Single

J. Phys. Chem. B 2006, 110, 23179-23191

23179

Trapping, Pattern Formation, and Ordering of Polyelectrolyte/Single-Wall Carbon Nanotube Complexes at the Air/Water and Air/Solid Interfaces Jose´ Luis Herna´ ndez-Lo´ pez,† Edgar Rogelio Alvizo-Pa´ ez,† Sergio Enrique Moya,‡ and Jaime Ruiz-Garcı´a*,† Instituto de Fı´sica, Dr. Manuel NaVa Martı´nez No. 6, Zona UniVersitaria, UniVersidad Auto´ noma de San Luis Potosı´, San Luis Potosı´ 78290, Mexico, and Centro de InVestigacio´ n en Quı´mica Aplicada, BlV. Enrique Reyna No. 240, Apdo. Postal 379, Saltillo, Coahuila 25100, Mexico ReceiVed: May 25, 2006; In Final Form: August 11, 2006

The Langmuir and Langmuir-Blodgett (LB) techniques have been applied in a novel approach to build structurally well-ordered, oriented, and organized assemblies of water-soluble single-wall carbon nanotubes (ws-SWCNTs) at the air/water and air/solid interfaces. The SWCNTs were rendered hydrophilic by complexing them with a quenched polyelectrolyte. We observed that the ws-SWCNT concentration at the air/water interface increases with time condensing into different patterns, among which are isolated soap-froths, rings, and the aggregation of cumuli-like 2D-structures. These patterns were recorded at different compression-expansion stages by Brewster angle microscopy (BAM). From the isotherm measurements, we are able to determine the diffusion process by which ws-SWCNT concentration builds up at the water surface. The corresponding LB films were very stable and could be transferred onto mica substrates easily. Atomic force microscopy (AFM) images revealed that the morphology of these films is surface-pressure dependent, and aligned structures with a nematic-like order formed closely packed mono- or multilayer films. The assembly of 2D-nanostructures by means of this approach offers a great potential for emergent technological applications using modified water-soluble SWCNTs.

Introduction The fabrication of uniform carbon nanotube films with controlled nanostructures, film thickness, and nanotube orientation is an important prerequisite for a number of optical and electrical nanodevices. Polydispersity, strong intertube interactions, hydrophobicity, and poor solubility in organic solvents1 represent serious drawbacks in the manipulation of single-wall carbon nanotubes (SWCNTs) and their alignment in organized films. Nanotubes can be deposited on substrates by spin-coating techniques, but for thin films (98%), which further suggests the formation of a closely packed monolayer on the water surface. Changes into the structure of the amphiphilic material wrapping the nanotubes have a marked effect upon the monolayer properties and hence on the shape and stability of the isotherm. For example, another value of surface concentration comparable to our ws-SWCNTs has been reported by Feng et al.6b working with crown ethermodified SWCNTs (Γ ) 8.55 × 10-4 mg/cm2 at 293 K). For further evaluation of the ws-SWCNT films, the Langmuir film was vertically transferred to freshly cleaved muscovite mica substrates, and then their morphology was examined by AFM. To observe orientation effects in LB films, higher compression areas were examined systematically to a monolayer of wsSWCNT on the water surface. Figure 7 compares tapping-mode AFM images of ws-SWCNT monolayers after 96 h of adsorption time, being studied at 250, 200, 150, and 100 cm2 compression areas. According to the surface profile of the images, the nanotube height distribution is found between 1 and 3 nm. Considering that HiPco SWCNTs have a diameter distribution between 0.7 and 1.4 nm, with an average diameter of approximately 1.1 nm, and that the polymer coating could add reasonably an additional nanometer, these height measurements suggest that the majority of the observed nanotubes are individual SWCNTs wrapped with PSS.22 Thereafter, the larger heights measured by AFM onto the monolayer should correspond to SWCNT bundles that are known to be present in our samples.12 The polydispersity of the ws-SWCNT complexes and its propensity to form bundles is captured in Figure 7. In this figure, the disordered bundles are easily observed surrounded by a more ordered monolayer of ws-SWCNTs, described in more detail below. A reduction in the isotherm compression area from 250 to 100 cm2 causes an increase in bundle surface density greater than 130%. As expected, the sample deposited at the higher compression areas (100 cm2) presented more densely packed and more uniform coverage of nanotubes (Figure 7d). The statistical analysis of bundle features in AFM images of ws-SWCNTs confirmed an increase in bundle surface density at lower compression areas. The average bundle surface density corresponding to each compression area is plotted in Figure 8. To establish a correlation between density and area, statistics were performed at two scan sizes: 10 × 10 µm2 and 25 × 25 µm2. For counting, only those bundles that showed a well-resolved

J. Phys. Chem. B, Vol. 110, No. 46, 2006 23185 and defined morphology were considered. By means of this analysis, an exponential decay behavior between bundle surface density and trough areas was found. Further AFM investigation performed on smaller scan sizes revealed that individual arrays of ws-SWCNTs are indeed present. At sufficient low densities (i.e., low surface pressure), the nanotubes form raftlike aggregates by aligning side-by-side due to directional interfacial energies and van der Waals attraction. The raft aggregates assume all possible orientations, that is, with a fluidlike isotropic arrangement (Figure 9a). As the density increases, it becomes increasingly difficult for the nanotubes/rafts to point in random directions, and one may expect the fluid to undergo a transition to a more ordered anisotropic phase having uniaxial symmetry, a 2D nematic-like phase (Figure 9b-d). This ordering occurs to maximize the entropy of the self-assembled structure by minimizing the excluded volume per particle in the array as predicted by the rigid-rod theory.23 Song et al.24 have reported experimental evidence on the phase behavior of multiwall carbon nanotube aqueous dispersions and found the isotropic-2D to nematic-2D phase transitions for nanotubes with an aspect ratio of ca. 30. This contribution, however, does not account for the tube-tube directional interactions such as interfacial energies and van der Waals attraction. The observation of transient, metastable nematic ordering in our nanotube system (with an aspect ratio much larger than 30) indicates the importance of both entropy and energetic consideration for the nonideal carbon nanotube system. Therefore, the most likely explanation for the wsSWCNT trapping and orienting mechanism at the air/water interface is being dictated by a competition between the minimization of the local surface free energy that the system possesses, or due to an entropic contribution, that is, the maximum entropy that the bulk (subphase) can gain due to the carbon nanotube adsorption process itself. The entropy loss by the adsorption of material at the air/water interface is overcompensated by the subphase entropy gain due to the water molecules freed by the interface. This mechanism is not exclusive for carbon nanotubes solely, but even it can be generalized for other molecular systems adsorbing at the air/water interface as well. Additional Π-A isotherms demonstrated that both adsorption time and surface pressure are critical parameters for the fabrication of good quality nanotube Langmuir films. For instance, LB films prepared at Π ) 5 mN/m but using a longer adsorption time (up to 173 h) partially feature multilayer domains, which are characterized by thicker films (2-4 nm), as depicted in Figure 10a. In contrast, when LB films are prepared by increasing the lateral pressure up to Π ) 10 mN/m and using shorter adsorption times (89 h), raftlike islands transform into denser close-packed monolayer domains with heights of about 0.7-0.9 nm (cf., Figure 10b). It is noteworthy to mention that below both the soap-froth and the raftlike structures exists a continuous and more ordered monolayer made of ws-SWCNTs. This is not shown clearly due to the z-range used in each AFM image. Instead, it helps to strengthen the different multilayer structures formed at higher pressures. During this process, a monolayer of nanotubes in a nematic-like arrangement is first obtained. The regularity of sideby-side intertube distance is reflected in the Fourier transform of the region (Figure 10b, inset). These observations suggest that, while the extent of the compression-induced orientation of ws-SWCNTs does depend on the surface pressure, it is a mechanism complicated by the involvement of other factors such as adsorption time, distribution in length, aggregation, and possible entanglement and bending of the nanotubes. In such

23186 J. Phys. Chem. B, Vol. 110, No. 46, 2006

Herna´ndez-Lo´pez et al.

Figure 9. AFM tapping-mode height images of ws-SWCNT LB monolayers vertically transferred to a freshly cleaved mica surface at different compression trough areas (A): (a) 250, (b) 200, (c) 150, and (d) 100 cm2. The inset in each image shows the 2D Fourier transform of the image, indicating how the monolayer evolves toward a 2D nematic-like ordering.

cases, the competition between bending elasticity and adhesion is sufficient to explain the shapes seen in filamentous aggregates.20 In Figure 11, a soap-froth collapses around ropes of wsSWCNTs, causing the ropes to form a ring. The images obtained by AFM show that the ring does not have a constant thickness and height around its circumference, indicating that it could be formed by separate ropes being curled together or by a coiling process. The occurrence of the initial buckling transition is determined by a competition between bending induced by buckling of the nanotube and the maximum compressive force that the interface can exert. The Euler buckling load for a simply supported nanotube of length l is25

Fb )

YIπ2 l2

(12)

where Y [N/m2] is the Young’s modulus of the carbon nanotubes, and I [m4] is the areal moment of inertia given by the second moment of the mass distribution in a cross-section perpendicular to the axis of symmetry.26 A liquid interface exerts a compression force Fc ) 2πrγ cos θ, where r is the radius of the nanotube, γ is the surface tension of the solid-vapor interface, and θ is the contact angle at the interface, assumed to be at its equilibrium value. If Fc > Fb, the nanotube buckles; otherwise, it remains straight. Balancing the two forces yields a critical length below which the nanotube remains straight and above which it buckles.

lc )

(2rγπYIcos θ)

1/2

(13)

For our ws-SWCNT ring, YI ≈ 1.95 × 10-25 N m2,27 r ≈ 0.7 nm, γ ) 71.89 mN/m,28 and θ ≈ 58°;29 therefore, lc ≈ 107

Polyelectrolyte/Single-Wall Carbon Nanotube Complexes

J. Phys. Chem. B, Vol. 110, No. 46, 2006 23187

Figure 10. (Upper figures) AFM tapping-mode height images of ws-SWCNT LB monolayers vertically transferred to a freshly cleaved mica surface at different conditions: (a) Π ) 5 mN/m, A ) 50 cm2, and 173 h adsorption time; (b) Π ) 10 mN/m, A ≈ 80 cm2, and 89 h adsorption time. The contrast shown covers height variations in the range of 0-5 nm. (Lower figures) Observed profiles along the black line of the upper figures.

nm. The value determined by Martel et al.30 for a typical SWCNT yields lc ) 155 nm, very close to the calculated value of 107 nm. The discrepancy observed between our prediction and the result of Martel et al. may be due to the fact that in Martel’s experiments take place some intertube shear, nonzero contact angle between the water and the rope, and a reduction of the surface tension of the water from dissolved H2SO4. Polyelectrolyte/SWCNT Adsorption at the Air/Water Interface: Diffusive Regime. The Lag Time. Because of the large molecular mass and high aspect ratios of carbon nanotubes (e.g., as compared to polyelectrolytes), it can take several days for carbon nanotubes to reach the equilibrium adsorption state described by Π ) f(Γ). Figure 12 represents a typical plot of surface pressure versus time Π(t) of ws-SWCNT dispersion on the subphase. To build this plot, each (Π,tads) point was obtained from independent Π-A isotherms, taken as the average of two surface pressure values reached at A ) 100 cm2. At this area value, it ensures that a single monolayer is being formed at the air/water interface. A lag time (tlat), during which the surface pressure is very low, seems to be characteristic of carbon nanotube adsorption curves. It is determined as the intersection between the time axis and the straight line following the beginning of the pressure increase. The surface concentration at the end of this initial period (Γlagtime) can be calculated in the same way from surface pressure-area isotherms. This plateau is seen as corresponding to the coexistence of a (LE+G) phase

of the carbon nanotube isotherm, the part for which the expected pressure appears to be of the same order as the noise of measurements. It is then consistent to consider that, at least during this regime, there is no energy barrier for adsorption, which means that the dynamics is dominated by the transport in the bulk. The crossover between diffusion and convection regimes has become a fundamental question in many mass transport-limited systems, and experimental evidence as in the case of carbon nanotube adsorption is particularly scarce. The first generalized results that treated to describe the diffusiveconvective transition behavior were achieved studying the adsorption of proteins at the air/water interface. For example, McRitchie and Alexander,14 and more recently Ybert and di Meglio,31 in their experiments on the bovine serum albumine (BSA) adsorption by the pendant drop method, found that there exists a correlation between the protein concentration and the lag time to model the transport in the bulk. The only degree of freedom is the width of the hydrodynamic layer δ imposed by free-convection. For that case in particular, the lag time exhibited in the diffusion limit (c0 < 10-3 wt %) a well-defined c-2 behavior, according to eq 9, and it changed in the convection limit (c0 > 10-3 wt %) to a c-1 behavior (eq 10). The crossover between those two regimes should occur when proteins from distances greater than δ are required to reach Γlagtime; that is, Dtlat g δ2. However, no experimental values for the carbon nanotube lag time have been reported previously. The best fitted

23188 J. Phys. Chem. B, Vol. 110, No. 46, 2006

Herna´ndez-Lo´pez et al. TABLE 2: Characteristic Adsorption Parameters of ws-SWCNTs at the Air/Water Interface at 298.15 K and at pH 6.0 D (m2/s)

δ (µm)

Γlagtime (mg/cm2)

(1.94 ( 0.36) × 10-11

135 ( 37

(1.31 ( 0.46) × 10-4

TABLE 3: Kinetic Parameters for the Adsorption of ws-SWCNTs at the Air/Water Interface at 298.15 K and at pH 6.0 m0 (µg)

Log Π vs (1/2) Log t intercept (mN/m‚s1/2)

13.4 ( 0.7

0.0204 (0.982)

tlata (s)

k2 × 105 (s-1) (LRb)

post-lag time (s)

461 411 1.46 (0.970) 645 263

a

Period during which diffusion controls the kinetics of adsorption of ws-SWCNTs at the air/water interface. b LR ) linear regression coefficient.

heat transport theory in fluids and adapted to the transport of molecules by free-convection,32 the overall mass-transfer coefficient kMT is given by

kMTL ) 0.069Gr0.33Sc0.407 D

(14)

where Gr and Sc are the Grashof and Schmidt numbers, respectively.

Gr )

Figure 11. Ring formation in ws-SWCNTs when a soap-froth collapsed around ropes of ws-SWCNTs, causing the ropes to coil. Section analysis: The cross-section gives an external diameter of 628.91 nm, an internal diameter of 457.03 nm, and height ) 9.893 nm. From these measurements, a rope diameter between 30 and 35 nm was estimated. Experimental conditions: Π ) 0.24 mN/m, A ) 200 cm2.

Figure 12. Rate of ws-SWCNT adsorption at the air/water interface. The data, Π versus t, showed a nonlinear relation. From the graph, two regimes are present. The sample mass deposited onto the subphase was m0 ) 13.4 ( 0.7 µg. T ) 298.15 K, pH ) 6.0. The error bars in Π correspond to 0.1 mN/m.

value for the term Dtlat of the diffusion layer taken from our data for ws-SWCNTs is ca. 3000 µm. Using models taken from

η gF2βL3 ∆T, Sc ) 2 FD η

(15)

In these two expressions, L is the characteristic length of the rectangular enclosure, D is the molecular diffusivity, g is the gravitational constant, F, η, and β are the density, the absolute viscosity, and the isothermal compressibility of the fluid, respectively, and ∆T is the overall temperature difference driving the free-convection. As kMT ) D/δ and using L ) 1 cm, D ) (1.94 ( 0.36) × 10-11 m2/s, ∆T ≈ 0.2 K, and for F, η, and β the values of water,33 we obtain the estimation δ ) 135 ( 37 µm, which is a much smaller value than the one fitted by Dtlat approximately by a factor of 22. The great overestimation found in δ layer between our prediction for ws-SWCNTs and the formula reported by Ybert and di Meglio for BSA could arise from differences in the solute concentration c0, kMT, L, and the method used in their experiments because BSA and wsSWCNTs have a very similar diffusion coefficient (BSA: L ) 2 cm, δ ) 400 µm, D ) 6.7 × 10-11 m2/s).31 Subsequently, because c0 for our ws-SWCNT nanotube system is assumed to be in the convective limit, kMT and tlat are then substituted in eq 11 to calculate the ws-SWCNT surface concentration at the lag time (Γlagtime). A summary of the main adsorption parameters obtained with our methodology is outlined in Table 2. Post-Lag-Time Dynamics. The induction period (ending at the lag time) is a region governed by diffusion and freeconvection transport processes. The presence of an induction time is found for the adsorption of carbon nanotubes from the bulk (cf., Table 3). However, to know the limits from one regime to another, comparison between experimental and calculated surface concentration and surface pressure relationships is necessary, and the influence of carbon nanotube concentration and fluid viscosity in the evolution of surface pressure Π(t) must be investigated. Surface Pressure as a Function of Γ. With our experimental setup it is possible to measure the surface pressure as a function of surface concentration by monitoring Π as function of time.

Polyelectrolyte/Single-Wall Carbon Nanotube Complexes

Figure 13. Relation of Π(Γ) versus Γ calculated from eq 7 assuming only diffusion and convection in the system for the mass transport of the ws-SWCNTs from the bulk to the air/water interface.

Combining the experimental Π(t) and the theoretical expression (eq 7) for Γ(t), it is possible to compute what would be the dependence of Π as a function of Γ if the dynamics were always been dominated by transport processes in the bulk. A Π(Γ) relationship, resulting from the above calculation, is plotted in Figure 13. The curve agrees very well from Π ≈ 0 to about 5 mN/m; then, the calculated Π(Γ) leads to higher surface pressures and a nonlinear behavior, indicating that the model estimates correctly the flux of carbon nanotubes toward the surface. At the initial stage in the formation of the adsorbed layer, solute molecules will diffuse from the bulk to the subsurface more slowly than they pass from the subsurface to the surface, on the assumption that equilibrium is established instantaneously between the latter two regions. This is very probable, because initially the surface is practically empty and every molecule arriving at the surface is likely to arrive at an empty “site” and become adsorbed. This can be understood as follows: when the surface is not covered enough, the time for carbon nanotubes to incorporate themselves to the surface is shorter than the time needed to diffuse from the bulk to the subsurface, and diffusion can describe the adsorption dynamics. From a kinetic point of view, the rate increase in surface pressure by ws-SWCNT adsorption after the lag period could be caused by different processes: (i) the ws-SWCNTs have to diffuse from the bulk to the subsurface (a layer immediately adjacent to the fluid interface, denoted by δ) by diffusion and/or convection; (ii) the first step is followed by the ws-SWCNT adsorption at the interface; and (iii) the adsorbed ws-SWCNTs (segments) rearrange at the fluid-fluid interface, a process caused by a balance between attractive van der Waals interactions and repulsive electrostatic interaction of the charged segments due to the polymer adsorbed on the nanotubes. Diffusion of ws-SWCNTs in Water. The most important step in the formation of 2D-structures in a continuum phase is the molecular condensation of the carbon nanotubes at the interface. At low surface concentrations, the surface pressure is low, and molecules adsorb irreversibly by diffusion. In the case of diffusion-controlled adsorption, the diffusion is driven by a concentration gradient.34 Thus, the first step of the adsorption process can be obtained from integration of Fick’s second law. The simple diffusion model accounts for the changes in surface pressure, Π, with time (eq 16). This equation indicates that a

J. Phys. Chem. B, Vol. 110, No. 46, 2006 23189

Figure 14. Diffusion of carbon nanotube adsorption at the air/water interface. The data, Log Π versus (1/2) Log t, show a linear relation in the interval of surface pressure corresponding to short times before the lag time. The sample mass deposited onto the subphase was m0 ) 13.4 ( 0.7 µg. T ) 298.15 K, pH ) 6.0.

Log-Log plot of surface pressure versus the square root of time will be linear.

[

Log Π(t) ) Log 2c0kBT

(Dπ) ] + 21Log t 1/2

(16)

The application of eq 16 to monitor the diffusion kinetics of carbon nanotube molecules at the air/water interface is shown in Figure 14. At this carbon nanotube concentration in water, we find a good fit of the experimental data with the Ward and Tordais’ equation in the interval of surface pressure corresponding to short times before the lag time is reached (cf., Table 3). Thus, it can be concluded that, during the initial period, the carbon nanotube adsorption kinetics at the air/water interface is controlled by a diffusion mechanism. The molecular diffusion coefficient D for the ws-SWCNTs is calculated from the intercept derived from the Log Π versus (1/2) Log t line in Figure 14. This value is reported in Table 3. The discrepancies observed at longer adsorption times, as surface pressure becomes higher than about 4 mN/m, could be due to an energy barrier for the nanotubes adsorption, related to a collapse (penetration) and organization/reorganization of the monolayer film at the interface after diffusion takes place, as will be treated in the next section. Penetration and Rearrangements of ws-SWCNTs at the Air/ Water Interface. The film stabilization depends to a great extend on the penetration and further rearrangements of previously diffused molecules at the air/water interface. To monitor induction processes at the interface and configurational rearrangements of adsorbed ws-SWCNT molecules, the following first-order equation can be used

Log

(

)

Π∞ - Π ) -kit Π∞ - Π 0

(17)

where Π∞, Π0, and Π are the surface pressures after an infinite time of adsorption, at time t ) 0, and at any time t, respectively, and ki is the first-order rate constant. In practice, a plot of eq 17 usually yields two or more linear regions. The initial slope is taken to correspond to a first-order rate constant due to an induction process (k1), while the second slope is taken to correspond to a first-order rate constant of molecular rearrange-

23190 J. Phys. Chem. B, Vol. 110, No. 46, 2006 ment (k2), occurring among a more or less constant number of adsorbed molecules. For all experiments of carbon nanotube adsorption, we found two linear regions in the plot of Log[(Π∞ - Π)/(Π∞ - Π0)] versus t. However, because carbon nanotube adsorption at fluid interfaces is very time-consuming, no attempt was made to analyze the experimental data for the induction step of previously adsorbed ws-SWCNT molecules. The ability of the carbon nanotube molecules to create space in the existing film and penetrate and rearrange at the interface is ratedetermining, when an activation energy barrier to adsorption exists (as deduced previously from the data in Figure 12). As a general trend, it can be assumed that the value of k2 increases with increasing c0 (cf., Table 3). That is, penetration of ws-SWCNTs at the interface will be facilitated by higher carbon nanotube concentration in the subphase. In this particular case, the pH does not play any significant role in the penetration of ws-SWCNTs at the air/ water interface. On the basis of molecular mass, the probability of penetration into the interface of previously diffused molecules should be higher for ws-SWCNTs with a lower molecular mass than the higher ones, but this was not observed in this study. Thus, the most likely explanation for these results is that, at almost neutral pH conditions (pH 6.0), both individual and bundles of carbon nanotubes are completely dissociated in their charged forms. Moreover, at this pH, the effect of their differences in aspect ratios is null. That is, at pH 6.0, dissociated ropes and bundles of polyelectrolyte/carbon nanotube composites penetrate at the interface at the same rate. Conclusions We have succeded in forming homogeneous thin films of ws-SWCNTs through an interfacial effect and using the Langmuir and LB techniques. It is possible to describe the dynamics of adsorption by only taking into account the diffusion of wsSWCNTs from the bulk solution to the surface and the thermal convection inside the trough. We report the formation of different patterns formed by the ws-SWCNTs at the air/water interface, such as soap-froths, rings, and cumuli-like. wsSWCNTs spontaneously assemble into ring shapes due to competition between elastic and interfacial energies. This feature can be explained with a simple mechanical model. By the assistance of the good film-forming properties of quenched polyelectrolytes, a stable monolayer of polyelectrolyte/SWCNT complexes can be realized at the air/water interface. The deposition can be performed via the vertical dipping method. AFM studies revealed that the morphology of these transferred films depends on the surface pressure, and aligned structures with a nematic-like order formed a closely packed mono- or multilayer films. In terms of applications, not only these wsSWCNT thin films but even pristine carbon nanotube films, fabricated according to this methodology, may contribute to developments in the design and construction of chemical and biological sensors, optical devices, and other interesting molecular and nanoscopic architectures. Acknowledgment. J.L.H.-L. gratefully acknowledges financial support from PROMEP through a postdoctoral fellowship. We acknowledge support from CONACyT through grant no. 45944 and FAI-UASLP. References and Notes (1) Bahr, J. L.; Mickelson, E. T.; Bronikowski, M. J.; Smalley, R. E.; Tour, J. M. Chem. Commun. (Cambridge) 2001, 2, 193.

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Yet the presence of both individual and small nanotube bundles has prompted us to term polyelectrolyte/SWCNT complexes in ethanol as “dispersion”. True solutions of ws-SWCNTs can be obtained by dilution in ethanol, shown to exhibit much better resolved optical absorption spectra. (10) Girifalco, L. A.; Hodak, M.; Lee, R. S. Phys. ReV. B 2000, 62 (19), 13104. (11) (a) Nikolaev, P.; Bronikowski, M. J.; Bradley, R. K.; Rohmund, F.; Colbert, D. T.; Smith, K. A.; Smalley, R. E. Chem. Phys. Lett. 1999, 313, 91. (b) Chiang, I. W.; Brinson, B. E.; Huang, A. Y.; Willis, P. A.; Bronikowski, M. J.; Margrave, J. L.; Smalley, R. E.; Hauge, R. H. J. Phys. Chem. B 2001, 105, 8297. (c) Zhou, W.; Ooi, Y. H.; Russo, R.; Papanek, P.; Luzzi, D. E.; Fischer, J. E.; Bronikowski, M. J.; Willis, P. A.; Smalley, R. E. Chem. Phys. Lett. 2001, 350, 6. (12) Moya, S. E.; Ilie, A.; Bendall, J. S.; Hernandez-Lopez, J. L.; RuizGarcia, J.; Huck, W. T. S., to be submitted. (13) (a) Ho¨ning, D.; Mo¨bius, D. J. Phys. 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Polyelectrolyte/Single-Wall Carbon Nanotube Complexes (25) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity, 3rd ed.; Pergamon Press: New York, 1986. (26) For molecular-sized objects such as nanotubes, Y and I are not separately well-defined, but their product is. It is often termed “flexural rigidity”. See, for example: Yakobson, B. I. Nanomechanics. In Handbook of Nanoscience, Engineering, and Technology; Goddard, W. A., III, Brenner, D. W., Lyshevski, S. E., Iafrate, G. J., Eds.; CRC Press LLC: Boca Raton, FL, 2003; pp 17-15. (27) Salvetat, J.-P.; Briggs, G. A. D.; Bonard, J.-M.; Bacsa, R. R.; Kulik, A. J.; Sto¨ckli, T.; Burnham, N. A.; Forro´, L. Phys. ReV. Lett. 1999, 82, 944. (28) This value was obtained from Π ) γ - γ*, where Π ) 0.24 mN/m at A ) 200 cm2 and γ ) 72.13 mN/m (T ) 298.15 K). See, for example:

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