3000
Langmuir 2006, 22, 3000-3006
Quasi-anomalous Diffusion Processes in Entangled Solutions of Wormlike Surfactant Micelles Anuj Shukla, Reiner Fuchs, and Heinz Rehage* Lehrstuhl fu¨r Physikalische Chemie II, UniVersita¨t Dortmund, Otto-Hahn-Strasse 6, D-44227 Dortmund, Germany ReceiVed December 19, 2005. In Final Form: February 6, 2006 In this article, we present a detailed analysis of the dynamic properties of entangled solutions of semi-flexible, threadlike surfactant micelles. These aggregates were formed by self-association processes in aqueous solutions of cationic surfactants such as cetylpyridinium chloride (CPyCl) or cetyltrimethylammonium bromide (CTAB) after the addition of different amounts of sodium salicylate (NaSal). We performed dynamic light scattering (DLS) experiments in combination with rheological measurements in order to investigate the dynamic properties of these viscoelastic surfactant solutions. In all samples, we observed three distinct relaxation regimes: initial monoexponential decay, followed by a power-law behavior at intermediate observation times. A second monoexponential region was detected at very long times, and this terminal regime described the viscoelastic features of the samples. The fast decay mode was induced by local cooperative motions in the gellike network. The intermediate and slowest decay modes point to the existence of quasi-anomalous diffusion processes. These phenomena are characterized by linear-diffusion properties at long times, and they obeyed anomalous logarithmic slow-dynamics behavior at intermediate time zones. The anomalous diffusion properties at intermediate time scales can be induced by the bending motions of the rodshaped micelles between two entanglement points. This regime, which was more extended at lower temperatures, was described by the power-law form of the correlation function. The power-law exponent depended on the chemical structure of the surfactants and the temperature. The power-law regime shifted toward earlier times as the gellike network evolved. The slowest mode of the correlation function coincided very well with the shear stress relaxation times of the three-dimensional, transient networks. We observed that the temperature dependence of the slowest mode followed Arrhenius laws. This result provides experimental evidence for thermally activated topological relaxation processes of random fluid phases. We obtained activation energies of approximately 30 kcal/mol, and these data coincided well with previously reported literature values, which were determined in similar surfactant solutions. Characteristic “screening lengths”, over which viscous effects became important, could also be determined from the activation energy. The elastic modulus G0, calculated from the slowest mode of the correlation function, was in pretty good agreement with rheological data. The light-scattering spectra were consistent with the theoretical model of dynamical coupling of the concentration fluctuations to viscoelasticity. Since only minute sample volumes are required for advanced DLS experiments, this method to extract viscoelasticity is well suited for advanced studies of gellike biomaterials.
Introduction In aqueous solutions, surfactant molecules self-assemble into various microstructures, such as spherical micelles, rod- or disklike aggregates, or different types of liquid crystals. The gellike samples studied in this publication contained an entangled, semidilute solution of wormlike micelles, often referred to as threadlike aggregates. It is easy to prepare such temporary networks by just solving cationic surfactants such as cetylpyridinium chloride (CPyCl) or cetyltrimethylammonium bromide (CTAB) and adding different amounts of the strongly binding counterion sodium salicylate (NaSal). Recently, viscoelastic wormlike micelles have drawn considerable interest in basic research and practical applications.1-12 Above some critical * Corresponding author. Phone: +49-(0)231-7553910. Fax: +49-(0)231-7555367. E-mail:
[email protected]. (1) Kern, F.; Lemarechal, P.; Candau, S. J.; Cates, M. E. Langmuir 1992, 8, 437. (2) Khatory, A.; Lequeux, F.; Kern, F.; Candau, S. J. Langmuir 1993, 9, 1456. (3) Vethamuthu, M. S.; Almgren, M.; Brown, W.; Mukhtar, E. J. Colloid Interface Sci. 1995, 174, 461. (4) Kim, W.-J.; Yang, S.-M.; Kim, M. J. Colloid Interface Sci. 1997, 194, 108. (5) Kim, W.-J.; Yang, S.-M. J. Colloid Interface Sci. 2000, 232, 225. (6) Imai, S.; Shikata, T. J. Colloid Interface Sci. 2001, 244, 399. (7) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712. (8) Soltero, J. F. A., Puig, J. E.; Manero, O. Langmuir 1996, 12, 2654. (9) Raghavan, S. R.; Kaler, E. W. Langmuir 2001, 17, 300. (10) Montalvo, G.; Rodenas, E.; Valiente, M. J. Colloid Interface Sci. 2000, 227, 171.
concentration, called the overlapping concentration c*, wormlike micelles start to entangle with each other, forming threedimensional, transient networks. The rheological properties of these samples are nowadays well explored and reminiscent of those observed in semidilute, entangled polymer solutions.1-12 Wormlike micelles are comparable to polymers in that they are quite flexible (typical persistence lengths on the order of 20 nm versus a diameter of about 5 nm), and they exhibit contour lengths on the order of several micrometers.13 It turns out that the stress relaxation of these structures is partly controlled by reptation. These so-called living polymers differ from classical macromolecules in that they are constantly breaking and recombining.14 Consequently, they do not exhibit a quenched contour-length distribution.14 The transient rheological properties of these entangled suspensions of wormlike aggregates can be described by a theory developed by Cates and co-workers.14 Because of the inertia effects of almost all mechanical measurements performed in rheometers, the plateau elastic modulus and the stress relaxation times of the gellike solutions are often insufficiently detectable, especially in the range of very short times or elevated angular frequencies. Unfortunately, these (11) Ponton, A.; Schott, C.; Quemada, D. Colloids Surf., A 1998, 145, 37. (12) Lin, Z.; Cai, J. J.; Scriven, L. E.; Davis, H. T. J. Phys. Chem. 1994, 98, 5984. (13) Cates, M. E.; Candau, S. J. J. Phys: Condens. Matter 1990, 2, 6869. (14) Turner, M. S.; Cates, M. E. Langmuir 1991, 7, 1590.
10.1021/la053435e CCC: $33.50 © 2006 American Chemical Society Published on Web 03/01/2006
Diffusion in Solutions of Wormlike Micelles
data are essential to describing the dynamic properties of the samples. In this article, we intend to use advanced dynamic light scattering (DLS) techniques in combination with rheological experiments in order to determine the weak elastic modulus of the gel G0 and the relaxation time ts. Previous investigators have already observed by DLS experiments a bimodal relaxation process in the concentration fluctuation spectrum of entangled solutions of threadlike micelles.15-19 Nemoto et al.,16 Buhler et al.,18 and Amin et al.19 analyzed the autocorrelation function by two methods: a double-exponential fit of the entire autocorrelation function and a single-exponential fit of the isolated fast and slow relaxation regimes. Amin et al.19 observed that simultaneous double-exponential fits were not applicable to all measured samples. Nemoto et al.16 described the bimodal distributions of the time correlation function on the basis of the dynamic coupling between stress and concentration fluctuations. This theory was developed by Doi and Onuki.20 This new theoretical model predicts the existence of a power-law regime at intermediate time scales. In this context, it is interesting to note that, in other types of gelling systems, sometimes three different relaxation modes occurred.21-27 On the basis of these previous observations, we shall examine in this publication the light-scattering autocorrelation function of viscoelastic surfactant solutions. The experimental data will be interpreted in terms of the theory of anomalous diffusion.28 In the framework of these investigations, we shall discuss the temperature-dependent scaling laws, the hydrodynamic correlation lengths, the activation energy, and the screening lengths determined by DLS techniques. The results thus obtained are systematically compared with the experimental data of rheological measurements. Experimental Section Preparation of the Solutions. CPyCl, CTAB and NaSal were purchased from Merck and Fluka and purified by recrystallization. The solutions were left standing for at least 2 days in order to reach equilibrium. After this time, we did not observe any time-dependent change in the storage and loss modulus, indicating that the solutions had reached their equilibrium structures (time sweep test). At these conditions, we also observed constant results of light-scattering data. We investigated surfactant solutions of 100 mM CPyCl and 250 mM NaSal (System I) or samples of 60 mM CTAB and 350 mM NaSal (System II) at different temperatures. DLS Measurements. The DLS experiments were performed in the commercially available instrument “Zetasizer-Nano” from Malvern. A 4 mW He-Ne laser (633 nm wavelength) with a fixed detector angle of 173° was used. It was observed that the investigated solutions scattered about 1000 times more light than did water. To reduce the effect of multiple scattering, we used noninvasive backscattering techniques. In a series of experiments, we measured the light scattering of the samples at different depths within the cuvette. The measurement position was changed manually to record (15) Koike, A.; Yamamura, T.; Nemoto, N. Colloid Polym. Sci. 1994, 272, 955. (16) Nemoto, N.; Kuwahara, M.; Yao, M.; Osaki, K. Langmuir 1995, 11, 30. (17) Shikata, T.; Imai, S.; Morishima, Y. Langmuir 1997, 13, 5229. (18) Buhler, E.; Munch, J.; Candau, S. J. Phys. II 1995, 5, 765 (19) Amin, S.; Kermis, T. W.; van Zanten, R. M.; Dees, S. J.; van Zanten, J. H. Langmuir 2001, 17, 8055. (20) Doi, M.; Onuki, A. J. Phys. II 1992, 2, 1631. (21) Nystro¨m, B.; Kjoniksen, A. J. Langmuir 1997, 13, 4520. (22) Martin, J. E.; Wilcoxon, J. P.; Odinek, J. Phys. ReV. A 1991, 43, 858. (23) Martin, J. E.; Adolf, D. Annu. ReV. Phys. Chem. 1991, 42, 311. (24) Martin, J. E.; Wilcoxon, J. P. Phys. ReV. Lett. 1988, 61, 373. (25) Adam, M.; Delsanti, M.; Munch, J. P.; Durand, D. Phys. ReV. Lett. 1988, 61, 706. (26) Ren, S. Z.; Shi, W. F.; Zhang, W. B.; Sorensen, C. M. Phys. ReV. A 1992, 45, 2416. (27) Lang, P.; Burchard, W. Macromolecules 1991, 24, 814. (28) Saichev, A. I.; Utkin, S. G. J. Exp. Theor. Phys. 2004, 99, 443.
Langmuir, Vol. 22, No. 7, 2006 3001 the effect of changing this parameter. We investigated all samples at at least five different positions. The minimum position was always selected near the cuvette wall. The maximum position was chosen at the center of the cuvette. In all these experiments, we did not observe any differences between these results. This indicates that multiple scattering did not give significant contributions in our measurements. For all forthcoming experiments, the measurement position was determined automatically through an optimization procedure of the signal-to-noise ratio of the scattered light. The measurements were performed in the temperature range between 25 and 50 °C. In all experiments, about 1 mL of the samples was transferred to a special dust-free light-scattering cell. To allow the temperature to equilibrate, we started the experiments 30 min after the cuvette was placed in the DLS apparatus. The temperature was controlled within ∆T ) (0.02 °C. The light-scattering process defines the wave vector q ) 4πn/λ‚sin(θ/2), where λ is the wavelength of the incident light in a vacuum, θ denotes the scattering angle, and n describes the refractive index of the solvent. Rheological Measurements. The rheological experiments were performed using an ARES rheometer (TA Instruments) fitted with an environmental chamber. The cone-and-plate device was made of titanium. Before starting dynamic mechanical measurements, we always performed strain-sweep experiments in order to explore the linear-viscoelastic regime. In these tests, a constant angular frequency was applied with varying strain amplitudes. In the regime of small deformations, one usually obtains linear-viscoelastic response, and, with increasing shear strain, nonlinear properties occur. The regime of linear viscoelasticity is characterized by the plateau value of the strain-sweep curve. For all measured viscoelastic surfactant solutions, the plateau region was terminated at strain amplitudes above 100%. To perform measurements in the linear-viscoelastic regime, for all subsequent frequency-sweep tests, we applied a constant shear strain amplitude of 20%. This value was selected on the grounds of the good signal-to-noise ratios of all the rheological curves. In this context, it is interesting to note that all rheological functions as the storage or loss modulus are defined in such a way that they do not depend on the shear strain amplitude. This holds, at least, for conditions of linear-viscoelastic response. In a series of frequency-sweep tests, a small amplitude oscillatory shear flow with a constant shear strain of about 20% was applied to the investigated viscoelastic solution. The magnitude of the complex viscosity |η*(ω)|, the storage modulus G′(ω), and the loss modulus, G′′(ω) were measured in periodical experiments in the angular frequency range between 0.001 and 300 rad/s.
Experimental Results DLS. The typical results of our measurements are summarized in Figure 1. It is worthwhile to mention that the correlation function exhibits three different relaxation regimes (see Figure 1). In contrast to former investigations, we did not observe a bimodal distribution or isolated single-exponential fit.16,18,19 To separate the three relaxation regimes, we used a piecewise description of the entire correlation function:
{
Af exp
g1(τ) )
( ) ( ) ( ) -τ + tf
Ap
1+
-τ As exp , τ g ts ts
, τ < ts
τd V
(1)
The parameters Af, Ap, and As are the amplitudes for the different relaxation processes. The time constants tf and ts denote the fast and slow relaxation time, respectively. Parameter V describes the time at which the power-law tail begins,22 and d is the power-law exponent. Close inspection of Figure 1 leads to the conclusion that the entire correlation function can be subdivided into monoexponential decay at short time, followed by a power-law
3002 Langmuir, Vol. 22, No. 7, 2006
Shukla et al.
Figure 1. Field correlation function g1(τ) as a function of the delay time τ at several temperatures of System I and System II.
regime. At elevated times, we observed a second zone of monoexponential decay. It is evident that the simultaneous double-exponential fit in combination with the power-law description, a technique often used to analyze the correlation function of semidilute polymer solution,21 did not accurately describe our viscoelastic surfactant systems. Therefore, all our results are based on the analysis of eq 1 at different time scales. For diffusive fast mode relaxation time tf, the hydrodynamic correlation length ξΗ can be obtained through use of the StokesEinstein relationship:
ξH )
kBT 6πηDc
(2)
Here, Dc denotes the cooperative diffusion coefficient: Dc ) (tfq2)-1. Parameter kB is Boltzmann’s constant, T denotes the absolute temperature, and η describes the solvent viscosity. For our solutions, we used the zero-shear viscosities of the NaSalwater systems as the solvent viscosity η.30 The characteristic features of the measured correlation functions can, in principle, be described by the theory of quasi-anomalous diffusion.28 These processes, which are characterized by waitingtime distributions, exhibit monoexponential decay at short and long times and anomalous diffusion at the intermediate regime.28 A possible explanation of such characteristic relaxation properties could be a fusing process of rod-shaped micelles at an entanglement point. This might lead to more stable junction points, which can only decay after reforming two separate micellar aggregates. If the shear stress relaxes by this special process, the characteristic time scale is determined by the waiting time for roughly one crossing reaction to occur within each entanglement length.29 Crossing reactions might then occur when the random (29) Cates, M. E. Phys. ReV. B 1992, 45, 12415. (30) Nemoto, N.; Kuwahara, M. Langmuir 1993, 9, 419.
Figure 2. G′′(ω) as a function of G′(ω) for both surfactant systems at T ) 25 °C (Cole-Cole plot). Solid-line semicircles describe the behavior of the simple Maxwell model.
motions of neighboring micellar sections acquire enough thermal energy to pass the activation barrier for topological changes. Dynamic Viscoelasticity. The Maxwell model predicts that the intersection point between G′(ω) and G′′(ω) is located at the angular frequency ωi ) 1/ts, where ts denotes the stress relaxation time.7 It is hence possible to calculate ts from the intersection point of the dynamic moduli.7 The zero-shear modulus G0 was calculated from the plateau value of the storage modulus. It is often convenient to represent the storage and loss moduli in the form of a Cole-Cole plot.14 In these diagrams, the loss modulus is plotted as a function of the storage modulus. The Cole-Cole plot provides a more precise determination of the relaxation behavior of samples than do simple frequency sweeps.14 For a Maxwell material, one obtains a semicircle. Figure 2 shows the Cole-Cole plots and the characteristic relaxation time together with the plateau modulus for both investigated systems at a temperature of T ) 25 °C. The zero-shear modulus G0 could also be determined from the simple extrapolation, as shown in Figure 2. According to Cates model, a suitable extrapolation is to take the point where the tangent to the Cole-Cole plot cuts the storage modulus axis.14 The G0 obtained from above-mentioned methods was similar within experimental error. Therefore, for the sake of simplicity, the G0 obtained from the plateau value of the storage modulus and the stress relaxation time ts obtained from the intersection point of the dynamic moduli are used for discussion. The zero-shear viscosity was calculated from the magnitude of the complex viscosity |η*(ω)| at very small oscillation frequencies (plateau regime).
Data Analysis and Discussion At all investigated temperatures (25-50 °C), we observed typical polymer solution dynamics. The existence of three different relaxation regimes has already been reported for several gelling systems21-27 in the postgel regime. However, for entangled solutions of rod-shaped micelles, such observations are new.
Diffusion in Solutions of Wormlike Micelles
Langmuir, Vol. 22, No. 7, 2006 3003
Recent publications16,18,19 on elongated wormlike micelles were analyzed in terms of an single-exponential decay followed by a slow mode. In analogy to our experiments, this slow mode may contain a combination of the power-law and exponential decay. The phenomenological theory developed by Doi and Onuki20 predicts a power-law behavior at intermediate time scales, just as it was observed in our experiments. One can see that the power-law behavior, which is linear on double-log scales (see Figure 1), increases with time as the sol evolves to a gel. It is easy to overlook this power-law regime at elevated temperatures because this region is then very small. During the evolution of the gellike network structure, however, it becomes much more extended at lower temperatures. We shall now consider these different modes in a more detailed manner. Figure 3 shows the dynamic correlation length obtained from the fast decay mode using eq 2. It turns out that ξH is independent of the temperature and takes a constant value of ξH ) 8.3 ( 0.70 nm for System I and ξH ) 14.82 ( 0.28 nm for System II. The smaller value of ξH for System I in comparison with that of System II is consistent with the rheological measurements because G0 is related to the average static correlation length, ξ, by the equation7
x 3
ξ≈
kBT G0
(3)
A larger value of G0 in System I (see Figure 2) corresponds to a smaller ξ (∼78% of ξ in System II). A smaller ξH value of the transient network in System I is probably due to a higher total surfactant content in System I. The static correlation length, ξ, can be identified using the simple approximation18
ξ≈
RG
x3
(4)
Here, RG denotes the radius of gyration. The ratio of RG to ξH is 8.3 for System I and 6.6 for System II. This value may be compared with the corresponding value of 5 obtained for semidilute solutions of polystyrene in benzene.31 The fast relaxation process is governed by the interplay between the osmotic restoring force (∂π/∂c) and the sedimentation coefficient s through the relationship32
∂π ∂c
Dc = s
(5)
A change in the value of the sedimentation coefficient is probably compensated by the corresponding change in osmotic compressibility. This leads to a constant reduced diffusion coefficient with temperature:20
kB Dcη = = constant T 6πξH
Figure 3. Temperature dependences of the dynamic correlation length ξH for both surfactant systems.
(6)
Equation 6 suggests that the short-time dynamics, described by the fast mode of the relaxation function, is hardly affected by the thermal agitation. This process is therefore not sensitive to the long-range connectivity. It turns out that the long time-scale dynamics are very different in the sol and gel states. The extension of the intermediate power-law relaxation behavior of the DLS spectrum depends very much on the (31) Nemoto, N.; Makita, Y.; Tsunashima, Y.; Kuwahara, M. Macromolecules 1984, 17, 2629. (32) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.
Figure 4. Illustration of the width of the time window of the powerlaw behavior of the correlation function for both surfactant systems as a function of temperature.
temperature. Figure 4 shows the width of the power-law regime. The second stage of the relaxation process, where the power-law behavior prevails, is probably associated with a crossover to small scale motions of the chains, which are not caused by breaking or reptation processes. This explanation is consistent with dynamic viscoelastic measurements that show deviations from the Maxwell regime at high frequencies and low temperatures (see Figure 2). Deviation from the semicircle in Figure 2 reflects the additional internal dynamics of the micelles.13,14 A Maxwell behavior can be explained by the fact that, before a given tube segment relaxes, the chain will undergo many breaking (tbreak) and fusing reactions. Consequently, there is no memory of the initial length and position of the chain in the tube on the time scale of the fast process τ > tbreak. At short times (τ e tbreak), memory is still present, and there are departures from the Maxwell behavior at high frequencies.14 At elevated frequencies (in the right part of the Cole-Cole plot), one expects to see a deviation due to reptation, tube length fluctuations (“breathing mode”), and molecular motions of pieces of chains on the order of the entanglement length (Rouse mode).13,14 The semicircular shape of the curve will deviate with an asymptotic slope of -1 in the case of reptation, which should occur at a frequency on the order of the inverse of the breaking time of the micelles (tbreak).13,14 Other possible deviations occurring at high frequencies are due to a crossover to small scale motions of the chains when the dominant motion is no longer breaking or reptation. If tbreak is reduced, one enters a regime in which tube length fluctuations or “breathing” is the dominant motion on the time scale of tbreak. In this case, the semicircular shape of the curve will deviate with an asymptotic slope of -0.41.13,14 At still smaller values of tbreak, a further regime arises in which the dominant motion on the chemical time scale is unentangled. In this regime, the relaxation no longer depends on the length or
3004 Langmuir, Vol. 22, No. 7, 2006
Shukla et al.
configuration of the micellar chains. It corresponds to Rouse modes, associated with molecular motions of pieces of chains on the order of the entanglement length. Crossover to Rouse mode requires a change in the sign of the gradient and a region of upward curvature at the right-hand end of the Cole-Cole plot. As shown in Figure 2, deviation from pure Maxwellian behavior is smaller in the case of System I than it is in System II. It can also be seen from Figure 2 that only the Rouse mode is present in System I at higher frequencies. In System II, both “breathing” and Rouse modes are present at elevated frequencies. These results are consistent with our finding that the power-law regime is smaller in the case of System I than it is in System II (see Figure 4). It is interesting to note that the power-law regime is maintained for frequencies above 1/ts until the fast mode 1/tf is reached. The fast mode describes the relaxation time of a single mesh in the flexible polymer network. This intermediate time regime, where the power-law behavior prevails, coincides well with the angular frequency region where internal dynamics is playing the dominant role. For example, at temperature 25 °C, the Rouse mode starts at a frequency of about 178 rad/s for System I, and the “breathing” mode starts at a lower frequency of about 32 rad/s for System II. These data coincide well with the upper limits of the power-law regimes of 5.6 ms (1/178 ) 5.61) and 30.6 ms (1/32 ) 31.3) for System I and System II, respectively (see Figures 2 and 4). To summarize, we can state that we can compare a light-scattering experiment, which was performed at time t with a rheological function measured at 1/ω. This suggest that, at intermediate time regimes, internal dynamics will be dominated by chain-length fluctuations caused by onedimensional Rouse-like motion and/or simple three-dimensional Rouse-like motion, which should lead to a continuum of relaxation times.20,33 Such hierarchies of time scales are well-known in molecular diffusion processes of complex macromolecules.34 The quasi-anomalous diffusion process is characterized by a continuous spectrum of relaxation times, and this regime exhibits normal diffusion properties at long times and anomalous diffusion properties at intermediate time scales. This interpretation is based on recent theoretical models,28 and it turns out that the results are in excellent accord with our experimental finding. The hierarchies of relaxation time scales may induce different activation barriers for topological changes. It was already theoretically proven that hierarchical potential barriers can induce anomalous logarithmic slow-dynamics behavior:35
〈R2〉 ∼ ln t d
(7)
For Gaussian diffusion models, eq 7 gives a power-law dependence of g1(τ) at intermediate time scales.36 In this context, it is interesting to note that the time window of power-law behavior is largest for the lowest temperatures, and it decreases with increasing temperature (see Figure 4). It is well-known that the lengths of the micellar aggregates depend on temperature.13 This phenomenon arises from the fact that the micelles are dynamic aggregates that undergo rapid breaking and recombination within characteristic time scales. An increase in temperature favors the breakage of the micelles and a subsequent decrease in the average micellar lengths.13 Therefore, the density of overlap junctions should decrease with temperature. At high temperatures, where there are few overlap junctions, internal dynamics loses its (33) Buchanan, M.; Atakhorrami, M.; Palierne, J. F.; MacKintosh, F. C.; Schmidt, C. F. Phys. ReV. E 2005, 72, 011504. (34) Austin, R. H.; Berson, K. W.; Eisenstein, L.; Frauenfelder, L. H.; Gunsalus, I. C. Biochemistry 1975, 14, 5355. (35) Havlin, S.; Weissman, H. Phys. ReV. B 1988, 37, 487. (36) Ren, S. Z.; Sorensen, C. M. Phys. ReV. Lett. 1993, 70, 1727.
Figure 5. Temperature dependences of the power-law exponent for both surfactant systems. Error bars for the experimental data correspond roughly to the size of the symbols. Solid lines show the scaling behavior with temperature.
meaning. This phenomenon should be reflected as a disappearance of anomalous diffusion (power-law regime) at higher temperatures, which we did indeed observe. Therefore, we expect that the number and strength of overlap junctions are related to the extension of the power-law regime. The present results suggest that the strength of the network is most pronounced at 25 °C. The narrow window of the power-law regime at higher temperatures (40-50 °C) indicates that the entanglement network has not yet developed or is still very weak. This result is consistent with the observed behavior that the viscoelastic surfactant solutions are gellike at lower temperatures, and they exhibit pronounced viscous properties at elevated temperatures. The effect of temperature on the power-law exponent (d) is illustrated in Figure 5. It is evident that the value of d decreases strongly with falling temperature. It turns out that the exponent d is smaller for System II in comparison to that for System I. These findings suggest that the power-law exponent decreases gradually as the strength of the micellar network increases. For a series of different gelling systems, it has been found that a deficiency in the entanglement density may give rise to a slight augmentation in the value of the relaxation exponent.37,38 The scaling behavior for the power-law exponent with temperature is given by d ∼ T2.33(0.25 for System I and d ∼ T0.93(0.05 for System II. Previous DLS studies on gelling systems of different natures have reported values of d in the approximate range of 0.1-0.5.19 The present results together with literature data suggest that the value of d may depend on several parameters, such as the nature of system, the temperature, and the polymer concentration. The phenomenological theory developed by Doi and Onuki20 showed an interesting relationship between the Winter’s viscoelastic exponent n and the power-law exponent d.39,40 This analysis was based on the dynamic coupling between the stress and composition of the entangled polymer solution, and this argument also holds for polymer blends. This simply leads to n ) d. Recently, the power-law behaviors of both real and imaginary parts of the high-frequency modulus were observed in the CPyCl/ NaSal/water system using microrheology, and the authors found an exponent of 0.67.33 This value lies in the range of measured values from DLS. In our rheological measurements, it was unfortunately not possible to measure the asymptotic scaling regime at high angular frequencies. Therefore, we could not (37) Nystro¨m, B.; Walderhaug, H. J. Phys. Chem. 1996, 100, 5433. (38) Koike, A.; Nemoto, M.; Takahashi, M.; Osaki, K. Polymer 1994, 35, 3005. (39) Winter, H. H.; Chambon, F. J. Rheol. 1986, 30, 367. (40) Winter, H. H.; Mours, M. AdV. Polym. Sci. 1997, 134, 167.
Diffusion in Solutions of Wormlike Micelles
Figure 6. Temperature dependences of the short-time cutoff of the power-law tail V, obtained by fitting eq 1 to the correlation functions (see text for details) for System I and System II. Solid lines show the scaling behavior with temperature.
Langmuir, Vol. 22, No. 7, 2006 3005
Figure 8. Zero-shear viscosity η0 for both investigated solutions at different temperatures. Solid lines show the scaling behavior with temperature.
scientists, as the structural relaxation of the wormlike micellar network.16,19 The slow decay of the shear stress is closely related to the terminal network relaxation time. As shown in Figure 7, the stress relaxation times in System I are smaller than those in System II. Similarly, it can be seen that the ratio of G′′min/G0 in System I (∼0.17) is higher than that in System II (∼0.12) (see Figure 2). According to Cate’s theory, the relation G′′min/G0 is related to the micellar contour length L:7
G′′min ξ5/3 ≈ G0 L
Figure 7. Temperature dependence of the slow relaxation mode ts for both investigated surfactant solutions. Solid lines show the scaling behavior with temperature. Solid symbols describe the mechanical relaxation time obtained from rheological experiments.
confirm the relationship between the viscoelastic exponent n and the DLS exponent d predicted by Doi and Onuki.20 The temperature dependence of the time-scale parameter V, associated with the start of the power-law tail, is summarized in Figure 6. As shown in Figure 6, the scaling behavior of V with temperature is almost similar to that of d. We observed that the value V was shifted toward shorter times as the micellar network evolved. For temperature-induced network formation, it is not unexpected that the power-law domain starts at earlier times when the entanglement process proceeds. If we consider the relaxation of density fluctuations in systems with strong local constraints, the appearance of a power-law regime suggests that these constraints decay on a time scale that is long, compared with the fast mode. We expect that, with increasing constraints, the onset of the power-law domain will be shifted toward earlier times.19 In solutions of threadlike micelles, local constraints can be induced by entanglements, which give local elastic contributions to the free energy. Modern theories of polymer viscoelasticity are focused on the relaxation of entanglements through diffusion and their influence on stress decay. Fluctuations in entanglement density may give rise to stress release at long times. The characteristic decay time of the slow mode is plotted as a function of temperature in Figure 7. We also compared these data with the shear stress relaxation times that were determined by means of rheological measurements (solid symbols). At elevated temperatures (>35 °C) the relaxation times for System I were too small to be determined by means of rheological measurements. We interpret the slow relaxation mode, in analogy to previous
(8)
The larger G′′min/G0 and smaller ξ suggest a significantly smaller L for System I than for System II. It can be expected that the micelles with shorter contour lengths require shorter times to reptate and therefore can undergo stress relaxation quickly (shorter ts). Similarly, the decrease in stress relaxation times with increasing temperature can be explained by the change in the micellar contour lengths. The stress relaxation time ts displays interesting temperature dependence, as shown in Figure 7. The scaling behavior for the relaxation time ts with temperature is given by ∼T-6 and this result is independent of the type of system. Moreover, the magnitude of the relaxation time ts is also in excellent accord with rheological data (see Figure 7). Additional information on the temperature dependence of the zero-shear viscosity η0 is summarized in Figure 8. The zero-shear viscosity describes the viscous resistance of the quiescent state of the surfactant solutions. The scaling behavior is η0 ∼ T-5.3 for System I and η0 ∼ T-5.9 for System II. A cross-checking of the scaling behavior of ts and η0 can be performed using the simple equation of the Maxwell material:
η 0 ≈ G 0t s
(9)
In former investigations, we noticed that G0 does not depend very much on temperature.41 It is hence possible to calculate the stress relaxation time from measurements of the zero-shear viscosity using eq 9. As the viscous resistance can be measured, even at elevated temperatures, this offers the possibility to determine the relaxation time above temperatures of T ) 35 °C. Relevant results calculated with G0 ) 49.2 Pa are summarized in Figure 7. On the other hand, eq 9 can also be used to calculate the zero-shear modulus from combined rheological and lightscattering measurements. Inserting the zero-shear viscosity η0 and the terminal relaxation time ts leads to G0 ) 24.9 Pa for System I and G0 ) 41.6 Pa for System II. From pure rheological (41) Fischer, P.; Rehage, H. Langmuir 1997, 13, 7012.
3006 Langmuir, Vol. 22, No. 7, 2006
Shukla et al.
Figure 9. Semilogarithmic representation of the variation in the terminal relaxation time ts with 1/T for both investigated surfactant solutions.
Figure 10. Temperature dependences of the amplitudes of slow relaxation mode ts for the investigated viscoelastic surfactant solutions.
experiments, we obtained G0 ) 23.7 Pa for System I and G0 ) 49.2 Pa for System II (Figure 2). Both results coincide pretty well within the limits of experimental error. The experimental data, plotted in Figure 9, suggest temperaturedependent Arrhenius behavior (eq 10). This observation provides experimental evidence for the thermally activated topological relaxation of random fluid phases, as predicted by Cates.29
In comparing this prediction with the data shown in Figure 10, we see that System II is in good accord with this theory. System I, however, shows large deviations at elevated temperatures. Large statistical error bars in the range between 40 °C < T < 50 °C and a very small intermediate regime might be the reason for this discrepancy.
ts ) t0‚exp
( ) Ea RT
(10)
The activation energy calculated from Figure 9 is 29 kcal/mol for System I and 32 kcal/mol for System II. It is interesting to note that these results are in excellent accord with measured values of about 30 kcal/mol, which were determined by means of forced-Rayleigh-scattering,42 T-jump,43 and dynamic viscoelastic measurements42 for similar surfactant solutions. Characteristic “screening length” δ, over which viscous effects become important, can be estimated from eq 11:29
δ = a‚exp
( )
Ea with a ) xlelp 2kBT
(11)
In this equation, lp is the persistence length, and le describes the average distance between entanglements. Viscous effects are important in the hydrodynamic range of length scales δ . a. For our observed activation energy, we obtained typical screening lengths δ = 1010a. This clearly indicates significant viscous effects in the systems under investigation. Figure 10 shows the slow-mode amplitudes as a function of the temperature. For the case of an entangled polymer solution under good solvent conditions, the Doi-Onuki 20 dynamic coupling theory predicts the slow-mode amplitude:16
As )
(
)
3.9πξH3G0 x with x ) 1+x kBT
(12)
The theory thus gives approximate values of 0.11 and 0.18 for System I and System II, respectively, at all measured temperatures. (42) Shikata, T.; Imai, S.-I.; Morishima, Y. Langmuir 1998, 14, 2026. (43) Kern, F.; Zana, R.; Candau, S. J. Langmuir 1991, 7, 1344.
Conclusions In this publication, we used a combined effort of DLS experiments and rheological measurements in order to explore the dynamic properties of entangled solutions of threadlike micelles. The results of these different experiments were found to be in fairly good agreement. In a series of experiments, we observed that DLS is a powerful technique to analyze the viscoelastic behavior of entangled solutions, especially under conditions of short relaxation times. The extension of many orders of time scales provided new insight into the dynamic processes that control the viscoelasticity of entangled surfactant solutions. In this study, we observed three different relaxation mechanisms. The light-scattering results were consistent with rheological measurements and Cates’s reptation theory for living polymers.13,14 The fast decay mode described local cooperative motions in the gellike network. The slowest mode characterized the relaxation of the transient network structure, and this regime did correspond to the relaxation process of shear stress, which was determined by means of rheological experiments. The intermediate time regime showed a power-law relaxation behavior, which has the same physical origin as Winter’s viscoelastic exponent.39,40 This relaxation regime reflected the additional internal dynamics of the micelles as bending motions of parts of the rod-shaped aggregates between two entanglement points. The power-law relaxation regime depended on the network density, and the measured exponent decreased gradually as the strength of the micellar network increased. The intermediate and slow relaxation process could be explained in terms of quasianomalous diffusion. These phenomena lead to linear diffusion at long time scales and they obey anomalous logarithmic slowdynamics behavior at intermediate times. It turned out that the slowest relaxation mode followed Arrhenius laws. This observation provided experimental evidence for the thermally activated topological relaxation processes of random fluid phases.28 LA053435E
