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
(a)
(b) Cross-section boundary curve
(c) Wall boundary
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.