**6. Diffusions of MWCNTs**

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

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 worm-

3/2 <sup>2</sup>

<sup>=</sup> ë û (12)


*sp* / *d*) + 0.3) for the translational motion and may

(14)

*sp*) as the

*R x f <sup>M</sup>*

( ) <sup>1</sup> 1/2 1 0.926 *<sup>b</sup> f D* q

( ) <sup>2</sup> ln 2.431 / *sp*

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‐

length of a single frictional element, the friction factor of the element in eq 14 may follow the

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

*<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

trinsic viscosity in the model. When we take the static bending persistence length (*l*

*sp* /(ln(*l*

*e a*

*sp* / *d*) −0.8)) for the end-over-end rotational motion. Translational-rota‐

like coil to the rigid random-coil, the following expressions are obtained,

é ù h

q

rigid-rod model such that *ζ<sup>T</sup>* =3*πηsl*

<sup>3</sup> /(3(ln(*l*

be *ζ<sup>r</sup>* =*πηsl*

*l sp* *sp*

<sup>21</sup> 2.20 10

*l*

æ ö = -+ ç ÷ ç ÷ è ø

*e*

depend on the order of magnitude of SBPL.

44 Syntheses and Applications of Carbon Nanotubes and Their Composites

**5. Intrinsic viscosity 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 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‐ shave shown that the dynamic bending persistence lengths (*l <sup>p</sup>*s) of carbon nanotubes (CNTs) are up to several millimetersdue to their exceptionally large Young's modulus of about 1.5 *TPa*. Real-time visualization technique revealed that *l <sup>p</sup>* s of singlewalled carbon nanotubes (SWCNTs) are between 32 and 174*μm*, indicating SWCNTs shorter than *l <sup>p</sup>*(=32*μm*) may be 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 100 times smaller than SWCNT.

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 been characterized by static bending persistence lengths (*l sp*s). Because both RRCs and WLCs are Gaussian, the models for the mobility of WLCs may also work for RRCs.

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

$$D = \frac{kT}{\zeta} \tag{15}$$

( ) <sup>111</sup> <sup>1</sup> <sup>1</sup> *ij ij*

molecule, and *N* is number of frictional element, *N* = *L* /*e*. Here, we can see that mathemati‐ cal expression for the mobility of RRCs is similar to that of WLCs. Equation 17 is widely used for the estimation of translational diffusion coefficient of macromolecules. Equation 18 has been solved by Hearst and Stockmayer using Riseman-Kirkwood theory and Daniel dis‐

element *i* and *j*. Then, *r*¯*ij* must depend on the conformation of the carbon nanotubes. Hearst

( ) ( ) ( ) ( ) ( ) <sup>1</sup> 1 10 1 10 1 10

where *F* (*r*, *n*) is the unknown distribution for all *n*, *f* (*r*, *n*) is the known distribution, *x* is the contour distance of the point of interest from one end of the carbon nanotube, *n*is the contour distance from the point of interest to the frictional element *n*, and *r* is the displace‐ ment of frictional element *n* from the point of interest. Hearst and Stockmayer chose the Daniels distribution which includes a first-order correction to a Gaussian distribution as

( ) ( ) ( ) ( )

where *N* = *L* /*e*, *L* is contour length, *e*is spacing between frictional elements along the con‐

like coil can be estimated by eqs 15, 17, and 20. Rotational diffusion coefficientis expressed

<sup>2</sup> 0.253 0.159ln 2 0.387 0.160

1/2

é ù æ öæ ö æ ö ê ú <sup>=</sup> ç ÷ç ÷ ç ÷ + -+ ç ÷ç ÷ê ú ç ÷ è øè øê ú è ø ë û

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

are so-called worm-like coils (WLCs). We see that the mobility is determined solely by the

4 *r p*

where *b* =2*l*

*kT <sup>L</sup> D L*


*ij* 2 ln 2 2.431 1.843 / 2 0.138 / 2 0.305 / 2 *p pp p rNL NL NL NL*

*N l N l N l*

*r dl dn F r n rdr dl dn F r n f r n rdr dl dn f r n rdr* ¥ ¥ ¥ - - = = -+ é ù åå òòò òòò ë û òòò (19)

<sup>1</sup> <sup>1</sup> , , , , <sup>2</sup>

by the Kirkwood-Riseman theory as eq 19.

*r*

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


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

47

is the distance between element *i* and *j*,*e* is solvated diameter of

is no more than a mean value of distance between the frictional

<sup>1</sup> 1/2 1/2 <sup>1</sup>

*<sup>p</sup>*.The semi-flexible rods in this coil limit, *L* > >*l*

*p* is persistence length. The translational diffusion coefficient of worm-

( )

(20)

(21)

*p* ,

d

*ij i j ij*

*R er e*

where *rij* ≡*Rij* /*e* and *Rij*

tribution. We notice that *r*¯*ij*

and Stockmayer obtained *r*¯*ij*

*xn n <sup>x</sup> x n* d

*f* (*r*, *n*), and obtained *r*¯*ij*

*<sup>p</sup>* /*e*, and *l*

h

random-coil scaling, *R* <sup>2</sup> = *N b* <sup>2</sup>

tour, *L <sup>p</sup>* =*l*

as eq 21.

as eq 20.

2

*s p p*

*l L l*

where *k* is Boltzman constant, *T* is temperature, and 1 / *ζ* is the mobility. By analogy to mac‐ romolecule, a MWCNT with static bend points can also be considered to be made up N identical structural elements with a frictional factor *ζe* per unit element and a spacing *e* be‐ tween elements along the contour of the coil.In this case, the mobility in eq 15 may be ex‐ pressed as the sum of free-draining contribution (1 / *N ζe*) and hydrodynamic interaction contribution which is called non-draining term.

$$\frac{1}{\zeta} = \frac{1}{N\zeta\_{\epsilon}} + \frac{1}{2!N^2} \sum\_{i} \sum\_{j} \frac{1}{\zeta\_{ij}} + \cdots \tag{16}$$

where *ζij* is the frictional factor by interaction between ith and jth element and *i* ≠ *j*.

**Figure 5.** CNT made up N identical frictional element.

When we choose a spherical bead having diameter of *a* as a frictional element, the frictional factor of each element follows Stokes-Einstein relation, *ζ<sup>e</sup>* =3*aπηs* where *η<sup>s</sup>* is the viscosity of solvent. The frictional factor by interaction between ith and jth element may also follow Stokes-Einstein relation since a mean value of distance between elements *i* and *j*is small. Then, Kirkwood expression is obtained such as eq 17.

$$\frac{1}{\zeta} = \frac{1}{N3a\pi\eta\_s} \left| 1 + \frac{a}{2Ne} \sum\_{i} \sum\_{j} \left( 1 - \delta\_{ij} \right) \left\langle r\_{ij}^{-1} \right\rangle \right| \tag{17}$$

Classification of Mass-Produced Carbon Nanotubes and Their Physico-Chemical Properties http://dx.doi.org/10.5772/52613 47

$$\frac{1}{\overline{R}\_{ij}} = \frac{1}{\sigma r\_{ij}} = \left(\frac{1}{\varepsilon}\right) \sum\_{i} \sum\_{j} \left\langle \left(1 - \delta\_{ij}\right) \left\langle r\_{ij}^{-1} \right\rangle \right\rangle \tag{18}$$

where *rij* ≡*Rij* /*e* and *Rij* is the distance between element *i* and *j*,*e* is solvated diameter of molecule, and *N* is number of frictional element, *N* = *L* /*e*. Here, we can see that mathemati‐ cal expression for the mobility of RRCs is similar to that of WLCs. Equation 17 is widely used for the estimation of translational diffusion coefficient of macromolecules. Equation 18 has been solved by Hearst and Stockmayer using Riseman-Kirkwood theory and Daniel dis‐ tribution. We notice that *r*¯*ij* is no more than a mean value of distance between the frictional element *i* and *j*. Then, *r*¯*ij* must depend on the conformation of the carbon nanotubes. Hearst and Stockmayer obtained *r*¯*ij* by the Kirkwood-Riseman theory as eq 19.

Translational diffusion coefficient is defined by the mobility of particle against thermal ener‐

where *k* is Boltzman constant, *T* is temperature, and 1 / *ζ* is the mobility. By analogy to mac‐ romolecule, a MWCNT with static bend points can also be considered to be made up N identical structural elements with a frictional factor *ζe* per unit element and a spacing *e* be‐ tween elements along the contour of the coil.In this case, the mobility in eq 15 may be ex‐ pressed as the sum of free-draining contribution (1 / *N ζe*) and hydrodynamic interaction

> 2 11 1 1 2! *N <sup>e</sup> N i j ij*

is the frictional factor by interaction between ith and jth element and *i* ≠ *j*.

When we choose a spherical bead having diameter of *a* as a frictional element, the frictional factor of each element follows Stokes-Einstein relation, *ζ<sup>e</sup>* =3*aπηs* where *η<sup>s</sup>* is the viscosity of solvent. The frictional factor by interaction between ith and jth element may also follow Stokes-Einstein relation since a mean value of distance between elements *i* and *j*is small.

> ( ) 1 1 <sup>1</sup> 1 1 3 2 *ij ij s i j a*

=+ - ê ú

*N a Ne*

 ph *r*

åå (17)

d


ë û

 z

=+ + åå <sup>L</sup> (16)

<sup>=</sup> (15)

*kT <sup>D</sup>* z

gy as Einstein relation, eq 15.

where *ζij*

contribution which is called non-draining term.

46 Syntheses and Applications of Carbon Nanotubes and Their Composites

**Figure 5.** CNT made up N identical frictional element.

Then, Kirkwood expression is obtained such as eq 17.

z

zz

$$\frac{1}{2} \sum\_{x} \sum\_{n} \left\{ (1 - \delta\_{xn}) \left< r\_n^{-1} \right>\_{x} \right\} = \int\_1^N dl \int\_1^l dn \int\_0^\alpha F\left(r, n\right) r dr = \int\_1^N dl \int\_1^l dn \int\_0^\alpha \left[ F\left(r, n\right) - f\left(r, n\right) \right] r dr + \int\_1^N dl \int\_1^l dn \int\_0^\alpha f\left(r, n\right) r dr \tag{19}$$

where *F* (*r*, *n*) is the unknown distribution for all *n*, *f* (*r*, *n*) is the known distribution, *x* is the contour distance of the point of interest from one end of the carbon nanotube, *n*is the contour distance from the point of interest to the frictional element *n*, and *r* is the displace‐ ment of frictional element *n* from the point of interest. Hearst and Stockmayer chose the Daniels distribution which includes a first-order correction to a Gaussian distribution as *f* (*r*, *n*), and obtained *r*¯*ij* as eq 20.

$$\tilde{r}\_{\vec{r}\vec{\eta}} = \left[2N\left(\ln\left(2L\_p\right) - 2.431 + 1.843\left(N / 2L\_p\right)^{1/2} + 0.138\left(N / 2L\_p\right)^{-1/2} - 0.305\left(N / 2L\_p\right)^{-1}\right)\right]^{-1} \tag{20}$$

where *N* = *L* /*e*, *L* is contour length, *e*is spacing between frictional elements along the con‐ tour, *L <sup>p</sup>* =*l <sup>p</sup>* /*e*, and *l p* is persistence length. The translational diffusion coefficient of wormlike coil can be estimated by eqs 15, 17, and 20. Rotational diffusion coefficientis expressed as eq 21.

$$D\_r = \left(\frac{kT}{\eta\_s}\right) \left[\frac{2}{l\_p L^2}\right] \left[0.253 \left(\frac{L}{4l\_p}\right)^{1/2} + 0.159 \ln\left(2L\_p\right) - 0.387 + 0.160\right] \tag{21}$$

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, *R* <sup>2</sup> = *N b* <sup>2</sup> where *b* =2*l <sup>p</sup>*.The semi-flexible rods in this coil limit, *L* > >*l p* , are so-called worm-like coils (WLCs). We see that the mobility is determined solely by the average conformation of particle with a given solvent viscosity and a contour length in eqs 13 and 14. We can reasonably surmise that the diffusion coefficients of RRCs are similar to those of WLCs with a given contour length, if the values of static bending persistence lengths (*l sp*s) of RRCs are the same as those of the dynamic bending persistence lengths (*l p*s) of WLCs.When hydrodynamic shielding effect is taken into account, the diffusion coeffi‐ cients of RRCs might be slightly larger than those of WLCs due to the static bent points. The root mean-squared end-to-end distance of RRCs are given by eq 22.

$$\left\langle \mathbf{R}^2 \right\rangle = \left( N^2 b^2 \right) \left( \sum\_{i=1}^k \phi\_i^2 \right) \left( \frac{1 + \cos\left(\theta\right)}{1 - \cos\left(\theta\right)} \right) = L^2 D\_b \tag{22}$$

Similary, expression for the rotational diffusion coefficient of RRCs can be obtained as fol‐

<sup>2</sup> 0.253 0.159ln 2 0.387 0.160

1/2

**R**

**Figure 6.** Size of carbon nanotube (R) is decreased to R2 at the elevated temperature.

é ù æ öæ ö æ ö ê ú <sup>=</sup> ç ÷ç ÷ ç ÷ + -+ ç ÷ç ÷ê ú ç ÷ è øè øê ú è ø ë û

cients of MWCNTs synthesized by a CVD method. In other words, eqs 25 and 26 give us the information of the shape and size of MWCNTs if we have the measured values of diffusion

4 *<sup>r</sup> ap*

*kT <sup>L</sup> D L*

2

*l L l*

h

**7. Dynamic light scattering**

autocorrelation function (*g* (2)

solution.

electric field autocorrelation function (*g* (1)

*s ap ap*

( )

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

**R2** 

(*t*)). The intensity autocorrelation function is connected to the

(1) *gt t* ( ) exp( ) = -G (27)

(*t*)) which is given by eq 27 for a monodisperse

q

q

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 us‐ ing dynamic light scattering (DLS) method, since time is correlated to obtain the intensity

*ap* /*e*.Equations 25 and 26 are promising for the estimation of diffusion coeffi‐

(26)

49

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

lowing.

where *L ap* =*l*

coefficients.

$$D\_b = \frac{\left\{\mathbf{R}^2\right\}}{L^2} \cong \left(\sum\_{i=1}^k \sigma\_i^2\right) \left(\frac{1+\cos\left(\theta\right)}{1-\cos\left(\theta\right)}\right) \cong \left(\frac{2l\_{p0}}{L}\right) \left(\frac{1+\cos\left(\theta\right)}{1-\cos\left(\theta\right)}\right) = \mathcal{C}\left(\frac{2l\_{p0}}{L}\right) = \frac{2l\_{sp}}{L} \tag{23}$$

where *Db* is bending ratio, *l sp* is static bending persistence length, *l <sup>p</sup>*<sup>0</sup> is an arbitrary constant segment length, *θ*is static bent angle from the MWCNT axis, *φ<sup>i</sup>* = *Ni* / *N* , *Ni* is the number of unit segment in i-direction segment, *N* is the total number of unit segment, *k* =*m* + 1, *m* is the number of static bending points on a coil. When RRCs have semi-flexibility by thermal ener‐ gy, the ensemble average of bent angle (*θ*2) always becomes larger in amount of *Δθ* than the static bent angle *θ*;*θ*<sup>2</sup> =*θ* + *Δθ*. This is due to the fact that the effective bending modulus to‐ ward the bent direction is smaller than that toward the opposite direction.This indicates that the overall size of RRCs may be decreased when they are fluctuated by thermal energy. Be‐ cause frictional elements of RRCs have Gaussian distribution by the definition of RRCs, those of semi-flexible RRCs also have Gaussian distrbution. Therefore, eqs 20 and 21 are also valid for semi-flexible RRCs when the persistence length is replaced by an apparent persis‐ tence length. The apparent persistence length (*l ap*) is determined by the static bent angle (*θ*) and dynamic bent angle due to thermal energy (*Δθ*) as following.

$$l\_{ap} = l\_{sp} \left( \frac{1 + \cos\left(\theta + \Delta\theta\right)}{1 - \cos\left(\theta + \Delta\theta\right)} \right) \left( \frac{1 - \cos\left(\theta\right)}{1 + \cos\left(\theta\right)}\right) \tag{24}$$

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

$$D\_{T} = \frac{kT}{3\pi\eta\_{s}L} \left[ 1 + \ln\left(2L\_{ap}\right) - 2.431 + 1.843\left(N/2L\_{ap}\right)^{1/2} + 0.138\left(N/2L\_{ap}\right)^{-1/2} - 0.305\left(N/2L\_{ap}\right)^{-1} \right] \tag{25}$$

Similary, expression for the rotational diffusion coefficient of RRCs can be obtained as fol‐ lowing.

$$D\_r = \left(\frac{kT}{\eta\_s}\right) \left[\frac{2}{l\_{ap}L^2}\right] \left[0.253\left(\frac{L}{4l\_{ap}}\right)^{1/2} + 0.159\ln\left(2L\_{ap}\right) - 0.387 + 0.160\right] \tag{26}$$

where *L ap* =*l ap* /*e*.Equations 25 and 26 are promising for the estimation of diffusion coeffi‐ cients of MWCNTs synthesized by a CVD method. In other words, eqs 25 and 26 give us the information of the shape and size of MWCNTs if we have the measured values of diffusion coefficients.

**Figure 6.** Size of carbon nanotube (R) is decreased to R2 at the elevated temperature.

### **7. Dynamic light scattering**

average conformation of particle with a given solvent viscosity and a contour length in eqs 13 and 14. We can reasonably surmise that the diffusion coefficients of RRCs are similar to those of WLCs with a given contour length, if the values of static bending persistence

of WLCs.When hydrodynamic shielding effect is taken into account, the diffusion coeffi‐ cients of RRCs might be slightly larger than those of WLCs due to the static bent points. The

22 2 2

*N b L D*

=

1 cos 1 cos

2 0 0

( ) ( )

1 cos 1 cos

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

*L L L L*

unit segment in i-direction segment, *N* is the total number of unit segment, *k* =*m* + 1, *m* is the number of static bending points on a coil. When RRCs have semi-flexibility by thermal ener‐ gy, the ensemble average of bent angle (*θ*2) always becomes larger in amount of *Δθ* than the static bent angle *θ*;*θ*<sup>2</sup> =*θ* + *Δθ*. This is due to the fact that the effective bending modulus to‐ ward the bent direction is smaller than that toward the opposite direction.This indicates that the overall size of RRCs may be decreased when they are fluctuated by thermal energy. Be‐ cause frictional elements of RRCs have Gaussian distribution by the definition of RRCs, those of semi-flexible RRCs also have Gaussian distrbution. Therefore, eqs 20 and 21 are also valid for semi-flexible RRCs when the persistence length is replaced by an apparent persis‐

> ( ) ( )

qq

qq

1 cos 1 cos *ap sp*

1 ln 2 2.431 1.843 / 2 0.138 / 2 0.305 / 2 <sup>3</sup> *<sup>T</sup> ap ap ap ap*

*kT <sup>D</sup> <sup>L</sup> NL NL NL*

1 cos 1 cos

Expression for the translational diffusion coefficient of RRCs can be obtained from eqs 15,

( ) ( ) ( ) ( ) 1/2 1/2 <sup>1</sup>


<sup>æ</sup> + +D - öæ ö <sup>=</sup> <sup>ç</sup> ÷ç ÷ <sup>ç</sup> - +D + <sup>è</sup> øè ø

æ öæ öæ ö + + æ ö æ ö

j

1

*i*

( ) ( )

q

segment length, *θ*is static bent angle from the MWCNT axis, *φ<sup>i</sup>* = *Ni* / *N* , *Ni*

=

q

*D C*

*k*

root mean-squared end-to-end distance of RRCs are given by eq 22.

*sp*s) of RRCs are the same as those of the dynamic bending persistence lengths (*l*

( )

q

æ öæ ö <sup>+</sup> = = ç ÷ç ÷ ç ÷ - è øè ø <sup>å</sup> **<sup>2</sup> <sup>R</sup>** (22)

( ) ( )

 q

 q

*p p sp*

*l l l*

*sp* is static bending persistence length, *l <sup>p</sup>*<sup>0</sup> is an arbitrary constant

( ) ( )

 q

 q

*ap*) is determined by the static bent angle (*θ*)

q

*i b*

1 cos 2 1 cos 2 2

*p*s)

(23)

(24)

is the number of

lengths (*l*

2

**R**

where *Db* is bending ratio, *l*

17,18, and 24.

*s*

ph*L* 2

*b i i*

1

tence length. The apparent persistence length (*l*

*l l*

and dynamic bent angle due to thermal energy (*Δθ*) as following.

j

48 Syntheses and Applications of Carbon Nanotubes and Their Composites

*k*

å

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 us‐ ing dynamic light scattering (DLS) method, since time is correlated to obtain the intensity autocorrelation function (*g* (2) (*t*)). The intensity autocorrelation function is connected to the electric field autocorrelation function (*g* (1) (*t*)) which is given by eq 27 for a monodisperse solution.

$$\left| \mathbf{g}^{(1)}(t) \right| = \exp(-\Gamma t) \tag{27}$$

$$\int\_0^s G\left(\Gamma\right)d\Gamma = 1$$

where *Γ* =*Dq* <sup>2</sup> with *D*, the translational diffusion coefficient of the molecules, and *q*, the scat‐ tering vector magnitude (*q* =4*πn*sin(*θ* / 2) / *λ*0 where *n* is the solution refractive index, *θ* is the scattering angle, and *λ*<sup>0</sup> is the incident light wavelength *in vacuo*). For polydisperse solu‐ tions, the electric field correlation function is given by a sum or distribution of exponentials,

$$\left| \mathbf{g}^{(1)}(t) \right| = \int\_0^\infty G\left(\Gamma\right) \exp\left(-\Gamma t\right) d\Gamma \tag{28}$$

and 12. And two equations for DLS measurement.Three unknown shape factors of static bending persistence length, contour length, and thermal fluctuation angle can be deter‐ mined from the measured diffusion coefficients and intrinsic viscosity using eqs 25, 26, and 12.This is uniqueness of carbon nanotubes compared to macromolecules, since macromole‐

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

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51

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

*s*

If we have measured values of translational diffusion coefficient, the viscosity of material of

*T*

where *a* is the radius of spherical colloidal particle; colloidal particles have usually average diameter between 1 nm and 1000 nm. It is comparable the ISO definition of nanoparticles those have average diameter between 1 nm and 100 nm. In this sense, nanoparticles are just some kinds of colloidal particles. When we have measured value of mean-squared displace‐

*s*

p h

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 nanoparti‐ cles 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 pol‐

*kT a D*

p

<sup>=</sup> (32)

<sup>=</sup> (33)

*<sup>a</sup>* = = (34)

of material of interest. The starting point is the Stokes-Einstein relation.

6 *<sup>T</sup>*

6 *<sup>s</sup>*

( ) <sup>2</sup> <sup>6</sup> *<sup>T</sup>*

ymers is determined from the entanglement lengths of polymer such as

*kT r t Dt*

h

ment of probe particle, the following equation can be utilized.

*kT <sup>D</sup> a*ph

cules have only two unknown shape factors of persistence length and contour length.

**8. Micro rheology**

interest can be easily obtained by

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

$$K\_1 = \left\langle \Gamma \right\rangle\_{av} = \int\_0^\infty \Gamma G \left(\Gamma \right) d\Gamma \tag{29}$$

where *K*1 is the first cumulant. When a solution is dilute enough neglecting the interaction between particles, the effective diffusion coefficient can be obtained by the intensity average. The intensity of light scattered by macromolecule species *i* is often proportional to the mo‐ lecular weight (*Mi* ) times the weight concentration (*ci* ). In this case, the intensity average dif‐ fusion coefficient equals to z average diffusion coefficient.

The average decay rate (*Γ*) of the electric field autocorrelation function can be obtained by using conventional DLS.The first cumulant generally fits the data well for carbon nanotube solutions. When the incident light and detector are both vertical, *Vv*, translational diffusion are characterized by eq 30.

$$
\Gamma\_{Vv} = q^2 D\_T \tag{30}
$$

When the incident light vertical and detector horizontal, *Hv* , the diffusion of anisotropic particle are characterized by eq 31

$$
\Gamma\_{Hv} = q^2 D\_T + \Theta D\_R \tag{31}
$$

where *DT* is translational diffusion coefficient and *DR* is rotational diffusion coefficient. This equation is valid if the particle rotates many times while diffusing a distance comparable to *q* <sup>−</sup><sup>1</sup> or if there is little anisotropy in particle dimension. MWCNTs solution meets with the former case in this work. Now we have three independent mathematical model eqs 25, 26, and 12. And two equations for DLS measurement.Three unknown shape factors of static bending persistence length, contour length, and thermal fluctuation angle can be deter‐ mined from the measured diffusion coefficients and intrinsic viscosity using eqs 25, 26, and 12.This is uniqueness of carbon nanotubes compared to macromolecules, since macromole‐ cules have only two unknown shape factors of persistence length and contour length.

#### **8. Micro rheology**

where *Γ* =*Dq* <sup>2</sup>

where

lecular weight (*Mi*

*q* <sup>−</sup><sup>1</sup>

are characterized by eq 30.

particle are characterized by eq 31

with *D*, the translational diffusion coefficient of the molecules, and *q*, the scat‐

= G -G G ò (28)

=G = G G G ò (29)

*Vv T* G = *q D* (30)

<sup>2</sup> <sup>6</sup> *Hv T R* G= + *qD D* (31)

). In this case, the intensity average dif‐

tering vector magnitude (*q* =4*πn*sin(*θ* / 2) / *λ*0 where *n* is the solution refractive index, *θ* is the scattering angle, and *λ*<sup>0</sup> is the incident light wavelength *in vacuo*). For polydisperse solu‐ tions, the electric field correlation function is given by a sum or distribution of exponentials,

( ) <sup>0</sup> *G d* <sup>1</sup> ¥ G G= ò

In many cases, a single exponent obtained from an average translational diffusion coefficient

where *K*1 is the first cumulant. When a solution is dilute enough neglecting the interaction between particles, the effective diffusion coefficient can be obtained by the intensity average. The intensity of light scattered by macromolecule species *i* is often proportional to the mo‐

The average decay rate (*Γ*) of the electric field autocorrelation function can be obtained by using conventional DLS.The first cumulant generally fits the data well for carbon nanotube solutions. When the incident light and detector are both vertical, *Vv*, translational diffusion

2

When the incident light vertical and detector horizontal, *Hv* , the diffusion of anisotropic

where *DT* is translational diffusion coefficient and *DR* is rotational diffusion coefficient. This equation is valid if the particle rotates many times while diffusing a distance comparable to

 or if there is little anisotropy in particle dimension. MWCNTs solution meets with the former case in this work. Now we have three independent mathematical model eqs 25, 26,

( ) ( ) ( ) (1) <sup>0</sup> *g t G td* exp ¥

( ) <sup>1</sup> *av* <sup>0</sup> *K Gd* ¥

value fits the decay rate of the electric field correlation function such as eq 29

) times the weight concentration (*ci*

fusion coefficient equals to z average diffusion coefficient.

50 Syntheses and Applications of Carbon Nanotubes and Their Composites

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.

$$D\_T = \frac{kT}{6a\pi\eta\_s} \tag{32}$$

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

$$
\eta\_s = \frac{kT}{6a\pi D\_T} \tag{33}
$$

where *a* is the radius of spherical colloidal particle; colloidal particles have usually average diameter between 1 nm and 1000 nm. It is comparable the ISO definition of nanoparticles those have average diameter between 1 nm and 100 nm. In this sense, nanoparticles are just some kinds of colloidal particles. When we have measured value of mean-squared displace‐ ment of probe particle, the following equation can be utilized.

$$\left\langle r^{2} \left( t \right) \right\rangle = 6D\_{T}t = \frac{kT}{\pi a \eta\_{s}} \tag{34}$$

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 nanoparti‐ cles 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 pol‐ ymers is determined from the entanglement lengths of polymer such as

$$G\_o^N = \frac{\rho RT}{M\_e} \tag{35}$$

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

53

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The plateau modulus of polymer is usually reported in the order of 10 6~7Pa. 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.
