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:
I=ER,
where I is the current, E is the voltage, R is the resistance. The resistance, R, in (1) is expressed in the form
R=kLA=kLπr2,
where A is the area
of the conductor, L is its length, r 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
RC∝LA∝L+1r−2.
So for the
Ohmic bulk conduction current, we have
Ic∝Rc−1∝Lr2,
which
corresponds to
Ic=kπr2LV,
where V is the applied electric field.

Resistance for Ohmic conductor: Rc=kL/A∝L+1r−2,
where r is the radius of the conductor.

The resistance for surface convection (see Figure 2), which occurs in electrospinning
and charged flow [7, 8], scales as
Rs∝r−1.
For the surface
convection current, we have
Is∝r,
which corresponds to [4]
Is=2πrσu,
where σ is surface density of the charge.

Resistance for surface
convection: R∝r−1.

For
SWNTs and other nonmetal materials, we suggest the following scaling relation [9]:
R∝Ldr−(1+D),
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 D = 1 (infinite smoothness
of the section perimeter) and d = 1 (infinite continuity of the wall), (9) turns
out to be (2). When D = 0 and d = 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:
R(L)∝exp(LL0)
which is different
from our scaling model, (9).

3. Fractal Dimension

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

Consider the
well-known Koch curve as illustrated in Figure 3, we have M=4 and N=3, so the
fractal dimension reads Df=ln4/ln3.

Koch curve.

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 M=2, and N=3 as illustrated in Figure 4(b), resulting in
D=ln2ln(3)=1.26.

Fractal boundary of carbon nanotube.

Cross-section boundary curve

Wall boundary

Similarly,
to calculate the fractal dimension of longitudinal length of the carbon nanotube,
we have M=4, and N=3 as illustrated in Figure 4(c), yielding the
following fractal dimension:
d=ln4ln(3)=2.52.
Our prediction, therefore, reads
R=aL2.52r2.26,
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 R=0 when L=0. But in
Sundqvist et al.’s experiment, we found that R(0)≈50 kΩ;
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.

Sundqvist et al.’s experiment [10]. Resistance (kΩ) versus length (μm) for SWNTs.

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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