Alignment Dynamics of Single-Walled Carbon Nanotubes in Pulsed Ultrahigh Magnetic Fields
J. Shaver, A. N. G. Parra-Vasquez, S. Hansel, O. Portugall, C. H. Mielke, M. von Ortenberg, R. H. Hauge, M. Pasquali, J. Kono
AAlignment Dynamics of Single-WalledCarbon Nanotubes in PulsedUltrahigh Magnetic Fields
October 30, 2018
Jonah Shaver , † , A. Nicholas G. Parra-Vasquez , † , Stefan Hansel , ,Oliver Portugall , Charles H. Mielke , Michael von Ortenberg ,Robert H. Hauge , Matteo Pasquali , and Junichiro Kono , ∗ Department of Electrical and Computer Engineering, Rice University,Houston, Texas 77005; Department of Chemical and BiomolecularEngineering, Rice University, Houston, Texas 77005; Institut f¨ur Physik,Humboldt-Universit¨at zu Berlin, Berlin, Germany; Laboratoire Nationaldes Champs Magn´etiques Puls´es, 31400 Toulouse, France; National HighMagnetic Field Laboratory, Los Alamos, New Mexico 87545; Departmentof Chemistry, Rice University, Houston, Texas 77005
October 30, 2018 † Jonah Shaver and A. Nicholas G. Parra-Vasquez contributed equally to thiswork. ∗ Please address all correspondence to [email protected]: Carbon nanotubes, Optical properties of carbon nanotubes, Dichro-ism of molecules, Absorption spectra of molecules, Light absorption andtransmission, Generation of high magnetic fields1 a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug bstract We have measured the dynamic alignment properties of single-walled carbon nanotube (SWNT) suspensions in pulsed high magneticfields through linear dichroism spectroscopy. Millisecond-durationpulsed high magnetic fields up to 56 T as well as microsecond-durationpulsed ultrahigh magnetic fields up to 166 T were used. Due to theiranisotropic magnetic properties, SWNTs align in an applied mag-netic field, and because of their anisotropic optical properties, alignedSWNTs show linear dichroism. The characteristics of their overallalignment depend on several factors, including the viscosity and tem-perature of the suspending solvent, the degree of anisotropy of nan-otube magnetic susceptibilities, the nanotube length distribution, thedegree of nanotube bundling, and the strength and duration of the ap-plied magnetic field. In order to explain our data, we have developed atheoretical model based on the Smoluchowski equation for rigid rodsthat accurately reproduces the salient features of the experimentaldata.
Single-walled carbon nanotubes (SWNTs), rolled up tubes ofgraphene sheets, are unique nano-objects with extreme aspect ra-tios, which lead to unusually anisotropic electrical, magnetic, andoptical properties. They can be individually suspended in aque-ous solutions with appropriate surfactants, and such suspendedSWNTs behave roughly as rigid rods undergoing Brownian mo-tion. In the absence of external fields, their orientation angles arerandomly distributed. However, when placed in a perturbing field,suspended SWNTs will align parallel to the field lines due to theiranisotropic properties. The steady state alignment of SWNTs inmagnetic, electric, flow, and strain fields has been char-acterized in many recent studies. Though mentions of dynamicalignment have been made, to date there are no comprehen-sive studies. Here we present the first combined experimental andtheoretical study that provides fundamental insight into the hy-drodynamic motion of these highly-anisotropic nano-objects. The magnetic susceptibilities of SWNTs of different diameters, chiralities,and types have been theoretically calculated using different methods.
Semiconducting SWNTs are predicted to be diamagnetic ( χ <
0) both par-allel ( (cid:107) ) and perpendicular ( ⊥ ) to their long axis, but the perpendicularsusceptibility is predicted to have a larger magnitude ( | χ ⊥ | > | χ (cid:107) | ), aligning2he SWNT parallel to the field. Metallic SWNTs are predicted to be para-magnetic (diamagnetic) parallel (perpendicular) to their long axes ( χ (cid:107) > χ ⊥ <
0) and thus also align parallel to the applied field. For ∼ χ = χ ⊥ − χ (cid:107) , calculatedby an ab initio method are between 1.2 and 1.8 × − emu/mol, depend-ing on the tube chirality, which are similar to the values calculated by a k · p method (1.9 × − emu/mol) and by a tight-binding method (1.5 × − emu/mol). These values are consistent with recently-reported exper-imental values, measured with steady-state optical methods.
The degree of alignment of SWNTs in a magnetic field can be convenientlycharacterized by the dimensionless ratio of the alignment potential energyand the thermal energy, ξ = (cid:115) B N ∆ χk B T (1)where B is the magnetic field, N is the number of carbon atoms in theSWNT, k B is the Boltzmann constant, and T is the temperature of thesolution. A significant fraction of nanotubes in the solution will align with B when the alignment energy is greater than the randomizing energy, i.e., when ξ >
1. Using u and the angle ( θ ) between a SWNT and the aligning magneticfield, an angular distribution function, P ( θ ), in thermal equilibrium can becalculated as dP ( θ ) dθ = e − ξ sin θ sin θ (cid:82) π/ e − ξ sin θ sin θdθ . (2)Many experiments have studied the equilibrium alignment of SWNTs in mag-netic fields. More recent experiments have explored the chirality de-pendence of SWNT alignment to extract the SWNT species specific magneticsusceptibilities. Linear dichroism spectroscopy has a well-developed history of applicationto both steady state and dynamic situations, such as the flow-induced align-ment of fibrils and the magnetic-field-induced alignment of polyethyleneand carbon fibers. However, to date no one has studied the dynamic effectsof alignment of SWNTs. Defined as the difference between the absorbanceof light polarized parallel ( A (cid:107) ) and perpendicular ( A ⊥ ) to the orientationaldirector of a system, ˆ n , linear dichroism ( LD ) is a measure of the degreeof alignment of any solution of anisotropic molecules. Experimentally, thesign of LD gives qualitative information about the relative orientation of3olecules, positive for alignment parallel to ˆ n and negative for perpendicular.Reduced LD , LD r , is normalized by the unpolarized, isotropic absorbance( A ) of the system, and gives a quantitative measure of the alignment. Themeasured LD r spectrum is related to both the polarization of the transitionmoment being probed and the overall degree of alignment of the moleculesbeing investigated: LD r = LDA = A (cid:107) − A ⊥ A = 3 (cid:18) α − (cid:19) S (3)where α is the angle between the transition moment and the long axis of themolecule and S is the nematic order parameter. S is a dimensionless quantitythat scales from 0 for an isotropic sample to 1 for a perfectly aligned sampleand is defined as S = 3 (cid:104) cos θ (cid:105) −
12 (4)where (cid:104) cos θ (cid:105) is averaged over the angular probability distribution functionand θ is the microscopic angle made between a SWNT’s long axis and thealignment director of the system.For the case of SWNTs, optical selection rules coupled with a strongdepolarization for light polarized perpendicular to the tube axis result inappreciable absorption features observed only when light is polarized parallelto the tube axis. Hence, we can simplify Eq. (3) using α = 0, to LD r = 3 S ,giving a direct link between the measured LD r and the orientation of theSWNTs.In this study, the dynamic effects of SWNT alignment in pulsed high mag-netic fields were investigated for the first time. We measured time-dependenttransmittance through individually-suspended SWNTs in aqueous solutionsin the Voigt geometry (light propagation perpendicular to the applied mag-netic field) in two polarization configurations, parallel and perpendicular tothe applied magnetic field. From this we calculated LD as a function oftime, both in millisecond (ms)-long pulsed high magnetic fields up to 56 Tand microsecond ( µ s)-long pulsed ultrahigh magnetic fields up to 166 T. Wedeveloped a theoretical model based on the Smoluchowski equation, whichextracts the length distribution of the SWNTs in suspension based on a fit totime-dependent LD . These results pave the way to further study of SWNTdynamics in solution. 4 esults Measured Transmittance
All ms-pulse data was taken using a spectrally resolved, near-infrared setup.To avoid any convolution with spectral lineshape broadening and splitting the data was integrated over the entire InGaAs range ( ∼
900 nm to 1800 nm).The benefit of removing ambiguity associated with spectral changes inducedby the Aharonov-Bohm effect coupled with the large number of nanotubechiralities present in our sample outweighs the possibility for any chiralityselective analysis (which has been performed at low magnetic fields ).Figure 1(a) displays spectrally-integrated, time-dependent transmittancethrough the sample and polarizer [in parallel (blue) and perpendicular (red)configurations] and the accompanying 56 T magnetic field trace (green). Theraw transmittance data is normalized to the zero-field value as T N (cid:107) , ⊥ ( t ) = T (cid:107) , ⊥ ( t ) T ( t = 0) (5)where T (cid:107) , ⊥ ( t ) denotes the raw transmittance as a function of time with therespective polarization configuration. Starting at time zero, before the fieldpulse, the transmittance in both polarization configurations is equal. As thefield increases, and the SWNTs start to align with the field, light polarizedparallel (perpendicular) to the magnetic field decreases (increases) in overalltransmittance.Similarly, Fig. 1(b) shows the optical response of suspended SWNTs toa µ s-pulse magnetic field produced by the Megagauss Generator in Berlin. This data was collected with an Ar + ion laser at 488 nm, which is in thesecond subband region of the SWNT optical spectra, and thus the Aharonov-Bohm-effect-induced spectral changes are small in relation to the linewidth,negating the need for spectral integration. As the field rises to 140 T ( ∼ µ srise time), the nanotubes align to their maximum value, which lags the peakfield by ∼ µ s. It should be noted that in this experiment the field returnsto zero at ∼ µ s and then increases in the negative direction , reaching aminimum of ∼ −
50 T at ∼ µ s. However, since only the magnitude of themagnetic field ( | (cid:126)B | ) is important in aligning the nanotubes, the transmit-tance shows a secondary peak at ∼ µ s. This is also clearly demonstratedby the parallel configuration data in Fig. 1(d) where we used the MegagaussGenerator to produce a rapidly oscillating field of ≈
65 T. Figure 1(c) shows5 .61.20.80.4 T ( t ) / T ( t = ) Time (ms) M agne t i c F i e l d ( T ) T || T ⊥ (a) T ( t ) / T ( t = ) M agne t i c F i e l d ( T ) T ⊥ T || (b) T ( t ) / T ( t = ) Time ( µ s) -60-3003060 M agne t i c F i e l d ( T ) T || (d) T ( t ) / T ( t = ) M agne t i c F i e l d ( T ) T ⊥ Detector overload (from coil break flash) (c)
Figure 1: (color online) Time-dependent traces of transmittance of lightpolarized parallel (red, left axis) and perpendicular (blue, left axis) to theapplied magnetic field (green, right axis) for (a) a 56 T, 50-ms-rise-time pulse,(b) a 140 T, 2.5- µ s-rise-time pulse (Megagauss), (c) a 166 T, 2.5- µ s-rise-time pulse (STP) in the perpendicular polarization geometry, and (d) 65 Toscillating µ s-field pulse and transmittance in parallel polarization geometry.At zero magnetic field, the transmittances are equal. As the field strengthgrows, the SWNTs align and decrease (parallel) or increase (perpendicular)the intensity of transmitted light. 6esults from the Los Alamos Single Turn Coil Project (STP) magnet in aperpendicular configuration with a 635 nm laser and a different sample. Atapproximately 6 µ s, when magnitude of the field was low, in part (c) the de-tector overloaded due to the arc flash from the routine disintegration of thecoil, this does not affect the data collected before the coil break. This dataconfirms our results from the Megagauss Generator with a different mag-net of similar design, different excitation wavelength, and different sample.Overall, the magnitude of the change in transmittance is less than the mspulse experiment due to the shorter field duration. Figures 1(a) and 1(d)are nearly the same magnitude, but the µ s-pulse in 1(d) shows an order ofmagnitude smaller response than the ms-pulse in 1(a). For our qualitativeanalysis we use ms-pulse data from Toulouse and µ s-pulse data from Berlin. Calculated Dynamic Linear Dichroism
The time-dependent (or dynamic) linear dichroism, LD ( t ), of SWNT align-ment is calculated directly from the normalized transmittances. Using therelationship between transmittance ( T ) and absorbance ( A ), LD ( t ) can berelated to the measured transmittances, T (cid:107) ( t ) and T ⊥ ( t ), as LD ( t ) = A (cid:107) ( t ) − A ⊥ ( t )= − ln T (cid:107) ( t ) T + ln T ⊥ ( t ) T = ln T ⊥ ( t ) T (cid:107) ( t ) (6)where the transmittance of the background medium, T , cancels out. This isof particular advantage in pulsed field experiments, where the induced changein transmittance is very straightforward to collect, but the background signalcan be cumbersome. As we are studying the dynamics of SWNT alignmentin pulsed fields, and not the magnitude of alignment, we can utilize LD ( t )normalized to its maximum value ( LD ( t ) ≡ LD ( t ) /LD max ). Although thisprocedure washes out the quantitative measure of the alignment as opposedto normalizing by isotropic absorption as in LD r = 3 S , it retains the dy-namics of the SWNTs in response to the magnetic field pulse.Figure 2 shows LD ( t ) (purple) for (a) ms and (b) µ s pulses calculatedfrom the transmittances of Fig. 1. The relationship of LD and LD r is such7 .000.750.500.250.00 N o r m a li z ed L D Time (ms) M agne t i c F i e l d ( T ) (a) N o r m a li z ed L D Time ( m s) M agne t i c F i e l d ( T ) (b) Figure 2: (color online) Time-dependent traces of calculated normalized dy-namic linear dichroism LD ( t ) (purple, left axis) and applied magnetic mag-netic field (green, right axis) for (a) a 56 T, ms-pulse and (b) a 140 T, µ s-pulse. As the sample is isotropic at zero magnetic field, the linear dichro-ism is zero. As the field strength grows in time and the SWNTs align withthe magnetic field, the dichroism increases, peaking at a time slightly laggedto the maximum of the magnetic field. After the magnetic field pulse, thesample gradually relaxes to its unaligned state. Comparison to normalizedlinear dichroism computed from our model is shown in solid black.that they share the same dynamic features. The positive sign of the sig-nal indicates that the SWNTs are aligning with the magnetic field. As themagnetic field increases to a strength greater than the randomization of theBrownian potential, the SWNTs feel a strong force to align. However, there8s a lag due to viscous drag, thus they always have a torque to align to thedirection of the applied magnetic field. As the magnetic field decreases, thereis a point where the tubes will no longer increase in alignment (the point ofmaximum LD ). As the field decreases further, and the Brownian term be-comes more significant, eventually the SWNTs randomize, slowed by viscousdrag. When the magnetic field is back to zero, starting from any residualalignment present in the sample, there is a competition between Brownianmotion and the viscosity of the solution; this gives the characteristic relax-ation time of the SWNTs. Theory Figure 3: A SWNT with a direction defined by the vector u at an angle θ to the magnetic field B ; the magnetic properties of the SWNT creates atorque N mag forcing the SWNT to align with the magnetic field. Its directionchanges as ˙ u , defining an angular velocity ω = u × ˙ u In order to understand the effect of the magnetic field on the overallalignment of the SWNTs in solution, we must understand the competitionbetween thermal agitation, or Brownian motion, which functions to random-ize the nanotube orientation and the magnetic field, which functions to alignthe nanotubes. Since the persistence length of a single SWNT is much greater9han the length of the SWNTs in our study, we consider SWNTs to behavelike rigid-rods in suspension of radius R and poly-disperse length L . Weexamine a dilute dispersion of non-interacting SWNTs, which enables us toconsider the orientation of each nanotube independently and determine thebulk orientation by summing the contributions from each nanotube in thedistribution.Figure 3 depicts a SWNT oriented in the direction u at an angle θ to themagnetic field B ; the SWNT orientation is dependent on the total torque, N tot = N Brown + N mag , (7)which is the sum of contributions from Brownian motion and the magneticfield. If Ψ( L ; u ; t ) is the probability distribution function of u and U ( L ; u ; t )is the external potential, then the Brownian motion contribution is includedby adding k B T ln Ψ to U . The angular velocity ω induced by the total torqueis ω = 1 ς r N tot = − ς r ( k B T (cid:60) ln Ψ + (cid:60) U ) , (8)where the rotational operator (cid:60) is defined as (cid:60) ≡ u × ∂∂ u (9)and the rotational friction constant ς r is defined as ς r = πη s L (cid:15)f ( (cid:15) ) , (10)where (cid:15) = (cid:18) ln LR (cid:19) − (11)and f ( (cid:15) ) = 1 + 0 . (cid:15) − . (cid:15) + 1 . (cid:15) . (12)The equation for the conservation of the probability distribution Ψ thenbecomes ∂ Ψ ∂t = −(cid:60) · ( ω Ψ) = D r (cid:60) · [ (cid:60) Ψ + Ψ k B T (cid:60) U ] , (13)where the rotational diffusion is defined as D r = k B Tς r . (14)10q. (13) is known as the Smoluchowski equation for rotational diffusion. In our system the external potential is the magnetic field’s effect on theorientation of an individual SWNT. This potential depends on the magneticsusceptibility anisotropy, ∆ χ , of the SWNT, the number of carbon atoms inthe SWNT, N ( L ), the strength of the magnetic field, B ( t ), and the orienta-tion of the SWNT as measured by the angle θ ( u ): U ( L ; u ; t ) = − ∆ χN ( L ) B ( t ) cos θ ( u ) . (15)To track the nematic order parameter S ( t ) of the SWNT suspension in atime-dependent magnetic field, we first solve the Smoluchowski equation byexpanding Ψ as a sum of spherical harmonics Y mn :Ψ( L ; u ; t ) = N (cid:88) n =0 , n (cid:88) m = − n, A mn ( L ; t ) Y mn ( u ) . (16)Spherical harmonics are ideal basis functions because they are eigenfunctionsof the highest derivative operator in Eq. (13). Note that only the even valuesof n are used because the system is symmetric about the alignment axis.Note also that only the even values of m are needed since the SWNTs haveno permanent magnetic moments (they have only induced magnetic dipoles),and so Ψ( u ) = Ψ( − u ). The energy can be expressed simply in terms of the second sphericalharmonic Y U = ∆ χN B cos θ = ∆ χN B (cid:20) (cid:114) π (cid:18) Y + 13 (cid:19)(cid:21) = κ (cid:18) Y + 13 (cid:19) (17)where κ ( L ; t ) = 43 (cid:114) π χN ( L ) B ( t ) (18)The partial differential equations, Eq. (13), are then converted into asystem of ordinary differential equations for A mn using Galerkin’s method.By multiplying Eq. (13) by each basis function Y pq and integrating over allspace, the time evolution of each corresponding coefficient, ddt A pq , can be11etermined as (cid:90) sin θdθ (cid:90) dφ Y pq d Ψ dt = (cid:90) sin θdθ (cid:90) dφ Y pq ddt N (cid:88) n =0 , n (cid:88) m = − n, A mn Y mn = ddt A pq (19)= − D r q ( q ) A pq − κ D r k B T N (cid:88) n =0 , n (cid:88) m = − n, A mn (cid:90) sin θdθ (cid:90) dφ Y qp Y mn Y − κ D r k B T N (cid:88) n =2 , n (cid:88) m = − n +2 , A mn (cid:114)
32 ( n − m )( n + m + 1) (cid:90) sin θdθ (cid:90) dφ Y qp Y m − n Y − κ D r k B T N (cid:88) n =2 , n − (cid:88) m = − n, A mn (cid:114)
32 ( n + m )( n − m + 1) (cid:90) sin θdθ (cid:90) dφ Y qp Y m +1 n Y − , (20)where the integrals of the multiplication of three spherical harmonics, i.e., (cid:82) sin θdθ (cid:82) dφ Y qp Y m +1 ,m,m − n Y − , , , are nonzero only when m = p = 0or p = − m . The initial values of the coefficients are determined fromthe initial orientation of the nanotubes; a random orientation is describedby A mn = 0 except for A = 1. The magnetic field is turned on at t =0 and varies with time. The coefficients at each time step are solved byusing a numerical ordinary differential equation integration technique – third-order Runge-Kutta, available in MATLAB (ODE23). S is related with thecoefficients, A mn ( L ), by averaging over cos θ ( L ), (cid:10) cos θ ( L ) (cid:11) = (cid:28) (cid:114) π Y + 13 (cid:29) = (cid:90) (cid:90) (cid:18) (cid:114) π Y + 13 (cid:19) N (cid:88) n =0 , n (cid:88) m = − n, A mn Ψ mn sin θ dθ dφ = 43 (cid:114) π A (cid:90) (cid:90) ( Y ) sin θ dθ dφ + 13 A (cid:90) (cid:90) sin θ dθ dφ = 43 (cid:114) π A + 13 . (21)By placing Eq. (21) into Eq. (4), we find S ( L, t ) to be S ( L ; t ) = 2 (cid:114) π A ( L ; t ) . (22)12he bulk solution’s nematic order parameter S ( t ) is determined by inte-grating S ( L ; t ) over the distribution of lengths S ( t ) = 2 (cid:114) π (cid:90) ∞ A ( L, t )Ω( L ) dL. (23)To compare with experimental data, we assume a lognormal probability dis-tribution, Ω( L ) = 1 Lσ √ π e − (ln L − µ )22 σ , (24)and vary the parameters µ and σ , the mean and standard deviation of log L ,respectively, to calculate LD ( t ) = max( S ( t )) /S ( t ), which is compared withour measured LD ( t ). Discussion
We can now use our model to calculate the dynamic response of SWNTsin time-varying magnetic fields and compare with the experimental data.Figure 4 compiles simulated LD for several lengths. Each simulated LD trace (dotted black) is offset vertically and plotted along side its appliedmagnetic field (green) and experimental LD (purple). In general, shorternanotubes have less viscous drag, and hence, align to the field pulse faster,but also randomize faster as they have less ∆ χ . Longer nanotubes takelonger to respond to the field, as they have more viscous drag in solution,but their overall alignment is larger due to their larger ∆ χ . These effects arealso convolved with the duration and strength of the field impulse. A shorterimpulse will more readily align short tubes than long tubes during the pulseduration. Figure 4(a) shows the ms-pulse data while Fig. 4(b) displays the µ s-pulse.Due to the fact we have a sample that is polydisperse in length, as ex-pected, no individual simulated length is able to reproduce all the featuresof the experimental data, as shown in Fig. 4. To describe a typical SWNTlength distribution, we use a log-normal form, which has been measuredand confirmed by AFM and rheology measurements on similarly preparedsamples. In Fig. 6 the lengths indicated by symbols are those that wereexplicitly calculated to determine the overall LD that best fit our experi-ment. Figure 5 compares the experimental LD signal with that obtainedfrom our simulation as a function of magnetic field. Our model shows a good13 Time (µs)(b) Length(µm)2.701.400.700.370.190.100.050.02
Time (ms) N o r m a li z ed L D Figure 4: (color online) The contributions from each length in the distributionnormalized to their maximum value (dotted black), the experimental LD (purple), and the accompanying magnetic field pulses (green). Traces areoffset for clarity, and the field and experimental traces are reproduced ateach offset for ease of comparison. Note that no single dotted trace cansuccessfully reproduce the experimental LD . Part (a): ms-pulse, part (b): µ s-pulse.overall match to the measured data using published values for ∆ χ , the cor-responding alignment potential from Eq. (15), and the length distribution,Ω( L ) from Eq. (24). These results were obtained by varying average, µ , and14tandard deviation, σ , of the natural log of L in a log-normal distribution(Fig. 6). N o r m a li z ed L D Magnetic Field (T)(a) N o r m a li z ed L D Magnetic Field (T)(b)
Figure 5: (color online) Magnetic field dependent traces of calculated (purple)and simulated (black) normalized linear dichroism vs. applied magnetic field.The hysteresis is indicative of the lag to the magnetic field produced by ourpoly-disperse length sample. Part (a) shows a 56 T, ms-pulse and (b) showsa 140 T, µ s-pulse.The comparisons in Fig. 2 are fit by the length distributions of Fig. 6.Figure 4 gives an indication of which population of SWNTs is responsiblefor each part of the simulated LD . Shorter nanotubes are the predominantsource of signal during the upsweep of the field and longer nanotubes forthe down sweep (and lag). As the samples were not from the same batchfor the different time duration pulses, a rigorous comparison between these15 .000.750.500.250.00 S W N T P r obab ili t y D en s i t y Length ( m m) SelectedLengths 56 T, ms-pulse 140 T, m s-pulseLog-normal Distributions 56 T ms-pulse 140 T, m s-pulse Figure 6: (color online) Histograms of log-normal length distributions usedto compute the simulated linear dichroism for each magnetic field pulse. Thecontributions from selected lengths in the distribution are noted by filledcircles and triangles.effects cannot be made. Nonetheless, it is feasible to conclude that a shorterduration pulse will be moving predominantly individual nanotubes as our fitlength distribution is close to published values. The µ s-pulse experiment isof too short duration to appreciably align very long SWNTs, so it is not sen-sitive to possible bundles in solution. The ms-pulse experiment on the otherhand is long enough to move large nanotubes but shows a slight mismatchon the upsweep of the magnetic field (Fig. 5). It is possible that a bi-modallength distribution exists in solution, a population of shorter individualizednanotubes and one of longer bundles of nanotubes. Further experiments onsamples of known length distribution, measuring LD r , are needed to inves-tigate this hypothesis. Conclusion
We have measured the magnetic-field-induced dynamic linear dichroism ofSWNT solutions. Our presented technique establishes a method for the ex-traction of the length distribution of the SWNTs present in solution basedon the Smoluchowski equation. However, future work is needed, specificallycomparison with other techniques for determining length distributions, such16s rheology and AFM measurements, allowing for refinement of publishedvalues of SWNT magnetic susceptibility and chirality dependence. It is alsopossible from this work to design experiments that will predominantly probecertain lengths of SWNTs in solution, and investigate the possibility of vary-ing length distributions with chirality.This work was supported by the Robert A. Welch Foundation (throughgrant Nos. C-1509 and C-1668), the National Science Foundation (throughGrants Nos. DMR-0134058, DMR-0325474, OISE-0437342, CTS-0134389,and CBET-0508498), and EuromagNET (EU contract RII3-CT-2004-506239).We thank the support staff of the Rice Machine Shop, Institut f¨ur Physik,NHMFL, and LNCMP. We also thank Scott Crooker and Erik Hobbie forhelpful discussions.
Methods
HiPco SWNTs were suspended in aqueous surfactant solutions of sodium dode-cylbenzene sulfonate (SDBS) using standard techniques. It is noted that theultracentrifugation step in our preparation procedure minimizes the presence offerromagnetic catalyst particles, which have been shown to have a strong effecton SWNT alignment in low DC magnetic field fields. Samples were loaded intohome-built cuvettes with path lengths of ∼ ∼ ∼
150 ms current pulse, using ∼ ∼
26 mm free bore reinforcedcopper coil cooled to liquid nitrogen temperature, designed for 60 T pulses. Asthe coil was at liquid nitrogen temperature before each experiment, a cryostat wasutilized to keep the samples maintained at room temperature.Megagauss measurements ( µ s-pulse) were performed at two installations: theMegagauss Generator ( ∼
140 T) at Humboldt-Universit¨at zu Berlin and theSingle Turn Coil Project (STP) magnet ( ∼
166 T) at the National High Mag-netic Field Laboratory (NHMFL) in Los Alamos. The Megagauss Generator andthe STP magnet are single-turn coil magnets of similar design. They each utilize ow inductance capacitor banks ( ∼
225 kJ in Berlin and 259 kJ in Los Alamos)capable of discharging ∼ µ s time-scale through a 15 mm or 10 mmsingle-turn copper coil. These experiments are deemed “semi-destructive,” as themassive amount of current and huge Lorentz force on the conductor causes anoutward expansion followed by explosion of the coil, ideally preserving the sampleand sample holder for repeated use. Oscillating fields were realized by preventingcoil expansion through reinforcement. Since the duration of the field in megagaussexperiments was ≈ − that of a long-pulse experiment, transmittance data wascollected with higher intensity, single wavelength lasers. An Ar + ion laser at488 nm was utilized in Berlin and a diode laser at 635 nm was used in Los Alamos.Light transmitted through a fiber coupled sample holder, cuvette, and polarizer,with similar geometries to the long pulse experiment, was collected on a Si pho-todiode (3 ns rise-time) connected to a fast oscilloscope using the sophisticatedsetup of reference. The measurements were done at room temperature, withoutthe need of a cryostat. eferences and Notes
1. O’Connell, M. J.; Bachilo, S. M.; Huffman, C. B.; Moore, V. C.; Strano, M. S.;Haroz, E. H.; Rialon, K. L.; Boul, P. J.; Noon, W. H.; Kittrell, C.; Ma, J.;Hauge, R. H.; Weisman, R. B.; Smalley, R. E.
Science , , 593–596.2. Duggal, R.; Pasquali, M. Phys. Rev. Lett. , , 246104.3. Fujiwara, M.; Oki, E.; Hamada, M.; Tanimoto, Y.; Mukouda, I.; Shimo-mura, Y. J. Phys. Chem. A , , 4383–4386.4. Zaric, S.; Ostojic, G. N.; Kono, J.; Shaver, J.; Moore, V. C.; Strano, M. S.;Hauge, R. H.; Smalley, R. E.; Wei, X. Science , , 1129–1131.5. Zaric, S.; Ostojic, G. N.; Kono, J.; Shaver, J.; Moore, V. C.; Hauge, R. H.;Smalley, R. E.; Wei, X. Nano Lett. , , 2219–2221.6. Islam, M. F.; Milkie, D. E.; Kane, C. L.; Yodh, A. G.; Kikkawa, J. M. Phys.Rev. Lett. , , 037404.7. Islam, M. F.; Milkie, D. E.; Torrens, O. N.; Yodh, A. G.; Kikkawa, J. M. Phys.Rev. B , , 201401.8. Torrens, O.; Milkie, D.; Ban, H.; Zheng, M.; Onoa, G.; Gierke, T.; Kikkawa, J. J. Am. Chem. Soc. , , 252–253.9. Fagan, J. A.; Bajpai, V.; Bauer, B. J.; Hobbie, E. K. Appl. Phys. Lett. , , 213105.10. Davis, V. A.; Ericson, L. M.; Parra-Vasquez, A. N. G.; Fan, H.; Wang, Y. H.;Prieto, V.; Longoria, J. A.; Ramesh, S.; Saini, R. K.; Kittrell, C.;Billups, W. E.; Adams, W. W.; Hauge, R. H.; Smalley, R. E.; Pasquali, M. Macromolecules , , 154–160.11. Hobbie, E. K. J. Chem. Phys. , , 1029–1037.12. Parra-Vasquez, A. N. G.; Stepanek, I.; Davis, V. A.; Moore, V. C.;H´aroz, E. H.; Shaver, J.; Hauge, R. H.; Smalley, R. E.; Pasquali, M. Macro-molecules , , 4043–4047.13. Casey, J. P.; Bachilo, S. M.; Moran, C. H.; Weisman, R. B. ACS Nano ,in press, doi/10.1021/nn800351n.14. Fagan, J. A.; Simpson, J. R.; Landi, B. J.; Richter, L. J.; Mandelbaum, I.;Bajpai, V.; Ho, D. L.; Raffaelle, R.; Hight Walker, A. R.; Bauer, B. J.; Hob-bie, E. K.
Phys. Rev. Lett. , , 147402.
5. Zaric, S.; Ostojic, G. N.; Shaver, J.; Kono, J.; Portugall, O.; Frings, P. H.;Rikken, G. L. J. A.; Furis, M.; Crooker, S. A.; Wei, X.; Moore, V. C.;Hauge, R. H.; Smalley, R. E.
Phys. Rev. Lett. , , 016406.16. Shaver, J.; Kono, J.; Hansel, S.; Kirste, A.; von Ortenberg, M.; Mielke, C. H.;Portugall, O.; Hauge, R. H.; Smalley, R. E. In Proceedings of the 12th Confer-ence on Narrow Gap Semiconductors ; Kono, J.; L´eotin, J., Eds.; Taylor andFrancis, New York, 2005, pp 273–278.17. Ajiki, H.; Ando, T.
J. Phys. Soc. Jpn. , , 2470–2480.18. Lu, J. P. Phys. Rev. Lett. , , 1123–1126.19. Ajiki, H.; Ando, T. J. Phys. Soc. Jpn. , , 4382–4391.20. Marques, M. A. L.; d’Avezac, M.; Mauri, F. Phys. Rev. B , , 125433.21. Walters, D. A.; Casavant, M. J.; Qin, X. C.; Huffman, C. B.; Boul, P. J.;Ericson, L. M.; Haroz, E. H.; O’Connell, M. J.; Smith, K.; Colbert, D. T.;Smalley, R. E. Chem. Phys. Lett. , , 14–20.22. Tsui, F.; Jin, L.; Zhou, O. Appl. Phys. Lett. , , 1452–1454.23. Adachi, R.; Yamaguchi, K.; Yagi, H.; Sakurai, K.; Naiki, H.; Goto, Y. J. Biol.Chem. , , 8978–83.24. Kimura, T.; Yamato, M.; Koshimizu, W.; Koike, M.; Kawai, T. Langmuir , , 858–861.25. Rodger, A.; Nord´en, B. Circular Dichroism & Linear Dichroism ; OxfordUniverisity Press: Oxford, 1997.26. Ajiki, H.; Ando, T.
J. Phys. Soc. Jpn. , , 1255–1266.27. Shaver, J.; Crooker, S. A.; Fagan, J. A.; Hobbie, E. K.; Ubrig, N.; Por-tugall, O.; Perebeinos, V.; Avouris, P.; Kono, J. Phys. Rev. B , ,081402(R).28. Ando, T. J. Phys. Soc. Jpn. , , 024707.29. Portugall, O.; Puhlmann, N.; M¨uller, H.-U.; Barczewski, M.; Stolpe, I.; vonOrtenberg, M. J. Phys. D: Appl. Phys. , , 2354–2366.30. Mielke, C. H.; R. D. McDonald, In Megagauss Magnetic Fields and High En-ergy Liner Technology, Proceedings of the International Conference on Mega-gauss Magnetic Field Generation ; IEEE Transactions, 2006, pp 227–231.
1. Doi, M.; Edwards, S. F.
The Theory of Polymer Dynamics ; Oxford UniversityPress: New York, NY, 1986.32. Larson, R. G.
The Structure and Rheology of Complex Fluids ; Oxford Univer-sity Press: New York, 1999.33. Stewart, W. E.; Sorensen, J. P.
J. Rheol. , , 1–13.34. Arfken, G. In Mathematical Methods for Physicists, 3rd ed.
Physica B , , 356–359., 356–359.