Biomedical Engineering Reference
In-Depth Information
is the total energy of the same CNT structure with
N
gas adsorbates,
X
and
. This is essentially equivalent to
Eq. (7.1), and the dependence on coverage and radius are obtained by
undertaking a series of simulations and fitting to a simple function,
as described later on.
Note that more complicated definitions of
μ
is the chemical potential of
X
X
may be derived
for any gas species, or indeed any molecules used in studies of
functionalization, although additional terms would be required to
account for adsorbate-adsorbate interactions.
μ
X
7.3.3
Rehybridization Energy
Next, the energy due to the re-hybridization of the C atoms
surrounding the adsorbates will be:
=
3
__
3
E
4
N
E
(sp
) +
N
E
(7.6)
3
sp
sp
DB
DB
3
3
where
N
is the number of sp
-bonded C atoms, and
E
(sp
) is the
sp
3
sp
cohesive energy. The numerical coefficient 3/4 appears since
each sp
3
3
-hybridized C atom is bound to only three other neighboring
C atoms, not four as
3
) assumes. The fourth bond is either bound
to the gas adsorbate (so that the energy is accounted for in
E
(sp
E
) or left
X
3
dangling. Therefore,
N
is the number of these dangling sp
bonds
DB
(accounting for the under-coordination) and
E
is the energy of the
DB
3
dangling bonds. Since all sp
atoms will have either a C-X bond or a
dangling bond,
N
=
N
-
N
, and
3
DB
sp
X
=
3
__
sp
3
E
4
N
E
(
) +
E
[
N
-
N
].
(7.7)
3
3
sp
sp
DB
sp
X
7.3.4
Curvature Dependent Strain Energy
Finally, the strain energy may be describe in terms of the bending
energy of a homogeneous, isotropic elastic (graphene) sheet of area
A
and thickness
h
, such that:
+
h
/2
2
3
A
z
Ah
E
=
E
(Θ,
R
,
X
) =
dz
(7.8)
S
S
2
2
2
R
24
R
-
h
/2
where
is the mean radius of
curvature of the CNT. This approach has been found to be successful
in describing the strain energy of nanotubes before [59]. If we assume
κ
is the in-plane bending modulus and
R
