Effect of grain size on gas bubble evolution in nuclear fuel:Phase-field investigations

Dan Sun(孙丹), Qingfeng Yang(杨青峰), Jiajun Zhao(赵家珺), Shixin Gao(高士鑫),Yong Xin(辛勇), Yi Zhou(周毅), Chunyu Yin(尹春雨), Ping Chen(陈平),†,Jijun Zhao(赵纪军), and Yuanyuan Wang(王园园),‡

1Science and Technology on Reactor System Design Technology Laboratory,Nuclear Power Institute of China,Chengdu 610213,China

2Key Laboratory of Materials Modification by Laser,Ion,and Electron Beams,Dalian University of Technology,Dalian 116024,China

Keywords: grain size,point defects,fission gas bubble

1.Introduction

Since the Fukushima accident occurs, the development of accident-tolerant fuels (ATFs) has been a more prominent topic of interest for improving the safety, competitiveness and economics of commercial nuclear power.[1]Large-grained UO2is a promising candidate to replace the traditional UO2fuel, as its larger grain size may reduce irradiation-induced swelling and fission gas release.[2]In addition, U3Si2fuel is also a potentially attractive substitute for commercial UO2fuel due to its higher thermal conductivity and uranium density.[2]These excellent properties can not only extend the refuelling cycle but also have a promising potential in reducing fuel enrichment.[3,4]Recently,more attention has been paid on the irradiation swelling behavior of large-grained UO2and U3Si2.

Noirotet al.[5]compared the microstructures of coarsegrained UO2and conventional-grained UO2at the burnups of 72 GWd/tHM and 76 GWd/tHM, respectively.The bubbles and grain refinement were observed in the standard-grained UO2,while in the large-grained UO2,bubbles only formed at the center of grains and on the grain boundaries (GBs), indicating that coarse grains can effectively improve the irradiation resistance of UO2.The irradiation experiments performed by AREVA[6,7]indicate that fission gas release and fuel swelling are reduced by doping Cr2O3in the large-grained UO2.In the Cr2O3-doped UO2with large grain size,fission gas is mainly dissolved in the grains rather than on the GBs, which is different from that in the conventional-grained UO2.It can be attributed to the fact that the larger grain size increases the diffusion distance of fission gas from grain interior to GB,improving irradiation swelling resistance.Cooperet al.[8]investigated the influence of Cr on the diffusion rate of Xe atom in UO2using the cluster dynamics method,which is called by the BISON program to study the behavior of fission gas release in the doped UO2.

Moreover, the irradiation properties of U3Si2are examined and analyzed.According to the microstructure of irradiated U3Si2/Al dispersion fuel, the fission bubbles are uniformly distributed within the U3Si2fuel particles.The mean diameter of bubbles is approximately 94 nm,leading to～11%increase of the volume fraction.[9]Taking account of the experimental results of irradiated U3Si2fuel in the light water reactor,[10,11]the bubble density and size are assessed by a fission gas model.It is assumed that fission gas release is controlled by the bubbles on the GB, and the sensitivity of the model parameters is additionally analyzed.On the other hand,a rate theory model of fission gas behavior is developed and applied to explore the swelling of U3Si2fuel.[12]The amorphous U3Si2fuel has also been observed under irradiation,and hence, Rest[13]established a model to study the behavior of fission gas bubble in the amorphous U3Si2.

Despite the extensive research has been carried out on the irradiation performance of large-grained UO2and U3Si2, to authors’knowledge,the evolution of fission gas bubble in UO2and U3Si2fuels associated with different grain sizes under different fission rates and temperatures is still waiting to be identified.Nowadays,at the atomic scale,ab initiois widely used to simulate the thermal dynamics behaviors of irradiationinduced point defects(i.e., vacancy and self-interstitial atom)and fission gas atom in both UO2and U3Si2.[14–16]The phasefield approach considered as a typical mesoscopic method is employed to understand the formation and growth of bubbles and the gas swelling in the nuclear fuel.[17–20]For examples,Aagesenet al.[18]established a phase-field model by tracking the concentrations of vacancy and gas atom to simulate the fission bubbles in UO2fuel.Wanget al.[19]investigated the effects of vacancy concentration,the generation rate of Xe atom and the temperature gradient on the evolution of bubbles in UO2by means of the phase-field model.In addition, the morphology structure of intergranular fission gas bubbles in U3Si2is also investigated using the phase-field method.[20]

Hence, in current work,ab initiocalculation is first performed to obtain the energetic values(e.g.,formation energy)of vacancy, self-interstitial and fission gas atom in U3Si2and UO2, respectively.A comparison of bubble density, size and porosity of U3Si2and UO2is presented under different temperatures and fission rates.The difference of bubble evolution characteristics under different grain sizes is shown.Moreover,the corresponding thermal conductivities of bubble-containing U3Si2are evaluated by the established steady heat transfer model.The predicted results can provide fundamental understanding on the mechanics of bubble formation and evolution,which is helpful for the future research on the irradiation swelling of U3Si2and UO2fuels.

2.Computational framework

2.1.Point defect energies from atomic simulation

All the calculations are performed within the framework of density functional theory (DFT) using the projector augmented wave method implemented in Viennaab initiosimulation package (VASP).[21]We adopt the projector-augmented wave (PAW) potentials for the ion–electron interaction.[22]The generalized gradient approximation(GGA)in the parameterization by Perdew, Burke, and Ernzerh (PBE) of functional is used to describe the electron exchange–correlation interactions.[23]Moreover, the effects due to the localization of 5f electrons of uranium in UO2and U3Si2were treated with the GGA+Uapproximation, the value ofUeffis set as 4 eV for both UO2and U3Si2.[24–26]The calculated equilibrium lattice constants of UO2(a=c=5.57 ˚A,b=5.50 ˚A)are in reasonable agreement with the experiment data[27]and previous theoretical values.[28]The lattice constants of U3Si2(a=b=7.33 ˚A,c=3.90 ˚A)are using the experiment data.[29]The structures of UO2and U3Si2have been presented in Fig.1,respectively.We use a 96-atom UO2supercell formed by 2×2×2 unit cells and a 160-atom U3Si2supercell formed by 2×2×4 unit cells.The cutoff energy of plane wave is set as 500 eV.The Brillion zone integrations for UO2and U3Si2are performed using 4×4×5 and 3×3×3 Monkhorst–Pack grid,respectively.All atomic positions are fully relaxed at the constant volume until the energy variation on each atom is less than 1×10−4eV and the total force on each atom is less than 0.01 eV/˚A.

Fig.1.Structures of UO2 and U3Si2 crystals.Red ball represents O atom,blue ball represents U atom,and yellow ball represents Si atom.

The formation energies (Ef) of vacancy and Xe atom in UO2and U3Si2are defined as

Here,Etotalis the energy of the supercell containing a defect,µdenotes the chemical potential of atom in the most stable site, and thereforeµXeis the energy of an isolated Xe atom in a large empty supercell,µOis one half of the energy of a gas-phase O2molecule in a large empty supercell,µUandµSiare the energy per U/Si atom in the corresponding bulk phase,E0is the total energy of UO2/U3Si2supercell.By definition,a system with the positive formation energy means its formation process is endothermic,while the negative value to formation energy means exothermic.

2.2.Phase-field model

To explore the evolution features of intra- and intergranular bubbles in both UO2and U3Si2, the phase-field model is developed.[30–32]The total free energy of the polycrystalline system is written as

Here,ηandφare the order parameters to distinguish the bubble phase and matrix phase and to identify different grains,respectively.Especially,ηequaled to 1 represents the bubble phase,andηequaled to 0 represents the matrix phase.Pis the number of possible orientations in space,andφi(i=1,...,p)are the orientation field variables.Across the GBs between the grainφiand its neighborhood grain,the absolute value ofφivaries continuously from 1 to 0.

The free energy of the matrix(fm)is expressed as

wherefbis the free energy of the gas bubble and is expressed by the simplified van der Waals equation of state in the following equation:

whereA'is a constant fitted based on the experimental data,p0is the reference pressure,cbgis the atomic concentration of gas atom in the bubble.The definition ofcbgis given as

wherenis the number of Xe atom in a bubble,Vbis the volume of bubble,NVis the number of vacancies,andVsiteis the volume of crystal lattice.

The reference pressure is expressed as

In current work,the simplified van der Waals equation of state is used and its expression is

According to Ref.[33], parametersa,b, andcare determined as 259780 J·cm3/mol2, 23.9276 J·cm3/mol2, and 55.6583 J·cm3/mol2,respectively.

To predict the intergranular bubble evolution,the effect of polycrystalline free energy is considered,andfpcis expressed as follows:

The concentrations of vacancy (cv), self-interstitial (ci),and gas atom(cg)as a function of time are given as

Here,Mis the migration rate of defect(Mm=Dmcm/kBTwithm=v i, and g), whereDis the diffusion coefficient,Pis the generation rate of defect, andRv,iis the recombination rate between the vacancy and the self-interstitial atom.All the performances are carried out in a 256Δx×256Δysimulation domain.The formation energies of defects are obtained by usingab initiocalculations in the present work and the other parameters are summarized in the following Table 1.The periodic boundary condition is applied to the simulation domain and we use the normalization parameters for the numerical calculation:l∗=1 nm,t∗=0.1 s.

Table 1.Parameters used for phase-field simulations.

3.Results and discussion

3.1.Formation energies of point defects and Xe atom

First,we compare the possible configurations and stabilities of intrinsic point defects in UO2and U3Si2host crystal.Four types of point defects in UO2(that is U vacancy (VU),O vacancy (VO), U interstitial (IU), O interstitial (IO)), and five types of point defects in U3Si2(that is two different U vacancies marked asVU1andVU2, where U1 and U2 represent the occupied positions in the minimum symmetric cell with U 4h(0.181, 0.681, 0.5) and U 2a(0, 0, 0), respectively, Si vacancy(VSi1), U interstitial(IU), and Si interstitial(ISi))are considered.The formation energies of these point defects are summarized in Fig.2.

Fig.2.Formation energies of irradiation-induced point defects in(a)UO2 and(b)U3Si2 crystals,respectively,and the incorporation energy of Xe atom in(c)UO2 and(d)U3Si2 crystals,respectively.

As shown in Fig.2(a), the stabilities of intrinsic defects in UO2increase in an order ofIO,IU,VO,andVU,and the corresponding formation energies are 0.34 eV, 3.27 eV, 5.40 eV,and 11.19 eV, respectively.While the stabilities of intrinsic defects in U3Si2increase in an order ofISi,VU2,VSi,IU, andVU1.Among the values of formation energies,Efof U vacancy in UO2is the highest one,and the value ofVU2in U3Si2is the lowest one.From the energetic point,O interstitial first forms in UO2than other intrinsic defects, while in U3Si2, the formations of U vacancy and Si interstitial are more energetically favorable than those of other intrinsic defects.

To further reveal different thermodynamic behaviors of fission gas bubbles between UO2and U3Si2, we analyze the incorporation energy of Xe and the interaction between vacancy and Xe.The Xe atom occupied the interstitial site(IXe)and the substitutional site(XeX,Xrepresents U,O or Si atoms)is considered here.As shown in Fig.2(c), the incorporation energy ofIXein UO2is lower than those of XeUand XeO.Nevertheless,in U3Si2,the opposite result is obtained that Xe on the U2 site shows the most stable status.Then,we compare the stabilities of Xe–vacancy complexes in UO2and U3Si2.It should be noted that there are many types of divacancy in UO2and U3Si2, and we only consider the most stable divacancy.The most stable double vacancy in UO2is composed of one O vacancy and one U vacancy (VUO) and that in U3Si2is composed of two Si vacancies (V2Si).In UO2, the incorporation energies of Xe occupied the sites of single vacancy (VOandVU) and divacancy (VUO) are much lower than that of Xe at the interstitial site,especially theVUOsite.On the contrary,in U3Si2, the energy of Xe atom incorporated with divacancy is higher than that with monovacancy.A comparison of the incorporation energies in UO2and U3Si2indicates that Xe atom is more stable in U3Si2than that in UO2.

3.2.Fission gas bubble and its effect on thermal conductivity

3.2.1.Intragranular bubble

Different temperatures and fission densities influence the formation characteristics of gas bubble (i.e., the density and size of gas bubbles) in the nuclear fuel.This is due to the temperature-dependent materials’parameters and the external irradiation environment.For UO2,we choose 1200 K,1773 K,and 2073 K as the representative temperatures to understand the effect of temperature on the bubble evolution, and three different temperatures at 473 K, 673 K, and 873 K are assumed to reveal the temperature impact on bubble formation in U3Si2.Figures 3 and 4 show the bubble morphologies in UO2and U3Si2as functions of time and temperature,respectively.The morphologies of UO2and U3Si2after irradiation show that the intragranular bubbles are generally formed.[35–37]Our simulation results are in good agreement with the experimental results.

Fig.3.Morphologies of intragranular bubbles as a function of temperature in UO2 under different time points.

Fig.4.Morphologies of intragranular bubble as a function of temperature in U3Si2 at different time points.

With time increasing,both number and size of bubble increase.At the same time,their differences determined by temperature are obviously captured.In Fig.5, the density, size and porosity of intragranular bubbles in UO2and U3Si2varied with time are plotted respectively.Generally speaking,three stages of bubble density can be divided, that is incubation stage, formation and growth stage and coarsening stage.In each stage, obvious features are shown.At the incubation stage,there is no bubble in the matrix.A dramatic increase of bubble number occurs at the formation and growth stage,and then the number of bubbles keep almost unchanged at a longer time.Compared with the incubation stage at a lower temperature,higher temperature leads to longer incubation time.As is known, the gas atoms, vacancies and self-interstitial atoms experience the generation,diffusion,aggregation,recombination and annihilation to form individual gas bubbles, and the bubble connection would occur with increasing time.The temperature-dependent material parameters influence the formation time of bubble.The formation of bubble becomes more difficult at higher temperature since the energetic value and the defect diffusion coefficient are higher.

Fig.5.Comparison of intragranular bubble density, size, and porosity at different temperatures in(a)–(c)UO2 and(d)–(f)U3Si2.

For UO2, the bubble density at 1200 K is higher than those at both 1773 K and 2073 K (Fig.5(a)), and the mean size of gas bubble at 1773 K shows the highest value than those at other temperatures (Fig.5(b)).To assess the porosity,the bubble size and density are simultaneously considered.The maximal porosity occurs at 1773 K and the minimal one presents at 2073 K if the time is in a range of 180 days to 220 days(Fig.5(c)).For U3Si2,lower temperature reaches to the coarsening stage earlier than that at higher temperature in Fig.5(d), and the maximal and minimal bubble densities appear at 873 K and 473 K, respectively.Overall speaking, in Fig.5(e), the mean size of bubble at 873 K shows the largest one,and the smallest size presents at 473 K.As for a comparison of porosity in Fig.5(f),the highest one and the lowest one appear at 473 K and 873 K,respectively.

Fig.6.Morphologies of intragranular bubble as a function of fission rate in UO2 at different time points.

Fig.7.Morphologies of intragranular bubble as a function of fission rate in U3Si2 at different time points.

In addition to temperature effect, higher fission rate indicates there will be more point defects generated in the fuel,facilitating the formation of more bubbles.In the following investigation,the fission rates of 1.09×1019m−3·s−1,2.09×1019m−3·s−1, and 3.09×1019m−3·s−1for UO2and 3.0×1020m−3·s−1, 5.0×1020m−3·s−1, and 7.0×1020m−3·s−1for U3Si2are assumed and performed on the phase-field simulation.As the microstructures shown in Figs.6 and 7,lower fission rate results in longer bubble incubation time in UO2and U3Si2.It can be attributed to the face that the number of point defects produced by lower fission rate is smaller than that by higher fission per unit time and per unit space.Yaoet al.[38]investigated the irradiation behavior of U3Si2under 300-keV Xe+ion beam bombardment, and the results shown that the formation and coalescence of Xe bubble increase with an increment of irradiation dose.

Fig.8.Comparison of intragranular bubble density, size, and porosity at different fission rates in(a)–(c)UO2 and(d)–(f)U3Si2.

To quantitatively investigate the influence of fission rate,the bubble density,size and porosity as a function of time are evaluated.The corresponding results of UO2and U3Si2are plotted in Fig.8, respectively.There is no doubt that bubble density,size and porosity in UO2increase with the fission rate increase.The value of bubble size at 3.09×1018m−3·s−1is much higher than those at other fission rates.However,for U3Si2, the bubble size increases with increasing fission rate if the evolution time is less than 316 days as shown in Fig.8(e).Once the time is greater than 316 days, the bubble size at 3.0×1020m−3·s−1increases dramatically as high as 58 nm.The bubble density increases with the increment of fission rate as shown in Fig.8(d).The porosity of U3Si2in Fig.8(f)illustrates that the increment of fission rate can lead to higher porosity.Compared with the experimentally measured data,[37]the predicted porosity of U3Si2is consistent with the evolution trend,which is shown in Fig.9.

Fig.9.Predicted porosity in U3Si2 versus previous experimentally measured data.[29] LEU represents low enriched uranium, MEU represents medium enriched uranium,HEU represents highly enriched uranium.

The bubble-containing microstructure can highly degrade the thermal property since the heat transfer capability of gas is much lower than that of solid matrix.To evaluate the effect of Xe gas-filled bubble on the thermal conductivity,the steady heat transfer according to the Fourier’s law is utilized.The effective thermal conductivity is expressed as

Here, ¯Jis the average heat flux per unit area, dT/dris the mean temperature gradient along therdirection.To numerically solve Eq.(15), the finite difference method is used.To authors’ knowledge, the temperature-dependent thermal conductivity of U3Si2is[39]

The temperature-dependent thermal conductivity of Xe is[40]

Therefore, the thermal conductivities of U3Si2at 473 K,673 K, and 873 K are 10.540 W/m·K, 13.198 W/m·K, and 15.704 W/m·K, respectively, and those of Xe at 473 K,673 K,and 873 K are 0.00861 W/m·K,0.01166 W/m·K,and 0.01452 W/m·K, respectively.From the comparison of thermal conductivities, it is found that the thermal conductivity of U3Si2is 3–4 orders of magnitude higher than that of Xe.According to the existing thermal conductivity model,[41]the variation rule of thermal conductivity can be described by the empirical formulaK=K0(1+P)ε, whereK0is the thermal conductivity of the matrix,Pis the porosity,εis the fitting parameter.Accordingly, the expression of thermal conductivity can be extended to

Figure 10(a)shows the comparison between the simulated results and the fitting data.It confirms that the empirical model can well describe the thermal conductivity evolution with changing porosity.From the predicted results, the thermal conductivity decreases linearly with the increase of porosity, and the overall thermal conductivity increases with increasing temperature as expected.

The thermal conductivity of U3Si2under different fission rates (e.g., 3.0×1020m−3·s−1, 5.0×1020m−3·s−1, and 7.0×1020m−3·s−1) can also be fitted by the empirical formulaK=K0(1+P)ε, and the corresponding expression is shown in the following format:

To further consider the effect of fission rate on thermal conductivity, the formula can be derived intoK=K0(1+P)(A˙f+B),whereAandBare fitting parameters and ˙fis the fission rate.On a basis of the simulated thermal conductivity, the thermal conductivity can be fitted as follows:

Figure 10(b)presents our predicted effective thermal conductivity of U3Si2using the classical Fourier’s law,compared with the data calculated from Eq.(22).It shows that the results match well with each other from these two simulation and modeling approaches, and the effective thermal conductivity as a function of fission rate decreases linearly since a higher fission rate leads to more generation of gas bubble.

Fig.10.Thermal conductivity predictions of bubble-containing U3Si2 vary with(a)porosity and(b)fission rate,and predicted thermal conductivity as functions of(c)grain size and(d)porosity in polycrystalline U3Si2.

3.2.2.Intergranular bubble

GB is usually considered as the trapping position to vacancies, self-interstitial and gas atoms.Hence, the grain size plays an important role in the evolution of bubble.In this section, we investigated the interaction between GB and gas bubble in UO2and U3Si2with three different grain sizes(i.e.,10 µm, 25 µm, and 40 µm).The evolutions of intergranular bubbles in UO2and U3Si2are shown in Figs.11 and 12,respectively.

Fig.11.Morphologies of intergranular bubble as a function of grain size in UO2 under different time points.

A remarkable characteristic is captured that gas bubbles preferentially form at the trigonal GBs regardless of fuel type.This is because the incorporation energies of defects at the trigonal GBs are much lower,where point defects tend to cluster.On the other hand, bubbles formed on the straight GBs tend to be elliptical shape, which is closely related to minimize the interface energy between bubble surface and matrix.

Fig.12.Morphologies of intergranular bubble as a function of grain size in U3Si2 under different time points.

Fig.13.Grain size effect on the fraction of intergranular bubbles in(a)UO2 and(b)U3Si2.

Figure 13 shows the GBs covered by bubble as a function of time in UO2and U3Si2, respectively, under different grain sizes.Relatively speaking, small grain size could induce more bubbles to form on the GBs.The porosity in UO2and U3Si2with the smaller grain size is higher than that with the larger grain size, which is more obvious in the case of U3Si2(Fig.13(b)).The results of in-pile irradiation test also demonstrate that the large-grained UO2fuel can improve the irradiation swelling performance.[6,7]The intergranular porosity in the large-grained UO2is much lower than that in the small-grained one.[5]As aforementioned, GB is the trapping location of point defects and gas atom.More GB proportions mean more bubble formation on the GB.In addition, previously formed bubbles are preferred to trap point defects and fission gas atoms to accelerate bubble growth or shrinkage.In summary,UO2and U3Si2with the mean grain size of 10µm show higher bubble coverage than the microstructures with the grain sizes of 25µm and 40µm.

As is known, the thermal conductivity depends on the morphology of microstructure.On a basis of the phase-field simulated structure,we calculate the effective thermal conductivity of polycrystalline U3Si2.The effective thermal conductivity model and the thermal conductivities of the U3Si2matrix and Xe bubble at 473 K,673 K,and 873 K have been described in Subsection 3.2.1.The thermal conductivity of GB is used as 5 W/m·K in this study,which is the same as the previous work on polycrystalline UO2and U–Mo alloy.[42,43]Millettet al.[44]presented a general formula for estimating the effective thermal conductivity of polycrystals, and the specific expression is as follows:

wherePmis the intergranular bubble occupancy,αandnare the fitting parameters,dis the mean grain size,G0kis the Kalcha conductivity coefficient.IfPmis equal to zero, the values ofβ,G0k, andmare 0.0158, 156.76 MW/m2·K, and 0.192,respectively,according to the fitted thermal conductivities related to different grain sizes.Consequently,the thermal conductivity considering grain size effect is given as

Figure 10 plots the effective thermal conductivity as functions of(a)grain size and(b)porosity.In Fig.10(a),the effective thermal conductivity increases rapidly if the grain size is less than 10 µm; when the grain size exceeds 10 µm, the effective thermal conductivity increases in a slow rate.Different from grain size effect,the effective thermal conductivity varies with increasing porosity in a decrement.As expected, more gas bubbles are created in the microstructure with a smaller grain size,leading to a lower effective thermal conductivity as shown in Fig.10(b).To quantitatively evaluate the porosity effect,the thermal conductivities of polycrystalline U3Si2under the grain sizes of 10µm,25µm,and 40µm are described by the following expressions:

4.Conclusions

In this study,ab initiocalculations are carried out to assess the stabilities of point defects and fission gas atom, and the phase-field model are established to simulate gas bubble evolution in both UO2and U3Si2.The stabilities of vacancy,interstitial atom and fission gas atom,the interaction between vacancy and fission gas atom,the role of temperature, fission rate,and grain size on the bubble evolution as well as the corresponding effective thermal conductivities of irradiated microstructures have systematically investigated.The main findings can be summarized as follows:

(i) The formation energy of vacancy and the incorporation energy of Xe atom in UO2are higher than those in U3Si2,and the vacancies can decrease the incorporation energy of Xe atom in both UO2and U3Si2.

(ii) Higher fission rate indicates that more point defects are generated in the fuel, leading to more bubbles formation.Hence, the porosity increases with the increment of fission rate.

(iii) The porosities of UO2and U3Si2with 40-µm grain size are much lower than those in the smaller one, since the diffusion distance of fission gas atom from the interior grain to the GB increases in larger grain size.

(iv) The effective thermal conductivity strongly depends on the grain size,which influence the size and density of intergranular bubble.Higher porosity in the irradiated microstructure with smaller grain size can obviously degrade the thermal transfer capability.

Our comprehensive calculations enhance the understanding on the intra- and inter-granular gas bubble evolutions in UO2and U3Si2, which is critical for the prediction of gas swelling of large-grained fuel pellet.

Acknowledgments

Project supported by the National Natural Science Foundation of China (Grant Nos.U2167217, 12205286, and 11905025) and the National MCF Energy Research and Development Program of China(Grant No.2018YFE0308105).