Langmuir 1994,10, 1714-1723
1714
Dynamical Properties of Salt-Free Viscoelastic Micellar Solutions F. Kern,? F. Lequeux,t R. Zana,S and S. J. Candau’tt Laboratoire d’Ultrasons et de Dynamique des Fluides Complexes, Unit6 de Recherche Associee au CNRS no. 851, Universit6 Louis Pasteur, 4 rue Blaise Pascal, 67070 Strasbourg Cedex, France, and Institut Charles Sadron (CRM-EAHP), 6 rue Boussingault, 67083 Strasbourg Cedex, France Received January 3, 1994. I n Final Form: March 7,1994” Rheological experiments were carried out on aqueous micellar solutions of the “dimeric”surfactant ethanediyl-a,w-bis(dodecyldimethy1a”onium bromide) as a function of surfactant volume fraction and temperature. The aim was to investigatethe effect of electrostaticinteractionson the micellar size through the linear viscoelastic properties of the system. The zero-shear viscosity results indicate a sharp crossover from a dilute regime in which the micelle growth is very weak to a regime of rapid growth, in agreement with the theoretical predictions. However, upon increasing further the volume fraction, a maximum of the zero-shear viscosity is observed which suggests that the micellar length also goes through a maximum. This is confirmed by the analysis of the viscoelastic spectra on the basis of the recently developed models taking into account both reptation and Rouse processes. The decrease of micellar length observed in the high volume fraction range has been interpreted as due to the decrease of the effective end-cap energy associated with the theoreticallypredicted increase of the micelle ionization degree with volume fraction. The effect of the electrostatic interactions also appears in the behavior of the plateau modulus that exhibits a larger volume fraction dependence than in highly screened micelles. NaC103, respectively, even though they form extremely long wormlike micelles, exhibit an amazingly high fluidity Recent studies of elongated micelles of ionic surfactant and also a weak dependence of the viscosity on 9. Such in water show that in some systems the amphiphiles a behavior cannot be accounted for by the coupled assemble reversibly into flexible wormlike micelles that reptation-reaction model developed by Cates,lJoJ1and it can form a viscoelastic fluid reminiscent of an entangled was suggested that it may be due to the formation of polymer solution.l12 Such micelles tend to grow with intermicellar cross-links,leading to a solution of entangled increasing sdrfactant and/or electrolyte concentrations. branched micelles or even to a multiconnected network. Most studies have focused on highly screened micelles. A recent model of reptation of branched micelles underFor such systems, the equilibrium between different going reversible scissionshows that the viscosity is reduced micelle lengths L results in a broad distribution of sizes. with respect to that of entangled linear micelles.12 ConThe average micellar length L grows with increasing trary to the case of ordinary branched polymers where the surfactant volume fraction CP. Thisgrowth is characterized branch points have fixed positions on the polymer by a simple power law increase with exponent 1/2,and for backbone, here the connections can move along the some systems the scaling of micellar lifetime,3vis~osity,~ cylindrical part of the micelles, which speeds up the and self-diffusions with CP was found to be in accord with diffusion of the micelle within the tube. theoretical predictions based on this growth law. In the opposite limit where the amount of added salt is insufficient to screen out the electrostatic interactions, However, significant deviations from this picture were the variation of the viscosity of semidilute micellar observed for systems in both limits of very low and very solutions with surfactant volume fraction was found to be high salt concentrations. In particular, it was shown that larger than that of neutral ~ystems.~J3 This was intermicellar solutions of hexadecyltrimethylammoniumbropreted as due to an abrupt increase of micellar size resulting mide (CTABP and hexadecylpyridinium chlorate from electrostatic interactions. Light scattering studies (CPC103)7-9 in the presence of an excess of KBr and supported a rapid micellar growth at low salt content.14 In this paper we present an experimental study of the t Universit.6 Louis Paeteur. linear viscoelastic properties of micellar solutions of t Institut Charles Sadron. *Abstract published in Advance ACS Abstracts, May 1, 1994. ethanediyl-a,w-bis(dodecyldimethy1ammoniumbromide), (1) See for instance: Cates, M. E.; Candau, S. J. J . Phys.: Condens. referred to as 12-2-12, in pure water. This “dimeric” or Matter 1990, 2, 6869 and references therein. “gemini” surfactant is the homologue in the series of (2) See for instance: Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, alkanediyl-a,w-bis(dodecyldimethy1ammoniumbromide) 933 and references therein. (3) Candau, S. J.; Merikhi, F.; Waton, G.;Lemarhhal, P.J. Phys. with the shortest alkanediyl spacer group. In the as(Paris) 1990, 51, 977. semblies formed by this surfactant and its homologues (4) Candau, S.J.; Hirsch, E.; Zana, R.; Delsanti, M. Langmuir 1989, with a longer, but not too long, spacer, the arrangement 5, 1525. (5) Mesaager,R.;Ott,A.;Chatenay,D.;Urbach,W.;Langevin,D.Phys. of the charged groups at the interface between water and
Introduction
Rev. Lett. 1988, 60, 1410. (6) Khatory, A.; Lequeux, F.; Kern, F.; Candau, S. J. Langmuir 1993, 9, 1456. (7) Appell, J.; Porte,G.; Khatory, A.; Kern, F.; Candau, S. J. J. Phys. I r i992,z, 1045. (8) Khatory, A.; Kern, F.;Lequeux, F.; Appell, J.; Porte,G.;Morie, N.; Ott, A.; Urbach, W. Langmuir 1993, 9, 933. (9) Candau, S.J.; Khatory, A.;Lequeux, F.;Kern, F.J. Phys.IV1993, 3, 197.
(IO)Cates, M. E. J. Phys. (Paris) 1988,49, 1593. (11) Cates, M. E.Macromolecules 1987,20, 2289. (12) Lequeux, F. Europhys. Lett 1992,19(8), 675. (13) Candau, S.J.;Hirsch, E.;Zana, R.; Adam, M. J. CoZloidInterface Sci. 1988., 122. ---,430. ~ - (I& Delsanti, M.; Moussaid, A.; Munch, J. P. J . Colloid Interface Sci. 1993,157, 285.
0743-7463/94/2410-1714$04.50/00 1994 American Chemical Society
Salt-Free Viscoelastic Micellar Solutions the assembly hydrophobic core is very different from that with conventional surfactants (one charged group-one alkyl chain). In the latter the charged groups are randomly distributed, with an average distance ranging between 7 and 10 A, as indicated by the values of the surface areas per head group. In the dimeric surfactants, the spontaneous arrangement of the head groups is strongly perturbed by the linking of the head groups two by two. The distribution of distances between head groups can be bimodal for short spacers with a peak at the "chemical distance" determined by the carbon number of the alkanediyl spacer and another peak, in the 7-10-A range for the thermodynamic equilibrium distance. This fact can strongly affect the self-assemblyof dimeric surfactants with respect to conventionalones. Also, the end-capenergy in these systems can be rather large due to the bulkiness of the twin hydrophobic tails that pack more easily in the cylindrical part of the micelles than in the hemispherical end caps. Indeed, a cryotransmissionelectron microscopy study showed that the 12-2-12 surfactant and some of its homologues assemble into wormlike micelles at fairly low O even in the absence of salt, contrary to the corresponding dodecyltrimethylammoniumbromide which can be considered as the monomer of 12-2-12.15 Various physicochemical aspects of the solutions of dimeric surfactants have been recently studied.15 In the present study we turn to the rheological aspects of the aqueous solutions of 12-2-12. These solutions visually show shear-thickening at volume fractions as low as 1% and are viscoelastic at a volume fraction of 4%. This behavior, already reported by Rehage and Hoffman2for other surfactant systems, is the signature of a shearinduced transition. A thorough investigation of this phenomenon is underway. This paper deals only with linear viscoelasticity. The experimental viscoelastic spectra have been determined in salt-free aqueous solutions and analyzed on the basis of the recently developed reptation models.18-18 The results are compared to the predictions of the theory of Mackintosh et al. relative to the micellar growth in the presence of electrostatic interactions.19*20 Theoretical Background (1) Micellar Growth. In a mean field approach the total free energy F per unit volume of micellar solution may be written as1
where k g is the Boltzmann constant, T the temperature, and p~ the number of micelles composed of Nsurfactants per unit volume. The first term in eq 1corresponds to the entropy of mixing, and ENis the energy of scission of the micelle. For neutral micelles, the scission energy is equal to the end-cap energy E,, energy required to create two end caps where there were none before. The minimization of eq 1with respect to P N with the constraint of constant (15) (a) Zana, R;Talmon,Y.Nature 1993,362,228. h a , R.;Talmon, Y. Manuscript in preparation. (b) h a , R.; Benrraou, M.; Rueff, P. Langmuir 1991, 7, 1072. (c) Alami, E.;Beinert, G.; Marie, P.; Zana, R. Langmuir 1993,9,1465. (d) Alami, E.;Levy, H., h a , R.; Skoulioe, A. Langmuir 1993, 9,940. (16) Catea, M. E.;Turner, M. Europhys. Lett. 1 9 W , l l , 681. (17) Turner, M.; Cabs, M. E. J . Phys. (Paris) 1990,51,307. (18)Granek, R.; Catea, M. E. J. Chem. Phys. 1992, W,4768. (19) Safran. S.:Pmcus..P.:.Catea, M. E.:Mackintosh, F. J.Phvs. (Paris) 1990,5l, 503.. . (20) Mackintosh, F.;Safran, S.; Pincus, P. Europhys. Lett. 1990,12,
697.
Langmuir, Vol. 10, No. 6,1994 1715 volume fraction leads to a broad distribution of micellar sizes and to a mean micelle aggregation number N that increases as the square root of the surfactant volume fraction O according to
For charged micelles the energy of scission is composed of the repulsive energy of the surface charges that favors the breaking of micelles and of the end-cap energy that favors micellar growth. In addition to these two effects, the micellar growth is also modified by the increased entropy of the counterions near the end caps. This leads to a complicated behavior where one can distinguish three regimes.19920 (i) A dilute regime characterizedby a Debye length larger than the mean micelle size. In this regime the micelles are practically monodisperse, and their mean aggregation number increases very slowly with O according to
fl = (1/lBaU2)(EdkBT+ lOg(@/m)
(3)
Ig is the Bjerrum length, a the radius of the cylindrical micelle, and Y an effective charge per unit length. (ii)A semidilute regime characterized by a Debye length smaller than the mean micelle size. As in the case of neutral micelles, the distribution of micellar sizes is wide, but the effective scission energy is reduced by the repulsion of the surface charges, leading to a mean aggregation number given by
N
20'/2 exp[(l/P)(EJk~T- 1~aU~/'@"~)](4)
The growth rate depends on both Ec and the micelle ionization degree CY that is related to the Debye screening length K - ~by
where p is the number density of polar heads. The growth is rapid and cannot be represented by a power law. (iii) A concentrated regime. This regime corresponds to the case where the dominant electrostatic contribution is that of the entropy of the counterions near the end caps. The owth may be characterized by an effective power where A is related to the net charge of law f a 01/2(1+A) an end cap and depends only weakly (logarithmically)on O. The crossover between regimes i and ii is expected to be rather sharp and to occur in the absence of salt at the overlap volume fraction O* that, for cylinders, is given by
a* N ( k B ~ 1 B ~ ~ 2 / ~ , ) 2
(6)
On the contrary, the transition between regimes ii and iii is rather smooth. To illustrate the growth behavior, we have reported in Figure 1 a typical O dependence of the length of charged micelles as calculated by Mackintosh et al.9 It must be noted that, for this calculation, the value taken for u is about 10 times smaller than the value calculated for a semidilute solution of cylinders. This point is discussed below. The purpose of Figure 1is to show the signature of the electrostatic effects, i.e., the emergence of a sharp between a crossover at the overlap volume fraction 0*, regime of weak micellar growth and a regime of rapid growth.
Kern et al.
1716 Langmuir, Vol. 10, No. 6,1994
lo
i 00000 G‘
ooooo
G“
Figure 1. Micellar length versus surfactant volume fraction as calculatedfrom the minimizationof eq 1 with the constraint of constant The parameters used for the calculations are E, = 20 ~ B Ta ,= 20 A, and u 2 1 ~= 0.05 (from ref 20).
(2) Stress Relaxation. The main features of the theoretical models describing the dynamical properties of wormlike micelles can be found in refs 1,10, 11,and 1 6 18. Here, we simply recall the theoretical results needed to discuss our experiments, more specifically regarding the shape of the stress relaxation and the effect of surfactant concentration. Typical frequency dependences of the storage modulus G’(w)and of theloss modulus G”(o) and the corresponding Cole-Cole plot G”(G’) taken from the theoretical work of Granek and CatesI8 are represented in Figure 2. At low frequencies the behavior is Maxwellian as ascertained by the semicircular shape of the Cole-Cole plots. A deviation from the half-circle occurs at a circular frequency w of the order of the inverse of the breaking time Tbr& of the micelles. It is generally assumed that the chemical relaxation process is the reversible unimolecular scission, characterized by a temperature-dependent rate constant kl per unit time and per unit arc length, which is the same for all elongated micelles and is independent of time and of volume fraction. This assumption results in
In the Cole-Cole representation, the departure from the Maxwellian behavior is characterized by a linear dependence of G” on G’with a slope of -1.l6J7 This is followed by a regime in which micelle diffusion between scission events is dominated by breathing modes and where G” decays faster (with a slope of -2.4).18 Eventually, there is an upturn associated with the occurrence of the Rouse modes, thus creating a clear dip in the Cole-Cole plot. It was found that, provided Tbre& >> T , (where T , is the Rouse time of an entanglement length le), the value of G” at the dip obeys G”,,/G‘,
= AlJL
(8)
where G’, is the plateau modulus and A is a constant of the order of 1. Thus, the frequency dependence of the complex shear modulus can be described with two parameters: the ratios = T b r e d T r e p and Ze/L; T , , ~ is the reptation time of the micelle of mean length. However, it must be noted that the calculationsof Granek and Cates involve an adjustable parameter, namely, the amplitude of the tube length fluctuations.
Figure 2. Viscoelasticbehavior of a solution with @ = 0.078 at 20 OC: (a,top) frequency dependenceof the normalized complex shear modulus, (b, bottom) Cole-Cole representation. The dashed lines represent best-fitting calculated curves. The parameters of the fits are f = 0.1 and l d t = 0.03.
The model of Cates, derived for linear flexible micelles,’ was recently extended to the case of branched flexible micelles.6J2 It was shown that the general features of the stress relaxation and more specifically eqs 7 and 8 ars maintained, provided one replaces the average length L of the micelle by a new length L, defined as n2 L, = - n, + 2n3 where 1, is the micelle persistence length, nl the concentration of end caps, n2 the number density of persistence lengths, proportional to the surfactant volume fraction 9, and n3 the concentration of 3-fold connections. For linear micelles1 n3 = 0 and L, = L a @Iz. For saturated networks, i.e., networks for which the arc length between cross-links is equal to _the correlation length? nl = 0 and L, = E, 0: @-It2, where L,is the average length of the strand of the network. In the general case of the coexistence of end caps and connectionsthe variation of L,with the surfactant concentration is in between those of L and L,. The dynamic relaxation of stiff rodlike micelles has also been studied.21 Such structures can be encountered in low-salinity aqueous surfactant solutions when polyelectrolyte effects cause an increase in stiffness. For such systems, one expects the shape of the stress relaxation to (21)Cates, M.E.;Marques, C.; Boucheud,J.P. J . Chem. Phys. 1991, 94.8629.
Langmuir, Vol. 10, No. 6, 1994 1717
Salt-Free Viscoelastic Micellar Solutions
be qualitatively similar to that for flexible micelles. Only the scaling behavior to dilution of some rheological parameters, discussed below, is modified because of both the stiffness of the micelles and the effect of electrostatic interactions on micellar growth. (3) Scaling Behavior to Dilution. We consider first the case of flexible micelles. In the fast-breaking limit (Tbreak/Treep 0.095 to the entire viscoelastic spectrum, and therefore we cannot determine Gtm,' T b d , and 1. However, Figure 14 shows that the temperature dependence of the zero-shear viscosity becomes Arrhenian in this domain. The activation energy E,, found for the viscosity is -32 kBT, in good agreement with the values obtained for other micellar systems in the presence of salt.* This suggests that the temperature dependence of E in this concentration range becomes also similar to that of highly screened micelles. The possible causes for the high values of E, inferred from those of E remain to be identified. Such high values can be expected owing to the dimeric nature of the surfactant. Whereas the 'doublets" of head groups can easily accommodate in the cylindrical part of the micelles,for instance parallel to the axis, their arrangement in the hemispherical caps involves necessarily the formation of topological defects that cost energy. One must also take into account the bulkiness of the tails,which also increases the end-cap energy. In the absence of salt, the electrostatic interactions lower the _effectiveend-cap energy, leading to reasonable values of L. However, if the interactions are screened by addition of a salt having the same counterion as the surfactant, the effective end-cap energy becomes excessively high and would lead to unrealistic values of the micellar length. This brings us to the case for saturated micellar networks (systems with very few hemispherical end caps) where the formation of cross-links between threadlike micelles is predi~ted.6*~*~ In fact, preliminary experiments showed that a 4% 12-2-12 solution separated into a gel phase and a fluid phase of comparable volume in the presence of KBr at concentrations below 0.1 M. Such a behavior is indeed predicted for saturated micellar netw0rks.3~ It must be noted that the kinetic properties of the micelles are also sensitive to the electrostatic effect. This can be ascertained from the behavior of the rate constant kl = (rbre&L)-'. The variation of such a parameter with (35)Cabs,M.E.;Drye, T.J. J . Chem. Phys. 1991,96, 1367.
Salt-Free Viscoelastic Micellar Solutiom
Langmuir, Vol. 10, No. 6,1994 1723
10 ’
10 3 : 1
1 102;
I 6
8
0.1
Figure 15. Variations of the product q , d L with at different temperatures. The lines are guides for the eyes.
the surfactant volume fraction is shown in Figure 15. One observes an increase of kl upon increasing 9 in the range where L is found to decrease. In the presence of salt, kl was found to be independent of
Conclusions The results reported in this paper provide strong support
to the model of micellar growth in the presence of electrostatic interactions, developed by Mackintosh et al.
In particular, the earlier observation by Hoffmann et of a very sharp transition between two regimes for the rheological properties is confirmed. The temperature dependence of the crossover volume fraction agrees with the theoretical prediction. The very rapid micellar growth in the semidilute range is ascertained by the behavior of the zero-shear viscosity. However,the regime of rapid micellar growth is followed by a rather complex behavior characterized by a maximum of the zero-shear viscosity and a minimum of the ratio G”,iJG’,. The experimental results suggest that the micellar length goes through a maximum when the volume fraction is varied. This behavior is explained by an increase of the effective micelle ionization degree with surfactant predicted theoretically. In this concentration at 9 > 9*, regime, the shear modulus is found to increase with 9 faster than predicted for flexible chains. We attribute such a behavior to the effect of orientational correlations due to electrostatic interactions, which reduces the shear modulus to an extent varying with 9. Several studies are under way in order to check these different conjectures: (i) effect of the addition of salt to the rheological and structural properties, (ii) nonlinear viscoelasticity, and (iii) rheooptical behavior. These experiments should allow us in particular to give deeper insight into the role of the orientational correlations in the structure and the rheology of these systems.
Acknowledgment. We wish to thank P. Pincus for stimulating discussions.
