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Fe3+ ion irradiated EUROFER97
Journal of Nuclear Materials 484 (2017) 59e67
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TEM study and modeling of bubble formation in dual-beam Heþ/Fe3þ ion irradiated EUROFER97 B. Kaiser a, *, C. Dethloff a, E. Gaganidze a, D. Brimbal b, M. Payet b, P. Trocellier b, L. Beck b, J. Aktaa a a b
Karlsruhe Institute of Technology, Institute for Applied Materials (IAM), P.B. 3410, 76021, Karlsruhe, Germany CEA, DEN, Service de Recherches de M etallurgie Physique, Laboratoire JANNUS, F-91191, Gif-sur-Yvette, France
h i g h l i g h t s Investigation of He and dpa effects in EUROFER97 by irradiation with He/Fe-ions at 330 C, 400 C and 500 C. TEM analysis of the size distribution of helium bubbles as a function of irradiation temperature. Modeling of helium bubble formation with a kinetic rate model. Influence of two different thermodynamic descriptions of helium bubbles on the rate model. Comparison between experimental and numerical results.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 June 2016 Received in revised form 30 September 2016 Accepted 13 November 2016 Available online 15 November 2016
The Reduced Activation Ferritic/Martensitic (RAFM) steels are promising structural materials for the first wall and blanket components of future fusion reactors. To obtain further insight into the temperature dependence of helium effects induced by 14 MeV energy neutrons under fusion like conditions, EUROFER97 was exposed to Heþ/Fe3þ dual-beam ion irradiation at the JANNUS laboratory at Saclay. The implantation was carried out at temperatures of 330 C, 400 C and 500 C and induced a damage and helium concentration up to 26 dpa and 450 appm He, respectively. TEM microstructure analysis indicates a spatially homogeneous distribution of helium bubbles at 330 C and 400 C whereas a coexistence of homogeneous and heterogeneous nucleation of bubbles is found at 500 C. An increasing mean bubble diameter and decreasing concentration of bubbles with rising irradiation temperature, as predicted by numerical results of a kinetic rate model for diffusion governed homogeneous nucleation of helium bubbles, are mostly confirmed by the irradiation experiment. Furthermore, within the rate model two approaches for the determination of the thermodynamic properties of helium filled voids in a-iron are applied. With respect to the final bubble size distribution, the commonly used surface energy of a void in the iron matrix is compared to the “variable gap model” of [1], J. Nucl. Mater. 418 (2011), which includes additionally the interaction between the helium atoms themselves, the energy at the helium-iron interface and the elastic deformation of the iron matrix. © 2016 Published by Elsevier B.V.
Keywords: Radiation effects Ion irradiation Cluster dynamics Fusion Helium bubbles RAFM steels
1. Introduction The reduced activation ferritic-martensitic steels are considered as potential structural materials for the first wall and blanket components of future fusion reactors due to their promising low irradiation induced swelling and well balanced physical and
* Corresponding author. E-mail address: [email protected] (B. Kaiser). http://dx.doi.org/10.1016/j.jnucmat.2016.11.014 0022-3115/© 2016 Published by Elsevier B.V.
thermo-mechanical properties [2]. Nevertheless, when exposed for several years to a high flux of 14 MeV neutrons in fusion reactors these materials are still expected to sustain severe damage. Accordingly, a detailed knowledge of the evolution of microstructure damage is crucial for the estimation of the mechanical degradation of structural components and the further development of accurate design limits. Due to the absence of facilities with fusion reactor relevant neutron spectra substitutional irradiation programs are executed to simulate the mechanisms of void nucleation under simultaneous
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production of atomic displacement cascades and helium. Aside from the here discussed dual-beam irradiation with iron and helium ions, helium effects in EUROFER97 were studied in the fission neutron irradiation programs SPICE and ARBOR by using boron doping technique [3e6] and in spallation proton irradiation programs STIP-II and STIP-III [7,8] including also protons and helium ions in addition to neutrons. For the modeling of helium effects in a-iron a couple of theoretical methods is applied, providing insight on different time and length scales. On atomic scales density functional theory [9] and molecular dynamics [10,11] provide insight into properties of interstitial helium and small helium-vacancy clusters embedded in an iron-matrix, like migration and binding energies, and support the transfer to macroscopic thermodynamic description of helium bubbles [1,12]. Based on the knowledge of these physical quantities, usually object kinetic Monte Carlo calculations, kinetic rate theory or the Fokker-Planck equation are used to trace the evolution of microstructural damage on times and length scales typical for radiation experiments [4,13e16]. At the moment, the damage of 14 MeV neutrons as well as its reasonable imitation are not sufficiently understood and of great importance for the development and verification of suitable materials. Particularly, it was found that helium in EUROFER97 is responsible for embrittlement [17] and for an increase of the nanohardness [18]. Accordingly, the temperature dependence of helium bubble formation induced by Heþ/Fe3þ dual-beam ion irradiation of EUROFER97 is analyzed with TEM investigations of the irradiated samples. Beside the experimental part, the understanding of helium effects is supported by a kinetic rate model for diffusion governed homogeneous nucleation of helium bubbles. In this manner, we will evaluate the qualitative and quantitative capabilities of the model for nucleation of helium bubbles in EUROFER97 in order to assess the need for extending the model or adjusting parameters. 2. Dual-beam ion irradiation and sample preparation €hler The 25 mm thick EUROFER97 plates were produced by Bo Austria GmbH. The final heat treatment applied by the manufacturer includes a normalization at 980 C for 0.5 h and tempering at 760 C for 1.5 h. For the ion irradiation discs of 3 mm diameter and approximately 90 mm thickness were prepared. The irradiation with 1.2 MeV Heþ and 3.0 MeV Fe3þ ions was performed at the JANNUS Facility at Saclay [19,20] at temperatures of 330 C, 400 C and 500 C. We achieved a broadened implantation profile of helium by inserting aluminum energy degraders into the beam. In order to choose ion doses which lead to damage and helium
Fig. 1. SRIM calculation of the damage and helium concentration profile achieved by 6 h of 1.2 MeV Heþ and 3.0 MeV Fe3þ ion irradiation with respective doses of 3.2$1016 Fe3þ cm2 and 1.1$1016 Heþ cm2. The desired damage and helium content of approximately 26 dpa and 450 appm He is achieved at implantation depths in between 500 nm and 800 nm.
concentration ratios similar to conditions expected in future fusion reactors, SRIM [21] calculations were performed. Fig. 1 displays the expected depth profiles of damage and helium concentration for the applied doses of 3.2$1016 Fe3þ cm2 and 1.1$1016 Heþ cm2. Thus, fusion like conditions are obtained at depths between 500 nm and 800 nm from the surface where the applied fluxes lead to 26 dpa/450 appm He and a helium-dpa ratio of approximately 15 appm He/dpa. TEM samples were prepared via electrolytic-polishing or focused-ion-beam (FIB) machining. In particular, attention has to be paid to the electrolytic-polishing procedure for the creation of a thin film at a depth with the desired damage and helium content. Accordingly, the disks were thinned separately from the front and back side in an asymmetric manner. A more detailed description of the specimen preparation, the irradiation experiment and the TEM sample preparation can be found in Ref. [22]. The preparation of TEM-samples out of the uppermost irradiated 1 mm thick layer with asymmetric electrolytic-polishing and FIB machining may fail due to thinning at the wrong depth and the additional ion irradiation, respectively. In order to avoid features in the microstructure introduced by the sample preparation for all irradiation temperatures samples were produced in both ways. The observation of the same qualitative results in case of both preparation methods therefore ensures that they do not dependent on the preparation method. Nevertheless, for the subsequent detailed quantitative evaluation of size and density of helium bubbles we used FIB samples to guarantee for the results depths of 500 nme800 nm. Thus, we assure a damage and helium content of 26 dpa and 450 appm He, respectively. 3. TEM-analysis of the helium bubble distribution The irradiation damage is analyzed via TEM bright-field technique to obtain size distributions of voids. As well accepted, voids are identified by changing of their contrast from a bright spot with a dark fresnel fringe in under-focus condition to a dark spot with a bright fresnel fringe in over-focus condition. Under the assumption of spherical voids, their size is determined by measuring the corresponding diameter in an under-focused micrograph. The subsequent results and micrographs stem all from samples prepared with FIB machining and refer to the region of interest at a depth in between 500 nm and 800 nm, as marked in Fig. 1. Besides cavities other defects of the microstructure as dislocation loops and precipitates characterize the irradiation damage. The in depth investigation of these types of defects will be subject of future publications. With respect to the application of rate-model we limit the present TEM study to the size and density of helium bubbles. For characteristics of dislocations necessary for modeling we will refer on existing studies on EUROFER97 for dual-beam irradiation [22] and neutron irradiation [5,23,24]. In the case of Tirr ¼ 330 C we find homogeneously distributed bubbles as displayed in the two micrographs of Fig. 2. Similar observations are obtained for specimens irradiated at a temperature of Tirr ¼ 400 C, shown in Fig. 3. The bubbles are distributed homogeneously in the matrix without large variations in size. Remarkably, rising the irradiation temperature from 330 C to 400 C does not qualitatively influence the formation of helium bubbles. In contrast, further increasing the irradiation temperature up to Tirr ¼ 500 C strongly influences the nucleation of bubbles, concerning their size as well as spatial distribution. In contrast to the low temperature irradiation a heterogeneous nucleation is observed at Tirr ¼ 500 C. A major part of the bubbles is still homogeneously distributed in the matrix of the steel, as shown in Fig. 4. In addition, a remarkable fraction of the bubbles are now nucleated at different extended microstructural sinks such as grain
B. Kaiser et al. / Journal of Nuclear Materials 484 (2017) 59e67
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Fig. 2. Homogeneously distributed bubbles in EUROFER97 irradiated at Tirr ¼ 330 C up to 26 dpa/450 appm He and defocused to ±0.5 mm.
Fig. 3. Homogeneously distributed bubbles in EUROFER97 irradiated at Tirr ¼ 400 C up to 26 dpa/450 appm He and defocused to ±0.5 mm.
Fig. 4. Homogeneously distributed bubbles in EUROFER97 irradiated at Tirr ¼ 500 C up to 26 dpa/450 appm He and defocused to ±1.0 mm.
boundaries, dislocation lines and precipitates, see Fig. 5. Note, that Fig. 5 is not representative of the frequency or relative contribution of each sink type. At Tirr ¼ 500 C line dislocations and grain boundaries are found to be the major sink types whereas precipitates reveal only a minor role. Fig. 6 displays the concentration histograms of bubbles as a function of their diameter for the three applied irradiation temperatures. Representative micrographs prepared via FIB machining and used for measuring the diameter of bubbles are shown in Fig 2e4 and are defocused to ±0.5 mm for 330 C and 400 C and to ±1.0 mm in the case of 500 C. The sample thickness was determined by electron energy loss spectroscopy. Bubble diameters for 330 C are plotted in Fig. 6a and are narrow and symmetrically distributed with a mean bubble diameter of 1.3 nm and a total bubble density of r ¼ 2.0$1023 m3. For an increased temperature Tirr ¼ 400 C the quantitative analysis results in a total bubble concentration of r ¼ 9.4$1022 m3 whereas the shape and the mean diameter of 1.2 nm remain unchanged compared to 330 C irradiation within the experimental uncertainties, as shown in Fig. 6b.
Finally, in case of the highest applied irradiation temperature Tirr ¼ 500 C in Fig. 6c the mean diameter increases to 2.0 nm and the total density r drops to 2.5$1022 m3. Interestingly, also the shape of the distribution becomes broader and reveals a tail including bubbles with diameters up to 4.0 nm. The diameter distribution displayed for 500 C contains only spatially homogeneous distributed bubbles nucleated in the matrix well separated from any heterogeneous features. Bubbles at grain boundaries, line dislocations or precipitates indicated only minor differences in size. For all three irradiation temperatures the swelling does not exceed 0.02%. Of course, the estimated swelling with help of TEM relies only on visible bubbles. Nevertheless, with the below presented rate model we found that bubbles smaller then 0.5 nm do not have a large enough density to affect swelling significantly. These observations are in agreement with the prediction given in Ref. [25], which correlates the bubble size and the appearance of either heterogeneous or homogeneous distributions to the temperature dependence of the helium diffusion. Thus, our findings can be explained as follows. The homogeneous bubble distribution
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Fig. 5. Heterogeneous nucleation of bubbles at a) precipitates, b) dislocation lines and c) grain boundaries observed in EUROFER97 at Tirr ¼ 500 C and defocused to 2.5 mm. Note, that the figure does not represent the general proportion of helium absorption by each sink type. Line dislocations and grain boundaries are found to be the major sink types whereas precipitates revealed only a minor role.
homogeneous nucleation. Therefore the concentration of bubbles in the grain decreases at higher irradiation temperature. Concerning the impact of the elevated temperature of Tirr ¼ 500 C on the size of the bubbles we discriminate two features. First, the shift of the peak bubble size to larger bubble radii is also caused by the increased mobility of helium and vacancies, leading to a higher growth rate of the bubble radii. Second, the small contribution with a roughly constant concentration for diameters between 2.5 nm and 4.0 nm indicate the onset of a ripening process i.e. the redistribution from small to large bubbles. A meaningful comparison with the observations on boron doped specimens of the neutron irradiation program SPICE is difficult because besides the primarily homogeneous distribution of boron also the formation of BN and M23(C,B)6 precipitates was found [26]. It was further analyzed that the void nucleation is critically influenced by the vicinity of these precipitates [27]. Indeed the bubble diameters for SPICE [6] of up to 15 nm for 350 C at similar irradiation conditions of 16.3 dpa and 415 appm helium content are much larger than the size found in the present dualbeam irradiation experiment. In case of 450 C the mean diameter of bubbles in boron doped neutron irradiated specimens is approximately 4 nm and comes closer to the 2.0 nm of the present Heþ/Fe3þ dual-beam experiment at 500 C, but the bubble size distribution is still much broader covering bubbles up to 10 nm. In addition to the influence of BN and M23(C,B)6 particles, a further reason for the disagreement in the results between the two experiments could be found in the non constant helium production in SPICE, providing most helium (95%) after only 1 dpa in the first two months and the subsequent ripening of bubbles [16]. Furthermore, the damage and helium production rates differ up to four orders of magnitude, often interpreted as a temperature shift. In the SPICE experiment, a transition from homogeneous to a preferred heterogeneous bubble nucleation at grain boundaries and dislocation lines is observed for increasing the irradiation temperature from 350 C of 450 C [6,28]. In comparison, rising the irradiation temperature from 400 C to 500 C leads in the present dual-beam irradiation only to the coexistence between homogeneous and heterogeneous distribution of bubbles. Of course, the precise estimation of the temperature shift between the neutron and ion irradiation with the help of this rather qualitative feature requires more irradiation experiments in the temperature interval of the transition. However, in contrast to the dominating nucleation at dislocations and grain boundaries for neutrons at 450 C, we interprete the observed coexistence in the case of ions at 500 C as an early stage of the tendency for heterogeneous nucleation with rising irradiation temperature. Therefore, we indicate a temperature shift larger than 50 C in agreement with reference [22]. 4. Kinetic rate model
Fig. 6. The figure displays the bubble size distribution at implantation depths from 500 nm to 800 nm for a): Tirr ¼ 330Tirr¼ 330 C with micrographs defocused to ±0.5 mm±0:5, b): Tirr ¼ 400 C with micrographs defocused to ±0.5 mm and c): Tirr ¼ 500 C with micrographs defocused to ±1.0 mm. The data of subfigure c) contains only bubbles distributed homogeneously in the matrix.
at low temperature is enabled by the lower helium and vacancies mobility. The lower mobility increases the number of sources for nucleation in the matrix and depresses the diffusion to grain boundaries but also slows down the growth of bubbles. On the contrary, during the incubation at high temperatures the introduced helium and vacancies exhibit an increasing tendency to reach grain boundaries and thus decrease the possibilities for
The applied rate model is an extended version of that described in Ref. [16] and was originally developed for bubble formation under neutron irradiation in SPICE and ARBOR experiments and future fusion reactors. It estimates size and density distribution of homogeneous distributed bubbles due to high helium and vacancy production during irradiation. Helium atoms diffuse rapidly through the crystal, but are trapped effectively by microstructure defects e.g. vacancies, voids, dislocations and grain boundaries, resulting either in spatially homogeneous or heterogeneous bubble formation. Under irradiation the occupation of substitutional lattice sites by helium is preferred due to the high binding energy and the elevated concentration of vacancies. Subsequently, as a result of diffusive motion, these helium-vacancy clusters join and form stable nuclei for spatially homogeneous distributed bubble formation by the subsequent accumulation of helium and vacancies. In
B. Kaiser et al. / Journal of Nuclear Materials 484 (2017) 59e67
comparison with vacancies we consider SIA-defects to play a minor role in the nucleation process due to the high mobility of SIA. The expected fast diffusion leads to the formation of the interstitial defect clusters which evolve into the dislocation loops. The experimentally observed helium bubble microstructure however does not give any hint on the spatial correlation with the dislocation loop microstructure. Note also that the present model does not take into account the influence of solid solution carbon atoms or SIAs in the process of bubble nucleation and growth. The model is centered on helium-enhanced bubble nucleation and growth. In the case of heterogeneous nucleation, rate-theory is commonly not suited for the determination of bubble growth and the respective bubble size distribution. Thus, the model only accounts for the fraction of mobile defects absorbed by sinks and thus contributing to heterogeneous nucleation of bubbles. As usually, these sinks are described by averaged effective sink terms [29,30]. The model is based on the emission and absorption of mobile unit clusters from bubbles consisting of i mobile unit clusters and traces the corresponding concentrations Ci. The probability of these reactions is determined by the rate coefficients ki for absorbing and gi for emitting single mobile unit clusters. One of the approximations of this model is the assumption of a constant helium-vacancy ratio xHe accounting only for void configurations of i helium atoms and i/xHe vacancies and thus consequently leads to HeV1/xHe as mobile unit cluster. The equations of motion for cluster concentrations Ci read as follows:
X X vC1 ¼ GHe LSinks ki C1 Ci þ gi Ci ; vt
(1a)
vCi ¼ ½ki C1 þ gi Ci þ giþ1 Ciþ1 þ ki1 C1 Ci1 : vt
(1b)
i1
i2
The concentration of the mobile cluster type C1 is determined by equation (1a) which includes the helium generation rate GHe(t), losses at sinks LSinks and the absorption and emission of mobile units by clusters. The evolution of clusters of size i 2 described by equation (1b) couples with the concentration for clusters of size i1 and iþ1 via absorption and emission of mobile units. Therefore, in addition to the time independent absorption and emission coefficients the evolution of Ci(t) is strongly influenced by the concentration of the mobile type C1(t). The sink term Lsink accounts for the accumulation of helium at grain boundaries. Of course, in this description the individual character of helium atoms and vacancies is lost. However, we assume that due to the supersaturation of vacancies during the irradiation most of the helium atoms are trapped by vacancies and the subsequent dynamics are well approximated by the accumulation of mobile vacancy-helium clusters. The concentration distribution function of bubbles as a function of the bubble radius CðRÞ is obtained from the discrete quantity Ci as follows.
CðRÞ ¼ CiðRÞ
di : dR
(2)
Properties like the bubble mean and peak size of the distribution C(R) as well as the total concentration r obtained by the integration over R can be directly compared with those of the histograms of the TEM analysis plotted in Fig. 6. Note that in order to convert C(R) into a histogram one determines bin heights either approximately by the multiplication of C(R) with the bin width DR or exactly by the integration of C(R) over the respective bin intervals. The numerical solution of the differential equation (1) is calculated with the python wrappers of the odespy package [31] for the odepack algorithms [32] which can profit from the sparse Jacobi matrix.
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In the next step, the coefficients ki and gi must be defined. For the determination of ki a diffusion controlled absorption is assumed, implying a capture process if a mobile type hits a nonmobile cluster. Thus, the rate ki depends obviously on the cluster eff radius Ri and the helium diffusion constant DHe . eff
ki ¼ 4pRi DHe ðTirr Þ:
(3)
eff DHe
For we apply a temperature dependent formula extracted from Monte-Carlo results by Ref. [33] for the substitutional helium diffusion via the divacancy-mechanism. As usually, the emission probability is derived from a steady state argumentation resulting in a relation between the equilibrium cluster concentration and the kinetic coefficients ki and gi. Using thermodynamic expressions for the equilibrium concentrations introduces the free binding energy Fib of the mobile type to a cluster in the formula for gi, explicitly written as
"
1
gi ¼ ki U
# Fib exp : kB Tirr
(4)
Here U denotes the atomic volume and kB the Boltzmann constant. The losses of mobile clusters to sinks are defined by 2 Lsink ¼ Deff He C1 ðtÞ ksink ;
(5)
with the total sink strength k2sink . Usually, k2sink is the sum of the respective sink strengths for different sink types as grain boundaries, dislocation lines, dislocation loops and precipitates. With respect to the present results of the microstructure analysis we omit precipitates and dislocation loops as possible sink sites in the model. As mentioned, we observe mainly heterogeneous bubble nucleation at grain boundaries and dislocation lines. Also, reference [24] reported that in EUROFER97 precipitates preferentially occur at grain boundaries. Consequently, they do not represent an additional interface to the defects in the matrix, reducing further the need to include precipitates as sinks. Besides, dislocation loops sometimes lead to heterogeneous formation of bubbles. In model alloys dislocation loops caused a spatially heterogeneous distribution of bubbles [34,35] due to a preferential bubble formation at the loops. Nevertheless, the latter was experimentally not observed in the present study even at an elevated irradiation temperature of 500 C as well as in other studies on EUROFER97 [6,22,23]. The derivation of expressions for diffusion controlled sinks was intensively studied in the past and formulas can be found for all kinds of sinks [29,30]. In particular we will apply for the diffusion controlled sink strength of grain boundaries k2gb and dislocation lines k2dl the following formulas [29]:
. k2gb ¼ 15 R2gb ;
(6a)
k2dl ¼ Z rdl :
(6b)
Input parameters for the sink strengths are the grain radius Rgb, the dislocation line density rdl and the bias factor Z. Experimental measured values for the grain size of approximately 15 mm for EUROFER97 are found in literature [36]. However, this value concerns the size of former austenitic grains and does not represent the lath structure of the ferritic microstructure. Plausible values for the dislocation line density are 9.0$1013 m2 for unirradiated EUROFER97 and 6.7$1014 m2 in case of the present dual beam irradiation at 400 C, provided by Ref. [22]. Even though [29], estimated Z z 1, they mentioned the challenges in the derivation of the dislocation bias factor and outlined that its experimental determination of the
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dislocation bias appears reasonable. Therefore, we continue the discussion of a suitable parameter set for the losses at sinks at section 5 with respect to the results of the model in comparison to the TEM-analysis. The most critical assumption of the rate model is the constant helium-vacancy ratio xHe. As a consequence we only account for one mobile cluster type HeV1/xHe and omit elementary types as single helium atoms and vacancies as well as other small clusters. Thus, the possible pathways for nucleation schemes are strongly reduced and to a certain degree already defined with the choice of xHe. For example, bubble to void transformations can not be described by the model. Nevertheless, the usage of the model is justified in particular by the vacancy supersaturation due to the high dpa in case of the dual beam and neutron irradiation. Under this condition helium atoms are most likely trapped by vacancies and form clusters similar to the mobile unit cluster introduced by the present rate model. For the results discussed in this paper a helium-vacancy ratio xHe ¼ 1 was applied, because molecular dynamics predict a minimum close to the formation energy for bubbles of these configurations [37]. The requested application of the kinetic rate model consists of predicting tendencies whether bubbles occur spatially homogeneous or heterogeneous distributed. In the case of a homogeneous bubble formation in the matrix the size distribution of bubbles additionally is determined. Of course, further types of radiation damage like self interstitial clusters and dislocation loops evolve under irradiation and can influence the bubble formation and even more the mechanical properties. However, for an increased number of effect types the kinetic rate equations become computationally demanding in particular for the large bubbles and long irradiation duration of several years. Moreover, increasing the number of defect types require many input parameters and, consequently, the adjustment of the model on EUROFER97 will be more prone to errors. From this point of view, the here applied approach including effective parameters appears appropriate for the defined purposes. 4.1. Thermodynamic description of helium bubbles So far the macroscopic rate equation approach depends on microscopic attributes as the binding energy Fib and the void radius Ri. In Ref. [16] the bubble radius was directly linked to the spherical volume created by the corresponding number of vacancies. Thus, elastic deformation of the surrounding matrix due to over- or under-pressurized insert gas is neglected. With the knowledge of the void radius Ri the binding energy of the mobile type to a cluster was calculated by the difference in the surface energy when lowering the cluster size from i to i1. In this manner, one can straightforwardly determine the absorption and emission coefficient ki and gi with respect to equations (3) and (4). Another possibility to receive the quantities necessary for ki and gi was introduced by Ref. [1], denoted as “variable gap model” [1]. It is based on the partitioning of the free formation energy of a bubble in contributions for helium, the iron matrix and a helium-iron interface term as suggested by Ref. [38]. In order to provide a suitable representation of the interface term a spatial gap between helium and iron is introduced. Subsequently, these terms are parametrized via molecular dynamics as a function of the number of helium atoms and vacancies and at first also depend on the radii of the helium gas and the void. The equilibrium state of a helium filled void is found by minimizing the free formation energy with respect to the two radii. In spite of the reliable results of the variable gap model for medium to large cluster sizes the involved assumption of a volume-surface ratio for spheres usually is not adaptable for very small clusters. Therefore we have combined the formation energy calculated for fluid helium by the variable gap
model with the results of density functional theory for small clusters of Ref. [39,40]. In comparison to the thermodynamic model used by Ref. [16] in Ref. [16] the “variable gap model” accounts for several additional aspects. Obviously, the formation energy is changed since the “variable gap model” also includes interaction between the helium atoms themselves and between helium atoms and the iron matrix. In addition, the free energy term representing the iron matrix also consists of an elastic deformation term besides the surface term. The elastic deformation is in competition with the helium pressure and therefore can induce deviations compared with the former model in which the cluster radius depends only on the number of vacancies. 4.2. Comparison of microscopic parameters In order to evaluate the impact of the variable gap model on the results of the rate-theory we first discuss in which manner the cluster radius and the binding energies are quantitatively affected. Since the present rate model is restricted to constant heliumvacancy ratios only the corresponding subsets of the cluster configuration are discussed, even though the variable gap model provides quantities for all cluster configurations. As discussed in Ref. [16] the helium-vacancy ratio is assumed to lie within the interval 0.5
B. Kaiser et al. / Journal of Nuclear Materials 484 (2017) 59e67
beam experiment. Also, the numerical results indicated only a negligible absorption of helium at grain boundaries. Although the experimental part of the present study does not include a quantitative estimation of the contribution of each sink type, the measured size distributions allow a rough approximation of the fractions of helium captured by homogeneously nucleated bubbles in the matrix or absorbed by sinks at Tirr ¼ 500 C. We assume that at Tirr ¼ 300 C the measured size distribution of Fig. 6a for homogeneously distributed bubbles includes almost all of the helium which is represented by the volume fraction of these bubbles. In the distribution of Fig. 6c at Tirr ¼ 500 C the volume occupied by homogeneously distributed bubbles is reduced to 55% of the volume of bubbles at Tirr ¼ 300 C. We associate the missing volume with helium absorbed by line dislocations and grain boundaries. Therefore the fraction contributing to heterogeneous nucleation at Tirr ¼ 500 C can be approximated by 45%, which agrees qualitatively with the observed coexistence between heterogeneous and homogeneous nucleation. An improvement with respect to the experimental observation requires an increased sink strength of the grain boundaries and a decreased sink strength of dislocation lines. For this reason we set the grain size to Rgb ¼ 2.5 mm to get closer to the ferritic lath structure and chose a dislocation line density of unirradiated EUROFER97 rdl ¼ 9.0$1013 m2 [22]. Note that the dislocation line density of unirradiated EUROFER97 is a more adequate value, since nucleation occurs during the first seconds of the irradiation at all temperatures. Thus, the initial dislocation line density of unirradiated EUROFER97 obviously has a stronger influence on the appearance of either homogeneous or heterogeneous nucleation than the final density of dislocation lines after the irradiation. However, further studies for a suitable parametrization of the bias factor did not accomplish a satisfying description of the transition between homogeneous and heterogeneous bubble nucleation. Exemplary, the bias factor for the mobile clusters is reduced to 0.025. With this choice of parameters, we obtain an increasing accumulation of mobile cluster types at dislocation lines and grain boundaries with rising irradiation temperature. The relative amount of helium captured by sinks evolves from 1.8% at 330 C to 4.0% at 500 C. Although the final contribution to heterogeneous nucleation at 300 C seems reasonable on the base of the TEM analysis, the sink strength at 500 C is strongly underestimated. However, adjusting a higher bias factor closer to predictions of literature aiming at explanation of heterogeneous nucleation at 500 C leads simultaneously to an overestimation of heterogeneous nucleation at lower temperatures. Further investigations of a reasonable treatment of sinks in the rate model is dedicated to future work. For the following results of size distributions of homogeneously distributed bubbles determined with the rate model we apply Rgb ¼ 2.5 mm, rdl ¼ 9.0$1013 m2 and Zeff ¼ 0.025, bearing in mind that in particular at 500 C the helium absorption by sinks is underestimated. Results of the rate model for the cluster concentration C(d) as a function of the bubble diameter d are displayed in Fig. 7. The colors of the curves represent the temperatures 330 C (red), 400 C (blue) and 500 C (green) during irradiation. The curves clearly exhibit an increasing bubble diameter the higher the temperature is. In order to provide a comparison between the different thermodynamic descriptions of the helium bubbles solid lines show the results using the “variable gap model” whereas dotted lines denote results with binding energies and void radii following Ref. [16]. In spite of the marginal deviation in the bubble sizes between the two approaches at 330 C, large differences occur at higher temperatures. The radii corresponding to the maximum of the distribution shrinks from 5.2 nm to 4.6 nm at 400 C (dotted and solid blue curve) and from 10.3 nm to 5.7 nm at 500 C (dotted and
65
Fig. 7. Concentration distribution of bubbles as a function of the bubble diameter C(d) calculated with the rate-model for dual beam irradiation at 330 C, 400 C and 500 C. Results using the variable gap model introduced by Ref. [1] are plotted with solid lines. The dotted lines correspond to the results obtained using binding energies as defined by Ref. [16].
solid green curve) as indicated by the arrows in Fig. 7. As discussed above, in case of the variable gap model the emission of mobile unit cluster types is reduced due to the increased binding energy. One has to recall the coupling of the concentration of clusters with the concentration of mobile unit clusters in equation (1) as well as the size dependence of the emission and absorption coefficients. A high emission probability from smaller clusters together with the increasing capture rate of large clusters leads effectively to the redistribution of mobile unit clusters from small to larger clusters. Thus, a decreased emission suppresses this redistribution process and consequently leads to smaller clusters. This explanation is further confirmed at Tirr ¼ 330 C where, when emission becomes negligible small, the rate model results in almost identical bubble distributions for both energetic descriptions of bubbles. Recalling equation (1a) for the mobile clusters we see that they are generated by the emission from i 2 clusters and the ion-implantation rate. Therefore, the rate model predicts in case of the present ion irradiation experiment at low temperature that the bubble growth is almost exclusively determined by the implantation rate and redistribution processes play a minor role. Finally, we compare the predictions of the rate model with the observations of the microstructure analysis. All values concerning the size of bubbles obtained via numerics or TEM are summarized in Table 1. They obviously reveal the similar qualitative temperature dependence for bubble size and density, but still disagree in absolute values. In particular, the maxima of the calculated bubble distributions are found at larger diameters than in the histograms in Fig. 6 of the TEM analysis whereas the calculated densities are too small. The discrepancy in bubble sizes of the rate model was already reported in Ref. [16] in the case of neutron irradiated samples. Also, the model predicts that the bubble diameter increases in the whole applied temperature interval. Instead, we found this behavior experimentally only for rising the temperature from 400 C to 500 C. Bearing in mind that the bubble sizes are overestimated by the rate model and that the predicted change in
Table 1 Summary of results for the helium bubble peak diameter and total concentrations at different irradiation temperatures. The listed numerical results of the rate equations were obtained using binding energies of the variable gap model [1]. Tirr
TEM analysis b d
330 C 400 C 500 C
1.2 nm 1.2 nm 1.7 nm
Rate-theory
r 23
3
2.0$10 m 9.4$1022 m3 2.5$1022 m3
b d
r
3.8 nm 4.6 nm 5.7 nm
2.2 $1022 m3 1.3 $1022 m3 6.6 $1021 m3
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diameter between 330 C and 400 C is only 0.8 nm, the real differences in diameter might be even smaller and experimentally not resolved. Thus, it is close at hand to exclude a general failure of the model as reason for the qualitative inconsistence of the bubble size at low temperatures. Defining the reasons for differences between the TEM-analysis and the rate-theory requires to reconsider the approximations of the model. Responsible for the density of bubbles is mainly the number of nuclei at the beginning of the irradiation, which is controlled in our model by the production rate of mobile heliumvacancy clusters and the sink strength. Thus, vacancies which are not occupied by a helium atom are neglected, resulting in the underestimation of nucleation sites and consequently in a lower density of bubbles. Of course, the same reason holds for neglected SIA-clusters and intrinsic point defects as isolated carbon atoms. A more complex situation is found for the bubble growth and final size, which depend on the diffusion constant, defect production and emission rate. Concerning the emission rate examined by the two energetic descriptions of helium bubbles, we found that the results of the rate calculations applying the variable gap model get much closer to the microstructure analysis due to the increased binding energies. The final discrepancies in size probably remain due to the influence of omitted SIA. SIAs captured by bubbles obviously reduce the size of the bubbles. However, the general good qualitative agreement of the rate model is remarkable although the contribution of interstitial-type defects appears necessary to overcome the remaining quantitative discrepancies. 6. Conclusion We have presented results of the TEM microstructure analysis of Heþ/Fe3þ dual-beam irradiated EUROFER97 with respect to the nucleation of helium bubbles for different temperatures. The micrographs show a spatially homogeneous bubble distribution in the matrix at Tirr ¼ 330 C and Tirr ¼ 400 C whereas at a higher temperature of Tirr ¼ 500 C in addition to a homogeneous contribution bubbles nucleate also at dislocation lines and grain boundaries. Thus, dual beam ion irradiation successfully reproduced the transition from homogeneous to heterogeneous bubble formation with rising temperature as observed for the neutron irradiation of EUROFER97. The bubble sizes for Tirr ¼ 330 C and Tirr ¼ 400 C are distributed around to 1.2 nm, but increases for the step from Tirr ¼ 400 C to Tirr ¼ 500 C. At Tirr ¼ 500 C the size distribution of bubbles shows a maxima at 1.7 nm and bubbles up to 4 nm are observed. The density of bubbles monotonically decreases with rising temperature. For all three irradiation temperatures the swelling due to visible bubbles does not exceed 0.02%. The rate model predicts no significant contribution to the swelling of bubbles which are too small for detection by means of TEM. In addition, the findings of the microstructure analysis for the temperature dependence of bubble size and density are in good qualitative agreement with the numerical results of the kinetic rate model for the formation of helium bubbles. The rate model also predicts an increasing accumulation of helium at the grain boundaries and line dislocations with rising temperature. However, the gradient is to low to explain the experimentally observed transition from spatially homogeneous to heterogeneous nucleation. Further investigations for a suitable treatment of sinks responsible for heterogeneous nucleation are necessary. Furthermore, the extension of the rate model with the molecular dynamic based “variable gap model” for the binding energies helium bubbles improved the results for the bubble sizes. For high temperatures the calculated mean diameter gets much closer to the experimentally estimated values than the approach originally used in Ref. [16]. Even though calculated values for the diameter and
bubble density differ from the experiment, adjusting the parameters like the diffusion coefficient or including SIA defects in order to match quantitatively experimental results of homogeneously bubble nucleation appears promising. Acknowledgement € ber to The authors acknowledge the contribution of Oliver Tro the execution of the dual beam irradiation and to the sample preparation. The authors thank the Jannus staff for the irradiation experiments. This work has been carried out within the EFDA Goal Oriented Trainee Program WP12 RadEff. References [1] T. Jourdan, J.-P. Crocombette, J. Nucl. Mater 418 (2011) 98e105, http:// dx.doi.org/10.1016/j.jnucmat.2011.07.019. [2] N. Baluc, Plasma Phys. Control. Fusion 48 (2006) B165. http://stacks.iop.org/ 0741-3335/48/i¼12B/a¼S16. €slang, R. Rolli, H.-C. Schneider, J. Nucl. Mater [3] E. Materna-Morris, A. Mo 386e388 (2009) 422e425, http://dx.doi.org/10.1016/j.jnucmat.2008.12.157. [4] E. Gaganidze, C. Dethloff, O.J. Weiß, V. Svetukhin, M. Tikhonchev, J. Aktaa, Eff. Radiat. Nucl. Mater. 25 (2012) 123e142, http://dx.doi.org/10.1520/stp103972. [5] O.J. Weiß, E. Gaganidze, J. Aktaa, J. Nucl. Mater 426 (2012) 52e58, http:// dx.doi.org/10.1016/j.jnucmat.2012.03.027. € slang, E. Materna-Morris, J. Nucl. Mater 453 (2014) [6] M. Klimenkov, A. Mo 54e59, http://dx.doi.org/10.1016/j.jnucmat.2014.05.001. http://www. sciencedirect.com/science/article/pii/S0022311514002657. [7] Z. Tong, Y. Dai, J. Nucl. Mater 398 (2010) 43e48, http://dx.doi.org/10.1016/ j.jnucmat.2009.10.008. http://dx.doi.org/10.1016/j.jnucmat.2009.10.008. [8] T. Zhang, C. Vieh, K. Wang, Y. Dai, J. Nucl. Mater 450 (2014) 48e53, http:// dx.doi.org/10.1016/j.jnucmat.2013.12.007. http://dx.doi.org/10.1016/j. jnucmat.2013.12.007. [9] C.-C. Fu, F. Willaime, Phys. Rev. B 72 (2005), http://dx.doi.org/10.1103/ physrevb.72.064117. €ublin, J. Nucl. Mater 386e388 (2009) 360e362, http:// [10] G. Lucas, R. Scha dx.doi.org/10.1016/j.jnucmat.2008.12.128. [11] K. Morishita, R. Sugano, B. Wirth, T. Diaz de la Rubia, Nucl. Instrum. Meth. B 202 (2003) 76e81, http://dx.doi.org/10.1016/s0168-583x(02)01832-3. [12] A. Caro, J. Hetherly, A. Stukowski, M. Caro, E. Martinez, S. Srivilliputhur, L. Zepeda-Ruiz, M. Nastasi, J. Nucl. Mater 418 (2011) 261e268, http:// dx.doi.org/10.1016/j.jnucmat.2011.07.010. [13] R. Stoller, S. Golubov, C. Domain, C. Becquart, J. Nucl. Mater 382 (2008) 77e90, http://dx.doi.org/10.1016/j.jnucmat.2008.08.047. http://dx.doi.org/10.1016/j. jnucmat.2008.08.047. [14] T. Jourdan, G. Bencteux, G. Adjanor, J. Nucl. Mater 444 (2014) 298e313, http:// dx.doi.org/10.1016/j.jnucmat.2013.10.009. [15] D. Xu, B.D. Wirth, J. Nucl. Mater 403 (2010) 184e190, http://dx.doi.org/ 10.1016/j.jnucmat.2010.06.025. http://dx.doi.org/10.1016/j.jnucmat.2010.06. 025. [16] C. Dethloff, E. Gaganidze, V.V. Svetukhin, J. Aktaa, J. Nucl. Mater 426 (2012) 287e297, http://dx.doi.org/10.1016/j.jnucmat.2011.12.025. [17] E. Gaganidze, H.-C. Schneider, B. Dafferner, J. Aktaa, J. Nucl. Mater. 355 (2006) 83e88, http://dx.doi.org/10.1016/j.jnucmat.2006.04.014. http://dx.doi.org/10. 1016/j.jnucmat.2006.04.014. €gler, G. Müller, A. Ulbricht, [18] C. Heintze, F. Bergner, M. Hernandez-Mayoral, R. Ko J. Nucl. Mater. 470 (2016) 258e267, http://dx.doi.org/10.1016/j.jnucmat.2015.12.041. http://dx.doi.org/10.1016/j.jnucmat.2015.12.041. ^tre, D. Brimbal, [19] L. Beck, Y. Serruys, S. Miro, P. Trocellier, E. Bordas, F. Lepre T. Loussouarn, H. Martin, S. Vaubaillon, et al., J. Mater. Res. (2015) 1e12, http://dx.doi.org/10.1557/jmr.2014.414. ~ [20] Y. Serruys, P. Trocellier, S. Miro, E. Bordas, M. Ruault, O. KaAtasov, S. Henry, O. Leseigneur, T. Bonnaillie, S. Pellegrino, S. Vaubaillon, D. Uriot, J. Nucl. Mater 386e388 (2009) 967e970. http://www.sciencedirect.com/science/article/pii/ S0022311508010301. http://dx.doi.org/10.1016/j.jnucmat.2008.12.262 (proceedings of the Thirteenth International Conference on Fusion Reactor Materials). [21] J.F. Ziegler, Nucl. Instrum. Meth. B 219e220 (2004) 1027e1036. http://www. sciencedirect.com/science/article/pii/S0168583X04002587. http://dx.doi.org/ 10.1016/j.nimb.2004.01.208 (proceedings of the Sixteenth International Conference on Ion Beam Analysis). [22] D. Brimbal, L. Beck, O. Troeber, E. Gaganidze, P. Trocellier, J. Aktaa, R. Lindau, J. Nucl. Mater 465 (2015) 236e244, http://dx.doi.org/10.1016/j.jnucmat.2015.05.045. http://www.sciencedirect.com/science/article/pii/ S0022311515300246. €slang, J. Nucl. Mater 417 (2011) [23] M. Klimenkov, E. Materna-Morris, A. Mo 124e126, http://dx.doi.org/10.1016/j.jnucmat.2010.12.261. http://www. sciencedirect.com/science/article/pii/S0022311510011086 (proceedings of ICFRM-14). [24] C. Dethloff, E. Gaganidze, J. Aktaa, J. Nucl. Mater. 454 (2014) 323e331, http://
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TEM study and modeling of bubble formation in dual-beam Heþ/Fe3þ ion irradiated EUROFER97 B. Kaiser a, *, C. Dethloff a, E. Gaganidze a, D. Brimbal b, M. Payet b, P. Trocellier b, L. Beck b, J. Aktaa a a b
Karlsruhe Institute of Technology, Institute for Applied Materials (IAM), P.B. 3410, 76021, Karlsruhe, Germany CEA, DEN, Service de Recherches de M etallurgie Physique, Laboratoire JANNUS, F-91191, Gif-sur-Yvette, France
h i g h l i g h t s Investigation of He and dpa effects in EUROFER97 by irradiation with He/Fe-ions at 330 C, 400 C and 500 C. TEM analysis of the size distribution of helium bubbles as a function of irradiation temperature. Modeling of helium bubble formation with a kinetic rate model. Influence of two different thermodynamic descriptions of helium bubbles on the rate model. Comparison between experimental and numerical results.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 June 2016 Received in revised form 30 September 2016 Accepted 13 November 2016 Available online 15 November 2016
The Reduced Activation Ferritic/Martensitic (RAFM) steels are promising structural materials for the first wall and blanket components of future fusion reactors. To obtain further insight into the temperature dependence of helium effects induced by 14 MeV energy neutrons under fusion like conditions, EUROFER97 was exposed to Heþ/Fe3þ dual-beam ion irradiation at the JANNUS laboratory at Saclay. The implantation was carried out at temperatures of 330 C, 400 C and 500 C and induced a damage and helium concentration up to 26 dpa and 450 appm He, respectively. TEM microstructure analysis indicates a spatially homogeneous distribution of helium bubbles at 330 C and 400 C whereas a coexistence of homogeneous and heterogeneous nucleation of bubbles is found at 500 C. An increasing mean bubble diameter and decreasing concentration of bubbles with rising irradiation temperature, as predicted by numerical results of a kinetic rate model for diffusion governed homogeneous nucleation of helium bubbles, are mostly confirmed by the irradiation experiment. Furthermore, within the rate model two approaches for the determination of the thermodynamic properties of helium filled voids in a-iron are applied. With respect to the final bubble size distribution, the commonly used surface energy of a void in the iron matrix is compared to the “variable gap model” of [1], J. Nucl. Mater. 418 (2011), which includes additionally the interaction between the helium atoms themselves, the energy at the helium-iron interface and the elastic deformation of the iron matrix. © 2016 Published by Elsevier B.V.
Keywords: Radiation effects Ion irradiation Cluster dynamics Fusion Helium bubbles RAFM steels
1. Introduction The reduced activation ferritic-martensitic steels are considered as potential structural materials for the first wall and blanket components of future fusion reactors due to their promising low irradiation induced swelling and well balanced physical and
* Corresponding author. E-mail address: [email protected] (B. Kaiser). http://dx.doi.org/10.1016/j.jnucmat.2016.11.014 0022-3115/© 2016 Published by Elsevier B.V.
thermo-mechanical properties [2]. Nevertheless, when exposed for several years to a high flux of 14 MeV neutrons in fusion reactors these materials are still expected to sustain severe damage. Accordingly, a detailed knowledge of the evolution of microstructure damage is crucial for the estimation of the mechanical degradation of structural components and the further development of accurate design limits. Due to the absence of facilities with fusion reactor relevant neutron spectra substitutional irradiation programs are executed to simulate the mechanisms of void nucleation under simultaneous
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production of atomic displacement cascades and helium. Aside from the here discussed dual-beam irradiation with iron and helium ions, helium effects in EUROFER97 were studied in the fission neutron irradiation programs SPICE and ARBOR by using boron doping technique [3e6] and in spallation proton irradiation programs STIP-II and STIP-III [7,8] including also protons and helium ions in addition to neutrons. For the modeling of helium effects in a-iron a couple of theoretical methods is applied, providing insight on different time and length scales. On atomic scales density functional theory [9] and molecular dynamics [10,11] provide insight into properties of interstitial helium and small helium-vacancy clusters embedded in an iron-matrix, like migration and binding energies, and support the transfer to macroscopic thermodynamic description of helium bubbles [1,12]. Based on the knowledge of these physical quantities, usually object kinetic Monte Carlo calculations, kinetic rate theory or the Fokker-Planck equation are used to trace the evolution of microstructural damage on times and length scales typical for radiation experiments [4,13e16]. At the moment, the damage of 14 MeV neutrons as well as its reasonable imitation are not sufficiently understood and of great importance for the development and verification of suitable materials. Particularly, it was found that helium in EUROFER97 is responsible for embrittlement [17] and for an increase of the nanohardness [18]. Accordingly, the temperature dependence of helium bubble formation induced by Heþ/Fe3þ dual-beam ion irradiation of EUROFER97 is analyzed with TEM investigations of the irradiated samples. Beside the experimental part, the understanding of helium effects is supported by a kinetic rate model for diffusion governed homogeneous nucleation of helium bubbles. In this manner, we will evaluate the qualitative and quantitative capabilities of the model for nucleation of helium bubbles in EUROFER97 in order to assess the need for extending the model or adjusting parameters. 2. Dual-beam ion irradiation and sample preparation €hler The 25 mm thick EUROFER97 plates were produced by Bo Austria GmbH. The final heat treatment applied by the manufacturer includes a normalization at 980 C for 0.5 h and tempering at 760 C for 1.5 h. For the ion irradiation discs of 3 mm diameter and approximately 90 mm thickness were prepared. The irradiation with 1.2 MeV Heþ and 3.0 MeV Fe3þ ions was performed at the JANNUS Facility at Saclay [19,20] at temperatures of 330 C, 400 C and 500 C. We achieved a broadened implantation profile of helium by inserting aluminum energy degraders into the beam. In order to choose ion doses which lead to damage and helium
Fig. 1. SRIM calculation of the damage and helium concentration profile achieved by 6 h of 1.2 MeV Heþ and 3.0 MeV Fe3þ ion irradiation with respective doses of 3.2$1016 Fe3þ cm2 and 1.1$1016 Heþ cm2. The desired damage and helium content of approximately 26 dpa and 450 appm He is achieved at implantation depths in between 500 nm and 800 nm.
concentration ratios similar to conditions expected in future fusion reactors, SRIM [21] calculations were performed. Fig. 1 displays the expected depth profiles of damage and helium concentration for the applied doses of 3.2$1016 Fe3þ cm2 and 1.1$1016 Heþ cm2. Thus, fusion like conditions are obtained at depths between 500 nm and 800 nm from the surface where the applied fluxes lead to 26 dpa/450 appm He and a helium-dpa ratio of approximately 15 appm He/dpa. TEM samples were prepared via electrolytic-polishing or focused-ion-beam (FIB) machining. In particular, attention has to be paid to the electrolytic-polishing procedure for the creation of a thin film at a depth with the desired damage and helium content. Accordingly, the disks were thinned separately from the front and back side in an asymmetric manner. A more detailed description of the specimen preparation, the irradiation experiment and the TEM sample preparation can be found in Ref. [22]. The preparation of TEM-samples out of the uppermost irradiated 1 mm thick layer with asymmetric electrolytic-polishing and FIB machining may fail due to thinning at the wrong depth and the additional ion irradiation, respectively. In order to avoid features in the microstructure introduced by the sample preparation for all irradiation temperatures samples were produced in both ways. The observation of the same qualitative results in case of both preparation methods therefore ensures that they do not dependent on the preparation method. Nevertheless, for the subsequent detailed quantitative evaluation of size and density of helium bubbles we used FIB samples to guarantee for the results depths of 500 nme800 nm. Thus, we assure a damage and helium content of 26 dpa and 450 appm He, respectively. 3. TEM-analysis of the helium bubble distribution The irradiation damage is analyzed via TEM bright-field technique to obtain size distributions of voids. As well accepted, voids are identified by changing of their contrast from a bright spot with a dark fresnel fringe in under-focus condition to a dark spot with a bright fresnel fringe in over-focus condition. Under the assumption of spherical voids, their size is determined by measuring the corresponding diameter in an under-focused micrograph. The subsequent results and micrographs stem all from samples prepared with FIB machining and refer to the region of interest at a depth in between 500 nm and 800 nm, as marked in Fig. 1. Besides cavities other defects of the microstructure as dislocation loops and precipitates characterize the irradiation damage. The in depth investigation of these types of defects will be subject of future publications. With respect to the application of rate-model we limit the present TEM study to the size and density of helium bubbles. For characteristics of dislocations necessary for modeling we will refer on existing studies on EUROFER97 for dual-beam irradiation [22] and neutron irradiation [5,23,24]. In the case of Tirr ¼ 330 C we find homogeneously distributed bubbles as displayed in the two micrographs of Fig. 2. Similar observations are obtained for specimens irradiated at a temperature of Tirr ¼ 400 C, shown in Fig. 3. The bubbles are distributed homogeneously in the matrix without large variations in size. Remarkably, rising the irradiation temperature from 330 C to 400 C does not qualitatively influence the formation of helium bubbles. In contrast, further increasing the irradiation temperature up to Tirr ¼ 500 C strongly influences the nucleation of bubbles, concerning their size as well as spatial distribution. In contrast to the low temperature irradiation a heterogeneous nucleation is observed at Tirr ¼ 500 C. A major part of the bubbles is still homogeneously distributed in the matrix of the steel, as shown in Fig. 4. In addition, a remarkable fraction of the bubbles are now nucleated at different extended microstructural sinks such as grain
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Fig. 2. Homogeneously distributed bubbles in EUROFER97 irradiated at Tirr ¼ 330 C up to 26 dpa/450 appm He and defocused to ±0.5 mm.
Fig. 3. Homogeneously distributed bubbles in EUROFER97 irradiated at Tirr ¼ 400 C up to 26 dpa/450 appm He and defocused to ±0.5 mm.
Fig. 4. Homogeneously distributed bubbles in EUROFER97 irradiated at Tirr ¼ 500 C up to 26 dpa/450 appm He and defocused to ±1.0 mm.
boundaries, dislocation lines and precipitates, see Fig. 5. Note, that Fig. 5 is not representative of the frequency or relative contribution of each sink type. At Tirr ¼ 500 C line dislocations and grain boundaries are found to be the major sink types whereas precipitates reveal only a minor role. Fig. 6 displays the concentration histograms of bubbles as a function of their diameter for the three applied irradiation temperatures. Representative micrographs prepared via FIB machining and used for measuring the diameter of bubbles are shown in Fig 2e4 and are defocused to ±0.5 mm for 330 C and 400 C and to ±1.0 mm in the case of 500 C. The sample thickness was determined by electron energy loss spectroscopy. Bubble diameters for 330 C are plotted in Fig. 6a and are narrow and symmetrically distributed with a mean bubble diameter of 1.3 nm and a total bubble density of r ¼ 2.0$1023 m3. For an increased temperature Tirr ¼ 400 C the quantitative analysis results in a total bubble concentration of r ¼ 9.4$1022 m3 whereas the shape and the mean diameter of 1.2 nm remain unchanged compared to 330 C irradiation within the experimental uncertainties, as shown in Fig. 6b.
Finally, in case of the highest applied irradiation temperature Tirr ¼ 500 C in Fig. 6c the mean diameter increases to 2.0 nm and the total density r drops to 2.5$1022 m3. Interestingly, also the shape of the distribution becomes broader and reveals a tail including bubbles with diameters up to 4.0 nm. The diameter distribution displayed for 500 C contains only spatially homogeneous distributed bubbles nucleated in the matrix well separated from any heterogeneous features. Bubbles at grain boundaries, line dislocations or precipitates indicated only minor differences in size. For all three irradiation temperatures the swelling does not exceed 0.02%. Of course, the estimated swelling with help of TEM relies only on visible bubbles. Nevertheless, with the below presented rate model we found that bubbles smaller then 0.5 nm do not have a large enough density to affect swelling significantly. These observations are in agreement with the prediction given in Ref. [25], which correlates the bubble size and the appearance of either heterogeneous or homogeneous distributions to the temperature dependence of the helium diffusion. Thus, our findings can be explained as follows. The homogeneous bubble distribution
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Fig. 5. Heterogeneous nucleation of bubbles at a) precipitates, b) dislocation lines and c) grain boundaries observed in EUROFER97 at Tirr ¼ 500 C and defocused to 2.5 mm. Note, that the figure does not represent the general proportion of helium absorption by each sink type. Line dislocations and grain boundaries are found to be the major sink types whereas precipitates revealed only a minor role.
homogeneous nucleation. Therefore the concentration of bubbles in the grain decreases at higher irradiation temperature. Concerning the impact of the elevated temperature of Tirr ¼ 500 C on the size of the bubbles we discriminate two features. First, the shift of the peak bubble size to larger bubble radii is also caused by the increased mobility of helium and vacancies, leading to a higher growth rate of the bubble radii. Second, the small contribution with a roughly constant concentration for diameters between 2.5 nm and 4.0 nm indicate the onset of a ripening process i.e. the redistribution from small to large bubbles. A meaningful comparison with the observations on boron doped specimens of the neutron irradiation program SPICE is difficult because besides the primarily homogeneous distribution of boron also the formation of BN and M23(C,B)6 precipitates was found [26]. It was further analyzed that the void nucleation is critically influenced by the vicinity of these precipitates [27]. Indeed the bubble diameters for SPICE [6] of up to 15 nm for 350 C at similar irradiation conditions of 16.3 dpa and 415 appm helium content are much larger than the size found in the present dualbeam irradiation experiment. In case of 450 C the mean diameter of bubbles in boron doped neutron irradiated specimens is approximately 4 nm and comes closer to the 2.0 nm of the present Heþ/Fe3þ dual-beam experiment at 500 C, but the bubble size distribution is still much broader covering bubbles up to 10 nm. In addition to the influence of BN and M23(C,B)6 particles, a further reason for the disagreement in the results between the two experiments could be found in the non constant helium production in SPICE, providing most helium (95%) after only 1 dpa in the first two months and the subsequent ripening of bubbles [16]. Furthermore, the damage and helium production rates differ up to four orders of magnitude, often interpreted as a temperature shift. In the SPICE experiment, a transition from homogeneous to a preferred heterogeneous bubble nucleation at grain boundaries and dislocation lines is observed for increasing the irradiation temperature from 350 C of 450 C [6,28]. In comparison, rising the irradiation temperature from 400 C to 500 C leads in the present dual-beam irradiation only to the coexistence between homogeneous and heterogeneous distribution of bubbles. Of course, the precise estimation of the temperature shift between the neutron and ion irradiation with the help of this rather qualitative feature requires more irradiation experiments in the temperature interval of the transition. However, in contrast to the dominating nucleation at dislocations and grain boundaries for neutrons at 450 C, we interprete the observed coexistence in the case of ions at 500 C as an early stage of the tendency for heterogeneous nucleation with rising irradiation temperature. Therefore, we indicate a temperature shift larger than 50 C in agreement with reference [22]. 4. Kinetic rate model
Fig. 6. The figure displays the bubble size distribution at implantation depths from 500 nm to 800 nm for a): Tirr ¼ 330Tirr¼ 330 C with micrographs defocused to ±0.5 mm±0:5, b): Tirr ¼ 400 C with micrographs defocused to ±0.5 mm and c): Tirr ¼ 500 C with micrographs defocused to ±1.0 mm. The data of subfigure c) contains only bubbles distributed homogeneously in the matrix.
at low temperature is enabled by the lower helium and vacancies mobility. The lower mobility increases the number of sources for nucleation in the matrix and depresses the diffusion to grain boundaries but also slows down the growth of bubbles. On the contrary, during the incubation at high temperatures the introduced helium and vacancies exhibit an increasing tendency to reach grain boundaries and thus decrease the possibilities for
The applied rate model is an extended version of that described in Ref. [16] and was originally developed for bubble formation under neutron irradiation in SPICE and ARBOR experiments and future fusion reactors. It estimates size and density distribution of homogeneous distributed bubbles due to high helium and vacancy production during irradiation. Helium atoms diffuse rapidly through the crystal, but are trapped effectively by microstructure defects e.g. vacancies, voids, dislocations and grain boundaries, resulting either in spatially homogeneous or heterogeneous bubble formation. Under irradiation the occupation of substitutional lattice sites by helium is preferred due to the high binding energy and the elevated concentration of vacancies. Subsequently, as a result of diffusive motion, these helium-vacancy clusters join and form stable nuclei for spatially homogeneous distributed bubble formation by the subsequent accumulation of helium and vacancies. In
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comparison with vacancies we consider SIA-defects to play a minor role in the nucleation process due to the high mobility of SIA. The expected fast diffusion leads to the formation of the interstitial defect clusters which evolve into the dislocation loops. The experimentally observed helium bubble microstructure however does not give any hint on the spatial correlation with the dislocation loop microstructure. Note also that the present model does not take into account the influence of solid solution carbon atoms or SIAs in the process of bubble nucleation and growth. The model is centered on helium-enhanced bubble nucleation and growth. In the case of heterogeneous nucleation, rate-theory is commonly not suited for the determination of bubble growth and the respective bubble size distribution. Thus, the model only accounts for the fraction of mobile defects absorbed by sinks and thus contributing to heterogeneous nucleation of bubbles. As usually, these sinks are described by averaged effective sink terms [29,30]. The model is based on the emission and absorption of mobile unit clusters from bubbles consisting of i mobile unit clusters and traces the corresponding concentrations Ci. The probability of these reactions is determined by the rate coefficients ki for absorbing and gi for emitting single mobile unit clusters. One of the approximations of this model is the assumption of a constant helium-vacancy ratio xHe accounting only for void configurations of i helium atoms and i/xHe vacancies and thus consequently leads to HeV1/xHe as mobile unit cluster. The equations of motion for cluster concentrations Ci read as follows:
X X vC1 ¼ GHe LSinks ki C1 Ci þ gi Ci ; vt
(1a)
vCi ¼ ½ki C1 þ gi Ci þ giþ1 Ciþ1 þ ki1 C1 Ci1 : vt
(1b)
i1
i2
The concentration of the mobile cluster type C1 is determined by equation (1a) which includes the helium generation rate GHe(t), losses at sinks LSinks and the absorption and emission of mobile units by clusters. The evolution of clusters of size i 2 described by equation (1b) couples with the concentration for clusters of size i1 and iþ1 via absorption and emission of mobile units. Therefore, in addition to the time independent absorption and emission coefficients the evolution of Ci(t) is strongly influenced by the concentration of the mobile type C1(t). The sink term Lsink accounts for the accumulation of helium at grain boundaries. Of course, in this description the individual character of helium atoms and vacancies is lost. However, we assume that due to the supersaturation of vacancies during the irradiation most of the helium atoms are trapped by vacancies and the subsequent dynamics are well approximated by the accumulation of mobile vacancy-helium clusters. The concentration distribution function of bubbles as a function of the bubble radius CðRÞ is obtained from the discrete quantity Ci as follows.
CðRÞ ¼ CiðRÞ
di : dR
(2)
Properties like the bubble mean and peak size of the distribution C(R) as well as the total concentration r obtained by the integration over R can be directly compared with those of the histograms of the TEM analysis plotted in Fig. 6. Note that in order to convert C(R) into a histogram one determines bin heights either approximately by the multiplication of C(R) with the bin width DR or exactly by the integration of C(R) over the respective bin intervals. The numerical solution of the differential equation (1) is calculated with the python wrappers of the odespy package [31] for the odepack algorithms [32] which can profit from the sparse Jacobi matrix.
63
In the next step, the coefficients ki and gi must be defined. For the determination of ki a diffusion controlled absorption is assumed, implying a capture process if a mobile type hits a nonmobile cluster. Thus, the rate ki depends obviously on the cluster eff radius Ri and the helium diffusion constant DHe . eff
ki ¼ 4pRi DHe ðTirr Þ:
(3)
eff DHe
For we apply a temperature dependent formula extracted from Monte-Carlo results by Ref. [33] for the substitutional helium diffusion via the divacancy-mechanism. As usually, the emission probability is derived from a steady state argumentation resulting in a relation between the equilibrium cluster concentration and the kinetic coefficients ki and gi. Using thermodynamic expressions for the equilibrium concentrations introduces the free binding energy Fib of the mobile type to a cluster in the formula for gi, explicitly written as
"
1
gi ¼ ki U
# Fib exp : kB Tirr
(4)
Here U denotes the atomic volume and kB the Boltzmann constant. The losses of mobile clusters to sinks are defined by 2 Lsink ¼ Deff He C1 ðtÞ ksink ;
(5)
with the total sink strength k2sink . Usually, k2sink is the sum of the respective sink strengths for different sink types as grain boundaries, dislocation lines, dislocation loops and precipitates. With respect to the present results of the microstructure analysis we omit precipitates and dislocation loops as possible sink sites in the model. As mentioned, we observe mainly heterogeneous bubble nucleation at grain boundaries and dislocation lines. Also, reference [24] reported that in EUROFER97 precipitates preferentially occur at grain boundaries. Consequently, they do not represent an additional interface to the defects in the matrix, reducing further the need to include precipitates as sinks. Besides, dislocation loops sometimes lead to heterogeneous formation of bubbles. In model alloys dislocation loops caused a spatially heterogeneous distribution of bubbles [34,35] due to a preferential bubble formation at the loops. Nevertheless, the latter was experimentally not observed in the present study even at an elevated irradiation temperature of 500 C as well as in other studies on EUROFER97 [6,22,23]. The derivation of expressions for diffusion controlled sinks was intensively studied in the past and formulas can be found for all kinds of sinks [29,30]. In particular we will apply for the diffusion controlled sink strength of grain boundaries k2gb and dislocation lines k2dl the following formulas [29]:
. k2gb ¼ 15 R2gb ;
(6a)
k2dl ¼ Z rdl :
(6b)
Input parameters for the sink strengths are the grain radius Rgb, the dislocation line density rdl and the bias factor Z. Experimental measured values for the grain size of approximately 15 mm for EUROFER97 are found in literature [36]. However, this value concerns the size of former austenitic grains and does not represent the lath structure of the ferritic microstructure. Plausible values for the dislocation line density are 9.0$1013 m2 for unirradiated EUROFER97 and 6.7$1014 m2 in case of the present dual beam irradiation at 400 C, provided by Ref. [22]. Even though [29], estimated Z z 1, they mentioned the challenges in the derivation of the dislocation bias factor and outlined that its experimental determination of the
64
B. Kaiser et al. / Journal of Nuclear Materials 484 (2017) 59e67
dislocation bias appears reasonable. Therefore, we continue the discussion of a suitable parameter set for the losses at sinks at section 5 with respect to the results of the model in comparison to the TEM-analysis. The most critical assumption of the rate model is the constant helium-vacancy ratio xHe. As a consequence we only account for one mobile cluster type HeV1/xHe and omit elementary types as single helium atoms and vacancies as well as other small clusters. Thus, the possible pathways for nucleation schemes are strongly reduced and to a certain degree already defined with the choice of xHe. For example, bubble to void transformations can not be described by the model. Nevertheless, the usage of the model is justified in particular by the vacancy supersaturation due to the high dpa in case of the dual beam and neutron irradiation. Under this condition helium atoms are most likely trapped by vacancies and form clusters similar to the mobile unit cluster introduced by the present rate model. For the results discussed in this paper a helium-vacancy ratio xHe ¼ 1 was applied, because molecular dynamics predict a minimum close to the formation energy for bubbles of these configurations [37]. The requested application of the kinetic rate model consists of predicting tendencies whether bubbles occur spatially homogeneous or heterogeneous distributed. In the case of a homogeneous bubble formation in the matrix the size distribution of bubbles additionally is determined. Of course, further types of radiation damage like self interstitial clusters and dislocation loops evolve under irradiation and can influence the bubble formation and even more the mechanical properties. However, for an increased number of effect types the kinetic rate equations become computationally demanding in particular for the large bubbles and long irradiation duration of several years. Moreover, increasing the number of defect types require many input parameters and, consequently, the adjustment of the model on EUROFER97 will be more prone to errors. From this point of view, the here applied approach including effective parameters appears appropriate for the defined purposes. 4.1. Thermodynamic description of helium bubbles So far the macroscopic rate equation approach depends on microscopic attributes as the binding energy Fib and the void radius Ri. In Ref. [16] the bubble radius was directly linked to the spherical volume created by the corresponding number of vacancies. Thus, elastic deformation of the surrounding matrix due to over- or under-pressurized insert gas is neglected. With the knowledge of the void radius Ri the binding energy of the mobile type to a cluster was calculated by the difference in the surface energy when lowering the cluster size from i to i1. In this manner, one can straightforwardly determine the absorption and emission coefficient ki and gi with respect to equations (3) and (4). Another possibility to receive the quantities necessary for ki and gi was introduced by Ref. [1], denoted as “variable gap model” [1]. It is based on the partitioning of the free formation energy of a bubble in contributions for helium, the iron matrix and a helium-iron interface term as suggested by Ref. [38]. In order to provide a suitable representation of the interface term a spatial gap between helium and iron is introduced. Subsequently, these terms are parametrized via molecular dynamics as a function of the number of helium atoms and vacancies and at first also depend on the radii of the helium gas and the void. The equilibrium state of a helium filled void is found by minimizing the free formation energy with respect to the two radii. In spite of the reliable results of the variable gap model for medium to large cluster sizes the involved assumption of a volume-surface ratio for spheres usually is not adaptable for very small clusters. Therefore we have combined the formation energy calculated for fluid helium by the variable gap
model with the results of density functional theory for small clusters of Ref. [39,40]. In comparison to the thermodynamic model used by Ref. [16] in Ref. [16] the “variable gap model” accounts for several additional aspects. Obviously, the formation energy is changed since the “variable gap model” also includes interaction between the helium atoms themselves and between helium atoms and the iron matrix. In addition, the free energy term representing the iron matrix also consists of an elastic deformation term besides the surface term. The elastic deformation is in competition with the helium pressure and therefore can induce deviations compared with the former model in which the cluster radius depends only on the number of vacancies. 4.2. Comparison of microscopic parameters In order to evaluate the impact of the variable gap model on the results of the rate-theory we first discuss in which manner the cluster radius and the binding energies are quantitatively affected. Since the present rate model is restricted to constant heliumvacancy ratios only the corresponding subsets of the cluster configuration are discussed, even though the variable gap model provides quantities for all cluster configurations. As discussed in Ref. [16] the helium-vacancy ratio is assumed to lie within the interval 0.5
B. Kaiser et al. / Journal of Nuclear Materials 484 (2017) 59e67
beam experiment. Also, the numerical results indicated only a negligible absorption of helium at grain boundaries. Although the experimental part of the present study does not include a quantitative estimation of the contribution of each sink type, the measured size distributions allow a rough approximation of the fractions of helium captured by homogeneously nucleated bubbles in the matrix or absorbed by sinks at Tirr ¼ 500 C. We assume that at Tirr ¼ 300 C the measured size distribution of Fig. 6a for homogeneously distributed bubbles includes almost all of the helium which is represented by the volume fraction of these bubbles. In the distribution of Fig. 6c at Tirr ¼ 500 C the volume occupied by homogeneously distributed bubbles is reduced to 55% of the volume of bubbles at Tirr ¼ 300 C. We associate the missing volume with helium absorbed by line dislocations and grain boundaries. Therefore the fraction contributing to heterogeneous nucleation at Tirr ¼ 500 C can be approximated by 45%, which agrees qualitatively with the observed coexistence between heterogeneous and homogeneous nucleation. An improvement with respect to the experimental observation requires an increased sink strength of the grain boundaries and a decreased sink strength of dislocation lines. For this reason we set the grain size to Rgb ¼ 2.5 mm to get closer to the ferritic lath structure and chose a dislocation line density of unirradiated EUROFER97 rdl ¼ 9.0$1013 m2 [22]. Note that the dislocation line density of unirradiated EUROFER97 is a more adequate value, since nucleation occurs during the first seconds of the irradiation at all temperatures. Thus, the initial dislocation line density of unirradiated EUROFER97 obviously has a stronger influence on the appearance of either homogeneous or heterogeneous nucleation than the final density of dislocation lines after the irradiation. However, further studies for a suitable parametrization of the bias factor did not accomplish a satisfying description of the transition between homogeneous and heterogeneous bubble nucleation. Exemplary, the bias factor for the mobile clusters is reduced to 0.025. With this choice of parameters, we obtain an increasing accumulation of mobile cluster types at dislocation lines and grain boundaries with rising irradiation temperature. The relative amount of helium captured by sinks evolves from 1.8% at 330 C to 4.0% at 500 C. Although the final contribution to heterogeneous nucleation at 300 C seems reasonable on the base of the TEM analysis, the sink strength at 500 C is strongly underestimated. However, adjusting a higher bias factor closer to predictions of literature aiming at explanation of heterogeneous nucleation at 500 C leads simultaneously to an overestimation of heterogeneous nucleation at lower temperatures. Further investigations of a reasonable treatment of sinks in the rate model is dedicated to future work. For the following results of size distributions of homogeneously distributed bubbles determined with the rate model we apply Rgb ¼ 2.5 mm, rdl ¼ 9.0$1013 m2 and Zeff ¼ 0.025, bearing in mind that in particular at 500 C the helium absorption by sinks is underestimated. Results of the rate model for the cluster concentration C(d) as a function of the bubble diameter d are displayed in Fig. 7. The colors of the curves represent the temperatures 330 C (red), 400 C (blue) and 500 C (green) during irradiation. The curves clearly exhibit an increasing bubble diameter the higher the temperature is. In order to provide a comparison between the different thermodynamic descriptions of the helium bubbles solid lines show the results using the “variable gap model” whereas dotted lines denote results with binding energies and void radii following Ref. [16]. In spite of the marginal deviation in the bubble sizes between the two approaches at 330 C, large differences occur at higher temperatures. The radii corresponding to the maximum of the distribution shrinks from 5.2 nm to 4.6 nm at 400 C (dotted and solid blue curve) and from 10.3 nm to 5.7 nm at 500 C (dotted and
65
Fig. 7. Concentration distribution of bubbles as a function of the bubble diameter C(d) calculated with the rate-model for dual beam irradiation at 330 C, 400 C and 500 C. Results using the variable gap model introduced by Ref. [1] are plotted with solid lines. The dotted lines correspond to the results obtained using binding energies as defined by Ref. [16].
solid green curve) as indicated by the arrows in Fig. 7. As discussed above, in case of the variable gap model the emission of mobile unit cluster types is reduced due to the increased binding energy. One has to recall the coupling of the concentration of clusters with the concentration of mobile unit clusters in equation (1) as well as the size dependence of the emission and absorption coefficients. A high emission probability from smaller clusters together with the increasing capture rate of large clusters leads effectively to the redistribution of mobile unit clusters from small to larger clusters. Thus, a decreased emission suppresses this redistribution process and consequently leads to smaller clusters. This explanation is further confirmed at Tirr ¼ 330 C where, when emission becomes negligible small, the rate model results in almost identical bubble distributions for both energetic descriptions of bubbles. Recalling equation (1a) for the mobile clusters we see that they are generated by the emission from i 2 clusters and the ion-implantation rate. Therefore, the rate model predicts in case of the present ion irradiation experiment at low temperature that the bubble growth is almost exclusively determined by the implantation rate and redistribution processes play a minor role. Finally, we compare the predictions of the rate model with the observations of the microstructure analysis. All values concerning the size of bubbles obtained via numerics or TEM are summarized in Table 1. They obviously reveal the similar qualitative temperature dependence for bubble size and density, but still disagree in absolute values. In particular, the maxima of the calculated bubble distributions are found at larger diameters than in the histograms in Fig. 6 of the TEM analysis whereas the calculated densities are too small. The discrepancy in bubble sizes of the rate model was already reported in Ref. [16] in the case of neutron irradiated samples. Also, the model predicts that the bubble diameter increases in the whole applied temperature interval. Instead, we found this behavior experimentally only for rising the temperature from 400 C to 500 C. Bearing in mind that the bubble sizes are overestimated by the rate model and that the predicted change in
Table 1 Summary of results for the helium bubble peak diameter and total concentrations at different irradiation temperatures. The listed numerical results of the rate equations were obtained using binding energies of the variable gap model [1]. Tirr
TEM analysis b d
330 C 400 C 500 C
1.2 nm 1.2 nm 1.7 nm
Rate-theory
r 23
3
2.0$10 m 9.4$1022 m3 2.5$1022 m3
b d
r
3.8 nm 4.6 nm 5.7 nm
2.2 $1022 m3 1.3 $1022 m3 6.6 $1021 m3
66
B. Kaiser et al. / Journal of Nuclear Materials 484 (2017) 59e67
diameter between 330 C and 400 C is only 0.8 nm, the real differences in diameter might be even smaller and experimentally not resolved. Thus, it is close at hand to exclude a general failure of the model as reason for the qualitative inconsistence of the bubble size at low temperatures. Defining the reasons for differences between the TEM-analysis and the rate-theory requires to reconsider the approximations of the model. Responsible for the density of bubbles is mainly the number of nuclei at the beginning of the irradiation, which is controlled in our model by the production rate of mobile heliumvacancy clusters and the sink strength. Thus, vacancies which are not occupied by a helium atom are neglected, resulting in the underestimation of nucleation sites and consequently in a lower density of bubbles. Of course, the same reason holds for neglected SIA-clusters and intrinsic point defects as isolated carbon atoms. A more complex situation is found for the bubble growth and final size, which depend on the diffusion constant, defect production and emission rate. Concerning the emission rate examined by the two energetic descriptions of helium bubbles, we found that the results of the rate calculations applying the variable gap model get much closer to the microstructure analysis due to the increased binding energies. The final discrepancies in size probably remain due to the influence of omitted SIA. SIAs captured by bubbles obviously reduce the size of the bubbles. However, the general good qualitative agreement of the rate model is remarkable although the contribution of interstitial-type defects appears necessary to overcome the remaining quantitative discrepancies. 6. Conclusion We have presented results of the TEM microstructure analysis of Heþ/Fe3þ dual-beam irradiated EUROFER97 with respect to the nucleation of helium bubbles for different temperatures. The micrographs show a spatially homogeneous bubble distribution in the matrix at Tirr ¼ 330 C and Tirr ¼ 400 C whereas at a higher temperature of Tirr ¼ 500 C in addition to a homogeneous contribution bubbles nucleate also at dislocation lines and grain boundaries. Thus, dual beam ion irradiation successfully reproduced the transition from homogeneous to heterogeneous bubble formation with rising temperature as observed for the neutron irradiation of EUROFER97. The bubble sizes for Tirr ¼ 330 C and Tirr ¼ 400 C are distributed around to 1.2 nm, but increases for the step from Tirr ¼ 400 C to Tirr ¼ 500 C. At Tirr ¼ 500 C the size distribution of bubbles shows a maxima at 1.7 nm and bubbles up to 4 nm are observed. The density of bubbles monotonically decreases with rising temperature. For all three irradiation temperatures the swelling due to visible bubbles does not exceed 0.02%. The rate model predicts no significant contribution to the swelling of bubbles which are too small for detection by means of TEM. In addition, the findings of the microstructure analysis for the temperature dependence of bubble size and density are in good qualitative agreement with the numerical results of the kinetic rate model for the formation of helium bubbles. The rate model also predicts an increasing accumulation of helium at the grain boundaries and line dislocations with rising temperature. However, the gradient is to low to explain the experimentally observed transition from spatially homogeneous to heterogeneous nucleation. Further investigations for a suitable treatment of sinks responsible for heterogeneous nucleation are necessary. Furthermore, the extension of the rate model with the molecular dynamic based “variable gap model” for the binding energies helium bubbles improved the results for the bubble sizes. For high temperatures the calculated mean diameter gets much closer to the experimentally estimated values than the approach originally used in Ref. [16]. Even though calculated values for the diameter and
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