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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

addressed in section 2.2. After establishing an ade-

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)dτ, (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) =A1e−τ/t1+A2e−τ/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) =A1e−τ/t1+A2e−τ/t2

+(1 −A1−A2)e−τ/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=ED−EO

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