## 1. Introduction

Carbon nanotube (CNT) may be classified into single walled (SWCNT), double walled (DWCNT), and multiwalled carbon nanotube (MWCNT) according to the number of graphene layers. In some cases, bamboo-shaped multiwalled carbon nanotubes were also synthesized. Among these carbon nanotubes, multiwalled carbon nanotubes have been mass-produced in hundreds metric tons level. Many researchs on multiwalled carbon nanotubes point to an electrode, polymer composites, coating, and others. The number of graphene layers, purity, and crystallinity are the main features of multiwalled carbon nanotubes, which need to be characterized. We have proposed one important characteristic of multiwalled carbon nanotubes, the mesoscopic shape of MWCNT, of which many industrial applications may be comprised. According to our suggestion, one can determine the degree of tortuousness of MWCNT, quantitatively. (* see ref.1-5 and sections 1-6 in this chapter*)In this chapter, we will describe the mesoscopic shape factor of MWCNT in detail. Various physical properties as well as toxicity may strongly depend on the mesoscopic shape factor of MWCNT. Our suggestion has also been published as an international standard ISO/TS11888 by international organization for standardization (ISO) in 2011.

I hope readers enjoy the concepts and expressions shown in this chapter. Especially, this chapter shall be helpful to whom may want to develop a commercial application by selecting a proper CNT.

## 2. Static bending persistence length (SBPL, l s p )

If MWCNTs have no defect along their axis, their appearance would be straight to several hundred micro meter. Persistence length is the maximum straight length that is not bent by thermal energy. The persistence length of MWCNT is expected to be several hundred micro meter due to its exceptional high modulus. Static bending persistence length (SBPL) has been proposed in our earlier work to quantify the mesoscopic shape of MWCNT. SBPL is the maximum straight length that is not bent by permanent deformation. Fig. 1 shows the concept of SBPL. When a length considered is longer than SBPL, the shape of MWCNT looks tortuous. On the contrary, the shape of MWCNT looks straight as a length considered is shorter than SBPL.

If length considered is longer than SBPL, the shape of MWCNT looks tortuous. On the contrary, the shape of MWCNT looks straight as a length considered is shorter than SBPL.

## 3. Mathematical expression of SBPL (l s p )

The end-to-end vector can be obtained such as eq 1 when the distribution of bending points (

The spatial average of end-to-end distance

where * N*is the total number of unit segment,

*is the number of static bending points on a coil, and*m

*and (*ith

*th segments is a fixed small angle. The spatial average of the square end-to-end vector is obtained as following*i+1)

Equation 7 can also be renormalized into the coil that has a constant segment length,

where

## 4. Measurement methods for SBPL

The plot of eq 10 is presented in Fig. 3. Given data, the SBPL can be obtained by eq 11.

In this method, one need to have experimental data for

The mean radius of curvature approximates the SBPL. One can easily obtain the mean value of the radius of curvatures of MWCNTs from any SEM images as seen in Fig. 4. The approximation method is convenient because SEM images of as-synthesized or as-received MWCNT can be directly used. The SBPL obtained by the approximation could have an error up to 200% compared to those obtained by exact method. However, the approximated value of SBPL still has physical significant in many applications, since many applied properties depend on the order of magnitude of SBPL.

## 5. Intrinsic viscosity of MWCNTs

From the molecular weight, the contour length, and the persistence length, the intrinsic viscosity of MWCNTs can be calculated. If we apply the intrinsic viscosity model of a worm-like coil to the rigid random-coil, the following expressions are obtained,

where

## 6. Diffusions of MWCNTs

Not only the toxicological issues but also researches on novel hybrid materials or nano-scale devices points to the need for the understanding of overall shape and mobility of carbon nanotube particles in a solution or in atmosphere.The degree of flexibility of carbon nanotubes is the major ingredient for the shape and mobility, however it is also puzzling.The persistence lengths of single-walled carbon nanotubes are expected to be in the order of tens to hundreds of micrometers due to their exceptionally large modulusand to have longer persistence lengths for muliwalled nanotubes, indicating currently prepared several-micrometer long nanotubes behave like rigid rods. Elastic fluctuations of semi-rigid particles by thermal energy have been described exactly by the worm-like coil model proposed more than 50 years ago by Kratky and Porod. The model describes the stiffness of molecules by dynamic bending persistence lengths (mean radius of curvatures) which are determined by effective bending modulus (^{-3}

Both MWCNTs and SWCNTs discussed above are no more than worm-like coils (WLCs) where ensemble average of overall size (end-to-end distance) scales with the square root of molecular weight (contour length) in asymptotic limit. Our recent work has revealed that the spatial average of overall size of MWCNTs also follows the same scaling as WLCs in spite of their static bent points. We designated these MWCNTs as rigid random-coils (RRCs).The only difference between RRCs and WLCs is whether the bending points are static or dynamic by thermal energy.The relationship between the shape and size of RRCs has been characterized by static bending persistence lengths (

Translational diffusion coefficient is defined by the mobility of particle against thermal energy as Einstein relation, eq 15.

where

where

When we choose a spherical bead having diameter of

where

where

where

Equations 20 and 21 are valid for a semi-flexible rod when the contour length of rod is much longer than its persistence length such that the mean squared end-to-end distance follows random-coil scaling,

where * N*is the total number of unit segment,

*is the number of static bending points on a coil. When RRCs have semi-flexibility by thermal energy, the ensemble average of bent angle (*m

Expression for the translational diffusion coefficient of RRCs can be obtained from eqs 15, 17,18, and 24.

Similary, expression for the rotational diffusion coefficient of RRCs can be obtained as following.

where

## 7. Dynamic light scattering

The translational and rotational Brownian motions lead to fluctuation in the intensity of scattered light.The velocity of particles in Brownian motion can be directly measured by using dynamic light scattering (DLS) method, since time is correlated to obtain the intensity autocorrelation function (

where * in vacuo*). For polydisperse solutions, the electric field correlation function is given by a sum or distribution of exponentials,

where

In many cases, a single exponent obtained from an average translational diffusion coefficient value fits the decay rate of the electric field correlation function such as eq 29

where

The average decay rate (

When the incident light vertical and detector horizontal,

where

## 8. Micro rheology

The terminology of “microrheology” is used, to distinguish the technique from conventional (macro) rheology. In the microrheology, colloidal particles are used for probing the rheology of material of interest. The starting point is the Stokes-Einstein relation.

If we have measured values of translational diffusion coefficient, the viscosity of material of interest can be easily obtained by

where

This seemingly simple idea has done a great impact on various research fields, indeed. One example is the nanoparticles dispersed in a polymer melt. It is often reported that nanoparticles seems diffuse faster than expected. The origin of this phenomenon lies in the “Nano” size. The viscosity of polymer melt is well described by integral constitutive equations such as reptation model. In this model, the viscosity is determined by the stress relaxation time of polymer chain from the constraint of entanglement. When the observation time is much shorter than any relaxation time of polymer in rheometry of frequency sweep, the polymers behave like a crosslinked rubber, exhibiting a plateau modulus. The plateau modulus of polymers is determined from the entanglement lengths of polymer such as

The plateau modulus of polymer is usually reported in the order of 10 ^{6~7}Pa. Entanglement molecular weight of polymer is about 1000~2000 g/mole. The entanglement length is about 10~100 nm.The particles having comparable size to the entanglement length of a polymer would feel less frictional force than expected from the melt viscosity in macrorheology. Therefore, viscosity of polymer melt is much lower for the nanoparticles. This may lead to the faster thermal motion of nanoparticle compared to a larger particles.

## 9. Applications

In our previous works, we demonstrated that MWCNT having shorter SBPL have a certain merit in a polymer composite for electrical conductive application. When MWCNTs are needle-like, polymer composites comprised of them exhibit higher electrical conductivity compared to those comprised of tortuous MWCNTs. The situation changed drastically when the composites were molded into a specimen by injection molding machine. The needle like CNTs aligned to the flow direction, which broke the electrical conductive networks, then the composites lose the electrical conductivity. However, this problem was not observed when the composites contained tortuous MWCNTs which have a short SBPL. We also showed that the electrical percolation threshold depends on the length of MWCNT when MWCNT are needle-like. But, the electrical percolation threshold depends on the SBPL for a tortuous MWCNT. Thermal conductivity and linear thermal expansivity are also strongly dependant properties on the SBPL of MWCNT. Especially, thermal shrinkable material can be fabricated as well as thermal expansive material by controlling SBPL of MWCNT.