Effects of Temperature, Salt, and Deuterium Oxide on the Self

The effects of salt, temperature, and deuterium oxide on the self-aggregation ofn-tetradecyl-β-D-maltoside. (C14G2) in dilute solution have been inve...
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Langmuir 2005, 21, 1507-1515

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Effects of Temperature, Salt, and Deuterium Oxide on the Self-Aggregation of Alkylglycosides in Dilute Solution. 2. n-Tetradecyl-β-D-maltoside Caroline A. Ericsson,† Olle So¨derman,† Vasil M. Garamus,‡ Magnus Bergstro¨m,§ and Stefan Ulvenlund*,†,| Department of Physical Chemistry 1, Centre of Chemistry and Chemical Engineering, Lund University, P.O. Box 124, S-221 00 Lund, Sweden, GKSS Research Centre, Max Planck Street, D-215 02 Geesthacht, Germany, Department of Pharmacy, Uppsala University, Box 580, S-751 23 Uppsala, Sweden, and AstraZeneca R&D Lund, S-221 87 Lund, Sweden Received September 21, 2004. In Final Form: November 24, 2004 The effects of salt, temperature, and deuterium oxide on the self-aggregation of n-tetradecyl-β-D-maltoside (C14G2) in dilute solution have been investigated by static light scattering, dynamic light scattering (DLS), small-angle neutron scattering (SANS), tensiometry, and capillary viscometry. SANS data show that the micelles can be described as relatively flexible polymer-like micelles with an elliptical cross section, at least at temperatures between 35 and 50 °C. The micelles grow in one dimension with increasing temperature and concentration. DLS and viscometry data suggest that the micelle size reaches a maximum at 60-70 °C. Comparison of DLS data in D2O and H2O shows that the micelles are larger in the former case. The effect of salt on the micelle size was found to follow the Hofmeister series. Thus, at constant salt concentration, the micelle size decreases according to the sequence SO42- > Cl- > NO3- > I- > SCN-, where I- and SCNact as salting-in anions. From tensiometric data, it can be concluded that the temperature effects on micelle morphology do not correlate directly with those on unimer solubility. Rather, the temperature effect on the hydrocarbon chain conformation seems to be decisive for the micelle morphology. At constant temperature, on the other hand, the effect of salt and deuterium isotope is attributable to changes in effective headgroup area, including intermolecular interactions and water of hydration.

Introduction 1

In a previous paper, we reported on the effects of temperature, salt, and deuterium oxide on the selfaggregation of n-nonyl-β-D-glucoside (C9G1). The main conclusions drawn from that work are that C9G1 micelles can be described as relatively stiff, elongated structures with a circular cross section. Upon decreasing temperature, the micelles were found to grow in one dimension, although the critical micelle concentration (cmc) shows a concomitant increase. Similarly, substituting D2O for H2O was found to induce substantial micellar growth, without affecting cmc. This deuterium effect can be rationalized based on the shorter length of the O-D bond, as compared with the O-H one, and the resulting decrease in effective headgroup area.2 From the temperature and deuterium effects, it was therefore concluded that the size of C9G1 micelles is governed primarily by the effective headgroup size (including water of hydration), rather than by unimer solubility. This conclusion was lent further support by studies of salt effects on the micellization. In the present paper, we extend our studies of the interplay between headgroup size, surfactant solubility, and micelle morphology in alkylglycoside systems by investigating the behavior of n-tetradecyl-β-D-maltoside (C14G2). One of the main objectives is to use C14G2 as a model maltoside in order to test to what extent the * To whom correspondence should be addressed. E-mail: [email protected]. Fax: +46-46-337128. † Physical Chemistry 1. ‡ GKSS Research Centre. § Uppsala University. | AstraZeneca R&D Lund. (1) Ericsson, C. A.; So¨derman, O.; Garamus, V. M.; Bergstro¨m, M.; Ulvenlund, S. Langmuir 2004, 20, 1401. (2) Whiddon, C. R.; So¨derman, O. Langmuir 2001, 17, 1803.

conclusions drawn from our previous study of the glucoside C9G1 are applicable to alkylglycosides as a class. The micellization of C14G2 has not been studied in detail previously. However, the qualitative differences between C14G2 and C12G2 micelles have been studied by smallangle neutron scattering (SANS) in the low q range at constant temperature and a concentration of 50 mM. This investigation shows the formation of rodlike micelles for C14G2.3 The literature also provides data on some of the CnG2 congeners with n e 12, where C12G2 is by far the most investigated one. SANS4,5 and small-angle X-ray scattering (SAXS)4 data show that C12G2 micelles form monodisperse ellipsoidal aggregates with an aggregation number in the range of 80-130.4-7 According to fluorescence quenching measurements, viscometry,7 and dynamic light scattering studies,8 the micellar size is independent of the temperature and only shows a slight dependence on salt.5 Zhang et al. found that the packing of C12G2 molecules at the air/water interface is unaffected by the nature of salt, whereas the cmc and surface activity depend on cation and anion character.9 Similarly, C8G2 has been investigated by SANS and SAXS, and the data suggest (3) von Minden, H. M.; Brandenburg, K.; Seydel, U.; Koch, M. H. J.; Garamus, V. M.; Willumeit, R.; Vill, V. Chem. Phys. Lipids 2000, 106, 157. (4) Cecutti, C.; Focher, B.; Perly, B.; Zemb, T. Langmuir 1991, 7, 2580. (5) Bucci, S.; Fagotti, C.; Degiorgio, V.; Piazza, R. Langmuir 1991, 7, 824. (6) Dupuy, C.; Auvray, X.; Petipas, C.; Rico-Lattes, I.; Lattes, A. Langmuir 1997, 13, 3965. (7) Warr, G. G.; Drummond, C. J.; Grieser, F.; Ninham, B. W.; Evans, D. F. J. Phys. Chem. 1986, 90, 4581. (8) Focher, B.; Savelli, G.; Torri, G.; Vecchio, G.; McKenzie, D. C.; Nicoli, D. F.; Bunton, C. A. Chem. Phys. Lett. 1989, 158, 491. (9) Zhang, L.; Somasundaran, P.; Maltesh, C. Langmuir 1996, 12, 2371.

10.1021/la047651j CCC: $30.25 © 2005 American Chemical Society Published on Web 01/19/2005

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that the micelles form spherical aggregates with no temperature, concentration, or deuterium isotope effect.10 The phase diagram of C14G2 has not been published, but the diagrams of the shorter homologues nevertheless provide valuable information about the behavior of maltosides as a class. C12G2 displays a wide micellar phase up to 47 wt %.11 The micellar phase is followed by a hexagonal phase, which extends up to 75 wt %. The boundary between the micellar and hexagonal phase is largely independent of temperature. At higher concentrations, the hexagonal phase is followed by a centered rectangular phase (MR). Upon increasing temperature at constant concentration, the MR phase is followed first by a cubic phase (QR) and then by a lamellar phase (LR). The behavior illustrates the general principle that whereas the phase diagrams of alkylglucosides are rather simple, an increase of the headgroup size favors formation of more complicated ensembles of lyotropic phases.11 The presence of the wide micellar region may imply that the aggregateaggregate interactions are short-ranged, and the vertical phase boundaries in the phase diagram may serve as a proof for small temperature dependence of the morphology of the aggregates.12 Furthermore, the fact that the micellar region is followed by a hexagonal phase indicates that the micelles at high concentrations are threadlike, rather than branched. For short-chain alkylglucosides (n e 9), the micellar region is instead followed by a bicontinuous cubic phase. NMR diffusion data show that there is no discontinuity when going from the micellar region to the cubic phase, which may serve as a proof for similar microstructure in the two phases, at least close to the phase boundary.12 In other words, the phase behavior suggests the presence of branched aggregates in the micellar phase of alkylglucosides and threadlike ones in alkylmaltosides. The insensitivity of maltoside micelles to temperature, salt, and other factors implied by the literature data contrasts with the corresponding results on C9G1. However, comparison between C8G1 and C9G1 shows that temperature, salt, and deuterium have much more dramatic effects on the micelle morphology for the latter surfactant. By analogy, C14G2 would therefore be expected to be more sensitive to these factors than the shorterchained maltosides investigated so far. Detailed understanding of the micellization of longchain alkylglycosides such as C14G2 is also important from an applied point of view. When solubilizing bulky aromatic molecules in micellar systems, it is advantageous to use surfactants comprising long hydrocarbon chains for the obvious reason that a longer hydrocarbon chain results in a larger hydrocarbon region in the micelle. Furthermore, the cmc decreases as the alkyl chain length increases, so that at a given total surfactant concentration, a longer chain surfactant generally has a larger concentration of surfactant in the micellar state.13,14 However, for a given headgroup, increasing hydrophobic chain length usually also results in a rise of the Krafft boundary. Our studies of the solid-state behavior of C14G2 indeed show that its Krafft point is slightly above room temperature (31 °C for the crystalline anhydrate and 26 °C for the crystalline hemihydrate).15 However, amorphous C14G2 can be dis(10) He, L.-Z.; Garamus, V. M.; Funari, S. S.; Malfois, M.; Willumeit, R.; Niemeyer, B. J. Phys. Chem. B 2002, 106, 7596. (11) Auvray, X.; Petipas, C.; Anthore, R.; Rico-Lattes, I.; Lattes, A. Langmuir 1995, 11, 433. (12) Nilsson, F.; So¨derman, O.; Hansson, P.; Johansson, I. Langmuir 1998, 14, 4050. (13) So¨derlind, E.; Wollbratt, M.; von Corswant, C. Int. J. Pharm. 2003, 252, 61. (14) Yalkowsky, S. H. Solubility and Solubilization in Aqueous Media, 1st ed.; Oxford University Press: New York, 1999.

Ericsson et al.

solved in water at room temperature, and the resulting metastable solutions can be stored for extended periods at temperatures below the Krafft temperature without any precipitation. Experimental Details Materials and Sample Preparation. The surfactant n-tetradecyl-β-D-maltoside (C14G2) was purchased from Anatrace Inc. (Maumee, OH) and was of Anagrade quality. Three different batches were used. In all cases, the materials were used without further purification and no batch-to-batch variations were observed. The supplier states a purity of >99.5%, and high performance liquid chromatograms provided by the supplier confirm this claim. Solutions were prepared by simply dissolving carefully weighed amounts of surfactant powder in doubledistilled water, D2O, or salt solutions at room temperature. All concentrations are given in units of grams of solute per liter of solution. The D2O used for dynamic light scattering experiments was purchased from Aldrich (Milwaukee, WI) and had a claimed isotopic purity of 100%. Surface tension, SANS, and static light scattering experiments were conducted using D2O with an isotopic purity of 99%, purchased from Cambridge Isotope Laboratories. The salts, NaCl, NaSCN, Na2SO4, NaI, and NaNO3, were obtained from Merck (Darmstadt, Germany). They were of analytical grade, except NaSCN which was of purum quality. Methods. Dynamic Light Scattering (DLS). The effective hydrodynamic diameter (dH) of surfactant micelles was determined on a Brookhaven ZetaPALS instrument equipped with a laser operating at 532 nm and a thermostated sample cell with a temperature range of 5-75 °C. For comparison, selected systems were also investigated on an ALV/DLS/SLS-5000 compact goniometer system fitted to a diode-pumped solid-state laser from Coherent (532 nm, 400 mW). In both cases, the scattered light was measured at an angle of 90° relative to the primary beam. The two different instruments gave consistent results within the limits of experimental error. DLS measurements at temperatures above 75 °C were performed on a third instrument, a Malvern High Performance Particle Sizer (HPPS). This instrument was equipped with a thermostated sample cell with a temperature range of 8-95 °C and a laser operating at 633 nm. The HPPS instrument differed from the other two in the sense that it measures the scattered light at an angle of 173° relative to the primary beam (“backscattering mode”). All the samples intended for DLS studies were filtered through an Acrodisc filter with a pore size of 0.1 or 0.2 µm prior to measurements. Analysis of DLS data is described elsewhere.1 Since the micelles studied in the present work are elongated objects, the effective hydrodynamic diameter must not be confused with the “real” hydrodynamic diameter of a sphere. Rather, the effective hydrodynamic diameter is to be considered merely as an alternative way to represent the primary DLS data, namely, the diffusion coefficient D. The DLS data did not reveal any bimodality in the micelle size distribution. Rather, the data were found to be consistent with one polydisperse micelle population for all conditions investigated. Tensiometry. A KSV Sigma 70 instrument equipped with a DuNouy ring made of platinum was used to determine the air/ liquid surface tension as a function of surfactant concentration. The probe was temperature-controlled by a circulating water bath. At each measurement, the surface tension was recorded after proper equilibration. The initial drift tended to be fairly substantial, but stable surface tension readings were generally observed after 500 s of equilibration. Capillary Viscometry. The capillary viscometer was of CannonFenske type, fitted in a mantle through which thermostated water was circulated by means of an external water bath. The specific constant (K) of the capillary was 0.00381 mm2/s2. Static Light Scattering (SLS). SLS measurements were performed with a BI-200SM goniometer system connected to a BI-9000AT digital correlator from Brookhaven Instruments and a water-cooled Lexel 95-2 laser with maximum power of 2 W and (15) Ericsson, C. A.; Aaro¨e, C.; So¨derman, O.; Kocherbitov, V.; Ulvenlund, S. To be published.

Self-Aggregation of Tetradecylmaltoside wavelength of 514 nm. The temperature was controlled to within (0.2 K. Experiments were performed at 29 different angles in the range of 15° e θ e 155°, corresponding to q values in the range of 4.26 × 10-4 Å-1 e q e 33.2 × 10-4 Å-1. For each angle, five individual measurements were performed and subsequently averaged. The data were then normalized to absolute scale intensities using toluene as a reference standard. The surfactant was dissolved in D2O. Small-Angle Neutron Scattering. Neutron scattering data were collected on SANS-1 at the Geesthach Neutron Facility GeNF, Geesthacht, Germany. The range of scattering vectors q (q ) 4π sin θ/λ, where 2θ is the scattering angle and λ is the wavelength) from 0.005 to 0.25 Å-1 was covered by four sample-to-detector distances (0.7-9.7 m). In all experiments, the neutron wavelength was 8.5 Å with a wavelength resolution, ∆λ/λ, of 10% (full width at half-maximum value). The samples were enclosed in a quartz cuvette with a path length of 2 mm. To ensure isothermal conditions, the cuvettes were placed in a thermostated sample holder. To enhance the scattering contrast and reduce the incoherent background, D2O was used as the solvent in the SANS studies. The surfactant concentration was 10 or 50 g/L, and data were recorded at 35 and 50 °C. Analysis of SANS and SLS Data. The raw SANS spectra were corrected for backgrounds from the solvent, sample cell, and other sources by conventional procedures. The two-dimensional isotropic scattering spectra were azimuthally averaged, converted to an absolute scale, and corrected for detector efficiency by dividing the incoherent scattering spectra of pure water, which was measured with a 1 mm path length quartz cell. In the subsequent data analysis, corrections were made for instrumental smearing.16,17 For each instrumental setting, the ideal model scattering curves were smeared by the appropriate resolution function when the model scattering intensity was compared with the experimental data by means of least-squares methods. The parameters in the model were optimized by means of conventional least-squares analysis.18 In a final procedure, the static light scattering data were “converted to neutron units” by multiplication by (∆Fm)2/KSLS, where KSLS is contrast for light scattering and (∆Fm)2 is the neutron contrast. Further details of the data analysis are presented in the results section.

Results Temperature Effects on the Micellization in Water. As described in the Introduction, the Krafft temperature of C14G2 has previously been determined to be 31 °C by differential scanning calorimetry (DSC).15 Due to the sluggishness of precipitation at temperatures below the Krafft boundary, DLS and surface tension measurements on solutions of C14G2 can be performed at temperatures as low as 10 °C. Nevertheless, it should be kept in mind that the solutions are metastable at temperatures below 31 °C. The surface tension of C14G2 in water as a function of the surfactant concentration at 10, 20, and 40 °C is displayed in Figure 1. The data suggest that the critical micelle concentration shows a modest increase with increasing temperature (Figure 1, Table 1). The only literature report states a cmc of 0.015 mM (0.0081 g/L) at 20 °C.19 The area per molecule was estimated from the tensiometric data by means of the Gibbs adsorption isotherm (Table 1). The calculations suggest that the area per molecule at the air/water interface increases from 32 ( 4 Å2 to 45 ( 2 Å2 when the temperature is raised from 10 to 40 °C. The values may be compared with the area per headgroup at the air/liquid interface for homologues n-decyl-β-D-maltoside (49 Å2)20 and n-dodecyl-β-D-malto(16) Pedersen, J. S. J. Phys. IV (Paris) Colloq. C8 1993, 3, 491. (17) Pedersen, J. S.; Posselt, D.; Mortensen, K. J. Appl. Cryst. 1990, 23, 321. (18) Bevington, B. R. Data Reduction and Error Analysis for Physical Sciences; McGraw-Hill: New York, 1969. (19) Bo¨cker, T.; Thiem, J. Tenside, Surfactants, Deterg. 1989, 26, 318.

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Figure 1. Tensiometric determination of cmc for C14G2 in H2O at 10 °C (open circles), 20 °C (filled triangles), and 40 °C (open squares). Table 1. Critical Micelle Concentration (cmc), Surface Excess Concentration (Γmax), Molecular Area (A0), and Surface Pressure at the cmc (πcmc) for C14G2 in H2O and D2O at Different Temperatures, as Determined by Tensiometrya Γmax × 103 πcmc cmc T (°C) solvent (g L-1) (mmol m-2) A0 (Å2) (mN m-1) 10 20 20 40

H2O H2O D2O H2O

0.0070 0.0075 0.0075 0.0095

5.3 4.6 5.0 3.7

32 ( 4 36 ( 2 33 ( 1 45 ( 2

38.6 37.5 37.5 37.1

cpp 0.65 0.58 0.63 0.46

a A and Γ 0 max are calculated from the Gibbs adsorption isotherm. The critical packing parameter is calculated from the length of an extended C14 alkyl chain, assuming that A0 represents the headgroup area.

side (50 Å2),21 both determined by tensiometry at 25 °C in 0.1 M NaCl and H2O, respectively. As can be seen from Figure 1, the surface tension at concentrations above the cmc is also deceasing with increasing temperature. However, when compensated for the temperature effect on the surface tension of water (i.e., when expressed as a change of the surface pressure πcmc ) γwater - γcmc), the effect is observed to be small and amounts to a decrease of πcmc from 39 mN/m at 10 °C to 37 mN/m at 40 °C. The effect of temperature on the micelle size of C14G2 in water was studied by means of DLS and capillary viscometry. The DLS measurements reveal that the temperature effect on micellar size is the same as the temperature effect on poly(ethylene glycol)-based surfactants, i.e., the micelle size increases with increasing temperature (Figure 2). However, the data suggest that the size of the C14G2 micelles reaches a maximum at 6070 °C. DLS data also reveal that the micelle size of C14G2 displays a pronounced concentration dependence (Figure 3). The micelles are suggested to grow with increasing concentration, but the rate of growth seems to decrease at higher concentrations, particularly at elevated temperatures. However, as clearly shown by SANS data (see below), there is substantial intermicellar interaction at higher surfactant concentrations, which would be expected to affect DLS data. The data in Figure 3 should therefore be interpreted with considerable care. (20) Rosen, M. J.; Sulthana, S. B. J. Colloid Interface Sci. 2001, 239, 528. (21) Drummond, C. J.; Warr, G. G.; Grieser, F.; Ninham, B. W.; Evans, D. F. J. Phys. Chem. 1985, 89, 2103.

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Figure 2. Effective hydrodynamic diameter vs temperature for H2O and D2O solutions with 10 g/L of C14G2 as determined by means of DLS.

Ericsson et al.

Figure 4. Temperature dependence of the relative viscosity (η/η0) for 10 g/L C14G2 in H2O measured with capillary viscometry. The error bars denote the standard deviation of three measurements.

Figure 3. Concentration dependence of the effective hydrodynamic diameter measured with DLS for C14G2 in H2O. The lines are included merely as a guide to the eye.

The character of micellar solutions was further investigated by capillary viscometry. Solutions of C14G2 in water show a Newtonian behavior up to at least 60 g/L (as determined by rheology; results not shown). Solutions subjected to study by means of capillary viscometry had a concentration of 10 g/L and were thus well within the Newtonian regime. Figure 4 clearly shows that the viscosity is strongly influenced by temperature and closely parallels the DLS results. In particular, the viscometric data lend strong support to the idea that the micelle size actually reaches a maximum at ca. 60 °C. Salt Effects on Micellization in Water. The salt dependence on the micelle size was investigated by means of DLS. Five different sodium salts were investigated with respect to their influence on the effective hydrodynamic diameter of micelles in solutions containing 10 g/L of C14G2 (Figure 5). The effect on the micellar size follows the Hofmeister series,22,23 in the sense that the micelle size decreases according to the sequence SO42- > Cl- > NO3> I- > SCN-. Here, I- and SCN- act as salting-in anions that give rise to a decrease of the micellar size as compared with neat water. Although the salt effect on micelle size is quite substantial, the results illustrate the high salt tolerance of alkylglycosides in the sense that up to 1.5 M salt could be added without inducing any phase separation. (22) Collins, K. D.; Washabaugh, M. W. Q. Rev. Biophys. 1985, 4, 323. (23) Hofmaister, F. Arch. Exp. Pathol. Pharmakol. 1888, 24, 247.

Figure 5. The effect of different sodium salts on the effective micellar hydrodynamic diameter for a 10 g/L solution of C14G2 at 40 °C, as measured with DLS.

Micellization in D2O. As previously observed in other systems of alkylglucosides, the deuterium oxide effect on micelle size is quite large.1,2,24 Consistent with this general observation, DLS data show that a substitution of D2O for H2O causes a large increase in effective hydrodynamic micellar diameter for C14G2 (Figure 2). On the other hand, surface tension measurements in D2O and H2O at 20 °C reveal that the cmc is identical in the two solvents, within the experimental error of the method (Figure 6, Table 1). Small-angle neutron and static light scattering were performed in order to gain further insight into micellar morphology and dynamics. Figure 7 shows the q dependence of measured neutron and static light scattering intensity divided by the concentration of surfactant for micelles [dΣ(q)/dΩ/(c - cmc)] formed in 10 g/L C14G2 solution at 35 °C. In quantitative terms, the scattering curves include information on many important parameters: overall size and mass (i.e. apparent radius of gyration Rg,app and apparent molar mass Mapp) at the low q range, the flexibility (i.e. persistence length, lp) in the intermediate q range, and the local structure at higher q (cross-section radius of gyration RCS,g and the mass per length ML of the cylindrical micelle). The q dependence (24) Zhang, R.; Marone, P. A.; Thiyagarajan, P.; Tiede, D. M. Langmuir 1999, 15, 7510.

Self-Aggregation of Tetradecylmaltoside

Langmuir, Vol. 21, No. 4, 2005 1511 Table 2. Cross-Section Radius of Gyration, RCS,g, and Mass per Unit Length, ML, Obtained by the Indirect Fourier Transformation Method for Polymer-like Micelles Formed by C14G2 in D2Oa concn (g L-1)

T (°C)

RCS,g (Å)

ML (10-13 g cm-1)

Rg,app (Å)

Mapp (105 g mol-1)

10.1 50.4 10.1 50.4

35 35 50 50

15 ( 1 15 ( 1 15 ( 1 15 ( 1

1.26 1.25 1.16 1.15

280 ( 30 190 ( 10 300 ( 30 180 ( 10

9.8 ( 1.4 4.6 ( 0.3 11.4 ( 1.6 4.1 ( 0.2

a Also given are the apparent radius of gyration, R g,app, and apparent molar mass, Mapp, from the analysis of the low q part.

Figure 6. Surface tension vs surfactant concentration at 20 °C for C14G2 in H2O (filled circles) and D2O (open circles).

Figure 8. The cross-section distance distribution function p˜ CS(r) obtained for a 10 g/L solution of C14G2 in D2O at 35 °C.

Figure 7. SLS (open squares) and SANS (filled squares) data for a 10 g/L C14G2 solution in D2O at 35 °C. The solid line is the theoretical model for polymer-like micelles with an elliptical cross section.

for 10 g/L solutions of C14G2 is similar to that observed, for example, for inverse micelles of lecithin in isooctane25 and for the nonionic surfactant hexaethylene glycol monohexadecyl ether (C16E6) in D2O.26 At low q, there is a typical Guinier behavior originating from the overall size of the micelles. At higher q a power-law behavior characteristic for self-avoiding chains is observed. This power-law scattering crosses over to a 1/q behavior at scattering vectors where the local rodlike behavior is observed. At the highest q, one observes a cross-section Guinier behavior due to the finite cross-section radius of the micelles. At 10 g/L of C14G2, one thus observes the scattering which is expected for individual well-separated micelles. At higher concentrations, the micelles are entangled and this results in a decrease of the forward scattering (q ) 0) due to the reduced concentration fluctuations in this region (result not shown). Concomitantly, the Guinier behavior moves to a higher q value, which demonstrates that the apparent radius of gyration becomes smaller (see Table 2). One can extract information on the local structure of the micelles by applying the indirect Fourier transforma(25) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Phys. Rev. E. 1997, 56, 5772. (26) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Langmuir 1998, 14, 6013.

tion (IFT)27 to the experimental data from the high q range. On the length scales where the scattered intensity is controlled by the local stiffness of the micelles, it is possible to decouple the scattering into two contributions: one that originates from the overall chain structure and another that reflects the cross-section structure.28 The asymptotic behavior of the scattering function for q . 1/Rg (where Rg is the radius of gyration of the cylindrical micelles) can be expressed as

dΣ(q)/dΩ )

(πq)2π∫

∞

0

p˜ CS(r)J0(qr)r dr )

(πq)I

CS(q)

(1)

where J0 is the zeroth-order Bessel function and ICS(q) is the cross-section scattering intensity. The normalized cross-section distance distribution function p˜ CS(r) is given by

p˜ CS(r) )

c - cmc 2πML

∫∆F(r′)∆F(r + r′) dr′

(2)

where the vectors r and r′ are lying in the cross-section plane.29 We obtained an estimate of the distance distribution function p˜ CS(r) by applying the IFT method. Experimental data and fitted curves coincide perfectly within the fit range of q > 0.03 Å-1. The distance distribution function exhibits a nonsymmetrical shape that is characteristic of some elliptical cross section of polymer-like micelles, and we obtained a first estimate of the crosssection diameter from the maximum distance of p˜ CS(r), which is approximately 50 Å (Figure 8). From p˜ CS(r) we can calculate integral parameters of the micellar cross section30 such as the mass per length ML

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Ericsson et al.

and the cross-section radius of gyration RCS,g, which is given by

RCS,g )

[

]

∫0∞ r2p˜ CS(r) dr 1/2 ∞ 2∫0 p˜ CS(r) dr

(3)

The cross-section forward scattered intensity ICS(0) is given by

ICS(0) ) 2π

∫0∞ p˜ CS(r) dr ICS(0) ∆Fm2

(5)

(6)

where x ) q2Rg,app2. Using this, the scattering intensity is given by

dΣ(q)/dΩ/(c - cmc) ) Mapp∆Fm2 fDebye((qRg,app)2) (7) The experimental results for the concentration dependence of the apparent molar mass and radius of gyration are shown in Table 2. The concentration dependence of Mapp can be analyzed by considering both a concentration dependence of the micellar mass given by a power law in the form M ∼ cR as well as the intermicellar interaction described by the results from conformation space renormalization-group theory originally developed for semidilute polymer solutions.32 In the limit of q f 0 the scattered intensity is related to the molar mass M through

dΣ(q)/dΩ/(c - cmc)/∆Fm2 ) MS(0) ) Mapp

10.1 50.4 10.1 50.4

35 35 50 50

1.3 2.9 1.6 3.6

L (Å)

b (Å)

A (Å)

B (Å)

Rg (Å)

1700 ( 100 3800 ( 100 2300 ( 100 5200 ( 200

650 ( 20 650 (fixed) 700 ( 20 700 (fixed)

16 ( 1 16 ( 1 16 ( 1 16 ( 1

26 ( 1 26 ( 1 25 ( 1 25 ( 1

340 600 430 750

An explicit functional form for S(0) has been calculated using the renormalization-group method:35

The values obtained by the analysis are presented in Table 2. The apparent radius of gyration Rg,app and apparent molar mass Mapp were obtained from the scattering data by fitting the lowest q range,