The unexpected thermal conductivity from graphene disk,

carbon nanocone to carbon nanotube

Dengke Ma1,2, Han Meng1,2, Xiaoman Wang1,2, Nuo Yang1,2,*, Xing

Zhang3,*

1State Key Laboratory of Coal Combustion, Huazhong University of Science and

Technology (HUST), Wuhan 430074, P. R. China

2Nano Interface Center for Energy (NICE), School of Energy and Power Engineering,

Huazhong University of Science and Technology (HUST), Wuhan 430074, P. R.

China

3Key Laboratory for Thermal Science and Power Engineering of Ministry of

Education, Department of Engineering Mechanics, Tsinghua University, Beijing, P. R.

China

*Corresponding authors: N.Y.(nuo@hust.edu.cn) and X.Z.(x-zhang@tsinghua.edu.cn)

ABSTRACT

Graphene and single-wall carbon nanotubes (SWCNTs) have attracted great attention

because of their ultra-high thermal conductivity. The carbon nanocone (CNC) is a

graded structure fall in between graphene disk (GD) and SWCNTs. And yet, there are

few reports on the thermal properties of CNC. We perform non-equilibrium molecular

dynamics (NEMD) simulation to study the thermal conductivity of CNC with

different apex angles, and then compare them with that of GD and SWCNTs. The

SWCNTs, which can be looked as a CNC with 0° apex angle, has a homogeneous

thermal conductivity. Our results show that, similar to the GD, the CNC also has a

graded thermal conductivity. Unexpectedly, the graded rate shows a surprising

increase when the apex angle decreases from 113° to 19°. We find that except the

difference in phonon mode number, a brand new mechanism, the phonon power

spectra mismatch are responsible for this novel change from GD, CNC to SWCNTs.

INTRODUCTION

Over the past decades, most of the researches have focused on graphene1 and

single-wall carbon nanotubes (SWCNTs)2 due to their unusually high thermal

conductivity. It is found that the thermal conductivity of the suspended graphene

exceeds (2500 + 1100/-1050) W/m-K near 350 K,3-5 and the thermal conductivity of

SWCNTs exceeds 2000 W/m-K at room temperature.6, 7 Although carbon nanocone

(CNC) has been observed just after the SWCNTs,8, 9 it drew less attention. Researches

towards CNC mainly concentrate on its growth mechanism,9-13 mechanical

properties,14-16 and electronic properties17-19. It is found that CNC can be used for

atomic force microscopy,20 as thermal rectifier,21 ammonia sensor,22 sorbent

materials,23 and interface materials.24 However, few works have paid attention to its

thermal properties21 even though thermal properties are of fundamental and practical

significance.

Different from previous studies on divergence of thermal conductivity in

nano-structures with different size,25-28 Yang et al found a novel graded thermal

conductivity along the radial direction in one graphene disk (GD) in 2015 based on

molecular dynamics simulation.29 And the similar phenomenon was observed by

Wang et al through Monte Carlo simulation.30 This new finding broadens our

understanding of thermal transport property of materials in nanoscale, and opens up a

new door for us to explore the connection of thermal conductivity between GD, CNC

and SWCNTs.

Graphene and SWCNTs are strongly correlated with each other in structure. A

SWCNTs can be described as graphene that is rolled up into a seamless cylinder.

Generally, the phonon dispersion relations in a SWCNTs can be obtained from those

of the 2D graphene sheet by using the zone folding approach. However, zone-folding

of the graphene phonon branches does not always give the correct dispersion relation

for a carbon nanotube.31 CNC is characterized by the apex angle (as shown in

Figure1c). The extreme situations when the apex angle goes to 180° and 0° are

corresponding to GD and SWCNTs, respectively. So CNC has an outstanding

structure advantage to relate GD and SWCNTs together by changing its apex angle

(as shown in Figure1a)。

In this letter, the thermal conductivity of GD, CNC and SWCNTs along the

radial direction were calculated by using the non-equilibrium molecular dynamics

(NEMD) simulation. The dependence of thermal conductivity on apex angle was

investigated. The results of CNCs were compared with those of GD and SWCNTs. To

understand the underlying physical mechanism, we analyzed the atomic position

density (APD) and the phonon power spectra of the top and bottom atomic layers in

GD, CNC and SWCNTs.

Structure and Methods

CNC is rolled up by cutting the GD a sector off (as shown in Figure 1b).9 When

the apex angle decreases, its degree of asymmetry decreases (as shown in Figure 1c).

The lattice constant (a) and thickness (d) of GD, CNC and SWCNTs are 0.1418 nm

and 0.334 nm, respectively. The GD and CNC have 73 layers, and their interlayer

spacing is 0.1418 nm (as shown in Figure 1b). The SWCNTs has 84 layers, and its

interlayer spacing is 0.1228 nm.

When studying the thermal conductivity of CNC, we focus on the CNCs with the

apex angle of 113°,84°,60°,39° and 19°, which are rolled up by cutting a GD

60°,120°,180°,240° and 300° off respectively (as shown in Figure 1b). As the thermal

conductivity depends on system size26, 28 and tube diameter32, we keep the length L

and the top radius rtop (as shown in Figure 1d) the same for different apex angle case.

Moreover, the same top radius makes it possible to realize all the structures by merely

changing the apex angle from 180° to 0°. L is 10nm, and rtop is approximately the

same with the radius of a (10,10) SWCNTs.

As we study the thermal conductivity of CNC by using the classical NEMD

method, a temperature gradient is built in CNC from top to bottom along the radial

direction. The CNC is coupled with Langevin heat bathes33 at the 2nd to 4th layers

and (N−1)th layers with temperatures are Ttop and Tbottom, respectively. And atoms at

boundaries (the 1st and Nth layers) are fixed.

The potential energy is described by a Morse bond and a harmonic cosine angle

for bonding interaction, which includes both two-body and three-body potential

terms34, 35. Although this force field potential is developed by fitting experimental

parameters for graphite, the accuracy of this potential is demonstrated in MD

simulation by Yang.21 And our result of the thermal conductivity of a 10nm long (10,

10) SWCNTs (as shown in Figure S1) is 75.06 W/m-K, which is very close to the

values reported by others.36, 37 To integrate the discretized differential equations of

motions, the velocity Verlet algorithm is used.

Simulations are performed long enough such that the system reaches a stationary

state. All results given here are obtained by averaging 8106 time steps. The time step

is 0.5 fs. In general, the temperature TMD is calculated from the kinetic energy of

atoms( i iivmT 2/2). The heat flux J along the nanocone can be calculated at the

heat bath region as:

inTT

in

in

N

i

i

TT

TTt

ε

NJ

out

out

out

1 Δ2

Δ1 (1)

where Δε is the energy added to/removed from each heat bath (Tin or Tout) at each step

Δt. The thermal conductivity ( ) is calculated based on the Fourier definition as:

dr

dTθrπκJ )2sin(d2 (2)

where r is the distance to the vertex of cone. We use a combination of time and

ensemble sampling to obtain better average statistics. The results represented averages

from 12 independent simulations with different initial conditions.

Results and Discussion

The temperature profiles of CNC along the radial direction from top to bottom for

different apex angles are plotted in Figure 2. The symbols are direct MD results. As is

shown in Figure 2a, the temperature profile of GD is a curve rather than a line.

Clearly, the thermal conductivity of GD in our result is graded, similar to that of Yang

et al.29

Same as Figure 2a, the temperature profiles of CNC are also curves in Figure 2b,

2c, 2d, 2e, 2f. It implies that, instead of being homogeneous, the thermal

conductivities of CNCs with different apex angles are graded, too. It is because that

the CNC forms a similar heat transfer to that in GD, streaming from inside to outside

(as shown in Figure 1d, the top view). The difference between GD and CNC is that

the heat flux of GD is in a two dimensional flat, while heat flux in CNC has a vertical

component. This difference makes CNC has an unexpected graded thermal

conductivity which we will discuss later. The temperature profile of SWCNTs is

obviously a line (as shown in Figure S1 supporting information). Thus, the thermal

conductivity of SWCNTs is homogeneous, which matches well with previous work.38

Our previous analytical result showed that the non-homogeneous steady state

temperature distributions and thermal conductivity of GDs satisfies: 29

1,)ln(/)ln(1

)ln(/)ln(rR)]()([)(

1,)ln(/)ln(1

)ln(/)ln(rR)]()([)(

)(

1

11

αCrCr

CrCrrTrTrT

αCrCr

CrCrrTrTrT

rT

outin

outinnormalinoutin

α

outin

α

outin

α

normal

inoutin

(3)

)ln(

)ln(r R

]rR[)ln(

)ln()( 00

out

normal

α

normal

α

out

rC

rC

κrC

rCκrκ

(4)

where rin is the distance from the vertex of cone to the top of the cone, rout is the

distance from the vertex of cone to the bottom of the cone (as shown in Figure 1c).

is the exponential factor, which would depend on the apex angle, temperature and

length. 0 and C are constants.

We use Eq. (3) to fit the temperature profile data obtained from MD, which may

overcome the problem of fluctuation in temperature profile. As shown in Figure 2, the

numerical data can be well fitted by using Eq. (3). The adjusted R square for fitted

lines are 1.000, 0.999,0.999,0.999,1.000 and 1.000 corresponding to apex angle of

180°, 113°, 84°, 60°, 39° and 19°, respectively. Using the fitted parameters, we get

the thermal conductivity of GD and CNC by Eq. (2) and Eq. (4). The calculation of

the thermal conductivity of SWCNTs is shown in SI of the supporting information.

As shown in Figure 3 on a log-log plot, the thermal conductivity of SWCNTs is

homogeneous, while for GD and CNC, their thermal conductivity increases linearly

from top to bottom along the radial direction. Moreover, when decreasing the apex

angle from 180° to 19°, the thermal conductivity increases obviously. Since we keep

the top radius the same (as shown in Figure1a), the CNC becomes thinner when the

apex angle decreases. This tendency is similar to that in the SWCNTs with different

tube diameters.39 Since CNC can be treated as SWCNTs connected with different tube

diameters, the thermal conductivity of CNC will increase when the apex angle

decreases.

What is more interesting, the slope of the thermal conductivity lines show an

increase tendency when the apex angle decreases, which demonstrates that the graded

rate of the thermal conductivity for different apex angle increases. Since the

exponential factor in Eq. (2), indicates the graded rate of thermal conductivity for

each angle, we focus on it to get a better understanding of this change. When equals

zeroit corresponds to a homogeneous thermal conductivity. As shown in Figure 3b,

the lowest value presented by SWCNTs is zero. However, when the apex angle

decreases from 113° to 19°, experiences an unexpected monotonic increase.

Previous understanding of the graded thermal conductivity works well in GD and

SWCNTs.29 They imply that because of the geometric asymmetry, there are more

phonon modes in the large layer, leading to the higher thermal conductivity of the

larger layer than the small layer. While in CNC, when the apex angle decreases from

113° to 19°, the geometric asymmetry decreases, but the graded rate increases

surprisingly. And when the apex angle changes from 19° to 0°, the graded rate has a

sharp decline, and turns to zero. When the apex angle changes from 180°to 113°, the

graded rate has a slight decrease. Thus the transition from GD to SWCNTs through

CNCs experiences some complicated processes. And the more phonon modes view

alone can’t govern the explanation of the graded thermal conductivity.

To better comprehend the graded thermal conductivity and explore the reason of

the unexpected change of with the apex angle. We remove the heat bath, and keep

the whole system in equilibrium state. Then we record the position and velocity of the

atoms at the top (10th) and bottom (63th) atomic layer of GD (inner and outside for

GD), CNC and SWCNTs (11th and 71th atomic layer for SWCNTs), and calculate the

APD (as shown in Figure 4) and the phonon power spectra (as shown in Figure 5).

Here, the phonon power spectra is exhibited along three directions, two in-plane

directions and one out-plane direction, as illustrated in Figure 1c.

In order to quantify the above APD and the power spectrum analysis, the average

mismatch (M) of the APD and the phonon power spectra of two different layers are

calculated as:

max

min

max

min

))()(()()(

x

x

bottomtop

x

x

bottomtop dxxfxfdxxfxfM (5)

here ftop(x) is the APD or the power spectrum of the top atomic layer and fbottom(x) is

the APD or the power spectrum of the bottom atomic layer. xmin is the lower limit of

the integration, and xmax is the upper limit of the integration. In Figure 4 and Figure5,

the value of M for the APD and the normal direction phonon power spectra of CNC

are also illustrated.

For SWCNTs, as shown in Figure 4d and Figure 5d, the APD and the phonon

power spectra along three directions of the top and bottom atomic layers are the same.

This claims the same heat carriers of the top and bottom atomic layers, which agree

well with the homogeneous thermal conductivity of SWCNTs.

For GD, it is found that there is an obvious difference in APD between atoms at

the inside and outside atomic layers (as shown in Figure 4a). The atoms at the outside

atom layers have a large spread of vibrations, which corresponds to more phonon

modes and thus a higher thermal conductivity at the outside layer. This result agrees

with Yang et al.29 And the phonon power spectra is almost the same of the inner and

outside atomic layers (as shown in Figure 5). Therefore in GD, it is the phonon modes

difference that solely induces the graded thermal conductivity.

While for CNC, the APD of the top and bottom atomic layers show a completely

different result with that of GD (as shown in Figure 4b and 4c). When the apex angle

is 113°, the atoms at the top and bottom atomic layers have almost the same spread of

vibrations. When the apex angle is smaller than 113°, the atoms at the top atomic

layers have the larger spread of vibrations than the bottom atomic layers (as shown in

Figure 4c and Figure SII). It suggests that the top atomic layers is supposed to have a

higher thermal conductivity than the bottom atomic layers which is contrary to our

result. So different to GD, the APD mismatch in CNC imposes a restrain on the

graded thermal conductivity. And clearly, the APD alone can’t govern the explanation

of graded thermal conductivity of CNC. But contrary to GD, the phonon power

spectra of the atoms at the top and bottom atomic layers is different for CNC (as

shown in Figure 5 and Figure SIII). The normal direction phonon power spectra have

an obvious shift from low frequency to high frequency, while the longitudinal

direction and transverse direction phonon power spectra have a slight shift from high

frequency to low frequency. It claims that the top and bottom atomic layers have the

different heat carries, which will result in the different thermal conductivity of the top

and bottom atomic layers. So this phonon power spectra mismatch is another reason

for graded thermal conductivity.

To justify the unexpected change in the graded rate of thermal conductivity, a

more noteworthy phenomenon should be mentioned. When the apex angle decreases

from 113° to 39°, the phonon power spectra mismatches are 0.09, 0.16, 0.20 and 0.22

respectively, and the APD mismatches are 0.05, 0.19, 0.19 and 0.22 respectively.

Although the increase in the APD mismatch suggests that the restrain caused by the

phonon mode number difference on graded thermal conductivity of CNC becomes

strong. The increase in the phonon power spectra mismatch demonstrates the enlarged

difference between the heat carriers of the top and bottom atomic layers, which result

in the increase of the graded rate (). More remarkable, when the apex angle

decreases from 84° to 60°, the APD mismatch keeps almost the same, while the

phonon power spectra mismatch increase from 0.16 to 0.20. This different change in

APD and phonon power spectra mismatch agrees well with the sharp increase in the

graded rate from 84° to 60° (as shown in Figure 3). Moreover, when the apex angle

decreases from 39° to 19°, the phonon power spectra mismatch decreases from 0.22 to

0.14. This mismatch decreases will lead to the decrease of , which is contrary to our

result. But the mismatch of APD here experiences a larger decreases from 0.22 to

0.10. This decreases of APD mismatch has a stronger effect and result in the increase

of These inverse change in APD and phonon power spectra mismatch in 39° means

that the APD and the phonon power spectra mismatch reach a maximum value when

the apex angle is 39°.

Conclusion

We study the thermal conductivity of SWCNTs and GD together by changing the

apex angle of CNC. Our NEMD simulation results indicate that the thermal

conductivity of CNC is graded rather than homogeneous from top to bottom along the

radial direction of cone. By studying the dependence of thermal conductivity on apex

angle, we find that the thermal conductivity of CNC experiences an increase when the

apex angle decreases from 180° to 19°. It is attributed to the decrease of the average

diameter of the CNC. What’s more interesting, we find that the graded rate of the

thermal conductivity (exponential factor ) unexpectedly increases when the apex

angle decreases from 113° to 19°. When the apex angle changes from 19° to 0°, the

graded rate has a sharp decline to zero. When the apex angle changes from 180° to

113°, the graded rate has a slight decrease. We infer that the previous phonon modes

difference alone can’t govern the explanation of graded thermal conductivity. After

analyzing the APD and the phonon power spectra of the top and bottom (inner and

outside for GD) atomic layers in equilibrium state, we find the absolute difference in

the APD of CNC and GD. For GD, the atoms at the outside atomic layers have a large

spread of vibrations. While for CNC, the atoms at the top atomic layers have the large

spread of vibrations, which may lead to the inverse graded thermal conductivity of

CNC. We propose that a new mechanism, the phonon power spectra mismatch among

the top and bottom atomic layers, is another reason responsible for the graded thermal

conductivity. By combining the APD and the phonon power spectra, we successfully

explain the novel change from GD to SWCNTs through CNC.

Our work relates GD, CNC and SWCNT together, presents the unusual thermal

conductivity of CNC and finds a new mechanism which induces the graded thermal

conductivity.

Acknowledgments

The project was supported by the National Natural Science Foundation of China

51327001 (XZ) and 51576067 (NY). The authors are grateful to Shiqian Hu for useful

discussions. The authors thank the National Supercomputing Center in Tianjin

(NSCC-TJ) for providing help in computations.

Additional Information Supplementary information accompanies this paper.

Competing financial interests: The authors declare no competing financial interests.

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Figure 1. (Color online) (a) Schematic picture of the relationship of graphene disk,

CNC and SWCNTs. (b) A graphene disk which is cutted a sector off, is the angle

of the sector cutted off. d is the distance from the atom to the center point in cone

sheet. We define the atoms in the ith layer as the atoms whose ]ir,r)i[(d cc1 . cr is

the equilibrium carbon-carbon bond length. (c) The side view of CNC. inr and outr

are the distance from the vertex of cone to the top and bottom of the cone along the

radial direction. The length along the side face of nanocone is cniout Nr)rr(L ,

where N is the number of layers. is the angle of the cone. (d) The top view of

CNC. topr and bottomr are the radius of the top and bottom circle of the cone

respectively.

Figure 2. The temperature profiles along the radial direction of graphene disk (a) and

CNC with different apex angle as 113° (b), 84° (c), 60° (d), 39° (e) and 19° (f) at

300K. The symbols are numerical data obtained by MD simulation, and the lines are

fitted lines based on Eq.(3). The normalized radius, Rnormal , is defined in Eq.(4). The

temperature profiles of SWCNTs is shown in Figure S1 (supplementary information).

.

Figure 3. (a) The thermal conductivity of graphene disk (180°), CNCs with different

apex angle and SWCNTs (0°) at 300K. (b) The exponential factor (graded rate)

versus apex angle. The red dashed line is a linear fitting to the data from 19° to 113°.

Figure 4. The APD profiles around its equilibrium positions between the top (inner for

GD) and bottom (out for GD) atomic layers for different apex angle in equilibrium

state, where re is the distance to the atom equilibrium position. The average mismatch

number M is defined according to Eq. (5). The apex angle is 180° (GD), 113°, 39°

and 0° (SWCNTs) respectively. The rest cases when the apex angle is 84°, 60° and

19° are shown in Figure S2 (supplementary information).

Figure 5. The phonon power spectra along three directions between the top (inner for

GD) and bottom (out for GD) atomic layers according to the Figure 1c in equilibrium

state. The average mismatch number M is defined according to Eq. (5). The apex

angle is 180° (GD), 113°, 39° and 0° (SWCNTs) respectively. The rest cases when

the apex angle is 84°, 60° and 19° are shown in Figure S3 (supplementary

information).

Supplementary Information

The unexpected thermal conductivity from graphene disk,

carbon nanocone to carbon nanotube

Dengke Ma1,2, Han Meng1,2, Xiaoman Wang1,2, Nuo Yang1,2,*, Xing

Zhang3,*

1State Key Laboratory of Coal Combustion, Huazhong University of Science and

Technology (HUST), Wuhan 430074, P. R. China

2Nano Interface Center for Energy (NICE), School of Energy and Power Engineering,

Huazhong University of Science and Technology (HUST), Wuhan 430074, P. R.

China

3Key Laboratory for Thermal Science and Power Engineering of Ministry of

Education, Department of Engineering Mechanics, Tsinghua University, Beijing, P. R.

China

*Corresponding authors: N.Y.(nuo@hust.edu.cn) and X.Z.(x-zhang@tsinghua.edu.cn)

SI. The temperature profile of (10,10) single-wall carbon nanotubes (SWCNTs).

As the temperature profile of SWCNTs is line obviously, we use linear fit to get

the slope of the line, and calculate the thermal conductivity of the (10,10) SWCNTs

according to Eq. (S1).

TdC

J

(S1)

where x is distance along the axis direction of the tube. C is the perimeter of the tube.

And d is the thickness.

Figure S1. The temperature profiles of a (10,10) SWCNTs. The symbols are MD

results directly. The red line is fitted line.

SII. The atomic position density (APD) for graphene disk (GD), carbon nanocone

(CNC) and single wall carbon nanotubes (SWCNTs) in equilibrium state.

Figure S2 The APD around its equilibrium positions between the top and bottom

atomic layers for different apex angle in equilibrium state, where re is the distance to

the atom equilibrium position. The average mismatch number M is defined according

to Eq. (5). The apex angle is 180° (GD), 113°, 84°, 60°, 39°, 19°and 0° (SWCNTs)

respectively.

SIII. The phonon power spectra for GD, CNC and SWCNTS in equilibrium

state.

Figure S3. The phonon power spectra along three directions between the top and

bottom atomic layers according to Figure 1c in equilibrium state. The average

mismatch number M is defined according to Eq. (5). The apex angle is 180° (GD),

113°, 84°, 60°, 39°, 19°and 0° (SWCNTs) respectively.