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Perfecting the Carbon Nanotube Forest
James Harper, Robert L. Mifflin *
MAE 268
University of California San Diego, Department of Bioengineering1 and Department of Mechanical
Engineering2, La Jolla, CA 92093-0412 USA
*
Correspondence to [email protected] or [email protected]
Abstract:
Carbon nanotubes can be used to enhance materials and sensors, improving everyday life.
However, techniques for effective sorting or generating carbon nanotubes of similar physical
geometries have not been developed in current literature. The fundamental concepts of carbon
nanotubes and their sorting are examined and analyzed. Then, enhancements to current sorting
processes, namely flow fractionalization, are examined and a new method of creating carbon nanotube
forests of known chirality is proposed.
Moral Justification:
Carbon nanotubes provide a tool that can be used for the enhancement of materials and sensors
that touch on the everyday life of the individual. Improving the performance of sensors allows more
detailed information to be gathered in the area of biological systems interaction, material processing,
and insight into the basic physical properties of materials at the nanoscale. All of these research areas
are central to moving the world forward and enhancing the quality of life.
INTRODUCTION
While new concepts for applications of purified carbon nanotubes (CNTs) abound in the
literature, the actual implementation of these concepts has often been hindered or blocked by the lack
of the technology to overcome the sorting and placement of this novel one dimensional material. The
CNT is derived from several processes resulting in a sheet of graphene that forms a hollow continuous
tube. CNTs may be formed with a single wall (SWCNT), multiple walls (MWCNT) and may behave
as a metal (M-SWNT) or semiconductor (S-SWNT) with their metallic and semiconducting properties
based on the chiral vector (n,m) formed from the unit vectors of the carbon lattice (Figure 1).
Figure 1: Chiral Definition of CNTs [1]
The structure of the CNT is related the process which was used to generate tubes and while
efforts to grow CNTs of a consistent chirality are ongoing, the current state of the art produces
mixtures of tubes that differ in length, diameter, number of concentric sheets, and chirality [2]. Thus
the problem that needs to be solved is to quickly sort or generate CNTs with the same intrinsic physical
geometry.
A Proposed Solution
A solution presents itself if a page is taken from the biosciences and one observes what is
involved in generating lines of cells. A single organism is either engineered or selected and then cloned
and monitored to eliminate drift in the genetic code of the organism. These steps then reduce the
problem to selecting a set of organisms, pick one, clone the organism and make a unique line of
organisms with one genetic code. This allows the line to be used as a source for reference material and
to manage the genetic drift while being able to generate the material needed for experiments or
production. . In the arena of CNTs, the equivalent to genetic drift is contamination and defect
inclusion.
The overall process would be as follows:
1. Separate the CNTs to a subset that may contain the desired target.
2. Isolate the CNTs in a large array of dielectrophoretic capacitive electrodes (CEs)
3. Verify which CNTs match the parameters desired using Raman Spectrometry and measuring
conduction properties
4. Releasing only the desired CNTs.
5. Isolating the desired CNTs into another preparation well for cloning.
The above process shows that the perfect sorting of CNTs may not be required in order to
achieve the desired goal, if all of the above operations can be performed successfully.
Material Preparation
The actual process starts with the unbundling of the CNTs. By suspending the CNTs in a
surfactant such as sodium benzene sulfate (SBS). This is then followed by sonication to separate the
CNTs thus allowing the surfactant to surround the CNTs which keeps them from rebundling [3]. The
resulting solution is then ultra-centrifuged and the desired fractions removed for further processing. At
this stage the major bundles have been eliminated and the CNTs are ready for further separation and
sorting.
Sorting Techniques
Many techniques have been developed to attempt to separate the CNTs with the properties
desired. Sorting techniques include optical trapping, fluid flow fractionalization, ultra-centrifugation,
electrophoresis, and dielectrophoresis (DEP). All of these methods depend on properties of the CNT
including length, type (SWCNT vs MWCNT), defects, and/or external modifiers (chemical or
physical).
Optical trapping – Based on the interaction between the electric field of the laser and the dipole
of the carbon nanotubes, varying forces can be generated on the nanotubes.
Fluid flow fractionalization – The difference in terminal velocities between CNTs of different
geometries caused by a large difference in drag induced by the medium and the
orientation of the CNT as it travels through the medium is exploited.
Ultra-centrifugation – Different fractions of CNTs are separated from the solution via
centrifugation.
Electrophoresis – DC electric fields are used to move the CNT through a flow, gel or pore. The
CNTs terminal velocity and final position is determined by the drag induced by moving
through the medium.
Dielectrophoresis (DEP) – Based on non-uniform AC electric fields and the dipole formed by
the CNT, surfactant and physical geometry of the CNT, varying frequencies allow the
sorting of CNTs with different geometries.
Each of these processes allow for a partial separation of the desired CNTs from the bulk
solution. Unfortunately, when used alone, these techniques do not allow for the discrimination of
materials that may have different geometries while presenting the same selection parameters to the
sorting system. For example, CNTs of different lengths and different diameters that present the same
amount of drag to a fluid flow fractionalization have overlapping parameters and would be
indistinguishable to the sorting system. Thus, while any single sorting technique will allow for sorting
a bulk solution to a limited range of materials, multiple techniques will have to be used to further
isolate a still smaller set of CNTs, again with possible overlapping characteristics of the remaining
subset (Figure 2).
Figure 2: Plot of dielectrophoretic force as a function of diameter for nanowires of different lengths [4]
For our implementation we have chosen to use DEP as it offers the ability to take advantage of
integrating field flow fractionalization, electrorotation, and individual CNT placement and release. In
addition to standard DEP, a slight but significant modification to DEP was suggested by the team
which closely paralleled current research being performed in Heller Labs. Note that Dr. Heller has
given approval for use in this paper of this concept as patents have been recently filed. The team
concept was to pulse the dielectrophoresis using a duty cycle while reversing direction of the
asymmetrical field. This allows for finer separation of materials that have a similar Clausius–Mossotti
factor.
PRINCIPLES
Dielectrophoresis and Carbon Nanotubes:
Distinguished by Pohl, the phenomenon of dielectrophoresis (DEP) describes the force exerted
on an uncharged dielectric particle in the presence of a non-uniform electric field due to an induced
uneven charge distribution [5, 6]. The uneven charge distribution causes strongly polarized particles to
move towards regions of strong electric field, while less polarizable particles move towards regions of
weaker electric field, respectively termed positive and negative dielectrophoresis [7]. The total force
acting on a particle of net charge Q in a non-uniform electric field E is
F  QE  ( p) E
(1)
where  is the del vector operator and p is the dipole moment present on the particle, which is a
function of size of the particle, the applied electric field and the complex dielectric constants of the
particle and suspending medium [8, 9]. Because the particles undergoing dielectrophoresis are
assumed to be uncharged (Q=0), then the right hand term will dominate so that the time averaged force
will reduce to
F     m Re( K f ) E
2
(2)
where  is a factor dependent on geometry, εm is the real part of the permittivity of the suspending
medium, Re denotes the real component of value Kf, which depends on the complex permittivities of
both the particle and the medium [7, 8]. This equation also applies to charged particles if the
frequency of the field is approximately 1 kHz, where electrophoretic effects are negligible [8].
When dielectrophoresis is used to manipulate single-walled carbon nanotubes (SWNTs),
Equation 2 can be applied to describe the force exerted on them. For an elongated object with the long
axis aligned with the field, the value of Kf , a version of the Clossius – Mosati Factor, becomes
Kf 
 *p   m*

*
m
,
where  *    i


(3)
where the indices p and m refer to the particle and the medium, respectively, ε is the real permittivity, σ
is the conductivity, and ω is the angular frequency of the applied electric field. For cylindrical objects,
such as carbon nanotubes, the geometrical factor  is given by

  r 2l
(4)
6
where r is the radius of the cylinder and l is the length of the cylinder [7]. Given the above equations,
the DEP force can be calculated for a given SWNT.
As shown in the above equations, the DEP force acting on a SWNT depends on the
permittivities and conductivities of both the CNT and the medium in which it is suspended. Because
different types of SWCNTs (M-SWNTs and S-SWNTs) have different permittivities and
conductivities, different types of SWNTs experience different DEP forces under the same non-uniform
electric field. This is the driving principle behind the sorting of CNTs with dielectrophoresis: both the
magnitude and the direction of the force on a given CNT depends strongly on the properties of the
CNT. The sorting of CNTs in this manner has been demonstrated by Dimaki and Bøggild [7].
Therefore, utilizing the concepts of dielectrophoresis describes above and the experimentation
performed by Dimaki and Bøggild, the intricacies of dielectrophoretic sorting of SWNTs and what
influences it can be examined.
Because the DEP force is dependent on conductivity, as shown in Equations 2 and 3, the
conductivity of the SWNT of interest must be known in order to control it. The conductivity of
SWNTs is strongly dependent on what type of SWNT, metallic or semi-conducting, is being
considered. The value of the conductivity can be calculated by observing the resistance of the CNTs at
zero gate voltage using the relation   Gl / r 2 , where G is the measured conductance and Δr is the
thickness of the SWNT wall. Quoted from unity to infinity in the literature, the permittivity of SWNTs
also varies greatly between the two types of CNTs, but can be determined to be certain values in
practice [7, 10-12].
Another important characteristic affecting the DEP force is the frequency of the electric field
applied to the SWNTs. Dimaki and Bøggild showed that at low frequencies, when the medium has a
lower conductivity than the CNT, as is usually the case with metallic CNTs, the DEP force is high on
the SWNT. Conversely, when the medium has a relatively higher conductivity, the force on the
SWNT is lower. At higher frequencies, it was observed that the relative permittivity, not the relative
conductivity, of the SWNT and medium influences the dielectrophoretic force on the SWNT [7].
Therefore, the selection of frequency is very important in the sorting of SWNTs.
Considering the above determinations about the DEP force on SWNTs, some conclusions can
be drawn about dielectrophoretic sorting of CNTs. Because metallic and semiconducting CNTs
experience different DEP forces, direction sorting of CNTs is possible by selecting a frequency where
semiconducting tubes are slightly repelled from the electrodes, areas of high electric field (negative
dielectrophoresis), while metallic tubes are strongly attracted to the electrodes (positive
dielectrophoresis), and thereby are removed from the solution [7]. This concept has been proven in
many studies, such as the proof-of-principle experiment by Krupke et al [13] and the experiment
performed by Bachtold et al [14].
Therefore, it has been shown both in concept and in experimentation that CNTs can be sorted
using dielectrophoresis. However, additionally, Dimaki and Bøggild have shown that even if both
types of CNTs experience the same direction of DEP force, each type of CNT will have radically
different terminal velocities within a fluid flow in the presence of the same electric field [7]. This
difference in the terminal velocities of different types of CNTs could be a concept worth investigating
in order to enhance the dielectrophoretic sorting of CNTs. By maximizing the difference in terminal
velocities of different types of CNTs within a fluid flow, sorting of CNTs of different types could be
exploited more easily, allowing better and more efficient sorting of CNTs. This concept, called flow
fractionalization in literature, was investigated further by the authors.
DISCUSSION
An Analysis of Flow Fractionalization and a Potential Improvement:
It is shown in literature that sorting of CNTs with dielectrophoresis is possible due to the fact
that CNTs with different conductances and morphologies acquire different terminal velocities in a fluid
flow while under the influence of the same electric field [7]. The equation describing the velocity of a
particle in a fluid flow of velocity u and under a deterministic force Fdet is
f
 t 
 Fdet

m 

(5)
v  
 u 1  e 
 f


where m is the mass of the particle and f is the friction factor [15]. In the case of dielectrophoretic
sorting of CNTs, Fdet would be equal to the dielectrophoretic force, Fdep, on the particle, which is
described in the previous section. For times much greater than the characteristic time constant τ = f/m,
the particle will move with a terminal velocity equal to
Fdep
(6)
vT 
u
f
When considering CNTs as the particles within the fluid flow, it is desired that the terminal
velocities of the different types and morphologies be as different as possible in order to exhibit
exploitable characteristic differences within the fluid flow, which would allow better sorting. Because
the friction factor f is dependent on the geometry of the tube and its orientation within the fluid flow, it
was conceived that the difference in the terminal velocities of CNTs with different morphologies could
be maximized by controlling their orientation within the fluid flow. As described by Morgan and
Green [15], the friction factor of a randomly moving, prolate ellipsoid with length l and radius r within
a fluid of viscosity η is
6l
(7)
f 
ln( 2l / r )
This is the friction factor used by Dimaki and Bøggild in their study [7]. However, by orienting the
CNTs in the flow by an external force, such as electrorotation, as described by Pohl [6], Arnold and
Zimmerman [16], Mischel et al [17], and Edwards et al. [18], the expression describing the friction
factor changes. If oriented perpendicularly to the flow, the friction factor of the nanotubes can be
described by
16l
.
(8)
f 
2 ln( 2l / r )  1
And if the tubes were oriented parallel to the flow, then their friction factors could be described by
8l
.
(9)
f 
2 ln( 2l / r )  1
By assuming a constant dielectrophoretic force Fdep and a constant fluid flow velocity u, it is
possible to compare these three orientations for CNTs in a fluid flow by plotting the inverse of the
friction factor, a value that is directly proportional to the terminal velocity of the carbon nanotubes
within the fluid flow, as a function of the length of the tubes. From this plot, it is possible to determine
whether a certain orientation offers a larger difference of terminal velocities than the others. It is
assumed that the widths of the carbon nanotubes within the flow are similar to simplify this
comparison. The resulting plot of the inverse of the friction factor as a function of the length of the
CNTs for each orientation is shown below in Figure 3.
Figure 3: A plot of the inverse of the friction factor as a function of length for carbon nanotubes of
length from 100 nm to 2 μm and width of 2 nm oriented in three different ways within a fluid flow
As expected, the terminal velocities of CNTs aligned perpendicularly to the fluid flow are
lower for all lengths than when oriented otherwise. However, the important comparison to evaluate is
the relative differences between the terminal velocities of the longer and shorter tubes in each
orientation. A large difference in the terminal velocities of CNTs of different lengths will cause a
larger separation of the different lengths distributed in the flow, leading to better sorting. By
calculating the difference between the inverse of the friction factors for tubes of length 100 nm and 2
μm for each orientation, the relative advantage of each orientation can be qualified. These values are
calculated and displayed in Figure 4 below.
Inverse of Friction Factor (s/kg)
Difference (s/kg)
100 nm length
2 μm length
Perpendicular 4.5 × 105
51.3 × 105
46.8 × 105
Random
6.4 × 105
76.8 × 105
70.4 × 105
Parallel
8.9 × 105
103.0 × 105
94.1 × 105
Figure 4: Calculations of inverse of friction factor for 100 nm and 2 μm long carbon nanotubes, and the
differences between them in three different orientations with the fluid flow. The inverse of the friction
factor is directly proportional to the terminal velocity of particles in a fluid flow.
Orientation
As shown in Figure 4, the difference between the inverse of the friction factors of the 100 nm
and 2 μm long CNTs is maximized when oriented parallel to the flow. This implies that the maximum
difference between the terminal velocities of these lengths of CNTs will be achieved when oriented
parallel to the fluid flow. The difference in terminal velocities in the parallel orientation will exceed
the difference in the random orientation, used by Dimaki and Bøggild [7]. It is also shown that by
orienting the tubes perpendicular to the flow, the difference is at a minimum. Therefore, it is shown
that the parallel orientation of CNTs in a fluid flow will enhance sorting using flow fractionalization
above that achievable by the currently used random orientation.
However, the magnitude of the enhancement to the dielectrophoretic sorting technique brought
by this new orientation of the CNTs is difficult to calculate. The fact that the difference in the inverse
of the friction factors for the currently used random orientation is only 75% of the potential difference
achievable with the parallel orientation gives a certain qualitative conclusion to the enhancement in
sorting possible by orienting the tubes parallel to the flow.
Additionally, the application of this orientation may be difficult in practice. Because the
dielectrophoretic force subjected on the tubes within the flow produces a dipole moment and causes a
reorientation of the nanotubes within the flow, maintaining a parallel orientation of the tubes within the
flow may prove difficult. It may be necessary to alternately pulse the dielectrophoretic force and an
electrorotative force to keep the nanotubes aligned within the flow. It is also conceivable that a
dielectrophoretic system could be constructed that keeps the nanotubes in a parallel orientation by
using the dipole moment of the dielectrophoretic force itself. This setup would also make the above
analysis of flow fractionalization, which neglected any changes in the dielectrophoretic force due to
the reorientation of the CNTs, more valid, as the dielectrophoretic force in this setup would not be
changing due to the orientation of the CNTs. Even if the dielectrophoretic force was not used to orient
the CNTs in the fluid flow, the analysis above still allows for a conclusion about the qualitative
difference between the terminal velocities of CNTs of different lengths in a fluid flow in each
orientation.
Despite the potential problems with this enhancement to the flow fractionalization of CNTs, its
effects on the dielectrophoretic sorting of CNTs has been shown to be an enhancement worth
investigation.
Isolation Technique
Having undergone the sorting techniques describes above, the CNTs are ready to be
individually captured. To perform this operation DEP is again used to bring the CNTs down to an array
of probes, as describes in Reference 19 and shown in Figure 5 below.
A
Figure 5: CNTs being sorted to individual sites using DEP
The concept is based on performing DEP while changing the low field and high field positions
by reconfiguring the electrodes on a PWM basis and adding a frequency shift component. Though the
net result of this process will still have CNTs with overlapping parameters present, a smaller subset of
candidates will be ready to submit to a DEP based capture array. The capture array allows the CNTs to
be drawn to the surface and as the single CNT (and occasional bundle) lands on the contacts, the
induced DEP field is modified, keeping other CNTs from being drawn to the same landing location
[19]. The CNTs are then analyzed using Raman scattering and correlated with conduction data from
the pads to determine the chiral configuration of the CNT that has landed at that pad. The end result is
a cartridge of several million CNTs that have been identified on an individual basis and can be used (or
eliminated) based on the desired properties required for the final application.
Cloning Technique
CNTs with the desired chirality are now released and moved into the next chamber for
sonication to create shorter lengths to be used as seeds. When sonication is performed on the CNTs,
the CNTs are accelerated to speeds up to 2500 M/s. This motion induces stress at the center of the tune
which exceeds the maximum strength of the tube causing the tube to break. Depending on the energy
and amount of time, the CNTs will be cut to a terminal length Lt, as described in Figure 6 below [2].
Figure 6: Length of CNT as a function of sonication time [2].
Once the seeds have been generated, they are placed in a surface built using layer by layer
electrophoresis to provide a vertical orientation. The seeds are then used as a starting stock for
generating a uniform set of CNTs for end applications. Thus, a perfect carbon nanotube forest has
been created.
CONCLUSION
While the purification of CNTs with a specified chirality using current technology appears
problematic, we have proposed a process consisting of enhancements in sorting and isolation
technologies that results in the ability to generate CNT forests of a desired chirality. Thus, a relatively
direct and scaleable path to create carbon nanotubes with a specific chirality on demand is possible.
These technologies are still in their infancy and will require further development before this process
can be mainstreamed.
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