On this webpage I present information concerning the thermal properties of carbon nanotubes. I have attempted to provide the necessary background information to understand these concepts though no single source can begin to cover all the fundamentals of this topic. If you are interested, I encourage you to look at the Additional References listed at the end of each section as they will provide much more detailed lessons of many of the topics I briefly review.
Carbon
Carbon exists in many different forms such as diamond and graphite. In all of these materials, the carbon atoms form directional covalent bonds, sharing their electrons. Due to the similar energy of the 2s and 2p orbitals, orbital hybridization often occurs. These lead to strong bonds with very small distances between each carbon atom (~1.41 - 1.44 A).
Nanoscale
Structures that range in size from 1 nanometer (10-9 meters) to 100 nanometers are considered to be on the nanoscale. 1 nanometer is only 14 times larger than a single Carbon atom.
Description of Carbon Nanotubes
A carbon nanotube can be thought of as a single layer graphite sheet, one carbon atom thick, that is rolled into a tube. Each graphite sheet is comprised of hexagonal sets of carbon atoms as shown below.
Carbon nanotubes are described using the (n, m) notation. These numbers correspond to the way the graphite sheet is rolled. One hexagon at the corner of the graphite sheet is designated as the origin (0, 0). This origin is then superimposed onto another hexagon which is n hexagons along the circumference and m hexagons up from the edge.
The way in which a nanotube is rolled affects its properties. There are three types of nanotubes formed depending on the (n, m) values. They are the armchair (n = m) nanotube, the zigzag (m = 0) nanotube, and the chiral nanotube formed by any other (n, m) pair. The naming is based on the physical appearance of the nanotube. Along the circumference of an armchair nanotube is a structure that looks like an armchair, while zigzag nanotubes have an edge with a zigzag pattern. Chiral nanotubes have an overall helical structure.
Moreover, if n - m = 3k, where k = 0, 1, 2…, then the nanotube is metallic. If this relation does not hold, then the nanotube is semiconducting. The ability for a nanotube to have different electronic properties based solely on its structure is very interesting.
Carbon nanotubes can be made up of a single rolled graphite layer, which is called a single-walled nanotube (SWNT) or a few layers rolled up together into a series of concentric tubes called a multi-walled nanotube (MWNT).
History of Carbon Nanotubes
Carbon Nanotubes were discovered by Sumio Iijima of the NEC Corporation in 1991. The nanotubes pictured below are MWNT varying in diameter. The nanotube forming the inner tube of the MWNT in c has a diameter of only 2.2nm.
Fabrication
Today, carbon nanotubes are fabricated using three primary methods.
The first is the spark method. This involves sending a charge through two graphite rods placed a few millimeters apart. The charge vaporizes the carbon atoms and when the recondense they form nanotubes. This method produces SWNT and MWNT at low yield.
The second method is chemical vapor deposition. In this process, a substrate is heated to 600C and exposed to a Carbon gas such as methane. The carbon atoms are freed and recombine into nanotubes. CVD is capable of producing very high yields of MWNT, but the defect concentration tends to be higher.
The third method uses a laser to produce the same effect as the spark in the first method. This process produces SWNT and can be tuned in order to control nanotube diameter, but it is the most expensive of the three.
These methods often create "mats" consisting of millions of nanotubes
Models
Software is available for free on the internet which allows you to model various nanotube structures. A few in particular are:
http://www.photon.t.u-tokyo.ac.jp/~maruyama/wrapping3/wrapping.html
http://www.ugr.es/~gmdm/contub.htm
Some Properties of Carbon Nanotubes
Carbon nanotubes have a tensile strength of 45 billion Pascals (22 times greater than high strength steel alloys). A Young's modulus of 1 to 1.8 TPa has been measured (20% better than graphite fibers). The current capacity is estimated to be 1 billion amps/cm2 (1,000 times greater than copper wire).
Additional References
- Iijima, S., Ichihashi, T., Single-shell Carbon Nanotubes of 1-nm Diameter, Letters to Nature, 1993
- Endo, M., Iijima, S., and Dresselhaus, M.S., Carbon Nanotubes, 1996
- http://en.wikipedia.org/wiki/Carbon_nanotube
- http://www.pa.msu.edu/cmp/csc/ntproperties/
- Wang, Z. L., Characterization of Nanophase Materials, 2000
- Baughman, R., et al., Carbon Nanotubes—the Route Toward Applications, Science, 297, 787, 2002
- Gogotsi, Y., Carbon Nanomaterials, 2006
- Dai, L., Carbon Nanotechnology, 2006
- O'Connell, M., Carbon Nanotubes - Properties and Applications, 2006
There is a lot of background information that is necessary to understand the various properties of nanotubes. This includes the principles of Quantum Mechanics, Materials Science and much more. I will attempt to give a brief overview of some of the ideas necessary to understand the rest of the material I present, but more detailed sources for this information can be found in the Additional References section below.
Wave Particle Duality
One of the fundamental concepts of quantum mechanics is wave particle duality. This is the idea that an object such as a phonon, a photon, or an electron, can be thought of as both a wave and a particle. This is difficult to conceptualize largely because we have difficultly thinking of something with both a position and a wavelength. This is put into mathematical terms by Heisenberg's Uncertainty Principle which states that the standard deviation of the position times the standard deviation of the momentum (related to wavelength by de Broglie's formula $\lambda = \frac{\hbar}{p}$) must be greater or equal to a constant $\sigma_x \sigma_p \geq \frac{\hbar}{2}$ When we think of a particle, we naturally assign it a specific location and it is for this reason that we then cannot think of it as having a wavelength. Instead, it is better to think of a piece of a wave, one that doesn't have a very well defined wavelength, but also doesn't have a specific location.
Phonon
A phonon is just a wave moving through a crystal as a vibration. If you consider a system of masses connected by springs as shown below, a fictitious thermal force would correspond to displacing one of the masses. This would in turn cause surrounding masses to be displaced and would propagate as a wave throughout the system.
The potential between the atoms, and thus the propagation of the phonons, will vary depending on the properties of the atoms. For carbon atoms a good model that can be used is the Brenner Potential. This potential is derived experimentally, specifically for hydrocarbons and is based on a many body potential using Tersoff's covalent bonding formalism.
Carbon Nanotube Materials Science
There are various effects that can occur within a crystal structure that affect the thermal properties. Two such things are grain boundaries and dislocations. Grain boundaries are the result of the formation of a crystal from multiple "seeds" as shown below
A dislocation is an error within the crystal lattice. One example of a dislocation is the edge dislocation which is the placement of an "extra set" of atoms within the lattice as shown below.
As was discussed of all Carbon-based materials, the potential for very close, tight bonds is high. This has the effect of reducing the possibility for defects such as edge dislocations in the carbon nanotube structure. Also, because carbon nanotubes often form from one "seed" it is possible to select nanotubes that have few or even no grain boundaries.
One other important Materials concept is the definition of the Brillouin zone. This is the area around an atom in which the solution to the wave equation is completely characterized. It corresponds to the K-space of the crystal as is physically the volume whose sides bisect the vector from one atom to the next as shown below.
The physical effect of this zone is that it limits the possible phonons that can exist. A phonon cannot physically be created that is too large to "fit" within the Brillouin zone.
Phonon Scattering
There are two primary types of phonon scattering: Normal Process Scattering and Umklapp Scattering.
Normal Process Scattering involves the collision of two phonons which simply produce a third resultant phonon within the Brillouin zone as shown below. These processes conserve total momentum and cause no direct thermal resistance.
Umklapp Scattering is the result of two phonons colliding to produce a third resultant phonon that is "too large" for the Brillioun zone. In other words, the wavevector K of the resultant phonon is not within the K-space of the Brillouin zone as shown below. When this occurs, the produced phonon is offset by another vector G, which has the magnitude of one side of the Brillouin zone, and is therefore in a completely different direction. Umklapp scattering is the source of phonon thermal resistance.
Mean Free Path
The mean free path of a particle is defined as the average distance the particle can travel before colliding with another particle. Matthieseen's Rule states that the mean free path can be defined as $\frac{1}{l} = \frac{1}{l_{defect}} + \frac{1}{l_{boundary}} + \frac{1}{l_{phonon}}$ where the defect path corresponds to phonon-defect collisions, the boundary path corresponds to phonons colliding with the ends of the nanotube and so is just the length of the nanotube, and the phonon path relates to phonon-phonon collisions such as those described above. As most nanotubes have a large aspect ratio (their length is much greater than their diameter), the boundary mean free path is large and thus is often neglected. As was discussed above, the defect concentration of nanotubes is often very low so the defect mean free path will also be large. As a result, the limit to the overall mean free path is often a result of phonon-phonon scattering and in particular Umklapp scattering. These also, will be limited for small carbon nanotubes because of reasons that are detailed next.
Density of States
The density of states $D \left( \epsilon \right)$ is a function which describes the number of phonon states between energy $\epsilon$ and $\epsilon + d \epsilon$. Consider once again, the system of masses and springs:
In such a configuration, it is possible to think of any number of waves propagating through the system. There are no external conditions that must be satisfied by these waves. However, if you now consider the system of masses and springs as it would exist in a carbon nanotube, an interesting effect becomes apparent:
Now, as can be seen, there is an imposed boundary condition in this system. If a wave propagates through the loop starting at atom A shown above, when it goes around the loop and comes back to atom A, the wave must be defined such that atom A has a certain position. This is because atom A is both the beginning of the wave and the end of the wave. If this imposition is not made, it would be assuming that atom A could have two different positions at the same time, which is obviously an impossibility. The effect that this boundary condition has is that only certain waves (or states) are possible in the system. Thus, the density of states becomes quantized. This effect is only seen in a 1D system. If you compare it to the density of states for a 2D graphene sheet and a 3D graphite crystal, you can see that the spikes, or van Hove singularities, only exist in the 1D nanotube.
The overall density of states for a 2D graphene sheet is higher than that for a nanotube or a 3D graphite crystal because the allowable waves in a sheet contain those of the sheet oscillating vertically as well, which isn't physically possible for the chunk of graphite or a tube. The distance between the van Hove singularities shown above is defined as $E_{sub} \approx \frac{ \hbar v}{R}$ where $R$ is the nanotube radius. Thus, as $R$ increases, $E_{sub}$ will decrease until the singularities merge and cease to exist. This corresponds to the radius of the nanotube becoming large enough that the nanotube no longer exhibits 1D behavior.
The effect that this quantized density of states has on the thermal properties relates directly to the Umklapp scattering. When two phonons collide and a third is produced, if the density of states is quantized it means that only certain phonons are physically allowable. Thus, Umklapp scattering is limited in a 1D system. This in turn implies that the phonon mean free path is large.
Additional References
- Hill, T., An Introduction to Statistical Thermodynamics, Dover Publications, Inc., New York, 1986
- Tien, C., Majumdar, A., Gerner, F., Microscale Energy Transport, Taylor and Francis, 1998
- Griffiths, D., Introduction to Quantum Mechanics, Prentice Hall, New Jersey, 2005.
The initial predicted value for the thermal conductivity of carbon nanotubes was $\sim 6000 \frac{W}{mK}$ at 300K. As can be seen below, this corresponds to a thermal conductivity equivalent to high-end diamond
Heat Transfer
Classical heat transfer is derived from kinetic theory, which is based on the idea that macroscale thermodynamic properties such as temperature and pressure are the result of the motion of molecules. It is from these principles that Fourier's Law is developed where the heat flux q is defined as $q = k\nabla T$ where k is the thermal conductivity $k = \frac{Cvl}{3}$, $C$ is the specific heat, $v$ is the velocity, and $l$ is the mean free path.
However, kinetic theory assumes that for a substance there exists a local thermodynamic equilibrium. When considering length scales in the nanometer range, this is not a valid assumption. So instead, a common method is to develop heat transfer properties from the Boltzmann Transport Equation (BTE):
$\frac{\partial f}{\partial t} + v \cdot \nabla f + F \cdot \frac{\partial f}{\partial p} = \left( \frac{\partial f}{\partial t} \right)_{scat}$
where $\frac{\partial f}{\partial t}$ is the change over time of the statistical distribution function of the ensemble of particles, $p$ is the momentum of the particle, $v$ is the velocity of the particle, $F$ is any external forces acting on the particles, and $\left( \frac{\partial f}{\partial t} \right)_{scat}$ represents the rate of change of the distribution due to collisions and scattering.
The Boltzmann Transport Equation is often linearized under the assumption that collisions and scattering will restore the distribution back to some equilibrium with an exponential decay behavior $f - f_0 \approx e^{\frac{-t}{\tau}}$ where $f_0$ represents the equilibrium distribution such as a Maxwell-Boltzmann distribution for a gas, a Fermi-Dirac distribution for electrons, or a Bose-Einstein distribution for photons and phonons. The effect that this has on the BTE is that $\left( \frac{\partial f}{\partial t} \right)_{scat} = \frac{\left( f_0 - f \right)}{\tau}$ where $\tau$ is the relaxation time, related to the mean free path by $\tau = \frac{l}{v}$
The BTE becomes $\frac{\partial f}{\partial t} + v \cdot \nabla f + F \cdot \frac{\partial f}{\partial p} = \frac{\left( f_0 - f \right)}{\tau}$
We can then define the heat flux as $q = \sum_p v \cdot f \cdot \epsilon$ which can be transformed into an integral using the density of states to $q = \int v \cdot f \cdot \epsilon \cdot D \left( \epsilon \right) d\epsilon$
We can also write an expression for the phonon energy density as $E_{ph} = \sum_p \int \left( f_0 + \frac{1}{2} \right) \cdot \hbar \cdot \omega \cdot D\left(\omega\right)d\omega$
The specific heat is defined as $C = \frac{dE}{dT}$ which is then $C = \sum_p \int \frac{df_0}{dT} \cdot \hbar \cdot \omega \cdot D\left(\omega\right)d\omega$
To reduce the complication of this equation, we can use an approximation. If we consider very long wavelength (low frequency) vibrations compared to the interatomic spacing a ($\lambda >> a$) then the wave does not "see" the individual atoms, but instead "sees" a continuum. The Debye approximation assumes that this is valid for all wavelengths, rather than just large ones as shown below:
Based on this approximation the specific heat can be written as $C = 3Nk \left( 4D \left( u \right) - \frac{3u}{e^u - 1} \right)$ where $u = \frac{\Theta_D}{T}$ where $\Theta_D = \frac{\hbar v_s}{k}$ is the Debye Temperature.
The specific heat for a 3-dimensional system then displays the following behavoir:
$C \rightarrow \frac{12Nk \pi^4}{5} \left( \frac{T}{\Theta_D} \right)^3$ as $T \rightarrow 0$
$C \rightarrow 3Nk$ as $T \rightarrow \infty$
Specific Heat for Carbon Nanotubes
Technically, the specific heat is defined as $C = C_e + C_{ph}$ where $C_e$ is the specific heat contribution from electrons and $C_{ph}$ is the specific heat contribution from phonons. However, if we consider the form of each of these for semiconducting nanotubes we see that $C_{ph} = \frac{D*\pi^{\frac{D}{2}}\Omega\lambda k_B^{D+1} T^D}{\left( 2\pi \right)^D \left( \frac{D}{2} \right)! \hbar^D v^D} \int_0^\infty \frac{x^{D+1}e^x}{\left( e^x - 1 \right)^2}dx$ where $D$ is the dimension of the system and $\Omega$ is the volume. Similarly, $C_e = \frac{2A k_B^3 T^2}{\pi \hbar^2 v_F^2} \int_0^\infty \frac{x^{3}e^x}{\left( e^x + 1 \right)^2}dx$ which yields $\frac{C_{ph}}{C_e} \approx \left( \frac{v_F}{v} \right)^2 \approx 10^4$. The same can be done for metallic tubes yielding $\frac{C_{ph}}{C_e} \approx \frac{v_F}{v} \approx 10^2$. Thus, regardless of the structure of the tube, the contribution from electrons to the specific heat is negligible.
So, for a 1D system, $C = C_{ph} = 3.292 \frac{3Lk_B^2T}{\pi \hbar v}$ for $T << \Theta_D$
As can be seen, in the 1D case low temperature case, specific heat and T are linearly related. This is very different for the bulk case where $C \propto T^3$. This linear relationship can therefore be looked for experimentally to verify the predicted 1D behavior or carbon nanotubes. In the Modeling and Experiments Sections, this effect is further discussed.
Mean Free Path and High Thermal Conductivity
The thermal conductivity is directly related to the mean free path. As discussed in the Background Information Section, carbon nanotubes have a relatively large mean free path because of their large aspect ratio, their low defect concentration, and their quasi-1D nature, which reduces phonon scattering. As a result, the thermal conductivity of carbon nanotubes is high. See the Modeling and the Experiments Sections for the current estimated and measured values of thermal conductivity.
Additional References
- Debye Model: http://en.wikipedia.org/wiki/Debye_model
- Hill, T., An Introduction to Statistical Thermodynamics, Dover Publications, Inc., New York, 1986
- Dresselhaus, M.S., Eklund, P.C., Phonons in Carbon Nanotubes, Advances in Physics Vol. 49, No. 6, p. 705-814, 2000
- Dresselhaus, M.S., Dresselhaus, G., Jorio, A., Unusual Properties and Structure of Carbon Nanotubes, Annual Reviews, 2004
Two excellent sources for much of the information presented here are the websites for Dr. Li Shi's Nanoscale Thermal Science Course and Dr. Gang Chen's Nanoscale Energy Transfer Course
http://www.me.utexas.edu/~lishi/ME381R_MTF.html
http://ocw.mit.edu/OcwWeb/Mechanical-Engineering/2-57Fall-2004/CourseHome/
In this section I present very brief descriptions of the methods and results of a few different papers published on the computational modeling of carbon nanotubes. Take note of the dates of these papers in order to see how the methods have developed. Be aware though that there are thousands of papers published on this topic using many different methods.
Most of these models assume that a carbon nanotube can be modeled as a 1 atom thick graphene sheet that is seamlessly rolled into a tube. They assume that there are no grain boundaries and no defects unless specifically noted. Also, most ignore the effect of tube length and assume that Umklapp scattering will be the dominant limitation to the mean free path.
Cohen calculated the specific heat for a 1D system as it was presented earlier.
$C = C_{ph} = 3.292 \frac{3Lk_B^2T}{\pi \hbar v}$ for $T << \Theta_D$
He discusses the fact that a linear relationship between C and T is valid for small R and low T. Putting this into numbers, if T = 300K then R must be 2.5A and if T = 7.5K, R must be 100A. As nanotubes with radii of 10A can be produced, this linear behavior is attainable. Cohen also describes the dominant effect of the phonon contribution to the specific heat over the electron contribution as was described previously.
In 2000, Tomanek presents the predicted thermal conductivity of a carbon nanotube using Green-Kubo equilibrium and non-equlibrium molecular dynamics. This method assumes the existence of a fictitious thermal force with drives the heat flow through the tube. From this method, Tomanek predicts a thermal conductivity of $\sim 6000 \frac{W}{mK}$ at 300K.
Also in 2000, Goddard used a fluctuation-dissipation relation developed from linear response theory along with the Brenner Potential to analyze thermal conductivity. It is noted that this method is developed from first principles and thus does not suffer from some of the erroneous dynamics models used by others. Goddard verifies the prediction that the phonon boundary scattering will be dominated by the Umklapp scattering for long enough tube lengths and estimates a thermal conductivity for long enough tube lengths of $\sim 2900 \frac{W}{mK}$.
He then considers the thermal conductivity as a function of defect concentration.
In 2000 and 2002, Hone presents work that considers the bonding between nanotubes in a carbon nanotube rope and looks at the effect this has on the 1D nature of carbon nanotubes , [[footnote Hone, J., et al., Thermal Properties of Carbon Nanotubes and Nanotube-based Materials, Applied Physics A, 74, p. 339-343, 2002 [ He discusses that for weak intertube bonding where the perpendicular energy transfer ([[$ E_{\perp} $]) is small, a 1D system describes the nanotube rope well. However, for stronger bonding, the system never displays 1D behavior.
Hone then models the difference of a isolated SWNT and a SWNT rope in which he enforces strong coupling. The dots which are also plotted are experimental data points showing that in reality, there is weak coupling between nanotubes.
Finally, Hone also points out that there is a discrepancy found when comparing the experimental specific heat (black dots) with that predicted using only the acoustic phonon modes (blue), which is the common practice. However, when optical modes (red) are added to the model, the predicted values (black line) and the data seem to be well-aligned.
In 2005, Broido published his results from a ballistic conductance study. Broido describes that high thermal conductivities can be a result of either high ballistic conductance over short lengths or low ballistic conductance over long lengths. He sets a theoretical limit to the possible conductance of a carbon nanotube. Previous results that have violated this limit, such as one describing a 100K thermal conductivity of $\sim 10000 \frac{W}{mK}$ are likely due to the erroneous use of classical molecular dynamic models. Broido points out that such a high thermal conductivity would require an unreasonably long ballistic length.
Chen, in 2007, takes a deeper look into the idea proposed by Hone that the use of phonon optical modes play a large role in the thermal conductivity of carbon nanotubes. He notes that this inclusion adds large amounts of complexity to the models. The following plot shows the phonon dispersion relations. Individual phonon branches (120 total) can be viewed by taking 20 cuts along the j axis of the plot.
Including the effect of all of these modes, Chen then models the mean free path and the thermal conductivity. As can be seen this creates a much more complicated mean free path and estimates a more modest room temperature thermal conductivity of $\sim 600 \frac{W}{mK}$.
As in the Modeling section, the following is a very brief description of the experimental results of a few different papers but be aware that there are thousands of papers published on this topic using many different methods.
Initial experiments on carbon nanotubes were limited to working on "mats" or "forests" of millions of nanotubes clumped together. It was not until new techniques were developed that individual carbon nanotubes could be investigated.
In 1999, Lu presented the results from a test on a carbon nanotube "mat" using the $3 \omega$ method. This method involves applying a voltage to two points on the mat and looking for $3 \omega$ harmonics. By measuring the frequency dependence of the amplitude and phase shift of the harmonics the thermal conductivity and specific heat, along with other properties, can be calculated. Qualitatively, these results show the existence of a linear relationship between specific heat and temperature:
Quantitatively though, a very low thermal conductivity of $\sim 25 \frac{W}{mK}$ is measured. The low value obtained from this measurement is likely due to the intertangling of the nanotubes in the mat sample.
This shows the need for the ability to measure a single nanotube. However, the devices to carry out such a measurement had to be microfabricated. This was successfully done by Kim in 2001,,,,. The device consisted of two resistors suspended on silicon nitride beams. The carbon nanotube could then be placed between the two resistors for measurements.
Using this device, Kim measured a single SWNT and a single MWNT both 14nm in diameter and 2.5 microns in length and obtained a room temperature thermal conductivity of $\sim 3000 \frac{W}{mK}$ and $\sim 2000 \frac{W}{mK}$, respectively.
Thus, after all of this there is still a discrepancy between experiments and models. It is not yet known what the actual thermal conductivity of a single carbon nanotube is. Models are continually being refined and new devices that can be used to experimentally measure the properties of nanotubes are continually be fabricated. However, in 2002, Hone did a simple check of the effectiveness of carbon nanotube thermal conductivity by loading one epoxy sample with carbon nanotubes and another with larger carbon fibers. Carbon fibers are often used today to enhance the thermal conductivity of epoxies so this test is a good way to check if the thermal conductivity of carbon nanotubes can provide the same, or better, enhancement. The results show that for equivalent loadings of carbon nanotubes, there is a very significant thermal conductivity enhancement, beyond what carbon fibers can provide.
Additional References
- Hone, J., Piskoti, C., Whitney, M., Zettl, A., Thermal Conductivity of Single-Walled Carbon Nanotubes, Vol. 59, No. 4, 1999
- Hone, J., Quantized Phonon Spectrum of Single-Wall Carbon Nanotubes, Science, 289, 1730, 2000
