18882
J. Phys. Chem. C 2007, 111, 18882-18887
Persistence Length of Multiwalled Carbon Nanotubes with Static Bending Heon Sang Lee,*,† Chang Hun Yun,† Heon Mo Kim,† and Cheol Jin Lee‡ LG Chemical Company Limited, Technology Center, Jang-Dong 84, Yusung-Gu, Daejeon 305-343, Korea, and School of Electrical Engineering, Korea UniVersity, Anam-Dong, Seongbuk-Gu, Seoul 136-701, Korea ReceiVed: June 29, 2007; In Final Form: October 1, 2007
Persistence length of multiwalled carbon nanotubes (MWCNTs) is studied using three-dimensional analysis of MWCNT individual particles, employing scanning electron microscopy with various view angles and a capillary viscometer. The root-mean-squared end-to-end distance of MWCNTs follows random-coil scaling in spite of their static bending points. The static bending persistence length (lsp) of the MWCNTs is estimated to be 271 nm. The intrinsic viscosity of the MWCNTs follows the wormlike coil model when the dynamic bending persistence length (lp) is replaced by the static bending persistence length (lsp).
Introduction Carbon nanotubes have been studied intensively due to their novel potential as a polymer additive or filler.1-4 A carbon nanotube (CNT) would be a straight shape, since the CNT consists of rolled graphene layer or layers. However, nonhexagon structures can be introduced to the graphene layer such as the heptagon-pentagon pair and others, which bend the tube permanently.5 The bending angle from the axis is measured to be 20-30° by the heptagon-pentagon pair at the bend point.5 Other possible defects in graphene layers are also reported.6 Absolutely speaking, the chemical formula of a bent CNT is different from that of a straight CNT. The nature of bent CNTs is possibly different from that of so-called perfect CNTs in properties such as chemical reactivity, electrical properties, solubility, and mechanical properties as well as hydrodynamic properties. The modulus of polymer-multiwalled carbon nanotube (MWCNT) composites depends on the degree of bent curvature of the MWCNT.7 The field emission properties of MWCNTs also depend on the bent shape.8 The solution behavior of tortuously bent MWCNT differs from that of straight MWCNT which gives rise to liquid crystalline dilute solution.9,10 A bent structure of a MWCNT has a certain merit in formation of a less anisotropic composite of a polymer and MWCNTs which may produce good dimensional stability. For MWCNTs synthesized by a chemical vapor deposition (CVD) method, the bend direction and the bend point distribution along the tube axis may be random. The dynamic bending persistence lengths of CNTs are known to be up to several millimeters.11 The dynamic bending persistence length of single-walled carbon nanotube (SWCNT) is reported to be 32-174 µm by real-time visualization.12 So, most of the bent shape of MWCNTs in the micrometer scale may be from static bending and may not be from the tube’s semiflexibility. A semiflexible rigid rod can be considered as a random coil having Kuhn lengths equal to twice the dynamic bending persistence length, when the contour length of the rigid rod is much longer than its dynamic bending persistence length, such that 〈R2〉 ) 2lpL, where L is the contour length and lp is the dynamic bending persistent length.13-16 * To whom correspondence should be addressed. Tel.: +82-42-8608250. Fax: +82-42-863-1084. E-mail:
[email protected]. † LG Chem. ‡ Korea University.
MWCNTs grown by CVD methods are of growing interest for both academic7-10 and industrial applications for polymer composites and also for solutions for conductive coating. The shapes of MWCNTs are generally shown to be varied. In this paper, we present a method to determine the shape and size of MWCNTs. We propose a rigid random coil where bend points are static but the overall size relation follows random-coil scaling. We demonstrated that the MWCNTs produced by a CVD method are the typical examples of rigid random coils. The intrinsic viscosity of rigid random coils of MWCNTs follows the wormlike coil model17 when the dynamic bending persistence length (lp) is replaced by the static bending persistence length (lsp). The method for the determination of the static bending persistence length (lsp) is promising for the characterization of MWCNT shape. Experimental Section Multiwalled carbon nanotubes were synthesized by the catalytic reaction of C2H4 over Fe/Mo/Al2O3 catalyst at 923 K in an Ar/H2 atmosphere. In order to fabricate the catalyst, a mixture of Fe(NO3)3‚9H2O (99%, Aldrich) and Mo (ICP standard solution) was dissolved 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 a 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 flow rates 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. 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-necked 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 continuously stirring. After the acid treatment, the product was filtered on a membrane filter (PTFE 0.5 µm), washed to neutral pH, and dried at 393 K for
10.1021/jp075062r CCC: $37.00 © 2007 American Chemical Society Published on Web 11/21/2007
Rigid Random Coils of MWCNTs
Figure 1. SEM image of the as-synthesized carbon filaments by catalytic chemical vapor decomposition.
J. Phys. Chem. C, Vol. 111, No. 51, 2007 18883
Figure 3. MWCNTs solutions in DMF. Picture was taken after 1 year at rest.
Figure 2. Raman spectrum of the as-synthesized MWCNTs by catalytic chemical vapor decomposition.
12 h. The purified MWCNTs were dispersed in DMF by an ultrasonic homogenizer (Bandelin, HD2200) for 3 h, followed by centrifugation under 3000 rpm. The sediment was separated by filtration, and the residue solution was taken for analysis of this work. The morphology and 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 MWCNT solutions was measured by an automatic Ubbelohde viscometer (Schott Instrument) with a capillary diameter of 0.46 mm at 298 K. Results and Discussion Figure 1 shows an SEM image of the as-synthesized MWCNTs in this work, indicating high-purity MWCNTs without amorphous carbon materials or catalyst metal particles. Figure 2 shows Raman analysis of the as-synthesized MWCNTs. The MWCNTs indicate an IG/ID ratio of about 1.08 revealing fairly good crystalline structure of graphite sheets. Thus, according to SEM and Raman analysis, we suggest that the produced MWCNTs have a high-purity and high-crystalline carbon nanostructure. Figure 3 shows well-dispersed MWCNTs solution in DMF. The solutions are stable without any surfactant and showed no
Figure 4. SEM images of MWCNTs on ceramic filter (top view): (a) low- and (b) high-magnification SEM image.
sign of precipitation for 1 year. A volume of 1 mL of 0.001 wt % fresh solution was filtered on a ceramic filter with a pore size of 20 nm. Figure 4 shows the top view SEM images. Individually dispersed MWCNTs are observed. This indicates that the MWCNTs are also individually dispersed in the solution state, since aggregated particles are not likely separating during the filtering process. We can see the bent shape of the MWCNTs on the top view in Figure 4. The cumulative distribution of contour length and end-to-end distance are obtained from many top view micrographs. However, it is not clear in Figure 4 if the MWCNTs are flat on the surface. For the latter case, the lengths from the top view tend to be underestimated, since the top view is the trajection image. Figure 5 shows the side view with various view angles. The side view clearly shows the MWCNTs are not flat on the surface. This indicates the effect of surface force on the bent shape is small. A surface force in a flat surface is large enough holding elastic deformation in
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Figure 6. Three-dimensional images of MWCNT with various angles.
Figure 7. Contour length distribution of MWCNTs.
Figure 5. SEM image of MWCNTs on ceramic filter (side view). The angles indicate the observation positions from the vertical line.
spite of the high bending modulus of CNTs.18 But the side view images in Figure 5 show very small contact area between the porous surface and the MWCNTs. The MWCNTs in a stressfree state show bent structures in Figure 5. So, we can surmise that the bent structures are mainly static and are not from elastic bending in Figures 4 and 5. The coordinate values (xi, yi, zi) of arbitrary points along the MWCNT axis are determined in threedimensional (3-D) rectangular coordinate from the data obtained by side view images with various angles. From the 3-D coordinate values of several MWCNTs, it is found that the errors of contour lengths from the top view are less than 15% even though the contour lengths are longer than 2000 nm. The 3-D images of MWCNTs are obtained by plotting the 3-D coordinate data by using AutoCAD. Figure 6 shows the 3-D images of MWCNT on the ceramic filter surface with various view angles. The diameter in Figure 6 is arbitrary. The cumulative size distributions obtained from many top view micrographs are corrected by the data obtained from 3-D image analysis. From
the first derivative of the cumulative size distribution, the normalized contour length distributions are obtained as shown in Figure 7. In this study, we designate a randomly grown tortuous MWCNT as a rigid random coil (RRC) where the static bend points are randomly distributed along its axis. The spatial average of the square end-to-end vector can be obtained such as eq 1 when the distribution of bending points ({φ} ≡ (φ1, φ2, ..., φk)) is given. k
k
(φiri)‚(φjrj) ) N2b2Db ∑ ∑ i)1 j)1
〈R2〉 ) N2
(1)
k
Db ≡ 〈R2〉/L2 =
φi2 ∑ i)1
(2)
where Db is a bending ratio, φi ) Ni/N, Ni is the number of unit segment in the i-direction segment, N is the total number of unit segments, k ) m + 1, m is the number of static bending points on a coil, and ri is the i-direction segment vector with the length of b. Equation 2 holds only if a probability of the fold-back conformation is the same as that of the straight
Rigid Random Coils of MWCNTs
J. Phys. Chem. C, Vol. 111, No. 51, 2007 18885
Figure 8. End-to-end distance with respect to contour length. The solid line represents the calculation result by eq 5. The dashed line is the Kratky-Porod model, eq 6. Vertical lines indicate error bars.
Figure 10. TEM image of MWCNT.
Figure 9. Bending ratio with respect to contour length. The solid line represents the calculation result by eq 1. The dashed line is the KratkyPorod model, eq 6. Vertical lines indicate error bars.
conformation. By using the scaling law,15 the coil expressed in eqs 1 and 2 can be renormalized into the coil that has constant segment length, 2lp0. Then we can obtain eq 2 with φi ) 2lp0/L and k ) L/2lp0. We can also consider a case where the bend 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 expressed as eq 3.
(
φi2) ∑ 1 - cos(θ) i)1
Db ≡
〈R2〉 L2
)
1 + cos(θ)
k
〈R2〉 ) (N2b2)(
k
(
) L2Db
)
1 + cos(θ)
φi2) ∑ 1 - cos(θ) i)1
=(
(3)
(4)
Equation 3 can also be renormalized into the coil that has a constant segment length, 2lsp. The bending ratio (Db) is expressed as eq 5
( )(
) ( )
2lsp 2lp0 1 + cos(θ) 2lp0 〈R 〉 )C ) Db ≡ 2 = L 1 - cos(θ) L L L 2
(5)
where lsp ) Clp0 is the static bending persistence length and C should be a constant for a fixed bend 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 curvature, a more accurate statement is that the static bending persistence length is the mean radius of curvature of the rigid random coil due to static bending. The same quantity arising from dynamic bending instead of static bending is the dynamic bending persistence length (lp). The
Figure 11. Specific viscosity with respect to solution concentration. The solid line is the linear regression result which is coincident with the calculation result by the Hearst eq 7.
dynamic bending persistence length represents the stiffness of the molecules as determined by the effective bending modulus against thermal energy in Brownian motion. Equation 5 is valid when L . lsp, the coil limit. Db ) 〈R2〉/L2 ) 1 when L < lsp, the rod limit. If we know the values of end-to-end distance and contour length, the bending ratio can be obtained from the meansquared 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 difference can be compromised by using an arbitrary unit segment length which is similar to the scaling of a polymer chain.15,16 The mean-squared endto-end distance by the Kratky-Porod expression14 is given by eq 6 when the dynamic bending persistence length (lp) is replaced by the static bending persistence length (lsp) and twice lsp equals the Kuhn length.
〈R2〉 ) 2lspL + 2lsp2(e-L/lsp - 1)
(6)
The root-mean-squared end-to-end distance is plotted with respect to contour length in Figure 8. The size distributions are segmented in each size range as shown in Figure 8. The average values of bending ratio (Db) and contour length are calculated at each size range. It is striking that the measured root-meansquared end-to-end distances follow the random-coil scaling, eq 5, in Figure 8. The Kratky-Porod expression also fits the data well. A more effective way to see the size scaling is to plot the bending ratio (Db) with respect to the reciprocal contour length as shown in Figure 9. Equation 5 fits the data well with the static bending persistence length of 271 nm. The Kratky-
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Lee et al.
TABLE 1: Contour Length (L), End-to-end Distance (R), and Bending Ratio (Db) of the MWCNTs data pointsa contour length (L) [nm] end-to-end distance (R) [nm] bending ratio (Db) a
1
2
3
4
5
6
7
8
9
10
578
752
894
1178
1546
1924
2344
3345
4923
8229
512
626
629
702
765
885
1078
1400
1680
2131
0.785
0.694
0.578
0.355
0.245
0.212
0.212
0.175
0.117
0.067
Data points in Figures 8 and 9.
Porod expression (eq 6) with the same static persistence length fits the data at large L (>1000 nm) but significantly deviates from the data at small L. The contour length, end-to-end distance, and bending ratio (Db) are listed in Table 1. The static bending persistence length determined in this work is much smaller than the dynamic bending persistence length expected for a CNT which is reported up to several millimeters.11 Because the intrinsic viscosity is a function of molecular weight, end-to-end distance, and contour length, we need to determine average molecular weight. Average molecular weights were calculated from the measured length, interlayer distance, and diameter by using Materials Studio v4.0 (Accelrys, U.S.A.) with a C-C bonding distance of 0.142 nm. For the calculations, the average diameter and interlayer spacing was determined from the TEM images. To measure the interlayer spacing, we focused on an area where less defect was observed in Figure 10. The interlayer spacing is measured to be 0.35 nm. From several TEM images, the average inner and outer diameters were found to be 6.5 and 21 nm, respectively. The TEM image in Figure 10 shows a straight shape, since the length scale in Figure 10 is much smaller than the static bending persistence length. The weight-average molecular weight is calculated to be 1.34 × 109 g/mol. From the molecular weight, the contour length, and the persistence length, the intrinsic viscosity of MWCNTs can be calculated. From Figures 7 and 8, the weight-average end-toend distance (R) and contour length (L) are 1397 and 3602 nm, respectively. If we apply the intrinsic viscosity model of a wormlike coil to the rigid random coil, the following expressions are obtained:17
〈R 〉 f M
(7)
f = [1 + 0.926θ(Db)1/2]-1
(8)
2 3/2
[η] ) 2.20 × 1021
θ ) ln
( )
2lsp - 2.431 + (e/a) e
(9)
where M is molecular weight, e is the spacing between frictional elements along the contour, a ) ζ/3πηs, ζ is the friction factor for a single frictional element, and ηs is the solvent viscosity. Because the weight-average contour length of the MWCNTs is much longer than that of the static bending persistence length, both eq 5 and Kratky-Porod eq 6 give similar calculation results in intrinsic viscosity. For the nondraining limit for the random coil, f ) 1, giving the maximum value of intrinsic viscosity in the model. For the nondraining limit for the random coil, the intrinsic viscosity is calculated to be 4.2 dL/g using the data obtained from the micrographs. The intrinsic viscosity of the MWCNTs solution is experimentally measured by using a capillary viscometer. Figure 11 shows that the inherent viscosity (ηinh ) (1 - η/ηs)/c) is not dependent on the solution concentration. This indicates that the solution is dilute enough neglecting the hydrodynamic interaction between the particles. This has also been shown in Figure 3. The measured value of
intrinsic viscosity is 3.75 dL/g as shown in Figure 11. To obtain the exact value of the friction factor in eq 9 is difficult. When we take the static bending persistence length (lsp) as the length of a single frictional element, the friction factor of the element in eq 9 may follow the rigid-rod model such that ζT ) 3πηslsp/ (ln(lsp/d) + 0.3) for the translational motion and may be ζr ) πηslsp3/(3(ln(lsp/d) - 0.8)) for the end-over-end rotational motion.16 Translational-rotational coupling and hydrodynamic shielding may also be considered for the evaluation of the friction factor in eq 4.19-21 In this case, we can surmise that the friction factor in eq 9 is scaled with lsps, where s is larger than unit value. We can reasonably neglect e/a in eq 9; then the intrinsic viscosity value of 3.75 dL/g is obtained by using eq 7 with lsp ) 271 nm and e ) 28 nm. This is consistent with the persistence length obtained by SEM analysis. The spacing between frictional elements (e) should be close to the hydrated diameter of the molecules.22 Therefore, we can conclude that the wormlike coil model by Hearst17 successfully describes the intrinsic viscosity of the rigid random coil when the dynamic bending persistence length (lp) is replaced by the static bending persistence length (lsp). The intrinsic viscosity result reported in this work is analyzed where the solution is dilute enough neglecting the interparticle interaction. It is noticeable that the intrinsic viscosity of the concentrated MWCNTs solution has been reported to exhibit polymeric behavior, in terms of an entanglement-like transition.23 Conclusions To summarize, the size of a randomly grown MWCNTs by the CVD method follows the random-coil model in spite of their static bending points. The static bending persistence length (lsp) of the MWCNTs is estimated to be 271 nm. The intrinsic viscosity of the rigid random coils can also be described by using the wormlike coil model when the dynamic bending persistence length (lp) is replaced by the static bending persistence length (lsp). The lsp estimated from the intrinsic viscosity is consistent with that estimated from the micrographs 3-D analysis. Acknowledgment. This study was supported in part by Research Grants from the Ministry of Commerce, Industry, and Energy through the Materials and Component Technology Development Program. H. S. Lee thanks Professor W. N. Kim, Dr. Min Park, Professor H. Yu, Professor C. D. Han, Professor H.J. Jin, and Professor C. R. Park for their advice on this work. H. S. Lee thanks Mr. S. H. Han and Mr. C. H. Hwang for their help on the AutoCAD program. References and Notes (1) Iijima, S. Nature 1991, 354, 56-58. (2) Breuer, O.; Sundararaj, U. Polym. Compos. 2004, 25, 630-645. (3) Liu, T.; Phang, I. Y.; Shen, L.; Chow, S. Y.; Zhang, W. D. Macromolecules 2004, 37, 7214-7222. (4) Gao, C.; Vo, C. D.; Jin, Y. Z.; Li, W.; Armes, P. Macromolecules 2005, 38, 8634-8648.
Rigid Random Coils of MWCNTs (5) Nakayama, Y.; Nagataki, A.; Suekane, O.; Cai, X.; Akita, S. Jpn. Soc. Appl. Phys. 2005, 44, 720-722. (6) Hashimoto, A.; Suenaga, K.; Gloter, A.; Urita, K.; Iijima, S. Nature 2004, 430, 870-873. (7) Fisher, F. T.; Bradshaw, R. D.; Brinson, L. C. Compos. Sci. Technol. 2003, 63, 1689-1703. (8) Jang, H. S.; Lee, H.-R.; Kim, D.-H. Thin Solid Films 2006, 500, 124-128. (9) Song, W.; Kinloch, A.; Windle, A. H. Science 2003, 302, 13631363. (10) Somoza, A. M.; Sagui, C.; Roland, C. Phys. ReV. B 2001, 63, 081403(R)(1-4). (11) 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. (12) Duggal, R; Pasquali, M. Phys. ReV. Lett. 2006, 96, 246104 (1-4). (13) Mason, T. G.; Dhople, A.; Wirtz, D. Macromolecules 1998, 31, 3600-3603. (14) Kratky, O.; Porod, G. Recl. TraV. Chim. Pays-Bas 1949, 68, 11061122.
J. Phys. Chem. C, Vol. 111, No. 51, 2007 18887 (15) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: New York, 1979; Part A. (16) Doi, M; Edward, S. J. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986; Chapter 2. (17) Hearst, J. E. J. Chem. Phys. 1964, 40, 1506-1509. (18) Hertel, T.; Martel, R.; Avouris, P. J. Phys. Chem. B 1998, 102, 910-915. (19) Wegener, W. A. J. Chem. Phys. 1982, 76, 6425-6430. (20) Fernandes, M. X.; de la Torre, J. G. Biophys. J. 2002, 83, 30393048. (21) Koehler, S. A.; Power, T. R. Phys. ReV. Lett. 2000, 85, 48274830. (22) Hearst, J. E.; Stockmayer, W. H. J. Chem. Phys. 1962, 37, 14251433. (23) Shaffer, M. S. P.; Windle, A. H. Macromolecules 1999, 32, 68646866.
