Characterizing the Effect of Salt and Surfactant Concentration on the

Mar 31, 2016 - Separation of double-wall carbon nanotubes by electronic type and diameter. J. K. Streit , S. Lam , Y. Piao , A. R. Hight Walker , J. A...
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Characterizing the Effect of Salt and Surfactant Concentration on the Counterion Atmosphere around Surfactant Stabilized SWCNTs Using Analytical Ultracentrifugation Stephanie Lam, Ming Zheng, and Jeffrey A. Fagan* Materials Science and Engineering Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, United States

Langmuir 2016.32:3926-3936. Downloaded from pubs.acs.org by UNIV OF NEW ENGLAND on 08/20/18. For personal use only.

S Supporting Information *

ABSTRACT: Accurate characterization of dispersed-phase nanoparticle properties such as density, size, solvation, and charge is necessary for their utilization in applications such as medicine, energy, and materials. Herein, analytical ultracentrifugation (AUC) is used to quantify bile salt surfactant adsorption on length sorted (7,6) single-wall carbon nanotubes (SWCNTs) as a function of bulk surfactant concentration and in the presence of varying quantities of a monovalent saltsodium chloride. These measurements provide high precision adsorbed surfactant density values in the literature for only the second SWCNT structure to date and report the quantity of adsorbed surfactant across a broad range of bulk surfactant concentrations utilized in SWCNT dispersion processing. Second, the measurements presented herein unambiguously demonstrate, via AUC, a direct relation between the size of the counterion cloud around a surfactant-stabilized SWCNT and solution ionic strength. The results show that changes in the size of the counterion cloud around surfactant-stabilized SWCNT are attributable to electrostatic phenomenon and not to changes in the quantity of adsorbed surfactant with salt addition. These results provide important reference values for projecting SWCNT dispersion behavior as a function of solution conditions and extend the range of nanoparticle properties measurable via AUC.

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INTRODUCTION Nanoparticles (NPs) have made their way into our everyday lives (e.g., sunscreens and coatings) and have many potential applications in medicine, energy, and materials science.1,2 It is well-known that NP interactions with each other and with organismsas well as their distribution in dispersions depend not only on their size, shape, and composition but also on their surface functionality as well as the effect of the surrounding environment on the properties of the solution− particle interface.2−5 Thus, the development of robust, high throughput characterization methods capable of studying such properties of nanoparticles in different solution environments is very important. Analytical ultracentrifugation (AUC) is a powerful technique capable of providing detailed information, including particle size, molecular mass, anisotropy, and density for such nanoparticle systems. It can be applied to polydisperse samples and mixtures, a primary limitation of light and neutron scattering methods, as well as to analysis of a statistically significant population of particles in a single experiment, a limitation of microscopy.1,6,7 We utilize the capabilities of AUC herein to first measure the quantity of bound surfactant and solvated water per unit length of nanotube for only the second single-wall carbon nanotube (SWCNT) structure to date. We also extend this information to include the dependence of the adsorbed surface concentration of surfactant on its bulk concentration. We further demonstrate that the size of the solvation shell around a SWCNT−surfactant complex as This article not subject to U.S. Copyright. Published 2016 by the American Chemical Society

measured in the AUC, a piece of information that cannot easily be provided by most scattering methods or microscopy,8−10 is correlated to the decay length of the counterion cloud surrounding the charged (anionic) surfactant-coated SWCNT. To measure these values with high precision, we utilize precisely defined length-sorted, single-chirality, SWCNTs dispersed with the highly effective small molecule surfactant sodium deoxycholate (DOC). Although AUC is one of few methods capable of simultaneously fractionating and analyzing samples containing mixtures of particles, minimizing population parameters with polydispersity is still beneficial.1,11 Chirality and length sorted SWCNTs are an ideal nanoparticle system for measurement in AUC: (1) they have very well-defined optical transitions, allowing ready concentration measurements in the AUC using absorbance optics,12 (2) they have known crystalline structures leading to calculable and length independent densities, and finally (3) they are industrially significant materials.13−15 Each structure of SWCNT has special properties including high tensile strength, thermal conductivity, and unique optical properties.13 Depending on whether a specific SWCNT structure is metallic or semiconducting, they can be attractive for applications such as nanoelectronic devices Received: February 16, 2016 Revised: March 29, 2016 Published: March 31, 2016 3926

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SWCNT Sample Length Sorting. SEC was performed on a GE Ä TKA Purifier HPLC system equipped with three SEC columns arranged in series from largest to smallest pore size (Sepax Technologies Inc., Newark, DE). The columns are packed with 20 μm silica beads with pore sizes of 2000, 1000, and 300 Å with a pore volume of ≈1.0 mg/L. 5 mL of (7,6) SWCNT dispersed in 10 g/L DOC was injected into the feed stream. SWCNTs were eluted with DOC (10 g/L) pumped through the column at a flow rate of 5 mL/ min. Fractions of 5 mL were collected. Each fraction was rinsed with DOC (10 g/L) and concentrated using a pressurized stir cell with ultrafiltration membranes (Millipore Ultracel Ultrafiltration Discs, 100 or 30 kDa MW cutoff). Fractions 3−6 were utilized in this study. It will be indicated which fraction was used in which type of experiment. SWCNT Sample Characterization. Absorbance spectra of nanotube solutions were measured using a Cary Varian 5000 UV− vis−NIR spectrophotometer. Spectra were measured from 1880 to 200 nm in a 1 mm quartz cuvette at 1 nm data intervals with a 0.1 s integration time and a 2 nm bandpass slit width. Calibration for 100% transmission of the light beam was performed before each set of measurements. The spectra of reference buffers were measured as separate samples at the same conditions and manually subtracted. Atomic Force Microscopy (AFM). AFM was used to measure the length distributions of the separated SWCNT samples. Aliquots of multiple (7,6) length-sorted fractions eluted from SEC were deposited onto Si wafers functionalized with 3-(ethoxydimethylsilyl)propylamine (APDMES, Sigma) and imaged using a Bruker Dimension Icon AFM in peak force tapping (ScanAsyst) mode. ScanAsyst-Air probes were used for imaging. AFM wafer and sample preparation is described in detail in Silvera-Batista et al.8 The length distributions of the different fractions were obtained by manually measuring the lengths of several hundred to 1000 SWCNTs using ImageJ. AFM images showing the (7,6) sample before and after SEC sorting are presented in the Supporting Information. Analytical Ultracentrifugation (AUC). Sedimentation velocity (SV) experiments on the DOC-coated SWCNT populations were measured in the AUC at different DOC and NaCl concentrations. AUC experiments were performed in a Beckman Coulter XL-I analytical ultracentrifuge using an AN-50 8 cell titanium rotor loaded with two-sector Epon-charcoal centerpieces. The AUC cells used here have an optical path length of 12 mm. Experiments were performed, and data were collected using both absorbance and interference optics. Sample absorbance was measured at λ = 376 nm (the E33 optical transition of (7,6)). A centrifugation speed of 2094 rad/s (20 000 rpm ≈ 30 700 g in the center of AUC cell) was chosen so that the sedimentation of the SWCNT−DOC complex in the AUC would occur over a time scale which could be sufficiently monitored by the instruments’ optical system. Radial scans of the absorbance/ interference pattern were measured every 5 min, and 150−200 scans were taken per sample to ensure that the entire sedimentation process was captured. Samples were measured at 20 °C after a minimum of a 2 h temperature equilibration. Reference cell buffers were of the same composition as the sample dispersions minus the nanotubes. A concentration series was performed using fraction 3 from SEC sorting (L = 563 ± 266 nm). SV experiments performed on (7,6) samples having an OD between 0.2 and 0.8 at λ = 376 nm in the AUC showed no sign of concentration-dependent sedimentation in the regime evaluated, agreeing well with the theoretical calculations shown in the Supporting Information that the dispersions are in the dilute limit. This data can be found in the Supporting Information. For all other experiments presented in this article, samples were diluted to have an OD between 0.1 and 0.4 at λ = 376 nm to ensure a system of dilute, noninteracting particles. Of the SEC separated fractions, only the most concentrated fractions (4−6) were used for densitometry measurements in the AUC. This restriction was necessary as only ≈13.8 μg of SWCNTs was eluted in each fraction; (7,6) nanotubes from one length fraction were used to perform a full set of experiments for a density measurement. Data from samples of different length fractions were not mixed to extract particle densities. This can be done because while the sedimentation coefficient of a particle will depend on its length, the

or as electrodes for batteries. The primary limitation for their adoption to such applications is that synthesis methods typically produce many of these structures simultaneously. Much effort, described in detail elsewhere,16 has been spent to determine methods for the separation of SWCNT structures via aqueous-phase processing. Each of these methods, such as aqueous two phase extraction (ATPE),17−20 rely on the modification of the nanotubes’ surface by an adsorbed dispersant to control the separation of the nanotubes by diameter or metallicity.21,22 The ability to characterize the adsorbed surfactant layer and associated hydration layer around SWCNTs at different solution conditions is thus important to understanding SWCNT separation as well as their general stability and colloidal characteristics. Several previous studies following this logic have thus characterized surfactant adsorption on SWNCTs using AUC.8−10,23 However, chirality and length sorted nanotubes enabling SWCNT structureresolved conclusions were only used in one case.9 This work contributes multiple new findings on the dependencies of surfactant adsorption onto nanotubes, including quantification of the DOC surfactant structure on the (7,6) SWCNT as well as measuring and describing the changes in the size of the hydration shell for the same SWCNT structure as a function of ionic strength and surfactant concentrationneither of which have been studied using AUC until this report. Additionally, to the knowledge of the authors, this is the first report in which counterion cloud collapse around a charged colloid has been observed and quantified using densitometry measurements in AUC.

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EXPERIMENTAL SECTION

Materials. Certain equipment, instruments or materials are identified in this paper in order to adequately specify the experimental details. Such identification does not imply recommendation by the National Institute of Standards and Technology (NIST) nor does it imply the materials are necessarily the best available for the purpose. Single-wall carbon nanotube powder was donated by Southwest Nanotechnologies (SG76 grade, lot # 0024). Fully individualized (water-filled) nanotubes were obtained using typical sonication/ centrifugation processing followed by a rate-zonal centrifugation step.24 From this fraction, the semiconducting (7,6) population used was obtained by iterative application of the ATPE18 separation methodology. This chirality sorted population of SWCNTs was also length sorted using size exclusion chromatography (SEC).25 Each separation step is useful for enabling better AUC measurements: chirality sorting narrows the diameter (and thus density) distribution and concentrates the absorbance signal from the desired SWCNT species relative to the total sample concentration; length sorting reduces the length (and friction coefficient) polydispersity; rate-zonal separation reduces variations in SWCNT morphology and density that improve the effectiveness of the other two separations. AUC data comparing the sedimentation coefficient distributions of the different nanotube populations through the sequence of purification steps are shown in the Supporting Information. Sodium deoxycholate (DOC) was purchased from Sigma (BioXtra, >98%, Lot #BCBN4417V). Stock solutions of the desired concentration of DOC were prepared in ultrapure water and then filtered using vacuum filtration through a membrane filter (Advantec, 0.2 μm pore) to remove light scattering impurities. Ultrapure water was obtained from a Barnstead Nanopure water purification unit (18.1 MΩ·cm resistivity, Thermo Scientific). Deuterium oxide (D2O, D >99.8%) used in the AUC experiments for anhydrous density measurements was purchased from Cambridge Isotopes. Iodixanol (sold as Opti-Prep) used in measurements for buoyant density was purchased from Sigma. Sodium chloride was purchased from VWR and used without further purification. 3927

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density of a crystalline cylinder geometry is length independent. The mass-weighted average lengths for fractions 4−6 from SEC are as follows: (LF4 = 456 ± 177 nm, LF5 = 342 ± 150 nm, and LF6 = 282 ± 154 nm). Densities and viscosities of buffer solutions were measured using an Anton Paar 5000 M densitometer/Lovis ME viscometer combination instrument. Solutions were allowed to temperature equilibrate in the instrument for 5 min prior to measurement. Three measurements were taken per sample, and the values were averaged. Densities and viscosities of intermediate H2O/D2O, H2O/iodixanol buffers, and H2O/D2O or H2O/iodixanol buffers containing intermediate concentrations of NaCl were obtained by linear interpolation between measured values. Tabulated values for the measured density (ρs) and viscosity (ηs) of all buffers are reported in the Supporting Information. Sedimentation data were analyzed using SEDFIT version 14.6e, which corrects for an instrument time stamp error previously discovered.26,27 The c(s) model was used to fit the data and extract sedimentation coefficient distribution. It was previously shown that there was no difference between using the c(s) model and the c(s, ff0) analysis, which enables variation of frictional ratio ( f/f 0) with the s value to solve for SWCNT s-distribution.9 This is the case because all of the samples used in this work have narrow length distributions and thus narrow f/f 0 value ranges. Accounting for this variation in the numerical analysis is thus unnecessary to achieve good fits to the data. Values for s were fitted over a range from 0 to 20 Sv at a resolution of 0.1 Sv (200 points). The frictional coefficient was allowed to float during analysis. In the analysis, the regularization parameter P was set to 0.68. Initial guesses for the partial specific volume, v ̅ , of the particle were made based on past results from Fagan et al.9 This value was iterated for in the data analysis of the densitometry experiments until convergence. Since v ̅ = ρp−1, it is through this iterative analysis that we obtain the density of the effective particle. Similar s-distributions and densities were calculated using the two-dimensional spectrum analysis (2DSA) in UltraScan28 as with the c(s) model in SEDFIT. Details for the UltraScan analysis as well as a comparison of the results obtained are presented in the Supporting Information. Unless otherwise noted, uncertainty in this contribution is represented by error bars equal to one standard deviation of the reported value.

Figure 1. UV−vis spectra of length sorted (7,6) SWCNTs (purple line) as compared with that of the non-length sorted (7,6) sample (yellow line), the non-chirality sorted SG76 water-filled sample (blue line) as well as the non-vertically sorted SG76 sample (black line). The spectra are normalized to the absorbance of the sample at 800 nm. The peak to baseline ratio of the E11 and E22 optical transitions specific to the (7,6) nanotube increase with sorting, indicative of a purer sample. The inset photograph shows aqueous solutions of (7,6) nanotubes in 10 g/L DOC.

center of rotation, u the sedimentation velocity, Vp the particle volume, and ω the angular velocity. For a population of particles of consistent composition, size, and shape (i.e., constant Vp, a, and f/f 0), s is thus directly proportional to the difference between the particle density and the solution density. With an appropriate choice of contrast agents to modify the solution density, it is thus possible to measure the sedimentation coefficient as a function of the solution density to solve a set of eq 1 relations for the unknown particle density. For SWCNTs, the two most useful particle densities are the anhydrous and buoyant densities.6,29 For anhydrous density measurements, the density of the bulk medium is modulated by varying the concentration of H2O/ D2O in the solution. Assuming that the H2O/D2O molecules distribute evenly around and throughout any hydrated portions of the SWCNT-DOC complex, the only contrast of the particle density to the density of the bulk solution should come from the particle complex composed of the nanotube, the adsorbed deoxycholate surfactant, and the associated counterions (to the deoxycholate) (Figure 2a). Note that for the water-filled SWCNTs used in this contribution the endohedral water does not contribute to the anhydrous density because its isotopic composition is in equilibrium with the external water isotope composition and thus does not contribute a density contrast to the bulk. We describe the sample used herein as water-filled to discriminate from empty SWCNTs (which have closed ends and an inaccessible core volume), for which the core volume does contribute to the anhydrous density. For buoyant density measurements, the density of the bulk medium is modulated by varying the ratio of H2O to iodixanol in the solution. In this formulation, the nonionic iodixanol molecules do not penetrate into the layer of tightly associated water and counterions around the SWNCT−DOC complex (Figure 2b), altering the effective particle contrasted to the bulk solution to include the region of water around and inside the nanotube into which the iodixanol does not penetrate. In this

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RESULTS AND DISCUSSION Densitometry of (7,6) SWCNTs: Data Analysis and Extraction of Anhydrous and Buoyant Densities. Absorbance spectra demonstrating the effects of pre-AUC nanotube sample purification are presented in Figure 1. The (7,6) SWCNT species has well-defined optical transitions at 1129, 651, and 376 nm. From elimination of impurities, unwanted nanotube species, and defective nanotubes, these optical transitions become narrower, and the peak to baseline ratio improves with each separation step. The final samples display dramatically stronger absorbance signals on a per mass basis and thus enable AUC measurements at much lower concentrations, and with greater specificity, than nonseparated samples. The measurable parameter in a SV experiment is the sedimentation coefficient, s, which is the terminal velocity of the probed particle per applied acceleration. The Svedberg equation is derived from a balance of friction, drag, and buoyancy forces and is valid in the dilute and noninteracting particle limit. These conditions were met in our experiments. s≡

Vp(ρp − ρs ) u = 6πaηs(f /f0 ) ω 2R

(1)

wherein ρs is the solution density, ρp the particle density, a the equivalent hard sphere hydrodynamic radius, f the friction coefficient of the sedimenting particle, f 0 the friction coefficient of a sphere, ηs the solution viscosity, R the radial distance from 3928

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using the c(s) model in SEDFIT. The Lamm equation is derived from the mass balance on a sector-shaped volume in the absence of convection: ⎛ ∂ 2c ⎞ 1 ∂c ⎞ ∂c 2 ⎛ ∂c = D⎜ 2 + ⎟ − ω s⎜r + 2c ⎟ ⎝ ⎠ r ∂t ⎠ ∂t ∂r ⎝ ∂r

(2)

where c is the solute concentration, D the rotationally averaged diffusion coefficient, r the radial coordinate, s the sedimentation coefficient, ω the angular velocity, and t the time. Using SEDFIT, eq 2 is numerically solved to obtain the sedimentation coefficient distribution for the sample. Sedimentation coefficient distributions for (7,6) dispersed in 10 g/L DOC and buffers of varying H2O/D2O content are shown in Figure 3a. As expected, as the density of the bulk medium increases (with increasing D2O concentration), the sdistribution of the (7,6)−DOC complex shifts to lower values. As seen in eq 1, s is directly proportional to (ρp − ρs). Thus, as the density differential between the bulk phase and the particle decreases, so does s. The sedimentation coefficient distributions

Figure 2. Schematic explaining (a) anhydrous and (b) buoyant densities and radii. The dotted circle indicates the position of the anhydrous radius (ran), and the solid circle indicates the position of the buoyant radius (rb). The locations of ran and rb demarcated by the dotted and solid lines, respectively, show an average position for these radii. As can be seen in the diagrams above, if the lines were removed, the boundaries of the anhydrous and buoyant particle would be rough. Note that the volume around and inside the nanotube not occupied by the surfactant or contrast agent is filled by water molecules (not shown) and that the deoxycholate molecule is shown in two perspectives.

case, the difference in density between the sedimenting effective particle and the bulk medium will have contributions from the hydration shell and endohedral water as well. Sedimentation coefficient distributions for the densitometry measurements were determined through fitting the radial absorbance scans acquired by AUC to the Lamm equation

Figure 3. (a) s-distributions of (7,6) in solutions of varying H2O/D2O content (10 g/L DOC). (b) s-distributions of (7,6) in solutions of varying iodixanol/H2O content (10 g/L DOC). As seen in both sets of data, the s-distribution of the nanotube sample left-shifts as the density of the bulk solution is increased. These experiments were performed using fraction 4 from SEC elution. 3929

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where mSWCNT is the mass/nm of the (7,6) carbon nanotube, VSWCNT the volume of the carbon nanotube annulus, rSWCNT‑O the outer radius of the carbon nanotube, rcore the inner radius of the carbon nanotube, and l the length of the carbon nanotube. The width of the graphene lattice, i.e. (rSWCNT‑O − rcore), is 0.34 nm, equal to the interlayer spacing in graphite.30,31 A schematic describing this can be found in the Supporting Information. The value for SWCNT density calculated using this method corroborates with values which have been reported in the literature.32 The density of DOC, ρDOC = ν̅DOC−1, was determined using a Kratky density balance to find the partial specific volume of DOC in water.6 This plot can be found in the Supporting Information. Knowing the density of the (7,6) SWCNT and the density of the DOC molecule itself, the contribution of the DOC to the measured density of the SWCNT−DOC complex can be determined through a simple mass balance. Similarly, with the contribution of the SWCNT and DOC to the density of the hydrated particle known, the mass of water associated with the complex per unit length can be determined. From the anhydrous and buoyant densities presented above, the mass of DOC bound to the (7,6) SWCNT is calculated to be ≈3.5 × 10−21 g/nm of SWCNT and mass of water associated with the SWCNT−DOC complex is ≈2.2 × 10−20 g/nm of SWCNT (these values have been determined on a per length basis because the density of the SWCNT is independent of its length). Calculation of Anhydrous and Buoyant Radii. From the measured densities, the anhydrous and buoyant radii of the effective particle can be estimated. Assuming a radially symmetric core−shell model for the SWCNT−DOC system (Figure 2) with the DOC fully filling the radial volume beyond the SWCNT exclusion volume, the anhydrous radius is calculated using eq 4 and the buoyant radius is calculated using eq 5. The derivations for both these equations are shown in the Supporting Information.

for (7,6) SWCNTs dispersed in 10 g/L DOC and buffers of varying H2O/iodixanol content are shown in Figure 3b. It can also be seen here that the s-distribution for the (7,6) left-shifts to lower values as the density of the bulk phase is increased by increasing the iodixanol content in the sample. From each distribution, we extract an average sedimentation coefficient for the particle by taking a thresholded average to exclude particles having a concentration less than 20% of the sedimenting species with the highest concentration. This is done so that the average s is representative of the (7,6)−DOC complex rather than any aggregates or dust which may be present in the sample. The density of the particle was determined by correcting the average s for solution viscosity, plotting it against solution density, and extrapolating to find the density at which s = 0. At this solution density, where the particle no longer sediments, the density of the particle is equal to the density of the solution. The anhydrous and buoyant density results for (7,6) SWCNTs dispersed in 10 g/L DOC are presented in Figure 4. From this analysis, we obtain an

ran 2 = Figure 4. Solution density versus viscosity-corrected sedimentation coefficient for extraction of the anhydrous and buoyant densities of the (7,6)−DOC complex. These experiments were performed using fraction 4 from SEC elution. The concentration of DOC in the solutions here is 10 g/L. Lines drawn to guide the eyes.

rcore 2(ρan − ρSWCNT ) + rSWCNT‐O2(ρSWCNT − ρDOC ) ρan − ρDOC (4)

2

rb =

anhydrous density, ρan, of 1521 ± 16 kg/m3 and a buoyant density, ρb, of 1074 ± 2.2 kg/m3 for the effective average particle in the applied environments. Again, the anhydrous density is the density of the (7,6) SWCNT, adsorbed DOC, and bound sodium counterions. The buoyant density is the density of the anhydrous particle, the diffuse counterion cloud, as well as any tightly associated water. These measurements were repeated several times and were performed for the different (7,6) length fractions (fractions 4−6). This data can be found in the Supporting Information. From the anhydrous density, the mass of DOC adsorbed on the (7,6) nanotube can be determined via a simple mass balance. Since the carbon structure of the (7,6) SWCNT is a well-defined crystalline lattice, the density of the nanotube, ρSWCNT, can be calculated. It was found to be 2244 g/cm3 by applying the equation m mSWCNT ρSWCNT = SWCNT = VSWCNT π (rSWCNT‐O2 − rcore 2)l (3)

ran 2(ρan − ρhydro ) + rcore 2(ρcore − ρan ) ρbuoy − ρhydro

(5)

where ran is the anhydrous radius, rcore the radius of the nanotube core, rSWCNT‑O the outer radius of the SWCNT, rb the buoyant radius, and ρhydro the density of the DOC in H2O solution. By applying these equations with the anhydrous and buoyant densities of the particle as determined in the AUC experiments, we estimate the thickness of the adsorbed DOC layer on the (7,6) SWCNT, the linear packing density of DOC (i.e., the number of DOC per nanometer of SWCNT), the amount of the SWCNT surface covered by DOC, as well as the thickness of and amount of water in the hydration shell around the particle. These values are reported in Table 1 and are compared with those of another semiconducting length and chiralitysorted SWCNT previously studied by Fagan et al.9 It should be noted that this core−shell model assumes a radially homogeneous adsorption of surfactant on the nanotube and also assumes an average cutoff distance from the nanotube surface for the adsorbed surfactant layer and hydration shell. In reality, the boundaries of the anhydrous and buoyant shells will 3930

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Table 1. Comparison of Densitometry Results for (7,6) and (6,5) SWCNTs Dispersed in 10 g/L DOC in H2Oa d (nm) ρan (kg/m3) ρb (kg/m3) ran (nm) rb (nm) Δ(rb − ran) DOC/nm tDOC (nm) % coverage free H2O/nm

(7,6)

(6,5)

0.8829 1521 ± 16 1074 ± 2.2 1.11 ± 0.04 2.87 ± 0.16 1.76 ± 0.16 5.21 ± 0.3 0.502 ± 0.04 66 ± 2 708 ± 69

0.7474 1574 ± 20 1068 ± 2.3 0.96 ± 0.04 2.74 ± 0.16 1.78 ± 0.16 3.75 ± 0.2 0.417 ± 0.04 54 ± 2 672 ± 66

a

(7,6) data based on F4 fraction from SEC. The anhydrous and buoyant densities of the (6,5) SWCNT dispersed in 10 g/L DOC were extracted from Fagan et al.9 Measured parameters are boldface, calculated values are not boldface, and the diameter of the nanotube is a constant. The calculated values for % coverage of the nanotube surface by DOC depends on the projected cross-sectional area of the molecule used. Information about the dimensions of the DOC molecule used in the calculations can be found in the Supporting Information.

Figure 5. Plot of solution density versus viscosity-corrected sedimentation coefficient for extraction of the anhydrous and buoyant densities of the (7,6)−DOC complex in solutions of various DOC concentrations. Filled symbols denote anhydrous density measurements, and open symbols denote buoyant density measurements. All data presented here are for fraction F4 from SEC. F5 was used for measurement of the anhydrous density of (7,6) SWCNT dispersed in 10 mmol/L DOC. The results for these measurements may be found in the Supporting Information.

be rough (fuzzy) rather than smooth due to shape effects limiting the packing efficiency of the surfactant and molecular fluctuations.33,34 It should also be noted that the calculated size of ran is the smallest possible size of the adsorbed surfactant layer. In actuality, it will be slightly larger due to voids in the DOC layer created from jamming effects during surfactant adsorption and, possibly, the stacking of surfactant molecules on top of each other. The calculated size of rb, however, is a representative physical radius for the system in which ran is fully contained; in previous work by Silvera-Batista et al.8 it was shown that rb is equivalent to the radial hydrodynamic radius for surfactant-coated SWCNTs. As seen in Table 1, the anhydrous density of the (7,6)−DOC complex is lower than that of the (6,5)−DOC complex when both species of nanotubes are dispersed in 10 g/L DOC. This tells us that there is more surfactant adsorbed on the (7,6) SWCNT than the (6,5) SWCNT. This most likely results from the fact that the (7,6) nanotube (dc‑c = 0.8829 nm) is larger in diameter than the (6,5) tube (dc‑c = 0.7474 nm) and therefore has more surface area available for surfactant adsorption. The buoyant densities of the (7,6) and (6,5) nanotubes measured in the AUC were very similar (although readily distinguishable on the scale of isopycnic density gradient ultracentrifugation).21,35 From this set of experiments, it is clear that AUC can be used to characterize differences in the adsorption of a specific surfactant on two different types of carbon nanotubes andon a more general scaleon two different types of nanoparticles. Densitometry of (7,6) in the Presence of Varying DOC Concentration. Densitometry experiments were conducted for (7,6) SWCNTs in the presence of varying bulk concentrations of DOC dispersant to quantitatively probe the sensitivity of the adsorbed surface concentration of DOC to the bulk surfactant concentration. Two concentrations of DOC, which were lower than 10 g/L (24 mmol/L), were chosen for this part of the study: 6.25 g/L (15 mmol/L) and 4.2 g/L (10 mmol/L). The densitometry data for these experiments are shown in Figure 5. In the figure, it can be seen that a reduction of the DOC concentration clearly results in an increase in the anhydrous density of the (7,6)−DOC complex. This occurs due to a decrease in the quantity of DOC adsorbed on the

nanotube and is what is expected for an adsorbed layer in equilibrium with the bulk. Decreasing the DOC concentration can also be seen to shift the sedimentation coefficient distributions of the samples to higher values. Using eq 1, it is clear that much of this effect is due to the higher anhydrous particle density as well as to a small reduction in the density of the bulk solution with decreasing DOC concentration. Plots of sedimentation coefficient distribution for the (7,6)−DOC particle dispersed in solutions of varying bulk DOC content may be found in the Supporting Information. Results from buoyant density measurements are also shown in Figure 5. As seen here, the buoyant density of the (7,6)−DOC complex also increases with a decreasing concentration of DOC in the bulk. This effect results from (1) an increase in the anhydrous density of the SWCNT−DOC complex with decreasing DOC concentration and (2) a decrease in the number of (hydrated) sodium ions in the counterion cloud around the SWCNT− DOC complex with a decrease in DOC adsorption to the (7,6) SWCNT. The anhydrous and buoyant densities and radii, the linear packing density of DOC on (7,6), the thickness of the adsorbed surfactant layer, the percentage of nanotube surface area covered by DOC, and the quantity of water associated with the nanoparticle−surfactant complex are tabulated for (7,6) dispersed in solutions of varying bulk surfactant concentrations in Table 2. Densitometry of (7,6) in the Presence of Added Electrolyte. Buoyant density measurements in the AUC9 as well as analyses of density gradient ultracentrifugation (DGU) results for surfactant-dispersed SWCNTs10,21,31 have demonstrated that the hydrated SWCNT−surfactant particle can effectively be contrasted using a solution density-modifying molecule that does not interact with the effective particle. The excess water around the SWCNT−surfactant complex has been widely referred to in the SWCNT literature as the “hydration shell”. However, the origin and composition of this hydration shell has never been fully described in the literature. Since 3931

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Table 2. Particle Densities and Properties of the Adsorption Layer Calculated for (7,6)−DOC Complex at Varying DOC Concentrationsa ρan (kg/m ) ρb (kg/m3) ran (nm) rb (nm) Δ(rb − ran) DOC/nm tDOC (nm) % coverage free H2O/nm 3

24 mmol/L DOC

15 mmol/L DOC

10 mmol/L DOC

1513 ± 12 1068.5 ± 1.7 1.13 ± 0.04 3.01 ± 0.16 1.88 ± 0.17 5.45 + 0.30 0.52 ± 0.04 69 ± 2 789 ± 74

1556 ± 46 1070.6 ± 0.7 1.05 ± 0.09 2.85 ± 0.25 1.80 ± 0.27 4.33 ± 0.52 0.44 ± 0.09 55 ± 4 714 ± 110

1562 1096 ± 1.8 1.04 ± 0.02 2.43 ± 0.09 1.39 ± 0.09 4.21 ± 0.11 0.43 ± 0.02 53 ± 1 484 ± 32

a

Measured parameters are boldface, and calculated values are not boldface.

sodium deoxycholate is an anionic surfactant, the adsorption of DOC on the (7,6) SWCNT renders each nanotube negatively charged.36−40 This implies that a system of SWCNTs dispersed in ionic surfactant should be surrounded by a counterion cloud whose diffusiveness will depend on the total ionic content of the bulk medium. From knowing this, we hypothesize that the hydration shell which has been previously deduced from DGU analysis and which is measured in the densitometry experiments for buoyant density in the AUC arises from the counterion cloud surrounding the charged SWCNT−surfactant complex in solution.21 Using AUC, we attempt to characterize the nature of this hydration shell by measuring the anhydrous and buoyant densities of the (7,6) SWCNTs dispersed in 10 g/L DOC in the presence of varying concentrations of a common monovalent salt−sodium chloride (NaCl). Results from densitometry measurements for the (7,6)−DOC system at varying NaCl concentrations are shown in Figures 6 and 7. Figure 6a, and plots presented in the Supporting Information, show that as the concentration of NaCl in the system is increased, the sedimentation coefficient right-shifts to larger values for both the anhydrous and buoyant density measurements. The explanation for this effect will be revisited after the results for buoyant density are presented. However, despite the shift in s with increasing NaCl concentration, the anhydrous densities of the (7,6) in 10 g/L DOC remain the same (within measurement uncertainty) up to ≈50 mmol/L of added salt (Figure 6b). This implies that up until this point there is no significant modulation to DOC adsorption on the (7,6) SWCNT by the addition of the salt. These results are consistent with results from Doorn et al.,41 in which photoluminescence (PL) measurements were used to detect changes in surfactant packing structure on SWCNTs (including the (6,5) and (7,6)) dispersed in solutions of sodium dodecyl sulfate and sodium deoxycholate. Their results indicated no changes to DOC adsorption on a (7,6) SWCNT dispersed in 10 g/L DOC with salt addition up to 40 mmol/L NaCl. In our experiments, the anhydrous density of the (7,6)−DOC complex begins to increase at 70 mmol/L NaCl. We hypothesize that this effect is likely due to a decrease in DOC adsorption on the surface of the SWCNT resulting from a change in micellization dynamics at this ionic strength. The growth of DOC micelles as well as the reduction of the critical micelle concentration (CMC) in the presence of salt (usually >100 mmol/L NaCl for DOC) has been observed in several studies.42,43 The reduction in the concentration of free

Figure 6. (a) Plots of solution density versus viscosity-corrected sedimentation coefficient for extraction of anhydrous density of the (7,6)−DOC complex at various salt conditions. (b) Anhydrous density of the (7,6)−DOC complex in solutions of 10 g/L DOC and various salt concentrations. These experiments were performed using fraction 5 from SEC elution. As seen here, the anhydrous density of the (7,6)− DOC complex fluctuates around 1515 kg/m3 (indicated by the dotted black line) for experiments performed in solutions containing 0−53.2 mmol/L added salt.

surfactanteither through lowering of the CMC or growth of micellescausing a depletion of surfactant from surfaces has been observed in many other fields as well as for SWCNTs.44 Figure 7 shows how the addition of NaCl affects the buoyant density of the SWCNT−DOC complex. With the addition of NaCl, the sedimentation coefficient and the buoyant density of the SWCNT−DOC complex increase. For samples where the anhydrous density of the SWCNT−DOC complex remains constant with salt addition, an increase in buoyant density can only be attributed to a decrease in the quantity of water associated with the sedimenting SWCNT−DOC complex. This likely occurs as a result of a collapse of the counterion cloud toward the anhydrous surface of the nanotube−surfactant complex with the addition of NaCl,45−47 causing the buoyant radius to shrink. Additionally, the ability of the ions to approach closer together with increasing electrolyte content will result in them shedding water from their outer hydration shells, causing the buoyant density of the hydrated SWCNT−DOC complex to increase. 3932

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Figure 7. (a) Solution density versus viscosity-corrected sedimentation coefficient for extraction of buoyant density of the (7,6)−DOC complex in solutions containing varying amounts of NaCl. Arrow points in the direction of increasing NaCl content. (b) Buoyant density of the (7,6)−DOC complex at various salt conditions. Unlike the anhydrous density of the particle, the buoyant density of the SWCNT−DOC complex changes drastically with increasing NaCl content. These experiments were performed using fraction 5 from SEC.

Figure 8. (a) Anhydrous and (b) buoyant radii of (7,6)−DOC complex in solutions containing varying concentrations of added salt. All samples were dispersed in 10 g/L DOC.

divide it by the volume occupied by one water molecule, we obtain that there is a difference of 445 molecules of water/nm of SWCNT between the system with 0 mmol/L added salt and 70 mmol/L added salt. From the AUC data, the linear packing density of DOC on the nanotube, the thickness of the adsorbed DOC layer, the surface coverage of SWCNT by surfactant, and the number of water molecules associated with the hydrated SWCNT−DOC complex were calculated on a mass basis. This data are presented in the Supporting Information. From these values, the difference in number of free water molecules associated with the SWCNT−DOC complex between SWCNT−DOC systems at these two salt conditions is 419 ± 88 waters. Thus, the difference in the number of waters calculated on a mass basis from the AUC results between the 0 mmol/L NaCl and 70 mmol/L NaCl cases corroborates very well with the difference in the number of waters obtained from the volume-based “back-of-the-envelope calculation” presented above. Separate from the buoyant density measurements, the anhydrous density measurements independently confirm the reduction of water in the effective particle with salt addition. While only the density of the sedimenting anhydrous particle is measured in the anhydrous density intercept, a hydration layer still exists around the anhydrous particle and thus contributes

The sizes of the anhydrous and buoyant radii as a function of NaCl concentration are shown in Figure 8. These values were calculated using eqs 4 and 5. Since the anhydrous density of the (7,6)−DOC complex does not change significantly with the addition of salt at [NaCl] < 70 mmol/L, the calculated anhydrous radii also remain more or less constant at 1.13 ± 0.02 nm for samples dispersed in solutions containing less than 70 mmol/L NaCl. With the addition of 70 mmol/L NaCl, the anhydrous radius decreases to 1.06 ± 0.04 nm. The buoyant radius of the (7,6)−DOC complex, however, shrinks by ≈0.82 nm between 0 and 70 mmol/L added salt. This equates to a difference of 13.35 nm2 H2O/nm SWCNT associated with the particle. For the F5 fraction of (7,6), which has a mass-weighted average length of 342 nm, this gives a difference in volume of ≈4566 nm3. Based on a molar volume of 18.048 cm3 for water at 20 °C, one molecule of water occupies a volume of 2.997 × 10−23 cm3 or 0.029 97 nm3. If we take the change in buoyant volume for (7,6)−DOC calculated above for the particle in the presence of 0 mmol/L of NaCl and 70 mmol/L NaCl and 3933

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to the absolute sedimentation rate through hydrodynamic friction. As determined from the buoyant density measurements, this layer contains the counterion cloud as well as the water which is dragged along with the counterion cloud during the sedimentation process. It is only not captured in the anhydrous density because this layer has (more or less) the same distribution of water as the bulk solution and thus can be mathematically eliminated using eq 1, as s = 0. However, because this hydration layer is still present and still shrinks in the presence of salt, the drag on the sedimenting cylindrical particle decreases with increasing salt content, causing the complex to sediment faster. If we calculate the friction coefficient for the sedimenting particle using an expression derived from slender body theory for rigid rods by Batchelor, and which was previously shown by Silvera-Batista et al. to describe well the sedimentation behavior of surfactant stabilized nanotubes,8 we predict that the friction coefficient should decrease from ≈7.75 × 10−10 to ≈7.2 × 10−10 kg/s in a system with 0 mmol/L added salt to that having 70 mmol/L added salt, respectively. This corresponds to an ≈8% increase in the sedimentation rate of the particle, which agrees very well with the measured increase from 9.78 ± 0.78 Sv in 10 g/L DOC and 0 mmol/L NaCl to 10.54 ± 0.78 Sv in 10 g/L DOC and 70 mmol/L NaCl. A description for the calculation of friction coefficient and the calculated friction coefficients of the SWCNT−surfactant complex in 10 g/L DOC solutions containing varying concentrations of salt can be found in the Supporting Information. Using AUC To Probe Electrostatic Interactions. From our measurements, it is clear that the buoyant density of the SWCNT−surfactant complex is highly affected by the concentration of salt in the system. In the section above, we demonstrated that the increase in buoyant density and decrease in the volume of the hydrated SWCNT−DOC complex most likely results from a loss of water with salt addition due to the collapse of the counterion cloud toward the anhydrous surface of the SWCNT−DOC complex. To more explicitly model this phenomenon for a system of charged colloids, the shrinking of the counterion cloud around the charged particle in the presence of increasing electrolyte content is compared with the expected change in Debye length (κ−1) with increasing ionic strength. Although not a perfect comparison due to granularity and excluded volume packing effects, the displacement of the buoyant radius from the anhydrous radius should be closely correlated to this scale when κ−1 is small. Plotted as a function of solution ionic strength, one can see that the value of Δ(rb − ran), the size difference between the anhydrous and buoyant radii, increases exponentially with a decrease in the ionic content (I) of the solution until I < 0.04 mol/L (Figure 9). The Debye length calculated for a system containing charged particles in the presence of a 1:1 electrolyte also increases exponentially as a function of decreasing ionic strength in a similar manner. The Debye length for a solution containing a 1:1 electrolyte was calculated for the different experimental conditions using the equation47 −1/2 ⎛ ρi ∞e 2zi 2 ⎞ ⎟⎟ k (m) = ⎜⎜∑ ⎝ i ε0εk bT ⎠

Figure 9. Δ(rb− ran) as a function of solution ionic strength. This data are compared to the Debye length for a charged colloid dispersed in a solution containing a 1:1 electrolyte at the same ionic strength. Empty circles represent the calculated Debye length, the blue (squares) and purple (triangles) symbols represent values measured in buffers containing 10 g/L DOC and varying NaCl concentrations, and the yellow and green symbols represent values measured in buffers containing 0 mmol/L NaCl and varying DOC concentrations. The ionic strength of the DOC solution was calculated based on a CMC of 10 mmol/L for DOC. Δ(rb − ran*) denotes values calculated using a fixed value of 0.57 nm (height of DOC molecule) for ran rather than the value extracted from anhydrous density measurements. Information about the dimensions of the DOC molecule used in the calculations can be found in the Supporting Information.

10−23 J/K), and T the temperature. Figure 9 shows that the value for Δ(rb − ran) is the same as that for κ−1 in the SWCNT−DOC system from I = 0.084 mol/L until I ≈ 0.04 mol/L, where Δ(rb− ran) begins to diverge from the expected trend. The likely reason for this divergence of the data from the expected trend for Debye length is that the increasing diffusiveness of the counterion cloud in the SWCNT−DOC dispersions with less than 25 mmol/L added NaCl allows for deeper penetration of the iodixanol molecules into the cloud.46 This means that at low ionic strength conditions the buoyant radius extracted from the AUC measurements corresponds to the radius of exclusion of iodixanol from the SWCNT−DOC complex, whereas at I > 0.04 mol/L, the buoyant radius corresponds to the location of the outer plane of the counterion cloud around the SWCNT−DOC particle. A graphical depiction of this explanation is shown in the TOC graphic as well as in the Supporting Information. Another possible explanation, but not as likely, for the deviation of the experimental data from the expected trend for Debye length vs ionic strength at low ionic strengths is that at the particle size scales encountered in this studywhere the radius of the nanoparticle, the size of the molecular stabilizer, and the sizes of hydrated co- and counterions begin to approach each other the applicability of continuum approximations begin to break down.48 The length scales and locations of such breakdowns in the use of continuum descriptions have been realized as being of significant importance to the continued development of nanoparticle science, and AUC as demonstrated above is likely to be a valuable technique for examining such phenomena.

−1

(6)

where ρi∞ is the concentration of ion of species i in the bulk, e the charge of an electron (1.6 × 10−19 C), zi the charge of an ion of species i, ε0 the permittivity of vacuum, ε the dielectric constant of the medium, kb Boltzmann’s constant (1.381 × 3934

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Understanding Biophysicochemical Interactions at the Nano-Bio Interface. Nat. Mater. 2009, 8, 543−557. (3) Sperling, R. A.; Parak, W. J. Surface Modification, Functionalization and Bioconjugation of Colloidal Inorganic Nanoparticles. Philos. Trans. R. Soc., A 2010, 368, 1333−1383. (4) Cho, E. J.; Holback, H. Nanoparticle Characterization: State of the Art, Challenges, and Emerging Technologies. Mol. Pharmaceutics 2013, 10, 2093−2110. (5) Pfeiffer, C.; Rehbock, C.; Hühn, D.; Carrillo-Carrion, C.; de Aberasturi, D. J.; Merk, V.; Barcikowski, S.; Parak, W. J. Interaction of Colloidal Nanoparticles with Their Local Environment: The (ionic) Nanoenvironment around Nanoparticles Is Different from Bulk and Determines the Physico-Chemical Properties of the Nanoparticles. J. R. Soc., Interface 2014, 11, 20130931. (6) Mächtle, W.; Börger, L. Analytical Ultracentrifugation of Polymers and Nanoparticles; Mächtle, W., Börger, L., Eds.; Springer: New York, 2006. (7) Walter, J.; Löhr, K.; Karabudak, E.; Reis, W.; Mikhael, J.; Peukert, W.; Wohlleben, W.; Cö lfen, H. Multidimensional Analysis of Nanoparticles with Highly Disperse Properties Using Multiwavelength Analytical Ultracentrifugation. ACS Nano 2014, 8, 8871−8886. (8) Silvera Batista, C. A.; Zheng, M.; Khripin, C. Y.; Tu, X.; Fagan, J. A. Rod Hydrodynamics and Length Distributions of Single-Wall Carbon Nanotubes Using Analytical Ultracentrifugation. Langmuir 2014, 30, 4895−4904. (9) Fagan, J. A.; Zheng, M.; Rastogi, V.; Simpson, J. R.; Khripin, C. Y.; Silvera Batista, C. A.; Hight Walker, A. R. Analyzing Surfactant Structures on Length and Chirality Resolved (6,5) Single-Wall Carbon Nanotubes by Analytical Ultracentrifugation. ACS Nano 2013, 7, 3373−3387. (10) Arnold, M. S.; Suntivich, J.; Stupp, S. I.; Hersam, M. C. Hydrodynamic Characterization of Surfactant Encapsulated Carbon Nanotubes Using an Analytical Ultracentrifuge. ACS Nano 2008, 2, 2291−2300. (11) Planken, K. L.; Cölfen, H. Analytical Ultracentrifugation of Colloids. Nanoscale 2010, 2, 1849−1869. (12) Weisman, R. B.; Bachilo, S. M. Dependence of Optical Transition Energies on Structure for Single-Walled Carbon Nanotubes in Aqueous Suspension: An Empirical Kataura Plot. Nano Lett. 2003, 3, 1235−1238. (13) Futaba, D. N.; Hata, K.; Yamada, T.; Hiraoka, T.; Hayamizu, Y.; Kakudate, Y.; Tanaike, O.; Hatori, H.; Yumura, M.; Iijima, S. ShapeEngineerable and Highly Densely Packed Single-Walled Carbon Nanotubes and Their Application as Super-Capacitor Electrodes. Nat. Mater. 2006, 5, 987−994. (14) Landi, B. J.; Ganter, M. J.; Cress, C. D.; DiLeo, R. A.; Raffaelle, R. P. Carbon Nanotubes for Lithium Ion Batteries. Energy Environ. Sci. 2009, 2, 638−654. (15) Lin, Y.; Taylor, S.; Li, H.; Fernando, K. A. S.; Qu, L.; Wang, W.; Gu, L.; Zhou, B.; Sun, Y.-P. Advances toward Bioapplications of Carbon Nanotubes. J. Mater. Chem. 2004, 14, 527−541. (16) Tambraparni, M. B.; Wang, S. Separation of Metallic and Semiconducting Carbon Nanotubes. Recent Pat. Nanotechnol. 2010, 4, 1−9. (17) Khripin, C. Y.; Fagan, J. A.; Zheng, M. Spontaneous Partition of Carbon Nanotubes in Polymer-Modified Aqueous Phases. J. Am. Chem. Soc. 2013, 135, 6822−6825. (18) Fagan, J. A.; Khripin, C. Y.; Silvera Batista, C. A.; Simpson, J. R.; Hároz, E. H.; Hight Walker, A. R.; Zheng, M. Isolation of Specific Small-Diameter Single-Wall Carbon Nanotube Species via Aqueous Two-Phase Extraction. Adv. Mater. 2014, 26, 2800−2804. (19) Ao, G.; Khripin, C. Y.; Zheng, M. DNA-Controlled Partition of Carbon Nanotubes in Polymer Aqueous Two-Phase Systems. J. Am. Chem. Soc. 2014, 136, 10383−10392. (20) Subbaiyan, N. K.; Parra-Vasquez, A. N. G.; Cambré, S.; Cordoba, M. A. S.; Yalcin, S. E.; Hamilton, C. E.; Mack, N. H.; Blackburn, J. L.; Doorn, S. K.; Duque, J. G. Bench-Top Aqueous TwoPhase Extraction of Isolated Individual Single-Walled Carbon Nanotubes. Nano Res. 2015, 8, 1755−1769.

CONCLUSIONS In summary, we utilized AUC to characterize properties associated with a nanoparticle−molecule complex (density, hydration) as well as the properties of the molecular adsorption layer itself (density, thickness) as formed in the presence of a perturbing analyte. The measurable changes, such as quantifying the mass of a dispersant adsorbed on the surface of a nanoparticle or detecting differences in the quantity of surfactant adsorbed on two different species of single-wall carbon nanotubes, are extremely precise; in this example, AUC is able to discriminate differences in adsorption for cylindrical particles differing by only 0.136 nm in diameter. Moreover, use of methodologies such as density contrast sedimentation velocity enable direct measurement of parameters of high importance to nanoparticle science, such as the thickness and extent of surfactant adsorption on nanoparticle (nanotube) surfaces even in the absence of total monodispersivity. We demonstrate the use of AUC for measuring the effect of modifying the nanoparticle’s environment, such as the effect of salt addition, on surfactant packing as well as on the size of the electric double layer around charged particles. Future work will likely see an expansion from the simple monovalent salt to larger or multivalent ions in a consistent manner. It can be expected that measurements of the buoyant and anhydrous radii may enable critical evaluation between experimentally observed effects and conventional theory.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b00605. Additional AUC data, AFM images for the (7,6) SWCNTs, buffer densities and viscosities, equation derivations, and calculations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (J.A.F.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS S.L. acknowledges the National Research Council for her postdoctoral fellowship. The authors also acknowledge the computing time which was utilized for AUC data analysis performed in UltraScan. Calculations were performed on the UltraScan LIMS cluster at the Bioinformatics Core Facility at the University of Texas Health Science Center in San Antonio and XSEDE resources supported by NSF XSEDE Grant #MCB070038 (to Borries Demeler). The Gateway is made possible by the use of XSEDE resources and the Extended Collaborative Support Service (ECSS) Program funded by the NSF through Award OCI-1053575.

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