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J.H. Hafner et al.rChemical Physics Letters 296 1998 195–202
thought to be limited by the diffusion of carbon
the energy per carbon atom as a function of structure
size. For SWNTs, the caps were neglected in favor
of the vastly greater number of carbon atoms in the
side walls. The energies of carbon atoms at the
nanotube–metal interface and nanotube ends were
neglected since we are only considering the final
product energies and not nucleation or growth mech-
anisms. We also considered a graphene capsule en-
tirely surrounding the catalyst particle. For all cata-
lyst particle diameters, one expects the graphene
cylinder to be lower in energy than the capsule since
the cylinder has only simple curvature compared
with the complex curvature of the capsule. However,
the attractive interaction between the graphene cap-
sule and the metal particle will lower the energy per
atom of the capsule. We used simple formulas for
the energies of curved graphene sheets for the nan-
w x
through the catalyst particles 9 .
In the reaction of CO to produce nanotubes, the
slow carbon supply rate arises because CO decompo-
sition is a bimolecular disproportionation that in-
volves the breaking of two strong CO triple bonds:
such a reaction is expected to proceed very slowly
except at very high pressures, much greater than our
reaction pressure of ;1 atm. The catalytic decom-
position of C2 H4 proceeds quickly at 1 atm, so in
this case we slow the reaction down by limiting the
partial pressure of C2 H4 to 0.5 Torr.
Evidence that we have successfully changed the
rate-limiting step from carbon diffusion through the
catalytic particle to carbon supply to the catalytic
particle can be found in three aspects of the C2 H4
system. First, it has been found that the product mass
increase rate varies linearly with the C2 H4 partial
pressure. Second, ignoring termination, the mass
growth rate is independent of the reaction tempera-
ture from 700 to 8508C. If the reaction were limited
by diffusion of carbon through the metal, the rate
would double from 700 to 8508C assuming an Arrhe-
nius temperature dependence and the activation en-
w
x
w x
otubes 10 and large fullerenes for the capsules 11 .
An estimate for the graphene–metal interaction was
taken from an experimental measurement of the en-
w
x
ergy of the graphite–ferrite interface 12 . The result,
displayed in Fig. 5, shows that the nanotube energy
is lower than that of the capsule in a diameter range
similar to the SWNT diameters we find in our
experiments. These calculations lend support to our
hypothesis that in our experiments supply-limited
growth allows more time to anneal to the lowest-en-
ergy structure so that smaller particles produce nan-
otubes while larger particles are encapsulated. This
model could give further insight into the presence or
absence of DWNTs and multiwalled nanotubes if the
relative graphene–graphene and graphene–metal in-
teraction strengths were well known.
w
x
ergy of carbon diffusion through iron 10 . Admit-
tedly, the current experiments only measure a bulk
growth rate as opposed to the microscopic growth
rate of an individual nanotube. However, assuming
that the same number of nanotubes nucleate per unit
mass of catalyst, the two rates are proportional.
Finally, we find that the bulk growth rate of carbon
on the catalyst equals 5% of the mass of carbon in
ethylene that flows over the catalyst. Although this is
not 100% as would be expected of a supply limited
reaction, a simple model assuming laminar flow
suggests that only 5% of the ethylene molecules
strike the catalyst bed.
For experiments in which the reaction time is
short, we have observed that the SWNTs grow with
a particle at one end and closed at the other. This
supports nucleation of these nanotubes by the
w x
Changing from a diffusion-limited to a supply-
limited reaction by lowering the carbon supply to the
catalyst reduces the carbon concentration in the cata-
lyst particles as they form nanotubes. A lower carbon
concentration will likely allow the structures to form
more slowly, giving each carbon atom more time to
anneal to its lowest energetic configuration. There-
fore, the observed products should match those which
are predicted to have the lowest energy.
yarmulke mechanism, described previously 8 , in
which a hemispherical graphene cap forms on the
catalyst particles and lifts off to nucleate closed
nanotubes.
As shown in Fig. 2, both CO disproportionation
over Mo catalyst particles at 8508C and the reaction
of C2 H4 with FerMo particles at 7008C appear to
generate SWNTs that grow continuously without
termination of the growth reaction. These results
constitute the first demonstration of continuous gen-
eration of SWNTs with lengths that are, in principle,
We considered the energetics of structures that
could grow off of the catalyst particles by calculating