Directly Measured Optical Absorption Cross Sections for Structure

Feb 6, 2014 - We have measured peak and spectrally integrated absolute absorption cross sections for the first (E11) and second (E22) optical transiti...
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Directly Measured Optical Absorption Cross Sections for StructureSelected Single-Walled Carbon Nanotubes Jason K. Streit, Sergei M. Bachilo, Saunab Ghosh, Ching-Wei Lin, and R. Bruce Weisman* Department of Chemistry and Richard E. Smalley Institute for Nanoscale Science and Technology, Rice University, 6100 Main Street, Houston, Texas 77005, United States S Supporting Information *

ABSTRACT: We have measured peak and spectrally integrated absolute absorption cross sections for the first (E11) and second (E22) optical transitions of seven semiconducting single-walled carbon nanotube (SWCNT) species in bulk suspensions. Speciesspecific concentrations were determined using short-wave IR fluorescence microscopy to directly count SWCNTs in a known sample volume. Measured cross sections per atom are inversely related to nanotube diameter. E11 cross sections are larger for mod 1 species than for mod 2; the opposite is found for E22. KEYWORDS: SWCNT, single-walled carbon nanotubes, absorption cross section, molar absorptivity, quantitative analysis, oscillator strength

S

transition.11−14 These discrepancies clearly indicate the need for further studies that can measure SWCNT absorption cross sections of various species with enough accuracy to reveal structural trends and allow reliable quantitative analyses. Determining a SWCNT absorption cross section requires combined measurements of optical attenuation and (n,m) species concentration. In experiments on single nanotubes, the sample’s identity and “concentration” are well-known but attenuation measurements are challenging. Conversely, in bulk experiments the attenuation measurements are rather straightforward but concentrations are elusive. Direct gravimetric measurements of concentration are hampered by the need for substantial quantities of well-sorted fractions and by large contents of surfactant or polymer coatings. Indirect methods have been devised for assessing bulk SWCNT concentrations using smaller sample sizes, but their intrinsic assumptions and approximations induce uncertainties. We report here results from a new direct scheme for reliably measuring absolute absorption cross sections of semiconducting SWCNTs in bulk samples. Our method finds number concentrations by using short-wave IR (SWIR) fluorescence microscopy to count individual SWCNTs of specific species within known volumes. The measured number concentration is then converted into a carbon atom concentration using the mean nanotube length determined by nanotube diffusion analysis (LAND method) and confirmed by AFM image analysis. We have applied this approach to determine the absolute peak and spectrally integrated absorption cross sections for the first and second optical transitions of (6,5) SWCNTs, using a

ingle-walled carbon nanotubes (SWCNTs) exist in a variety of discrete structural forms that differ in diameter and roll-up (chiral) angle. Each such structural species is labeled by a pair of integers, (n,m), and has a distinct electronic structure with specific optical transition energies and cross sections.1 Although the structure-dependence of SWCNT optical transition energies is well established,2−4 much less information is available for cross sections. This knowledge gap is significant because optical absorption cross sections are central photophysical quantities. In addition to their fundamental relation to electronic excitations, absorption cross sections have great practical value for measuring the concentrations of individual (n,m) species in structurally sorted and unsorted SWCNT samples. Islam et. al made the first report of a SWCNT absorption cross section: 1.7 × 10−18 cm2 per C-atom for the E22 (second electronic) transition of semiconducting species in an unsorted sample.5 The broad and structureless spectrum of their sample suggests that this value reflected contributions from multiple unresolved absorption peaks and a significant background. A recent femtosecond transient absorption study estimated a cross section of ∼5.0 × 10−18 cm2 per C-atom for absorption at the E22 peak of the (6,4) species.6 Another group used Rayleigh scattering intensity measurements in combination with AFM to deduce a resonant absorption cross section of ∼2.5 × 10−17 cm2 per C-atom for E33 transitions of nanotubes of different diameters.7 Because of its reasonable abundance, short transition wavelength, and ease of enrichment from CoMoCAT and HiPco samples, (6,5) SWCNTs have been the focus of several absorption studies. Reported values for the orientationally averaged (6,5) absorption cross section range from 2.9 × 10−18 to 1.7 × 10−17 cm2 per C-atom for the E11 transition,8−10 and 2.5 × 10−19 to 1.3 × 10−17 cm2 per C-atom for the E22 © 2014 American Chemical Society

Received: December 26, 2013 Revised: January 28, 2014 Published: February 6, 2014 1530

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selectively extracted SWCNT samples is essential in our study, both to allow selective counting of the targeted species in SWIR fluorescence microscopy and also to permit secure analysis of the bulk absorption spectrum without interference from broad backgrounds or overlapping transitions of other species. We found nanotube concentrations in our samples by selectively counting SWCNTs within known volumes under a SWIR fluorescence microscope. This instrument, described previously,23 is a customized inverted microscope equipped with a tunable Ti:Sapphire laser and fixed wavelength diode lasers for sample excitation and an InGaAs photodiode array (Princeton Instruments 2D-OMA V) for imaging. To slow down nanotube motion and assist particle counting, we increased the solution viscosity to ∼40 mPa-s by diluting the nanotube suspension into a 65% sorbitol solution.24 This sample, diluted by a known factor from the initial suspension, was then loaded into a custom fabricated microfluidic channel whose precisely measured depth (see Supporting Information) nearly matched the optical depth of field of our microscope objective. This ensured that all SWCNTs remained in focus in each SWIR image. Selective imaging of (6,5) nanotubes was achieved by spectral discrimination in excitation and detection. We used 840 nm light to specifically excite the (6,5) E11 vibronic sideband transition. The resulting emission was filtered through a 975 nm long-pass filter combined with a 1000 nm short-pass filter to select for the (6,5) fluorescence band centered at 984 nm. We recorded sequences of 50 frames, each with a 250 ms exposure time, for every specific 75 × 75 μm imaged area. The observed sample volume was simply computed by multiplying the image area by the channel depth. We used custom image analysis software, written using Matlab 2011a (Mathworks), to average the 50 recorded frames (during which SWCNT diffusion was nearly absent because of high viscosity) and to count the total number of emitting centers per volume. Figure 1b shows the sequence-averaged image for a typical analyzed microfluidic volume. Emission from each individual SWCNT is easily distinguished from the background, making nanotube counting straightforward. We analyzed 30 different regions of the sample in this fashion and averaged the results to determine the number concentration of nanotubes in the diluted sample. That concentration was then multiplied by the dilution factor to find the number concentration of (6,5) nanotubes in the original suspension. This procedure was applied to three diluted samples containing a total of 2430 counted SWCNTs. Averaging the results gave us the bulk (6,5) number concentration with a statistical uncertainty (relative standard error of the mean) of ∼4%. To convert number concentration to the desired carbon atom concentration, it was necessary to find the mean length of SWCNTs in the sample. We measured length distributions using the LAND method,24 which tracks diffusional motions of individual SWCNTs in our SWIR fluorescence microscope and deduces lengths from diffusion coefficients. The LAND results were independently checked by analyzing AFM images. As shown in Figure 2, the two methods gave consistent results and found mean lengths of 460 and 483 nm, respectively (a 5% relative difference). In addition, comparison of the distributions at small lengths shows no evidence that the LAND observations undercounted short nanotubes, which would be dimmer and more difficult to image than others. This gives confidence in the validity of our counting results, which had even greater sensitivity than the LAND measurements because of much longer effective exposure times. We used the average measured

structurally sorted sample. The method was then extended to measure cross sections of six other (n,m) species relative to the known (6,5) values. Our findings reveal important structuredependent optical absorption patterns. In addition, they provide key information that can enable absolute concentration measurements of common SWCNT species using simple absorption spectroscopy. Another optical method, SWIR fluorimetry, was shown ten years ago to be a sensitive and selective method for qualitative analysis of semiconducting SWCNTs.2,15 Subsequent studies of structure-dependent quantum yields and photoluminescence action cross sections have extended this capability and enabled quantitative fluorimetric determinations of relative species concentrations.16−21 However, absolute species concentrations are nearly impossible to deduce from fluorescence data because nanotube quantum yields vary substantially with environment and sample condition. By contrast, absorption oscillator strengths are believed to be nearly insensitive to those factors, so measurements of integrated absorption features should allow reliable quantitative analyses if structure-specific cross sections are known. We note that the longest wavelength (E11) transitions are best suited to such analyses because they show the least spectral congestion and lowest background absorptions. The E11 absorption cross sections are therefore the essential properties needed for quantitative optical analysis of SWCNT samples. The first measurements in our current study were performed on a sorted dispersion of HiPco SWCNTs in aqueous sodium cholate that had been enriched to ca. 90% purity in (6,5) by nonlinear density-gradient ultracentrifugation (NDGU).22 The absorption spectrum of that sample, displayed in Figure 1a, shows a low background and only minor impurity peaks, mainly from the (9,1) and (6,4) species. The use of sorted or

Figure 1. (a) Absorption spectrum of the (6,5)-enriched sample in aqueous sodium cholate. (b) SWIR spectrally selected fluorescence image of the sample, averaged over 50 exposures of 250 ms each. The image frame is 75 μm wide. 1531

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milder assumptions, it seems likely to provide higher reliability. We note that the σ22 cross section measured in the careful single-particle study of Oudjdi et al. agrees with our value within the combined uncertainties of the two determinations after adjustment for orientational averaging and background absorption. Our E11 cross section value for (6,5) is larger than all prior estimates. A likely source for this discrepancy is a narrower E11 absorption band in our sample. The integrated absorption cross section, which is proportional to the transition oscillator strength, f, should be nearly insensitive to nanotube environment, so a decreased spectral width will cause an increased peak cross section. Our bulk (6,5) sample shows a relatively narrow resonant full width at half-maximum (fwhm) of 226 cm−1. Its spectrally integrated absorption cross section, ∫ σ dν,̅ equals 9.2 × 10−15 cm/C-atom ( f = 0.0100) for the E11 transition, and 5.6 × 10−15 cm/C-atom ( f = 0.0061) for the E22 transition. This value for the integrated E11 cross section is nearly identical to that reported by Schoeppler et al.,25 even though our peak cross section is ∼1.5 times larger than theirs because of their sample’s broader spectral transition. We concur with previous suggestions that spectrally integrated absorption cross sections are more robust parameters than peak values.26,27 We extended our study by using a related counting-based approach to find the absorption cross sections of six additional SWCNT structures relative to the (6,5) cross section. We prepared a sample of HiPco SWCNTs (Rice University, batch HPR 166.12) dispersed in a toluene solution of the organic polymer poly(9,9-di-n-octylfluorenyl-2,7-diyl) (PFO), which is known to selectively extract the (7,5), (7,6), (8,6), (8,7), and (9,7) near-armchair species.28 One reason PFO-coated samples were chosen for this study is that they emit more brightly than typical aqueous dispersions, making the nanotubes easier to locate and count. Microscope samples were prepared by dropcasting a diluted SWCNT suspension onto a glass coverslip, which was fastened to a PMMA microscope slide after evaporation of the toluene. We imaged 30 separate areas of the sample through the coverslip, in each area changing the experimental configuration to selectively capture images of each of the five (n,m) species listed above. The changes to acquire species-selective images were 1) tuning the excitation wavelength to an absorption peak of the targeted species; 2) inserting narrow band emission filters to selectively pass fluorescence from that species; and 3) adjusting the microscope focus to image the targeted emission while leaving other wavelengths defocused by objective lens chromatic aberration.29 This method allowed us to optically distinguish the five SWCNT species and to count their abundances with high statistical precision. The results, averaged over the 30 observation regions, were normalized to the abundance of (8,6) and plotted as the “Measurement 1” points in Figure 3a. This entire procedure was repeated twice more with separate samples to obtain the other plotted data points. In total, more than 10 000 counted SWCNTs are represented in this figure. It can be seen that the three replicate determinations agree well, with a relative precision in abundances of only ∼6%. Applying the mild assumption that each SWCNT species in the sample has the same distribution of lengths and accounting for the diameter-dependence in carbon atoms per unit length, we thus obtained the relative carbon atom concentrations for all five nanotube species in the sample. Using a procedure similar to the one described above, we spectrally simulated the absorption spectrum of the PFO SWCNT sample, shown in Figure 3b, to determine the relative peak and integrated absorption cross

Figure 2. Length distributions of the enriched (6,5) sample as measured by (a) the LAND method and (b) analysis of AFM images.

nanotube length of 472 nm and the structural factor of 88 carbon atoms per nanometer to convert the (6,5) SWCNT number density into carbon concentration in the sample. The absorption cross section per atom was then found by analyzing the corresponding bulk absorption spectrum. After subtracting a small background (see Supporting Information Figure S3), we simulated the resonant absorption peaks as Voigt (E11) or Lorentzian (E22) functions using width and shape parameters found from fluorescence emission and excitation spectra. Our fitting process is described in greater detail in the Supporting Information. This fitting gave values for the peak and integrated absorbance values of the E11 and E22 electronic transitions. The absorption cross section per atom, σ, and the corresponding molar absorptivity, ε, were calculated from the following relations σ (ν ̅ ) =

2.303 A(ν ̅ ) lρ

and

ε (ν ̅ ) =

A (ν ̅ ) lc

(1)

where A(ν)̅ is the resonant part of the frequency-dependent decadic absorbance, l is the optical path length, ρ is the number density of carbon atoms, and c is molar concentration of carbon. We obtain an orientationally averaged peak σ11 value of 2.5 × 10−17 cm2/C-atom (ε11 = 6700 MC−1 cm−1) for the resonant (background excluded) component of the E11 transition of (6,5) SWCNTs. The corresponding resonant peak value for the E22 transition is σ22 = 0.60 × 10−17 cm2/Catom (ε22 = 1600 MC−1 cm−1). The ratio of peak resonant optical cross section to physical cross section (calculated from the diameter between atomic centers) is then 0.29 for the E11 transition and 0.07 for the E22 transition. Prior values reported for the (6,5) absorption cross section per carbon atom (adjusted when appropriate to give orientationally averaged results) range from 2.9 × 10−18 to 1.7 × 10−17 cm2 for σ11 and 2.5 × 10−19 to 1.3 × 10−17 cm2 for σ22.8−14 All but one of these results came from indirect bulk studies in which required assumptions may have introduced unrecognized systematic errors. Because our technique is more direct and involves 1532

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challenging to determine. To address this problem, we took advantage of the fact that different HiPco batches contain different relative concentrations of the five (n,m) species in PFO-extracted samples. By correlating the magnitudes of the well resolved E11 peaks in different samples to the spectral structure in the E22 region, we disentangled the separate E22 absorption contributions from specific species and calculated their corresponding peak and integrated absorbance values. This let us find the absolute peak and integrated E11 and E22 cross sections for the five (n,m) species (see Table 1). Our spectral deconvolution procedure is described in greater detail in the Supporting Information. Because the polymers PFO and PDDF selectively disperse near-armchair SWCNT species, the studied samples contained only a small range of roll-up angles (24.5 to 27.8°). To clarify the dependence of absorption cross sections on roll-up angle, we extended the study to include the (8,3) species (15.3°). Gel chromatography was used to prepare two different samples, one highly enriched in (8,3) and the other in (7,5).31 These were then mixed in comparable concentrations to allow (7,5) to serve as an internal cross section standard for the (8,3) determination. Both sorted fractions originated from the same bulk suspension, and we assume that the two species had the same average length. Counting procedures described above were applied to find the relative concentrations of (8,3) and (7,5), and the sample’s absorption spectrum was measured and analyzed to calculate the relative E11 peak and integrated cross sections. However, strong spectral overlap of the (8,3) and (7,5) E22 transitions prevented us from directly finding σ22 for (8,3). Instead, as explained in greater detail below, we deduced the spectrally integrated E22 cross section based on the corresponding E11 value and the integrated E11/E22 absorption ratio measured in a separate (8,3)-enriched sample. Table 1 summarizes our findings for E11 and E22 absolute cross sections of the seven studied (n,m) species. We find that values for σ and ∫ σ dν̅ vary with SWCNT structure, with the largest σ11 value a factor of 2.4 greater than the smallest. As illustrated in Figure 4a, the integrated absorption cross sections show a systematic dependence on nanotube diameter. For all species, the first (E11) electronic transition is more intense than the second (E22) and both cross sections increase with decreasing nanotube diameter. Another significant observation is a dependence on the value of mod(n − m, 3), which for semiconducting SWCNTs equals either 1 (“mod 1”) or 2 (“mod 2”). Figure 4a shows that ∫ σ11 dν̅ is larger for mod 1 than mod 2 species, while the reverse is found for ∫ σ22 dν.̅ Figure 4b reveals similar structural trends for peak cross sections σ. However, due to variations in spectral width, the difference between σ11 and σ22 is found to be larger and the

Figure 3. (a) Abundances of five (n,m) species in a PFO-extracted sample, as measured by counting emissive objects in spectrally selected fluorescence images. Three separate determinations are plotted as Measurements 1, 2, and 3, annotated with the total numbers of counted nanotubes of each species. (b) Absorption spectrum of the PFO-extracted sample whose composition is shown in (a).

sections of each species. This analysis benefitted from the well separated E11 transitions and low absorption backgrounds in PFO-extracted SWCNT samples. To confirm our cross section results, we repeated the entire measurement process with three additional SWCNT samples, two prepared with PFO using different HiPco batches (Rice University HPR 188.4 and HPR 195.1), and a third prepared with the related polymer poly(9,9didodecylfluorenyl-2,7-diyl) (PDDF) using HiPco batch HPR 188.4. When present in excess, PDDF extracts the (6,5) species in addition to the five species extracted by PFO,30 allowing us to use (6,5) as an internal standard for calibrating the absolute cross sections of the five PFO species. Although the E11 absorption features in these extracted samples are readily analyzed, the E22 features show significant spectral overlap, making their separate cross sections Table 1. Absorption Cross Sections and Oscillator Strengths (n,m)

σ11 (10−17 cm2 C−1)a

(8,3) (6,5) (7,5) (7,6) (8,6) (8,7) (9,7)

1.72 2.54 1.62 2.03 1.38 1.33 1.08

∫ σ11 dν̅ (10−15 cm C−1)b 7.28 9.24 5.90 7.39 5.00 6.03 3.91

f11

σ22 (10−17 cm2 C−1)

0.0079 0.0100 0.0064 0.0080 0.0054 0.0065 0.0042

0.61 0.62 0.53 0.54 0.45 0.46

∫ σ22 dν̅ (10−15 cm C−1)b 5.61 5.61 5.35 4.36 4.53 3.36 3.31

f 22 0.0061 0.0061 0.0058 0.0047 0.0049 0.0036 0.0036

a

Peak absorption cross section per carbon atom, excluding background component. bIntegral of the absorption cross-section spectrum versus frequency in cm−1, excluding background component. 1533

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species. We calculated this ratio for the seven SWCNT species described above using the values in Table 1, and supplemented those results by analyzing absorption spectra of the (6,4), (9,1), (8,3), (7,3), and (10,2) species in sorted samples prepared by nonlinear DGU or gel chromatography. As shown in Figure 5, a

Figure 5. Ratio of spectrally integrated absorption cross sections for the E11 and E22 transitions plotted versus roll-up angle for 12 semiconducting (n,m) species. Blue and red symbols represent mod 1 and mod 2 structures, respectively. The smooth curve is a guide for the eye.

strong pattern is found relating roll-up angle to the integrated ratios. The ratio values reach a minimum of approximately 1.0 near 24° and increase sharply for larger angles between 25 and 28°. This unusual shape seems related to a combination of a diameter dependence in the cross sections and the clustering of roll-up angles for near-armchair species of the same mod value. Our cross section results can be compared with findings from recent studies based on different experimental approaches. In one, Malapanis et al. measured photocurrent spectra from twelve individual p-n diode devices fabricated from airsuspended SWCNTs with diameters ranging from 1.0 to 1.9 nm.27 They obtained products of absorption cross section and exciton dissociation efficiency and then deduced cross sections using dissociation efficiencies estimated through theoretical modeling of oscillator strengths. The results found σ11 values larger than σ22 (in qualitative agreement with our findings), but the reported σ11 values exceed ours by more than an order of magnitude and show a diameter dependence opposite to that in our Figure 4. Another recent cross section study by Blancon et al. used spatial modulation spectroscopy to measure broadband absorption spectra of four individual semiconducting airsuspended SWCNTs with diameters between 1.8 and 2.5 nm.32 Two of the measured nanotubes had identifiable (n,m) indices. The resonant parallel-polarized peak absorption cross sections for E33 through E55 transitions were reported to be 0.3 to 3.7 × 10−17 cm2/C-atom. These values are smaller than the polarized cross sections implied by our orientationally averaged results, although they represent higher transitions of much larger diameter nanotubes. Some evidence of opposite diameter dependences for cross sections of mod 1 and mod 2 species was found within the limited set of measured data. A larger group of SWCNTs, including most of the (n,m) species measured in our project, were studied by Vialla et al. using a sensitized excitation method applied to unsorted bulk samples of nanotubes that had been noncovalently complexed with tetraphenylporphyrin.14

Figure 4. (a) Integrated absorption cross sections (from cm−1 spectra) plotted as a function of SWCNT diameter for E11 and E22 transitions. Closed and open symbols represent mod 1 and mod 2 species, respectively. Curves show guides for the eye. (b) Same as in (a) except peak absorption cross sections are plotted. (c) Spectrally integrated cross section data plotted against a composite parameter reflecting rollup angle and mod type. The vertical dashed line marks the armchair boundary between mod 1 and mod 2 structures.

dependence on diameter slightly weaker. Figure 4c plots our ∫ σ dν̅ data versus the parameter q cos(3θ), where q = 1 or −1 for mod 1 or mod 2 SWCNTs, respectively, and θ is the roll-up angle. A smooth dependence on this parameter is characteristic of many angle-dependent SWCNT properties. In contrast to the diameter plot of Figure 4a, Figure 4c shows no monotonic patterns for either ∫ σ11 dν̅ or ∫ σ22 dν.̅ Instead, the ∫ σ22 dν̅ data appear roughly symmetric with respect to reflection about the armchair angle (q cos(3θ) = 0), as is consistent with a property correlated more with diameter than with roll-up angle. However, we note that our data set includes only one (n,m) species with a roll-up angle below 24°, and additional results will be needed to reach firm conclusions about the angle dependence of SWCNT absorption cross sections. Another interesting photophysical parameter is the E11/E22 ratio of spectrally integrated cross sections for individual (n,m) 1534

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The results from this study should prove very useful for the practical task of assessing absolute carbon atom concentrations of specific (n,m) species, particularly in sorted samples. In Table 2, we list the peak decadic molar absorptivities for E11

Nanotube cross section values were found relative to that of the porphyrin sensitizer by comparing SWCNT fluorescence intensities induced by direct and sensitized excitation. For the set of five (n,m) species common to both studies, the σ22 values reported by Vialla et al. are consistently larger than ours by a factor of approximately 2.1. We suggest that this discrepancy reflects inadequacies in some of the assumptions needed to deduce nanotube absorption cross sections from relative fluorescence intensities and/or contributions from nonresonant absorptions that were deliberately subtracted in our analysis but could not be identified in the fluorescence-based study. Although we do not observe the systematic dependence of cross section (per unit length) on roll-up angle deduced by Vialla et al., our data set is too limited in SWCNT angle to confirm or contradict that finding. Theoretical modeling of structure-dependent SWCNT absorption strengths has been reported by several investigators.33−37 Among these, Oyama et al.33 found that E22 absorptions are stronger for mod 2 nanotubes than for mod 1, in agreement with our experimental findings. A similar result was reported recently by Verdenhalven and Malic,37 who also found that E11 absorptions, conversely, are stronger for mod 1 SWCNTs. In addition, these workers predicted E11 transitions to be more intense than E22 for all species, as is seen in our experimental results in Figure 4. In another recent study, Choi et al.36 developed a theoretical formula that predicts oscillator strengths to decrease with increasing SWCNT diameter. This qualitatively agrees with our experimental dependence shown in Figure 4a. Their model also predicts ∫ σ11 dν̅ to be slightly larger than ∫ σ22 dν̅, although with a different and weaker dependence on roll-up angle than our Figure 5 results show. Their theoretically predicted oscillator strengths lie above the experimental values we obtained by integrating resonant spectral features. Compared to previous experimental methods used to estimate SWCNT absorption cross sections in bulk samples, ours is more direct and involves fewer assumptions and potential sources of systematic errors. Our major potential systematic error is overlooking weakly emitting SWCNTs, including short, damaged, or bundled nanotubes, in the fluorescence microscopy counting. However, the close agreement between sample length distributions measured by AFM image analysis and our fluorescence-based LAND method shows that short SWCNTs were not systematically undercounted. Furthermore, our fluorescence imaging sensitivity was even higher for the counting process than for those LAND measurements. We can also discount undercounting of bundles because our (6,5) reference sample was prepared by careful density gradient ultracentrifugation, a method known to provide strong discrimination against nanotube aggregates.38 In addition, a previous study from our laboratory demonstrated that semiconducting SWCNTs could be counted with at least 96% efficiency using SWIR fluorescence microscopy.39 We therefore estimate systematic counting errors to be in the low percent range. Random counting errors are also low (from 3 to 5%) because of the large numbers of counted nanotubes. The relative uncertainties in peak cross sections contributed by subtracting nonresonant backgrounds in the (6,5) analysis are estimated as ∼1% for E11 and ∼3% for E22. Similar uncertainties are introduced in fitting overlapped E22 resonant features. Considering all of these error sources, we estimate overall relative uncertainties of approximately 10% for the (6,5) cross sections and up to 13% for others.

Table 2. Peak and Spectrally Integrated Molar Absorptivity Values Useful for Sample Analysis (n,m)

peak ε11 (MC−1 cm−1)a

E11 fwhm (nm)

(8,3) (6,5) (7,5) (7,6) (8,6) (8,7) (9,7)

4500 6700 4200 5300 3600 4300 2800

20.9 21.8 18.2 20.3 21.9 23.9 26.4

∫ ε11 dλ peak ε22 (105 MC−1 cm−1 nm)b (MC−1 cm−1)a 1.74 2.34 1.67 2.48 1.86 2.59 1.85

1600 1600 1400 1400 1200 1200

a

Peak molar absorptivity, expressed in molarity of carbon atoms, excluding background component. bIntegral of the molar absorptivity versus wavelength in nm, excluding background component.

transitions along with the corresponding resonant full-widths at half-maximum measured in our samples. (These widths are expressed in wavelength units to match typical spectrophotometer data.) To analyze samples in different environments, for which the E11 transitions may be broader (or sharper), the tabulated peak absorptivities should be proportionally scaled down (or up). In addition, broad background absorption should be subtracted to find the resonant peak height before dividing by the absorptivity. Table 2 also includes values for wavelength-integrated E11 molar absorptivities and E22 peak molar absorptivities for resonant components above broad backgrounds. As a practical matter, SWCNT absorption analysis is significantly more reliable for E11 than for E22 or higher transitions because of less spectral congestion and lower backgrounds. In summary, we have measured absolute bulk absorption cross sections for the resonant E11 and E22 transitions of seven small diameter semiconducting (n,m) species. Concentrations were determined for the first time by direct nanotube counting. The results show that cross sections per carbon atom decrease with increasing nanotube diameter. E11 absorptions are stronger than E22 in all species studied. E11 is more intense for mod 1 than for mod 2 species, whereas that pattern is reversed for the E22 transitions. Our numerical results enable one to find absolute concentrations of commonly studied SWCNT species using simple bulk absorption spectroscopy. We expect that our methods can be extended to measure cross sections of additional species and reveal detailed structural dependences that will guide refinement of SWCNT photophysical models and further improve the capabilities for optical sample characterization.

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ASSOCIATED CONTENT

S Supporting Information *

Description of the custom fabricated sample chamber; details of sample preparation; absorption spectral data and fitting procedures; integrated cross section ratios versus SWCNT diameter; examples of (n,m)-selective SWIR fluorscence images; and experimental spectral widths found for E11 and E22. This material is available free of charge via the Internet at http://pubs.acs.org. 1535

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: 713-348-3709. Fax: 713-3485155. Notes

The authors declare the following competing financial interest(s): R.B.W. has a financial interest in Applied NanoFluorescence, LLC, which manufactures one of the instruments used in this study.

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ACKNOWLEDGMENTS This research was supported by grants from the National Science Foundation (CHE-1112374) and the Welch Foundation (C-0807).

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dx.doi.org/10.1021/nl404791y | Nano Lett. 2014, 14, 1530−1536