3.20 Modeling of Fission-Gas-Induced Swelling of
Nuclear Fuels*
J. Rest
Argonne National Laboratory, Argonne, IL, USA

ß 2012 Elsevier Ltd. All rights reserved.

580

3.20.1

Introduction

3.20.2

Intragranular Bubble Nucleation: Uranium-Alloy Fuel in the High-Temperature
Equilibrium g-Phase
Introduction
A Multiatom Nucleation Mechanism
Calculation of the Fission-Gas Bubble-Size Distribution
Bubble Coalescence
Analysis of U–10Mo High-Temperature Irradiation Data
Conclusions
Intergranular Bubble Nucleation: Uranium-Alloy Fuel in the Irradiation-Stabilized
g-Phase
Introduction
Calculation of Evolution of Average Intragranular Bubble-Size and Density
Calculation of Evolution of Average Intergranular Bubble-Size and Density
Calculation of Intergranular Bubble-Size Distribution
Comparison Between Model Calculations and Intragranular Data
Comparison Between Model Calculations and Intergranular Data

Calculation of Gas-Driven Fuel Swelling Safety Margins
Conclusions
Irradiation-Induced Re-solution
Introduction
Flux Algorithm
Grain-Boundary-Bubble Growth
Analysis of Bubble Growth on Grain Boundaries
Discussion and Conclusions
Irradiation-Induced Recrystallization
Introduction
Model for Initiation of Irradiation-Induced Recrystallization
Model for Progression of Irradiation-Induced Recrystallization
Theory for the Size of the Recrystallized Grains
Calculation of the Cellular Network Dislocation Density and Change in
Lattice Parameter
Calculation of Recrystallized Grain Size
Evolution of Fission-Gas Bubble-Size Distribution in Recrystallized U–10Mo Fuel
Effect of Irradiation-Induced Recrystallization on Fuel Swelling
Discussion and Conclusions
Final Thoughts

3.20.2.1
3.20.2.2
3.20.2.3
3.20.2.4
3.20.2.5
3.20.2.6
3.20.3
3.20.3.1
3.20.3.2

3.20.3.3
3.20.3.4
3.20.3.5
3.20.3.6
3.20.3.7
3.20.3.8
3.20.4
3.20.4.1
3.20.4.2
3.20.4.3
3.20.4.4
3.20.4.5
3.20.5
3.20.5.1
3.20.5.2
3.20.5.3
3.20.5.4
3.20.5.5
3.20.5.6
3.20.5.7
3.20.5.8
3.20.5.9
3.20.6
References

*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 royaltyfree license to publish or reproduce the published form of this
contribution, or allow others to do so, for US Government
purposes.


581
581
582
584
586
586
591
591
591
591
593
593
595
596
597
599
601
601
601
603
604
608
610
610
610
611
614
614
616

620
621
624
625
625

Abbreviations
ATR
EOS
PIE

Advanced test reactor
Equation of state
Postirradiation examination

579


580

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

RERTR Reduced Enrichment for Research and
Test Reactors
SEM
Scanning electron microscope
TEM
Transmission electron microscope

3.20.1 Introduction

This chapter addresses various aspects of modeling
fission-gas-induced swelling in both oxide and metal
fuels. The underlying theme underscores the similarities 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 critical 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 irradiation environment. Examples of such properties
are gas-atom and bubble diffusion coefficients, bubble 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 possibility 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 realistic 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., stoichiometry). As the grain boundaries move, they encounter 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 to
obtain a model that can accurately predict fission-gas
release.
On a different level, below temperatures at which
defect annealing occurs, at relatively high doses, fuel
materials such as UO2 and uranium alloys such as
U–10Mo undergo irradiation-induced recrystallization wherein the as-fabricated micron-size polycrystalline grains are transformed to submicron-sized
grains. As a result of this transformation, fission
gases are moved from within the grain to the grain
boundaries, transferring the materials response to gasdriven swelling from intragranular to intergranular. In
addition, gas-bubble/precipitate complexes can act as
pinning sites that immobilize potential recrystallization nuclei, and thus affect the dose at which recrystallization is initiated. The synergy between these
different forces needs to be realistically captured in
order to accurately model the phenomena.
Given the current uncertainties in materials properties, critical parameters, and proposed behavioral
mechanisms, a key issue in modeling of fission-gas
behavior in nuclear fuels is realistic validation. In
general, most of the model validation is accomplished
by adjusting/predicting these properties and parameters to achieve agreement with measured gas
release and swelling, and with mean values of the
bubble-size distribution. However, the uncertainties
in these properties and parameters generate an inherent uncertainty in the validity of the underlying
physics and the physical reality of proposed behavioral mechanisms. This inherent uncertainty clouds

the predictive aspects of any mechanistic approach to
describe the phenomena. Thus, more detailed data
are required to help clarify these issues.
The shape of bubble-size distribution data contains information on the nature of the behavioral
mechanisms 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
distributions are few and far between, and measured
intergranular bubble distributions are all but nonexistent. In Sections 3.20.2 and 3.20.3, recently
measured intra- and intergranular bubble-size distributions obtained from U–Mo alloy fuel are used
for model validation, and the robustness of this
technique in reducing uncertainties in proposed
mechanisms and materials properties as compared


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

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 parameters, 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, intragranular multiatom nucleation is the dominant nucleation
mechanism at high temperature. Although the discussion on gas-bubble nucleation focuses on uranium
alloys (because of the availability of measured bubble-size distributions), there is no reason to believe
that they would not be applicable to oxide fuel as well.
Section 3.20.4 presents an analysis of irradiationinduced re-solution. Specifically, the analysis presents 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 intergranular bubble density of 1 Â 1013 mÀ2 is equivalent
to a bubble density of 2 Â 1018 mÀ3 for a grain size of
5 Â 10À6 m. This is to be compared to observed intragranular bubble densities that are on the order of
1023 mÀ3. In addition, typical intergranular bubble
sizes of tenths of a micron are to be compared to
nanometer-sized intragranular bubbles. This consideration is supported not only by the experimental
results presented in Section 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 initiation and progression of irradiation-induced recrystallization 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

581

recrystallization story is clarified. Calculations are
compared to data for the dislocation density and
change in lattice displacement in UO2 as a function
of burnup. In addition, calculations are compared to
available data for the recrystallized grain-size distribution in UO2 and in U–10Mo.
Models such as those described in this chapter are
stories that remain just stories until validated by
experiment. Fission-gas behavior in nuclear fuels
has been studied since the early 1950s, and although
almost 60 years have elapsed, a definitive picture of
these phenomena is still unavailable today. The reason, as stated above, is the difficulty in obtaining
reliable single-effects data in a multivariate irradiation environment, coupled with the highly synergistic nature of the beast.

3.20.2 Intragranular Bubble
Nucleation: Uranium-Alloy Fuel in
the High-Temperature
Equilibrium g-Phase
3.20.2.1

Introduction

Figure 1 shows scanning electron microscope
(SEM) micrograph of g-U–Zr–Pu alloy fuel.1 The
microstructure shown in Figure 1 is 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.


582

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

uranium metal alloys irradiated in the equilibrium
g-phase. Swelling of this material is predominantly
due to the growth of fission-gas bubbles. Its fissiongas behavior is characterized by high mobility at
relatively high temperatures at which it exists at the
equilibrium g-U–Zr–Pu phase. As seen in Figure 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 apparent, and some of the larger bubbles appear to be
growing into the smaller neighboring bubbles.
Most attempts at describing intragranular
gas-bubble nucleation in nuclear fuels at higher
temperatures have relied on a homogeneous2 or
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 resolution, 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.2 In 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 evolution of the density of gas in intragranular bubbles is
given by
d½mb ðt Þcb ðt ފ
¼ 16pfn Dg rg cg ðt Þcg ðt Þ
dt
þ 4prb ðt ÞDg cg ðt Þcb ðt Þ À bmb ðt Þcb ðt Þ

½1Š

where cg , cb are the densities of gas atoms and bubbles, respectively, mb is the average number of gas
atoms per bubble, Dg is the gas-atom diffusion coefficient, b is the gas-atom re-solution rate from bubbles,
and fn , the so-called nucleation factor, is the probability 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 example, for heterogeneous bubble nucleation along fission tracks in UO2, fn is approximately the average
volume fraction of fission tracks %10À4. The three
terms on the right-hand-side of eqn [1] represent,

respectively, the change in the density of gas in intragranular bubbles because of bubble nucleation, gasatom diffusion to bubbles of radius, rb , and the loss of
gas atoms from bubbles because of irradiationinduced re-solution.
An implicit assumption in eqn [1] is that once a
two-atom nucleus forms, it grows instantaneously to
an m-atom bubble. Values of fn ranging from 10À7 to

10À2 have been proposed, which makes the nucleation factor little more than an adjustable parameter.5
A substantial contribution to the spread of reported
values for fn is that most models describe the time
evolution of mean values of cb and rb which are
compared to the respective mean values of the measured quantities (comparing model predictions with
average quantities is by far the dominant validation
technique reported in the literature). In this regard,
as will be demonstrated in the following section,
the use of bubble-size distributions goes a long way
toward the reduction of such uncertainties.6
As an approach to circumventing the deficiencies
thus described, in what follows a multiatom bubble
nucleation mechanism is proposed and implemented
into a mechanistic calculation of the intragranular
fission-gas bubble-size distribution. The results of
the calculations are compared to a measured bubblesize distribution in U–10Mo irradiated at relatively
high temperature to 4% U-atom burnup. The multiatom nucleation model is compared to the two-atom
model within the context of the data, and the implications of each mechanism for the observable quantities are discussed.
In the next section, a multiatom nucleation mechanism is formulated. Section 3.20.3 presents an outline for a calculation of the time evolution of the
bubble-size distribution. In Section 3.20.4, a discussion is presented of processes that lead to coarsening
of the as-nucleated bubble distribution. In Section
3.20.5, model calculations are used to interpret a
measured distribution in U–8Mo uranium alloy fuel
irradiated to 4% U burnup at 850 K. In addition, in
this section, a comparison between the multiatom
and two-atom nucleation mechanisms is attempted.
Finally, conclusions are presented in Section 3.20.6.
3.20.2.2 A Multiatom Nucleation
Mechanism
Fission gases Xe and Kr are generated in a nuclear

fuel at a rate of about 0.25–0.30 atoms per fission as a
result of decay of the primary fission products. About
seven times more Xe is produced than Kr. These gas


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

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.7 Given enough energy via thermal fluctuations,
and/or via irradiation, the gas atoms can hop randomly from one site to another and thus diffuse
through the material. The gas atom/vacancy complexes 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 uranium 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 10À10.8 This
figure may be increased up to %10À5 in 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 nucleation event repeats. Thus, assuming that all the gas
precipitates into bubbles of equal size r 0, the concentration of gas in the bubble at nucleation is given by
mðr 0 Þ ¼

bv cgcrit
4=3pr 03 cb ðr 0 Þ

½2Š

583


where cgcrit is the concentration of gas at the solubility
limit, bv is the volume per atom (van der Waals constant), and cb ðr 0 Þ is the concentration of bubble nuclei
at the unrelaxed radius r 0, that is, the initial stage of
bubble nucleation is a volume-conserving process.
Subsequently, in order to lower the free energy of the
system, the overpressurized nuclei relax by absorbing
vacancies until the bubbles reach equilibrium. At equilibrium, the bubble radius is r and, in the absence of
significant external stress, the pressure in the bubble is
given by
2g
½3Š
r
where g is the surface energy per unit area. If it is
assumed that the average gas bubble size r 0 is a function
of the equilibrium bubble size, then differentiating eqn
[2] with respect to the equilibrium radius r and rearranging terms yields
Pe ¼

1 dcb ðr 0 Þ
1 dmðr 0 Þ 3 dr 0
¼À
À 0
0
cb ðr Þ dr
mðr 0 Þ dr
r dr

½4Š

Let us assume that during the relaxation phase

there is no interaction between the nucleated bubbles,
that is,
r 0 ! r ; mðr 0 Þ ! mðr Þ ¼ mðr 0 Þ; cb ðr 0 Þ ! cb ðr Þ ¼ cb ðr 0 Þ
½5Š
The nucleation problem thus consists of determining
the two terms on the RHS of eqn [4]. The first term on
the RHS of eqn [4] can be determined from the equation of state (EOS), the capillarity relation, and the
conditions expressed in eqn [5]. Using the van der
Waals EOS,
PðV À mbv Þ ¼ mkT

½6Š

where V ¼ 4/3pr 0 3 is the bubble volume. Recognizing
that at nucleation the bubble size is small such that
2g/r 0 ) s , where s is the external stress, and differentiating eqn [6] with respect to the equilibrium
radius r one obtains
!
1 dmðr 0 Þ 3
rkT
½7Š
¼ 1À
mðr 0 Þ dr
r
3ðrkT þ 2gbv Þ
The remaining term on the RHS of eqn [4] can
be determined by invoking energy minimization as
the driving force for bubble equilibration. The change
in the Gibbs free energy due to bubble expansion is
given by

4
ÁG ¼ pr 03 ÁGv þ 4pr 02 g
3

½8Š


584

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

where ÁGv is the free energy driving bubble equilibration, which, in analogy with the treatment of the nucleation of liquid droplets in a vapor,9 can be expressed as
kT
½9Š
lnðPðr Þe =Pðr 0 ÞÞ
O
where O is the atomic volume. The critical bubble
radius at equilibrium is given by the condition
ÁGv ¼

@ÁG
À2g
¼ 0 ! r ¼ rcrit ¼
@r
ÁGv

½10Š

Inserting the expressions for Pe and P from eqns [3] and
[6], respectively, into eqn [10], differentiating with

respect to the bubble radius r, and applying a little
algebra results in


dr 0
1 O kT
¼À
þ
½11Š
4pr 02
dr
X r 2g

3.20.2.3 Calculation of the Fission-Gas
Bubble-Size Distribution

where
X¼

kTr


4 03
2g pr À mbv
3

½12Š

Making use of eqn [2] in eqn [11] results in



!
3 dr 0
m2 c b 2gO=rkT rkT dm kT O
bv dm
¼
À
þ
À
þ
e
bv c g
m dr
r 0 dr
2gm dr 2g r
½13Š
Finally, substituting eqn [5], eqn [7], and eqn [13] into
eqn [4] yields
1 dcb
1 dm mcb
¼À
þ
cb dr
m dr bv cg


!
rkT dm mkT mO
dm
þ

À
þ bv
e2gO=rkT
2gm dr
2g
r
dr

Definition of variables in eqn [15],

The model consists of a set of coupled nonlinear differential equations for the intragranular concentration
of fission product atoms and gas bubbles of the form10
dCi
¼ Àai Ci Ci À bi Ci þ ci ði ¼ 1; . . . ; N Þ ½15Š
dt
where Ci is the number of bubbles in the ith size class
per unit volume; and the coefficients ai , bi , and ci obey
functional relationships of the form
ai ¼ ai ðCi Þ
bi ¼ bi ðC1 ; . . . ; CiÀ1 ; Ciþ1 ; . . . ; CN Þ

½14Š

The as-nucleated bubble-size distribution is then
obtained by the simultaneous solution of eqns [7]
and [14].

Table 1

Subsequent to the nucleation event, the asnucleated bubble-size distribution evolves under the

driving forces of gas diffusion to bubbles, gas-atom
re-solution from bubbles, and bubble coalescence
due to bubble–bubble interaction via bubble motion
and geometrical contact. As stated earlier, additional
nucleation events are delayed because of the gas in
solution remaining below the solubility limit, as the
gas generated by continuing fission events is trapped
within the existing bubble-size distribution. This last
point is facilitated by the relatively high gas-atom diffusivities at the temperatures of interest (i.e., those
under which the equilibrium g-phase of the alloy
exists). Eventually, the gas in solution may again
reach the solubility limit at which time the nucleation event repeats.

The 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 coalescence
with bubbles in that class; bi represents the rate at
which bubbles are lost from the ith size class because
of coalescence with bubbles in other size classes and

dCi
¼ Àai Ci Ci À bi Ci þ ci ði ¼ 1; . . . ; NÞ
dt

i

Ci

aiCiCi


biCi

ci

1

Concentration of
intragranular gas atoms

2,. . .,N

Concentration of
intragranular gas bubbles

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

Rate of gas atom loss
due to diffusion into
gas bubbles
Rate of gas bubble
loss due to
coalescence with
bubbles in other

size classes

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


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

re-solution; and ci represents the rate at which bubbles
are being added to the ith size class because of fissiongas generation, bubble nucleation, bubble growth
resulting from bubble coalescence, and bubble shrinkage 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 Si denotes
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

585


given by
Tijk ¼

Skþ1 þ Si À Sj
Sj
¼1À
Skþ1 À Sk
Skþ1 À Sk

½20Š

and the probability that the coalescence will result in
a k þ 1 bubble is given by
Tijkþ1 ¼

S i þ S j À Sk
Sj
¼
Skþ1 À Sk
Skþ1 À Sk

½21Š

The array Tijk may be considered the probability that
an i-bubble will become a k bubble as a result of its
coalescence with a j-bubble. The rate Nijk at which
i-bubbles become k bubbles is given by
X
Àij Tijk

½22Š
Nijk ¼
j i

Si ¼ nSiÀ1

½16Š

pffiffiffiffiffiffiffiffiffi
where the integer n ! 0:5 þ 1:25, 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 coalescence Àij of i-bubbles with j-bubbles is given by
Àij ¼ Pij Ci Cj

½17Š

where Pij is the probability in m3 sÀ1 of an i-bubble
coalescing with a j-bubble. For i ¼ j, Àij becomes
1
½18Š
Àii ¼ Pii Ci Ci
2
so that each pair-wise coalescence is counted only
once.
Coalescence between bubbles results in bubbles
growing from one size class to another. The probability 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 Si þ Sj . The array Tijk is defined

by three conditions:
P
1.
k Tijk ¼ 1 (the total probability of producing a
bubble
is unity).
P
2.
k Tijk Sk ¼ Si þ Sj (the number of gas atoms, on
average, is conserved).
3. For a given pair ij, only two of the Tijk array elements are nonzero. These elements correspond to
k and k þ 1, where Sk Si þ Sj Skþ1 .

The j-bubble is assumed to disappear because gas
atoms are absorbed into the i-bubble. The rate of
disappearance wj is given by
X
Àij
½23Š
wj ¼
j !i

The rate Nik at which i-bubbles become k bubbles,
with k ¼ i þ 1, is reduced by various processes such as
the re-solution of gas atoms. Re-solution is the result
of collisions (direct and/or indirect) between fission
fragments and gas bubbles. From eqns [21] and [22],
X
Àij Tijk
Nik ¼

j i

¼

X
j i

¼

Sj
S k À Si
X
Pij Cj Sj

Pij Ci Cj

Ci
S k À Si

½24Š

j i

P
The expression j i Pij Cj Sj is the rate at which gas
atoms are added to an i-bubble. Re-solution causes
an i-bubble to lose gas atoms at a rate given by bi Si,
where bi is the probability that a gas atom in an
i-bubble is redissolved. The reduced Nik becomes
Ci X

ðPij Cj Sj À bi Si Þ
½25Š
Nik ¼
S k À Si j i

½19Š

If the expression within the parentheses is negative,
then Nik is zero, and Nik 0 , the rate at which i-bubbles
become i À 1 bubbles, with k 0 ¼ i À 1, is defined as
!
X
Ci
Nik 0 ¼
bi S i À
Pij Cj Sj
½26Š
S i À Sk
j i

Thus, the probability that a coalescence between an
i-bubble and a j-bubble will result in a k bubble is

Equations [25] and [26] are proportional to the probabilities that any particular i-bubble will become an

From these three conditions, it follows that k ¼ i,
and
Tijk Sk þ ð1 À Tijk ÞSkþ1 ¼ Si þ Sj



586

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

i þ 1 or an i À 1 bubble, respectively; the ratio of
the probabilities is equal to the ratio of the rates.
The aforementioned definition of Nik and Nik 0 is
consistent with the conservation of the total number
of gas atoms.
3.20.2.4

Bubble Coalescence

gc

The bubbles are assumed to diffuse randomly
through the solid alloy by a volume diffusion mechanism. The bubble diffusion coefficient Di of a bubble
having radius Ri is given by
Di ¼

3a03 Dvol
4pR3i

j 6¼i

The interaction cross-section represented in eqn [28]
is based on an analysis of colloidal suspensions within
the framework of the continuum theory.11 Fission-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 from eqn [27] for the volume diffusion mechanism, 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 continual 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 uniformly distributed, contact is reached when
2Rb ð2Cb =3Þ1=3 ¼ 1

½29Š

In analogy with percolation theory, the probability of
an i-bubble contacting a j-bubble is given by

h
pffiffiffiffiffiffi i
gc
1=3
Pij ¼ 0:5 1 À erf 1 À Rij Cij
0:5=s
½30Š
where
Rij ¼ Ri þ Rj ;

2
¼ ðCi þ Cj Þ

3

ai ¼ 4pDvol Ri Pii
X
gc
bi ¼ 4pDvol
Ri Cj Pij

½32Š

j 6¼i

where Pij is given by eqn [30].
In what follows, it is assumed that DXe ¼ Dvol .

½27Š

where a0 is the lattice constant and Dvol is 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 of eqn [15]) are represented, respectively, by
X
ðRi þ Rj ÞðDi þ Dj ÞCj ½28Š
ai ¼ 16pRi Di ; bi ¼

1=3
Cij

and s is the width of the distribution that characterizes divergences from spherical bubbles and the
uniform distribution assumption. In principle, s is a

measurable parameter.
The ai and bi coefficients in eqn [28] now have an
additional term given by

!1=3
½31Š

3.20.2.5 Analysis of U–10Mo
High-Temperature Irradiation Data
Figure 2 shows the as-nucleated bubble-size distribution made with the simultaneous solution of eqns [7]
and [14] for a gas solubility of 10À7 at a fuel temperature
of 850 K. At a fission rate of 1 Â 1020 fissions mÀ3 sÀ1,
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 trapping of gas in solution by the nucleated gas bubbles
damps the rate at which the generated gas increases the
gas concentration in dynamic solution. It is important to
point out that here the solubility limit is an unknown
parameter. If the solubility limit was 10À6 or 10À5, the
initial bubble nucleation event would occur after 1400
or 14 000 s of irradiation, respectively.
Figure 3 shows m versus r obtained from the
solution of eqn [7] for T ¼ 850 K and g ¼ 0:5 JmÀ2 .
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 a
function of bubble size corresponding to Figures 2
and 3. As is evident from Figure 3, although the
bubble-size distribution shown in Figure 2 is relatively broad, the majority of the gas generated prior
to the nucleation event (i.e., within the first 140 s of

irradiation) exists in bubbles having radii <1 nm. As
discussed earlier, subsequent to the multiatom bubble
nucleation event, the concentration of gas in solution
stays below the solubility limit due to the trapping
effect of the nucleated gas bubbles such that additional multiatom nucleation events are delayed.
Thus, until the solubility limit is again exceeded, for
the situation shown in Figures 2–4, for irradiation
times >140 s, the bubble distribution follows from the
evolution of the as-nucleated distribution shown in
Figure 2 because of bubble–bubble coalescence and
diffusion of generated gas to the existing bubble


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

587

1E + 21
1E + 20
1E + 19

Cb (m−3)

1E + 18
1E + 17
1E + 16
1E + 15
1E + 14
1E + 13
1E + 12

1E + 11

2

0

4

6

8

10

12

r (nm)
Figure 2 As-nucleated bubble-size distribution made with the simultaneous solution of eqns [7] and [14] for a gas
solubility of 10À7. Reproduced from Rest, J. J. Nucl. Mater. 2010, 402(2–3), 179–185.

50 000

40 000

m

30 000

20 000


10 000

0

0

2

4

6

8

10

12

r (nm)
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.

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, gas-atom diffusion to, and gasatom re-solution from bubbles.
Figure 5 shows the calculated bubble-size distribution 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 3.20.2 for
three values of the rare-gas solubility. The calculations shown in Figure 5 were made using a gas-atom
diffusivity, and re-solution rate given by
Dvol ¼ 2 Â 10À4 eÀ33000=kT cm2 sÀ1
b ¼ 1 Â 10À18 f_

½33Š


588

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

0.12

Fraction of generated gas in bubbles

0.12
0.10

0.10
0.08

0.08
0.06
0.04

0.06

0.02

0.04

0.00
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

0.02

0.00

0

4

2

6

8

10

12

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.

1E + 20
U–8Mo data
Multiatom nucleation

solubility limit = 2.5 ϫ 10−9
Multiatom nucleation
solubility limit = 2.5 ϫ 10−8
Multiatom nucleation
solubility limit = 2.5 ϫ 10−7

1E + 19
1E + 18

Cb (r) (m−3)

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

2

4

6
8
Bubble radius (µm)

10


12

14

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.

where f_ is the fission rate. The value for Dvol given 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 than
the U self-diffusion coefficient.13 In addition, it is
not at all clear how these diffusion couple measurements extrapolate to lower temperatures (lowest diffusion couple temperature was 1073 K) and to an
irradiation environment. The value for b is


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

approximately an order of magnitude less than estimated for UO2.14 This value is consistent with estimated irradiation-enhanced creep rates in U–10Mo,
which are approximately an order of magnitude less
than for UO2.15 These effects can be traced to a higher
thermal conductivity in the metal alloy as compared
to the metal oxide.
The irradiation data shown in Figure 5 were

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 from %5 to 12 mm. As shown in Figure 4,
a solubility of %2.5 Â 10À8 provides a plausible interpretation of the data.
Figure 6 shows the dependence of the calculated
bubble-size distribution on the value of Dvol for a gas
solubility of 2.5 Â 10À8 compared with the measured
quantities. As seen from Figure 6, not surprisingly,
the value of Dvol has 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 the
RHS of eqn [1]. Figure 7 shows the calculated bubble-size distributions for an irradiation in U–8Mo at
850 K to 4% U-atom burnup using eqn [15] and the
two-atom nucleation model for three values of the
nucleation factor compared with irradiation data.
Also shown are results for two different values of
the volume diffusion coefficient for fn ¼ 10À3. It is
clear from Figure 7 that the two-atom nucleation
model does not satisfactorily interpret the measured

bubble-size distribution over a 6 orders of magnitude
range in fn and 2 orders of magnitude range in Dvol .
Thus, comparing Figures 5 and 7, the multiatom
nucleation model provides a better interpretation of
the data than the two-atom model. This becomes a
stronger statement when the relative insensitivity of
the calculated tail of the distribution to the value
of the nucleation factor and the volume diffusion
coefficient for the two-atom model are compared
to the ‘bracketing’ of the data by commensurate
changes in solubility and diffusion coefficient for the
multiatom model.
A more definitive differentiation between these
two models requires data at a much lower burnup
where the effects of bubble diffusion and coalescence
are minimal. Unfortunately, such data are currently
unavailable. Figure 8 shows a comparison of multiatom and two-atom nucleation mechanisms for an
irradiation to 0.04% U burnup of U–8Mo fuel at

1E + 20
U–8Mo data
Dvol = 6.5 ϫ 10−20 m2 s−1
Dvol = 6.5 ϫ 10−19 m2 s−1

1E + 19
1E + 18

Dvol = 6.5 ϫ 10−18 m2 s−1

Cb (r) (m−3)


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

2

4

589

6
8
Bubble radius (µm)

10

12

14

Figure 6 Dependence of the calculated bubble-size distribution on the value of Dvol for a gas solubility of 2.5 Â 10À8
compared with the measured quantities. Reproduced from Rest, J. J. Nucl. Mater. 2010, 402(2–3), 179–185.



590

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

1E + 20
U–8Mo data
fn = 10-6

Cb (r) (m-3)

1E + 19
1E + 18

fn = 10-3
fn = 1

1E + 17

fn = 10-3
Dvol = 6.5 ϫ 10-18 m2 s-1

1E + 16

fn = 10-3
Dvol = 6.5 ϫ 10-20 m2 s-1

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

1E + 12
1E + 11
0

2

4

6

8

10

12

14

Bubble radius (mm)
Figure 7 Calculated bubble-size distributions for an irradiation in U–8Mo at 850 K to 4% U-atom burnup using eqn [15] and
the two-atom nucleation model for three values of the nucleation factor compared with irradiation data. Also shown are
results for two different values of the volume diffusion coefficient for fn ¼ 10À3 . Reproduced from Rest, J. J. Nucl. Mater.
2010, 402(2–3), 179–185.

1E + 21
1E + 20

Multiatom nucleation
Solubility limit = 2.5 ϫ 10-8
Diatom nucleation

fn = 10-3

1E + 19
1E + 18
1E + 17
Cb (r) (m-3)

1E + 16
1E + 15
1E + 14
1E + 13
1E + 12
1E + 11
1E + 10
1E + 9
1E + 8
1E + 7
1E + 6
0.0

0.2

0.4

0.6

0.8

1.0


1.2

1.4

Bubble radius (mm)
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.

850 K. As shown in Figure 8, the two-atom nucleation model leads to a substantially broader distribution than the multiatom model. This feature is carried
on to high burnup and, on comparing Figures 5 and 7,
is one of the key differences between these two

nucleation models. It is anticipated that low burnup
bubble distribution data will become available in the
relatively near future.18 Once this data become available, a more definitive differentiation between these
two models can be undertaken.


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

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.

3.20.3 Intergranular Bubble
Nucleation: Uranium-Alloy Fuel in the
Irradiation-Stabilized g-Phase
3.20.3.1

Introduction

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
high-temperature g-phase described in Section
3.20.2 needs to be complemented with a model for
gas-bubble behavior in the low-temperature irradiation-stabilized g-regime. The swelling at high temperature is primarily intragranular, whereas at low
temperature, intergranular swelling becomes appreciable. As discussed in the previous section, a multiatom 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 capability 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 mechanisms 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 distributions6,20 obtained from U–Mo alloy aluminum
dispersion fuel developed as part of the Reduced
Enrichment for Research and Test Reactor (RERTR)

591

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 is
described wherein the calculation of the time evolution
of the average intergranular bubble radius and number
density is used to set the boundary condition for the
calculation of the intergranular bubble-size distribution based on differential growth rate and sputtering
coalescence processes. Sputtering coalescence, or bubble coalescence without bubble motion, is a relatively
new phenomenon observed heretofore in implantation
studies in pure metals.21 In particular, the sputtering
coalescence mechanism is validated on the basis of the
comparison of model calculations with the measured
distributions. Recent results on transmission electron
microscope (TEM) analysis of intragranular bubbles
in U–Mo were used to set the irradiation-induced
diffusivity and re-solution rate in the bubble-swelling
model. Using these values, a good agreement was
obtained for intergranular bubble distribution compared against measured postirradiation examination

(PIE) data using grain-boundary diffusion enhancement factors of 150–850, depending on the Mo
concentration. This range of enhancement factors is
consistent with values obtained in the literature.
3.20.3.2 Calculation of Evolution of Average
Intragranular Bubble-Size and Density
The model presented here considers analytical solutions to coupled rate equations that describe the
nucleation and growth of inter- and intragranular
bubbles under the simultaneous effect of irradiation-induced gas-atom re-solution. The aim of the
formulation is to avoid a coupled set of nonlinear
equations that can only be solved numerically, using
instead a simplified, physically reasonable hypothesis
that makes the analytical solutions viable. The gasinduced swelling rate is then assessed by calculating
the evolution of the bubble population with burnup
and subsequently the amounts of gas in bubbles
and lattice sites. Uncertain physical parameters of
the model are determined by fitting the calculated
bubble populations at given burnups with measured
bubble size and density data.
At the irradiation temperatures of interest
(T < 500 K), in analogy with UO2, the diffusion of
fission-gas atoms is assumed to be athermal with the
gas-atom diffusivity Dg proportional to the fission
rate f_ . The gas-atom re-solution rate b is also
assumed proportional to the fission rate.


592

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels


The rate equation describing the time evolution of
the density of gas in intragranular bubbles is given by
d½mb ðt Þcb ðt ފ
¼ 16pfn Dg rg cg ðt Þcg ðt Þ
dt
þ 4pr b ðt ÞDg cg ðt Þc b ðt Þ À bmb ðt Þcb ðt Þ

½34Š

The three terms on the RHS of eqn [34] represent,
respectively, the change in the density of gas in intragranular bubbles due to bubble nucleation, the gas-atom
diffusion to bubbles of radius r b , and the loss of gas
atoms from bubbles because of irradiation-induced resolution. Equation [34] can also be represented as the
sum of two equations denoting, respectively, the time
evolution of the fission-gas bubble density cb and of the
gas content in bubbles mb as follows:
dc b ðt Þ 16pfn Dg rg cg ðt Þcg ðt Þ
À obc b ðt Þ
¼
mb ðt Þ
dt

½35Š

dmb
¼ 4pr b ðt ÞDg cg ðt Þ À ð1 À oÞbmb ðt Þ
dt

½36Š


In eqn [35], fn is the bubble nucleation factor, and cg and
rg are the gas-atom concentration and radius, respectively. In general, the value of fn is 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., %10À4).
The first term on the RHS of eqn [35] can be
interpreted to represent the generation rate of ‘average’ size bubbles of radius r b . For every two-atom
bubble that is nucleated, 2=mb of a bubble of radius r b
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) in eqns [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., bubbles smaller than a critical size) and others that are
only partially damaged (e.g., bubbles larger than a
critical size). Including b in both eqns [35] and [36]
is an attempt to depict these processes using a simplified 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 À oÞbmb was
not included in eqn [36], the number of atoms in a
bubble could never decrease. However, the partition
of gas-atom re-solution between these two mechanisms, where o is the partitioning fraction, is an
assumption that remains to be tested experimentally.
In what follows, equal partition is also assumed, that
is, o ¼ 1=2.
Because of the strong effect of irradiation-induced
gas-atom re-solution, in the absence of geometric contact, the bubbles stay in the nanometer size range. The
density of bubbles increases rapidly early in the irradiation. At longer times, the increase in bubble concentration occurs at a much-reduced rate. On the basis
of the above considerations, a quasi steady-state solution for the average bubble density cb and the average
number of gas atoms per bubble mb as a function of
the density of gas in solution cg and the gas-atom
radius rg is given by Spino et al.22
cb ¼

mb ðt Þ ¼

16pfn rg Dg cg2

3bv
4p

bmb ðt Þ
1=2 

4pDg cg ðt Þ 3=2
b

½37Š


½38Š

In eqn [37], fn is the bubble nucleation factor, and in
eqn [38], bv is the van der Waals constant. In general,
the value of fn is 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 average bubble radius rb is related
to mb through the gas law and the capillarity relation.
Imposing gas-atom conservation, that is, requiring that
the sum of the gas in solution, in intragranular bubbles,
and on the grain boundary is equal to the amount of
gas generated (there is no gas released from the U–Mo
fuel), the term cg ðt Þ is determined as
h
i1=2
À1 þ 1 þ 64pð1 À fs Þfn rg Dg f_ bt =b
½39Š
cg ðt Þ ¼
32pfn rg Dg =b
where b is the number of gas atoms produced per
fission event and fs is the fraction of gas released to
the grain boundaries of grains of diameter dg, where,
following Speight23

1=2
8
b
6

b
fs % pffiffiffi
Dg
À 2 Dg
t
t
½40Š
bþg
dg b þ g
pdg
where g ¼ 4pDg r b c b .


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

3.20.3.3 Calculation of Evolution of
Average Intergranular Bubble-Size and
Density
Following the work of Wood and Kear,24 grainboundary 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
given by

1=2
8zaK
½41Š
Cb ¼
121=3 p2 xDg d

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 enhancement 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 radiationinduced 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
dg _ dð fs t Þ
bf
½42Š
3
dt
In general, in an irradiation environment where bubble nucleation, gas-atom diffusion to bubbles, and
irradiation-induced re-solution are operative, a differential growth rate between bubbles of different size
results in a peaked monomodal size distribution.25 The
position of the peak in the bubble-size distribution that
occurs under these conditions is defined by the balance 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 bubbles 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.
K ¼


3.20.3.4 Calculation of Intergranular
Bubble-Size Distribution
Let nðr Þdr be the number of bubbles per unit volume
on the grain boundaries with radii in the range r to
r þ dr . Growth by gas-atom collection from fission

593

gas diffusing from the grain interior removes bubbles
from this size range, but these are replaced by the
simultaneous growth of smaller bubbles. The distribution of intragranular gas consists primarily of fission-gas atoms because of the strong effect of
irradiation-induced gas-atom re-solution. Bubbles
appear on the grain boundaries due to the reduced
effect of re-solution, ascribed to the strong sink-like
property of the boundary, as well as to the altered
properties of bubble nucleation. In addition, nðr Þdr is
affected by bubble–bubble coalescence. A differential
growth rate between bubbles of different sizes leads
to a net rate of increase in the concentration of
bubbles in the size range r to r þ dr . This behavior
is expressed by
!
!
!
dnðr Þ
d
dr
d
dr
nðr Þ

dr À
dr
dr ¼ À nðr Þ
dt
dr
dt d
dr
dt c
½43Š
where the subscripts d and c refer to growth by gasatom diffusion and bubble coalescence, respectively.
The growth rate (dr =dt ) of a particular bubble is
related to the rate (dm=dt ) at which it absorbs gas
from the boundary, either by diffusion of single gas
atoms, or by coalescence with another bubble. The
rate of growth due to gas-atom precipitation is controlled by the grain-boundary gas-atom diffusion
coefficient xDg and the average concentration Cg of
fission gas retained by the boundary.
Studies on the evolution of helium bubbles in
aluminum during heavy-ion irradiation at room temperature have shown that bubble coarsening can take
place by radiation-induced coalescence without bubble motion.21 This coalescence is the result of the net
displacement of Al atoms out of the volume between
bubbles initially in close proximity. The resulting
nonequilibrium-shaped bubble evolves toward a
more energetically favorable spherical shape whose
final size is determined by the equilibrium bubble
pressure.
Bubble coalescence without bubble motion (sputtering coalescence) can be understood on the basis of
the difference in the probability for an atom to be
knocked out of the volume between a pair of bubbles
and the probability of an atom to be injected into this

interbubble volume. If the bubbles contained the
same atoms as that comprising the interbubble volume, the net flux of atoms out of the interbubble
volume would be zero. However, as the gas bubbles
contain fission gas and not matrix atoms, the flux of
atoms into the interbubble volume is reduced by the


594

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

bubble volume fraction, that is, the net flux out of
volume is equal to lV À lðV À VB Þ, 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
dms
¼ Àbs ms
dt

½44Š

Inserting eqns [44]–[46] into the second term
on the RHS of eqn [43] and differentiating with
respect to r,



dnðr Þ
6
2
dnðr Þ
dr ¼ lds f_ pr 2 nðr Þdr þ lds f_ pr 3
dr
dt
d
d
dr
g
g
c
½49Š
Subsequent to intergranular bubble nucleation, gas
arriving at the boundary will be adsorbed by the
existing bubble population. The rate at which a
grain-boundary bubble adsorbs gas is approximately
given by

Using the van der Waals EOS,

dm=dt ¼ 12pr xDg Cg =dg

2

dr
3ðrkT þ 2gbv Þ
dm

¼
dt 16pgðkTr 3 þ 3gbv r 2 Þ dt

½45Š

In continuation, the sputtering rate bs can be
expressed as
bs ¼

6
lds f_ pr 2
dg

½46Š

where the effective interbubble volume is assumed
to be disk-shaped with volume = ds pr 2 , and where
ds is the thickness of the material undergoing sputtering. For a lenticular bubble with radius of curvature r, the equivalent radius of a spherical bubble is
given by
!1=3
3
1
2
½47Š
r ¼ r 1 À cos y þ cos y
2
2
where
cosðyÞ ¼


ggb
2g

½48Š

and ggb is the grain-boundary energy.
It is assumed that bubble coalescence is
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.

½50Š

As mentioned in Section 3.20.3.3, re-solution of
grain-boundary bubbles is not explicitly considered,
for example, in eqn [50]. The rationale for this is that
because of the very strong sink-like nature of the
grain boundary, gas-atoms ejected from a gas bubble

located on the boundary that land within the steep
portion of the concentration gradient are ‘sucked
back’ into the boundary and quickly reenter the
bubble such that the ‘effective’ re-solution rate is
relatively small.26
Combining eqns [9] and [14]
dr =dt ¼

9r xDg Cg ðrkT þ 2gbv Þ2 3bv xDg Cg
%
4gdg ðkTr 3 þ 3gbv r 2 Þ
dg r

½51Š

Using the approximation on the RHS of eqn [51], the
first term on the RHS of eqn [43] becomes


3bv xDg Cg 3bv xDg Cg dnðr Þ
dnðr Þ
¼ nðr Þ
À
dr
dg r 2
dg r
dt
dr
d
½52Š

The overall net rate of change of the concentration of
bubbles in a given size range is given by the sum of
eqns [49] and [52]


3bv xDg Cg 3bv xDg Cg dnðr Þ
dnðr Þ
À
dr ¼ nðr Þ
dg r 2
dg r
dt
dr
À

6
2
dnðr Þ
lds f_ pr 2 nðr Þdr À lds f_ pr 3
dr
dg
dg
dr

½53Š

The equilibrium population of bubbles is obtained by
setting eqn [53] to zero
3bv xDg Cg 3bv xDg Cg dnðr Þ
À

dg r 2
dg r
dr
6
À lds f_ pr 2 nðr Þdr ¼ 0
dg

nðr Þ

½54Š


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

where the last term in eqn [53] has been omitted
3bv xDg Cg
2
because
) lds f_ pr 3 for the conditions
dg r
dg
explored in this paper.
Equation [54] must be solved subject to the relevant boundary condition. In general, this boundary
condition concerns the rate at which bubbles are
formed at their nucleation size r0 . From a consideration of freshly nucleated bubbles25


Cb
nðr0 Þdr ¼
dr =ðdr =dt Þr ¼r0

½55Š
tb
The rate of bubble nucleation is provided by the
Wood–Kear nucleation mechanism24 where on 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
1
tb ¼
pxDg Cb

½56Š

Thus, from eqn [20], it follows that the bubble nucleation rate is given by
dCb
Cb
¼
dt
tb

½57Š

where  is a proportionality constant that is determined by imposing the conservation of gas atoms.
The observed grain-boundary bubbles are a combination of lenticular-shaped objects whose size is substantially 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 intragranular 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
ðdm=dt Þr ¼ r0 ¼ bv Cg =ð4tb Cb pr03 =3Þ

½58Š

where Cg is the gas concentration on the boundary
given by
dg _
fs bf t
½59Š
3
As described by eqn [50], subsequent to bubble nucleation gas solubility on the boundary will drop to a
relatively low value and gas arriving at the boundary
Cg ¼

595

will be adsorbed by the existing bubble population.
Combining eqns [45] and [58]
ðdr =dt Þr ¼ r0 ¼

3Cg bv ðrkT þ 2gbv Þ2
16pgð4tb Cb pr03 =3ÞðkTr 3 þ 3gbv r 2 Þ
½60Š

The solution of eqn [54] subject to the boundary condition expressed by eqns [55] and [60] is
nðr Þ ¼


64gCb2 p2 r 3 ðkTr 3 þ 3gbv r 2 Þexp½Àkðr 4 À r04 ފ
3bv Cg dg ðrkT þ 2gbv Þ2
½61Š

where
k¼

p f_ lds
2bv xDg Cg

½62Š

3.20.3.5 Comparison Between Model
Calculations and Intragranular Data
One of the major challenges in the field of fission-gas
behavior in nuclear fuels is the quantification of
critical materials properties. There is a direct correlation between the accuracy of the values of critical
properties and the confidence level that the proposed
underlying physics is realistic.
The values of the key parameters used in the
model are given in Table 2. Many of them are
known or estimated from the literature27; the values

Table 2

Values of parameters used in the calculations

Parameter


Value

References

b
x

0.25
125
850
UO2: 2 Â 10À23 m3
U–10Mo: 2 Â 10À24 m3
UO2: 10À39 m3 sÀ1
U–10Mo: 10À40 m3 sÀ1
0.216 nm
0.6 J mÀ2
0.5 J mÀ2
0.2
8.5 Â 10À23 m3 per
atom
1
1 Â 10À9 m
1.8 Â 10À8 m
4
2.5745 Â 10À8 m
1 Â 10À9 m

27
Annealed28
Nonannealed6,28

22
28
29

_
b0 ðb ¼ b0 fÞ
_
D0 ðDg ¼ D0 fÞ
rg
g
cosðyÞ
bv
fn
ds
l
z
Á0
Ái

27
Annealed28
Nonannealed6,28
30
27
28
6
6
24
6
6


Source: Rest, J. J. Nucl. Mater. 2010, 407, 55–58.


596

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

of the others (e.g., x) result from a comparison of the
present theory with measured data for bubble populations. As an example of estimated parameters, the
values of Dg and 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 irradiationenhanced 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 compared 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 in eqns [37]–[39], it is reasonable to assume
that this ratio of critical properties is the same for
both materials.
The calculated intragranular bubble-size distribution 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 in Figure 3 are in reasonable accord with the observed estimates of the average bubble density and size. However, it should

be noted that highly over pressurized solid gas

bubbles with diameters of 1–2 nm were observed to
form a superlattice in the U–Mo with a relatively
close spacing (6–7 nm) and having an approximate
monomodal-like distribution.32 For this reason, as
listed in Table 1, the gas-bubble nucleation factor
was taken to be equal to unity. In any event, the
physics presented in this section is not compatible
with the formation of a bubble superlattice.
3.20.3.6 Comparison Between Model
Calculations and Intergranular Data
The calculated distributions are obtained by integrating eqn [61] over the bin sizes Ái , that is, the bubble
density N ðÁi Þ in units of mÀ3 is given by
ÁZ0 þiÁ

N ðÁi Þ ¼

nðr Þdr

½63Š

Á0 þðiÀ1ÞÁ

where Á0 is the minimum bubble size. The intergranular bubble size depends on the value of x (see
eqn [41] and Table 2), which is a grain-boundary
gas-atom diffusion enhancement factor that reflects
the fact that grain-boundary diffusion is decidedly
faster than grain lattice diffusion.33,34 The effect of x
on the intergranular bubble nucleation is visible in

eqn [41]. By increasing x the intergranular bubble
density is reduced with a commensurate increase in
bubble size. The larger value used for x for the

1E + 19

Theory
Estimated data

Bubble density (cm−3)

1E + 18

1E + 17

1E + 16

1E + 15
0

2

4
6
Bubble diameter (nm)

8

10


Figure 9 Calculated intragranular bubble-size distribution for Z03 (fully annealed) contrasted with data32 for 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.


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

nonannealed miniplates reflects the increase in diffusivity with decreased molybdenum content.
The experimental database consists of both asatomized 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 Â 1014 f cmÀ3 sÀ1, temperature from
66 to 191  C, and Mo content from 6 to 10 wt%.20
Table 2 shows the value of the key physical parameters used in the model. The remaining critical
parameter x was determined by best overall interpretation 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.

4E + 8
Theory
Data Z03

Bubble density (cm-2)

3E + 8


2E + 8

1E + 8

0

0.04

0.06

0.08
0.10
0.12
Bubble diameter (mm)

0.14

0.16

4E + 8
Theory
Data Y01

Bubble density (cm-2)

3E + 8

2E + 8

1E + 8


597

These miniplates were fully annealed and as such
have a uniform distribution of molybdenum across
the fuel region. Z03 was fabricated by atomization,
whereas Y01 was made from a ground powder. The
calculated distribution is in very good agreement
with the measured quantities.
Figure 11 shows calculated and measured intergranular bubble-size distribution for U–10Mo asatomized plates. As is evident from the comparisons
in Figure 11, in general, the model calculations are in
remarkable agreement with the data. Figure 12 shows
calculated and measured intergranular bubble-size
distribution for U–6Mo and U–7Mo as-atomized
plates, respectively. The deviation between calculated
and measured results shown in Figure 12 is most
likely due to the lower Mo content and, therefore,
requires different (larger) values for Dg and x.
The results of calculations shown in Figure 13
demonstrate the increased validation leverage
secured with the use of bubble-size distributions
compared with the use of mean values (i.e., average
quantities such as bubble density and diameter). Comparing model predictions with average quantities
is by far the dominant validation technique reported
in the literature. The graph on the left hand side
of Figure 13 shows the sensitivity of the calculated
distributions to the value of the gas-atom knockon distance. The right-hand graph in Figure 13
shows the results of a series of calculations made
with paired values for the grain-boundary-diffusion
enhancement factor and the thickness of the grain

boundary chosen such that the calculation of average quantities remains unchanged. These calculated
results demonstrate that the calculated distribution
undergoes significant changes in shape as well as position and height of the peak. As such, the capacity to
calculate bubble-size distributions along with the availability of measured distributions (as has been obtained
from RERTR irradiated fuel plates) goes a long way in
validating not only values of key materials properties
and model parameters, but also proposed fuel behavioral mechanisms.

0

0.04

0.06

0.08
0.10
0.12
Bubble diameter (mm)

0.14

0.16

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.


3.20.3.7 Calculation of Gas-Driven Fuel
Swelling Safety Margins
The model presented here, taken together with the
analysis of fuel swelling in the high-temperature
equilibrium g-phase presented in Section 3.20.2,
enables the calculation of gas-driven fuel swelling
safety margins. Figure 14 shows the calculated percentage of unrestrained fuel swelling as a function


598

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

2.5E + 8

2E + 8
Theory
Data V03

2E + 8

Theory
Data V07

2E + 8
Bubble density (cm−2)

−2

Bubble density (cm )


2.0E + 8
1.5E + 8
1.0E + 8
5.0E + 7

1E + 8
1E + 8
1E + 8
8E + 7
6E + 7
4E + 7
2E + 7

0.0

0.0
0.05

0.10

0.15

0.20

0.25

0.30

0.35


0.05

0.10

0.15

Bubble diameter (μm)

0.25

0.30

0.35

1.8E + 8

2.5E + 8

1.6E + 8

Theory
Data V8005B

Theory
Data V002

1.4E + 8

−2


Bubble density (cm−2)

2.0E + 8
Bubble density (cm )

0.20

Bubble diameter (μm)

1.5E + 8
1.0E + 8
5.0E + 7

1.2E + 8
1.0E + 8
8.0E + 7
6.0E + 7
4.0E + 7
2.0E + 7

0.0

0.0
0.05

0.10

0.15


0.20

0.25

0.30

0.35

0.05

0.10

Bubble diameter (μm)

0.15

0.20

0.25

0.30

0.35

Bubble diameter (μm)

2.5E + 8
Theory
Data V6019G


Bubble density (cm−2)

2.0E + 8
1.5E + 8
1.0E + 8
5.0E + 7
0.0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Bubble diameter (μm)

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.

of burnup for U–8Mo fuel irradiated at various temperatures. The calculated swelling is a strong function of the irradiation 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 in Figure 14.
The curves in Figure 14 do 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

at %33% swelling. However, in general, the calculation
of the swelling at which the bubbles interconnect
is complicated by a relatively broad distribution of
nonspherical bubbles, nonuniformly distributed within
the fuel regions (e.g., such as that in Figure 14).
The maximum gas release in these high swelling fuels
approaches 80%. There are many small bubbles
between the larger interconnected bubbles that continue to drive the swelling even at high gas release
values. However, even so, the calculated swelling
curves in Figure 8 are typical of those that have been


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

5E + 8

1.6E + 8
Theory
Data S03

1.4E + 8

Data

l = 1.26 × 10-8
l = 1.80 × 10-8
l = 2.34 × 10-8

4E + 8
Bubble density (cm-2)

1.2E + 8
Bubble density (cm-2)

599

1.0E + 8
8.0E + 7
6.0E + 7
4.0E + 7
2.0E + 7

3E + 8
2E + 8
1E + 8
0

0
0.05

0.10

0.15
0.20

0.25
Bubble diameter (mm)

0.30

0.04

0.35

Theory
Data R6007F

0.14

0.16

1.5E + 8
1.0E + 8
5.0E + 7
0.0

Data
x = 21; d = 0.70 × 10-9
x = 15; d = 1.00 × 10-9
x = 10; d = 1.43 × 10-9

6E + 8
Bubble density (cm-2)

Bubble density (cm-2)


2.0E + 8

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

0.10

0.15
0.20
0.25
Bubble diameter (mm)

0.30

0.35

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.

measured. The key here is that Figure 14 shows unrestrained 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 margins can be calculated using the results of Figure 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 15 is the 50% unrestrained swelling threshold
obtained from Figure 14. Also shown in Figure 15
is the fission density and fuel temperature for
RERTR-9. As shown in Figure 15, the calculated
safety margin for RERTR-9 is $150 K.
3.20.3.8

0.08
0.10
0.12
Bubble diameter (μm)

7E + 8

2.5E + 8

0.05

0.06

Conclusions

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


0.04

0.06

0.08
0.10
0.12
Bubble diameter (μm)

0.14

0.16

Figure 13 Sensitivity of the calculated distributions to the
value of the gas-atom knock-on distance (left-hand graph).
Results of a series of calculations made with paired values
for the grain-boundary-diffusion enhancement factor and
the thickness of the grain boundary, chosen such that the
calculation of average quantities remains unchanged
(right-hand graph). Reproduced from Rest, J.; Hofman,
G. L.; Kim, Y. S. J. Nucl Mater. 2009, 385(3), 563–571.

obtained for the rate equations, thereby providing
for increased transparency and ease of use. The results
support a multiatom gas-bubble nucleation mechanism on grain boundaries that have substantially
higher gas solubility than that in the grain interior.
The gas-atom diffusion enhancement factor on the
grain boundaries was determined to be $125–850 in
order to obtain agreement with the measured distributions. The enhancement factor is about 8 times

higher for as-fabricated powder plates than for the
annealed plates because of the lower Mo content on
the boundaries. This range of values for the enhancement factor is consistent with values obtained in the
literature.30 The largest deviation between calculated
and measured results (Figure 12) is most likely due to
several fuel plates that have a lower Mo content (6 and
7 wt% vs. 10 wt%) and, thus, require different (larger)
values for Dg and x.


600

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

500
684 K
700 K
716 K
732 K
748 K
764 K
780 K

Swelling (%)

400

300

200


100

0
2.0E + 20 4.0E + 20 6.0E + 20 8.0E + 20 1.0E + 21 1.2E + 21 1.4E + 21
Fission density (cm−3)

0.0

Figure 14 Calculated unrestrained fuel swelling as a function of burnup and temperature. Reproduced from Rest, J. J. Nucl.
Mater. 2010, 407, 55–58.

1E + 22

Fission density (cm−3)

8E + 21

50% unrestrained swelling
threshold
RERTR-9

147 K

6E + 21
Safety margin
4E + 21

2E + 21


0

400

500

600
700
Fuel temperature (K)

800

900

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 agreement between the model and the
measured distributions for the 10 wt% Mo fuel supports 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 model
based on the growth of bubbles in a regular array35
have not been successful.
A number of the critical parameters listed in
Table 2 are assumed to be a factor of 10 less than
those listed in the literature for UO2. However, it is



Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

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 supported 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 distributions 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 primarily tied to uncertainties in the values of the volume
and Xe diffusion coefficients.

3.20.4 Irradiation-Induced
Re-solution
3.20.4.1

Introduction

After a short period of irradiation, the intragranular
structure of UO2 is populated with a high-density
(%1023 mÀ3 ) of small (r % 10À9 m) bubbles,19 separated by %5–10 bubble diameters. In general, observations 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 bubbles grow to sizes observable with a SEM.38 Most calculations on intergranular gas behavior found in the
literature have focused on the condition for grain-face
saturation and have not addressed the specific mechanics 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.43–45 This assumption has relied on heuristic arguments23 that the strong sink-like nature of

601

a grain boundary provides a relatively short recapture distance for gas that has been knocked out of a
bubble, and as such neutralizes the ‘shrinking’ effect
of the re-solution process. These grain-boundary
bubbles grow at an enhanced rate as compared
to those in the bulk material. The importance of
understanding the physics underlying intergranular
bubble growth is underscored by the rim region of
high-burnup fuels which are characterized by an
exponential growth of intergranular porosity toward
the pellet edge: a narrow band of fully recrystallized

porous material exists at the pellet periphery, and
a rather wide adjacent transition zone with partially
recrystallized porous areas appears dispersed within
the original matrix structure.46 In particular, the
understanding of the dynamics of irradiation-induced
recrystallization and subsequent gas-bubble swelling
requires a quantitative assessment of the nucleation
and growth of grain-boundary bubbles.45,46
A mechanistic model is described, for the growth of
grain-boundary bubbles during irradiation at relatively
low temperatures (i.e., where gas-atom diffusion is
athermal) in order to quantify the effect of gas-atom
re-solution on their growth. A variational method is
used 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-boundary
bubble density on the growth of intergranular bubbles is studied, and the conditions under which intergranular bubble growth occurs are elucidated.
3.20.4.2

Flux Algorithm

The flux of gas atoms diffusing to the grain boundaries in a concentration gradient is obtained by solving the concentration of gas atoms Cg within a
spherical grain that satisfies the equation




@Cg

1 @
2 @Cg
¼ 2
þ a f þ f ðlÞboundary ½64Š
Dg r
@t
@r
r @r
where Dg is the gas-atom diffusion coefficient, f_ is
the fission rate, a is the average number of rare-gas
atoms produced per fission, r is the radial distance
from the grain center, and f ðlÞboundary is the flux of
gas atoms from the boundary bubbles because
of irradiation-induced re-solution. This back flux of
gas can be thought of as an additional matrix gasatom generation mechanism and is assumed to be
distributed uniformly within a spherical annulus of


602

Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

thickness l, where l is the gas-atom knock-out distance. In eqn [64], intragranular bubble trapping of
fission gas has been neglected. However, this effect
can be modeled by using an effective diffusion coefficient given by Turnbull3
Dgeff ¼

b
Dg
bþg


½65Š

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 concentrations %1023 mÀ3 of intragranular bubbles of
%1 nm radius3 with b ¼ 2 Â 10À4 sÀ1 yields a value
for g ¼ 2:5 Â 10À4 sÀ1 and Dgeff ¼ 0:44Dg .
In general, eqn [64] is solved with the boundary
conditions
Cg ¼ 0 at t ¼ 0 for 0

r

Cg ¼ 0 at r ¼ dg =2 for t0

t

dg =2

½66aŠ

t0 þ dt

½66bŠ

@Cg
¼ 0 at r ¼ 0 for t0 t t0 þ dt
½66cŠ
@r

where dt is an increment of time and dg is the grain
diameter. For an increment of time dt the concentration of gas atoms in a spherical grain described in eqn
[64] is




Cg Cg0
1 d
2 dCg
À
þ þ af þ f ðlÞboundary ¼ 0
r
D
g
2
dr
dt dt
r dr
½67Š
Euler’s theorem may now be used to obtain a variational principle equivalent to eqn [3]:
1=2d
ð g "  2
Cg2
Dg dCg
4p
þ
d
2 dr
2dt

0

! !
Cg0


þ af þ f ðlÞboundary Cg r 2 dr ¼ 0
À
dt

½68Š

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 function 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 of eqn [64] for long times. Matthews
and Wood47 obtained a realistic level of accuracy

with a minimum of computer storage and running
time by splitting the spherical grain into two concentric regions of approximately equal volume. In each
region, the gas concentration was represented by a
quadratic function. In the inner region, the concentration function was constrained to have dCg =dr ¼ 0
at r ¼ 0. In the outer region, the concentration function was constrained to a value of Cg ¼ 0 at r ¼ dg/2.
The two functions were also constrained to be continuous at the common boundary of the two regions.
This left three free parameters: the concentrations

g
g
g
C1 , C2 , and C3 , respectively, for the radius ratio
r1 ¼ 0:2, r2 ¼ 0:4, and r3 ¼ 0:45, where r ¼ r =dg .
These positions are the midpoint radii of the inner
region, the boundary between the regions, and the
midpoint radius of the outer region, respectively.
However, this method is too crude if one is interested in an accurate representation of the concentration gradient in the presence of irradiation-induced
re-solution from grain-boundary bubbles where the
gas atom knock-out distance l is on the order of
100 A˚.5 In this case, it is necessary to formulate the
problem in terms of l. The radius ratio at the interface of the two regions is now expressed as
rl ¼ 1=2 À l=dg

½69Š

The trial functions are as follows:
For the inner region,
 gÁÃ
Á
gÀ
Cg ðrÞ ¼ 4 C1 r2l À r2 þ C2 r2 À r2l =4 =3r3l ½70Š
For the outer region;
g

Cg ðrÞ ¼

C2

½8r2 À 2ð2rl þ 3Þr þ 2rl þ 1Š
ð2rl À 1Þ2
g

þ

8C3
½ð2rl þ 1Þr À 2r2 À rl Š
ð2rl À 1Þ2

½71Š

Equations [69] and [70] are substituted for Cg in eqn
[68] and an extremum is found by differentiating
g
g
g
with respect to C1 , C2 , and C3 . This results in a set
of three coupled algebraic equations that can be
g
g
directly solved to obtain the concentrations C1 , C2 ,
g
and C3 , as follows:
g

g

C1 ¼


X1 À F2 C2
F1

F2
F4
X1 þ X3 À X2
F
F5
g
C2 ¼ 1
F2
F4
F2 þ F4 À F3
F1
F5

½72Š

½73Š


Modeling of Fission-Gas-Induced Swelling of Nuclear Fuels

and
g
g
C3 ¼ ðX3 À F4 C2 Þ=F5

½74Š


where
F1 ¼ q1 Dg dt =dg2 þ q2
F2 ¼ q3 Dg dt =dg2 þ q4
F3 ¼ q5 Dg dt =dg2 þ q6
F4 ¼ q7 Dg dt =dg2 þ q8
F5 ¼ q9 Dg dt =dg2 þ q10

½75Š

and


X1 ¼ af dtq11 þ C10 q2 þ C20 q4
½76Š

C10 , C20 , and C30 are the values of the concentrations at
the evaluation points at the start of the time increment. The various q coefficients are integrals that
depend on r and are given in Rest.26
The flux of gas atoms to the boundary (in units of
atoms mÀ3 sÀ1) is given by

4Dg @C 
J ¼À
½77Š
dg @r r¼1=2
or
J¼

4Dg
g

g
ð4C À C2 Þ
dg ð1 À 2rl Þ 3

½78Š

When rl ¼ 0:4 eqns [70], [71], and [78] reduce to
those derived in Rest and Hofman45 for the special
case of fixed spatial nodes. (Rest and Hofman45 define
r ¼ 2r =dg . Thus, r ! r=2 to convert from the present treatment to the one described in Rest.14) The
variational method described here has been compared to finite difference calculations. Suitable
choice of step length d results in insignificant timestep sensitivity with comparable accuracy to the finite
difference approach with one-fifth to one-tenth computer time.48
3.20.4.3

where g is the surface tension, T is the temperature
in K, k is Boltzman’s constant, bv is the van der Waals
constant, and n is the number of gas atoms in a grainboundary bubble, that is,
X
ðfc J ðt Þ=Nb þ zð1 À fc ÞDg Cgb À bnðt ÞÞdt
nðt Þ ¼
dt
½80Š
where b is the gas-atom re-solution rate (sÀ1),
fc % prb2 Nb is the fractional coverage of the grain
boundary by bubbles, z is a grain-boundary diffusion
enhancement factor, Nb is the total number of bubbles
on the boundary (bubbles mÀ2), and Cgb is the gas-atom
concentration on grain boundaries (atoms mÀ2), that is,


X
Cgb ¼
ð1 À fc ÞJ ðt Þ À zDg Cgb Nb dt
½81Š
dt



X2 ¼ af dtq12 þ C10 q4 þ C20 q6 þ C30 q8
  

X3 ¼ af þ fboundary dtq13 þ C20 q8 þ C30 q10

603

Grain-Boundary-Bubble Growth

The bubble radius Rb is calculated using the van der
Waals EOS, that is,


2g 4 3
pRb À bv n ¼ nkT
½79Š
Rb 3

When fc is small (e.g., during the initial stages of
boundary bubble growth) most of the gas reaching
the boundary exists as single gas atoms and diffuses
by random walk to the boundary bubbles. This is

analogous to gas-atom accumulation by bubbles in
the grain interior. When fc is large, the majority of
the gas reaching the boundary flows directly into
boundary bubbles. The grain-boundary enhancement
factor, z, accounts for the general view that gas-atom
diffusion on the boundary is more rapid than in the
matrix because of the existence of more space and
sites (e.g., ledges) from which and to which the gas
atoms can hop.
In general, the gas-atom re-solution rate, b, is
dependent on the damage rate and on the bubble
size; b is calculated under the assumption that gasatom re-solution from a spherical bubble is isotropic
and proceeds by the ejection of single gas atoms. Thus,


3b0 f
b¼ 3
Rb

ð Rb



Rb Àl


1 þ cos ’ 2
r dr
2


½82Š

where b0 is a constant and
cos ’ ¼ ðR2b À l2 À r 2 Þ=2r l

½83Š

Gas atoms that are knocked out of grain-boundary
bubbles (the second term on the RHS of eqn [80]) are
evenly dispersed within an annulus of thickness l
adjacent to the grain boundary. This backward flux
of gas atoms affects the concentration gradient of gas
atoms from the matrix to the boundary, and thus
the overall flux of gas atoms to the boundary. Thus,
this backward flux of gas atoms, in atoms mÀ2 sÀ1,
can be thought of as an additional matrix gas-atom