^{1,a)}and Richard L. Pober

^{2}

### Abstract

The thermal conductivities of nanotube and liquid dispersions are described with a simple physical model without any adjustable parameters. The geometric model uses periodic lattices with equal surface-to-surface distances between the nanotubes and their nearest neighbors in all directions. A combination of series model in the direction of the thermal gradient and a parallel model perpendicular to it is used along with averaging over all orientations to derive an effective bulk thermal conductivity. This model closely matches the experimental data to date.

INTRODUCTION

SPACING OF HIGH ASPECT RATIO NANOTUBES

PARALLEL AND SERIES THERMAL CONDUCTION

COMBINED MODEL

ORIENTATION AVERAGING

CONCLUSIONS

### Key Topics

- Thermal conductivity
- 75.0
- Nanotubes
- 54.0
- Carbon nanotubes
- 50.0
- Thermal models
- 26.0
- Graphite
- 6.0

## Figures

Data of the thermal conductivity of carbon nanotubes in oil, see Refs. 1 and 2. The standard models in (a) of mixtures of nanotubes in liquids show very little increase and vastly underestimate the data as shown. Models by Nan *et al.* in (b) significantly overestimate the data when the correct thermal conductivities are used.

Data of the thermal conductivity of carbon nanotubes in oil, see Refs. 1 and 2. The standard models in (a) of mixtures of nanotubes in liquids show very little increase and vastly underestimate the data as shown. Models by Nan *et al.* in (b) significantly overestimate the data when the correct thermal conductivities are used.

Models from Chen *et al.* (Ref. 11) along with data, see Ref. 1. These models do not capture the magnitude or the shape of the experimental data.

Models from Chen *et al.* (Ref. 11) along with data, see Ref. 1. These models do not capture the magnitude or the shape of the experimental data.

The sphere swept out by a nanotube rotating on its axis has a diameter equal to the length of the nanotube.

The sphere swept out by a nanotube rotating on its axis has a diameter equal to the length of the nanotube.

Nanotubes shown on square and hexagonal lattices and a cylindrical approximation. The surface-to-surface spacing for the nearest neighbors is denoted by , and for next nearest neighbors.

Nanotubes shown on square and hexagonal lattices and a cylindrical approximation. The surface-to-surface spacing for the nearest neighbors is denoted by , and for next nearest neighbors.

Surface-to-surface spacing for cylindrical nanotubes on square and hexagonal lattices and a cylindrical approximation as a function of volume fraction. The aspect ratio used is 2000.

Surface-to-surface spacing for cylindrical nanotubes on square and hexagonal lattices and a cylindrical approximation as a function of volume fraction. The aspect ratio used is 2000.

Schematic of parallel and series conductivities between plates kept at temperatures and with conductivities for the fluid and the nanotube .

Schematic of parallel and series conductivities between plates kept at temperatures and with conductivities for the fluid and the nanotube .

Pure series (a) and pure parallel (b) thermal conductivity models graphed with data for carbon nanotubes in oil. The series model underestimates and the parallel model overestimates the measured data.

Pure series (a) and pure parallel (b) thermal conductivity models graphed with data for carbon nanotubes in oil. The series model underestimates and the parallel model overestimates the measured data.

The composite can be broken into regions parallel to the direction of the thermal gradient. There are layers containing pure liquid in parallel with layers with both nanotubes and liquid.

The composite can be broken into regions parallel to the direction of the thermal gradient. There are layers containing pure liquid in parallel with layers with both nanotubes and liquid.

The normalized effective thermal conductivity along different directions as a function of volume fraction, plotted with data, see Ref. 1. The aspect ratio is 2000, the liquid thermal conductivity is , the nanotube thermal conductivity is , and there is no boundary resistance. Even if magnified as in the insert, all the nearest and next nearest neighbor models overlap and show very little increase in thermal conductivity.

The normalized effective thermal conductivity along different directions as a function of volume fraction, plotted with data, see Ref. 1. The aspect ratio is 2000, the liquid thermal conductivity is , the nanotube thermal conductivity is , and there is no boundary resistance. Even if magnified as in the insert, all the nearest and next nearest neighbor models overlap and show very little increase in thermal conductivity.

Interface resistance changes the model results only slightly. The calculations with and without resistance overlap one another. The nanotube diameter used was and the value for was , the largest reported in the literature. Other parameter values used are the same as in Fig. 9.

Interface resistance changes the model results only slightly. The calculations with and without resistance overlap one another. The nanotube diameter used was and the value for was , the largest reported in the literature. Other parameter values used are the same as in Fig. 9.

The orientation average thermal conductivity, normalized by the thermal conductivity of the liquid, shown for both the square and hexagonal lattice approximations.

The orientation average thermal conductivity, normalized by the thermal conductivity of the liquid, shown for both the square and hexagonal lattice approximations.

Data for three different liquids, see Ref. 2, compared with the orientation average model, showing that the model (square lattice) overestimates the thermal conductivity by a large margin for all three liquids.

Data for three different liquids, see Ref. 2, compared with the orientation average model, showing that the model (square lattice) overestimates the thermal conductivity by a large margin for all three liquids.

Data, see Ref. 2, compared with the orientation model using an aspect ratio of 200, much lower than the reported value of 2000.

Data, see Ref. 2, compared with the orientation model using an aspect ratio of 200, much lower than the reported value of 2000.

Data, see Ref. 2, with orientation average model using a carbon nanotube thermal conductivity of , the value for graphite perpendicular to the basal plane.

Data, see Ref. 2, with orientation average model using a carbon nanotube thermal conductivity of , the value for graphite perpendicular to the basal plane.

Data, see Ref. 2, with orientation average model using a carbon nanotube thermal conductivity of , the value for graphite perpendicular to the basal plane, and no interface thermal resistance. This lower bound on the interface resistance shows a slightly better fit with the data.

Data, see Ref. 2, with orientation average model using a carbon nanotube thermal conductivity of , the value for graphite perpendicular to the basal plane, and no interface thermal resistance. This lower bound on the interface resistance shows a slightly better fit with the data.

## Tables

Effective volume fractions.

Effective volume fractions.

.

.

Experimental Parameters.

Experimental Parameters.

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