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Growth mechanisms of interstitial loops in α-doped UO2 samples
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NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 266 (2008) 3008–3012 www.elsevier.com/locate/nimb
Growth mechanisms of interstitial loops in a-doped UO2 samples J. Jonnet a,*, P. Van Uffelen b, T. Wiss b, D. Staicu b, B. Re´my c, J. Rest d a Delft University of Technology, Reactor Institute Delft, Mekelweg 15, 2629JB Delft, The Netherlands European Commission, Joint Research Centre, Institute for Transuranium Elements, P.O. Box 2340, 76125 Karlsruhe, Germany c Laboratoire d’Energe´tique et de Me´canique The´orique et Applique´e, CNRS-UMR 7563, ENSEM-INPL, 2 avenue de la Foreˆt de Haye, 54506 Vandoeuvre le`s Nancy, France d Argonne National Laboratory, Nuclear Engineering-208, Argonne, IL 60439, USA b
Available online 25 March 2008
Abstract New experimental size distributions are presented for interstitial-type dislocation loops in (U, Pu)O2 samples after 4 and 7 years of self-irradiation along with a new model. For this model, the work of Hayns has been extended to doped materials and to take into account additional phenomena, namely re-solution and coalescence as applied to gas bubble precipitation. The calculations are compared to the experimental size distributions obtained by means of Transmission Electron Microscopy for two different storage times. The role of re-solution and coalescence is discussed based on this model. This constitutes a basis for modelling the high burn-up structure (HBS) formation which is a consequence of the accumulation of radiation damage in nuclear fuels. Ó 2008 Elsevier B.V. All rights reserved. PACS: 61.72.Ff; 61.72.Ji; 61.72.Lk; 23.60.+e; 61.80.Az Keywords: Nuclear fuel; UO2; a-emitters; Interstitial loops; Electron microscopy; Interstitial re-solution; Loop coalescence
1. Introduction An increase of the discharge burn-up of UO2 nuclear fuels in light water reactors results in the emergence of a microstructural change. This structure, called HBS for ‘‘high burn-up structure”, is characterised by fission gas depletion from the matrix, appearance of a micrometric porosity, and a sub-division of the original grains. Although well characterised experimentally, important points have still to be clarified, among which are HBS formation mechanisms. In order to answer this issue, a study of the contribution of dislocation-type defects was conducted [1]. Since irradiated fuel is very complex, a simpler system reproducing some of the features observed in the HBS, was chosen in the form of UO2 sintered pellets doped with 10 wt% 238Pu and stored for different times at room temperature, during which a-damage resulting from the *
Corresponding author. E-mail address: [email protected] (J. Jonnet).
0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.03.154
decay of 238Pu was produced. In this way, the single growth mechanisms of interstitial-type dislocation loops can be studied as these loops are directly formed by aggregation of interstitials, generated by the self-irradiation. The objective of the present paper is to clarify the role of two basic mechanisms, namely irradiation-induced re-solution and migration and coalescence coarsening, involved in the damage evolution in a simplified system being self-irradiated fuel on the basis of a comparison with experimental results. It therefore contributes to the understanding of defect evolution in UO2, rather than proposing a comprehensive model for the simulation of the behaviour of UO2 during its irradiation in a reactor. 2. Experiments The present samples were produced by incorporating 10 wt% 238PuO2 (half-life = 87 years) to natural UO2 by a sol–gel process [2], producing a homogeneous solid solution.
J. Jonnet et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3008–3012
3009
Fig. 1. TEM bright field image evidencing the formation of dislocation loops at t = 4 years (a) and t = 7 years (b), respectively.
The a-activity of the doped material was calculated to be 3.76 1010 Bq/g, from which the damage rate was found to be 9.06 109 dpa/s. TEM examinations were carried out on samples after storage times of 4 and 7 years at room temperature, which exhibit a dpa value of 1.1 and 2, respectively. Fragments of a few milligrams of the original (U0.9, Pu0.1)O2 pellets were crushed in ethanol. The resulting suspension was dropped on a copper grid filmed with carbon. The TEM used in this study is a Hitachi H700 HST operating at 200 kV and modified to handle radioactive materials. Fig. 1 shows two bright field pictures of the sample after 4 and 7 years, respectively. The main defect that was observed consists of interstitial-type dislocation loops, whose contrast is similar to the ones simulated by Marian et al. [3] in a-Fe, homogeneously distributed within the material (Fig. 1). The dislocation loop size and density have been determined by
manual image analysis. The loops size-distributions were obtained and are shown in Fig. 2. In order to enable a comparison with the computations in Section 4, the experimental distributions are described in terms of statistical values, more precisely the first four moments: the loop mean radius, standard deviation, skewness and normalised kurtosis (Table 1). 3. A comprehensive model of defects in a-doped UO2 In the following, the diffusion of uranium interstitials and vacancies is only considered. Because of its low mobility, uranium diffusion in UO2 is rate-controlling [4]. The most straightforward manner to model the behavior of uranium interstitials is by using the chemical rate theory [5]. Hayns [6] proposed a set of equations describing the nucleation and growth of a dislocation loop distribution in irradiated graphite and steel: eq X dcv ðtÞ ¼ K ar cv ðtÞci ðtÞ Ln cni ðtÞcv ðtÞ dt n¼2
n
Dv qd cv ðtÞ; dci ðtÞ ¼ K ar cv ðtÞci ðtÞ dt
Fig. 2. Loop size histogram from the sample observed at t = 4 years and t = 7 years.
ð1Þ neq X
Rn cni ðtÞci ðtÞ
n¼2
þ L2 c2i ðtÞcv ðtÞ; dc2i ðtÞ 1 ¼ K 2 ci ðtÞci ðtÞ þ L3 c3i ðtÞcv ðtÞ dt 2 L2 c2i ðtÞcv ðtÞ R2 c2i ðtÞci ðtÞ; dc3i ðtÞ ¼ R2 c2i ðtÞci ðtÞ þ L4 c4i ðtÞcv ðtÞ dt R3 c3i ðtÞci ðtÞ L3 c3i ðtÞcv ðtÞ; ... dcni ðtÞ ¼ Rn1 cðn1Þi ðtÞci ðtÞ þ Lnþ1 cðnþ1Þi ðtÞcv ðtÞ dt Rn cni ðtÞci ðtÞ Ln cni ðtÞcv ðtÞ:
ð2Þ
ð3Þ
ð4Þ
ð5Þ
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J. Jonnet et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3008–3012
Table 1 Statistical values of the experimental distributions, and obtained from the histogram in Fig. 3 Experimental results
Loop mean radius Standard deviation Modal value Corresponding loop density Skewness Normalised kurtosis
Numerical results
4 years old
7 years old
4 years and 5 months old
3.05 nm 2.03 nm 2 nm 8 1015 cm3 ± 3 1015 cm3 1.65 3.01
3.71 nm 1.69 nm 3 nm 23 1015 cm3 ± 6 1015 cm3 2.04 6.34
3.52 nm 1.58 nm 3 nm 284 1015 cm3 1.69 8.19
In these equations, cv, ci and cni are the concentrations of vacancies, interstitials and dislocation loops containing n interstitials, respectively; K is the damage rate (considered to be constant); ar the interstitial-vacancy recombination coefficient; K2 is the rate constant for the production of di-interstitials (interstitial loop nucleus); Dv and Di are the vacancy and interstitial diffusivities; qd is the density of pre-existing dislocations, neglected in our analysis as almost no dislocation line was observed by means of TEM in the samples studied. Ln and Rn are the reaction rate constants of a loop containing n atoms with a vacancy and interstitial, respectively. neq is the number of equations of the system, which corresponds to the maximum number of interstitials involved in a loop. The rate constants Rn and Ln are defined by [6]:
2p Rn ¼ z i D i X
ffi rffiffiffiffiffiffiffiffiffi nc2cell ; p
ffi rffiffiffiffiffiffiffiffiffi 2p nc2cell Ln ¼ zv Dv ; X p
ð6Þ
where c2cell is the area per UO2 Schottky trio in a loop (preserving electrical neutrality), assuming that interstitial-type dislocation loops form in the (1 1 0) planes [1]. The temperature dependence of cv and ci is contained in the interstitial and vacancy diffusivities [7]: Dv ¼ x2i a2 mv expðvm =kT Þ;
ð7Þ
2 Di ¼ x2i a2 mi expðim =kT Þ; 3
ð8Þ
where vm and im are the vacancy and interstitial migration enthalpies, mv and mi the vacancy and interstitial jump frequencies, a is the lattice parameter and xi a factor that takes into account the deviation from stoichiometry. Most values of the various parameters used in Eqs. (1)– (5) are taken from [7–10], except for x2i and im due to the uncertainty in their exact value. Their impact has been studied in a sensitivity analysis [1]. It is proposed in the present paper that existing dislocation loops may be subjected to re-solution of their interstitials back into the matrix due to the impact with a 234U recoil atom, which in turn is produced by the a-decay of 238 Pu. For this purpose, Eq. (2) for the interstitial concentration in the matrix and the generic Eq. (5) for the concentration of (n)-interstitial loops are replaced by Eqs. (9) and (10), respectively:
dci ðtÞ ¼ K ar cv ðtÞci ðtÞ þ L2 c2i ðtÞcv ðtÞ dt neq neq X X Rn cni ðtÞci ðtÞ þ cn ðtÞF la ðnÞc2cell minðn; na Þ; n¼2
n¼2
ð9Þ dcni ðtÞ ¼ Rn1 cðn1Þi ðtÞci ðtÞ þ Lnþ1 cðnþ1Þi ðtÞcv ðtÞ dt Rn cni ðtÞci ðtÞ Ln cni ðtÞcv ðtÞ þ cðnþna Þi ðtÞF la ðn þ na Þc2cell cni ðtÞF la ðnÞc2cell :
ð10Þ
The additional summation in the right-hand side of Eq. (9) is the sum of all interstitials that were driven back into the matrix. The fifth term in the right-hand side of Eq. (10) represents the creation of a (n)-interstitial loop from a (n + na)-interstitial loop having lost na interstitials after impact with a recoil atom. The parameter na has been calculated by considering that all U and O displacements created by the recoil atom constitute a cylinder of length la, which is the penetration depth of the recoil atom, enabling the calculation of the impact section with an interstitial loop [1]. In addition, it is assumed that the interstitial loops are able to move in the absence of temperature or stress gradients along the normal of their cylindrical plane, and then to coalesce with another loop. This mechanism, called Migration and Coalescence (M&C), is one of the two well-known coarsening mechanisms, together with the Ostwald Ripening (OR). As a consequence of this mechanism, Eq. (5) for the concentration of (n)-interstitial loops contains two more terms: dcni ðtÞ ¼ Rn1 cðn1Þi ðtÞci ðtÞ þ Lnþ1 cðnþ1Þi ðtÞcv ðtÞ dt n X W j;nj ðtÞ Rn cni ðtÞci ðtÞ Ln cni ðtÞcv ðtÞ þ j¼2
nX eq 1
W n;j ðtÞ;
ð11Þ
j¼2
where W n;j ðtÞ ¼
8ðrn þ rj ÞðDn þ Dj Þ cni ðtÞcji ðtÞ; X
ð12Þ
J. Jonnet et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3008–3012
is the rate of collisions between loops with radii rn and rj. The first summation in Eq. (11) represents the creation of a loop containing (n)-interstitials from a (n j)-interstitial loop colliding with a (j)-interstitial loop. The second summation is the loss of (n)-interstitial loops that underwent coalescence. Both surface and volume diffusion of dislocation loops in the matrix has been considered [11]. The coefficients were derived from the expressions for the atomic mechanism of bubble motion due to surface and volume diffusion [11]. However, it appeared that surface diffusion is negligible at the temperature considered (room temperature) [1]. Moreover, the volume diffusion coefficient Dvolume of a loop l of radius rn reads: a ’ Di ; ð13Þ Dvolume l prn where a is the UO2 lattice parameter. It is noted that the volume diffusion coefficient of loops has the same migration energy as that of the uranium interstitial and a sizedependent prefactor. 4. Model application and discussion The aim of this section is to study the effect of the independent parameters that were not set in Eqs. (1)–(5), as well as the main parameter characterising the re-solution process, la. To this end, the Taguchi [12] method was applied on im, x2i and la. These parameters have been set to a minimum and a maximum value: 102 < x2i < 1 [13], 10 nm < la < 30 nm [14,15], 0.65 eV < im < 0.8 eV. Concerning the uraniuminterstitial migration enthalpy in UO2, the range of values is narrow in comparison with that in the literature: Matzke determined experimentally im in UO2 to be around 2 eV [16], whereas Veshchunov et al. [9] used a value of 0.3 eV, Rest and Hofman [17,7] used 0.6 eV, and Morelon et al. [18] calculated im in UO2 to be 5.0 eV, by using molecular dynamics simulations. In a first step, Eqs. (1)–(5) completed by Eqs. (9) and (10) were solved in order to take into account the re-solution process alone. Eight cases were considered with the orthogonal array L8(23), which means that eight runs are performed to investigate the effect of three different parameters, each of which is set at two predetermined levels (min/max). From the calculated concentrations of interstitial loops, the associated histograms have been built and the same statistical values tabulated in Table 1 were derived for all cases. A level-average analysis by applying the Taguchi method led to the following conclusions: – All characteristics of the loop size-distribution are sensitive to im and x2i . The effect of im is twofold. First, a decreasing value of im enhances the diffusion of a uranium-interstitial (like increasing x2i ). Second, this enhances the Frenkel pair recombination coefficient. It
3011
is shown that for the highest value of im, the recombination is small and a very high peak in the maximum loop concentration is induced, conflicting with experimental observations in spite of the experimental uncertainties. This suggests lowering the value of im. Because im and x2i have opposite effects on Di, the increase in the latter due to im can be compensated by using a small value for x2i . – The re-solution process has an important effect on the loop mean size. Without re-solution, the size-distribution is steadily moving towards higher radii. With resolution, the big loops are destroyed, resulting in a distribution which does not significantly evolve between 4 and 7 years when the value of Di is not so high. This means that there is competition between loop growth by trapping of diffusing interstitials and loop destruction through re-solution. The parameter la has little effect compared to x2i and im. As a result, even if la may be subjected to uncertainties, the overall effect is small and a value of 20 nm, as determined by means of TRIM calculations [15], can be used. In a second step, only the migration and coalescence of interstitial loops has been introduced into the equations (i.e. re-solution is disregarded), hence the two mechanisms can be studied separately. The single effect of coalescence can be studied if and only if Di is the only parameter that is varied, which means varying x2i because im changes the recombination coefficient as well. Moreover, in view of the conclusions above, an adequate value for im was chosen in order to avoid both an extreme computation time (induced by Eqs. (11) and (12)) and an irrelevant low Frenkel pair recombination coefficient: im = 0.7 eV. An application of this case shows that the calculated distribution evolves very rapidly after a simulated time of 4 years. Further computation of this run would result in an unstable distribution of interstitial loop concentrations. After 4 years and 5 months (with im = 0.7 eV, and Di = 5 1019 cm2 s1 imposed), a tail typical of the volume diffusion and coalescence of loops appeared. As a first conclusion, these results show that once a distribution of interstitial loops forms, loop coalescence is very effective. Large loops form more rapidly from the small ones by coalescence than the small ones form from di-interstitial nucleation and trapping of interstitials. Fig. 3 shows the loop size-histogram obtained after 4 years and 5 months, from which the four moments are derived and shown in Table 1. All moments are in good agreement with the experimental values, except for the maximum loop concentration which is too high, even when accounting for the experimental uncertainty. As a consequence, the volume diffusion and coalescence of interstitial-type dislocation loops must be accounted for in the behavior of a-produced radiation damage in UO2 in order to obtain the correct skewness in the loop size histogram.
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J. Jonnet et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3008–3012
A further analysis must be carried out in order to study the combined effects of re-solution and coalescence of interstitial loops. Acknowledgements The work presented in the paper has been performed during the PhD Thesis of the first author at JRC-ITU, and was financially supported by the European Commission (Euratom PhD Grant 21080). References
Fig. 3. Loop size histograms at 4 years and 5 months.
5. Summary and conclusions The model developed by Hayns [6] has been extended for the study of radiation damage in a-doped UO2 samples, for which new experimental results are presented. Two additional mechanisms have been considered. The first one is re-solution of interstitials. A sensitivity analysis revealed that the re-solution process slows down the growth of interstitial loops, as observed experimentally. The second mechanism proposed is the coalescence of diffusing interstitial-type dislocation loops. A second sensitivity analysis showed that coalescence is very effective and strongly dependent on the uranium-interstitial diffusion coefficient. Furthermore the analysis revealed that the simulated loop size-histograms contain a tail oriented towards the larger loop sizes (positive skewness), which is typical for this coarsening mechanism and similar to the trend observed experimentally.
[1] J. Jonnet, Thesis, Institut National Polytechnique de Lorraine, Nancy, France, 2007. [2] V.V. Rondinella, Hj. Matzke, J. Cobos, T. Wiss, Radiochim. Acta 88 (2000) 527. [3] J. Marian, B.D. Wirth, R. Scha¨ublin, J.M. Perlado, T. Diaz de la Rubia, J. Nucl. Mater. 307 (2002) 871. [4] Hj. Matzke, J. Less-Common Metal 121 (1986) 537. [5] A.D. Brailsford, R. Bullough, J. Nucl. Mater. 44 (1972) 121. [6] M.R. Hayns, J. Nucl. Mater. 56 (1975) 267. [7] J. Rest, G.L. Hofman, J. Nucl. Mater. 277 (2000) 231. [8] M.A. Krivoglaz, Springer, Berlin, 1995. [9] M.S. Veshchunov, V.D. Ozrin, S.E. Shestak, V.I. Tarasov, R. Dubourg, G. Nicaise, in: Proc. Int. ANS Top. Meet. on LWR Fuel Perf., Orlando, Florida, USA, September 19–22, 2004. [10] J. Rest and G.L. Hofman, in: Proc. of the Materials Research Society Meeting, Symposium R, Volume 650, Boston, 2000. [11] D.R. Olander, Fundamental Aspects of Nuclear Reactors Fuel Elements, Energy Research and Development Administration, 1976. [12] E.P. Box, S. Bisgaard, The scientific context of quality improvement, Quality Progress, 1987. [13] A.B. Lidiard, J. Nucl. Mater. 19 (1966) 106. [14] Hj. Matzke, Solid State Phenom. 30 (1993) 355. [15] Hj. Matzke, T. Wiss, ITU annual report (eur 19812), European Commission, Institute for Transuranium Elements, Karlsruhe, 2000. [16] Hj. Matzke, J. Chem. Soc., Faraday Trans. 2 (83) (1987) 1121. [17] J. Rest, G.L. Hofman, J. Nucl. Mater. 210 (1994) 187. [18] N.D. Morelon, D. Ghaleb, J.M. Delaye, L. Van Brutzel, Philos. Mag. 83 (2003) 1533.
NIM B Beam Interactions with Materials & Atoms
Nuclear Instruments and Methods in Physics Research B 266 (2008) 3008–3012 www.elsevier.com/locate/nimb
Growth mechanisms of interstitial loops in a-doped UO2 samples J. Jonnet a,*, P. Van Uffelen b, T. Wiss b, D. Staicu b, B. Re´my c, J. Rest d a Delft University of Technology, Reactor Institute Delft, Mekelweg 15, 2629JB Delft, The Netherlands European Commission, Joint Research Centre, Institute for Transuranium Elements, P.O. Box 2340, 76125 Karlsruhe, Germany c Laboratoire d’Energe´tique et de Me´canique The´orique et Applique´e, CNRS-UMR 7563, ENSEM-INPL, 2 avenue de la Foreˆt de Haye, 54506 Vandoeuvre le`s Nancy, France d Argonne National Laboratory, Nuclear Engineering-208, Argonne, IL 60439, USA b
Available online 25 March 2008
Abstract New experimental size distributions are presented for interstitial-type dislocation loops in (U, Pu)O2 samples after 4 and 7 years of self-irradiation along with a new model. For this model, the work of Hayns has been extended to doped materials and to take into account additional phenomena, namely re-solution and coalescence as applied to gas bubble precipitation. The calculations are compared to the experimental size distributions obtained by means of Transmission Electron Microscopy for two different storage times. The role of re-solution and coalescence is discussed based on this model. This constitutes a basis for modelling the high burn-up structure (HBS) formation which is a consequence of the accumulation of radiation damage in nuclear fuels. Ó 2008 Elsevier B.V. All rights reserved. PACS: 61.72.Ff; 61.72.Ji; 61.72.Lk; 23.60.+e; 61.80.Az Keywords: Nuclear fuel; UO2; a-emitters; Interstitial loops; Electron microscopy; Interstitial re-solution; Loop coalescence
1. Introduction An increase of the discharge burn-up of UO2 nuclear fuels in light water reactors results in the emergence of a microstructural change. This structure, called HBS for ‘‘high burn-up structure”, is characterised by fission gas depletion from the matrix, appearance of a micrometric porosity, and a sub-division of the original grains. Although well characterised experimentally, important points have still to be clarified, among which are HBS formation mechanisms. In order to answer this issue, a study of the contribution of dislocation-type defects was conducted [1]. Since irradiated fuel is very complex, a simpler system reproducing some of the features observed in the HBS, was chosen in the form of UO2 sintered pellets doped with 10 wt% 238Pu and stored for different times at room temperature, during which a-damage resulting from the *
Corresponding author. E-mail address: [email protected] (J. Jonnet).
0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.03.154
decay of 238Pu was produced. In this way, the single growth mechanisms of interstitial-type dislocation loops can be studied as these loops are directly formed by aggregation of interstitials, generated by the self-irradiation. The objective of the present paper is to clarify the role of two basic mechanisms, namely irradiation-induced re-solution and migration and coalescence coarsening, involved in the damage evolution in a simplified system being self-irradiated fuel on the basis of a comparison with experimental results. It therefore contributes to the understanding of defect evolution in UO2, rather than proposing a comprehensive model for the simulation of the behaviour of UO2 during its irradiation in a reactor. 2. Experiments The present samples were produced by incorporating 10 wt% 238PuO2 (half-life = 87 years) to natural UO2 by a sol–gel process [2], producing a homogeneous solid solution.
J. Jonnet et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3008–3012
3009
Fig. 1. TEM bright field image evidencing the formation of dislocation loops at t = 4 years (a) and t = 7 years (b), respectively.
The a-activity of the doped material was calculated to be 3.76 1010 Bq/g, from which the damage rate was found to be 9.06 109 dpa/s. TEM examinations were carried out on samples after storage times of 4 and 7 years at room temperature, which exhibit a dpa value of 1.1 and 2, respectively. Fragments of a few milligrams of the original (U0.9, Pu0.1)O2 pellets were crushed in ethanol. The resulting suspension was dropped on a copper grid filmed with carbon. The TEM used in this study is a Hitachi H700 HST operating at 200 kV and modified to handle radioactive materials. Fig. 1 shows two bright field pictures of the sample after 4 and 7 years, respectively. The main defect that was observed consists of interstitial-type dislocation loops, whose contrast is similar to the ones simulated by Marian et al. [3] in a-Fe, homogeneously distributed within the material (Fig. 1). The dislocation loop size and density have been determined by
manual image analysis. The loops size-distributions were obtained and are shown in Fig. 2. In order to enable a comparison with the computations in Section 4, the experimental distributions are described in terms of statistical values, more precisely the first four moments: the loop mean radius, standard deviation, skewness and normalised kurtosis (Table 1). 3. A comprehensive model of defects in a-doped UO2 In the following, the diffusion of uranium interstitials and vacancies is only considered. Because of its low mobility, uranium diffusion in UO2 is rate-controlling [4]. The most straightforward manner to model the behavior of uranium interstitials is by using the chemical rate theory [5]. Hayns [6] proposed a set of equations describing the nucleation and growth of a dislocation loop distribution in irradiated graphite and steel: eq X dcv ðtÞ ¼ K ar cv ðtÞci ðtÞ Ln cni ðtÞcv ðtÞ dt n¼2
n
Dv qd cv ðtÞ; dci ðtÞ ¼ K ar cv ðtÞci ðtÞ dt
Fig. 2. Loop size histogram from the sample observed at t = 4 years and t = 7 years.
ð1Þ neq X
Rn cni ðtÞci ðtÞ
n¼2
þ L2 c2i ðtÞcv ðtÞ; dc2i ðtÞ 1 ¼ K 2 ci ðtÞci ðtÞ þ L3 c3i ðtÞcv ðtÞ dt 2 L2 c2i ðtÞcv ðtÞ R2 c2i ðtÞci ðtÞ; dc3i ðtÞ ¼ R2 c2i ðtÞci ðtÞ þ L4 c4i ðtÞcv ðtÞ dt R3 c3i ðtÞci ðtÞ L3 c3i ðtÞcv ðtÞ; ... dcni ðtÞ ¼ Rn1 cðn1Þi ðtÞci ðtÞ þ Lnþ1 cðnþ1Þi ðtÞcv ðtÞ dt Rn cni ðtÞci ðtÞ Ln cni ðtÞcv ðtÞ:
ð2Þ
ð3Þ
ð4Þ
ð5Þ
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J. Jonnet et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3008–3012
Table 1 Statistical values of the experimental distributions, and obtained from the histogram in Fig. 3 Experimental results
Loop mean radius Standard deviation Modal value Corresponding loop density Skewness Normalised kurtosis
Numerical results
4 years old
7 years old
4 years and 5 months old
3.05 nm 2.03 nm 2 nm 8 1015 cm3 ± 3 1015 cm3 1.65 3.01
3.71 nm 1.69 nm 3 nm 23 1015 cm3 ± 6 1015 cm3 2.04 6.34
3.52 nm 1.58 nm 3 nm 284 1015 cm3 1.69 8.19
In these equations, cv, ci and cni are the concentrations of vacancies, interstitials and dislocation loops containing n interstitials, respectively; K is the damage rate (considered to be constant); ar the interstitial-vacancy recombination coefficient; K2 is the rate constant for the production of di-interstitials (interstitial loop nucleus); Dv and Di are the vacancy and interstitial diffusivities; qd is the density of pre-existing dislocations, neglected in our analysis as almost no dislocation line was observed by means of TEM in the samples studied. Ln and Rn are the reaction rate constants of a loop containing n atoms with a vacancy and interstitial, respectively. neq is the number of equations of the system, which corresponds to the maximum number of interstitials involved in a loop. The rate constants Rn and Ln are defined by [6]:
2p Rn ¼ z i D i X
ffi rffiffiffiffiffiffiffiffiffi nc2cell ; p
ffi rffiffiffiffiffiffiffiffiffi 2p nc2cell Ln ¼ zv Dv ; X p
ð6Þ
where c2cell is the area per UO2 Schottky trio in a loop (preserving electrical neutrality), assuming that interstitial-type dislocation loops form in the (1 1 0) planes [1]. The temperature dependence of cv and ci is contained in the interstitial and vacancy diffusivities [7]: Dv ¼ x2i a2 mv expðvm =kT Þ;
ð7Þ
2 Di ¼ x2i a2 mi expðim =kT Þ; 3
ð8Þ
where vm and im are the vacancy and interstitial migration enthalpies, mv and mi the vacancy and interstitial jump frequencies, a is the lattice parameter and xi a factor that takes into account the deviation from stoichiometry. Most values of the various parameters used in Eqs. (1)– (5) are taken from [7–10], except for x2i and im due to the uncertainty in their exact value. Their impact has been studied in a sensitivity analysis [1]. It is proposed in the present paper that existing dislocation loops may be subjected to re-solution of their interstitials back into the matrix due to the impact with a 234U recoil atom, which in turn is produced by the a-decay of 238 Pu. For this purpose, Eq. (2) for the interstitial concentration in the matrix and the generic Eq. (5) for the concentration of (n)-interstitial loops are replaced by Eqs. (9) and (10), respectively:
dci ðtÞ ¼ K ar cv ðtÞci ðtÞ þ L2 c2i ðtÞcv ðtÞ dt neq neq X X Rn cni ðtÞci ðtÞ þ cn ðtÞF la ðnÞc2cell minðn; na Þ; n¼2
n¼2
ð9Þ dcni ðtÞ ¼ Rn1 cðn1Þi ðtÞci ðtÞ þ Lnþ1 cðnþ1Þi ðtÞcv ðtÞ dt Rn cni ðtÞci ðtÞ Ln cni ðtÞcv ðtÞ þ cðnþna Þi ðtÞF la ðn þ na Þc2cell cni ðtÞF la ðnÞc2cell :
ð10Þ
The additional summation in the right-hand side of Eq. (9) is the sum of all interstitials that were driven back into the matrix. The fifth term in the right-hand side of Eq. (10) represents the creation of a (n)-interstitial loop from a (n + na)-interstitial loop having lost na interstitials after impact with a recoil atom. The parameter na has been calculated by considering that all U and O displacements created by the recoil atom constitute a cylinder of length la, which is the penetration depth of the recoil atom, enabling the calculation of the impact section with an interstitial loop [1]. In addition, it is assumed that the interstitial loops are able to move in the absence of temperature or stress gradients along the normal of their cylindrical plane, and then to coalesce with another loop. This mechanism, called Migration and Coalescence (M&C), is one of the two well-known coarsening mechanisms, together with the Ostwald Ripening (OR). As a consequence of this mechanism, Eq. (5) for the concentration of (n)-interstitial loops contains two more terms: dcni ðtÞ ¼ Rn1 cðn1Þi ðtÞci ðtÞ þ Lnþ1 cðnþ1Þi ðtÞcv ðtÞ dt n X W j;nj ðtÞ Rn cni ðtÞci ðtÞ Ln cni ðtÞcv ðtÞ þ j¼2
nX eq 1
W n;j ðtÞ;
ð11Þ
j¼2
where W n;j ðtÞ ¼
8ðrn þ rj ÞðDn þ Dj Þ cni ðtÞcji ðtÞ; X
ð12Þ
J. Jonnet et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3008–3012
is the rate of collisions between loops with radii rn and rj. The first summation in Eq. (11) represents the creation of a loop containing (n)-interstitials from a (n j)-interstitial loop colliding with a (j)-interstitial loop. The second summation is the loss of (n)-interstitial loops that underwent coalescence. Both surface and volume diffusion of dislocation loops in the matrix has been considered [11]. The coefficients were derived from the expressions for the atomic mechanism of bubble motion due to surface and volume diffusion [11]. However, it appeared that surface diffusion is negligible at the temperature considered (room temperature) [1]. Moreover, the volume diffusion coefficient Dvolume of a loop l of radius rn reads: a ’ Di ; ð13Þ Dvolume l prn where a is the UO2 lattice parameter. It is noted that the volume diffusion coefficient of loops has the same migration energy as that of the uranium interstitial and a sizedependent prefactor. 4. Model application and discussion The aim of this section is to study the effect of the independent parameters that were not set in Eqs. (1)–(5), as well as the main parameter characterising the re-solution process, la. To this end, the Taguchi [12] method was applied on im, x2i and la. These parameters have been set to a minimum and a maximum value: 102 < x2i < 1 [13], 10 nm < la < 30 nm [14,15], 0.65 eV < im < 0.8 eV. Concerning the uraniuminterstitial migration enthalpy in UO2, the range of values is narrow in comparison with that in the literature: Matzke determined experimentally im in UO2 to be around 2 eV [16], whereas Veshchunov et al. [9] used a value of 0.3 eV, Rest and Hofman [17,7] used 0.6 eV, and Morelon et al. [18] calculated im in UO2 to be 5.0 eV, by using molecular dynamics simulations. In a first step, Eqs. (1)–(5) completed by Eqs. (9) and (10) were solved in order to take into account the re-solution process alone. Eight cases were considered with the orthogonal array L8(23), which means that eight runs are performed to investigate the effect of three different parameters, each of which is set at two predetermined levels (min/max). From the calculated concentrations of interstitial loops, the associated histograms have been built and the same statistical values tabulated in Table 1 were derived for all cases. A level-average analysis by applying the Taguchi method led to the following conclusions: – All characteristics of the loop size-distribution are sensitive to im and x2i . The effect of im is twofold. First, a decreasing value of im enhances the diffusion of a uranium-interstitial (like increasing x2i ). Second, this enhances the Frenkel pair recombination coefficient. It
3011
is shown that for the highest value of im, the recombination is small and a very high peak in the maximum loop concentration is induced, conflicting with experimental observations in spite of the experimental uncertainties. This suggests lowering the value of im. Because im and x2i have opposite effects on Di, the increase in the latter due to im can be compensated by using a small value for x2i . – The re-solution process has an important effect on the loop mean size. Without re-solution, the size-distribution is steadily moving towards higher radii. With resolution, the big loops are destroyed, resulting in a distribution which does not significantly evolve between 4 and 7 years when the value of Di is not so high. This means that there is competition between loop growth by trapping of diffusing interstitials and loop destruction through re-solution. The parameter la has little effect compared to x2i and im. As a result, even if la may be subjected to uncertainties, the overall effect is small and a value of 20 nm, as determined by means of TRIM calculations [15], can be used. In a second step, only the migration and coalescence of interstitial loops has been introduced into the equations (i.e. re-solution is disregarded), hence the two mechanisms can be studied separately. The single effect of coalescence can be studied if and only if Di is the only parameter that is varied, which means varying x2i because im changes the recombination coefficient as well. Moreover, in view of the conclusions above, an adequate value for im was chosen in order to avoid both an extreme computation time (induced by Eqs. (11) and (12)) and an irrelevant low Frenkel pair recombination coefficient: im = 0.7 eV. An application of this case shows that the calculated distribution evolves very rapidly after a simulated time of 4 years. Further computation of this run would result in an unstable distribution of interstitial loop concentrations. After 4 years and 5 months (with im = 0.7 eV, and Di = 5 1019 cm2 s1 imposed), a tail typical of the volume diffusion and coalescence of loops appeared. As a first conclusion, these results show that once a distribution of interstitial loops forms, loop coalescence is very effective. Large loops form more rapidly from the small ones by coalescence than the small ones form from di-interstitial nucleation and trapping of interstitials. Fig. 3 shows the loop size-histogram obtained after 4 years and 5 months, from which the four moments are derived and shown in Table 1. All moments are in good agreement with the experimental values, except for the maximum loop concentration which is too high, even when accounting for the experimental uncertainty. As a consequence, the volume diffusion and coalescence of interstitial-type dislocation loops must be accounted for in the behavior of a-produced radiation damage in UO2 in order to obtain the correct skewness in the loop size histogram.
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J. Jonnet et al. / Nucl. Instr. and Meth. in Phys. Res. B 266 (2008) 3008–3012
A further analysis must be carried out in order to study the combined effects of re-solution and coalescence of interstitial loops. Acknowledgements The work presented in the paper has been performed during the PhD Thesis of the first author at JRC-ITU, and was financially supported by the European Commission (Euratom PhD Grant 21080). References
Fig. 3. Loop size histograms at 4 years and 5 months.
5. Summary and conclusions The model developed by Hayns [6] has been extended for the study of radiation damage in a-doped UO2 samples, for which new experimental results are presented. Two additional mechanisms have been considered. The first one is re-solution of interstitials. A sensitivity analysis revealed that the re-solution process slows down the growth of interstitial loops, as observed experimentally. The second mechanism proposed is the coalescence of diffusing interstitial-type dislocation loops. A second sensitivity analysis showed that coalescence is very effective and strongly dependent on the uranium-interstitial diffusion coefficient. Furthermore the analysis revealed that the simulated loop size-histograms contain a tail oriented towards the larger loop sizes (positive skewness), which is typical for this coarsening mechanism and similar to the trend observed experimentally.
[1] J. Jonnet, Thesis, Institut National Polytechnique de Lorraine, Nancy, France, 2007. [2] V.V. Rondinella, Hj. Matzke, J. Cobos, T. Wiss, Radiochim. Acta 88 (2000) 527. [3] J. Marian, B.D. Wirth, R. Scha¨ublin, J.M. Perlado, T. Diaz de la Rubia, J. Nucl. Mater. 307 (2002) 871. [4] Hj. Matzke, J. Less-Common Metal 121 (1986) 537. [5] A.D. Brailsford, R. Bullough, J. Nucl. Mater. 44 (1972) 121. [6] M.R. Hayns, J. Nucl. Mater. 56 (1975) 267. [7] J. Rest, G.L. Hofman, J. Nucl. Mater. 277 (2000) 231. [8] M.A. Krivoglaz, Springer, Berlin, 1995. [9] M.S. Veshchunov, V.D. Ozrin, S.E. Shestak, V.I. Tarasov, R. Dubourg, G. Nicaise, in: Proc. Int. ANS Top. Meet. on LWR Fuel Perf., Orlando, Florida, USA, September 19–22, 2004. [10] J. Rest and G.L. Hofman, in: Proc. of the Materials Research Society Meeting, Symposium R, Volume 650, Boston, 2000. [11] D.R. Olander, Fundamental Aspects of Nuclear Reactors Fuel Elements, Energy Research and Development Administration, 1976. [12] E.P. Box, S. Bisgaard, The scientific context of quality improvement, Quality Progress, 1987. [13] A.B. Lidiard, J. Nucl. Mater. 19 (1966) 106. [14] Hj. Matzke, Solid State Phenom. 30 (1993) 355. [15] Hj. Matzke, T. Wiss, ITU annual report (eur 19812), European Commission, Institute for Transuranium Elements, Karlsruhe, 2000. [16] Hj. Matzke, J. Chem. Soc., Faraday Trans. 2 (83) (1987) 1121. [17] J. Rest, G.L. Hofman, J. Nucl. Mater. 210 (1994) 187. [18] N.D. Morelon, D. Ghaleb, J.M. Delaye, L. Van Brutzel, Philos. Mag. 83 (2003) 1533.