multilayer vesicles and vesicle clusters formed by the fullerene-based surfactant c60(ch3)5k

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Journal of Colloid and Interface Science 275 (2004) 632–641www.elsevier.com/locate/jcis

Multilayer vesicles and vesicle clusters formed by the fullerene-bassurfactant C60(CH3)5K

Christian Burger,a Jingcheng Hao,a Qicong Ying,a Hiroyuki Isobe,b Masaya Sawamura,b

Eiichi Nakamura,b and Benjamin Chua,∗

a Department of Chemistry, State University of New York at Stony Brook, Stony Brook, NY 11794-3400, USAb Department of Chemistry, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Received 17 October 2003; accepted 23 February 2004

Available online 9 April 2004

Abstract

The self-assembly behavior of a fullerene-based surfactant, C60(CH3)5K, in water was studied using a combination of static andnamic light scattering, as well as transmission electron microscopy, and compared to that of the compound C60(C6H5)5K. Both fullerenesurfactant systems spontaneously assemble into large vesicles consisting of closed spherical shells formed by bilayers, with critical aggregation concentrations (CAC) lower than 10−6 g ml−1. At low concentrations, the aggregate sizes of C60(CH3)5K (radiusR ∼ 26.8 nm) andC60(C6H5)5K (R ∼ 17.0 nm) were found to be substantially different from each other, showing that the change of the substituents surrounding the polar cyclopentadienide head group makes it possible to control the size of the resulting aggregates. Furthermore, the C60(CH3)5Kvesicles were found to exist in two qualitatively different types of aggregation with a critical reaggregation concentration (CRC) located a3.30× 10−6 g ml−1. Above the CRC, larger aggregates were observed (R ∼ 37.6 nm), showing a more complex form of supramolecuaggregation, e.g., in terms of multi-bilayer vesicles and/or of clusters of bilayer vesicles. 2004 Elsevier Inc. All rights reserved.

Keywords: Fullerenes; Surfactants; Aggregation; Vesicles; Dynamic laser light scattering; Static laser light scattering

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1. Introduction

Assemblies of organic components with dimensionsthe range of about ten to a few hundred nanometersganic nanoparticles”) are ubiquitous in nature and pa central role in both the structure and function of ling organisms[1]. After billions of years of evolution, thsizes and shapes of natural organic nanoparticles arecisely controlled with respect to their functions. On tother hand, development of artificial organic nanopartiwith a regulated size and shape is still a challenging sject in the current scientific community. The developmof self-assembled nanostructures of organic amphiphwith sizes in the lower to intermediate nanometric ra(50–500 nm) has drawn particular attention[2]. Among thevarious forms of molecular self-assembly of amphiph

* Corresponding author. Second address: Department of Materials Scence and Engineering, State University of New York at Stony Brook, SBrook, NY 11794, USA.

E-mail address: [email protected] (B. Chu).

0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2004.02.048

-

molecules in aqueous solution, closed bilayer membr(vesicles) have been established as one of the best conrations for such nanoparticles[3]. Studies of synthetic lipidshowed that one can control the size and shape of aggreby changing the structures of the hydrophobic hydrocarchains[4].

Fullerenes are not soluble in aqueous solution due toextremely hydrophobic nature of the C60. In recent yearsthe synthesis of amphiphilic fullerene derivatives containg polar functions, such as carboxylic acid groups[5–9]and amines[10–14], or starlike fullerene derivatives, suchC60[(CH2)nSO3Na]6 [15], which are soluble in water, havbroadened the scope of functionalized fullerenes. Howefullerenes can also be made water-soluble by pure hydrobon pentasubstitution stabilizing the cyclopentadienideion, as shown inFig. 1. The conformations of such modifiefullerene molecules are uniquely different from those ofditional chain-based surfactants such as lipids[16], replac-ing their flexible hydrocarbon chains with rigid hydrocarbballs. This should not be confused with other fullere

http://www.elsevier.com/locate/jcis

C. Burger et al. / Journal of Colloid and Interface Science 275 (2004) 632–641 633

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modified surfactants or phospholipids that still contain loalkyl chains as the main hydrophobic parts of the aphiphilic molecule[17–19]. In our case, the fullerene baitself is the main hydrophobic part of the molecule. Suamphiphilic fullerene surfactants are capable of spontaneoformation of large spherical aggregates in aqueous stion [20]. As studied recently[21] in detail for the com-pound C60Ph5K (Fig. 1 with R = Ph = C6H5), they formlarge vesicles consisting of bilayers similar to those of chbased surfactants and lipids. Considering the significantferences in their respective molecular geometries, it is noself-evident that fullerene-based surfactants would packbilayer membranes; cf.Fig. 2.

Due to the unique properties of fullerene compounsuch as their photochemical and photophysical properthe possibility of regulating assembled fullerene structuredrawing the attention of scientists in various fields[22–24].

Because of their similarities to cell membranes and tpotential for encapsulating and segregating water-sol

Fig. 1. Chemical structure of the pentasubstituted fullerene potassiumIn the present study, the methyl-substituted compound has been used=Me = CH3).

materials from a bulk solution[25], together with non-toxicity relative to chain-based surfactants, fullerene surtant vesicles could also have important pharmaceuticaplications[26].

In the present study, we have investigated the aggrtion behavior of a differently substituted fullerene surfactanamely C60Me5K, Fig. 1, by laser light scattering (LLSand transmission electron microscopy (TEM). As will be dscribed below, we have found that, unlike their lipid-basnanoparticle counterparts, the size of the fullerene-bavesicles can be controlled by the choice of substituentssurround the polar head group. Furthermore, the effectpears not to be based simply on steric interactions wthe bulkiness of the substituents would induce spontaneocurvature, but is more subtle, since the smaller metsubstituted fullerene compounds generate the larger vesWe have also found that, together with the concentrationchoice of the substituents can lead to a clustering of whvesicles.

2. Experimental and evaluation methods

2.1. Sample preparation

The cyclopentadienes are readily synthesized throfivefold addition of the organocopper reagent to the C60fullerene [27–29]. A tetrahydrofuran (THF)-free aqueousolution of the modified fullerenes, C60Me5K, at a concen-tration of c = 3 × 10−3 mol l−1 was purified as describein Ref. [21]. Finally, a small amount of precipitation waremoved by centrifugation (10 min at 1.5 × 104 rpm or1.3× 104 g). The resultant clear stock solution at a conctration ofc = (1.01± 0.03) × 10−3 mol l−1 had a blood redcolor. After preparation, all samples were kept for at le

Fig. 2. Comparison of a traditional lipid bilayer vs a fullerene surfactant bilayer (schematic representations). The hydrophilic head groups are sketched in blue(dark), the hydrophobic parts in green (light), and thelikewise hydrophobic substituents in yellow (lighter).

634 C. Burger et al. / Journal of Colloid and Interface Science 275 (2004) 632–641

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1 week at room temperature (∼23◦C) before measurementThe sample preparation is described in more detail in[21].

2.2. Laser light scattering (LLS)

To prepare dust-free solutions for the laser light-scattemeasurements, all solutions were filtered directly into dfree light-scattering cells through a 0.1-, 0.22-, or 0.45-Millipore sterile membrane filter, depending on the conctrations and the sizes of the aggregates. The light-scattcells had been rinsed inside and outside with distilled (dfree) acetone to ensure a dust-free condition before use

The refractive index increments (dn/dc) of the C60Me5Ksolutions were determined with a Brice–Phoenix differtial refractometer (Model BP-2000-V) at a wavelength546 nm (green) and at room temperature (∼23◦C). In wa-ter, a value ofdn/dc = (−0.235± 0.002) ml g−1 was ob-tained over a concentration range from 8.00 × 10−5 to2.00× 10−4 g ml−1. The value was very similar to that oC60Ph5K, which was determined over a concentration raof 4.6× 10−4 to 7.6× 10−4 g ml−1 [21].

A standard laboratory-built laser light-scattering sptrometer equipped with a Coherent Radiation 200-mW dipumped solid-state (DPSS 532) laser, operating at 532and a Brookhaven Instruments (BI-9000AT) correlator wused for the LLS measurements. The spectrometer ispable of making measurements both of the angular dedence of the absolute integrated scattered intensity (slight scattering, SLS) over a scattering angular range of◦to 140◦ and of the intensity–intensity digital photon corretion over a similar angular range (dynamic light scatteriDLS, and depolarized DLS). About 2 to 3 ml of samplelutions was transferred into a dust-free light-scatteringfor the light-scattering measurements. The scatteringwere held in a brass thermostat block filled with refracindex-matching silicone oil. The temperature was controto within ±0.05◦C.

Dynamic light scattering (DLS) measures the intensintensity time correlation functionG(2)(τ ) in the self-beating mode, whereτ is a relaxation time.G(2)(τ ) can berelated to the electric field time correlation functiong(1)(τ ),

(1)G(2)(τ ) = A(1+ b

∣∣g(1)(τ )∣∣2),

whereA andb are, respectively, the background (baseliand a coherence factor (a parameter depending on the dtion coherence). The electric field time correlation functg(1)(τ ) was analyzed by the constrained regularizedCONTIN

method[30,31]to yield the characteristic linewidth distribution G(Γ ) by inversion of

(2)∣∣g(1)(τ )

∣∣ =∞∫

0

G(Γ )exp(−Γ τ) dΓ,

whereΓ is the characteristic linewidth. The first and seond moments ofG(Γ ) are 〈Γ 〉 = ∫ ∞

0 Γ G(Γ )dΓ andµ2 = ∫ ∞

(Γ − 〈Γ 〉)2G(Γ )dΓ , respectively. The value o
0
,

--

c-

µ2/〈Γ 〉2 is a measure of the particle polydispersity. If trelaxation is diffusive,Γ can be related to the translationdiffusion coefficientD:

(3)D = Γ/q2.

The apparent hydrodynamic radius,Rh, can be obtained vithe Stokes–Einstein equation,

(4)Rh = kBT

6πηD,

wherekB is the Boltzmann constant andη is the solvent vis-cosity at absolute temperatureT . Based onEqs. (3) and (4),a characteristic linewidth distributionG(Γ ) corresponds toa distribution of apparent hydrodynamic radii from whice.g., the average apparent hydrodynamic radius〈Rh〉 canbe determined. The DLS measurements were performfinite concentrations and interparticle interactions wereglected.

The Rayleigh–Gans–Debye equation forms the basstatic light scattering (SLS). The angular dependence oexcess absolute time-averaged scattered intensity, for snoninteracting particles in the limit of dilute concentratiohas the form[32]

(5)Hc

Rvv(q)∼= 1

Mw

(1+

〈R2g〉

z

3q2

)+ 2A2c,

whereH = 4πn2(dn/dc)2/NAλ4 is an optical parameten is the solution refractive index (for dilute concentratioit can be replaced with that of the solvent,n0), dn/dc isthe specific refractive index increment,NA is Avogadro’sconstant,λ is the laser wavelength of the incident beamvacuum (in our case,λ = 532 nm),q = 4πnsin(θ/2)/λ isthe absolute value of the scattering vector,θ is the scattering angle,〈R2

g〉z is the z-average of the squared radiusthe gyration,Mw is the weight-average molecular weigof the aggregates, andA2 is the second virial coefficienRvv(q) denotes the excess Rayleigh ratio of the samplelution using vertically polarized incident and scattered li(subscriptvv).

The scattering intensity can be normalized byRvv(q) =RBz,90n[I (q) − I0]/(nBzIBz,90), whereRBz,90 is the Ray-leigh ratio of benzene at scattering angleθ = 90◦ and has avalue of 2.0× 10−5 cm−1 atλ = 532 nm[33]. I (q), I0, andIBz,90 are the scattered intensities of the solution atq , thesolvent, and benzene, respectively, andnBz is the refractiveindex of benzene. Absorption effects of the lightly coloredsolutions were neglected.

2.3. Transmission electron microscopy (TEM)

A 4-µl sample solution filtered by a 0.22-µm Milliposterile membrane filter was dropped onto a TEM grid (cper grid, 3.02 mm, 200 mesh, coated with Formvar film)stained with∼4 µl 2% uranyl acetate aqueous solution. Afthe solution was dried in air, TEM images were taken oJEOL JEM1200ex (Japan) transmission electron microsat an accelerating voltage of 80 kV.


r,

(a) (b)

(c) (d)

(e) (f)

Fig. 3. Apparent hydrodynamic radius distributions of C60Me5K as obtained from DLS experiments in water; concentrations as indicated. (a) Yellow colo(b–d) slight yellow color, (e–f) colorless.

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3. Results and discussion

3.1. Dynamic light scattering

Based on theCONTIN method of analysis,Fig. 3 showsplots of the relative intensity contributionΓ × G(Γ ) as afunction of the apparent hydrodynamic radiusRh for aque-ous solutions of C60Me5K at different concentrations anddifferent scattering angles, as indicated. Even at a highlyluted concentration ofc = 6.68× 10−7 g ml−1, the scatteredintensity was sufficiently high to permit reliable DLS me

surements, as seen inFig. 3f. The angular dependence of tDLS was small at all concentrations.

At a concentration ofc = 8.35× 10−5 g ml−1, Fig. 3a,the aqueous solution had a clear yellow color. DLS measments showed that the size distribution was unimodal wan average hydrodynamic radius〈Rh〉 of about 37.1 nm anda polydispersity index ofµ2/〈Γ 〉2 ∼= 0.24. At diluted con-centrations between 1.50× 10−5 and 1.67× 10−6 g ml−1,the yellow color became less intense. The DLS measments forc = 4.01×10−6 g ml−1 are shown inFig. 3b, witha large peak centered about〈Rh〉 ∼= 37.5 nm and a polydis


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Fig. 4. Plot of the relative excess scattered intensity atθ = 90◦ vs con-centration in water at 25◦C. A change of slope is observed at a transitconcentration ofc = 3.30×10−6 g ml−1 (critical reaggregation concentration, CRC). At the transition point, single bilayer vesicles (top left) stablbelow the CRC are transformed into larger more complicated aggre(e.g., the four-layer vesicle shown onthe right). The single bilayer vesiclon the left consists of 2.06× 104 molecules distributed over two shellsradii 27.4 and 26.2 nm, respectively. For the double bilayer vesicle onright, 8.67× 105 molecules were distributed over four shells with radii39.7, 38.4, 36.8, and 35.5 nm, respectively. In both models, a sectobeen cut out to enhance the visibility.

persity indexµ2/〈Γ 〉2 ∼= 0.27. At c = 3.34× 10−6 g ml−1

we observe〈Rh〉 ∼= 37.7 nm andµ2/〈Γ 〉2 ∼= 0.21 to 0.37;seeFig. 3c. The differences between the results ofFigs. 3a,3b, and 3care small.

At a concentration ofc = 3.00× 10−6 g ml−1, Fig. 3d,a 〈Rh〉 of ∼26.7 nm indicates an abrupt change in agggate size. The value ofµ2/〈Γ 〉2 was ∼0.3. At and belowc = 1.67× 10−6 g ml−1, Fig. 3e, the sample became essetially colorless. The unimodal size distribution was peaaround〈Rh〉 ∼= 26.6 nm with µ2/〈Γ 〉2 ∼= 0.21 to 0.35. Atc = 6.68× 10−7 g ml−1, Fig. 3f, the scattered intensity wastill high enough that the measurements at lower scatteangles could be performed. The unimodal size distribuwas again peaked around〈Rh〉 ∼= 26.5 nm and the polydispersitiesµ2/〈Γ 〉2 ∼= 0.35 to 0.45 became larger. Upofurther decrease in concentration, the scattered intensity bcame so low that DLS measurements could not be perforwith reasonable precision.

Fig. 4shows a plot of the relative excess scattered insity at θ = 90◦ vs concentration, where the transition cocentration is observed at the change of slope atc = 3.30×10−6 g ml−1. This type of plot is normally used to determithe critical aggregation concentration (CAC), which womark the transition from monomers to aggregates. Howein our case we already observe aggregates at concentrabelow this transition concentration. Therefore, we aredently dealing with some form of reaggregation into laraggregates. We designate the critical reaggregation concentration (CRC) as the concentration of the transition betwtwo qualitatively different types of aggregates. BasedFig. 3f, we can conclude that the actual CAC for the tr

s

(a)

(b)

Fig. 5. SLS data of C60Me5K in water atT = 25◦C, evaluated as Zimmplot (a) above the CRC and over a concentration range of 9.98× 10−6 to1.83× 10−5 g ml−1; (b) below the CRC and over a concentration range1.06× 10−6 to 1.71× 10−6 g ml−1.

sition from C60Me5K monomers to aggregates is lower th6.68× 10−7 g ml−1 and outside of our experimental rang

3.2. Static light scattering

Fig. 5a shows a Zimm plot over a scattering angurange of 20◦ to 140◦ above the CRC of 3.30× 10−6 g ml−1

up to concentrations of 1.83 × 10−5 g ml−1 at a tem-perature of (25± 0.05)◦C. From the extrapolations olimθ→0,(c−cCRC)→0 H(c − cCRC)/Rvv(θ), the quantitiesMw = (7.23± 0.78) × 107 g mol−1, Rg = (37.6± 3.9) nm,andA2 = (−4.94±0.60)×10−5 ml mol g−2 were obtained

The SLS experiments were also carried out at ccentrations below the CRC down to a concentration1.06 × 10−6 g ml−1. Fig. 5b shows the correspondinZimm plot, resulting in values ofMw = (1.72 ± 0.19) ×107 g mol−1, Rg = (26.8 ± 3.1) nm, andA2 = (−6.00±0.70) × 10−4 ml mol g−2. It should be noted that this valuof Mw was much lower than the one above the CRC.

3.3. Aggregate microstructures

The ratio of radius of gyration and hydrodynamicdius,ρm = Rg/Rh, is a useful structure-sensitive parame


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which depends on the particle shape (and the density prif it is not uniform). For impermeable solid spheres,Rg =(3/5)1/2R and Rh = R, so thatρm = (3/5)1/2 ∼= 0.775;for impermeable spherical shells (hollow spheres) inlimit of infinitely thin wall thickness,Rg = Rh = R, so thatρm = 1 (polydispersity effects on this ratio are discussedlow and in Appendix A); for disk micelles in the limit ofinfinitely thin disk thickness,ρm = π/

√8 ∼= 1.11, indepen-

dent of the disk radius[34]; for monodisperse random coilρm = 8/(3

√π) ∼= 1.5 [34]; for rodlike micelles,ρm depends

on the length and the diameter; for wormlike micelles,ρm

values are expected to be between those of random coilrodlike micelles (higher than 1.5). An increase in chain stness as well as in polydispersity leads to largerρm [35]. Insummary, for elongated micelles,ρm can vary in the rangeof 1.35 to 4.01[36]. Herzog et al.[37] obtainedρm = 1.5 to1.7 for wormlike mixed micelles. Alargova et al.[38–40]re-ported wormlike micelles of ionic surfactants in the preseof salts from LLS measurements and obtainedρm = 1.96 foran ionic strength at 24 mM KBr andρm = 2.00 for an ionicstrength at 64 mM KBr, respectively.

From our DLS and SLS data for C60Me5K in water, weobtained the ratio of radius of gyration to hydrodynamicdius,ρm = (37.6± 3.9)/37.0= 1.02± 0.11 above the CRCof 3.30×10−6 g ml−1, and(26.8±3.1)/26.6= 1.01±0.12below the CRC. Even though the aggregate sizes characized byRg andRh are substantially different above and blow the CRC, the ratioρm = Rg/Rh remains close to unity ineither case, suggesting the presence of spherical shells[21].

We can now estimate the average aggregation numbeN ,i.e., the average number of C60Me5K molecules per aggregate, on the basis of the LLS results,Mw , Rg , andRh, together with the molecular weight of C60Me5K, M =834.9 g mol−1. Using Mw = (7.23± 0.78) × 107 g mol−1

above the CRC andMw = (1.72± 0.19) × 107 g mol−1 be-low the CRC, the average aggregation numbers areN =Mw/M = (8.66±0.93)×104 andN = (2.06±0.23)×104,above and below the CRC, respectively.

Assuming the thickness of the shell of a spherical vesto be very small compared to its radius, we consider a hospherical shell of radius ofR ∼= Rg

∼= Rh. Within this ap-proximation, we can estimate the average area per moleaM as a function of the number of stacked layers and thedetermine the most probable multilayer geometry[21].

Below the CRC, assuming a spherical bilayer ves(Fig. 4 top left), we calculate anaverage area per molecule aM = 0.88 nm2, which is in good agreement with ouprevious findings[21] and experimental Langmuir–Blodgedata[41]. However, above the CRC the corresponding spical vesicle consisting of a single bilayer would only leaan area per molecule ofaM = 0.42 nm2, not enough to packthe fullerene bodies. Therefore, we propose that a posmodel in agreement with our experimental results is giby a multilayer vesicle consisting of two individual bilayeas sketched on the right ofFig. 4. In this case, the approx

d

mate area per molecule ofaM = 0.84 nm2 again assumes aacceptable value.

It should be noted that polydispersity effects complicthis straightforward analysis. As discussed in detail inAp-pendix A, an increase in polydispersity of the particle sdistribution leads to an increase of the ratioρm beyond unity.However, if the particle system does not consist of agle species with a sufficiently narrow size distribution, ligscattering alone cannot be expected to provide an unamous result, and additional input from complementary expmental techniques will be welcome.

While the single bilayer vesicle discussed in Ref.[21]and in the present work below the CRC is a simple, content, and straightforward concept that can be postulateda high level of confidence, the multilayer vesicle discusabove is not as unambiguous a description of the largegregates above the CRC, and alternative models needconsidered.

Transmission electron microscopy is an experimetechnique that provides direct real-space morphologicaformation, which can prove helpful either to confirm ostructural models proposed based on the light-scatteanalysis or to disprove them and give hints for alternastructures.

3.4. Transmission electron microscopy

Fig. 6 shows transmission electron micrographsC60Me5K for two concentrations, both above the CRC.a concentration ofc = 4.17× 10−6 g ml−1, slightly higherthan the CRC, we observe large isolated circular objepresumably due to collapsed spherical vesicles,Fig. 6a,which would be in agreement with our large multilayer vecle. The exact number of stacked bilayers, as well as accquantitative size information, is difficult to assess fromTEM micrographs.

At a much higher concentration ofc = 4.17 × 10−4

g ml−1, the picture has changed, andFig. 6bshows distinc-tive arrangements of several spherical vesicles into clusof various shapes and geometries (dimers, trimers, tetrametc.).

3.5. Light scattering from clusters of hollow spheres

In order to decide whether the observed clusters aragreement with our light scattering results, we will calculthe ratioρm = Rg/Rh and check if it is close to unity. Thidealized case of identical clusters with regular arrangemenof monodisperse vesicles can be treated quantitatively.Fig. 7shows an example of such an idealized cluster in a regtetrahedral geometry.

As shown inAppendix B, the radius of gyrationRg of acluster of hollow spheres is given in closed form,Eq. (B.3).Together with the hydrodynamic radiiRh listed in Ref.[42],we can obtain theρm = Rg/Rh ratios for various regular cluster geometries as shown inTable 1. It should be


(a) (b)

Fig. 6. TEM micrographs of isolated vesicles and clusters: (a) spherical vesicles with diameters 2R of 61 to 78 nm atc = 4.17× 10−6 g ml−1; (b) variouscluster geometries atc = 4.17× 10−4 g ml−1.

vesi-

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Fig. 7. Idealized cluster model consisting of four monodisperse bilayercles (each with radiusR = 23.0 nm) in a tetrahedral arrangement.

Table 1Radii of gyrationRg (Eq. (B.3)), hydrodynamic radiiRh (Ref. [42]), andtheir ratiosρm for regular clusters of monodisperse hollow spheres wgeometries as indicated

N Geometry Rg/R Rh/R ρm = Rg/Rh

1 Monomer 1.00 1.00 1.002 Dimer

√2= 1.41 1.38 1.02

3 Triangle√

7/3 = 1.52 1.61 0.953 Linear

√11/3 = 1.91 1.71 1.12

4 Tetrahedron√

5/2 = 1.58 1.77 0.894 Square

√3= 1.73 1.82 0.95

4 Linear√

6= 2.45 2.00 1.226 Octahedron

√3= 1.73 2.02 0.86

8 Cube√

4= 2.00 2.31 0.87

mentioned that our approach differs from the one giveRef. [43] in that (1) our spheres are hollow, (2) we do nuse the Kirkwood approximation forRh, and (3) our clus-ters are not on a cubic lattice.

It can be seen that the ratiosρm = Rg/Rh of these clusterare close to unity in all cases. They are somewhat smfor compact clusters and somewhat larger for extended

ters. Since the actual samples consist of mixtures of clubuilt from polydisperse spheres in irregular arrangemeit can be concluded that any observed ratioρm

∼= 1 ± 15%is reasonably consistent with a mixture of different clustIt should also be noted that, for broad distributions, a cparison of averaged quantities with different weightingsquickly lead to discrepancies of this magnitude.

Furthermore, the individual vesicles inside the clusmay consist of one or more bilayers, accommodating a wrange of possible areas per molecule.

4. Conclusions

The aggregation behavior of C60Me5K in water was in-vestigated by static and dynamic light scattering, reveaa transition (reaggregation) to larger aggregates whichnot simple spherical bilayers, which was not observedC60Ph5K [21]. Furthermore, the size of the single bilayvesicles was found to be considerably larger for C60Me5K(radiusR ∼ 27 nm and aggregation numberN ∼ 2.1× 104)vs C60Ph5K (R ∼ 17 nm,N ∼ 1.2 × 104). It is not obvi-ous why the smaller methyl group should apparently inda larger radius of curvature than the phenyl group.

The aggregates formed above the critical reaggregatioconcentration (CRC) could be multilayer vesicles, as emated from the light-scattering data, which are compatwith transmission electron micrographs at moderate ccentrations above the CRC. At higher concentrations, Tpictures show a mixture of clusters built from spherical vecles in various sizes and geometries. Although the manyparameters inherent to these clusters make a definitivement difficult, they were found to be in agreement withlight-scattering data on the basis of idealized models wwell-defined geometries and monodisperse componentsaggregation into clusters is surprising and suggests tharearrangement of the surfactants and especially of the cterions results in an electrostatic attraction of the vesicle

Double bilayer membranes and clustered vesicles arten found in cellular compartments, such as the nuclmitochondria, and Golgi, and the functions of each compar


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ment are elegantly controlled by these complex supramcular structures[1]. The present study shows that the aficial membranes ofπ -rich molecules can generate similcomplex structures. Further study on the geometrical trformations of fullerene vesicles may give us some hintthe understanding of natural systems and to the construof functional nanoparticles possessing double bilayer mbranes.

Acknowledgments

Acknowledgment is made to the Donors of the Amecan Chemical Society Petroleum Research Fund, for pasupport of this research. This work was further suppoby the US Department of Energy (DEFG0286ER45237.0and the National Science Foundation (DMR9984102). Eacknowledges support from the Ministry of Education, Cture, Sports, Science and Technology (Grant-in-Aid for Sentific Research, Specially Promoted Research). We thanMr. N. Nagahama for initial experimental work.

Appendix A. Polydispersity effects

There has been some confusion as to whether the cparison ofRg andRh employed throughoutthis paper is stillvalid in the polydisperse case sinceRg andRh are based ondifferent averages over the size distribution. It is the purpof this appendix to shed light on this question.

Starting fromEq. (5), we note that the light-scatterinintensity per unit volume of a monodisperse solution ofgregates (or polymers) is proportional to the weight conctrationc of the aggregate/polymer (in mass per volume)to the molecular weightM of the aggregate (polymer). Using a number concentrationcM (in moles per volume) resultin a proportionality toM2,

(A.1)I ∼ cMI0 ∼ cMM2I0.

HereI0 is O(1) in c or M, i.e., varies only via higher viriacoefficients. For a polydisperse system, we consider paconcentrations,

(A.2)cM ∼ ctotpM(M),

wherectot is the total number concentration andpM(M) isthe molecular weight number distribution, i.e., a normized probability density distribution with the property thpM(M)dM describes the probability that a given moleular weight lies betweenM and M + dM. The averag-ing is carried out by summing over allM, constituting az-average:

(A.3)I =∫ ∞

0 pM(M)M2I0 dM∫ ∞0 pM(M)M2 dM

= 〈I0〉z.

This type of averaging applies to the intensity itself, as was to the terms of its series expansion, e.g.,R2

g .

-

Since we are ultimately interested in the comparisonaverages of radii (of gyration and hydrodynamic), weformulate(A.3) using a radius number distributionp(R).We postulate that our aggregates consist of hollow sphcal vesicles of radiusR, so that their molecular weight iproportional to the area of the spherical shell,M ∼ R2. To-gether withdM ∼ R dR, we can rewrite(A.3) as

(A.4)I =∫ ∞

0 p(R)R5I0 dR∫ ∞0 p(R)R5 dR

= 〈R5I0〉〈R5〉 ,

where here and below a not further qualified average〈. . .〉is understood to be taken overp(R). It should be notedthat a generalization of(A.4) to particles with different scaling behavior such as solid spheres (M ∼ R3) or even arbi-trary exponents (M ∼ Rα , polymer coils, mass fractals)straightforward.

The proper average for the radius of gyration as demined by static light scattering can now be deduced at oFor a spherical shell of radiusR in the limit of small shellthickness, we haveRg = R, so that the root mean square rdius of gyration is given by

(A.5)“Rg” = (R2

g

)1/2 = 〈R7〉1/2

〈R5〉1/2.

For dynamic light scattering, let us consider the integkernel ofEq. (2),

(A.6)exp(−Γ τ) = exp(−aτ/R),

where we introduced a constanta = Γ R defined based o(3) and(4). Averaging(A.6) overp(R) and backsubstitutingto Γ we find

〈R5 exp(−aτ/R)〉〈R5〉 = 〈R5〉−1

∞∫0

p(R)R5 exp(−aτ/R)dR

(A.7)= 〈R5〉−1a6

∞∫0

Γ −7p(a/Γ )exp(−Γ τ) dΓ.

A comparison ofEqs. (2) and (A.7)shows that the relationship between the characteristic linewidth distributionG(Γ )

(as experimentally determined by theCONTIN inversion) andthe radius number distributionp(R) for spherical shells isgiven by

(A.8)G(Γ ) = 〈R5〉−1a6Γ −7p(a/Γ ).

Based on(A.8), the moments of the distributionG(Γ ) canbe readily related to those of the radius number distribup(R).

(A.9)〈Γ n〉 =∞∫

0

Γ nG(Γ )dΓ = an 〈R5−n〉〈R5〉 .

Accordingly, an averaged apparent hydrodynamic radiusRh

based on〈Γ 〉−1 is given by

(A.10)Rh = 〈R5〉4 .
〈R 〉


we

en-tionityd

esenra-

enet inbe

rvedat-tual

nin-on-ral

ce

ier

t-e of

ion,

61.e,

..

shi,)

.

ura,

Ru-

uge,

ka-

ew.

tt.

37

ng,

ker,

y-

3–

i,

obe,

noh,947.

,

11

ms-

For the experimentally determined polydispersity index,find

(A.11)〈Γ 2〉〈Γ 〉2

− 1 = 〈R5〉〈R3〉〈R4〉2

− 1.

UsingEqs. (A.5) and (A.10), the ratioRg/Rh in the poly-disperse case is given by

(A.12)Rg

Rh

= 〈R7〉1/2〈R4〉〈R5〉3/2 .

The treatment presented so far is completely indepdent of the actual shape of the radius number distribup(R). In order to quantitatively estimate the polydisperseffect on the ratioRg/Rh, explicit distribution shapes neeto be considered, which goes beyond the scope of the pranalysis. It can be estimated that the deviation of thetio Rg/Rh from unity is small for sufficiently narrow sizdistributions, and that it will grow to values larger than owith increasing polydispersity as the higher order momenthe numerator will dominate the quotient. It should alsonoted that there is a contribution to the apparently obseG(Γ ) distribution width due to experimental and mathemical artifacts that leads to an overestimation of the acpolydispersity effects.

Appendix B. Radius of gyration of regular clusters ofhollow spheres

The spherically averaged scattering intensity of a noteracting (“dilute”) system of monodisperse particles csisting of identical rigid arrangements of isotropic structuunits is given by the following form of Debye’s formula,

(B.1)I (q) = 1

N2

∣∣f (q)∣∣2 N∑

m=1

N∑n=1

sin(qrmn)

qrmn

,

wherermn = |rm − rn| is the absolute value of the differenvector of the two position vectorsrm and rn of the struc-tural units with indicesm andn, respectively, and|f (q)|2 isthe “form factor” of the structural unit (squared 3D Fourtransform of its density distribution).Equation (B.1)holdsfor static light scattering (below the Mie limit), X-ray scatering, and neutron scattering, provided the relevant typdensity is identified properly.

For hollow spheres of radiusR with infinitely thin shells,we insert the form factor|f (q)|2 = [sin(qR)/(qR)]2 (forsolid spheres it would have been{3(qR)−3[sin(qR) −qR cos(qR)]}2), expand into powers ofq ,

I (q) = 1

N2

[sin(qR)

qR

]2 N∑m=1

N∑n=1

sin(qrmn)

qrmn

= 1−(

R2 + 1

2N2

N∑m=1

N∑n=1

r2mn

)q2

3+ O(q4)

(B.2)≡ 1− R2g

q2+ O(q4),

3

t

and compare equal powers ofq , finding

(B.3)Rg =(

R2 + 1

N2

N∑m=1

N∑n=m+1

r2mn

)1/2

,

where the simplification of the double sum is based onrnn =0 andrmn = rnm. For the actual calculation ofRg for regulargeometric clusters like the tetrahedron shown inFig. 7, con-sisting of monodisperse spheres of radiusR touching eachother, most of the center-to-center distancesrmn are simplemultiples ofR.

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multilayer vesicles and vesicle clusters formed by the fullerene-based surfactant c60(ch3)5k

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