Journal of Nuclear Materials 430 (2012) 181–189

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Journal of Nuclear Materials

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Modeling Deep Burn TRISO particle nuclear fuel q

T.M. Besmann a,⇑, R.E. Stoller a, G. Samolyuk a, P.C. Schuck a, S.I. Golubov a, S.P. Rudin c, J.M. Wills c,J.D. Coe c, B.D. Wirth b, S. Kim d, D.D. Morgan d, I. Szlufarska d

a Oak Ridge National Laboratory, P.O. Box 2008, Oak Ridge, TN 37831, United Statesb University of Tennessee, Knoxville, TN 37996-0750, United Statesc Los Alamos National Laboratory, P.O. Box 1663, Los Alamos, NM 87545, United Statesd University of Wisconsin, 1509 University Ave., Madison, WI 53706, United States

a r t i c l e i n f o

Article history:Received 19 March 2012Accepted 25 June 2012Available online 4 July 2012

0022-3115/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.jnucmat.2012.06.041

q This manuscript has been authored by UT-BattelleAC05-00OR22725 with the U.S. Department of Energment retains and the publisher, by accepting the artiedges that the United States Government retainirrevocable, world-wide license to publish or reprodumanuscript, or allow others to do so, for United State⇑ Corresponding author. Tel.: +1 865 574 6852; fax

E-mail addresses: besmanntm@ornl.gov (T.M. BesmStoller), samolyukgd@ornl.gov (G. Samolyuk), schugolubovsi@ornl.gov (S.I. Golubov), srudin@lanl.gov (SWills), jcoe@lanl.gov (J.D. Coe), bdwirth@utk.edu (B.Ddu (S. Kim), ddmorgan@engr.wisc.edu (D.D. MorgaSzlufarska).

a b s t r a c t

Under the DOE Deep Burn program TRISO fuel is being investigated as a fuel form for consuming pluto-nium and minor actinides, and for greater efficiency in uranium utilization. The result will thus be todrive TRISO particulate fuel to very high burn-ups. In the current effort the various phenomena in theTRISO particle are being modeled using a variety of techniques. The chemical behavior is being treatedutilizing thermochemical analysis to identify phase formation/transformation and chemical activitiesin the particle, including kernel migration. Density functional theory is being used to understand fissionproduct diffusion within the plutonia oxide kernel, the fission product’s attack on the SiC coating layer, aswell as fission product diffusion through an alternative coating layer, ZrC. Finally, a multiscale approachis being used to understand thermal transport, including the effect of radiation damage induced defects,in a model SiC material.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

A U.S. Department of Energy program is underway to utilize theTRISO fuel form to obtain substantially higher burn-up levels inthermal reactor systems. It is being considered for a potentiallyvery high burn-up modular high temperature gas-cooled reactor(MHTGR) that would consume transuranics from reprocessedconventional light water reactor (LWR) used fuel. An additionalconcept is to obtain significantly higher burnups in LWRs bysubstituting the particulate fuel embedded in a ceramic matrixfor current homogeneous oxide pellets. This concept termed ‘‘DeepBurn’’ where a burn-up of 60% fission of initial metal atoms (FIMA)or more can be achieved in a single irradiation will require excep-tional fuel capability. The envisioned fuel form is the TRISO coatedparticle with an oxide fuel kernel and standard sequence of buffer

ll rights reserved.

, LLC, under Contract No. DE-y. The United States Govern-cle for publication, acknowl-s a non-exclusive, paid-up,ce the published form of thiss Government purposes.: +1 865 574 4913.ann), stollerre@ornl.gov (R.E.

ckpc@ornl.gov (P.C. Schuck),.P. Rudin), jxw@lanl.gov (J.M.. Wirth), sungtae@cae.wisc.e-n), izabela@engr.wisc.edu (I.

or low density carbon layer followed by a high density, isotropicpyrolytic carbon layer, a SiC or ZrC layer and final outer high den-sity, isotropic pyrolytic carbon layer. The fuel kernel will consist ofplutonium with up to 30 at.% uranium and contain the minor actin-ides neptunium and americium. Alternative strategies include useof separate target particles containing minor actinides. To accom-plish the goal of achieving high burn-up will require significantunderstanding of fuel behavior utilizing advanced modeling andsimulation tools coupled with experimentally-derived informa-tion, and hence a program to model fuel with respect to thermo-chemistry, thermal properties, and fission product transport ofthe Deep Burn fuel has been underway and initial results obtained.

The purpose of the TRISO fuel configuration is to contain actin-ides and fission products within the particle to prevent leakage andcontamination of the coolant. To understand fuel behavior it is nec-essary to have a sufficient understanding of the thermochemistryto allow determination of phase formation and properties, and ulti-mately the activities and therefore driving forces for various pro-cesses. There are a number of effects and processes that conspireto allow release of actinide and fission product elements, andwhich are exacerbated with increasing burnup. These are de-scribed schematically in Fig. 1. They include kernel migration inwhich the oxide kernel is transported with increasing burn-up to-ward the hotter side of a particle in a thermal gradient, ostensiblydue to carbon from the buffer and inner pyrolytic carbon layerbeing transported to the cooler side. This is an issue as penetrationof the inner pyrolytic carbon layer and contact of the kernel withthe SiC layer can cause failure of particle integrity. Actinide and

Fig. 1. Schematic of processes in a TRISO particle that are observed during highburn-up.

182 T.M. Besmann et al. / Journal of Nuclear Materials 430 (2012) 181–189

fission product transport through the kernel and release to the buf-fer layer, and transport through the high density carbon and SiClayers are also sources of radionuclide contamination. Potentialmechanisms for this transport include bulk diffusion and transportalong grain boundaries and cracks. Finally, understanding how thethermal transport of particle constituents evolve over time will beimportant in accurate simulations of fuel behavior and predictionof over-temperature. All these issues are being addressed in thecurrent program.

Fig. 2. Computed molar amounts of equilibrium condensed phases in an initial(Pu0.7U0.3)O1.81 fuel as a function of burn-up assuming carbon does not react withthe constituents.

2. TRISO fuel phase equilibria and thermochemistry

The exceptionally complex phase equilibria and thermochemis-try of nuclear fuel is far from characterized. There exist reasonablesublattice models applying the compound energy formalism [1] forUO2±x and PuO2�x [2,3] which have been combined to create amodel for (U, Pu)O2±x, however addition of minor actinides or fis-sion products dissolved in the fluorite-structure phase have hardlybeen addressed. Recent efforts to represent AmO2�x using the com-pound energy formalism have produced a usable model [4], andthis has been combined with that for (U, Pu)O2±x yielding a repre-sentation for (U, Pu, Am)O2±x. The model lacks any interactionterms that include americium constituents, however it is a reason-able first estimate for the phase. More accurate formulations forthe more complex phases containing additional minor actinidesare being generated, however they await likely ab initio computa-tional results coupled with experimental efforts to fully definethermochemical values.

To demonstrate global fuel calculations the elemental composi-tion of TRISO fuel undergoing significant burn-up was computed asfunction of burn-up using SCALE for a starting fuel stoichiometry of(Pu0.64, U0.32, Np0.04)O1.81. A representative data file for actinidesand fission products for oxide fuels was generated utilizing a num-ber of simplifying assumptions. The fluorite-structure fuel phasewas represented by the (U, Pu, Am)O2±x model described above,with the original and bred in neptunium and curium contentaggregated with americium. Rare earths were represented by twophases, although it is likely that they will form a single solidsolution. Those with 3+ and 4+ possible valences were aggregatedas ceria (either Ce2O3 or CeO2�x after Zinkevich et al. [5]), and theremaining solely 3+ valence rare earths aggregated as La2O3.Typically seen in oxide fuel is the white or five metal alloy phase

containing ruthenium, rhodium, molybdenum, palladium, andtechnicium, and the various structures of the metal phase wererepresented by the models of Kaye et al. [6]. Although the fuel rep-resents a TRISO particle, for this example no carbon was includedin the system. Unless otherwise indicated above, the thermody-namic values for all the phases and species considered were takenfrom the SGTE database [7].

The equilibrium state in fuel was computed using the data filedescribed above and the FactSage thermodynamic equilibriumcode [8] at 1400 K and results plotted in Fig. 2 as a function ofburn-up with only the more abundant (above �1 mol%) phasesindicated. The fluorite structure fuel phase content is seen to de-crease as actinides are consumed. While the phase may dissolvesome fission products, most notably rare earths, these were notconsidered as phase constituents. The next most abundant phaseis the white phase; i.e., five metal alloy, which under the presentconditions has the hcp structure. The remaining significant phasesare rare earth oxides, barium and cesium zirconates, and the inter-metallic UPd3 which likely would contain other actinides, but werenot included in the representation of the phase as there is thermo-chemical data only for the uranium-containing phase. As is gener-ally true for oxide fuels, oxygen released as the result of fissioningof the actinide oxide will oxidize the susceptible fission products,however these cannot accommodate all the released oxygen andthus the oxygen-to-metal (O/M) ratio of the fluorite structure fuelphase will also increase.

An issue in high burn-up oxide kernel TRISO fuel is kernelmigration, which appears to be the transport of carbon in the buf-fer and inner pyrolytic carbon layer from the hotter side to thecooler side of the particle in a thermal gradient such that the kernelis allowed to eventually contact the SiC layer causing failure [9].There is evidence that this is driven by high oxygen partial pres-sures, possibly through a combination of solid state transportand CO diffusion through the buffer layer [10–14]. Kernel migra-tion is mitigated by lowering the oxygen pressure either throughstarting with low O/M fuel and restricting burn-up, by utilizingpartially carbide fuel or by including gettering phases in or nearthe kernel that buffer the oxygen pressure. Two gettering phasesthat have been successfully demonstrated to prevent kernel migra-tion are SiC and ZrC where the oxygen pressure is maintained atvery low values via the equilibria:

SiCþ 1:5O2 ¼ SiO2 þ CO

Fig. 3. Equilibrium calculational results as a plot of the log of the total, CO and CO2

pressure at 1673 K versus burn-up (FIMA) for the ungettered and gettered oxidekernel (initially PuO1.7) TRISO fuel system. Reaction with carbon is, of course,included.

Fig. 4. Calculated energy barrier to Ag diffusion in a PuO2 matrix, stepping along asubstitutional-vacancy path on the Pu sublattice. The calculated energy barrier ofapproximately 27 meV per atom equals the thermal energy available at roomtemperature, indicating that this energy barrier is easily overcome at hightemperatures.

T.M. Besmann et al. / Journal of Nuclear Materials 430 (2012) 181–189 183

or

ZrCþ 1:5O2 ¼ ZrO2 þ CO:

A set of thermodynamic equilibrium calculations demonstrat-ing the oxygen and total pressure behavior within a particle ofungettered and gettered fuel was performed, and the extensive re-sults can be seen in Besmann [15]. The effects are reproduced inFig. 3 where calculations similar to those described above wereperformed at 1673 K for an initial PuO1.7 fuel with elemental con-tents taken from SCALE calculations and assumptions about phaseformation identical to those for the mixed oxide fuel describedabove. The total pressure assumed complete release of noble fis-sion product gases from the kernel into the buffer layer void space.As is evident at high burn-up CO contributes significantly to the to-tal pressure in the ungettered kernel. Gettering with SiC or ZrC lar-gely eliminates the contribution of CO to the total pressure,decreasing the value a factor of two at the highest burn-up. Bes-mann [15] using the model of Choi and Lee [12] also confirmedthe experimental observations that gettering phases do mitigatekernel migration through reducing the oxygen partial pressureand thus the CO pressure in the particle.

3. Diffusion of fission products (e.g., silver, palladium) in a PuO2

kernel

In recent years a major tool in ab initio computing of the funda-mental thermodynamic properties of phases has been densityfunctional theory (DFT). Application of DFT to actinides, however,frequently requires moving beyond the standard approximationsso successful in describing other systems. In particular, the varyingdegrees of electron correlation apparent in the 5f electronic statesof plutonium must somehow be included for an accurate descrip-tion of its electronic structure.

While DFT calculations on pure plutonium must include elec-tron correlation in order to yield accurate structural data, DFT ap-plied to systems such as PuO2 – where the Pu atoms are not nearestneighbors – gives results in good agreement with experiment evenunder standard approximations (e.g., local or generalized gradientdensities). In addition, some simpler treatments of electron corre-lation introduce spurious tetragonal distortions of the PuO2 crystal,while more sophisticated approximations become computationallytoo costly for prediction of fission product and actinide transport inthe fuel kernel.

Investigation of PuO2 using DFT even with standard approxima-tions remains limited in terms of tractable system sizes. Most

calculations use simulation cells with periodic boundary condi-tions in all three dimensions, and a persistent problem is makingthese cells large enough that a fission product diffusing in one celldoes not interact with its image in a neighboring cell. Still, applica-tion of DFT to fission products in PuO2 can shed light on generaltrends regarding differences in activation energy or distortion ofthe PuO2 matrix. For example, replacing an oxygen atom in aPuO2 matrix with silver or cesium pushes the neighboring oxygenatoms away from the substitutional defect. An accurate evaluationof this displacement would require large simulation cells, but evenwith small simulation cells our calculations can predict that the ef-fect of the cesium substitution is much stronger than that of thesilver substitution. Similarly, the calculations provide estimatesof energy barriers to silver or cesium diffusion on the plutoniumsublattice in a PuO2 matrix (Fig. 4). The simulation cells used inthese calculations remain too small for accurate prediction ofexperimentally measured barrier heights, but their sizes sufficeto predict trends across various fission products; in particular, withthe barrier for cesium determined to be three times that for silver.An examination of the diffusion path provides a rationale for thisdisparity: Silver atoms tend to attract oxygen atoms, whereas ce-sium atoms tend to repel them. Cesium atoms encounter a largerbarrier because the forced diffusion path reduces by �15% the dis-tance to the nearest oxygen atom (Fig. 5), raising significantly therepulsive contribution to the energy.

Despite large differences in energies and bonding preferencesbetween silver and cesium in a PuO2 matrix, these two fissionproducts behave similarly when compared to, e.g., palladium. Cal-culation of activation barriers for palladium diffusion yields valuesan order of magnitude smaller than those for silver or cesium. Withan energy barrier only a fraction of the thermal energy available atroom temperature, our results suggest that palladium diffusesthrough the PuO2 matrix with almost no resistance and thus a verylarge diffusion constant.

In addition to energy barriers, calculation of diffusion constantscharacterizing movement of fission products through the matrixrequires vibrational spectra at the minimum- and maximum-en-ergy points along paths connecting adjacent hopping sites. DFTevaluation of full vibrational spectra would require an immensecomputational effort, and given the limited size of the simulationcells and the associated focus on trends rather than absolute num-bers, it is preferable to use a vibrational spectrum proxy requiringless computational effort. For this purpose the elastic constants

Fig. 5. Calculated distances between fission products and nearest oxygen atoms ina PuO2 matrix along a substitutional-vacancy path on the Pu sublattice.

184 T.M. Besmann et al. / Journal of Nuclear Materials 430 (2012) 181–189

which were selected can be calculated with significantly fewer re-sources. For a pure PuO2 crystal at the experimental lattice con-stant the calculated values are C11 = 344 GPa, C12 = 102 GPa, andC44 = 48 GPa; the bulk modulus B0 under the same conditions is183 GPa.

In order to access greater spatial (and temporal, in the case ofmolecular dynamics) length scales, a more efficient but still accu-rate alternative to DFT is needed. For this purpose the Los AlamosTransferable Tight-Binding for Energetics (LATTE) [16] code isbeing extended for use with f-electron materials such as PuO2.LATTE implements a set of very efficient algorithms for calculationat the self-consistent-charge density functional tight-binding (SCC-DFTB) [17] level. SCC-DFTB is based on second-order expansion ofthe charge density about that of the Kohn–Sham electronic groundstate. The self-consistent character of the charge distribution per-mits polarization and charge transfer in addition to covalent bondrearrangement, all of which would be impossible to describe with-in the fixed-partial-charge framework of standard force fields. SCC-DFTB calculations in LATTE employ a (minimal) basis of orthonor-mal atom-centered orbitals, and leverage novel algorithms for O(N)scaling [18] in addition to rigorously time-reversible Born–Oppen-heimer [19] or extended Lagrangian [20] dynamics. Although plu-tonium has been modeled with conventional tight-binding [21],there appear to be no previous attempts to model PuO2 with anintermediate level of theory such as SCC-DFTB. This combinationof realism and tractability should provide better estimates of en-ergy barriers in addition to permitting extension to phonon spec-tra, the combination of which allows prediction of diffusionconstants as well as thermal conductivity.

Fig. 6. Atomic configuration of PdTC and the energy barrier for the migrati

4. Palladium attack on SiC in TRISO fuel

Palladium is known to react with SiC to form a palladium sili-cide and carbon. Multiple experimental studies have been com-pleted where palladium has been deposited on the surface of SiCand palladium silicides have formed at temperatures as low as773 K [22–25]. The palladium silicides generally form islands onthe surface of SiC and the carbon forms graphite around the islands[26]. Experiments have provided important information on thereaction of Pd with SiC, but at the present time, detailed mecha-nisms are unknown.

Ab initio calculations can help identify important mechanismsin the reaction of palladium with SiC that might be elusive forexperimental techniques. In general, SiC is represented by rebuild-ing the atomic structure and placing the atoms in a simulation cell.For many ab initio programs the cells have periodic boundary con-ditions which create an infinite super-cell as the cells are repeatedin all three Cartesian directions. The size of the cell is important foraccuracy in calculations because defects should not interact withtheir mirror image in an adjacent cell. By using this approach,one can model intrinsic defects (vacancy or interstitial) or impuri-ties (palladium substitutional or interstitial) and gain informationon the energy of these defects. Multiple configurations are createdand relaxed to an energy minimum in order to find the lowest en-ergy configurations. Once enough stable configurations have beenfound, migration pathways between the stable configurations areinvestigated and the energy barrier discovered. In addition, sur-faces can be modeled by including vacuum layers in the simulationcell.

Here are reported two results found by using first principle cal-culations with palladium and cubic SiC. The calculations are per-formed with the Vienna ab initio simulation package (VASP) andmore details on the setup of the calculations are found in Schucket al. [27]. The first result is the migration pathway and energy pro-file for a palladium interstitial migrating through SiC. The palla-dium atom travels through the SiC via tetrahedral interstitialsites, alternating between being coordinated with four C (PdTC) orfour Si (PdTSi) atoms.

Fig. 6 shows the energy barrier for palladium migration and theatomic configuration of PdTC. The formation energy is high (7–9 eV), but the energy barrier is much lower (2 eV). Since SiC willbe irradiated, clustered interstitials will exist and the energy bar-rier is thus more critical for understanding how palladium behaves.In this case, palladium traveling via tetrahedral interstitials sites isa viable pathway, with others being explored as well.

The second result describes the energy of palladium clusters onthree different surfaces in SiC. We identified three typical surfacesfrom the literature [28,29], {100} C terminated surface, {100} Siterminated surface, and {111} surface. Fig. 7 shows the energy

on pathway of palladium through SiC via tetrahedral interstitial sites.

Fig. 7. Average energy of palladium in a cluster of size n on three different surfacesof SiC, silicon terminated {100} surface, carbon terminated {100} surface, and a{111} surface.

T.M. Besmann et al. / Journal of Nuclear Materials 430 (2012) 181–189 185

results for the palladium clusters on the surfaces. On each of thesurfaces, palladium atoms were added individually and relaxedto an energy minimum. The energy difference from a clean surfaceand one that has palladium atoms was averaged by the number ofpalladium atoms in the cluster. The results show how the averageenergy changes as each palladium is added to the cluster. There aretwo distinct results. For the {100} silicon terminated surface, thereis very little change to the overall energy as palladium atoms areadded. This occurs because the palladium binds very strongly tothe silicon and does not form tight clusters. For the other two sur-faces, palladium binds more strongly to the other palladium atomsin the cluster. Therefore, as atoms are added to the cluster, theybind more strongly to each other. When the cluster has threeatoms or more, 0.5 eV is gained in energy for each atom that isadded to the cluster. Thus, on some surfaces the palladium willpredominantly cluster, while on others the palladium will bindto the surface.

Fig. 8. Ab initio predicted ZrC defect formation energies for different chemicalreservoirs of Zr and C. A Zr rich reservoir is bulk Zr for a Zr defect and ZrC for a Cdefect. A C rich reservoir is defined analogously [44].

5. Atomistic modeling of fission product transport in ZrC

Silicon carbide provides the primary barrier layer in presentTRISO fuel particles for retaining the fission products. However,as noted above SiC has known issues with degradation from palla-dium-SiC interaction (palladium attack) [30], permeability to cer-tain fission products (e.g., silver), and rupture at highertemperatures [31,32]. ZrC has the potential to be an improved bar-rier layer as compared to SiC for a number of reasons [33], whichinclude resistance to palladium attack [34–36], higher meltingtemperature (3693 K) for ZrC (3123 K carbon eutectic) vs. 2818 Kfor SiC [37], reduced particle rupture rates at high temperatures[35,38,39], and apparently increased retention of some metal fis-sion products (e.g., silver) [35,38,40,42–45]. In this work we areparticularly interested in the potential for ZrC to reduce fissionproduct release through providing an improved barrier to fissionproduct diffusion as compared to SiC.

Experimental studies to date suggest that ZrC has the potentialto significantly reduce fission product release compare to SiC, butthe mechanisms and extent of the effects are far from clear. Bullock[38] studied a range of fuels with SiC and SiC plus ZrC coatings. Inparticular, he annealed UO2 kernels with varying levels of ZrC inthe fuel particle for over 1.25 y at 1773 K. Bullock found that a par-ticle with a thin overcoating of ZrC was the only type of fuel thatreleased no fission products during his study. In addition, Bullockshowed that for otherwise very similar fuel particles the inclusionof ZrC dispersed in the buffer layer surrounding the kernel couldreduce the fractional silver release from �100% to �27% over the

annealing time. More recently, Minato and collaborators have per-formed a number of studies which generally suggest that ZrC canimprove retention for cesium [41,42]. A detailed examination ofannealed particles [44] suggested that ZrC is itself a good barrierto cesium transport (this conclusion was based on particles whoseinner carbon layer had fractured, leaving ZrC as the major barrierlayer), but could not establish detailed mechanisms by which ZrCreduces cesium release.

Current work is using ab initio methods to elucidate the funda-mental properties of ZrC related to fission product diffusion. Allcalculations discussed here are performed with DFT using VASP[46] and the projector-augmented plane-wave (PAW) method[47,48]. The exchange–correlation was treated in the GeneralizedGradient Approximation, as parameterized in Perdew et al. [49].The approach taken in this work is to determine the basic energet-ics of intrinsic and fission product defects in ZrC, including theirformation and migration energetics. These values are then usedin thermodynamic and statistical mechanical models to predicthow fission products can enter and diffuse through ZrC. Initialstudies are focused on bulk diffusion, but similar approaches canbe adapted to predict grain boundary diffusivity.

As a first step in studying fission products in ZrC the intrinsicpoint defect formation energetics in bulk ZrC have been explored.These defect energies are necessary for fission product transportmodeling (e.g., vacancy formation energetics), as well as of interestin their own right for better understanding ZrC. Furthermore, thesepoint defect properties form the foundation for understandingradiation effects in ZrC, which is essential for controlling propertiesin the fuel particle.

The basic point defect energies are summarized in Fig. 8 [50],which demonstrate a number of interesting properties of ZrC. First,the VaC (vacancy on a carbon site) is very stable, which explainswhy ZrCy is easily made off-stoichiometric and zirconium rich(y < 1). The predicted value for VaC (carbon rich refer-ence) = 0.93 eV agrees quite well with the experimental value of0.91 eV, which was estimated from the thermodynamic model ofGuillermert [51] according to the approach of Kim et al. [50]. Inter-estingly, VaZr are much less stable than VaC, suggesting that the zir-conium sublattice will play a smaller role than the carbonsublattice in transport, at least in the absence of radiation. AfterVaC, interstitial carbon is generally the next most stable defect,which is not surprising given the much smaller size of carbon com-pared to zirconium. Zirconium interstitials and zirconium and car-bon antisite defects are all quite unstable, with defect formationenergies over 7 eV. These defects will therefore be present in verylow quantities from thermal excitations, but could play a role

186 T.M. Besmann et al. / Journal of Nuclear Materials 430 (2012) 181–189

under irradiation. Further ab initio studies will focus on transportand will provide an increased understanding of the mechanismsand magnitude of diffusion of fission products in ZrC.

6. Mesoscale modeling of fission product transport in coatinglayers

A mesoscale kinetic Monte Carlo model has been developed toinvestigate the diffusion of silver through the pyrolytic carbonand SiC containment layers of a TRISO fuel particle. The modelatomically resolves silver, but provides a non-atomic, mesoscalemedium of carbon and SiC that includes grain boundaries, the car-bon – SiC interfaces, cracks, precipitates and nano-cavities that canserve as either fast diffusional pathways or traps for the migratingsilver. The model consists of a two-dimensional slab geometryincorporating the pyrolytic carbon and SiC layers, with incident sil-ver atoms placed at the innermost pyrolytic carbon layer, as shownin Fig. 9.

Within the kinetic Monte Carlo model, the atomic position ofsilver is fully resolved, although a mesoscale medium is used forthe carbon and SiC layers. These layers can include a variety of de-fects including grain boundaries, reflective interfaces, cracks, andradiation-induced cavities that can either accelerate silver diffu-sion or slow diffusion by acting as traps for silver. The key inputparameters to the model (diffusion coefficients, trap binding ener-gies, interface characteristics) are determined from availableexperimental data, or parametrically varied, until more precise val-ues become available from lower length scale modeling. The pre-dicted results, in terms of the time/temperature dependence ofsilver release during post-irradiation annealing and the variabilityof silver release from particle to particle, have been compared toavailable experimental data from the German High Temp ReactorFuel Program [52] and studies performed by the Japan Atomic En-ergy Research Institute [53]. A variety of mechanisms have beenproposed to explain the experimental data, including a vapor phasetransport mechanism through nanocracks [54], trapping mediatedsolid state diffusion [55], and/or grain boundary diffusion; but nosingle mechanism can provide a fully self-consistent explanationof the experimental data, and thus, the exact transport mecha-nism(s) remain unknown.

The present study of silver fission product transport in TRISOfuel particles is focused on the effects of the irradiated microstruc-ture within the TRISO SiC layer on silver release at 1873 K, 1973 Kor 2073 K. The microstructural parameters investigated include:the silver diffusion coefficients in PyC and SiC; the number densityof irradiation-induced cavities that act as traps for silver in SiC; thepresence of cracks in SiC; and the presence and geometry of grain

Fig. 9. Kinetic Monte Carlo simulations of the fractional release of silver throughthe pyrolytic carbon and silicon carbide layers at 1973 K, which demonstrate theinfluence of the grain geometry in SiC on silver release.

boundaries in SiC. Select results that highlight the effect of micro-structure presented here with a more detailed description of themodel and provided in another publication [56].

Fig. 9 shows the effect of different grain geometries in SiC on sil-ver release during post-irradiation annealing. In this model, thegrains are considered to have a rectangular geometry. The smallerdimension is parallel to the interfaces and has a fixed length of1 lm. The longer dimension, parallel to the radial direction in aTRISO fuel particle, has a variable length that is uniformly distrib-uted among grains over a range from 1 lm to 40 lm, as shown inthe upper plot of Fig. 9. Such a grain distribution mimics a highlycolumnar structure, as is often observed experimentally [57].

The diffusion coefficient for silver transport within the grainboundaries has been assumed to be three orders of magnitudehigher than in bulk SiC to demonstrate the likely difference inbehavior between bulk and grain boundary diffusion. As expectedwithin this model, the presence of grains provides fast diffusionpaths for silver transport and accelerates the released fraction.Adding a columnar structure in those grains further increases therelease of silver; the released fraction increases from 50% to 80%when the grain distribution shifts from an isotropic structure(grains are 1 lm squares) to a highly columnar structure (lengthof grains uniformly distributed from 1 lm to 40 lm), after 270 hof heating at 1973 K.

A substantial modeling effort has focused on reproducing thesilver release behavior observed in the German experiments [52],and a highlight of that effort is presented here, with more exten-sive discussion of the KMC modeling approach, assumptions used,and results obtained described by de Meric de Bellefon and Wirth[56]. The results of KMC simulations of the transport of silverthrough a given PyC/SiC/PyC microstructure during post-irradia-tion thermal annealing at 1873 K, 1973 K and 2073 K is presentedin Fig. 10, as well as results from three German experimental re-lease measurements performed at annealing temperatures of1873 K, 1973 K and 2073 K. The simulated microstructures includereflective interfaces, trapping cavities and a grain boundary struc-ture in the SiC layer. The microstructures for the 1873 K and1973 K simulations (Particle 23 and 24) are identical and containan isotropic grain geometry in SiC that consists of 1 lm long squaregrains with a grain boundary diffusivity 100X higher than in thebulk. In the 2073 K simulation (Particle 25) the SiC grain character-istics are varied in order to match the measured release at 2023 K.Faster transport through grain boundaries is required to match the

Fig. 10. Kinetic Monte Carlo simulations of the fractional release of silver, ascompared with the measured fractional release of silver at 1873 K, 11973 K, and2073 K.

T.M. Besmann et al. / Journal of Nuclear Materials 430 (2012) 181–189 187

experimental results, which is obtained by implementing a highlycolumnar structure in which the grains are 0.5 lm-wide and havea radial length between 10 lm and 40 lm, as well as a much high-er grain boundary diffusivity, i.e. 2,000 times higher than in thebulk.

Fig. 11b. Thermal conductivity as a function of temperature, experimental andcalculated using RNEMD and GK approaches.

Fig. 12. Comparison of thermal resistance increase due to vacancies/void volume inSiC as calculated by MD and analytical methods.

7. Thermal properties and radiation damage

Thermal conductivity plays an important role in determiningfuel performance. The dominant carrier of the thermal energy inceramic materials is phonons. Therefore, thermal resistance arisesprimarily from phonon–phonon interactions and from scattering ofphonons by crystal imperfections: point defects (impurities andvacancies), linear defects (dislocations, dislocation loops) and twoand three-dimensional defects (voids, grain boundaries, secondaryphase precipitates, cracks, etc.). Some models have been developedto account for the impact of the defects using certain assumptionsand simplifications but their accuracy has not yet been verifiedproperly, thus the current models have to be critically analyzed,corrected, and properly verified using more fundamental theoryand by specially designed experiments. The ultimate objective ofthis research is to provide an improved model of thermal conduc-tivity that is practicable for use in a modern fuel performance code.

The following analytical approach is used in modeling thermalconductivity. In the relaxation time approximation the lattice ther-mal conductivity is given by:

KðTÞ ¼ 13

ZSðxÞv2ðxÞsðxÞdx; ð1Þ

where x is the phonon frequency, S(x) is the specific heat, v(x) isthe phonon velocity, s(x) is the phonon relaxation time [58–62]. Inthe framework of the Debye-Callaway model, relaxation time is gi-ven by

1sðxÞ ¼

1suþX

i

1siðxÞ

; ð2Þ

where 1=su ¼ vx2T=ax2DTm corresponds to an umklapp process and

1=siðxÞ -scattering on lattice defects (xD, v and Tm are the Debyefrequency, sound velocity and melting temperature, respectively).In Figs. 11a, 11b and 12 this approach is compared with results ob-tained by molecular dynamics (MD) simulations.

MD simulations are ideal for addressing issues such as the con-tribution to thermal transport properties of individual defects orgroups of defects. Thermal conductivity can be calculated througheither non-equilibrium MD (NEMD) or equilibrium MD (EMD)

Fig. 11a. Thermal conductivity, K, calculated by GK at T = 500, 1000 and 1500 K as afunction of upper limit of integration.

methods. Below are presented preliminary results obtained byusing both methods.

The EMD Green–Kubo (GK) method [63] uses heat flux fluctua-tions to compute the thermal conductivity via a fluctuation–dissi-pation theorem thus there is no temperature gradient and a systemis always in the linear response regime. The coefficient K is calcu-lated as an integral of the auto-correlation function:

KabðsÞ ¼1

3VkBT2

Z s

0dthJaðtÞJbð0Þi; Ja ¼

ddt

Xi

Eirai ; ð3Þ

where J is the heat flux, <> denote an ensemble average and the sumover i runs over all particles. The Large-scale Atomic/MolecularMassively Parallel Simulator (LAMMPS, http://lammps.sandia.gov)code has been used for atomistic molecular dynamics simulations.The interatomic interactions were described by the Stillinger & We-ber potential [64] and modified embedded atom method (MEAM)potential for silicon and MEAM potential for SiC [65].

In the reverse non-equilibrium MD (RNEMD) method [66], tworegions of the cell are designated the source and sink. At each timestep, the value of the kinetic energy of the most energetic particlein the sink region and the particle with least energetic particle inthe source region are exchanged. As a result, a temperature gradi-ent is imposed by the heat flux. The strengths of this method arethat it gives better accuracy compared to the EMD method, andit can be applied to a systematic study of interfacial effects.

188 T.M. Besmann et al. / Journal of Nuclear Materials 430 (2012) 181–189

However, the results are sensitive to the size of the sample andgradient of the temperature.

All the RNEMD simulations were done on systems of120 � 4 � 4, 240 � 4 � 4 and 320 � 4 � 4 unit cells of b-SiC, and6 � 6 � 6 with the GK method, using a time step of 0.0002 picosec-onds. As can be seen in Fig. 11b, reasonable agreement with exper-iment is reached for the sample of 320 � 4 � 4 unit cells fortemperatures of 1000 and 1500 K. However the value of K is stilllower than that of experimental observations at 500 K which isprobably attributable to the increase of the phonons mean-freepath at this temperature.

Although the result obtained from the RNEMD method for thesample with 320 � 4 � 4 unit cells is close to convergence, it isquite different from the GK results shown. A similar difference be-tween RENMD and GK results was reported by Schelling et al. [67].The disagreement between the two methods is possibly related toa significant increase in the phonon free path at low temperatures,i.e. a switch from diffusion to the ballistic regime in the NEMDmethod. However, both methods give similar results for tempera-tures of practical interest to nuclear fuels.

The GK calculations have been done for SiC with imperfections,i.e. when one, two, or eight Si/C vacancies or a void of 8 vacanciesintroduced into a 6 � 6 � 6 unit cell sample. The values of K andvacancy concentrations/swelling are 150 W/m K for pure material,116 W/m K for 0.06% vacancy concentration, 96 W/m K for 0.12%and 30 W/m K for 0.46% concentration, for one void – 57 W/m K.The results indicate a significant reduction of K value with increas-ing vacancy concentration that agrees with previous results ob-tained by another group for vacancy concentrations of 0.5% and0.2% with a modified Tersoff potential [68].

These MD results provide an opportunity to verify the analyticalapproach. One can find that the relative change in the thermalresistance, k ¼ DW=W (W = 1/K(T)), related to vacancies and voidsare given by:

kvac ¼3p3Cv

4TmT ;

9p3Cv Tm4T � 1

6ffiffiffiffiffiffiffiffiffiffiffiffiffiffipCv

TmT

q; 9p3Cv Tm

4T � 1

8<: ; kvoid ¼

p2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3S

ar

Tm

T

� �s; ð4Þ

where S = 4pNr3/3 is the volume fraction of the voids. Note that inthe case of vacancies there is a critical concentration Ccr

v ¼ 4T=9p3

for Tm ¼ 4:7� 10�3; which separates two different regimes: a linearone when the vacancy concentration is smaller than Ccr

v and squareroot dependence when it is larger than that.

The concentration dependence of thermal resistivity (W = 1/K)described by Eq. (4) is presented in Fig. 12, together with the MD re-sults (shown as individual data points). As can be seen from Fig. 12the simulation data quite closely fits a linear power law. There is notransition to a square-root power law such as takes place for the De-bye-Calloway theory for the highest vacancy concentrations. TheMD result for the case of a void, in agreement with the theory, issmaller than that for the equivalent number of randomly distrib-uted vacancies. However, the MD result is significantly higher thanthe analytical result. The origin of this discrepancy is not clear.Work is ongoing to report a quantitative comparison.

A more realistic description of thermal conductivity that can bedeployed in a fuel performance code has to include improvementto the analytical models. A way to accomplish this is to use theMD results and a realistic phonon density of states instead of theDebye approximation that is typically used. This density of statesmay be obtained from ab initio electronic structure calculations.

8. Summary

A large number of processes occur simultaneously in TRISO fuelundergoing high burn-up, including secondary phase formation,

kernel migration (if ungettered), interactions with SiC, and trans-port of species through a variety of mechanisms. All these affectthe life of the fuel and influence properties such as thermal trans-port and particle failure rate. Initial thermochemical equilibriumanalysis of the fuel has identified likely secondary phase formationduring burn-up and their influence on properties such as pressurein the particle. It is observed that the addition of gettering phasessuch as SiC and ZrC minimize CO pressure and thus reduce stresson the particle coatings as well as eliminate kernel migration.

The application of DFT to study fission products in PuO2 servedto shed light on general trends regarding differences in activationenergy or distortion of the plutonia matrix. The calculations pre-dicted that the effect of cesium substitution is much stronger thanthat of the silver substitution, and that palladium diffuses throughthe plutonia matrix with a very large diffusion constant.

A KMC model was developed to simulate the atomistic trans-port of fission product silver through a mesoscale two-dimensionalTRISO fuel particle medium. A variety of microstructural featureshave been introduced into the PyC/SiC/PyC layers, allowing aninvestigation of how silver permeation during annealing dependson microstructure. As expected, the presence of radiation-inducedtrapping features decreases the amount of silver release, while thepresence of grain boundary networks in the SiC layer can acceler-ate silver release. The model is able to reproduce the experimen-tally observed characteristics of silver release in irradiated TRISOfuel particles during thermal annealing between 1873 K and2073 K through a variation of the assumed microstructure of theirradiated particles. Similarly, to access greater spatial and tempo-ral length scales for fission product diffusion in the kernel, theLATTE code is being extended for use with f-electron materials.

First principles calculations have provided insights into theinteractions of palladium and SiC. Palladium interstitials migratethrough bulk SiC via tetrahedral interstitial sites with an energybarrier of 2 eV. Palladium atoms cluster on the {111} surface and{100} C-terminated surface and gain approximately 0.5 eV per pal-ladium atom added to the cluster.

Ab initio methods are being used to explore the potential of ZrCas a fission product barrier in TRISO fuels. The point defect energet-ics and structures have been calculated, predicting dramaticallygreater stability for carbon vacancies and interstitials than forequivalent zirconium defects. These results provide a foundationfor modeling fission product diffusion, fitting interatomic poten-tials, and understanding radiation damage effects in ZrC.

A multi-scale approach that included both atomistic (MD) andanalytical methods has been applied to study the impact of defectson thermal conductivity in SiC. Good agreement was obtained be-tween the calculated temperature dependence of the thermal con-ductivity and experiment data for a perfect crystal at elevatedtemperatures.

The diverse subjects related to TRISO fuel undergoing signifi-cant burn-up are providing both valuable insights into behavioras well as potential models for inclusion in a fuel simulation code.Significant work remains for these areas, including validation withexperimental information and the use of experiments to providebasic data.

Acknowledgments

This work was funded under the U.S. Department of Energy –NE Deep Burn program. The authors wish to thank T.R. Allen, S.L.Voit, and Y. Katoh for helpful discussions. K.T. Clarno performedthe SCALE calculations.

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