Linear Viscoelasticity of Wormlike Micellar Solutions Found in the

Sep 1, 1997 - (11) Mackintosh, F.; Safran, S.; Pincus, P. Europhys. Lett. 1990, 12,. 697. (12) Cates, M. E. Macromolecules 1987, 20, 2289; J. Phys. (P...
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Langmuir 1997, 13, 5235-5243

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Linear Viscoelasticity of Wormlike Micellar Solutions Found in the Vicinity of a Vesicle-Micelle Transition Janaky Narayanan,†,‡ C. Manohar,§ F. Kern,† F. Lequeux,† and S. J. Candau*,† Laboratoire d’Ultrasons et de Dynamique des Fluides Complexes, Unite´ de Recherche Associe´ e au C.N.R.S. No. 851, Universite´ Louis Pasteur, 4, rue Blaise Pascal, 67070 Strasbourg Cedex, France, and Chemistry Division, Bhabha Atomic Research Centre, Trombay, Bombay-400 085, India Received February 27, 1997. In Final Form: July 7, 1997X Linear viscoelastic behavior of micellar solutions of cetyltrimethylammonium hydroxynaphthalene carboxylate is studied at temperatures above the vesicle-micelle transition under salt-free conditions and also in the presence of salt (sodium bromide and potassium acetate). The salt-free solutions, beyond a crossover volume fraction φ*, obey the micellar growth model of Mackintosh et al. The low values of φ* indicate that the effective charge on the micelles is very low. The decrease of the terminal relaxation time upon increasing surfactant concentration for volume fractions larger than ∼6φ* is attributed to the formation of intermicellar connections, favored by the diminution of the electrostatic contributions. A similar effect is found by the addition of a salt that does not bind with the surfactant. The phase diagrams shown in this study suggest that the branched micelles constitute the intermediate structures between linear micelles and bilayers.

Introduction In the dilute range, the morphology of the surfactant self-assemblies is controlled by the spontaneous curvature that itself depends on the packing parameter p ) v/al, where v is the effective hydrophobic chain volume, a the area per polar head, and l the surfactant alkyl chain length.1 The area per polar head depends strongly on the nature of the counterions and more specifically on their ability to bind to the surfactant molecules. Therefore the structure of the aggregates can be controlled by an adequate choice of counterions. It is now well-known that the binding of counterions to a cationic surfactant will increase according to the Hofmeister series,2 i.e., CH3COO< NO3- < Br- < ClO3- < Sal- < HNC-. For some surfactants like the cetyltrimethylammonium or cetylpyridinium cation the increasing degree of binding upon scanning the Hofmeister series is revealed by a more and more favored micellar growth of cylindrical micelles upon addition of corresponding salts.3-5 In the limit where the quasi-totality of counterions is bound to the surfactants, the area per polar head becomes small enough that bilayers are formed. This was observed recently in solutions of cetyltrimethylammonium 3-hydroxynaphthalene-2-carboxylate (CTAHNC), which is obtained by mixing equimolar amounts of cetyltrimethylammonium bromide (CTAB) and sodium hydroxynaphthalenecarboxylate (SHNC) and washing out the sodium bromide.6 A vesicular phase is obtained at room temperature, and it was suggested that †

Universite´ Louis Pasteur. Permanent address: Department of Physics, R. J. College, Bombay-400 086, India. § Bhabha Atomic Research Center. X Abstract published in Advance ACS Abstracts, September 1, 1997. ‡

(1) Israelachvili, J.; Mitchell, D. J.; Ninham, B. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (2) Hofmeister, F. Arch. Exp. Pathol. Pharmakol.1888, 24, 247. (3) Anacker, E. W.; Ghose, H. M. J. Am. Chem. Soc.1968, 90, 3105. (4) Hoffmann, H. In Structure and Flow in Surfactant Solutions; Herb, C., Prud’homme, R., Eds.; ACS Symp. Ser. No. 578; American Chemical Society: Washington, DC, 1994; Chapter 1. (5) Porte, G.; Appell, J.; Poggi, Y. J. Phys. Chem.1980, 84, 3105. (6) Hassan, P. A.; Narayanan J.; Menon, S. V. G.; Salkar, R. A.; Samant, S. D.; Manohar, C. Colloid Surf., A 1996, 117, 89.

S0743-7463(97)00215-1 CCC: $14.00

like in the other mixtures of anionic and cationic surfactants previously investigated, the CTA+-HNC- ion pairs act as lipids and favor the formation of vesicles.7 It was also found that upon increasing the temperature, the system undergoes a phase transition to form a viscoelastic and clear phase, consisting of entangled wormlike micelles. Fluorescence anisotropy, NMR, and neutron scattering experiments gave evidence of the vesicle-micelle transition for a 12 mM CTAHNC solution and suggested the existence of a domain of temperatures where cylindrical micelles and vesicles coexist.8-10 In the present study we report on an investigation of the viscoelastic properties of these systems in the higher temperature micellar phase as a function of surfactant concentration. The experimental data are confronted to the predictions inferred from the model of micellar growth for salt-free micellar systems11 and the theory of Cates12 for the dynamics of equilibrium polymer solutions. We have compared the results with those reported for other systems in order to obtain a general description of the viscoelasticity of salt-free micellar systems. We have also performed experiments in the presence of an added salt to study the effect of electrostatic screening on the rheological behavior of these systems. Sodium bromide and potassium acetate have been chosen because the Br- and CH3COO- ions are not expected to bind to the surfactant molecules when competing with the HCN- ions, and therefore they contribute only to increase the ionic strength of the solution. Theoretical Background 1. Micellar Growth. In a mean field approach the total free energy F per unit volume of micellar solution (7) Menon, S. V. G.; Manohar, C.; Lequeux, F. Chem. Phys. Lett. 1996, 263, 727. (8) Salkar, R. A.; Hassan, P. A.; Samant, S. D.; Valaulikar, B. S.; Kumar, V. V.; Kern, F.; Candau, S. J.; Manohar, C. J. Chem. Soc., Chem. Commun. 1996, 1223. (9) Hassan, P. A.; Valaulikar, B. S.; Manohar, C.; Kern, F.; Bourdieu, L.; Candau, S. J. Langmuir 1996, 12, 4350. (10) Mendes, E.; Narayanan J.; Oda, R.; Kern, F.; Manohar, C.; Candau, S. J. Submitted to J. Phys. Chem. (11) Mackintosh, F.; Safran, S.; Pincus, P. Europhys. Lett. 1990, 12, 697. (12) Cates, M. E. Macromolecules 1987, 20, 2289; J. Phys. (Paris) 1988, 49, 1593.

© 1997 American Chemical Society

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may be written as13

F)

FN [kBT log FN + EN] ∑ N

(1)

At very high surfactant concentration, or in the presence of salt, this entropic contribution as well as the electrostatic term in eq 3 vanish and one recovers the classical expression derived for nonionic surfactant, i.e.

where kB is the Boltzmann constant, T the temperature, and FN the number of micelles composed of N surfactants per unit volume. The first term in eq 1 corresponds to the entropy of mixing and EN is the energy of scission of the micelle. For ionized surfactants the energy of scission is composed of the repulsive energy of the surface charges Ee that favors the breaking of micelles and of the end-cap energy Ec that favors micellar growth. The variation of the mean micellar length with surfactant concentration exhibits three characteristic regimes.11,14 (i) A dilute regime in which the range of electrostatic interactions is larger than the mean micelle size. In this regime the micelles are practically monodisperse, and their mean aggregation number increases very slowly with the volume fraction Φ according to

2. Stress Relaxation. The main features of the theoretical models describing the dynamical properties of wormlike micelles can be found in refs 12, 13, and 15-17. Here, we simply recall the theoretical results needed to discuss our experiments, more specifically regarding the shape of the stress relaxation and the effect of surfactant concentration. The shape of the stress relaxation depends strongly on the ratio ζ ) τbreak/τrep, where τbreak is the characteristic breaking time of the micelles and τrep is the reptation time of the micelle of mean length L h . The scaling theories for semidilute solutions of flexible polymers predict18

N h = (1/lBaν*2 (Ec/kBT + log(Φ/N h ))

h 3 Φ3/2 τrep ∝ L

(2)

lB is the Bjerrum length, a the radius of the cylindrical micelle, and ν* is an effective charge per unit length. (ii) A semidilute regime in which the range of electrostatic interactions is smaller than the mean micelle size. As in the case of neutral micelles, the distribution of micellar sizes is wide but the effective scission energy is reduced by the repulsion of the surface charges, leading to a mean aggregation number given by

N h = 2Φ1/2 exp

[

]

1 (E - Ee) 2kBT c

(3)

where Ee represents the electrostatic contribution given by

Ee = kBT lBav*2 Φ-1/2

(4)

In this range of surfactant concentration, the effective scission energy is reduced because of this contribution, which leads to a smaller mean micellar length as compared to a neutral system with same end-cap energy. However the micellar growth is much more rapid and cannot be represented by a power law. The crossover between regimes i and ii is expected to be rather sharp and to occur in the absence of salt at the overlap volume fraction Φ* that, for cylinders, is given by

Φ* = (kBTlBaν*2/Ec)2

(5)

By combining the above expression with eq 3, one obtains the following expression for the micellar growth in the semidilute regime

N h = 2Φ1/2 exp[(Ec/2kBT)(1 - (Φ*/Φ)1/2]

(6)

(iii) A Concentrated Regime. This regime corresponds to the case where the dominant electrostatic contribution is that of the entropy of the counterions near the end-caps. The growth may be characterized by an effective power law N h ∝ Φ1/2(1+∆), where ∆ is related to the net charge of an end-cap and depends only weakly (logarithmically) on Φ. (13) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (14) Safran, S.; Pincus, P.; Cates, M. E.; Mackintosh, F. J. Phys. (Paris) 1990, 51, 503.

N h = Φ1/2 exp[(Ec/2kBT)

(7)

(8)

It is generally assumed that the chemical relaxation process is the reversible unimolecular scission, characterized by a temperature-dependent rate constant k1 per unit time, per unit arc length, which is the same for all elongated micelles and is independent of time and of volume fraction. This assumption results in

τbreak ) (k1 L h )-1

(9)

Then τbreak represents simply the time between two consecutive events, scission or recombination. In the fastbreaking limit (τbreak/τrep , 1) the long time behavior of the stress relaxation can be described by a single exponential decay with the relaxation time given by

TR ) (τbreak τrep)1/2

(10)

Typical Cole-Cole plots G′′(G′) where G′(ω) and G′(ω) are respectively the frequency dependent storage and loss moduli and ω is the circular frequency taken from the theoretical work of Granek and Cates17 are represented in Figure 1. At low frequencies the behavior is Maxwellian as ascertained by the semicircular shape of the ColeCole plots. A deviation from the half-circle occurs at a circular frequency ω of the order of the inverse of the breaking time τbreak of the micelles. The departure from the Maxwellian behavior is characterized by a linear dependence of G′′ on G′ with a slope of -1. This is followed by a regime in which micelle diffusion between scission events is dominated by breathing modes and where G′′ decays faster (with a slope of -2.4). Eventually, there is an upturn associated with the occurrence of the Rouse modes, thus creating a clear dip in the Cole-Cole plot. The depth of the dip depends strongly on the relative values of the reptation time τrep, of the breaking time τbreak, and also of τe the Rouse time of the entanglement length le, which is given by18

τe ∝ le2

(11)

(15) Cates, M. E.; Turner, M. Europhys. Lett. 1990, 11, 681. (16) Turner, M.; Cates, M. E. J. Phys. (Paris) 1990, 51, 307. (17) Granek, R.; Cates, M. E. J. Chem. Phys. 1992, 96, 4758. (18) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986.

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length, n1 ) 0 and again in the absence of electrostatic h s ∝ Φ-1/2, where L h s is the average length interactions L hc ) L of the strand of the network. In the general case of the coexistence of end-caps and connections, the variation of L h c with the surfactant concentration is between those of L h and L h s. When the electrostatic interactions are not screened out, the variation of L h c with concentration is complex as the concentration of connections depends on the electrostatic energy Ee that itself depends on concentration (cf. eq 4). This will be discussed later. 3. Scaling Behavior to Dilution. We consider first the case of flexible linear micelles. The scaling behavior to dilution of the terminal time of the stress relaxation in the fast breaking limit is obtained by combining eqs 8-10

TR ∝ k1-1/2 L h Φ3/4

Figure 1. Theoretical Cole-Cole plots for le/L h ) 0.03 and three values of ζ ) τbreak/τrep. The data were normalized by dividing G′ and G′′ by G′′max, the maximum value of G′′ (from ref 17).

It was found that in the limit τrep . τbreak . τe, the value of G′′ at the dip obeys

G′′min/G′∞ ) Ale/L h

(12)

where G′∞ is the plateau modulus, A is a constant of the order of 1, and the circular frequency ωmin at which the dip appears is of the order of τe-1. Thus, the frequency dependence of the complex shear modulus can be described with two parameters: the ratios h . However, it must be noted that the ζ ) τbreak/τrep and le/L calculations of Granek and Cates17 involve an adjustable parameter, namely, the amplitude of the tube length fluctuations. Also, if either one of the conditions τbreak/τrep , 1 (cf. Figure 1) or τe/τbreak , 1 is not fulfilled, then the dip becomes shallower and the value of G′′min has a different meaning than that given by eq 12. The situation τbreak ∼ τe that can destroy the Maxwellian behavior, can be encountered in particular in systems which undergo a micellar growth at very low surfactant concentration since then τe is rather large (cf. eq 11). Furthermore, if τbreak is very short, TR is short also because of eq 10. In that case, and contrary to the case of ordinary polymers for which one has always TR . τe, the coupled reptation reaction and the Rouse process give overlapping contributions. The model of Cates,12 derived for linear flexible micelles, was recently extended to the case of branched flexible micelles. It was shown19,20 that the general features of the stress relaxation and more specifically eqs 9 and 12 are maintained, provided one replaces the average length L h of the micelle by a new length L h c defined as

L hc )

n2 l n1 + 2n3 p

(13)

where lp is the micelle persistence length, n1 the concentration of end-caps, n2 the number density of persistence lengths, proportional to the surfactant volume fraction Φ, and n3 the concentration of 3-fold connections. For linear micelles n3 ) 0 and, in the case where the h ∝ Φ1/2. electrostatic interactions are screened out, L hc ) L For saturated networks, i.e., networks for which the arc-length between cross-links is equal to the correlation (19) Lequeux, F. Europhys. Lett. 1992, 19 (8), 675. (20) Elleuch, K.; Lequeux, F.; Pfeuty, P. J. Phys. I 1995, 5, 465.

(14)

with the variation of L h given by eq 6. Thus for a given kinetics, characterized by k1, TR is an increasing function of Φ. In the limit where the electrostatic interactions are screened out (cf. eq 7) TR is given by

TR ∝ k1-1/2 Φ5/4

(15)

For systems containing predominantly connections, where L h is replaced by L h c (cf. eq 13) one has

TR ∝ k1-1/2 Φ1/4

(16)

The zero-shear viscosity η0 is related to the terminal time TR and the plateau modulus G′∞ through18

η0 = G′∞ TR

(17)

For semidilute solutions it has been shown that

G′∞ ∝ kBT Φ9/4

(18)

Therefore, in the fast-breaking limit the zero-shear viscosity varies with L h and Φ according to

h Φ3 k1-1/2 η0 ∝ L

(19)

Combining the above expression with the law of micellar growth (eq 6) in the semi-dilute regime leads to

η0 ∼ k1-1/2 Φ7/2 exp[(Ec/2kBT)(1 - (Φ*/Φ)1/2)]

(20)

In the slow breaking limit, TR ) τrep given by eq 8 and one has

h 3 Φ15/4 η0 ∝ L

(21)

Experimental Section 1. Materials and Methods. CTAHNC was prepared from CTAB and SHNC by removing the counterions (Na+ and Br-) using a solvent extraction technique. Methyl isobutyl ketone was used as the solvent. The product was purified by recrystallization. CTAB was purchased from Sigma Chemicals and SHNC from Atul products, Bombay. The salts used in this study were sodium bromide (Prolabo product) and potassium acetate (Sigma product). The samples were prepared by weighing CTAHNC and deionized water/salt solution of required salinity directly into the sample vial. To get a homogeneous solution, heating at 70 °C for sufficient time and simultaneous stirring was necessary. After thorough mixing, the samples were kept in an oven at 40 °C for 1 day to reach equilibrium. The rheological experiments were conducted on the following systems: (a) CTAHNC solutions in water of concentrations varying from 2 to 20 mM at temperatures varying from 50 to 70

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Figure 2. Viscoelastic spectra of a 15 mM CTAHNC solution at 65 °C: real and imaginary parts of the shear modulus and real part of the complex viscosity versus angular frequency. °C; (b) 12 mM CTAHNC in sodium bromide and potassium acetate solutions of salinity 0.5, 1, and 2 mM at temperatures 60, 65, and 70 °C; (c) CTAHNC solutions of concentrations 4-15 mM in 1 mM sodium bromide solution at 70 °C. The phase diagram for 12 mM CTAHNC in sodium bromide solutions of salinity varying from 0.2 mM to 1 M was evaluated by equilibrating the samples for 1 day at the set temperature and visually observing the turbidity/phase separation. The rheological experiments were done on a Rheometrics RFS II fluid spectrometer using Couette geometry which consisted of a 34 mm diameter aluminum-coated cup and a titanium bob of diameter 32 mm and height 33.3 mm. Experiments were carried out with imposed strain equipment with appropriate correction for inertia effects in the high-frequency range. Proper care was taken to reduce the evaporation of the sample to the minimum. The frequency range investigated was from 0.1 to 100 rad/s. The lower limit 0.1 rad/s was chosen to restrict the duration of each experiment to 15 min. By this protocol it was ascertained that no evaporation takes place during the experiment even at 70 °C. Beginning the experiment at 0.01 rad/s would increase its duration by a factor of 8, and the evaporation of the sample in the temperature range studied would seriously affect the reproductibility of the data. Furthermore, at these frequencies, because of the weakness of the measured signals, the dispersion of the data points would be too large. 2. Data Analysis. Figures 2 and 3 show examples of the variations of the storage modulus G′, the loss modulus G′′, and the real part of the complex viscosity η′ as a function of frequency for a sample with CTAHNC concentration C ) 15 mM. The data have been taken at two temperatures T ) 65 °C and T ) 70 °C. It can be seen that G′′(ω) goes through a maximum whereas G′(ω) exhibits an ill-defined plateau. The curves representative of G′(ω) and G′′(ω) cross each other at a frequency that corresponds roughly to the maximum of G′′(ω). Despite the lack of data in the low-frequency range, due to the reasons invoked in the preceding paragraph, one can infer from these results that the behavior of the complex shear modulus in the low-frequency range is close to a Maxwellian behavior. This is confirmed by the ColeCole representation G′′(G′) shown in Figure 4, which resembles that obtained for other wormlike micelles, characterized by halfcircle in the low-frequency range and an upturn at high frequency, associated with the occurrence of the Rouse modes. However, contrary to many other systems previously investigated, the dip in the Cole-Cole plot is here rather shallow and there is no clear separation between the two main relaxation processes. That is why, in this study, we have determined an estimate of the terminal relaxation time TR from the frequency ωR at which G′(ω) and G′′(ω) cross each other : TR ) (ωR)-1. As for the plateau modulus G′∞ we have estimated it from G′∞ ) 2 G′(ωR) = 2 G′′max. Despite the limited range of frequency available below ωR, we still can, at least for T g 65 °C, obtain a reasonable extrapolation of η′(ω) to ω f 0 to get the zero frequency η0, which is also equal

Narayanan et al.

Figure 3. Viscoelastic spectra of a 15 mM CTAHNC solution at 70 °C: real and imaginary parts of the shear modulus and real part of the complex viscosity versus angular frequency.

Figure 4. Experimental Cole-Cole plots for a 15 mM CTAHNC solution at three different temperatures. The semicircle depicts the behavior of a Maxwellian liquid. to the zero-shear viscosity. The values thus obtained for η0 coincide within 10% with those obtained from G′∞ and TR, using eq 17. At lower temperatures, as for instance at T ) 60 °C where we still have a pure micellar phase, the relaxation times become too long and G′(ω) and G′′(ω) cross each other at a frequency lower than the experimentally available range (cf. Figure 5). One can still obtain approximate values of TR, G′∞, and therefore of η0. However, as the behavior at 60 °C does not differ significantly from that obtained at higher temperatures, in the following we report only the results obtained at T g 65 °C.

Results 1. Salt-Free Systems. The vesicular phase observed at room temperature in dilute solutions of CTAHNC has a low viscosity and is turbid due to the vesicle size (1-10 µm). With increase of the temperature, the system undergoes a phase transition to form a viscoelastic and clear phase consisting of entangled wormlike micelles. The transition has a width of about 20 °C around 50 °C for a solution at a concentration of 12 mM with coexisting vesicles and wormlike micelles, as revealed by SANS and rheological experiments.9,10 Figure 6 shows the variation with surfactant concentration of the temperature of transition between the

Linear Viscoelasticity of Wormlike Micellar Solutions

Figure 5. Viscoelastic spectra of a 15 mM CTAHNC solution at 60 °C: real and imaginary parts of the shear modulus and real part of the complex viscosity versus angular frequency.

Figure 6. Variation of the vesicle-micelle transition temperature with surfactant concentration.

coexistence phase and the micellar one as obtained from visual observation. Videomicroscopy experiments show the presence of large vesicles in the low-temperature range but they do not allow us to see the cylindrical micelles when they coexist with vesicles. It can be seen in Figure 6 that the transition temperature increases significantly with surfactant concentration. The HNC- counterion is only weakly soluble in water and therefore has a strong tendency to bind to the surfactant, which explains the formation of vesicles at low temperature. With an increase of temperature, the solubility of HNC- increases and the dissociation of the surfactant molecules permits the formation of the micelles. It is the same limited solubility that is likely to be the origin of the increase of the transition temperature with surfactant concentration. The rheological experiments were performed in a range of temperature (T g 60 °C) and of surfactant concentration (C e 20 mM) rather far from the transition line of Figure 6. Figure 7 shows the variations of the zero-shear viscosity taken at 0.1 rad/s with the surfactant concentration at T ) 65 °C and T ) 70 °C. One observes a very rapid increase of the viscosity in the concentration range 4-20 mM. At lower concentrations, the viscosity, which is of the order

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Figure 7. Variation of the real part of the complex viscosity at an angular frequency of 0.1 rad/s with surfactant concentration at 65 and 70 °C. The triangles are for [NaBr] ) 1 mM and T ) 70 °C with a best fit slope of 2.3 to the data. The dashed lines are guides for the eyes. The two horizontal lines represent the viscosity of the water at the two temperatures considered.

of a few millipascals seconds, cannot be accurately measured with our setup, but obviously its variation with concentration has to be rather weak compared to that observed at higher concentration. A similar behavior has been reported by Hoffmann et al. for tetradecylammonium salicylate and hexadecyloctyldimethylammonium bromide4,21 and by Kern et al. for the dimeric 12-2-12 surfactant,22 and seems to be characteristic of wormlike micellar solutions at low ionic strength. Also an abrupt change of both viscosity and the CottonMouton constant was observed in hexadecylpyridinium bromide by Porte et al.5 at a surfactant concentration labeled as “second cmc.” The comparison of the data of Figure 7 with the predicted micellar growth (eqs 2 and 6) strongly suggests that this change of regime corresponds to the fast micellar growth regime. For concentrations C g 6 mM the viscosity of the solutions is sufficiently high to allow us to characterize the ensemble of the viscoelastic behavior. The variations of TR with surfactant concentration for two temperatures are reported in Figure 8. Unexpectedly, the relaxation time exhibits a maximum. The increase of TR observed in the low-concentration range becomes less pronounced upon decreasing temperature. For T = 60 °C, the data, not reported here, are not very accurate for the reasons indicated in the paragraph Data Analysis, but they indicate that TR is roughly independent on concentration. Figure 9 shows the concentration dependence of G′∞ in log-log scale. The data can be fitted by a straight line with a slope of 2.27. This value is very close to 9/4, which is the value of the exponent of the power law of G′∞ for semidilute flexible polymer solutions. Also, as in polymer solutions, G′∞ is independent of temperature to the first approximation. 2. Effect of Salt. The addition of NaBr to aqueous CTAHNC systems modifies strongly the phase diagram as shown in Figure 10. At temperature larger than ∼50 °C, a transition between micelles and vesicles occurs at a salt concentration, which is an increasing function of temperature. As the Br- ions are unlikely to exchange (21) Wunderlich, I.; Hoffmann, H.; Rehage, H. Rheol. Acta 1987, 26, 532. (22) Kern, F.; Lequeux, F.; Zana, R.; Candau, S. J. Langmuir 1994, 10, 1714.

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Figure 8. Variation of the terminal stress relaxation time with surfactant concentration at 65 and 70 °C.

Figure 10. Phase diagram for a 12 mM CTAHNC solution as a function of temperature and salt (NaBr) concentration.

Figure 9. Variation of the plateau modulus with surfactant concentration at 65 and 70 °C. The solid line is the best fit to the data at 65 °C (slope ) 2.26) and the dashed line is the best fit to the data at 70 °C (slope ) 2.28).

Figure 11. Variations of real and imaginary parts of the shear modulus with angular frequency at 65 °C for [CTAHNC] ) 12 mM and [NaBr] ) 0, 1, and 2 mM.

with the HNC- ions, the primary effect of salt is a screening of the charges carried by the micelles. At higher salt concentration, one observes a phase separation, but the visual aspect of the system depends on the temperature range. At low temperatures (T e 50 °C) one observes turbid lumps floating in the bottom of the test tube, and optical microscopy between crossed polarizers reveals textures typical of a lamellar phase. The top phase, clear, isotropic, and very fluid, is likely to be mainly water. At high temperatures, T g 50 °C, one has an isotropic phase in the bottom with a supernatant, which is reddish and oily, and must be mainly SHNC resulting from a recombination of the HNC- and of the Na+ ions. Figure 11 shows the effect of salt on the complex modulus. One observes a large shift of the crossing frequency ωR of G′(ω) and G′′(ω) toward higher frequencies upon addition of salt. At the same time the minimum of G′′(ω) disappears. Figures 12-14 show the effect of the addition of salt, sodium bromide, or potassium acetate, on the terminal time TR, the plateau modulus G′∞, and the zero-shear viscosity η0, respectively, of a 12 mM, CTAHNC solution at T ) 65 °C. One observes a large decrease of TR and η0

upon increasing the salt content whereas the plateau modulus is only slightly affected. The presence of salt also modifies the scaling behavior to dilution. This is illustrated in Figure 7 where the variation of the zero-shear viscosity with surfactant concentration in the presence of 1 mM NaBr at T ) 70 °C has been reported. The increase in viscosity is much smoother than in the absence of salt and can be fitted by a power law with an exponent 2.3. Discussion 1. Micellar Growth at Φ > ∼Φ*. The observation in Figure 7 of two well-defined regimes in the variation of the viscosity with surfactant volume fraction is in agreement with the theory that predicts a very fast micellar growth beyond Φ*. According to the theories developed for polyelectrolyte solutions, it can be speculated that the micellar conformation in the semidilute regime is that of a semiflexible chain with an electrostatic persistence length of the order of the mesh size of the entangled chains.23,24 This holds as long as the mesh-size is larger than the natural persistence length of the micelles. The (23) Barrat, J. L.; Joanny, J. F. Europhys. Lett. 1993, 24, 333.

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Figure 12. Variation of the terminal relaxation time of a 12 mM CTAHNC solution at 65 °C with salt concentration.

Figure 14. Variation of the zero-shear viscosity of a 12 mM CTAHNC at 65 °C with salt concentration.

Figure 13. Variation of the plateau modulus of a 12 mM CTAHNC solution at 65 °C with salt concentration.

Figure 15. Variation of the zero-shear viscosity with the surfactant volume fraction for CTAsal at 20 °C from ref 4, CTAHNC at 70 °C, 12-2-12 at 20 °C (from ref 22) and equimolar CTAHNC + CTAB at 55 °C (from ref 9). The theoretical fits of (eq 20) were adjusted to give η ) ηs at Φ ) Φ*. The horizontal lines represent the viscosity of the water at the temperatures corresponding to the different sets of data.

latter has been determined for several systems and found to be e200 Å,5,25,26 which is of the order of the mesh size in the range of volume fraction around a few percent. The viscoelasticity of semidilute polymer solutions is mainly controlled by the density of entanglements. It has been found experimentally that this density is much less than the density of interchain contact points, so that the volume fraction, Φe, beyond which the entanglements show up is larger than Φ*. For neutral polymers, one has Φe = (5-10)Φ*,27 but for polyelectrolyte systems, the regime of semidilute unentangled chains characterized by Φ* e Φ e Φe is much more extended.23,24 However, in the case of cylindrical micelles, the sharp micellar growth leads to a rapid interpenetration of the wormlike micelles at Φ > ∼Φ*, which explains that one does not observe an intermediate regime of viscometric behavior between the dilute one and the entangled one, as in the case of classical polyelectrolyte chains.28 (24) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. Macromolecules 1995, 28, 1859. (25) Porte, G.; Poggi, Y.; Appell, J.; Maret, G. J. Phys. Chem. 1984, 88, 5713. (26) Appell, J.; Marignan, J. J. Phys. II 1991, 1, 1447. (27) Kavassalis, T. A.; Noolandi, J. Phys. Rev. Lett. 1987, 59, 2674. (28) Yamaguchi, M.; Wakutsu, M.; Takahashi, Y.; Noda, I. Macromolecules 1992, 25, 470.

The viscosity behavior of wormlike micelles also depends on the ratio τbreak/τrep, as discussed in the theoretical section. The viscoelastic behavior, to be discussed later, of systems at low concentration does not allow one to infer an estimate of τbreak, but the analysis of the ensemble of the results concerning the complex shear modulus suggests that we are in the fast breaking limit, that is τbreak/τrep , 1. Then eq 20 applies for the description of the concentration dependence of the shear viscosity in the semidilute regime. As for the dilute regime, the viscosity is too low to provide information on the micellar growth that according to the model is very limited, due to the electrostatic interactions. In Figure 15 we have fitted that expression to the experimental data, obtained in this study as well as those reported previously for other surfactants. The parameters were Ec and Φ*, with the supplementary condition that at Φ* the viscosity is that of the water ηs. This is justified by the fact that Φ* is very low. Using the Einstein law for hard spheres η0 ) ηs (1 + 2.5Φ) leads to a negligible correction. Furthermore the rapid variation of viscosity

5242 Langmuir, Vol. 13, No. 20, 1997

Narayanan et al.

Table 1. Comparison of the Values of the Crossover Concentration (C* or Φ*) the End-Cap Energy, Ec, and the Linear Charge Density, ν*, Obtained for Different Salt-Free Micellar Solutions surfactant

T in °C

C* in mM

Φ*

Ec in kBT

ν*, e/Å

CTAHNC CTAsal 12-2-12 CTAHNC + CTAB

70 20 20 55

2.3 3.0 24.4 82.0

0.0011 0.00126 0.015 0.0343

8 13 35 36

0.04 0.05 0.178 0.20

a For these determinations l has been taken equal to 7 Å, a ) B 2 nm for 12-2-12 and a ) 2.5 nm for the other surfactants.

is mainly governed by Ec, and taking a higher value of the viscosity at Φ* does not affect the obtained values of Ec and modifies only slightly the values of Φ*. One observes a quite universal behavior, with a good fit of the data at the initial stage of growth, followed by a lesser increase of viscosity and eventually in some cases a decrease of the viscosity. The values of Ec obtained from the fits are reported in Table 1. As noted above, these values rest on the fast breaking assumption. We could also fit the data with the theoretical expressions meant for the slow breaking limit in the case of both stiff and flexible micelles. In both cases, this leads to very small unrealistic values of Ec (ekBT). From the experimental values of Φ* and the obtained values of Ec, one can estimate the linear charge density ν* by using eq 5. These values reported in Table 1 must be taken only as orders of magnitude, as there can be a prefactor in the right size of eq 5. One sees that the value of Φ* is highly sensitive to the charge density and, to a much lesser extent, to the endcap energy. According to eq 5, Φ* is controlled by the ratio ν*4/Ec2. This means that the rapid micellar growth of salt-free systems can be observed only for surfactants that exhibit a good combination of ν* and Ec, keeping in mind that the experimental window for the observation of such a growth is typically 10-4 e Φ e 10-1. For surfactants characterized by a high charge density and a small end-cap energy, the crossover volume fraction Φ* will be very large and the system will turn into a liquid crystalline phase at Φ > ∼Φ*. This is what is observed for instance for CTAB solutions for which Φ* is found to be ∼0.12 and nematic phase forms at Φ = 0.23.29 Conversely, surfactants with low charge density and high end-cap energy will overlap at very low concentration below the critical micelle concentration (cmc), which is unrealistic. Inspection of Table 1 reveals that the increase of Φ* with the ratio ν*4/Ec2 is accompanied by a simultaneous increase of ν* and Ec. The explanation of this effect is not trivial. According to the phenomenological approach of Safran et al.30 Ec is related to the curvature parameters through

Ec )

[

]

2π K h F 14 - 10 - 32/3 K 27 K F0

(22)

The above expression, derived for microemulsions which exhibit small curvature with respect to the molecular size, does not apply strictly to the micelles. It is however given as an indication to stress that the end-cap energy might vary in a subtle way with the microscopic parameters (29) Cappelaere, E.; Cressely, R.; Decruppe, J. P. Colloids Surf. 1995, 104, 353. (30) Safran, S. A.; Turkevich, L. A.; Pincus, P. A. J. Phys. Lett. 1984, 45, L69.

Figure 16. Variation of ωmin/ωc with surfactant concentration at 65 °C.

depending on how these affect the spontaneous radius of h , and the bending curvature F0, the saddle-splay energy K energy K. The increase of charge density is expected to decrease F0 which would in turn decrease Ec, but conversely a decrease of K h or an increase of K would tend to increase Ec. In particular one expects for CTAHNC systems that tend to form vesicles the saddle splay energy to be rather large and lowered by the introduction of charges. Also an increase of rigidity resulting from the charge effect is likely to increase K. 2. Viscoelastic Behavior. The shape of the stress relaxation function or equivalently of the frequency dependent complex modulus G*(ω) for semidilute solutions of wormlike micelles depends mainly on the relative values of the characteristic times τrep, τbreak, and τe. In particular, it will depend on the ratio τe/TR, which according to eqs 11 and 14 can be expressed in terms of surfactant volume fraction and of micellar length

TR/τe ∼ Φ13/4 L h k1-1/2

(23)

The time τe, which is approximately the inverse circular frequency corresponding to the upper end of the plateau modulus, is not attainable experimentally. One can however infer a qualitative behavior of TR/τe from that of the ratio ωmin/ωc of the frequencies corresponding respectively to the minimum and the maximum of G′′(ω). The ratio TR/τe is a strongly increasing function of the volume fraction (cf. eq 23). In the system considered here, the viscoelastic behavior is observed at very low surfactant volume fraction, so that one expects to obtain a rather small value of TR/τe and therefore of ωmin/ωc compared to systems with larger Φ* and k1 of same order of magnitude. Inspection of Figures 2, 3, and 5 shows that ωmin/ωc = 4 (for a concentration of 15 mM CTAHNC) that is a value 10 to 100 times smaller than those reported for other saltfree systems with higher Φ* or for systems in presence of salt. This ratio is found to increase rapidly with surfactant concentration initially (cf. Figure 16) as could be expected from eq 23. The anomalous decrease of this ratio at high surfactant concentration can be entirely accounted for by the decrease of the terminal time TR observed in Figure 8. The ratio ωmin/ωc is also expected to decrease upon increasing temperatures, because of the decrease of TR. The resulting overlapping of the two relaxation processes might contribute to smooth down the dip in the Cole-

Linear Viscoelasticity of Wormlike Micellar Solutions

Cole plot (cf. Figure 4), which leads to an underevaluation of the mean micellar length L h as the temperature is decreased if we want to use eq l2. In a previous study,9 we have attempted to determine the end-cap energy Ec from the temperature dependence of the dip in the ColeCole plot for a given surfactant concentration. This led us to a much larger value of Ec than that reported in this study. The effect described above is likely to be at the origin of an overevaluation of Ec. Let us discuss now the behavior to dilution of the different rheological parameters. The most significant result is the decrease of TR observed in Figure 8 at high surfactant concentration. This decrease occurs roughly in the same range of concentration where the zero-shear viscosity deviates from the theoretical curve (cf. Figure 15). This behavior is quite general for all salt-free systems investigated so far. This is not a simple screening effect since the deviation is observed as well in the CTAHNC systems characterized by a large Debye-Hu¨ckel length (small ionization degree and low volume fraction) as in the 12-2-12 solutions for which the Debye-Hu¨ckel length is much smaller (higher charge density and higher Φ). In fact, the inspection of Figure 15 reveals that the deviations of η0 from theoretical results occur at values of Φ/Φ* of the same order of magnitude: Φ/Φ* = 3-6. Referring back to eqs 4 and 5, one obtains Φ/Φ* ) (Ec/Ee)2. Therefore it seems that the anomalous decrease of TR upon increasing Φ occurs when the end-cap energy overcomes the electrostatic energy. A possible explanation of this effect is a decrease of the mean micellar length upon increasing concentration. Such a behavior would violate the mass action law and could be interpreted only by a change of the ionization degree with concentration which would affect Ee and possibly k1. In fact, an enhanced condensation of counterions is predicted from the solution of the Poisson-Boltzmann equation for semidilute solutions of cylinders, in the absence of any specific binding effect, between counterions and surfactants.11 However, this calculation cannot be reliably applied to the systems considered here where strong binding effects occur. In fact, the limited solubility of the counterion is likely to be a more important factor that would tend to decrease the ionization degree upon increasing the concentration. A more likely interpretation rests on the formation of intermicellar connections. The presence of intermicellar branching speculated by Porte et al.,31 then suggested by rheological experiments,32 was finally revealed by cryoTEM experiments.33 The formation of connections is favored when the scission energy, i.e., Ec - Ee becomes much larger than the energy of formation of a 3-fold crosslink.34 As the electrostatic energy varies with Φ as Φ-1/2, more and more cross-links are formed as the surfactant volume fraction increases beyond Φ*. According to the Lequeux model,19 the reptation process is speeded up by the formation of cross-links and therefore the terminal time is reduced. It was also suggested that the stress relaxation could occur through “ghostlike” crosslinks of the micelles, but there is no quantitative prediction for such a model.35 One expects this process to be dominant in the limit of high density of cross-links. 3. Effect of Salt. The behavior of the plateau modulus, which is, within the experimental error, independent on (31) Appell, J.; Porte, G.; Khatory, A.; Kern, F.; Candau, S. J. J. Phys. II 1992, 2, 1045. (32) Khatory, A.; Kern, F.; Lequeux, F.; Appell, J.; Porte, G.; Morie, N.; Ott, A.; Urbach, W. Langmuir 1993, 9, 933. (33) Danino, D.; Talmon, Y.; Levy, H.; Beinert, G.; Zana, R. Science 1995, 269, 1420. (34) Drye, T. J.; Cates, M. E. J. Chem. Phys. 1992, 96, 1367. (35) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081; 1988, 4, 354; 1989, 5, 398.

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the salt concentration (cf. Figure 13) indicates that the structure of entangled micelles is maintained. The major feature is the decrease of the terminal time induced by addition of salt (cf. Figure 12). At the same time the two main relaxation processes tend to overlap, as shown by the behavior of the complex shear modulus (cf. Figure 11). As a consequence, the minimum of G′′(ω) disappears. The latter effect is likely to be due to a decrease of TR, while τe remains constant. In fact the effect of adding salt is qualitatively the same as that of increasing surfactant concentration at Φ . Φ*. Again the most probable explanation is an increasingly branching occurring in the presence of salt due to the screening by the salt of electrostatic interactions. The decrease of the effective length L h c upon increasing salt concentration would lead to an increase of τbreak and a decrease of τrep, but one cannot discard also an effect on k1. The observed effect here is a purely electrostatic screening one. Indeed, one knows that the Br- ions bind much less than the HNC- ones. As for the CH3COOions, they do not exhibit any binding effect, so that one can exclude any exchange between the counterions when adding salt. If one refers back to the phenomenological approach of Safran et al.,30 one can again think of a salt effect on the short range interactions and consequently on K h (eq 22). It is interesting to note in this respect that beyond a relatively small salt content (as well as beyond a given surfactant concentration) the micelles transform into vesicles (cf. Figures 6 and 10). Conclusion Salt-free solutions of CTAHNC micelles formed at temperatures above the vesicle-micelle transition exhibit beyond a crossover volume fraction Φ* a micellar growth which is well described by the theoretical model of Mackintosh et al.11 The fit of the theory to the data provide an estimate of the end-cap energy which is of the order of 10kBT. The very low value of the crossover volume fraction indicates that the effective electric charge carried by these micelles is very low. At surfactant volume fractions larger than ∼6Φ* the rheological behavior shows deviations from the theoretical prediction, in particular a decrease of the terminal time of the stress relaxation upon increasing surfactant concentration. This effect can be explained by the formation of intermicellar connections, favored by the diminution of the electrostatic contribution. The same effect is obtained by addition of a salt that does not bind with the surfactant. A comparison of the obtained results with those reported for other salt-free systems suggests that the rheological behavior is quite universal, with the following features: the crossover surfactant volume fraction Φ* is mainly controlled by the charge density of the micelles; the larger the end-cap energy, the sharper the increase of viscosity in the semidilute regime; the lesser increase or the decrease of the viscosity observed for Φ/Φ* > ∼3-6 is attributed to the presence of intermicellar connections. The partial phase diagrams shown in this study suggest that the branched micelles constitute intermediate structures between linear micelles and bilayers, whose formation may be associated with an effect of K h , the saddlesplay energy favoring the equality of the two local radii of curvature and therefore the formation of lamellae and spheres. Acknowledgment. This work was performed under the project 1007-1 sanctioned by the Indo-French Centre for the Promotion of Advanced Research. LA970215V