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Dynamic light scattering studies of rod-like micelles in dilute and semi-dilute regime
Colloids and Surfaces A: Physicochem. Eng. Aspects 275 (2006) 161–167
Dynamic light scattering studies of rod-like micelles in dilute and semi-dilute regime Gunjan Garg, P.A. Hassan ∗ , S.K. Kulshreshtha Novel Materials and Structural Chemistry Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India Received 18 May 2005; received in revised form 12 September 2005; accepted 4 October 2005 Available online 11 November 2005
Abstract The effect of anionic hydrophobic salts, sodium p-toluenesulphonate (Na-PTS) and sodium salicylate (Na-Sal) on the growth behavior of cationic surfactant micelles, cetyltrimethyl ammonium bromide (CTAB) has been investigated in the presence as well as absence of sodium chloride (NaCl) using dynamic light scattering measurements. In the absence of NaCl, a progressive decrease in the diffusion coefficient of CTAB (50 mM) micelles was observed with increasing concentration of hydrophobic salts, Na-PTS and Na-Sal, suggesting an increase in the average dimension of the micelles. The micellar growth is explained in terms of a prolate ellipsoidal transition of the micelles. When the axial ratio of the micelles becomes long enough to drive the system into semi-dilute regime, a slow mode of long relaxation time begins to evolve and shows bimodal distribution of relaxation time. In the presence of NaCl, the micellar growth is enhanced and the appearance of slow mode is evident at lower hydrophobic salt concentration than that is observed in the absence of NaCl. The onset of semi-dilute regime is consistent with the modified Doi–Edwards model of rod-like polymers. © 2005 Elsevier B.V. All rights reserved. Keywords: Rod-like micelles; Dynamic light scattering; Sphere-to-rod transition; Living polymers; Semi-dilute regime
1. Introduction Surfactants are known to form thread-like or rod-like micelle in dilute aqueous solutions containing suitable salts. The addition of inorganic or organic salts to an ionic surfactant solution facilitates the transition from spherical to rod-like micelles by shielding the repulsions between the charged headgroups [1–7]. In recent years, considerable interest has been developed in the solution properties of the extended micellar structures formed when salts are added to solutions of cationic surfactants such as cetyl trimethyl ammonium bromide (CTAB). When the salt is an inorganic electrolyte (e.g., NaBr, NaCl), long flexible, threadlike micelles are formed with the molar ratio of salt to surfactant typically above 1, as inorganic counterions (e.g., Cl− , Br− ) bind moderately to cationic micelles and thus lead to gradual micellar growth. In presence of organic salts, such as sodium salicylate (Na-Sal) and sodium p-toluenesulphonate (Na-PTS), worm-like micellar structures are formed in cationic micelles at
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substantially lower concentrations of added salt due to the strong binding of hydrophobic counterions (salicylate and p-toluene sulphonate) to the micellar surface [8–10]. When the length of these micelles becomes sufficiently large, they become entangled and are known to exhibit properties similar to semi-dilute polymer solutions. The worm-like micelles are similar to polymers in that they are quite flexible and exhibit contour lengths of the order of micrometers [11]. These so-called equilibrium polymers differ from classical polymers in that they are constantly breaking and making and, therefore, do not exhibit a quenched contour length distribution [12]. During the past few years, several groups have made efforts to explore the dynamics of these worm-like systems using various techniques such as nuclear magnetic resonance (NMR) spectroscopy [13–15], dynamic light scattering (DLS) [16–19], forced rayleigh scattering (FRS) [20], fluorescence spectroscopy [21–23], electrochemical route [24–28] and dynamic viscoelastic (DVE) [29–32] techniques. Nemoto and co-workers [16,17,20,33] have carried out systematic studies on dynamics of the CTAB/Na-Sal micelles using DLS technique. According to their investigations, a bimodal distribution of the decay rate Γ was observed over the semidilute regime of thread-like micelles. Similar studies have been
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performed by Brown et al. [34] on CTAB and sodium naphthalene sulphonate in aqueous solutions. Recently, Amin et al. [35] have reported a detailed dynamic light scattering study of CTAB/Na-Sal solution over a wide range of CTAB concentration and Na-Sal:CTAB mole ratio. According to their studies, the hydrodynamic correlation length, associated with the fast relaxation mode is observed to depend on both the CTAB concentration and Na-Sal:CTAB mole ratio. Previously, Magid et al. studied the counterion-mediated micellar growth for cetyltrimethylammonium micelles in presence of 2,6-dichlorobenzoate and chloride counterions using small angle scattering techniques. A variation in counterion composition at constant ionic strength induces large changes in micelle size. The aromatic ion 2,6dichlorobenzoate increases the surfactant-packing parameter by increasing the average volume per surfactant monomer, a cosurfactant-like effect, and decreasing the area per headgroup. Counterions such as chloride, which show only surface adsorption, affect only headgroup area and are much less effective at driving micellar growth [36,37]. Recent rheological studies on CTAB/Na-Sal system in presence of added inorganic electrolytes revealed the occurrence of a shear-thickening behavior beyond a critical shear rate [38]. This phenomenon is ascribed to the formation of a shear-induced structure and is very sensitive to the concentration of inorganic electrolytes. Thus, comparison of the microscopic structural parameters of the CTAB/Na-Sal micelles in the presence and the absence of NaCl is relevant for the understanding of this shear-induced phenomenon. The objective of the present study is to carry out a semiquantitative estimate of the parameters of the rod-like micelles in CTAB/Na-Sal and CTAB/Na-PTS systems using DLS technique and compare the transitions from dilute to semi-dilute regime using models of rod-like polymers. Here we report the results of DLS measurements made on dilute and semi-dilute solutions of thread-like micelles of (50 mM) CTAB in presence of different concentrations of Na-PTS and Na-Sal, in presence as well as absence of 0.5 M NaCl.
130◦ . All the measurements were taken at 25 ◦ C. The measured intensity correlation functions were analyzed by two methods: a double exponential fit where bimodal distribution of relaxation time is assumed and by the method of cumulants [39] where unimodal distribution of relaxation time is considered. 3. Results and discussion 3.1. Effect of hydrophobic salts on micelle size To understand the microstructure evolution in CTAB micelles with addition of hydrophobic salts (Na-Sal and Na-PTS), DLS measurements were performed at different concentrations of hydrophobic salts. The concentration of surfactant was kept constant (50 mM) and the concentration of hydrophobic salts (csalt ) was varied from 10 to 50 mM. Fig. 1 shows a representative plot of variation of the intensity correlation function, g2 (τ), for 50 mM CTAB with cNa-PTS = 10, 20 and 50 mM at a scattering angle of 90◦ . The solid line is a fit to the data by the method of cumulants with a mean relaxation rate (Γ ) and polydispersity index [P.I. = variance/(mean)2 ] as the fitted variables [39]. At low salt concentration for both the salts (Na-Sal and Na-PTS), the cumulants method gives a reasonably good fit to the data with the correlation coefficient (r) greater than 0.999 and random distribution of residuals. However, at csalt = 50 mM, the cumulants fit slightly deviates from the data at larger times (see solid line in Fig. 1), due to the evolution of a slow mode of long relaxation time. This is consistent with the earlier reports by Amin et al. [35] and others [17,20,33,34] that show a bimodal relaxation of correlation function of polymer-like micelles in the semi-dilute regime. The appearance of a slow mode at csalt = 50 mM though small in amplitude, is an indication of uniaxial growth of the micelles thereby approaching the semi-dilute regime. Since the amplitude of the slow mode is very small (∼0.02) and is observed only at csalt = 50 mM, the data was analyzed by the method of cumulants. Analysis using the constraint regularization method, CONTIN, [40,41] also revealed a unimodal distribution of
2. Experimental 2.1. Chemicals CTAB and Na-PTS were obtained from Sigma Chemicals and Na-Sal was obtained from Fluka. All chemicals were used as received. Deionized water from a Millipore-MilliQ system (resistivity ∼18 M cm) was used in all cases to prepare aqueous solutions. 2.2. Dynamic light scattering Dynamic light scattering (DLS) measurements were performed using a Malvern 4800 Autosizer employing 7132 digital correlator. The light source was Ar-ion laser operated at 514.5 nm with maximum power output of 2 W. The samples of micellar solutions were filtered through 0.2-m filters (Acrodisc) to avoid interference from dust particles. Measurements were made at five different angles ranging from 50 to
Fig. 1. Representative plot of the intensity correlation function for 50 mM CTAB with cNa-pts = 10, 20 and 50 mM at a scattering angle of 90◦ . The solid line is fit to the data using method of cumulants.
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Fig. 2. The variation of (a) apparent diffusion coefficient (Da ) obtained from the slope of the plots of Γ vs. q2 and (b) length (L) calculated from Perrin’s formula for 50 mM CTAB at different concentrations of Na-PTS in presence as well as absence of 0.5 M NaCl.
Fig. 3. The variation of (a) apparent diffusion coefficient (Da ) obtained from the slope of the plots of Γ vs. q2 and (b) length (L) calculated from Perrin’s formula for 50 mM CTAB at different concentrations of Na-Sal in presence as well as absence of 0.5 M NaCl.
relaxation rate supporting the validity of cumulants results. The shift of the correlation function to larger times with increase in salt concentration is an indication of the growth of the micelles. The angular variation of the average decay rate (Γ ) of the intensity correlation function obtained from the fit at different concentrations of Na-PTS and Na-Sal was found to vary linearly as a function of q2 indicating translational diffusion of the scatterers and negligible contribution from rotational diffusion in case of anisotropic micelles. This is expected for micelles due to very small dimension as compared to the wavelength of the light. The apparent diffusion coefficient (Da ) of the micelles was calculated from the slope of Γ versus q2 plot. The variation of apparent diffusion coefficient, Da , of CTAB micelles at different concentrations of Na-PTS and Na-Sal is depicted in Figs. 2(a) and 3(a), respectively. The open circles in Figs. 2(a) and 3(a) show the data for CTAB/Na-PTS and CTAB/Na-Sal in presence of 0.5 M NaCl, which will be discussed further. At very low concentration of the salts (csalt = 10 mM), Da and the corresponding hydrodynamic diameter, calculated using Stokes–Einstein relation, is in the same range as that expected for small globular micelles. With successive addition of Na-PTS and Na-Sal, Da was found to drop steeply reflecting a sudden increase in the average dimension of the micelles. At cNa-PTS = 15 mM, the Da value is 63.4 × 10−8 cm2 /s corresponding to an equivalent sphere diameter of 7.7 nm, which increases further upon increasing the concentration of the hydrophobic salts. However, it is unrealistic to have an isotropic spherical micelle having a diameter of 7.7 nm for CTAB molecules, since the maximum extended length of
a dodecyl chain being only 2.17 nm [42]. Moreover, previous reports on CTAB micelles in the presence of hydrophobic salts reveals the formation of long thread-like micelles [8–10,43–46]. Thus, the observed drastic decrease in Da could be attributed to the formation of prolate ellipsoidal micelles. The DLS data at high salt concentration were analyzed in terms of prolate ellipsoidal structure. To get a quantitative insight in to the growth of the micelles, the lengths of the micelles were estimated using Perrin’s formula [47] that relates the average diffusion coefficient to the axes of the ellipsoid. The minor axis of the ellipsoid is taken as constant at 4.3 nm and only the semi-major axis is varied. Corrections to the measured Da for possible hydrodynamic and thermodynamic contributions were applied during the data analysis as reported earlier [52]. The average length (L) of the micelles as obtained from the major axis of the ellipsoid for Na-PTS and Na-Sal is depicted in Figs. 2(b) and 3(b), respectively. The open circles in Figs. 2(b) and 3(b) show the length of the micelles for CTAB/Na-PTS and CTAB/Na-Sal in presence of 0.5 M NaCl, which will be discussed further. A marked increase in the length of the micelles was observed with increasing concentration of hydrophobic salts, viz. from 11 ± 1 to 94 ± 9 nm for Na-PTS and 20 ± 2 to 93 +9 nm for Na-Sal. It is true that when the micelles are sufficiently long, they become flexible beyond certain length scale; known as the persistence length, lp , and the rigid-rod approximation is questionable. The total persistence length for a polyelectrolyte chain can be represented as the sum of an intrinsic and an electrostatic length, lP,t + lp ,e [48]. In the past several years, there has been a number of small angle neutron scattering (SANS) studies of
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worm-like micelles in which values of lp were estimated using Holtzer (bending rod) plots [49]. Magid [50] made a survey on persistence lengths determined for different micellar systems using analyses of light scattering and SANS data. The results suggest that micellar flexibility is apparently highly dependent on surfactant monomer structure, with values ranging from 100 ˚ for nonionic surfactants to a few hundred nanometers. to 200 A The lp ’s reported for the penetrating counterion salicylate with C16 TA and C16 Py, suggest that its effect relative to a nonpenetrating counterion is to rigidify the micelles. For C16 TA Sal/NaSal system, the lp was found to be 137 nm [51]. In the present study, the flexibility of the micelles is not taken in to account due to the limited number of parameters that can be estimated from DLS. However, the length calculated for CTAB/Na-Sal and CTAB/Na-PTS micelles, in the present study, is comparable or smaller than the value of lp reported for C16 TA Sal/Na-Sal system. Therefore, the rigid-rod model is a good approximation to be used to analyze the data.
Fig. 4. Representative plot of the intensity correlation function for 50 mM CTAB with cNa-pts = 10 and 20 mM in presence of 0.5 M NaCl at a scattering angle of 90◦ . The solid line is fit to the data using method of cumulants and the dotted line shows the fit to the data (20 mM Na-PTS) by biexponential decay.
3.2. Effect of hydrophobic salts on micelle size in presence of NaCl As the micellar growth is very sensitive to the nature and the amount of salt added to the solution, it is important to investigate the effect of electrolytes on rod-like micelles. Inorganic electrolytes such as NaCl, NaBr, NaNO3 and NaF can influence significantly the formation of shear-induced structures in CTAB/Na-Sal micelles [38]. In particular, addition of small amounts of NaCl to CTAB/Na-Sal micelles imparts shearthickening behavior to the fluid as opposed to shear thinning in the absence of NaCl. To envisage the effect of NaCl on the growth of CTAB/Na-Sal and CTAB/Na-PTS systems, DLS measurements were carried out on aqueous solution of CTAB (50 mM) at different concentrations of Na-PTS and Na-Sal in presence of 0.5 M NaCl. The concentration of hydrophobic salts was varied from 2 to 50 mM for Na-PTS and 2 to 30 mM for Na-Sal. Fig. 4 shows the intensity correlation function for 50 mM CTAB in presence of 0.5 M NaCl at Na-PTS concentrations of 10 and 20 mM at a scattering angle of 90◦ . At small concentration of Na-PTS (≤10 mM), a unimodal distribution of the relaxation times was observed and the data could be fitted by the method of cumulants (r > 0.999). However, for csalt ≥ 15 mM, the slow mode of long relaxation time begins to evolve. This is evident from the poor quality of fit to the data by the method of cumulants (r < 0.999). The cumulants fit (solid line) deviates significantly from the data at larger times (Fig. 4). A similar behavior is observed for CTAB/Na-Sal system also in presence of NaCl. The difference between the dynamical behavior of CTAB/Na-PTS and CTAB/Na-Sal micelles in absence and presence of NaCl is that in absence of NaCl, the amplitude of slow relaxation rate is very small as compared to the one observed in presence of NaCl at the same concentration (Figs. 1 and 4). It indicates that presence of NaCl in addition to hydrophobic salts plays an important role in the growth behavior of CTAB micelles. The growth of CTAB micelles in presence of hydrophobic salts (Na-Sal and Na-PTS) is much faster with addition of NaCl than that is observed in absence of NaCl for the same concen-
tration of Na-PTS and Na-Sal. This is due to the shielding of the surface charge by the addition of NaCl. In addition to the hydrophobic as well as electrostatic interactions of hydrophobic salts, the presence of NaCl helps in further reducing the electrostatic repulsion and makes the aggregation easier. Due to which, the micelles grow faster and the topological interactions become significant at lower concentrations of the hydrophobic salts. Previous reports indicated that the slow mode of relaxation starts evolving when the axial ratio of the micelles becomes long enough to drive the system into semi-dilute regime [17,34]. According to Nemoto et al. [17], the normalized time correlation function of intensity scattered from the samples exhibited the bimodal distribution of the decay rate Γ in a semi-dilute regime of thread-like micelles. The fast mode, Γ f , was ascribed to the diffusion of micelles with a cooperative diffusion coefficient (Dc ). On the other hand, the characteristic decay rate, Γ s , of the slow mode is related to the coupling between concentration fluctuation and stress. The existence of two characteristic decay times in presence of NaCl is an indication of two dynamical processes. To account for the existence of two relaxation modes in the correlation function for csalt ≥ 15 mM, the data were analyzed by biexponential decay, given by the relation: g2 (τ) − 1 = Af e−Γf τ + As e−Γs τ
(1)
where Af and As are the amplitude of the fast and slow components with relaxation rates, Γ f and Γ s , respectively. For csalt ≥ 15 mM, biexponential decay function gives a better fit to the data with r > 0.999 in comparison to that obtained by the method of cumulants and the incorporation of additional parameters in the fit is validated from the F-test [53]. The dashed line in Fig. 4 shows the fit with biexponential decay. The parameters of the fit for CTAB/Na-PTS system obtained from biexponential decay of the correlation function at 90◦ are summarized in Table 1.
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Table 1 Parameters of intensity correlation function obtained from the fit by biexponential decay analysis for CTAB in presence of different concentrations of Na-PTS and 0.5 M NaCl at a of scattering angle 90◦ : correlation coefficient (r), amplitude of fast mode (Af ), fast decay rate (Γ f ) and amplitude of slow mode (As ), slow decay rate (Γ s ) Concentration (Na-PTS) (mM)
r
Af
Γ f (s−1 )
As
Γ s (s−1 )
15 20 25 30 35 40 45 50
0.99996 0.99992 0.99990 0.99983 0.99983 0.99990 0.99990 0.99982
0.774 (3) 0.764 (3) 0.764 (4) 0.806 (2) 0.799 (2) 0.757 (2) 0.785 (4) 0.786 (2)
0.0143 (1) 0.0140 (1) 0.0123 (1) 0.0142 (2) 0.0118 (1) 0.0139 (1) 0.0139 (4) 0.0144 (2)
0.051 (3) 0.076 (3) 0.072 (4) 0.039 (2) 0.054 (2) 0.071 (2) 0.056 (4) 0.049 (2)
0.00098 (8) 0.00065 (4) 0.00063 (5) 0.00008 (1) 0.00016 (2) 0.00018(1) 0.00060 (6) 0.00008 (1)
The variation of apparent diffusion coefficient, Da (obtained from the slope of Γ f versus q2 plot) of CTAB micelles (50 mM) at different concentrations of Na-PTS and Na-Sal in presence of 0.5 M NaCl is shown in Figs. 2(a) and 3(a), respectively. As salt concentration (cNa-PTS and cNa-Sal ) increases, Da decreases initially and then takes a more or less constant value. The observed diffusion coefficient was used to calculate the apparent length of the micelles, assuming a prolate ellipsoid structure as discussed earlier. The average length (L) of the micelles as obtained from the major axis of the ellipsoid for Na-PTS and Na-Sal is depicted in Figs. 2(b) and 3(b), respectively. Initially, the length of the micelles increases with increasing cNa-Sal and cNa-PTS until reaching a maximum value and then takes a nearly constant value of ∼170 nm. It should be noted that this observed saturation in the calculated length of the micelles does not indicate a limiting growth of the micelles. This is merely a reflection of entering the semi-dilute regime of rod-like micelles. Beyond the semi-dilute concentration, topological interactions become significant and the diffusion of the micelles is severely hindered. This leads to an underestimation of the calculated length of the micelles. Theoretical treatment by Doi and Edwards [54,55] provides the basis for an understanding of dynamical behavior of a system of strongly interacting rod-like polymers whose length (L) is much larger than the rod diameter (d). In the present study, we used Doi–Edwards model of rod-like polymers to compare the dynamics of rod-like micelles in semi-dilute regime. For rod-like polymers of length (L), the semi-dilute regime is identified as the concentration at which the number density (ν) of the polymer becomes much greater than 1/L3 . According to the tube model of Doi and Edwards [54–56], the rotational diffusion constant in semi-dilute (Dr ) and infinite dilute solutions (Dr,0 ) are related to the expression: −2
Dr = βDr,0 (νL3 )
(2)
where β is a numerical factor that indicates the concentration at which topological interactions become significant. Mori et al. [57] measured the Dr of rod-like polymers of different lengths over a wide range of concentration using dynamic electric birefringence and showed that the value of β is of the order of 103 . By a detailed study of tube statistics and comparison with experimental results Hayakawa et al. [58] suggested that the semi-dilute regime starts at νL3 ∼ 30.
The number density (ν) of the rod-like micelles having length L, and radius r can be calculated with the equation: ν = (c − cmc)
Na v πr 2 L
(3)
where c is the concentration of surfactant, cmc is the critical micelle concentration, Na is Avogadro’s number and v is the volume of the surfactant monomer. In the present study, knowing the volume fraction of the micelles, one can show that the condition νL3 ∼ 30 is satisfied when the axial ratio of the micelles becomes 42. This corresponds to a length of the micelles of the order of 185 nm, considering the diameter of the micelle as 4.4 nm. The observed saturated value of length 170 nm is close to the expected value of 185 nm based on the molecular theory of entanglement of rod-like polymers [58]. This further supports the fact that the biexponential relaxation arises from a transition of the system to semi-dilute regime. A comparative study of lengths obtained for CTAB micelles in presence as well as absence of NaCl at different concentrations of Na-PTS and Na-Sal is given in Table 2. The micellar parameters reveal that, the striking difference in the rheology of CTAB/Na-Sal in presence of NaCl and the formation of shearinduced structures [38] could be due to the strong growth and longer length of the micelles as compared to that in absence of NaCl. In the present study, the amplitude of the slow mode As is fairly small, typically less than 0.1 in all the cases and is found to be almost independent of the concentration of hydrophobic salts. However, in the earlier studies by Amin et al. [35], the amplitude of the slow relaxation mode was observed to be a strong function of NaSal:CTAB mole ratios. This difference in the behavior arises from the limited range of concentration studied here and smaller length of the micelles. At Na-Sal:CTAB molar ratio <1, at CTAB concentration of 50 mM, the amplitude is less than 0.1 and almost independent of Na-Sal:CTAB molar ratio, within the experimental errors. Moreover, the amplitude will be further reduced when one considers the intensity correlation, instead of electric field correlation. Due to very small amplitude, (As ), inherent polydispersity in Γ s and associated standard deviation of the measured data, the accuracy of Γ s obtained from the fit is limited. Thus, no attempts were made to extract stress relaxation parameters of the micelles from the slow mode.
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Table 2 Comparison of the length of CTAB micelles in presence as well as absence of NaCl at different concentrations of the salts, Na-PTS and Na-Sal csalt (mM)
2 5 10 15 20 25 30 35 40 45 50
Length of CTAB micelles in presence of Na-PTS (nm)
Length of CTAB micelles in presence of Na-Sal (nm)
CTAB/Na-PTS
CTAB/Na-PTS/NaCl
CTAB/Na-Sal
CTAB/Na-Sal/NaCl
– – – 11 ± 1 25 ± 2 40 ± 4 42 ± 4 50 ± 5 53 ± 5 60 ± 6 94 ± 9
10 45 127 142 161 200 167 195 167 167 163
– – – 20 ± 2 24 ± 2 34 ± 3 40 ± 4 51 ± 5 63 ± 6 75 ± 7 93 ± 9
10 ± 1 58 ± 6 153 ± 15 167 ± 16 169 ± 17 – 167 ± 17 – – – –
± ± ± ± ± ± ± ± ± ± ±
1 4 13 14 16 20 17 20 17 17 16
4. Conclusion This report investigates the growth behavior of CTAB micelles in presence as well as absence of NaCl on addition of different concentrations of hydrophobic salts like, Na-PTS and Na-Sal, by dynamic light scattering measurements. From DLS results, it was found that the growth of the CTAB micelles is much more pronounced at the same concentration of hydrophobic salts, Na-PTS and Na-Sal, when 0.5 M NaCl is added to the solution. The length of the micelles was estimated assuming a prolate ellipsoid structure, ranging from dilute to semi-dilute regime. When the micelles become long enough to entangle each other with addition of hydrophobic salt, the system reaches to semi-dilute regime and shows a biexponential decay behavior. In presence of NaCl, a faster growth of the micelles in the dilute regime and corresponding transition from dilute to semidilute regime at lower concentration of Na-Sal and Na-PTS is observed. Comparison of the micellar lengths in the presence and absence of NaCl in the dilute regime suggests that the biexponential relaxation arises possibly from topological interaction of the micelles. This was confirmed by calculating the expected length of the micelles for topological interaction to occur based on Doi–Edwards model for rod-like polymers. It is indeed observed that the length of the micelles at the transition from dilute to semi-dilute regime is consistent with the Doi–Edward model. The transition point shift to lower hydrophobic salt concentration when the micellar growth is faster. All the evidences are in favor of the conjecture that the slow relaxation mode arises from topological interaction of the micelles. Moreover, the above studies suggest that the observed difference in the rheology of the CTAB-Na-Sal micelles in presence and absence of NaCl can be correlated to the difference in the length of the micelles and the growth behavior. The larger length of the micelles in presence of NaCl could be responsible for the shear-thickening behavior of the micelles, as compared to the shear-thinning behavior in absence of NaCl. Acknowledgements The authors are thankful to Dr. C. Manohar of Indian Institute of Technology, Mumbai for many fruitful discussions. G. Garg
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Dynamic light scattering studies of rod-like micelles in dilute and semi-dilute regime Gunjan Garg, P.A. Hassan ∗ , S.K. Kulshreshtha Novel Materials and Structural Chemistry Division, Bhabha Atomic Research Centre, Trombay, Mumbai 400085, India Received 18 May 2005; received in revised form 12 September 2005; accepted 4 October 2005 Available online 11 November 2005
Abstract The effect of anionic hydrophobic salts, sodium p-toluenesulphonate (Na-PTS) and sodium salicylate (Na-Sal) on the growth behavior of cationic surfactant micelles, cetyltrimethyl ammonium bromide (CTAB) has been investigated in the presence as well as absence of sodium chloride (NaCl) using dynamic light scattering measurements. In the absence of NaCl, a progressive decrease in the diffusion coefficient of CTAB (50 mM) micelles was observed with increasing concentration of hydrophobic salts, Na-PTS and Na-Sal, suggesting an increase in the average dimension of the micelles. The micellar growth is explained in terms of a prolate ellipsoidal transition of the micelles. When the axial ratio of the micelles becomes long enough to drive the system into semi-dilute regime, a slow mode of long relaxation time begins to evolve and shows bimodal distribution of relaxation time. In the presence of NaCl, the micellar growth is enhanced and the appearance of slow mode is evident at lower hydrophobic salt concentration than that is observed in the absence of NaCl. The onset of semi-dilute regime is consistent with the modified Doi–Edwards model of rod-like polymers. © 2005 Elsevier B.V. All rights reserved. Keywords: Rod-like micelles; Dynamic light scattering; Sphere-to-rod transition; Living polymers; Semi-dilute regime
1. Introduction Surfactants are known to form thread-like or rod-like micelle in dilute aqueous solutions containing suitable salts. The addition of inorganic or organic salts to an ionic surfactant solution facilitates the transition from spherical to rod-like micelles by shielding the repulsions between the charged headgroups [1–7]. In recent years, considerable interest has been developed in the solution properties of the extended micellar structures formed when salts are added to solutions of cationic surfactants such as cetyl trimethyl ammonium bromide (CTAB). When the salt is an inorganic electrolyte (e.g., NaBr, NaCl), long flexible, threadlike micelles are formed with the molar ratio of salt to surfactant typically above 1, as inorganic counterions (e.g., Cl− , Br− ) bind moderately to cationic micelles and thus lead to gradual micellar growth. In presence of organic salts, such as sodium salicylate (Na-Sal) and sodium p-toluenesulphonate (Na-PTS), worm-like micellar structures are formed in cationic micelles at
∗
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substantially lower concentrations of added salt due to the strong binding of hydrophobic counterions (salicylate and p-toluene sulphonate) to the micellar surface [8–10]. When the length of these micelles becomes sufficiently large, they become entangled and are known to exhibit properties similar to semi-dilute polymer solutions. The worm-like micelles are similar to polymers in that they are quite flexible and exhibit contour lengths of the order of micrometers [11]. These so-called equilibrium polymers differ from classical polymers in that they are constantly breaking and making and, therefore, do not exhibit a quenched contour length distribution [12]. During the past few years, several groups have made efforts to explore the dynamics of these worm-like systems using various techniques such as nuclear magnetic resonance (NMR) spectroscopy [13–15], dynamic light scattering (DLS) [16–19], forced rayleigh scattering (FRS) [20], fluorescence spectroscopy [21–23], electrochemical route [24–28] and dynamic viscoelastic (DVE) [29–32] techniques. Nemoto and co-workers [16,17,20,33] have carried out systematic studies on dynamics of the CTAB/Na-Sal micelles using DLS technique. According to their investigations, a bimodal distribution of the decay rate Γ was observed over the semidilute regime of thread-like micelles. Similar studies have been
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performed by Brown et al. [34] on CTAB and sodium naphthalene sulphonate in aqueous solutions. Recently, Amin et al. [35] have reported a detailed dynamic light scattering study of CTAB/Na-Sal solution over a wide range of CTAB concentration and Na-Sal:CTAB mole ratio. According to their studies, the hydrodynamic correlation length, associated with the fast relaxation mode is observed to depend on both the CTAB concentration and Na-Sal:CTAB mole ratio. Previously, Magid et al. studied the counterion-mediated micellar growth for cetyltrimethylammonium micelles in presence of 2,6-dichlorobenzoate and chloride counterions using small angle scattering techniques. A variation in counterion composition at constant ionic strength induces large changes in micelle size. The aromatic ion 2,6dichlorobenzoate increases the surfactant-packing parameter by increasing the average volume per surfactant monomer, a cosurfactant-like effect, and decreasing the area per headgroup. Counterions such as chloride, which show only surface adsorption, affect only headgroup area and are much less effective at driving micellar growth [36,37]. Recent rheological studies on CTAB/Na-Sal system in presence of added inorganic electrolytes revealed the occurrence of a shear-thickening behavior beyond a critical shear rate [38]. This phenomenon is ascribed to the formation of a shear-induced structure and is very sensitive to the concentration of inorganic electrolytes. Thus, comparison of the microscopic structural parameters of the CTAB/Na-Sal micelles in the presence and the absence of NaCl is relevant for the understanding of this shear-induced phenomenon. The objective of the present study is to carry out a semiquantitative estimate of the parameters of the rod-like micelles in CTAB/Na-Sal and CTAB/Na-PTS systems using DLS technique and compare the transitions from dilute to semi-dilute regime using models of rod-like polymers. Here we report the results of DLS measurements made on dilute and semi-dilute solutions of thread-like micelles of (50 mM) CTAB in presence of different concentrations of Na-PTS and Na-Sal, in presence as well as absence of 0.5 M NaCl.
130◦ . All the measurements were taken at 25 ◦ C. The measured intensity correlation functions were analyzed by two methods: a double exponential fit where bimodal distribution of relaxation time is assumed and by the method of cumulants [39] where unimodal distribution of relaxation time is considered. 3. Results and discussion 3.1. Effect of hydrophobic salts on micelle size To understand the microstructure evolution in CTAB micelles with addition of hydrophobic salts (Na-Sal and Na-PTS), DLS measurements were performed at different concentrations of hydrophobic salts. The concentration of surfactant was kept constant (50 mM) and the concentration of hydrophobic salts (csalt ) was varied from 10 to 50 mM. Fig. 1 shows a representative plot of variation of the intensity correlation function, g2 (τ), for 50 mM CTAB with cNa-PTS = 10, 20 and 50 mM at a scattering angle of 90◦ . The solid line is a fit to the data by the method of cumulants with a mean relaxation rate (Γ ) and polydispersity index [P.I. = variance/(mean)2 ] as the fitted variables [39]. At low salt concentration for both the salts (Na-Sal and Na-PTS), the cumulants method gives a reasonably good fit to the data with the correlation coefficient (r) greater than 0.999 and random distribution of residuals. However, at csalt = 50 mM, the cumulants fit slightly deviates from the data at larger times (see solid line in Fig. 1), due to the evolution of a slow mode of long relaxation time. This is consistent with the earlier reports by Amin et al. [35] and others [17,20,33,34] that show a bimodal relaxation of correlation function of polymer-like micelles in the semi-dilute regime. The appearance of a slow mode at csalt = 50 mM though small in amplitude, is an indication of uniaxial growth of the micelles thereby approaching the semi-dilute regime. Since the amplitude of the slow mode is very small (∼0.02) and is observed only at csalt = 50 mM, the data was analyzed by the method of cumulants. Analysis using the constraint regularization method, CONTIN, [40,41] also revealed a unimodal distribution of
2. Experimental 2.1. Chemicals CTAB and Na-PTS were obtained from Sigma Chemicals and Na-Sal was obtained from Fluka. All chemicals were used as received. Deionized water from a Millipore-MilliQ system (resistivity ∼18 M cm) was used in all cases to prepare aqueous solutions. 2.2. Dynamic light scattering Dynamic light scattering (DLS) measurements were performed using a Malvern 4800 Autosizer employing 7132 digital correlator. The light source was Ar-ion laser operated at 514.5 nm with maximum power output of 2 W. The samples of micellar solutions were filtered through 0.2-m filters (Acrodisc) to avoid interference from dust particles. Measurements were made at five different angles ranging from 50 to
Fig. 1. Representative plot of the intensity correlation function for 50 mM CTAB with cNa-pts = 10, 20 and 50 mM at a scattering angle of 90◦ . The solid line is fit to the data using method of cumulants.
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Fig. 2. The variation of (a) apparent diffusion coefficient (Da ) obtained from the slope of the plots of Γ vs. q2 and (b) length (L) calculated from Perrin’s formula for 50 mM CTAB at different concentrations of Na-PTS in presence as well as absence of 0.5 M NaCl.
Fig. 3. The variation of (a) apparent diffusion coefficient (Da ) obtained from the slope of the plots of Γ vs. q2 and (b) length (L) calculated from Perrin’s formula for 50 mM CTAB at different concentrations of Na-Sal in presence as well as absence of 0.5 M NaCl.
relaxation rate supporting the validity of cumulants results. The shift of the correlation function to larger times with increase in salt concentration is an indication of the growth of the micelles. The angular variation of the average decay rate (Γ ) of the intensity correlation function obtained from the fit at different concentrations of Na-PTS and Na-Sal was found to vary linearly as a function of q2 indicating translational diffusion of the scatterers and negligible contribution from rotational diffusion in case of anisotropic micelles. This is expected for micelles due to very small dimension as compared to the wavelength of the light. The apparent diffusion coefficient (Da ) of the micelles was calculated from the slope of Γ versus q2 plot. The variation of apparent diffusion coefficient, Da , of CTAB micelles at different concentrations of Na-PTS and Na-Sal is depicted in Figs. 2(a) and 3(a), respectively. The open circles in Figs. 2(a) and 3(a) show the data for CTAB/Na-PTS and CTAB/Na-Sal in presence of 0.5 M NaCl, which will be discussed further. At very low concentration of the salts (csalt = 10 mM), Da and the corresponding hydrodynamic diameter, calculated using Stokes–Einstein relation, is in the same range as that expected for small globular micelles. With successive addition of Na-PTS and Na-Sal, Da was found to drop steeply reflecting a sudden increase in the average dimension of the micelles. At cNa-PTS = 15 mM, the Da value is 63.4 × 10−8 cm2 /s corresponding to an equivalent sphere diameter of 7.7 nm, which increases further upon increasing the concentration of the hydrophobic salts. However, it is unrealistic to have an isotropic spherical micelle having a diameter of 7.7 nm for CTAB molecules, since the maximum extended length of
a dodecyl chain being only 2.17 nm [42]. Moreover, previous reports on CTAB micelles in the presence of hydrophobic salts reveals the formation of long thread-like micelles [8–10,43–46]. Thus, the observed drastic decrease in Da could be attributed to the formation of prolate ellipsoidal micelles. The DLS data at high salt concentration were analyzed in terms of prolate ellipsoidal structure. To get a quantitative insight in to the growth of the micelles, the lengths of the micelles were estimated using Perrin’s formula [47] that relates the average diffusion coefficient to the axes of the ellipsoid. The minor axis of the ellipsoid is taken as constant at 4.3 nm and only the semi-major axis is varied. Corrections to the measured Da for possible hydrodynamic and thermodynamic contributions were applied during the data analysis as reported earlier [52]. The average length (L) of the micelles as obtained from the major axis of the ellipsoid for Na-PTS and Na-Sal is depicted in Figs. 2(b) and 3(b), respectively. The open circles in Figs. 2(b) and 3(b) show the length of the micelles for CTAB/Na-PTS and CTAB/Na-Sal in presence of 0.5 M NaCl, which will be discussed further. A marked increase in the length of the micelles was observed with increasing concentration of hydrophobic salts, viz. from 11 ± 1 to 94 ± 9 nm for Na-PTS and 20 ± 2 to 93 +9 nm for Na-Sal. It is true that when the micelles are sufficiently long, they become flexible beyond certain length scale; known as the persistence length, lp , and the rigid-rod approximation is questionable. The total persistence length for a polyelectrolyte chain can be represented as the sum of an intrinsic and an electrostatic length, lP,t + lp ,e [48]. In the past several years, there has been a number of small angle neutron scattering (SANS) studies of
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worm-like micelles in which values of lp were estimated using Holtzer (bending rod) plots [49]. Magid [50] made a survey on persistence lengths determined for different micellar systems using analyses of light scattering and SANS data. The results suggest that micellar flexibility is apparently highly dependent on surfactant monomer structure, with values ranging from 100 ˚ for nonionic surfactants to a few hundred nanometers. to 200 A The lp ’s reported for the penetrating counterion salicylate with C16 TA and C16 Py, suggest that its effect relative to a nonpenetrating counterion is to rigidify the micelles. For C16 TA Sal/NaSal system, the lp was found to be 137 nm [51]. In the present study, the flexibility of the micelles is not taken in to account due to the limited number of parameters that can be estimated from DLS. However, the length calculated for CTAB/Na-Sal and CTAB/Na-PTS micelles, in the present study, is comparable or smaller than the value of lp reported for C16 TA Sal/Na-Sal system. Therefore, the rigid-rod model is a good approximation to be used to analyze the data.
Fig. 4. Representative plot of the intensity correlation function for 50 mM CTAB with cNa-pts = 10 and 20 mM in presence of 0.5 M NaCl at a scattering angle of 90◦ . The solid line is fit to the data using method of cumulants and the dotted line shows the fit to the data (20 mM Na-PTS) by biexponential decay.
3.2. Effect of hydrophobic salts on micelle size in presence of NaCl As the micellar growth is very sensitive to the nature and the amount of salt added to the solution, it is important to investigate the effect of electrolytes on rod-like micelles. Inorganic electrolytes such as NaCl, NaBr, NaNO3 and NaF can influence significantly the formation of shear-induced structures in CTAB/Na-Sal micelles [38]. In particular, addition of small amounts of NaCl to CTAB/Na-Sal micelles imparts shearthickening behavior to the fluid as opposed to shear thinning in the absence of NaCl. To envisage the effect of NaCl on the growth of CTAB/Na-Sal and CTAB/Na-PTS systems, DLS measurements were carried out on aqueous solution of CTAB (50 mM) at different concentrations of Na-PTS and Na-Sal in presence of 0.5 M NaCl. The concentration of hydrophobic salts was varied from 2 to 50 mM for Na-PTS and 2 to 30 mM for Na-Sal. Fig. 4 shows the intensity correlation function for 50 mM CTAB in presence of 0.5 M NaCl at Na-PTS concentrations of 10 and 20 mM at a scattering angle of 90◦ . At small concentration of Na-PTS (≤10 mM), a unimodal distribution of the relaxation times was observed and the data could be fitted by the method of cumulants (r > 0.999). However, for csalt ≥ 15 mM, the slow mode of long relaxation time begins to evolve. This is evident from the poor quality of fit to the data by the method of cumulants (r < 0.999). The cumulants fit (solid line) deviates significantly from the data at larger times (Fig. 4). A similar behavior is observed for CTAB/Na-Sal system also in presence of NaCl. The difference between the dynamical behavior of CTAB/Na-PTS and CTAB/Na-Sal micelles in absence and presence of NaCl is that in absence of NaCl, the amplitude of slow relaxation rate is very small as compared to the one observed in presence of NaCl at the same concentration (Figs. 1 and 4). It indicates that presence of NaCl in addition to hydrophobic salts plays an important role in the growth behavior of CTAB micelles. The growth of CTAB micelles in presence of hydrophobic salts (Na-Sal and Na-PTS) is much faster with addition of NaCl than that is observed in absence of NaCl for the same concen-
tration of Na-PTS and Na-Sal. This is due to the shielding of the surface charge by the addition of NaCl. In addition to the hydrophobic as well as electrostatic interactions of hydrophobic salts, the presence of NaCl helps in further reducing the electrostatic repulsion and makes the aggregation easier. Due to which, the micelles grow faster and the topological interactions become significant at lower concentrations of the hydrophobic salts. Previous reports indicated that the slow mode of relaxation starts evolving when the axial ratio of the micelles becomes long enough to drive the system into semi-dilute regime [17,34]. According to Nemoto et al. [17], the normalized time correlation function of intensity scattered from the samples exhibited the bimodal distribution of the decay rate Γ in a semi-dilute regime of thread-like micelles. The fast mode, Γ f , was ascribed to the diffusion of micelles with a cooperative diffusion coefficient (Dc ). On the other hand, the characteristic decay rate, Γ s , of the slow mode is related to the coupling between concentration fluctuation and stress. The existence of two characteristic decay times in presence of NaCl is an indication of two dynamical processes. To account for the existence of two relaxation modes in the correlation function for csalt ≥ 15 mM, the data were analyzed by biexponential decay, given by the relation: g2 (τ) − 1 = Af e−Γf τ + As e−Γs τ
(1)
where Af and As are the amplitude of the fast and slow components with relaxation rates, Γ f and Γ s , respectively. For csalt ≥ 15 mM, biexponential decay function gives a better fit to the data with r > 0.999 in comparison to that obtained by the method of cumulants and the incorporation of additional parameters in the fit is validated from the F-test [53]. The dashed line in Fig. 4 shows the fit with biexponential decay. The parameters of the fit for CTAB/Na-PTS system obtained from biexponential decay of the correlation function at 90◦ are summarized in Table 1.
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Table 1 Parameters of intensity correlation function obtained from the fit by biexponential decay analysis for CTAB in presence of different concentrations of Na-PTS and 0.5 M NaCl at a of scattering angle 90◦ : correlation coefficient (r), amplitude of fast mode (Af ), fast decay rate (Γ f ) and amplitude of slow mode (As ), slow decay rate (Γ s ) Concentration (Na-PTS) (mM)
r
Af
Γ f (s−1 )
As
Γ s (s−1 )
15 20 25 30 35 40 45 50
0.99996 0.99992 0.99990 0.99983 0.99983 0.99990 0.99990 0.99982
0.774 (3) 0.764 (3) 0.764 (4) 0.806 (2) 0.799 (2) 0.757 (2) 0.785 (4) 0.786 (2)
0.0143 (1) 0.0140 (1) 0.0123 (1) 0.0142 (2) 0.0118 (1) 0.0139 (1) 0.0139 (4) 0.0144 (2)
0.051 (3) 0.076 (3) 0.072 (4) 0.039 (2) 0.054 (2) 0.071 (2) 0.056 (4) 0.049 (2)
0.00098 (8) 0.00065 (4) 0.00063 (5) 0.00008 (1) 0.00016 (2) 0.00018(1) 0.00060 (6) 0.00008 (1)
The variation of apparent diffusion coefficient, Da (obtained from the slope of Γ f versus q2 plot) of CTAB micelles (50 mM) at different concentrations of Na-PTS and Na-Sal in presence of 0.5 M NaCl is shown in Figs. 2(a) and 3(a), respectively. As salt concentration (cNa-PTS and cNa-Sal ) increases, Da decreases initially and then takes a more or less constant value. The observed diffusion coefficient was used to calculate the apparent length of the micelles, assuming a prolate ellipsoid structure as discussed earlier. The average length (L) of the micelles as obtained from the major axis of the ellipsoid for Na-PTS and Na-Sal is depicted in Figs. 2(b) and 3(b), respectively. Initially, the length of the micelles increases with increasing cNa-Sal and cNa-PTS until reaching a maximum value and then takes a nearly constant value of ∼170 nm. It should be noted that this observed saturation in the calculated length of the micelles does not indicate a limiting growth of the micelles. This is merely a reflection of entering the semi-dilute regime of rod-like micelles. Beyond the semi-dilute concentration, topological interactions become significant and the diffusion of the micelles is severely hindered. This leads to an underestimation of the calculated length of the micelles. Theoretical treatment by Doi and Edwards [54,55] provides the basis for an understanding of dynamical behavior of a system of strongly interacting rod-like polymers whose length (L) is much larger than the rod diameter (d). In the present study, we used Doi–Edwards model of rod-like polymers to compare the dynamics of rod-like micelles in semi-dilute regime. For rod-like polymers of length (L), the semi-dilute regime is identified as the concentration at which the number density (ν) of the polymer becomes much greater than 1/L3 . According to the tube model of Doi and Edwards [54–56], the rotational diffusion constant in semi-dilute (Dr ) and infinite dilute solutions (Dr,0 ) are related to the expression: −2
Dr = βDr,0 (νL3 )
(2)
where β is a numerical factor that indicates the concentration at which topological interactions become significant. Mori et al. [57] measured the Dr of rod-like polymers of different lengths over a wide range of concentration using dynamic electric birefringence and showed that the value of β is of the order of 103 . By a detailed study of tube statistics and comparison with experimental results Hayakawa et al. [58] suggested that the semi-dilute regime starts at νL3 ∼ 30.
The number density (ν) of the rod-like micelles having length L, and radius r can be calculated with the equation: ν = (c − cmc)
Na v πr 2 L
(3)
where c is the concentration of surfactant, cmc is the critical micelle concentration, Na is Avogadro’s number and v is the volume of the surfactant monomer. In the present study, knowing the volume fraction of the micelles, one can show that the condition νL3 ∼ 30 is satisfied when the axial ratio of the micelles becomes 42. This corresponds to a length of the micelles of the order of 185 nm, considering the diameter of the micelle as 4.4 nm. The observed saturated value of length 170 nm is close to the expected value of 185 nm based on the molecular theory of entanglement of rod-like polymers [58]. This further supports the fact that the biexponential relaxation arises from a transition of the system to semi-dilute regime. A comparative study of lengths obtained for CTAB micelles in presence as well as absence of NaCl at different concentrations of Na-PTS and Na-Sal is given in Table 2. The micellar parameters reveal that, the striking difference in the rheology of CTAB/Na-Sal in presence of NaCl and the formation of shearinduced structures [38] could be due to the strong growth and longer length of the micelles as compared to that in absence of NaCl. In the present study, the amplitude of the slow mode As is fairly small, typically less than 0.1 in all the cases and is found to be almost independent of the concentration of hydrophobic salts. However, in the earlier studies by Amin et al. [35], the amplitude of the slow relaxation mode was observed to be a strong function of NaSal:CTAB mole ratios. This difference in the behavior arises from the limited range of concentration studied here and smaller length of the micelles. At Na-Sal:CTAB molar ratio <1, at CTAB concentration of 50 mM, the amplitude is less than 0.1 and almost independent of Na-Sal:CTAB molar ratio, within the experimental errors. Moreover, the amplitude will be further reduced when one considers the intensity correlation, instead of electric field correlation. Due to very small amplitude, (As ), inherent polydispersity in Γ s and associated standard deviation of the measured data, the accuracy of Γ s obtained from the fit is limited. Thus, no attempts were made to extract stress relaxation parameters of the micelles from the slow mode.
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Table 2 Comparison of the length of CTAB micelles in presence as well as absence of NaCl at different concentrations of the salts, Na-PTS and Na-Sal csalt (mM)
2 5 10 15 20 25 30 35 40 45 50
Length of CTAB micelles in presence of Na-PTS (nm)
Length of CTAB micelles in presence of Na-Sal (nm)
CTAB/Na-PTS
CTAB/Na-PTS/NaCl
CTAB/Na-Sal
CTAB/Na-Sal/NaCl
– – – 11 ± 1 25 ± 2 40 ± 4 42 ± 4 50 ± 5 53 ± 5 60 ± 6 94 ± 9
10 45 127 142 161 200 167 195 167 167 163
– – – 20 ± 2 24 ± 2 34 ± 3 40 ± 4 51 ± 5 63 ± 6 75 ± 7 93 ± 9
10 ± 1 58 ± 6 153 ± 15 167 ± 16 169 ± 17 – 167 ± 17 – – – –
± ± ± ± ± ± ± ± ± ± ±
1 4 13 14 16 20 17 20 17 17 16
4. Conclusion This report investigates the growth behavior of CTAB micelles in presence as well as absence of NaCl on addition of different concentrations of hydrophobic salts like, Na-PTS and Na-Sal, by dynamic light scattering measurements. From DLS results, it was found that the growth of the CTAB micelles is much more pronounced at the same concentration of hydrophobic salts, Na-PTS and Na-Sal, when 0.5 M NaCl is added to the solution. The length of the micelles was estimated assuming a prolate ellipsoid structure, ranging from dilute to semi-dilute regime. When the micelles become long enough to entangle each other with addition of hydrophobic salt, the system reaches to semi-dilute regime and shows a biexponential decay behavior. In presence of NaCl, a faster growth of the micelles in the dilute regime and corresponding transition from dilute to semidilute regime at lower concentration of Na-Sal and Na-PTS is observed. Comparison of the micellar lengths in the presence and absence of NaCl in the dilute regime suggests that the biexponential relaxation arises possibly from topological interaction of the micelles. This was confirmed by calculating the expected length of the micelles for topological interaction to occur based on Doi–Edwards model for rod-like polymers. It is indeed observed that the length of the micelles at the transition from dilute to semi-dilute regime is consistent with the Doi–Edward model. The transition point shift to lower hydrophobic salt concentration when the micellar growth is faster. All the evidences are in favor of the conjecture that the slow relaxation mode arises from topological interaction of the micelles. Moreover, the above studies suggest that the observed difference in the rheology of the CTAB-Na-Sal micelles in presence and absence of NaCl can be correlated to the difference in the length of the micelles and the growth behavior. The larger length of the micelles in presence of NaCl could be responsible for the shear-thickening behavior of the micelles, as compared to the shear-thinning behavior in absence of NaCl. Acknowledgements The authors are thankful to Dr. C. Manohar of Indian Institute of Technology, Mumbai for many fruitful discussions. G. Garg
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