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Fusion Engineering and Design 82 (2007) 2666–2670
Kinetic Monte Carlo modelling of neutron irradiation damage in iron L. G´amez a,b,∗ , E. Mart´ınez a,d , J.M. Perlado a , P. Cepas a , M.J. Caturla c , M. Victoria a , J. Marian d , C. Ar´evalo a , M. Hern´andez e , D. G´omez e a
Instituto de Fusi´on Nuclear, UPM, Madrid, Spain Departamento de F´ısica Aplicada, ETSII, UPM, Madrid, Spain Departamento de F´ısica Aplicada, Universidad de Alicante, Alicante, Spain d Lawrence Livermore National Laboratory, LLNL, CA 94550, USA e CIEMAT, Madrid, Spain b
c
Received 31 July 2006; received in revised form 25 April 2007; accepted 25 April 2007 Available online 18 June 2007
Abstract Ferritic steels (FeCr based alloys) are key materials needed to fulfill the requirements expected in future nuclear fusion facilities, both for magnetic and inertial confinement, and advanced fission reactors (GIV) and transmutation systems. Research in such field is actually a critical aspect in the European research program and abroad. Experimental and multiscale simulation methodologies are going hand by hand in increasing the knowledge of materials performance. At DENIM, it is progressing in some specific part of the well-linked simulation methodology both for defects energetics and diffusion, and for dislocation dynamics. In this study, results obtained from kinetic Monte Carlo simulations of neutron irradiated Fe under different conditions are presented, using modified ad hoc parameters. A significant agreement with experimental measurements has been found for some of the parameterization and mechanisms considered. The results of these simulations are discussed and compared with previous calculations. © 2007 Elsevier B.V. All rights reserved. PACS: 02.50.Ng; 61.72; 61.80; 61.82.Bg; 81.05.Bx Keywords: Multiscale modelling; Radiation damage; Nuclear fusion experiments
1. Introduction
∗
Corresponding author at: Instituto de Fusi´on Nuclear, UPM, Madrid, Spain. E-mail address:
[email protected] (L. G´amez). 0920-3796/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2007.04.040
The development of fusion reactors requires of materials that are able to sustain high radiation levels maintaining their good mechanical properties. In order to achieve such materials it is necessary to
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have a good understanding of the underlaying mechanisms responsible for material degradation under irradiation. Multiscale modelling has proved to be an important tool in this understanding, in particular when models are developed closely linked to experiments. One of the materials of interest for fusion reactors, both for magnetic confinement and for inertial confinement facilities, are low activation ferritic steels. Therefore, a large effort in the modelling community is devoted to these type of materials. In particular, it is considered that a good understanding of pure bcc Fe is needed before going to complex alloy structures. Therefore, within the last few years a large number of experiments and simulations in pure Fe have appeared in the literature. There is now a much better understanding about the basic intrinsic defects in Fe from ab initio calculations [1,2] and empirical potentials [3,4]. With respect to experiments, information about the damage produced in Fe by electron irradiation [5] as well as neutron irradiation [6,7] is known. The link between the fundamental properties of defects and the experimental measurements has been attempted through kinetic Monte Carlo (kMC) simulations by several authors, in particular under electron irradiation [1] and neutron irradiation [8,9]. These models however still include some assumptions, in particular regarding the mobility of self-interstitial clusters and their interactions with impurities. In this study, kinetic Monte Carlo simulations of damage accumulation under neutron irradiation in Fe are presented. The goal of these calculations is to reproduce the experimental data by Eldrup et. al [7]. In this experiment positron annihilation was used to measure the damage produced in Fe under neutron irradiation as a function of irradiation dose. First a description of the kinetic Monte Carlo used in these calculations is presented as well as the input parameters. Then the results of the calculations for different simulation conditions are shown and compared with previous calculations and with the experimental measurements.
2. Kinetic Monte Carlo: description and input parameters BIGMAC [10] is a computational efficient kMC program which tracks the locations of defects, impu-
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rities and clusters as a function of time. The starting point of these simulations is the primary damage state, that is the spatially correlated locations of vacancy and interstitials, obtained from simulations of displacement cascades. Each defect produced (including the clusters) has an activation energy for diffusion that can be extracted from simulations or in some cases from experiments. The defects are allowed to execute random diffusion jumps (in one-, two- or three-dimensions depending on the nature of the defect) with a probability (rate) proportional to their diffusivity. Similarly, cluster dissociation rates are governed by a dissociation probability that is proportional to the binding energy of a particle to the cluster. The temperature dependence of the defect diffusivity can be written as −Em D = D0 exp , (1) kT where D is the defect diffusivity, D0 the preexponential factor, Em the migration energy, T the temperature of the crystal, and k is the Boltzmann’s constant. A similar form applies for dissociation rates from clusters, with Em replaced by a dissociation energy that includes the binding energy, Eb , of a particle to the cluster. More specifically, because the particle must migrate at least one jump distance away from the cluster to be free, the effective diffusivity for a free particle leaving a cluster is approximated to be −(Em + Eb ) D = D0 exp . (2) kT The BIGMAC program requires input tables of D0 and Em for all mobile species, as well as the pre-factors and binding energies Eb for all possible clusters. The input tables can become rather large, but the program is very flexible, as only the input tables need to be changed to study another set of conditions or even another material system. During the simulation various kinetic processes are allowed to take place. Possible events are: (i) the dissociation of a particle from a cluster, (ii) the diffusive jump of a particle, (iii) recombination of two defects of opposite types, (iv) aglomeration of two defects of the same type, (v) annihilation of a defect and a sink, (vi) trapping or detrapping of a defect at an impurity, and (vii) the introduction of a new cascade, that is, a
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new PKA (primary knock-on atoms) and all its associated vacancies and SIAs (sel-interstitials atoms). The dissociation and migration rates are given by 2dD R= , (3) δ2 where δ is the jump distance set by the lattice, d the dimensionality of the migrational mechanism, and the diffusivity is obtained from Eqs. (1) and (2). At each time step, it is randomly chosen among all possible events, ensuring that events occur at the proper rate by assigning each event a probability proportional to its rate. Following each chosen event, all events that occur spontaneously are performed as a result of that event. Finally, another input in these simulations is the capture radius for defect interaction. The interaction radius between defects has been defined as r = rsph + δ,
(4)
where δ is the jump distance set by the lattice and 3 3N rsph = (5) 4π with the atomic volume and N the number of defects in the cluster. Since the stress field of single interstitials is larger than the one of single vacancies, a larger capture radius is considered for interstitials (rI ) interacting with loops (vacancy or interstitial clusters) than for single vacancies (rV ), rI = 1.15 × rV , where rV is defined in Eq. (5) above. This includes a bias for the interaction of interstitials and vacancies with the microstructure. The code has been modified to include the modelling of defects in Fe with mechanisms that consider the nature of the different interstitial dislocation loops observed in ␣-Fe and ferritic materials, assess the effect of substitutional impurities on migrating (1/2)1 1 1 clusters, and apply atomistic modelling to investigate the mechanisms of formation and growth of 1 0 0 loops from smaller cascade-produced (1/2)1 1 1 clusters. The proposed mechanisms reconciles experimental observations with continuum elasticity theory and recent MD modelling of defect production in displacement cascades. In addition, the interaction of screw dislocations, known to control the lowtemperature plastic response of bcc materials to external stress, with 1 0 0 dislocation loops is investi-
Table 1 Migration energies (Em ) and diffusivity pre-factors (D0 ) for Fe defects, interstitials (I) and vacancies (V) Species
Em (eV)
D0 (cm2 /s)
V V2 I I2 I3
0.55 0.55 0.34 0.42 0.45
1.37 × 10−3 1.37 × 10−3 2.0941 × 10−3 7.321 × 10−4 5.5673 × 10−4
gated with MD, where the main physical mechanisms are identified, cutting angles estimated and a firstorder estimation of the induced hardening is provided [11,12]. The results obtained in this studies were compared with the experimental findings revealed by Eldrup et al. [7]. The experiment performed by Eldrup had the objective to study the influence of the dose rate in the accumulation of defects in Fe and Cu irradiated by neutrons. In this study is only considered pure Fe case with an impurity concentration of carbon atoms. The values of migration energies and pre-factors for diffusion used in this study are given in Table 1. The interstitial values are taken from ab initio calculations [13]. On the other hand, the vacancy values are taken from experimental [14] values which have been used in earlier studies. These values are introduced in the code like an input. Concerning neutron irradiation under REVE (Reactor for Virtual Experiments), a complete modelling from neutron interaction to defects evolution in the materials has been implemented. That procedure includes linked consideration of primary knock-on atoms spectra ranges, cascades formation from Molecular Dynamics and finally evolution by kinetic Monte Carlo. The final conclusions are actually working out and they will be published in a following article.
3. Results Fig. 1 shows the results obtained by comparison of Eldrup experimental values (PAS) and those obtained with BIGMAC simulations. It can be observed that the cluster density obtained from simulations is higher than the experimental one with enclose approach up to low doses (< 0.1 dpa). Also the cluster density to higher
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Fig. 1. Cluster density vs. dose for Fe under conditions of Eldrup experiment.
dose in these simulations shows a linear dependence with the dose while the experimental cluster density approaches to a certain saturation value. This simulation was performed considering a box of 30 nm edge, irradiation temperature of 343 K, and a dose rate of 1 × 10−6 dpa/s, which is the same that is considered in Eldrup experiment. Also it has been included an impurity concentration of carbon atoms 100 appm, equivalent to a concentration of 8.372 × 1018 C atoms for the 30 nm edge box. It has been used a combination of cascades of energy 5, 10 and 20 keV consequent with a final cascade with average energy of 11.67 keV. In order to obtain better results it has been considered the inclusion of sinks in the diffusion conditions, introducing them in a similar way to carbon impurities. The interaction between a sink and any type of defects results with the annihilation of the defect. From a physical point of view, sinks play the role of the dislocations and grain boundaries, these defects can be found in every crystal and they show the behaviour previously described. Fig. 2 shows the results of calculations considering the existence of sinks. It can be noticed that the cluster density obtained by BIGMAC calculations is lower than the experimental one and it seems that a certain saturation value can be achieved when dose is increased. For lower doses, simulations results are found between estimated uncertainty bands for clus-
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Fig. 2. Cluster density vs. dose for Fe under conditions of Eldrup experiment considering sinks.
ter density derived from PAS data. Cluster density increases at higher doses, but it does not show a linear dependence with dose when it increases. It is the main difference between results obtained with and without the inclusion of sinks. Also kMC results approach to experimental ones, showing a similar behaviour. Results for doses upper 0.01 dpa are now in progress.
4. Conclusions In this study have been compared the results obtained in Eldrup experiment with those obtained by kMC simulations for iron by different assumptions. The inclusion of sinks which play the role of the dislocations and grain boundaries has been revealed fundamental to evaluate all the possible interactions between the different species present in the material. At lower doses a good agreement is obtained between experimental and simulation results when sinks are included. At higher doses, the inclusion of sinks reduce the cluster densities obtained by kMC simulations showing a similar behaviour with the experimental case.
Acknowledgements This work has been performed under the auspices of the US Department of Energy and Lawrence Livermore
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National Laboratory under contract W-7405-Eng-48, the CSN-UNESA Coordinated Research Programme under contract P000531499 and National R&D Programme under contract C0503002.
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