*2.1.2. C70 peapods*

The optimal interlayer C60-SWCNT distance is calculated around 0.30–0.33 nm, which is close to gaps commonly observed in carbon systems. For all optimized configurations, the interfullerene C60-C60 distance varies from 0.998 to 1.01 nm. This result is in good agreement with the previously reported peapod interball separation of 0.97 nm from electron-diffraction profiles [18] and 0.95 nm from HRTEM data [7]. The predicted phases of C60s inside SWCNTs have been experimentally observed by Kholoystov et al. HRTEM micrographs

**Table 1.** Optimized linear and zigzag structural parameters of the C60 molecules inside SCNTs for different nanotubes

**Tube index (n,m) Diameter (nm) C60-tube distance (nm) Angle θ (deg) C60-C60 distance (nm)**

(10,10) 1.36 0.323 180 1.003 (15,4) 1.41 0.355 180 1.003 (11,11) 1.49 0.321 164 1.006 (18,4) 1.59 0.309 150 1.003 (12,12) 1.63 0.306 145 1.008 (21,0) 1.64 0.307 143 1.008 (15,10) 1.70 0.30 134 1.006 (22,0) 1.72 0.302 132 1.006 (13,13) 1.76 0.301 127 1.007 (23,0) 1.80 0.30 121 1.003 (18,9) 1.86 0.298 112 1.003 (14,14) 1.90 0.30 108 1.006 (25,0) 1.96 0.298 99 1.006 (19,10) 2.00 0.298 90 1.008 (15,15) 2.03 0.3025 88 1.004 (17,14) 2.11 0.296 70 0.998

72 Fullerenes and Relative Materials - Properties and Applications

**Figure 2.** (a) Schematical representation of carbon peapods showing some of the parameters used for the geometrical optimization of the C60 molecules inside the nanotube (see text). (b) and (c) Schematic view of ordered phases resulting

from C60 packing in SWCNTs: (b) double helix and (c) two-molecule layers.

[19].

diameter and chirality.

Fullerenes with ellipsoidal shape-like C70 are of particular interest. Unlike the spheroidal molecules such as C60, there are several geometrically distinct orientations possible for the C70 molecule within a nanotube. Experimentally, depending on the nanotube diameter, two different orientations (with regard to the nanotube axis) of a C70 molecule encapsulated into SWCNTs are observed by Chorro et al.: the lying down orientation where the long axis of C<sup>70</sup> molecules is parallel to the nanotube long axis, and the standing up orientation where the C<sup>70</sup> long axis is perpendicular to the nanotube axis [19, 20]. The value of the nanotube diameter beyond which the change from the lying to standing orientation occurs is experimentally estimated to ∼1.42 nm. Besides, HRTEM measurements showed that there is no SWCNT containing C70 in both orientations.

These results indicate that only one type of orientation can exist for a given nanotube and confirm that the C70 alignment is associated with the geometrical parameters of the nanotube host. Theoretical studies showed that the configuration of C70s inside SWCNT depends primarily on tube diameter and does not significantly depend on tube chirality [21, 22]. In order to obtain the optimal structure of the C70 inside the nanotubes, an energy minimization procedure was performed by Fegani et al. (for more detail, see reference [21]). The authors found that the C70 molecules adopt a lying orientation for small SWCNT diameters (below 1.356 nm), whereas a standing orientation is preferred for large diameters (above 1.463 nm). Between these diameters, an intermediate tilted regime where C70 are tilted have been obtained (see **Figure 3**).

Structural parameters issued from the energy minimization are listed in **Table 3**. The optimum fullerene packing can be characterized by the inclination θ of the molecule long axis with respect to the nanotube axis: θ = 0° for lying orientation, θ = 90° for standing orientation, and 0° < θ < 90° for tilted orientation.

#### **2.2. Elements of lattice dynamics**

An extensive review on phonon dynamics is out of the scope of this chapter. The lattice dynamical properties comprise the phonon modes, which provide fundamental information regarding the interatomic interaction within the material, and direct connections can be made to macroscopic phenomena and properties. The lattice dynamics theory was established by Born in the 1920s and further developed by Debye, Einstein, and others. For a comprehensive description and discussion regarding lattice dynamic theory, readers are referred to the book by Born and Huang [23]. We restrict the presentation to vibrations within the harmonic approximation. In this case, the expression of the hamiltonianis:

$$H = \frac{1}{2} \sum\_{\forall a} M\_{\mid} \dot{\mu}\_a \text{ (i)}^2 + \mathcal{Q} \tag{1}$$

where ∅ is the potential energy of particles in interaction, while *uα*(*i*) is the displacement of

**Table 3.** Optimized structural parameters of the C70 molecules inside SCNT for different diameters and chiralities.

**SWNT index (n,m) SWNT diameter (nm)** *C***70-SWNT distance (nm) Angle** *θ* **(deg)** *C***70-***C***70 distance (nm)**

Structural and Vibrational Properties of C60 and C70 Fullerenes Encapsulating Carbon Nanotubes

(17,0) 1.330 0.307 0 1.125 (14,5) 1.335 0.309 0 1.125 (10,10) 1.356 0.320 0 1.125 (18,0) 1.409 0.307 5 1.122 (17,2) 1.415 0.305 9 1.118 (12,9) 1.428 0.307 41 1.106 (13,8) 1.437 0.309 47 1.101 (17,3) 1.463 0.323 60 1.082 (19,0) 1.487 0.329 90 1.003 (16,5) 1.487 0.329 90 1.003 (11,11) 1.491 0.331 90 1.003

<sup>2</sup> ∑*ij*, ∅(*i*, *j*) *u<sup>α</sup>*

is the static potential energy of the system (molecule or crystal). The starting point is the

√ \_\_\_\_\_ *Mi Mj*

The symmetry of the system permits to reduce the number of independent ∅*αβ*(*i*, *j*) coefficients. The dynamical matrix is the key component for the computation of accurate vibrational frequencies and normal modes and contains all the information required for vibrational analysis within the harmonic approximation. In the case of C60 and C70 peapods, the dynamical matrix is calculated by block by using the coupling between the density functional theory (C60 and C70), Saito force field (SWCNT) [24], and van der Waals potential (fullerene − SWCNT and

> \_\_ σ r ) 12 − ( \_\_ σ r ) 6

where the depth of the potential well and the finite distance at which the interparticle poten-

where ∅*αβ*(*i*, *j*) are the force constants between i and j atoms, the mass *Mi* (*Mj*

tial is zero are given by ϵ = 2.964 meV and *σ* = 0.3407 nm, respectively.

(*i*) *uβ*(*j*) (2)

http://dx.doi.org/10.5772/intechopen.71246

75

∅(*i*, *j*) (3)

] (4)

) of the *i*(*j*)*th*atom.

atom *i*. It is given by:

∅0

<sup>∅</sup> <sup>=</sup> <sup>∅</sup><sup>0</sup> <sup>+</sup> \_\_1

fullerene-fullerene interactions) type:

VLJ(r) = 4ϵ[(

calculation of the dynamical matrix D given by:

*<sup>D</sup>*(*i*, *<sup>j</sup>*) <sup>=</sup> \_\_\_\_\_ <sup>1</sup>

**Figure 3.** Orientations of a C70 inside a SWCNT: (a) lying, (b) tilted, and (c) standing.

Structural and Vibrational Properties of C60 and C70 Fullerenes Encapsulating Carbon Nanotubes http://dx.doi.org/10.5772/intechopen.71246 75


These results indicate that only one type of orientation can exist for a given nanotube and confirm that the C70 alignment is associated with the geometrical parameters of the nanotube host. Theoretical studies showed that the configuration of C70s inside SWCNT depends primarily on tube diameter and does not significantly depend on tube chirality [21, 22]. In order to obtain the optimal structure of the C70 inside the nanotubes, an energy minimization procedure was performed by Fegani et al. (for more detail, see reference [21]). The authors found that the C70 molecules adopt a lying orientation for small SWCNT diameters (below 1.356 nm), whereas a standing orientation is preferred for large diameters (above 1.463 nm). Between these diameters, an intermediate tilted regime where C70 are tilted have been obtained (see

Structural parameters issued from the energy minimization are listed in **Table 3**. The optimum fullerene packing can be characterized by the inclination θ of the molecule long axis with respect to the nanotube axis: θ = 0° for lying orientation, θ = 90° for standing orientation,

An extensive review on phonon dynamics is out of the scope of this chapter. The lattice dynamical properties comprise the phonon modes, which provide fundamental information regarding the interatomic interaction within the material, and direct connections can be made to macroscopic phenomena and properties. The lattice dynamics theory was established by Born in the 1920s and further developed by Debye, Einstein, and others. For a comprehensive description and discussion regarding lattice dynamic theory, readers are referred to the book by Born and Huang [23]. We restrict the presentation to vibrations within the harmonic

<sup>2</sup> ∑*<sup>i</sup> Mi u<sup>α</sup>*

̇ (*i*)2 + ∅ (1)

approximation. In this case, the expression of the hamiltonianis:

**Figure 3.** Orientations of a C70 inside a SWCNT: (a) lying, (b) tilted, and (c) standing.

**Figure 3**).

and 0° < θ < 90° for tilted orientation.

74 Fullerenes and Relative Materials - Properties and Applications

*H* = \_\_1

**2.2. Elements of lattice dynamics**

**Table 3.** Optimized structural parameters of the C70 molecules inside SCNT for different diameters and chiralities.

where ∅ is the potential energy of particles in interaction, while *uα*(*i*) is the displacement of atom *i*. It is given by:

$$\mathfrak{Q} = \mathfrak{Q}\_0 + \frac{1}{2} \sum\_{(i,a\emptyset)} \mathfrak{Q}\_{a\emptyset}(i,j) \,\,\mu\_a(i) \,\,\mu\_\emptyset(j) \tag{2}$$

∅0 is the static potential energy of the system (molecule or crystal). The starting point is the calculation of the dynamical matrix D given by:

$$D\_{\ast \phi}(i, j) = \frac{1}{\sqrt{M\_i M\_j}} \oslash\_{\ast \phi}(i, j) \tag{3}$$

where ∅*αβ*(*i*, *j*) are the force constants between i and j atoms, the mass *Mi* (*Mj* ) of the *i*(*j*)*th*atom. The symmetry of the system permits to reduce the number of independent ∅*αβ*(*i*, *j*) coefficients.

The dynamical matrix is the key component for the computation of accurate vibrational frequencies and normal modes and contains all the information required for vibrational analysis within the harmonic approximation. In the case of C60 and C70 peapods, the dynamical matrix is calculated by block by using the coupling between the density functional theory (C60 and C70), Saito force field (SWCNT) [24], and van der Waals potential (fullerene − SWCNT and fullerene-fullerene interactions) type:

$$\mathbf{V}\_{\perp}(\mathbf{r}) = 4\mathbf{e} \left[ \left( \frac{\mathbf{e}}{\mathbf{r}} \right)^{\parallel 2} - \left( \frac{\mathbf{e}}{\mathbf{r}} \right)^{\delta} \right] \tag{4}$$

where the depth of the potential well and the finite distance at which the interparticle potential is zero are given by ϵ = 2.964 meV and *σ* = 0.3407 nm, respectively.
