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Modelling of pore coarsening in the high burn-up structure of UO2 fuel
Journal of Nuclear Materials 488 (2017) 191e195
Contents lists available at ScienceDirect
Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Modelling of pore coarsening in the high burn-up structure of UO2 fuel M.S. Veshchunov, V.I. Tarasov* Nuclear Safety Institute (IBRAE), Russian Academy of Sciences, 52, B. Tulskaya, Moscow, 115191, Russian Federation
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 January 2017 Received in revised form 5 March 2017 Accepted 6 March 2017 Available online 8 March 2017
The model for coalescence of randomly distributed immobile pores owing to their growth and impingement, applied by the authors earlier to consideration of the porosity evolution in the high burnup structure (HBS) at the UO2 fuel pellet periphery (rim zone), was further developed and validated. Predictions of the original model, taking into consideration only binary impingements of growing immobile pores, qualitatively correctly describe the decrease of the pore number density with the increase of the fractional porosity, however notably underestimate the coalescence rate at high burn-ups attained in the outmost region of the rim zone. In order to overcome this discrepancy, the next approximation of the model taking into consideration triple impingements of growing pores was developed. The advanced model provides a reasonable consent with experimental data, thus demonstrating the validity of the proposed pore coarsening mechanism in the HBS. © 2017 Elsevier B.V. All rights reserved.
1. Introduction It is generally understood that when the pellet burn-up exceeds 40e45 GWd/tM, the UO2 grains in the outer region of the fuel pellet commence to recrystallize [1e5]. The micro-structural changes are characterised by simultaneous sub-division of the grains, increase in the porosity and depletion of fission gas in the UO2 matrix close to the pellet edge. The transformed microstructure, which is referred to as the high burn-up structure (HBS), consists of small, recrystallized grains of average size 0.1e0.2 mm and a high concentration of gas filled pores of typical diameter 1e2 mm in the cold, high burn-up periphery region of the fuel pellet. This type of the HBS is first formed at the pellet rim (at a local burn-up of 60e70 GWd/tM), since the local burn-up is higher (and temperature is lower) in these positions than in radial positions closer to the pellet centre, and is therefore usually referred to as the “rim effect”. Furthermore, it has been found that coalescence, or coarsening, of very large pores might be important for the late stage of the HBS formation in the rim zone of irradiated fuel. In particular, the porosity is observed to increase with burn-up and the presence of ‘extra-large’ pores (with sizes of 4e10 mm or more) in very high burn-up samples has been detected [6e8]. In particular, detailed
* Corresponding author. E-mail address: [email protected] (V.I. Tarasov). http://dx.doi.org/10.1016/j.jnucmat.2017.03.013 0022-3115/© 2017 Elsevier B.V. All rights reserved.
experimental findings of the pore coarsening in the athermal, burnup dominated region of the fuel pellet periphery for the fuel with 98 GWd/tM average burn-up were deduced by Spino et al. [7] from optical and SEM micrographs using the Saltykov method. The calculated burn-ups at various radial positions of the pellet agreed within 95% with the values derived from the Neodymium concentrations as measured by EPMA. This allowed describing evolution of the pore populations across the radius in the fuel pellet as a function of the local burn-up. The remarkable coincidence of the porosity and the local burn-up radial profiles allowed attributing unambiguously a fully burn-up controlled process in this zone. As a result, for burn-ups above z100 GWd/tM, or, on exceeding porosity fractions of z10%, a sustained increase of the mean pore size was verified. On the other hand, the maximum in the pore number density achieved at about 100 GWd/tM at the pellet radial position r=r0 z 0.9 (where r0 is the pellet radius) signalled the onset of the pore coarsening process anticipated for the peripheral region. In this region the pore size distributions became markedly flatter and widened towards larger sizes, until they finally split into bimodal or tri-modal size distributions at the outmost positions (r=r0 ¼ 0.99 and r=r0 ¼ 1), with the mean size of each new pore population appearing at roughly a factor 2 or 3 of that of the previous population. In particular, at radial position r=r0 ¼ 1 (where the local burn-up was evaluated as z 250 GWd/tM), the mean pore size attained z 2.5 mm and doubled the value (z1.1 mm) of that at
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position r=r0 ¼ 0.9. At the outmost radial positions the major contribution to the fractional porosity aroused thus from the largest pores (sizes > 4 mm). Formation of extra-large gas pores of diameters up to 15 mm in the outmost region of the rim zone with burn-up z 300 GWd/tM was observed by Romano et al. in Ref. [8], which was interpreted as the result of a growth process, responsible for the formation of 3e4 mm pores, followed by a phase where coalescence of pores can lead to the formation of the extra-large pores, in agreement with consideration in Ref. [7]. Overpressurization of pores was associated with the continuous flow of gas atoms into the pores by fission, which can be counterbalanced by the vacancy flux only partly. Recently, a new methodology was introduced to analyse porosity data in the high burn-up structure by Cappia et al. [9]. Image analysis was coupled with the adaptive kernel density estimator to obtain a detailed characterisation of the pore size distribution, without a-priori assumption on the functional form of the distribution. Subsequently, stereological analysis was carried out. The method showed advantages compared to the classical approach based on the histogram in terms of detail in the description and accuracy within the experimental limits. Results were compared to the approximation of a log-normal distribution. In the investigated local burn-up range (80e200 GWd/tM), the agreement of the two approaches was satisfactory. From the obtained total pore number density and mean pore diameter as a function of local burn-up, pore coarsening was observed starting from z100 GWd/tM, in agreement with the previous investigation [7]. The evaluated test data [7] on the evolution of the fractional porosity, pore number density and mean size in explicit dependence on the local radial position and burn-up were collected in tables and presented in the graphical form. This provides necessary information for development and validation of numerical models for coalescence mechanisms of growing pores in the HBS of fuel pellets.
penetrable mono-size spherical cavities. The kinetic theory of this coalescence mechanism for polydisperse system of cavities was developed by Mansur et al. [13] in application to swelling of irradiated structural materials. The simplified mean-field (or monodisperse) approximation for Mansur's general model was proposed by White [14] in application to two-dimensional arrays of intergranular bubbles, which was corrected and further developed in the author's paper [15]. Basing on this approach, the coalescence rate equation for the three-dimensional system of pores was deduced in the mean-field approximation and applied to preliminary analysis (missing data [7]) of the HBS pore coarsening in the authors' recent paper [16]. Following this approach it is assumed that point defects in the irradiated fuel matrix diffuse to the randomly distributed overpressurized pores, resulting in their continuous growth. In the mean field approximation for the pores with the number density NðtÞ, mean radius RðtÞ and volume V ¼ 4pR3 =3, each pore is surrounded by a sphere with the radius 2R, in which no other pore centres can reside. Any pore centre located in this exclusion zone would find its perimeter within the perimeter of the parent pore and coalescence would occur. Any further growth of the mean volume of pores by an amount dV ¼ 4pR2 dR in the time step dt, effectively increases the volume of the exclusion zone by 8dV and opens the possibility that 8NdV pores may be swept out by the parent pore, under assumption of random (uncorrelated) spatial distribution of pores (which can be well substantiated for pores of the typical size ~ 1 mm, relatively large in comparison with the average grain size of 0.1e0.2 mm in the HBS). In that event the pore perimeters will interact with the probability dP12 ¼ 8NdV and coalescence occurs, presumably, instantaneously (on the time scale of dt). In this approximation, the total number of binary impingements is ð1=2ÞdP12 ,N ¼ 4N2 dV, where the factor of 1/2 is introduced to avoid counting each interaction twice, and the total loss of pores by coalescence following an increase in volume is given by
2. Model formulation
dN vV ¼ 4N 2 ; dt vt
Several cavity coarsening mechanisms in application to the HBS of UO2 were discussed in the literature, including coalescence of migrating cavities and thermal mechanism of Ostwald ripening. However, as noticed in Ref. [7], the first hypothesis of cavity mobility would be inapplicable for large 1 mm-size rim pores (since gas bubbles in UO2 with diameters > 2 nm become virtually immobile at temperatures below 1500 C). Similarly, the second hypothesis of Ostwald ripening might be valid only for small nanometre bubbles (with extremely high internal gas pressure), owing to very small solubility of gas atoms in the UO2 matrix. Indeed, following estimation in Ref. [10], the equilibrium gas pressure Pe for solid solution of gas atoms under typical reactor irradiation conditions attains several GPa (comparable to the internal pressure of nanometre bubbles), whereas characteristic values from 50 to 70 MPa are reasonable estimations for the internal pressure P of pores in the high-burnup fuel (cf. [8]). This leads to exponentially strong suppression of thermal effects by a factor of ðP=Pe ÞexpðbðP Pe Þ=kTÞ (cf. [10]), where b z 8.5,1029 m3/atom is the Van der Waals constant for Xe gas, which makes the Ostwald ripening mechanism completely insignificant for rim pores. The alternate possibility of coalescence of randomly distributed immobile pores owing to their growth and impingement was also discussed in Ref. [7]. It was observed in Ref. [7] that pore contacts occurred at porosity fractions between 0.10 and 0.11, very closely to theoretical estimations of Chandrasekhar [11] and Torquato [12] of the porosity limit (z0.085) for an idealized system of random
(1)
where vV=vt denotes continuous growth of a pore (by diffusion sinking of point defects) up to the moment of its coalescence with a neighbour. However, Eq. (1) does not take into consideration discontinuous growth (jump) of the pore size owing to pore instantaneous coalescence. After averaging over the pore distribution, jumps of the pore sizes will be washed away; however the growth rate of the mean pore size will be effectively enhanced. In the mean field (or monodisperse) approximation such averaging procedure can be performed following the author's approach [17], applied to coalescence of growing pores owing to their migration. At first, in the lack of pore growth (e.g., by diffusion sinking of point defects into pores), one can notice that the total pore volume due to pores collisions (by any mechanism) and coalescence cannot change, dðNVÞ=dt ¼ 0. Therefore, the fractional porosity S ¼ NV can increase only owing to diffusion growth of the pores. In this case of non-zero diffusion growth, the total pore volume balance takes the form
dS vV ¼N : dt vt
(2)
Substituting Eq. (2) into Eq. (1), one derives
dS 1 dN 0 ¼ dt 4N dt or
(3)
M.S. Veshchunov, V.I. Tarasov / Journal of Nuclear Materials 488 (2017) 191e195
dN ; dS ¼ 4N
(4)
which has a solution
N ¼ N0 expð 4ðS S0 ÞÞ;
(5)
realized in the assumption that variation of the pore number density and fractional porosity from N0 and S0 to N and S, respectively, at this stage of the pore coarsening process takes place in neglect of new pore nucleation. This equation, Eq. (5), can be used for analysis of the experimental observations [7] of the evolution of the fraction porosity and the pore number density (the number of pores per unit volume) in explicit dependence on the local burn-up in the HBS at local burnups above 100 GWd/tM, where a sustained increase of the mean pore size and decrease of the pore number density owing to the onset of the pore coarsening process was verified in the pellet radial zone r=r0 0.93 (and for this reason, the values of the pore number density and the fractional porosity at r=r0 ¼ 0.93 can be chosen as N0 and S0 , i.e. N0 z 7.6,107 mm3 and S0 z 9%). Predictions of the simplified model, Eq. (5), based on consideration of binary impingements of growing pores, are presented in Fig. 1 as a dashed line, in comparison with the experimental data [7] for the sample with the average burnup of 98 GWd/tM at different radial positions r=r0 , characterized by different burn-ups. From this figure one can see that the model qualitatively correctly describes the decrease of the pore number density with the increase of the fractional porosity; however, it notably underestimates the coalescence rate at high burn-ups, 200e250 GWd/tM (i.e. in the outmost region of the HBS, r=r0 0.98). In order to overcome this disagreement, the next approximation of the model, considering triple impingements of the growing pores, should be developed. 3. Advanced model for triple impingements In derivation of Eq. (1) it was implicitly assumed that for small enough dt the probability of any third pore to be swept out by the parent pore in dt is negligibly small. Indeed, the event corresponding to coalescence in dt among three pores has the probability, P 0 fðdP12 Þ2 ¼ ð8NdV Þ2 fðdtÞ2 , which can be neglected in
Fig. 1. Comparison of predictions of the simplified model for binary impingements (dashed line) and of the advanced model with consideration of triple impingements (solid line) with the experimental data of Spino et al. [7] (symbols for various radial positions r=r0 ).
193
comparison with the binary coalescence probability, dP12 ¼ 8NdVfdt, if dt/0 (cf. [13]). However, one should take into additional consideration that in the moment of the binary coalescence (assumed as instantaneous) pffiffiffi and formation of a new pore of twice volume (and radius 3 2R), an instantaneous increase of the exclusion zone will take place, which results in the enhancement of the probability of a third pore to be swept out. This is schematically illustrated in Fig. 2, where the cross-section of the two pores (of radii R) in the momentpof ffiffiffi their collision and the newly formed pore (a circle of radius 3 2R) are shown. The exclusion zones around the colliding pores are shown as two circles of radii 2R, and the cross-hatched area corresponds to the instantaneous increment of the exclusion zone. The total volume of the pffiffiffiadditional (cross-hatched) zones is DV ¼ xV, where x ¼ 9ð2 3 2Þ=4z1:67 (see Appendix), and the probability for a third pore to be swept out is P3 ¼ xNV; therefore, the probability of a triple impingement is dP123 ¼ P3 dP12 ¼ dP12 xNV, whereas the total number of triple impingements in unit volume per unit time becomes equal to 1 dP12 xNV N ¼ 4N 2 xS vV . vt 2 dt Since the number of pores after each triple coalescence is reduced by 2 (from 3 to 1), the equation for the total loss of pores by coalescence, Eq. (1), takes the form
dN vV vV ¼ 4N 2 2 4N 2 xS : dt vt vt
(6)
Excluding vV=vt in the right hand side of Eq. (6) using Eq. (2) one obtains
dN ; ð1 þ 2xSÞdS ¼ 4N
(7)
which has a solution
N ¼ N0 exp 4ðS S0 Þ 4x S2 S20 zN0 exp 4ðS S0 Þ 6:68 S2 S20 : (8) It is straightforward to show that the higher order, quadruple impingements contribute to Eq. (8) proportionally to the third
Fig. 2. To calculation of the instantaneous increase of the exclusion zone after collision and coalescence of two pores (cross-hatched areas).
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power of the small parameter S, and thus can be neglected with a good accuracy in the experimentally observed range of the fractional porosity, S 0.24. The new solution, Eq. (8), represented in Fig. 1 as a solid line provides a much better agreement with the experimental data [7] for high burn-ups attained in the outmost region of the rim zone. This allows a conclusion on validity of the proposed coalescence mechanism for pores in the HBS of UO2 fuel, despite the mean-field (monodisperse) consideration applied in the current analysis is rather rough and can be considered only as the first approximation. For more detailed and precise analysis, the full polydisperse scheme taking into consideration evolution of the pore size distribution function, should be applied. This approach is foreseen as the next step of the model development, which is assumed to be carried out on the base of the most recent experimental data [9], including more accurate measurements of the pore distribution function in the HBS. Currently validation of the model predictions, Eq. (8), against the new measurements was not attempted, lacking in Ref. [9] explicit information on dependency of the fraction porosity from burn-up. This stage of the model development and validation will be performed, when these additional data are published. 4. Discussion In the analysis of the pore coarsening mechanism only relatively large pores contributing to the fractional porosity should be taken into consideration. For instance, as noticed in Ref. [7], at the outmost radial positions, r=r0 ¼ 0.99 and r=r0 ¼ 1, the major contribution to the fractional porosity arises from the largest pores (sizes > 4 mm); whereas pores with diameters around or below 1 mm, although more abundant, contribute only slightly to the cavity volume fraction. For this reason, for evaluation of the number density of pores, N, contributing to the pore coalescence rate, Eq. (6), a corresponding cut-off limit for pore sizes should be determined from the experimental data. In particular, for the above mentioned outmost zone this cut-off limit could be roughly evaluated as ~1 mm, which fairly corresponds to the resolution limit of optical microscopy in Ref. [7]. More precise evaluation depends on the accuracy of the porosity measurements, i.e. small pores with the contribution to the total void volume less than the evaluated accuracy of the porosity measurements should be excluded from consideration of the coalescence mechanism. On the other hand, the true distribution of pore sizes could extend to the atomic scale, which requires an increasing resolution of scanning electron microscopy (SEM) [9]. For instance, as noticed in Ref. [9], small pores might be relevant for development of bubble nucleation models in the HBS. For this reason, both ranges (micron and submicron) of the pore size distribution should be thoroughly investigated. However, for analysis of the pore coarsening mechanism the proper cut-off limit should be determined and used either for evaluation of the pore number density in Eq. (8), or of the truncated size distribution function in a more accurate polydisperse model. In particular, this will allow reducing (or avoiding) some discrepancy between results of the two tests [7] and [9] concerning the estimated pore number density (which was notably higher in Ref. [9]) and the mean pore radius (which was smaller in Ref. [9]). As noted in Ref. [9], this discrepancy was likely to arise from the different magnifications used to acquire the data, which resulted in different resolution limits. Higher magnifications used in Ref. [9] in order to increase the accuracy in the measurements of small pores, resulted in higher pore number density and smaller mean diameter for similar fractional porosities in the two tests. Therefore, it seems rather important for the current analysis to determine the pore size
cut-off limits in the two tests and, on this base, to re-evaluate the pore number density, NðrÞ, entering the solution, Eq. (8), of the coalescence rate equation. 5. Conclusions The kinetic model for the pore coalescence in the rim zone of a high burn-up UO2 fuel pellet proposed by the authors in Ref. [16] was further developed and validated using the experimental data [7]. The rate equation, deduced in the mean-field (monodisperse) approximation from the general polydisperse model of Mansur et al. [13], for coalescence of randomly distributed immobile pores owing to their growth and impingement, was applied to analysis of the HBS. Predictions of the simplified model, taking into consideration only binary impingements of growing pores, qualitatively correctly describe the decrease of the pore number density with the increase of the fractional porosity observed in Ref. [7], however notably underestimate the coalescence rate at high burn-ups, 200e250 GWd/tM, attained in the outmost region of the HBS, r=r0 0.98. In order to overcome this divergence, the next approximation of the model considering triple impingements of growing pores was developed. The advanced model provides much better agreement with the experimental data [7], thus demonstrating the validity of the proposed coarsening mechanism in the HBS, despite the meanfield approach applied in the current analysis can be considered only as the first approximation. For more detailed and precise consideration, the full polydisperse scheme considering evolution of the pore size distribution function taking into account triple impingements should be applied. This approach is foreseen as the next step of the model development, which is assumed to be carried out on the base of the most recent experimental data [9] with more accurate measurements of the pore distribution function in the HBS, when additional experimental data, required for the model validation, are published. Acknowledgements The authors thank Ms. F. Cappia and Dr. V. V. Rondinella (JRCITU) for valuable discussions and information exchanges, and Mr. P. Polovnikov (IBRAE) for his kind assistance in analysis of experimental data. Appendix Any pore of radius R with its centre located in the additional exclusion zone, instantly formed after coalescence of two neighbouring pores of radii R and corresponding to the cross-hatched areas in Fig. 2, could not interact with each of the two parent pores and would find its perimeter within the perimeter of the pffiffiffi newly formed pore of radius 3 2R, resulting in a triple coalescence. The volume of this additional exclusion zone can be calculated as
DV ¼ V12 V1∩12 V2∩12 þ V1∩2 pffiffiffi 3 ffiffiffi 4pp 3 3 ¼ 2 þ 1 R3 2u 2R; 2 þ 1 R; R þ uð2R; 2R; 2RÞ 3 p p ffiffiffi. ffiffiffi. 4p 3 3 3 R ,9 2 2 4 ¼ V,9 2 2 4; ¼ 3 (A.1) where pffiffiffi V12 is the volume of the newly formed sphere ‘12’ of radius ð 3 2 þ 1ÞR, V1 and V2 are the volumes of two spheres ‘1’ and ‘2’ of equal radii 2R, VA∩B is the volume of intersection of spheres A and B;
M.S. Veshchunov, V.I. Tarasov / Journal of Nuclear Materials 488 (2017) 191e195
it is also taken into account that the intersection zone of the three spheres ‘1’, ‘2’ and ‘12’ coincides with the intersection zone of the two spheres ‘1’ and ‘2’;
uðx; y; zÞ≡
pðx þ y zÞ2 ðx þ y þ zÞ2 4 x2 xy þ y2 12z
; (A.2)
is the volume of the intersection zone of two spheres of radii x and y which are from each other at distance z [18]. References [1] C.T. Walker, T. Kameyama, S. Kitajima, M. Kinoshita, Concerning the microstructure changes that occur at the surface of UO2 pellets on irradiation to high burnup, J. Nucl. Mater 188 (1992) 73. [2] M.E. Cunningham, M.D. Freshley, D.D. Lanning, Development and characteristics of the rim region in high burnup UO2 fuel pellets, J. Nucl. Mat. 188 (1992) 19e27. [3] Hj Matzke, On the rim effect in high burnup UO2 LWR fuels, J. Nucl. Mater 189 (1992) 141e146. [4] K. Une, K. Nogita, S. Kashibe, M. Imamura, Microstructural change and its influence on fission gas release in high burnup UO2 fuel, J. Nucl. Mater 188 (1992) 65e72. [5] J. Spino, K. Vennix, M. Coquerelle, Detailed characterisation of the rim microstructure in PWR fuels in the burn-up range 40e67 GWd/tM, J. Nucl. Mater 231 (1996) 179.
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[6] R. Manzel, C.T. Walker, EPMA and SEM of fuel samples from PWR rods with an average burn-up of around 100 MWd/kgHM, J. Nucl. Mater 301 (2002) 170. [7] J. Spino, A.D. Stalios, H. Santa Cruz, D. Baron, Stereological evolution of the rim structure in PWR-fuels at prolonged irradiation: dependencies with burn-up and temperature, J. Nucl. Mater 354 (2006) 66. [8] A. Romano, M.I. Horvath, R. Restani, Evolution of porosity in the high-burnup fuel structure, J. Nucl. Mater 361 (2007) 62e68. [9] F. Cappia, D. Pizzocri, A. Schubert, P. Van Uffelen, G. Paperini, D. Pellottiero, R. Maci_an-Juan, V.V. Rondinella, Critical assessment of the pore size distribution in the rim region of high burnup UO2 fuels, J. Nucl. Mater 480 (2016) 138e149. [10] M.S. Veshchunov, On the theory of fission gas bubble evolution in irradiated UO2 fuel, J. Nucl. Mater 277 (2000) 67e81. [11] S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys. 15 (1943) 1. [12] S. Torquato, Random Heterogeneous Materials e Microstructure and Macroscopic Properties, Springer, 2002. [13] L.K. Mansur, Void swelling in metals and alloys under irradiation: an assessment of the theory, Nucl. Technol. 40 (1978) 5e34. [14] R.J. White, The development of grain-face porosity in irradiated oxide fuel, J. Nucl. Mater 325 (2004) 61e77. [15] M.S. Veshchunov, Modelling of grain face bubbles coalescence in irradiated UO2 fuel, J. Nucl. Mater 374 (2008) 44e53. [16] V.I. Tarasov, M.S. Veshchunov, Models for fuel porosity evolution in UO2 under various regimes of reactor operation, Nucl. Eng. Des. 272 (2014) 65e83. [17] M.S. Veshchunov, A new model of grain growth kinetics in UO2 fuel pellets. Part 2. Normal grain growth kinetics controlled by pore migration, J. Nucl. Mater 346 (2005) 220e225. [18] L.S. Chkhartishvili, Analytical optimization of the lattice parameter using the binding energy calculated in the quasi-classical approximation, Phys. Solid State 48 (2006) 846e853.
Contents lists available at ScienceDirect
Journal of Nuclear Materials journal homepage: www.elsevier.com/locate/jnucmat
Modelling of pore coarsening in the high burn-up structure of UO2 fuel M.S. Veshchunov, V.I. Tarasov* Nuclear Safety Institute (IBRAE), Russian Academy of Sciences, 52, B. Tulskaya, Moscow, 115191, Russian Federation
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 January 2017 Received in revised form 5 March 2017 Accepted 6 March 2017 Available online 8 March 2017
The model for coalescence of randomly distributed immobile pores owing to their growth and impingement, applied by the authors earlier to consideration of the porosity evolution in the high burnup structure (HBS) at the UO2 fuel pellet periphery (rim zone), was further developed and validated. Predictions of the original model, taking into consideration only binary impingements of growing immobile pores, qualitatively correctly describe the decrease of the pore number density with the increase of the fractional porosity, however notably underestimate the coalescence rate at high burn-ups attained in the outmost region of the rim zone. In order to overcome this discrepancy, the next approximation of the model taking into consideration triple impingements of growing pores was developed. The advanced model provides a reasonable consent with experimental data, thus demonstrating the validity of the proposed pore coarsening mechanism in the HBS. © 2017 Elsevier B.V. All rights reserved.
1. Introduction It is generally understood that when the pellet burn-up exceeds 40e45 GWd/tM, the UO2 grains in the outer region of the fuel pellet commence to recrystallize [1e5]. The micro-structural changes are characterised by simultaneous sub-division of the grains, increase in the porosity and depletion of fission gas in the UO2 matrix close to the pellet edge. The transformed microstructure, which is referred to as the high burn-up structure (HBS), consists of small, recrystallized grains of average size 0.1e0.2 mm and a high concentration of gas filled pores of typical diameter 1e2 mm in the cold, high burn-up periphery region of the fuel pellet. This type of the HBS is first formed at the pellet rim (at a local burn-up of 60e70 GWd/tM), since the local burn-up is higher (and temperature is lower) in these positions than in radial positions closer to the pellet centre, and is therefore usually referred to as the “rim effect”. Furthermore, it has been found that coalescence, or coarsening, of very large pores might be important for the late stage of the HBS formation in the rim zone of irradiated fuel. In particular, the porosity is observed to increase with burn-up and the presence of ‘extra-large’ pores (with sizes of 4e10 mm or more) in very high burn-up samples has been detected [6e8]. In particular, detailed
* Corresponding author. E-mail address: [email protected] (V.I. Tarasov). http://dx.doi.org/10.1016/j.jnucmat.2017.03.013 0022-3115/© 2017 Elsevier B.V. All rights reserved.
experimental findings of the pore coarsening in the athermal, burnup dominated region of the fuel pellet periphery for the fuel with 98 GWd/tM average burn-up were deduced by Spino et al. [7] from optical and SEM micrographs using the Saltykov method. The calculated burn-ups at various radial positions of the pellet agreed within 95% with the values derived from the Neodymium concentrations as measured by EPMA. This allowed describing evolution of the pore populations across the radius in the fuel pellet as a function of the local burn-up. The remarkable coincidence of the porosity and the local burn-up radial profiles allowed attributing unambiguously a fully burn-up controlled process in this zone. As a result, for burn-ups above z100 GWd/tM, or, on exceeding porosity fractions of z10%, a sustained increase of the mean pore size was verified. On the other hand, the maximum in the pore number density achieved at about 100 GWd/tM at the pellet radial position r=r0 z 0.9 (where r0 is the pellet radius) signalled the onset of the pore coarsening process anticipated for the peripheral region. In this region the pore size distributions became markedly flatter and widened towards larger sizes, until they finally split into bimodal or tri-modal size distributions at the outmost positions (r=r0 ¼ 0.99 and r=r0 ¼ 1), with the mean size of each new pore population appearing at roughly a factor 2 or 3 of that of the previous population. In particular, at radial position r=r0 ¼ 1 (where the local burn-up was evaluated as z 250 GWd/tM), the mean pore size attained z 2.5 mm and doubled the value (z1.1 mm) of that at
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position r=r0 ¼ 0.9. At the outmost radial positions the major contribution to the fractional porosity aroused thus from the largest pores (sizes > 4 mm). Formation of extra-large gas pores of diameters up to 15 mm in the outmost region of the rim zone with burn-up z 300 GWd/tM was observed by Romano et al. in Ref. [8], which was interpreted as the result of a growth process, responsible for the formation of 3e4 mm pores, followed by a phase where coalescence of pores can lead to the formation of the extra-large pores, in agreement with consideration in Ref. [7]. Overpressurization of pores was associated with the continuous flow of gas atoms into the pores by fission, which can be counterbalanced by the vacancy flux only partly. Recently, a new methodology was introduced to analyse porosity data in the high burn-up structure by Cappia et al. [9]. Image analysis was coupled with the adaptive kernel density estimator to obtain a detailed characterisation of the pore size distribution, without a-priori assumption on the functional form of the distribution. Subsequently, stereological analysis was carried out. The method showed advantages compared to the classical approach based on the histogram in terms of detail in the description and accuracy within the experimental limits. Results were compared to the approximation of a log-normal distribution. In the investigated local burn-up range (80e200 GWd/tM), the agreement of the two approaches was satisfactory. From the obtained total pore number density and mean pore diameter as a function of local burn-up, pore coarsening was observed starting from z100 GWd/tM, in agreement with the previous investigation [7]. The evaluated test data [7] on the evolution of the fractional porosity, pore number density and mean size in explicit dependence on the local radial position and burn-up were collected in tables and presented in the graphical form. This provides necessary information for development and validation of numerical models for coalescence mechanisms of growing pores in the HBS of fuel pellets.
penetrable mono-size spherical cavities. The kinetic theory of this coalescence mechanism for polydisperse system of cavities was developed by Mansur et al. [13] in application to swelling of irradiated structural materials. The simplified mean-field (or monodisperse) approximation for Mansur's general model was proposed by White [14] in application to two-dimensional arrays of intergranular bubbles, which was corrected and further developed in the author's paper [15]. Basing on this approach, the coalescence rate equation for the three-dimensional system of pores was deduced in the mean-field approximation and applied to preliminary analysis (missing data [7]) of the HBS pore coarsening in the authors' recent paper [16]. Following this approach it is assumed that point defects in the irradiated fuel matrix diffuse to the randomly distributed overpressurized pores, resulting in their continuous growth. In the mean field approximation for the pores with the number density NðtÞ, mean radius RðtÞ and volume V ¼ 4pR3 =3, each pore is surrounded by a sphere with the radius 2R, in which no other pore centres can reside. Any pore centre located in this exclusion zone would find its perimeter within the perimeter of the parent pore and coalescence would occur. Any further growth of the mean volume of pores by an amount dV ¼ 4pR2 dR in the time step dt, effectively increases the volume of the exclusion zone by 8dV and opens the possibility that 8NdV pores may be swept out by the parent pore, under assumption of random (uncorrelated) spatial distribution of pores (which can be well substantiated for pores of the typical size ~ 1 mm, relatively large in comparison with the average grain size of 0.1e0.2 mm in the HBS). In that event the pore perimeters will interact with the probability dP12 ¼ 8NdV and coalescence occurs, presumably, instantaneously (on the time scale of dt). In this approximation, the total number of binary impingements is ð1=2ÞdP12 ,N ¼ 4N2 dV, where the factor of 1/2 is introduced to avoid counting each interaction twice, and the total loss of pores by coalescence following an increase in volume is given by
2. Model formulation
dN vV ¼ 4N 2 ; dt vt
Several cavity coarsening mechanisms in application to the HBS of UO2 were discussed in the literature, including coalescence of migrating cavities and thermal mechanism of Ostwald ripening. However, as noticed in Ref. [7], the first hypothesis of cavity mobility would be inapplicable for large 1 mm-size rim pores (since gas bubbles in UO2 with diameters > 2 nm become virtually immobile at temperatures below 1500 C). Similarly, the second hypothesis of Ostwald ripening might be valid only for small nanometre bubbles (with extremely high internal gas pressure), owing to very small solubility of gas atoms in the UO2 matrix. Indeed, following estimation in Ref. [10], the equilibrium gas pressure Pe for solid solution of gas atoms under typical reactor irradiation conditions attains several GPa (comparable to the internal pressure of nanometre bubbles), whereas characteristic values from 50 to 70 MPa are reasonable estimations for the internal pressure P of pores in the high-burnup fuel (cf. [8]). This leads to exponentially strong suppression of thermal effects by a factor of ðP=Pe ÞexpðbðP Pe Þ=kTÞ (cf. [10]), where b z 8.5,1029 m3/atom is the Van der Waals constant for Xe gas, which makes the Ostwald ripening mechanism completely insignificant for rim pores. The alternate possibility of coalescence of randomly distributed immobile pores owing to their growth and impingement was also discussed in Ref. [7]. It was observed in Ref. [7] that pore contacts occurred at porosity fractions between 0.10 and 0.11, very closely to theoretical estimations of Chandrasekhar [11] and Torquato [12] of the porosity limit (z0.085) for an idealized system of random
(1)
where vV=vt denotes continuous growth of a pore (by diffusion sinking of point defects) up to the moment of its coalescence with a neighbour. However, Eq. (1) does not take into consideration discontinuous growth (jump) of the pore size owing to pore instantaneous coalescence. After averaging over the pore distribution, jumps of the pore sizes will be washed away; however the growth rate of the mean pore size will be effectively enhanced. In the mean field (or monodisperse) approximation such averaging procedure can be performed following the author's approach [17], applied to coalescence of growing pores owing to their migration. At first, in the lack of pore growth (e.g., by diffusion sinking of point defects into pores), one can notice that the total pore volume due to pores collisions (by any mechanism) and coalescence cannot change, dðNVÞ=dt ¼ 0. Therefore, the fractional porosity S ¼ NV can increase only owing to diffusion growth of the pores. In this case of non-zero diffusion growth, the total pore volume balance takes the form
dS vV ¼N : dt vt
(2)
Substituting Eq. (2) into Eq. (1), one derives
dS 1 dN 0 ¼ dt 4N dt or
(3)
M.S. Veshchunov, V.I. Tarasov / Journal of Nuclear Materials 488 (2017) 191e195
dN ; dS ¼ 4N
(4)
which has a solution
N ¼ N0 expð 4ðS S0 ÞÞ;
(5)
realized in the assumption that variation of the pore number density and fractional porosity from N0 and S0 to N and S, respectively, at this stage of the pore coarsening process takes place in neglect of new pore nucleation. This equation, Eq. (5), can be used for analysis of the experimental observations [7] of the evolution of the fraction porosity and the pore number density (the number of pores per unit volume) in explicit dependence on the local burn-up in the HBS at local burnups above 100 GWd/tM, where a sustained increase of the mean pore size and decrease of the pore number density owing to the onset of the pore coarsening process was verified in the pellet radial zone r=r0 0.93 (and for this reason, the values of the pore number density and the fractional porosity at r=r0 ¼ 0.93 can be chosen as N0 and S0 , i.e. N0 z 7.6,107 mm3 and S0 z 9%). Predictions of the simplified model, Eq. (5), based on consideration of binary impingements of growing pores, are presented in Fig. 1 as a dashed line, in comparison with the experimental data [7] for the sample with the average burnup of 98 GWd/tM at different radial positions r=r0 , characterized by different burn-ups. From this figure one can see that the model qualitatively correctly describes the decrease of the pore number density with the increase of the fractional porosity; however, it notably underestimates the coalescence rate at high burn-ups, 200e250 GWd/tM (i.e. in the outmost region of the HBS, r=r0 0.98). In order to overcome this disagreement, the next approximation of the model, considering triple impingements of the growing pores, should be developed. 3. Advanced model for triple impingements In derivation of Eq. (1) it was implicitly assumed that for small enough dt the probability of any third pore to be swept out by the parent pore in dt is negligibly small. Indeed, the event corresponding to coalescence in dt among three pores has the probability, P 0 fðdP12 Þ2 ¼ ð8NdV Þ2 fðdtÞ2 , which can be neglected in
Fig. 1. Comparison of predictions of the simplified model for binary impingements (dashed line) and of the advanced model with consideration of triple impingements (solid line) with the experimental data of Spino et al. [7] (symbols for various radial positions r=r0 ).
193
comparison with the binary coalescence probability, dP12 ¼ 8NdVfdt, if dt/0 (cf. [13]). However, one should take into additional consideration that in the moment of the binary coalescence (assumed as instantaneous) pffiffiffi and formation of a new pore of twice volume (and radius 3 2R), an instantaneous increase of the exclusion zone will take place, which results in the enhancement of the probability of a third pore to be swept out. This is schematically illustrated in Fig. 2, where the cross-section of the two pores (of radii R) in the momentpof ffiffiffi their collision and the newly formed pore (a circle of radius 3 2R) are shown. The exclusion zones around the colliding pores are shown as two circles of radii 2R, and the cross-hatched area corresponds to the instantaneous increment of the exclusion zone. The total volume of the pffiffiffiadditional (cross-hatched) zones is DV ¼ xV, where x ¼ 9ð2 3 2Þ=4z1:67 (see Appendix), and the probability for a third pore to be swept out is P3 ¼ xNV; therefore, the probability of a triple impingement is dP123 ¼ P3 dP12 ¼ dP12 xNV, whereas the total number of triple impingements in unit volume per unit time becomes equal to 1 dP12 xNV N ¼ 4N 2 xS vV . vt 2 dt Since the number of pores after each triple coalescence is reduced by 2 (from 3 to 1), the equation for the total loss of pores by coalescence, Eq. (1), takes the form
dN vV vV ¼ 4N 2 2 4N 2 xS : dt vt vt
(6)
Excluding vV=vt in the right hand side of Eq. (6) using Eq. (2) one obtains
dN ; ð1 þ 2xSÞdS ¼ 4N
(7)
which has a solution
N ¼ N0 exp 4ðS S0 Þ 4x S2 S20 zN0 exp 4ðS S0 Þ 6:68 S2 S20 : (8) It is straightforward to show that the higher order, quadruple impingements contribute to Eq. (8) proportionally to the third
Fig. 2. To calculation of the instantaneous increase of the exclusion zone after collision and coalescence of two pores (cross-hatched areas).
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power of the small parameter S, and thus can be neglected with a good accuracy in the experimentally observed range of the fractional porosity, S 0.24. The new solution, Eq. (8), represented in Fig. 1 as a solid line provides a much better agreement with the experimental data [7] for high burn-ups attained in the outmost region of the rim zone. This allows a conclusion on validity of the proposed coalescence mechanism for pores in the HBS of UO2 fuel, despite the mean-field (monodisperse) consideration applied in the current analysis is rather rough and can be considered only as the first approximation. For more detailed and precise analysis, the full polydisperse scheme taking into consideration evolution of the pore size distribution function, should be applied. This approach is foreseen as the next step of the model development, which is assumed to be carried out on the base of the most recent experimental data [9], including more accurate measurements of the pore distribution function in the HBS. Currently validation of the model predictions, Eq. (8), against the new measurements was not attempted, lacking in Ref. [9] explicit information on dependency of the fraction porosity from burn-up. This stage of the model development and validation will be performed, when these additional data are published. 4. Discussion In the analysis of the pore coarsening mechanism only relatively large pores contributing to the fractional porosity should be taken into consideration. For instance, as noticed in Ref. [7], at the outmost radial positions, r=r0 ¼ 0.99 and r=r0 ¼ 1, the major contribution to the fractional porosity arises from the largest pores (sizes > 4 mm); whereas pores with diameters around or below 1 mm, although more abundant, contribute only slightly to the cavity volume fraction. For this reason, for evaluation of the number density of pores, N, contributing to the pore coalescence rate, Eq. (6), a corresponding cut-off limit for pore sizes should be determined from the experimental data. In particular, for the above mentioned outmost zone this cut-off limit could be roughly evaluated as ~1 mm, which fairly corresponds to the resolution limit of optical microscopy in Ref. [7]. More precise evaluation depends on the accuracy of the porosity measurements, i.e. small pores with the contribution to the total void volume less than the evaluated accuracy of the porosity measurements should be excluded from consideration of the coalescence mechanism. On the other hand, the true distribution of pore sizes could extend to the atomic scale, which requires an increasing resolution of scanning electron microscopy (SEM) [9]. For instance, as noticed in Ref. [9], small pores might be relevant for development of bubble nucleation models in the HBS. For this reason, both ranges (micron and submicron) of the pore size distribution should be thoroughly investigated. However, for analysis of the pore coarsening mechanism the proper cut-off limit should be determined and used either for evaluation of the pore number density in Eq. (8), or of the truncated size distribution function in a more accurate polydisperse model. In particular, this will allow reducing (or avoiding) some discrepancy between results of the two tests [7] and [9] concerning the estimated pore number density (which was notably higher in Ref. [9]) and the mean pore radius (which was smaller in Ref. [9]). As noted in Ref. [9], this discrepancy was likely to arise from the different magnifications used to acquire the data, which resulted in different resolution limits. Higher magnifications used in Ref. [9] in order to increase the accuracy in the measurements of small pores, resulted in higher pore number density and smaller mean diameter for similar fractional porosities in the two tests. Therefore, it seems rather important for the current analysis to determine the pore size
cut-off limits in the two tests and, on this base, to re-evaluate the pore number density, NðrÞ, entering the solution, Eq. (8), of the coalescence rate equation. 5. Conclusions The kinetic model for the pore coalescence in the rim zone of a high burn-up UO2 fuel pellet proposed by the authors in Ref. [16] was further developed and validated using the experimental data [7]. The rate equation, deduced in the mean-field (monodisperse) approximation from the general polydisperse model of Mansur et al. [13], for coalescence of randomly distributed immobile pores owing to their growth and impingement, was applied to analysis of the HBS. Predictions of the simplified model, taking into consideration only binary impingements of growing pores, qualitatively correctly describe the decrease of the pore number density with the increase of the fractional porosity observed in Ref. [7], however notably underestimate the coalescence rate at high burn-ups, 200e250 GWd/tM, attained in the outmost region of the HBS, r=r0 0.98. In order to overcome this divergence, the next approximation of the model considering triple impingements of growing pores was developed. The advanced model provides much better agreement with the experimental data [7], thus demonstrating the validity of the proposed coarsening mechanism in the HBS, despite the meanfield approach applied in the current analysis can be considered only as the first approximation. For more detailed and precise consideration, the full polydisperse scheme considering evolution of the pore size distribution function taking into account triple impingements should be applied. This approach is foreseen as the next step of the model development, which is assumed to be carried out on the base of the most recent experimental data [9] with more accurate measurements of the pore distribution function in the HBS, when additional experimental data, required for the model validation, are published. Acknowledgements The authors thank Ms. F. Cappia and Dr. V. V. Rondinella (JRCITU) for valuable discussions and information exchanges, and Mr. P. Polovnikov (IBRAE) for his kind assistance in analysis of experimental data. Appendix Any pore of radius R with its centre located in the additional exclusion zone, instantly formed after coalescence of two neighbouring pores of radii R and corresponding to the cross-hatched areas in Fig. 2, could not interact with each of the two parent pores and would find its perimeter within the perimeter of the pffiffiffi newly formed pore of radius 3 2R, resulting in a triple coalescence. The volume of this additional exclusion zone can be calculated as
DV ¼ V12 V1∩12 V2∩12 þ V1∩2 pffiffiffi 3 ffiffiffi 4pp 3 3 ¼ 2 þ 1 R3 2u 2R; 2 þ 1 R; R þ uð2R; 2R; 2RÞ 3 p p ffiffiffi. ffiffiffi. 4p 3 3 3 R ,9 2 2 4 ¼ V,9 2 2 4; ¼ 3 (A.1) where pffiffiffi V12 is the volume of the newly formed sphere ‘12’ of radius ð 3 2 þ 1ÞR, V1 and V2 are the volumes of two spheres ‘1’ and ‘2’ of equal radii 2R, VA∩B is the volume of intersection of spheres A and B;
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it is also taken into account that the intersection zone of the three spheres ‘1’, ‘2’ and ‘12’ coincides with the intersection zone of the two spheres ‘1’ and ‘2’;
uðx; y; zÞ≡
pðx þ y zÞ2 ðx þ y þ zÞ2 4 x2 xy þ y2 12z
; (A.2)
is the volume of the intersection zone of two spheres of radii x and y which are from each other at distance z [18]. References [1] C.T. Walker, T. Kameyama, S. Kitajima, M. Kinoshita, Concerning the microstructure changes that occur at the surface of UO2 pellets on irradiation to high burnup, J. Nucl. Mater 188 (1992) 73. [2] M.E. Cunningham, M.D. Freshley, D.D. Lanning, Development and characteristics of the rim region in high burnup UO2 fuel pellets, J. Nucl. Mat. 188 (1992) 19e27. [3] Hj Matzke, On the rim effect in high burnup UO2 LWR fuels, J. Nucl. Mater 189 (1992) 141e146. [4] K. Une, K. Nogita, S. Kashibe, M. Imamura, Microstructural change and its influence on fission gas release in high burnup UO2 fuel, J. Nucl. Mater 188 (1992) 65e72. [5] J. Spino, K. Vennix, M. Coquerelle, Detailed characterisation of the rim microstructure in PWR fuels in the burn-up range 40e67 GWd/tM, J. Nucl. Mater 231 (1996) 179.
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