pubs.acs.org/Langmuir © 2009 American Chemical Society
Catanionic Surfactants as Nanospring Suspensions: A Model Janaky Narayanan,*,† P. A. Hassan,‡ and C. Manohar† †
Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai-400076, India, and ‡ Chemistry Division, Bhabha Atomic Research Centre, Mumbai-400085, India Received January 16, 2009. Revised Manuscript Received May 22, 2009
A model for a dilute suspension of nanosprings, whose equilibrium configuration and extension are controlled by electrical double layer forces, is presented along with a model for changing packing parameter. The dependence of viscosity on surface charge is calculated. The possibility of shear-thickening is demonstrated. It is argued that mixtures of cationic and anionic surfactants which show two peaks in viscosity and a minimum along with shear-thickening are likely candidates for this model.
Introduction
*To whom correspondence should be addressed. E-mail: janaky_n@ iitb.ac.in.
The first thing is to note that the existence of two peaks even at low concentrations of 10-5 M, much lower than the overlap concentration (C*), implies that this feature is due to changes occurring in a single WLM and is not due to WLM-WLM interactions (entanglements). This would imply that conformational changes in single WLMs are responsible for the existence of two peaks. Since the WLMs are flexible, a WLM can fold or bend to form contacts on itself provided Coulomb repulsion due to charge on surface does not forbid such formation. This folding reduces the effective length of micelles, thus decreasing the viscosity. On hydrodynamic length scales, this folded micelle would look like a cylinder. Therefore the formation of coils with springlike structures (elastic helices) is a reasonable possibility in between the two viscosity peaks. When an anionic surfactant is added to a cationic micellar solution, the packing parameter of the mixed micelle changes as a function of mole fraction of the additive, leading to sphererod-sphere transition. This will result in an initial increase and a final decrease in viscosity of the solution. However, as shown in the subsequent section, consideration of only the packing parameter for the linear growth of micelles cannot explain the occurrence of two peaks in viscosity. When SS is continuously added to CPC solution, not only the packing parameter changes, but also the positive surface charge is reduced, is brought to zero, and then is made negative. When the charge is zero, one should expect the formation of tightly bound springs and this should be at the minimum in viscosity. By invoking the concept of nanospring model for the micelles and considering suitable interaction potentials, we predict a viscosity minimum at the equimolar concentration and elucidate the dependence of viscosity on the surface charge of the micelles. Another natural consequence of this model is that a suspension of springs is expected to show shear-thickening as the springs would stretch under a velocity gradient. These ideas are elaborated in the following sections, and predictions are made hoping that this would motivate further experiments.
(1) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712. (2) Zana, R., Kaler, E. W., Eds. Giant Micelles: Properties and Applications; CRC Press: Boca Raton, FL, 2007; cover page. (3) Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933. (4) Mishra, B. K.; Samant, S. D.; Pradhan, P.; Mishra, S. B.; Manohar, C. Langmuir 1993, 9, 894. (5) Kaler, E. W.; Murthy, A. K.; Rodriguez, B. E.; Zasadzinski, J. A. N. Science 1989, 245, 1371. (6) Lequeux, F. Europhys. Lett. 1992, 19, 675.
(a) Model for Packing Parameter. The shape of a micelle is decided by the packing parameter v ð1Þ p ¼ al
The motivation for this work came from a beautiful publication of Hoffmann and Rehage1 where they demonstrated that if one added continuously sodium salicylate (SS) to an aqueous solution of cetylpyridinium chloride (CPC) the zero shear viscosity showed two peaks as a function of concentration of SS.2 The main features of the observations from their work and subsequent papers3 were as follows: 1. The existence of two peaks occurred for all the concentrations of CPC investigated, low and high. 2. The first peak (CPC-rich) had wormlike micelles (WLMs) with positive surface charge, while the other peak (SS-rich) had WLMs with negative surface charge. 3. The viscosity minimum in between the peaks occurred approximately at equimolar concentrations of SS and CPC. 4. The existence of two peaks occurred in most of the systems (cationic-anionic mixtures) studied, indicating that it was a common phenomenon. 5. In some systems, the solutions became turbid in between the two peaks most probably forming vesicles.4,5 The existence of a minimum in between the two peaks and the above features have never been explained satisfactorily, though some attempts were made by Lequeux6 arguing that networks with flexible contacts occurred, but this did not explain the above features, in particular the existence of two peaks in dilute region and vesicle formation. The main aim of the current paper is to propose a semiquantitative explanation for all the observations mentioned above and to predict new phenomena based on a conjecture.
7260 DOI: 10.1021/la901368t
The Model
Published on Web 06/11/2009
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Figure 1. Schematic representation of catanionic complex formation. Surfactants 1 and 2 with packing parameters p1 and p2 and of mole fraction x for surfactant 2 are mixed, and the complex formed has a packing parameter p12 and a mole fraction x12.
where v is the volume of the surfactant hydrophobic part, a is the surface area occupied by the surfactant headgroup, and l is the extended length of the hydrophobic part.7 p e 1/3 for spherical micelles and 1/3 e p e 1/2 for rodlike micelles. When we mix two surfactants 1 and 2 with packing parameters p1 and p2 and of mole fraction x for surfactant 2, the average packing parameter of the mixture is p ¼ ð1 -xÞp1 þ xp2
ð2Þ
A similar idea is conventionally used with success in industries for hydrophilic lipophilic balance (HLB).8 The same idea is used for the packing parameter too.9 This concept can be generalized to ternary mixtures. When we mix a cationic surfactant and an anionic surfactant, a complex is formed (Figure 1). This complex forms the third surfactant with a packing parameter p12 given by p12 ¼ p1 ð1 þ v2 =v1 Þ
ð3Þ
In eq 3, it is assumed that surfactant 1 has a longer hydrophobic part and both surfactants occupy same headgroup area.10 The mole fraction of the complex x12 is given by the law of mass action: x12 ¼ Rxð1 -xÞ
ð4Þ
where R is a constant (R e 4). The average packing parameter is now given by (taking into account double counting) P ¼ ð1 -x -x12 =2Þp1 þ x12 p12 þ ðx -x12 =2Þp2
ð5Þ
Here, x is the mole fraction of surfactant 2 (anionic surfactant). Substituting eqs 3 and 4 in eq 5, P ¼ ð1 -xÞp1 þ Rxð1 -xÞ½p1 v2 =v1 þ ðp1 -p2 Þ=2 þ xp2
ð6Þ
This complex formation is the key concept needed to explain the presence of pairs of viscoelastic nanodiscs and vesicle phases. This is illustrated by taking two examples: mixtures of cetyltrimethylammonium bromide (CTAB) and sodium n-alkyl sulfonate (CnSO3Na, n = 7, 10). Using eq 1, assuming a = 0.632 nm2, calculating v from v = 0.0269m + 0.0274 nm3 and l from l = 0.1265m + 0.15 nm,11 where m is the number of carbon atoms in the saturated hydrocarbon chain, the packing parameters for (7) Israelachvili, J.; Mitchell, D. J.; Ninham, B. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (8) Becher, P. Emulsions: Theory and Practice; Krieger: New York, 1977. (9) Kumar, V. V. Proc. Natl. Acad. Sci. U.S.A. 1991, 88, 444. (10) Mishra, S.; Mishra, B. K.; Chokappa, D. K.; Shah, D. O.; Manohar, C. Bull. Mater. Sci. 1994, 17, 1103. (11) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes, 2nd ed.; John Wiley and Sons: New York, 1980.
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Figure 2. Average packing parameters for the mixtures of CTAB-C7SO3Na (circle; curve 1) and CTAB-C10SO3Na (plus; curve 2) calculated using eq 6. R = 4. The solid lines are only guides to the eye. P < 1/3 is the region of spherical micelles, 1/3 < P < 1/2 is of cylindrical micelles, and P > 1/2 is of vesicles. Curve 1 represents cylindrical micelles, and curve 2 in addition represents vesicles.
CTAB, C7SO3Na, and C10SO3Na are, respectively, 0.3333, 0.33, and 0.332. With R = 4, the average packing parameters for the mixtures of CTAB-C7SO3Na and CTAB-C10SO3Na are calculated using eq 6 and depicted in Figure 2. For the CTAB-C7SO3Na mixture, P e 0.5 for all values of x, suggesting that this mixture forms only spherical/cylindrical micelles, whereas for the CTAB-C10SO3Na mixture P exceeds 0.5 around x = 0.5, suggesting the possibility of vesicle formation in this region. Similar arguments hold, respectively, for CTABSS and CTAB-sodium 3-hydroxy 2-naphthalene carboxylate (SHNC) mixtures with the latter forming spontaneous vesicles on either side of equimolar concentrations.4 Now, we discuss the example of CTAB-C7SO3Na mixture which forms end-capped rodlike micelles. If L is the length and R is the radius of these micelles, then from packing constraints P ¼
L þ ð4=3ÞR 2ðL þ 2RÞ
ð7Þ
From eqs 6 and 7, L is calculated as a function of x and plotted in Figure 3. For a suspension of randomly oriented, long, slender, rodlike bodies, the intrinsic viscosity is given by12 η ¼
4ar 2 15ln ar
ð8Þ
where the axial ratio ar = L/2R. This is the formula for a suspension of rods which maintains a random orientation distribution in shearing flow and is applicable for dilute solutions independent of concentration. Figure 4 gives the plot of the intrinsic viscosity versus mole fraction and shows only one peak in viscosity while experiments show two peaks as mentioned above, indicating the need for a new concept. This is done below by invoking the concept of a nanospring. (b) Model for Nanospring. We consider a system of flexible wormlike micelles which when condensed into closely packed springs will be of length ð9Þ Ls ¼ nDm where n is the number of rings on the spring and Dm is their mean distance apart. The length Ls of the spring is decided by the (12) Powell, R. L. J. Stat. Phys. 1991, 62, 1073.
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The attractive van der Waals energy is given by13 VvW ¼
-A 12πD2
ð15Þ
A is the Hamaker constant.The repulsive hydration potential is given by14 -D Vhyd ¼ W0 exp ð16Þ λ0
Figure 3. Length of rodlike micelles formed by the mixture of CTAB-C7SO3Na calculated using eqs 6 and 7. R = 4; R = 2.5 nm. The solid line is a guide to the eye.
W0 and λ0 are, respectively, the strength and range of the hydration potential. In the absence of any shear, the separation between the rings D is given by the potential energy minimum, dV/dD = 0 and D = Dm. Using eqs 10-16, 2εK2 ψ0 2 expð -KDm Þ þ
W0 -Dm A 1 exp ¼ ð17Þ 6π Dm 3 λ0 λ0
For various surface potentials ψ0, using eq 17, Dm can be found, and substituting in eq 9 the spring length Ls can be calculated. If d is the diameter of the spring, the axial ratio ars = Ls/d and the intrinsic viscosity ηs (related to the zero-shear viscosity) can be calculated using eq 8. As |ψ0| increases from zero, Dm increases and along with it ηs also increases. Only those surface potentials are chosen for which |ψ0| e |ψC|, where ψC corresponds to the condition dV/dD = 0 and V = 0 for which D = DC. For |ψ0| e |ψC|, the potential energy at the minimum is negative, thus ensuring stability of the rings. DC and ψC are given by the following equations: Figure 4. Intrinsic viscosity of rodlike micelles formed by the mixture of CTAB-C7SO3Na calculated using eqs 6-8. R = 4; R = 2.5 nm. The solid line is a guide to the eye.
balance of the various potential energies brought into play. Modeling the adjacent rings to be charged hydrated plates separated by a distance D, the net potential energy per unit area V ¼ VCou þ VvW þ Vhyd
ð10Þ
ð11Þ
Here, κ is the reciprocal Debye-Huckel screening length, which for z-z electrolytes is given by 2e2 z2 n¥ K ¼ εkB T 2
ð12Þ
T is the absolute temperature, ε is the permittivity of the solution, kB is the Boltzmann constant, n¥ is the bulk concentration (number density) of the ions of charge ze, and Γ0 is related to the surface potential ψ0 by zeψ0 Γ0 ¼ tanh 4kB T
zeψ0 4kB T
ð13Þ
ð14Þ
(13) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997.
7262 DOI: 10.1021/la901368t
" # expðKDC Þ A 1 W0 -DC exp ¼ 2εK2 6π DC 3 λ0 λ0
ð19Þ
ψ0 ¼ ð1 -xÞψ1 þ xψ2
ð20Þ
For equimolar mixtures, at x = 0.5, ψ0 = 0. Further, corresponding to ψC, x = xC. Substituting these in eq 20, ψ0 ¼ ψC
1 -2x 1 -2xC
ð21Þ
Using the above equations, the equilibrium length Ls of the nanosprings can be calculated for different surface potentials ψ0 and the corresponding viscosity ηs can be evaluated as a function of mole fraction x.
Results and Discussion
For small ψ0 and z (mono- or divalent ions), Γ0 =
ψC
2
ð18Þ
When surfactants 1 and 2 (oppositely charged) are mixed,
The Coulomb repulsive energy13 VCou ¼ 64kB Tn¥ K -1 Γ0 2 expð -KDÞ
! -DC K -1 -λ0 A ð2K -1 -DC Þ ¼ W0 exp 12π λ0 λ0 DC 3
We consider a typical case of a wormlike micelle modeled as a condensed close packed spring with the number of rings n = 25 and diameter d = 12 nm. The Debye-Huckel screening length κ-1=10 nm for a 1:1 electrolyte of concentration 1 mM. The Hamaker constant A=5 10-21 J for typical surfactant solutions, and W0 =0.5 mJ m-2; λ0 =1 nm. Substituting these values in eqs 10-16, the net potential energy V is calculated and plotted as a (14) Israelachvili, J. N. Intermolecular and surface forces; Academic Press: London, 1992.
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function of D for a surface potential ψ0 =5 mV in Figure 5. The potential energy is repulsive at low D values, and passes through a minimum at Dm=6.4 nm and then a maximum at Dmax = 15 nm. For zero-shear rate, the WLMs will have minimum potential energy and will have a viscosity ηs corresponding to distance Dm characteristic of the surface potential. DC and |ψC| calculated using eqs 18 and 19, respectively, are 6.58 nm and 5.74 mV. For -ψC e ψ0 e ψC, using eq 17, Dm is found and ηs is calculated. Figure 6 is a plot of viscosity as a function of surface potential. As |ψ0| increases, Dm and hence the viscosity increase symmetrically about ψ0=0. This result explains, qualitatively, the existence of a minimum between two peaks in viscosity. The ηC values corresponding to |ψC| in Figure 6 are compared with η values in Figure 4. Assuming xC to be 0.445 and using eq 21, x values corresponding to ψ0 values in Figure 6 are calculated and used in Figure 7 in the plot of viscosity versus mole fraction. For comparison, the η values from Figure 4 are also plotted in this range of x. It is clear from Figure 7 that introduction of the nanospring model and the effect of double layer potential on the stability of the nanosprings leads to the viscosity minimum at equimolar ratio of the catanionic mixture that is observed experimentally, whereas consideration of only a linear growth of rodlike micelles in catanionic mixtures, constrained by only packing fraction, suggests a viscosity maximum at equimolar ratio. The above discussion applies to dilute solutions, and hence, the effect of concentration has not been considered. Mixtures of cationic and anionic surfactants form complexes, and therefore, micelle breaking time is the longest for equimolar solutions in view of maximum number of tightly bound catanionic pairs.2 Further, if the catanionic micelle is modeled as polyelectrolyte, the rodlike behavior is justified as polyelectrolyte chains in a dilute salt-free (large Debye length) solution are always extended with end-to-end distance ≈ their extended size.15 Shear-thickening has been observed in surfactant solutions at low concentrations, as low as 100 ppm by weight (0.01 wt %).16,17 In fact, the shear-thickening effect is not usually observed for surfactant concentrations exceeding a few thousand ppm. For the equimolar tris(2-hydroxyethyl) tallowalkyl ammonium acetate (TTAA)-SS solutions, Hu et al.16 observed shear-thickening in the concentration range 1-10 mM, both below and above the overlap concentration C* of 3.3 mM for this system. In the equimolar CPC-SS solutions (C*=7.5 mM), Sung et al.17 observed shear-thickening at 4 and 8 mM. The shear viscosity of semidilute CPC-SS solutions (C>C*) showed almost a constant value until the critical shear rate, and then shear-thinning began, followed by shear-thickening at higher shear rate.17,18 For dilute solutions (C e C*), negligible shear-thinning was observed before the shear-thickening.17 The shear-thickening was observed in the shear rate range 10-100 s-1 with its occurrence at lower shear rates for lower concentrations. The fractional increase in viscosity was largest for the lowest concentrations. As discussed in detail by Hu et al.16 and Fischer et al.,18 ideas of hydrodynamic instabilities and shear-induced phase transitions are invoked in explaining the shear-thickening phenomenon. The micelles are believed to align and extend under shear flow, grow longer, interconnect and form shear-bands, resulting in viscosity increase. However, for concentrations less than the overlap concentration, we believe (15) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. Macromolecules 1995, 28, 1859. (16) Hu, Y. T.; Boltenhagen, P.; Pine, D. J. J. Rheol. 1998, 42, 1185. Hu, Y. T.; Boltenhagen, P.; Matthys, E.; Pine, D. J. J. Rheol. 1998, 42, 1209. (17) Sung, K.; Han, M. S.; Kim, C. Korea-Australia Rheol. J. 2003, 15, 151. (18) Fischer, P.; Wheeler, E. K.; Fuller, G. G. Rheol. Acta 2002, 41, 35.
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Figure 5. Typical potential energy curve for surface potential ψ0 = 5 mV with Dm at 6.4 nm.
Figure 6. Viscosity versus surface potential. Note the minimum at zero surface potential. Experimentally minimum is observed in between two peaks in several catanionic surfactant mixtures at approximately equimolar concentrations.1-3
Figure 7. Viscosity ηs as a function of surface potential plotted in Figure 6 replotted as a function of mole fraction x using eq 21 and xC = 0.445 (triangle; curve 1). The viscosity versus mole fraction η plotted in Figure 4 is replotted here for comparison (circle; curve 2). The solid lines are guides to the eye. Randomly oriented nanosprings are shown in the inset. As x increases from its value at zero, viscosity follows curve 2 until it hits curve 1 where nanosprings are formed and it follows this curve until it hits the curve 2.
that the following analogy with very dilute polymer solutions may apply. In the finitely extensible nonlinear elastic (FENE) spring model, the polymer molecule is modeled by beads connected by nonlinear spring chains.19 Rheological properties of an infinitely (19) Owens, R. G.; Phillips, T. N. Computational Rheology; Imperial College Press: London, 2002.
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dilute polymer solution modeled by bead-nonlinear spring chains under Peterlin approximation were obtained numerically.20 The steady shear viscosity was predicted to show shear-thinning, while strain stiffening was predicted under uniaxial elongation; that is, elongational viscosity was higher at higher elongation rates before reaching saturation. The amount of stiffening was larger for longer chains (more number of beads) and for longer springs. In analogy with the dilute polymer solutions, the catanionic micelles modeled as springs at very low concentrations may be stretched under shear due to velocity gradient with an increase in D and a corresponding increase in viscosity. Therefore, the system, in this range of concentration, should show strain stiffening. Each shear rate will correspond to a D value (not necessarily linear relation), but the exact relation, to the authors’ knowledge, is not known. Though eqs 8 and 9 predict an increase in the viscosity due to elongation, this model is unable to predict a limited extension of the spring as in FENEtype models. Sheared single polymers undergo both rotational and extensional deformations, and double-stranded DNA molecules are used as model systems to directly observe polymer dynamics in flow.21,22 At large shear rates, a free chain approaches a mean extension of 50% of its contour length21 while a tethered chain approaches complete extension though very slowly.22 As polymer models are used in the study of wormlike micelles, shear-thickening in dilute catanionic nanospring suspensions seems to be a natural process caused by elongation under shear flow. Shear-thickening in dilute polymer solutions at very high shear rates has been investigated.23-26 The underlying mechanism behind shear-thickening has been shown to be flow-induced structure formation, that is, formation of macromolecular associations at high shear rates. However, since catanionic micelles are “living polymers” with a fast breaking and making mechanism, they may require lower energy input for shear-thickening which may be the reason for its observation within a much reduced shear rate window compared to the polymer solutions. Shear-thickening is observed in electrostatically or sterically stabilized colloidal suspensions and also in some weakly flocculated systems.27 The balance between the ordering action of the interparticle repulsion and the disordering effect of the hydrodynamic forces is considered to be responsible for the shear-thickening in weakly flocculating suspensions of sufficient concentration. The critical shear rate at which shear-thickening occurs is controlled by pH, salt content, thickness of the grafted layer on the colloidal particles, and so on. Thus, intramolecular rearrangements and intermolecular interactions controlled by experimental and solution parameters are responsible for the shear-thickening in dilute polymer and micellar systems as well as weakly flocculating colloidal (20) Wiest, J. M.; Tanner, R. I. J. Rheol. 1989, 33, 281. (21) Smith, D. E.; Babcock, H. P.; Chu, S. Science 1999, 283, 1724. (22) Ladoux, B.; Doyle, P. S. Europhys. Lett. 2000, 52, 511. (23) Dupuis, D.; Wolff, C. J. Rheol. 1993, 37, 587. (24) Vrahopoulou, E. P.; McHugh, A. J. J. Non-Newtonian Fluid Mech. 1987, 25, 157. (25) Kishbaugh, A. J.; McHugh, A. J. Rheol. Acta 1993, 32, 9. (26) Hatzikiriakos, S. G.; Vlassopoulos, D. Rheol. Acta 1996, 35, 274. (27) Mewis, J. In Structure and Dynamics of Polymer and Colloidal Systems, Proceedings of the NATO Advanced Study Institute, Les Houches; Borsali, R., Pecora, R., Eds.; Springer: Heidelberg, 2002; Vol. 568.
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dispersions. A complete understanding of the underlying principles warrants more experimental and theoretical efforts.
Consequences 1. Formation of a tightly bound spring and thus the viscosity minimum occurs at equimolar concentration and is independent of total surfactant concentration. This seems to be consistent with the observation of Hoffmann and Rehage where they observed the two peaks in viscosity for all concentrations with minimum occurring at equimolar concentration.1 Recently, Shukla and Rehage28 observed a double-peak behavior in the apparent molecular weight of CTAB + SS solutions as a function of SS concentration with a striking similarity to the viscosity double peaks. The molecular weight was calculated from the rod length of the mixed micelles and the minimum in molecular weight supports our prediction of minimum spring length at the viscosity minimum. 2. The rise in viscosity on either side of equimolar concentration is a natural consequence of the formation of nanosprings leading to two peaks, one with a positive charge and the other with a negative charge consistent with experiments. The magnitude of the surface potential at the two peaks is expected to be the same and is a check on the present model. The zeta potential of CTAB + SS solutions as a function of SS concentration28 shows a sign reversal at a much higher concentration compared to equimolar concentration, and also the magnitude is not the same on either side of the zero in potential. This is at variance with our model. The reason may be that not all SS molecules get adsorbed on the CTAB micellar surface and many remain in the bulk. Thus, zero surface charge occurs at a much larger SS concentration than the equimolar value. Also, large SS concentration may push the solution into the semidilute regime which will affect the mobility and zeta potential measurements. One possibility is that by choosing a more hydrophobic molecule such as SHNC rather than SS one may observe a charge reversal closer to equimolar concentration. 3. A suspension of noninteracting springs is expected to show shear-thickening, as the springs would stretch under velocity gradients decided by the surface charge. Cetyltrimethylammonium tosylate (CTAT), being an equimolar mixture of positive cetyltrimethylammonium ion and negative tosylate ion, is a candidate for shear-thickening around C*. This has been observed.29 Similarly, CTAB+ SS,30 CPC+ SS,17 and in fact any mixture of+and - surfactants (one of them with a short chain length) should show shear-thickening if the concentration is between the two peaks in viscosity. 4. Springlike structures should give a Bragg reflection corresponding to the pitch of the spring just like lamellar structures and should show a tendency for a vesicle formation on slight change of curvature.29 Acknowledgment. We thank the reviewers for useful comments and correcting some of our concepts. (28) Shukla, A.; Rehage, H. Langmuir 2008, 24, 8507. (29) Bandyopadhyay, R.; Basappa, G.; Sood, A. K. Phys. Rev. Lett. 2000, 84, 2022. Soltero, J. F. A.; Bautista, F.; Puig, J. E.; Manero, O. Langmuir 1999, 15, 1604. (30) Liu, C.-H.; Pine, D. J. Phys. Rev. Lett. 1996, 77, 2121.
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