Rheological Behavior of Locally Cylindrical Micelles in Relation to

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Rheological Behavior of Locally Cylindrical Micelles in Relation to Their Overall Morphology Cl. Oelschlaeger, G. Waton, and S. J. Candau* Laboratoire de Dynamique des Fluides Complexes, U.M.R. No. 7506, Universite´ Louis Pasteur, C.N.R.S., 4 rue Blaise Pascal, 67070 Strasbourg, Cedex, France Received June 19, 2003. In Final Form: September 4, 2003 The kinetics of wormlike micelles formed in aqueous solutions of binary mixtures made of cetylpyridinium chloride and sodium salicylate in the presence of 0.5 M NaCl has been investigated by means of T-jump and rheology experiments. The micellar lifetime was found to follow power laws of the concentration with exponents in agreement with the theoretical models. The scission energy of the micelles was determined by combining T-jump and linear viscoelasticity measurements as a function of temperature. This paper also reviews previously reported rheological studies and presents a general discussion on the correlation between different dynamical mechanisms and counterion binding.

I. Introduction In the past two decades, evidence has accumulated about the formation in aqueous surfactant solutions of long flexible cylindrical micelles with a large spread in length and a behavior similar to that of polymer chains in solution.1,2 In the semidilute range, that is, at surfactant concentrations large enough that the elongated micelles overlap, the systems exhibit a viscoelastic behavior very reminiscent of that of transient polymeric networks. These wormlike micelles can break and recombine on a time scale which is dependent on the system and on the physicochemical conditions. A theory based on the tube model of polymer dynamics but including the effects of reversible scission kinetics has been derived by Cates to describe the viscoelastic properties of these systems.3,4 In the linear regime, the model predicts several rheological regimes depending on the relative rates of diffusive polymer motion and the reversible breakdown process. In particular, a nearly single-exponential stress decay function is predicted in the limit where the micelle breaking time is short, compared to the reptation time of a micelle of length equal to the average micellar length. Detailed results were obtained from a computer simulation by Turner and Cates5-7 and from a Poisson renewal model by Granek and Cates8 for various values of the ratio ζ ) τbreak/τrep, where τbreak is the time taken for a micelle of the mean length to break and τrep is its reptation time. The calculated Cole-Cole plots, in which the imaginary part G′′(ω) of the frequency-dependent shear modulus is plotted against the real part G′(ω), can be compared to the experimental ones providing a direct estimate of the parameter ζh ) τbreak/TR where TR is the terminal stress relaxation time. The Poisson renewal model was also applied to study the regimes that arise for small breaking times and/or small time scales when the dominant polymer motion is not pure reptation but either breathing (in which (1) See for instance: Cates, M. E.; Candau, S. J. Europhys. Lett. 2001, 55, 887. (2) See for instance: Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933. (3) Cates, M. E. J. Phys (Paris) 1988, 49, 1593. (4) Cates, M. E. Macromolecules 1987, 20, 2289. (5) Turner, M. S.; Cates, M. E. J. Phys. France 1990, 51, 307. (6) Cates, M. E.; Turner, M. S. Europhys. Lett. 1990, 7, 681. (7) Turner, M. S.; Cates, M. E. Langmuir 1991, 7, 1590. (8) Granek, R.; Cates, M. E. J. Chem. Phys. 1992, 96, 4758.

tube length fluctuations play a dominant role) or local Rouse motion.8 A typical signature of the latter effect is a turn-up of both G′(ω) and G′′(ω) at high frequency, whereas the simple reptation picture predicts a constant asymptote for G′(ω) and a decreasing G′′(ω).5-8 This results in a minimum in the Cole-Cole representation of the loss modulus, whose depth can be used to estimate the number of entanglements per chain in the system. The above model applies strictly to systems of entangled wormlike micelles with entanglement length le (i.e., the contour length between two successive entanglements) much larger than the persistence length lp and in a regime where the breaking time is much larger than the Rouse time of a chain with the entanglement length. The overall features of the stress relaxation described above stand whatever the kinetics mechanism. Three different model schemes for the kinetics of micellar fusion and breakdown have been analyzed;5-9 these are reversible scission, end interchange, and bond interchange. For each scheme, there is a characteristic time τbreak for changes in micellar size. In a T-jump experiment, which preserves the form of the length distribution but changes the average length, the chain length distribution is predicted to decay monoexponentially with relaxation time τT-J ) τbreak/2 for reversible scission whereas there is no decay at all when end-interchange or bond-interchange reactions are present.5 Thus, complementary T-jump and rheological experiments allow for a full characterization of the micellar kinetics. For electrostatically neutral systems, the model of Cates predicts scaling behaviors to dilution of different rheological, kinetic, and structural parameters: zero-shear viscosity, plateau modulus, breaking time, and minimum of the loss modulus. It also expresses the temperature dependences of these parameters in terms of the scission free energy Esciss and of the energy Ebreak associated with the Arrhenius behavior of the breaking process. Only few experimental data are in agreement with the scaling predictions of the Cates model.10 The deviations from the theoretical predictions were assessed to the (9) Cates, M. E. Europhys. Lett. 1987, 4, 497. (10) Candau, S. J.; Lequeux, F. In Structure and Flow in Surfactant Solutions; Herb, C. A., Robert, K., Eds.; ACS Symposium Series 578; American Chemical Society: Washington, DC, 1994; Chapter 3 and references therein.

10.1021/la035082u CCC: $25.00 © 2003 American Chemical Society Published on Web 11/12/2003

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formation of branched micelles10,11 or even of networks of multiconnected micelles.12-16 It must be stressed that generally the experimental studies are only partial and do not allow one to check the self-consistency of the scaling behaviors obtained from the various rheological, structural, and kinetics parameters. One exception is provided by the experiments performed by Berret et al. on cetylpyridinium salicylate at high ionic strength.17 This study shows that the rheological and structural parameters follow scaling laws to dilution with exponents in satisfactory agreement with the Cates predictions. Still, no results on the kinetics and energy parameters are given. In the first part of the paper, we report on complementary rheological and T-jump experiments on the same system as that investigated by Berret at al. as a function of temperature and surfactant concentration. The results obtained provide a full description of the dynamic properties of this system that can be considered as a model one as far as the Cates model is concerned. The second part of this paper is a review of the various rheological behaviors reported for different systems at high ionic strength. A classification is proposed that establishes a correspondence between the rheological behavior observed and the topology of the micellar aggregates. To support this classification, some rheological measurements on a few new systems are also presented.

Oelschlaeger et al. aluminum-coated cup and a titanium bob of 32 mm diameter and 33.31 mm height). For dynamical measurements, the investigated frequency range was 0.002 Hz e f e 16 Hz.

III. Dynamical Properties of a “Model” System of Entangled Wormlike Micelles (a) Theoretical Reminder. We recall below the scaling behavior to dilution of various parameters for electrostatically screened solutions of wormlike micelles undergoing reversible scission. The theoretical predictions have been obtained from the models derived for “classical” polymer solutions but taking into account the two specific features of equilibrium polymers, namely: (i) The average length of the micelles is determined by the thermodynamical equilibrium of the solution. (ii) The micelles continuously break and recombine. The length distribution of equilibrium polymers has been calculated using statistical mechanics. In the limit of large aggregation numbers which correspond to a concentration much higher than the critical micelle concentration (cmc), the model predicts an exponential distribution of length C(L):20

( LLh )

C(L) ∝ exp -

with the average length given by

( )

II. Materials and Methods The surfactant solutions investigated here are the binary mixtures made of cetylpyridinium chloride (CPy+, Cl-) and sodium salicylate (Na+, Sal-), diluted in brine 0.5 M (NaCl). The ratio R ) [Sal]/[CPy] has been kept to a constant, R ) 0.5. We have also studied solutions of tetradecyltrimethylammonium nitrate (TTA+NO3-) in the presence of 1 M NO3Na. This surfactant was obtained by ion exchange with TTA+Br- (commercial product). We also investigated solutions of a cationic fluorinated surfactant, namely, perfluorooctylbutane trimethylammonium chloride (C8F17(CH2)4N+(CH3)3Cl-, hereafter called C8F17+Cl-), in the presence of 300 mM NaCl. This surfactant was obtained by ion exchange with C8F17+Br- whose synthesis was previously reported.18 The solutions were prepared by weighing the components. As these investigated surfactants have a density very close to 1, we will take in the following the weight concentration equal to the volume fraction φ. (a) T-Jump Device. The T-jump device has been described elsewhere.19 For T-jump measurements, the surfactant solutions were filtered through a 0.22 µm Millipore filter. The T-jump is produced by means of a discharge of a capacitor through the sample. The rise time of the T-jump is 1 µs, and its amplitude, calculated from the values of the stored electrical energy and the heat capacity of the sample, can be varied between 0.1 and 2 °C. For all the investigated systems, the relaxation curve of the signal shows an exponential-like decay of the scattered light intensity with time. The decrease is well described by a monoexponential function which gives a characteristic relaxation time τT-J. (b) Rheological Properties. The rheological experiments were performed with an imposed strain Rheometrics RFS II rheometer. Couette geometry was used (a 34 mm diameter (11) Porte, G.; Gomati, R.; El Haitamy, O.; Appell, J.; Marignan, J. J. Phys. Chem. 1986, 90, 5746. (12) Khatory, A.; Lequeux, F.; Kern, F.; Candau, S. J. Langmuir 1993, 9, 1456. (13) Candau, S. J.; Khatory, A.; Lequeux, F.; Kern, F. J. Phys. IV 1993, 3, 197. (14) Appell, J.; Porte, G.; Khatory, A.; Kern, F.; Candau, S. J. J. Phys. (France) II 1992, 2, 1045. (15) Khatory, A.; Kern, F.; Lequeux, F.; Appell, J.; Porte, G.; Morie, N.; Ott, A.; Urbach, W. Langmuir 1993, 9, 933. (16) Drye, T. J.; Cates, M. E. J. Chem. Phys. 1992, 96, 1367. (17) Berret, J. F.; Appell, J.; Porte, G. Langmuir 1993, 9, 2851. (18) Kotora, M.; Ha´jek, M.; Ameduri, B.; Boutevin, B. J. Fluorine Chem. 1994, 68, 49. (19) Faetibold, E.; Waton, G. Langmuir 1995, 11, 1972.

(1)

L h = 2lpφR exp

Esciss 2kBT

(2)

where lp is the persistence length and the exponent R is 0.5 in a mean-field theory3,4,20 or ∼0.6 in a scaling approach.3,4 The micellar growth resulting from an increase of concentration can be monitored by studying the behavior of τbreak determined either by T-jump experiments in the dilute regime or by rheological experiments in the semidilute regime. It is generally assumed that the scission of a micelle occurs with equal probability per unit time per unit arc length on all chains. The rate of this reaction is a constant k which is the same for all elongated micelles and is independent of the surfactant concentration, so that

τbreak )

(

)

Esciss 1 = 2k-1lpφ-R exp kL h 2kBT

(3)

The scaling behaviors to dilution of τbreak and L h (eqs 2 and 3) allow one to predict that of the terminal time of the stress relaxation. In the regime τbreak , τrep, the stress relaxation function is a nearly single exponential with a terminal time TR given by

TR ) (τbreakτrep)1/2

(4)

By combining eqs 2-4 and using the prediction τrep ∼ L h 3φ3/2, for semidilute polymer solutions in good solvent,21 one obtains

1 3 E + E TR ∼ φ((3/4)+R) exp 2kBT break 2 sciss

[

]

(5)

where Ebreak represents the activation energy associated with the Arrhenian temperature dependence of τbreak (20) Israelachvili, J.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (21) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.

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(including temperature effects of both L h and k). In the mean-field model TR ∼ φ 1.25, and in the scaling approach TR ∼ φ 1.35. The zero-shear viscosity η0 is related to the terminal time TR and the plateau modulus G′∞ through

η0 ) G′∞TR

(6)

For semidilute solutions in good solvent, it has been shown that22

G′∞ ∼ kBTφ9/4

(7)

Whereas the mean-field model predicts G′∞ ∼ kBTφ2. In summary, the behaviors of the various experimentally measurable parameters as a function of surfactant volume fraction and temperature are described by the scaling laws given below (with R ) 0.6). Rheological Parameters

Figure 1. Normalized Cole-Cole plot (O) for a solution of CPy+Cl-/Na+Sal- (2/1) with T ) 20 °C, φ ) 4 × 10-2 g cm-3, and [NaCl] ) 0.5 M (data from ref 17). The dotted line is the calculated Cole-Cole plot for a value of the parameter ζh ) 1.23. The solid line represents the osculating semicircle at the origin.

3 1 E + E η0 ∼ φ3.6 exp 2kBT break 2 sciss

(

)

1 3 TR ∼ φ1.35 exp E + E 2kBT break 2 sciss

(

)

G′∞ ∼ φ2.25kBT Structural Parameters

(

)

Esciss G′′min le ∼ ∼ φ-1.85 exp G′∞ L h 2kBT ξ ∼ φ-0.75

(ξ ) correlation length)

Kinetic Parameter

( )

τbreak ∼ φ-0.6 exp

Ebreak 2kBT

Furthermore, by combining different measurements as a function of temperature, one can obtain both Esciss and Ebreak. (b) Results and Discussion. We have first performed a more detailed analysis of the rheological data of ref 17 in order to obtain the breaking time τbreak of the micelles as a function of the surfactant volume fraction. Figure 1 gives a typical normalized Cole-Cole plot, where both G′ and G′′ have been divided by Gosc, the radius of the osculating circle at the origin. Following the procedure described in ref 23, we have superimposed on the experimental data the numerically calculated Cole-Cole plots disregarding the nonreptative effects for different values of ζh ) τbreak/TR. The best fit is obtained for ζh ) 1.23. The terminal time TR of the stress relaxation is given by the inverse angular frequency corresponding to the maximum of G′′. From the values of ζh and TR, one determines τbreak. The variation of τbreak with the surfactant concentration in the entangled regime is given in Figure 2. T-jump experiments were performed on the same system in the dilute regime. The T-jump responses are well described by a single-exponential decay with a relaxation time τT-J. The variations of τbreak ) 2τT-J with the surfactant (22) Adam, M.; Delsanti, M. J. Phys. (Paris) 1985, 44, 1760. (23) Kern, F.; Lemarechal, P.; Candau, S. J.; Cates, M. E. Langmuir 1992, 8, 437.

Figure 2. Variations of the T-jump times and of the breaking time (at 20 °C) with the volume fraction φ of CPy+Cl-/Na+Sal(2/1); [NaCl] ) 0.5 M. (a) (b) T-Jump measurements at 20 °C; (O) breaking time measured by rheology at 20 °C. (b) T-Jump measurements at 25 °C (9) and 30 °C (2).

concentration obtained in the dilute regime at three different temperatures are reported in Figure 2. One observes a very good agreement between the results obtained by T-jump and rheology respectively at T ) 20 °C. This shows that the reversible scission is the effective kinetic mechanism for this system. The variation of τbreak with concentration is described by the following power law:

τbreak ∼ φ-0.67 at T ) 20 °C τbreak ∼ φ-0.51 at T ) 25 °C τbreak ∼ φ-0.59 at T ) 30 °C The above results agree within the experimental accuracy with the scaling expressions describing the micellar growth (cf. eq 3). They are also consistent with the results of Berret et al.17 concerning the surfactant concentration dependences of the viscoelastic and structural parameters. We have also investigated the temperature dependences of 2τT-J obtained by T-jump in the dilute regime and of τbreak and TR determined from rheological experiments in the semidilute regime. The results are reported in Figure 3. Both τbreak and TR follow Arrhenian behaviors, from which activation energies are inferred, allowing one to estimate the enthalpic part Hsciss of the scission energy Esciss which is found to be equal to

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L hc )

n2 n1 + 2n3

(8)

where n1 is the concentration of end-caps, n2 is the number density of persistence lengths, proportional to the volume fraction φ of surfactant, and n3 is the concentration of 3-fold connections. For linear micelles, n3 ) 0 and L hc ) L h ∝ φ0.6 in the scaling approach. In the limit n3 . n1, L h c ∝ φ-0.6.10,12,29 Therefore, for a system of branched micelles the relevant length and breaking time to be used in eqs 2 and 3 scale with concentration according to

L h c ∼ φβ Figure 3. Variations of the T-jump times, of the breaking time, and of the terminal time with the temperature for solutions of CPy+Cl-/Na+Sal- (2/1) in the presence of 0.5 M NaCl. (a) Breaking time τbreak (4) and terminal time TR (O) measured by rheology in the semidilute regime at φ ) 4 × 10-2 g cm-3. (b) T-Jump measurements in the dilute regime for solutions with φ ) 0.2 × 10-2 g cm-3 (b), φ ) 0.4 × 10-2 g cm-3 (2), and φ ) 0.7 × 10-2 g cm-3 (9).

∼(38 ( 2)kBT. This value is of the same order as that found for several other systems.24 Such values of Hsciss might appear to be unrealistically large for the scissionrecombination process as they would lead to unrealistic values of the micellar length in the absence of an entropic contribution. The latter might be associated with the rearrangement of the counterions upon formation of an end-cap.25,26 Thus, the Arrhenius plots would lead to a large scission free energy but when the micellar length is calculated by using eq 2, a large positive entropic term would lead to reasonable micellar lengths. IV. Rheological Behaviors in Relation to the Intermicellar Connectivity The behavior reported in the preceding paragraph is, by far, not universal. Many other highly screened systems exhibit viscoelastic properties that are markedly different from the coupled reptation-reaction predictions. In particular, the terminal time of the stress relaxation shows a much lesser increase than predicted by eq 512 or even a decrease upon increasing the surfactant volume fraction.27 Such deviations from the classical picture were generally interpreted by assuming the formation of equilibrium cross-links arising through local fusion of the micelle. It is at present understood that an increase of the curvature energy of surfactant molecules in the end-cap relative to the cylindrical body of the micelles will increase the scission energy. This will in turn decrease the cross-linking energy ECL since in a crosslink the curvature change has the opposite sense to that of an end-cap. A model based on the coupled reptation/reaction for branched micelles was developed by Lequeux.28 It was shown that the general features of the stress relaxation remain, provided one replaces the average length L h of the micelle by a new length L h c defined as (24) Oda, R.; Panizza, P.; Schmutz, M.; Lequeux, F. Langmuir 1997, 13, 6407. (25) Kern, F.; Zana, R.; Candau, S. J. Langmuir 1991, 7, 1344. (26) Cates, M. E. Private communication. (27) Buhler, E.; Munch, J. P.; Candau, S. J. Europhys. Lett 1996, 34, 251. (28) Lequeux, F. Europhys. Lett. 1992, 19, 675.

τbreak ∼ φ-β

(9)

with -0.6 e β e 0.6. Under these conditions, eqs 5 and 6 become

TR ∼ φγ η0 ∼ φγ′

with 0.15 e γ e 1.35

(10)

with 2.4 e γ′ e 3.6

(11)

For some systems, γ and γ′ were found to be smaller than the lower limits given above, γ being sometimes negative.27 This behavior was attributed to the formation of a saturated network of multiconnected micelles for which no reptation process can be invoked. It was suggested that the flow mechanism is based on a process similar to that proposed to describe the rheological properties of the bicontinuous L3 phase30 and that involves a sliding of the branch points along the cylindrical micelles.14,15 Figure 4 shows an illustration of the different behaviors encountered for the concentration dependences of the zeroshear viscosity of highly screened semidilute solutions of wormlike micelles. The power law exponent of η0(φ) is 3.57 for the (C8F17+Cl-) in 0.3 M NaCl, 3 for the mixture (12-2-12)(2Br-)/CTA+Br- with a molar ratio of 0.4 in 0.15 M KBr,31 and 2 for the TTA+NO3- in 1 M NO3Na. A review of the results reported in the literature for high ionic strength solutions of cationic surfactants reveals a complex effect of the nature of both the surfactant and the added salt on the viscoelastic properties. The scission energy that controls the micellar growth has been expressed theoretically as follows:

Esciss ) Ec - Ee

(12)

where Ec represents the end-cap energy associated with surfactant packing near the cylinder end-caps and Ee accounts for the electrostatic energy associated with the surface charges on the micelles and counterions in solutions, as well as the entropy of counterions. The electrostatic term has been analyzed by Mackintosh et al.32 through the use of a Poisson-Boltzmann model of semidilute systems of charged cylinders. The electrostatic self-energy was found to depend only on the Debye-Hu¨ckel screening length and the effective charge density. Therefore, at high surfactant and/or salt concentration the electrostatic term in eq 13 vanishes and one should recover (29) The mean-field approach predicts that L h c ∼ φ-0.5. We have assumed that the excluded volume correction calculated by Cates (ref 4) for linear micelles applies also for branched micelles (F. Lequeux, private communication). (30) Snabre, P.; Porte, G. Europhys. Lett. 1990, 13, 641. (31) Oelschlaeger, C.; Waton, G.; Candau, S. J. Eur. Phys. J. E, in press. (32) Mackintosh, F.; Safran, S.; Pincus, P. Europhys. Lett. 1990, 12, 697.

Rheological Behavior of Cylindrical Micelles

Figure 4. Variations of the zero-shear viscosity with φ for three different surfactants at T ) 20 °C: (b) C8F17+Cl- + 0.3 M NaCl; (O) 12-2-122+(2Br-)/CTA+Br- (2/3) + 0.15 M KBr; (2) TTA+NO3- + 1 M NO3Na.

the behavior of neutral micelles. As a matter of fact, the splitting of Esciss into end-cap energy and electrostatic energy is questionable considering the two following experimental observations: (i) The electrostatics appears to play more through binding than screening. In many systems, the viscoelastic behavior is still dependent on salt concentration even in a concentration range where the screening length is less than the Bjerrum length. (ii) The counterions (or co-ions) modify the end-cap energy. This is particularly true for counterions that penetrate into the hydrophobic interior of the micelles, like hydroxynaphthalenecarboxylate, or salicylate. The cationic moiety and counterion form ion pairs, decreasing the areas per headgroup and increasing the volume of the hydrophobic part of the surfactant. This increases the curvature energy of surfactant molecules in the end-cap relative to the cylindrical part, to lead sometimes to the formation of vesicles even in the absence of salt.33 On the other hand, we would expect that the nonpenetrating counterions (or co-ions) that bind through the polarizability effect should not modify strongly the endcap energy. Still the viscoelastic properties of such systems are strongly dependent on the salt concentration, even at high salt concentration. The efficiency of nonpenetrating counterions to promote micellar elongation increases according to the sequence34

F- < Cl- < Br- < NO3- < ClO3The above sequence is what is expected from the Hofmeister series.34 Thus, the structural and dynamical properties will depend on the combination of two factors: the effective hydrophobicity of the surfactant, linked to both the cationic moiety and the nature of the counterions, and their binding efficiency that depends on the nature of the counterion but also on the concentration of the added salt. One can arbitrarily classify the surfactants into highly and moderately hydrophobic surfactants. Surfactants with high hydrophobicity are those with fluorinated, gemini, or long unsaturated tails, or also single hydrocarbon tail surfactants coupled with a penetrating counterion like CPySal or CTAHNC, or TTA tosilate. Surfactants with moderate hydrophobicity are those with a single chain hydrocarbon tail coupled with a nonpenetrating counterion. (33) Narayanan, J.; Manohar, C.; Kern, F.; Lequeux, F.; Candau, S. J. Langmuir 1997, 13, 5235.

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Of course, this classification must be modulated. Microscopic details concerning the insertion of the counterion in the micellar film play a major role. For instance, manipulation of the chlorine substitution pattern in the cetyltrimethylammonium mono- and dichlorobenzoates allowed tuning of the micellar morphology and solution properties.36 Also, the mixing in different proportions of two surfactants such as for instance a gemini one and a single hydrocarbon tail surfactant allows us to vary progressively the hydrophobic character within the micelles.31 As for the binding efficiency of nonpenetrating counterions, one can also classify it, still arbitrarily, as low, moderate, high, and very high, respectively, depending on the position of the counterion (co-ion) within the sequence given above and on the concentration of added salt. These considerations lead us to propose the classification given in Table 1. The systems are roughly classified in three categories depending on the relative values of the scission energy and the cross-linking energy. The rheological properties of systems in category I combining a high hydrophobicity with a low binding of the counterions (or moderate and moderate) are satisfactorily described by the reptation-reversible scission of linear micelles. T-Jump studies were reported for the three last systems of this category.31,39 The results were in agreement with the predictions of the reversible scission models for linear micelles concerning the concentration dependence of τbreak in the dilute regime. Systems in category II are less hydrophobic but exhibit an enhanced binding of the counterions. The rheological behavior is that of branched micelles undergoing a reptation-reversible scission process. T-Jump experiments performed in dilute solutions of the third system in category II showed a complex behavior with a bimodal T-jump response.31 Finally, by combining a moderate hydrophobicity with a very high binding efficiency, one obtains systems whose viscoelastic behavior cannot be described by a reptation process. We suspect that in that case the surfactant selfassembles in saturated networks. One can make several remarks concerning this classification. (a) The case of highly hydrophobic surfactants with strongly binding counterions is not included in the table because such surfactants generally self-assemble into vesicles even at low ionic strength. This is the case of 12-2-122+‚2Br- and C8F17+Cl- in the presence of more than 30 mM of KBr and of 12-2-122+‚2Br- with NaNO3. This effect is even enhanced for penetrating counterions. Thus the CTA+HNC- (cetyltrimethylammonium hydroxynaphthalene carboxylate) and 12-2-12 tosilate form vesicles at room temperature even under salt-free conditions.33 (b) If the end interchange or bond interchange are the kinetic processes involved in the rheological properties, then the exponent γ′ is slightly decreased. These processes are likely to be the relevant ones for systems in category II, where intermicellar branching occurs. (34) Porte, G.; Appell, J. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1983; Vol. 2, p 805. (35) Hofmeister, F. Arch. Exp. Pathol. Pharmakol. 1888, 24, 247. (36) Carver, M.; Smith, T. L.; Gee, J. C.; Delichere, A.; Caponetti, E.; Magid, L. J. Langmuir 1996, 12, 691. (37) Couillet, I., et al. To be published. (38) Candau, S. J.; Hirsh, E.; Zana, R.; Delsanti, M. Langmuir 1989, 5, 1525. (39) Candau, S. J.; Merikhi, F.; Waton, G.; Lemare´chal, P. J. Phys. France 1990, 51, 977.

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Table 1. Classification Aiming at Establishing a Correspondence between the Rheological Behavior Observed and the Intermicellar Organization of Cylindrical Micelles category

surfactant

salt

surfactant hydrophobicity

binding efficiency

γ′expa

relaxation process

I

C8F17+ClEHA+Cl- c CPy+Cl-/Na+Sal- (2/1) CTA+Br12-2-122+‚2Br-/CTA+Br- (1/4)d

0.3 M NaCl 0.4 M KCl 0.5 M NaCl 0.25 M KBr 0.15 M KBr

H H H M M

L L L M M

3.57e 3.40f 3.30g 3.6h 3.2i

Esciss , ECL reptation/reversible scission of linear micelles γ′th ) 3.2-3.6b

II

CTA+BrCPy+ClO312-2-122+‚2Br-/CTA+Br- (2/3)

1.5 M KBr 0.1 M ClO3Na 0.15 M KBr

M M M-H

H H M

2.55j 2.2k 2.98i

Esciss ∼ ECL reptation/reversible scission of branched micelles ∼2.4 e γ′th e 3.2

III

TTA+NO3CPy+ClO3-

1 M NO3Na 1 M ClO3Na

M M

VH VH

2e 1k

Esciss . ECL no reptation

a Experimental exponent of the scaling law to dilution of the zero-shear viscosity. b Theoretical exponent of the scaling law to dilution of the zero-shear viscosity. The scaling approach value is 3.6. The mean-field approach with R ) 0.5 and G′∞ ∼ φ 2 leads to γ′th ) 3.2. c EHA+ Cl- ) erucyl bis(hydroxyethyl) methylammonium chloride. d 12-2-122+ 2Br- ) gemini (ethandiyl-1,2-bis(dodecyldimethylammonium bromide)). e This study. f I. Couillet et al. (ref 37). g J. F. Berret et al. (ref 17). h S. J. Candau et al. (ref 38). i C. Oelschlaeger et al. (ref 31). j A. Khatory et al. (ref 12). k S. J. Candau et al. (refs 13 and 14).

(c) The observed decrease of γ′exp upon decreasing the micellar charge density (in passing from systems in categories I to II to III) suggests that the effect of the repulsive energy between surface charges is a key factor to prevent the formation of cross-links. However, a decrease of γ′exp can also be induced by increasing the endcap energy at constant ionic strength, as illustrated by the results obtained in the mixture 12-2-122+‚2Br-/ CTA+Br-.31 (d) The above classification is valid in given ranges of salt concentration. An increase of salt concentration would likely induce a vertical shift of the systems toward the bottom of Table 1. This might explain the maximum of viscosity observed in many systems upon a variation of salt concentration. (e) Also, many salt-free semidilute solutions show a maximum of the terminal time as the surfactant concentration increases. It has been suggested that the increase of concentration favors the binding because of the limited solubility of the counterion and/or the increase of the polarity of the medium.24 Again, increasing the surfactant concentration would amount to vertically shifting the systems toward the bottom of Table 1. V. Conclusion The linear viscoelasticity of micellar solutions at high ionic strength is described by the coupled reactionreptation of linear micelles only under very specific

conditions. These conditions combine a high hydrophobicity of the cationic moiety with a quite large concentration of a weakly binding salt. Typical examples are provided by solutions of fluorocarbon surfactants, gemini surfactants, or surfactants with long unsaturated tails in the presence of a large amount of sodium chloride. Such systems can be used as efficient thickeners of brines and are relatively insensitive to variations of the salt concentration, which makes them interesting for some applications. For this class of micellar solutions, the scaling behaviors to dilution of the various rheological, structural, and kinetic parameters are self-consistent and follow the predictions of the Cates model. As the micellar charge density is decreased and/or the end-cap energy is increased, the zero-shear viscosity at a given volume fraction and the exponent of the scaling law to dilution of the zero-shear viscosity decrease, likely because of the occurrence of the intermicellar branching. As a consequence, strongly binding counterions are not efficient to produce a large thickening at high ionic strength. Acknowledgment. The authors thank J. Appell for having communicated the frequency dependences of the complex shear modulus concerning the results published in ref 17. LA035082U