2654
Langmuir 1996, 12, 2654-2662
Rheology of the Cetyltrimethylammonium Tosilate-Water System. 2. Linear Viscoelastic Regime J. F. A. Soltero and J. E. Puig* Departamento de Ingenierı´a Quı´mica, Universidad de Guadalajara, Boulevard M. Garcı´a Barraga´ n # 1451, Guadalajara, JAL 44430, Me´ xico
O. Manero Instituto de Investigaciones en Materiales, Universidad Nacional Auto´ noma de Me´ xico, Apdo. Postal 70-360, Me´ xico, D.F. 04510, Me´ xico Received May 1, 1995. In Final Form: February 13, 1996X In this work, the linear viscoelastic properties of the cetyltrimethylammonium tosilate (CTAT)-water system are examined in detail. This system forms elongated micelles at low and intermediate concentrations, and it yields a hexagonal phase above 27 wt % CTAT at 25 °C. Rheological behavior at low frequencies in a small-amplitude oscillatory shear experiments or at long times in stress relaxation measurements is governed by a single dominant relaxation time, although deviations from the limiting slope of the elastic modulus in the terminal region are observed at high CTAT concentrations. For higher frequencies, however, there is an additional mechanism whose dependence on frequency is analyzed with several rheological models. Analysis of data in terms of the theory of Cates demonstrates that the system consists of flexible micelles in the slow-breaking limit and it exhibits a constant entanglement density along the whole micellar region, even though the average micellar length decreases monotonically with concentration. Under these conditions, reptation speed up by the kinetics process of breaking and re-forming is the controlling relaxation mechanism.
Introduction The rheological properties of aqueous solutions of cetyltrimethylammonium with different binding counterions have been examined extensively.1-6 Most of these papers have focused on the relationship between the viscoelastic properties and the conformation of the structural units. Elongated flexible micelles have been observed in these systems by transmission electron microscopy6-9 and the measured rheological properties (i.e., strong non-Newtonian behavior and viscoelasticity) are explained in terms of network formation.6,10 In some cases, the rheological behavior is similar to that found in transient polymer networks, and in other cases, Maxwell type behavior with a single relaxation time is observed at high counterion concentrations.1-6,11,12 Studies reported elsewhere on mixtures of cetyltrimethylammonium bromide (CTAB)-sodium salicylatewater indicate that at constant surfactant concentration and increasing ionic strength the viscosity shows a complicated behavior with a maximum followed by a minimum and a second maximum.2 The viscoelastic behavior of the low to intermediate counterion concentra* To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, April 15, 1996. (1) Wunderlich, I.; Hoffmann, H. Rheol. Acta 1987, 26, 532. (2) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712. (3) Shikata, T.; Hirata, H. Langmuir 1987, 3, 1081. (4) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988, 4, 354. (5) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1989, 5, 398. (6) Clausen, T. M.; Vinson, P. K.; Minter, J. R.; Davis, H. T.; Talmon, Y.; Miller, W. G. J. Phys. Chem. 1992, 96, 474. (7) Sakaiguchi, Y.; Shikata, T.; Urakami, H.; Tamura, A.; Hirata, H. Colloid Polym. Sci. 1987, 265, 750. (8) Sakaiguchi, Y.; Shikata, T.; Urakami, H.; Tamura, A.; Hirata, H. J. Electron Microsc. 1987, 119, 291. (9) Shikata, T.; Sakaiguchi, Y.; Urakami, H.; Tamura, A.; Hirata, H. J. Colloid Interface Sci. 1987, 119, 291. (10) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (11) Rehage, H.; Wunderlich, L.; Hoffmann, H. Prog. Colloid Polym. Sci. 1986, 72, 51. (12) Shikata, T.; Hirata, H.; Takatori, E.; Osaki, K. J. Non-Newtonian Fluid Mech. 1988, 28, 171.
S0743-7463(95)00368-4 CCC: $12.00
tions is similar to that of solutions of flexible polymer chains with a spectrum of relaxation processes, where the polymer chains have just begun to entangle.1-6 When the salicylate counterion concentration (Cci) is larger than the surfactant concentration (Cs), i.e., after the first maximum in the viscosity curve, the system behaves as if the micelles are fully entangled, showing a plateau modulus G0 which is proportional to Cs2.2.2,12 The same dependence has been reported in concentrated solutions of high-molecular weight polymers.13 However, these micellar solutions (with Cci > Cs) exhibit a Maxwell-type behavior with a single modulus, G0, and a single relaxation time, τR, in contrast to concentrated solutions of flexible polymer chains.2 A single relaxation mechanism has also been observed in other cationic surfactant-salt-water systems, depending strongly on the surfactant/salt molar ratio.4,6 At high counterion concentrations, the dynamics of micellar solutions are dominated by the kinetics of breaking and re-forming of the micelles and the relaxation behavior is described by the Maxwell model with a single relaxation time.10,14-17 However, this model does not take into account the effect of stresses in flow that may disturb the kinetics of chain breaking and re-forming. In part 1 of this series, the phase behavior of the CTATwater system and the relation of structure and structural changes to rheological behavior were examined.18 Here the linear viscoelastic behavior of CTAT-water micellar solutions (low and intermediate concentrations) and of the hexagonal phase that forms at higher concentrations (>27 wt % at 25 °C) is reported. This system was chosen because it forms elongated micelles and exhibits strong (13) Ferry, J. D. Viscoelastic Properties of Polymers; Wiley: New York, 1980. (14) Cates, M. E. Macromolecules 1987, 20, 2289. (15) Cates, M. E.; Turner, M. Europhys. Lett. 1990, 11, 681. (16) Turner, M. E.; Cates, M. J. Phys. (Paris) 1990, 51, 307. (17) Granek, R.; Cates, M. E. J. Chem. Phys. 1992, 96, 4758. (18) Soltero, J. F. A.; Puig, J. E.; Manero, O.; Schulz, P. C. Langmuir 1995, 11, 3337.
© 1996 American Chemical Society
Rheology of the CTAT-Water System
Langmuir, Vol. 12, No. 11, 1996 2655
viscoelasticity at low concentrations (1 wt %) without the addition of electrolytes and because it has a low critical micelle concentration (cmc) and a Krafft temperature close to room temperature. The rheological results are analyzed in terms of the recent theory of Cates for “living polymer” solutions.10,14-17 It is important to stress that the rheological measurements were made in the absence of additional electrolytes that cationic surfactants such as DTAB require to form cylindrical micelles instead of globular ones at low concentration. This is an important difference with most papers published on the subject1-6 because the rheological behavior of such systems depends strongly on the level of extraneous electrolyte concentration. In fact, Kern et al.19 demonstrated in a recent paper the importance of electrostatic interactions on the rheological behavior of elongated micellar solutions. Experimental Section Cetyltrimethylammonium p-toluensulfonate or CTAT (Sigma) had a purity of 98%. It was further purified by recrystallization from a chloroform-ethyl ether mixture (50:50 by volume). Doubly distilled and deionized water was used. Samples were prepared by weighing appropriate amounts of CTAT and water in glass vials. These vials were placed in a water bath at 60 °C for a week where they were frequently shaken by hand to accelerate homogenization. Then samples were placed in a water bath for 3 days at the measurement temperature. All samples were centrifuged to remove suspended air bubbles before being tested. Rheological properties were measured with a Rheometrics Dynamical Spectrometer RDS-II using a cone-and-plate geometry. The cone angle was 0.1 rad, and its diameter was 2.5 cm. The transducer sensitivity was 2-2000 g-cm. To prevent changes in composition for water evaporation during measurements, a humidification chamber containing wetted sponges was placed around the cone-and-plate fixture. To maintain constant the temperature of measurement, the humidification chamber contains a coil to circulate thermostated fluids. Except where indicated, the dynamical measurements were made at strain amplitude levels where the dynamic moduli are strain independent.18
Theoretical Considerations The modeling of the two relaxation processes present in micellar networks or living polymers, i.e., one associated with reptation of long polymer-like chains and another related to the breaking and fusing of the constitutive units, has been treated by Cates et al.10,14-16 and more recently by Granek and Cates17 to include breathing and Rouse motions of the chains for describing the overall stress relaxation function. The model considers that at low frequencies the behavior is Maxwellian as ascertained by the semicircular shape of the Cole-Cole plots. A deviation from the semicircle occurs at a frequency corresponding to the inverse of the breaking time of the micelles, τb. At higher frequencies, there is an upturn associated with the presence of Rouse modes with time scales corresponding to that of molecular motions between entanglements. The characteristic time in this region (τe) is shorter than τb, with the dynamic moduli given by17
π G′ ) G′′ ) G0 ωτe 2
1/2
( )
for ωτe > 1
(1)
G′′min le )A G0 L h
(2)
In eqs 1 and 2, G0 is the plateau modulus, ω is the frequency, A is a constant of order unity, le is an entanglement length associated with the Rouse time τe, and L h is the average micellar length. For flexible micelles, le can be estimated from the relation19
G0 ) kBT/le9/5
(3)
where kB is the Boltzmann constant and T is the temperature. The Cates model depends strongly on the parameter ζ ) τb/τrep, where τrep is the reptation time of micelle of mean length L h . In the fast breaking limit (τb/τrep , 1), the long time behavior of the stress relaxation can be described by a single exponential decay with a relaxation time given by
τR ) (τbτrep)1/2
(4)
τR is also equal to the reciprocal of the crossover frequency between G′ and G′′, as we demonstrate later. According to the Doi-Edwards model,20 the reptation time of a micelle of mean length L h is given by
h 3c3/2 τrep ≈ L
(5)
where c is the surfactant concentration in weight percent. The zero-shear rate viscosity η0 is related to the terminal time τR and to the plateau modulus G0 through
η0 ) G0τR
(6)
The dependence of G0 on concentration is given by
G0 ≈ kBTc9/4
(7)
Therefore, in the fast breaking limit η0 varies with L h and c according to
η0 ∝ L h c3
(8)
On the other hand, in the slow breaking limit, (τb ≈ τrep), which from eq 4 implies that τR ) τrep, η0 varies with L h and c according to
h 3c15/4 η0 ∝ L
(9)
For solutions of monodisperse flexible molecules, the expressions that govern the frequency dependence of the dynamic moduli in the terminal zone are given by the Doi-Edwards expressions:20
G′ ) G0
∑
p odd
G′′ ) G0
∑
p odd
8 1
(ωτR/p2)2
π2 p2 1 + (ωτR/p2)2 8 1
(ωτR/p2)
π2 p2 1 + (ωτR/p2)2
for ωτe < 1
(10)
for ωτe < 1
(11)
The occurrence of the Rouse regime creates a dip in the Cole-Cole plot, and provided τb . τe, the value of G′′ at the dip (G′′min) is equal to17
where p is an odd integer counter. In this paper, we show that, for elongated micellar systems of constant entanglement density, eqs 10 and 11 can be used to predict their
(19) Kern, F.; Lequeux, F.; Zana, R.; Candau, S. J. Langmuir 1994, 10, 1714.
(20) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Claredon Press: Oxford, 1986.
2656
Langmuir, Vol. 12, No. 11, 1996
Soltero et al.
Figure 1. Stress relaxation as a function of CTAT concentration measured at 30 °C after an instantaneous strain deformation (γ) of 50%.
viscoelastic behavior even though they are polydisperse.21 Only the first two terms of the expansions in (10) and (11) are necessary, since the remaining terms are negligible. Similarly, the stress relaxation modulus is given by
( )
8 1 -p2t G(t) ) G0 exp τR π2 p odd p2
∑
(12)
Another model that accounts for the behavior of viscoelastic surfactant solutions at high frequencies is the Hess model.22-24 This model predicts a plateau in G′ and a dependence of G′′ in frequency with a slope of 1 within the region of times shorter than a characteristic time τ. The expressions for the dynamic moduli are
η0 - η∞ ω2τ2 G′ ) τ 1 + ω2τ 2
(13)
G′′ )
η0 - η∞ ωτ + η∞ω τ 1 + ω2τ 2
(14)
1 6D(1 - /kBT)
(15)
τ )
Figure 2. Relaxation modulus of a 20 wt % CTAT micellar solution measured at 30 °C as a function of instantaneous strain deformation. Solid lines represent the predictions of the Maxwell (M) and the Doi-Edwards (D) model.
In these expressions, τ is the main relaxation time of the system, D is the rotational diffusion coefficient of the particles, and is the energy associated with alignment in the flow of the particles which itself is influenced by the presence of the surrounded particles. In this work, τ ) τ R. Results Figure 1 shows stress relaxation experiments at 30 °C after an instantaneous deformation (γ ) 50%) for 5, 10, and 20 wt % CTAT. Stress decays exponentially with time with a single relaxation time for the three samples. The decaying slopes depend on surfactant concentration and become steeper as the CTAT concentration increases, demonstrating faster relaxation times. Figure 2 depicts the relaxation modulus of a 20 wt % CTAT micellar solution after an instantaneous deformation as a function of the applied deformation. The solid lines represent the predictions of the Maxwell model with a single relaxation time (M) and of the Doi-Edwards model (D). Stress relaxes monoexponentially up to deformations (21) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1992. (22) Thurn, H.; Lo¨bl, M.; Hoffmann, H. J. Phys. Chem. 1985, 89, 517. (23) Hess, S. Z. Naturforsch. A 1975, 30A, 728, 1224. (24) Hess, S. Physica A (Amsterdam) 1977, 87A, 273.
Figure 3. Stress versus time at the inception of shear flow for a 20 wt % CTAT micellar solution at 30 °C. The applied shear rate was 0.5 s-1. The solid line is the prediction of the Maxwell model using the relaxation time (τ ) 0.32 s) obtained from stress relaxation experiments.
of 50%. For higher applied deformations, overshoots and deviations from linearity are observed in agreement with oscillatory strain deformation sweeps reported elsewhere.18 Figure 3 shows stress versus time at the inception of shear flow for a 20 wt % CTAT sample at 30 °C. The shear stress applied here was 0.5 s-1. Steady state is achieved rapidly ( τrep. However, theory was developed only for micellar solutions, so the analysis may not be valid for the hexagonal phase. Notice that, in the whole concentration range, ζ > 0.1 (see Table 1), suggesting that the system is in the slow-breaking limit.16,17 Figure 8 reports G′′min/G0 at 30 °C as a function of surfactant concentration. This ratio was obtained from the value of the dip in the Cole-Cole plots.17,19 This ratio
is constant with surfactant concentration within the micellar region and increases with temperature (see Table 1). At higher concentrations where the hexagonal phase forms, G′′min/G0 increases with increasing CTAT concentrations, but, again, the analysis may not be valid outside the micellar region. Using these results and eqs 2 and 3, the mean micellar length, L h , was estimated as a function of surfactant concentration (Figure 9). Clearly, mean micellar length decreases monotonically with surfactant concentration (with a slope of -1.3 ( 0.1 in the log-log plot) within the micellar region. Figure 10 shows plots of the logarithm of the mean micellar length, L h , versus the reciprocal of the absolute temperature. Micellar length decreases with increasing temperature. Also, plots are linear with similar slopes at all concentrations within the micellar region, indicating an Arrhenius-type behavior. From the slope of these plots an activation energy, Ec ) 10.0 ( 2 kcal/mol, was estimated. Kern et al. also reported Arrhenius dependence for the length of micellar solutions of the “dimeric” surfactant ethanediyl-R,ω-bis(dodecyldimethylammonium) bromide.19
Rheology of the CTAT-Water System
Figure 10. Semilogarithmic plot of the mean micellar length, L h , versus 103/T at different surfactant concentrations. The straight lines are the best least-squares fits to experimental data.
Langmuir, Vol. 12, No. 11, 1996 2659
Figure 12. Arrhenius dependence of the main relaxation time (τR ) 1/ωco) with absolute temperature. The arrows are the relaxation times obtained from the crossover of G′ and G′′ in temperature sweeps done at 10 rad/s.18
it remains constant upon increasing CTAT concentration. The plateau modulus, on the other hand, is fairly independent of temperature, but it follows a correlation with CTAT concentration of the form
G0 ∝ [CTAT]2.0(0.1
(16)
A similar correlation has been reported in other cationic surfactant-salt-water micellar solutions2,12 and in polymer solutions.13 Figure 12 presents plots of log τR versus the reciprocal of the absolute temperature for various CTAT concentrations. Plots are linear and demonstrate that the main relaxation time follows an Arrhenius-type behavior of the form
τR ∝ exp(Eτ/kBT)
Figure 11. Dependence on surfactant concentration of relaxation time (τR ) 1/ωco) (top), zero-frequency complex viscosity (η0) (middle), and plateau elastic modulus (G0) (bottom) at various temperatures.
Figure 11 presents the dependence of the main relaxation time (τR), of the complex viscosity at the terminal region (η0), and of the plateau modulus (G0) with CTAT concentration and temperature. The main relaxation time increases with increasing concentration up to between 2 and 3 wt % CTAT, depending on temperature, where it reaches a maximum and then it diminishes monotonically with concentration up to 27 wt %. Above this concentration, it remains fairly constant with increasing surfactant concentration. τR also depends strongly on temperaturesit increases up to 10-fold with an increase of 5 °C. The zerofrequency complex viscosity, in turn, also varies strongly with concentration and temperature. It increases rapidly with CTAT concentration up to 2-3 wt % CTAT, and then
(17)
where Eτ is an activation energy. All of the plots have similar average slopes, suggesting that the main structure of the samples is the same. From the slope of these plots a value of Eτ of 21.8 kcal/mol was estimated. A value of 25.4 kcal/mol was reported by Candau et al. for CTAB in 0.25 M KBr brine.29 The arrows in the figure correspond to the temperature at which G′ and G′′ cross over in the temperature sweeps done at a frequency of 10 rad/s (Figure 11 of part 1 of this series).18 Time-temperature superposition (Master curves) for the dynamic moduli and for the complex viscosity are depicted in Figure 13. The shifting factors employed are those of the WLF equation.13 The overlapping of the data is excellent except for the viscous modulus in the highfrequency region. However, by using a shifting factor aT(τe) instead of aT(τR), where τe is the onset time for Rouse behavior (see Table 1), it is possible to overlap G′′ values at various temperatures in the high-frequency region (see inset in Figure 13b). Time-concentration superposition is shown in Figure 14 for G′, G′′, and complex viscosity. Again, superposition is excellent for the three dynamic functions. The shifting factors used are ac ) τR and bc ) (τR/η0). The master curve for complex viscosity follows the Maxwell model:
()
G′
τR x2 ) η0 1 + x2
(18)
where x ) τRω. Similar plots were reported elsewhere for micellar solutions made of a mixture of tetradecylpyridinium salicylate and tetradecyltrimethylammonium (29) Candau, S. J.; Hirsch, E.; Zana, R.; Delsanti, M. Langmuir 1989, 2, 1225.
2660
Langmuir, Vol. 12, No. 11, 1996
Figure 13. Time-temperature superposition master curves for (from a (top) to c (bottom)) G′, G′′, and complex viscosity. WLF shifting factors (aT and bT) were used.13 Inset: Timesuperposition master curves using the value of τe of the DoiEdwards model.
salicylate in the presence of sodium bromide.22 Also, the Hess model reproduces well the master curve for G′′. Discussion In part 1 of this series,18 the relationship between structure, structural changes, and rheology of the CTATwater system was examined. At temperatures below the Krafft temperature (23 °C), the system is a viscous dispersion of crystals where a linear viscoelastic regime was not found. Above 23 °C, the system looks like a transparent gellike material. Elastic moduli are independent of strain deformation for deformations near 100%. The linear viscoelastic regime decreases with concentration but increases with temperature (Figures 9 and 10 of part 1).18 Here the linear viscoelastic response of CTATwater mixtures from low to high concentrations as a function of temperature is investigated. The stress relaxation behavior (Figure 1-3) as well as the frequency dependent moduli at low frequencies (Figures 4 and 5) can be predicted from the relaxation time obtained from the crossing of G′ and G′′ and the experimental plateau modulus. Figure 6 shows, however, that deviations from a single relaxation time (Maxwell behavior) occur at high frequencies. In fact, the viscous modulus is proportional to ω-1 at high frequencies, but an upturn corresponding to additional relaxation mechanisms is observed at even higher frequencies (>100 rad/s) for
Soltero et al.
Figure 14. Time-concentration superposition master curves for (from a (top) to c (bottom)) G′, G′′, and complex viscosity. Shifting factors were aC ) τR and bC ) τR/η0.
the low concentrated samples (Figure 4). This upturn was not detected for samples with higher CTAT concentrations, probably because we did not have access to the entire viscoelastic spectrum (Figure 5). However, these departures are not observable in the stress relaxation experiments because the time scale where upturn is observed in frequency sweeps is rather short (of the order of 0.01 s or smaller). The linear viscoelastic behavior of the CTAT-water system can be analyzed with the theory developed by Cates and collaborators.10,14-17 However, it is interesting to examine it with well-known rheological models. The Maxwell model with a single relaxation time predicts moderately well the frequency dependence G′ and |η*| (see Figures 4 and 5). The predicted values deviate from the experimental data because the experimental values of G0 and τR were used instead of fitting data to the model. The Maxwell model also predicts well the behavior of G′′ at low and moderate frequencies, indicating a fast breaking regime in this frequency range. However, at higher frequencies the model underestimates G′′. Departures from experimental values occur at frequencies that shift to lower values (i.e., longer times) with increasing CTAT concentration (see Figure 6) and decreasing temperature. Inasmuch as the reciprocal of the frequency at which deviations from the Maxwell model occur is the breaking time of the micelles,17 this result implies that τb decreases with increasing CTAT concentrations or decreasing temperatures (see Table 1).
Rheology of the CTAT-Water System
The Doi-Edwards model, in turn, predicts quite well all of the viscoelastic functions in a larger frequency range that the Maxwell model does (Figures 4 and 5). Deviations in G′′ are observed for samples with low concentrations at high frequencies (ωτR > 1), where G′′ goes through a minimum followed by an increase at higher frequencies (Figure 4). This upturn has been interpreted as a crossover to the Rouse or breathing regime.17 The Doi-Edwards model was developed to predict the dynamic behavior of monodisperse polymer systems formed by flexible macromolecules at high concentrations.20 Elongated micelles can be considered as flexible macromolecules with an equilibrium molecular weight (or size) distribution. This distribution is known to be broad.21 However, CTAT micelles behave rheologically as monodisperse systems that relax by reptation speeded up by the breaking and re-forming processes. Figure 7 and Table 1 show that the ratio of the breaking time to the reptation time (ζ) is larger than 0.1 and exceeds 1 at high concentrations. In this case the dynamic behavior corresponds to the slowbreaking limit of flexible micelles where τR ≈ τrep.17 This is further corroborated by the dependence of the average micellar length on concentration, shown in Figure 9 (L h∝ c-1.3(0.1), which implies, from eq 4, that η0 should be independent of CTAT concentration, as shown in Figure 11b for CTAT concentrations above 3 wt %. Also, the ratio of the entanglement length to the average micellar h , remains constant with CTAT concentration length, le/L up to 25 wt % CTAT (Figure 8), which means that the h )-1, is also constant. micellar entanglement density (le/L This condition approaches that of the reptation dynamics of monodisperse systems of flexible units. Further evidence that CTAT micelles respond rheologically as monodisperse systems is the sharp transition to the shear thinning region in viscosity vs shear rate curves.26 Finally, the Hess model, which was used by Thurn et al.22 to reproduce the rheological behavior of elongated micelles of a mixture of cationic surfactants, predicts the behavior of G′ and G′′ quite well over the whole range of frequencies for samples of low concentrations (Figure 4). In the Hess model (eq 13), there is an additional term to the Maxwell expression for the loss modulus which corresponds to the influence of the alignment tensor of the constitutive units.23,24 This quantity (η∞ω) is proportional to the energy associated with the alignment of the particles or units influenced by their neighboring particles, where η∞ represents the viscosity at high frequencies which has chosen here as a fitting parameter in the predictions of the experimental data. The consequence of the presence of this term is the linear variation of G′′ with ω at high enough frequencies, which follows closely the observed upturn at higher frequencies. This effect leads to a stronger frequency dependence than that in the Rouse regime (where these units are presumably behaving as free chains). In fact, G′′ goes as ω1/2 in the Rouse region;13 here G′′ increases as ωsbehavior which is only predicted by the Hess model. The upturn of G′′ at high frequencies has also been interpreted in terms of a crossover between the regimes of reversible scission and of breathing of the polymer-like micelles.30 Relaxation time increases sharply with concentration up to 2-3% CTAT, and then it decreases steadily with CTAT concentration (Figure 11). This behavior has been observed elsewhere in the CTAB-sodium salicylatewater system,12 and it was explained by the response of the plateau modulus and the zero-frequency viscosity to temperature and concentration (Figure 11). Since τR ) η0/G0, the increase in τR at low concentrations is caused (30) Berret, J. F.; Appell, J.; Porte, G. Langmuir 1993, 9, 2851.
Langmuir, Vol. 12, No. 11, 1996 2661
by the rapid increase of the zero-shear viscosity with concentration. However, after 2-3 wt % CTAT, η0 becomes constant for the reasons discussed above, but G0 continues to increase, resulting in a continuous decrease in the relaxation time. Notice also that, above the phase transition, which occurs around 27-30 wt %sdepending on temperature18sthe main relaxation time becomes constant since both the complex viscosity and the plateau modulus remain fairly constant. A decrease in the relaxation time with increasing concentration has also been observed and predicted elsewhere in systems with a strong level of orientation of the constitutive units.31,32 Other authors have reported the decrease of τR with concentration in the CTAB-sodium salicylate-water system in terms of an specific 1:1 complexation process that leads to a highly nontrivial dependence of the decay time on salt and surfactant concentrations.12 The relaxation time follows an Arrhenius law with temperature with an activation energy on the order of 22 kcal/mol, which is similar to the ones reported in other living polymer systems.29 For typical polymer solutions, the main relaxation time (or τ/ηs, where ηs is the solvent viscosity) is a weak function of T, in contrast to the results reported here (Figure 12). The strong dependence of τR on T has been interpreted to occur as a consequence of the decrease in the average micellar lengthsL ∼exp(E/kT), where E is scission energy (Figure 10)sand the increase of the rate constant of micellar breakage upon increasing temperature.10 Figure 12 also includes relaxation times obtained from the crossover of G′ and G′′ in temperature sweeps done at a frequency of 10 rad/s (Figure 11 in part 1).18 These values fall within their respective line for each concentration (arrows in Figure 12), demonstrating that the crossover temperature is the temperature at which the main relaxation time of the system (inverse of the applied frequency) is equal to 0.1 s. The plateau modulus increases with concentration up to the phase transition with a slope of 2 ( 0.1, and then it remains constant for higher concentrations (Figure 11c). This indicates that the same structure is preserved throughout the concentration range under these conditions of shear and that only the density of effective elastic chains is varying. Slopes of the order of 2.3 ( 0.2 are observed in most polymeric systems.13 This result is consistent with scaling laws in semidilute systems of elongated units and also appears to be empirically valid in relatively dense systems.10 Time-temperature superposition for G′, G′′, and η0 is achieved remarkably well using the WLF shifting factors except for G′′ at high frequencies (Figure 13). However, by defining the shifting factor, aT in terms of τe instead of τR, the data at high frequencies collapse in a single line (see inset in Figure 13). Likewise, time-concentration superposition works well for the three dynamic functions (Figure 14), which corroborates that the dynamic behavior of these micellar solutions can be determined by measuring only two steady-state quantities, η0 and τR, for each solution. It is important to stress here that there are important differences with the rheological behavior of another electrolyte-free micellar system reported in the literature.19 In the “gemini” surfactant-water system reported by Kern et al.,19 the entanglement density does not vary monotonously with surfactant concentration (Figure 8 of ref 19). Similarly, micellar length goes through a maximum and then it decreases with concentration at higher (31) Doi, M. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 229. (32) Valdes, M.; Manero, O.; Soltero, J. F. A.; Puig, J. E. J. Colloid Interface Sci. 1993, 160, 59.
2662
Langmuir, Vol. 12, No. 11, 1996
concentrations. Also, and contrary to what is observed for highly screened ionic micelles, the temperature dependence of η0 in the vicinity of its maximum is non-Arrhenian and the overall variation is much larger than that obtained for screened micelles. Finally, the activation energies obtained by these authors for such micelles are anomalously high. The explanation to these rheological differences is, as pointed out by Kern and collaborators, that the dimeric surfactants are rather bulky so they pack more easily in the cylindrical part of the micelles than in the hemispherical end caps; i.e., the end-cap energy can be rather large. This large energy together with the increase in micelle ionization degree with surfactant concentration should result in a rapid micellar growth in the semidilute range followed by a decrease at high surfactant volume fractions. Conclusions Cetyltrimethylammonium forms elongated micelles in the presence of counterions such as salicylate or tosilate ions. The presence of such constitutive units leads to strong viscoelastic effects in the linear regime.
Soltero et al.
Analysis of the data with the Cates theory indicates that the dynamic behavior of the micellar system deviates from that of a single relaxation time mechanism, corresponding to a kinetic process of breakage and re-formation of the structural units. Instead, the system is in the slowbreaking limit, and it exhibits constant entanglement density in the whole concentration range where micelles form. Under these conditions, reptation speeded up by the kinetic process is the main relaxation mechanism and the Doi-Edwards model reproduces well the rheological behavior. At low concentration and high frequencies, on the other hand, an upturn in G′′ is observed where G′′ increases with frequency with a slope of 1. This dependence can be predicted only by the Hess model. At higher concentrations, where the hexagonal phase forms, strong deviations are observed which may be attributed to limitations of the Cates model for organized, long-ordered phases. Acknowledgment. This work was supported by the Consejo Nacional de Ciencia y Tecnologı´a de Me´xico (Grants No. 161-E92 and 465100-5-3397-E). LA950368N
