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The relationship between the adjacent walls of the multiwall carbon nanotube (MWCNT) has been investigated and geometrical formulations have been derived. We have provided the relative index for each wall, owing to the diameter of the MWCNT. The index is higher for the smaller inner diameter MWCNT, and smaller for the higher inner diameter MWCNT. In the second formulation, we have put an empirical relation between the ratio of number of hexagons in the adjacent walls of two different sub-lattices. One can speculate the properties of inner walls from these relations by obtaining the accurate diameter of the concentric walls of a MWCNT.

Multiwall carbon nanotubes (MWCNTs) are concentric graphene layers, rolled with nanometer size diameter [^{2} hybrid orbitals. Additionally the p bond results between adjacent layers, by van der Waals force. There are many relations among different parameters of CNT, and a pair of integers, (n,m) are the most important one, governing them. The unit basis vectors along with these integers describes most of the parameters like chiral vector, chiral angle, length and diameter of the CNT [2-4].

Depending on the nanotube chirality and diameter, a CNT can be either metallic or semiconducting. It is theoretically predicted that 1/3 of as prepared CNTs are metallic and 2/3 are semiconducting. Moreover, the band gap of the nanotube is tuned by adjusting the diameter of the nanotube. Hence electronic structure of a CNT depends only on its geometrical dimension without any doping [

Aiming to understand the properties of concentric walls of MWCNT, we here derive two relations based on their diameter only. The first relation depicts the CNT wall index, owing to the ratio of diameters of two adjacent walls of the MWCNT. In the second, it relates the ratio of no. of hexagons in two adjacent walls of different sub lattices of MWCNT.

There are various preparation processes like arc discharge, laser ablation, chemical vapor deposition (CVD), plasma enhance CVD, which are routinely used to synthesize CNTs [

In one setup, benzene (5 ml) is pyrolised (at 750˚C) along with ferrocene (50 mg) to obtain carbon and iron species in a closed quartz tube (diameter 10 mm, length 500 mm), by the following reactions.

The quartz tube which is on the resistive heater, is closed at one end containing precursor and the other end is closed by a rubber bladder (see

seed, and the active carbon species react with it to form CNTs (see ref. [8,9] for details). CNTs formed in this way deposit on the inner wall of the quartz tube. At room temperature the quartz tube is taken out and the bladder is removed to collect CNTs for characterization.

In another setup, methane is pyrolised with ECR plasma (500 W) to obtain carbon species, in a home-built ECRPECVD chamber (see

Nickel-coated Si substrates (at 500˚C) were used on which CNTS were grown. Initially, a nickel film (~40 nm) was deposited on a Si(111) substrate at room temperature by magnetron sputtering at a working pressure of 5 × 10^{−3} mbar (argon) while the base pressure of the system was 1 × 10^{−5} mbar. During CNT preparation, the working gas pressure was maintained at 6 × 10^{−4} mbar by using a mass-flow controller with a flow rate 10 sccm. Vertically aligned CNTs were grown with −200 V DC substrate bias. The growth time for CNTs was 30 min. The details of the preparation process has been describe elsewhere [9,11].

Field emission SEM (FEI SIRION, XL40) was used to study the morphology of the CNTs.

Field emission TEM (FEI TECNAI, G2 F30) was used to confirm the MWCNT. Figures 3(a) and (b) show the TEM images of the CNTs prepared by pyrolysis of benzene. It is clearly seen that the CNTs are multiwall in nature. Similarly the CNTs are multiwall in nature, which are prepared by decomposing methane in ECRPECVD setup as shown in the Figures 3(c) and (d).

Based on the observation of SEM images from both experimental setup, it is concluded that, the CNTs are grown, parallel to each other. The CNT diameters observed from both SEM and TEM images are consistent, for each experimental growth condition.

The preparation of CNTs by pyrolysis of benzene is argued to be base growth.

The synthesis of CNTs by ECRPECVD method is considered to be tip growth.

In the base growth, the catalyst particle is at the base,

through which carbon atom diffuses from the bottom of the catalyst to the top side where the nucleation of the CNTs happens. The diffused carbon atom joins with the CNTs, pushing it forward (see

Formulation 1: MWCNT can be modeled as concentric cylinders (see

(a)

(b)

_{0} is the inner radius and t is the inter wall thickness).

“N_{w}” is the N^{th} no. of nanotubes from the center, “R_{0}” is the inner radius of the MWCNT and “t” is the average interlayer separation of the MWCNT. Plotting the wall index versus the no. of walls, we obtain the graph as shown in the

Formulation 2: The no. of hexagons, N_{h} per unit cell of a chiral nanotube, is given by

where, if is not a multiple of 3d, or, if is a multiple of 3d, where d is defined as the largest common divisor of [

Let there be a certain no. of unit cells be present in the unit length of the MWCNT. Taking the ratio between the different stacking in CNTs, we obtain, where M, N are the no. of unit cells in the sub-lattice A and B respectively. P and Q are the excess no. of unit cells in the unit length of the penultimate sub-lattice of A and B respectively. If the excess no. of unit cell in the inner side wall of the A and B sub-lattice of the can be represented as, and, then we obtain. Hence the ratio of no of hexagons in the adjacent walls is

Directly taking the ratio of no of hexagons, in the different sub-lattice of MWCNT, one can obtain

In the reality both the Equations (3) and (4) represent the same quantity. Hence putting them equal we obtain

The pair of integers are the two independent parameters of a particular wall of the nanotube. The no. of hexagons in a finite length nanotube is also fixed, for a certain diameter. But the increase in no. of hexagons, moving radially outward of a particular MWCNT, provides the value of P, Q, I and J. In future, (we expect with the development of sophisticated TEM), with careful measurements, one can obtain the accurate value of the diameter nanotube. Hence for a unit length the value of n and m can be predicted. Obtaining the value of n and m, one can estimate the value of no. of unit cells in the sub lattices (i.e. the values of M, N, P, Q, I, J can be calculates), and hence the no. of hexagon in a particular wall.

The inner diameter of MWCNT is mainly governed by the size of the catalyst particle, and is treated as independent parameter for this modeling. Again in this process, we take the average distance between the adjacent walls of the CNT. However the inter wall spacing can actually range from 0.34 to 0.39 nm, and the inter wall spacing increases with decreasing CNT diameter, and this effect is more pronounced in small diameter nanotubes due to high curvature [6,18]. The interlayer separation is related to the diameter of the CNT with the following relation

where R is the inner radius of the CNT [

However factors like Peierls distortion [

In summary, we elucidate two relations between the walls of the MWCNT. In the first relation we have proposed wall index for CNT, based on the ratio of the radius of the concentric walls. The index is higher for the smaller inner diameter MWCNT and smaller for the higher inner diameter MWCNT. In the second formulation, we have put an empirical relation between the ratio of no. of hexagons in the adjacent two different sub lattices. We expect the nature of inner walls can be predicted by calculating the no. of unit cells in the sub lattices that can be obtained from the accurate diameter of the concentric walls of a MWCNT.

We are thankful to nano-centre, Indian Institute of Science for providing the facilities for characterizing the samples.