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Thermal resistance from irradiation defects in graphite

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An atomistic level understanding of how varying types and numbers of irradiation induced defects affect thermal resistance in graphite is vital in designing accident tolerant fuels for next-generation nuclear reactors. To this end we performed equilibrium molecular dynamics simulations and computed the change to thermal conductivity due to a series of clustering and non-clustering point defects using the Green–Kubo method. In addition, we present a comprehensive discussion of several approaches to converge the integral of the heat current autocorrelation function. Our calculations show that more energetically favorable clustering defects exhibit fewer low frequency modes and increase the anisotropic nature of graphite selectively exerting a significant effect on thermal resistance along the c-axis.
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Thermal Resistance from Irradiation Defects in Graphite
Laura de Sousa Oliveira and P. Alex Greaney
School of Mechanical, Industrial & Manufacturing Engineering
Oregon State University
Corvallis, OR, 97331
Published in June 2015
Abstract
An atomistic level understanding of how varying
types and numbers of irradiation induced defects af-
fect thermal resistance in graphite is vital in designing
accident tolerant fuels for next-generation nuclear re-
actors. To this end we performed equilibrium molecu-
lar dynamics simulations and computed the change to
thermal conductivity due to a series of clustering and
non-clustering point defects using the Green–Kubo
method. In addition, we present a comprehensive
discussion of several approaches to converge the inte-
gral of the heat current autocorrelation function. Our
calculations show that more energetically favorable
clustering defects exhibit fewer low frequency modes
and increase the anisotropic nature of graphite selec-
tively exerting a signiﬁcant eﬀect on thermal resis-
tance along the c-axis.
1 Introduction
In the early 1940s, polycrystalline graphite was the
only abundantly produced material with the required
purity to be used as a moderator in nuclear reactors
[1]. While other reactor materials have since been
adopted, at the present time, graphite is still in high
demand for the development of high-tech fuel ele-
ments for next-generation nuclear reactors. Graphite
or pyrolytic carbon is included in many nuclear fuel
assemblies to encapsulate the ﬁssile material. In these
applications, in addition to utilizing its high temper-
ature strength the graphite acts as a neutron mod-
erator and reﬂector. In some fuels graphite encapsu-
lates the ﬁssile materials in which case all the heat
produced by ﬁssion in a fuel pin must be conducted
out through the graphite. As the moderating prop-
erties of graphite are temperature dependent, accu-
rately predicting the thermal conductivity of graphite
and other fuel assembly materials — including how
their thermal conductivity evolves under irradiation
— is vitally important for the design of accident tol-
erant fuels.
The thermal conductivity (κ) of graphite is exper-
imentally found to change with synthesis conditions
and while in service as a direct result of radiation [2].
This indicates that κis not an intrinsic property
and is instead governed by the defect morphology of
the graphite. Simulations typically measure intrin-
sic properties, but we aim to determine an atomistic
level understanding of scattering processes from col-
lections of irradiation induced point defects and to
establish a systematic understanding of how defect
type, number and diﬀerent defect-type ensembles af-
fect thermal resistance and phonon mean free path
in graphite. We do so with the goal that the insight
that we gain can be incorporated into approaches for
quantitatively predicting the lattice thermal conduc-
tivity that are based on solving the Boltzmann trans-
port equation. Such a tool would be useful to nuclear
engineers and materials scientists in the process of
designing new reactors and fuel systems that are ac-
cident tolerant. As the ﬁrst step along this path, we
have computed the energy and structure of a zoo of
point defects and determined their separate eﬀects
on thermal conductivity along and across the basal
plane.
In section 2 we establish and validate our method
for computing thermal conductivity of defect-free
graphite. More speciﬁcally, we discuss advantages
and challenges associated with the Green–Kubo for-
malism: in section 2.1 we discuss diﬀerent approaches
to converge the heat current autocorrelation function
(HCACF) and propose a solution based on our ﬁnd-
ings; the issue of size convergence is explained and
quate system size, we introduce defects and compute
their formation energies in Section 3. Values are ob-
tained using classical molecular dynamics and com-
1
pared with density functional theory (DFT) calcula-
tions. Interstitial defects are also annealed to ﬁnd the
most energetically favorable conﬁguration. In section
4 we compare the perfect crystalline system, where
transport is limited by crystal lattice anharmonic-
ity and the acoustic phonons carrying the bulk of
heat are only scattered by other phonons, with sys-
tems with point defects, where defect scattering is
expected to play a crucial role in thermal transport.
Concluding remarks are presented in section 5.
2 Computational method and
validation
Molecular dynamics modeling captures the anhar-
monic interactions of atomic vibrations that carry
heat and both equilibrium and non-equilibrium sim-
ulations can be used to predict thermal conductiv-
ity [3]. The Green–Kubo formalism [4, 5] is a well es-
tablished equilibrium molecular dynamics approach
that has been used successfully to compute thermal
conductivity in a wide range of materials from sili-
con [6] to metal-organic–frameworks [7]. This method
is derived from the ﬂuctuation-dissipation theorem
and computes the thermal conductivity, κ, from the
equilibrium ﬂuctuations in the heat current vector, J,
by:
κxx =V
kBT2Z
0
CJxx(τ), (1)
where kB,Tand Vare the Boltzmann’s constant,
temperature and volume of the simulated region re-
spectively. The term CJ(τ) = hJ(t)J(t+τ)i, and
is the non-normalized heat current autocorrelation
function (HCACF). The net ﬂow of heat ﬂuctuates
about zero at equilibrium and the thermal conductiv-
ity is related to how long it takes for the ﬂuctuations
to dissipate. Both equilibrium and non-equilibrium
molecular dynamics (NEMD) simulations suﬀer from
size artifacts that must be mitigated. In NEMD, the
simulated system size must be larger than the intrin-
sic mean-free path of the phonons in order to elimi-
nate ballistic transport between the heat source and
sink [3]. Equilibrium MD aﬀords one a smaller system
size as phonons may move through periodic bound-
aries unhindered.
Simulations were performed with the large-scale
equilibrium classical molecular dynamics software
LAMMPS [8] in the microcanonical ensemble (NVE)
at 300 K for 0.6 ns with a 0.2 fs time step and periodic
boundary conditions. Note that this is well below the
Debye temperature for graphite (approximately 2500
K in the basal plane and 950 K along the c-axis [2]).
However, our goal is a comparative analysis of phonon
scattering from and around the defect. As scatter-
ing from classically occupied high frequency modes
is present with and without the defect this has little
contribution to the change in κ. The adaptive inter-
molecular reactive empirical bond-order (AIREBO)
potential function formulated by Stuart et al. [9] was
used for all simulations. The AIREBO potential in-
cludes anharmonic terms in the carbon bonds, an
adaptive treatment of the non-bonded and dihedral
angle interactions and has the capability to model
the interaction between layers in graphite [9]. Two
main challenges result from using the Green–Kubo:
(1) determining an appropriate system size and (2)
converging the HCACF. We shall ﬁrst address the
later challenge and propose a solution based on the
work of Chen et al. [10].
2.1 HCACF convergence
There is no average net heat ﬂux, hJi, for a sys-
tem in equilibrium, and the HCACF, i.e. the term
inside the integral in Eqn. (1), is therefore expected
to decay to zero given suﬃcient time. Instead, long
lived oscillations with a signiﬁcant contribution to
the computed thermal conductivity have been ob-
served [11–14]; this behavior is illustrated in Fig.
1. The HCACF is crucial in computing κusing the
Green–Kubo method and yet there is little consensus
among researchers on whether these oscillations are
signiﬁcant to thermal transport or a result of noise,
and on what approach to take. A discussion of this
behavior and of possible approaches is essential in un-
derstanding the limitations of the Green–Kubo and
validating thermal transport calculations.
Figure 1(a) shows the accumulation of the averaged
HCACF along a basal direction over a typical sim-
ulation. It can be seen that the tail of the HCACF
contains many ﬂuctuations, but rather than these de-
caying smoothly as more data is averaged there oc-
cur sporadic events that can overwhelm the average
to add new ﬂuctuations to CJ(τ) and signiﬁcantly
change the initial value CJ(0). These large events
show up in the majority of simulations and for all
simulated system sizes. Long lasting oscillations are
prevalent along the basal plane and diﬀerent in char-
acter to oscillations along the c-axis (see Fig. 1).
Fluctuations along the c-axis exhibit a much higher
frequency and oscillate around zero with the HCACF
converging to zero with only minor instabilities aﬀect-
ing its integral. Fluctuations along the basal plane,
on the other hand, do not fade away during compu-
tation time and signiﬁcantly aﬀect κ. In graphite,
2
κcalculations in the c-direction are not aﬀected by
HCACF ﬂuctuations as much as basal plane calcu-
lations are. This makes results perpendicular to the
basal plane easier to compute and more reliable.
Along the basal plane the HCACF exhibits a two-
stage decay: a rapid decay associated with high fre-
quency phonons and a slower decay associated with
lower frequency phonons. Similar two-stage decay
(or three-stage decay) is observed in many single el-
ement materials and diﬀerent authors have modeled
κby ﬁtting the HCACF to the sum of two or more
exponentials [13–15]. This is a more physically mean-
ingful approach than a single exponential ﬁt in that
it captures multiple relaxation processes, but it ap-
pears to neglect the contribution of the HCACF tail
and to play a part in the systematic underestimate
of κ[3, 14]. When addressing the issue of conver-
gence in the HCACF we have examined a wide vari-
ety of strategies. These strategies included direct in-
tegration of the HCACF truncated to various cutoﬀs,
ﬁts of varying sums of exponentials to the truncated
HCACF, and ﬁts in the frequency domain. Here we
present only a few of the best or otherwise insightful
ﬁndings and a brief discussion of our approach.
(i–iv) Direct numerical integration of the truncated
HCACF up to (i) 50 ps, (ii) 5 ps, (iii-iv) and
a noise dependent cut oﬀ time, tc, proposed
by Chen et al. and described below [10]. For
(iv) individual cut-oﬀs were computed for each
HCACF as shown in Figs. 1(b)–1(d) , and for
(iii) an average tcwas used for each simulation
set.
(v) Single exponential ﬁts to the ﬁrst 5ps of the
HCACF.
(vi) The ﬁtting procedure proposed by Chen et al.,
which includes a ﬁxed oﬀset term in the ﬁtting
function:
CJ(τ)
CJ(0) =A1eτ/t1+A2eτ/t2+Y0,(2)
such that κis computed as
κxx =V CJxx(0)
kBT2(A1t1+A2t2+Y0tc),(3)
where A1,A2,Y0,t1and t2are ﬁtting parame-
ters. Chen et al. argue that including the oﬀset
Y0reduces the computational error. In our im-
plementation of this we used the simplex method
to optimize the ﬁt variables. It is physically
meaningless to have negative Y0and this term
was weighed with a Heaviside function to pro-
hibit negative Y0terms. We also imposed the
condition that A1+A2+Y0= 1.
(vii) Double exponential of the form in (vi) with Yo
set to zero.
(viii) Triple exponential of the form:
CJ(τ)
CJ(0) =A1eτ/t1+A2eτ/t2
+(1 A1A2)eτ/t3,(4)
ﬁt to each HCACF.
The issue of the cut-oﬀ time should now be discussed,
before analyzing the results in Fig. 2. The necessity
to truncate the HCACF is illustrated in Fig. 1 in
which it can be seen that after roughly 2–5 ps the
integrals of the autocorrelations diverge even though
the HCACF is almost zero. This divergence arises
from the integration of random ﬂuctuations in CJ(τ)
eﬀectively adding a random walk to the integral of
CJ(τ). The error from this random walk grows over
time, while the systematic error from omitting the
long tail of slow decay processes in the HCACF di-
minishes over time. There exists an optimal trun-
cation point that minimizes the error in the integral
of CJ(τ), but there is little consensus in the litera-
ture on how to determine it [3,12]. While selecting a
consistent cut-oﬀ may often suﬃce to obtain a com-
parative analysis, it introduces a systematic error in
the estimation of the HCACF, potentially neglecting
the contribution to kof lower phonon modes. Chen
et al. [10] propose obtaining a quantitative descrip-
tion of the numerical noise in the relative ﬂuctuation
of the HCACF, F(t), deﬁned as
F(t) =
σ(CJ)
E(CJ)
,(5)
where σis the standard deviation and Ethe expected
value of the HACF in an interval (t,t+δt). The cut-
oﬀ point is determined to be above an F(t) of 1 (see
Figs. 1(b)–1(d)), i.e. when the ﬂuctuations become
the same scale as the mean. Chen suggests that F(t)
is insensitive to the choice of δ. We ﬁnd this is the
case for only small variations and between a δof 1,
3, and 5 ps the best results correspond to the 1 ps in-
terval. Both 3 and 5 ps intervals resulted in outliers
with a signiﬁcant eﬀect on κ. The variability we ob-
served with the choice of δsuggests that to obtain a
good ﬁt using this method requires a balance between
having suﬃcient data points to compute the local av-
erages while maintaining enough temporal resolution
to reasonably determine when in time the noise ex-
ceeds F(t) = 1. A cut-oﬀ point was computed for
each run and the average cut-oﬀ point for a given
system was then obtained. Each system was simu-
lated 10 times. We compared κfor the cases when δ
3
was 1, 3 and 5 ps with κbeing computed using both
each independent simulation’s cut-oﬀ (as in Chen et
al.) and using the average cut-oﬀ for all simulations.
We found that using the average cut-oﬀ yielded sim-
ilar results with error bars signiﬁcantly smaller than
using the corresponding systems’ individual run cut-
oﬀs to compute κfor each simulation within a cell
size. In theory, if we could consider the average local
ﬂuctuations in the heat ﬂux over an inﬁnite amount
of time, we should be able to ﬁnd a “true” thermal
conductivity of a given system. It is then reasonable
to assume that each HCACF is an approximation to
an HCACF obtained over inﬁnite time and that there
is a “true” cut-oﬀ point, thus providing an argument
for using the average cut-oﬀ on each individual run to
compute κ. When only the ﬁrst two terms of the
HCACF were computed, as in (vii), Y0contributed
up to nearly 100 W/(mK) in the most extreme case.
This illustrates the insuﬃciency of the two exponen-
tial ﬁts to estimate κ. The sum of three exponential
ﬁts yields results very similar to the strategy adopted
by Chen et al. with the added modiﬁcation of using
the average cut-oﬀ instead of each individual simula-
tion’s cut-oﬀ. However, as the number of ﬁtting vari-
ables increases, results are expected to mimic those
of a full integration and the ﬁt loses its physical sig-
niﬁcance. This correspondence nevertheless suggests
Eqn. (4) to be an adequate ﬁt and substantiates the
cut-oﬀ method. More strikingly, simply using the av-
erage cut-oﬀ as the HCACF integration limit yields
similar results with error bars comparable to the ﬁt.
The correct behavior of the HCACF along the basal
plane is thus more accurately explained by the ﬁt type
suggested by Chen at al. than merely the sum of ex-
ponentials, but in order to compute actual κvalues,
the ﬁt introduces an unnecessary hassle to no gain.
Furthermore, the nature of the HCACF along the c-
axis is very diﬀerent than that of the basal plane, as
can be seen by looking at Figs. 1 and 2 and this ﬁt
type is not adequate to explain the HCACF perpen-
dicular to the basal plane. That said, the error bars
are noticeably smaller when the HCACF is integrated
only up to tcthan when they are integrated over the
total HCACF time. The simplest, most eﬀective ap-
proach is to select the cut-oﬀ for each simulation by
setting F(t) = 1, but to use the average cut-oﬀ of
all simulations when computing each simulation’s in-
dividual κ. This method is adequate to compute κ
along any direction for highly oriented graphite. In
light of this analysis, a similar simulation time with
a lower HCACF is likely to yield more accurate κ
results, as it would allow more time for convergence
and not necessarily lower the cut-oﬀ.
While there is no consensus on the best method to
reduce noise and capture the nature of the HCACF of
graphite and other materials, the approach selected
in this paper yields κestimates higher than a sum of
exponentials, with moderately small error bars and
without the need of a complicated ﬁt. This method
was used for all defect calculations along the basal
plane and along c, taking into account that the cut-
oﬀ along c must neglect the ﬁrst values of F(t) = 1
which take place in the initial decay stages (see Fig.
1). Being consistent with the choice of method is
often suﬃcient for a signiﬁcant comparative analysis
and this method allows us to do that.
2.2 Size convergence
Periodic boundary conditions allow simulations of
a small number of particles to mimic the behavior
of an inﬁnite solid; however, they limit the number
and wavelength of the vibrational modes available
to carry heat. Thus, when using the Green–Kubo
method it is ﬁrst necessary to establish size conver-
gence. Thermal conductivity values were computed
for perfectly crystalline systems of varying size, as
can be observed in Fig. 2. An 8 atom unit cell was
deﬁned and 7 systems ranging between 3x3x3 and
15x15x15 super cells were simulated (again 10 times
each). Along the basal plane the systems’ size was
asymmetric in the x and y dimensions with x smaller
than y — this was done to better gauge potential
size artifacts. While there was a large variability in
the thermal conductivity, the values are scattered be-
tween 300 and 400 W/(mK) along the x direction and
350 and 450 W/(mK) along the y. This suggests a size
artifact not evident just looking at the system size in-
crease within each direction and that only comes into
play within each system. For this reason diﬀerent x
and y values were maintained when computing ther-
mal conductivity in defective systems as well. For
computations performed with defects, the 11x11x11
super cell was selected to allow for a big enough com-
pute cell with a feasible computational expense as-
sociated. The 11x11x11 super cell corresponds to a
10648 atom system in the perfect graphite, with a
270.5×468.6×737.9nm3volume in the x,y, and
zdirections, respectively.
3 Identifying defect structures
In irradiated graphite carbon atoms are displaced
due to cascade reactions giving rise to many point
defects. We categorize these into defects that have
a strong driving force for clustering, such as vacan-
cies and interstitials, and defects that are less driven
to cluster such as bond rotation defects, and isotopic
4
defects. The following clustering defects were con-
sidered: a single interstitial (Fig. 3(b)), a single va-
cancy (Fig. 3(j)), clusters of 2–8 interstitials (Figs.
3(c)–3(i)) and clusters of 2–3 vacancies (Figs. 3(k)
and 3(l)). For a single interstitial, three interstitial
locations were considered, as depicted in Fig. 4(a).
Similarly, four conﬁgurations were simulated for 2-
interstitial clusters, as shown in Fig. 4(b). The single
vacancy site is between the centers of hexagonal voids
on the planes adjacent to the plane of the vacancy,
i.e. where the type A single interstitial is positioned
in Fig. 4(a), but in the lower, less visible layer. The
added vacancies lie directly between atom sites on
the adjacent layers. The non-clustering defects con-
sidered were a Stone-Wales defect (Fig. 3(a)) and
an isotope. The C14 isotope was selected for hav-
ing a higher mass than C13, another common carbon
isotope, and thus being expected to have a higher
contribution to changes in κ. The defects were intro-
duced to the center of the selected 11x11x11 perfect
system; the interstitial defects were placed between
the 11th and 12th layer of the 22 layer cell, and the
remaining defects within the 11th layer, as shown in
Fig. 5.
Formation energies were computed using classical
MD for all defects. These calculations were used to
estimate the likelihood of formation of each defect,
where the energy per defect is given by
Ed=EDEO
NO
ND.(6)
NDand NOare the number of atoms in the defective
system and the corresponding non-defective system,
in that order. EDcorresponds to the total energy of
the system and EOto the total energy of the perfect
system of the same size.
The optimization process for the classical calcula-
tions is described in the ﬂowchart in Fig. 6 as was per-
formed using the FIRE scheme [16] as implemented
in LAMMPS. As part of the process to optimize the
geometry of the interstitial defects, low energy inter-
stitials (type Ain Fig. 4(a) and type Cin Fig. 4(b)
for one and two-interstitial defects respectively) were
also annealed and subsequently cooled. The defects
were annealed to 1500 K for 500 ps and cooled to 300
K for 1 ns, in the canonical (NVT) ensemble. By do-
ing this we allowed the already low energy interstitial
defects to migrate and rearrange themselves into po-
tentially lower energy conﬁgurations.
Energy values for the diﬀerent defect types and num-
bers are depicted in Fig. 7 and in Table 1. Classical
interstitial defect energies were computed for the op-
timized structures before and after annealing. It is
notable in Fig. 7, that the annealing process often
yielded defect structures with considerably lower en-
ergy than those reached by direct relaxation using
the FIRE algorithm — even for very simple defects
such as a lone interstitial. There is, however, good
agreement in the overall trend of defect energies as
modeled with the AIREBO empirical potential and
those from Li et al., computed using density func-
tional theory (DFT) with the local density approxi-
mation (LDA) [17]. Furthermore, the type Asingle
interstitial when annealed becomes structurally sim-
ilar to Li et al.’s 5.5 eV formation energy “free” in-
terstitial, computed with DFT. Stone-Wales defects
have the lowest formation energy of all intrinsic de-
fects in graphenic systems [18], calculated with DFT
at 5.2 eV [17].
4 Thermal resistance from de-
fects
While the thermal conductivity of near-perfect
graphite has been reported to be as high as 4180
W/(mK) along the basal plane [19], the experimen-
tally measured anisotropy ratio (κa/κc) of near-ideal
graphite has been found to be just below 210 at 300
K [20]. This suggests the Green–Kubo calculations
to be an order of magnitude below experimental val-
ues in the basal plane, but within the expected or-
der of magnitude for κalong the c-axis. Finally, κ
for nine defects including the more energetically fa-
vorable ones was computed using the Green–Kubo
method as with the perfect crystal. Thermal conduc-
tivity for the hexagonal platelet was also computed
using diﬀerent super cell sizes (Fig. 10). While κis
within the error bars along the basal plane for all de-
fects, the overall trend suggests a decrease in thermal
conductivity with the presence of defects, as would
be expected. More distinctly, the systems with in-
terstitial clusters exhibit a clear decrease in the ther-
mal conductivity along c (see Fig. 8(c)). Note that
these defects correspond to low conﬁguration ener-
gies as well and are therefore more likely to occur.
Frenkel pairs is one type of defect that is expected to
emerge from exposure to radiation due to knock-out
reactions; the added eﬀect of vacancy and interstitial
clusters would signiﬁcantly reduce κperpendicular to
the basal plane.
Performing a discrete cosine transform (DCT) of
the HCACF reveals the presence of localized modes
exclusively associated with the lower thermal conduc-
tivity defect types (see Fig. 9). We performed DCTs
for the defect systems both along x,yand z, and
found two notable diﬀerences between systems in the
5
Defect Type Single: A Single: B Single: C Two: A Two: B Two: C Two: D
LAMMPS 3.57 eV 4.73 eV 4.46 eV 4.99 eV 3.27 eV 2.95 eV 2.98 eV
Literature (DFT, LDA) [17] 6.7 eV 7.7 eV 7.4 eV - - - -
Table 1: Classical MD energy calculations for single and double interstitial defect types based on location.
The values obtained for a single interstitial are compared with available density functional theory (DFT)
calculations using the local density approximation (LDA) from Ref. [17].
DCT of the c-axis HCACF. Systems containing in-
terstitial platelets develop a series of peaks at 1.3,
2.5 and 3 THz. We attribute these to rattling of
the platelets in the c-direction and the defects being
large enough to have relatively low frequency vibra-
tional modes. More interestingly, there is a dramatic
reduction in the intensity of low frequency modes in
the HCACF of the systems with diminished thermal
conductivity.
There is little diﬀerence in terms of how the number
of interstitials (between 5 and 8) in a cluster aﬀect the
overall thermal conductivity in the system, but there
is a noticeable change as the system size increases —
the systems containing an hexagonal platelet increase
in κwith system size (see Fig. 10). This seems to
suggest that the defect concentration has an eﬀect on
the total thermal conductivity as well. The last sys-
tem corresponds to an 11x11x11 super cell with two
hexagonal defects equally spaced and, as expected,
it shows a lower thermal conductivity than the same
system size with a single defect.
If instead of considering κwe assume that defects
make an additive contribution to the systems’ ther-
mal resistance, r, then we might expect rdefect =
rdefective rperfect, and that thus the thermal re-
sistance for a system containing two defects would
be r2defects =rperfect + 2 ·rdefect , or r2def ects =
2.32 ±0.32W/(mK) for the hexagonal platelet. It
appears from Fig. 10(b) that adding a defect does
not double its thermal resistance, but reducing the
size to half does; a ﬁt through a system with varying
defect numbers may shed light into how κscales with
defect concentration for each defect type.
5 Conclusions
In this work we have reported calculations of the
reduction in thermal conductivity of graphite due to a
series of point defects typical under irradiation. The
calculations reveal three important conclusions:
Clustered interstitial defects are stable (with re-
spect to lone interstitials) and strongly detrimen-
tal to the thermal transport in both the in-plane
and c-axis directions.
In addition to lowering the thermal conductiv-
ity they also increase the thermal conductivity
anisotropy.
Although the noise in the calculations of κis
large, it is clear that the platelets create larger
thermal resistance than the constituent number
of lone interstitials.
In pebble bed reactors graphite is used to encapsulate
the ﬁssile materials and thus the graphite experiences
an extremely large neutron dose. The average fuel
temperatures in such a reactor is 1200 K (with peak
temperatures expected to stay below 1500 K) [21].
At these temperatures interstitials are highly mobile
and readily condensing to interstitial platelets. These
platelets are responsible for c-axis swelling under ir-
radiation [2]. Our work indicates that this has a dou-
bly negative eﬀect on thermal conductivity; elongat-
ing grains along their thermally resistive directions
while also increasing the thermal resistance in these
directions.
In addition to computing the reduction in ther-
mal conductivity due to defects we have performed
a systematic comparison of various numerical strate-
gies for reducing uncertainty in the integration of the
HCACF. Our simulations reveal infrequent large heat
current ﬂuctuations that are large enough to over-
whelm the averaged HCACF. The origin of these ﬂuc-
tuations is unclear to us at this stage and we specu-
late two possible causes. It is possible that the ﬂuc-
tuations are a manifestation of Fermi–Pasta–Ulam–
Tsingou recurrence [22] or some related breakdown of
ergodicity over the time period accessible to simula-
tion. An alternative explanation is that the ﬂuctua-
tions are physically realistic processes similar to rogue
ocean waves and caused by amplitude dependence of
the phonon dispersion in graphite. It has been pro-
posed that carbon nanotubes possess soliton-like heat
carriers [23] and it is possible that similar conditions
may arise in graphite. These two potential explana-
tions are incompatible and would require one to treat
the ﬂuctuations diﬀerently: in the ﬁrst case remov-
ing their eﬀect from computed thermal conductivity,
and in the latter case performing enough simulations
to obtain a statistically signiﬁcant sampling of these
infrequent ﬂuctuations.
6
6 Acknowledgments
This work used the Extreme Science and Engineer-
ing Discovery Environment (XSEDE), which is sup-
ported by National Science Foundation grant number
OCI-1053575.
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Figure 1: In Fig. 1(a), the HCACF is computed as
the simulation progresses along x for the perfectly
crystalline 11x11x11 super cell. At ﬁrst only a few
values contribute to the ensemble average and the
initial HCACFs are noisy. As the averaging time
progresses the HCACF becomes smoother with the
exception of well deﬁned crests and troughs, most of
which do not fade away during the total simulation
time. Figures 1(b), 1(c) and 1(d) correspond to the
HCACF noise (computed as F(t)), the ﬁnal HCACF,
and the integral of the HCACF, respectively, for all
simulations of the perfect 11x11x11 super cell system
along y and z (or c).
8
Figure 2: These ﬁgures correspond to κmeasured for
diﬀerent super cells along x (2(a)) and y (2(b)) in the
basal plane and along the c-axis (2(c)). In addition
to establishing size convergence, the ﬁgures illustrate
a set of diﬀerent approaches (labeled in the legend)
considered to converge the HCACF and the corre-
sponding standard error. Method (iii) was selected.









Figure 3: Illustration of the defects examined in this
study: Stone-Wales defect (3(a)); single interstitial
(3(b)); 2-8 interstitials (3(c) -3(i)); single vacancy
(3(j)), di-vacancy (3(k)), and 3 vacancies (3(l)). The
interstitial defects are shown in their annealed con-
ﬁgurations.
Figure 4: Possible defect types for single (Fig. 4(a))
and two-interstitial defects (Fig. 4(b)).
Figure 5: Slice of a graphite system with an hexag-
onal platelet, indicating the location of the defect.
There are 22 total layers in the system.
9
Figure 6: Schematic of the optimization procedure
applied to classically simulated defects before com-
puting formation energies.
Figure 7: These energies correspond to the defects
depicted in Fig. 3. In the case of the interstitial
defect-types, values were computed both for annealed
and non-annealed systems.
10
Figure 8: Anisotropy ratio (κa/κc) computed for both x and y for diﬀerent defect types, including defects
found to be most energetically favorable (Fig. 8(a)); κobtained for diﬀerent defect types along x and y (Fig.
8(b)) and in the basal plane (Fig. 8(c))
Figure 9: Discrete cosine transform applied to the
c-axis HCACF for diﬀerent defect types.
Figure 10: Hexagonal platelet κand corresponding
standard error computed along x and y in the basal
plane (Fig. 10(a)) and along the c-axis (Fig. 10(b))
for 4 diﬀerent super cell sizes including the 10648 base
atom system, and for two hexagonal platelets in the
same base system.
11
... This framework would allow the modeling of phonon transport in any material where phonons are the dominant mechanism of heat transfer, such as graphite, silicon, lithium aluminate, silicon carbides, or new types of nuclear fuel [5,[21][22][23][24]. For nuclear fuel in particular, this method of prediction must consider processes which occur on an atomistic scale, the production of fission products, the formation of defects in the microstructure of the fuel [6,12,14,18,25], the evolution of burned-in plutonium [7][8][9]11], and the formation and migration of gas bubbles due to fission or alpha decay of plutonium [26][27][28][29]. ...
... We investigated both pristine graphite, and graphite in the presence of interstitial defects-the motivation is the effect of these defects on κ. The dominant mechanism which contributed to an increase in thermal resistance along the z axis were the multiple types of interstitial cluster defects [22]. Using molecular dynamics, The Green-Kubo method was employed to compute equilibrium thermal conductivity. ...
... We generated two types of FEM mesh for graphite-with and without defect presence. The geometry is a collection of eight simulation cells used by Oliveira [22] and recreated for this purpose. Each sub-cell has dimensions of x, y, z ð Þof 2.71 nm Â 4.69 nm Â 7.38 nm and consisted of 22 layers of graphite, with the simulated defect placed in between the 11th and 12th layer. ...
Chapter
We present a review, demonstration, and simulation of phonon transport for the purposes of predicting materials performance at the mesoscale. We focus primarily on the development and implementation of a unified methodology to enable predictive heat transport. We report on the current state of the art as it pertains to deterministic phonon transport methodologies, discussing various topics concerning phonons. In application, we focus on the self-adjoint angular flux (SAAF) formulation of the Boltzmann transport equation for phonons, and develop the spatial, angular, and material property discretization required to accurately simulate the predictive physics of heat transport in dielectrics. We discuss thermal interfacial resistance and present our formulation of the diffuse mismatch model for simulating phonon interactions at internal boundaries. We recently developed a deterministic, spectral phonon transport method for predicting effective thermal conductivity, using Bose–Einstein source terms coupled through an average material temperature. This method provides a way of obtaining temperature using the linearized Boltzmann transport equation, without the necessary nonlinear outer iteration on temperature used in many approaches. Our thermal conductivity and heat flux results are consistent with existing research. We introduce a closure term to the phonon transport system which acts as a redistribution function for the total energy of the system and serves as an indicator of the amount of nonequilibrium behavior occurring in the system. We predict thermal conductivity and equilibrium temperature distributions in homogeneous and heterogeneous materials using data generated by ab initio density functional theory methods. We employ polarization, density of states and full dispersion spectra to resolve thermal conductivity with numerous angular and spatial discretizations. The equations associated with this method are solved via a modification of traditional source iteration. We compare the performance of source iteration applied to an existing uncoupled, traditional SAAF method to our new method and comment on the iterative performance of each. We observe ballistic and diffusive phonon scattering as acoustic thickness of the domain changes, and are able to make comparisons between the accuracy and efficiency of both methods.
... At the fundamental level, effects such as fuel cracking, physical strain, plutonium migration and void formation among others are being computationally modeled at the nanoscale. These efforts are necessary for the continued development of nuclear fuel and advanced Generation IV nuclear reactors [6][7][8][9][10]. ...
... Two variations of MD phonon transport simulations can be used to determine thermal conductivity or a host of other quantities [6,[16][17][18][19][20] -equilibrium molecular dynamics (EMD) and non-equilibrium molecular dynamics (NEMD). EMD relies on a constant temperature, measuring fluctuation decay in a variable, e.g., heat flux, measured to compute thermal conductivity. ...
... These calculations are carried out at a constant temperature, relying on potential models to determine system characteristics. The Green-Kubo method, used by Oliveira [6] and Hori [18] is an EMD approach which is applied to measure lattice thermal conductivities. It relies on oscillations of the heat current autocorrelation function (HCACF) about equilibrium, and thermal conductivity is inferred from the time taken for the oscillations to vanish. ...
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approved: Todd S. Palmer The Boltzmann transport equation derived in the Self-Adjoint Angular Flux (SAAF) formulation is applied to simulate phonon transport. The neutron transport code Rattlesnake is leveraged in this fashion, slightly modified to accept input from variables consistent with phonon transport simulations. Several benchmark problems are modeled to assess the potential of this application to predict thermal conductivity in materials with heterogeneity and isotopic fission products affecting thermal transport. The 1-D, SAAF formulation of the Boltzmann transport equation for phonons is derived along with associated boundary conditions. Comparisons to phonon transport problems solved via deterministic, Monte Carlo (MC) and molecular dynamics (MD) methods are shown. Phonon intensity and heat flux are used to compute thermal conductivity in materials. Phonon transport with Rattlesnake using similar input conditions compares well to test problems in open literature. Transport is simulated in one and two element systems, with special emphasis on uranium dioxide (UO 2) with xenon cluster defects. Rattlesnake solutions show thermal conductivity in UO 2 decreasing by up to a factor of 4 at elevated temperatures. Transport behavior for these problems appears qualitatively correct, though lack of data for xenon properties yields results which deviate from MD simulations.
... Instead, large oscillations with a significant contribution to the integral have been observed [15,[19][20][21][22]. Figure 1(a) depicts an example of fluctuating HCACFs and the growing error in the corresponding integrals, and Fig. 1(b) shows the longevity of the fluctuations. If we were able to sample an infinite system for infinite time, we should find the system's true ACF and thus a fixed true transport quantity. ...
... While it has been shown that the Green-Kubo approach can be successfully used with quantum-based calculations [23,24], simulation size FIG. 1. Panel (a) shows the HCACFs (the decaying functions) plotted along side their integrals (the curves that rise to a plateau) computed from nine separate simulations of a 10 648-atom, perfectly crystalline, and periodically contiguous block of graphite. The data were taken from a study to determine the influence of Wigner defects on thermal transport in graphite [22]. The dashed lines correspond to the heat flux along the [2110] direction and the solid lines correspond to the heat flux along [0110]. ...
... The system depicted in Fig. 1 exhibits a rapid decay associated with high-frequency phonons and a slower decay associated with lower frequency phonons; similar two-or three-stage decay is observed in many single-element materials and different authors have modeled κ by fitting the HCACF to the sum of two or more exponentials [20,21,27]. This approach captures multiple relaxation processes and is therefore more physically meaningful than a single exponential fit, but it is ineffective when the HCACF cannot be represented by an exponential fit [12,22,28] and it forces a behavior description of the HCACF that might not be accurate. The same is true of shear relaxation times in viscosity calculations. ...
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The Green-Kubo method is a commonly used approach for predicting transport properties in a system from equilibrium molecular dynamics simulations. The approach is founded on the fluctuation dissipation theorem and relates the property of interest to the lifetime of fluctuations in its thermodynamic driving potential. For heat transport, the lattice thermal conductivity is related to the integral of the autocorrelation of the instantaneous heat flux. A principal source of error in these calculations is that the autocorrelation function requires a long averaging time to reduce remnant noise. Integrating the noise in the tail of the autocorrelation function becomes conflated with physically important slow relaxation processes. In this paper we present a method to quantify the uncertainty on transport properties computed using the Green-Kubo formulation based on recognizing that the integrated noise is a random walk, with a growing envelope of uncertainty. By characterizing the noise we can choose integration conditions to best trade off systematic truncation error with unbiased integration noise, to minimize uncertainty for a given allocation of computational resources.
... Given the motivation to improve data used in reactor physics simulations, the development of a computational framework to model phonon transport would allow researchers to more accurately characterize processes affecting heat transport in UO 2 fuel in nuclear reactors. This framework would allow the modeling of phonon transport in any material where phonons are the dominant mechanism of heat transfer, such as graphite, semiconductors, thermoelectric devices, or new types of fuel under development for Generation IV nuclear reactors [30][31][32][33][34], and would not be restricted only to nuclear fuels. ...
... Bulk property prediction can be achieved using molecular dynamics (MD) or density functional theory (DFT) numerical models. These methods assume a known electronic potential and use this in either a fluctuation model (Green-Kubo method with MD or non-equilibrium MD [31,34,42]) or solve the full Schrödinger equation (DFT) [35,36,39,40,113,114] and resolve the full phonon dispersion and density of states. MD and DFT methods are effective in predicting bulk properties, but are limited to very small problems; because of the computational restraint imposed by solving all the degrees of freedom in a perturbed atomic lattice, simulations are restricted to about 100,000 atoms, nowhere near enough to simulate a macroscopic quantity of interest. ...
Thesis
Full-text available
We present a deterministic spectral method to predict equilibrium temperature distributions, heat flux, and thermal conductivity in homogeneous and heterogeneous media. We solve the Boltzmann transport equation in a second order spatial, self-adjoint angular flux formulation. We implemented this method into the radiation transport code Rattlesnake, built using the MOOSE (Multiphysics Object Oriented Simulation Environment) framework. The spatial variable is discretized using the continuous finite element method with unstructured meshes, and the angular variable is discretized with the discrete ordinates method. We implemented the diffuse mismatch model in a general geometry to simulate phonon interfacial resistance, using the grey approximation of the Boltzmann transport equation. Using material properties generated by density functional theory and molecular dynamics methods, we were able to elucidate properties of xenon (Xe) at temperatures and pressures experienced in irradiated nuclear fuel. We found Xe to undergo phase change from liquid to solid, and were able to compute coefficients of phonon transmission and reflection at the Xe-UO$_2$ interface. We found κ to decrease by about a factor of 4 with increasing temperature, agreeing with other trends and research in the open literature. We developed a new method for simulating deterministic, spectral phonon transport to predict heat flux, thermal conductivity, and equilibrium temperature distributions in homogeneous and heterogeneous materials. All the spectral phonon groups are coupled through a local average material temperature, and a new term, $\beta$, is derived and is used as a closure term in the phonon transport equation. $\beta$ acts to redistribute the fraction of total energy which is exchanged between the transport system and equilibrium distribution of phonons. This method predicts thermal conductivity trends in materials spanning geometric domain sizes from nanometers to micrometers, and exhibits the correct asymptotic heat flux behavior as domain size increases. We observed $\beta$ to be the most infuential at smaller domain sizes, where equilibrium is difficult to establish due to the proximity of the boundary phonon sources; as domain size increased, $\beta$ diminished in size, and nearly vanished at the maximum domain size of 10 microns. This further makes the case to perform BTE simulations for nanometer to micrometer heat transfer, as Fourier's law will not accurately capture the heat transfer in such small domain sizes, e.g., thermoelectric devices, heat transfer around defects and heterogeneities in reactor fuel. Additionally, we developed a novel material property discretization scheme which is consistent with the discretization of the angular variable in the transport equation. We performed convergence studies to test the efficiency of the spectral method, which used a modifed source iteration (MSI) to solve the linear system of equations. We compare the performance of traditional source iteration (SI) of the uncoupled self-adjoin angular flux method we previously implemented to the new method and comment on the iterative performance of each. We capture ballistic and diffusive phonon scattering, and are able to make comparisons between the accuracy and effciency of both methods. We find that MSI outperforms SI in most cases, especially as the spatial domain becomes acoustically thick.
... The obtained results for ion-beam induced thermal conductivity degradation are also important for future model calculations, because they can be used as a benchmark for computer models such as finite element and molecular dynamics simulations [35][36][37][38][39] . ...
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Isotropic polycrystalline graphite samples were irradiated with ∼1 GeV 197Au and 238U ions of fluences up to 5 × 1013 ions/cm2. Beam-induced changes of thermophysical properties were characterized using frequency domain photothermal radiometry (PTR) and the underlying structural transformations were monitored by Raman spectroscopy. The ion range (∼60 µm) was less than the sample thickness, therefore thermal diffusivity contributions of the irradiated as well as non-irradiated layer were considered when analyzing the PTR data. At the highest applied fluences, the thermal effusivity of the damaged layer degrades down to 20% of the pristine value and the corresponding calculated values of thermal conductivity decrease from 95 Wm−1K−1 for pristine material to 4 Wm−1K−1, a value characteristic for the glassy carbon allotrope. This technique provides quantitative data on thermal properties of ion-irradiated polycrystalline graphite and is very valuable for the prediction of lifetime expectancy in long-term applications in extreme radiation environments.
... Other researchers have taken different approaches to computing heat flux, incorporating quantities obtained through MD simulations (phonon group velocity, wave vectors, and angular frequencies [17][18][19]). We compute an effective thermal conductivity by taking the ratio of the average heat flux to the end-to-end temperature gradient (which includes effects at the boundaries) in the system ...
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We present a method for solving the Boltzmann transport equation (BTE) for phonons by modifying the neutron transport code Rattlesnake which provides a numerically efficient method for solving the BTE in its self-adjoint angular flux (SAAF) form. Using this approach, we have computed the reduction in thermal conductivity of uranium dioxide (UO2) due to the presence of a nanoscale xenon bubble across a range of temperatures. For these simulations, the values of group velocity and phonon mean free path in the UO2 were determined from a combination of experimental heat conduction data and first principles calculations. The same properties for the Xe under the high pressure conditions in the nanoscale bubble were computed using classical molecular dynamics (MD). We compare our approach to the other modern phonon transport calculations, and discuss the benefits of this multiscale approach for thermal conductivity in nuclear fuels under irradiation.
... Besides the pore size, a good irradiation behavior is also essential to the candidate graphite for MSR. Irradiationinduced defect [14], such as vacancy clusters, interstitial clusters and dislocations [15], may cause changes in properties [16], such as the Young's modulus [17], fracture strength [18] and thermal conductivities [19]. The irradiation of isostatic-molded graphite of fine grain, such as IG-110, was studied extensively for its application in hightemperature gas-cooled reactor [20][21][22][23]. ...
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... The thermal conductivity is shown to be directly proportional to the operation temperature (T) for irradiated graphite but the opposite is true for unirradiated graphite. Without using an atomistic physical approach for calculating of the variation of thermal conductivity for the graphite such as the one proposed in [16] , the thermal conductivity dependence on dose and temperature can be deducted from experimental data curves or include empirical terms obtained from the experiments using Material Test Reactor (MTR). In the current paper, thermal conductivity variation results of MTR from Marsden et al. [12] are used to generate the empirical equation. ...
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