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Diameterdependent bending dynamics of singlewalled carbon nanotubes in liquids

Edited by Harry L. Swinney, University of Texas, Austin, TX, and approved June 30, 2009 (received for review April 15, 2009)
Abstract
By relating nanotechnology to soft condensed matter, understanding the mechanics and dynamics of singlewalled carbon nanotubes (SWCNTs) in fluids is crucial for both fundamental and applied science. Here, we study the Brownian bending dynamics of individual chiralityassigned SWCNTs in water by fluorescence microscopy. The bending stiffness scales as the cube of the nanotube diameter and the shape relaxation times agree with the semiflexible chain model. This suggests that SWCNTs may be the archetypal semiflexible filaments, highly suited to act as nanoprobes in complex fluids or biological systems.
Singlewalled carbon nanotubes (SWCNTs) are tubular nanostructures of covalently bonded carbon atoms, with typical diameters near 1 nm and lengths in the micrometer range. Their remarkable combination of very large aspect ratios, high mechanical strength, and versatile electronic properties makes SWCNTs highly suited for many applications, e.g., in material engineering, soft matter science, and biomedicine (1, 2). However, advances in applications are presently limited by scant fundamental understanding of their mechanical and dynamical behavior in fluid environments. When embedded in a viscous medium, the dynamics of these 1D nanoobjects are essentially dominated by the balance of Brownian forces, which tend to bend them, and elastic forces, which oppose this curvature. The simplest model for such filaments is that of an inextensible elastic beam with bending stiffness κ = EI, where E is the elastic modulus and I is the area moment of inertia about the filament axis. The ratio of bending stiffness to thermal energy yields a characteristic persistence length L_{p} = κ/k_{B}T, where k_{B} is the Boltzmann constant, and T is the absolute temperature (3). This is the length scale over which the filament shows significant curvature induced by thermal fluctuations. Hence, “short” filaments (L ≪ L_{p}) essentially appear as rigid rods, whereas “long” filaments (L ≫ L_{p}) display significant bending.
SWCNT persistence length is important in a variety of applications such as organization on patterned surfaces (4), the ability to form liquid crystals (5), and the stiffening of polymeric networks (6, 7). Although various methods have been used to infer or measure these persistence lengths, there is still controversy about whether, in the absence of flow, individual SWCNTs in liquids should be considered rigid (L ≪ L_{p}) or semiflexible (L ≈ L_{p}). A study based on the size of closed SWCNT rings estimated a persistence length of 800 nm (8); experimental data from neutron scattering indicated that SWCNTs behave as rigid rods on length scales of at least ≈150 nm (9), and Xray scattering data suggested that SWCNTs do not display rodlike behavior at any length (10). Fluorescence video microscopy studies of fluorescently labeled SWCNTs deduced persistence lengths ranging between 32 and 174 μm (11), although this method could not distinguish between individual SWCNTs and small bundles. Theory based on a continuum model for a hollow cylinder of radius R predicts (12–14) the bending stiffness of SWCNTs to be κ = πCR^{3}, where the “inplane stiffness” C is estimated as ≈345 J/m^{2} from ab initio calculations of the elastic energy of a tube under axial strain (12). This model suggests that bending stiffness, and hence persistence length, should depend strongly on nanotube diameter. To date, however, no experimental test of such a relationship has been reported.
Fluorescence microscopy is a powerful tool for studying the mechanical properties of onedimensional nanoobjects by directly measuring their individual shape fluctuations. Variants of this technique have been used to measure the bending stiffness of biopolymers such as actin and microtubules (15–18). Here, we exploit the intrinsic nearinfrared fluorescence of semiconducting SWCNTs to image individual nanotubes moving freely in aqueous surfactant suspension as they undergo Brownian motion in a quasitwodimensional sample chamber (19, 20). The use of intrinsic fluorescence provides three benefits. First, because nanotubes are imaged by using their own characteristic emission rather than that of an added fluorophore, all observed objects are confidently identified as SWCNTs. Second, aggregation of SWCNTs into bundles strongly quenches the emission, so there is good discrimination against nanotube bundles. Finally, the wavelength of a nanotube's nearinfrared fluorescence is characteristic of its exact transverse structure (21), so spectral analysis reveals individual SWCNT diameters. Fig. 1 shows the thermally induced undulations of a SWCNT visualized by this technique. The bending rigidity (persistence length) of each individual SWCNT can be determined from the amplitude of such shape fluctuations (16, 22).
We analyzed 34 individual nanotubes with diameters between 0.7 and 1.2 nm. Measurements were restricted to those with lengths >3 μm to minimize systematic errors in nanotube shape determinations due to pixilation and diffraction. Moreover, we used only SWCNTs that showed identical emission spectra along their lengths, indicating an absence of major structural defects (e.g., chirality switch) or bundling with another nanotube (20). In a typical experiment, 1,000 images of each SWCNT were acquired. The backbone coordinates of the SWCNTs were determined with subpixel accuracy by an intensityweighted analysis (see Methods below and Fig. 2A). Treating SWCNTs as wormlike chains (22) and following the procedure of Gittes et al. (16), the shape of the SWCNT in each image was expressed as a sum of cosine bending modes (which form an orthogonal basis); mode amplitudes were computed by projecting the image shape onto these basis functions.
Fig. 2B shows the results compiled from applying this procedure to all 1,000 images in a sequence recorded for one nanotube. Here, the variance in the amplitude of each bending mode is plotted vs. the mode number. The measured mean amplitude is essentially zero, confirming that this SWCNT had no static curvature. The longer wavelength (low n) modes show the expected n^{−2} dependence of amplitude on mode number, indicating that the measured variance is due to thermal fluctuations. This n^{−2} dependence breaks down for higher modes (typically n > 5) because systematic errors due to image pixilation and diffraction effects start to dominate those mode amplitudes (16). The analysis of mode amplitudes also provides an internal consistency test. For example, a structural defect in a nanotube should yield a locally lower stiffness, which would increase the amplitudes of specific modes and break the n^{−2} decay law. The fact that we observe no anomalies in the plots of amplitude variance vs. mode number suggests an absence of significant mechanical defects in the studied nanotubes.
Fig. 3A shows how SWCNT persistence length (or bending stiffness) depends on SWCNT diameter. The experimental values of L_{p} range from 26 to 138 μm. We find that the measured bending stiffness scales with the cube of nanotube diameter, as predicted by the elastic continuum model (12). However, the bending stiffness relation κ = πCR^{3} implies a value for inplane stiffness C of 678 ± 22 J/m^{2}. This is approximately a factor of two larger than the value predicted theoretically (12). According to Odijk–Skolnick–Fixman theory (23, 24), the discrepancy cannot be explained by a stiffening effect from the ionic surfactant that coats the nanotube because the Debye length is only 1.7 nm under our experimental conditions. Moreover, the persistence length normalized by the nanotube diameter cubed shows no significant correlation with nanotube length (correlation coefficient of 0.022), as one would expect in such systems (Fig. 3B). There is also no apparent correlation between persistence length and SWCNT chiral angle (see Fig.S1).
Fourier modes provide a convenient basis for describing dynamic as well as static SWCNT shapes. The relaxation times of the SWCNT deformations can be extracted from the exponential decay of the autocorrelation of Fourier mode amplitudes (see Methods). However, in this analysis, exposure times that are long relative to the relaxation time can blur images and depress the apparent shape fluctuations, thereby hampering the interpretation of nanotube dynamics. To minimize this source of error, we slowed the nanotube dynamics by raising the solvent viscosity with a mixture of sucrose and glucose. Fig. 4A shows experimental autocorrelation functions for the first and second bending modes. There is good consistency for data acquired with different exposure times, indicating that the image acquisition rate was adequate to capture undistorted dynamics. Brangwynne et al. (15) have shown that, although the cosines are not exactly normal modes, the relaxation of the first two modes is well approximated by a single exponential. The data in Fig. 4A are well fit by an exponential decay model, which gives the results shown as solid curves and provides the relaxation times τ_{1} and τ_{2} (for the first and second modes, respectively). Because all other relevant parameters (fluid viscosity, temperature, and SWCNT length) are known independently, the persistence length of each SWCNT can be extracted from the dynamic data by using the theoretical relationship between persistence length and bending relaxation times (25–28), L_{p} ≈ γ/τ_{n}k_{B}T((n + ½)π/L)^{4}, where γ ≈ 4πη/ln(L/d) is the friction coefficient, and τ_{n} is the relaxation time of mode n (see Methods). Persistence length values extracted from the relaxation times of the first bending mode for five SWCNTs are in excellent agreement with our equilibrium bending analysis for the same nanotubes (see Fig. 4B). We consider such agreement remarkable, considering the simplified nature of the semiflexible chain model.
In summary, we have studied the Brownian bending dynamics of diameterresolved SWCNTs in ordinary aqueous surfactant suspensions. The measured persistence lengths range from 26 to 138 μm for SWCNTs of diameter from 0.77 to 1.15 nm. Persistence length is found to scale with the cube of diameter, as expected from an elastic continuum model (12), but the deduced nanotube inplane stiffness significantly exceeds a theoretically predicted value. Smaller values of persistence length reported in previous studies might reflect the presence of other forces in addition to Brownian ones, such as the intertube van der Waals attractions that control the structure of buckypaper. The persistence length of a SWCNT within our diameter range is comparable with that of actin filaments (L_{p} ≈ 17 μm) but somewhat shorter than that of microtubules (L_{p} on the order of millimeters)—because the higher elastic modulus of the SWCNTs is counterbalanced by larger diameters of actin (≈7 nm) and microtubules (≈25 nm). Interestingly, based on the scaling of persistence length with CNT diameter, SWCNTs should be several orders of magnitude less stiff than their multiwalled counterparts (MWCNTs, diameter 10–50 nm) (13). The ultimate mesoscopic and macroscopic properties of functional materials such as fibers are dictated by the behavior of the constituent single molecules (13, 29). Understanding the relationship between singlemolecule mechanics and the collective properties of their macromolecular assemblies has implications for the rational design of carbon nanotubebased functional materials (29, 30). We have shown that even small changes in diameter affect significantly the stiffness of SWCNTs. This result implies that larger diameter SWCNTs could yield stiffer fibers of low density—the density of SWCNTs scales inversely with their diameter. Interestingly, recent experimental results show that largediameter single and doublewalled carbon nanotubes yield fibers with the best mechanical properties achieved so far (30, 31)—although there are limitations, because excessively large fewwalled carbon nanotubes are prone to buckle radially into flattened ribbons (32, 33). Because stiffness is an important property in biological systems, the diameterdependent rigidity of SWCNTs should also be considered as a potentially important variable in studies of interactions of carbon nanotubes with cells.
When only Brownian forces are present, we observe that relatively short SWCNTs (below ≈3 μm) behave as rigid rods in ambient fluid suspension and show translational and rotational diffusion but essentially no shape fluctuations (11, 34). We find that the thermal shape fluctuations of longer SWCNTs accurately follow the semiflexible filament model. Therefore, existing models for semiflexible networks should be applicable to the viscoelastic properties of concentrated SWCNT suspensions (35), possibly by constructing model networks with tunable stiffness by using diametersorted SWCNT samples (36). In addition, the varying persistence lengths within the family of SWCNT structures could be exploited to develop rodlike nano and microscale probes for microrheological studies of complex fluids and biological systems and to tailor nanotube mobility in confined environments (37) such as living cells and tissues.
Methods
SWCNTs with diameters between 0.7 and 1.2 nm were obtained from the Rice University HiPco reactor (batches 125.2, 161.1, and 162.5). Dilute aqueous suspensions of SWCNTs (1–10 ng/mL) in SDBS (sodium dodecylbenzenesulfonate, 1 wt %) were prepared by mild ultrasonication (7 W, 5–6 s) to minimize SWCNT breaking. Suspended SWCNTs were imaged in a quasitwodimensional chamber ≈1.0 ± 0.5μm thick made by sandwiching ≈0.7 μL of sample (1.0 μL for samples with higher viscosity) between a microscope slide and coverslip. The sample was sealed with vacuum grease to prevent convective flow due to evaporation. Enough time was allowed between sealing of the sample and image acquisition to ensure the cessation of any convective flow induced during sample preparation. All experiments were performed at room temperature (23 °C).
Nearinfrared fluorescence images of individual nanotubes were recorded under a Nikon TE2000 inverted microscope using 150× magnification (Nikon Plan Apo oilimmersion 100×, 1.4 N.A. objective and 1.5× auxiliary magnification), equipped with a liquid nitrogencooled NIR InGaAs camera (OMAV 2D; Roper) (19). The microscope was fiberoptically coupled via one of its output ports to the input slit of a JY C140 spectrograph with a cryogenically cooled 512element InGaAs array (OMAV; Roper) for measuring spatially resolved emission spectra. Samples were excited with focused diode laser beams at 658 and 785 nm. The excitation intensity at the target was adjusted between ≈0.1 and 10 kW/cm^{2} to capture highquality image sequences. We confirmed that at these irradiation levels the excitation laser had no noticeable effects on SWCNT dynamics. To slow down the SWCNT dynamics for bendingmode relaxation analysis, the sample viscosity was increased by adding 1 μL of SWCNT suspension to 10 μL of a 60:40 sucrose/glucose solution (40% total sugar by mass in 1 wt % SDBS) to yield a final viscosity of ≈13 mPas. SWCNT images were acquired with exposure times of 50 ms for shape analysis and 10, 20, or 50 ms for relaxation time analysis.
An image analysis algorithm was developed to track the fluctuating shape of SWCNTs with high accuracy. First, the position of the SWCNT was estimated by thresholding the image. The coordinates of the backbone of the SWCNT were determined with an intensityweighted centerofmass method. A twodimensional fixedarea kernel moved along the contour of the nanotube in the original image in 1pixel steps. By measuring the center of intensity for each position of the kernel, the realspace coordinates of that point on the backbone were extracted to obtain a full set of SWCNT backbone coordinates (x_{i},y_{i}). Following the procedure developed by Gittes et al. (16), the shape (tangent angle) of the nanotube, θ(s_{i}) = tan^{−1}(y_{i+1} − y_{i})/(x_{i+1} − x_{i}), was calculated from the backbone coordinates and decomposed into Fourier modes: where n is the mode number, a_{n} is the mode amplitude, and L is the SWCNT contour length. The amplitude of each mode was extracted by taking the Fourier inverse of this equation. In the absence of external forces, the equipartition theorem dictates that the variance in the amplitude of the bending modes is inversely proportional to the bending rigidity: where the angular brackets denote an ensemble average, which is equivalent to the time average because the system is ergodic. The cosine modes are not exactly normal modes for the dynamics of semiflexible filaments with free ends (15, 28), yet their dynamics is well approximated by singleexponential relaxation (15), and the relaxation times can be extracted from the autocorrelation of mode amplitudes: It can be shown that the mode relaxation times τ_{n} are given by (16) τ_{n} ≈ γ/κq^{4}. Here, q ≈ (n + ½)π/L and the hydrodynamic friction coefficient γ for a rod of length L and diameter d confined in a gap comparable to L can be approximated as twice the bulk friction coefficient (38), γ ≈ 4πη/ln(L/d), where η is the bulk viscosity.
Acknowledgments
We thank Rajat Duggal, Boris Yakobson, Howard Schmidt, and Fred MacKintosh for useful discussions. This work was supported by National Science Foundation (NSF) Grants CTSCAREER 0134389 and CHE0809020, NSF Center for Biological and Environmental Nanotechnology Grants EEC0118007 and EEC0647452, Robert A. Welch Foundation Grants C0807 and C1668, Applied NanoFluorescence, LLC, Welch Foundation Postdoctoral Fellowship LC0004 (to D.A.T.), the Fullbright Foundation (L.C.) and Délégation Générale pour ľArmement Grant ERE060016 (to L.C.).
Footnotes
 ^{1}To whom correspondence should be addressed. Email: mp{at}rice.edu

Author contributions: N.F., D.A.T., L.C., R.B.W., and M.P. designed research; N.F., D.A.T., and L.C. performed research; N.F., D.A.T., and L.C. analyzed data; and N.F., D.A.T., L.C., R.B.W., and M.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/cgi/content/full/0904148106/DCSupplemental.
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