Micellar Growth in Salt-Free Aqueous Solutions of a Gemini Cationic

We study salt-free aqueous solutions of a cationic gemini surfactant in the concentration range between the critical micellar concentration and the on...
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Langmuir 2003, 19, 992-1000

Articles Micellar Growth in Salt-Free Aqueous Solutions of a Gemini Cationic Surfactant: Evidence for a Multimodal Population of Aggregates V. Weber,† T. Narayanan,‡ E. Mendes,† and F. Schosseler*,† Laboratoire de Dynamique des Fluides Complexes, UMR 7506, (CNRS-ULP), 3 rue de l’Universite´ , 67084 Strasbourg Cedex, France, and ESRF, BP 220, 38043 Grenoble Cedex, France Received June 7, 2002 We study salt-free aqueous solutions of a cationic gemini surfactant in the concentration range between the critical micellar concentration and the onset of the semidilute regime by combining SAXS, static and dynamic light-scattering techniques. The results give strong evidence for the coexistence of small, intermediate, and very large micellar sizes in the whole range of concentration. The qualitative evolution of the different micellar populations could be followed as a function of concentration and is found in good qualitative agreement with recent cryo-TEM observations on the same system by Bernheim-Grosswasser et al.

I. Introduction Above the critical micellar concentration (cmc), amphiphilic molecules in aqueous solutions self-associate and form small spheroidal aggregates to decrease the contacts between the hydrophobic parts of the molecules and the surrounding solvent. A small fraction of the surfactant molecules remain free in the solution, with a concentration close to the cmc value. As the concentration increases, the number of aggregates increases with an average constant size until, above a second critical concentration, referred to as the second micellar concentration, the micelles start to increase their average size and form disclike or threadlike aggregates.1 Micellar growth has attracted considerable interest since it modifies strongly the properties of the solutions and in particular their rheological behavior.2 The simpler phenomenological models for the growth of threadlike micelles assume a cylindrical body and hemispherical end caps with the same radius and a lower standard chemical potential for the surfactant molecules in the cylindrical body than in the hemispherical caps.3 They predict a monotonic growth of the micelles above the cmc and a smooth distribution of sizes with a single maximum shifting to larger sizes with increasing concentration. In the case of ionic surfactants, the Coulombic interactions play a major role in defining the standard chemical potential of the molecules but the overall picture remains the same,4,5 in contradiction with the experimentally observed existence of a second cmc. † Laboratoire de Dynamique des Fluides Complexes, UMR 7506, (CNRS-ULP). ‡ ESRF.

(1) Israelachvili, J. N.; Mitchell, D.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (2) Cates, M.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (3) Ben-Shaul, A. In Micelles, Membranes, Microemulsions, and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer: Berlin, 1994. (4) Mackintosh, F. C.; Safran, S. A.; Pincus, P. A. J. Phys.: Condens. Matter 1990, 2, 359; Europhys. Lett. 1990, 12, 697.

As first shown by Porte et al.,6 taking into account the very different local structure of the surfactant molecules in their local environment (i.e., spherical vs cylindrical geometry as well as the junction zones between)1 can result in the energy barrier, associated with the creation of unfavorable end caps and the subsequent micellar growth, that is necessary to explain the second cmc. They proposed that the growing micelles, although retaining a cylindrical symmetry, should have spherical end caps with a larger radius than the cylindrical body. Recently, the progress in the cryo-TEM technique allowed Bernheim-Grosswasser et al. to investigate very dilute solutions of a cationic gemini surfactant at concentrations between about 5 and 25 cmc.7 The cryo-TEM images show distinctly the coexistence of small spheroidal aggregates along with much longer threadlike micelles. At the limit of the experimental resolution, the micellar end caps appear also bulkier than the cylindrical body. Motivated by this de visu demonstration of the existence of the second cmc, May and Ben-Shaul proposed very recently8 a molecular theory of the sphere to rod transition in dilute micellar solutions taking into account the molecular arrangement as a function of local curvature of the micelles. In particular, within these conditions, they were able to calculate the size distribution function as a function of concentration and showed distinctly that the onset of micellar growth takes place at a critical concentration and that there is a gap in the size distribution above this concentration; i.e., small aggregates coexist with much larger micelles. The objective of this paper is to investigate by scattering techniques samples at thermal equilibrium and to compare (5) Safran, S. A.; Pincus, P. A.; Cates, M.; Mackintosh, F. C. J. Phys. (Paris) 1990, 51, 503. (6) Porte, G.; Poggi, Y.; Appel, J.; Maret, G. J. Phys. Chem. 1984, 88, 5713. (7) Bernheim-Grosswasser, A.; Zana, R.; Talmon, Y. J. Phys. Chem. B 2000, 104, 4005. (8) May, S.; Ben-Shaul, A. J. Phys. Chem. B 2001, 105, 630.

10.1021/la0205295 CCC: $25.00 © 2003 American Chemical Society Published on Web 01/24/2003

Evidence for a Multimodal Population of Aggregates

the results with the observations on frozen samples made by other authors with the cryo-TEM technique. Our experimental conditions are slightly different from those in ref 7. We use the same surfactant but in D2O instead of H2O to match the conditions where shear-thickening behavior has been studied in detail for this system.9-11 The intriguing increase of the steady-state viscosity above a critical shear rate has attracted much interest12 and is still not understood. Therefore, it is critical to get a good picture of the equilibrium state solutions at rest before trying to understand their behavior under shear. We use small-angle X-ray scattering as well as static and dynamic light scattering to cover a broad range of characteristic sizes for solutions with concentrations between about the cmc and the onset of semidilute regime. SAXS results are first presented, analyzed, and discussed (section III). We then proceed along the same lines with the light-scattering results (section IV). The final discussion tries to summarize these results into a consistent picture (section V).

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Figure 1. SAXS intensity as a function of scattering wavevector for micellar solutions with varying concentration. The curves are normalized by the concentration of bound surfactant.

III. SAXS II. Experimental Section Materials. The gemini surfactant ethanediyl-1,2-bis(dodecyl dimethylammonium bromide),13 hereafter called 12-2-12, has been synthesized in our laboratory. Solutions are prepared by weighing the surfactant molecules in D2O. They are stirred at 50 °C for a few hours to ensure a complete dissolution and then filtered through 0.45 µm filters. Heavy water has been used as the solvent throughout this study to allow comparison with earlier studies.9-11 The critical micellar concentration of 12-2-12 in H2O at 25 °C is φcmc ) 0.84 ( 0.04 mM,13 and we have used this value as a good approximation for D2O since deuteration of the solvent has in general only a weak effect (about a few percent) on the cmc values.14,15 Small-Angle X-ray Scattering. SAXS measurements were performed at the beam line ID02, European Synchrotron Radiation Facility, Grenoble, France. The wavelength of the incident photons was λ ) 1 Å, and two sample-detector distances were used, 1 and 5 m, allowing for scattering wavevectors in the range 8 × 10-3 < q (Å-1) < 0.7, where q is the defined as q ) 4π/λ sin(θ/2), θ being the scattering angle. The measurements were performed using a two-dimensional multiwire proportional gas counter. Samples were studied at room temperature in a quartz capillary. Because of the very small level of scattering intensities ( 0.2 Å-1). In the second step, data points for q values below 1.1 × 10-2 Å-1 were discarded for the analysis because of the very poor signal-to-noise ratio in this region and we fitted the remaining scattering intensity curves with eqs 1-3 and 5, while allowing R and L as free parameters and keeping the Fi’s and Ri’s to fixed values, either specific to each concentration or averaged over the four concentrations investigated. The resulting values of R and L thus obtained have been averaged and the mean values are displayed in Table 2, together with useful quantities discussed in the previous sections. Figure 2 shows the fits obtained with the average values of Fi’s and Ri’s together with the experimental curves. Discussion The agreement between the measured and the calculated curves is rather satisfactory. However the number of fitting parameters and uncertain experimental quantities is rather large and the physical consistency of the parameters has to be checked. As noted above, the fitted Fij’s can be considered in good agreement with their estimated values. They do not appear to depend clearly on the surfactant concentration and this would favor a minor effect of the degree of ionization on the contrast density. In fact, our estimation for F2 would yield a moderate 10% variation for the R values in Table 2, consistent with the numerical results. Our confidence in the estimations of the contrast densities is also supported by the Ri values that are consistent with published values in the literature. Moreover, the fact that the absolute intensities can be fitted in the high q range without any additional numerical factor indicates that our set of values for the Fi’s and Ri’s is very close to the actual one. It can be noted that the oscillations in the form factor at large q values are not present in the experimental data. The same feature was already observed for spherical micelles27 and has currently no clear explanation as far as we know.

It is however likely that the finite smoothness of the interface between the solvent and the cylinders averages the oscillations of the form factor. Indeed the theoretical curves oscillate around the measured ones. The situation is not as clear for the parameters L and R that are strongly dependent on the model used to calculate S(q). The general trend with increasing φ, i.e., L increases and R decreases, is consistent with the expected behavior for these cationic surfactants forming entangled micellar solutions above a critical overlap concentration. The inspection of the quantity φvmL/R2 (Table 2) shows that it remains much smaller than 1 for the three most dilute solutions and begins to be comparable to 1 only for the most concentrated one. Thus, our criterion for the decoupling of rotational and positional correlations seems fulfilled. On the other hand, the range of the interaction potential is never large compared to the length of the cylinders. We obtain a (σ + κ-1)/L value close to 1 already for φ ) 14.48 mM, and this parameter becomes even smaller than 1 for the largest concentration. This is confirmed by a s ) 1 value, which means that the electrostatic repulsion is not large enough to forbid close contact configurations of the initial equivalent hard spheres. We expect then a breakdown for the approximation of the interaction potential between the cylinders by a spherical potential for concentrations in the range 1518 mM, which is precisely the overlap concentration range estimated from rheology experiments and from SANS experiments for this system.9-11 This breakdown could explain why the interaction peak is not well described by the fit for the largest concentration. Our values for the degree of ionization R are consistent with published estimations (measurements) by other techniques.28 Nevertheless, in the present case, we have to emphasize that our values depend on the definition used for κ-1 (eq 5). For example, if one neglects the electrostatic screening by the free surfactant, the main effect is to decrease the R values. However, this does not affect significantly the overall quality of the fits, the L values and the general trend for R to decrease with increasing φ. This is due to the fact that L is essentially fixed by the position of the interaction peak, while R and κ-1 have only a slight effect on its intensity and a strong effect on the intensity at small q values. This q range could not be accessed due to the very poor signal-to-noise ratio. Therefore, the range of R variation can be arbitrarily shifted by changing the definition of κ-1 without affecting the L values and the quality of the fits. Only accurate data at smaller q values could help to clarify this point. One has however to remember that, even in this case, the R value is an effective degree of ionization in the approximation of the spherical interaction potential. Its relationship to the true degree of ionization of the cylinders is probably not easy to establish. Finally we note that the influence of the polydispersity in the lengths of the micelles is difficult to estimate. The fitting procedure is able to return an average value for the length of the cylinders. This average value seems physically consistent but it is not possible to determine which moment of the distribution of lengths is evaluated through the use of a mean spherical interaction potential between identical equivalent spheres. Some hints will be given in the next section devoted to light-scattering experiments. (27) Cantu`, L.; Corti, M.; Zemb, T.; Williams, C. J. Phys. IV Fr. 1993, 3, C8-221. (28) Zana, R. J. Colloid Interface Sci. 2002, 246, 182.

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Figure 3. Light-scattering intensity as a function of scattering wavevector for micellar solutions with varying concentration (from bottom to top: φ (mM) ) 1.81, 7.21, 18.1, 36.2).

Figure 5. Scattering angle dependence of the distribution of relaxation times in the electric field autocorrelation functions measured at φ ) 7.21 mM.

Figure 4. Typical examples for the measured electric field autocorrelation function (closed symbols) and the distribution of relaxation times (open samples) obtained through the constrained regularization method (CONTIN) with an ajustable baseline (φ ) 7.1 mM, θ ) 45°).

Figure 6. Scattering wavevector dependence of the relaxation times measured for different micellar concentrations. The straight lines have a slope of -2.

IV. Static and Dynamic Light Scattering Results. Figure 3 shows the variation of the static lightscattering intensity as a function of scattering wavevector q. The most striking feature is the existence of a marked upturn at small q values. Considering the results from the SAXS experiments, one would have expected that the scattering intensities at low q values, in the range of lightscattering experiments, would be dominated by interparticle correlation effects and exhibit very small q dependence. The complexity of the system is confirmed by the shape of the electric field autocorrelation functions g(1)(q,t) that are obviously not single-exponential functions (Figure 4). The analysis by the CONTIN software allows one to obtain P(τ), the distribution of relaxation times, and shows the presence of three distinct decay modes (Figure 4) for the three concentrations 1.81, 7.21 and 18.1 mM. We note that for most samples, it was necessary to include a floating baseline in the CONTIN analysis to take into account the small offset of g(1)(q,t) (see Figure 4) at long delay times. This offset arises from strong but rare fluctuations in the scattering intensities that are always found in our experiments. Since the solutions are filtered, these fluctuations are likely due to very large aggregates present in the solutions; For the largest concentration φ ) 36.2 mM, the CONTIN analysis yields one distinct peak in the range 10-5-10-3 s, depending on the scattering angle, and a large number of weak peaks at longer delays times.

For this last concentration, the long time tail of g(1)(q,t) is best described by a stretched exponential function A exp(-(t/τ0)b), which yields an average decay time τ ≈ Γ(1/b) τ0/b, where Γ(x) is Euler’s gamma function and b ≈ 0.31. The characteristic times and the amplitudes of the decay modes vary in a consistent way with scattering angle (Figure 5), indicating that the peaks in P(τ) are not generated by an artifact in the CONTIN analysis. We note τi (i ) 1, 2, 3) in increasing order the relaxation times measured for the three most diluted samples and τ1 (respectively τ3) the characteristic times measured for the fast (respectively slow) decay mode in the most concentrated solution. These notations are justified by Figure 6, which shows the q dependence of these relaxation times for the four concentrations investigated. The time τ1 corresponds to a diffusive process that is nearly independent of concentration in the range investigated. The time τ2 is also associated with a diffusive relaxation and seems to exhibit a nonmonotonic variation with concentration. Finally the time τ3 displays a marked increase with concentration and a crossover from diffusive behavior at small concentrations to a more pronounced q dependence, consistent with a q-3 decay, for the largest concentration. The values of the corresponding diffusion coefficients are given in Table 3. Analysis. To proceed further with the analysis of the scattering intensities, we use the fact that several distinct decay modes are systematically found in the decay of g(1)(q,t) and we write the scattered electric field as a sum of independently fluctuating components

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Table 3. Diffusion Coefficient Calculated from the Dynamic Light Scattering Results φ (mM)

D1 (10-6 cm2 s-1)

D2 (10-7 cm2 s-1)

D3 (10-8 cm2 s-1)

1.81 7.21 18.1 36.2

0.85 ( 0.05 1.33 ( 0.14 0.84 ( 0.075 1.08 ( 0.09

1.32 ( 0.27 2.33 ( 0.40 0.70 ( 0.15

1.87 ( 0.43 1.78 (0.35

Es(q,t) )

∑i Ei(q,t)

(6)

where the number of components is 2 or 3 depending on the concentration. Then, by definition of g(1)(q,t),

g(1)(q,t) )

∑i 〈Ei(q,0)E/i (q,t)〉

(7)

I(q) since the cross-correlation terms vanish in the time average due to the hypothesis of independent fluctuations. Defining the intensity Ii(q) as the intensity associated with the fluctuations of surfactant concentration with decay time τi, and using the normalization condition

g(1)(q,0) )

∑i ai(q)

(8)

we identify29-33

Ii(q) ) ai(q)I(q)

(9)

where ai(q) is the area of the peak of p(τ) associated with time τi (Figure 5). Thus, this analysis allows one to split the scattering intensity according to the contributions by the fluctuations of concentration with different decay modes. Figure 7 shows the results for two of the investigated concentrations. Discussion. As shown in Figure 7, within experimental accuracy, the contributions from the two fastest decay modes to the total intensity show no dependence on the scattering wavevector and can be associated with an average intensity value Ii that depends on surfactant concentration. This is not the case for the contribution of the slowest decay mode. It is then clear that the evolution of the shape of the total scattering intensity in Figure 3 reflects the evolution of the weights of the different contributions with surfactant concentration. Figure 8 summarizes the variation with φ of the different quantities determined so far. From Figure 8a, it appears that the fastest mode could be attributed to small aggregates with constant size and increasing concentration as φ increases. Various attempts to extract a characteristic length from the experimental D1 values all converge to small values in the range of the diameter of the aggregates, independent of the refinement of the models used.34-37 We can therefore conclude that the fastest decay mode is most likely due to the trans(29) Zhou, Z.; Chu, B.; Peiffer, D. G. Macromolecules 1993, 26, 1876. (30) Raspaud, E.; Lairez, D.; Adam, M.; Carton, J. P. Macromolecules 1994, 27, 2956. (31) Zhang, Y.; Wu, C.; Fang, Q.; Zhang, Y.-X. Macromolecules 1996, 29, 2494. (32) Sun, T.; King, H. E. Macromolecules 1996, 29, 3175. (33) Klucker, R.; Munch, J. P.; Schosseler, F. Macromolecules 1997, 30, 3839. (34) Lin, S. C.; Lee, W.; Schurr, J. M. Biopolymers 1978, 17, 1041. (35) Tirado, M. M.; Garcı´a de la Torre, J. J. Chem. Phys. 1979, 71, 2581. (36) Liu, H.; Skibinska, L.; Gapinski, J.; Patkowski, A.; Fisher, E. W.; Pecora, R. J. Chem. Phys. 1998, 109, 7556. (37) Eimer, W.; Pecora, R. J. Chem. Phys. 1991, 94, 2324.

Figure 7. Scattering wavevector dependence of the intensities Ii(q) associated with the different relaxation modes in the solutions (see text): (a) φ ) 7.21 mM; (b) φ ) 18.1 mM.

lational diffusion of small spherical or nearly spherical micelles, which are then still present for the largest concentration where a large increase of the zero shear viscosity is already measured.9,10 The average intensity and the diffusion coefficient associated with the second decay mode display a nonmonotonic variation as a function of φ (Figure 8b). If the variation of D2 is interpreted in terms of a characteristic size, then the variation of the average intensity I2 would imply a nonmonotonic variation with φ of the concentration of the corresponding aggregates. It must be emphasized that these are only conjectures since the average intensities as well as the diffusion coefficients are measured at finite concentration and include the effects of electrostatic interactions. If we try nevertheless to estimate a characteristic size corresponding to a typical value D2 ≈ 10-7 cm2/s, we can choose between different approaches. If we first assume boldly that D2 is the translation diffusion coefficient of a rigid rod, we get24,35 L ≈ 1800 Å, which would correspond to a radius of gyration (L2/12 + R2/2)1/2 ≈ 520 Å, i.e., a value too large to be consistent with the absence of angular variation for I2(q). Furthermore, the hypothesis of rigid rods becomes then inconsistent with a natural persistence length lp ≈ 200-300 Å for wormlike micelles.38,39 On the other hand, we can use an other extreme hypothesis and assume that D2 corresponds to the translation diffusion of wormlike chains. To our knowledge, there is no calculation for the friction factor of such objects and we assume simply that the expression for random coils is suitable, i.e., D2 ) kT/(6πη0RH), where (38) Zana, R. Private communication. (39) Sommer, C.; Pedersen, J. K.; Egelhaaf, S. U.; Cannavacciuolo, L.; Kohlbrecher, J.; Schurtenberger, P. Langmuir 2002, 18, 2495.

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Figure 9. Scattering wavevector dependence of the lightscattering intensity associated with the third relaxation mode. Lines are guides for the eye.

values out of the experimental range. By using the form factor of a Gaussian chain with RG ≈ 1660 Å, we can estimate the zero q extrapolation of the data for φ ) 1.81 mM and φ ) 7.21 mM in Figure 9 to be about twice the value measured at the smallest scattering angle. Again if we assume I3(0) ∼ φ3M3, we then estimate φ3/φ1 ≈ 4 × 10-4. The continuous crossover to a q-3 dependence of τ3 as φ increases could then be interpreted as the appearing contribution of internal modes of relaxation for still larger aggregates. Figure 8. Variation with the bound surfactant concentration of the average light-scattering intensity and of the diffusion coefficient associated with the first (a) and the second (b) relaxation modes of the electric field autocorrelation function.

the hydrodynamic radius is about 0.66 RG, with RG being the radius of gyration (Gaussian coils). Using standard relationships for wormlike chains,40 this estimation yields RH ≈ 220 Å, RG ≈ 330 Å, and L ≈ 1800 Å for lp ≈ 200 Å. It is not clear that the friction factor of such a short wormlike object can readily be approximated by that of a random coil. Nevertheless, the orders of magnitude thus obtained for RH and RG are consistent with the experimental behaviors of D2 and I2(q). Although we do not believe that the micellar aggregates actually have randomcoil or rodlike shapes, the above calculations clearly demonstrate that the order of magnitude for the contour length does not depend strongly on the approximations involved. If the respective magnitudes of I1(q) and I2(q) are then interpreted solely on the basis of concentration φi and molecular weights Mi of the associated objects, i.e., Ii ∼ φiMi, the concentration of the short semiflexible cylinders would be about 10 times smaller than the concentration of the small spherical micelles for the two smallest surfactant concentrations. It would then decrease and progressively vanish as φ is increased. Finally, for the slowest decay mode, we can only define a diffusion coefficient for the two most dilute solutions (Table 3). The shape of the associated intensity changes with φ while the average level is approximately constant for the four concentrations (Figure 9). Using a typical value D3 ≈ 2 × 10-8 cm2/s (Table 3) and the same approximation of random coils as above, we obtain RH ≈ 1100 Å, RG≈ 1660 Å, and L ≈ 42 000 Å. These surprisingly high values are nevertheless consistent with the fact that the intensity I3(q) shows no saturation at low q values for the two most dilute solutions (Figure 9), indicating that the Guinier range for the corresponding objects is at still smaller q (40) Rawiso, M. J. Phys. IV Fr. 1999, 9, Pr1-147.

V. General Discussion and Conclusions We try now to reconcile the somewhat divergent pictures emerging from our SAXS and light-scattering experiments. In the concentration range investigated by SAXS, three distinct relaxation modes are present in the electric field autocorrelation functions measured by dynamic light scattering at smaller q values. Our analysis of the associated intensities and of their q dependence shows that at larger q values, the dominant contribution to the intensity measured in the SAXS q range should be I1(q) (Figure 7) since I2(q) and I3(q) arise from larger objects that are present in much smaller concentrations. The SAXS analysis has been performed with the form factor of cylinders and, as noted above, the use of the form factor of spheres gives unsatisfying results. This appears in contradiction with the various estimates of the length L of the micelles from the experimental D1 values, which point consistently to small objects. At that point, the SAXS results appear more robust than these estimations and we would be inclined to discard the latter and to favor a cylindrical symmetry for the small aggregates. However it must be kept in mind that polydispersity effects have been neglected in the SAXS analysis and that in practice, size dispersity and shape dispersity can hardly be distinguished.41 We will come back to that point later on. The L values derived from the SAXS analysis could be the right order of magnitude, although there are still some pending questions regarding the use of a spherical interaction potential to model electrostatic interaction between charged rods. In particular, it is not clear that the increase of L with φ (Table 2) reflects only a micellar growth since, on the other hand, D1 is approximately constant. It is however true that, even for a constant D1, virial effects leave some place for a moderate increase of L with φ.34 Nevertheless we think some caveats should be issued if these L values are used to test models of micellar growth.19 (41) Hayter, J. B.; Penfold, J. J. Colloid Polym. Sci. 1983, 261, 1022.

Evidence for a Multimodal Population of Aggregates

Considering these difficulties for the objects associated with the fastest decay mode, we have no reason to have a high confidence in the values roughly estimated above for the characteristic dimensions of the objects responsible for the two other relaxation modes. However we believe that the orders of magnitude are correct since they are consistent with the q variations of Ii(q) and τi(q). Thus, it seems that three different populations of aggregates with characteristic contour lengths L1 ∼200 Å, L2 ∼2000 Å, and L3 > 104 Å coexist in the solutions. Furthermore, the data suggest that the smaller aggregates are still present at the overlap concentration (∼15-20 mM depending on the criterion used) while the intermediate size aggregates progressively disappear as the population of very large aggregates increases when the semidilute regime is reached. The q dependence of the total scattering intensity should then vanish as the correlation length of the solution decreases with increasing φ. We notice also that earlier transient electric birefringence measurements on the same system, in H2O at a concentration around the overlap concentration, have shown42 a complex multimodal relaxation of the induced electric birefringence, which could be related to the multimodal population of aggregates evidenced here. Similar behavior was also observed for CTAT solutions.43 This picture appears consistent with the recent TEM experiments performed by Bernheim-Grosswasser et al. on 12-2-12/H2O solutions.7 They observed “the coexistence of spheroidal micelles and long, threadlike micelles, the number and length of the latter increasing with the concentration at the expense of the former”, with “very few elongated micelles of intermediate sizes”. These observations have been made for concentrations in the range 4.2-24.5 mM with the samples being quenched from T ) 25 °C. Since the overlap concentration of 12-2-12 in D2O is smaller than in H2O,10 i.e., deuteration of the solvent promotes the micellar growth, it appears that we are able to detect long threadlike aggregates at smaller concentrations than in ref 7. Thus, our scattering results on samples at thermal equilibrium are in good general agreement with those obtained by the direct observation of quenched samples. There are however some differences. The first one concerns the population of small aggregates that disappear progressively with increasing concentration in the quenched samples while the small aggregates appear to be still present at 36.2 mM in our samples. This feature was predicted very recently by May and Ben-Shaul who calculated explicitly the evolution of the size distribution function as a function of concentration,8 while taking into account the different packing energy of the surfactant molecules in the end caps and in the cylindrical body of the elongated micelles. As shown first by Porte et al.,6 this implies an energy barrier to overcome for the growth of the micelles to occur and the existence of a second cmc beyond which the aggregates start to grow significantly. Thus, spheroidal micelles are predicted to coexist with larger aggregates beyond the second cmc as free surfactant molecules coexist with spherical micelles above the first cmc. Bernheim-Grosswasser et al. justify the progressive disappearance of small spheroidal micelles with increasing concentration by the application of the mass action law, the continuous increase of free counterions with φ and the higher degree of ionization of the spheroidal micelles.7 On the other hand, the effects of ionization were not considered in the theoretical approach (42) Oda, R.; Mendes, E.; Lequeux, F. J. Phys. (Paris) II 1996, 6, 1429. (43) Narayanan, J.; Manohar, C.; Mendes, E. J. Phys. Chem. 1996, 100, 18524.

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of ref 8 that neglected also the effects of intermicellar interactions on the growth. These interactions are clearly important in the concentration range investigated in our experiments as revealed by the structure factor peak (Figure 1), but their relevance to the growth law has yet to be investigated. It remains that some experimental discrepancy exists concerning the evolution with surfactant concentration of the population of small micelles in the quenched samples and in the samples measured at thermal equilibrium. The second discrepancy concerns the shape of the small micelles, which appear nearly spherical in the quenched samples, while, as noted above, a spherical form factor does not describe correctly the measured scattering intensity in the large q range where the structure factor is close to 1. However, on one hand, the TEM observations are close to the resolution limit of the technique and, on the other hand, it is very difficult to distinguish shape dispersity and size dispersity in the SAXS analysis. Also it must be kept in mind that some minor modification of the shape could occur upon the freezing of the samples if the characteristic time of the shape fluctuations is shorter than the quenching time. More experiments are needed before reaching a firm conclusion on this point. Finally we note that the location of the second cmc is not clear in our experiments. According to May and BenShaul,8 the leveling off of the concentration of the small aggregates should be a good criterion. If we assume simply I1 ≈ φ1M1, then this would correspond to concentrations of the order of 40 mM (Figure 8a), i.e., well in the semidilute regime for our system. However, we are able to detect very large aggregates at much smaller concentrations, slightly above the cmc. This is also true in the cryo-TEM images, which show a few elongated micelles with L ≈ 1000 Å at a concentration as low as 4.2 mM. Although the calculated size distributions8 display a tail of long micelles even below the second cmc, their concentration is too small to account for the respective amplitudes of I1 and I3 in our experiments and for the observation of long micelles in the cryo-TEM experiments. This discrepancy might be due to the fact that intermicellar interactions are neglected in the model. To conclude, our scattering experiments give strong evidence for the coexistence of micelles with different sizes in the range of concentration between the cmc and the semidilute regime, in good qualitative agreement with the results obtained by the cryo-TEM technique. Some divergence has been found concerning the evolution of the concentration of small micelles with total concentration as well as the shape of these small micelles. The results display some similarities with the behavior predicted by May and Ben-Shaul but a number of questions are waiting for more detailed experiments to be settled. We have also shown that the elucidation of these questions requires the careful combination of several techniques to compensate for the lack of fully reliable models to analyze the scattering experiments. In particular, the validity of the RMSA approach should be investigated in details with solutions of rigid rods of known length before its results can be used with confidence to test micellar growth models. Upon completion and submission of this paper, we became aware of two very recent papers covering studies on different dilute systems with the use of static light and neutron scattering44 or dynamic light scattering and rheology.45 In ref 44, it is very clear that dilute salt free (44) Truong, M. T.; Walker, L. M. Langmuir 2002, 18, 2024. (45) Oelschlaeger, Cl.; Waton, G.; Buhler, E.; Candau, S. J.; Cates, M. Langmuir 2002, 18, 3076.

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CTAT solutions also exhibit a strong upturn in the lightscattering q range while the SANS curves are completely consistent with our SAXS results. In particular, the use of the RMSA model in that q range would produce a fitting curve with no q dependence in the light-scattering q range. Moreover transient birefringence experiments on CTAT solutions have consistently shown three relaxation modes in the decay of the induced birefringence.43 In ref 45, an upturn in the total static scattering intensity and two modes of relaxation in the electric field autocorrelation function are observed for the dilute salt free solutions of a fluorinated surfactant suggesting the presence of large aggregates, although no analysis of the scattering intensity associated with the different relaxation modes is given.

Weber et al.

Thus, there are some indications that the behavior reported here is not specific to our system. Acknowledgment. We gratefully acknowledge R. Zana for his kind availability to answer precisely our questions. We also thank G. Porte and M. Delsanti for useful comments made on an early version of this manuscript.46 We are indebted to O. Gavat for the kind synthesis of the samples. LA0205295 (46) Weber, V. The`se Universite´ Louis Pasteur, Strasbourg, France, July 2001.