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Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels
Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels M. Chiapetto a,b,⇑, C.S. Becquart b,d, C. Domain c,d, L. Malerba a a
SCKCEN, Nuclear Materials Science Institute, Boeretang 200, B-2400 Mol, Belgium Unité Matériaux Et Transformations (UMET), UMR 8207, Université de Lille 1, ENSCL, F-59600 Villeneuve d’Ascq Cedex, France c EDF R&D, Département Matériaux et Mécanique des Composants, Les Renardières, F-77250 Moret sur Loing, France d Laboratoire commun EDF-CNRS Etude et Modélisation des Microstructures pour le Vieillissement des Matériaux (EM2VM), France b
a r t i c l e
i n f o
Article history: Received 8 July 2014 Received in revised form 20 October 2014 Accepted 29 November 2014 Available online xxxx Keywords: Object kinetic Monte Carlo Radiation-induced defects Neutron irradiation Fe–C FeMnNi
a b s t r a c t Radiation-induced embrittlement of bainitic steels is one of the most important lifetime limiting factors of existing nuclear light water reactor pressure vessels. The primary mechanism of embrittlement is the obstruction of dislocation motion produced by nanometric defect structures that develop in the bulk of the material due to irradiation. The development of models that describe, based on physical mechanisms, the nanostructural changes in these types of materials due to neutron irradiation are expected to help to better understand which features are mainly responsible for embrittlement. The chemical elements that are thought to influence most the response under irradiation of low-Cu RPV steels, especially at high fluence, are Ni and Mn, hence there is an interest in modelling the nanostructure evolution in irradiated FeMnNi alloys. As a first step in this direction, we developed sets of parameters for object kinetic Monte Carlo (OKMC) simulations that allow this to be done, under simplifying assumptions, using a ‘‘grey alloy’’ approach that extends the already existing OKMC model for neutron irradiated Fe–C binary alloys [1]. Our model proved to be able to describe the trend in the buildup of irradiation defect populations at the operational temperature of LWR (300 °C), in terms of both density and size distribution of the defect cluster populations, in FeMnNi model alloys as compared to Fe–C. In particular, the reduction of the mobility of point-defect clusters as a consequence of the presence of solutes proves to be key to explain the experimentally observed disappearance of detectable point-defect clusters with increasing solute content. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Low-alloy bainitic steels are used to fabricate the reactor pressure vessel (RPV) of most commercial light water nuclear power plants. The vessel is an irreplaceable component and its capability to maintain its integrity determines the lifetime of the installation. Prolonged exposure to neutron flux impinging on the metal creates a large amount of point-defects (vacancies, V, and self-interstitial atoms, SIA) and relevant clusters within displacements cascades, which may evolve to form voids and dislocation loops (so-called matrix damage). These radiation-induced defects are also the main responsible for the enhanced diffusion of solute atoms within the matrix, leading to the formation of solute-rich clusters, generally denoted as precipitates. The link between changes in the microor nanostructure of the material and its mechanical properties is ⇑ Corresponding author at: SCKCEN, Nuclear Materials Science Institute, Boeretang 200, B-2400 Mol, Belgium. Tel.: +32 (0)14 33 31 81. E-mail address: [email protected] (M. Chiapetto).
therefore strong: point-defect clusters and/or precipitates constitute obstacles to the movement of dislocations, increasing yield strength and reducing ductility, thereby contributing to increase the probability of brittle fracture. Computer simulation models are an effective way to understand these kinds of degradation processes and, in this context, FeMnNi systems containing up to a maximum of 1.5–2 wt.% of these two solutes are archetypal model materials for low-alloy ferritic steels, able to highlight the solute influence on the nanostructural evolution under irradiation. The present work is a continuation of the ongoing efforts to develop a model for the nanostructure evolution in RPV steels under irradiation, in terms of density and size of defects versus neutron dose, expressed in displacements per atom (dpa), using the object kinetic Monte Carlo (OKMC) simulation technique [1–3]. We present a comparison between an already validated model for Fe–C alloys neutron irradiated at 560 K [1] and a further developed model able to describe the nanostructural evolution of both V and SIA defect populations under neutron irradiation when also Mn and Ni are present. This model is based on assumptions that translate the
http://dx.doi.org/10.1016/j.nimb.2014.11.102 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
2
M. Chiapetto et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
physical effect of these solutes on the irradiation-induced defect formation and accumulation into, mainly, modified defect mobility parameters, without explicitly introducing the solute atoms in the model [3].
conditions applied in all three directions, as discussed in [9]. Simulation temperature, dpa rate, grain size and dislocation density were dictated by the reference experiment [6,10,11]. 3. Parameterization
2. Computational method For our object kinetic Monte Carlo (OKMC) simulations we used the LAKIMOCA code, thoroughly described in [4], while the approach is explained in detail in [2]. The OKMC is a stochastic method that can be used to describe the nanostructural evolution in materials subjected to neutron irradiation. The production of point-defects like V and SIAs, which also form clusters, is reproduced by a random introduction of cascade debris containing V and SIA objects of different sizes in the system, at a certain rate for the given volume. These are the ‘‘objects’’ of the system, the evolution of which we want to study: they are described in the model by a set of parameters that define their stability, their diffusion properties, and their possibility of interacting with each other and with other defects. Objects of importance are e.g. carbon (C) atoms and C–V complexes, that play the role of traps for mobile point-defect clusters and are characterized by a certain trapping energy that depends on the size and type of the trapped cluster [2]. The pre-existing microstructure, i.e. grain boundaries and dislocations, is taken into account explicitly [4]: dislocations are described as spherical sinks, the radius and number of which are defined in such a way that their sink strength equals the one corresponding to the dislocation density of the reference material [5]. However, since SIA clusters are observed by TEM (transmission electron microscopy) to decorate dislocations [6], meaning that they are not absorbed if large enough, we only allowed clusters smaller than the core of the dislocation, i.e. size 1–4, to be absorbed by the sinks. Every object has an associated reaction volume, generally spherically shaped, with the exception of large dislocation loops, which are represented by toroids. When the reaction volumes of two objects overlap, a predefined reaction, like clustering or recombination between a V and an SIA, takes place. The events in the OKMC simulation determine the dynamics of the system. The probability for the objects to perform an event are given in terms of Arrhenius frequencies for thermally activated processes:
Ai kB T
Ci ¼ mi exp
ð1Þ
Here mi is the attempt frequency (or prefactor) of the event i, Ai is the corresponding activation energy (eV), which must embody both the thermodynamics and the kinetics of the system, kB is Boltzmann’s constant (eV/K) and T is the irradiation temperature (K). Due to their dependence on the type and size of the objects, a very large number of parameters is needed for a standard simulation. In this work we use, besides modifications described below, the parameters from [1–3]. For every simulation step an event is chosen, based on the corresponding probabilities in the parameterization and according to the stochastic Monte Carlo algorithm. The simulated time is increased following the residence time algorithm [7,8]:
Dt ¼
lnR
RCi
ð2Þ
where R is a uniform random number between 0 and 1, included to take into account correctly the stochasticity, according to the Poisson distribution. A non-cubic simulation box of size 600 750 880 a30, where a0 = 2.871010 m is the lattice parameter of a-Fe, was used in order to avoid potential anomalies from 1D-migrating defects entering a migration trajectory loop, due to the periodic boundary
In the present work we compare the nanostructural evolution under irradiation of Fe(C)MnNi alloys (Fe-1.2%Mn-0.7%Ni) and Fe–C binary alloys. The link between the two systems is that the concentration of C in the model material remains the same, i.e. 134 appm, but in the first case we also consider the effect of two different solute atoms, namely Mn and Ni, on the diffusivity of defect clusters. These solute atoms are not explicitly present in the model, which therefore cannot describe their redistribution (segregation, precipitation, . . .) under irradiation, but their effect is introduced by modifying the parameters that govern the mobility of both SIA and V clusters. For the Fe–C alloy the parameterization presented in [1] was adopted. For what concerns FeMnNi alloys, on the other hand, density functional theory (DFT) calculations indicate that Mn has a strong binding energy with the crowdion [12,13]. We therefore reduced the diffusion coefficient DFeMn of all SIA clusters as comn pared to pure Fe, DFe n , using an explicit function of Mn concentration, according to the following equation [14]: bD F n DFeMn ¼ DFe n n e
ð3Þ
where n is the number of SIA in the cluster and b is 1/kBT. We translated this reduction of the diffusion coefficient in a decrease of the attempt frequency mi in Eq. (1) (temperature being fixed). In dilute alloys the cluster diffusivity decreases with increasing Mn concentration simply because of the increasing amount of solute atoms interacting with the cluster. Under this assumption, the variation of the cluster free energy, DFn, can be expressed as [14]:
DF n K n Eb1
ð4Þ
Eb1 being the effective binding energy between a crowdion and a solute Mn atom, i.e. the decrease in free energy between the condition without and with associated solute atoms, here set to 0.65 eV consistently with DFT calculations [12], while Kn represents the mean number of Mn atoms interacting with the cluster,
K n ¼ nnI xMn
ð5Þ
Here, nI is the range of the solute-crowdion interaction in atomic positions (nI 9) and xMn is the concentration of solute atoms [14]. Recent molecular dynamics studies [15] show that Ni has also the effect of reducing the mobility of SIA clusters, especially for relatively large ones. This effect, however, is not explicitly and separately accounted for in the model, because the slowing down predicted by Eqs. (3)–(5) and here associated with Mn is already strong. Finally, for what concerns the V cluster population, we assumed that single V are slowed down significantly by the presence of Mn and Ni, the effective migration energy having been calibrated to 1.2 eV [16]. This migration energy is extended to all clusters of size <10 V, while above this size complete immobility is postulated: the introduction of this threshold, the value of which was opportunely calibrated (as discussed in Section 4), represents a simplification and hides a lack of knowledge on the actual effect of the presence of Mn and Ni on the mobility and stability of V clusters. For all other parameters we refer to [1,3]. 4. Results and discussion The simulation conditions were dictated by the reference irradiation experiment [17], performed at 563 K. In the experiment
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
M. Chiapetto et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
several model alloys of increasing complexity, from Fe to a real steel, were irradiated up to 0.2 displacements per atom (dpa) at relatively high flux, namely (8.6 ± 0.5)1013 n cm2 s1 (En > 1 MeV), i.e. 107 dpa/s [10]. After irradiation, the materials were examined using positron annihilation spectroscopy (PAS), smallangle neutron scattering (SANS) [11], transmission electron microscopy (TEM) [6], and atom probe tomography (APT) [18]: the combination of the results of these different experimental techniques gave a thorough description of the nanostructure of the materials in terms of defect densities and average sizes of both V and SIA cluster populations. The main effect observed in the experiment is the fact that, with the addition of solutes, there is a progressive decrease in the density and size of the point-defect cluster populations, until the complete disappearance of visible SIA loops and of any V cluster in the steel. Here we focus on the comparison between Fe–C and FeMnNi model alloys. Fig. 1 shows the V size distribution at 0.2 dpa in both: the formation of big V clusters and nanovoids is suppressed in dilute FeMnNi alloys as compared to Fe–C, while single- or di-vacancies appear in relatively high density (up to 1025 m3 after irradiation to 0.2 dpa at 290 °C, to be compared with a number density of 1023 m3 in the case of Fe–C under the same irradiation conditions). These results are in agreement with the experimental PAS values [10,11]. The formation of a limited number of visible voids in FeMnNi in the simulations, not reported in either TEM or PAS studies [6], might be a limitation of our model, in which immobile traps used to reproduce the effect of mobile C atoms and potentially mobile C–V complexes might be more efficient than they should and act as sort of ‘‘artificial’’ nucleation points for TEM visible voids. Their number density, however, remains very small if compared to those of mono- and di-vacancies, so their existence in reality cannot be excluded based purely on PAS studies, while TEM was not performed to specifically search for them. In Fig. 2 the TEM-visible loop number density evolution with dpa for both model alloys is shown. Based on indications from experimentalists, loops are considered visible in TEM if their radius is larger than 1.3 nm (100 SIA). TEM observations showed that the number density should be about a factor 4 smaller in the FeMnNi alloy, while the mean size at the same dose was estimated to decrease from 10 nm in Fe–C to 5 nm in the presence of Mn and Ni solute atoms: in the latter case most loops were found to have sizes at the
3
Fig. 2. Number density of TEM-visible interstitial loops versus dose according to model (lines) and reference experiment (dots) from TEM [6] for FeMnNi (brown) and Fe–C (green). Loops are considered visible in the transmission electron microscope (TEM) if their radius is larger than 1.3 nm (>100 SIA). Chosen parameters for FeMnNi: Eb = 0.7 eV, Nth = 10. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
threshold for visibility. In the model the sizes do not grow significantly with dose in FeMnNi, which is qualitatively not far from the experimental observation. Moreover, the shift in the dose at which loops become visible in the experiment is well reproduced by the model: no loops were detected by TEM below 0.1 dpa in FeMnNi, while in Fe–C some loops could be seen already at 0.025 dpa. Fig. 3 shows another important difference between the SIA size distribution in Fe–C and FeMnNi model alloys: here, the fraction of SIA in both TEM visible and invisible clusters at 0.2 dpa is reported and it is possible to notice the outstanding prevalence of non-visible clusters (i.e. <100 SIA) when the effect of Mn and Ni solute atoms on the mobility of both defect cluster populations is taken into account; on the contrary, in Fe–C virtually all interstitial clusters are in the visibility range. Figs. 4 and 5 correspond to a parametric study of how the number density of TEM-visible interstitial loops versus dose changes by
Fig. 1. Size distribution of vacancy clusters at 0.2 dpa for Fe–C (dark grey) and FeMnNi (pink) model alloys. In FeMnNi the largest majority of clusters contains less than 7 vacancies (chosen parameters: Eb = 0.7 eV, Nth = 10). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
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M. Chiapetto et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
Fig. 3. Fraction of SIA in both TEM visible and invisible clusters in Fe–C (blue) and FeMnNi (red) at 0.2 dpa. Criterion for loop visibility and parameters as in Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Number density of TEM-visible interstitial loops versus dose according to model (lines) and reference experiment (dots) [6]. Different values of the threshold for the immobility of bigger vacancy clusters have been investigated, while the binding energy between crowdion and Mn is set to 0.65 eV. Criterion for loop visibility as in Fig. 2.
hides subtle effects of solute distribution and redistribution under irradiation (e.g. segregation of solutes on loops) that the model cannot catch, because of the ‘‘grey alloy’’ approach adopted here as first step towards more complete models. This parametric study leads to the conclusion that a binding energy of 0.65–0.7 eV between the crowdion and the Mn solute atoms, together with postulating immobility of vacancy clusters about size 10, is a suitable combination of parameters to provide agreement between experimental observations and OKMC predictions in FeMnNi alloys, implicitly allowing for effects that the model cannot fully include. Further investigations are needed to make the model parameterization more physical.
Fig. 4. Number density of TEM-visible interstitial loops versus dose according to model (lines) and reference experiment (dots) [6]. Different values of binding energy between crowdion and Mn have been investigated, while the threshold for the immobility of bigger vacancy clusters is set here deliberately to size 7. Criterion for loop visibility as in Fig. 2.
choosing different values for two of the calibrated parameters of the model, namely the binding energy between the crowdion and Mn, and the threshold size above which vacancy clusters are immobile. In Fig. 4 the TEM-visible loops number density predicted by the OKMC model decreases progressively and approaches the experimental values while the binding energy grows to 0.5 eV; from this binding energy on, there is a sort of saturation of the influence of this parameter. The difference between the experimental values and the curves for >0.5 eV in Fig. 4 is due to the choice of the threshold for vacancy cluster immobility, as is proven by Fig. 5, which was obtained for a fixed Mn-crowdion binding energy of 0.65 eV (a value consistent with DFT calculations [12,16]). The agreement between the experimental values and the predictions of our model for the visible loop number density improves by making the vacancy mobility threshold increase. Such threshold is obviously an oversimplification that hides our limited knowledge of the actual effect of the presence of Mn and Ni on the mobility and stability of V clusters. First studies aimed at better understanding this effect are presented in this volume [19]. Likewise, the effective value for the crowdion-Mn binding energy
5. Conclusions This work shows that both V and SIA cluster mobility is strongly affected by Mn and Ni, and probably other solutes, as compared to pure Fe and that this is the origin of the ‘‘disappearance’’ of sizeable point-defect clusters with increasing alloy complexity. The present OKMC model is therefore suitable to provide rough indications about the effect of the alloying elements present in RPV steels on the nanostructural evolution under irradiation, allowing for example studies of the effect of variables such as neutron flux and temperature. Key in our model was the identification of a binding energy of 0.65 eV between the crowdion and a solute Mn atom, thereby affecting the diffusivity of SIA objects: a clear physical explanation is found in terms of strong interaction between Mn and the crowdions that form the SIA clusters, similarly to what happens in Fe–Cr alloys [14], although with a higher binding energy. Sensitivity studies have shown that the assumption that vacancy and vacancy cluster mobility should be reduced by the presence of Mn and Ni is also a key ingredient of the model: specifically, an effective migration energy of 1.2 eV extended to all vacancy clusters of size smaller than 10 defects was assumed, while above this size vacancy clusters were made completely immobile. However, in this second case further investigations are needed to clearly identify the origin of this effect and better quantify it. The next step will be the introduction in the model of explicit transport of solute atoms via point-defects.
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
M. Chiapetto et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
V. Jansson, M. Chiapetto, L. Malerba, J. Nucl. Mater. 442 (2013) 341–349. V. Jansson, L. Malerba, J. Nucl. Mater. 443 (2013) 274–285. M. Chiapetto, L. Malerba, C.S. Becquart, J. Nucl. Mater. (2014). C. Domain, C. Becquart, L. Malerba, J. Nucl. Mater. 335 (2004) 121–145. F. Nichols, J. Nucl. Mater. 75 (1978) 32–41. M. Hernández-Mayoral, D. Gómez-Briceño, J. Nucl. Mater. 399 (2010) 146– 153. W. Young, E. Elcock, Proc. Phys. Soc. 89 (1966) 735. A.B. Bortz, M.H. Kalos, J.L. Lebowitz, J. Comput. Phys. 17 (1975) 10–18. L. Malerba, C. Becquart, C. Domain, J. Nucl. Mater. 360 (2007) 159. M. Lambrecht, L. Malerba, A. Almazouzi, J. Nucl. Mater. 378 (2008) 282–290. M. Lambrecht, PhD thesis, University of Gent, 2009. P. Olsson, T.P.C. Klaver, C. Domain, Phys. Rev. B 81 (2010) 054102. R. Ngayam-Happy, P. Olsson, C.S. Becquart, C. Domain, J. Nucl. Mater. 407 (2010) 16–28.
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[14] D. Terentyev, L. Malerba, A.V. Barashev, Philos. Mag. Lett. 85 (2005) 587–594. [15] N. Anento, N. Castin, A. Serra, L. Malerba, Technical Report, Public Deliverable Nr. D1–3.16, FP7/Perform60 Collaborative Project, Grant Agreement Nr. FP7232612, 2013. [16] A. Bakaev, D. Terentyev, G. Bonny, T.P.C. Klaver, P. Olsson, D. Van Neck, J. Nucl. Mater. 444 (2014) 237–246. [17] L. Malerba, E. van Walle, C. Domain, S. Jumel, J.-C. Van Duysen, Proceedings of the 10th International Conference on Nuclear Engineering (ICONE-10): Arlington, VA-USA, April 14–18, 2002 (on CD-ROM), The American Society of Mechanical Engineers (2002). [18] E. Meslin, B. Radiguet, P. Pareige, C. Toffolon, A. Barbu, Exp. Mech. 51 (2011) 1453. [19] L. Messina, P. Olsson, L. Malerba, Nucl. Instr. Meth. B (2014), http://dx.doi.org/ 10.1016/j.nimb.2014.11.102. accepted for publication.
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels M. Chiapetto a,b,⇑, C.S. Becquart b,d, C. Domain c,d, L. Malerba a a
SCKCEN, Nuclear Materials Science Institute, Boeretang 200, B-2400 Mol, Belgium Unité Matériaux Et Transformations (UMET), UMR 8207, Université de Lille 1, ENSCL, F-59600 Villeneuve d’Ascq Cedex, France c EDF R&D, Département Matériaux et Mécanique des Composants, Les Renardières, F-77250 Moret sur Loing, France d Laboratoire commun EDF-CNRS Etude et Modélisation des Microstructures pour le Vieillissement des Matériaux (EM2VM), France b
a r t i c l e
i n f o
Article history: Received 8 July 2014 Received in revised form 20 October 2014 Accepted 29 November 2014 Available online xxxx Keywords: Object kinetic Monte Carlo Radiation-induced defects Neutron irradiation Fe–C FeMnNi
a b s t r a c t Radiation-induced embrittlement of bainitic steels is one of the most important lifetime limiting factors of existing nuclear light water reactor pressure vessels. The primary mechanism of embrittlement is the obstruction of dislocation motion produced by nanometric defect structures that develop in the bulk of the material due to irradiation. The development of models that describe, based on physical mechanisms, the nanostructural changes in these types of materials due to neutron irradiation are expected to help to better understand which features are mainly responsible for embrittlement. The chemical elements that are thought to influence most the response under irradiation of low-Cu RPV steels, especially at high fluence, are Ni and Mn, hence there is an interest in modelling the nanostructure evolution in irradiated FeMnNi alloys. As a first step in this direction, we developed sets of parameters for object kinetic Monte Carlo (OKMC) simulations that allow this to be done, under simplifying assumptions, using a ‘‘grey alloy’’ approach that extends the already existing OKMC model for neutron irradiated Fe–C binary alloys [1]. Our model proved to be able to describe the trend in the buildup of irradiation defect populations at the operational temperature of LWR (300 °C), in terms of both density and size distribution of the defect cluster populations, in FeMnNi model alloys as compared to Fe–C. In particular, the reduction of the mobility of point-defect clusters as a consequence of the presence of solutes proves to be key to explain the experimentally observed disappearance of detectable point-defect clusters with increasing solute content. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction Low-alloy bainitic steels are used to fabricate the reactor pressure vessel (RPV) of most commercial light water nuclear power plants. The vessel is an irreplaceable component and its capability to maintain its integrity determines the lifetime of the installation. Prolonged exposure to neutron flux impinging on the metal creates a large amount of point-defects (vacancies, V, and self-interstitial atoms, SIA) and relevant clusters within displacements cascades, which may evolve to form voids and dislocation loops (so-called matrix damage). These radiation-induced defects are also the main responsible for the enhanced diffusion of solute atoms within the matrix, leading to the formation of solute-rich clusters, generally denoted as precipitates. The link between changes in the microor nanostructure of the material and its mechanical properties is ⇑ Corresponding author at: SCKCEN, Nuclear Materials Science Institute, Boeretang 200, B-2400 Mol, Belgium. Tel.: +32 (0)14 33 31 81. E-mail address: [email protected] (M. Chiapetto).
therefore strong: point-defect clusters and/or precipitates constitute obstacles to the movement of dislocations, increasing yield strength and reducing ductility, thereby contributing to increase the probability of brittle fracture. Computer simulation models are an effective way to understand these kinds of degradation processes and, in this context, FeMnNi systems containing up to a maximum of 1.5–2 wt.% of these two solutes are archetypal model materials for low-alloy ferritic steels, able to highlight the solute influence on the nanostructural evolution under irradiation. The present work is a continuation of the ongoing efforts to develop a model for the nanostructure evolution in RPV steels under irradiation, in terms of density and size of defects versus neutron dose, expressed in displacements per atom (dpa), using the object kinetic Monte Carlo (OKMC) simulation technique [1–3]. We present a comparison between an already validated model for Fe–C alloys neutron irradiated at 560 K [1] and a further developed model able to describe the nanostructural evolution of both V and SIA defect populations under neutron irradiation when also Mn and Ni are present. This model is based on assumptions that translate the
http://dx.doi.org/10.1016/j.nimb.2014.11.102 0168-583X/Ó 2014 Elsevier B.V. All rights reserved.
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
2
M. Chiapetto et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
physical effect of these solutes on the irradiation-induced defect formation and accumulation into, mainly, modified defect mobility parameters, without explicitly introducing the solute atoms in the model [3].
conditions applied in all three directions, as discussed in [9]. Simulation temperature, dpa rate, grain size and dislocation density were dictated by the reference experiment [6,10,11]. 3. Parameterization
2. Computational method For our object kinetic Monte Carlo (OKMC) simulations we used the LAKIMOCA code, thoroughly described in [4], while the approach is explained in detail in [2]. The OKMC is a stochastic method that can be used to describe the nanostructural evolution in materials subjected to neutron irradiation. The production of point-defects like V and SIAs, which also form clusters, is reproduced by a random introduction of cascade debris containing V and SIA objects of different sizes in the system, at a certain rate for the given volume. These are the ‘‘objects’’ of the system, the evolution of which we want to study: they are described in the model by a set of parameters that define their stability, their diffusion properties, and their possibility of interacting with each other and with other defects. Objects of importance are e.g. carbon (C) atoms and C–V complexes, that play the role of traps for mobile point-defect clusters and are characterized by a certain trapping energy that depends on the size and type of the trapped cluster [2]. The pre-existing microstructure, i.e. grain boundaries and dislocations, is taken into account explicitly [4]: dislocations are described as spherical sinks, the radius and number of which are defined in such a way that their sink strength equals the one corresponding to the dislocation density of the reference material [5]. However, since SIA clusters are observed by TEM (transmission electron microscopy) to decorate dislocations [6], meaning that they are not absorbed if large enough, we only allowed clusters smaller than the core of the dislocation, i.e. size 1–4, to be absorbed by the sinks. Every object has an associated reaction volume, generally spherically shaped, with the exception of large dislocation loops, which are represented by toroids. When the reaction volumes of two objects overlap, a predefined reaction, like clustering or recombination between a V and an SIA, takes place. The events in the OKMC simulation determine the dynamics of the system. The probability for the objects to perform an event are given in terms of Arrhenius frequencies for thermally activated processes:
Ai kB T
Ci ¼ mi exp
ð1Þ
Here mi is the attempt frequency (or prefactor) of the event i, Ai is the corresponding activation energy (eV), which must embody both the thermodynamics and the kinetics of the system, kB is Boltzmann’s constant (eV/K) and T is the irradiation temperature (K). Due to their dependence on the type and size of the objects, a very large number of parameters is needed for a standard simulation. In this work we use, besides modifications described below, the parameters from [1–3]. For every simulation step an event is chosen, based on the corresponding probabilities in the parameterization and according to the stochastic Monte Carlo algorithm. The simulated time is increased following the residence time algorithm [7,8]:
Dt ¼
lnR
RCi
ð2Þ
where R is a uniform random number between 0 and 1, included to take into account correctly the stochasticity, according to the Poisson distribution. A non-cubic simulation box of size 600 750 880 a30, where a0 = 2.871010 m is the lattice parameter of a-Fe, was used in order to avoid potential anomalies from 1D-migrating defects entering a migration trajectory loop, due to the periodic boundary
In the present work we compare the nanostructural evolution under irradiation of Fe(C)MnNi alloys (Fe-1.2%Mn-0.7%Ni) and Fe–C binary alloys. The link between the two systems is that the concentration of C in the model material remains the same, i.e. 134 appm, but in the first case we also consider the effect of two different solute atoms, namely Mn and Ni, on the diffusivity of defect clusters. These solute atoms are not explicitly present in the model, which therefore cannot describe their redistribution (segregation, precipitation, . . .) under irradiation, but their effect is introduced by modifying the parameters that govern the mobility of both SIA and V clusters. For the Fe–C alloy the parameterization presented in [1] was adopted. For what concerns FeMnNi alloys, on the other hand, density functional theory (DFT) calculations indicate that Mn has a strong binding energy with the crowdion [12,13]. We therefore reduced the diffusion coefficient DFeMn of all SIA clusters as comn pared to pure Fe, DFe n , using an explicit function of Mn concentration, according to the following equation [14]: bD F n DFeMn ¼ DFe n n e
ð3Þ
where n is the number of SIA in the cluster and b is 1/kBT. We translated this reduction of the diffusion coefficient in a decrease of the attempt frequency mi in Eq. (1) (temperature being fixed). In dilute alloys the cluster diffusivity decreases with increasing Mn concentration simply because of the increasing amount of solute atoms interacting with the cluster. Under this assumption, the variation of the cluster free energy, DFn, can be expressed as [14]:
DF n K n Eb1
ð4Þ
Eb1 being the effective binding energy between a crowdion and a solute Mn atom, i.e. the decrease in free energy between the condition without and with associated solute atoms, here set to 0.65 eV consistently with DFT calculations [12], while Kn represents the mean number of Mn atoms interacting with the cluster,
K n ¼ nnI xMn
ð5Þ
Here, nI is the range of the solute-crowdion interaction in atomic positions (nI 9) and xMn is the concentration of solute atoms [14]. Recent molecular dynamics studies [15] show that Ni has also the effect of reducing the mobility of SIA clusters, especially for relatively large ones. This effect, however, is not explicitly and separately accounted for in the model, because the slowing down predicted by Eqs. (3)–(5) and here associated with Mn is already strong. Finally, for what concerns the V cluster population, we assumed that single V are slowed down significantly by the presence of Mn and Ni, the effective migration energy having been calibrated to 1.2 eV [16]. This migration energy is extended to all clusters of size <10 V, while above this size complete immobility is postulated: the introduction of this threshold, the value of which was opportunely calibrated (as discussed in Section 4), represents a simplification and hides a lack of knowledge on the actual effect of the presence of Mn and Ni on the mobility and stability of V clusters. For all other parameters we refer to [1,3]. 4. Results and discussion The simulation conditions were dictated by the reference irradiation experiment [17], performed at 563 K. In the experiment
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
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several model alloys of increasing complexity, from Fe to a real steel, were irradiated up to 0.2 displacements per atom (dpa) at relatively high flux, namely (8.6 ± 0.5)1013 n cm2 s1 (En > 1 MeV), i.e. 107 dpa/s [10]. After irradiation, the materials were examined using positron annihilation spectroscopy (PAS), smallangle neutron scattering (SANS) [11], transmission electron microscopy (TEM) [6], and atom probe tomography (APT) [18]: the combination of the results of these different experimental techniques gave a thorough description of the nanostructure of the materials in terms of defect densities and average sizes of both V and SIA cluster populations. The main effect observed in the experiment is the fact that, with the addition of solutes, there is a progressive decrease in the density and size of the point-defect cluster populations, until the complete disappearance of visible SIA loops and of any V cluster in the steel. Here we focus on the comparison between Fe–C and FeMnNi model alloys. Fig. 1 shows the V size distribution at 0.2 dpa in both: the formation of big V clusters and nanovoids is suppressed in dilute FeMnNi alloys as compared to Fe–C, while single- or di-vacancies appear in relatively high density (up to 1025 m3 after irradiation to 0.2 dpa at 290 °C, to be compared with a number density of 1023 m3 in the case of Fe–C under the same irradiation conditions). These results are in agreement with the experimental PAS values [10,11]. The formation of a limited number of visible voids in FeMnNi in the simulations, not reported in either TEM or PAS studies [6], might be a limitation of our model, in which immobile traps used to reproduce the effect of mobile C atoms and potentially mobile C–V complexes might be more efficient than they should and act as sort of ‘‘artificial’’ nucleation points for TEM visible voids. Their number density, however, remains very small if compared to those of mono- and di-vacancies, so their existence in reality cannot be excluded based purely on PAS studies, while TEM was not performed to specifically search for them. In Fig. 2 the TEM-visible loop number density evolution with dpa for both model alloys is shown. Based on indications from experimentalists, loops are considered visible in TEM if their radius is larger than 1.3 nm (100 SIA). TEM observations showed that the number density should be about a factor 4 smaller in the FeMnNi alloy, while the mean size at the same dose was estimated to decrease from 10 nm in Fe–C to 5 nm in the presence of Mn and Ni solute atoms: in the latter case most loops were found to have sizes at the
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Fig. 2. Number density of TEM-visible interstitial loops versus dose according to model (lines) and reference experiment (dots) from TEM [6] for FeMnNi (brown) and Fe–C (green). Loops are considered visible in the transmission electron microscope (TEM) if their radius is larger than 1.3 nm (>100 SIA). Chosen parameters for FeMnNi: Eb = 0.7 eV, Nth = 10. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
threshold for visibility. In the model the sizes do not grow significantly with dose in FeMnNi, which is qualitatively not far from the experimental observation. Moreover, the shift in the dose at which loops become visible in the experiment is well reproduced by the model: no loops were detected by TEM below 0.1 dpa in FeMnNi, while in Fe–C some loops could be seen already at 0.025 dpa. Fig. 3 shows another important difference between the SIA size distribution in Fe–C and FeMnNi model alloys: here, the fraction of SIA in both TEM visible and invisible clusters at 0.2 dpa is reported and it is possible to notice the outstanding prevalence of non-visible clusters (i.e. <100 SIA) when the effect of Mn and Ni solute atoms on the mobility of both defect cluster populations is taken into account; on the contrary, in Fe–C virtually all interstitial clusters are in the visibility range. Figs. 4 and 5 correspond to a parametric study of how the number density of TEM-visible interstitial loops versus dose changes by
Fig. 1. Size distribution of vacancy clusters at 0.2 dpa for Fe–C (dark grey) and FeMnNi (pink) model alloys. In FeMnNi the largest majority of clusters contains less than 7 vacancies (chosen parameters: Eb = 0.7 eV, Nth = 10). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
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M. Chiapetto et al. / Nuclear Instruments and Methods in Physics Research B xxx (2014) xxx–xxx
Fig. 3. Fraction of SIA in both TEM visible and invisible clusters in Fe–C (blue) and FeMnNi (red) at 0.2 dpa. Criterion for loop visibility and parameters as in Fig. 2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Fig. 5. Number density of TEM-visible interstitial loops versus dose according to model (lines) and reference experiment (dots) [6]. Different values of the threshold for the immobility of bigger vacancy clusters have been investigated, while the binding energy between crowdion and Mn is set to 0.65 eV. Criterion for loop visibility as in Fig. 2.
hides subtle effects of solute distribution and redistribution under irradiation (e.g. segregation of solutes on loops) that the model cannot catch, because of the ‘‘grey alloy’’ approach adopted here as first step towards more complete models. This parametric study leads to the conclusion that a binding energy of 0.65–0.7 eV between the crowdion and the Mn solute atoms, together with postulating immobility of vacancy clusters about size 10, is a suitable combination of parameters to provide agreement between experimental observations and OKMC predictions in FeMnNi alloys, implicitly allowing for effects that the model cannot fully include. Further investigations are needed to make the model parameterization more physical.
Fig. 4. Number density of TEM-visible interstitial loops versus dose according to model (lines) and reference experiment (dots) [6]. Different values of binding energy between crowdion and Mn have been investigated, while the threshold for the immobility of bigger vacancy clusters is set here deliberately to size 7. Criterion for loop visibility as in Fig. 2.
choosing different values for two of the calibrated parameters of the model, namely the binding energy between the crowdion and Mn, and the threshold size above which vacancy clusters are immobile. In Fig. 4 the TEM-visible loops number density predicted by the OKMC model decreases progressively and approaches the experimental values while the binding energy grows to 0.5 eV; from this binding energy on, there is a sort of saturation of the influence of this parameter. The difference between the experimental values and the curves for >0.5 eV in Fig. 4 is due to the choice of the threshold for vacancy cluster immobility, as is proven by Fig. 5, which was obtained for a fixed Mn-crowdion binding energy of 0.65 eV (a value consistent with DFT calculations [12,16]). The agreement between the experimental values and the predictions of our model for the visible loop number density improves by making the vacancy mobility threshold increase. Such threshold is obviously an oversimplification that hides our limited knowledge of the actual effect of the presence of Mn and Ni on the mobility and stability of V clusters. First studies aimed at better understanding this effect are presented in this volume [19]. Likewise, the effective value for the crowdion-Mn binding energy
5. Conclusions This work shows that both V and SIA cluster mobility is strongly affected by Mn and Ni, and probably other solutes, as compared to pure Fe and that this is the origin of the ‘‘disappearance’’ of sizeable point-defect clusters with increasing alloy complexity. The present OKMC model is therefore suitable to provide rough indications about the effect of the alloying elements present in RPV steels on the nanostructural evolution under irradiation, allowing for example studies of the effect of variables such as neutron flux and temperature. Key in our model was the identification of a binding energy of 0.65 eV between the crowdion and a solute Mn atom, thereby affecting the diffusivity of SIA objects: a clear physical explanation is found in terms of strong interaction between Mn and the crowdions that form the SIA clusters, similarly to what happens in Fe–Cr alloys [14], although with a higher binding energy. Sensitivity studies have shown that the assumption that vacancy and vacancy cluster mobility should be reduced by the presence of Mn and Ni is also a key ingredient of the model: specifically, an effective migration energy of 1.2 eV extended to all vacancy clusters of size smaller than 10 defects was assumed, while above this size vacancy clusters were made completely immobile. However, in this second case further investigations are needed to clearly identify the origin of this effect and better quantify it. The next step will be the introduction in the model of explicit transport of solute atoms via point-defects.
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
V. Jansson, M. Chiapetto, L. Malerba, J. Nucl. Mater. 442 (2013) 341–349. V. Jansson, L. Malerba, J. Nucl. Mater. 443 (2013) 274–285. M. Chiapetto, L. Malerba, C.S. Becquart, J. Nucl. Mater. (2014). C. Domain, C. Becquart, L. Malerba, J. Nucl. Mater. 335 (2004) 121–145. F. Nichols, J. Nucl. Mater. 75 (1978) 32–41. M. Hernández-Mayoral, D. Gómez-Briceño, J. Nucl. Mater. 399 (2010) 146– 153. W. Young, E. Elcock, Proc. Phys. Soc. 89 (1966) 735. A.B. Bortz, M.H. Kalos, J.L. Lebowitz, J. Comput. Phys. 17 (1975) 10–18. L. Malerba, C. Becquart, C. Domain, J. Nucl. Mater. 360 (2007) 159. M. Lambrecht, L. Malerba, A. Almazouzi, J. Nucl. Mater. 378 (2008) 282–290. M. Lambrecht, PhD thesis, University of Gent, 2009. P. Olsson, T.P.C. Klaver, C. Domain, Phys. Rev. B 81 (2010) 054102. R. Ngayam-Happy, P. Olsson, C.S. Becquart, C. Domain, J. Nucl. Mater. 407 (2010) 16–28.
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[14] D. Terentyev, L. Malerba, A.V. Barashev, Philos. Mag. Lett. 85 (2005) 587–594. [15] N. Anento, N. Castin, A. Serra, L. Malerba, Technical Report, Public Deliverable Nr. D1–3.16, FP7/Perform60 Collaborative Project, Grant Agreement Nr. FP7232612, 2013. [16] A. Bakaev, D. Terentyev, G. Bonny, T.P.C. Klaver, P. Olsson, D. Van Neck, J. Nucl. Mater. 444 (2014) 237–246. [17] L. Malerba, E. van Walle, C. Domain, S. Jumel, J.-C. Van Duysen, Proceedings of the 10th International Conference on Nuclear Engineering (ICONE-10): Arlington, VA-USA, April 14–18, 2002 (on CD-ROM), The American Society of Mechanical Engineers (2002). [18] E. Meslin, B. Radiguet, P. Pareige, C. Toffolon, A. Barbu, Exp. Mech. 51 (2011) 1453. [19] L. Messina, P. Olsson, L. Malerba, Nucl. Instr. Meth. B (2014), http://dx.doi.org/ 10.1016/j.nimb.2014.11.102. accepted for publication.
Please cite this article in press as: M. Chiapetto et al., Nanostructure evolution under irradiation of Fe(C)MnNi model alloys for reactor pressure vessel steels, Nucl. Instr. Meth. B (2014), http://dx.doi.org/10.1016/j.nimb.2014.11.102