322
12 Carbon-Nanotube Reinforced Polymers
Σ
r
α (E) = ˆ
R α
Σ r
α↑ 0
0 Σ r
α↓
ˆ
R
†
α , ˆ
R α =
⎡
⎢
⎣
cos
θ α
2
sin
θ α
2
− sin
θ α
2
cos
θ α
2
⎤
⎥
⎦ ,
(12.6)
where α = L, R. The nanotube Hamiltonian, H tube , is a nearest neighbor π −orbital
tight-binding model with bond potential, V ppπ = 2.75 eV:
H tube = −V ppπ
a
†
i a j + c.c. .
(12.7)
This is another example of a tight-binding calculation that could be executed
using the NRL Slater-Koster framework. The general class of bianisotropic materials is defined in (12.8) [135]:
D(x) = 0 bi
· E(x) + (
0 /μ 0 )α bi
· B(x)
H(x) = (
0 /μ 0 )β
bi
· E(x) + (1/μ 0 )χ
bi
· B(x) .
(12.8)
The dispersive (frequency dependent) parameters, α bi
and β
bi
, in addition to
bi
and χ
bi
, are needed in VIC-3D® in order to work with bianisotropic materials.
12.7 Modeling Paramagnetic Effects in Carbon Nanotubes
Carbon nanotube reinforced polymers (cnrp) have excellent electromagnetic properties [55, 56, 126, 139], which means that one should be able to characterize
and evaluate them nondestructively in much the same way as for carbon fiber
reinforced polymers [30, 54, 70, 117]. There is a significant difference between
the two advanced composites, however, and that is that CNRPs display significant
magnetic effects that can be used to characterize them, and that are missing in
CFRPs [55, 56, 126, 139].
A single-wall carbon nanotube (SWCN) can have either a paramagnetic or
diamagnetic response to an applied magnetic field depending upon the tube’s
diameter, chirality, Fermi energy level, and the direction of the magnetic field
relative to the tube axis [126].
The more interesting magnetic effect occurs because nanotubes, and their
constituents can form with sizes smaller than the smallest ferromagnetic domain that
can occur in these materials, as suggested in Fig. 12.7 [126]. Hence, the result is that
the material behaves paramagnetically as a ‘single-domain particle’, and because
there will be a number of atoms with unpaired spins in the lattice structure, we say
that superparamagnetism results [6, 11, 16, 23, 25, 28, 30, 49, 55, 56, 67, 70–72, 74–
76, 97, 113, 116–119, 124, 126, 136, 139, pp.410–418]. The magnetic particles
12 Carbon-Nanotube Reinforced Polymers
Σ
r
α (E) = ˆ
R α
Σ r
α↑ 0
0 Σ r
α↓
ˆ
R
†
α , ˆ
R α =
⎡
⎢
⎣
cos
θ α
2
sin
θ α
2
− sin
θ α
2
cos
θ α
2
⎤
⎥
⎦ ,
(12.6)
where α = L, R. The nanotube Hamiltonian, H tube , is a nearest neighbor π −orbital
tight-binding model with bond potential, V ppπ = 2.75 eV:
H tube = −V ppπ
a
†
i a j + c.c. .
(12.7)
This is another example of a tight-binding calculation that could be executed
using the NRL Slater-Koster framework. The general class of bianisotropic materials is defined in (12.8) [135]:
D(x) = 0 bi
· E(x) + (
0 /μ 0 )α bi
· B(x)
H(x) = (
0 /μ 0 )β
bi
· E(x) + (1/μ 0 )χ
bi
· B(x) .
(12.8)
The dispersive (frequency dependent) parameters, α bi
and β
bi
, in addition to
bi
and χ
bi
, are needed in VIC-3D® in order to work with bianisotropic materials.
12.7 Modeling Paramagnetic Effects in Carbon Nanotubes
Carbon nanotube reinforced polymers (cnrp) have excellent electromagnetic properties [55, 56, 126, 139], which means that one should be able to characterize
and evaluate them nondestructively in much the same way as for carbon fiber
reinforced polymers [30, 54, 70, 117]. There is a significant difference between
the two advanced composites, however, and that is that CNRPs display significant
magnetic effects that can be used to characterize them, and that are missing in
CFRPs [55, 56, 126, 139].
A single-wall carbon nanotube (SWCN) can have either a paramagnetic or
diamagnetic response to an applied magnetic field depending upon the tube’s
diameter, chirality, Fermi energy level, and the direction of the magnetic field
relative to the tube axis [126].
The more interesting magnetic effect occurs because nanotubes, and their
constituents can form with sizes smaller than the smallest ferromagnetic domain that
can occur in these materials, as suggested in Fig. 12.7 [126]. Hence, the result is that
the material behaves paramagnetically as a ‘single-domain particle’, and because
there will be a number of atoms with unpaired spins in the lattice structure, we say
that superparamagnetism results [6, 11, 16, 23, 25, 28, 30, 49, 55, 56, 67, 70–72, 74–
76, 97, 113, 116–119, 124, 126, 136, 139, pp.410–418]. The magnetic particles