Thermal conductivity of carbon nanocoilsJunhua Zhao, Jianyang Wu, Jin-Wu Jiang, Lixin Lu, Zhiliang Zhang, and Timon Rabczuk

Citation: Applied Physics Letters 103, 233511 (2013); doi: 10.1063/1.4839396 View online: http://dx.doi.org/10.1063/1.4839396 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/103/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Sensitivity of thermal conductivity of carbon nanotubes to defect concentrations and heat-treatment J. Appl. Phys. 113, 034312 (2013); 10.1063/1.4778477 Limited thermal conductance of metal-carbon interfaces J. Appl. Phys. 112, 094904 (2012); 10.1063/1.4764006 Size dependent thermal conductivity of single-walled carbon nanotubes J. Appl. Phys. 112, 013503 (2012); 10.1063/1.4730908 Effect of bending buckling of carbon nanotubes on thermal conductivity of carbon nanotube materials J. Appl. Phys. 111, 053501 (2012); 10.1063/1.3687943 Thermal conductivity of deformed carbon nanotubes J. Appl. Phys. 109, 074317 (2011); 10.1063/1.3573509

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Thermal conductivity of carbon nanocoils

Junhua Zhao,1,2,a) Jianyang Wu,3,a) Jin-Wu Jiang,2 Lixin Lu,1 Zhiliang Zhang,3,b)

and Timon Rabczuk2,4,c)

1Jiangsu Province Key Laboratory of Advanced Manufacturing Equipment and Technology of Food,Jiangnan University, 214122 Wuxi, China2Institute of Structural Mechanics, Bauhaus-University Weimar, 99423 Weimar, Germany3Nanomechanical Lab, Norwegian University of Science and Technology (NTNU), Richard Birkelands vei 1a,N-7491 Trondheim, Norway4School of Civil, Environmental and Architectural Engineering, Korea University, South Korea

(Received 1 October 2013; accepted 17 November 2013; published online 5 December 2013)

The extreme reduction of the thermal conductivity by defects and folds in carbon nanocoils

(CNCs) are first demonstrated using nonequilibrium molecular dynamics simulations. The thermal

conductivity for two different defect types with five different folds in the CNCs is extensively

studied and the maximum reduction can be up to 70% at both room temperature and 600 K by

comparison of the corresponding straight single-walled carbon nanotubes. We reveal that the

phonon scattering by coupled defects and folds can govern the reduction of the thermal

conductivity by calculating phonon polarization vectors. VC 2013 AIP Publishing LLC.

[http://dx.doi.org/10.1063/1.4839396]

Carbon nanocoils (CNCs) (or coiled carbon nanotubes

(CNTs)) were synthesized experimentally by Zhang’s group

in 1994.1,2 Due to their unique helical structures, CNCs ex-

hibit some special electronic,3,4 mechanical,5,6 and magnetic

properties7,8 and possess significant potentials to make reso-

nators, nanosprings, electromagnetic wave absorbers, etc.

Great effort was made in the past years to identify the electri-

cal, mechanical, and thermal properties. Experiments by

Hayashida et al.9 revealed that the electric conductivity of

the CNCs ranges from 107 to 180 s/cm. These results suggest

that carbon nanocoils have good conductivity and can be

applied to nanosized electronic devices. Akita et al.10 found

that CNCs could be semimetal by tight-binding (TB) calcula-

tions. Lu4 studied their electronic transport properties and

observed a gap opening in the conductance spectrum based

on a -orbital TB model incorporated with the Greens func-

tion method. Chen et al.6 measured the mechanical property

of a CNC and obtained the spring constant of 0.12 N/m

under a low strain. Subsequently, Wu et al.11,12 reported the

high energy absorbing capacity, the giant stretchability, and

reversibility of CNCs using molecular dynamics simulations.

Recently, Ma et al.13 have predicted the thermal conductivity

of a CNC by a one-dimensional thermal conduction model

based on the field emission experiments, in which the ther-

mal conductivity of the CNC is evaluated to be around 38 W

m�1 K�1. However, the numerical study on the thermal

properties of CNCs has not been reported and the mechanism

for their low thermal conductivity is not clear yet. In this pa-

per, we first perform nonequilibrium molecular dynamics

(NEMD) simulations to investigate the thermal conductivity

of the CNCs ð7; 5ÞNDT , where DT and N denote the incorpo-

rated defect type and the segment length measured by the

number of hexagons between any two neighboring defect

groups (see Fig. 1). We only examine single-walled CNCs

adopting either pentagon k heptagonjheptagon k pentagon

groups-hereafter represented as 5 k 7j7 k 5 defect type or

pentagon k octagon k pentagon groups—5 k 8 k 5 defect

type. The thermal conductivity for two different defect types

(5 k 7 k 7 k 5 and 5 k 8 k 5) with five different segment

lengths (N¼ 5, 7, 11, 15, 19) in the CNCs is extensively

studied and the maximum reduction can be both up to 70%

at room temperature and 600 K compared to the correspond-

ing straight single-walled carbon nanotubes. The reduction

mechanism is revealed by analyzing the phonon polarization

vectors.

In order to obtain the thermal conductivity of CNCs

which is close to the real size of the experiments,13,14 the

total number of carbon atoms in all initial structures of

CNCs is close to 15 000, and the length is around 155 nm for

the corresponding straight single-walled CNT (7,5). The MD

simulations are carried out using Brenners second generation

reactive empirical bond order potential,15 which has been

widely used to study the mechanical and thermal properties

of carbon nanotubes and other carbon allotropes.16–19 The

long range van der Waals (vdW) interaction is calculated by

FIG. 1. The configuration for the two defect types of (a) pentagon kheptagonjheptagon k pentagon 5 k 7j7 k 5 and (b) pentagon k octagon kpentagon 5 k 8 k 5 in the carbon nanocoils of ð7; 5ÞNDT , where DT and Ndenote the incorporated defect type and the segment length measured by the

number of hexagons between any two neighboring defect groups, respectively.

a)J. Zhao and J. Wu contributed equally to this work.b)Electronic mail: zhiliang.zhang@ntnu.noc)Electronic mail: timon.rabczuk@uni-weimar.de

0003-6951/2013/103(23)/233511/4/$30.00 VC 2013 AIP Publishing LLC103, 233511-1

APPLIED PHYSICS LETTERS 103, 233511 (2013)

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the Lennard-Jones 12–6 potential with well-depth energy of

e ¼ 2:8437 meV and equilibrium distance of r¼ 3.4 A.17

Each initial structure is run for 2 ns at 300 K and 4 ns at

600 K until the pressure and energy of the system is stable,

respectively, keeping both the temperature T¼ 300 K (or

600 K) and the pressure P¼ 1 atm (the time step

�t ¼ 0:5 fs), in the NPT ensemble (constant number of

atoms N, constant temperature T and constant pressure P en-

semble) controlled by the Nose-Hoovers thermostat.20,21 The

system is then kept in the NVE ensemble for 4 ns and the

thermal conductivity is calculated using NEMD simulations.

The simulation box is uniformly divided into 50 slabs along

the heat flow direction, and the hot region is kept in the thin

slab of the center. The two cold regions are kept at the

boundary of the box. Periodic boundary conditions are

applied in all axial directions. A snapshot of the ð7; 5Þ55k7j7k5system and the simulation setup for thermal conductivity cal-

culation are shown in Fig. 2(a). All the MD simulations have

been performed using LAMMPS software.22 The heat flux Jin the steady state is evaluated as23

J ¼ 1

2tA

X

transf ers

m

2v2

hot � v2cold

� �; (1)

where t is the duration of the simulation, A is the cross-

sectional area of the corresponding straight CNT (7,5), vhot

and vcold denote the velocities of the hot and the cold atom of

equal mass m, and the factor 2 arises from the periodicity.

The thermal conductivity can be calculated by k ¼ �J=�T,

where �T is the temperature gradient along the flow direc-

tion. To calculate the temperature gradient, the temperature

of each slab is averaged over the last 2 ns of the simulations.

It should be noted that the length of CNCs are defined as

NCNC/NCNT� LCNT, where NCNC, NCNT, and LCNT are the

number of carbon atoms in the CNC, the number of carbon

atoms in the corresponding straight CNT, and the length of

the straight CNT, respectively.

Fig. 2(b) shows the typical temperature profile of current

NEMD simulations for thermal conduction. The temperature

gradient is equal to the average value by fitting the data from

0:15–0:35 and 0:65–0:85 since the temperatures exhibit linear

profiles in these regions. Fig. 3 shows the relative and absolute

thermal conductivity of the current two different defect types

ð5 k 7j7 k 5 and 5 k 8 k 5Þ with five different segment lengths

(N¼ 5, 7, 11, 15, 19) in the CNCs, in which the relative ther-

mal conductivity is defined as the absolute values of the

CNCs divided by the one of the corresponding straight CNT

(7,5) at the same temperature. The thermal conductivity of the

straight CNT (7,5) at 300 K is equal to 226.7 W m�1 K�1,

which is in good agreement with values from the previous

work.14,18,24 The thermal conductivity of a CNC from a

one-dimensional thermal conduction model based on the field

emission experiments is around 38 W m�1 K�1 in the previ-

ous work.13 This value is slightly lower than our results here,

because of the impurity scattering in the experiments.

The relative thermal conductivity strongly decreases

with decreasing N for two different temperatures, see Fig. 3.

The relative thermal conductivity decreases to only 24.3%

and 23.1% at room temperature (27.2% and 26.3% at 600 K)

for the two defect types as N decreases from 1 (N ¼ 1denotes the straight CNT) to 5. For N¼ 19, the relative ther-

mal conductivity of 5 k 7j7 k 5 and 5 k 8 k 5 is 37.6% and

34.2% at room temperature (43% and 40.3% at 600 K),

respectively. From N¼ 19 to N¼ 5, the reduction is about

13% and 11% at room temperature (16% and 14% at 600 K)

for the two defect types of 5 k 7j7 k 5 and 5 k 8 k 5, respec-

tively. The results indicate that the effect of N � 19 on the

thermal conductivity is more pronounced than 5 � N � 19.

For the same defect type and segment lengths N, the relative

thermal conductivity at room temperature is slightly lower

than those at 600 K. A similar phenomenon can also be

found in other materials.24,25 For the same N and tempera-

ture, the relative thermal conductivity for the two defect

types is also weakly distinct. Therefore, the thermal

FIG. 2. (a) A snapshot of the ð7; 5Þ55k7j7k5 system and the simulation setup

for thermal conductivity calculation. The kinetic energy of each atom is dif-

ferent colored for clarity, (b) temperature profiles obtained from NEMD sim-

ulations on CNC s of ð7; 5ÞN5k7j7k5 and ð7; 5ÞN5k8k5 at room temperature, (c)

temperature profiles obtained from NEMD simulations on CNCs of

ð7; 5ÞN5k7j7k5 and ð7; 5ÞN5k8k5 at 600 K.

FIG. 3. The relative thermal conductivity of the current CNCs ð7; 5ÞNDT (inset

represents the absolute thermal conductivity of the CNCs).

233511-2 Zhao et al. Appl. Phys. Lett. 103, 233511 (2013)

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conductivity only strongly depends on the segment lengths N.

The higher the length is, the higher the thermal conductivity

is. Due to almost same total number of carbon atoms in all ini-

tial CNC structures, the higher the segment length N is, the

smaller the number of defects and folds are. From the previ-

ous work, the defects and folds have a large effect on the ther-

mal conductivity of the single-walled CNTs and graphene (or

silicon), respectively.25–27 Depending on the defect and fold

density, the maximum reduction by the defects or folds can be

up to around 70%.25–27 The reason is that the thermal conduc-

tivity decreases with decreasing N due to the scattering of

phonons at the position of the defects and folds. The effect of

the coupled defects and folds on the reduction of the thermal

conductivity is caused by the special CNC structures.

To further reveal the microscopic mechanism for the

reduction of the thermal conductivity in the CNCs, we study

the phonon modes in one folded segment of the CNCs (see

Fig. 4), which are the only energy carrier in the molecular

dynamics simulation. There are four acoustic modes in the

quasi-one-dimensional structure: longitudinal acoustic (LA),

twisting (TW), and two transverse acoustic (TA) modes. In

Fig. 4, we show the vibration morphology of these four

acoustic modes in the CNCs with 5 k 8 k 5 defects. For the

TAz mode, atoms at the fold also have a large vibration am-

plitude, so the thermal energy can be passed from one end to

the other by this mode.

The TW mode has been found to be pinched at the kink

in the kinked silicon nanowires, where nanowires are kinked

without helical rotational defect.25 However, Fig. 4(a) shows

that the TW mode is not pinched by the defect and fold.

Atoms at the defect and fold are also involved in the vibra-

tion of the TW mode. This fact reveals that the helical rota-

tional defect in the CNC here helps to protect the TW mode

from being interrupted by the defects, since the helical rota-

tional defect is similar to the vibration morphology of the

TW mode.

Fig. 4(b) shows that the TAy and LA modes in the CNC

are coupled due to the defected and folded interface. Both

modes are pinched at the defect and fold, i.e., atoms at the

fold have zero vibration amplitude in these modes. As a

result, the heat energy cannot be passed from one arm to the

other. The analysis shows that half of the four acoustic

modes are able to contribute to the thermal conductivity of

the CNC, while the other two have no contribution. This is

the major underlying mechanism for the observed reduction

in the thermal conductivity of the CNC.

Fig. 5 reveals another possible mechanism from the

defect and fold effect on the thermal conductivity. The figure

shows two localized modes in the CNC with 5 k 8 k 5

defects. In these modes, only atoms at the fold have large

vibration amplitude, while the other atoms are almost not

involved in the vibration of this mode. The localization has

no contribution to the thermal conductivity, as it only local-

izes the heat energy at the defect (or fold).28

For the CNC with 5 k 7 k 7 k 5 defects, we find similar

two mechanisms as shown in Figs. 6 and 7. Both TAz and

FIG. 4. Comparison between the four low-frequency acoustic modes in a

segment of the CNCs ð7; 5ÞN5k8k5. Arrows in the figures represent the polar-

ization vector of the phonon mode. Blue color indicates good heat transport

capability for the mode. (a) TA mode vibrating in z direction and TW mode.

(b) The coupled TA mode vibrating in y direction and LA mode. The unit of

x is cm�1.

FIG. 5. Two localized modes in ð7; 5ÞN5k8k5. (a) x ¼ 1494 cm�1, (b)

x ¼ 1495 cm�1.

FIG. 6. Comparison between the four low-frequency acoustic modes in a

segment of the CNCs ð7; 5ÞN5k7j7k5. Arrows in the figures represent the polar-

ization vector of the phonon mode. Blue color indicates good heat transport

capability for the mode. (a) TA mode vibrating in z direction and TW mode.

(b) The coupled TA mode vibrating in y direction and LA mode. The unit of

x is cm�1.

233511-3 Zhao et al. Appl. Phys. Lett. 103, 233511 (2013)

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TW modes are not affected by the defect and fold, while the

other two acoustic modes are pinched at the defect and fold.

There are also some localized modes in this structure.

In summary, we first performed NEMD simulations to

demonstrate the extreme reduction of the thermal conductivity

by defects and folds in CNCs. The higher the segment length

is, the higher of the thermal conductivity is. The maximum

reduction can be up to 70% at room temperature and 600 K

compared to the corresponding straight single-walled carbon

nanotubes. We reveal that the phonon scattering by the

coupled defects and folds can govern the reduction of the ther-

mal conductivity by calculating phonon polarization vectors.

We gratefully acknowledge support by the Germany

Science Foundation (DFG), the Research Council of Norway

under NANOMAT KMB Project (MS2MP) No. 187269, and

National Natural Science Foundation of China (Grant No.

11302084).

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FIG. 7. Two localized modes in ð7; 5ÞN5k7j7k5. (a) x ¼ 1492:8 cm�1, (b)

x ¼ 1500:6 cm�1.

233511-4 Zhao et al. Appl. Phys. Lett. 103, 233511 (2013)

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