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

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.

( ) ( ) <sup>2</sup> 2 2

 j

22 2

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

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‐

( ) 2 2

*n m n np n np*

2 2 2 2

æ ö æ ö æ ö æ ö + ×= + = ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ è ø - è ø è ø è ø å åå å **r r** (6)

*i b*

1 cos 1 cos

 = æ öæ ö +

q

q

*p*

11 1

2 22 2 2 1 1 cos( ) ( )( )( ) 1 cos( ) *k*

f

*N b L D*

=

2

º @ ç ÷ç ÷ ç ÷ - è øè ø <sup>å</sup>

j

1

*k*

*Nb Nb*

*k k*


**R rr** = ×= × @ × åå å å å å **r r r r** (5)

1 cos 1 2 cos

qj

cos *<sup>k</sup> n m*

 q-

**r r** å (4)

+ +

<sup>+</sup> = = - *<sup>R</sup>* <sup>å</sup> (7)

( ) ( )

q

q

1 cos

q

q

(8)

1

= æ ö × = ç ÷ ç ÷ è ø

1 1 1 1 1

= = = =- + = =-¥

*n m npn n p*

*n np i i*

*i*

*p ip i*

=-¥ = = =

2

2

*L*

**R**

*b i i*

j

¥ ¥

*D*

*k k k kn k*

*n m i i N b* j

1

j

*i*

=

*k*

*ij b*

*NbD*

= ×= åå **<sup>2</sup> R rr i j** (2)

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

is i-direction segment vector with the length of b. The expression

º @ **<sup>R</sup>** å (3)

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

41

1 1

*D L*

j

/

*b i*

*i j N*

points on a coil, and *ri*

tained as following

2

+

= =

*k k*

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

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.
