited void nucleation pulse, the yield of which represents the experimentally observed saturated void density.

Absence of saturation of void growth in rate theory with anisotropic diffusion

We present a first attempt at solution the problem of the growth of a single void in the presence of anisotropically diffusing radiation induced self-interstitial atom (SIA) clusters. In order to treat a distribution of voids we perform ensemble averaging over the positions of centres of voids using a mean-field approximation. In this way we are able to model physical situations in between the Standard Rate Theory (SRT) treatment of swelling (isotropic diffusion), and the purely 1-dimensional diffusion of clusters in the Production Bias Model. The background absorption by dislocations is however treated isotropically, with a bias for interstitial cluster absorption assumed similar to that of individual SIAs. We find that for moderate anisotropy, unsaturated void growth is characteristic of this anisotropic diffusion of clusters. In addition we obtain a higher initial void swelling rate than predicted by SRT whenever the diffusion is anisotropic.

Progress in modelling the microstructural evolution in metals under cascade damage conditions

In recent years, it has been shown that intra-cascade clustering of vacancies and self-interstitial atoms (SIAs), differences in the thermal stability and mobility of the resulting clusters and one-dimensional (1-D) diffusional glide of SIA clusters play a key role in damage accumulation in metals under cascade damage conditions. The model taking these aspects into account (production bias model, PBM) succeeded in rationalising striking features in the microstructural evolution in pure metals, where the conventional rate theory model failed: the high overall swelling even at low dislocation densities, the enhanced swelling near grain boundaries, the decoration of dislocations with SIA loops, saturation of void growth and void lattice formation. In the present paper, the main ideas and results of these considerations are reviewed. We discuss recent work on possible effects of deviations of SIA cluster diffusion from strictly 1-D by direction changes and/or self-climb and formulate a general reaction kinetics including 1-D and 3-D cluster diffusion. Such reaction kinetics may be considered to form the basis for a general description of cascade damage accumulation in metals and complex technical alloys.

Microstructure of Cu–Ni alloys neutron irradiated at 210°C and 420°C to 14 dpa

Transmission electron microscope (TEM) disks of pure copper and copper containing 0.17–10% Ni were neutron irradiated at 210°C and 420°C in Hf-shielded capsules in the High Flux Isotopes Reactor to doses of 13.5 and 14.9 displacements per atom (dpa), respectively. Void swelling was not observed in any of the specimens irradiated at 210°C. Instead, a high density of stacking fault tetrahedra (SFTs) and a moderate density of dislocation loops were observed. There was no evidence for defect cluster patterning (wall formation) in any of the specimens irradiated at 210°C. The SFT density was independent of Ni content, whereas the loop density was highest in the alloy containing ∼2% Ni. Pronounced void swelling was observed in all of the specimens irradiated at 420°C. A void denuded zone of ∼2 μm width was observed adjacent to grain boundaries. The void swelling in copper containing ∼2% Ni showed a pronounced maximum in a 10-μm wide band adjacent to the grain boundary denuded zones. Matrix voids aligned along 〈110〉 directions were observed in Cu–10%Ni irradiated at 420°C. The occurrence of the peak swelling zone and the partial ordering of voids are both consistent with the predictions of the recent production bias model which includes 1-D diffusional transport of clusters of self-interstitial atoms.

On recoil energy dependent void swelling in pure copper Part I. Experimental results

Abstract In recent years, the problem of void swelling has been treated within the framework of the production bias model (PBM). The model considers the intracascade clustering of vacancies and self-interstitial atoms (SIAs), their thermal stability and the resulting asymmetry in the production of free and mobile vacancies and SIAs. The model also considers the influence of onedimensional diffusional transport of glissile clusters of SIAs on damage accumulation in the form of voids and defect clusters. One of the major predictions of the PBM is that, at a given irradiation temperature and damage rate, the void swelling should depend sensitively on the recoil energy, since it affects strongly the intracascade clustering of SIAs and vacancies, particularly at lower recoil energies. In order to test the validity of this prediction directly by experiment, pure and annealed copper specimens were irradiated with 2.5 MeV electrons, 3 MeV protons and fission neutrons at about 520 K. All three sets of irradiation ...

Features of Vacancy Interaction with Interstitial Clusters in BCC Metals

AbstractRecent theoretical calculations and atomistic computer simulations have shown that one-dimensional (1D) glissile clusters of self-interstitial atoms (SIAs) play an important role in the evolution of the microstructure in metals and alloys under cascade damage conditions. Theoretical treatments based on the production bias model (PBM)and cascade-induced source hardening clarify many features of materials behaviour under irradiation by considering 1D glissile SIA clusters. However, further progress in understanding irradiation phenomena demands a detailed knowledge of different reactions where glissile SIA clusters take part. The latest version of the PBM has demonstrated that there is a significant lack of knowledge on mechanisms such as the interaction between mobile clusters and other microstructure components, changes on glide directions or the reduction of cluster mobility. Some aspects of interactions have been recently studied and the formation of immobile complexes of glissile clusters in the bulk and near dislocations has been shown. However, in addition to the cluster - clusters and cluster – dislocation interactions, the interactions of clusters with point defects should be considered. Here we present the first results on interaction between a SIA cluster and vacancy in bcc-iron. The interactions were studied by both atomistic simulations and elasticity theory. Some qualitative differences between the results obtained by these two methods were found, thereby demonstrating the need for the atomistic approach. It is shown that the interaction with vacancies can reduce and, in some cases, suppress the mobility of SIA clusters. The results are discussed in the light of both experimental data and theoretical predictions.

Defect accumulation in fcc and bcc metals and alloys under cascade damage conditions – Towards a generalisation of the production bias model

In recent years, it has been shown that the problem of defect accumulation under cascade damage conditions can be properly addressed within the framework of production bias model (PBM) based on intracascade clustering of point defects, differences in the thermal stability of the resulting clusters and one-dimensional (1-D) diffusion of interstitial clusters. Within this framework, different aspects of defect accumulation such as the high swelling rate at low dislocation densities, enhanced swelling near grain boundaries, effects of grain size, irradiation dose and recoil energy on void swelling have been treated quantitatively. In the present work we have attempted to address the problem of differences in the defect accumulation behaviour between fcc and bcc metals under cascade damage conditions. In this analysis we have chosen copper and molybdenum to represent fcc and bcc metals, respectively. This choice was suggested by the fact (a) that a large amount of experimental information exists on these metals and (b) that the damage accumulation behaviour in copper can be fully accounted for in terms of the PBM. An analysis of the existing experimental observations in terms of the PBM raises the question about differences between the reaction kinetics of the one-dimensionally diffusing interstitial clusters with sinks in fcc and bcc metals. In the present paper, the impact of different frequencies of changes in the 1-D diffusion direction of such clusters on their reaction kinetics is discussed. The present considerations provide an appropriate framework for generalising the PBM in order to describe damage accumulation behaviour in metals and alloys in general, including the formation of void superlattices.

Interactions between mobile dislocation loops in Cu and α-Fe

The relevance of one-dimensionally gliding clusters in the understanding of the damage accumulation produced by the displacement cascades has been underlined by the production bias model. The properties and mobility of isolated clusters of vacancies and self-interstitials have been recently studied by molecular dynamics and valuable information about their diffusional characteristics is obtained. The next step in the understanding of radiation damage should include the possible reactions of these clusters with other clusters, dislocations and other sinks. In this paper we present the first results of a molecular dynamics study of interactions between glissile interstitial clusters and small dislocation loops in α-Fe and Cu. Different types of interactions have been studied between clusters of different sizes (from 12 to 91 defects) in the temperature range from 300 to 1000 K. As a result of the inter-cluster interactions both glissile and sessile clusters can be obtained and this depends on the metal, reaction type and size of the clusters. In general the probability to form sessile clusters increases for larger clusters and it is higher in Cu. The results obtained are discussed from the point of view of the difference in radiation damage effects in fcc Cu and bcc Fe.

Stability and mobility of defect clusters and dislocation loops in metals

According to the production bias model, glissile defect clusters and small dislocation loops play an important role in the microstructural evolution during irradiation under cascade damage conditions. The atomic scale computer simulations carried out in recent years have clarified many questions about the structure and properties of glissile clusters of self-interstitial atoms that are formed directly in the cascade volume. It has been found that such clusters consist of sets of crowdions and are highly mobile in the crowdion direction. Very recently, one-dimensional glide of similar character has been observed in the computer simulation of small vacancy loops in α-Fe. In the present paper we summarise results obtained by molecular dynamics simulations of defect clusters and small dislocation loops in α-Fe(bcc) and Cu(fcc). The structure and stability of vacancy and interstitial loops are reviewed, and the dynamics of glissile clusters assessed. The relevance and importance of these results in establishing a better understanding of the observed differences in the damage accumulation behaviour between bcc and fcc metals irradiated under cascade damage conditions are pointed out.

Defect production and damage evolution in Al: a molecular dynamics and Monte Carlo computer simulation

Predicting microstructure evolution of metals under irradiation requires an in depth understanding of the production and properties of point defects and their clusters. Although, analytical descriptions based on the production bias model have recently been applied successfully to explain the behaviour of metals under different irradiation conditions, they still lack an important ingredient, namely that of the diffusion mechanism of the defects produced. In the present investigation, a hierarchical computer simulation model which takes into account spatial and temporal correlation in the production and diffusion of the defects is used to investigate the production of freely migrating defects in irradiated aluminium. The approach uses classical molecular dynamics with the embedded atom method potential to investigate the primary damage state from displacement cascades and a kinetic Monte Carlo simulation to follow defect kinetics over macroscopic time scales. In the present paper, we have used this approach to investigate the primary damage state and the fraction of freely migrating defects in pure A1 as a function of irradiation particle energy and crystal temperature. The results are compared to other FCC metals and to the experiments when available.

Beyond the mean-field formulation of the production bias model

The Production Bias Model was first conceptualized in the Silkeborg Workshop 10 years ago, to incorporate effects due to intracascade recombination, clustering and thermal dissociation of the clusters into the theory of microstructure evolution. Comparison of the theory with experimental results clearly supports the model, which has since been thoroughly reviewed. However, several important issues related to the probabilistic nature of cascade initiation and the statistical nature of the microstructure development have not received adequate attention. The most important feature of PBM is the production of damage in the form of point-defect clusters, both mobile and immobile, in cascades. Under most circumstances, the cluster population is one of the most important, if not dominant, components of the microstructure, the evolution of which controls the behaviour of the system under irradiation. Due to the small size of the point-defect clusters, fluctuations in the point-defect flux, caused by the stochastic nature of point-defect production and transport, govern their size distribution and number densities. Taking into account the effects of the fluctuating fluxes requires the formulation of PBM beyond the mean-field approximation. In this keynote paper, the readers are introduced to investigations in this vital area and the significance of the results obtained to the application of the production bias model in the analysis of microstructure development in solids during irradiation by neutrons and heavy ions.

Impacts of damage production and accumulation on materials performance in irradiation environment

Aspects of generation of atomic displacements and surviving defect fractions and their sensitivity to recoil energy have been considered. These considerations show that the nature of the primary damage production plays a vital role in the evolution of the damage accumulation as well as mechanical properties. The problem of damage accumulation on a global scale still remains a theoretical challenge! A substantial amount of theoretical effort is required to develop a realistic model to describe the damage accumulation at low temperatures (<stage V) and to establish a reliable relationship between the damage accumulation, irradiation hardening and the loss of ductility. The recently developed production bias model together with one-dimensional glide of interstitial clusters produced in the cascades has been shown to describe the damage accumulation at temperatures above stage V for pure metals and needs to be extended to temperatures below stage V and to materials of practical interests. This requires, however, that the information regarding the effects of alloying elements and impurity atoms on the nature of the primary damage state are available from molecular dynamics and kinetic Monte Carlo type of simulations.

Glissile and Sessile Vacancy and Self-Interstitial Clusters in BCC and FCC Metals

AbstractThe production bias model of radiation damage has focused attention on the production of glissile clusters (GCs) in the displacement cascades. The formation of glissile SIA clusters in molecular dynamics simulations of cascades has been reported frequently whereas information about sessile clusters (SCs) is rather sparse. Comprehensive models of microstructure evolution in metals under irradiation conditions should include all possible defects (and clusters) and interactions between them. Detailed information of this nature can be obtained by means of atomic-scale computer simulation. In the present paper we review some recent results of atomistic simulation of sessile and glissile defect clusters in metals, dealing with their mechanisms of formation and their properties. In particular, we discuss the formation of SCs and GCs in the cascade region and also their creation in bulk metal as a result of cluster growth.

Mobility of Self-Interstitial Clusters in FE and CU

ABSTRACTMolecular dynamics (MD) simulation has been used to study the thermally activated mobility of clusters of self-interstitial atoms (SIAs) in Fe and Cu. Such clusters are formed in metals during irradiation with energetical particles and, according to the cascade production bias model, they play an important role in the microstructure evolution of metals under irradiation. An extensive simulation of clusters from 2 to 30 interstitials has been carried out for the temperature range ≍360-1200K using long-range interatomic pair potentials. The results show that clusters bigger than two SIAs are one-dimensionally mobile. Di-interstitials have two migration mechanisms depending on the temperature. At low temperature the mechanism is one-dimensional whereas at high temperature the probability of rotation and three-dimensional migration increases. It was found that in both metals the effective migration energy of clusters estimated via their jump frequency does not depend on the cluster size, although the cluster jump frequency decreases as the cluster size increases. The mechanism of cluster migration and problems of the treatment of one-dimensional mobility are discussed.

Aspects of microstructure evolution under cascade damage conditions

The conventional theoretical models describing the damage accumulation, particularly void swelling, under cascade damage conditions do not include treatments of important features such as intracascade clustering of self-interstitial atoms (SIAs) and one-dimensional glide of SIA clusters produced in the cascades. Recently, it has been suggested that the problem can be treated in terms of ‘production bias’ and one-dimensional glide of small SIA clusters. In the earlier treatments a ‘mean size approximation’ was used for the defect clusters and cavities evolving during irradiation. In the present work, we use the ‘size distribution function’ to determine the dose dependence of sink strengths, vacancy supersaturation and void swelling as a function of dislocation density and grain size within the framework of production bias model and glide of small SIA clusters. In this work, the role of the sessile-glissile loop transformation (due to vacancy supersaturation) on the damage accumulation behaviour is included. The calculated results on void swelling are compared with the experimental results as well as the results of the earlier calculations using the ‘mean size approximation’. The calculated results agree very well with the experimental results.

Recent advances in the understanding of damage production and its consequences on void swelling, irradiation creep and growth

The consequences of displacement damage produced by energetic particles on physical and mechanical properties of metals and alloys have been investigated both experimentally and theoretically for several decades. Over the years, a number of theoretical models have been proposed to rationalize the rate and magnitude of defect accumulation under different irradiation conditions. In recent years, significant advances have been made in understanding the nature of the damage produced in this form of multi-displacement cascades. The new knowledge regarding the intra-cascade recombination and clustering of self-interstitial atoms and vacancies during the cooling-down phase of cascades makes it necessary to re-examine the appropriateness of the available models for describing the accumulation of damage under cascade damage conditions. In this paper, recent advances in the understanding of damage production and its consequences are reviewed. A historic perspective is adopted. A comprehensive analysis of the effects of temperature, dose rate and particle type on multi-phenomena (swelling, creep, growth, microstructure evolution, RED, RIS) is presented to discuss the strength and weakness of various models, as they have evolved with the understanding of the damage processes. It has been shown that the irradiation damage modeling has progressed from the standard rate theory model to the BEK model to the production bias model with an increasing degree of sophistication as increasingly more realistic features of the irradiation damage production process were incorporated. It is shown that the newly proposed production bias model uniquely includes the necessary features of cascade damage production in its treatment of the damage accumulation.