^{1}, Takuma Shiga

^{2}, David Nicholson

^{1}, Junichiro Shiomi

^{2,a)}and Leonid V. Zhigilei

^{1,a)}

### Abstract

The effect of bendingbuckling of carbon nanotubes(CNTs) on thermal conductivity of CNT materials is investigated in atomistic and mesoscopic simulations. Nonequilibrium molecular dynamics simulations of the thermal conductance through an individual buckling kink in a (10,10) single-walled CNT reveal a strong dependence (close to inverse proportionality) of the thermal conductance of the buckling kink on the buckling angle. The value of the buckling kink conductance divided by the cross-sectional area of the CNT ranges from 40 to 10 GWm^{−2} K^{−1} as the buckling angle changes from 20 to 110°. The predictions of the atomistic simulations are used for parameterization of a mesoscopic model that enables calculations of thermal conductivity of films composed of thousands of CNTs arranged into continuous networks of bundles. The results of mesoscopic simulations demonstrate that the conductivity of CNTfilms is sensitive to the angular dependence of the buckling kink conductance and the length of the individual CNTs. For a film composed of 1 *µ*m-long CNTs, the values of the in-plane filmconductivity predicted with a constant conductance of 20 GWm^{−2} K^{−1} and the angular-dependent conductance obtained in atomistic simulations are about 40 and 20% lower than the conductivity predicted for the same film with zero thermal resistance of the buckling kinks, respectively. The weaker impact of the angular-dependent buckling kink conductance on the effective conductivity of the film is explained by the presence of a large fraction of kinks that have small buckling angles and correspondingly large values of conductance. The results of the simulations suggest that the finite conductance of the buckling kinks has a moderate, but non-negligible, effect on thermal conductivity of materials composed of short CNTs with length up to 1 *µ*m. The contribution of the buckling kink thermal resistance becomes stronger for materials composed of longer CNTs and/or characterized by higher density of buckling kinks.

A.N.V., D.N., and L.V.Z. acknowledge financial support by AFOSR (Grant FA9550-10-10545) and NSF (Grant CBET-1033919), as well as computational support by NSF through TeraGrid resources (project TG-DMR110090) and NCCS at ORNL, USA (project MAT009). T.S. and J.S. acknowledge partial support by KAKENHI 23760178 and 22226006, and Global COE program, “Global Center of Excellence for Mechanical System Innovation” from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

I. INTRODUCTION

II. ATOMISTIC SIMULATIONS OF THERMAL CONDUCTANCE OF BUCKLING KINKS

III. MESOSCOPIC SIMULATIONS OF THERMAL CONDUCTIVITY OF CNTFILMS

A. Mesoscopic modeling of the formation of CNTnetwork materials

B. Structure of CNTfilms

C. Method of calculation of thermal conductivity of mesoscopic samples

D. Thermal conductivity of CNTfilms

IV. SUMMARY

### Key Topics

- Carbon nanotubes
- 170.0
- Buckling
- 129.0
- Thermal conductivity
- 50.0
- Thermal conduction
- 40.0
- Networks
- 20.0

##### B82B1/00

## Figures

(Color online) A schematic of the simulation process of bending (10, 10) CNT to a bending angle *Θ* = 60° (a) and snapshots of buckled structures at various values of *Θ* (b).

(Color online) A schematic of the simulation process of bending (10, 10) CNT to a bending angle *Θ* = 60° (a) and snapshots of buckled structures at various values of *Θ* (b).

(Color online) An illustration of the definition of the local angle *Ψ* (a), the dependence of *Ψ* on the unit cell index *p* counted from the center of the CNT (b), and the dependence of the buckling angle *χ* on the bending angle *Θ* (c). In (a), *P* and *A* are the positions of the centers of mass of *p*th and (*p*-1)th unit cells, respectively. The error bars show the standard deviation for 50 different configurations at equilibrium.

(Color online) An illustration of the definition of the local angle *Ψ* (a), the dependence of *Ψ* on the unit cell index *p* counted from the center of the CNT (b), and the dependence of the buckling angle *χ* on the bending angle *Θ* (c). In (a), *P* and *A* are the positions of the centers of mass of *p*th and (*p*-1)th unit cells, respectively. The error bars show the standard deviation for 50 different configurations at equilibrium.

(Color online) The dependence of the thermal conductance of the bending buckling kink on the buckling angle predicted for (10,10) CNTs in atomistic simulations. Square symbols show the values found in atomistic simulations performed in the present work and the triangle symbol shows the value recalculated from Ref. 22. Solid curve represents a power law fit to the data points given by Eq. (3).

(Color online) The dependence of the thermal conductance of the bending buckling kink on the buckling angle predicted for (10,10) CNTs in atomistic simulations. Square symbols show the values found in atomistic simulations performed in the present work and the triangle symbol shows the value recalculated from Ref. 22. Solid curve represents a power law fit to the data points given by Eq. (3).

(Color online) Schematic sketch of the mesoscopic representation of a nanotube *i* as a chain of cylindrical segments defined by nodes *j* (a) and the distribution of temperature along the nanotube in the vicinity of the buckling kink (b). The solid circles show the mesoscopic nodes, with nodes where buckling occurs colored red. In the mesoscopic model, the buckling angle is defined as an angle between axes of segments adjacent to the buckling kink. Buckling kinks and CNT ends divide the nanotubes into several buckling-free parts. Temperature is assumed to be constant within any buckling-free part of the nanotube and exhibits a jump at buckling kinks.

(Color online) Schematic sketch of the mesoscopic representation of a nanotube *i* as a chain of cylindrical segments defined by nodes *j* (a) and the distribution of temperature along the nanotube in the vicinity of the buckling kink (b). The solid circles show the mesoscopic nodes, with nodes where buckling occurs colored red. In the mesoscopic model, the buckling angle is defined as an angle between axes of segments adjacent to the buckling kink. Buckling kinks and CNT ends divide the nanotubes into several buckling-free parts. Temperature is assumed to be constant within any buckling-free part of the nanotube and exhibits a jump at buckling kinks.

(Color online) Fraction of nanotubes with a given number of buckling kinks, , in samples *A* (red open squares) and *B* (green open circles) (a) and a part of sample *A* shown in Fig. 6 with positions of buckling kinks marked by dark (red online) color (b). The average number of buckling kinks per buckled nanotube, , and the fraction of buckled nanotubes, , are marked in (a) by the solid red square and green circle for samples *A* and *B*, respectively.

(Color online) Fraction of nanotubes with a given number of buckling kinks, , in samples *A* (red open squares) and *B* (green open circles) (a) and a part of sample *A* shown in Fig. 6 with positions of buckling kinks marked by dark (red online) color (b). The average number of buckling kinks per buckled nanotube, , and the fraction of buckled nanotubes, , are marked in (a) by the solid red square and green circle for samples *A* and *B*, respectively.

(Color online) Steady-state temperature distribution established in a mesoscopic sample *A* and used in the calculation of the thermal conductivity of the network structure. The sample has dimensions of 500 × 500 × 100 nm^{3} and is composed of 7829 (10,10) CNTs with nm and material density of 0.2 g cm^{−3}. The nanotubes are colored by their temperature. The temperature distribution is shown for , but it is visually indistinguishable from the one calculated with .

(Color online) Steady-state temperature distribution established in a mesoscopic sample *A* and used in the calculation of the thermal conductivity of the network structure. The sample has dimensions of 500 × 500 × 100 nm^{3} and is composed of 7829 (10,10) CNTs with nm and material density of 0.2 g cm^{−3}. The nanotubes are colored by their temperature. The temperature distribution is shown for , but it is visually indistinguishable from the one calculated with .

(Color online) Ratio of thermal conductivity calculated for different values of constant conductance of buckling kinks (.) to the thermal conductivity calculated with zero thermal resistance of buckling kinks (). The results are shown for samples *A* ( nm; red squares and solid curve) and *B* ( nm; green circle and dashed curve).

(Color online) Ratio of thermal conductivity calculated for different values of constant conductance of buckling kinks (.) to the thermal conductivity calculated with zero thermal resistance of buckling kinks (). The results are shown for samples *A* ( nm; red squares and solid curve) and *B* ( nm; green circle and dashed curve).

(Color online) A part of sample *A* shown in Fig. 6 with nanotubes colored by their total heat flux sum , Eq. (7), calculated with (a) and (b). and mark two characteristic bundles of CNTs. Bundle splits into many thinner bundles and includes multiple buckled CNTs. As a result, the values of for nanotubes in this bundle are visibly different in panels (a) and (b). Bundle mostly consists of nonbuckled CNTs, and the values of for nanotubes in this bundle exhibit only small difference between panels (a) and (b).

(Color online) A part of sample *A* shown in Fig. 6 with nanotubes colored by their total heat flux sum , Eq. (7), calculated with (a) and (b). and mark two characteristic bundles of CNTs. Bundle splits into many thinner bundles and includes multiple buckled CNTs. As a result, the values of for nanotubes in this bundle are visibly different in panels (a) and (b). Bundle mostly consists of nonbuckled CNTs, and the values of for nanotubes in this bundle exhibit only small difference between panels (a) and (b).

(Color online) The values of the total heat flux sum averaged over individual nanotubes that have the same number of buckling kinks . The results are calculated for sample *B* with (red squares) and (green triangles). Data for exhibit strong scattering due to the small number of corresponding nanotubes in the sample. The curves show least-squares quadratic polynomial approximation of data points with .

(Color online) The values of the total heat flux sum averaged over individual nanotubes that have the same number of buckling kinks . The results are calculated for sample *B* with (red squares) and (green triangles). Data for exhibit strong scattering due to the small number of corresponding nanotubes in the sample. The curves show least-squares quadratic polynomial approximation of data points with .

(Color online) The results of the calculation of thermal conductivity performed for mesoscopic samples composed of nanotubes of different length with constant values of thermal conductance of the buckling kinks (red squares for and green triangles for GWm^{−2} K^{−1}) and with angular-dependent thermal conductance given by Eq. (3) (blue circles). The data in (a) are normalized to the thermal conductivity predicted for zero resistance of the buckling kinks (). In panel (b), the corresponding dimensional values of and are plotted. The black dashed line in (b) shows the approximate scaling law obtained in Ref. 21 for the (*L _{T} *) dependence. The values of shown by diamonds in (b) are slightly different from the ones in Ref. 21 because of a small difference in the methods used in the calculations of the temperature gradient .

(Color online) The results of the calculation of thermal conductivity performed for mesoscopic samples composed of nanotubes of different length with constant values of thermal conductance of the buckling kinks (red squares for and green triangles for GWm^{−2} K^{−1}) and with angular-dependent thermal conductance given by Eq. (3) (blue circles). The data in (a) are normalized to the thermal conductivity predicted for zero resistance of the buckling kinks (). In panel (b), the corresponding dimensional values of and are plotted. The black dashed line in (b) shows the approximate scaling law obtained in Ref. 21 for the (*L _{T} *) dependence. The values of shown by diamonds in (b) are slightly different from the ones in Ref. 21 because of a small difference in the methods used in the calculations of the temperature gradient .

Probability density function of the thermal conductance of the buckling kinks in sample *B* ( nm) calculated with the angular dependence of the conductance on the buckling angle, , given by Eq. (3). The dots show the values obtained from the actual distribution of the buckling angles and the curve is least-squares 10th order polynomial approximation of the data points with GWm^{−2} K^{−1}. σ_{ b max} and are the most probable and mean values of the conductance .

Probability density function of the thermal conductance of the buckling kinks in sample *B* ( nm) calculated with the angular dependence of the conductance on the buckling angle, , given by Eq. (3). The dots show the values obtained from the actual distribution of the buckling angles and the curve is least-squares 10th order polynomial approximation of the data points with GWm^{−2} K^{−1}. σ_{ b max} and are the most probable and mean values of the conductance .

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