Nanotechnology’s Wonder Material: Synthesis of Carbon Nanotubes - Hierarchical Nanostructures for Energy Devices - page 43

where r e : pore radius, E v : porosity, t: tortuosity, T: temperature, and
M: molecular mass. The above two diffusion mechanisms can be combined
into an effective diffusion coecient (D e ):

1
D e

¼ 1
D g

þ 1
D K

d n 3 r 4 n g | 6

(3 : 6)

Xiang et al. evaluated possible growth deceleration by a VACNT diffusion
barrier. 58 Based on a quantitative analysis with the reported diffusion and
surface reaction parameters, they suggested that the growth deceleration
could be caused by the aforementioned diffusional barrier. However, the es-
timated surface reaction parameters can be significantly modified depending
on possible prior gas-phase reactions and the identity of the corresponding
active carbon precursors. Therefore, a more thorough analysis seems to be
necessary for the rate-limiting effect of gas-phase diffusion to be assured.

3.2.2.4 Adsorption and Surface Reactions
After undergoing a series of conversions in the gas phase, an active carbon
precursor species impinges on the surface and forms a transient adsorbate.
As typical physisorption energy is small compared to the thermal energy at a
typical growth temperature, most of the species that reside on the surface
must be chemisorbed. In the case of CNT growth, chemisorption involves
dehydrogenation of the hydrocarbon adsorbate and scission of an inter-
carbon bond to produce an atomic carbon. Carbon-to-catalyst bond for-
mation then compensates for the high cost of the reconfiguration of the
molecular structure of the adsorbate.
This adsorption-related argument can explain qualitatively the differential
reactivity of various carbon precursor species. For instance, methane (CH 4 )
is a saturated hydrocarbon that has a very stable molecular structure
(4 s-bonds), thus the only way it can interact with the surface is through
dissociative adsorption. In contrast, the p-bonds of unsaturated hydrocarbons
(ethylene and acetylene) enable adsorption with no need for dissociation.
Indeed, it has been consistently reported that methane needs a higher
temperature than ethylene or acetylene to grow CNTs by thermal CVD.
The surface coverage of adsorbates is an important parameter that can
account for the reaction rate and the adsorption rate occurring on the
catalyst surface. The Langmuir isotherm provides a fundamental model to
describe equilibrium surface coverage on a catalyst. 59 According to gas
kinetics, the molecular impingement is expressed by:

.

N A P
2pMRT

F ¼ p (3 : 7)

where N A : Avogadro's number, P: precursor partial pressure, M: molecular
mass, R: ideal gas constant, T: temperature. Then the surface coverage (y)by