Accurate determination of the length of carbon nanotubes using multi-angle light scattering

ORIGINAL PAPER

Accurate determination of the length of carbon nanotubesusing multi-angle light scattering

Julien Gigault & Bruno Grassl & Isabelle Le Hécho &

Gaëtane Lespes

Received: 26 April 2011 /Accepted: 22 August 2011 /Published online: 2 September 2011# Springer-Verlag 2011

Abstract We have investigated and compared variousapproaches for calculation and modeling (using multi-anglelight scattering; MALS) in order to determine the length ofcarbon nanotubes (CNTs) following asymmetrical flow field-flow fractionation. Various fit methods and shape models forlight scattering were considered with respect to their accuracyand robustness of the data obtained. Both single- and multi-walled CNTs (SWCNTs and MWCNTs) were analyzed tobetter understand these approaches in view of the differentstructure of CNTs (i.e., single- andmulti-walled). The effect ofthe angular interval used for extrapolation was investigated byconsidering the uncertainties in the intensities of MALS atsmall angles. The determination of the lengths of SWCNTandMWCNT using MALS can be accurately made using twospecific approaches. The ellipsoid shape model using thegyration radius as determined by the Zimm method and therod-like fitting method were found to be most suitable for thedetermination of the length of SWCNTs and MWCNTsrespectively.

Keywords Light scattering . Carbon nanotubes . Lengthdetermination

Introduction

The size characterization of carbon nanotubes (CNTs) is animportant step for controlling CNT synthesis, proliferation

and interaction with their environment [1]. Multi-angle lightscattering (MALS) is a powerful technique that is oftenused to characterize polymers and nanoparticles. For largepolydisperse analytes such as carbon nanotubes, MALS isoften combined with size fractionation techniques, such assize-exclusion chromatography (SEC) or field-flow frac-tionation (FFF), to characterize the sizes in each elutedfraction directly [2–4]. If the angular dependence of thescattered light can be measured, it is then possible todetermine the size of the analyte. The information providedby MALS is the gyration radius (Rg), which is a measure ofthe analyte’s equivalent physical radius weighted by themass distribution around its center of mass. For CNT lengthdetermination, it is necessary to use a shape model totransform the Rg into length (L) [5–7]. In the literature, fewpotentially usable shape-based transformation models areoften applied with no legitimate consideration of theparticular CNT, its shape and its structure. Nevertheless,specific transformation models for CNTs that have onlybeen studied theoretically do exist. Currently, no compre-hensive comparative studies on the relevance of variousfitting methods and shape models for various CNTstructures and shapes (i.e., single vs. multi-walled and shortvs. long) have been performed. It is clear that the choice ofthese fitting methods and shape models will have an effecton the accuracy of CNT length determination. Indeed, it iswell known in classical light scattering that fitting methodsbecome more complex for large nanoparticles and macro-molecules, and the extrapolation procedures must becarefully considered.

Consequently, the aim of this work was to study andcompare various fit formalisms and shape models forMALS measurements to obtain an accurate CNT lengthcharacterization depending on its specific structure andshape.

J. Gigault :B. Grassl : I. Le Hécho :G. Lespes (*)Laboratoire de Chimie analytique Bio-Inorganiqueet Environnement, UMR IPREM 5254 -Technopôle Hélioparc,Université de Pau et des Pays de l’Adour (UPPA)/CNRS,Av. du Président Angot,64053 Pau Cedex, Francee-mail: [email protected]

Microchim Acta (2011) 175:265–271DOI 10.1007/s00604-011-0687-z

Theory

MALS (or static light scattering) involves the measurementof the total scattered intensity as a function of theobservation angle θ. This is commonly summarized by theZimm equation, which is described by [8]:

Rq ¼ K»:Mw:c:P qð Þ:b1$2A2:Mw:c:P qð Þc ð1Þ

In this expression, Rθ is the Rayleigh ratio, which dependson the measured intensities of the scattered and incidentlight, c is the concentration of the scattering species, K* is anapparatus constant, M is the weight-average molar mass, A2

is the second virial coefficient, and Pθ is the theoreticallyderived form factor expressing the angular dependence of thescattered light via the term sin(θ/2).

If Pθ is known, Eq. 1 can be used to obtain analyte sizeinformation. However, the mathematical formulation of Pθ isextremely complex. Therefore, various “more easily usable”expressions of Pθ that make assumptions about either thespecific analyte shape or the particular conditions of thescattered light measurements without considering analyteshape have been proposed [9]. These expressions arepresented in Table 1 [8, 10–14]. Using these simplifiedexpressions for Pθ, two main approaches for calculating CNTlengths using light scattering measurements currently exist.

In the first approach, Rg is transformed into length using anappropriate shape model. For CNTs, either prolate ellipsoid orhollow cylinder models can be used. Prolate ellipsoidmodeling has previously been used and studied theoretically[5, 6]. A prolate ellipsoid is formed by rotating an ellipsearound its major axis. If a sphere is defined as an ellipsoidwith its half axes a=b=c, a prolate ellipsoid is such that a>b=c, where 2a corresponds to the CNT length (L), and 2bcorresponds to the CNT diameter (d). The general relationshipbetween Rg and the half axes of an ellipsoid is [15, 16]:

Rg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ b2 þ c2

5

rð2Þ

This equation can be also expressed in terms of CNTlength and diameter as:

Rg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL2 þ 2d2

20

rð3Þ

The “hollow” cylinder shape model assumes that CNTsare rigid, hollow cylinders, which comes from the fact thatRg depends on the mass distribution of the analyte [7, 17].In this model, Rg can be expressed in terms of thecylinder’s length and diameter as [7, 17]:

R2g ¼

L2

12þ d2

8ð4Þ

For CNT samples, it is acknowledged that d<< L.Therefore, d is negligible compared to L, and Eqs. 3 and 4can be rewritten as

L ¼ Rg

ffiffiffiffiffi20

pð5Þ

for the prolate ellipsoid model and

L ¼ Rg

ffiffiffiffiffi12

pð6Þ

for the hollow cylinder model.The second approach for size calculation involves

extrapolating the intensity of the scattered light from a plotof K*c/Rθ as a function of sin(θ/2). The CNTs are thenassumed to be shaped like rigid-rod particles or fullcylinders. The CNT rod length can be obtained directlyfrom the rod-like fitting method (provided by Astrasoftware) with no preliminary Rg determination, and thiscorresponding expression of Pθ is shown in the last line ofTable 1.

Experimental

Chemicals and samples

High-purity CNTs were provided and characterized fromNanocyl and are presented in Table 2 [18–24]. To comparethe average individual CNT lengths determined by MALS,aqueous dispersions of SWCNTs and MWCNTs in thepresence of SDS were used. Preliminary such dispersionshave been demonstrated to correspond to nanotubesindividually dispersed and to be stable. All of the experi-ments were performed on CNT samples that were passedthrough a 5.0-μm filter (cellulose regenerated). Thisfiltration cut-off permitted the removal of possible bulkimpurities while covering the entire CNT length rangegiven by the manufacturer. The final CNT concentrationbefore filtration was 0.05 gL−1 for a convenient MALSsignal, and the dispersions were prepared with an SDSconcentration that has been previously found to providesatisfactory dispersion (i.e., greater than the CMC value)[25].

Instruments

The asymmetrical flow field-flow fractionation (A4F) systemwas an Eclipse 3 (Wyatt Technology, Dernbach, Germany).The detection sequence consisted of a DAWN HELEOSMALS detector (Wyatt Technology, Santa Barbara, USA).The data from the MALS detector were collected at 15 angles(from 4 to 18) and analyzed using Astra 5.3.1.5 (WyattTechnology). Calculations of the gyration radius from the

266 J. Gigault et al.

experimental data, which are presented in the figures andtables, were made using Astra 5.3.4.19 (Wyatt Technology,Santa Barbara, CA). Astra first normalized the data and thensubtracted the baseline, and the data was then imported into aspreadsheet. The error limit of values given in this work isbased on the average and standard deviation from multipleinjections (8 injections).

A4F setup

A4F was hyphenated to MALS to remove all chemicalspecies, which can introduce bias into the measurements asit was previously studied [2]. The run sequences firstcontained two short, consecutive elution and focusing stepsprior to injection to equilibrate the system and identify thesignal’s baseline. During these steps, all of the impuritiesand surfactant micelles in the samples were removed by thecross-flow through the membrane. Next, elution of indi-vidually dispersed CNTs began without fractionation (i.e.,no cross-flow) to obtain average values of the CNT sizeparameters. Note that no CNT fractionation occurred viaA4F under these fractionation conditions.

Results and discussion

To evaluate the relevance and efficiency of the CNT lengthcalculation approaches, various criteria were investigated.First, the deviations inferred from the Debye-plot extrapo-lation procedures at specific angles corresponding to the

MALS instrument angles were examined. Indeed, thisdeviation represented the first interesting criterion fordetermining an adequate fit for CNT length calculationbased on structure, because deviations introduce errors intoCNT size determination. Extrapolation was performed vialinear or polynomial curve-fitting on all of the data pointsusing the least-squares method. All of the fitting methods(Zimm, Debye, Berry and rod-like; see Table 1) wereevaluated using identical sin2(θ/2) and sin(θ/2) ranges andthe calculation models presented above. The secondcriterion for evaluating the relevance and efficiency of theCNT length calculation methods was the length valuedetermined from the various shape models. Finally, targetdata (Table 2) were then compared to the calculated CNTlengths of the SWCNTs and MWCNTs.

Debye plots

Debye plots of the various fitting methods (Zimm, Debye,Berry and rod-like) demonstrated whether or not they areappropriate for the determination of CNT Rg and length.Table 3 summarizes the Rg values measured for bothSWCNTs and MWCNTs using the Zimm, Berry and Debyeformations.

First, Fig. 1 shows the Debye plots for SWCNTs andMWCNTs using the Zimm and Berry methods. These twomethods generally gave similar fits, which was probablydue to similarities in the fitting methods (i.e., the onlydifference between the two fits was plotting in terms of

Table 3 CNT (SWCNT and MWCNT) Rg values (nm) determinedusing the various fitting methods

Fitting method SWCNT MWCNT

Zimm 196±7 91±1

Debye 58±4 54±2

Berry 108±5 77±1

Table 1 Presentation of the various light scattering fitting methods studied in this work according to their respective Debye plots and sizeinformation

Fitting method Debye plot Pθ u Reference

Zimm K»c

Rq¼ f sin2 q

2

" #$ %1$ u2

34pl n0Rg sin q

2

" #[8, 10–13, 29]

Debye Rq

K»c¼ f sin2 q

2

" #$ %

BerryffiffiffiffiffiffiK

»c

Rq

q¼ f sin2 q

2

" #$ %

Rod-like K»c

Rq¼ f sin q

2

" #$ %1u

R2u

t¼0

sin tt dt $ sin2u

u2pl n0L sin

q2

" #[8, 14, 29]

l is the wavelength of incident light in the medium, θ is the angle between the incident light and the scattered light (with θ=0° being straightforwardfrom the perspective of the incident light), n0 is the solvent refractive index, Rg is the gyration radius, and L is the length of the rod.

Table 2 Size of the CNTs used in this work

SWCNT MWCNT

Average length (nm) 800±50 600±50

Average diameter (nm) 1.5±0.5 9.5±5.5

References [18–20] [21–24]

Accurate determination of the length of carbon nanotubes using multi-angle light scattering 267

K*c/Rθ or its square root; see Table 1). However, a slightdeviation in the curve-fitting of the data appeared at smallerangles using the Berry method, while the values fit betterusing the Zimm method. Furthermore, the CNT Rg valuescalculated using the Berry method were inferior to thoseobtained using the Zimm method (Table 3). This differencewas more pronounced for SWCNTs. Using the Zimmmethod, it appeared that data fit well at all angles. Thisobservation agrees with a previous report demonstratingthat, for large macromolecules, the Berry and Debyemethods significantly underestimate Rg, while the Zimmmethod slightly overestimates Rg [8]. Additionally, itshould be noted that the Berry method is the mostappropriate fitting method for polymers when there is noa priori knowledge regarding the polymer’s structure or size[9].

Figure 2 illustrates the Debye plot according to theDebye fitting method for both SWCNTs and MWCNTs.The results were inspected to identify the angles at whichthe fit deviated the most from the experimental data. Forboth SWCNTs and MWCNTs, it appears that the curvefitting deviated from the data at the smallest angles (i.e.,sin2(θ/2)<0.3). It has previously been observed that theDebye fitting method is more convenient for small particles(i.e., Rg<100 nm) [9]. This could explain why the

corresponding CNT Rg values that were determined forboth SWCNTs and MWCNTs did not exceed 58 nm(Table 3).

These results concerning Rg determination demonstratedthat the Zimm method was the most accurate fittingmethod. Moreover, because the Debye and Berry fits arebetter adapted to small spherical particles, their use led toan underestimation of the CNT Rg values and introducedbias into the length measurements [8]. Consequently, theZimm method was chosen for studying and comparing thevarious shape model calculations.

Figure 3 shows a Debye plot for SWCNTs andMWCNTs using the rod-like fitting method. The SWCNTfit deviates from the experimental data at the smallestangles (i.e., 14° to 90°). Therefore, this fit could not beused for accurate SWCNT length measurements; however,the MWCNTs were reasonably fit using the rod-likemethod.

Length calculation method

Next, a comparison and evaluation of the various shapemodel calculations was performed to accurately determinethe average CNT length. This included using both theprolate ellipsoid and cylinder models following preliminary

Fig. 1 SWCNT and MWCNTDebye plots for the Zimm andBerry fitting methods as a func-tion of various angles (θ)


Rg determinations and directly using the rod-like fittingmethod. Table 4 summarizes the CNT lengths determinedfrom the various approaches described above.

The cylinder method is the model that is most frequentlyused to simulate the structure of CNTs [7, 17]. However,the length determined for the SWCNTs was less thanexpected. Indeed, SWCNTs are the most flexible CNTs andare known to be similar to wires on the nanoscale (i.e.,nanowires) [26, 27]. The cylinder model does not considerthis flexibility; therefore, a significant SWCNT size factorwas not taken into account for the final length distributiondetermination. The prolate ellipsoid model appeared to bemore suitable for simulating SWCNT shape and, conse-quently, determined the length accurately. Thus, theSWCNT length calculated from this model (875±31 nm)corresponded to the expected value (800±50 nm). Usingthe rod-like particle model, even though the average lengthdetermined from the rod-like particle model calculation(886 nm) corresponded to the expected value (800 nm), itwas associated with significant calculation uncertainty (i.e.,the standard deviation was equal to 151 nm) compared tothe other calculation methods. This result can be explainedby the fact that this method was based on the Debye plot.Indeed, the length calculation uncertainty increased withincreasing deviation in the Debye-plot fitting curves.

For MWCNTs, it is possible that the cylinder method inconjunction with the Zimm method could be a suitableapproach for length evaluation [7]. MWCNTs could bemore satisfactorily assumed to resemble a full cylinder(rod). In the present case, the average MWCNT lengthdetermined using the cylinder method (314±3 nm) did notagree with the expected value (600±50 nm), confirmingthat the hollow cylinder approach does not accuratelysimulate CNT structure (Table 2). Additionally, the lengthvalues obtained from the cylinder model also depend on theRg values. Thus, by using the hollow cylinder method,significant bias arose from considering the MWCNT massdistribution inside the tube. Similar comments could be

Fig. 2 SWCNT and MWCNT Debye plots for the Debye fittingmethod as a function of various angles (θ)

Fig. 3 SWCNT and MWCNT Debye plots for the rod-like fittingmethod as a function of various angles (θ)

Table 4 CNT (SWCNT and MWCNT) lengths (nm) determinedusing the various fitting methods and shape models

Shape model Fitting method SWCNT MWCNT

Prolate ellipsoid Zimm 875±31 405±4

Debye 257±19 241±8

Berry 484±24 342±5

Cylinder Zimm 678±24 314±3

Debye 199±15 186±6

Berry 375±19 265±4

Rod-like particle (full cylinder) 886±151 606±11


made for the prolate ellipsoid model. These resultsdemonstrate that, for MWCNTs, methods based on prelim-inary Rg determination in conjunction with either hollowcylinder or prolate ellipsoid shape models are inadequatefor obtaining accurate length values.

In contrast, the MWCNT data were satisfactorily fitusing the theoretical rod-like model curve at all angles(Fig. 3). The average MWCNT length that was determinedwas precise (606±11 nm) and corresponded to the expectedvalue (600±50 nm). Because MWCNTs are composed ofseveral graphene sheets (several SWCNTs) and have asignificant bulk density, they should be treated as fullcylinders (or similar to rod-like particles), which alsoexplains why the rod-like model appeared more suitable.

Conclusion

To determine SWCNT and MWCNT lengths using MALS,two different calculation approaches should be followed.For SWCNTs, the prolate ellipsoid shape model using theRg values determined from the Zimm method providedaccurate length determinations. Moreover, SWCNTs can betreated as nanowires, and prolate ellipsoid modeling wasapplicable for simulating this particular shape. This mod-eling was also preferable over the rod-like method, whichwas unable to provide the necessary precision. This resultconfirmed that significant differences exist betweenSWCNTs and rigid-rod objects, and rod-like methods areinappropriate for modeling SWCNTs [28].

For MWCNTs, the rod-like particle method led to anaccurate MWCNT length determination directly from lightscattering measurements. These results confirmed thesignificance of treating MWCNTs as rod-like particles.

Finally, the choice of the fitting method and shape modelhas a strong influence on the accuracy of the obtained CNTlength value, depending on the structure and shape(SWCNTs and MWCNTs). Carefully and critically exper-imentally evaluating the Debye plots obtained using theZimm, Berry and Debye methods for obtaining accurateCNT Rg values appears important.

Acknowledgements The authors wish to thank Dr Julien Amadoufrom NANOCYL (Belgium) for providing us the SWCNT andMWCNT samples, and also for his technical assistance.

References

1. Brar SK, Verma M (2011) Measurement of nanoparticles by light-scattering techniques. TrAC - Trends in Anal Chem 30:4–17

2. Gigault J, Le Hécho I, Dubascoux S, Potin-Gautier M, Lespes G(2010) Single walled carbon nanotube length determination by

asymmetrical-flow field-flow fractionation hyphenated to multi-angle laser-light scattering. J Chromatogr A 1217:7891–7897

3. Chun J, Fagan JA, Hobbie EK, Bauer BJ (2008) Size separationof single-wall carbon nanotubes by flow-field flow fractionation.Anal Chem 80:2514–2523

4. Bauer BJ, Fagan JA, Hobbie EK, Chun J, Bajpai V (2008)Chromatographic fractionation of SWNT/DNA dispersions withon-line multi-angle light scattering. J Phys Chem C 112:1842–1850

5. Phelan FR, Bauer BJ (2007) Simulation of nanotube separation infield-flow fractionation (FFF). Chem Eng Sci 62:4620–4635

6. Phelan FR, Bauer BJ (2009) Comparison of steric effects in themodeling of spheres and rodlike particles in field-flow fraction-ation. Chem Eng Sci 64:1747–1758

7. Branca C, Magazù V, Mangione A (2005) Determination ofMWNTs length-to-diameter ratio by static and dynamic lightscattering. Diam Relat Mater 14:846–849

8. Wyatt PJ (1993) Light scattering and the absolute characterizationof macromolecules. Anal Chim Acta 272:1–40

9. Andersson M, Wittgren B, Wahlund KG (2003) Accuracy inmultiangle light scattering measurements for molar mass andradius estimations. Model calculations and experiments. AnalChem 75:4279–4291

10. Debye P (1947) Molecular-weight determination by light scatter-ing. J Phys Colloid Chem 51:18–32

11. Guinier A (1939) La Diffraction des rayons X aux Tres PetitsAngles: Application a L’etude de Phenomenes Ultramicroscopi-ques. Ann Phys Onzieme Serie 12:161–237

12. Berry GC (1966) Thermodynamic and conformational propertiesof polystyrene. I. Light-scattering studies on dilute solutions oflinear polystyrenes. J Chem Phys 44:4550–4564

13. Zimm BH (1948) Apparatus and methods for measurement andinterpretation of the angular variation of light scattering; Prelim-inary results on polystyrene solutions. J Chem Phys 16:1099–1116

14. Van De Hulst H (1982) Light scattering by small Particles(Structure of Matter Series.), Dover Pubn Inc

15. Whittaker ET (1998) A treatise on the analytical dynamics ofparticles and rigid bodies. Cambridge University Press, New York

16. Elias HG (2008) Macromolecules: physical structures andproperties. Wiley-VCH, Weinheim

17. Ortega A, Garcia De La Torre J (2003) Hydrodynamic propertiesof rodlike and dislike particles in dilute solution. J Chem Phys119:9914–9919

18. Pettersson M, Rintala J, Herranen O, Johansson A, Ahlskog M(2009) Raman spectroscopy and low-temperature transport meas-urements of individual single-walled carbon nanotubes withvarying thickness. J Phys Chem C 113:15398–15404

19. Nanocyl (2009), Single Wall isolated and bundled carbon nano-tubes, Products datasheets (www.nanocyl.com), Sambreville,Belgium

20. Sinani VA,GheithMK,Yaroslavov AA, RakhnyanskayaAA, Sun K,Mamedov AA, Wicksted JP, Kotov NA (2005) Aqueous dispersionsof single-wall and multiwall carbon nanotubes with designedamphiphilic polycations. J Am Chem Soc 127:3463–3472

21. Tessonnier JP, Rosenthal D, Hansen TW, Hess C, Schuster ME,Blume R, Girgsdies F, Pfänder N, Timpe O, Su DS, Schlögl R(2009) Analysis of the structure and chemical properties of somecommercial carbon nanostructures. Carbon 47:1779–1798

22. Krause B, Boldt R, Pötschke P (2011) A method for determinationof length distributions of multiwalled carbon nanotubes beforeand after melt processing. Carbon 49:1243–1247

23. Krause B, Mende M, Pötschke P, Petzold G (2010) Dispersabilityand particle size distribution of CNTs in an aqueous surfactantdispersion as a function of ultrasonic treatment time. Carbon48:2746–2754

24. Nanocyl (2009) Thin multi-wall carbon nanotubes. Productsdatasheets, Sambreville, Belgium


http://www.nanocyl.com

25. Islam MF, Rojas E, Bergey DM, Johnson AT, Yodh AG (2003)High weight fraction surfactant solubilization of single-wallcarbon nanotubes in water. Nano Lett 3:269–273

26. Hoa ND, Van Quy N, Kim D (2009) Nanowire structured SnOx-SWNT composites: high performance sensor for NOx detection.Sensors Actuat B-Chem 142:253–259

27. Buzea C, Pacheco II, Robbie K (2007) Nanomaterials andnanoparticles: Sources and toxicity, Biointerphases 2:MR17

28. Mansfield ML, Douglas JF (2008) Transport properties of rodlikeparticles. Macromolecules 41:5422–5432

29. Xu R (2000) Particle characterization: light scattering methos.Scarlett B, Netherlands


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Accurate determination of the length of carbon nanotubes using multi-angle light scattering