Download Three-dimensional modelling of dc arc discharges for carbon nanostructure production

Three–dimensional Modelling of dc Arc Discharges for Carbon
Nanostructure Production
E. Tam and A. B. Murphy
CSIRO Material Science and Engineering, P.O. Box 218, Lindfield NSW 2070, Australia
Abstract: Discharges in helium between carbon electrodes at atmospheric or sub–
atmospheric pressure have proved to be excellent sources of carbon
nanostructures, such as graphene and carbon nanotubes. However, the formation
mechanisms are poorly understood. A three–dimensional fluid mode of a dc arc
discharge in helium that includes the carbon electrodes and their vaporization
self–consistently has been developed; preliminary results are presented in this
paper. This model allows the determination of the concentration of carbon
species at all points in the arc. Results obtained by including and neglecting
electrode vaporization are compared.
Keywords: arc discharge, carbon nanotubes, graphene, modeling, computational
fluid dynamics
1. Introduction
2. Model
Carbon nanostructures, such as carbon nanotubes
and graphene nanoribbons, have unique properties
that have motivated many researchers to attempt
their integration into advanced new devices.
Potential applications include, but are not limited to,
drug and gene delivery, hydrogen storage and
electron field emission[1,2].
Arc discharges generally produce large volumes of
high quality nanostructures (The ratio of intensity
G–band peaks over the D–band peaks are higher
when carbon nanotubes produced by arcs are
examined using Raman spectroscopy), because the
manufacturing process occurs at very high
temperatures when compared to other methods [3,4].
However, even though arc discharges are already
used to produce nanostructures commercially for
scientific purposes, there are still challenges that
must be overcome for applications outside the
laboratory. These challenges include controlling the
environment to minimize the large temperature
gradients typically seen in an arc discharge,
optimizing the energy inputs so there the arc
produces just enough heat to ablate the electrodes
and nucleate the nanostructures and, as with all other
fabrication methods, controlling the growth of the
desired nanostructure (such as diameter and chirality
of single–walled carbon nanotubes).
Figure 1. Schematic of the system modeled.
A schematic of the system modeled in these
simulations can be seen in Fig. 1. The computation
domain is a rectangular prism. Room temperature
helium is pumped into the system at a constant rate
through the hollow graphite anode. The graphite
cathode is roughly twice the diameter of the anode.
The plasma is assumed to be incompressible and can
be approximated with local thermal equilibrium
(LTE). The heat transfer boundary conditions given
by Tanaka and Lowke for a thermionic cathode and
an anode were used [5]. In addition, the cooling
effects associated with the latent heat of vaporization
were taken into account.
The standard conservation and continuity equations,
adapted from those of conventional computational
fluid dynamics to take into account thermal plasma
phenomena, are used. These include the mass
continuity equation, charge continuity, the Navier–
Stokes equation and energy conservation [6-11].
√
(5)
In this binary gas system, the diffusive mass flux of
carbon, relative to the mass–average velocity in a
helium plasma, was determined using the combined
diffusion coefficient method [9-11]:
(
)
An equation for conservation of the carbon vapor
mass fraction is also required. This is [10]
(
)
(1)
where is the mass density of the plasma, is the
plasma velocity,
is the mass fraction of carbon in
the plasma, is mass flux of the carbon vapor,
is
the source term for the ablated carbon vapor and is
the time.
The carbon mass source term used in Eqn (1) is
approximated to be [12]
(
)
(2)
at the plasma–electrode interface and 0 elsewhere.
The variables
,
and
are the mass of a
carbon atom, the vaporization (or ablation) flux and
the deposition flux respectively.
The Hertz–Knudsen relation [12]
√
(3)
is used to determine the vaporization flux, where
is the saturation vapor pressure of carbon,
determined by Clausius–Clapeyron relation[12]
(4)
where is the latent heat of the graphite and
is
the
specific
gas
constant
for
carbon
⁄
(
).
The deposition flux is calculated using [12]
(6)
The finite difference method described by Patankar
is used to numerically solve these equations in three
dimensions [13]. These equations are solved self
consistently and the electrodes are included in the
computational
domain.
The
thermophysical
properties of the helium–carbon mixtures were
determined by Murphy [10,11]. Net radiative
emission coefficients were determined using the
methods of Cram, assuming a 1 mm absorbing
region [14].
3. Results and Discussion
Figure 2 shows the velocity streamlines and the
current density in a system with and without carbon
ablating from the electrodes. When carbon ablation
is not considered, eddies form in the plasma around
the electrodes, centered close to a local maximum of
the current density. As there is no ablation or
deposition occurring, the plasma flow must travel
parallel to the solid surfaces (i.e. both electrodes and
the chamber walls) in their immediate proximity.
The two main driving forces of the plasma flow in
this system are the magnetic pinch force and the
influx of helium through the hollow anode. This
influx determines the overall flow of the plasma.
The magnetic pinch force, on the other hand,
accelerates the plasma very strongly in the localized
regions in which the current density is high. The
magnetic pinch force contributes little to the overall
trend of the plasma flow entering through the anode
and leaving at the outlet; rather it is the cause of the
eddies that form at the electrodes and force the
plasma to move at very high local speeds.
significantly increases the electrical conductivity in
the cooler regions of the plasma, spreading the
current over a larger region.
Figure 3: The temperature distrbution of a dc arc discharge in
He gas with graphite electrodes excluding carbon vapor
evaporation from the electrodes. Scale bar is in kelvin.
Figure 2: Current density and plasma stream lines of (a) arc
with no carbon ablating from the electrodes and (b) arc with
carbon ablating and depositing on to the electrodes. The colors
represent current density on a logarithmic scale, with scale bar
to the left of each figure (units in A/cm2). Note the
discontinuities in current density and the kinks in the streamline
that appear in (b) are artifacts due to the low number of iteration
used in this particular run.
When vaporization and deposition of carbon vapor is
included, we can see some significant changes in the
plasma flow. Whereas the plasma flow was parallel
to the electrodes in their immediate vicinity, it is
now perpendicular to the surface in regions where
the electrode is hot. This shows that the large
volume of vaporized carbon disrupts the flow close
to the electrodes. In addition, the eddies that had
formed when no carbon was being ablated are
reduced in size when vaporization is taken into
account. This is most likely due to the fact that the
current density is not as high when carbon is ablated.
The addition of carbon vapors to the plasma
The temperature distribution in the plasma with no
carbon vaporization is shown in Figure 3. When no
carbon vaporization is considered, a global
temperature maximum can be seen adjacent to the
anode. The large plasma velocities and the eddies
that form under these conditions increases the effects
of convective cooling, creating very large
temperature gradients in the immediate vicinity of
the electrodes. While final results have not yet been
obtained for the case in which the vaporization of
the electrodes is included, it is expected that the
carbon vapor that is introduced into the plasma will
cool the arc region, as carbon has significantly
higher radiative emission and higher electrical
conductivity (leading to lower current densities).
The size reduction of the eddies means there is little
convective transport of heat, leading to a reduction
in the temperature gradients.
Figure 4 shows the mass fraction of carbon due to
vaporization. Clearly the carbon concentration is
greatest in the regions immediately below the anode.
Inclusion of diffusion in the calculation is expected
to smear out the minimum in carbon mass fraction
on the arc axis.
The lower maximum temperature (which is still
larger than carbon’s sublimation point), the smaller
temperature gradients and the reduction in
convective motion of the plasma are the key to the
growth
of
high
quality
nanostructures.
Nanostructures of different morphologies all
nucleate and grow at specific temperatures. In the
case of carbon, no nanostructure will nucleate at
temperature greater than ~3915 K. Without the
presence of catalysts, the temperatures at which
various useful nanostructures can be nucleated are in
the range from around 1500 to 2500 K, after which a
narrow temperature range is optimal for their growth
[15,16]. Assuming that the nanoparticles that form in
the arc travel with plasma flow, ideally the
maximum arc temperature should be just above
optimum nucleation temperatures for the required
nanostructures, and the temperature of the plasma
through which the nanoparticles travel as they grow
should stay fairly close to the optimum growth
temperature of the desired nanostructure. This will
ensure high the production of high volumes of
quality nanostructures.
the plasma from vaporization of the anode. The
decrease in the eddy size weakens the convective
cooling, which will reduce the temperature
gradients. This improves the suitability of such arcs
for the growth of carbon nanostructures. However,
there is a large amount of further work require to
tailor the arc conditions so that selective growth of
nanostructures can be achieved.
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Figure 4: Mass fraction of He in the plasma when carbon
ablation from the electrodes is included. Note that these results
were obtained with only convective mixing taken into account
(i.e. neglecting diffusion).
4. Conclusion
The effect of the inclusion of carbon mass
conservation on a dc arc discharge has been
investigated. Comparisons to results obtained for a
pure helium plasma with no carbon vaporization
show differences in the plasma flow, current
densities and temperature distribution of the plasma.
One of the strongest differences between the two
systems is the size reduction of the eddies when
ablation is included. This is a consequence of the
reduction of the current densities near the electrodes
and of the relatively large volume of carbon entering
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