Journal of Nanomaterials

Volume 2008, Article ID 954874, 3 pages

http://dx.doi.org/10.1155/2008/954874

## A New Resistance Formulation for Carbon Nanotubes

^{1}Key Laboratory of Science & Technology of Eco-Textile, Donghua University, Ministry of Education, Shanghai 200051, China^{2}Modern Textile Institute, Donghua University, 1882 Yan'an Xilu Road, Shanghai 200051, China

Received 1 February 2008; Accepted 5 May 2008

Academic Editor: Xuedong Bai

Copyright © 2008 Ji-Huan He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A new resistance formulation for carbon nanotubes is suggested using fractal approach. The new formulation is also valid for other nonmetal conductors including nerve fibers, conductive polymers, and molecular wires. Our theoretical prediction agrees well with experimental observation.

#### 1. Introduction

We know from Ohm’s law
that the current
flows down a voltage gradient in proportion to the resistance in the circuit.
Current is therefore expressed in the following form:
where *I* is the current, *E* is the voltage, *R* is the resistance. The resistance, *R*, in (1) is expressed in the form
where *A* is the area
of the conductor, *L* is its length, is the radius of the conductor, and *k* is the resistance
parameter.

Equation (2) is actually valid only for metal conductors where there are plenty of electrons in the conductor. The exponent, 2, in (2) can be interpreted as the fractal dimension of the section.

For nonconductors (e.g., nerve fibers [1, 2], conductive polymers [3], charged electrospun jets [4–6]), we suggested a modified resistance formulation discussed in the next section.

#### 2. Allometric Model

The resistance for Ohm conductor (see
Figure 1) scales as
So for the
Ohmic bulk conduction current, we have
which
corresponds to
where *V* is the applied electric field.

The resistance for surface convection (see Figure 2), which occurs in electrospinning and charged flow [7, 8], scales as For the surface convection current, we have which corresponds to [4] where is surface density of the charge.

For
SWNTs and other nonmetal materials, we suggest the following scaling relation [9]:
where *D* is the fractal dimension of its perimeter
of the section of the carbon nanotubes, *d* is the fractal dimension of longitudinal length. When = 1 (infinite smoothness
of the section perimeter) and = 1 (infinite continuity of the wall), (9) turns
out to be (2). When = 0 and = 1, (9)
is valid for the surface
convection current (Figure 2). Sundqvist et al. [10] found the
resistance of SWNTs does not follow what metal conductors do, and suggested the
following formulation:
which is different
from our scaling model, (9).

#### 3. Fractal Dimension

The fractal dimension is defined as [11, 12] where is the number of new units within the original unit with a new dimension, is the ratio of the original dimension to the new dimension.

Consider the well-known Koch curve as illustrated in Figure 3, we have and , so the fractal dimension reads .

For single-walled carbon nanotubes, we consider a special case of (6,6) CNTs as illustrated in Figure 4. To calculate the fractal dimension of its perimeter of the section of the carbon nanotube, we have , and as illustrated in Figure 4(b), resulting in

Similarly,
to calculate the fractal dimension of longitudinal length of the carbon nanotube,
we have , and as illustrated in Figure 4(c), yielding the
following fractal dimension:
Our prediction, therefore, reads
where *a* is a material constant, just like *k* in (2).

In order to verify our theoretical prediction, we have to reanalyze Sundqvist et al.’s experiment data [10] . It is obvious that when . But in Sundqvist et al.’s experiment, we found that k; this is the error due to the contact resistance at the tip, so the initial error (the contact resistance) is taken away from every obtained data, the modified experimental data is illustrated in Figure 5.

#### 4. Conclusion

In conclusion, the paper represents a novel attempt to characterize the relationship between the resistance and length of carbon nanotubes using fractal approach. We find our prediction agrees well with the experimental data, and the results might find some potential applications in future.

#### Acknowledgments

The work is supported by National Natural Science Foundation of China under Grand nos. 10772054 and 10572038, the 111 project under the Grand no. B07024, and by the Program for New Century Excellent Talents in University under Grand no. NCET-05-0417.

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