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Rheological Properties of Semidilute and Concentrated Aqueous Solutions of Cetyltrimethylammonium Bromide in the Presence of Potassium Bromide F. Kern,t P. Lemarecha1,t S. J. Candau,’??and M. E. Catesx Laboratoire d’Ultrasons et de Dynamique des Fluides Complexes, Unit6 de Recherche AssociBe au CNRS No. 851, Universitb Louis Pasteur, 4 rue Blaise Pascal, 67070 Strasbourg Cedex, France, and Department of Physics, Cavendish Laboratory, University of Cambridge, Madingley Road, Cambridge CB3 OHE,U.K. Received July 24,1991. In Final Form: October 23, 1991
The frequency-dependent shear modulus of semidilute aqueous solutions of cetyltrimethylammonium bromide (CTAB)in the presence of 0.25M KBr has been measured as a function of surfactant concentration and temperature. The comparison between the experimental data and the recent results of a computer simulation provides an estimate of 7bre&, the time taken for a micelle of the mean length to break. In the moderately low concentration range, the values obtained for 7break from such an analysis are found to be in good agreement with those measured previously by means of a temperature-jump technique. At high concentration one observes an unexpected increase of Tbre& that is tentatively attributed to the fact that the micelles are likely to behave as stiff rods on the scale of the geometrical mesh-size of the entangled micellar network.
Introduction Under appropriate conditions of concentration, salinity, temperature, presence of counterions, etc., small aqueous spherical micelles can undergo uniaxial growth to form flexible wormlike micelles.14 These are, in favorable cases, extremely long (many thousand angstroms) and undergo reversible breakdown processes. The micellar breaking time can be measured by temperature-jump meth0ds.~*6 The viscoelastic properties of systems of flexible and entangled micelles have been extensively investigated experimentally these last year~.l-~J-’~ Most of these experiments concern the frequency-dependent shear modulus G*(o) and the results are generally presented in the form of a Cole-Cole plot, in which the imaginary part G”(o) of the complex modulus is plotted against the real part G’(w). Depending on the conditions of surfactant
* To whom correspondence should be addressed. t Univeraitk Louis Pasteur. t University of Cambridge.
(1) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (2) Candau, S. J.; Hirsch, E.; Zana, R.; Delsanti, M. Langmuir 1989, 5, 1225, and references therein. (3) Jindal, V.; Kalus, J.; Pilsi, H.; Hoffmann, H.; Lindner, P. J.Phys. Chem. 1990,94, 3129, and references therein. (4) Porte, G.; Appell, J. Europhys. Lett. 1990, 12, 190. (5) Turner, M. S.; Cates, M. E. J. Phys. (Paris) 1990, 51, 977. (6) Candau, S. J.; Merikhi, F.; Waton, G.; Lemarechal, P. J. Phys. (Park) 1990,51,977. (7) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987,3,1081; 1988, 4, 354; 1989, 5, 398. (8) Shikata, T.; Hirata, H.; Takatori, E.; Osaki,K. J.Non-Newtonian Fluid Mech. 1988,28, 171. (9) Imae, T.; Abe, A.; Ikeda, S. J.Phys. Chem. 1988, 92, 1548. (lO)Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988,92,4712. (11) Hoffmann, H.; Ldbl, H.; Rehage, H.; Wiinderlich, I. Tenside Deterg. 1986, 22, 290. (12) Thurn, H.; Ldbl, M.; Hoffmann, H. J.Phys. Chem. 1985,89,517. (13) Ldbl, M.; Thurn, H.; Hoffmann, H. Ber. Bunsen-Ges. Phys. Chem. 1984,88, 1102. (14) (a) Hoffmann, H.; Platz, G.;Rehage, H.; Schorr, W. Ber. BunsenGes. Phys. Chem. 1981,25,877. (b)Ado. Colloid Interface Sci. 1982,17, ““e 41.3.
(15) Candau, S. J.; Hirsch, E.; h a , R. In Physics of Complex and Supermolecular Fluids; Safran, S., Clark, N., Eds.; Wiley: New York, 1987; p 569. (16) Candau, S. J.; Hirsch, E.; Zana, R.; Adam, M. J. ColloidZnterface Sci. 1988, 122, 430. (17) Kern, F.; Zana, R.; Candau, S. J. Langmuir 1991, 7, 1344.
and salt concentration, the Cole-Cole plots were found to vary from a semicircular shape characteristic of a single exponential stress relaxation to a somewhat flattened shape, indicative of a broad distribution of relaxation times.7&10,17 A simple theoretical model, based on the reptation theory which describes the rheological properties of entangled polymeric chains, has been derived by Cates.18 This model predicts several rheologicalregimes depending on the relative rates of diffusive polymer motion and reversible breakdown processes. In particular, a nearly single exponential stress decay function is predicted in the linear viscoelastic response, in the limit where the micelle breaking time is short compared to the reptation time of a micelle of length equal to the average micellar length. Recently, Turner and Cateslg presented detailed results from a computer simulation for various values of the ratio { = Tbreak/Trep where Tbre& is the time taken for a micelle of the mean length to break and 7rep is its reptation time. The calculated Cole-Cole plots can be compared to the experime_ntal ones providing a direct estimate of the parameter { = Tbreak/TR, where TRis the terminal stress relaxation time. In this paper we present measurements of the frequencydependent shear modulus for semidilute aqueous solutions of cetyltrimethylammonium bromide (CTAB) in the presence of 0.25M KBr. The same systems were previously investigated by means of a magneto rheometer that provides directly the stress relaxation function.2 However, in contrast to oscillatory devices this apparatus does not have the accuracy required to see deviations from single exponential behavior that are expected to occur in the short time range. Theory The main features of the theoretical models describing both equilibrium and dynamic properties of wormlike micelles can be found in ref l. Here we simply recall the (18) Cates, M. E. Macromolecules 1987, 20, 2289. Europhys. Lett. 1987, 4, 497. J. Phys. (Paris) 1988, 49, 1593. J. Phys. Chem. 1990, 94, 371. (19) Turner, M. S.; Cates, M. E. Langmuir 1991, 7, 1590.
0 1992 American Chemical Society
Kern et al.
438 Langmuir, Vol. 8,No. 2, 1992
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Figure 1 : Calculated Cole-Cole plots from ref 19 for three values of = Q,&TR. The imaginary and real parta of the shear modulus have been normalized by G, the radius of the osculating circle at the origin. G" (Pa)
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Figure 2. Experimental C o l d o l e plot (open circles) for a solution at C = 0.35 M and T = 30 OC. The crosses represent the best fit of the calculated C o l d o l e plots to the experimental data. The broken line is the osculating semicircle at the origin. From this plot one determines the following parameters: qo = 23.9 Pa-s;G. = 263 Pa (by linear extrapolation); , G = 203 Pa; TR= 91 ms; Tb&TR = 1.23; 7b& = 112 ma.
theoretical results that are needed to discuss our experiments, more specifically regarding the shape of the stress relaxation and the effect of surfactant concentration. Shape of the Stress Relaxation. The results of the numerical study of Turner and Cates19are shown in Figure 1. In the original paper the results were presented under the form of Cole-Cole plots G"/Ge versus G'/Ge, where Ge is the plateau modulus. Experimentally there is often incomplete data at very high frequenciesso that Ge cannot be measured accurately. To overcome this difficulty, Turner and Cates have proposed an extrapolation procedure, but the latter is based on assumptions that disregard nonreptative high-frequency effects. Here we have adopted a somewhat different normalization procedure by dividing both G' and G" by G, the radius of the osculating circle of the G' vs G" Cole-Cole plot at the origin. The corresponding curves are given in Figure 1for three values o f t = 7 b d TR. One clearly sees the departure from exponentiality in the high frequency side of the plot, that becomes more and more pronounced as 5 increases. At that point, it is important to recall the assumptions made for the calculations of G'(w).19 (i) Firstly, it was assumed that when a chain breaks, the two daughter chains become uncorrelated and evolve separately. This impliesthat each micellar end recombines with a randomly chosen micellar end and not the original partner end. This assumption will be discussed further below. (ii) The second assumption is that the only mode of stress relaxation is a reptation process. (iii) The chemical relaxation process is the reversible unimolecular scission, characterized by a temperaturedependent rate constant k per unit time per unit arc length, which is the same for all elongated micelles and is independent of time and of volume fraction. This assumption says that' where L is the average micellar length and C the surfactant concentration.
Effsct of Surfactant Concentration, The modified reptation model developed by Cates predicts the following dependence of the terminal time TR on the surfactant concentration: 112 TR
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The zero-shear viscosity qo is related to the terminal time and the plateau modulus Ge through
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Ge kBTC2 (4) Combining eqs 2 to 4 leads to the following behavior of the zero-shear viscosity: qo
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It must be noted that the exponents of the above scaling laws correspond to a mean-field approximation. If excluded volume interactions are taken into account, one obtains slightly different exponents. The exponente of the power laws of TR, G,, and qo as a function of concentration are 1.4, 2.3, and 3.7, respectively. These exponents are modified further if the breaking time is short enough that the dominant chain motion on that time scale is not reptation but local chain motion ("breathing" or Rouse modes).l8 However, these modes are unlikely to be relevant in determining the power laws for the systems studied here, which have breaking times in the 50-1000 ms range.
Materials and Methods The CTAB samples were the same as in our previous investigatiom.2J6J6The rheologicalproperties were investigated by performing oscillatory experiments with a Weissenbergrheogoniometer (Carrimed) modified in order to avoid evaporation and to realize a good temperature control. We used a cone (50 mm diameter, 4O angle) and plate geometry. A small amplitude oscillatory shear flow is applied to the investigated viscoelastic solution. From the phase lag of the corresponding shear stress and the ratio of the amplitudes of the imposed oscillation and of the response of the solution are calculated the complexviscosity q*(w), the storage modulus G'(w), and the loas modulus GN(u).21
Data Analysis Figure 2 shows an experimental Cole-Cole plot represented in its classical form, i.e. G"(w) versus G'(o). On (20) Adam, M.;Deleanti, M.J. Phys. (Pa&) 1986,44,1185. (21) Walters, K. Rheometry; Chapman and Hall Ltd.: London, 1976.
Langmuir, Vol. 8, No.2, 1992 439
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