Flexibility of Charged and Uncharged Polymer-like Micelles

Langmuir 1998, 14, 6013-6024

6013

Flexibility of Charged and Uncharged Polymer-like Micelles Go¨tz Jerke,† Jan Skov Pedersen,‡ Stefan Ulrich Egelhaaf,§,| and Peter Schurtenberger*,† Institut fu¨ r Polymere, ETH Zu¨ rich, CH-8092 Zu¨ rich, Switzerland, Department of Solid State Physics, Risø National Laboratory, 4000 Roskilde, Denmark, and Institut Laue-Langevin, Large Scale Structures Group, B.P. 156, 38042 Grenoble, France Received April 8, 1998. In Final Form: July 16, 1998 We have performed a series of small-angle neutron scattering (SANS) and static light scattering (SLS) experiments with dilute and semidilute solutions of polymer-like micelles formed by C16E6 in D2O at two different temperatures (T ) 26 °C, 35 °C). The local structure has been investigated by applying indirect Fourier transformation and square-root deconvolution techniques. We demonstrate that the micelles maintain their locally cylindrical structure in this temperature regime despite the significant change in the spontaneous curvature of the surfactant. Detailed information on the micellar flexibility has been obtained from the SANS data by applying a nonlinear least-squares fitting procedure based upon a numerical expression for the single-chain scattering function of a worm-like chain with excluded volume effects. Particular attention has been given to the determination of quantitative numbers for the “intrinsic” persistence length as well as for the contribution from electrostatic interactions upon the addition of a small fraction of an ionic surfactant. The results have been compared with predictions from theoretical models for polyelectrolytes.

1. Introduction It has been demonstrated in various reports that it is possible to find conditions where micelles or microemulsion particles grow dramatically with increasing surfactant concentration into giant cylindrical aggregates.1 These worm-like micelles can then entangle and form a transient network above a crossover concentration c* with static properties comparable to those of semidilute polymer solutions. These systems have attracted considerable attention because they can be considered as a prime example of equilibrium polymers. The term equilibrium or living polymer originates from the equilibrium polymerization process that governs the concentration and temperature behavior of the supramolecular aggregates.2,3 Due to their transient nature, equilibrium polymers exhibit novel static and dynamic properties on time scales both long and short compared to their finite lifetime (for reviews see refs 1 and 4-9). The obvious structural analogy between these systems and classical polymers has suggested the idea that theoretical concepts from polymer physics could be applied in order to achieve a deeper understanding of polymer-like aggregates. * To whom correspondence should be addressed. † Institut fu ¨ r Polymere. ‡ Risø National Laboratory. § Institut Laue-Langevin. | Present address: The University of Edinburgh, James Clerk Maxwell Building, Mayfield Road, Edinburgh EH9 3JZ, U.K. (1) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (2) Cates, M. E. Macromol. 1987, 20, 2289. (3) Cates, M. E. J. Phys. (France) 1988, 49, 1593. (4) Magid, L. In Dynamic Light Scattering: The Method and some Applications; Brown, W., Ed.; Clarendon Press: Oxford, U.K., 1993. (5) Schurtenberger, P. In Light Scattering: Principles and Development; Brown, W., Ed.; Clarendon Press: Oxford, U.K., 1996. (6) Structure and Flow in Surfactant Solutions; Herb, C. A., Prud’homme, R. K., Eds.; American Chemical Society: Washington, DC, 1994. (7) Odijk, F. Curr. Opinion Colloid Interface Sci. 1996, 1, 337. (8) Lequeux, F. Curr. Opinion Colloid Interface Sci. 1996, 1, 341. (9) Cates, M. E. J. Phys.: Condens. Matter 1996, 8, 9167.

Despite the considerable experimental and theoretical attention given to the characterization and understanding of polymer-like micelles and microemulsions, precise knowledge of fundamental properties is still missing. For example, we lack detailed and quantitative information on the flexibility of the micelles as a function of composition, ionic strength, or temperature. The persistence length lp and bending modulus κ [which are related for one-dimensional objects via the thermal energy kBT through lp ) κ/(kBT)10] are key parameters in a more fundamental description of fluid membrane phases provided by the flexible surface model.11 Evidence for a flexible rod-like structure was, for example, already given by Ikeda et al. in their light scattering study of dodecyldimethylammonium chloride (DDAC) solutions at high salt concentrations.12 A detailed study of the flexibility of cetylpyridinium bromide (CPBr) micelles at high ionic strength was then conducted by Porte et al., who combined dynamic light scattering (DLS), magnetic birefringence and NMR measurements. Their results supported a model of semiflexible cylindrical micelles with a persistence length lp ≈ 100-200 Å.13-15 Later on they were able to verify this number from a more direct measurement at high-q values using small-angle neutron scattering.16 Several other groups have subsequently tried to estimate the persistence length of other surfactant systems using a combination of SLS and DLS, i.e., by comparing hh the radius of gyration R h g and the hydrodynamic radius R or the apparent molar mass Mapp and R h g.17-19 However, (10) Landau, L. D.; Lifshitz, E. M. Statistical Physics; AddisonWesley: Reading, MA, 1958. (11) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain; VCH: New York, 1994. (12) Ikeda, S.; Ozeki, S.; Tsunoda, M.-A. J. Colloid Interface Sci. 1980, 73, 27. (13) Porte, G.; Appell, J.; Poggi, Y. J. Phys. Chem. 1980, 84, 3105. (14) Appell, J.; Porte, G. J. Colloid Interface Sci. 81 1981 85. (15) Porte, G.; Appell, J. J. Phys. Chem. 1981, 85, 2511. (16) Marignan, J.; Appell, J.; Bassereau, P.; Porte, G.; May, R. P. J. Phys. (France) 1989, 50, 3553. (17) Imae, T. Colloid Polym. Sci. 1989, 267, 707.

S0743-7463(98)00390-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 09/22/1998

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it is important to point out that in such an attempt one has to take into account micellar polydispersity and intermicellar interaction effects. The influence of polyh h has, for example, been considered dispersity on R h g and R for worm-like chains by Schmidt, who obtained an hh analytical expression for R h g and numerical results for R based on a Schulz-Zimm distribution.20 Neglecting polydispersity in model calculations for semi-flexible particles will lead to an overestimation of lp due to the h g2〉z1/2, and 〈1/ characteristic intensity weighting (〈M〉w, 〈R -1 R h h〉z , where the subscript “w” stands for weight-average and “z” for z-average, respectively) of the experimental quantities determined in SLS and DLS experiments.21,22 Furthermore, the experimental quantities Mapp, R h g,app, and R h h,app exhibit quite different dependencies on intermicellar interactions, which will also result in an incorrect value of the persistence length if not considered correctly.22 It is clear that a more precise determination of lp can be achieved for example with SANS or SAXS experiments, where the much smaller wavelength leads to an enhanced structural resolution even down to the length scale of the cross-section radius of the micelle. The characteristic crossover in the q dependence of the scattered intensity from a scattering pattern typical for rigid rods to one typical for flexible coils then becomes accessible and permits a precise measurement of lp that will only be weakly affected by polydispersity.16,23-25 However, both the influence of excluded volume effects26 as well as the incorporation of intermicellar interactions in the intermediate-q range are usually not included in the interpretation of small-angle scattering data. A major obstacle in experimental studies employing scattering methods has always been the problem of how to incorporate growth, polydispersity, flexibility, and intermicellar interactions in a consistent interpretation of scattering data. Therefore, we initiated a Monte Carlo simulation study in order to obtain accurate expressions for the scattering function of worm-like chains with excluded volume effects on all length scales.27 We subsequently demonstrated that these parametrized scattering functions28 are indeed capable of fitting the experimental data from polymer-like reverse micelles of lecithin in deuterated isooctane over more than three decades in q with very good agreement.29 The use of numerical expressions for the full scattering function allowed us to incorporate polydispersity and to determine apparent values of the contour length and the persistence length with high precision. These accurate values for lp,app yielded clear evidence that the SANS data at intermediate q, from where we extracted the information on the persistence length, are influenced by intermicellar interaction effects even at relatively low concentrations c < c*. From a direct comparison with a recent Monte Carlo (18) Mishic, J. R.; Fisch, M. R. J. Chem. Phys. 1990, 92, 3222. (19) van de Sande, W.; Persoons, A. J. Phys. Chem. 1985, 89, 404. (20) Schmidt, M. Macromolecules 1984, 17, 553. (21) Grob, M. C.; Schurtenberger, P.; Suter, U. W. Makromol. Chem. Phys. 1995, 196, 1391. (22) Schurtenberger, P.; Cavaco, C. Langmuir 1994, 10, 100. (23) Kirste, R. G.; Oberthu¨r, R. C. In Small-Angle X-ray Scattering; Glatter, O., Kratky, O., Eds.; Academic Press: London, 1982. (24) Schurtenberger, P.; Magid, L. J.; King, S. M.; Lindner, P. J. Phys. Chem. 1991, 95, 4173. (25) Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. J. Phys. Chem. 1995, 99, 1299. (26) Schurtenberger, P.; Cavaco, C. J. Phys. Chem. 1994, 98, 5481. (27) Pedersen, J. S.; Laso, M.; Schurtenberger, P. Phys. Rev. E 1996, 54, R5917. (28) Pedersen, J. S.; Schurtenberger, P. Macromolecules 1996, 29, 7602. (29) Jerke, G.; Pedersen, J. S.; Egelhaaf, S. U.; Schurtenberger, P. Phys. Rev. E. 1997, 56, 5772.

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simulation study of many chain systems30 we were able to identify and eliminate these interaction effects and determine precise values for the persistence length of worm-like lecithin reverse micelles. These findings now open new opportunities to study micellar flexibility and investigate the influence of compositional changes, temperature, or charges on lp. While the lecithin water-in-oil microemulsions used in our previous experiments,22,24,26,29,31-35 served as good model systems for an initial investigation of static and dynamic properties of equilibrium polymers due to the fact that they are oil-continuous and no complicating effects arise due to additional contributions from electrostatic interactions or salt effects, they are not ideal candidates for a detailed investigation of micellar flexibility and growth.36 At very low concentrations c , c* slightly different partitioning of the surfactant and the water into the organic solvent could become important, leading to a change in the micellar composition (the molar ratio of water to surfactant in the aggregate) and thus to a possible change in the micellar size superimposed on the concentration dependence. In addition small variations in the trace amounts of water added in order to adjust the extent of micellar growth caused dramatic changes in the micellar properties. This has considerable consequences for the reproducibility of the experimental results on an absolute scale. Moreover, most of the experimental model systems for equilibrium polymers are aqueous solutions of ionic surfactants at high ionic strength, and a very important question is thus the influence of (screened) electrostatic interactions on the micellar flexibility. We have thus attempted to reexamine our previous findings on the micellar flexibility with a simpler model system. Prime candidates for such a study are aqueous solutions of nonionic surfactants of the alkyl oligoethylene oxide type with a composition CH3(CH2)m-1(OCH2CH2)nOH (abbreviated by CmEn). For our investigation we have chosen hexaethylene glycol monohexadecyl ether, C16E6, which has a low CMC of approximately 5 × 10-7 g/cm3 37 and forms giant polymer-like micelles in water. This system has already been studied extensively using SANS38-40 and cryogenic transmission electron microscopy (cryo-TEM).41 In the SANS experiments, measurements were made on micellar samples subjected to an orientating shear flow, in which the micelles were expected to align with their long axis in the direction of the flow. This experimental approach was chosen in order to avoid problems in the interpretation of scattering data from an ensemble of polydisperse and interacting rod-like micelles due to the orientational averaging that takes place in isotropic solutions. The authors then argued that for fully aligned micelles the scattered intensity should reflect the (30) Pedersen, J. S.; Schurtenberger, P. Manuscript in preparation. (31) Schurtenberger, P.; Scartazzini, R.; Magid, L. J.; Leser, M. E.; Luisi, P. L. J. Phys. Chem. 1990, 94, 3695. (32) Schurtenberger, P.; Magid, L. J.; Penfold, J.; Heenan, R. Langmuir 1990, 6, 1800. (33) Schurtenberger, P.; Cavaco, C. J. Phys. II (France) 1993, 3, 1279. (34) Schurtenberger, P.; Cavaco, C. J. Phys. II (France) 1994, 4, 305. (35) Schurtenberger, P.; Jerke, G.; Cavaco, C.; Pedersen, J. S. Langmuir 1996, 12, 2433. (36) Schurtenberger, P.; Cavaco, C.; Tiberg, F.; Regev, O. Langmuir 1996, 12, 2894. (37) Balmbra, R. R.; Clunie, J. S.; Corkill, J. M.; Goodman, J. F. Trans. Faraday Soc. 1964, 60, 979. (38) Cummins, P. G.; Hayter, J. B.; Penfold, J.; Staples, E. Chem. Phys. Lett. 1987, 138, 436. (39) Cummins, P. G.; Staples, E.; Penfold, J.; Heenan, R. K. Langmuir 1989, 5, 1195. (40) Penfold, J.; Staples, E.; Cummins, P. G. Adv. Colloid Interface Sci. 1991, 34, 451. (41) Lin, Z.; Scriven, L. E.; Davis, H. T. Langmuir 1992, 8, 2200.

Flexibility of Polymer-like Micelles

micelle symmetry, and the existence of rod-like micelles should be visible with much less ambiguity than for the isotropic solution at rest. They indeed clearly demonstrated that C16E6 forms strongly anisotropic micelles in aqueous solutions, which could easily be aligned at moderate shear rates. The scattering data were consistent with locally cylindrical structures, and an estimate of the cross-section diameter of the micelles was made, which indicated that the local structure of the micelles was independent of temperature and salinity. To obtain quantitative information on the micellar contour length as a function of solution composition, a closed form of the orientational distribution function was used that had been derived previously by Hayter and Penfold42 on the basis of earlier birefringence work by Peterlin and Stuart.43 Assuming that the micelles are monodisperse rigid rods, they came to the conclusion that the micellar contour length L should be of the order of 1600-4000 Å, depending on temperature and salt concentration, but with only a very weak dependence on surfactant concentration. In contrast to these results, the cryo-TEM pictures revealed the existence of worm-like micelles with much longer contour length under comparable conditions. Neither the rigidity and the magnitude of the micellar lengths nor its trend with temperature appeared to agree with the SANS results published by Cummins et al.38,39 Unfortunately, cryo-TEM images can generally not be used reliably to quantitatively determine the micellar size distribution and the persistence length in solutions of polymer-like micelles. In a first step we thus performed a detailed characterization of the concentration dependence of the micellar size using static light scattering experiments.36 Measurements were made at two different temperatures in order to test for possible contributions from critical scattering, which could become important in the vicinity of the critical point. We found no measurable temperature dependence for the apparent molecular weight and the correlation length of the micelles for the given range of concentrations and temperatures. We then used an application of conformation space renormalization group theory originally developed for semidilute polymer solutions in order to analyze the measured apparent molar mass of the micelles and combined the concentration dependence of the micellar size distribution and intermicellar interaction effects in a self-consistent way.36 Our results indicated a much stronger micellar growth than previously estimated from the SANS experiments with shear-aligned micelles. The resulting values for the average micellar contour length are actually in reasonable agreement with the cryo-TEM observations by Lin et al.41 Having established the concentration dependence of the micellar size and the influence of interactions on the initial q dependence of the scattered intensity, we now present a detailed characterization of the local micellar structure and the flexibility using SANS experiments. In the interpretation of the SANS data, we profit from the recent progress in our understanding of the static structure factor of worm-like chains with excluded volume effects at arbitrary concentrations which has emerged from systematic Monte Carlo simulation studies.27,28,30 We are thus in an ideal position to not only determine the intrinsic persistence length of nonionic micelles formed by C16E6 but also to demonstrate that the micelles can be charged by the addition of small amounts of ionic surfactant without changing their locally cylindrical structure. It (42) Hayter, J. B.; Penfold, J. J. Phys. Chem. 1984, 88, 4589. (43) Peterlin, A.; Stuart, H. Hand- und Jahrbuch der Chemischen Physik; Akad. Verlag Becker und Erler: Leipzig, 1943; Vol. 8.

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then becomes possible to precisely measure the increase in lp due to electrostatic interactions. 2. Material and Methods 2.1. Materials. The surfactant hexaethylene glycol monohexadecyl ether CH3(CH2)15(OCH2CH2)6OH (abbreviated by C16E6) was obtained from Nikkol Ltd., Tokyo, the charged surfactant 1-hexadecane sulfonic acid (CH3(CH2)15SO3Na) (abbreviated by C16SO3 Na) was purchased from TCI, and D2O (99.9% isotopic purity) was delivered from Cambridge Isotope Laboratories. The SLS and SANS concentration series were done by diluting a 2 mg/mL stock solution. 2.2. Methods. Static Light Scattering. Static light scattering experiments were made with a modified Malvern PS/ MW spectrometer, equipped with an argon ion laser (Coherent, Innova 70, λ0 ) 488 nm) and a computer controlled and stepping motor driven variable fiber optics based detection system (for details on the use of few mode fibers, see ref 44). Measurements were usually performed at a temperature of 26.0 ( 0.2 °C. Approximately 1 mL of the solution was transferred into the cylindrical scattering cell (10 mm diameter). The scattering cell was then stoppered and centrifuged for approximately 30 min at 5000g and 26 °C in order to remove dust particles from the scattering volume. Experiments were performed at 35 different angles between 15.2° e θ e 145°, and 30-60 individual measurements were taken and averaged for each angle. The data were then corrected for background (cell and solvent) scattering and converted into absolute scattered intensities dσ(θ)/dΩ using toluene as a reference standard. The absolute scattered intensity per volume was calculated using45

( )( )

∆〈I(θ)〉 dσ dσ (θ) ) dΩ 〈Iref(θ)〉 dΩ

n nref

ref

2

(1)

where ∆〈I(θ)〉 and 〈Iref(θ)〉 are the average scattered intensity of the solution and the average scattered intensity of the reference solvent toluene, (dσdΩ)ref ) 39.6 × 10-4 m-1 is the absolute scattered intensity of toluene (per volume), and n and nref are the index of refraction of the solution and the reference solvent, respectively. The apparent molar mass, Mapp, was determined from the intercept of cKSLS/[dσ(q)/dΩ] versus q2 using a Lorentzian scattering law of the form

cKSLS 1 (1 + q2ξs2) ) dσ Mapp (q) dΩ

(2)

where q ) (4πn/λ0)sin(θ/2) is the magnitude of the scattering vector and ξs is the static correlation length. The contrast term is given by

KSLS :)

4π2n2 dn NAλ04 dc

2

( )

(3)

where dn/dc is the refractive index increment (with dn/dc ) 1.36 × 10-4 m3/kg), c is the surfactant concentration, and NA is the Avogadro number. The accessible range of scattering angles results in 4.5 × 10-4 Å-1 e q e 3.3 × 10-3 Å-1. The fit range was restricted to scattering angles of 15.2° e θ e 47.7° so that only the low-q part was included in the fit, for which we can use the approximation given in eq 2. Small-Angle Neutron Scattering. The small-angle neutron scattering (SANS) experiments were performed at the D22 instrument of the Institut Laue Langevin, France.46 An extended range for the magnitude of the scattering vector from 2.3 × 10-3 Å-1 e q e 0.38 Å-1 was covered by three combinations of neutron (44) Gisler, T.; Ru¨ger, H.; Egelhaaf, S. U.; Tschumi, J.; Schurtenberger, P.; Ricka, J. Appl. Opt. 1995, 34, 3546. (45) Schurtenberger, P.; Augusteyn, R. C. Biopolymers 1991, 31, 1229. (46) The yellow book: Guide to the neutron research facilities at the ILL; Ibel, K., Ed.; ILL: Grenoble, 1994. Further information: http:// www.ill.fr/d22/d22.html.

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Figure 1. Scattered intensity [dσ(q,c)/dΩ]/c versus scattering vector q and concentration c for polymer-like micelles of C16E6 in D2O. Data shown were obtained from SLS (low-q range) and SANS measurements (intermediate and high-q range). In the limit q f 0, the scattered intensity (given by b) is mainly dominated by the concentration-induced micellar growth at low concentrations and by intermicellar interactions at high concentrations. wavelength (λ ) 9 and 12 Å) and sample-to-detector distances (d ) 1.4, 10, and 17.9 m). All experiments at D22 were done with a 40 cm detector offset. The wavelength resolution was 10% (full-width-at-half-maximum value). The samples were kept in stoppered quartz cells (Hellma, Germany) with a path length of 2 mm. The neutron spectra of water used for calibration were measured with a 1 mm pathlength quartz cell. The raw spectra were corrected for background from the solvent, sample cell, and electronic noise by conventional procedures. Furthermore, the two-dimensional isotropic scattering spectra were azimuthally averaged, converted to an absolute scale, and corrected for detector efficiency by dividing with the incoherent scattering spectra of pure water.47-50 The average excess scattering length density per unit mass ∆Fm of polymer-like micelles formed by C16E6 (plus C16SO3Na) in D2O was determined from the known chemical composition. The corresponding values are ∆Fm ) -6.52 × 1010 cm/g for C16E6 and ∆Fm ) -6.57 × 1010 cm/g for C16SO3Na. The smearing induced by the different instrumental setups is included in the data analysis discussed below. For each instrumental setting the ideal model scattering curves were smeared by the appropriate resolution function when the model scattering intensity was compared to the measured one by means of least-squares methods,51,52 The parameters in the models were optimized by conventional least-squares analysis, and the errors of the parameters were calculated by conventional methods.53

3. Results and Discussion 3.1. General Features. Figure 1 shows the q dependence of the scattered intensity [dσ(q,c)/dΩ] from polymer-like micelles formed by C16E6 in D2O. The combination of data from static light and small-angle neutron scattering experiments results in an extended q (47) Jacrot, B.; Zaccai, G. Biopolymers 1981, 20, 2413. (48) Ragnetti, M.; Oberthu¨r, R. C. Colloid Polym. Sci. 1986, 264, 32. (49) Wignall, G. D.; Bates, F. S. J. Appl. Crystallogr. 1987, 20, 28. (50) Cotton, J. P. In Neutron, X.-Ray and Light Scattering: Introduction to an Investigative Tools for Colloidal and Polymeric Systems; Lindner, P., Zemb, T., Eds.; North-Holland: Amsterdam, 1991. (51) Pedersen, J. S.; Posselt, D.; Mortensen, K. J. Appl. Crystallogr. 1990, 23, 321. (52) Barker, J. G.; Pedersen, J. S. J. Appl. Crystallogr. 1995, 28, 105. (53) Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969.

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Figure 2. Apparent molar mass Mapp versus concentration c for polymer-like micelles of C16E6 in D2O (b, concentration series used in this study; O, former measurements, see ref 36), compared to a previously investigated nonaqueous system of lecithin reverse micelles in deuterated isooctane with a molar water-to-lecithin ratio, w0 ) 1.5 (0, see ref 29). Also shown are the best fits to the data according to the renormalization group theory approach, which result in a growth exponent of R ) 1.2 (solid line, C16E6; dotted line, reverse micellar system). The concentration dependence of the molar mass 〈M〉w ) Mapp/S(0) for the nonionic micellar solution is given as the dashed line.

range of approximately three decades so that structural properties on length scales from a few angstroms up to 1000 Å are resolved. The chosen range of surfactant concentrations from 0.4 mg/mL e c e 2 mg/mL allows us to follow the concentration-induced micellar growth at low values of c as well as the strong effect of intermicellar interactions on the scattered intensity at concentrations above the overlap threshold c*. Figure 1 clearly reveals the polymer-like structure of the aggregates; it also shows the limitation of the polymer analogy because of the selfassembling nature of the aggregates. The very strong concentration-induced growth of the micelles becomes even more apparent in a plot of the forward scattered intensity versus concentration. This is demonstrated in Figure 2, where we have plotted the values from the current study (filled circles) as well as those obtained in a recent light scattering investigation of the same system (open circles) that covered an even more extended range of concentrations.36 The data from the two studies overlap within experimental accuracy and thus provide us with a decisive test of the reproducibility of our measurements on an absolute scale. We have already previously shown33 that one can quantitatively describe the combined effects of micellar growth and intermicellar interactions on the apparent molar mass Mapp within the framework of conformation space renormalization group theory by appropriately taking into account the equilibrium nature of the micellar size distribution. The crucial idea behind this approach is the incorporation of the concentration dependent micellar growth via 〈M〉w ) B1cR. 〈M〉w is the weight average molar mass, c is the concentration, R is the growth exponent, and B1 is a proportionality constant. At any arbitrary concentration the influence of interaction effects, i.e., the value of the structure factor S(0) at q ) 0, can then be calculated by using the corresponding expression for “quenched” polymers for this particular weight-average


size and concentration given by renormalization group theory. The resulting theoretical curve for the concentration dependence of Mapp as well as the corresponding values of the ideal molar mass 〈M〉w using a growth exponent of R ) 1.2 as determined previously in reference36 are plotted as the solid and dashed lines, respectively. Figure 2 thus provides us with an estimate of the effects of intermicellar interactions on the scattering data at low values of q for the concentration regime used in the present study. We clearly see that interactions start to play a dominant role already at concentrations as low as c ) 1 mg/mL, where we approach the overlap threshold c*. This demonstrates the importance of taking into account interaction effects in any attempt to quantitatively interpret scattering data for systems that exhibit the formation of giant polymerlike micelles at low concentrations. It is thus crucial not only to consider the actual concentration but also to use effective concentrations X ∼ c/c* when estimating whether a sample is truly in a dilute regime. This point is made more clearly from a comparison with a system that also exhibits the formation of polymer-like micelles, but where the onset of considerable micellar growth occurs at higher values of c. Therefore we have also included the results for the concentration dependence of the apparent molar mass of polymer-like reverse micelles of lecithin in deuterated isooctane for comparison.29 We see that the qualitative features are the same for both systems, but the actual values for the micellar size and the location of c* are shifted considerably. In aqueous solutions of C16E6 the micellar growth is extremely pronounced, and the resulting micellar sizes are very large. The system is thus ideally suited to point out the close correspondence to classical polymers. The important characteristic length scales for the overall size (i.e., radius of gyration) and the flexibility (i.e., persistence length) are well separated, and the scattered intensity exhibits all the features observed for very large polymers. For classical polymers, one would expect to find the following three distinct regions in the scattering data: At low q, the Guinier region associated with the overall size of the chain (apparent molar mass Mapp, static correlation length ξs) is observed. At slightly higher q, the scattered intensity of a single coil crosses over to a power-law behavior with an exponent of -2 in Θ solvents and about -5/3 in good solvents. These exponents are characteristic of the random walk and self-avoiding walk configuration of the chain in Θ and good solvents, respectively. At higher q, one probes shorter length scales and the local stiffness of the chain shows up as a crossover to a q-1 behavior. At even higher q the local cross-section structure of the chain gives rise to a cross-section Guinier behavior and a strong decrease in the scattered intensity. Figure 1 indeed reveals that all these features can be seen for the data obtained with C16E6, and in particular close to the overlap threshold the resulting static correlation length (or apparent radius of gyration) is so large that we hardly reach the Guinier regime even at the lowest values of q accessible in our experiments. We have not only investigated the influence of concentration but also looked at the effect that a change in temperature and the addition of a small number of ionic surfactants have on the micellar size and structure. Figure 3 summarizes the temperature and charge dependence of dσ(q)/dΩ. Figure 3A demonstrates for a fixed concentration of c ) 0.6 mg/mL that dσ(q)/dΩ remains nearly unchanged upon an increase of the temperature from T ) 26 °C to T ) 35 °C. Slight deviations become visible in the low-q part of the data only, whereas at intermediate and high q both curves completely coincide.

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Figure 3. Scattered intensity dσ(q)/dΩ versus scattering vector q for polymer-like micelles of C16E6 in D2O as a function of temperature (Figure A: T ) 26 °C; 35 °C) and added charges (Figure B: [C16SO3Na]/ctot ) 0.06). Figure A: Increase of the temperature from T ) 26 °C (0) to T ) 35 °C (O). The surfactant concentration is c ) 0.6 mg/mL. Figure B: Scattered intensity of polymer-like micelles with added charges ([C16SO3Na]/ctot) 0.06) at three different surfactant concentrations (O, c ) 0.6 mg/mL; 0, c ) 1.2 mg/mL; 4, c ) 1.8 mg/mL).

This finding indicates that the local arrangement of the surfactants is not influenced by a temperature change, and a detailed analysis will be given in the following subsection. Figures 1, 2, and 3A summarize the behavior of C16E6 micelles as a function of concentration and temperature. Under these conditions, the conformational properties of the nonionic surfactant micelles should closely resemble those of a classical worm-like chain with excluded volume interactions, and from an application of classical polymer theory we should be able to extract quantitative information on the micellar size, flexibility, and local structure. However, aqueous solutions of C16E6 cannot only be used as simple model systems for equilibrium polymers, but allow us to go one step further and investigate the influence of charges on the conformational properties of the micelles;

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i.e., we can mimic “equilibrium polyelectrolytes”. The addition of trace amounts of a charged surfactant on the q dependence of the scattered intensity is shown in Figure 3B, which contains the SANS data sets of polymer-like micelles with [C16SO3Na]/ctot ) 0.06 at three different values of the total surfactant concentration. (In contrast to Figure 1 the data shown in Figure 3 are not normalized to concentration. This causes the displacement of the curves shown in Figure 3B.) An inspection of Figure 3B immediately reveals that the overall pattern of the q dependence of the scattered intensity has not changed upon the addition of the ionic surfactant, and we still observe all the essential features of a polymer-like structure. As will be demonstrated in the detailed discussion of the local properties and of the flexibility given below, it is primarily the overall size and the flexibility that are sensitive to the presence of charges. 3.2. Data Analysis: Local Structure. In Figures 1-3 we have seen that the scattered intensity of polymerlike micelles is strongly influenced by surfactant concentration in the range of low-q values, reflecting the pronounced dependence of the micellar size on concentration. This situation appears to be quite different if one focuses on the scattering curves at high-q values where local structural properties are resolved. Figure 3A also indicates that a temperature increase of nearly 10 °C does not alter the local cylindrical arrangement. We can now aim at a quantitative determination of local properties such as the cross-section distance distribution function p˜ cs(r) and the cross-section scattering length density profile ∆F(r) as a function of temperature (Figure 4) and added charges (Figure 5). As described in detail previously,29,35,54 we can extract information on the local structure of the micelles by applying the indirect Fourier transformation (IFT),55,56 and the square-root deconvolution (SQDEC) method57 to the experimental data from the high-q regime. On length scales where the local stiffness of the micelles becomes visible, the asymptotic behavior of the scattered intensity can be expressed by

dσ π (q) ) 2π dΩ q

()

∫0∞p˜ cs(r) J0(qr) dr ) (πq)Ics(q)

(4)

where J0 is the zeroth-order Bessel function. The normalized cross-section distance distribution function p˜ cs(r) is given by

p˜ cs(r) )

2πc r ML

∫0∞ ∆F(r′) ∆rho(r + r′) dr′

(5)

As described previously,35,54 we obtain a parametrized form of the distance distribution function p˜ cs by applying the IFT method. The resulting distance distribution functions and a comparison of fitted and experimental values of the scattered intensity are shown in Figure 4A,B for the data obtained from a dilute sample (c ) 0.6 mg/ mL) at T ) 26 °C. The lower limit qmin ) 0.03 Å-1 (indicated by arrows in Figure 4A) of the fitted q range was chosen close to the limit where the crossover from stiff to flexible behavior occurs. As one can easily see, experimental data and fitted curves coincide perfectly within the fit range of q g 0.03 Å-1, and the expected strong deviations in the low-q regime due to the crossover from locally stiff to (54) Pedersen, J. S.; Schurtenberger, P. J. Appl. Crystallogr. 1996, 29, 646. (55) Glatter, O. J. Appl. Crystallogr. 1977, 10, 415. (56) Glatter, O. J. Appl. Crystallogr. 1980, 13, 577. (57) Glatter, O. J. Appl. Crystallogr. 1981, 14, 101.

flexible coil structure are clearly visible. The distance distribution function exhibits a shape that is quite characteristic of an almost homogeneous locally cylindrical structure,54 and we obtain a first estimate of the crosssection diameter from the maximum distance of approximately 60 Å seen in Figure 4B. Having determined p˜ cs(r), we are able to calculate integral parameters of the micellar cross section58 such as the mass per length, ML, and the cross-section radius h cs,g of gyration R h cs,g. The cross-section radius of gyration R is given by

R h cs,g )

[

]

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

(6)

and the cross-section forward scattered intensity Ics(0) by

Ics(0) ) 2π

∫0∞p˜ cs(r) dr

(7)

The mass per length ML of the cylindrical micellar core (in units g/cm) can be calculated via

ML )

Ics(0) ∆Fm2

(8)

The resulting values are summarized in Table 1. The cross-section radius of gyration provides us with an additional estimate of the cross-section diameter of the micelles, and using the most simple model of a locally cylindrical structure with a uniform scattering length density distribution, we obtain a value of approximately 56 Å. As a next step we can also look at the effect of temperature on the local micellar structure. A comparison of the pair distribution functions obtained for c ) 0.6 mg/ mL at T ) 26 °C and T ) 35 °C, respectively, is given in Figure 4C. The figure shows that p˜ cs(r) measured at two different temperatures virtually coincide, which already provides very strong evidence that the increase in temperature has no measurable effect on the local structure of the micelles. This is also clearly visible when looking at the integral parameters given in Table 1, which again agree within experimental accuracy. We can also obtain a quantitative estimate of the radial scattering length density profile ∆F(r); i.e., we can get quite detailed information on the extension of the hydrocarbon chains and the hydrophilic headgroups and also on the degree of water penetration into the headgroup layer. This can be achieved from a deconvolution of p˜ cs(r) using the square-root deconvolution method.54,57 The resulting radial dependence of ∆F(r) is summarized in Figure 4D for two temperatures (T ) 26 °C and T ) 35 °C). The quality of the fit to the experimentally determined distance distribution function p˜ cs(r) is demonstrated in Figure 4B for the sample at T ) 26 °C. Within the fit range of 12 Å < r < 55 Å both the p˜ cs(r) obtained via IFT and the fitted one by using the deconvoluted scattering length density distribution shown as the solid line are in reasonable agreement. As already indicated by the excellent agreement of the distance distribution functions shown in Figure 4C, the deconvoluted profiles obtained at both temperatures coincide perfectly. Following the deconvolution procedure, we now have access to the cross(58) Glatter, O. In International Tables for Crystallography. Volume C. Mathematical, Physical and Chemical Tables; Wilson, A. J. C., Ed.; Kluwer Academic Publishers: Dordrecht, 1992.


Langmuir, Vol. 14, No. 21, 1998 6019

Figure 4. Local structure of polymer-like micelles formed by C16E6 in D2O at a surfactant concentration of c ) 0.6 mg/mL for different temperatures (T ) 26 °C and T ) 35 °C) as obtained from IFT and square-root deconvolution. Figure A: Comparison of experimental (0) and fitted (solid line) scattered intensity [dσ(q)/dΩ]/c by applying indirect Fourier transformation (IFT) to the high-q part of the SANS data measured at T ) 26 °C. The lower cutoff value of the q range used in the IFT is qmin ) 3 × 10-2 Å (indicated by an arrow). Figure B: Distance distribution function p˜ cs(r) as determined by IFT of data shown in figure A. Solid lines represent the fits in the range of 12 Å e r e 55 Å (indicated by arrows) by a square-root deconvolution of p˜ cs(r). Figure C: Comparison of distance distribution functions p˜ cs(r) as a function of temperature (0, T ) 26 °C; O, T ) 35 °C). Figure D: Radial excess scattering length density profile ∆F(r) calculated by deconvoluting p˜ cs(r) shown in figure C (0, T ) 26 °C; O, T ) 35 °C). Schematic representation of the expected excess cross-section scattering length density profiles (solid line) following a simple geometrical model of two box functions (dotted line) representing the inner core (surfactant tail region) and outer shell (surfactant headgroup) that is modified due to solvent penetration (solid line).

section profile on an absolute scale (in units cm-2, Figure 4D), which can be compared with a simple geometrical model of a cylindrical cross section.35 The model density profile shown in Figure 4D is approximated by an inner core, which starts at the hydrocarbon chain value of ∆F ) -6.81 × 1010 cm-2, combined with a modified box function for the hydrophilic headgroup layer, which takes into account the solvent penetration into the surfactant head region.35 We see that the cross-section scattering length density distributions obtained for both temperatures agree very well with the simple geometrical model on an absolute scale, and no evidence for any temperatureinduced modification of the local structure can be observed.

This absence of any temperature effect on the local cylindrical structure of the micelles is not trivial in the case of nonionic surfactants of the ethylene oxide type. For these surfactants the spontaneous curvature is known to decrease monotonically with increasing temperature.59 Ultimately, one could thus expect to find a change from a locally cylindrical to a locally lamellar structure. However, our results clearly demonstrate that any deviation from a cylindrical cross section, such as, for example, some ellipticity, has to be very small. This can be seen when comparing our experimental results with those from (59) Olsson, U.; Wennerstro¨m, H. Adv. Colloid Interface Sci. 1994, 49, 113.

6020 Langmuir, Vol. 14, No. 21, 1998

Jerke et al. Table 1. Cross-Section Radius of Gyration, R h cs,g, Cross-Section Forward Scattered Intensity, Ics(0), and Mass per Unit Length, ML, Obtained by Indirect Fourier Transformation Method (IFT) for Polymer-like Micelles Formed by C16E6 in D2O at a Concentration of c ) 0.6 mg/mL charge, wt % C16SO3Na 6

Figure 5. Local structure of polymer-like micelles formed by C16E6 in D2O as a function of added charges ([C16SO3Na]/ctot) 0.06, total surfactant concentration c ) 1.2 mg/mL) at a temperature of T ) 26 °C. (A) Comparison of experimental (O) and fitted (solid line) scattered intensity [dσ(q)/dΩ]/c by applying indirect Fourier transformation (IFT) to the high-q part of the SANS data measured at T ) 26 °C. The lower cutoff value of the q range used in the IFT is qmin ) 3 × 10-2 Å (indicated by an arrow). (B) Distance distribution function p˜ cs(r) as determined by IFT of data shown in figure A. Solid lines represent the fits in the range of 12 Å e r e 55 Å (indicated by arrows) by a square-root deconvolution of p˜ cs(r). (C) Comparison of calculated radial excess scattering length density profiles ∆F(r) by deconvoluting p˜ cs(r) shown in Figure 5B (O, 6 wt % charged surfactants) and in Figure 4B (0, uncharged surfactants).

temp, °C

R h cs,g, Å

Ics(0), 108 cm/g

ML, 10-13 g/cm

26 26 35

19.3 19.5 19.6

6.73 6.55 6.61

1.58 1.54 1.56

a detailed study of the influence of effects such as crosssection polydispersity or ellipticity on the analysis of scattering data using IFT and SQDEC based on simulated scattering data.54 Our findings are in agreement with the earlier measurements by Cummins et al.,38 where they also observed no indications for changes in local structure in a temperature range of 30-36 °C. The reason for the absence of changes in the local structure upon a variation of the temperature can be understood when taking into account the additional constraint imposed on the system by the required shielding of the hydrophobic tails from contact with water. A change of the spontaneous curvature thus induces a curvature stress that the system cannot match, as it has to shield the hydrophobic tails, and it eventually will lead to a phase transition rather than to a continuous change in local structure. In addition to the effect of temperature, we have also analyzed the influence of adding small amounts of ionic surfactants to the nonionic micelles on the cross-section structure. This is illustrated in Figure 5, where we show the scattered intensity, the distance distribution function, and the scattering length density profile of a micellar solution formed by C16E6 and C16SO3Na, where the weight ratio of ionic to total surfactant concentration was set to cionic/ctot) 0.06. Figure 5A contains the comparison of experimental and fitted scattered intensity by applying the IFT method, whereas Figure 5B shows the extracted distance distribution function in combination with the fitted one based on a square-root-deconvolution. Finally in Figure 5C the resulting scattering length density profile is given for the system both with and without charges. The two different profiles agree within 10%, and they closely follow the profile from the simple “hairy cylinder” model. Again, the fact that the addition of a small amount of an ionic surfactant, which corresponds to a ratio of ionic to nonionic surfactant of approximately 9.3% by number, has no measurable influence on the local micellar structure is an interesting finding. While we have specifically chosen a surfactant with the same hydrocarbon chain length in order to minimize the chain packing distortions in the hydrocarbon core, we nevertheless change the overall spontaneous curvature due to the additional electrostatic interactions between the charged headgroups. However, while this has a considerable effect on the micellar flexibility and the overall size, it does not change the cross-section structure. This is also evident from the integral parameters mass per length and cross-section radius of gyration given in Table 1. 4. Data Analysis: Flexibility We now proceed and investigate the intermediate-q range of the scattered intensity in order to achieve a quantitative evaluation of the flexibility that is given in terms of either persistence length, lp, or Kuhn length, b, where b ) 2lp. This will be done using a nonlinear leastsquares fitting procedure based on a model cross section for a single chain with excluded volume effects.28 We will


explicitly use the information obtained by IFT and SQDEC techniques on local length scales to reduce the number of adjustable parameters. An example of the data in the q regime that contains the necessary information is given in Figure 6A, which shows the scattered intensity for C16E6 in D2O at a concentration of c ) 0.6 mg/mL and a temperature T ) 26 °C in the so-called bending rod or Holtzer plot qI(q) versus q.60,61 For semi-flexible polymers, the Holtzer plot is ideally suited to visualize the distinct scaling regions discussed in context with Figure 1, and it directly demonstrates the crossover from a rigid rod-like to a flexible chain-like behavior at the length scale of the persistence length. Moreover, the Holtzer plot not only yields a first estimate of lp from the onset of the crossover, but it provides a sensitive way to test the agreement between theoretical and experimental scattered intensity for semiflexible chains as it significantly amplifies all deviations between data and theoretical curve.25,28,29 The data plotted in Figure 6 already show the importance of flexibility in the description of the structure of giant nonionic micelles and are in disagreement with the earlier findings deduced from SANS experiments on shearaligned micelles by Cummins et al.38,39 For rigid cylinders of finite length L, we would expect to find a plateau in the Holtzer plot at higher values of q due to the asymptotic q-1 behavior of an infinitely thin rigid rod, followed by a decrease to zero for q < 1/L due to the finite length, and an exponential decay caused by the finite thickness of the cylinder in the cross-section Guinier regime at higher values of q. However, Figure 6A clearly shows the upturn in qI(q) at lower values of q, which is characteristic for semi-flexible chains and should occur at qlp ≈ 1.9.23,61 In contrast to the physical properties on local length scales, which are not significantly influenced by interaction effects, we have previously demonstrated that the situation is more complicated on intermediate length scales.29 The evaluation of the scattered intensity is hindered by the fact that a clear distinction between the intrinsic stiffness and interaction effects is difficult to achieve. Even without interactions, the precise determination of the flexibility causes problems, because the often used crossover relationships or asymptotic expansions from the coil or rod limit are too uncertain to provide a good estimate of the persistence length. To overcome these problems, an extensive Monte Carlo simulation study has been performed,27 which aimed to determine a suitable model cross section that could be applied in a least-squares analysis of real worm-like micelles.28 The model used in the simulations is a discrete representation of a Kratky-Porod worm-like chain model with excluded volume interactions in the pseudocontinuous limit. Details of the simulations and the evaluation of the simulated scattering data have been given elsewhere.27,28 A numerical parametrization of the scattering function of semiflexible chains with excluded volume effects was performed using the following expression, which follows an approach described by Yoshizaki and Yamakawa62

Swc(q,L,b) ) [(1 - χ(q,L,b))Schain(q,L,b) + χ(q,L,b) Srod(q,L)]Γ(q,L,b) (9) where Schain(q,L,b) is the scattering function of a flexible (60) Holtzer, A. J. Polym. Sci. 1955, 17, 432. (61) Denkinger, P.; Burchard, W. J. Polym. Sci. B: Polym. Phys. 1991, 29, 589. (62) Yoshizaki, T.; Yamakawa, H. Macromolecules 1980, 13, 1518.

Langmuir, Vol. 14, No. 21, 1998 6021

chain with excluded volume effects, Srod(q,L) is the scattering function of a rod, χ(q,L,b) is a crossover function, and the function Γ(q,L,b) corrects the crossover region. Numerical approximations to these functions have been determined and these provide interpolations between the simulated functions.28 The finite size of the local cross section is included in a separate scattering function Scs(q) of a simplified two-shell cross section, i.e., a decoupling approximation has been used.28,29,54 It is known from previous SLS26 and SANS25 experiments that size polydispersity has to be included due to the equilibrium nature of the micelle. The scattering function of polydisperse worm-like micelles is given as the z-average

〈Swc(q,L,b)〉z )

∫N(L)L2Swc(q,L,b) dL ∫N(L)L2 dL

(10)

where N(L) is the number distribution of worm-like chains with contour length L, and Swc(q,L,b) is the normalized scattering function given by eq 9. In our data analysis we apply the Schulz-Zimm distribution,63,64 and the polydispersity is fixed to 〈M〉w/〈M〉n ) 2.1 In the data evaluation we follow the same scheme as previously established in the analysis of SANS data from polymer-like reverse micelles.29 A typical fit procedure involves the optimization of seven parameters. Five of them are determined by the high-q part of the data and were obtained from an independently performed analysis of the local properties with the help of IFT and SQDEC techniques, as described in the previous section. On the local length scale we have the inner and outer radius of the cylindrical cross section (R1, R2), the mass per length (ML), the ratio of the scattering length densities of inner and outer core (F1/F2), and the background. On the intermediate and global scale we have the persistence length lp and the contour length L. As mentioned before, we have fixed the polydispersity to the value 〈M〉w/〈M〉n ) 2. In a first step of our analysis we check and optimize the input values of the local physical parameters via a grid search, after that we continue the least-squares analysis with all parameters until the fitting procedure converges. As shown in Figure 6, the resulting fit based on this procedure provides an excellent description of the experimental data. Having established that the numerical expression for the scattering function given in eq 9 is capable of reproducing the experimental data, we can proceed to analyze the data from samples with different concentrations. The resulting (apparent) persistence lengths lp,app are shown in Figure 7, and we observe a linear dependence of the calculated persistence length on concentration. We have previously demonstrated,29,30 that this is caused by the application of a least-squares fitting procedure, which is based on the single-coil scattering function, where interaction effects are neglected, to scattering data from samples at finite concentrations. Results from a many chain Monte Carlo simulation study30 of the worm-like chain model have indeed demonstrated that interaction effects do influence the structure factor even in the q region used to determine lp. The simulations clearly revealed that the change in lp,app with concentration is not a result of a change in the intrinsic chain flexibility, but a structure factor effect on a relatively local scale. To estimate the true persistence length, we thus have to make an (63) Schulz, G. V. Z. Phys. Chem. 1939, B43, 25. (64) Zimm, B. H. J. Chem. Phys. 1948, 16, 1099.

6022 Langmuir, Vol. 14, No. 21, 1998

Jerke et al.

Figure 7. Apparent persistence length lp,app versus concentration for charged [(0) [C16SO3Na]/ctot ) 0.06, T ) 26 °C] and uncharged [(O) T ) 26 °C; b: T ) 35 °C] surfactants. In the limit of c f 0 the linear fits result in lp ≈ 170 Å for uncharged surfactants and lp ≈ 280 Å for charged surfactants.

Figure 6. Comparison of experimental SLS data (3) and experimental SANS data [(O) λ ) 12 Å, d ) 17.9 m; (0) λ ) 9 Å, d ) 10 m; (4) λ ) 9 Å, d ) 1.4 m] for polymer-like micelles of C16E6 in D2O with theoretical curves (solid line) based upon a least-squares fitting procedure of a single-chain scattering function with excluded volume effects in the Holtzer or bending rod representation. (A) c ) 0.6 mg/mL, T ) 26 °C. (B) c ) 0.6 mg/mL, T ) 35 °C. (C) c ) 1.2 mg/mL, T ) 26 °C, charged with 6 wt % C16SO3Na.

extrapolation to c f 0. In this limit we find lp ≈ 170 Å. Having eliminated the effects of finite concentrations on lp,app, we thus provide a precise value for the persistence length of polymer-like nonionic micelles formed by C16E6 in D2O. Moreover, our results again clearly demonstrate, that estimates made from single measurements at finite concentrations or using crossover relations can only be used with caution. A comparison with previous estimates reveals, that the value of lp ≈ 170 Å obtained for C16E6 in D2O is of the same magnitude as those reported from SANS studies of polymer-like reverse micelles of lecithin in isooctane (lp ≈ 150 Å29) or cyclohexane (lp ≈ 120 Å24) and of aqueous solutions of cetylpyridinium bromide at high ionic strength (lp ≈ 190 Å16). On the other hand, the published values of the persistence length are generally considerably larger both for nonionic and for ionic systems (lp ≈ 300-500 Å,17,19,65) if a combined analysis of static and dynamic light scattering experiments is done. The most likely reason for this discrepancy is the neglect of polydispersity and intermicellar interaction effects, which is known to result in too high values of lp. Our results can also be compared with the earlier findings of Cummins et al.,38,39 where they analyzed SANS experiments on shear-aligned micelles. They concluded that the micelles only possess limited flexibility and that the effective rod length determined from the SANS pattern approximates the true geometry of the micelles. However, it is important to point out that the data analysis procedure chosen by Cummins et al. is not appropriate for wormlike chains or flexible polymers at arbitrary concentrations. It relies on the assumption that the angular distribution function originally derived from optical birefringence work on dilute rigid and monodisperse rods is also valid for rod-like micelles aligned in a shear field. While these authors have attempted to study the effect of flexibility and polydispersity in an ad-hoc fashion by assuming that the micellar orientations are distributed over a cone of angles about the mean orientational direction of the socalled equivalent rigid rod,66 this approach does not reproduce the actual structural properties of linear chains deformed by a homogeneous solvent flow. This becomes (65) Imae, T.; Ikeda, S. J. Phys. Chem. 1986, 90, 5216. (66) Cummins, P. G.; Staples, E.; Hayter, J. B.; Penfold, J. J. Chem. Soc., Faraday Trans. 1987, 83, 2773.


quite obvious when looking at the anisotropic scaling behavior observed in the static structure factor from linear polymers in shear flow,67,68 Moreover, it is dangerous to directly compare orientation angles (or orientation resistance) measured by flow birefringence and light or neutron scattering experiments, since these two techniques define orientational properties on different length scales and generally lead to different results,68,69 To achieve a self-consistent interpretation of the SANS results, this would probably require the use of nonequilibrium molecular dynamics (NEMD) simulations, as recently performed in order to study the static structure factor of chain molecules in shear flow.68 We can therefore conclude that the earlier SANS experiments provided a very important and quite unambiguous proof of the existence of large anisotropic micelles in a number of different systems with a quite accurate determination of the local cross-section dimensions but that the more global properties such as the overall contour length as a function of solution composition have to be interpreted with care. It would certainly be worthwhile to reanalyze the data using a more appropriate model that is based on our current knowledge on polymer dynamics. In contrast to these studies, we have directly taken advantage of the enormous progress made in the calculation of scattering functions from semiflexible chains in recent years and included excluded volume interactions, polydispersity, and flexibility in a self-consistent way in order to extract local and global quantities of the micelles. We have also analyzed the effects of temperature and electrostatic intramicellar interactions on the micellar flexibility. The experimental data from measurements of C16E6 in D2O at a concentration of c ) 0.6 mg/mL and two temperatures (T ) 26 °C and T ) 35 °C) given in Figure 3A already indicate that the scattering intensity in the intermediate-q regime is insensitive to temperature under these conditions. This is confirmed by the quantitative analysis of the persistence length, which is identical within the experimental accuracy for both temperatures (see Figure 7). Moreover, the close agreement between the fitted curve and the experimental data at T ) 35 °C shown in Figure 6B reveals that the micellar conformation has not changed and is still well described by the corresponding scattering function for worm-like chains with excluded volume effects. This indicates that the increase of the temperature has not resulted in a significant change of the solvent quality. A quite dramatic effect, however, can be achieved upon the addition of a small amount of ionic surfactants. The resulting change in the apparent persistence length is given in Figure 7. We see that lp increases by almost 70%, while the corresponding q dependence of the intensity is still very well reproduced by the scattering function for worm-like chains with excluded volume effects given in eq 9. A similar effect has already been observed in aqueous solutions of lecithin and bile salt, where a decrease of the bile salt to lecithin molar ratio in the micellar aggregates was accompanied by a decrease in the apparent persistence length.76 This was interpreted as a decrease of lp,e due to (67) Lindner, P.; Oberthu¨r, R. Physica B 1989, 156 & 157, 410. (68) Pierleoni, C.; Ryckaert, J.-P. Phys. Rev. Lett. 1993, 71, 1724. (69) Bossart, J. C. Ph.D. Thesis, No. 12085; ETH Zu¨rich, Switzerland, 1997. (70) Dautzenberg, H.; et al. Polyelectrolytes: Formation, Characterization and Application; Hanser: Mu¨nchen, 1994. (71) Odijk, T. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 477. (72) Skolnick, J.; Fixman, M. Macromolecules 1977, 10, 944. (73) Fixman, M.; Skolnick, J. Macromolecules 1978, 11, 863. (74) Barrat, J. L.; Joanny, J. F. Adv. Chem. Phys. 1995, 94, 1. (75) Stevens, M.; Kremer, K. J. Chem. Phys. 1995, 103, 1669.

Langmuir, Vol. 14, No. 21, 1998 6023

the decrease in the linear charge density. We can try to relate this increase in lp to an electrostatic intrachain interaction effect that has, for example, been considered in the theory of polyelectrolytes. For polyelectrolytes it has become common practice to divide the total persistence length into an electrostatic lp,e and an intrinsic or “bare” part lp,0 such that lp,tot ) lp,0 + lp,e.70 Several theoretical models that account for the influence of electrostatic interactions on the persistence length and on the effect of screening from added salt have been presented in the past.71-75 A frequently used model to calculate lp,e is the so-called OSF theory, which has independently been derived by Odijk,71 Skolnick, and Fixman.72,73 In this theory, electrostatic contributions to the bending energy are calculated for a rather stiff worm-like chain using a screened Coulomb potential (i.e., a Debye-Hu¨ckel approximation) for the electrostatic interactions. In this context a worm-like chain is considered to be stiff if the condition lp,0 . λD holds, where

λD )

x

0rkBT 2e2I

(11)

is the Debye screening length. I is the ionic strength, 4π0r is the relative permittivity, and e is the elementary charge. The OSF theory predicts the following expression for the electrostatic contribution to the persistence length,71,72

lp,e )

()

λ B λD 4 λC

2

(12)

under the following two assumptions: The contour length L exceeds the Debye screening length λD, and the distance between charges along the contour, λC, is large compared to the Bjerrum length,

λB )

1 e2 4π0r kBT

(13)

defined as the distance where thermal and electrostatic energy of two elementary charges are comparable, which in our case leads to λB ≈ 7 Å. For λC , λB the electrostatic contribution to the persistence length is given by 2

1 λD lp,e ) 4 λB

(14)

For our measurements, we have added approximately 9.3% by number of ionic surfactant. We have specifically chosen the ionic surfactant such as to minimize the number of free monomers and thus we can safely assume that all of the ionic surfactant is indeed incorporated in the micelles. On the basis of the known mass per length of the micelles, this results in approximately λC ) 5.8 Å. In our experiments with charges, we have added a salt concentration of 0.01 M NaCl in order to work with a controlled value of the ionic strength, which yields a value of the Debye screening length of λD ) 30 Å. For our system, lp,e should thus be bracketed by the values derived from eqs 12 and 14, i.e., 33 Å < lp,e < 45 Å. If we compare this to the measured difference of the persistence length of C16E6 micelles with and without added charges of approximately 100 Å (see Figure 7), we see that we indeed find the right magnitude for the effect of electrostatic (76) Egelhaaf, S. U.; Schurtenberger, P. J. Chem. Phys. 1994, 98, 8560.

6024 Langmuir, Vol. 14, No. 21, 1998

interactions. The value of the measured lp,e should also be interpreted with some caution since the actual concentration of the added ionic surfactant and the ionic strength contains a rather large uncertainty due to the relatively small sample volumes with which we worked. However, it is important to point out that with our data analysis approach the actual values of the measured persistence length can be determined with very high precision for a given system due to the strong scattering of the micelles, which results in data of remarkable accuracy even at low concentrations. This opens up very interesting possibilities to investigate the effect of electrostatic interaction on the micellar flexiblity as a function of parameters such as the linear charge density or the ionic strength, which both can be varied continuously within a considerable range for the present system. 5. Conclusions A quantitative knowledge of the micellar persistence length is of vital importance as it provides a direct link to the bending modulus of the micelles as one of the key parameters of the flexible surface model. While numerous attempts have been made to determine lp in different systems, in most cases the thus obtained experimental values have to be considered with caution, as effects from intermicellar interactions and polydispersity were not included. We believe that the approach described in this paper now allows for a precise determination of lp as we have demonstrated that it is possible to include excluded volume interactions, polydispersity, and flexibility in a self-consistent way in the calculation of scattering functions from semiflexible chains in order to extract local and global quantities of the micelles.

Jerke et al.

However, we believe that this not only provides us with new information about micellar properties but also has consequences that reach beyond surfactant science. Previously, the application of concepts from polymer physics has had an enormous impact on the understanding of the structural and dynamic properties of micellar solutions. We believe that we are now in a position to “return this favor” and use micelles as ideal model system for “equilibrium polyelectrolytes”. The effect of charges on the flexibility of polyelectrolytes has been the subject of intense experimental and theoretical investigations and resulted in highly controversial results. Most experimetal studies with classical polyelectrolyte systems using SANS or SAXS have suffered from their rather weak scattering power, which makes it almost impossible to produce data with a sufficient accuracy over the required q range from dilute solutions, where one still can resolve single-coil properties. This is clearly not the case for the experimental system presented in this study, and we believe that aqueous solutions of polymer-like nonionic micelles doped with ionic surfactants can serve as ideal model systems for “equilibrium polyelectrolytes” and help to clarify some of the open questions in the polyelectrolyte literature. Acknowledgment. The support by the Swiss National Science Foundation (Grant 20-40339.94, 20-46627.96) is gratefully acknowledged. We thank John Daicic for his careful reading of the manuscript. The neutron scattering experiments were performed at the instrument D22 of the Institut Laue-Langevin in Grenoble. We also gratefully acknowledge the support from Risø National Laboratory, where preliminary SANS experiments were done. LA980390R

Flexibility of Charged and Uncharged Polymer-like Micelles

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