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Using solution thermodynamics to describe the dispersion of rod-like solutes: application to

dispersions of carbon nanotubes in organic solvents

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2012 Nanotechnology 23 265604

(http://iopscience.iop.org/0957-4484/23/26/265604)

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IOP PUBLISHING NANOTECHNOLOGY

Nanotechnology 23 (2012) 265604 (8pp) doi:10.1088/0957-4484/23/26/265604

Using solution thermodynamics todescribe the dispersion of rod-like solutes:application to dispersions of carbonnanotubes in organic solvents

J Marguerite Hughes1, Damian Aherne1, Shane D Bergin2,Arlene O’Neill1, Philip V Streich3, James P Hamilton3 andJonathan N Coleman1

1 School of Physics and CRANN, Trinity College Dublin, University of Dublin, Dublin 2, Ireland2 Department of Materials, Imperial College London, SW7 2AZ, UK3 Department of Chemistry and Engineering Physics and Nanotechnology Center for CollaborativeR&D, University of Wisconsin—Platteville, Platteville, WI 53818, USA

E-mail: hamiltoj@uwplatt.edu and colemaj@tcd.ie

Received 6 January 2012, in final form 30 April 2012Published 15 June 2012Online at stacks.iop.org/Nano/23/265604

AbstractWe report a simple model describing the solubility of rods in solvents, expressing the finalresult explicitly in terms of the surface entropy and the enthalpy of mixing. This model can becombined with any expression for the mixing enthalpy depending on the requirements. Forexample, in one instance it predicts the dispersed concentration of rods to decreaseexponentially with the Flory–Huggins parameter of the dispersion. Using a different enthalpyfunction, it predicts a Gaussian peak when concentration is plotted versus solvent surfaceenergy. The model also suggests specific solvent–rod interactions to be important and showsthe dispersed concentration to be very sensitive to ordering at the solvent–rod interface. Wehave used this model to describe experimental results for the concentration of dispersednanotubes in various solvents. Qualitative agreement with these predictions is observedexperimentally. However, we suggest that the fact that quantitative agreement is not foundmay be explained by solvent ordering at the nanotube surface.

(Some figures may appear in colour only in the online journal)

1. Introduction

Since the discovery of carbon nanotubes, it has been clear thatpurification and processing would be dramatically simplifiedif nanotubes could be dispersed in solvents. Over the lastdecade, a number of papers have appeared describing thepreparation of stable suspensions of single walled nanotubes(SWNTs) in a range of common solvents. The earliest reportdates from 1999, when Liu et al showed that individualSWNTs could be deposited from N,N-dimethylformamide(DMF) dispersions [1]. Shortly afterwards Ausman et al [2]demonstrated dispersion of SWNTs in a number of solvents

including N-methyl-2-pyrrolidone (NMP). Similarly, Bahret al demonstrated meta-stable dispersions of SWNT in arange of common solvents [3], while Mickelson showed thatfluorinated SWNT could be dispersed in various alcohols [4].In 2004, Furtado et al [5] showed that SWNTs can bedebundled to a significant degree in both DMF and NMP.In the same year, Landi et al [6] followed this up with aquantitative study of SWNT dispersion in a range of amidesolvents. Maeda et al [7] established that SWNTs could bedispersed in mixtures of tetrahydrofuran and various amines.Later, Giordani et al [8] observed concentration-dependentdebundling of SWNTs in NMP, which Bergin et al [9]

10957-4484/12/265604+08$33.00 c© 2012 IOP Publishing Ltd Printed in the UK & the USA

Nanotechnology 23 (2012) 265604 J M Hughes et al

explained by demonstrating that the enthalpy of mixing couldin fact be negative in certain solvents. More recently, Detricheet al [10], Ham et al [11] and Bergin et al [12] suggested thatgood solvents for nanotubes are determined by the values oftheir Hansen solubility parameters [13]. One common factorin most of these papers is that amides such as NMP are widelyconsidered to be the best solvents for pristine nanotubes.Furthermore, it appears that the pool of other successfulsolvents is relatively small.

Because of the practical importance of nanotubedispersion, it is essential to understand the nature ofthe dispersion process. Recently, a number of groupshave approached this problem from a thermodynamicstandpoint [9, 12, 10, 11]. In general, this has involveddemonstrating that good solvents for nanotubes are thosewith solubility parameters [13–15] within a certain range.This is consistent with solution thermodynamics, whichpredicts that the enthalpy of mixing (i.e. the energetic costof dispersing the solute in a solvent) is minimized whensolute and solvent have the same solubility parameters (theseparameters are usually related to the cohesive energy densityof the solute/solvent) [13–16]. However, this approach doesnot rigorously test solution thermodynamics as the main factorcontrolling the dispersion process. To begin to do this, itis necessary to understand how the dispersed concentrationscales with thermodynamic parameters. This would then allowexperiments to be designed to test the predicted relationships.Such an approach is crucial as it would test the ability ofsolution thermodynamics both to describe the existing dataand to make predictions. Such knowledge would allow us toimprove on our ability to use solvents to disperse not onlynanotubes but other nano-materials such as graphene [17] andinorganic layered compounds [18].

In this work we derive an expression relating thedispersed concentration of rod-like solutes to the enthalpyof mixing. Experimentally, we measure both the maximumdispersed concentration of nanotubes and the Flory–Hugginsparameter χ (proportional to the enthalpy of mixing)in a range of solvents. We find that the nanotubedata agree qualitatively with theory for rod-like solutes.In addition, the model demonstrates that the nanotubeconcentration is extremely sensitive to solvent ordering at thenanotube–solvent interface.

2. Theoretical section

The aim of this section is to develop a model relating theconcentration of a dispersion of rods to the thermodynamicproperties of the mixture. To do this, we considerthe rods as rigid and characterized by an aspect ratio(i.e. length/diameter), x = L/D. We then imagine a saturatedsolution of rods in a solvent. In this case, the dispersed rodswill be in equilibrium with any undispersed rods at the bottomof the vessel. This means the chemical potential of dispersedrods will equal that of the undispersed rods:

µundispersed − µdispersed = 0. (1)

In the simplest case, where we regard the undispersed rods as aliquid (i.e. ignoring the contribution of melt enthalpy) [14], we

can relate the difference in chemical potentials in equation (1)to the free energy of mixing,1Gmix (the change in free energygoing from undispersed to dispersed rods), and can hencewrite equation (1) as

∂

∂n(1Gmix) = 0 (2)

where n is the number of moles of dispersed rods. By thenwriting1Gmix in terms of the enthalpy and entropy of mixing,1Hmix and1Smix respectively, and by changing variable fromn to the rod volume fraction φ, we obtain

∂1Hmix

∂φ= T

∂1Smix

∂φ. (3)

There are a number of expressions in the literature forthe entropy of mixing of one-dimensional objects such aspolymer chains [19, 15]. These tend to be differentiated bythe rigidity of the dispersed one-dimensional object. Theentropy of mixing per unit volume for a mixture for rigid rodsdispersed isotropically in a liquid is given by [9, 20]

1Smix

V= −

R

vsol

[(1− φ) ln(1− φ)

+φ

vrod/vsol

(ln(φ

x

)+ (x− 1)

)]+1S0

V. (4)

Here vrod and vsol are the molar volumes of rods and solventrespectively and1S0 represents any additional entropic terms.We note that such terms are usually accounted for by addingan entropic term into the Flory–Huggins parameter [13]. Wefeel the method described here is advantageous as it allowsa clear differentiation of entropic from energetic terms. Weremark that this expression is quite general. If we write x = 1,equation (4) reduces to that appropriate to a non-rigid rod(i.e. a polymer) [15]. If we take x = 1 and equate vrod tovSol, the expression becomes that for a mixture of smallmolecules, where solvent and solute molecular volumes areapproximately identical [15].

Substituting equation (4) into equation (3), rearranging itand assuming that the volume fraction will be small enoughthat ln(1− φ) ≈ 0 gives:

φ ≈ K exp[

vrod

R

∂(1S0/V)

∂φ

]exp

[−

vrod

RT

∂(1HMix/V)

∂φ

](5)

where 1HMix/V is the enthalpy of mixing per unit volumeof mixture. In a more sophisticated approach, where thedissolved rods are treated as being packed in a crystal, theconstant K would include a fluidization term describing themelting of the crystal [14, 21]. Additionally, in a full treatmentK would include terms describing other effects such ashydrogen bonding or hydrophobicity (where appropriate) [21,22]. Through these terms, K depends on the solute size,which means that for dispersed rods, K will depend on therod length and diameter. We note that while K does notexplicitly depend on the solvent, it is important to considerany possible indirect solvent dependence. For example, realrod-like solutes such as carbon nanotubes are generallydispersed in solvents by sonication. This is known to shortenthe nanotube length via sonication-induced scission. The

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Nanotechnology 23 (2012) 265604 J M Hughes et al

resultant length has been predicted to depend on solventviscosity [23], although such solvent dependence has not beenreported experimentally. Specific to the experiments describedhere, we do not expect any significant length variation, asprevious reports on nanotubes dispersed using solvents andsonication regimes similar to those used here have not shownany significant difference in lengths [24, 25, 8].

We will examine the first exponential term in equation (5)first. One example of additional entropic contributions wouldbe those associated with solvent ordering at the surface ofthe rod—i.e. formation of a solvation shell. Here we expectthe configurational entropy associated with solvent moleculesat the rod surface to be different from that associated withmolecules in the vicinity of an imaginary surface within thebulk solvent. If we write this entropy difference per unitarea of surface as 1SA, then the entropy difference per rod(diameter D and length L) is πDL1SA, giving:

φ ≈ K exp[1SAArod

R

]exp

[−

vrod

RT

∂(1HMix/V)

∂φ

](6a)

where Arod is the surface area per mole of rods (Arod =

4vrod/D = πDLNA). This expression can be rewritten slightlyas

φ ≈ K exp[1SM

k

Arod

AM

]exp

[−

vrod

RT

∂(1HMix/V)

∂φ

](6b)

where1SM is the difference in entropy per molecule betweena solvent molecule at the rod surface versus the solvent bulk(i.e. 1SM = SI − SB where SI and SB are the entropy permolecule at the interface and in the bulk respectively) andArod and AM are the average rod and solvent molecule surfaceareas respectively. This means that ordering of the solvent atthe interface, i.e. negative 1SA, results in a reduction in theconcentration of dispersed rods. Because of the large valueof Arod/AM (e.g. ∼5–10 000 for the case of small solventmolecules, and rods with D = 1 nm and L = 1 µm), onlysmall entropy changes are required to produce observablereductions in the first exponential term in equation (6b). In thespecific case of dispersed nanotubes, computer simulationshave suggested that such entropy loss upon adsorption is onereason why toluene is such a poor nanotube solvent [26].This effect may partially explain why nanotubes cannotbe dispersed in so many solvents which otherwise appearpromising (e.g. benzene, toluene etc). In fact, we can go as faras to propose that those solvents which do disperse nanotubesare, by definition, those which do not have excessively largevalues of 1SM.

Now, we focus on the second term, writing (6b) as

φ ≈ K′ exp[−

vrod

RT

∂(1HMix/V)

∂φ

](7)

for simplicity. We note that K′ is not solvent independent asit will depend on 1SM and AM, both of which are explicitlycontrolled by the details of the rod–solvent interaction.However, we make a first-order approximation that solventswhich can disperse rods are those with similar (relativelysmall) values of 1SM, and so that K′ varies only weakly forgood solvents. We will revisit this approximation later.

Equation (7) is quite general, describing dispersionsof rods, flexible one-dimensional objects or small solutemolecules in solvents. We have even shown that it can be usedto describe data for dispersed two-dimensional objects such asMoS2 nanosheets [27].

An important feature of equation (7) is that it depends onthe expression we choose for1Hmix/V . The advantage here isthat a number of models exist for 1Hmix/V , with some of themost common being those of Flory [28, 15], Hildebrand [14,15] and Hansen [13]. Thus, writing equation (7) in thisform allows one to use the model most appropriate fortheir circumstances. For example, inserting Hildebrand’sexpression gives an equation close to that commonly usedto predict solubilities [14, 21]. The simplest expression for1Hmix/V commonly in use is [9, 19, 15]

1Hmix

V≈χφRT

vsol(8)

where χ is the Flory–Huggins parameter and the approxi-mation holds when φ � 1. This expression is particularlyuseful because χ contains information about all solute–solute,solute–solvent and solvent–solvent interaction energies. Im-portantly, χ can also be experimentally determined [9]. Wecan substitute equation (8) into equation (7) to give

φ = K′ exp[−

vrod

vsolχ

]. (9)

If we inspect equation (9), we can see that a plot of lnφ versusχ/vsol should be linear. Experimental observation of such arelationship would clearly support the validity of equation (7).

3. Experimental section

Returning to dispersions of nanotubes in solvents, it wouldbe of interest to test the relevance of equation (7) tosuch systems. To investigate this, we dispersed SWNTs(purified HiPCO, lot P2150, Unidym) in a range of solventsknown to be successful nanotube dispersants (aprotic amidesolvents) [9, 12, 29, 8]. The solvents used were 1,3-dimethyl-3,4,5,6-tetrahydro-2(1H)-pyrimidinone (DMPU), 1-formylpiperidine (NFP), 1-cyclohexyl-2-pyrrolidone (CHP), 1-octyl-2-pyrrolidone (N8P), 1-dodecyl-2-pyrrolidone (N12P),1-ethyl-2-pyrrolidone (NEP) and 1-methyl-2-pyrrolidone(NMP), all purchased from Sigma-Aldrich. With theexception of NMP (anhydrous, in a Sure-SealTM bottle),all solvents were stored over 3 A molecular sieves (Fluka,69 832) before use. Nanotubes were added to each solventat a concentration of 2 mg ml−1 and subjected to 30 mintip sonication (Vibracell VCX, 750 W, 20 kHz) at 30%amplitude using a Sonics four-element probe, with icecooling. The dispersions were then placed in Eppendorfcentrifuge tubes (1.5 ml capacity) before being centrifugedat 13 000 rpm (∼16 400 g) for 10 min. At this point itwas noticed that sonication in the more viscous solvents(e.g. N8P, N12P) visibly failed to break up large aggregatesas efficiently as the less viscous ones (e.g. NMP), thus leadingto unreliable starting concentrations. We found that this couldbe resolved by removing the supernatant from each sample

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Nanotechnology 23 (2012) 265604 J M Hughes et al

and collecting the remaining sediment. This sediment wasredispersed in fresh solvent at a concentration >0.8 mg ml−1.A similar approach has recently been reported to achieve highconcentration graphene dispersions [30]. These samples werethen sonicated for a further 40 min using the settings detailedabove. Immediately after sonication, the concentration ofeach sample was determined using absorbance spectroscopy(in a Cary 6000i UV–vis-NIR spectrophotometer, assumingαNT = 3264 ml mg−1 m−1 at 660 nm) [8] and all sampleswere diluted to a common concentration of 0.8 mg ml−1.Each sample then received a further 5 min sonication at 20%amplitude with ice cooling using a single tapered (3 mm)microtip. Samples were then permitted to equilibrate for 120 hbefore being centrifuged for 10 min at 13 000 rpm in order toremove any large aggregates. Absorbance spectroscopy wasemployed to determine the post-centrifugation concentrationof each dispersion, CNT. This concentration is of coursedirectly proportional to the dispersed volume fraction: φNT =

CNT/ρNT, where ρNT is the nanotube mass density.Samples for TEM were prepared by depositing a few

drops of dispersion onto carbon TEM grids which were driedon tin foil in air. TEM measurements were taken using anFEI Titan microscope operated at 300 kV. Samples wereprepared for AFM by soaking silanized silicon in the SWNTdispersion before rinsing in methanol and immediately dryingwith compressed air. AFM measurements were acquired usinga Veeco Nanoscope IIIa (Digital Instruments) operated intapping mode.

We measured χ via its relationship with the second virialcoefficient (B2) in the expansion of osmotic pressure in termsof solute content [16, 9]:

χ = 12 − B2VSρ

2NT. (10)

The second virial coefficient was determined by measuringthe elastically scattered light intensity from our nanotubedispersions, using a low detection limit static light scattering(SLS) photon counting spectrometer [9]. The scatteringintensity, usually expressed via the Rayleigh ratio, is relatedto CNT via B2 through the Debye light scattering equation [31,28], which for experimental convenience can be expressedas [9]

CNT

S− S0=

B2

A′CNT +

1MwA′′

(11)

where S and S0 are dimensionless numbers proportionalto the solution and pure solvent scattering intensitiesrespectively [16]. The quantities A′ and A′′ are instrumentalconstants unique to our SLS instrument and were obtainedfrom plots of C/(S − S0) versus C for the known molecularweight (Mw) standards and literature values for B2 to be A′ =1.0325(5) m3 mol kg−1 and A′′ = 9.0570(5) m3 mol kg−2.Subsequent measurements of B2s and χs for known materialswere uniformly within a few per cent of the literaturevalues [32]. We note that for technical reasons, we wereunable to measure χ for DMPU and NFP. For the solventsstudied here, we measured values of χ from 0.41 for N12P to−0.33 for CHP.

Zeta potential measurements were performed using aBrookhaven Instruments Corporation PALS Zeta PotentialAnalyser. The field frequency was 2 Hz and the voltage 150 V.We used Smoluchowski’s model to calculate the zeta potentialfrom the electrophoretic mobility. It is worth noting thatmeasuring the zeta potential for nanotubes in organic solventsis more difficult than in aqueous systems because the smallsolvent conductance results in a low count rate for typicalapplied voltages.

4. Results and discussion

Firstly, it is worth considering the state of the dispersednanotubes in order to verify that they can be thought of asrigid rods. Carbon nanotubes are extremely stiff objects withpersistence lengths of tens of microns [33, 34], suggestingthat we can model them as rigid rods. We can confirm thisby examining the state of nanotubes in two of the solventsstudied in this work, 1-dodecyl-2-pyrrolidone (N12P) and1-methyl-2-pyrrolidone (NMP). We deposited a few dropsof the dispersions onto TEM grids and Si substrates foranalysis. Shown in figure 1(A) is a TEM image of SWNTsdeposited from NMP, while figure 1(B) shows an AFM imageof SWNTs deposited from N12P. In each case the nanotubesappear very straight, suggesting that they can be treated asrigid rods. It should be noted that the images in figure 1are not taken in situ but are of dried, deposited nanotubes.However, recent cryo-TEM measurements have shown thatsingle walled nanotubes retain their rigidity in the liquidphase, reinforcing the idea that we can consider dispersednanotubes as rigid rods [35].

Shown in figure 2(A) is a plot of the measured datafor CNT versus χ which demonstrates that dispersions withlower χ tend to be more concentrated. We believe itwould be of interest to analyse these data in the light ofequations (7) and (9). However, for such a thermodynamicanalysis of solvent–nanotube dispersions to be appropriate,such dispersions must be at thermodynamic equilibrium.There is some evidence that this is the case. Previous studieshave shown that SWNTs dispersed in solvents such asNMP display spontaneous aggregation and de-aggregationas the concentration is increased (by solvent evaporation) ordecreased (by dilution) respectively [24, 9, 8]. This behaviouris reversible and occurs without any external energy input. Webelieve this to be strong evidence that these dispersions are atequilibrium.

In addition, before testing the veracity of equation (9),it is worth confirming that stabilization of nanotubes is con-trolled by the energetics of solvent–solvent, solvent–nanotubeand nanotube–nanotube interactions. If this were not the case,the only other explanation for the stabilization of nanotubesin the solvent would be if charge were transferred betweennanotube and solvent resulting in an electrostatic repulsionbetween nearby nanotubes, thus preventing aggregation. Infact, simulations have predicted a very small degree of chargetransfer between some solvents and nanotubes [34]. To testthe extent of this we performed zeta potential measurementson dispersions of SWNTs in NEP. The zeta potential is the

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Nanotechnology 23 (2012) 265604 J M Hughes et al

Figure 1. (A) TEM image of SWNTs deposited on a carbon gridfrom an NMP dispersion. (B) AFM image of SWNTs deposited on asilanized Si wafer from an N12P dispersion.

electrical potential close to the nanotube–solvent interface andscales with the amount of charge (if any) on the nanotube [36].We measured a mean zeta potential of 0.8 mV. This isextremely small compared with zeta potentials of∼25–30 mVwhich are usually considered the threshold for electrostaticstabilization [36] and negligible compared to the 50–70 mVtypically observed for ionic surfactant stabilized SWNTs [37].Thus, we conclude that the nanotubes are not charged insolvents such as NEP, confirming that the stabilization isrelated to the solvent–nanotube interaction.

This allows us to use our experimental data to determineif solution thermodynamics, as expressed by equation (9),can be used to describe nanotube dispersions. We do thisby plotting lnφNT versus χ/vsol for each solvent, displayedin figure 2(B) (here, φNT is the dispersed nanotube volumefraction, φNT = CNT/ρNT, taking ρNT = 1800 kg m−3). Wefind reasonable linearity, thus supporting equations (7) and(9). This is an important result for a number of reasons.Critically, it demonstrates that equilibrium thermodynamicscan qualitatively explain the dispersion of SWNTs in organicsolvents. This builds on the work of Pasquali et al whichshowed that nanotubes can be thermodynamically soluble insuperacids [38]. It is also worth noting that these experimentalresults confirm the previous observation that χ , and so1Hmix,can be negative for nanotube dispersions (i.e. solutions) [9].This is important, as dispersions of molecules which interactsolely via London forces (i.e. where the geometric meanapproximation is accurate) cannot result in negative χ [14,

Figure 2. (A) Dispersed concentration of nanotubes as a functionof χ and (B) natural log of nanotube volume fraction, lnφNT, as afunction of χ/vsol (here the volume fraction is found fromφNT = CNT/ρNT, where ρNT is the nanotube mass density, taken as1800 kg m−3). In (B), the line is a fit to equation (9) and isconsistent with vNT = 0.6 l mol−1.

39]. This suggests that specific solvent–nanotube interactionsare at play in ‘good’ solvents and are clearly important forgood solubility. The importance of such specific interactionsmay explain why aprotic amide solvents are the most commongood solvents for nanotubes [12]. By extension, the absenceof such specific interactions in other solvents, coupled withsolvent ordering at the nanotube–solvent interface, may partlyexplain the extremely poor solubility of nanotubes in mostcommon organic solvents.

It is possible to estimate K′ and vNT (i.e. the value of vrodin the specific case of dispersed nanotubes) from the fit line infigure 2(B). This gives values of K′ = 1.1× 10−4 and vNT =

0.6 l mol−1. This is an unrealistically low value for vNT. Fora SWNT with D = 1 nm and L = 1 µm, we can estimatevNT ∼ 500 l mol−1. For a bundle, D would be significantlylarger (at the high concentrations used in this work meanbundle diameters in excess of 3 nm are expected) [25] leadingto significantly larger values of vNT. This large discrepancybetween measured and expected values emphasizes that theexperimental data and theoretical model agree in a qualitative,rather than a quantitative manner.

As described above, one advantage of equation (7) is thatwe can use whichever enthalpy function is most suited tothe situation. Recently, we showed that for both nanotubesand graphene, the enthalpy of mixing could be expressed

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Nanotechnology 23 (2012) 265604 J M Hughes et al

Figure 3. Dispersed concentration as a function of solvent surfaceenergy; note that solvent surface energy, ES,Sol, is related to surfacetension, γ , by γ = ES,Sol − TSS,Sol where SS,Sol is the surfaceentropy and TSS,Sol ≈ 29 mJ m−2 for most liquids. The line is a fitto equation (14).

as a function of surface energy. Within this framework theenthalpy of mixing per volume is given by [9]

1Hmix/V ≈ 4(√

ES,NT −√

ES,Sol)2φNT/D (12)

where D is the dispersed nanotube (or bundle) diameterand ES,NT and ES,Sol are the nanotube and solvent surfaceenergies respectively. Intuitively, this means that the enthalpyof mixing is minimized and so the dispersed concentration ismaximized when ES,NT = ES,Sol. Substituting equation (12)into (7) gives

φNT ≈ K′ exp[−

4vNT

RTDBun(√

ES,NT −√

ES,Sol)2]. (13)

Writing vNT = NAπD2L/4 and by using the approximation(x− a)2 ≈ 4a(

√x−√

a)2 (for x ∼ a), we get:

φNT ≈ K′ exp[−

πDL

4ES,NTkT(ES,NT − ES,Sol)

2]. (14)

This predicts that the dispersed concentration should bedescribed by a Gaussian function in ES,Sol. We can testthis by plotting the measured concentration as a functionof solvent surface energy in figure 3 [40, 41]. (Solventsurface energy, ES,Sol, is related to surface tension, γ , byγ = ES,Sol − TSS,Sol, where SS,Sol is the surface entropyand TSS,Sol ≈ 29 mJ m−2 for most liquids.) This graphclearly shows a peak in concentration as has been observedpreviously [9, 29, 12]. (In the literature, some solvents tendto display concentrations which fall below that predicted byequation (14). This is a manifestation of the limitations ofsurface energies as solubility parameters and can be partiallyresolved by using Hansen’s approach [12, 13].) The solid linerepresents a Gaussian centred at 68 mJ m−2 and with width3.5 mJ m−2 and fits the experimental data very well. Thisclearly suggests that ES,NT = 68 mJ m−2. However, the widthof the Gaussian is also important.

The width, 0, of the Gaussian described by equation (14)is given by 0 ≈ 1.6

√4ES,NTkT/πDL. Calculating DL from

the measured width gives a value of 7.5 × 10−17 m2,significantly smaller than the minimum expected value ofDL = 10−15 m2 calculated for individually dispersed SWNTs(we actually expect the dispersion to consist of small bundles,which would give a substantially larger value). Thus, like thesituation with vNT, solution thermodynamics describes thedata but predicts parameters associated with nanotube sizeto be smaller than reality. Similar quantitative disagreementhas been observed when applying this model to dispersionsof inorganic layered compounds [27]. Thus, like the situationwith vNT, solution thermodynamics describes the data butpredicts parameters associated with nanotube size to besmaller than in reality. We note that the values of vNTand DL estimated from the data imply effective values ofdiameter and length to be 13 pm and 5.6 µm, far from thetrue values. This discrepancy again underlines the qualitativeagreement between experiment and model and emphasizesthat nanotubes do not behave as simple rod-like solutes.However, the implications of this discrepancy are not allnegative. Taking values of D = 1 nm and L = 1 µm, wecan estimate that if our model was quantitative, then wewould expect 0 = 1 mJ m−2. This is significantly smallerthan the measured value of 3.5 mJ m−2, a discrepancy thathas important practical implications. If the width of theGaussian described by equation (14) really was 1 mJ m−2,very few solvents would exist with the correct surface energiesto disperse nanotubes at all. This would obviously havesignificant practical disadvantages.

We can attempt to tentatively identify a possibleexplanation for the fact that the observed peak in figure 3is broader than expected from theory. Previously, we madethe approximation that K′ and so (1SM/k)(Arod/AM) didnot significantly vary with solvent. However, we note that1SM can never be positive as intuition would suggest thatthe solvent molecules at the nanotube surface can never beless ordered than those in the bulk. This potential for solventordering at the interface (but no loss of order) means that1SM must always be ≤ 0. Thus, we can consider the scenarioof a very good solvent, which interacts strongly with thenanotube surface, possibly by specific interactions. In such ascenario, the molecules at the nanotube surface may undergosome ordering, resulting in non-zero negative1SM. However,for poorer solvents, the interfacial ordering might not be sosignificant, resulting in less negative1SM. Very poor solventsmay have virtually no ordering at the nanotube surface atall, resulting in 1SM close to zero. Such a situation wouldresult in (1SM/k)(Arod/AM) varying with ES,Sol, in a formas shown in red in figure 4(A) (the quantitative nature ofthis curve has no physical basis and has been plotted onlyfor example). This phenomenon would certainly affect thedispersed concentration predicted by equation (6b). To seethis, we first plot the dispersed concentration of nanotubes,calculated using equation (7), in figure 4(B) (i.e. assuming theentropy difference, 1SM, to be invariant with solvent type).This gives a peak with width of ∼1 mJ m−2 (black curve).If we then calculate the dispersed concentration of nanotubes

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Nanotechnology 23 (2012) 265604 J M Hughes et al

Figure 4. (A) Possible difference in entropy per molecule betweensurface and bulk solvent molecules, as proposed in the text (seeequation (6b) for a definition of parameters), plotted versus solventsurface energy (red curve). The black curve (i.e. the x axis)illustrates the case where the entropy difference is independent ofsolvent. (B) Calculated nanotube concentration plotted for constantentropy difference (black) and for the entropy difference proposedin the text (red).

with equation (6b), and apply the entropy difference asgiven in figure 4(A), a significantly broader Gaussian-likepeak is observed (red curve). This is a significant resultas it shows that indirect effects such as the dependence ofsurface entropy on solvent solubility parameter can introducebroadening as observed experimentally. We emphasize thatwe cannot reproduce all of the observed broadening fromthe mechanism described previous. However, we believe thatadditional entropic contributions can probably account forthe rest of the broadening. We note that the entropy changesinvoked in figure 4(A) are actually very small, equivalent to afraction of k (i.e. Boltzmann’s constant) per molecule. Thusthe degree of ordering for a good solvent required to givebroadening is expected to be much less than that which leadsto destabilization of the dispersion as described previouslyfor toluene. As such, it is worth emphasizing that while verygood solvents may have a small degree of interfacial ordering,excessive ordering will lead to non-solvent behaviour.

5. Conclusions

In conclusion, we have developed a model for the dispersedconcentration of rod-like solutes in organic solvents. Inparallel we have experimentally characterized dispersions ofcarbon nanotubes in organic solvents, measuring the dispersedconcentration and Flory–Huggins parameter. We have appliedour model to these nanotube dispersions, obtaining qualitative

agreement. This shows that a thermodynamic framework canbe used to understand the dispersion of carbon nanotubes inorganic solvents. This is important as it opens up a wholenew landscape for understanding the dispersion of otherlow-dimensional materials such as graphene and inorganiclayered compounds [18, 42, 17, 27]. We note that thisframework has limitations and cannot make quantitativepredictions about the nanotube dimensions; however, wesuggest that entropic effects associated with the ordering ofsolvent molecules at the nanotube surface can explain thisdiscrepancy.

Acknowledgments

JNC would like to acknowledge Science Foundation Ireland,(grant number 07/IN.7/I1772), for financial support. JPHacknowledges the Wisconsin Distinguished Professor fundingestablished under Wisconsin State Statute 36.14, the WiSysTechnology Foundation and multiple US Department ofEnergy, ERLE Grants, e.g. DOE grant no. DE-FG26-03NT41687.

References

[1] Liu J, Casavant M J, Cox M, Walters D A, Boul P, Lu W,Rimberg A J, Smith K A, Colbert D T andSmalley R E 1999 Controlled deposition of individualsingle-walled carbon nanotubes on chemicallyfunctionalized templates Chem. Phys. Lett. 303 125–9

[2] Ausman K D, Piner R, Lourie O, Ruoff R S andKorobov M 2000 Organic solvent dispersions ofsingle-walled carbon nanotubes: toward solutions ofpristine nanotubes J. Phys. Chem. B 104 8911–5

[3] Bahr J L, Mickelson E T, Bronikowski M J, Smalley R E andTour J M 2001 Dissolution of small diameter single-wallcarbon nanotubes in organic solvents? Chem. Commun.193–4

[4] Mickelson E T, Chiang I W, Zimmerman J L, Boul P J,Lozano J, Liu J, Smalley R E, Hauge R H andMargrave J L 1999 Solvation of fluorinated single-wallcarbon nanotubes in alcohol solvents J. Phys. Chem. B103 4318–22

[5] Furtado C A, Kim U J, Gutierrez H R, Pan L, Dickey E C andEklund P C 2004 Debundling and dissolution ofsingle-walled carbon nanotubes in amide solvents J. Am.Chem. Soc. 126 6095–105

[6] Landi B J, Ruf H J, Worman J J and Raffaelle R P 2004 Effectsof alkyl amide solvents on the dispersion of single-wallcarbon nanotubes J. Phys. Chem. B 108 17089–95

[7] Maeda Y et al 2004 Dispersion of single-walled carbonnanotube bundles in nonaqueous solution J. Phys. Chem. B108 18395–7

[8] Giordani S, Bergin S D, Nicolosi V, Lebedkin S,Kappes M M, Blau W J and Coleman J N 2006 Debundlingof single-walled nanotubes by dilution: observation of largepopulations of individual nanotubes in amide solventdispersions J. Phys. Chem. B 110 15708–18

[9] Bergin S D et al 2008 Towards solutions of single-walledcarbon nanotubes in common solvents Adv. Mater.20 1876–81

[10] Detriche S, Zorzini G, Colomer J F, Fonseca A andNagy J B 2007 Application of the hansen solubilityparameters theory to carbon nanotubes J. Nanosci.Nanotechnol. 8 6082–92

[11] Ham H T, Choi Y S and Chung I J 2005 An explanation ofdispersion states of single-walled carbon nanotubes in

7

Nanotechnology 23 (2012) 265604 J M Hughes et al

solvents and aqueous surfactant solutions using solubilityparameters J. Colloid Interface Sci. 286 216–23

[12] Bergin S D, Sun Z Y, Rickard D, Streich P V,Hamilton J P and Coleman J N 2009 Multicomponentsolubility parameters for single-walled carbonnanotube-solvent mixtures ACS Nano 3 2340–50

[13] Hansen C M 2007 Hansen Solubility Parameters—A User’sHandbook (Boca Raton, FL: CRC Press)

[14] Hildebrand J H, Prausnitz J M and Scott R L 1970 Regularand Related Solutions (Princeton, NJ:Van Nostrand-Reinhold)

[15] Rubinstein M and Colby R H 2003 Polymer Physics (Oxford:Oxford University Press)

[16] Sperling L H 1992 Introduction to Physical Polymer Science(New York: Wiley)

[17] Hernandez Y et al 2008 High-yield production of graphene byliquid-phase exfoliation of graphite Nature Nanotechnol.3 563–8

[18] Coleman J N et al 2011 Two-dimensional nanosheetsproduced by liquid exfoliation of layered materials Science331 568–71

[19] Donald A, Windle A H and Hanna S 2006 Liquid CrystallinePolymers (Cambridge: Cambridge University Press)

[20] Ciferri A 1991 Liquid Crystallinity in Polymers (New York:Wiley)

[21] Ruelle P, Buchmann M, Namtran H and Kesselring U W 1992The mobile order theory versus unifac and regular solutiontheory-derived models for predicting the solubility of solidsubstances Pharm. Res. 9 788–91

[22] Ruelle P, Reymermet C, Buchmann M, Ho N T,Kesselring U W and Huyskens P L 1991 A new predictiveequation for the solubility of drugs based on thethermodynamics of mobile disorder Pharm. Res. 8 840–50

[23] Hennrich F, Krupke R, Arnold K, Rojas Stutz J A, Lebedkin S,Koch T, Schimmel T and Kappes M M 2007 Themechanism of cavitation-induced scission of single-walledcarbon nanotubes J. Phys. Chem. B 111 1932–7

[24] Bergin S D, Nicolosi V, Giordani S, de Gromard A,Carpenter L, Blau W J and Coleman J N 2007 Exfoliationin ecstasy: liquid crystal formation andconcentration-dependent debundling observed forsingle-wall nanotubes dispersed in the liquid druggamma-butyrolactone Nanotechnology 18 455705

[25] Bergin S D, Sun Z Y, Streich P, Hamilton J andColeman J N 2010 New solvents for nanotubes:approaching the dispersibility of surfactants J. Phys. Chem.C 114 231–7

[26] Grujicic M, Cao G and Roy W N 2004 Atomistic simulationsof the solubilization of single-walled carbon nanotubes intoluene J. Mater. Sci. 39 2315–25

[27] Cunningham G, Lotya M, Cucinotta C S, Sanvito S,Bergin S D, Menzel R, Shaffer M S P andColeman J N 2012 Solvent exfoliation of transition metaldichalcogenides: dispersibility of exfoliated nanosheetsvaries only weakly between compounds ACS Nano6 3468–80

[28] Flory P J 1956 Statistical thermodynamics of semi-flexiblechain molecules Proc. R. Soc. Lond. A 234 60–73

[29] Coleman J N 2009 Liquid-phase exfoliation of nanotubes andgraphene Adv. Funct. Mater. 19 3680–95

[30] Khan U, Porwal H, O’Neill A, Nawaz K, May P andColeman J N 2011 Solvent-exfoliated graphene atextremely high concentration Langmuir 27 9077–82

[31] Debye P 1946 J. Phys. Chem. 51 18[32] Brandrup J, Immergut E H, Grulke E A, Akihiro A and

Bloch D R 1999 Polymer Handbook (New York: Wiley)[33] Duggal R and Pasquali M 2006 Dynamics of individual

single-walled carbon nanotubes in water by real-timevisualization Phys. Rev. Lett. 96 4

[34] Torrens F 2006 Calculations of organic-solvent dispersions ofsingle-wall carbon nanotubes Int. J. Quantum Chem.106 712–8

[35] Parra-Vasquez A N G et al 2010 Spontaneous dissolution ofultralong single- and multiwalled carbon nanotubes AcsNano 4 3969–78

[36] Israelachvili J 1991 Intermolecular and Surface Forces(New York: Academic)

[37] Sun Z, Nicolosi V, Rickard D, Bergin S D, Aherne D andColeman J N 2008 Quantitative evaluation ofsurfactant-stabilized single-walled carbon nanotubes:dispersion quality and its correlation with zeta potentialJ. Phys. Chem. C 112 10692–9

[38] Davis V A et al 2009 True solutions of single-walled carbonnanotubes for assembly into macroscopic materials NatureNanotechnol. 4 830–4

[39] Hughes J M, Aherne D and Coleman J N 2012 J. Appl. Polym.Sci. at press

[40] Tsierkezos N G and Filippou A C 2006 Thermodynamicinvestigation of N,N-dimethylformamide/toluene binarymixtures in the temperature range from 278.15 to 293.15 KJ. Chem. Thermodyn. 38 952–61

[41] Lyklema J 1999 The surface tension of pureliquids—thermodynamic components and correspondingstates Colloids Surf. A 156 413–21

[42] Hernandez Y, Lotya M, Rickard D, Bergin S D andColeman J N 2010 Measurement of multicomponentsolubility parameters for graphene facilitates solventdiscovery Langmuir 26 3208–13

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