Translational and Rotational Diffusions of Multiwalled Carbon

We show that apparent persistence lengths of multiwalled carbon nanotubes decrease drastically to hundreds of nanometers if and only if they have stat...
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J. Phys. Chem. C 2008, 112, 10653–10658

10653

Translational and Rotational Diffusions of Multiwalled Carbon Nanotubes with Static Bending Heon Sang Lee* and Chang Hun Yun LG Chem. Ltd., Technology Center, Jang-Dong 84, Yusung-Gu, Daejeon 305-343, Korea ReceiVed: February 13, 2008; ReVised Manuscript ReceiVed: June 5, 2008

We show that apparent persistence lengths of multiwalled carbon nanotubes decrease drastically to hundreds of nanometers if and only if they have static bend points. We find that the apparent persistence length of multiwalled carbon nanotubes with static bend points is also proportional to the reciprocal temperature such as the purely elastic worm-like chain around room temperature. Our results demonstrated that the mobility of multiwalled carbon nanotubes with static bend points must be enhanced by thermal fluctuation even at lowtemperature ranges from 278 to 363 K. We anticipate our paper to be a starting point for more sophisticated statistical mechanics models for the quantitative shape factor and the mobility of carbon nanotubes. Introduction Not only the toxicological issues but also research on novel hybrid materials or nanoscale devices points to the need for the understanding of the overall shape and mobility of carbon nanotube particles in a solution or in atmosphere. It is well known that he sizes and shapes of inorganic particles affects their toxicity. A recent paper1 has reported that long needlelike multiwalled carbon nanotubes (MWCNTs) result in asbestoslike, length-dependent, pathogenic behavior. The biological effect of tortuous MWCNTs seems to be different from that of long needle-like particles,1 depending on the shape and mobility. 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 on the order of tens to hundreds of micrometers because of their exceptionally large modulus.2 Multiwalled carbon nanotubes are expected to have longer persistence lengths,3 indicating that currently prepared severalmicrometer long nanotubes behave like rigid rods.4–9 Elastic fluctuations of semirigid particles by thermal energy have been described exactly by the worm-like chain model proposed more than 50 years ago by Kratky and Porod.10 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 calculations2 have shown that the dynamic bending persistence lengths (lp) of carbon nanotubes (CNTs) are up to several millimeters because of their exceptionally large Young’s moduli of about 1.5 TPa. A real-time visualization technique5 revealed that the lp values of singlewalled carbon nanotubes (SWCNTs) are between 32 and 174 µm, indicating that SWCNTs shorter than lp()32 µm) may be rigid around room temperature in a solution. However, rippling developed on the compressive side of the tube, leading to a remarkable reduction of the effective bending modulus,11 which is more pronounced for multiwalled carbon nanotubes (MWCNTs).11 Theoretical calculations12 have shown that the effective bending moduli of MWCNTs are around 0.5 nN nm2 when the radii of curvatures are around 150-500 nm. This * To whom correspondence should be addressed. Tel: +82-42-860-8250. Fax: +82-42-863-1084. E-mail: [email protected].

indicates that MWCNTs longer than 0.5 µm might be flexible in a solution around room temperature because the thermal energy is about 4.1 × 10-3 nN nm. It seems unlikely 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 those of SWCNTs.13 Both the MWCNTs and SWCNTs discussed above are no more than worm-like chains (WLCs) where the ensemble average of the overall size (end-to-end distance) scales with the square root of the molecular weight (contour length) in the asymptotic limit.3 Our recent work14 has revealed that the spatial average of the overall size of MWCNTs also follows the same scaling as WLCs in spite of their static bend 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 (lsp).14 Because both RRCs and WLCs are Gaussian,14 the models for the mobility of WLCs may also work for RRCs. It is worth noting that MWCNTs synthesized by a CVD method show randomly tortuous shape in many cases. In this work, we explore the mobility of RRCs by dynamic light-scattering (DLS) and depolarized dynamic light-scattering (DDLS). The reduction of hydrodynamic radii of RRCs was observed by increasing temperature, indicating that the static bend points of RRCs are fluctuated by thermal energy. By analogy to the WLC model,15,16 expressions for translational and rotational diffusion coefficients of RRCs are presented where thermal fluctuations are taken into account. The model describes the translational and the rotational diffusion coefficients of RRCs successfully. We show that the static bending persistence length (lsp) is a major factor for the diffusion coefficients of RRCs. The importance of the static bending persistence length (lsp) as a shape factor of MWCNTs is also demonstrated in a practical application of MWCNT-polymer composites. Theory The translational diffusion coefficient is defined by the ratio of particle mobility and thermal energy as shown in the Einstein relation, eq 1.

10.1021/jp803363j CCC: $40.75  2008 American Chemical Society Published on Web 06/26/2008

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Lee and Yun

kT ζ

(1)

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

1 1 1 + ) ζ Nζe 2 ! N2

∑ ∑ ζ1ij + · i

· ·

(2)

j

where ζij 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 the Stokes-Einstein relation, ζe ) 3aπηs where ηs is the viscosity of the solvent. The frictional factor by interaction between ithe ith and jth element may also follow the Stokes-Einstein relation because a mean value of distance between elements and j is small. Then, Kirkwood expression15 is obtained: see eq 3.

[

1 a 1 1+ ) ζ N3aπηs 2Ne

]

∑ ∑ (1 - δij)〈rij-1〉 i

j

(3)

( )∑ ∑ 〈(1 - δ )〈r

1 1 1 ) ) e Rij erjij

ij

i

-1 ij 〉〉

(4)

j

where rij ≡ Rij/e and Rij is the distance between element i and j, e is the solvated diameter of the molecule, and N is the number of frictional elements, N ) L/e. Here we can see that the mathematical expression for the mobility of RRCs is similar to that of WLCs. Equation 3 is used widely for the estimation of the translational diffusion coefficients of macromolecules. We notice that jrij is no more than a mean value of the distance between the frictional elements i and j. Then, jrij must depend on the conformation of the carbon nanotubes. Hearst and Stockmayer15 chose the Daniels distribution, which includes a first-order correction to a Gaussian distribution as the known distribution, and obtained jrijas eq 5.

rij ) [2N(ln(2Lp) - 2.431 + 1.843(N ⁄ 2Lp)1⁄2 + 0.138(N ⁄ 2Lp)-1⁄2 - 0.305(N ⁄ 2Lp)-1)]-1 (5) where N ) L/e, L is the contour length, e is the spacing between frictional elements along the contour, Lp ) lp/e, and lp is the persistence length. The translational diffusion coefficient of worm-like chain can be estimated by eqs 1, 3, and 5. The rotational diffusion coefficient16 is expressed as eq 6.

( )( )[ ( )

kT 2 L Dr ) 0.253 ηs l L2 4lp p

1⁄2

+

]

0.159 ln(2Lp) - 0.387 + 0.160 (6) Equations 5 and 6 are valid for a semiflexible rod when the contour length of the rod is much longer than its persistence length such that the mean squared end-to-end distance follows random-coil scaling, 〈R2〉 ) Nb2 where b ) 2lp. The semiflexible rods in this coil limit, L . lp, are so-called worm-like chains (WLCs). We see that the mobility is determined solely by the average conformation of a particle with a given solvent viscosity and contour length in eqs 3 and 4. 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 (lsp) of RRCs are the same as those of the dynamic bending persistence lengths (lp) of WLCs. When the hydrodynamic shielding effect is taken into account, the diffusion coefficients of RRCs might be slightly larger than those of WLCs because of the static bend points. The root meansquared end-to-end distance of RRCs are given by14

( ) k

〈R2 〉 ) (N2b2) Db ≡

〈R2 〉 = L2

( )

cos(θ) ∑ φi2 ( 11 -+ cos(θ) ) ) L2Db i)1

k

cos(θ) ∑ φi2 ( 11 -+ cos(θ) )= i)1

(7)

( )( ( )

2lp0 1 + cos(θ) ) L 1 - cos(θ) C

)

2lsp 2lp0 ) (8) L L

where Db is the bending ratio, lsp is the static bending persistence length, lp0 is an arbitrary constant segment length, θ is a static bent angle from the MWCNT axis, φi ) Ni/N, Ni is the number of unit segments in the i direction segment, N is the total number of unit segments, k ) m + 1, and m is the number of static bending points on a coil. When RRCs have semiflexibility by thermal energy, the ensemble average of the bent angle (θ2) always becomes larger in amount of ∆θ than the static bent angle θ; θ2 ) θ + ∆θ. This is due to the fact that the effective bending modulus toward the bend direction is smaller than that toward the opposite direction.12 This indicates that the overall size of RRCs may be decreased when they are fluctuated by thermal energy. Because frictional elements of RRCs have Gaussian distribution by the definition of RRCs, those of semiflexible RRCs also have Gaussian distribution. Therefore, eqs 5 and 6 are also valid for semiflexible RRCs when the persistence length is replaced by an apparent persistence length. The apparent persistence length (lap) is determined by the static bent angle (θ) and dynamic bent angle due to thermal energy θ(∆θ) as follows:

cos(θ + ∆θ) 1 - cos(θ) ( 11 -+ cos(θ + ∆θ) )( 1 + cos(θ) )

lap ) lsp

(9)

The expression for the translational diffusion coefficient of RRCs can be obtained from eqs 1, 3, 5, and 9.

DT )

kT [1 + ln(2Lap) - 2.431 + 1.843(N ⁄ 2Lap)1⁄2 + 3πηsL 0.138(N ⁄ 2Lap)-1⁄2 - 0.305(N ⁄ 2Lap)-1] (10)

Similary, the expression for the rotational diffusion coefficient of RRCs can be obtained as follows:

Dr )

( ) ( )[ ( ) 2 L kT 0.253 ηs l L2 4lap ap

1⁄2

+

]

0.159ln(2Lap) - 0.387 + 0.160 (11) where Lap ) lap/e. Equations 10 and 11 are promising for the estimation of diffusion coefficients of MWCNTs synthesized by a CVD method. In other words, eqs 10 and 11 give us the information of the shape and size of MWCNTs if we have the measured values of diffusion coefficients. Experimental Section Multiwalled carbon nanotubes (MWCNTs) were synthesized by the catalytic reaction of C2H4 over Fe/Mo/Al2O3 catalyst at 923 K

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Figure 1. Shapes of MWCNT individuals: (a) Top-view SEM image of MWCNTs on a ceramic filter. The angle indicates the observation position from the normal of surface. Scale bar, 1.0 µm. (b) Side-view SEM image of MWCNTs on a ceramic filter. The angle indicates the observation position from the normal of the surface. Scale bar, 300 nm. (c) Top view of the 3D plot using the coordination values of arbitrary points along the MWCNT in the three-dimensional rectangular coordinate system. (d) Side view of the 3D plot using the same coordination values as those in c.

Figure 2. Contour length distribution of MWCNTs.

in an Ar/H2 atmosphere. To fabricate the catalyst, we dissolved a mixture of Fe(NO3)3 · 9H2O (99%, Aldrich) and Mo (ICP standard solution) in ethanol, and this solution was added to aluminum isopropoxide dissolved in ethanol. The mixture was then subjected to evaporation on a water bath at 353 K. The catalyst was calcined at 923 K in O2 for 2 h, followed by mechanical grinding for several hours. It was then placed in a quartz boat and inserted into the center of a reactor. During raising the furnace temperature to the reaction temperature, 500 sccm of Ar gas was allowed to flow into the furnace. For the production of MWCNTs, C2H4, Ar, and H2 were introduced into the quartz tube at a flow rate of 300, 500, and 500 sccm, respectively. The reactor was maintained at the reaction temperature of 923 K for 60 min during the growth of MWCNTs. After synthesizing MWCNTs, the furnace was cooled to room temperature in Ar atmosphere. MWCNT-2 used in Figure 4 was supplied from a commercial source by NCT. The MWCNTs were purified by the treatment with 3 N nitric acid. The MWCNTs (1 g) and 3 N nitric acid (400 mL) were added into a two-neck round-bottomed glass flask, and the MWCNTs/acid mixture was then subject to sonication for 30 min, followed by heating at 333 K for 12 h with continuous stirring. After the acid treatment, the

Figure 3. Average decay rates (Γ) of the electric field autocorrelation function by dynamic light-scattering at 298 K: (a) Dependence of ΓHv (detector horizontal, incident light vertically polarized) and ΓVv (detector vertical, incident light vertically polarized) on squared scattering vector magnitude for MWCNTs in DMF. (b) Best fit by eqs 10, 11, and 17.

product was filtered on a membrane filter (PTFE 0.5 µm), washed to neutral pH, and dried at 393 K for 12 h. The purified MWCNTs were dispersed in DMF by an Ultrasonic Homogenizer (Bandelin, HD2200) for 3 h, followed by centrifugation under 5000 rpm. The suspension was filtered twice using a filter paper (Whatman no. 4) to remove the aggregated CNT particles, and the residue solution was taken for analysis of this work. The morphology and

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P)

Lw

(12)

Ln

Figure 4. Viscosity of DMF.

TABLE 1: Size and Shape Factors of the MWCNT j w (g/mol) weight-averaged molecular weight, M polydispersity, Lj w/Lj n average outer diameter (nm) average inner diameter (nm) interlayer spacing (Å) weight-averaged contour length (nm) static bending persistence length, lsp (nm) intrinsic viscosity (dL/g)

5 × 108 1.23 21 6.5 3.5 1326 271 2.1

j w is the weight-averaged contour length and jLn is the where L number-averaged contour length. The polydispersity (P) is determined to be 1.23 from the size distribution data shown in Figure 2. This is the most narrow size distribution on MWCNTs that we can provide currently, minimizing the polydispersity effect in the dynamic light-scattering analysis. The size and shape factors are listed in Table 1, which were determined by the earlier procedures.14 The weight-averaged contour length (L) and static bending persistence length (lsp) are 1326 and 271 nm, respectively. The intrinsic viscosity of the MWCNT solution was measured by a capillary viscometer. We obtained the same value of inherent viscosity at several solution concentrations between 0.002 and 0.007 g/dL, indicating that the solution was dilute enough neglecting any hydrodynamic interaction or particle aggregation. The measured intrinsic viscosity of the MWCNTs in DMF is 2.1 dL/g at 298 K. The intrinsic viscosity of RRCs can be expressed as the following.14,17

[η] ) 2.20 × 1021〈R2 〉3⁄2M-1f 1⁄2 -1

f = [1 + 0.926θ(Db) ] microscopic structure of MWCNTs were characterized by scanning electron microscopy (SEM, Hitachi S-4700) and transmission electron microscopy (TEM, JEOL JEM-3011, 300 kV). Raman measurement (Horiba Jobin-Yvon HR-800 UV) was done to evaluate the overall crystallinity of the nanotubes. The intrinsic viscosity of the MWCNT solutions were measured by an automatic Ubbelohde viscometer with a capillary diameter of 0.46 mm at 298 K. The translational and the rotational diffusion coefficients were measured by dynamic light-scattering apparatus. A diodepumped solid-state laser (DPSSL) supplied about 100 mW at λ0 ) 532 nm. A 256-channel digital autocorrelator was used to compute the scattered photon time autocorrelation function with a 480 nsec minimum delay time. Measurements were made at several scattering angles from 30° to 90°. The polarizer has a 1:100,000 extinction ratio and the detector also has a 1:100,000 extinction ratio. The detector was rotated in 1° resolution by a motor control unit. The temperature of the DLS cell was controllable to increase from 278 to 393 K by increments of 1 K. The measurements were made up to 363 K. Results and Discussion The MWCNT solution of 5.0 × 10-3 wt % was diluted with pure DMF. One mL of 1.0 × 10-4 wt % solution was filtered on a ceramic filter with pore size of 20 nm. Figure 1 shows SEM images of the top view (Figure 1a) and side view (Figure 1b) of MWCNTs on the ceramic filter, showing individually dispersed states. The MWCNTs in a stress-free state show bent structures in Figure 1b. This clearly indicates that the bent structures are mainly static and are not from elastic bending as reported earlier.14 The coordinate values (xi, yi, zi) of arbitrary points along a MWCNT axis are determined in a 3D rectangular coordinate from the data obtained by side-view images with various angles, which are plotted in Figure 1c and d. The cumulative size distributions obtained from many top-view micrographs are corrected by the data obtained from 3D image analysis. From the first derivative of the cumulative size distribution, the normalized contour length distributions are obtained as presented in Figure 2. The polydispersity (P) is defined as in eq 12.

θ ) ln(2lsp ⁄ e) - 2.431

(13) (14) (15)

where R is the root-mean-squared end-to-end distance and M is the weight-averaged molecular weight. From eqs 13, 14, and 15, the static bending persistence length (lsp) is determined to be 271 nm with measured intrinsic viscosity and weightaveraged molecular weight. This result indicates that the static bending persistence length (lsp) obtained from the intrinsic viscosity in the solution state is coincident with that obtained from 3D SEM analysis in the dried state as shown in Table 1. We used the well-established18 dynamic light-scattering (DLS) technique. We prepared a very dilute solution with nL3 ) 0.5 for dynamic light-scattering (DLS) analysis. Because the size of the carbon nanotubes is bigger than other particles such as polymers or biomolecules, we can get quite strong scattering intensity in this dilute regime; and, as is often the case, the intensity of light scattered is proportional to the molecular weight.19 This nature of the carbon nanotube solution is a strong point for DLS analysis because we can neglect any hydrodynamic interaction and exclusive volume effect in very dilute solution.18 The Raman effect is so small in general19 that we can also neglect it, although the length dependency of the optical properties has been observed in CNTs.4 The scattering vector magnitude (q ) 4πn sin(θ/2)/λ0 where n is the solution refractive index, θ is the scattering angle, and λ0 is the incident light wavelength in vacuo) was adjusted by changing the scattering angle. The average decay rate (Γ) of the electric field autocorrelation function at 298 K is shown in Figure 3. The first cumulant20 fits the data well for all measurements. When both the incident light and detector are vertical, Vv, the translational diffusion is characterized18,21 by eq 16.

ΓVv ) q2DT

(16)

When the incident light is vertical and the detector is horizontal, Hv, the diffusion of anisotropic particles is characterized16,19 by eq 17

ΓHv ) q2DT + 6DR

(17)

where DT is the translational diffusion coefficient and DR is the rotational diffusion coefficient. This equation is valid if the particle

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Figure 5. Dynamics of MWCNTs: (a) Translational diffusion coefficients (DT) measured at the temperature range from 278 to 363 K. (b) Hydrodynamic radii determined from the measured translational diffusion coefficient. (c) Schematic image for the thermal fluctuation of MWCNTs with static bend points. The overall size (R) of the MWCNTs is reduced to R2 at the elevated temperature. θ2 ) θ + ∆θ. (d) Effect of fluctuating angles (∆θ) and static bent angle (θ) on the reduction of apparent persistence length (lap) compared to the static bending persistence length (lsp).

rotates many times while diffusing a distance comparable to q-1 or if there is little anisotropy in the particle dimension.21 The MWCNT solution meets with the former case in this work. The slope of the ΓVv curve is identical to that of ΓHv in Figure 3. From the slope of the ΓVv or ΓHv curves in Figure 3a, the translational diffusion coefficient (DT) is determined to be 2.1 × 10-8 cm2/s. The intercept of the ΓHv curve across the y axis gives DR ) 27.5 s-1 in Figure 3a. The calculation results using eqs 10, 11, and 17 are shown in Figure 3b. The computed curve that best fits the slope and intercept of the ΓHv data is obtained with lap ) 170 nm at 298 K. This value of the apparent persistence length (lap) is smaller than that of the static bending persistence length (lsp), which is determined to be 271 nm by 3D SEM and intrinsic viscosity. This result may indicate MWCNTs with static bend points are fluctuated slightly by thermal energy in a solution, leading to the reduction of hydrodynamic volume. In a capillary viscometer, MWCNTs may be extended to the static shape by shear force where the apparent wall shear rate is 484 s-1 where hydrodynamic contribution dominates the Brownian contribution to particle pressure.22 The Peclet number (Pe ) γ˙(2Rh)2/DT) is over 10 in the viscosity measurement. It is worth noting that the static bending persistence length determined from the intrinsic viscosity is consistent with that determined from 3D SEM analysis in the dried state. To monitor the thermal fluctuation of MWCNTs, we measured DT at various temperatures from 278 to 363 K. The hydrodynamic radius can be obtained by the Stokes-Einstein equation, Rh ) (kT)/ (6πDTηs) with the experimental data of DT and the solvent viscosity (ηs). The viscosity of DMF (ηs) is measured by a Ubbelohde viscometer, which is shown in Figure 4. The temperature dependency of DMF is determined to be ηs ) A exp(-∆E /kT)Pa · s, where -∆E ) 8685 J and A ) 2.3 × 10-5 Pa · s. In order to remove any experimental uncertainties, we correct the change of refractive index by increasing the temperature despite the negligible variation.

Figure 5a shows the measured DT at the temperature range. Hydrodynamic radii of MWCNTs are determined by the StokesEinstein equation in Figure 5b, showing a 10% reduction by increasing the temperature from 278 to 363 K. This directly indicates that the overall sizes (end-to-end distances) of the MWCNTs are decreased by the increase of temperature. The spatial average of the square end-to-end distance of MWCNTs with static bend points can be obtained with eq 8. When MWCNTs start to fluctuate by thermal energy, the time average of the bent angle becomes larger compared to the static bent state, and eventually this fluctuation leads to the reduction in overall size as illustrated in Figure 5c. Therefore, the reduction of the hydrodynamic radius points to the conclusion that MWCNTs are flexible in a solution. In Figure 5b, calculation results using eq 10 lead to the moreinteresting fact that the apparent persistence length (lap) is proportional to the reciprocal temperature, which is similar to the WLC model, lp ) Eeff/kT. Equation 10 best fits the data with Eeff ) 0.7 nN nm2 in Figure 5a. The lap value is decreased by about 20% when increasing the temperature from 278 to 363 K. Calculation results using eq 9 indicate ∆θ ) 2° result in a 20% reduction of lap in most cases in Figure 5d. Now, the shape and dynamics of randomly grown MWCNTs are unveiled. In short, MWCNTs having permanent bending points are fluctuated by thermal energy in a solution. These results have a great impact on various research fields such as mobility study for toxicology, stability of MWCNT solutions, morphology of MWCNT-polymer hybrids, and dimensional stability of nanostructured applications. In this section, we show the effect of static bending persistence length (lsp) on the morphology of the MWCNTspolycarbonate composite. Conventional injection-molding provides avery high shear rate over 2000 s-1 in the polymer industry. It is generally known that most anisotropic particles, such as carbon fiber, liquid crystalline polymer, and glass fiber,

10658 J. Phys. Chem. C, Vol. 112, No. 29, 2008

Lee and Yun bending persistence length already has a value larger than 10. Therefore, we can clearly conclude that the difference in morphology between MWCNTs and MWCNT-2 is mainly ascribed to the difference of static bending persistence lengths between MWCNTs and MWCNT-2. This result also supports the fact that MWCNTs have permanent bend points, otherwise the MWCNTs with lsp ) 271 nm should align to the flow direction such as a liquid-crystalline polymer at a high shear rate. Conclusions

Figure 6. SEM images of MWCNTs: (a) As-synthesized MWCNT with lsp ) 271 nm. Scale bar, 200 nm. (b) As-synthesized MWCNT-2 with lsp > 10 µm, Scale bar, 2.0 µm. (c and d) Cross section of molded specimen by injection-molding machine. After microtoming, the polycarbonate is etched by oxygen plasma. The arrow indicates the flow direction during the injection-molding process. Granular particles shown in these images are metal oxide that was added to the polycarbonate-CNT composite during melt-extruding by twin-screw extruder as conventional additives in the polymer industry. (c) Cross section of the polycarbonate/MWCNT composite. Scale bar, 0.5 µm. (d) Cross section of the polycarbonate/ MWCNT-2 composite. Scale bar, 1.0 µm.

align to the flow direction in polymer melt flow, resulting in the anisotropic physical properties of composites: for example, linear thermal expansivity. Both dilute and concentrated composites containing anisotropic particles eventually exhibit anisotropic properties after high-shear-rate injection-molding. Straight MWCNTs may also align to the flow direction at a high shear rate. Here, we show that the morphology of MWCNTs in a polymer composite strongly depends on the static bending persistence length. Figure 6a shows an SEM image of an assynthesized MWCNT whose static bending persistence length (lsp) is 271 nm. Figure 6b also shows an SEM image of MWCNT-2 whose lsp is up to 10 µm. Figure 6a and b shows similar shape but with significantly different scale bars; 200 nm and 2 µm for Figure 6a and b, respectively. 2 wt % of assynthesized MWCNTs were dispersed into polycarbonate by twin screw extruder at 523 K because then the mixture was pelletized, and the mixed pellets were molded into the test specimen using an injection machine with a shear rate of 2000 s-1 at 553 K. After injection-molding, the morphology of the 2 wt % MWCNTs dispersed into polycarbonate is shown in Figure 6c and d. Unexpectedly, the morphology of MWCNTs with lsp ) 271 nm in the injection molded part is similar to their initial morphology before mixing with polycarbonate in Figure 6c, showing an isotropic tortuous shape. We can surmise that the stress relaxation time of the MWCNTs is faster than the cooling time during the injection-molding process. On the contrary, MWCNT-2 with lsp g 10 µm align to the flow direction after injection-molding in Figure 6d. Although we do not obtain the contour length distribution of MWCNTs in the MWCNTpolycarbonate composite, the aspect ratio within the static

The expressions for translational and rotational diffusion coefficients of MWCNTs are presented by analogy to WLC model. The model works successfully for MWCNTs synthesized by a CVD method. The MWCNTs with static bending points are fluctuated by thermal energy, resulting in the reduction of hydrodynamic radii of MWCNTs at elevated temperatures. The reductions of hydrodynamic radii of MWCNTs are expressed well by the apparent persistence lengths (lap) in the proposed model. The apparent persistence length is originally determined by the contributions of the static bending persistence length (lsp), static bent angle (θ), and dynamic bent angle (∆θ). The static bending persistence length (lsp) is found to be the major factor for the mobility and morphology of MWCNTs in a solution or in a polymer. Acknowledgment. This study was supported in part by research grants from the Ministry of Knowledge-based Economy through the Materials and Component Technology Development Program. H.S L. thanks Prof. C. J. Lee, Prof. W. N. Kim, Dr. Min Park, Prof. H. Yu, Prof. C. D. Han, Prof. H. J. Jin, and Prof. C. R. Park for their advice on this work. References and Notes (1) Poland, C. A.; Duffin, R.; Kinloch, I.; Maynard, A.; Wallace, W. A. H.; Seaton, A.; Stone, V.; Brown, S.; MacNee, W.; Donaldson, K. Nat. Nanotechnol. 2008, 111, 1–6; published online 20 May; doi:10.1038/ nnano. (2) Treacy, M. M. J.; Ebbesen, J. M.; Gibson, J. M. Nature 1996, 381, 678–680. (3) Ajayan, P. M.; Tour, J. M. Nature 2007, 447, 1066–1068. (4) Fagan, J. A.; Simpson, J. R.; Bauer, B. J.; Lacerda, S. H. D. P.; Becker, M. L.; Chun, J.; Migler, K. B.; Walker, A. R. H.; Hobbie, E. K. J. Am. Chem. Soc. 2007, 129, 10607–10612. (5) Duggal, R.; Pasquali, M. Phys. ReV. Lett. 2006, 96, 246104 (1-4) (6) Fry, D.; Langhorst, B.; Wang, H.; Becker, M. L.; Bauer, B. J.; Grulke, E. A.; Hobbie, E. K. J. Chem. Phys. 2006, 124, 054703 (1-9) (7) Badaire, S.; Poulin, P.; Maugey, M.; Zakri, C. Langmuir 2004, 20, 10367–10370. (8) Zhou, W.; Islam, M. F.; Wang, H.; Ho, D. L.; Yodh, A. G.; Winey, K. I.; Fischer, J. E. Chem. Phys. Lett. 2004, 384, 185–189. (9) Song, W.; Kinloch, I. A.; Windle, A. H. Science 2003, 302, 1363. (10) Kratky, O.; Porod, G. Recl. TraV. Chim. Pays-Bas 1949, 68, 1106–1122. (11) Poncharal, P.; Wang, Z. L.; Ugarte, D.; Heer, W. A. Science 1999, 283, 1513–1516. (12) Arroyo, M.; Belytschko, T. Phys. ReV. Lett. 2003, 91, 215505 (1-4) (13) Chang, T.; Hou, J. J. Appl. Phys. 2006, 100, 114327 (1-4) (14) Lee, H. S.; Yun, C. H.; Kim, H. M.; Lee, C. J. J. Phys. Chem. C 2007, 111, 18882–18887. (15) Hearst, J. E.; Stockmayer, W. H. J. Chem. Phys. 1962, 37, 1425–1433. (16) Hearst, J. E. J. Chem. Phys. 1963, 38, 1062–1065. (17) Hearst, J. E. J. Chem. Phys. 1964, 40, 1506–1509. (18) Berne, B.; Pecora, R. Dynamic Light Scattering; Wiley: New York, 1976; Chaper 8. (19) Tanford, C. Physical Chemistry of Macromolecules; John Wiley & Sons: New York, 1961; Chapter 5. (20) Koppel, D. E. J. Chem. Phys. 1972, 57, 4814–4820. (21) Cush, R.; Russo, P. S.; Kucukyavuz, Z.; Bu, Z.; Neau, D.; Shih, D.; Kucukyavuz, S.; Ricks, H. Macromolecules 1997, 30, 4920–4926. (22) Yurkovetsky, Y.; Morris, J. F. J. Rheol. 2008, 52, 141–163.

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