Determination of the Length, Diameter, Molecular Mass, Density and

Apr 18, 2014 - Surfactant Adsorption of SWCNTs in Dilute Dispersion by Intrinsic. Viscosity ... novel Mark−Houwink−Sakurada equation, [η]c = 1.17...
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Determination of the Length, Diameter, Molecular Mass, Density and Surfactant Adsorption of SWCNTs in Dilute Dispersion by Intrinsic Viscosity, Sedimentation, and Diffusion Measurements Yanbo Yao,§ Sida Luo, and Tao Liu* High-Performance Materials Institute, FAMU-FSU College of Engineering, Florida State University, 2525 Pottsdamer Street, Tallahassee, Florida 32310, United States ABSTRACT: A combination of intrinsic viscosity, sedimentation, and diffusion measurements is performed to characterize the structures of single-walled carbon nanotubes (SWCNTs) in dilute dispersion. It allows quantifying the bulk averaged length (L), diameter (d), molecular mass (M), and density (ρ) as well as the surfactant adsorption for SWCNTs. With this approach applied to the SWCNTs processed by sonication, a novel Mark−Houwink−Sakurada equation, [η]c = 1.17M0.33 is recovered. The less than 0.5 exponent is a result of the powerlaw relationship d ∝ L0.62 caused by the concurrently occurred cutting and exfoliation of SWCNT bundles in the sonication process. An in-depth analysis of the experimentally acquired d and ρ enables a quantification of sodium dodecylbenzenesulfonate (SDBS) adsorption on SWCNTs to reveal the curvature effect on the surface coverage of surfactant adsorption.

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INTRODUCTION Single-walled carbon nanotubes (SWCNTs) have been considered as the ultimate rigid-rod polymers or macromolecules.1 This novel material possesses exceptional electronic, optical, and mechanical properties2,3 and finds a wide range of applications, such as micro- and nanoscale electronics,4 thin film sensors,5−7 lightweight structural nanocomposites,8 and biosensors.9,10 To fully explore these emerging applications, it is important to be able to reliably characterize the structures of SWCNTs, e.g., length, diameter, density, and molecular mass. There are a variety of different techniques currently available for quantitative characterization of SWCNT length and/or length distribution and other related structures. These include the commonly practiced microscopic imaging techniques, such as atomic force microscopy (AFM) and transmission electron microcopy (TEM), as well as the sophisticated scattering based methods, e.g., small angle neutron scattering.11−14 The size determination of SWCNTs based on real-space imaging is quite appealing for the microscopy methods. Nevertheless, the associated issues, such as time-consuming, subjective and sensitive to sample preparations, are also apparent. To overcome the disadvantages of the microscopy techniques, Parra-Vasquez et al.15 relied on intrinsic viscosity measurement to determine the SWCNT length. With this approach, one needs to assume the SWCNT density as well as its diameter in order to be able to extract the length information. Given this limitation, the method developed by Parra-Vasquez et al. is only applicable to individualized SWCNTs, since their diameter and density have reasonably well-defined values. Nevertheless, due to the © 2014 American Chemical Society

strong van der Waals interactions, the SWCNTs typically exist as bundles rather than individual tubes.16 Moreover, the diameter of SWCNT bundle can be reduced through exfoliation in a sonication process.17 This makes it impossible in sizing the as-sonicated SWCNTs by solely measuring the intrinsic viscosity of the SWCNT dispersion. Similar difficulties also exist in microscopy techniques, e.g., AFM and TEM, because the rebundling of SWCNTs in sample preparation may induce the size change of samples being imaged. The aforementioned limitations can be overcome by the preparative ultracentrifugation method (PUM) developed in our previous work.18 The PUM relies on measuring the sedimentation coefficient of SWCNTs in the dispersion, which, in conjunction with dynamic light scattering (DLS) measurement, is able to provide both length and diameter information on SWCNTs. The PUM method is applicable to both SWCNT bundles and individualized tubes. Nevertheless, to apply the PUM method to determine the SWCNT length and diameter, one has to assume the density of SWCNT bundles in the dispersion. A typical value of 1.5 g/cm3 has been used for this purpose. As shown in the present studies, the density of SWCNT bundles in the dispersion varies as a result of surfactant adsorption. In certain cases, the density can deviate from the assumed value (1.5 g/cm3) by ∼20%. In this paper, we performed a combined measurement of intrinsic viscosity, sedimentation, and diffusion on SWCNT dispersions. With minimal assumptions that the Received: February 15, 2014 Revised: April 11, 2014 Published: April 18, 2014 3093

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decreases with the exfoliation process. Second, the density of SWCNT bundles is not known a priori and it may vary with the adsorption of surfactant molecules. This difficulty, as shown below, can be overcome by a combined measurement of intrinsic viscosity, sedimentation coefficient and diffusion coefficient of SWCNT dispersions. According to the hydrodynamic calculations by Tirado22 and Broersam,23 the translational friction coefficient, ζt, for a rodlike particle is given by

SWCNTs, either individual tubes or bundles in the dispersion, can be modeled as cylindrical rigid rods, the combined approach allows the determination of the bulk averaged length, L; diameter, d; molecular mass, M; and density, ρ, as well as the quantitative information regarding the surfactant adsorption on SWCNTs. The structural parameters thus acquired provide very rich information about the processing-structure relationship of SWCNTs produced by sonication process. In particular, due to the simultaneously occurred cutting and exfoliation of SWCNT bundles, a power-law relation d ∝ L0.62 was identified. As a consequence, a novel Mark−Houwink−Sakurada (MHS) equation, [η]c = 1.17M0.33, was recovered. The MHS equation with a less than 0.5 exponent is unique to SWCNT bundles, which has not been observed for any other synthetic and natural rigid-rod polymers. Furthermore, according to the diameter and density results acquired by the combined measurements of intrinsic viscosity, sedimentation coefficient, and diffusion coefficient, we can also quantitatively infer the surfactant adsorption information. An interesting curvature effect for the surfactant adsorption on SWCNTs was experimentally identified: the larger is the diameter of SWCNT bundles; the smaller is the surface coverage density of the surfactant molecules.

ζt = 1500πη0LQ (A) = 1500πη0L × (ln 2A)4 /{[500 ln(A) + 188](ln 2A)4 − 75(ln 2A)3 − 5400(ln 2A)2 + 13750 ln(2A) − 7750} (4)

By using eq 4 and the definition of sedimentation coefficient − s: s=

■

DG =

υ(ρ − ρ0 ) ζt

=

πd 2L(ρ − ρ0 ) 4ζt

(5)

kBT ζt

(6)

we can establish the relationship between the experimentally measured quantity s and DG with respect to [η]c as ⎞3⎡ [η]ϕ (A) ⎤ ⎛ k T ⎞2 ⎛ 1 ⎟⎟ ⎢ − ρ0 ⎥ sDG 2 = ⎜ B ⎟ ⎜⎜ ⎝ 2πA ⎠ ⎝ 1500η0Q (A) ⎠ ⎣ [η]c ⎦

(1)

(7)

In eq 7, ρ0 and η0 are respectively the solvent density and viscosity measured at temperature T; kB is the Boltzmann constant. The expressions for Q(A) and [η]ϕ(A) can be found in eq 4 and eq 3, respectively. With the help of eq 7, a combined measurement of [η]c, s, and DG can be used for uniquely determining the rod aspect ratio, A, from which the rod density, ρ, volume, υ, mass, m, length, L, and diameter, d can also be derived accordingly. The results are given as follows:

where the coefficient [η]c or [η]ϕ is the intrinsic viscositya measure of the hydrodynamic volume of the dissolved polymer or dispersed particle. [η] c (cm 3 /g) is related to [η] ϕ (dimensionless) through the density ρ of the polymer/particle by [η]ϕ = ρ[η]c

ζt

=

and the mass-of-center diffusion coefficient, DG:

THEORETICAL BACKGROUND When polymers or colloidal particles are dissolved or dispersed in a solvent, the viscosity of the solution/dispersion (η) relative to that of the neat solvent (η0) is increased. At very low polymer/particle concentration (C) or volume fraction (ϕ), the viscosity of the solution/dispersion is given by19,20 η ≈ η0(1 + [η]c C) = η0(1 + [η]ϕ ϕ)

m(1 − ρ0 /ρ)

(2)

By considering the analogy between electrostatics and hydrodynamics, Mansfield and Douglas accurately computed the intrinsic viscosity of the rodlike particles by path integral approach.21 According to their calculations, the dimensionless intrinsic viscosity for a cylindrical particle of length L and diameter d is given by

ρ=

υ=

−1 8A2 ⎡ ⎛ 4A ⎞⎤ [η]ϕ = ⎢ln⎜ 25/12 ⎟⎥ ⎠⎦ 45 ⎣ ⎝ e ⎡ 1 − 1.178t + 1.233t 1.86 + 1.925t 6.28 + 0.625t 12.67 ⎤ ×⎢ ⎥ ⎣ 1 − 1.094t + 0.757t 3.76 + 1.344t 3.83 + 1.978t 12.07 ⎦

(3)

where A = L/d is the aspect ratio of the rod; and t = 1/ln(A). As indicated by eq 2 and eq 3, if the particle density is known, the aspect ratio A for a rodlike particle can be determined by measuring the intrinsic viscosity [η]c. With a further assumption of the rod diameter d, the rod length L can be accordingly calculated from A. This is exactly the approach taken by ParraVasquez et al. for measuring the length of individualized SWCNTs.15 However, this approach does not apply to SWCNT bundles−a typical product of as-sonicated SWCNT dispersion. First, the bundle diameter is not a constant and it

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[η]ϕ (A) [η]c

(8a)

s kBT DG ρ − ρ0

(8b)

m = ρυ

(8c)

⎡ 4υ ⎤1/3 d=⎢ ⎥ ⎣ πA ⎦

(8d)

L = Ad

(8e)

EXPERIMENTAL SECTION

Preparation of SWCNT Dispersions. Purified SWCNT raw materials (HiPco SWCNT) were purchased from Carbon Nanotechnologies Inc. (PO No. 126) and used as-received. With sodium dodecylbenzenesulfonate (SDBS, CAS Registry No. 25155-30-10, Sigma−Aldrich) as the dispersing agent, SWCNT aqueous dispersion was prepared by sonicating a mixture of ∼16 mg SWCNTs in 100 mL SDBS/H2O solution (0.7 wt % SDBS in concentration) in an ice bath. 3094

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Table 1. Summary of the Sedimentation Coefficient, s, Diffusion Coefficient, DG, Hydrodynamic Radius, Rh, and Intrinsic Viscosity, [η]C of SWCNT Dispersions Prepared under Different Conditions samples

sedimentation coefficient s (×10−13 s)

B-2.5h B-4.5h B-9h B-11.5h B-14h P-2h P200kg

1979.2 542.2 1425.7 634.3 1097.0 152.0 13.8

diffusion coefficient DG (×10−8 cm2/s) 0.23 0.74 0.53 0.67 0.72 2.42 1.53

± ± ± ± ± ± ±

0.01 0.02 0.05 0.09 0.06 0.00 0.19

hydrodynamic radius Rh (nm) 1066.4 331.4 462.8 366.1 340.6 101.3 160.3

± ± ± ± ± ± ±

46.4 9.0 43.7 49.2 28.4 0.0 19.9

intrinsic viscosity [η]c (cm3/g) 2694.9 950.6 1249.4 1029.6 1057.7 424.7 3434.9

± ± ± ± ± ± ±

12.2 5.8 6.8 9.0 15.9 4.5 37.8

An Ubbelohde capillary viscometer (Cannon Instrument Company, no. 1B, calibration constant of 0.05) was used for measuring the viscosity of SWCNT dispersions at 25 °C with temperature controlled by an electro-thermostatic water cabinet (±0.3 °C specified accuracy). During the measurement, the sample flow was videoed at 25 frames per second, which was then image processed for accurately determining the flow time. As an error control protocol, the viscosity of pure water (ρ = 0.998 g/cm3 at 25 °C) was measured at the end of each sample test and compared with the standard value. With this practice, a total of 31 measurements on water were performed during the entire course of SWCNT dispersion viscosity test. It resulted in an averaged viscosity of 0.899 ± 0.007 cP. This value is slightly higher than the standard viscosity 0.8902 cP of water at 25 °C.27 This difference is attributed to the kinetic energy effect caused by the relatively short flow time of water (17.99 ± 0.15 s) in capillary no. 1B. With the standard value of 0.8902 cP as a calibration, the kinetic energy corrected Poiseulle relation for the no. 1B capillary used in this work was obtained as η (cP) = 0.05t − 0.168/t. This relation was further used to calculate the viscosity of a given SWCNT dispersion and its diluted samples according to the experimentally measured flow time. Typically, each of the as-prepared SWCNT dispersion of concentration C0 was diluted to 4−5 samples with concentration ranging from (0.2−1) × C0 for viscosity test. The corresponding SWCNT concentration (g/cm3) was determined by UV−vis−NIR spectroscopy according to the procedures described previously. According to eq 1, a linear fitting of the SWCNT concentration against the corresponding dispersion viscosity was applied to determine the intrinsic viscosity [η]c. The results are given in Table 1.

To acquire a reasonably large sample space (SWCNT dispersions with varied SWCNT size) for subsequent studies, two different types of sonicator were used for sonicating SWCNT/SDBS/H2O mixture at varied effective sonication duration. One sonciator is a Misonix 3000 probe sonicator operated at 20 kHz in pulsed mode (on 10 s, off 30 s) with power level set at 45 W. Another is a Bransonic 2510 bath sonicator operated at 37 kHz in continuous mode with power level set at 100 W. With the procedures described above, a total of six assonicated dispersions were prepared and denoted respectively as B2.5h, B-4.5h, B-9h, B-11.5h, B-14h, and P-2h. To prepare the individualized SWCNT dispersion, the 1 h probe-sonicated dispersion was subjected to an ultracentrifugation process at 200 000g force for 3 h by using an Optima MAX Ultracentrifuge (Beckman Coulter, MLS50 swinging bucket rotor). At the end of the centrifugation process, the supernatant was carefully decanted and collected. The SWCNT dispersion thus prepared are denoted as P200kg. Spectroscopic Characterization of SWCNT Dispersions. A Renishaw inVia Raman microscope (785 nm excitation laser in backscattering geometry) was used for simultaneously collecting the Raman scattering and photoluminescence (PL) spectra of SWCNT dispersions. The simultaneously acquired Raman scattering and PL spectra allow the examination of the exfoliation or bundling states of SWCNT dispersion.17 Attal et al.24 demonstrated the use of UV− visible absorption spectroscopy as a simple, accurate, convenient, and rapid tool for determining the SWCNT concentration in different surfactant-dispersed aqueous systems. By following a similar approach, in this study, we measured the optical absorbance, Abs, of the SWCNT dispersion with a Varian Cary 5000 UV−vis−NIR spectrometer (optical path l = 0.5 cm), which was then used for accurately determining the SWCNT concentration, C, according to Beer’s law: Abs = εCl. In this practice, the previously determined ε value,9 1.03 ± 0.15 × 10−18 cm2/C atom =0.052 ± 0.008 L/(mg cm), was taken as the absorption cross-section of SWCNTs at 785 nm. Sedimentation, Diffusion, and Intrinsic Viscosity Measurement of SWCNT dispersions. With the Optima MAX-XP ultracentrifuge (Beckman Coulter, Inc., TLA-100.3, 30° fixed angle rotor) and a Varian Cary 5000 UV−vis−NIR spectrometer, the PUM method developed previously 18 was applied to obtain the sedimentation functions of SWCNT dispersions (13 000g force for as-sonicated dispersions and 65 000g force for individualized dispersion). According to the analytical sedimentation solution derived by Mason et al.25 and Shiragami et al.,26 the experimentally acquired sedimentation function was then numerically fitted to extract the bulk averaged sedimentation coefficient, s, of SWCNTs. The s results are listed in Table 1. Dynamic light scattering (DLS) was conducted by using a Delsa Nano C Particle Size Analyzer (Beckman Coulter, Inc.) to determine the diffusion coefficient DG of SWCNTs in the dispersion. In DLS measurements, the time fluctuation of the scattered light intensity, Is(t), of the SWCNT dispersion was recorded at 25 °C at a fixed scattering angle of 165°. It was used for calculating the time correlation functions, g1(τ), of the scattered electric field, which was then fitted by the well-known CONTIN method to extract the particle diffusion coefficient DG. The values of DG for all the SWCNT dispersions are listed in Table 1. In the same table, the hydrodynamic radii, Rh, of SWCNT bundles calculated according to the Stokes−Einstein relation Rh = kBT/6πη0DG were also listed.

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RESULTS AND DISCUSSION Diffusion of SWCNT Bundles Examined by Multiangle DLS Measurement. To examine the diffusion nature of SWCNT bundles in the dispersion, the multiangle DLS measurements (20−150°) were performed for a 2-h probesonicated SWCNT dispersion (BI-200SM/TruboCorr scattering system, Brookhaven Instruments, 637 nm excitation laser). Figure 1a shows the experimentally measured and the CONTIN fitted time correlation functions, g1(τ), at the scattering angle of 90°. The major decay mode of the

Figure 1. (a) Field correlation function g1(τ) and (b) relationship between the decay rate Γ and square of the scattering vector q for a 2 h probe-sonicated SWCNT dispersion. 3095

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correlation function g1(τ) occurred at the delay time τ ∼ 10−3 s is attributed to the diffusion of SWCNT bundles. For Brownian particles of small aspect ratio, there is a well-known relation, Γ = DGq2, to correlate the particle mass-of-center diffusion coefficient DG with the decay rate Γ of g1(τ) at a given scattering vector q.19,20,28 This relation was examined for the 2h probe-sonicated SWCNT dispersion by plotting Γ versus q2. The result is shown in Figure 1b. Clearly, the Γ ∝ q2 relation is recovered at small q. However, for large q, the experimentally determined Γ positively deviates from the one predicted by Γ = DGq2. This deviation is due to the long rod nature of SWCNT bundles. As discussed in the previous studies,29−34 the anisotropic translational diffusion of rod-like particles and the resulted translation-rotation coupling play a critical role in dictating their DLS behavior. Because of these complications, the relation of Γ = DGq2 originally applicable to Brownian particles of small aspect ratio should be modified to Γ = DGq2f(qL) for long rigid rods.34 The function f(qL) takes into account the effect of translation-rotation coupling. It assumes a value of 1.0 for qL less than ∼4.0 when the coupling effect is weak and a value of ∼1.33 for qL greater than 20.0 when the coupling effect is strong. Given the complexity induced by the effect of translation-rotation coupling, the common practice used for determining DG with DLS by Γ/q2 may have an error varying from 0−33% dependent upon the value of qL. Since qL is not known a priori, it creates difficulties for accurately determining the DG of long rods by DLS. Because of such difficulties, the conventional practice (DG = Γ/q2) was used in the present work to determine the DG values of SWCNT dispersion as listed in Table 1. The future research efforts will focus on obtaining the accurate DG of SWCNT bundles by considering the translation−rotation coupling effect. Cutting and Exfoliation of SWCNT Bundles Revealed by Spectroscopic, Sedimentation and Diffusion Measurements. It has been well established that the cutting and exfoliation simultaneously occur in the probe sonication of SWCNTs in surfactant-dispersed aqueous systems.17,35 As a consequence, the length and diameter of SWCNT bundles are reduced at the same time. Similar to probe sonication, the bath sonication process can also induce shortening and exfoliation of SWCNT bundles. Parts a and b of Figure 2 present the spectroscopic evidence to show this point. In Figure 2a, the simultaneously acquired Raman and photoluminescence (PL) spectra normalized by the G-band of SWCNTs (1591 cm−1) are compared for the dispersions that were sonicated for a relatively short period of time (B-2.5h, B-4.5h, and P-2h). The PL features observed at 2986, 2269, and 1819 cm−1 are evident for all samples, and their G-band normalized intensity follows a trend of B-2.5h < B-4.5h < P-2h. The PL emission observed in a given SWCNT dispersion is due to the presence of individualized semiconducting tubes or their small bundles.36 In addition to the PL features, the Raman band located at 267 cm−1, which originates from the radial breathing mode of the (10, 2) tube being constrained in a SWCNT bundle, also provides an indication on the exfoliation of SWCNT bundles.37 The SWCNT dispersion with a smaller intensity of 267 cm−1 band corresponds to a more exfoliated state. The PL features and the 267 cm−1 band shown in Figure 2a all indicates that the mild bath sonication is sufficient to exfoliate the large sized SWCNT bundles, though the efficiency is not as high as the probe sonication. The UV−vis−NIR spectroscopy provides additional evidence to corroborate this observation. Figure 2b

Figure 2. (a) Simultaneously acquired Raman scattering and photoluminescence spectra and (b) UV−vis−NIR absorption spectra for SWCNT dispersions prepared by relatively short sonication duration. (c) The power-law relationship between the sedimentation coefficient, s, and the diffusion coefficient, DG, for the as-sonicated SWCNT dispersions from different sources.

compares the UV−vis−NIR spectra of B-2.5h, B-4.5h, and P-2h dispersions. As compared to B-2.5h and B-4.5h, the wellresolved absorption peaks in P-2h suggest the high efficiency of probe sonication in exfoliating SWCNT bundles.38,39 It should be noted that the peak features observed in the UV−vis−NIR spectra are due to the electronic energy transitions of SWCNTs, which mostly depend on their electronic structures and the surrounding dielectric environment. The influence of SWCNT concentration is negligible. The simultaneously occurred cutting and exfoliation during the sonication process of SWCNTs result in the concurrent decrease of SWCNT bundle diameter d and length L. As a result, one may expect a correlation between d and L. As shown in Figure 2c, such a correlation is apparent. It is manifested by an approximate power-law relationship between the sedimentation coefficient, s, and the diffusion coefficient, DG, for all the as-sonicated SWCNT dispersions. In the same figure, we also include the s and DG results reported in ref 35 for probesonicated SWCNT dispersions. A power-law fitting to all the s and DG data results in a relation of s ∝ DG−1.58. By inspecting eq 4, 5, and 6 and neglecting the effect of ln(A), one finds s ∝ d2 and DG∝ L−1. With these results and the experimental observation s ∝ DG−1.58, a strong correlation between d and L − d ∝ L0.79 for the sonication processed SWCNTs can be established. This result, as expected, indicates that the SWCNT bundles are cut and exfoliated concurrently in the sonication process. Mark−Houwink−Sakurada Equation of SWCNTs. Effects of Cutting and Exfoliation. Mark−Houwink− Sakurada (MHS) equation19,20 − [η]c = KMα is a wellestablished relationship useful for characterizing polymers. With this equation, the viscosity-averaged molecular weight M of a polymer can be estimated by simply measuring the viscosity of its solution in an appropriate solvent. Moreover, the exponent α in MHS equation is a reflection of the polymer chain rigidity. For a flexible linear chain in theta solvent, α is 0.5; in good solvent, it becomes 0.7−0.8. When α is greater than 1.0, the 3096

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experimental observations (Figure 3), and it indeed suggests the importance of the strong correlation between d and L in the interpretation of the novel MHS equation of SWCNTs. It is noted that, due to their partially permeable/porous structures, the MHS equation for hyperbranched polymer and dendrimer also has a characteristic exponent value of less than 0.5.46,47 Given this, another possibility to explain the novel MHS equation of SWCNTs might be attributed to the formation of SWCNT aggregates with hyperbranched and/or dendrimer-like structures in the dispersion. However, such hyperbranched and/or dendrimer-like structures do not expect to exist, at least not in large quantities, in the SWCNT dispersion prepared by the protocols used in this study, since the same or similar protocols have long been established through a vast body of scattering, microscopic and spectroscopic evidence11−13,15,17,18,24,37,44,45,51 as an efficient technique to well disperse, exfoliate and stabilize SWCNTs as individual tubes and/or bundles in water. As a matter of fact, these experimental protocols have been the key for successfully separating SWCNTs according to their electronic types with density column centrifugation method,51 which cannot be done if SWCNTs exist in aggregation form. With these arguments, we can exclude the formation of hyperbranched and/or dendrimerlike structures in the SWCNT dispersion as a plausible explanation for the novel MHS equation observed in this study. With the experimentally determined results of s, DG, and [η]c, the length L, diameter d, aspect ratio A, density ρ, and molecular mass M of the SWCNTs in different dispersions were calculated according to eq 7 and eq 8. The results are given in Table 2 and plotted in Figure 4. Figure 4a clearly

chain is considered rigid. The polymeric nature of SWCNTs has been well appreciated.1 However, there is still a lack of experimental studies in establishing the MHS equation for SWCNTs. With the combined measurements of intrinsic viscosity, sedimentation and diffusion, we are attempting to do so in this study. Before proceeding to a quantitative analysis, a qualitative discussion is helpful to understand the effects of SWCNT bundling and exfoliation on the related MHS equation. By inspecting eq 5 and 6, one notes that the ratio of s/DG is proportional to the molecular mass M of SWCNTs. Therefore, according to MHS equation, a power-law relationship should exist between [η]c and s/DG. This is indeed the case for the as-sonicated SWCNT dispersions. As shown in Figure 3, the power-law relationship of [η]c ∝ (s/DG)0.34 ∝ M0.34 is clearly born out. Nevertheless, the less than 0.5 exponent is a surprising result and worthy of further discussions.

Figure 3. Power-law relationship between the experimentally determined intrinsic viscosity [η]c and the ratio of sedimentation coefficient to diffusion coefficient, s/DG, for the as-sonicated SWCNT dispersions.

Table 2. Summary of the Structural Parameters of SWCNT Dispersions Prepared under Different Conditions

As mentioned previously, for rigid rod polymers, the exponent α in MHS equation is greater than 1.0. For example, the synthetic rodlike polymers−poly(p-benzanilide terephthalamide) and poly(p-phenylene terephthalamide)dissolved in concentrated sulfuric acid have an exponent value of 1.22− 1.23;40 the natural rodlike biopolymersPBLG in dimethylformamide41 and ds-DNA in water,42 respectivelyhave an exponent of 1.75 and 1.05. The very rigid nature of an individual SWCNT or a SWCNT bundle is manifested by their extreme large persistence lengtha few to a few tens and even hundreds of micrometers, which has been theoretically predicted and experimentally validated.43−45 Considering this fact and applying the results of Mansfield and Douglas (eq 3), one should expect [η]c ∝ (L/d)1.785 ∝ M1.785∝ (s/DG)1.785 for SWCNTs with aspect ratio between 100−1000, if the diameter d is a constant. Obviously, this result contradicts the experimental observation shown in Figure 3. To understand this contradictory result, one needs to consider the unique behavior of SWCNT bundling and its exfoliation in the sonication process. As discussed previously, the sonication induces simultaneous cutting and exfoliation of SWCNT bundles, which leads to a strong diameter-length correlation, d ∝ L0.79. With this result and considering M ∝ d2L, we can obtain the mass-length scaling relation M ∝ L2.58 for SWCNT bundles. It is drastically different from M ∝ L, which is commonly observed for the other synthetic and natural rigid rod polymers. This novel mass-length scaling relation, M ∝ L2.58, for SWCNT bundles in conjunction with [η]c ∝ (L/d)1.785 eventually give rise to [η]c ∝ M0.14 ∝ (s/DG)0.14. The qualitative discussion shown here agrees reasonably well with the

samples

aspect ratio, A

density, ρ (g/cm3)

B-2.5h B-4.5h B-9h B-11.5h B-14h P-2h P200kg

295.8 168.2 217.4 176.9 208.9 115.2 317.5

1.20 1.25 1.50 1.26 1.65 1.45 1.07

molecular weight, M (g/mol)

length, L (nm)

Diameter, d (nm)

× × × × × × ×

12538 3479 5104 3882 3782 989 1912

42.4 20.8 23.5 22.0 18.1 8.6 6.0

1.28 8.96 1.99 1.12 9.66 5.00 3.50

1010 108 109 109 108 107 107

shows the power-law relationship between d and L − d = 0.12L0.62a consequence of simultaneous cutting and exfoliation of SWCNT bundles as discussed previously. Because of

Figure 4. Power-law relationship between (a) SWCNT bundle length L and diameter d and (b) intrinsic viscosity [η]C and molecular weight M for the as-sonicated SWCNT dispersions. Scattered data are taken from Table 2. 3097

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With this formula and using the typical value of dt = 0.98 ± 0.21 nm for an individual HiPco SWCNT,53 we can estimate the density of an individual pristine HiPco tube. It is 3.11 ± 0.67 g/ cm3a value significantly higher than the experimentally determined 1.07 g/cm3. The reason for this is because the individualized SWCNT dispersed in water has surfactant molecules (SDBS) adsorbed on its surface, which makes the effective diameter of the tube larger and therefore a smaller density. If the adsorbed SDBS layer has a thickness lc and a surface coverage of nSDBS, by using eq 9, the density of an individual SWCNT stabilized by SDBS can be written as

this strong diameter-length correlation, a novel MHS equation [η]c = 1.17M0.33 for SWCNT bundles is obtained, and the results are shown in Figure 4b. The MHS equation of SWCNT bundles shows a less than 0.5 exponent. As explained earlier, however, this does not make the claim of SWCNTs as rigid rod invalid. The rigid rod behavior of SWCNTs can be recovered by considering the relationship between the dimensionless intrinsic viscosity [η]ϕ and the aspect ratio A = L/d. By using the density results listed in Table 2, the [η]ϕ for all the dispersions, SWCNT bundle and individualized SWCNTs were calculated according to eq 2. Figure 5 shows the plot of [η]ϕ against A = L/d. In the same

ρt − SDBS = =

figure, we also include the results of individualized SWCNTs reported in ref 15 for comparison. As can be seen from Figure 5, all the SWCNT samples−the as-sonicated bundles and the individual tubes from different workobey the theoretically predicted power-law relationship for rigid rods[η]ϕ ∝ (L/ d)1.76. It is also noted that the individualized SWCNTs reported in ref 15 have a much higher [η]ϕ and aspect ratio L/d than the one prepared in this work. The incorrect values of SWCNT density (1.45 g/cm3) and diameter (1.25 nm) assumed in ref 15 are considered to be the major factors responsible for the observed difference. As discussed next, due to surfactant adsorption, the density and diameter value used in ref 15 were respectively over and underestimated. Certainly, the different processing conditions (sonication power and duration, centrifugation g-force and time, and the types of dispersing agents) used in ref 15 and our work might also play some roles. Density of SWCNTs. Individualized Tube vs SWCNT Bundle and the Effects of SDBS Adsorption. As indicated in Table 2, the density of SWCNTs varies with the sample preparation conditions. In particular, the individualized SWCNT has a much smaller density than SWCNT bundles. This result is consistent with those measured by density gradient sedimentation/ultracentrifugation approach reported in the literature.48−51 In particular, the density of individualized SWCNT obtained in this work −1.07 g/cm3 has a quantitative agreement with the values of 1.06−1.12 g/cm3 as reported in refs 49−51. But all these values are significantly lower than the theoretically predicted density of a pristine individual SWCNT, which can be easily derived according to the repeating unit structure of a pristine individual SWCNT of diameter dt as52 3.045 × 10−21 (g/nm 3) dt

deff 2

(4nSDBSmSDBS + 3.045 × 10−21)dt (dt + 2lc)2

(10)

In eq 10, mSDBS = 5.41 × 10−22 g is the mass of a SDBS molecule without inclusion of the sodium atom contribution. Equation 10 can be used for explaining the reduced density of an individual SWCNT when stabilized by SDBS. Moreover, it allows an estimation of the adsorption parameters lc and nSDBS of SDBS on SWCNT by using the experimentally determined ρt‑SDBS and deff. With ρt‑SDBS = 1.07 g/cm3 and deff = 6.0 nm for P200 kg and by taking dt = 0.98 nm, we obtain lc = 2.51 nm and nSDBS = 16.7 molecules/nm2. Such determined surface coverage, nSDBS = 16.7, has a close agreement with the literature reported value, nSDBS = 18.3 molecules/nm2 and nSDBS = 22.5 molecules/ nm2, which were respectively measured through surface tension and adsorption isotherm.54 In the same work,54 a monolayer adsorption of SDBS molecules with vertical orientation has been proposed for the SDBS stabilized SWCNTs. An excellent agreement between our calculated result of lc = 2.51 nm and the length of the hydrocarbon tail of SDBS molecule, 2.427 nm, provides good evidence to corroborate this monolayer adsorption model. Similar to an individual SWCNT discussed above, the effect of SDBS adsorption on the density reduction of a SWCNT bundle can also be quantified. A SWCNT bundle can be considered as an assembly of individual tubes packed into a 2-D hexagonal lattice with a wall-to-wall distance of 0.34 nm.16 With this structure in mind, the density of a pristine SWCNT bundle, which is composed of individual tubes with a uniform diameter dt, can be derived as

Figure 5. Power-law relationship between the dimensionless intrinsic viscosity [η]ϕ and the rod aspect ratio (L/d) for SWCNT bundles and individualized tubes prepared at different conditions.

ρt =

4dt nSDBSmSDBS + ρt dt 2

ρb =

2.762 × 10−21dt (dt + 0.34)2

(g/nm 3) (11)

Again, by taking dt = 0.98 ± 0.21 nm, the density of a pristine HiPco SWCNT bundle is 1.55 ± 0.16 g/cm3. When this value is compared with those listed in Table 2, one notes that the density reduction for SDBS-stabilized SWCNT bundles is not as great as that for individualized SWCNTs. This is an expected result, since the SWCNT bundle has a much larger diameter than an individual tube. Consequently, the effect of density reduction due to diameter increase by SDBS adsorption in a SWCNT bundle is not as significant as that in an individual tube. With eq 11, the density for a SDBS stabilized SWCNT bundle can be written as

(9) 3098

dx.doi.org/10.1021/ma5003497 | Macromolecules 2014, 47, 3093−3100

Macromolecules ρb − SDBS = =

Article

L, diameter, d, molecular mass, M, density, ρ, and surfactant adsorption of SWCNTs in dispersion. This simple but powerful method has been applied to study the processing-structure relationship of SWCNT aqueous dispersion produced by sonication process. For the first time, a novel MHS equation, [η]c = 1.17M0.33, with a less than 0.5 exponent is established for SWCNT bundles, which has not been observed for any other synthetic and natural rigid rod-like polymers. This novel phenomenon is a result of the power-law relationship d ∝ L0.62 caused by the concurrently occurred cutting and exfoliation of SWCNT bundles in the sonication process. An in-depth analysis of the experimentally acquired SWCNT bundle diameter and its density enables a quantification of sodium dodecylbenzenesulfonate (SDBS) adsorption on SWCNTs, which further reveals the curvature effect of surfactant adsorption on the surface coverage density.

4dbnSDBSmSDBS + ρb db 2 deff 2

4dbnSDBSmSDBS + ρb db 2 (db + 2lc)2

(12)

Equations 10 and 12 suggest that the density of an individual SWCNT or SWCNT bundle in the dispersion varies with its diameter and the amount of SDBS molecules being adsorbed. This in turn may depend upon the detailed processing conditions. As a consequence, one finds a relatively large density variation (Table 2) for the SWCNTs prepared under different conditions. Certainly, as discussed previously, the experimental error on the diffusion coefficient DG introduced by neglecting the effect of translation-rotation coupling may also play a role, since the error transferred from DG may cause uncertainties in the determination of length, diameter, and molecular mass and therefore the density of SWCNTs. Previous studies54−56 suggested that the surface coverage nSDBS strongly depends upon the curvature of the substrate being adsorbed. For an individual tube, nSDBS can be as high as 22.5 molecules/nm2, and for a flat graphite surface, a very low value of nSDBS = 1.45 molecules/nm2 was observed. These early findings suggest the curvature effect of SDBS adsorption on SWCNT bundles. Namely, the value of nSDBS expects to decrease with increasing the SWCNT bundle diameter. To examine this speculation, we calculated the values of nSDBS for all the SWCNT bundle samples by using the experimentally determined ρt‑SDBS and deff. In the calculation, the density ρb of a pristine SWCNT bundle and the SDBS adsorption thickness lc were respectively taken as 1.55 g/cm3 and 2.51 nm. The results are shown in Figure 6. Within the experimental error, the

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AUTHOR INFORMATION

Corresponding Author

*(T.L.) Telephone: +1 850 410 6606. Fax: +1 850 410 6342. Email: [email protected]. Present Address §

(Y.Y.) Advanced Materials Laboratory, Key Laboratory of High Performance Ceramic Fibers of Ministry of Education, College of Materials, Xiamen University, Xiamen 361005, China Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Y.Y. acknowledges financial support from the China Scholarship Council (File No. 201206310069) for her graduate study abroad at Florida State University.

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Figure 6. Effect of SWCNT diameter, dt or db, on the surface coverage of SDBS adsorption. dt or db is calculated by deff − 2lc. The solid line is a guide to the eye. Inset is the adsorption model to depict a monolayer of SDBS adsorbed on SWCNT with thickness lc = 2.51 nm determined experimentally.

curvature effect of SDBS adsorption on SWCNTs is clearly born out. Specifically, the larger is the SWCNT bundle diameter, the smaller is the SDBS surface coverage.

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CONCLUSIONS A combined study of intrinsic viscosity, sedimentation, and diffusion measurements was performed to characterize the structures of SWCNTs in dispersion. With minimal assumptions that the SWCNTeither individual tubes or bundles in the dispersioncan be modeled as a cylindrical rigid rod, the current approach allows quantifying the bulk averaged length, 3099

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