C A R B O N 7 7 ( 2 0 1 4 ) 2 8 1 – 2 8 8

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Phonon transport in helically coiled carbonnanotubes

http://dx.doi.org/10.1016/j.carbon.2014.05.0310008-6223/� 2014 Elsevier Ltd. All rights reserved.

* Corresponding author.E-mail address: ivag@rcub.bg.ac.rs (I. Milosevic).

Zoran P. Popovic, Milan Damnjanovic, Ivanka Milosevic *

University of Belgrade, Faculty of Physics, Studentski trg 12, 11001 Belgrade, Serbia

A R T I C L E I N F O

Article history:

Received 31 March 2014

Accepted 13 May 2014

Available online 22 May 2014

A B S T R A C T

We perform theoretical studies on the phonon thermal transport in helically coiled carbon

nanotubes (HCCNTs). The Gruneisen parameter, as a function of the phonon wave vector

and phonon branch, is numerically evaluated for each vibrational mode, so that the

three-phonon Umklapp scattering rates can be calculated exactly by taking into account

all allowed phonon relaxation channels. We considered wide temperature range and heat

conductor lengths from nano- to macro-scale. We examine the crossover from ballistic to

diffusive transport regime and impact of HCCNT geometrical parameters on their heat con-

duction. Thermal conductivity in HCCNTs is found to be slightly lower than that in single

walled carbon nanotubes (SWCNTs). This is interpreted by the competition among three

factors. Firstly, threefold reduction of the Gruneisen parameter for the acoustic branches.

Secondly, lower phonon group velocities. Finally, availability of purely acoustic scattering

channels. Nevertheless, HCCNTs are predicted to be more suitable (than SWCNTs) for

thermal management applications due to their spring-like shape. HCCNTs are extremely

elastic, natural NanoVelcro material.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Since the early work of Ruoff and Lorents [1] where high ther-

mal conductivity of carbon nanotubes have been predicted on

the basis of the high in-plane thermal conductivity of graph-

ite and of bulk and thin film diamond, wealth of experimental

and theoretical research of heat transport in carbon nano-

tubes and related carbon nanostructured materials have been

done [2]. It has been experimentally evidenced that, at all

temperatures, thermal conductivity of single walled carbon

nanotubes (SWCNTs) is dominated by phonons rather than

electrons [3] and that in nanotube thermal conductors Fourier

empirical law of thermal conduction is violated even when

the phonon mean free path (MFP) is much shorter than the

nanotube length [4].

However, with the exception of the recent reports on the

molecular dynamics calculations of thermal conductivity [5]

and thermal expansion [6] of helically coiled carbon nano-

tubes (HCCNTs) [7], theoretical research of their thermal

properties is lacking, although the first synthesis [8,9] of the

coiled nanotubes followed soon after the discovery of the

straight ones [10]. Presumably, the lack of large-scale synthe-

sis method of coiled CNTs over the past decades has hindered

extensive research of their precious properties and further

realizations of their potentials for the nano-technological

applications [11–13]. However, several recent reports on the

catalytic chemical vapor deposition production of the

HCCNTs with controlled morphology [14,15] substantially

improve the prospects of fundamental and applied physics

research of these peculiar nanostructures.

282 C A R B O N 7 7 ( 2 0 1 4 ) 2 8 1 – 2 8 8

Here, using the upgraded Klemens treatment of heat con-

duction in basal planes of graphite [16–18] we perform a detail

theoretical study of the intrinsic thermal conductivity of

HCCNTs.

Used is the model of HCCNTs based on the topological

coordinate method [19,20] which has significant advantages

over the scheme proposed in the pioneering work of Ihara

Fig. 1 – Model of HCCNTs. (a) Symmetry generating elements (h

translation F and twofold horizontal rotational axis, U-axis, wh

geometrical parameters of HCCNTs (tubular diameter d, coil pitc

the eyes one half of the monomer is highlighted; (b) Cohesive e

(lower panel) of the corrugated and un-corrugated models of HC

side: Initial LR triple connected graph (n6=1, ((3,0), (0,3))) of pentag

Super-cell ((3,0), (0,3)) is shown; rectangle denotes its unit cell. O

Super-cell ((3,0), (0,2)) is shown; rectangle denotes its unit cell. I

cells), n5 ¼ 2 columns of hexagons (between the pairs of pentag

(between the pairs of heptagons of adjacent unit cells) [19].

and Itoh [21]. Firstly, it allows for almost continual variation

of a single geometrical parameter while all the other param-

eters are kept fixed. Secondly, it matches geometrical param-

eters of the as-synthesized HCCNTs [22,23]. Finally, apart

from the previously defined smooth helical structure [19] we

here introduce also corrugated model of HCCNTs (Fig. 1), as

both circular and polygon-like shapes of the cross-sections

elical rotational angle 2p=Q which is followed by a fractional

ich interlinks pairs of atoms within a monomer) and

h p, inclination angle v and helical diameter D); as a guide to

nergies (upper panel) and relative difference of tubular radii

CNTs as a function of the graph parameter nr; (c) On the left

ons (black), hexagons (white) and heptagons (dark grey) [20].

n the right side: Augmented graph (1,2,2,2, ((3,0), (0,2))).

nserted are nr ¼ 2 rows of hexagons (between initial unit

ons within a unit cell) and n7 ¼ 2 columns of hexagons

Fig. 2 – Low energy phonon dispersion relations for a

(2,2,0,0, ((1,0), (0,5))) HCCNT, with geometrical parameters

D ¼ 2:2 nm; d ¼ 0:5 nm; p ¼ 2:6 nm; v ¼ 210. Acoustic

branches (green), the lowest optical branch (blue) and

scattering channels AAA (green solid line) and AAO (blue

solid line) are highlighted. Three-phonon Umklapp

scattering of the type I (II) is shown by solid (dashed) line.

Red diamond marks the considered phonon. (A colour

version of this figure can be viewed online.)

C A R B O N 7 7 ( 2 0 1 4 ) 2 8 1 – 2 8 8 283

are evidenced in the as-synthesized samples of the coiled car-

bon nanotubes [7,24].

We perform calculations of thermal conductivity of

HCCNTs of finite lengths, from 10nm to 10mm. Apart from

the anharmonic three-phonon Umklapp processes which

are treated exactly, for all allowed phonon relaxation chan-

nels, backscattering is accounted for within relaxation time

approximation. We also examine temperature dependant

crossover from ballistic to diffusive transport regime, as well

as dependance of thermal conductivity on geometrical

parameters of HCCNTs.

The breakdown of the Fourier law of heat conduction

which was evidenced in SWCNTs [4] we also predict for

HCCNTs. Namely, we find that regularly distributed pentago-

nal and heptagonal ‘‘defects’’ do not to scatter the long-wave

length phonons to the extent enough to remove the thermal

conductivity divergence with nanotube length.

We predict that thermal conductivity of HCCNTs amounts

� 75% of that of the corresponding SWCNTs as the coil length

exceeds 500 lm. So good heat conducting properties in com-

bination with the unique mechanical qualities [13,25] make

HCCNTs promising candidates for thermal management

applications.

2. Phonon dispersion relations

We consider the model of HCCNTs which is obtained by roll-

ing up a triple connected graph of pentagons, hexagons and

heptagons, conventionally defined by a set of numbers

ðn6;nr;n7;n5; ðb1;b2ÞÞ, where ðb1;b2Þ are the super-cell vectors,

while the tiling pattern is given by the first four parameters,

Fig. 1. In addition to the relaxation procedure given in Ref.

[19], tubular radial coordinates of carbon atoms are also var-

ied and obtained is a model of corrugated HCCNTs. Compar-

ison of the cohesive energies of the corrugated and un-

corrugated HCCNTs together with the rate of corrugation

are given in Fig. 1. Geometrical structure of HCCNTs is charac-

terized by tubular diameter d, outer diameter of a helix D,

helical step p and inclination angle v.

Unlike the case of SWCNTs where phonon dispersion

curves are evaluated using the graphite force constants [26]

adopted to the nanotube geometry [27], dynamical sub-matri-

ces of HCCNTs are numerically derived out of Brenner inter-

atomic potential [28] by varying space coordinates of carbon

atoms and calculating the corresponding energy changes.

Since HCCNTs are complex systems which, in general, do

not have translational periodicity and contain thousands of

atoms within a single coil pitch, application of the line group

symmetry [29] and related symmetry based techniques [30] as

well as helical quantum number representation of phonon

modes were essential for carrying out numerical

computation1.

Symmetry of HCCNTs [19], described by a 5th family LG

L ¼ TQðFÞD1, which is generated by helical transformation

ðCQ jFÞ (i.e. rotation for 2p=Q followed by fractional translation

1 Quite recently, similar method which apply line group symmetryvibrations in MoS2 NTs [31].

2 In the literature, helical quantum numbers are, by convention, dennumbers. However, as herein we are dealing only with the helical qu

F) and by p-rotation around twofold horizontal axis (U-axis,

Fig. 1). Generally, parameters Q and F (which is determined

by the inclination angle v and monomer length a of the

HCCNT: F ¼ a sin v) are real and only in the special case of

the rational Q value (i.e. Q ¼ q=r, where q and r are co-primes

satisfying condition q > r), there is a translational symmetry

with period A ¼ qF. However, even in these special cases, heli-

cal quantum number representation of the phonon disper-

sions is more convenient since unit cell is always very large.

In addition, helical angular momentum m is conserved in

Umklapp processes, regardless of whether reduced or

extended Brillouin zone scheme is applied [29]. Helical wave

vector k reflects helical periodicity and runs over the irreduc-

ible domain ½0; p=F�. Full symmetry assigned dispersion

curves, when shown over the irreducible domain, correspond

to doubly degenerated phonon states. On the other hand, first

Brillouin zone is given by the interval ð�p=F;p=F� and C-point

and edge-point phonon states are characterised by a parity

with respect to the U-axis. Lack of the rotational symmetry

implies vanishing helical angular momentum (m ¼ 0).

Although helical quantum number representation2 is

used, total number of phonon branches is still very large:

three times the number of atoms within a monomer, nF.

Therefore, for clarity, in Fig. 2, displayed is only the low

energy part of phonon dispersions (from 0 to 50�1012 rad/s

out of roughly 300� 1012 rad=sÞ for the HCCNT with relatively

small number of atoms in a monomer: nF ¼ 60.

Being quasi-one dimensional systems, HCCNTs have four

acoustic branches: In helical quantum numbers representa-

to empirical potential calculations has been used to study lattice

oted as ~k and ~m, in order to distinguish from the linear quantumantum numbers, tilde symbol is omitted for convenience.

284 C A R B O N 7 7 ( 2 0 1 4 ) 2 8 1 – 2 8 8

tion, transverse acoustic (TA) modes are characterized by

finite wave number k ¼ 2p=Q (V point) and quadratic disper-

sion, while longitudinal acoustic (LA) and shear wave acoustic

modes (SWA) are represented by k ¼ 0 (C point) and show lin-

ear dispersions (Fig. 2) with the sound velocities of the same

intensity, vLA ¼ vSWA ¼ 3000 m/s, which is much lower than

the sound velocities in SWCNTs [27]. However, in contrast to

SWCNTs, the maximal acoustic band velocities in HCCNTs

are not the sound velocities. As illustrated in Fig. 2, over a

relatively large interval of the Brillouin zone, dispersions of

the acoustic phonons are linear with the band velocity of,

typically � 9000 m/s, corresponding to the intensity of the

TA sound velocity in SWCNTs [27] (whilst the TA acoustic

modes of HCCNTs have quadratic dispersion xTA ¼ ak2;

a ¼ 10�6 m2=s). Finally, the optical gap is an order of

magnitude lower than in the case of straight SWCNTs.

3. Phonon scattering

Different momentum non conserving scattering processes

contribute to the thermal resistance: anharmonic phonon

interaction, diffuse phonon scattering from rough bound-

aries, phonon-defect scattering, etc. In ideal cases (crystal

structures without defects and without boundaries, e.g.) the

intrinsic thermal conductivity is limited only by the pho-

non–phonon scattering due to the lattice anharmonicity

which can be theoretically interpreted by considering three-

phonon Umklapp processes.

Following the method of Klemens and Pedraza for evaluat-

ing theoretical values of phonon thermal conductivity (which

was applied originally to single crystals of graphite [16] and

later, by Balandin and co-workers [18], advanced and adapted

for calculation of thermal transport in graphene) we derive

expressions of intrinsic thermal conductivity of HCCNTs tak-

ing into account the three-phonon Umklapp scattering pro-

cesses in which: (I) a phonon xðkÞ leaves the state k as

being absorbed by another phonon from the heat flux, and

(II) a phonon comes to the state k, due to the decay of a pho-

non from the state k00. For these two types of Umklapp pro-

cesses energy conservation laws are: xðkÞ � xðk0Þ � xðk00Þ ¼ 0,

where the upper sign corresponds to the first and lower sign

to the second process.

In Umklapp processes, in general, quasi-momenta and

quasi-angular momenta are not conserved. However, unlike

the case of the straight SWCNTs, where the scattering phase

space is severely restricted [32] due to the chirality dependant

quasi-angular momenta selection rules [29], there is no such a

reduction of the scattering phase space of HCCNTs. Hence,

three phonons which pertain to any combination of the

branches can take part in an Umklapp process provided that

the energy is conserved and the condition imposed on the

quasi-momenta is fulfilled: k� k0 ¼ k00 þ K (where K is a reci-

procal lattice vector). This is in contrast to the achiral

SWCNTs where Umklapp scattering which involves three

acoustic phonons is not allowed [32]. Consequently, while

contribution of the normal scattering processes have consid-

3 In the previous theoretical studies of thermal conductivity of graphbetter agreement with experiment.

erable impact on the phonon transport in the straight

SWCNTs, they do not have the same effect in HCCNTs.

Namely, as normal processes indirectly affect thermal con-

ductivity through redistribution of the phonon modes, if they

would not be taken into account, many triples of phonons of

SWCNTs would be left trapped in the states which fulfill

momenta and energy conserving conditions but do not satisfy

the angular momenta selection rules. This however is not the

case with HCCNTs, where all the phonon scattering channels

are open.

Out of the general forms for matrix elements of the three-

phonon interaction [17], we evaluate the expressions for the

three-phonon Umklapp scattering rates in HCCNTs taking

into account all the phonon branches and their dispersions

as well as energy and momenta selection rules. At the tem-

perature T, Umklapp scattering rate s�1U (where sU is the pho-

non relaxation time) of a phonon from the state km of the

branch m and with energy �hxm is calculated according to the

following expression (upper/lower sign correspond to first/

second type of the Umklapp process):

s�1U ðm;km;TÞ ¼

2c2m �hxmðkmÞF

3nFMv2m ðkmÞ

Xa;b

Xka ;kb2X

xaðkaÞxbðkbÞjDx0�ðkaÞj

�

�ffT½xaðkaÞ� � fT½xbðkbÞ� þ12� 1

2g:

The summations are taken over the all 3nF phonon dispersion

branches and over the set

X ¼ fk 2 ð�p; p� jDx�ðkÞ ¼ 0 ^ Dx0�ðkÞ– 0g;

Dx�ðkaÞ ¼ xmðkmÞ � xaðkaÞ � xbðkbÞ;

cm is Gruneisen parameter of the branch m (calculated is mode-

dependant Gruneisen parameter which is then averaged over

the each phonon branch3); M is a carbon atom mass; vmðkÞ is

the phonon group velocity; fT is Bose–Einstein equilibrium

distribution function at temperature T.

Gruneisen parameter cmk of an individual phonon mode

xmðkÞ is defined as the negative logarithmic derivative of the

frequency of the mode with respect to the volume. Overall

Gruneisen parameter for the acoustic branches is calculated

as the weighted average of the individual Grunisen parame-

ters in which the contribution of each phonon is weighted

by its contribution to the specific heat [34,16] and the typical

value obtained for HCCNTs is �0.5 (which is for a factor 3

lower than in the case of SWCNTs).

The coiled nanotubes do not have surface which can pro-

vide boundary scattering except for the nanotube ends which

may lead to the backscattering. The backscattering rate s�1B ,

which is of relevance in the ballistic regime only, is calculated

within relaxation time approximation: s�1B ¼ vmð1� sÞ=L,

where vm is the group velocity of the phonon branch m; L is

NT length and s is adjustable parameter giving fraction of

phonons which scatter specularly from the boundaries (the

values zero and one correspond respectively to the absolutely

rough and ideally smooth boundaries). Value of the s-param-

eter depends on the quality of the end-contacts [35] and in

ene [18] and BN NTs [33] mode-specific Gruneisen parameter gave

Fig. 3 – Phonon MFP K vs. phonon wave-vector k at room

temperature. Dotted horizontal line indicates MFP threshold

K0. Arrows point to the particular values of the cut-off

frequencies, xCðLAÞ;xCðSWAÞ and xjðTAÞ, which correspond

to the given cut-off in the k-space.

Fig. 4 – Intrinsic MFP for the first two phonon branches of a

(1,3,2,0, ((1,0), (0,5))) HCCNT at temperatures of 100 K and

300 K. Dashed lines show the room temperature intrinsic

MFP of the acoustic modes of a (7,1) SWCNT. (A colour

version of this figure can be viewed online.)

C A R B O N 7 7 ( 2 0 1 4 ) 2 8 1 – 2 8 8 285

our calculations nearly ideal contacts are assumed by taking

s ¼ 0:98.

The scattering rates we consider to be mutually indepen-

dent and apply the Matthiessen’s rule, taking thus the overall

scattering rate as a sum of the particular scattering rates.

4. Thermal conductivity

Thermal conductivity j is the sum of the individual phonon

conductivities jmðkÞ:

j ¼X

m

Xk

jmðkÞ ¼1V

Xm

Xk

cmðkÞvmðkÞKmðkÞ ð1Þ

where cmðkÞ ¼ �hxmðkÞ @@ T f TðxmðkÞÞ is the phonon specific heat,

KmðkÞ ¼ jvmðkÞjsmðkÞ is the phonon MFP and the summation is

taken over the entire phonon dispersions m and over the

wave-vectors k of the Brillouin zone (�p=F; p=F].

Calculations of thermal conductivity we perform on a

sample of HCCNTs with helical diameters D � 5 nm, helical

— tubular radii ratio D : d from 4 to 5.5 and helical steps p

between 2 and 6.2 nm.

Taking for the volume of the monomer of HCCNT

V ¼ 2prda, where d ¼ 3:4 A and r are the nanotube wall thick-

ness and tubular radius (averaged over the radii of the all

orbits of the corrugated model of HCCNT, Fig. 1), replacing

the summation over the 1D Brillouin zone with the integra-

tion over the irreducible domain of the reciprocal space, Eq.

(1) takes the form:

j ¼ 14p2rda

Xm

Z p=F

0cmðkÞvmðkÞKmðkÞdk ð2Þ

In order to avoid the problem of vanishing scattering of the

long-wave length phonons [36] (which makes the thermal

conductivity to diverge as the heat conductor length

increases) we introduce the cut-off frequencies for the disper-

sions around C and V points, (xC and xV) exempting from the

heat flow the phonons with infinite MFP. The cut-off frequen-

cies are determined by analysing the wave-vector depen-

dance of the MFP, Fig. 3. In all the cases considered it holds

xCðLAÞ > xCðSWAÞ > xVðTAÞ.Frequency dependance of the phonon MFPs for the acous-

tic branches of a (2,2,0,0, ((1,0), (0,5))) HCCNT, and of the LA, TA

and TWA modes of a (7,1) SWCNT are presented in Fig. 4. In

the calculations, only the Umklapp scattering rates are

accounted for. Generally, phonon MFP in the coiled tubes is

of the same order of magnitude as in the straight tubes. If

long wave length phonons which are characterized by infinite

MFP are excluded, MFP of the acoustic modes of vibration at

room temperature spans to 10 lm. As the system is cooled

down to 100 K, acoustic phonon MFP is predicted to reach

1 mm.

According to the empirical Fourier law of heat conduction,

which refers to the diffusive regime of heat transport (i.e.

under condition that length of the conductor is considerably

larger than the phonon MFP), the thermal conductivity j

directly relates the heat energy flux U to the temperature gra-

dient: U ¼ �jrT. The Fourier law implicitly states that the

thermal conductivity is independent of the length of the con-

ductor. This was experimentally proved in all the cases of

three-dimensional (3D) materials considered thus far. How-

ever, there are many indications that the Fourier law in 1D

systems does not hold. Theoretical investigation of the heat

conduction in 1D chains of anharmonic oscillators suggests

that the chaotic behavior in 1D is not enough to attenuate

the free traveling long wave-length phonons [37]. From the

experimental side, the breakdown of the Fourier law in car-

bon and boron-nitride nanotubes has been evidenced [4]:

measured thermal conductivity data were fitted to a function

jðLÞ � Lb; and the b parameter, which measures the deviation

from the Fourier low behavior, was determined to be

b 2 ½0:6;0:8�. According to our calculations, HCCNTs are yet

another variety of 1D systems that violates the law of heat

conduction. Fit to the numerically results at zero cut-off fre-

quencies, (which is essentially equivalent to the long wave-

length approximation) predicts b 2 ½0:4;0:6� for HCCNTs and

b 2 ½0:3;0:5� for straight SWCNTs.

Fig. 6 – Length dependance of thermal conductivity of

(1,3,2,0, ((1,0), (0,5))) HCCNT (solid line) and (7,1) SWCNT

(dashed line) at T ¼ 200 K (navy), T ¼ 300 K (olive) and

T ¼ 400 K (wine). Dotted lines indicate thermal conductivity

of HCCNT along the helical axis. (A colour version of this

figure can be viewed online.)

286 C A R B O N 7 7 ( 2 0 1 4 ) 2 8 1 – 2 8 8

In Fig. 5 presented is length dependance of the room tem-

perature thermal conductivity of a coiled (1,3,2,0, ((1,0), (0,5)))

CNT and of a straight (7,1) CNT having similar tubular diame-

ters. Dashed lines show the results obtained in the long wave-

length approximation and the parameters giving deviation

from the Fourier low are b ¼ 0:3 (SWCNT) and b ¼ 0:5

(HCCNT). If the frequency cut-off is introduced, thermal con-

ductivity saturates to finite value: j ¼ 3300 W/Km (matching

the conductivity of the purified diamond [38]) at L > 500 lm

(for the HCCNT) and j ¼ 4300 W/Km at L > 100 lm (for the

SWCNT). The latter is in excellent agreement with the previ-

ously reported results [39] (j � 4000 W/Km, L > 100 lm,

T ¼ 316 K, calculated for (10,0) NT).

The relative thermal conductivity of HCCNTs with respect

to the corresponding straight SWCNTs is length dependant.

Heat transfer through the coiled NTs over the distance of

10nm is �; 50% lower than through the straight counterparts.

However, relative thermal conductivity of HCCNTs rapidly

increases with the conductor length, being 70% at �100nm

and reaching the plateau of 75% at the lengths of the order

of 1 mm.

For the conductors shorter than phonon MFPs, scattering-

free phonon transport leads to linear dependance of the ther-

mal conductivity with length. The lower the temperature, the

larger is the phonon MFP (Fig. 4) and thus the limit of the bal-

listic conductor length is larger (Fig. 6).

Modifications of the helical-tubular diameter ratio, D : d,

induce the modifications of the phonon dispersions of HCCNTs

which are further reflected in the thermal and mechanical

properties. Namely, the lower D : d is, the higher are the both

thermal conductivity and elastic modulus. As apart from the

high thermal conductance it is important that theraml inter-

face materials are also highly elastic, in order to quantify the

optimal helical-tubular diameter ratio for heat-sinking appli-

cations of HCCNTs, thorough calculations of the both proper-

Fig. 5 – Room temperature thermal conductivity j of (1,3,2,0,

((1,0), (0,5))) HCCNT (black) and (7,1) SWCNT (gray) as a

function of the conductor length L. Solid lines show the

results obtained introducing the cut-off frequencies (Fig. 3),

while long wave-length approximation is indicated by the

dashed lines. Dotted line gives thermal conductivity of the

HCCNT along the helical axis. Inset highlights the conductor

length range: 0–10 lm.

ties are to be performed on a large and representative sample

of the modeled HCCNTs. However, this task is beyond the scope

of this paper and will be presented elsewhere.

As illustrated in Fig. 7, HCCNTs with lower helical-tubular

diameter ratio are characterized by higher acoustic phonon

velocities which are further reflected in the higher thermal

conductivity (under condition that other structural parame-

ters are not changed notably).

4.1. Thermal conductance

The heat flow rate through a single ballistic phonon channel

at temperature T is given by the quantum of thermal conduc-

tance [40]: rthðTÞ ¼ p2k2BT=3h. In the ballistic limit, thermal

Fig. 7 – Thermal conductivity of HCCNTs: (3,3,0,0, ((1,0), (0,5)))

(red); (1,3,2,4, ((1,0), (0,5))) (green); (1,4,2,2, ((1,0), (0,5))) (blue);

with different helical-tubular diameter ratio D : d, as a

function of the length of the helix (solid line) and of the

length of the helical axis (dotted line). Inset shows the

corresponding low energy phonon dispersions. (A colour

version of this figure can be viewed online.)

Fig. 8 – Temperature dependance of thermal conductance r

for 1lm long (2,2,0,0, ((1,0), (0,5))) HCCNT. Ballistic r � T (red)

and diffusive r � T�1 (blue) regimes are highlighted. Arrow

indicates that two quanta of conductance do not extrapolate

to zero at T ¼ 0 K. (A colour version of this figure can be

viewed online.)

C A R B O N 7 7 ( 2 0 1 4 ) 2 8 1 – 2 8 8 287

conductance r is an integer multiple of rth and thus more

appropriate measure of the heat transport than thermal con-

ductivity j. The conductance is related to the conductivity as:

r ¼ ðS=LÞj, where S is the area of the cross-section of the heat

conductor. Temperature dependance of the thermal conduc-

tance of one micrometer long (2,2,0,0, ((1,0), (0,5))) HCCNT is

shown in Fig. 8.

The quantized ballistic conductance is characterized by a

linear temperature dependance rðTÞ � T and holds until

T � 30 K. At temperatures higher than several Kelvins,

HCCNTs carry two quanta of thermal conductance,

rHCCNT ¼ 2rthðTÞ, i.e. a half of the energy flux through the

straight SWCNTs [41] in a scattering-free transport regime.

Only in the zero temperature limit HCCNTs have four trans-

port channels, Fig. 8. The peak of the conductance is reached

at T ¼ 55 K when a balance between specific heat increase

and increased scattering rate is established. At higher tem-

peratures scattering processes dominate and at T > 200 K dif-

fusive transport regime rule rðTÞ � T�1 holds.

5. Conclusion

Presented is the theoretical study of phonon heat conduction

in helically coiled carbon nanotubes. Flexible atomistic model

in which the coiling arises as an intrinsic property of the

structure [42] and which matches the geometrical parameters

of the as-synthesized HCCNTs is proposed.

Phonon dispersions of HCCNTs are found to substantially

differ from that of SWCNTs: The coiled tubes are character-

ized by narrow acoustic bands and by relatively low phonon

group velocities. In addition, since lacking pure rotational

symmetry, they allow for three-phonon Umklapp scattering

of solely acoustic modes of vibration (by contrast to the

straight SWCNTs, where such kind of processes are forbidden

by selection rules). Due to the balance of the low anharmonic-

ity of acoustic modes, on the one side, and lack of the restric-

tions on the type of the scattering channels, on the other side,

MFP of the acoustic phonons in HCCNTs is of the same order

as in the case of straight SWCNTs: from one to ten microme-

ters at room temperature.

Room temperature thermal conductivity of HCCNTs is pre-

dicted to match that of the purified diamond as the nanotube

length exceeds 0.5 mm. For many applications, vertically

aligned CNT arrays are promising as thermal interface mate-

rials and thermal conductance is a crucial measure of their

performance. However the optimal interfaces need to have

a variety of other characteristics like e.g. high elasticity. In

this respect, the coiled nanotubes are superior to the straight

ones. Besides, HCCNTs naturally form VelcroTM-like contact

[43] where the tubes mechanically entangle, so that the load

transfer efficiency in composites [11] made of the coiled CNTs

(rather than of the straight ones) is expected to be substan-

tially larger (as the coiled shape spontaneously induces

mechanical interlocking when the composite is subjected to

loading).

In this paper we gave insight into the mechanisms which

govern intrinsic heat conduction in HCCNTs. However, in

order to come to real applications, many questions are still

to be resolved. One of these, a heat dissipation along the

nanotubes and from the nanotubes to the substrate, is critical

to developing and improving CNT-based electronic devices

[44,45].

Acknowledgments

The authors acknowledge funding of Serbian Ministry of Sci-

ence (ON171035) and Swiss National Science Foundation

(SCOPES IZ73Z0–128037/1).

Appendix A. Supplementary data

Supplementary data associated with this article can be found,

in the online version, at http://dx.doi.org/10.1016/

j.carbon.2014.05.031.

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