Growth Behavior of Mixed Wormlike Micelles: a Small-Angle

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Langmuir 2003, 19, 4096-4104

Growth Behavior of Mixed Wormlike Micelles: a Small-Angle Scattering Study of the Lecithin-Bile Salt System Lise Arleth* Danish Polymer Centre, Risø National Laboratory, Frederiksborgvej 399, DK-4000 Roskilde, Denmark

Rogert Bauer and Lars Holm Øgendal Department of Mathematics and Physics, Royal Veterinary and Agricultural University, Bu¨ lowsvej 17, 1870 Frederiksberg C, Denmark

Stefan U. Egelhaaf Department of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom

Peter Schurtenberger Physics Department, University of Fribourg, Pe´ rolles, 1700 Fribourg, Switzerland

Jan Skov Pedersen Department of Chemistry, Aarhus University, Langelandsgade 140, DK-8000 Aarhus C, Denmark Received November 6, 2002. In Final Form: January 29, 2003 Aqueous mixtures of egg-yolk lecithin and the bile salt glycochenodeoxycholic acid sodium salt are studied using small-angle neutron scattering. Upon dilution, the shape and size of the aggregates change dramatically. This is due to very different critical micellar concentrations and spontaneous curvatures of lecithin and bile salt. At high concentrations, cylindrical micelles with a length of a few hundred angstroms are formed. As the samples are diluted, the length of the micelles first decreases and then increases by a factor of 3, their flexibility becomes noticeable, and the micelles can be described as semiflexible cylindrical micelles, also known as wormlike micelles. We have developed a mathematical model for the scattering of the wormlike micelles, which takes into account the intermicellar interaction effects. By the simultaneous fitting of the scattering data from a range of concentrations, the concentration-dependent growth law of the micelles can be parametrized. The obtained growth law of the mixed micelles is compared to the growth laws observed in simple micellar systems.

1. Introduction The main constituents of human bile are lecithin, different bile salts [taurocholate (TC), taurochenodeoxycholate (TCDC), glycocholate (GC), and glycochenodeoxycholate (GCDC)], and cholesterol.1 The overall composition of the bile varies largely from individual to individual as well as over time.1,2 Simple model systems for human bile consisting of aqueous solutions of lecithin and a single type of bile salt have been the object of several experimental studies over the past decades.3-10 In dilute * Corresponding author. Present address: Manuel Lujan Jr. Neutron Scattering Center, Los Alamos National Laboratory, Mail Stop H805, Los Alamos, NM 87545. E-mail: [email protected]. (1) Zubay, G. Biochemistry; Addison-Wesley: Reading, MA, 1983. (2) Hedenborg, G.; Norman, A. Scand. J. Clin. Lab. Invest. 1985, 45, 151-156. Hedenborg, G.; Norlander, A.; Norman, A. Scand. J. Clin. Lab. Invest. 1985, 45, 157-164. (3) Duane, W. C. Biochem. Biophys. Res. Commun. 1977, 74 (1), 223229. (4) Mazer, N. A.; Benedek, G. B.; Carey, M. C. Biochemistry 1980, 19 (4), 601-615. (5) Schurtenberger, P.; Mazer, N.; Ka¨nzig, W. J. Phys. Chem. 1985, 89, 1042-1049.

aqueous solutions, the lecithin and the bile salt selfassemble into micelles or vesicles depending on the composition of the sample. There are several good reasons for studying such systems. First of all, an understanding of the processes that occur in human bile would be useful in the treatment of disorders such as gallstone formation. Furthermore, the systems of lecithin and bile salt show potential for applications in membrane reconstitution. Finally, there is a general academic interest in the system because it serves as a model system for mixed micellar systems. The critical micellar concentrations (cmc’s) of lecithin and bile salt are very different. For lecithin, the cmc in (6) Hjelm, R. P.; Thiyagarajan, P.; Alkan, H. J. Appl. Crystallogr. 1988, 21, 858-863. (7) Hjelm, R. P.; Thiyagaragan, P.; Sivia, D. S.; Lindner, P.; Alkan, H.; Schwahn, D. Prog. Colloid Polym. Sci. 1990, 81, 225-231. (8) Long, M. A.; Kaler, E. W.; Lee, S. P.; Wignall, G. D. J. Phys. Chem. 1994, 98, 4402-4410. (9) Egelhaaf, S. U.; Schurtenberger, P. J. Phys. Chem. 1994, 98, 85608573. (10) Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. J. Phys. Chem. 1995, 99, 1299-1305.

10.1021/la026808+ CCC: $25.00 © 2003 American Chemical Society Published on Web 04/10/2003

Lecithin-Bile Salt System

aqueous solutions is on the order of 10-10 M,11 whereas for bile salt, the cmc in aqueous solutions is on the order of 10-3 M.12 This has the consequence that the composition of the lecithin-bile salt aggregates changes as the samples are diluted toward the cmc of the bile salt, where an increasing fraction of the bile salt will leave the aggregates. The change in the composition of the aggregates leads to a shape transformation from wormlike micelles formed at high concentrations to bilayer vesicles formed at low concentrations. The length of the micelles is found to increase dramatically as the samples are diluted toward the cmc of the bile salt.6-9 This dilution-induced growth of the micelles is the inverse trend compared to that of simple micellar systems, where a growth of the micelles with increasing concentration is theoretically predicted and experimentally observed.13-16 The growth behavior of the mixed micelles can be understood in terms of the average spontaneous curvature of the micelles. A reduced amount of bile salt in the aggregates leads to a decreased spontaneous curvature of the mixed micelles and a growth of the micelles to avoid the energetically unfavorable highly curved end caps.7,9,14,15,17 In simple micellar systems, surface-tension measurements indicate that the cmc is well-defined (see, e.g., ref 18). Below the cmc, the amphiphilic solution consists mainly of dissolved monomers. Above the cmc, the freemonomer concentration stays about constant at the cmc while the remaining part of the amphiphilic molecules self-assemble into micelles. In mixed micellar systems, the situation is usually more complicated. It can generally not be assumed that the cmc can be determined by simple interpolation between the cmc’s of the single constituents. In many cases, synergism or cooperativity effects will play an important role.19 In the cases where the cmc’s of the constituents are not too different, the cmc of the mixed system as well as the composition of the aggregates and the free-monomer concentration close to the cmc can be estimated theoretically, taking these effects into account. This makes it possible to predict the shape of the resulting aggregates with some certainty (see, e.g., refs 20 and 21 and references therein). However, the lecithin-bile salt system has to be regarded from another point of view. As a result of the extremely large difference between the cmc’s of lecithin and bile salt, the cmc of the lecithin can generally be neglected in an analysis of the system. As the cmc of the bile salt is approached, the composition of the aggregates changes. This results in a change of the spontaneous curvature, which induces a change of the shape and size of the aggregates and, therefore, a change in the mean free energy of a molecule in the aggregates. As a response to this, the free-monomer concentration will change. The free-monomer concentra(11) Tanford, C. The hydrophobic effect: Formation of micelles and biological membranes; John Wiley & Sons: New York, 1980. (12) Small, D. M. The bile acids; Nair, P., Kritchevsky, D., Eds.; Plenum Press: New York, 1971; Vol. 1. (13) Stradner, A.; Glatter, O.; Schurtenberger, P. Langmuir 2000, 16, 5354-5364. (14) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869-6892. (15) Israelachvili, J.; Mitchell, J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525-1568. (16) Missel, P. J.; Mazer, N. A.; Benedek, G. B.; Young, C. Y.; Carey, M. C. J. Phys. Chem. 1980, 84, 1044-1057. (17) Mitchell, J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1981, 77, 601-629. (18) Chevalier, Y.; Zemb, T. Rep. Prog. Phys. 1990, 53 (3), 279-371. (19) Shiloach, A.; Blanckschtein, D. Langmuir 1997, 13 (15), 39683981. (20) Puvvada, S.; Blanckschtein, D. J. Phys. Chem. 1992, 96, 55675579. (21) Shiloach, A.; Blanckschtein, D. Langmuir 1998, 14 (7), 16181636.

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tion is, therefore, coupled to the overall concentration and composition of the sample. Instead of having a constant free-monomer concentration of the bile salt, there will be an equilibrium between the bile salt present in the aggregates and the free-monomer concentration. As the samples are diluted, this equilibrium is shifted toward a larger and larger fraction of the bile salt present as monomers. It is, thus, important to keep in mind that the free-monomer concentration of the bile salt in the mixed lecithin-bile salt system depends on the lecithin concentration and, in particular, that it is generally different from the cmc observed in a pure bile-salt-inwater system.3 In previous small-angle neutron scattering (SANS) and light scattering studies performed by some of us, we have shown that the scattering data from wormlike micelles can be analyzed in terms of a model for semiflexible cylindrical micelles with a core formed by the hydrocarbon chains of the lecithin and a shell consisting of bile salt and the hydrophilic headgroups of the lecithin.10 In the past few years, theory, models, and methodology based on the results of polymer physics and Monte Carlo simulations have made it possible to give a full interpretation of the scattering data from more concentrated solutions of wormlike micelles, where the intermicellar effects play a significant role.10,13,22-27 These newly developed methods allow for a more quantitative study of the growth law of the lecithin-bile salt wormlike micelles than those developed previously. In the present work, we will study the lecithin-GCDC system in a concentration range where cylindrical micelles are formed and a change of the micelles can clearly be observed. The intermicellar interactions and flexibility of the micelles are taken into account in the data analysis. When the scattering data from all concentrations are fitted simultaneously, a parametrization of the growth law for the micelles is obtained. 2. Experimental Section Egg-yolk lecithin was obtained from Nutfield Nurseries. The lecithin comes in ampules containing 500 mg of lecithin dissolved in methanol/chloroform. GCDC sodium salt was obtained from Fluka. The tris(hydroxymethyl)amino methane for the Tris buffer solution was also obtained from Fluka. The Tris buffer solution contains 0.05 M tris(hydroxymethyl)amino methane and 0.10 M NaCl in D2O. According to the manufacturer, this gives a pH of 8.15 at 25 °C. We use the following procedure for preparing the samples: The bile salt is dissolved in ethanol and mixed with the lecithinmethanol/chloroform solution. The mixture is evaporated without heating to obtain a film of the amphiphiles on the inside of a flask. The buffer solution is added to obtain a stock solution. The flask is flushed with nitrogen, sealed, and left at room temperature for 24 h. The stock solution is diluted to the desired concentration, and the obtained sample is flushed with nitrogen, sealed, and left at room temperature for 48 h. Then, the samples are ready and are used within 3 days. The stock solutions contained 0.064 M lecithin and 0.058 M GCDC, the lecithin to bile salt molar ratio was, thus, close to 1.1. The molar masses, M, molecular volumes, V, and scattering-length densities, F, of (22) Pedersen, J. S.; Schurtenberger, P. Macromolecules 1996, 29, 7602-7612. (23) Pedersen, J. S.; Schurtenberger, P. Europhys. Lett. 1999, 45 (6), 666-672. (24) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Phys. Rev. E: Stat. Phys. Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 56 (5), 5772-5788. (25) Garamus, V.; Pedersen, J. S.; Kawasaki, H.; Maeda, H. Langmuir 2000, 16 (16), 6431-6437. (26) Magid, L. J.; Li, Z.; Butler, P. D. Langmuir 2000, 16, 1002810036. (27) Arleth, L.; Pedersen, J. S. Langmuir 2002, 18, 5343-5353.

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Table 1. Molecular Mass, M, Molecular Volume, V, and Scattering-Length Density, G, of the Constituents sample

M (g/mol)

V (Å3)

F (cm-2)

lecithin, polar part lecithin, apolar part GCDC D2O D2O Tris buffer lecithin-GCDC micelles

336 434 471.6 20

377 889 600 30

1.57 × 1010 -0.289 × 1010 0.964 × 1010 6.38 × 1010 7.32 × 1010 a 0.48 × 1010

aValue

determined via the IFT.

the constituents are given in Table 1. The buffer solution used as the solvent for the micelles makes sure that the net charge at the surface of the aggregates is well-defined and sufficiently screened such that the electrostatic effects can be neglected in the later treatment of the system. Prior to the SANS experiments, light scattering measurements were made at the facility in the Department of Mathematics and Physics at the Royal Veterinary and Agricultural University in Denmark28 to determine the position of the micelle-to-vesicle transition and test the reproducibility of the sample preparations. The measurements showed that the position of the micelle-tovesicle transition is well-reproducible. The intensity of the scattering is also well-reproducible in the concentration range where micelles and wormlike micelles are formed. However, in the transition range and range where vesicles are formed, the intensity is only reproducible within 20%. This indicates that either the system is not in equilibrium or that the system is extremely sensitive to the sample preparation in this concentration range. In the present study, we will concentrate on the range where micelles and wormlike micelles are formed. SANS measurements were made on the SANS instrument at Risø National Laboratory in Denmark.29,30 This instrument has now been installed at the Paul Scherrer Institute in Switzerland. When different combinations of neutron wavelengths and sampleto-detector distances were used, a scattering-vector range from 0.003 to 0.53 Å-1 was covered. The scattering vector, q, is defined by q ) (4π/λ) sin θ, where λ is the wavelength of the incident neutrons and 2θ is the scattering angle. The wavelength spread was 24% (full width at half-maximum), as was determined by the broadening of the Bragg peaks from a silver behenate sample. The SANS data were azimuthally averaged and normalized by the standard approach by division of the scattering spectrum of H2O.31 Measurements of the buffer solutions are used for background subtractions. Instrumental resolution effects are taken into account in the data analysis by convolution by the appropriate resolution function at each setting.32 All measurements were performed at 25 °C.

3. Results and Analysis The scattering spectra obtained from measurements on the dilution series of the lecithin-GCDC system are shown in Figures 1 and 2. The monotonically decreasing scattering curve observed at high concentrations (Figure 1) is consistent with the presence of wormlike micelles. At low concentrations, vesicles are formed (Figure 2). The vesicles lead to an oscillatory behavior of the scattering curve, and the transition from wormlike micelles to vesicles is, therefore, easily recognized. In the present article, we will concentrate on the analysis of the scattering data from the wormlike micelles in Figure 1. 3.1. Indirect Fourier Transform (IFT) of the Cross Section of the Micelles. The cross-sectional structure of the wormlike micelles can be investigated in a model(28) Bauer, R.; Hansen, M.; Hansen, S.; Øgendal, L.; Lomholt, S.; Qvist, K.; Horne, D. J. Chem. Phys. 1995, 103, 2725-2736. (29) Mortensen, K. Nukleonika 1994, 39 (1-2), 169-184. (30) Pedersen, J. S. In Modern Aspects of Small Angle Scattering; Brumberger, H., Ed.; Klu¨wer Academic Publishers: Norwell, MA, 1995. (31) Jacrot, B. Rep. Prog. Phys. 1976, 39, 911-953. (32) Pedersen, J. S.; Posselt, D.; Mortensen, K. J. Appl. Crystallogr. 1990, 23, 321-333.

Figure 1. SANS measurements of a dilution series with lecithin-GCDC. In the plot, each data set is multiplied by 4n, where n runs from -3 to +4 starting from the lower-most spectrum. The sample volume fractions corresponding to the data are listed to the left of the data. The results of the simultaneous fit with the model for wormlike micelles are shown as the lines.

independent way by performing an IFT33 of the high-q part of the data. This provides the cross-sectional pairdistance distribution function, pCS(r), which contains information about the cross-sectional radius of gyration and scattering-length density of the cross section of the micelles. The pCS(r) of one of the samples (total volume fraction of lecithin-GCDC, φ ) 0.0090) is shown in Figure 3. It exhibits a shape that is typical for a locally rodlike structure with a homogeneous scattering-length density. This is in good agreement with the fact that the excess scattering-length densities of the hydrophobic core and hydrophilic shell only differ by 30-40%, with the highest excess scattering-length density in the core of the micelles (see Table 1). The tail of the pCS(r) at high r values (30-45 Å) indicate that the cross section of the micelles is either slightly elliptical or slightly polydispersed. The mean square radius of gyration of the cross section of the micelles can be calculated from the pCS(r) using the equation 2

RG

∫0Rr2pCS(r) dr ) ∫0RpCS(r) dr

(1)

where R denotes the outer radius of the cross section. From the scattering data, we obtain RG ) 13.76 ( 0.25 Å. For a circular cross section of homogeneous scattering-length density, the radius of gyration is related to the mean outer radius by R h ) x2RG, giving R h ) 19.46 ( 0.35 Å. (33) Glatter, O. J. Appl. Crystallogr. 1977, 10, 415-421.

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Figure 2. Data points same as those in Figure 1 but for lower concentrations where vesicles are formed.

The cross-sectional forward-scattered intensity ICS(0) is given by (see, e.g., ref 34)

ICS(0) ) 2π

∫0∞pCS(r) dr

(2)

and is related to the volume fraction, φ, the excess scattering-length density, ∆F, and the cross-sectional area of the rods, ACS, via

ICS(0)/φ ) ACS∆F2

(3)

Figure 3. (A) Scattering of the micelles in the lecithin-GCDC sample with φ ) 0.0090 and (B) cross-sectional IFT. The minimum q value used in the IFT is 0.06 Å-1, and a cylindrical geometry of the micelles is assumed. The fit to I(q) corresponding to this pCS(r) is shown in part A (line). Table 2. Cross-Sectional Radii of Gyration, RG, Determined by the Cross-Sectional IFT of the Scattering Spectra for Different Volume Fractions, O φ

For the above-mentioned sample, integration over pCS(r) leads to ICS(0)/φ ) 5.01 ((0.5) × 108 cm-2. When ACS ) πR h 2 ) 1190 ( 40 Å2, we obtain |∆F| ) 6.50 ((0.5) × 1010 cm-2. The average excess scattering-length density of the micelles, as is calculated from the difference between the average scattering-length densities of the micelle and pure D2O, is |∆FD2O| ) 5.62 × 1010 cm-2. The different values for the average excess scattering-length density suggest that the D2O Tris buffer solution has a scattering-length density somewhat larger than that of pure D2O. However, the difference may also partly be explained by a higher counterion density near the surface of the micelles, as this factor will also increase the apparent value of |∆F|. An effective FTris can be estimated from the value of |∆F|. We obtain FTris ) 7.32 ((0.6) × 1010 cm-2, which is used in the remaining part of the analysis. Similar IFT analyses have been carried out for the remaining samples with similar results. The obtained cross-sectional radii of gyration are given in Table 2. There is a weak tendency for a decrease of RG as the samples are diluted toward the cylinder-vesicle transition. However, (34) Glatter, O. In Small-angle X-ray scattering; Glatter, O., Kratky, O., Eds.; Academic Press: New York, 1982; Chapter 4.

RG (Å)

φ

RG (Å)

φ

RG (Å)

0.033 14.20 ( 0.25 0.012 13.76 ( 0.25 0.0045 13.43 ( 0.25 0.024 14.05 ( 0.25 0.0090 13.57 ( 0.25 0.0034 13.59 ( 0.25 0.016 13.96 ( 0.25 0.0067 13.55 ( 0.25

the tendency is only weak, and for the remaining part of the analysis, we will assume that the dimensions of the cross section of the micelles do not change with concentration within the concentration range where wormlike micelles are formed. 3.2. Structure of Single Micelles. A model for the scattering from the micelles that includes molecular constraints and mass conservation was used to fit the experimental data: For the modeling of the single micelles, we assume that the polar part is constituted by the lecithin headgroups and bile salt molecules, whereas the apolar part is constituted by the alkyl chains of the lecithin molecules. We used the molecular volumes given in Table 1 for the calculation of the volume of the single micelles. The scattering-length density of the micelles is calculated from the composition of the micelles, and the scatteringlength densities of the single constituents are given in Table 1. For the calculation of the scattering-length density profiles, it is assumed that the bile salt is randomly distributed in the polar part of the micelles and that it is

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impossible to distinguish single bile salt molecules. The free-monomer concentration of the bile salt is taken into account by assuming that it is constant and equal to the so-called intermicellar concentration (imc), represented by φBS,imc, which has been measured by Schurtenberger et al. using a combination of quasi-elastic light scattering and equilibrium dialysis.5 The imc of GCDC used in the data analysis is 0.65 mM,5 which corresponds to a volume fraction of approximately 0.000 24. The volume fraction of bile salt in the micelles, φBS,mic, is then given by

φBS,mic ) φBS - φBS,imc

(4)

where φBS denotes the total volume fraction of the bile salt in the sample. This implies that the modeled excess scattering-length density of the shell decreases toward the excess scattering length of the polar part of the lecithin as φBS,imc is approached. In agreement with the previous work on other systems of wormlike micelles,10,25,27,35 we have assumed that the micelles can be described as semiflexible cylindrical micelles with an elliptical cross section. The total volume of a single micelle is, thus, given by

Vmic ) πR2L

(5)

where R is the minor axis of the outer shell of the micelles, L is the length of the micelles, and  is the axis ratio of the cross section of the micelles. The mean radius of the cross section, R h , is calculated by R h ) xR2. When the micelles are sufficiently long, L > 10R, the scattering from the single wormlike micelles is wellapproximated by an expression that separates F(q) into a cross-sectional contribution, FCS(q), and a longitudinal contribution, FWC(q):

F(q) ) φVmicFCS(q) FWC(q)

(6)

where φ is the total volume fraction of the micelles and Vmic is the volume of the single micelles. Note that FWC(q) is normalized to unity at q ) 0, whereas FCS(q) is weighted by the excess scattering lengths of the core and shell of the micelles. FCS(q) is for the micelles with an elliptical cross section and different scattering-length densities of the core and shell, given by (see, e.g., ref 36)

FCS(q) )

[

2J1qRs(, θ)

∫0π/2 (Fs - Fw)

2 π

+ qRs(, θ) 2J1qRc(, θ) Ac (F - Fs) As c qRc(, θ)

]

2

dθ (7)

where Fc, Fs, and Fw are the scattering-length densities of the core, shell, and buffer, respectively. Rc and Rs are the minor axes of the apolar core and polar shell, respectively, and  denotes the axis ratio. The cross-sectional areas are given by Ac ) πRc2 and As ) πRs2, and R(, θ) is generally given by (R2 sin2 θ + 2R2 cos2 θ)1/2. J1(x) denotes the firstorder Bessel function of the first kind. The integration over θ has to be carried out numerically. A numerical expression for the longitudinal contribution from wormlike micelles is given by Pedersen and Schurtenberger.22 The expression is based on a series of Monte Carlo simulations of semiflexible polymers and wormlike micelles with excluded volume effects. The polymer (35) Bergstro¨m, M.; Pedersen, J. S. Phys. Chem. Chem. Phys. 1999, 1, 4437-4446. (36) Pedersen, J. S. Adv. Colloid Interface Sci. 1997, 70, 171-210.

configurations have been sampled, and the scattering functions have been calculated.22,37 Furthermore, the scattering functions have been parametrized numerically using the procedure by Yoshizaki and Yamakawa.38 This leads to expressions for the scattering functions that can be used in the analysis of SANS data from semiflexible polymers or wormlike micelles:22

FWC(q, L, b) ) {[1 - χ(q, L, b)]Fchain(q, L, b) + χ(q, L, b) Frod(q, L)}Γ(q, L, b) (8) where L is the contour length of the micelles and b is the Kuhn length. Fchain(q, L, b) denotes the scattering function for a flexible chain with excluded volume effects, which is given by an empirical expression (eq 13 in ref 22). Frod(q, L) denotes the scattering function of an infinitely thin rod (eqs 6 and 7 in ref 22). The mathematical expression for Frod(q, L) was originally derived by Neugebauer.39 At low angles, Fchain(q, L, b) dominates the scattering pattern, and at higher angles, Frod(q, L) dominates. In the crossover region, the scattering is given by a combination of Fchain(q, L, b) and Frod(q, L, b). In the expression, χ(q, L, b) is a crossover function and Γ(q, L, b) is a function that corrects the crossover region.22,38 The polydispersity of the length of the micelles was not included in the present analysis. Multiple chemicalequilibrium theories for cylindrical micelles predict a quite significant polydispersity of the length of the micelles.14 However, previous studies of dense solutions of wormlike micelles conclude that, in the SANS pattern, the effect of the length distribution of the micelles is very small compared to the effects of the intermicellar interactions.25,27 Because these two effects affect a similar q range, the inclusion of a length distribution of the micelles has a vanishing influence on the final model for the scattering pattern. As a result of the different local curvatures along the cylinder axis and around the end caps of the cylinder, we expect the compositions of these two parts of the micelles to differ. Energetically, it will be most favorable to have a relatively high fraction of the bile salt with its high spontaneous curvature in the end caps, while lecithin with its low spontaneous curvature is preferentially located in the cylindrical part of the micelles. However, with the present contrast the excess scattering-length densities of the bile salt and lecithin are very similar, and the effect of a nonhomogeneous distribution of lecithin and bile salt is not observable from the scattering pattern. Rounded end caps could be included in the model and would provide a more physical description. However, because the micelles are relatively long, the effect of sharply cut-off cylinder ends instead of rounded ends is not significant in the overall scattering pattern, and we, thus, use the more simple model of (flexible) cylinders with sharply cut-off ends. 3.3. Intermicellar Interactions. In a previous study of sodium dodecyl sulfate (SDS) wormlike micelles in aqueous NaBr solutions,27 we concluded that the intermicellar interactions were best taken into account in the modeling using an expression from the Polymer Reference Interaction Site Model (PRISM). This conclusion is in good agreement with the experimental results from other systems.13,25 Because the sample concentrations and (37) Pedersen, J. S.; Laso, M.; Schurtenberger, P. Phys. Rev. E: Stat. Phys. Plasmas, Fluids, Relat. Interdiscip. Top. 1996, 54 (6), 59175920. (38) Yoshizaki, T.; Yamakawa, H. Macromolecules 1980, 13, 15181525. (39) Neugebauer, T. Ann. Phys. (Leipzig) 1943, 42, 509-533.

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micellar lengths in the present study are comparable to the ones in the SDS study, we expect that the PRISM expression will also provide a good description of the intermicellar interactions of our lecithin-bile salt micelles. The PRISM expression calculated in the equivalent site approximation for thin chains23,40 leads to the following expression for the scattering from the wormlike micelles:

Icyl(q) )

F(q) 1 + ν(X) c(q) FWC(q)

(9)

where F(q) is the scattering function of the single chain (eq 6), ν(X) is related to the concentration of the micelles (see below), X refers to the reduced concentration (see below), and c(q) is the normalized Fourier transform of the direct correlation function corresponding to the correlation hole arising along the chain if the chain has a finite radius. It has been empirically found23 that c(q) resembles the form factor of an infinitely thin rod, for which reason we used the following expression in the data analysis:

F(q) Icyl(q) ) XS 1 + ν(X) Frod(q, Lc) FWC(q)

(10)

Lc denotes a length that is related to the correlation length, ξ, of c(q) by Lc ) 6ξ. In the fitting routine, the expression is multiplied by a scale factor, XS, which is expected to be close to unity. We use the explicit form for ν(X) calculated by Ohta and Oono using a renormalization group theory approach:41

ν(X) )

) {[ ( )

(

2 ln(1 + X) 1 1 1 + 9X - 2 + exp 8 X 4 X 1 1 - 2 ln(1 + X) X

]}

(11)

The argument of the exponential function in eq 11 is multiplied by 1/4 in ref 41 and in the eq 11. However, a better agreement with the results of Monte Carlo simulations23 is obtained with 1/2.565 instead of 1/4, so in the present work, we actually used 1/2.565. The reduced concentration, X, is related to the volume fraction, φ, and the osmotic second virial coefficient. It was shown in ref 25 that X can be expressed as

X)

(16/9)B2V3β-1 mic φ

(12)

where B2 is a constant that provides the link between the micellar volume and the osmotic second virial coefficient (see ref 25) and β is the scaling exponent from the scaling law Rg ∝ V βmic, where Rg is the radius of gyration of the micelles. For the fits, the scaling exponent is fixed at the value obtained when the excluded volume effects are taken into account, β ) 0.588.42 3.4. Growth Law for the Micelles. Simple micelles show a concentration-induced growth with L ∝ φR, where R is the growth exponent. Mean field theory predicts R ) 0.5 (see ref 14 and references therein), whereas different values for R have been found experimentally.13,24-27,43,44 However, for mixed micelles such as the lecithin-bile salt (40) Schweizer, K. S.; Curro, J. G. Adv. Polym. Sci. 1994, 116, 319377. (41) Ohta, T.; Oono, Y. Phys. Lett. A 1982, 89, 460-464. (42) Schurtenberger, P.; Cavaco, C. J. Phys. Chem. 1994, 98, 54815486.

micelles, the growth law for the size of the micelles is to our knowledge unknown. To obtain a growth law for the system, we assume that the end caps are made out of bile salt while, along the cylinder axis, both lecithin and bile salt is present at a constant composition. The volume fraction of bile salt, as is expressed by eq 4, is then further decomposed into

φBS ) φBS,cyl + φBS,cap + φBS,imc

(13)

where the first term denotes the total volume fraction of bile salt placed along the cylinder axes, the second term denotes the total volume fraction placed in the end caps, and the third term is the imc. Because a certain number of bile salt molecules is necessary to form a pair of end caps, we obtain

Ncap ∝ φBS,cap

(14)

where Ncap denotes the number density of the end caps. For a given concentration, the average length of the micelles is proportional to the volume fraction of lecithin and bile salt placed along the cylinder axis, divided by the number density of the end caps:

L∝

φBS,cyl + φLec,cyl φBS,cyl + φLec,cyl ∝ Ncap φBS,cap

(15)

Combining eqs 13 and 15, we obtain

L∝

φBS,cyl + φLec,cyl φBS - φBS,cyl - φBS,imc

(16)

We assume that all the lecithin is placed along the cylinder axis, that is, φLec,cyl ) φLec. Furthermore, we assume that the lecithin-bile salt ratio along the cylinder axis is constant such that φBS,cyl ∝ φLec. Because the overall lecithin-bile salt ratio is also constant, these assumptions imply that, above the cylinder-vesicle transition, φBS,cyl ∝ φ, where φ is the total volume fraction of the lecithin and bile salt. Equation 16 can now be transformed into

L(φ) ∝

φ Aφ - φBS,imc

(17)

where A is a positive constant, or

φ L(φ) ) L0 φ - φ0

(18)

where L0 is a positive, constant length and φ0 denotes the onset of the dilution-induced micellar growth and, thus, the micelle-to-vesicle transition. Note that L will approach L0 as φ becomes much higher than φ0. A growth law with the same functional form is obtained if it is assumed that the end caps consist of both lecithin and bile salt, however, at a lower lecithin-bile salt ratio than along the cylinder axis. This simple growth law was incorporated in the expression for the reduced concentration, eq 12, and in the calculation of the form factor of eq 6. The data from the dilution series, Figure 1, were fitted simultaneously with eq 10, assuming that the dimensions of the micellar cross section remain constant throughout the data series while the length of the micelles and the intermicellar (43) Schurtenberger, P.; Cavaco, C. J. Phys. II 1994, 4 (2), 305-317. (44) Schurtenberger, P.; Cavaco, C.; Tiberg, F.; Regev, O. Langmuir 1994, 12 (12), 2894-2899.

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interactions vary according to the growth law and sample composition. The quality of the fits obtained with this growth law was very poor. This asymptotic growth law gives rise to a faster growth in a much more limited concentration range than that which we observe experimentally. To improve the growth law, the model was multiplied by the factor φR:

φ L ) L0 φR φ - φ0

(19)

where R is set equal to 0.5, in agreement with mean field theory.14 In this way, a growth of the micelles with increasing concentration is simulated for φ . φ0. This modification improved the quality of the fits at high concentrations; however, at low concentrations, where the dilution-induced growth dominates, the fit quality was still very poor and the growth law given by eq 19 was dismissed. We, thus, turned to an empirical growth law with a double power law of the form

L ) B1φR + B3φγ

Table 3. Fitting Parametersa for the Fits Shown in Figure 1 φ

L (Å)

XS

φ

L (Å)

XS

0.033 0.024 0.016 0.012

395 ( 40 342 ( 40 300 ( 40 284 ( 40

1.31 1.23 1.16 1.11

0.0090 0.0067 0.0045 0.0034

294 ( 40 344 ( 40 531 ( 40 851 ( 40

1.13 1.12 1.15 1.15

a φ is the volume fraction of the micelles (not fitted), L is the length of the micelles, as is calculated from the fitted growth law, and XS is a fitted scale factor. The remaining fitting parameters are kept common for all concentrations in the simultaneous fitting procedure. The results are as follows: R ) 15.3 ( 0.1 Å, where R is the radius of the minor axis of the cross section of the micelles;  ) 1.59 ( 0.1, where  is the corresponding axis ratio; ξ ) 20.7 ( 3 Å, where ξ is the correlation length of c(q); B2 ) 0.002 05 ( 0.0002, where B2 is related to the calculation of the reduced concentration (see eq 12); and b ) 400 ( 100 Å, where b is the Kuhn length.

(20)

If the constants R, B1, and B3 are positive and γ is negative, the first term becomes dominant at high concentrations and the second term becomes dominant at low concentrations. Depending on the magnitude of γ, this gives rise to a fast decrease of the length of the micelles at low concentrations, whereas a constant or weakly increasing value is reached at high concentrations, depending on the value of R. Equation 20 was implemented in the model for the micelles used in the fitting routine in a way similar to that with eqs 18 and 19. In the fitting procedure, the value of R was first fixed to R ) 0 so that a constant, concentration-independent length of the micelles was approached at high concentrations. After a number of values for γ were tried, the value was fixed at γ ) -2. The quality of the fits obtained with this growth law was much better than that of the fits obtained with the growth law given by eq 19. However, the quality of the fits could be improved further by setting R ) 0.5 and, thus, assuming that a mean-field-theory type of growth law for the micelles takes over at high concentrations. This improvement gave rise to significantly better fits at high concentrations. The fit results in the following parametrization of the growth law:

Vmic(φ) ) 2.53 ((0.25) × 106φ 0.5 Å3 + 8.35 ((0.80)φ-2 Å3 (21) where Vmic(φ) is the volume of the micelles. Or, using a cross-sectional area of πR2 ) 1169 Å2

Lmic(φ) ) 2.16 ((0.21) × 103φ 0.5 Å + 7.14 ((0.68) × 10-3φ-2 Å (22) where Lmic(φ) is the length of the micelles. The corresponding fits are shown in Figure 1. As is seen from the plot, a good agreement between the model and the experimental data is achieved throughout the dilution series. The obtained values for the fitting parameters and deduced micellar lengths are given in Table 3. Furthermore, the growth law is plotted and compared to the growth law for a simple micellar system in Figure 4. 3.5. Scattering at Zero Angle. We can now compare the growth law and the corresponding scattering at zero angle to the experimentally determined scattering in-

Figure 4. (A) I(0)/φ as a function of φ for the lecithin-GCDC sample and a sample with SDS wormlike micelles in an aqueous 1.0 M NaBr solution. The lines are fits. (B) Growth laws corresponding to the fits.

tensity extrapolated to the zero angle. This I(0) is determined using a previously applied technique of fitting the low-q part (q e 0.01 Å-1) with a Debye function that models the scattering from a flexible chain:25,45

I(q) ) I(0)

2(e-x - 1 + x) x2

(23)

where x ) q2RG2 and RG is the apparent radius of gyration. (45) Debye, P. J. Phys. Colloid Chem. 1947, 51, 18-32.

Lecithin-Bile Salt System

Figure 4A is a plot of I(0)/φ as a function of φ, where φ is the volume fraction for the lecithin-GCDC sample. The experimental data are in good agreement with the results of the empirical growth law, eq 21. In the plot, I(0)/φ is multiplied by 1.15, in agreement with the average value of the obtained scale factors from the model fits. For comparison, we also plot similar data from measurements of wormlike micelles of SDS in an aqueous 1.0 M NaBr solution.27 For the SDS micelles, I(0) increases at low concentrations and decreases at high concentrations. The increase at low concentrations is due to a concentrationinduced growth of the micelle of the form L ∝ φR, with R ) 0.5.14 The decrease at high concentrations is due to the intermicellar interactions that suppress the scattering at low angles. The curve going through the data points is a fit of a model for the scattering at zero angle including the concentration-induced growth of the micelles and intermicellar interactions.27 The growth behavior of the micelles in the SDS system is typical for the relatively wellunderstood simple micellar systems, and a similar behavior, with different growth exponents, however, can be observed in a number of systems.13,24-26,42-44 As is clearly seen from Figure 4, the behavior of I(0)/φ in the lecithinbile salt system differs significantly from this behavior. Although I(0)/φ decreases monotonically with φ, we, nevertheless, observe two different slopes that correspond to the two regimes of dilution-induced and concentrationinduced growth. This can be clearly seen in a plot of the length as a function of φ in Figure 4B, which also shows these two regimes. In contrast, there is only one region of concentration-induced growth observed for SDS micelles. 4. Discussion We will now take a closer look at the results from the fit. Again, we start with the smallest length scale, that is, the cross section, corresponding to the high q values. As is seen from Figure 1, there is generally a good agreement between the scattering data and the model for the micelles at high q values where the information about the cross section is contained. This justifies the assumption of a constant cross section of the micelles throughout the data series. This assumption is also consistent with the results of the analysis via IFT. The obtained cross-sectional area of the micelles, as is calculated from the minor axis and axis ratio of the cross section (values given in Table 3), is 1170 ( 20 Å2. This is consistent with the value ACS ) 1190 ( 40 Å2 from the model-independent IFT. The value is significantly smaller than the value determined in a previous study of the lecithin-TCDC sodium salt system,10 where a cross-sectional area close to 2000 Å2 was reported; the same is found for the radii of gyration, which are 13.76 ( 0.25 and 18.3 ( 0.5 Å for the lecithin-GCDC (present study) and lecithin-TCDC, respectively. An even larger area, close to 2500 Å2, was determined in the lecithinGC and lecithin-GCDC systems by Hjelm et al.46 Although the molecular structures of the different bile salts are quite similar, the lecithin-bile salt ratios are not the same in the above-mentioned experiments. In the present case, a molar ratio of lecithin to bile salt of 1.1 in the stock solutions was used. In the study by Pedersen et al.,10 it was 0.9, and in the study by Hjelm et al.,46 it was 0.5. A lower lecithin-to-bile salt ratio implies a larger amount of bile salt in the polar layer and, thus, a larger total cross-sectional area. In the present analysis, we have assumed an elliptical cross section of the micelles. The actual cross section of (46) Hjelm, R. P.; Thiyagarajan, P.; Alkan-Onyuksel, H. J. Phys. Chem. 1992, 96, 8653-8661.

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the micelles is probably both polydispersed and elliptic. However, with the present data a model including both the polydispersity and the ellipticity of the cross section cannot be justified. Nevertheless, a model for the micelles with a polydispersed cross section instead of an elliptic cross section was also tested. It resulted in fits of very similar quality and a polydispersity index of the crosssection radius of ∼0.25-0.30. The scattering data are consistent with a Kuhn length, b, of 400 Å for the wormlike micelles. A value of this magnitude is in agreement with ref 10, where a Kuhn length of b ) 300 Å was found. In the present case, slightly better fits were obtained with b ) 400 Å rather than b ) 300 Å; however, the value cannot be determined with very high certainty because it is correlated to the value of B2. We generally obtain good fits with b > 250 Å. As is seen from the table, the size of the Kuhn length is comparable to the lengths of the micelles, and this implies that the micelles basically behave like rigid rods and that the scattering data do not contain sufficient information for a more exact determination of b. Thus, we did not attempt to determine the concentration dependence of b. As was mentioned previously, the values of the parameter B2 and the Kuhn length are correlated. Therefore, B2 cannot be determined with a very high precision. However, we notice that the obtained value is comparable to the value determined in a previous study.27 The correlation length, ξ, of c(q) is determined to be 20.7 Å (see Table 3). This value is in good agreement with the results of the analysis of SDS wormlike micelles of comparable sizes at similar concentrations.27 However, in the SDS system the value of ξ decreased from a value of 20 Å at low concentrations down to values of 4 Å at high concentrations. A similar concentration-dependent behavior of ξ could be expected in the present system, but because the model already fit the data well, we decided not to fit separate values for ξ for each data set. A theoretical prediction of ξ is generally not available. Monte Carlo simulations of semiflexible chains, which take into account excluded volume effects, are analyzed and reported in ref 23. Similar simulations can be performed for the chain lengths and concentrations relevant to the present study to estimate the ξ value. However, this has not yet been done. As seen from Table 3, the scale constants XS are constant within a few percent except for the samples at the highest concentration, where it is slightly larger. As is mentioned previously, the absolute scattering-length density of the Tris-NaCl buffer solution cannot be calculated theoretically with very high accuracy. For this reason, we have used the scattering-length density of the solution, which can be deduced from the excess scattering-length density determined by the IFT. This ∆F is also difficult to determine accurately, and an error of ∼8% is expected. This gives rise to an error in the final expression for the scattering function of ∼17%, which justifies the deviation of the scale constant from unity. The increase of XS at high concentrations is likely to be due to an inaccuracy of the growth law of the micelles. According to the values of the scale constants, the micellar volumes are slightly underestimated at high values of φ. This will be discussed in further detail in the following analysis. The growth law that includes a weak growth of the micelles at high concentrations gives rise to significantly better fits than the growth law that assumes a monotonic decrease in the length of the micelles. This indicates that, even with the segregation of the bile salt into the micellar end caps, the lecithin-bile salt system behaves like a simple micellar system at concentrations much higher

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than the cmc. However, the concentration range where we observe the mean-field-theory type of growth is very limited, and the micelles formed at the highest concentration (φ ) 0.033) are only a factor of 1.4 longer than the micelles formed at φ ) 0.012, where the length has its minimum. Furthermore, the increase of the scale factors at the two highest concentrations indicates that the model for the micelles is still not completely accurate at high concentrations. Apparently, the micellar volumes are underestimated at these concentrations, and the growth exponent R, which is different from the mean-field-theory prediction of 0.5, is likely. To investigate this effect in more detail, it will be necessary to perform additional studies in a larger concentration range. When investigating the different types of growth laws, satisfactory fits could not be obtained with, for example, a single-parameter growth law. Even the phenomenological growth law, eq 19, does not yield a good description of the data. A good description was only obtained using the growth law given by eq 20. If the scattering data were only fitted at the zero scattering angle, it was possible to fit the data with simpler expressions for the growth law. However, with a simultaneous fit to the full q range of all sets of scattering data, we are extremely sensitive to the concentration dependence of the length of the micelles. A growth law can, thus, be determined with high confidence. 5. Conclusion In the mixed micellar lecithin-bile salt system, vesicles are formed at low concentrations and wormlike micelles are formed at high concentrations. A method for analyzing the SANS data from the wormlike micelles formed at high concentrations has been developed. As was done in previous studies of wormlike micelles,13,24-27 the scattering data are analyzed using a model based on Monte Carlo simulations of semiflexible chains. The intermicellar interactions are taken into account in the same way as

Arleth et al.

was done in ref 27 by using an expression based on PRISM theory and renormalization group theory. An empirical growth law for the micelles, which allows for a fast dilutioninduced power-law growth of the micelles at low concentrations and a slower concentration-induced growth at high concentrations, is implemented in the model for the scattering data. When the the scattering data from all concentrations are fitted simultaneously, a parametrization of the power law can be obtained. The analysis indicates that the length of the micelles increases with the square root of the concentration at high concentrations. This is in contrast to the results of the analysis of an earlier study of a similar system, from which it was concluded that micelles formed at high concentrations were small and nearly spherical.6,7 The explanation of these different conclusions is partly that the early studies by Hjelm and co-workers did not investigate correspondingly high concentrations and partly that the methodology for analyzing the interactions between wormlike micelles has only been developed recently. So far, we have only tested the method on a single dilution series. To test the method further and determine the growth law with higher precision, it will be relevant to extend the concentration range that is studied. For high concentrations, this is done by preparing more concentrated samples, and at low concentrations, it is done by changing the lecithin-bile salt ratio. To test the structure factor in more detail and, especially, investigate the correlation between the growth law and the structure factor, it will be relevant not only to perform computer simulations of short semiflexible chains but also to move the cylinder-vesicle transition and the whole dilution series to higher concentrations by changing the bile salt to, for example, GC sodium salt, which has a cmc that is a factor of 6-8 times higher than GCDC. LA026808+