J. Phys. Chem. C 2007, 111, 11583-11589
11583
Critical Coagulation Concentrations for Carbon Nanotubes in Nonaqueous Solvent Andrea N. Giordano,† H. Chaturvedi,‡ and J. C. Poler*,†,‡ Department of Chemistry and Center for Optoelectronics and Optical Communications, UniVersity of North Carolina at Charlotte, 9201 UniVersity City BouleVard, Charlotte, North Carolina 28223-0001 ReceiVed: April 17, 2007; In Final Form: May 18, 2007
Critical coagulation concentrations for several inorganic and metallodendrimer coagulants have been determined for dispersion of single-walled carbon nanotubes in nonaqueous solvent. The behavior of the nanotubes is not consistent with previously reported coagulations from aqueous electrolytes and is not described using classical Derjaguin, Landau, Verwey, and Overbeek theory of lyophobic colloids. Nanoscale rigid dendrimers are studied as they interact and bind to the carbon nanotubes. With a diameter of 5.8 nm and a charge of +20, these metallodendrimers bind strongly and specifically to the nanotubes. Systematic studies of aggregating nanotubes and nanoparticles are required to more completely understand the complex interactions between the nanotubes and the matrix in which they are dispersed. Strategies for directed self-assembly of the nanotubes have implications for the potential three-dimensional nanomanufacturing of nanoscale sensors and actuators.
I. Introduction Three-dimensional nanomanufacturing at economical throughput implies a directed self-assembly paradigm. Using nanotubes and nanowires for mechanical elements and scaffolding can enable the construction of low-cost, functional systems. The work presented here focuses on the interactions of single-walled carbon nanotubes (CNTs) with each other and with molecular ions in solution. A fundamental understanding of these interactions is still needed. An accurate description of assembling and coagulating CNTs is still not available and is required to push forward potential applications of CNTs as functional mechanical structures. In general, CNT powder can be dispersed, using various methods including ultrasonication, into various solvents.1-6 Dispersed nanotubes can act as lyophobic colloids, in that they do not strongly interact with their solvent. All CNT dispersions are unstable and will eventually aggregate and coagulate. It is the directed control of the aggregation process that will facilitate the placement, alignment, and integration of nanotubes into devices. Molecular aggregation and self-assembly is a well-studied field and will not be reviewed here.7,8 It is recognized that the size, shape, charge distribution, and electron density of a molecular system leads to its thermodynamic self-assembly into aggregates of nontrivial geometry. Moreover, the study of the aggregation of colloidal systems is also a mature field that is well-described by the DLVO theory of Derjaguin, Landau, Verwey, and Overbeek.9 Often the stability of a dispersion is characterized by the Schulze-Hardy (SH) rule, where the critical coagulation concentration (CCC) is related to the valence of the charged coagulant: (CCC) ∝ Z-6 + . This behavior is predicted by the DLVO theory for certain particle geometries and some constrained conditions. Sano10 et al. have studied the rapid coagulation of CNTs in aqueous media as a function of valence * Corresponding author. Phone: 704 687-3064. Fax: 704 687-3151. e-mail:
[email protected]. † Department of Chemistry. ‡ Center for Optoelectronics and Optical Communications.
on several inorganic coagulants. Their analysis shows good agreement with the SH rule. This result is surprising, given the assumptions made in the DLVO model and the extraordinary properties of single-walled CNTs.11-13 The kinetics of coagulation depend on the surface functionality of the pristine or modified CNT and on the nature of the solvent. CNT dispersions can collapse in minutes or be “stable” for months. The physical state of dispersed CNTs is still not clear. Several solution-phase scattering experiments indicate that dispersions of nanotubes consist of intertwined particles forming fractal geometries.14,15 Only under excessive ultrasonication do these light scattering results imply a more rodlike morphology consistent with dispersed and isolated single-walled carbon nanotubes.16 When CNT dispersions are deposited onto a surface, both morphologies are observed. Direct measurements like transmission electron microscopy, scanning electron microscopy (SEM), and atomic force microscopy (AFM) can discern these morphologies. We have shown that deposited CNTs are found in mats of bundles and ropes, and as isolated and aligned single-walled nanotubes.17 It is unlikely that fractal objects in solution would separate into isolated CNTs upon deposition. We believe an accurate description of dispersed CNTs is still needed. It is however likely that a dispersion of rodlike isolated CNTs will eventually aggregate in solution into more complex morphologies and eventually form a floc and segregate from the liquid phase. It is this process that interests us, and its description is central to the concepts described below. While interacting plates and spheres are accurately described by DLVO theory, the geometry and surface properties of CNTs are not. The well-accepted method of treating interacting particles in solution is to sum the attractive dispersion forces between the particles and the repulsive electrical double-layer forces. When the Poisson-Boltzmann equation is linearized and integration is done properly, the net potential energy of interacting spheres is9,18
V(x)sphere-sphere ) πr
[
-H121 64kTNoΓo2 exp(-κx) + 12πx κ2
10.1021/jp0729866 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/13/2007
]
11584 J. Phys. Chem. C, Vol. 111, No. 31, 2007 where r is the radius of the sphere, H is the Hammaker constant for the system, Γ is a result of linearization and depends on the ζ potential on the particles and the charge valence, Z, of the solvated ions, and 1/κ is the Debye length, which depends on the number density of ions in solution, No, and the dielectric strength of the solvent, 1/κ ) {[rokT]/[∑i (Zie)2Nio]}1/2. This and the potential energy for interacting plates V(x)slab-slab are results of several approximations. To apply the Debye-Huckle approximation used in linearizing the electrostatics, the ζ potential on the particles must be less than 25 mV (kT). CNTs in aqueous solvent have ζ potentials in the range of -15 to -65 mV. In order to integrate these equations in the closed form, the second assumption is that the radius of the particle must be much larger than the Debye length, κd > 5. However, the diameter of a CNT is much smaller than the Debye length in the solutions we are using. This requires numerical integration to model the system properly. Obviously, the geometry of interacting CNTs is neither spherical nor planar. Recent work has derived the electrical double-layer repulsion for a sphere interacting with a cylinder.19 Much progress has been made on the van der Waals (vdW) attraction between CNTs using a universal graphite potential,20 resulting in a general attractive vdW potential between single-walled nanotubes.21,22 While progress on theoretical descriptions of CNT aggregation is made, further experimental work is required. Recent molecular dynamics experiments on CNTs in water show an interesting solventinduced nanotube-nanotube repulsive interaction that is not taken into account in other models.23 This interaction should be found for other nonaqueous solvents. II. Experimental Methods All single-walled CNTs were used as purchased (Carbon Nanotechnologies Inc. HiPco tubes purified Grade P). Stable dispersions of the CNTs were made in N,N-dimethylforamide (DMF) (Aldrich as-received). Dispersions were made by adding excess (∼1.0 mg) CNT powder into 100 mL of DMF and ultrasonicating the mixture for 30 min at 10 W (RMS) of power (Fisher Model 60 1/8” probe tip). The opaque black solution was filtered through glass wool to yield a clear and gray CNT solution. CNT solution concentrations of 3.2 mg/L were measured from UV-vis spectroscopy (Cary 5000) by measuring the optical density at 1000 nm and were used throughout this study. We used the extinction coefficient from previous studies6 of nonaqueous CNT dispersions measured at 500 nm and recalibrated the absorbace versus [CNT] at 1000 nm. We measure the absorption at 1000 nm (0.0179 A/(mg/L)) to determine the [CNT] because our coordination compounds do not absorb there and the spectra are relatively flat and free of van Hove singularities. These dispersions were stable for weeks to months. Water absorbs readily into DMF. The solutions were capped, and the concentration of water in the dispersions was estimated to be 0.5% (w/w) from H NMR analysis. All solutions were kept in dim room light throughout the study. All ruthenium reagents were used as available. Stock solutions of [Λ6∆3Λ-Ru10]20+[PF6-]20 (a gift from F. M. MacDonnell) and of [Ru(phenanthroline)3]2+[PF6-]2 were prepared in DMF. Using MacDonnell’s notation, we refer to the [Λ6∆3Λ-Ru10]20+[PF6-]20 compound as the “decamer” since it is composed from 10 units of modified [Ru(phenanthroline)3]2+[PF6-]2, which we refer to as the “monomer”. All inorganic coagulants were used without purification and dissolved into DMF. The molar conductivity of these solutions was measured using an Accumet AR20 conductivity meter. Ruthenium complexes are highly absorbing in the UV-vis spectrum and are easily monitored. Concentrations of the
Giordano et al.
Figure 1. Relative CNT concentration remaining dispersed in solution after centrifugation at low acceleration. Removal of CNT floc without affecting dispersed nanotubes was effective at 100 g for 5 min.
ruthenium complexes in dispersions were measured by calibration against known standards. Decamer concentrations of 2 × 10-9 M (0.2 ppb Ru) are above the sensitivity limit of our spectrometer and technique. After the CNTs coagulate, the flocs are separated from solution by centrifugation at 100 g for 5 min. The supernatant is analyzed by UV-vis-NIR spectroscopy to determine the [CNT]. Removing coagulated CNTs from the supernatant was critical toward increasing the precision of our measurements. It is required that the flocs are removed from the solution without also removing dispersed CNTs. Centrifugation will remove all of the nanotubes eventually. The data in Figure 1 show the relative changes in [CNT] in milligrams per liter as a function of acceleration. As the acceleration field and time increase, the concentration of pristine CNTs in solution decreases slightly. However, after 15 min, and up to 300 g, the decrease in [CNT] is negligible. Adding a charged coagulant like barium nitrate destabilizes the dispersion. The same CNT solution was treated with barium nitrate at concentrations below the CCC for that coagulant. While the [CNT] does fall off slightly faster, as expected, it only changes by 1-2% after 5 min of centrifugation at 100 g. Care was also taken when withdrawing the CNT dispersion for spectroscopic analysis. The same part of the centrifuged solution (100 µL) was withdrawn by pipet for all samples in this study. When a floc was inadvertently withdrawn into the supernatant, the experiment was discarded. III. Results and Discussion CNT dispersions were checked for isolated tubes using AFM and UV-vis-NIR spectroscopy. Deposited nanotubes were analyzed by contact AFM and had the expected diameter distribution of 1.5 ( 0.7 nm, indicating isolated tubes. Moreover, the clearly visible van Hove singularities shown in Figure 2 are consistent with isolated tubes found in previous work.24 Transitions centered around 0.87 eV are consistent with calculated absorptions from isolated semiconducting tubes with diameters of 1.0-1.2 nm.24,25 A preliminary analysis of the spectral region from 0.70-1.10 eV indicates 21 discrete transitions from the two semiconducting van Hove singularities SE and SE , demonstrating adequate dispersion of the isolated 11 22 tubes needed for the studies below. Important to the results below is the dispersal of pristine CNTs into the solvent without significant deterioration of the tube’s electronic and mechanical
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Figure 2. UV-vis-NIR spectroscopy of CNT dispersions in DMF after significant ultrasonication to ensure a dispersion of isolated tubes. Clearly resolved van Hove singularities are not affected under the sonication protocol used in this study.
structure. It is also required that the tubes are not inadvertently chemically functionalized or brought into solution with the use of a surfactant. The choice of solvent26 and use of a surfactant25 are determining factors to increase the amount of dispersion material. Our choice of solvent and omission of a surfactant are required to simplify the system under study. To disperse the CNTs into the DMF, we used ultrasonication. While sonication can disrupt the integrity of the nanotubes, we were careful not to perturb them too aggressively. The UV-visNIR spectra in Figure 2 show consecutive scans after additional sonication of the dispersion. There seems to be some minor degradation of some tubes as indicated in the inset of Figure 2. However, in general, there is no change in these spectra even after an additional hour of sonication.27 These CNTs are dispersed, isolated, and pristine. In order to determine the CCC for various coagulants, we measure the concentration of the CNTs left in solution after the addition of a coagulant. We are particularly interested in understanding how nanoscale metallodendrimers interact and bind to CNTs in solution. The decamer used in this study has a valence of Z+ ) +20, and the largest distance across the decamer is 5.85 nm. Moreover, the molecule has several potential binding sites that are suitable to host a single-walled CNT (1-2 nm clefts and pockets).17 This decamer has a very strong metal-to-ligand charge transfer (MLCT) absorption at 441 nm (441 ) 225 000 M-1 cm-1) and an even stronger π f π* (375 ) 257 600 M-1 cm-1) absorption from the phenazine bridging ligands between the ruthenium metal centers.28 UVvis-NIR spectra of the decamer in DMF are shown in Figure 3. The large optical cross-section in the UV spectrum allows us to detect the decamer at very low concentrations (∼2.5 × 10-9 M). Both the decamer and the monomer absorb strongly out to ∼700 nm. Therefore, we measure the concentration of CNTs at 1000 nm. After high concentrations of decamer are added to a CNT dispersion, the nanotubes rapidly coagulate out of solution. When using this technique to measure small amounts of nanotubes in a nonaqueous solvent, care must be taken not to confuse the SE11 nanotube bands with the overtone and combination absorption bands from dissolved water. DMF absorbs water from the air if left open. The absorption at 1425 nm is from water. The concentration of water in the samples shown in Figure 3 is 0.5%.
Figure 3. UV-vis-NIR spectroscopy used to determine the concentration CNT in solution and also the amount of metallodendrimer not associated with the aggregated nanotubes. At high concentrations of added decamer, all of the CNTs are removed from solution. The π f π* transition, indicated in B, of the decamer is used to show that almost all of the decamer is strongly bound to the CNTs.
Of the different coagulants we have studied, the decamer is the only one that strongly binds to the nanotubes. The spectra in Figure 3 are of the supernatant of a 3.2 mg/L CNT dispersion less than an hour after the decamer was added at a concentration of 0.225 µM. After the floc was centrifuged down at 0.1 kg for 5 min, the spectra showed a negligible amount of CNT (0.2 mg/L) left in solution. A subsequent centrifugation at 7.0 kg removes nearly all of the remaining nanotubes. More striking in these spectra is that the absorptions from the decamer are barely visible even in an expanded view shown in Figure 3B. In Figure 3B, we show the π f π* absorption of the phenazine bridge ligands that remain in solution. We subtracted off a second-order fit to the spectra to remove any contribution from residual CNT absorption. From this analysis, the concentration of decamer that remains in solution after the coagulation is between 2.4 and 16.3, where the solution initially had a decamer concentration of 225 nM. We conclude that the decamer is strongly bound to the nanotubes. In previous work, we showed that the decamer was not removed from CNT flocs after extraction with acetonitrile.17 This is in stark contrast to the behavior of the inorganic coagulants described below and the coordination compound Ru(phenanthroline)32+ monomer. When atomic absorption spectroscopy (data not shown) of filtered solutions is used, it is seen that 90% of the monomer stays in solution. From UV-vis-NIR spectra in Figure 4, we have determined that ∼70% of the ruthenium monomer stays
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Figure 4. UV-vis-NIR spectra of CNT dispersions as a function of added Ru monomer. As the coagulant concentration increases, the dispersion becomes unstable.
Figure 5. Coagulation curve for CNTs in DMF as a function of added NaBr coagulant. CNT dispersions are stable when the coagulant concentration is significantly below the CCC. Intersection of the linear fits to the log-log data are used to determine the CCC for each of the coagulants used in this study.
in solution after the CNTs have collapsed out of the dispersion. The monomer does not bind strongly to the CNTs. The absorption at 1000 nm in Figure 4 determines the concentration of CNTs left in solution after the floc is removed (0.0179 A/(mg/L)). As the concentration of monomer coagulant increases, the concentration of CNTs decreases. However, below a certain monomer concentration, there is no change in [CNT]. This critical coagulation concentration is determined by analyzing the concentration data as illustrated in Figure 5. By plotting the absolute [CNT] on a linear axis, the reproducibility of these experiments is evident. The concentration of nanotubes, [CNT], versus [NaBr] is constant until it abruptly decreases with a further increase in the concentration of NaBr. For a [NaBr] of 10, 20, 50, and 100 µM, the experiments were done in triplicate. As the CCC is approached, the dispersion becomes unstable, and small variations in experimental conditions result in larger changes in [CNT]. In general, the onset of rapid coagulation of colloidal dispersions is difficult to determine. Turbidity and increased light scattering are often used to determine coagulation onset.29 Our dispersions became inhomogeneous as the flocs formed,
Giordano et al.
Figure 6. Relative concentration of CNTs that remain in solution as a function of increased Al(NO3)3 concentration. Coagulation curves were measured as a function of time after coagulant was added. Results between 30 and 70 h were consistent. These data were used to develop the experimental protocol used throughout this study.
and therefore light scattering was inherently imprecise. When we analyze the [CNT] versus [coagulant] on a log-log plot, we can more accurately determine the coagulation onset, as is shown in Figure 5. Equating the linear fits through the data before the CCC and just after the CCC, we determined the CCC for a given coagulant. The uncertainty in the CCC is determined by the uncertainty of the slope from the regression of the postCCC data. For all of the coagulants analyzed, we plot the relative CNT concentration [CNT]/[CNT]o, where we normalize with the original pristine CNT solution used as a control. Pristine CNT dispersions in DMF are stable for weeks without significant coagulation. All of the coagulations used to model the SchulzeHardy rule below were made from the same CNT dispersion. All of the experiments included in the CCC analysis were taken over a period of a few weeks, where the original dispersion remained stable. However, many other coagulation studies were required prior to these, in order to develop the experimental protocol described above. For coagulant concentrations near the CCC, the formation of flocs was rapid, but not instantaneous. Normalized [CNT] versus [Al(NO3)3 ] in DMF data are shown in Figure 6. These data are a compilation of many experiments over many months and show far less precision then is observed in the rest of the data set. Given these variations, it is clear that the CCC determination is sensitive to systematic variations. Data shown in Figure 6 were used to determine the optimum time, after the addition of coagulant, to measure the residual [CNT]. After only 9 h of aggregation, only the sample with the highest concentration of Al(NO3)3 coagulated enough to be centrifuged out of solution. However, from 30 to 70 h post-coagulantaddition, the coagulation curves were all similar. Therefore, we choose the optimum aggregation time to be 48 h for the Z+ ) +1, +2, and +3 coagulants. CNT concentrations remained constant for well over a week when the concentration of added coagulant was an order of magnitude smaller than the CCC. Coagulation using the Z+ ) +20 decamer took much longer, and we therefore waited 180 h before analyzing the residual [CNT]. For all coagulants, the CNT dispersions collapsed within minutes when the concentration of coagulant was more than an order of magnitude above the CCC. Coagulation curves for Ba2+ and Al3+ are shown in Figure 7. The CCC for Al(NO3)3 is lower than the CCC for
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Figure 7. Coagulation curves for various inorganic coagulants. Choice of coagulant was limited by solubility and DMF compatibility issues. As the valence of the cation increases, the CCC decreases. Anion properties do not seem to significantly affect the results.
TABLE 1: Molar Conductivities of Ionic Coagulants in DMF molar conductivity (cm2/Ωmol) H2O/DMF ratio
0:1
1:100
1:10
1:1
Ba(ClO4)2 Al(NO3)3 AlCl3
110 183 70
109 175 71
92.3 154 60.8
94.4 171 104
both Ba(NO3)2 and Ba(ClO4)2 as predicted by the SchulzeHardy rule and DLVO theory. The CCCs for both Ba2+ compounds are nearly identical. All of the coagulants we studied had an anion valence of Z- ) -1 (PF6-, NO3-, and ClO4-). We have not observed any dependence on the type of anion added to the DMF. Experiments using AlCl3 as the coagulant found a significantly higher CCC than that for the Al(NO3)3. All of these coagulants must dissolve and dissociate into DMF, which is not as good a solvent for the ionic coagulants as is water. The molar conductivities of the ionic coagulants were measured. Each solution was diluted with water to induce further dissociation. These results are listed in Table 1. The molar conductivity of Ba(ClO4)2 in DMF is less than the literature value30 of 148 cm2/(Ω mol) but does not increase with added water, and we assume it and Al(NO3)3 are fully dissociated in 0.5% H2O in DMF. While AlCl3 dissolves in DMF, we found that it does not fully dissociate until a significant amount of water is added to the DMF. At a 1:1 H2O/DMF ratio, the molar conductivity is consistent with the literature value31 of ∼100 cm2/Ω mol. Since the AlCl3 does not fully dissociate, the CCC should be higher than that for the Al(NO3)3 as observed. Attempts to measure coagulation curves for Fe3+ and Ce4+ salts were unsuccessful due to poor solubility and incompatibility with DMF. Coagulation curves for the metallodendrimers are given in Figure 8. Clearly, the Z+ ) +2 monomer has a much higher CCC than the Z+ ) +20 decamer, as expected. The metallodendrimer coagulants differ significantly from the inorganic coagulants. The phenanthroline ligands screen the coordinating metal’s charge from the solvent, while the inorganic ions coordinate to both the DMF (which is a good coordinating solvent) and the water. If the coagulants behaved as noncoordinating point charges, and the nanotubes were truly lyophobic colloids, then we should be able to model the CCC versus Z+
Figure 8. Coagulation curves for two metallodendrimers. The monomer has a charge of +2, while the decamer’s charge is +20. According to the SH rule, the CCC for the decamer should be orders of magnitude smaller than measured here. The slope of the data after the CCC is inversely proportional to the size of the ions in solution.
using the classic DLVO theory. But as we alluded to earlier, these assumptions are not general for molecular ions in nonaqueous solvents interacting with single-walled CNTs. Moreover, it is incorrect to describe the decamer as a +20 point charge with regard to its effect on the electrical double layer (EDL) around the CNTs. Not only is the diameter of the decamer (ddec) bigger than the diameter of the nanotube (dtube), but both their diameters are smaller than the Debye screening length in the DMF at the CCC of the decamer, 1/κ ) 220 nm. In order to derive the interaction between the EDLs of two parallel plates, Verwey and Overbeek9 showed that the stringent constraint of κdtube . 1, needed for Derjaguin’s integration method,32 can be relaxed to only κdtube > 5. For the system we describe, κdtube ≈ 0.01. A compilation of all of the CCC data as a function of Z+ of the coagulants is described using a SH plot and is shown in Figure 9. According to the DLVO theory for two infinite planes repelled by their EDL in the presence of symmetric ionic coagulants, while Derjaguin’s method is valid and the surface potential is constant and larger than kT, the Langmuir approximation can be used to solve the Poison-Boltzmann equation, resulting in a CCC that is inversely proportional to the valence of the counterion raised to the sixth power CCC ∝ Z+-6: the SH rule.9 This relationship is graphed on the log(CCC) versus log(Z+) plot of Figure 9, yielding a straight line with a slope of -6. The CCCs of all of the coagulants are shown as closed symbols on the same graph. A linear regression of all the data yields a slope of -3.1 with a standard error of 0.3 (n ) 5). A linear regression fit of the inorganic coagulant data only (Z+ ) +1, +2, and +3) yields a slope of -3.9, while the slope for the metallodendrimers only (Z+ ) +2 and +20) is -2.7. Because we only have two charge states for the metallodendrimers, we do not have enough data to statistically differentiate between the two regions of interacting particles. We assume that the monomer may be more similar to the inorganic coagulants than it is to the decamer since the latter binds strongly to the CNTs, while the former do not. Further analysis of the data in Figure 3 shows that more than 200 pmols of decamer are bound to the 3.2 µg of CNTs that collapse out of the solution. We have shown that the decamer binds strongly to the ends of the CNTs.17 However, there are
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Figure 9. Schulze-Hardy plot for all coagulants studied. According to the SH rule, a log-log plot of CCC versus Z+ should be linear. For the high surface potential approximation, the slope is -6, and for the low surface potential approximation, the slope is -2. Open squares indicate the modeled behavior for asymmetric coagulants like the ones used in this present study. Experimental data (closed symbols) are not consistent with CNT behavior in aqueous electrolyte solutions.
only 5-50 pmols (assuming a CNT molar mass of 1 - 5 × 106 g/mol) of nanotube ends in the dispersion. A SEM analysis of deposited CNT flocs shows some decoration along the sidewalls of the tubes, and an AFM analysis shows some codeposition of the decamer and the nanotubes. This is not observed from monomer coagulations. Given the calculated loading density of decamer per nanotube, there are 4-40 decamer molecules bound per nanotube. If the surface potential on the CNTs is smaller than kT, then the Poisson-Boltzmann equation can be solved by the DebyeHuckel approximation, yielding a CCC ∝ Z+-2 dependence,9 which is also graphed in Figure 9. Preliminary ζ-potential measurements of CNTs in DMF indicate a large surface potential (ζCNT ) -80 mV) that varies slowly with added monomer below the CCC (ζCNT+monomer ) -60 mV). Clearly, the DebyeHuckel approximation is not valid for this system. Classical DLVO theory assumes symmetric ionic coagulants where Z+ ) Z-. Except for NaBr, this is not the case for our coagulants. Hsu33-35 et al. have eloquently rederived the results of the DLVO theory of lyophobic colloids for the case of asymmetric electrolyte solutions where Z+ * Z-. For systems like ours that are best modeled using the Langmuir approximation, they find CCC ∝ 1/Z+4(Z+2 + 1), and this relationship is plotted with open symbols in Figure 9. This relationship approximates the SH rule for large Z+ values. For completeness, we also plot CCC ∝ 1/(Z+2 + Z+) with open symbols, where this relationship is the extension of the Debye-Huckel approximation for asymmetric coagulants. IV. Conclusions The decamer and the monomer studied both have accessible optical and electron-transfer properties which distinguish their potential interactions with the CNTs from the interactions of the inorganic coagulants and the nanotubes. We are surprised that the inorganic coagulants do not show similar behavior as seen by Sano10 et al. even though their coagulations were in aqueous electrolyte solutions. We are also surprised that the slope of the SH plot for the metallodendrimer coagulants falls very close to the slope from the inorganic coagulants. Obviously,
Giordano et al. additional charge states for the metallodendrimer series are needed to more fully explore the interactions between these large nanoscale molecules and the dispersed nanoparticles. Further progress on the nature of the decamer binding to the CNTs is needed and is being pursued. The dependence of the CCC on the valence of the counterions depends on the overall interparticle potential energy. The condition for rapid coagulation is when VT(r) ) 0 and ∂VT(r)/ ∂r ) 0, where the particles are separated by 1/κ. While there have been many recent advances in the theoretic description of nonspherical particles in solution, there is still no analytic solution that describes CNT dispersions. The attractive portion of the total potential between single-walled CNTs has recently been derived on a per length basis assuming parallel orientation21,22 using a universal graphite potential.20 The CNTs we used in this study are longer than their persistence length in solution and must be shortened before they are modeled by a rigid-rod geometry. Moreover, we do not have any evidence that the aggregating nanotubes are uniformly aligned to each other, as assumed in some theoretical models.21 There are more shortcomings with describing the repulsive EDL potential for CNTs. For geometries of parallel plate-plate, sphere-sphere,9 and sphere-cylinder,19 the EDL repulsion has been described. Further work on CNT-related geometries is needed. As we diversify our library of molecular coagulants, this line of investigation should yield insight toward directed self-assembly processes for nanoparticles in solution. Independently varying the charge state and the morphology of these molecular coagulants should enable methods of practical manufacturing of nanoparticle-based devices. Acknowledgment. We are deeply indebted to Professor F. M. MacDonnell at the University of Texas at Arlington for his generous donation of the ruthenium metallodendrimer [Λ6∆3ΛRu10]20+[PF6-]20. This research was supported in part by an award from Research Corporation and the NSF (Grant # 0404193) and significantly from the ARL (Grant # W911NF05-2-0053). We also thank Dr. Cliff Carlin and Dr. Jon Merkert and Ms. Sarah Subaran for their help and expertise on the properties of the ruthenium complexes and various technical supports. References and Notes (1) Wang, Y. B.; Iqbal, Z.; Mitra, S. J. Am. Chem. Soc. 2006, 128, 95. (2) Chichak, K. S.; Star, A.; Altoe´, M. V. P.; Stoddart, J. F. Small 2005, 1, 452. (3) Zhu, J.; Yudasaka, M.; Zhang, M. F.; Iijima, S. J. Phys. Chem. B 2004, 108, 11317. (4) Zheng, M.; Jagota, A.; Semke, E. D.; Diner, B. A.; McLean, R. S.; Lustig, S. R.; Richardson, R. E.; Tassi, N. G. Nat. Mater. 2003, 2, 338. (5) Zhao, W.; Song, C. H.; Pehrsson, P. E. J. Am. Chem. Soc. 2002, 124, 12418. (6) Bahr, J. L.; Mickelson, E. T.; Bronikowski, M. J.; Smalley, R. E.; Tour, J. M. Chem. Commun. 2001, 193. (7) Israelachvili, J. Intermolecular & Surface Forces, 2nd ed.; Academic Press: London, 1992. (8) Hill, T. L. Thermodynamics of Small Systems; Benjamin: New York, 1964. (9) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (10) Sano, M.; Okamura, J.; Shinkai, S. Langmuir 2001, 17, 7172. (11) Syue, S.-H.; Lu, S.-Y.; Hsu, W.-K.; Shih, H.-C. Appl. Phys. Lett. 2006, 89, 163115. (12) Yanagi, K.; Iakoubovskii, K.; Kazaoui, S.; Minami, N.; Maniwa, Y.; Miyata, Y.; Kataura, H. Phys. ReV. B: Condens. Matter Mater. Phys. 2006, 74, 155420. (13) Tao, N. J. Nat. Nanotechnol. 2006, 1, 173. (14) Saltiel, C.; Manickavasagam, S.; Menguc, M. P.; Andrews, R. J. Opt. Soc. Am. A 2005, 22, 1546.
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