Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels

Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels Comprehensive nuclear materials 3 20 modeling of fission gas induced swelling of nuclear fuels

Nuclear Fuels *

J Rest

Argonne National Laboratory, Argonne, IL, USA

ß 2012 Elsevier Ltd All rights reserved.

3.20.2 Intragranular Bubble Nucleation: Uranium-Alloy Fuel in the High-Temperature

3.20.2.3 Calculation of the Fission-Gas Bubble-Size Distribution 584

3.20.5.2 Model for Initiation of Irradiation-Induced Recrystallization 6103.20.5.3 Model for Progression of Irradiation-Induced Recrystallization 611

3.20.5.5 Calculation of the Cellular Network Dislocation Density and Change in

3.20.5.7 Evolution of Fission-Gas Bubble-Size Distribution in Recrystallized U–10Mo Fuel 6203.20.5.8 Effect of Irradiation-Induced Recrystallization on Fuel Swelling 621

AbbreviationsATR Advanced test reactor EOS Equation of state PIE Postirradiation examination

*The submitted manuscript has been authored by a contractor of

the US Government under contract NO W-31-109-ENG-38.

Accordingly, the US government retains a nonexclusive

royalty-free license to publish or reproduce the published form of this

contribution, or allow others to do so, for US Government

purposes.

579

RERTR Reduced Enrichment for Research and

Test Reactors

SEM Scanning electron microscope

TEM Transmission electron microscope

This chapter addresses various aspects of modeling

fission-gas-induced swelling in both oxide and metal

fuels The underlying theme underscores the

simila-rities and differences in gas behavior between these

two classes of nuclear materials The discussion focuses

more on a description of key mechanisms than on a

comparison of existing models Three interrelated

crit-ical phenomena that dominate fission-gas behavior

are discussed: the role of intra- and intergranular

gas-bubble nucleation, irradiation-induced re-solution,

and irradiation-induced recrystallization on gas-driven

swelling in these materials The results of calculations

are compared to experimental observations

A clarifying comparison of existing models is

clouded by the fact that many of the models employ

different values for critical parameters and materials

properties This condition is fueled by the difficulty

in measuring these quantities in a multivariate

irra-diation environment Examples of such properties

are gas-atom and bubble diffusion coefficients,

bub-ble nucleation rates, re-solution rate, surface energy,

defect formation and migration enthalpies, creep

rates, and so on

The behavior of fission gases in a nuclear fuel is

intimately tied to the chemical and microstructural

evolution of the material The complexity of the

phenomena escalates when one considers the

possi-bility that microstructure is dependent on the fuel

chemistry Some of the key behavioral mechanisms,

such as gas-bubble nucleation, are affected by fuel

microstructure Likewise, mechanisms such as the

diffusion of gas atoms and irradiation-produced

defects are affected by fuel chemistry Thus, a

realis-tic description of the phenomena entails an accurate

representation of the evolving fuel chemistry and

microstructure A simple example of this is the

dependence of fission-gas release on the grain size:

the larger the grains, the lower the fractional release

at a given dose On the other hand, grain growth

occurs as a result of time at temperature as well as

by irradiation effects and fuel chemistry (e.g.,

stoichi-ometry) As the grain boundaries move, they

encoun-ter fission products and gas bubbles that impede their

motion All aspects of this synergistic process need to

be accounted for and modeled correctly in order toobtain a model that can accurately predict fission-gasrelease

On a different level, below temperatures at whichdefect annealing occurs, at relatively high doses, fuelmaterials such as UO2 and uranium alloys such asU–10Mo undergo irradiation-induced recrystalliza-tion wherein the as-fabricated micron-size polycrys-talline grains are transformed to submicron-sizedgrains As a result of this transformation, fissiongases are moved from within the grain to the grainboundaries, transferring the materials response to gas-driven swelling from intragranular to intergranular Inaddition, gas-bubble/precipitate complexes can act aspinning sites that immobilize potential recrystalliza-tion nuclei, and thus affect the dose at which recrys-tallization is initiated The synergy between thesedifferent forces needs to be realistically captured inorder to accurately model the phenomena

Given the current uncertainties in materials erties, critical parameters, and proposed behavioralmechanisms, a key issue in modeling of fission-gasbehavior in nuclear fuels is realistic validation Ingeneral, most of the model validation is accomplished

prop-by adjusting/predicting these properties and meters to achieve agreement with measured gasrelease and swelling, and with mean values of thebubble-size distribution However, the uncertainties

para-in these properties and parameters generate an para-ent uncertainty in the validity of the underlyingphysics and the physical reality of proposed behav-ioral mechanisms This inherent uncertainty cloudsthe predictive aspects of any mechanistic approach todescribe the phenomena Thus, more detailed dataare required to help clarify these issues

inher-The shape of bubble-size distribution data tains information on the nature of the behavioralmechanisms underlying the observed phenomenathat are not present in the mean or average values

con-of the distribution This is due to informationcontained in the first and second derivates of thebubble density with respect to bubble size Literaturedescriptions of measured intragranular bubble-sizedistributions are few and far between, and measuredintergranular bubble distributions are all but non-existent In Sections 3.20.2 and 3.20.3, recentlymeasured intra- and intergranular bubble-size distri-butions obtained from U–Mo alloy fuel are usedfor model validation, and the robustness of thistechnique in reducing uncertainties in proposedmechanisms and materials properties as compared

to employing average values is underscored In this

regard, it will be shown in Section 3.20.3 that a

substantial increase in validation leverage is secured

with the use of bubble-size distributions compared

with the use of mean values The results of a series of

calculations made with paired values of critical

para-meters, chosen such that the calculation of average

quantities remains unchanged, demonstrate that the

calculated distribution undergoes significant changes

in shape as well as position and height of the peak As

such, a capacity to calculate bubble-size distributions

along with the availability of measured distributions

goes a long way in validating not only values of key

materials properties and model parameters, but also

proposed fuel behavioral mechanisms

Sections 3.20.2 and 3.20.3 contain discussions of

gas-bubble nucleation in the high-temperature

equilibrium g-phase, and in the low-temperature

irradiation-stabilized g-phase of uranium alloy fuel,

respectively The connection between these regimes

is that while intergranular multiatom nucleation

appears to dominate at low temperature,

intragranu-lar multiatom nucleation is the dominant nucleation

mechanism at high temperature Although the

discus-sion on gas-bubble nucleation focuses on uranium

alloys (because of the availability of measured

bub-ble-size distributions), there is no reason to believe

that they would not be applicable to oxide fuel as well

Section 3.20.4presents an analysis of

irradiation-induced re-solution Specifically, the analysis

pre-sents a rationale for why gas-atom re-solution from

grain-boundary bubbles is a relatively weak effect as

compared to that for intragranular bubbles One of

the arguments is that intergranular bubble nucleation

results in bubble densities that are far smaller than

observed in the bulk material For example, an

inter-granular bubble density of 1
1013

m2is equivalent

to a bubble density of 2
1018

m3for a grain size of

5
106m This is to be compared to observed

intra-granular bubble densities that are on the order of

1023m3 In addition, typical intergranular bubble

sizes of tenths of a micron are to be compared to

nanometer-sized intragranular bubbles This

consid-eration is supported not only by the experimental

results presented inSection 3.20.3, but also by the

results of the multiatom nucleation theory that form

the basis of the analysis

Finally, in Section 3.20.5, models for the

initia-tion and progression of irradiainitia-tion-induced

recrys-tallization are reviewed, and a theory for the size of

the recrystallized grains is discussed The role of

bubble nucleation and gas-atom re-solution in the

recrystallization story is clarified Calculations arecompared to data for the dislocation density andchange in lattice displacement in UO2as a function

of burnup In addition, calculations are compared toavailable data for the recrystallized grain-size distri-bution in UO2and in U–10Mo

Models such as those described in this chapter arestories that remain just stories until validated byexperiment Fission-gas behavior in nuclear fuelshas been studied since the early 1950s, and althoughalmost 60 years have elapsed, a definitive picture ofthese phenomena is still unavailable today The rea-son, as stated above, is the difficulty in obtainingreliable single-effects data in a multivariate irradia-tion environment, coupled with the highly synergis-tic nature of the beast

Figure 1 shows scanning electron microscope

microstructure shown inFigure 1is typical of most

10.0 mm

Figure 1 Scanning electron microscope micrograph of g-U–Zr–Pu alloy fuel Reproduced by permission of the experimentor, G L Hofman.

uranium metal alloys irradiated in the equilibrium

g-phase Swelling of this material is predominantly

due to the growth of gas bubbles Its

fission-gas behavior is characterized by high mobility at

relatively high temperatures at which it exists at the

equilibrium g-U–Zr–Pu phase As seen inFigure 1,

the bubbles in this material comprise a relatively

broad size range Some of the larger bubbles have a

sinuous plastic-like appearance, indicative of high

mobility A number of coalescence events are

appar-ent, and some of the larger bubbles appear to be

growing into the smaller neighboring bubbles

gas-bubble nucleation in nuclear fuels at higher

heterogeneous3 two-atom mechanism In general, it

is assumed that two atoms that come together in the

presence of vacancies or vacancy clusters become a

stable nucleus At lower temperatures, because of the

relatively strong effect of irradiation-induced

re-solution, the number of nucleated bubbles increases

due to the increase in the effective gas generation

rate.4 In theory, the number of nuclei will increase

until newly created gas atoms are more likely to be

captured by an existing nucleus than to meet other

gas atoms and form new nuclei.2In practice, because

of the coarsening of the bubble-size distribution, the

two-atom nucleation process continues throughout

the irradiation

If both bubble motion and coalescence are

neglected, the rate equation describing the time

evo-lution of the density of gas in intragranular bubbles is

given by

d½mbðtÞcbðtÞ

dt ¼ 16pfnDgrgcgðtÞcgðtÞ

þ 4prbðtÞDgcgðtÞcbðtÞ  bmbðtÞcbðtÞ ½1

where cg, cb are the densities of gas atoms and

bub-bles, respectively, mb is the average number of gas

atoms per bubble, Dgis the gas-atom diffusion

coeffi-cient, b is the gas-atom re-solution rate from bubbles,

and fn, the so-called nucleation factor, is the

proba-bility that two gas atoms that come together stick

long enough to form a stable bubble nucleus Often,

fn is interpreted as the probability that there are

sufficient vacancies or vacancy clusters in the vicinity

of the two-atom to form a stable nucleus For

exam-ple, for heterogeneous bubble nucleation along

fis-sion tracks in UO2, fn is approximately the average

volume fraction of fission tracks 104 The three

terms on the right-hand-side of eqn [1] represent,

respectively, the change in the density of gas in granular bubbles because of bubble nucleation, gas-atom diffusion to bubbles of radius, rb, and the loss ofgas atoms from bubbles because of irradiation-induced re-solution

intra-An implicit assumption ineqn [1] is that once atwo-atom nucleus forms, it grows instantaneously to

an m-atom bubble Values of fnranging from 107to

102 have been proposed, which makes the ation factor little more than an adjustable parameter.5

nucle-A substantial contribution to the spread of reportedvalues for fn is that most models describe the timeevolution of mean values of cb and rb which arecompared to the respective mean values of the mea-sured quantities (comparing model predictions withaverage quantities is by far the dominant validationtechnique reported in the literature) In this regard,

as will be demonstrated in the following section,the use of bubble-size distributions goes a long waytoward the reduction of such uncertainties.6

As an approach to circumventing the deficienciesthus described, in what follows a multiatom bubblenucleation mechanism is proposed and implementedinto a mechanistic calculation of the intragranularfission-gas bubble-size distribution The results ofthe calculations are compared to a measured bubble-size distribution in U–10Mo irradiated at relativelyhigh temperature to 4% U-atom burnup The multi-atom nucleation model is compared to the two-atommodel within the context of the data, and the impli-cations of each mechanism for the observable quan-tities are discussed

In the next section, a multiatom nucleation anism is formulated.Section 3.20.3presents an out-line for a calculation of the time evolution of thebubble-size distribution InSection 3.20.4, a discus-sion is presented of processes that lead to coarsening

mech-of the as-nucleated bubble distribution In Section3.20.5, model calculations are used to interpret ameasured distribution in U–8Mo uranium alloy fuelirradiated to 4% U burnup at 850 K In addition, inthis section, a comparison between the multiatomand two-atom nucleation mechanisms is attempted.Finally, conclusions are presented inSection 3.20.6

MechanismFission gases Xe and Kr are generated in a nuclearfuel at a rate of about 0.25–0.30 atoms per fission as aresult of decay of the primary fission products Aboutseven times more Xe is produced than Kr These gas

atoms are very insoluble in the fuel in that they do

not react chemically with any other species Thus,

left in the interstices, because of their relatively large

size, they produce a strain in the material In order to

lower the energy of the system and to minimize the

strain, the gas atoms tend to relocate in areas of

decreased density, such as in vacancies and/or

vacancy clusters For example, in UO2, gas atoms

have been calculated to sit in neutral trivacancy

sites consisting of two oxygen ions and one uranium

ion.7Given enough energy via thermal fluctuations,

and/or via irradiation, the gas atoms can hop

ran-domly from one site to another and thus diffuse

through the material The gas atom/vacancy

com-plexes can combine forming clusters of gas atoms

and vacancies If enough gas atoms come together,

they become transformed into a gas bubble which,

under equilibrium conditions, sits in a strain-free

environment This process of forming gas bubbles is

termed gas-bubble nucleation

According to phase transition theory, at relatively

large supersaturations, a system transforms not by

atom-to-atom growth, but simultaneously as a

whole In other words, the system is unstable against

transformations into a low free energy state, and the

new phase will have a certain radius defined by the

supersaturation Solubility of rare-gas atoms in

ura-nium alloys or ceramics is so low that it has not been

measured In perfect crystals, the order of magnitude

of the solubility has been estimated to be 1010.8This

figure may be increased up to105in the vicinity of

dislocations In addition, there may be a substantial

effect from gas in dynamic solution, that is, as a result

of irradiation-induced re-solution Thus, in regions

of nuclear fuels that are near irradiation-produced

defects and/or various microstructural irregularities,

the solubility of the gas can be substantially higher

than in the bulk material The gas concentration in

these regions will increase until the solubility limit is

reached, whereupon the gas will precipitate into

bubbles Subsequently, nucleation is limited because

of the gas concentration in solution falling below

the solubility limit The trapping of the gas by the

nucleated bubble distribution damps the increase

in gas concentration Eventually, the gas in solution

may reach the solubility limit at which time the

nucle-ation event repeats Thus, assuming that all the gas

precipitates into bubbles of equal size r0, the

concen-tration of gas in the bubble at nucleation is given by

mðr0Þ ¼ bvc

crit g

where ccrit

g is the concentration of gas at the solubilitylimit, bv is the volume per atom (van der Waals con-stant), and cbðr0Þ is the concentration of bubble nuclei

at the unrelaxed radius r0, that is, the initial stage ofbubble nucleation is a volume-conserving process.Subsequently, in order to lower the free energy of thesystem, the overpressurized nuclei relax by absorbingvacancies until the bubbles reach equilibrium At equi-librium, the bubble radius is r and, in the absence ofsignificant external stress, the pressure in the bubble isgiven by

r0! r; mðr0Þ ! mðrÞ ¼ mðr0Þ; cbðr0Þ ! cbðrÞ ¼ cbðr0Þ

½5The nucleation problem thus consists of determiningthe two terms on the RHS ofeqn [4] The first term onthe RHS ofeqn [4]can be determined from the equa-tion of state (EOS), the capillarity relation, and theconditions expressed in eqn [5] Using the van derWaals EOS,

where V¼ 4/3pr0 3

is the bubble volume Recognizingthat at nucleation the bubble size is small such that2g/r0 s , where s is the external stress, and differ-entiating eqn [6] with respect to the equilibriumradius r one obtains

1mðr0Þ

be determined by invoking energy minimization asthe driving force for bubble equilibration The change

in the Gibbs free energy due to bubble expansion isgiven by

3pr

03Gvþ4pr02g ½8

whereGvis the free energy driving bubble

equilibra-tion, which, in analogy with the treatment of the

nucle-ation of liquid droplets in a vapor,9can be expressed as

Gv¼kT

where O is the atomic volume The critical bubble

radius at equilibrium is given by the condition

@G

@r ¼ 0 ! r ¼ rcrit¼G2g

Inserting the expressions for Peand P fromeqns [3] and

[6], respectively, into eqn [10], differentiating with

respect to the bubble radius r, and applying a little

algebra results in

4pr02dr

0

dr ¼ 1X

O

r þkT2g

þbv

m

dmdr

½14

The as-nucleated bubble-size distribution is then

obtained by the simultaneous solution of eqns [7]

and [14]

Subsequent to the nucleation event, the nucleated bubble-size distribution evolves under thedriving forces of gas diffusion to bubbles, gas-atomre-solution from bubbles, and bubble coalescencedue to bubble–bubble interaction via bubble motionand geometrical contact As stated earlier, additionalnucleation events are delayed because of the gas insolution remaining below the solubility limit, as thegas generated by continuing fission events is trappedwithin the existing bubble-size distribution This lastpoint is facilitated by the relatively high gas-atom dif-fusivities at the temperatures of interest (i.e., thoseunder which the equilibrium g-phase of the alloyexists) Eventually, the gas in solution may againreach the solubility limit at which time the nucle-ation event repeats

Bubble-Size DistributionThe model consists of a set of coupled nonlinear dif-ferential equations for the intragranular concentration

of fission product atoms and gas bubbles of the form10

dCi

dt ¼ aiCiCi biCiþ ci ði ¼ 1; ; NÞ ½15where Ciis the number of bubbles in the ith size classper unit volume; and the coefficients ai, bi, and ciobeyfunctional relationships of the form

ai¼ aiðCiÞ

bi¼ biðC1; ; Ci1; Ciþ1; ; CNÞThe variables in eqn [15] are defined in Table 1

ai represents the rate at which bubbles are lost from(grow out of) the ith size class because of coalescencewith bubbles in that class; bi represents the rate atwhich bubbles are lost from the ith size class because

of coalescence with bubbles in other size classes and

Table 1 Definition of variables in eqn [15] ,dCi

dt ¼ a i CiCi b i Ciþ c i ði ¼ 1; ; NÞ

1 Concentration of

intragranular gas atoms

Rate of gas atom loss due to gas-bubble nucleation

Rate of gas atom loss due to diffusion into gas bubbles

Rate of gas atom gain due to atom re-solution and fission

of uranium nuclei

2, ,N Concentration of

intragranular gas bubbles

Rate of gas bubble loss due to bubble coalescence with bubbles within the same size class

Rate of gas bubble loss due to coalescence with bubbles in other size classes

Rate of gas bubble gain due

to bubble nucleation and coalescence, and diffusion

of gas atoms into bubbles

Source: Rest, J J Nucl Mater 2010, 402(2–3), 179–185.

re-solution; and cirepresents the rate at which bubbles

are being added to the ith size class because of

fission-gas generation, bubble nucleation, bubble growth

resulting from bubble coalescence, and bubble

shrink-age due to gas-atom re-solution

The bubbles are classified by an average size,

where size is defined in terms of the number of gas

atoms per bubble This method of bubble grouping

significantly reduces the number of equations needed

to describe the bubble-size distributions The bubble

classes are ordered so that the first class refers to

bubbles that contain only one gas atom If Sidenotes

the average number of atoms per bubble for bubbles

in the ith class (henceforth called i-bubbles), then the

bubble-size classes are defined by

where the integer n 0:5 þpffiffiffiffiffiffiffiffiffi1:25, i 2, and

Si ¼ 1 The i ¼ 1 class is assumed to consist of a

single gas atom associated with one or more vacancies

or vacancy clusters In general, the rate of

coales-cenceij of i-bubbles with j-bubbles is given by

where Pij is the probability in m3s1of an i-bubble

coalescing with a j-bubble For i¼ j, ij becomes

ii¼1

so that each pair-wise coalescence is counted only

once

Coalescence between bubbles results in bubbles

growing from one size class to another The

probabil-ity that a coalescence between an i-bubble and a

j-bubble will result in a k bubble is given by the

array Tijk The number of gas atoms involved in one

such coalescence is Siþ Sj The array Tijkis defined

3 For a given pair ij, only two of the Tijk array

ele-ments are nonzero These eleele-ments correspond to

k and kþ 1, where Sk  Siþ Sj  Skþ1

From these three conditions, it follows that k¼ i,

and

TijkSkþ ð1  TijkÞSk þ1¼ Siþ Sj ½19

Thus, the probability that a coalescence between an

i-bubble and a j-bubble will result in a k bubble is

an i-bubble will become a k bubble as a result of itscoalescence with a j-bubble The rate Nijk at whichi-bubbles become k bubbles is given by

of collisions (direct and/or indirect) between fissionfragments and gas bubbles Fromeqns [21] and [22],

prob-iþ 1 or an i  1 bubble, respectively; the ratio of

the probabilities is equal to the ratio of the rates

The aforementioned definition of Nik and Nik0 is

consistent with the conservation of the total number

of gas atoms

The bubbles are assumed to diffuse randomly

through the solid alloy by a volume diffusion

mecha-nism The bubble diffusion coefficient Diof a bubble

having radius Riis given by

Di ¼3a30Dvol

where a0is the lattice constant and Dvolis the volume

self-diffusion coefficient of the most mobile species

in the alloy The coefficients ai and bi (e.g., the first

and second terms on the RHS ofeqn [15]) are

repre-sented, respectively, by

ai¼ 16pRiDi; bi¼X

j6¼i

ðRiþ RjÞðDiþ DjÞCj ½28

The interaction cross-section represented ineqn [28]

is based on an analysis of colloidal suspensions within

the framework of the continuum theory.11Fission-gas

bubbles can also interact due to mobility from biased

motion within a temperature gradient This aspect of

the problem is handled in an analogous manner and

will not be considered here

As the bubbles grow and interact, the average

spacing between bubbles shrinks In addition, as

seen fromeqn [27]for the volume diffusion

mecha-nism, bubble mobility falls off as the inverse of the

radius cubed such that, for all practical purposes,

relatively large bubbles are immobile As the larger

bubbles grow because of accumulation of the

contin-ual production of gas due to fission, the bubbles

intercept other bubbles and coalesce This process is

here termed geometrical coalescence For spherical

bubbles that are all the same size and that are

uni-formly distributed, contact is reached when

In analogy with percolation theory, the probability of

an i-bubble contacting a j-bubble is given by

charac-The aiand bicoefficients ineqn [28]now have anadditional term given by

where Pij is given byeqn [30]

In what follows, it is assumed that DXe¼ Dvol

High-Temperature Irradiation Data

Figure 2shows the as-nucleated bubble-size tion made with the simultaneous solution of eqns [7]and [14]for a gas solubility of 107at a fuel temperature

distribu-of 850 K At a fission rate distribu-of 1
1020

fissions m3s1,the solubility limit is reached in 140 s Subsequently,nucleation is limited as a result of the gas concentration

in solution falling below the solubility limit The ping of gas in solution by the nucleated gas bubblesdamps the rate at which the generated gas increases thegas concentration in dynamic solution It is important topoint out that here the solubility limit is an unknownparameter If the solubility limit was 106or 105, theinitial bubble nucleation event would occur after 1400

trap-or 14 000 s of irradiation, respectively

Figure 3 shows m versus r obtained from thesolution ofeqn [7]for T ¼ 850 K and g ¼ 0:5 Jm2.

As expected from the form of eqn [7], the number

of gas atoms grows exponentially with bubble size

Figure 4 shows the amount of gas in bubbles as afunction of bubble size corresponding to Figures 2

and 3 As is evident from Figure 3, although thebubble-size distribution shown in Figure 2 is rela-tively broad, the majority of the gas generated prior

to the nucleation event (i.e., within the first 140 s ofirradiation) exists in bubbles having radii<1 nm Asdiscussed earlier, subsequent to the multiatom bubblenucleation event, the concentration of gas in solutionstays below the solubility limit due to the trappingeffect of the nucleated gas bubbles such that addi-tional multiatom nucleation events are delayed.Thus, until the solubility limit is again exceeded, forthe situation shown in Figures 2–4, for irradiationtimes>140 s, the bubble distribution follows from theevolution of the as-nucleated distribution shown in

Figure 2because of bubble–bubble coalescence anddiffusion of generated gas to the existing bubble

population When the solubility limit is again

exceeded, additional nucleation events occur within

the evolving bubble population, and this complex of

bubbles again evolves under the driving forces of

bubble coalescence, atom diffusion to, and

gas-atom re-solution from bubbles

Figure 5shows the calculated bubble-size

distri-bution for an irradiation in U–8Mo at 850 K to 4%

nucleation model described in Section 3.20.2 forthree values of the rare-gas solubility The calcula-tions shown inFigure 5were made using a gas-atomdiffusivity, and re-solution rate given by

Dvol¼ 2
104e33000=kTcm2s1

0 0

Figure 3 Number of gas atoms in a freshly nucleated bubble versus bubble radius corresponding to Figure 1

Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185.

0 1E + 11

1E + 12 1E + 13 1E + 14 1E + 15

Cb

3 ) 1E + 16 1E + 17 1E + 18 1E + 19 1E + 20 1E + 21

where _f is the fission rate The value for Dvolgiven in

eqn [33]is about a factor of 10 less than the out-of-pile

measured U self-diffusion coefficient in U–10Mo.12

On the other hand, it is not clear what diffusion

mechanism dominates gas behavior in these alloys

For example, the Mo self-diffusion coefficient in

U–10Mo is about an order of magnitude less thanthe U self-diffusion coefficient.13 In addition, it isnot at all clear how these diffusion couple measure-ments extrapolate to lower temperatures (lowest dif-fusion couple temperature was 1073 K) and to an

2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0 0.00

0.02

0.04 0.06 0.08 0.10 0.12

0.00 0.02 0.04 0.06 0.08 0.10 0.12

4

r (nm)

Figure 4 Fraction of generated gas in bubbles versus bubble radius corresponding to Figures 2 and 3.

Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185.

Bubble radius ( µm)

0 1E + 10 1E + 11 1E + 12 1E + 13 1E + 14 1E + 15 1E + 16 1E + 17 1E + 18 1E + 19 1E + 20

Multiatom nucleation solubility limit = 2.5 ⫻ 10 −9

Figure 5 Calculated bubble-size distributions for an irradiation in U–8Mo at 850 K to 4% U-atom burnup using eqn [15]

and the multiatom nucleation model described in Section 72.2 for three values of the rare-gas solubility compared with irradiation data Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185.

approximately an order of magnitude less than

esti-mated for UO2.14This value is consistent with

esti-mated irradiation-enhanced creep rates in U–10Mo,

which are approximately an order of magnitude less

than for UO2.15These effects can be traced to a higher

thermal conductivity in the metal alloy as compared

to the metal oxide

converted to a volume density from the measured

areal density16 using the Saltykov method.17 The

error bars associated with the solid circle data points

are unknown, but they are most certainly substantial

for the smaller bubble sizes where undercounting

errors are typical In addition, the fuel experienced

an end-of-life constraint of 10 mp (the effect of

hydrostatic constraint on bubble size is included

in the calculations) Given these uncertainties, the

bubble-size distribution is relatively flat for bubbles

having radii from5 to 12 mm As shown inFigure 4,

a solubility of2.5
108provides a plausible

inter-pretation of the data

Figure 6shows the dependence of the calculated

bubble-size distribution on the value of Dvolfor a gas

solubility of 2.5
108compared with the measured

quantities As seen from Figure 6, not surprisingly,

the value of Dvolhas a reasonably strong effect on the

calculated distribution

It is of interest to compare the multiatom

nucleation model with conventional two-atom

nucleation as expressed by the first term on theRHS ofeqn [1].Figure 7shows the calculated bub-ble-size distributions for an irradiation in U–8Mo at

850 K to 4% U-atom burnup usingeqn [15]and thetwo-atom nucleation model for three values of thenucleation factor compared with irradiation data.Also shown are results for two different values ofthe volume diffusion coefficient for fn¼ 103 It is

clear from Figure 7 that the two-atom nucleationmodel does not satisfactorily interpret the measuredbubble-size distribution over a 6 orders of magnituderange in fn and 2 orders of magnitude range in Dvol

nucleation model provides a better interpretation ofthe data than the two-atom model This becomes astronger statement when the relative insensitivity ofthe calculated tail of the distribution to the value

of the nucleation factor and the volume diffusioncoefficient for the two-atom model are compared

to the ‘bracketing’ of the data by commensuratechanges in solubility and diffusion coefficient for themultiatom model

A more definitive differentiation between thesetwo models requires data at a much lower burnupwhere the effects of bubble diffusion and coalescenceare minimal Unfortunately, such data are currentlyunavailable.Figure 8 shows a comparison of multi-atom and two-atom nucleation mechanisms for anirradiation to 0.04% U burnup of U–8Mo fuel at

Bubble radius ( µm)

1E + 10 1E + 11 1E + 12 1E + 13 1E + 14 1E + 15 1E + 16 1E + 17 1E + 18 1E + 19 1E + 20

Figure 6 Dependence of the calculated bubble-size distribution on the value of Dvolfor a gas solubility of 2.5
10 8

compared with the measured quantities Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185.

850 K As shown in Figure 8, the two-atom

nucle-ation model leads to a substantially broader

distribu-tion than the multiatom model This feature is carried

on to high burnup and, on comparingFigures 5 and 7,

is one of the key differences between these two

nucleation models It is anticipated that low burnupbubble distribution data will become available in therelatively near future.18Once this data become avail-able, a more definitive differentiation between thesetwo models can be undertaken

Bubble radius ( mm)

1E + 11 1E + 12 1E + 13 1E + 14 1E + 15 1E + 16 1E + 17 1E + 18 1E + 19 1E + 20

2010, 402(2–3), 179–185.

Bubble radius ( mm)

1E + 6 1E + 7 1E + 8 1E + 9 1E + 10 1E + 11 1E + 12 1E + 13 1E + 14 1E + 15 1E + 16 1E + 17 1E + 18 1E + 19 1E + 20 1E + 21

fn= 10 -3

Figure 8 Comparison of multiatom and two-atom nucleation mechanism for an irradiation to 0.04% U burnup in U–8Mo fuel at 850 K Reproduced from Rest, J J Nucl Mater 2010, 402(2–3), 179–185.

3.20.2.6 Conclusions

Analysis of different nucleation mechanisms in the

light of measured bubble-size distributions in U–8Mo

fuel irradiated in the equilibrium g-phase indicates

that a multiatom nucleation mechanism is operative

The conventional two-atom nucleation model is not

consistent with the trends of the data A more definitive

test of the nucleation mechanism requires measured

bubble distributions at a very low burnup

Nucleation: Uranium-Alloy Fuel in the

In order to assess the temperature dependence of

fission-gas swelling in a material such as U–Mo, the

model for the gas-driven swelling behavior in the

3.20.2needs to be complemented with a model for

gas-bubble behavior in the low-temperature

irradia-tion-stabilized g-regime The swelling at high

tem-perature is primarily intragranular, whereas at low

temperature, intergranular swelling becomes

appre-ciable As discussed in the previous section, a

multi-atom gas-bubble nucleation mechanism in uranium

alloy nuclear fuel operating in the high-temperature

equilibrium g-phase was proposed on the basis of

interpretation of measured bubble-size distribution

data The multiatom nucleation mechanism is also

operative at low temperatures but primarily affects

bubble nucleation on the grain boundaries The

capa-bility to calculate swelling behavior in U–Mo fuel

across the entire temperature spectrum enables an

assessment of safety margins for stable swelling of

U–Mo alloy fuel

The shape of bubble-size distribution data contains

information on the nature of the behavioral

mechan-isms underlying the observed phenomena that are

not present in the mean or average values of the

distribution This is due to information contained in

the first and second derivates of the bubble density

with respect to bubble size Literature descriptions

of measured intragranular bubble-size distributions19

are few and far between, and measured intergranular

bubble distributions are all but nonexistent Here,

we use measured intergranular bubble-size

dis-tributions6,20 obtained from U–Mo alloy aluminum

dispersion fuel developed as part of the Reduced

Enrichment for Research and Test Reactor (RERTR)

program and irradiated in the Advanced Test Reactor(ATR) in Idaho

An analytical model for the nucleation and growth

of intra- and intergranular fission-gas bubbles isdescribed wherein the calculation of the time evolution

of the average intergranular bubble radius and numberdensity is used to set the boundary condition for thecalculation of the intergranular bubble-size distribu-tion based on differential growth rate and sputteringcoalescence processes Sputtering coalescence, or bub-ble coalescence without bubble motion, is a relativelynew phenomenon observed heretofore in implantationstudies in pure metals.21In particular, the sputteringcoalescence mechanism is validated on the basis of thecomparison of model calculations with the measureddistributions Recent results on transmission electronmicroscope (TEM) analysis of intragranular bubbles

in U–Mo were used to set the irradiation-induceddiffusivity and re-solution rate in the bubble-swellingmodel Using these values, a good agreement wasobtained for intergranular bubble distribution com-pared against measured postirradiation examination(PIE) data using grain-boundary diffusion enhance-ment factors of 150–850, depending on the Moconcentration This range of enhancement factors isconsistent with values obtained in the literature

Intragranular Bubble-Size and DensityThe model presented here considers analytical solu-tions to coupled rate equations that describe thenucleation and growth of inter- and intragranularbubbles under the simultaneous effect of irradia-tion-induced gas-atom re-solution The aim of theformulation is to avoid a coupled set of nonlinearequations that can only be solved numerically, usinginstead a simplified, physically reasonable hypothesisthat makes the analytical solutions viable The gas-induced swelling rate is then assessed by calculatingthe evolution of the bubble population with burnupand subsequently the amounts of gas in bubblesand lattice sites Uncertain physical parameters ofthe model are determined by fitting the calculatedbubble populations at given burnups with measuredbubble size and density data

At the irradiation temperatures of interest(T< 500 K), in analogy with UO2, the diffusion offission-gas atoms is assumed to be athermal with thegas-atom diffusivity Dg proportional to the fissionrate _f The gas-atom re-solution rate b is alsoassumed proportional to the fission rate

The rate equation describing the time evolution of

the density of gas in intragranular bubbles is given by

d½mbðtÞcbðtÞ

dt ¼ 16pfnDgrgcgðtÞcgðtÞ

þ 4prbðtÞDgcgðtÞcbðtÞ  bmbðtÞcbðtÞ ½34

The three terms on the RHS ofeqn [34]represent,

respectively, the change in the density of gas in

intra-granular bubbles due to bubble nucleation, the gas-atom

diffusion to bubbles of radius rb, and the loss of gas

atoms from bubbles because of irradiation-induced

re-solution.Equation [34]can also be represented as the

sum of two equations denoting, respectively, the time

evolution of the fission-gas bubble density cband of the

gas content in bubbles mbas follows:

Ineqn [35], fnis the bubble nucleation factor, and cgand

rg are the gas-atom concentration and radius,

respec-tively In general, the value of fnis less than 1 reflecting

the premise that gas-bubble nucleation within the fuel

matrix requires the presence of vacancies/vacancy

clusters in order to become viable The value of fn is

estimated on the basis of the hypothesis that gas-atom

diffusion occurs by a vacancy mechanism and that a

three gas-atom cluster is a stable nucleus In this case,

fn is approximately the bulk vacancy concentration

(i.e.,104)

interpreted to represent the generation rate of

‘aver-age’ size bubbles of radius rb For every two-atom

bubble that is nucleated, 2=mbof a bubble of radius rb

appears In other words, nucleation of mb two-atom

clusters leads to the gain of one bubble of radius rb

This ‘average size’ bubble is in the peak region of the

bubble-size distribution

Both ‘whole’ bubble destruction and gas-atom

‘chipping’ from bubbles are included (last terms on

RHS) ineqns [35] and [36] in order to capture the

behavior of an average size bubble (that characterizes

the full bubble-size distribution) Within the full

bubble-size distribution, there are bubbles that are

destroyed by one fission fragment collision (e.g.,

bub-bles smaller than a critical size) and others that are

only partially damaged (e.g., bubbles larger than a

critical size) Including b in botheqns [35] and [36]

is an attempt to depict these processes using a

sim-plified formulation that enables an analytical solution

for swelling If obcb was not included in eqn [35],then the density of bubbles could never decrease as

a result of irradiation Likewise, if ð1  oÞbmb wasnot included ineqn [36], the number of atoms in abubble could never decrease However, the partition

of gas-atom re-solution between these two isms, where o is the partitioning fraction, is anassumption that remains to be tested experimentally

mechan-In what follows, equal partition is also assumed, that

is, o¼ 1=2

Because of the strong effect of irradiation-inducedgas-atom re-solution, in the absence of geometric con-tact, the bubbles stay in the nanometer size range Thedensity of bubbles increases rapidly early in the irra-diation At longer times, the increase in bubble con-centration occurs at a much-reduced rate On the basis

of the above considerations, a quasi steady-state tion for the average bubble density cband the averagenumber of gas atoms per bubble mb as a function ofthe density of gas in solution cg and the gas-atomradius rgis given by Spino et al.22

solu-cb¼16pfnrgDgc

2 g

½38

Ineqn [37], fnis the bubble nucleation factor, and in

eqn [38], bvis the van der Waals constant In general,the value of fnis less than 1 reflecting the premise thatgas-bubble nucleation within the fuel matrix requiresthe presence of vacancies/vacancy clusters in order tobecome viable The average bubble radius rbis related

to mbthrough the gas law and the capillarity relation.Imposing gas-atom conservation, that is, requiring thatthe sum of the gas in solution, in intragranular bubbles,and on the grain boundary is equal to the amount ofgas generated (there is no gas released from the U–Mofuel), the term cgðtÞ is determined as

fs 8ffiffiffip

3.20.3.3 Calculation of Evolution of

Average Intergranular Bubble-Size and

Density

grain-boundary bubble nuclei of radius Rb are produced

until such time that a gas atom is more likely to be

captured by an existing nucleus than to meet another

gas atom and form a new nucleus An approximate

result for the grain-boundary bubble concentration is

where a is the lattice constant, z is the number of sites

explored per gas-atom jump, d is the width of the

boundary, x is a grain-boundary diffusion

enhance-ment factor, and K is the flux of gas atoms per unit

area of grain boundary

The intergranular bubble nucleation and growth

formulation incorporated here is on the basis of the

assumption that, although the effect of

radiation-induced re-solution on intergranular bubble behavior

is not negligible, a reasonable approximation can be

obtained by neglecting such effect in the governing

eqn [25] Under the above considerations, the flux K

of atoms at the grain boundary is given by

K ¼dg

3b _f

dð fstÞ

In general, in an irradiation environment where

bub-ble nucleation, gas-atom diffusion to bubbub-bles, and

irradiation-induced re-solution are operative, a

differ-ential growth rate between bubbles of different size

results in a peaked monomodal size distribution.25The

position of the peak in the bubble-size distribution that

occurs under these conditions is defined by the

bal-ance between diffusion of gas-atoms to bubbles and

irradiation-induced re-solution of atoms from bubbles

As more gas is added to the lattice (e.g., as a result of

continued fission), the gas-atom diffusion flux to

bub-bles increases and the peak shifts to larger bubble sizes

and decreases in amplitude, resulting in an increased

level of bubble swelling with increased burnup The

model presented in this section describes the average

behavior of this peak as a function of burnup

Bubble-Size Distribution

Let nðrÞdr be the number of bubbles per unit volume

on the grain boundaries with radii in the range r to

rþ dr Growth by gas-atom collection from fission

gas diffusing from the grain interior removes bubblesfrom this size range, but these are replaced by thesimultaneous growth of smaller bubbles The distri-bution of intragranular gas consists primarily of fis-sion-gas atoms because of the strong effect ofirradiation-induced gas-atom re-solution Bubblesappear on the grain boundaries due to the reducedeffect of re-solution, ascribed to the strong sink-likeproperty of the boundary, as well as to the alteredproperties of bubble nucleation In addition, nðrÞdr isaffected by bubble–bubble coalescence A differentialgrowth rate between bubbles of different sizes leads

to a net rate of increase in the concentration ofbubbles in the size range r to rþ dr This behavior

is expressed bydnðrÞdt

dr nðrÞdrdt

d

dr nðrÞdrdt

c

dr

½43where the subscripts d and c refer to growth by gas-atom diffusion and bubble coalescence, respectively.The growth rate (dr=dt) of a particular bubble isrelated to the rate (dm=dt) at which it absorbs gasfrom the boundary, either by diffusion of single gasatoms, or by coalescence with another bubble Therate of growth due to gas-atom precipitation is con-trolled by the grain-boundary gas-atom diffusioncoefficient xDg and the average concentration Cg offission gas retained by the boundary

Studies on the evolution of helium bubbles inaluminum during heavy-ion irradiation at room tem-perature have shown that bubble coarsening can takeplace by radiation-induced coalescence without bub-ble motion.21This coalescence is the result of the netdisplacement of Al atoms out of the volume betweenbubbles initially in close proximity The resultingnonequilibrium-shaped bubble evolves toward amore energetically favorable spherical shape whosefinal size is determined by the equilibrium bubblepressure

Bubble coalescence without bubble motion tering coalescence) can be understood on the basis ofthe difference in the probability for an atom to beknocked out of the volume between a pair of bubblesand the probability of an atom to be injected into thisinterbubble volume If the bubbles contained thesame atoms as that comprising the interbubble vol-ume, the net flux of atoms out of the interbubblevolume would be zero However, as the gas bubblescontain fission gas and not matrix atoms, the flux ofatoms into the interbubble volume is reduced by the

(sput-bubble volume fraction, that is, the net flux out of

volume is equal to lV  lðV  VBÞ, where l is the

atom knock-on distance and VB is the intergranular

bubble volume fraction In this case, the growth rate

(dr=dt) of a bubble being formed by the coalescence

of two adjacent bubbles (and the commensurate

effective shrinkage rate of the adjacent bubbles) is

related to the rate (dms=dt) at which the interbubble

material is being sputtered away, where

where the effective interbubble volume is assumed

to be disk-shaped with volume = dspr2, and where

dsis the thickness of the material undergoing

sput-tering For a lenticular bubble with radius of

curva-ture r, the equivalent radius of a spherical bubble is

and ggb is the grain-boundary energy

approached by the gradual erosion of the material

between the bubbles This bubble coarsening process

can be visualized as lenticular intergranular bubbles

separated by a distribution of solid disks As these

disks are sputtered because of fission damage, the

majority of the sputtered atoms are injected into the

adjacent bubbles, with the commensurate drawing

together of the bubbles until the joining process has

been completed In order for this process to be viable,

the gas atom knock-on distance should be sufficiently

large such that the majority of atoms sputtered from

the solid disk can enter the adjacent bubbles Because

of the geometry of the lenticular gas bubbles and

solid disks, this distance will be substantially less

than the interbubble spacing

Inserting eqns [44]–[46] into the second term

respect to r,dnðrÞ

As mentioned in Section 3.20.3.3, re-solution ofgrain-boundary bubbles is not explicitly considered,for example, ineqn [50] The rationale for this is thatbecause of the very strong sink-like nature of thegrain boundary, gas-atoms ejected from a gas bubblelocated on the boundary that land within the steepportion of the concentration gradient are ‘suckedback’ into the boundary and quickly reenter thebubble such that the ‘effective’ re-solution rate isrelatively small.26

Combiningeqns [9] and [14]

½52The overall net rate of change of the concentration ofbubbles in a given size range is given by the sum of

dg

where the last term in eqn [53] has been omitted

Equation [54]must be solved subject to the

rele-vant boundary condition In general, this boundary

condition concerns the rate at which bubbles are

formed at their nucleation size r0 From a

consider-ation of freshly nucleated bubbles25

nðr0Þdr ¼ Cb

tbdr

The rate of bubble nucleation is provided by the

grain boundary the average time tb for a gas atom

to diffuse to an existing bubble (as discussed above

this is the time at which bubble nucleation would

essentially cease) is given by

Thus, fromeqn [20], it follows that the bubble

nucle-ation rate is given by

dCb

dt ¼ Cb

where  is a proportionality constant that is

deter-mined by imposing the conservation of gas atoms

The observed grain-boundary bubbles are a

combi-nation of lenticular-shaped objects whose size is

sub-stantially larger than the estimated thickness of the

grain boundary.20 In general, the solubility of gas on

the grain boundary is substantially higher than in the

bulk material In analogy with the treatment of

intra-granular bubble nucleation in the high-temperature

equilibrium g-phase discussed in Section 3.20.2.2,

the gas concentration on the boundary will increase

until the solubility limit is reached (approximately

given by tb), whereupon the gas will precipitate into

bubbles Thus, the rate at which a grain-boundary

bubble adsorbs gas is approximately given by

As described byeqn [50], subsequent to bubble

nucle-ation gas solubility on the boundary will drop to a

relatively low value and gas arriving at the boundary

will be adsorbed by the existing bubble population.Combiningeqns [45] and [58]

ðdr=dtÞr¼ r0 ¼ 3CgbvðrkT þ 2gbvÞ2

16pgð4tbCbpr3=3ÞðkTr3þ 3gbvr2Þ

½60The solution ofeqn [54]subject to the boundary con-dition expressed byeqns [55] and [60]is

nðrÞ ¼64gCb2p2r3ðkTr3þ 3gbvr2Þexp½kðr4 r4

0Þ3bvCgdgðrkT þ 2gbvÞ2

½61where

k¼ p _f lds

2bvxDgCg

½62

Calculations and Intragranular DataOne of the major challenges in the field of fission-gasbehavior in nuclear fuels is the quantification ofcritical materials properties There is a direct corre-lation between the accuracy of the values of criticalproperties and the confidence level that the proposedunderlying physics is realistic

The values of the key parameters used in the

known or estimated from the literature27; the values

Table 2 Values of parameters used in the calculations

of the others (e.g., x) result from a comparison of the

present theory with measured data for bubble

popu-lations As an example of estimated parameters, the

values of Dgand b used for U–Mo are assumed to be

an order of magnitude less than those for UO2 On

the basis of irradiation-enhanced creep rates

measured in UO2, UN, and UC,31 the

irradiation-enhanced gas-atom diffusivity Dg is expected to be

lower in U–Mo than in UO2 In addition, as a result of

the higher thermal conductivity of the alloy as

com-pared to the oxide, b is also expected to be lower in

U–Mo than in UO2 This argument is on the basis of

the expected larger interaction cross-section in the

metallic alloy with conduction electrons However,

because of the (assumed) linear dependence of both

Dg and b on _f , and because it is the ratio Dg=b that

appears ineqns [37]–[39], it is reasonable to assume

that this ratio of critical properties is the same for

both materials

The calculated intragranular bubble-size

distribu-tion for Z03 (fully annealed) is contrasted with data32

for the average bubble size and density in irradiated

U–10Mo fuel (ground and atomized) as shown in

Figure 9 Values for Dg and b obtained from data

and analyses of UO2 are listed in Table 2 The

calculated results shown inFigure 3are in

reason-able accord with the observed estimates of the

aver-age bubble density and size However, it should

be noted that highly over pressurized solid gas

bubbles with diameters of 1–2 nm were observed toform a superlattice in the U–Mo with a relativelyclose spacing (6–7 nm) and having an approximatemonomodal-like distribution.32 For this reason, aslisted in Table 1, the gas-bubble nucleation factorwas taken to be equal to unity In any event, thephysics presented in this section is not compatiblewith the formation of a bubble superlattice

Calculations and Intergranular DataThe calculated distributions are obtained by integrat-ingeqn [61]over the bin sizesi, that is, the bubbledensity NðiÞ in units of m3is given by

on the intergranular bubble nucleation is visible in

eqn [41] By increasing x the intergranular bubbledensity is reduced with a commensurate increase inbubble size The larger value used for x for the

Figure 9 Calculated intragranular bubble-size distribution for Z03 (fully annealed) contrasted with data32for the average bubble size and density in irradiated U–10Mo fuel (ground and atomized) The calculated distribution is not consistent with the observed bubble superlattice.

nonannealed miniplates reflects the increase in

diffu-sivity with decreased molybdenum content

The experimental database consists of both

as-atomized and g-phase annealed specimens The range

of burnup is from 5.8 to 9.2 at.% U, with fission rate

from 2.3 to 6.8
1014

f cm3s1, temperature from

66 to 191 C, and Mo content from 6 to 10 wt%.20

Table 2 shows the value of the key physical

para-meters used in the model The remaining critical

parameter x was determined by best overall

interpre-tation of the measured intergranular bubble-size

distributions for the g-phase annealed and for the

as-atomized specimens, respectively In addition,

the reduced value for the grain-boundary energy g

for the nonannealed material reflects lower angle

boundaries as compared to the annealed specimens.20

Figure 10 shows calculated results compared

with RERTR-3 miniplates Z03 and Y01 data

These miniplates were fully annealed and as suchhave a uniform distribution of molybdenum acrossthe fuel region Z03 was fabricated by atomization,whereas Y01 was made from a ground powder Thecalculated distribution is in very good agreementwith the measured quantities

Figure 11 shows calculated and measured granular bubble-size distribution for U–10Mo as-atomized plates As is evident from the comparisons

inter-inFigure 11, in general, the model calculations are inremarkable agreement with the data.Figure 12showscalculated and measured intergranular bubble-sizedistribution for U–6Mo and U–7Mo as-atomizedplates, respectively The deviation between calculated

likely due to the lower Mo content and, therefore,requires different (larger) values for Dgand x.The results of calculations shown in Figure 13

secured with the use of bubble-size distributionscompared with the use of mean values (i.e., averagequantities such as bubble density and diameter) Com-paring model predictions with average quantities

is by far the dominant validation technique reported

in the literature The graph on the left hand side

ofFigure 13shows the sensitivity of the calculateddistributions to the value of the gas-atom knock-

shows the results of a series of calculations madewith paired values for the grain-boundary-diffusionenhancement factor and the thickness of the grainboundary chosen such that the calculation of aver-age quantities remains unchanged These calculatedresults demonstrate that the calculated distributionundergoes significant changes in shape as well as posi-tion and height of the peak As such, the capacity tocalculate bubble-size distributions along with the avail-ability of measured distributions (as has been obtainedfrom RERTR irradiated fuel plates) goes a long way invalidating not only values of key materials propertiesand model parameters, but also proposed fuel behav-ioral mechanisms

Swelling Safety MarginsThe model presented here, taken together with theanalysis of fuel swelling in the high-temperatureequilibrium g-phase presented in Section 3.20.2,enables the calculation of gas-driven fuel swellingsafety margins.Figure 14shows the calculated per-centage of unrestrained fuel swelling as a function

Bubble diameter ( mm)

0.04 0.06 0.08 0.10 0.12 0.14 0.16

Bubble diameter ( mm)

Theory Data Y01

Figure 10 Calculated and measured intergranular

bubble-size distribution for U–10Mo g-phase annealed

plates Z03 was fabricated by atomization, whereas Y01

was made from ground powder Reproduced from

Rest, J.; Hofman, G L.; Kim, Y S J Nucl Mater 2009,

385(3), 563–571.

of burnup for U–8Mo fuel irradiated at various

tem-peratures The calculated swelling is a strong

func-tion of the irradiafunc-tion temperature as well as the fuel

burnup It should be noted that the temperature

dependence of fuel that is under restraint (e.g., by

cladding) is much softer than exhibited inFigure 14

The curves in Figure 14do not reflect any gas

release that may occur Empirically, gas release begins

to occur when the swelling reaches 25–30% If all the

bubbles are spherical, of the same size, and randomly

distributed, then interconnection will be initiated

at33% swelling However, in general, the calculation

of the swelling at which the bubbles interconnect

is complicated by a relatively broad distribution ofnonspherical bubbles, nonuniformly distributed withinthe fuel regions (e.g., such as that in Figure 14).The maximum gas release in these high swelling fuelsapproaches 80% There are many small bubblesbetween the larger interconnected bubbles that con-tinue to drive the swelling even at high gas releasevalues However, even so, the calculated swellingcurves inFigure 8are typical of those that have been

0.0 5.0E+ 7

Figure 11 Calculated and measured intergranular bubble-size distribution for U–10Mo as-atomized plates V03, V07, V002, V8005B, and V6019G Reproduced from Rest, J.; Hofman, G L.; Kim, Y S J Nucl Mater 2009, 385(3), 563–571.

measured The key here is thatFigure 14shows

unre-strained swelling If the fuel is given enough room, it

will keep on deforming

If it is arbitrarily assumed that the maximum

allowable fuel swelling is 50%, then fuel safety

mar-gins can be calculated using the results ofFigure 14

As an example of this type of calculation,Figure 15

shows the calculated boundary between stable and

unstable unrestrained fuel swelling as a function of

fission density and fuel temperature The solid line in

Figure 15is the 50% unrestrained swelling threshold

obtained fromFigure 14 Also shown inFigure 15

is the fission density and fuel temperature for

safety margin for RERTR-9 is 150 K

Calculations of intergranular bubble-size distribution

made with a mechanistic model of grain-boundary

bubble formation kinetics are consistent with the

measured distributions Analytical solutions are

obtained for the rate equations, thereby providingfor increased transparency and ease of use The resultssupport a multiatom gas-bubble nucleation mecha-nism on grain boundaries that have substantiallyhigher gas solubility than that in the grain interior.The gas-atom diffusion enhancement factor on thegrain boundaries was determined to be 125–850 inorder to obtain agreement with the measured distri-butions The enhancement factor is about 8 timeshigher for as-fabricated powder plates than for theannealed plates because of the lower Mo content onthe boundaries This range of values for the enhance-ment factor is consistent with values obtained in theliterature.30The largest deviation between calculatedand measured results (Figure 12) is most likely due toseveral fuel plates that have a lower Mo content (6 and

7 wt% vs 10 wt%) and, thus, require different (larger)values for D and x

Theory Data R6007F

Figure 12 Calculated and measured intergranular

bubble-size distribution for as-atomized plates S03 (U–6Mo)

and R6007F (U–7Mo) Reproduced from Rest, J.; Hofman,

G L.; Kim, Y S J Nucl Mater 2009, 385(3), 563–571.

Bubble diameter ( μm) 0.04 0.06 0.08 0.10 0.12 0.14 0.16

Bubble diameter ( μm) 0.04 0.06 0.08 0.10 0.12 0.14 0.16

0 1E + 8 2E + 8 3E + 8 4E + 8 5E + 8 6E + 8 7E + 8

G L.; Kim, Y S J Nucl Mater 2009, 385(3), 563–571.

The agreement between the model and the

measured distributions for the 10 wt% Mo fuel

sup-ports the validity of a sputtering coalescence (bubble

coalescence without bubble motion) coarsening

mechanism on the grain boundaries In this regard,

attempts by this author to reproduce the shape of the

intergranular bubble-size distribution using a modelbased on the growth of bubbles in a regular array35have not been successful

A number of the critical parameters listed in

Table 2 are assumed to be a factor of 10 less thanthose listed in the literature for UO However, it is

Fission density (cm −3)0.0 2.0E + 20 4.0E + 20 6.0E + 20 8.0E + 20 1.0E + 21 1.2E + 21 1.4E + 21

0 100 200 300 400

1E + 22

147 K

Safety margin

50% unrestrained swelling threshold

RERTR-9

Figure 15 Calculated threshold between stable and unstable gas-driven fuel swelling Also shown in is the fission density and fuel temperature for RERTR-9 Reproduced from Rest, J J Nucl Mater 2010, 407, 55–58.

the ratio of these parameters (b=Dg, x=l) that appear

in the model solution; thus, the validity of their use

for U–Mo reduces to the ratios being approximately

the same for both materials This assumption is

sup-ported by the observed similarity (albeit remarkable)

in bubble behavior and microstructure evolution

between the two materials.36

The results demonstrate the increased validation

leverage secured with the use of bubble-size

dis-tributions compared with the use of mean values

(i.e., average quantities such as bubble density and

diameter) Model predictions are sensitive to various

materials and model parameters Improved prediction

capability requires an accurate quantification of these

critical materials properties and measurement data

The results of this analysis enable the calculation

of safety margins for unrestrained fuel swelling

These safety margins contain an uncertainty

primar-ily tied to uncertainties in the values of the volume

and Xe diffusion coefficients

Re-solution

After a short period of irradiation, the intragranular

structure of UO2 is populated with a high-density

(1023m3) of small (r  109m) bubbles,19

rated by5–10 bubble diameters In general, observa-

sepa-tions that bubbles confined to the bulk (lattice) material

of irradiated nuclear fuels do not grow to appreciable

sizes at low temperatures (fuel temperatures where the

gas-atom diffusivity is irradiation enhanced, i.e.,<0.5

melting temperature) are ascribed to the effect of

irradiation-induced re-solution (see Chapter 2.18,

Radiation Effects in UO2).3,37 Gas-atom re-solution

is a dynamic bubble-shrinkage mechanism wherein

fission fragments either directly or indirectly cause gas

atoms to be lost from a bubble Only when sinks, such as

grain boundaries, are present in the material can

bub-bles grow to sizes observable with a SEM.38Most

cal-culations on intergranular gas behavior found in the

literature have focused on the condition for grain-face

saturation and have not addressed the specific

mechan-ics of intergranular bubble growth in the presence

of irradiation-induced re-solution.39–42 Calculations

of grain-boundary bubble growth have been performed

under the assumption that the effective gas-atom

re-solution rate from grain-boundary bubbles is

negligible.43–45This assumption has relied on

heu-ristic arguments23that the strong sink-like nature of

a grain boundary provides a relatively short ture distance for gas that has been knocked out of abubble, and as such neutralizes the ‘shrinking’ effect

recap-of the re-solution process These grain-boundarybubbles grow at an enhanced rate as compared

to those in the bulk material The importance ofunderstanding the physics underlying intergranularbubble growth is underscored by the rim region ofhigh-burnup fuels which are characterized by anexponential growth of intergranular porosity towardthe pellet edge: a narrow band of fully recrystallizedporous material exists at the pellet periphery, and

a rather wide adjacent transition zone with partiallyrecrystallized porous areas appears dispersed withinthe original matrix structure.46 In particular, theunderstanding of the dynamics of irradiation-inducedrecrystallization and subsequent gas-bubble swellingrequires a quantitative assessment of the nucleationand growth of grain-boundary bubbles.45,46

A mechanistic model is described, for the growth ofgrain-boundary bubbles during irradiation at relativelylow temperatures (i.e., where gas-atom diffusion isathermal) in order to quantify the effect of gas-atomre-solution on their growth A variational method isused to calculate diffusion from a spherical fuel grain.The junction position of two trial functions is set equal

to the bubble gas-atom knock-out distance The effect

of grain size, gas-atom re-solution rate and diffusivity,gas-atom knock-out distance, and grain-boundarybubble density on the growth of intergranular bub-bles is studied, and the conditions under which inter-granular bubble growth occurs are elucidated

The flux of gas atoms diffusing to the grain aries in a concentration gradient is obtained by solv-

spherical grain that satisfies the equation

of irradiation-induced re-solution This back flux ofgas can be thought of as an additional matrix gas-atom generation mechanism and is assumed to bedistributed uniformly within a spherical annulus of

thickness l, where l is the gas-atom knock-out

dis-tance In eqn [64], intragranular bubble trapping of

fission gas has been neglected However, this effect

can be modeled by using an effective diffusion

coef-ficient given by Turnbull3

Deffg ¼ b

where b is the gas-atom re-solution rate and g is the

probability per second of a gas atom in solution being

captured by an intragranular bubble Observed

con-centrations 1023m3 of intragranular bubbles of

1 nm radius3

with b¼ 2
104s1 yields a value

for g¼ 2:5
104s1and Deff

where dt is an increment of time and dgis the grain

diameter For an increment of time dt the

concentra-tion of gas atoms in a spherical grain described ineqn

Euler’s theorem may now be used to obtain a

varia-tional principle equivalent toeqn [3]:

which assumes that Dirichlet boundary conditions

are to be applied An approximate solution to the

problem may now be obtained by choosing a trial

function that satisfies the boundary conditions and

minimizes the integral in eqn [68] in terms of free

parameters in the function Many types of trial

func-tion could be chosen, but it is easier to work with

piecewise functions than global functions Quadratic

functions are attractive because they allow an exact

representation ofeqn [64]for long times Matthews

and Wood47 obtained a realistic level of accuracy

with a minimum of computer storage and runningtime by splitting the spherical grain into two concen-tric regions of approximately equal volume In eachregion, the gas concentration was represented by aquadratic function In the inner region, the concen-tration function was constrained to have dCg=dr ¼ 0

at r¼ 0 In the outer region, the concentration tion was constrained to a value of Cg¼ 0 at r ¼ dg/2.The two functions were also constrained to be con-tinuous at the common boundary of the two regions.This left three free parameters: the concentrations

func-C1g, C2g, and C3g, respectively, for the radius ratio

r1¼ 0:2, r2¼ 0:4, and r3¼ 0:45, where r ¼ r=dg.These positions are the midpoint radii of the innerregion, the boundary between the regions, and themidpoint radius of the outer region, respectively.However, this method is too crude if one is inter-ested in an accurate representation of the concentra-tion gradient in the presence of irradiation-inducedre-solution from grain-boundary bubbles where thegas atom knock-out distance l is on the order of

100 A˚ 5

In this case, it is necessary to formulate theproblem in terms of l The radius ratio at the inter-face of the two regions is now expressed as

The trial functions are as follows:

For the inner region,

=3r3

l ½70For the outer region;

g 2

ð2rl1Þ2½8r22ð2rlþ3Þrþ2rlþ1

g 3

ð2rl1Þ2½ð2rlþ1Þr2r2rl ½71

Equations [69] and [70]are substituted for Cgineqn[68] and an extremum is found by differentiatingwith respect to C1g, C2g, and Cg3 This results in a set

of three coupled algebraic equations that can bedirectly solved to obtain the concentrations C1g, C2g,and C3g, as follows: