Chemistry Reference
In-Depth Information
Eqn (3.12) presents several interesting features. First, not surprisingly, it
predicts that the growth speed is linearly proportional to the driving force of
the CNT growth. Second, it can predict the maximum growth rate that is
theoretically achievable. For instance, when the adsorption resistance is
much smaller than the diffusion resistance by precursor activation of
plasma, and gas-phase diffusion is facilitated with a very short nanotube
forest, the maximum growth rate can be obtained as:
d
n
3
r
4
n
g
|
6
Dm
kT
n
0
oD
d
c
D
c
v
max
¼
(3
:
13)
Third, the general model predicts several kinetic regimes for the growth
process depending on the relative dominance of the denominator terms
of eqn (3.9). For instance, when R
1
cR
2
, R
3
, the equation reduces to the
Deal-Grove equation that is often used to describe diffusion-limited CNT
array growth.
71
When R
1
, R
3
{
R
2
, then the growth is primarily limited by
the adsorption process.
Note that the general model above assumes in its derivation that the
carbon coverage (y) on the catalyst is negligibly small. If this assumption
is not valid, an alternative model should be developed. Tibbetts et al.
72
proposed the following relation:
2
N
s
N
0
O
J
c
1
þ
J
c
d
c
D
c
N
s
v
g
¼
(3
:
14)
N
s
r
i
r
0
1
.
where the growth rate increases nonlinearly with respect to carbon flux,
resembling the Langmuir adsorption model.
Unfortunately, the models above do not explicitly account for GPRs in
their expression despite the critical kinetic effects on CNT growth. Recently,
In et al. proposed a simplified model to explain GPR kinetics based on their
in situ growth data.
46
Figure 3.9 shows the calculated high activation energy
that is indicative of GPRs. This high activation suggests that ethylene
pyrolysis consumes most of the driving force in their growth condition,
thereby the GPRs of ethylene are limiting CNT growth. They also showed a
straightforward relation between growth rates and partial pressure of
ethylene and hydrogen. The following model was suggested to effectively
describe the kinetics:
d C
½
dt
¼
k
1
C
2
H
4
½
k
2
H
2
½
C
½
(3
:
15)
C
½ ¼
K
C
2
H
4
½
v
g
B
1
e
k
2
t
r
H
2
½
(3
:
16)
½
H
2
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