**3. Mathematical expression of SBPL (***lsp***)**

The end-to-end vector can be obtained such as eq 1 when the distribution of bending points ({*φ*}≡(*φ*1, *φ*2,...,*φ<sup>k</sup>* )) is given.

$$\mathbf{R} = \mathbf{N} \sum\_{i=1}^{k} \varphi\_i \mathbf{r}\_i \tag{1}$$

The spatial average of end-to-end distance *R* should be zero, since probability to bend to one direction is the same as that to the opposite direction. Then spatial average of square end-to-end vector is obtained as eq 2

Classification of Mass-Produced Carbon Nanotubes and Their Physico-Chemical Properties http://dx.doi.org/10.5772/52613 41

$$\left\langle \mathbf{R}^2 \right\rangle = N^2 \sum\_{i=1}^k \sum\_{j=1}^k (\rho\_i \mathbf{r\_i}) \cdot \left(\rho\_j \mathbf{r\_j}\right) = N^2 b^2 D\_b \tag{2}$$

$$D\_b \equiv \left\langle \mathbf{R}^2 \right\rangle / L^2 \equiv \sum\_{i=1}^k \rho\_i^2 \tag{3}$$

where *Db* is a bending ratio, *φ<sup>i</sup>* = *Ni* / *N* , *Ni* is the number of unit segment in i-direction seg‐ ment, *N* is the total number of unit segment, *k* =*m* + 1, *m* is the number of static bending points on a coil, and *ri* is i-direction segment vector with the length of b. The expression shown in eq 3 is significant. This indicates that we can obtain the distribution function when we have enough data. This is often called as ill-posed problem. Regularization method in applied mathematics gives us the solution for solving the problems. Equation 3 holds only if a probability of the fold-back conformation is the same as that of the straight conforma‐ tion.By using the scaling law, the coil expressed in eq 2 and 3 can be renormalized into the coil that has constant segment length, 2*l <sup>p</sup>*0. Then we can obtain eq 3 with *φ<sup>i</sup>* =2*l <sup>p</sup>*<sup>0</sup> / *L* and *k* = *L* / 2*l <sup>p</sup>*0. We can also consider a case where the bent angle (*θ*) between the *ith* and (*i+1)*th segments is a fixed small angle. The spatial average of the square end-to-end vector is ob‐ tained as following

**2. Static bending persistence length (SBPL,** *lsp***)**

40 Syntheses and Applications of Carbon Nanotubes and Their Composites

**Figure 1.** The concept of static bending persistence length of MWCNT (*l*

**3. Mathematical expression of SBPL (***lsp***)**

({*φ*}≡(*φ*1, *φ*2,...,*φ<sup>k</sup>* )) is given.

end-to-end vector is obtained as eq 2

shorter than SBPL.

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

*sp*).

**R r** <sup>=</sup> å (1)

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

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

1

The spatial average of end-to-end distance *R* should be zero, since probability to bend to one direction is the same as that to the opposite direction. Then spatial average of square

*i N* j

=

*i i*

*k*

$$\left\{\mathbf{r}\_n \cdot \mathbf{r}\_m\right\} = \mathcal{N}\left(\sum\_{i=1}^k \rho\_i^2\right) b^2 \left(\cos\theta\right)^{|n-m|}\tag{4}$$

$$
\left\langle \mathbf{R}^2 \right\rangle = \sum\_{n=1}^{k} \sum\_{m=1}^{k} \left\langle \mathbf{r}\_n \cdot \mathbf{r}\_m \right\rangle = \sum\_{n=1}^{k} \sum\_{p=-n+1}^{k-n} \left\langle \mathbf{r}\_n \cdot \mathbf{r}\_{n+p} \right\rangle \cong \sum\_{n=1}^{k} \sum\_{p=-n}^{\infty} \left\langle \mathbf{r}\_n \cdot \mathbf{r}\_{n+p} \right\rangle \tag{5}
$$

$$\sum\_{p=-\infty}^{\infty} \left\langle \mathbf{r}\_n \cdot \mathbf{r}\_{n+p} \right\rangle = N \left( \sum\_{i=1}^k \phi\_i^2 \right) b^2 \left( 1 + 2 \sum\_{p=1}^n \cos^p \theta \right) = N \left( \sum\_{i=1}^k \phi\_i^2 \right) b^2 \left( \frac{1 + \cos \theta}{1 - \cos \theta} \right) \tag{6}$$

$$\left\langle \mathbf{R}^2 \right\rangle = (N^2 b^2) (\sum\_{i=1}^k \phi\_i^2) (\frac{1 + \cos(\theta)}{1 - \cos(\theta)}) = L^2 D\_b \tag{7}$$

$$D\_b \equiv \frac{\left\langle \mathbf{R}^2 \right\rangle}{L^2} \equiv \left( \sum\_{i=1}^k \rho\_i^2 \right) \left( \frac{1 + \cos\left(\theta\right)}{1 - \cos\left(\theta\right)} \right) \tag{8}$$

**Figure 2.** Tortuous MWCNT; bent points are distributed randomly along MWCNT axis.

Equation 7 can also be renormalized into the coil that has a constant segment length, 2*l sp*. The bending ratio (*Db*) is expressed as eq 9

$$D\_b \equiv \frac{\left\langle \mathbf{R}^2 \right\rangle}{L^2} \equiv \left(\frac{2l\_{p0}}{L}\right) \left(\frac{1+\cos\left(\theta\right)}{1-\cos\left(\theta\right)}\right) = \mathcal{C}\left(\frac{2l\_{p0}}{L}\right) = \frac{2l\_{sp}}{L} \tag{9}$$

**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.

*sp L dD <sup>l</sup>*

Db

**Figure 3.** Bending ratio (*Db*) with respect to reciprocal contour length.

0.0 0.2 0.4 0.6 0.8 1.0

1 2 ln lim *<sup>b</sup>*

**1/L [nm-1]** 10-5 10-4 10-3 10-2 10-1 100 101

In this method, one need to have experimental data for *R* **<sup>2</sup>** and *L* . In order to obtain these data, one have to cut MWCNTs into pieces with various *L* . Acid cutting or mechanical cut‐ ting method may be applied to obtain pieces of MWCNT. It is worth to note that *R* **<sup>2</sup>** are Gaussian, given contour length (*L* ). That is, various end-to-end distances may be measured

for a constant *L* . This method is exact, but hard to obtain the experimental data.

**Figure 4.** Approximation method to determine SBPL; the mean radius of curvature approximate SBPL.

®¥ *d L* <sup>=</sup> (11)

http://dx.doi.org/10.5772/52613

43

Classification of Mass-Produced Carbon Nanotubes and Their Physico-Chemical Properties

lsp=10 nm lsp=50 nm lsp=100 nm lsp=500 nm lsp=1000 nm lsp=5000 nm

where *l sp* =*Cl <sup>p</sup>*<sup>0</sup> is the static bending persistence length and *C*should be a constant for a fixed bent angle. The static bending persistence length is a statistical quantity, representing the maximum straight length that is not bent by static bending. In the case of continuous curva‐ ture, a more accurate statement is that the static bending persistence length is the mean radi‐ us of curvature of the rigid random-coil due to static bending. The same quantity arising from dynamic bending instead of static bending is dynamic bending persistence length (*l p*). The dynamic bending persistence length represents the stiffness of the molecules as deter‐ mined by the effective bending modulus against thermal energy in Brownian motion. Equa‐ tion 5 is valid when *L* > >*l sp*, the coil limit. *Db* <sup>=</sup> *<sup>R</sup>* <sup>2</sup> / *<sup>L</sup>* <sup>2</sup> =1 when *<sup>L</sup>* <sup>&</sup>lt;*<sup>l</sup> sp*, the rod limit.If we know the values of end-to-end distance and contour length, the bending ratio can be ob‐ tained from the mean-squared end-to-end distance divided by the mean-squared contour length. The end-to-end distance of RRC varies with the change of bending angle. The differ‐ ence can be compromised by using an arbitrary unit segment length which is similar to the scaling of polymer chain. The mean-squared end-to-end distance by the Kratky-Porod (KP) expression is given by eq 10 when the dynamic bending persistence length (*l <sup>p</sup>*) is replaced by the static bending persistence length (*l sp*) and the twice *l sp* equals to Kuhn length.

$$\left\langle \mathbf{R}^2 \right\rangle = 2l\_{sp}L + 2l\_{sp}^2 \left( e^{-L/l\_{sp}} - 1 \right) \tag{10}$$

### **4. Measurement methods for SBPL**

*sp* 2*l*

f1

where *l*

**N1r1** 

q

42 Syntheses and Applications of Carbon Nanotubes and Their Composites

f2

2

**R**

2

**N2r2** 

The bending ratio (*Db*) is expressed as eq 9

*b*

tion 5 is valid when *L* > >*l*

the static bending persistence length (*l*

**R** 

**N3r3** 

**Figure 2.** Tortuous MWCNT; bent points are distributed randomly along MWCNT axis.

**N4r4** 

f3

Equation 7 can also be renormalized into the coil that has a constant segment length, 2*l*

( ) ( )

2 1 cos 2 2

*l l l*

*p p sp*

*sp* =*Cl <sup>p</sup>*<sup>0</sup> is the static bending persistence length and *C*should be a constant for a fixed

0 0

q

bent angle. The static bending persistence length is a statistical quantity, representing the maximum straight length that is not bent by static bending. In the case of continuous curva‐ ture, a more accurate statement is that the static bending persistence length is the mean radi‐ us of curvature of the rigid random-coil due to static bending. The same quantity arising from dynamic bending instead of static bending is dynamic bending persistence length (*l*

The dynamic bending persistence length represents the stiffness of the molecules as deter‐ mined by the effective bending modulus against thermal energy in Brownian motion. Equa‐

know the values of end-to-end distance and contour length, the bending ratio can be ob‐ tained from the mean-squared end-to-end distance divided by the mean-squared contour length. The end-to-end distance of RRC varies with the change of bending angle. The differ‐ ence can be compromised by using an arbitrary unit segment length which is similar to the scaling of polymer chain. The mean-squared end-to-end distance by the Kratky-Porod (KP)

*sp*) and the twice *l*

( ) / <sup>2</sup> 22 1 *sp L l sp sp*

expression is given by eq 10 when the dynamic bending persistence length (*l*

*sp*, the coil limit. *Db* <sup>=</sup> *<sup>R</sup>* <sup>2</sup> / *<sup>L</sup>* <sup>2</sup> =1 when *<sup>L</sup>* <sup>&</sup>lt;*<sup>l</sup>*

*L L L L* q

æ öæ ö + æ ö

1 cos

º @ ç ÷ç ÷ = = ç ÷ ç ÷ - ç ÷ è øè ø è ø

*D C*

**N5r5** 

f4 f5

*sp*.

(9)

*p*).

*sp*, the rod limit.If we

*<sup>p</sup>*) is replaced by

*sp* equals to Kuhn length.

*lL l e*- =+ - **<sup>2</sup> <sup>R</sup>** (10)

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

$$d\_{sp} = \varprojlim\_{L \to \infty} \frac{1}{2} \frac{dD\_b}{d\ln L} \tag{11}$$

**Figure 3.** Bending ratio (*Db*) with respect to reciprocal contour length.

In this method, one need to have experimental data for *R* **<sup>2</sup>** and *L* . In order to obtain these data, one have to cut MWCNTs into pieces with various *L* . Acid cutting or mechanical cut‐ ting method may be applied to obtain pieces of MWCNT. It is worth to note that *R* **<sup>2</sup>** are Gaussian, given contour length (*L* ). That is, various end-to-end distances may be measured for a constant *L* . This method is exact, but hard to obtain the experimental data.

**Figure 4.** Approximation method to determine SBPL; the mean radius of curvature approximate SBPL.

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 approx‐ imation 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 vis‐ cosity of MWCNTs can be calculated. If we apply the intrinsic viscosity model of a wormlike coil to the rigid random-coil, the following expressions are obtained,

$$
\left[\eta\right] = 2.20 \times 10^{21} \frac{\left<\text{R}^2\right>^{3/2}}{M} f \tag{12}
$$

(*Pe* =*γ*˙(2*Rh* )<sup>2</sup> / *DT* ) is over 10.It is worth noting that the static bending persistence length de‐ termined from intrinsic viscosity is consistent with that determined from 3-D SEM analysis

Classification of Mass-Produced Carbon Nanotubes and Their Physico-Chemical Properties

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 nano‐ tubes is the major ingredient for the shape and mobility, however it is also puzzling.The per‐ sistence 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 persis‐ tence 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 (*Eeff* ) against thermal energy (*kT* ) in a solution. Theoretical calculation‐

are up to several millimetersdue to their exceptionally large Young's modulus of about 1.5

rigid around room temperature in a solution. However, rippling developed on the compres‐ sive side of the tube leading to a remarkable reduction of the effective bending modulus, which is more pronounced for multiwalled carbon nanotubes (MWCNTs). Theoretical calcu‐ lationshave shown that the effective bending moduli of MWCNTs are around 0.5 *nN nm*<sup>2</sup> when the radii of curvatures are around 150 ~ 500*nm*. This indicates MWCNTs longer than 0.5 *μm*might be flexible in a solution around room temperature, since thermal energy is about 4.1 x 10-3*nN nm*. It seems not likely that van der Waals interaction between graphene layers is the only reason that makes the effective bending moduli of MWCNTs more than

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 stat‐ ic or dynamic by thermal energy.The relationship between the shape and size of RRCs has

WLCs are Gaussian, the models for the mobility of WLCs may also work for RRCs.

*<sup>p</sup>*s) of carbon nanotubes (CNTs)

http://dx.doi.org/10.5772/52613

45

*sp*s). Because both RRCs and

*<sup>p</sup>*(=32*μm*) may be

*<sup>p</sup>* s of singlewalled carbon nanotubes

in dried state.

**6. Diffusions of MWCNTs**

100 times smaller than SWCNT.

shave shown that the dynamic bending persistence lengths (*l*

been characterized by static bending persistence lengths (*l*

(SWCNTs) are between 32 and 174*μm*, indicating SWCNTs shorter than *l*

*TPa*. Real-time visualization technique revealed that *l*

$$f \equiv \left[ 1 + 0.926 \theta \left( D\_b \right)^{1/2} \right]^{-1} \tag{13}$$

$$\theta = \ln\left(\frac{2l\_{sp}}{e}\right) - 2.431 + \left(e/a\right) \tag{14}$$

where *M* is molecular weight, *e*is spacing between frictional elements along the contour, *a* =*ζ* / 3*πηs*, *ζ*is the friction factor for a single frictional element, and *η<sup>s</sup>* is the solvent viscosi‐ ty. For the non-draining limit for the random coil, *f* =1 , giving the maximum value of in‐ trinsic viscosity in the model. When we take the static bending persistence length (*l sp*) as the length of a single frictional element, the friction factor of the element in eq 14 may follow the rigid-rod model such that *ζ<sup>T</sup>* =3*πηsl sp* /(ln(*l sp* / *d*) + 0.3) for the translational motion and may be *ζ<sup>r</sup>* =*πηsl sp* <sup>3</sup> /(3(ln(*l sp* / *d*) −0.8)) for the end-over-end rotational motion. Translational-rota‐ tional coupling and hydrodynamic shielding may also be considered for the evaluation of friction factor in eq 14.In this case, we can surmise that friction factor in eq 14 is scaled with *l sp <sup>s</sup>* , where *s* is larger than unit value. We can reasonably neglect *e* / *a* in eq 14. The measure‐ ment of intrinsic viscosity assumes the deformation rate is slow enough. The intrinsic viscos‐ ity is determined by the competition of tendency of orientation toward flow direction and tendency to random orientation due to thermal motion (Brownian motion). The measure‐ ment often performed at shear rate of several hundreds reciprocal second. At this regime, CNTs may be extended to the static shape by shear force where peclet number (*Pe* =*γ*˙(2*Rh* )<sup>2</sup> / *DT* ) is over 10.It is worth noting that the static bending persistence length de‐ termined from intrinsic viscosity is consistent with that determined from 3-D SEM analysis in dried state.
