Network properties of semidilute aqueous KBr solutions of cetyltrimethylammonium bromide

Network Properties of Semidilute Aqueous KBr Solutions of Cetyltrimethylammonium Bromide S. J. CANDAU,* E. H I R S C H , * : R. Z A N A , t AND M. ADAM~: *Laboratoire de Spectromdtrie et d'Imagerie Ultrasonores, Unitd Associ#e au CNRS, Universitd Louis Pasteur, 4, rue Blaise Pascal, 67070 Strasbourg Cedex, France," t lnstitut Charles Sadron and Greco Microemulsion (CRM-EAHP), CNRS/ULP, 6, rue Boussingault, 67083 Strasbourg Cedex, France, and ~Laboratoire Ldon Brillouin, CEN, Saclay, 91191 Gif-sur- Yvette Cedex, France

Received March 25, 1987; accepted May 13, 1987 Elastic and quasi-elastic light scattering and rheological (shear viscosity,viscoelasticrelaxation time) measurements have been performed on cetyltrimethylammoniumbromide (CTAB) aqueous solutions, in the presenceof 0.1 or 0.25 M K.Br, at surfactant concentrationswhere the micellesare very elongated, and between 20 and 60°C. The results provide additional evidence for the previouslynoted analogy of behavior between concentrated solutions of very elongated micellesand solutions of polymer chains in the semidilute range where the chains are entangled. The results confirm that the micelles keep growing when the surfactant concentration exceeds the cross-overconcentration where the micellar domains overlap. The power laws governingthe rheologicalbehavior of semidilute polymer solutions all apply to the micellarsystemsinvestigatedwhen the micellesare sufficientlyelongated. This appears to be the case for CTABsolutionsat concentrationsabove0.3 M, in the presenceof 0.1 MKBr or more, and temperatures below about 35°C. © 1988 Academic Press, Inc. INTRODUCTION It is now well established that the micelles present in aqueous solutions of surfactants tend to grow at high ionic strengths (1-9). This effect is enhanced by an increase in surfactant concentration which leads to very elongated (rod-like) aggregates, for most surfactants with a single linear alkyl chain (10, 12). Simple thermodynamic treatments for the sphere to rod transition in nonionic systems have been proposed by Mukerjee (13), Tausk and Overbeek (14), and Israelachvili et al. (15). A similar approach was used by Benedek et al. (l 6-18) to describe the growth of ionic micelles upon increasing surfactant concentration and/ or ionic strength. The numerous studies reported on these systems have assumed rigid rod-like cylindrical micelles, semiflexible cylindrical micelles, or 1permanent address: Laboratoire des Sciences de l'Imageet Ttltdttection, ENSPS, Universit6LouisPasteur, 7, rue de l'Universitt, 67000 Strasbourg, France. 0021-9797/88 $3.00 Copyright @ 1988 by Academic Press, Inc. All rights of reproduction in any form reserved.

highly flexible micelles (1-12). The magnetic birefringence and light scattering studies of Appell et al. (7-9) gave strong support to a model of very flexible micelles in dilute aqueous solutions of cetylpyridinium bromide at high NaBr concentrations. They also showed that there exist similarities in the behavior of solutions of such giant unidimensional micelles and of polymer solutions. The existence of polymer-like phases associated with the formation of long flexible cylinders has been predicted in a recent report of Safran et al. (19). The strongest evidence in favor of very flexible micelles was obtained in studies performed in the semidilute regime, i.e., at surfactant concentrations large enough that the elongated micelles overlap, forming a transient network. In this regime, the systems exhibit a viscoelastic behavior very reminiscent of that of transient polymeric networks (l l, 12, 20-23). A thorough characterization of these systems requires two sorts of experimental approaches.

430 Journal of Colloid and InteOraceScience, Vol. 122, No. 2, April 1988

431

SEMIDILUTE SOLUTIONS OF CTAB

The first aims at probing those properties of the transient network which depend only on the correlation length ~. This length can be visualized as the mesh size of the network of overlapping cylindrical micelles. The network properties can be characterized through the measurement of the intensity of scattered light Is and the cooperative diffusion coefficient De, which are both associated with collective modes of the network, and the high-frequency elastic moduli (measured at a time scale much shorter than the lifetime of the network). All these parameters are insensitive to the overall length of the micelles and depend mostly on ~. In previous studies (11, 12) we have shown that Is and D¢ obey power laws of the surfactant volume fraction with exponents close to those predicted for semidilute polymer solutions. However, these experiments cannot provide information on the evolution of the micellar size in the semidilute region. The second experimental approach deals with rheological properties. The reptation model, proposed by de Gennes (24) to describe the viscoelastic properties of semidilute polymer solutions, predicts a very strong dependence of the zero shear viscosity ~s and of the longest viscoelastic relaxation time TR on the polymer volume fraction and molecular weight. However, de Gennes' model applies to micellar systems only in the limit where the reptation time is shorter than the average time rBr taken by a micelle to break in two parts (25). Previous viscosity measurements for aqueous solutions of cetyltrimethylammonium bromide (CTAB) in the presence of 0.1 M KBr have shown that in a limited range of (high • values) the micellar growth induced by an increase in the surfactant volume fraction extends beyond the overlap volume fraction ~*. Our measurements however were of limited accuracy. In this paper we report measurements of ns and TR for aqueous CTAB solutions in the presence of 0.1 M KBr, for concentrations ranging from 0.3 to 0.8 M and at temperatures from 20 to 50°C, by means of a magnetorheometer. This apparatus allowed viscosity

measurements at a much lower shear rate than that in our previous study (26), thus leading to more accurate values ofns. The temperature dependence of ns and TR reveals a large decrease in miceUe size (length) upon increasing temperature, in good agreement with previous light scattering results obtained in the dilute concentration range (3, 7-9). Both the shear elastic modulus G calculated from a = •s/TR

[1]

and the hydrodynamic correlation length ~r~ obtained from quasi-elastic light scattering measurements have been found to exhibit a fairly weak dependence on temperature. Such behavior is consistent with a model of network of entangled flexible micelles for the investigated micellar solutions. THEORY

In this section we briefly review the main predictions of theories dealing with miceUar growth and micellar flexibility and we extend the analogy between giant micelles and polymers. The growth of micelles has been described by the so-called ladder model (16, 17) which is based on micelle formation through multiple chemical equilibria. This treatment was recently extended by Blankshtein et al. (18) who proposed a form of the free energy of micelle formation still based on multiple chemical equilibria but which includes, in addition to the usual contributions, a term for the entropy of mixing of the elongated micelles and the water molecules, and a term accounting for intermicellar interactions. This last contribution was often neglected in previous calculations. This new treatment permits the calculation of the micelle size distribution curve, and the prediction of the conditions where phase separation takes place. The predictions concerning the changes of micelle size depend on the choice of the mathematical form of the various contributions to the free energy and more specifically of the entropy of mixing which should reflect the size, shape, and flexibility of the micelles. For instance, the authors Journal of Colloid and Interface Science, Vol. 122,No. 2, April 1988

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CANDAU ET AL.

showed that a form similar to that used in regular solution theory leads to a ~I,°5 dependence of the weight average micellar molecular weight, whereas a • dependence is obtained by using a Flory-Huggins model. Note, however, that the micellar flexibility is not explicitly considered in this treatment. In this regard, a simple microscopic model for the elasticity of the interfacial film separating oil and water in microemulsions has been recently proposed by Safran et aL (19). Although this work specifically dealt with microemulsions its conclusions are likely to hold for micellar systems. A calculation of the curvature energy of the interfacial film predicts the existence of cylindrical structures in a region of the phase diagram. For these micelles, thermal fluctuations determine a temperaturedependent persistence length ~c below which the cylinders are rigid and above which the cylinder axis wanders randomly in the solution. This random wandering suggests a polymer-like description of the long cylindrical micelles. However, an essential difference exists between real polymer chains and polymerlike micelles, namely the degree of polymerization of the polymer chain is fixed by the polymerization process, whereas the number N of persistence lengths for such polymer-like micelles depends on both the temperature and the micelle volume fraction. It is obtained selfconsistently by minimizing the total free energy of the micelle with respect to N. Taking a Flory-Huggins form for the entropic contribution, Safran et al. (19) obtained N=~exp

+ 1 ,

[21

where K' is the energy difference per surfactant between the hemispherical endcap and the cylindrical region ofa micelle. In a more detailed treatment which takes into account the micelle size polydispersity these authors obtained o5

fl

Quasi-elastic light scattering experiments in dilute solutions of CTAB in the presence of Journal of Colloid and Interface Science, Vol. 122,No. 2, April 1988

0. I M KBr have shown a rather large polydispersity of the cylindrical micelles (11, 12). Moreover, there is some experimental evidence o f a ~0.5 dependence of the aggregation number (18, 27). Therefore in what follows we assume Eq. [3] to be valid as a basis for discussion. The major difficulty encountered in the experimental investigation of micellar growth in the dilute range arises from the fact that the changes of measured parameters with • depend on the micelle size and polydispersity and also on intermicellar interactions. However, we previously pointed out (26) that the results pertaining to the semidilute range, where the flexible micelles overlap, may be easier to interpret than those in the dilute range provided that micellar growth is not hindered by micellar overlapping (11, 12). Characteristic behavior can be expected for the viscoelastic parameters of semidilute micellar solutions if these systems do behave like semidilute polymer solutions, that is, if the micelles retain the same size during a time equal to or longer than the longest viscoelastic relaxation time. For instance, the theory of de Gennes (24) for the viscosity of semidilute polymer solutions in good solvents predicts the following scaling laws for the zero shear viscosity and the longest viscoelastic relaxation time: ~r T/s/r/0 OC N 3 ~ 3"92 [4] =

TR OC N301"6,

[5]

where nr represents the relative zero shear viscosity and n0 the viscosity of the solvent. For micellar solutions, Nwould then represent the number of persistence lengths per micelle (see above). As already pointed out, the distinctive feature of micellar systems with respect to polymer solutions is that for the former N is a function of both ¢ and T through Eq. [3], whereas it is constant for the latter. As a result, one can expect a very large increase in ~/r and TR upon increasing ,I, and/or decreasing temperature T. The case where TR is longer than the average time rBr taken by a micelle to break into two

SEMIDILUTE SOLUTIONS OF CTAB

parts has been considered in the very recent theoretical study of Cates (25). This work deals with the dynamics of the stress relaxation through reptation in a system of living polymers, that is, chain polymers that can break (fragmentation) and recombine (coagulation) on the experimental time scale, just like the investigated elongated micelles. In the regime TR "> rBr the equations given by Cates lead to n OC ~I~3'6

[6]

TR pc (I~1"27

[7]

provided one assumes that the micellar growth obeys Eq. [3]. An interesting feature of this regime is that the stress relaxation function should be a nearly pure exponential decay. Turning now to the network properties, they should be, as noted above, nearly independent of N and T, to the first approximation. Indeed, they depend only on the correlation length which in turn depends only very little on T and varies with cI, according to (28, 29) OC (I)-0'77.

[8]

In the limit of very large micelles the same power law applies to the hydrodynamic correlation length ~H which is related to the cooperative diffusion coefficient D~ according to D~ = kBT/61rno~H.

[9]

The elastic shear modulus G given by Eq. [1] is proportional to ~-3. It therefore varies with ~ according to (30) G ~ ,/~2.31

[10]

and should be nearly independent of N and T. MATERIALS AND METHODS

The sample of CTAB was the same as that in previous investigations (11, 12, 26). Most of the measurements were performed in H200.1 M KBr. Some quasi-elastic light scattering measurements in H20-0.25 M KBr are also presented. Elastic and quasi-elastic light scattering experiments were performed in the temperature

433

range from 30 to 60°C, using the same apparatus as in previous studies (11, 12). The measurements involved the intensity Is and the autocorrelation function of the intensity of the light scattered by the micellar solutions. This function was analyzed by means of the cumulant method (31, 32) to yield the variance v of the function and the first reduced cumulant (I')/2K 2, where ( P ) is the average decay rate of the autocorrelation function and K is the magnitude of the scattering wave vector, K = [4rn sin(~/2)]p,, [ 11 ] where Ois the scattering angle, n the refractive index of the scattering medium, and 9~ the wavelength of the incident light in vacuo. The extrapolation of (I')/2K 2 to K = 0 yields the value of the mutual diffusion constant D of the micelle in the dilute range and of the cooperative diffusion constant Dc in the semidilute range. The variance v yields information on the polydispersity of micelle size in the limit of low surfactant concentration. In general nearly spherical miceUes are characterized by low v values (<0.03) and elongated micelles by higher v values (>0.1) (11, 12, 16). The zero shear viscosity ~s and the longest relaxation time TR were measured using the magnetorheometer described elsewhere (30). In this apparatus a magnetic sphere immersed in the sample experiences a magnetic force created by the electrical current in a metal wire surrounding the sample cell. The current intensity i which is monitored by a feedback amplifier is such that it maintains the sphere at a fixed position. When the sample cell is displaced by a stepmotor at a speed re, the viscous force which exerts on the sphere is balanced by the magnetic force so that i - io ~- ~rv¢,

[12]

where i0 is the current intensity needed to balance the gravitational force acting on the sphere, and r is the sphere radius. All experiments were performed at a reduced shear rate v~ T R / r < 10-2. The absolute value of the sample viscosity was obtained by Journal of Colloid and Interface Science, Vol. 122,No. 2, April 1988

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CANDAU ET AL.

using a calibration curve established from measurements on a standard silicone oil. The accuracy in the determination of n~ depends on the sample viscosity but was always better than 2%. The TR measurements were made by monitoring the change o f intensity after stopping the motor which actuates the cell. The resulting recorded i vs time t curve was then fitted to an exponential lineshape

i = A exp(-t/T~) + B,

[13]

as is illustrated in Fig. 1. It should be stressed that for all of the systems investigated the intensity data could be always fitted with good accuracy to a single exponential curve. Hoffmann et aL (20b) have reported a single relaxation time for the flow birefringence behavior of systems of elongated micelles. The measurements were generally performed in the temperature range between 30 and 60°C, that is above the Krafft temperature TK of the surfactant solutions. However, some viscosity measurements were performed down to 20°C, (below TK), that is on supercooled solutions. Indeed we noted that the solutions then retained clarity at such temperatures for several hours. This allowed the viscosity measurements to be reliably and reproducibly performed. Note that critical micellization concentration (CMC) determinations and chem-

ical relaxation studies have been performed on supercooled dilute aqueous solutions of sodium dodecyl sulfate and showed no adverse effect (33). The measured parameters showed no discontinuity when the solution was supercooled below TI~. Had any change of the micellar structure taken place at TK it would have certainly shown on the measured value of the chemical relaxation time which is dramatically sensitive to such changes (33). The surfactant volume fraction • was calculated from its molar concentration C as done previously (12). RESULTS

1. Quasi-elastic Light Scattering

Figure 2 shows the variation of the first reduced cumulant ( F } / K 2 with the surfactant volume fraction • at different temperatures for CTAB solutions in 0.25 M K B r . As in previous studies (1 l, 12) the minimum of the various curves in Fig. 2 is interpreted as the crossover between the dilute and semidilute ranges• In the dilute range micellar growth leads to a decrease in ( F ) / K 2 upon increasing • . Opposite behavior is observed in the semidilute range due to the formation of a transient network of entangled micelles. The mesh size of the network decreases upon increasing ¢. As this size is related to the correlation length this • results in a decrease in ( F ) / K 2 upon increasing ,I~. As to be expected, the volume i fraction O*, where the ( F ) / K 2 vs ¢ plots go through a minimum, shifts to higher values as T increases. Indeed increasing T always results in a decrease in the size of ionic surfactant micelles (16, 22). Note that this effect is corl. rectly predicted by Eq. [3] which relates the number N of persistence lengths to the free energy difference K'. This quantity is assumed • O;-ooo ~°oo. ; to be temperature independent, in a first aptime proximation. FIG. 1. Exampleof determination of T~: fitting of the In the semidilute range, ( F } / K 2 becomes intensity data to Eq. [13] for a 0.84 MCTAB solution in independent of K a n d provides a measurement 0.1 M KBr at 23°C. The inset showsthe result on a much longer time scale (fourfold).The two plots are well fitted of the cooperative diffusion coefficient De. The plots o f ( r ) / K 2 of Fig. 2 show linear behavior, to a single exponentialcurve, with the same TRvalue.

Journal o f Colloid and Interface Science, Vol. 122, No. 2, April 1988

SEMIDILUTE SOLUTIONS OF CTAB

l I07
I

0.01

0.02 0.103 0.05

I

0.1

FIG. 2. Variation of the first reduced cumulant with the CTAB concentration at varying temperatures, in H200.25 M KBr (scattering angle 0 = 90°).

as to be expected from power laws. However, even if the network structure exists at • > ~* for the whole range of T investigated the correct value of the exponent, that is, -0.77 (see Eq. [5]), is obtained only at low T: below 30°C in H20-0.1 M KBr solutions and below 40°C in H20-0.25 M KBr solutions. It is probably below these temperatures that the micelles become long enough to obey the limiting power laws valid for infinite chains. The values of the hydrodynamic correlation length ~r~ calculated from Dc using Eq. [9] are plotted as a function of T in Fig. 3 for the 0.1 and 0.25 M KBr solutions, at the surfactant volume fraction 0.18 that is a CTAB concentration of about 0.5 M. The plots show that ~H is independent of T at low T and then decreases slightly as T is increased. This result indicates that at low T and large enough ~ the network formed by the elongated miceUes retains approximately the same mesh size, even though the length of the individual micelles decreases very much when Tincreases, as will clearly appear from the rheological measurements below.

435

tration C in a semilogarithmic representation. In order to check whether TR follows a power law with e~, the results of Fig. 4 have been plotted in Fig. 5 as log TR vs log • at T = 25, 30, and 35°C. The results are consistent with the predictions of the reptation model (i.e., TR oc ~3.~) although the plots seem to exhibit an upward curvature. Contrary to log TR, Fig. 6 shows that log ~Tr does not increase linearly with 1/Tat variance with what is expected on the basis of Eqs. [3] and [4]. In the low T range, however, the plots are nearly linear. The variations of log n, with log • at T = 25, 30, and 35°C are plotted in Fig. 7. The results can be fitted within the experimental error to power laws with exponents equal to 4.6, 5, and 5.4, respectively, that is, values rather close to the predictions of the simple reptation model (24) 7h oc ~s42

[ 14]

obtained by inserting N oc ffo.5in Eq. [4]. Finally the shear elastic modulus G has been calculated using Eq. [1 ] and plotted against T at constant CTAB volume fraction in Fig, 8 and against • at constant T in Fig. 9 using semilogarithmic and log-log representations, respectively. It is seen that G varies only slightly with T, as compared to 7/r and TR. On the other hand the log G vs log • plots at low T are linear and obey Eq. [ 10] within the ex-

60 o

50 40 30

"~

20 10

2. Rheological Measurements Figure 4 shows the variations of TR with 1/ T for CTAB solutions of increasing concen-

L

I

t

30

z~O

50

T/°Cl

60

lOG. 3. Temperature dependence of the hydrodynamic correlation length for CTAB solutions in H20-0.1 M KBr (e) and H20-0.25 M KBr (©) at ~, = 0.18. Journal of Colloid and Interface Science, Vol. 122, No. 2, April 1988

436

CANDAU ET AL. Ta(s}

i

i

I

I.C 0.5 0.3 0.2 0.1 0.05 0.03

0.02

O.Ol 3.1

T I

I

I

3.2

3.3

3.4-

3.5

FIG.4. Semilogarithmicvariationof the viscoelasticrelaxation time TRwith 1/T for CTAB solutions in H200.1 M KBr at 0.42 M (A), 0.56 M (tT), 0.7 M (V), and 0.84 M (O).

shear modulus depends on the number and efficiency of entanglements. In polymer solutions and melts, there is a critical polymer length Are below which elasticity-effective entanglements cannot form, and the material does not exhibit elastic behavior (24). As a matter of fact, it has been shown that in semidilute polymer solutions, the efficiency of the entanglements and, consequently, the shear modulus decrease as the temperature increases (30). For the micellar solutions investigated here a temperature rise produces a random breakup of the micelles as evidenced by light scattering measurements. If the system is polydisperse, as the temperature is increased, more and more micelles become too small to be effective for elasticity, while they still contribute to the osmotic compressibility.

2. Rheological Properties perimental error. Similar behavior was previously observed by Hoffmann et aL (23). Recall that the increase in G with • is due to the increased overlap of the micelles.

The results presented above, relative to the changes of TR and n, with ~, indicate that the micelles keep growing at • > ~* and thus support the conclusion of our previous investigation (26). Recent N M R studies led to the same conclusion (34).

DISCUSSION I I

1. Network Properties

I

i I

I

Ill

T~(s) 2

The observed dependences of D~ and G on confirm the picture of a transient network of overlapped flexible micelles in the semidilute range. As noted above, there are some similarities in the variations of ~r~ and G with T, at constant • . In both instances, the variations are small at low T and become appreciable only at higher T. The change of G with T is larger than that of ~H as expected from the fact that G oc ~-3 (see above). Another factor may be responsible for the comparatively large change of G. Pursuing the analogy with polymeric systems one can note that the cooperative fluctuations probed by light scattering are driven by the osmotic compressibility, mainly controlled by the screening length, while the Journal of CoUoidand InterfaceScience, VoL 122, No. 2, April 1988

I

o/

0.5 0.2 0.1 0.05 0.02

J09 I

0J53

0203 025/, 0305

t

t

0.3 o'.4 o'.s~o'.60.7o'.8'o'.9

elM/,,) FIG. 5. Semilogarithmicvariations of TR with • for CTAB solutions in H2O-0.1 M KBr at 25°C (O), 30°C (~7),and 35°C ([2).As a guideto the eye the straight lines havethe slope3.1 predictedby the simplereptationmodel.

437

SEMIDILUTE SOLUTIONS OF CTAB I

I

105 nr a

I

/

1

I

~

J//

10' 102

///// ]1iIII ,/ 3.1

10_' 3.'z

313 3! TA

FIG. 6. Semilogarithmic variation of the relative zero shear viscosity ~, with l I T for CTAB solutions in H200.1 M KBr at concentrations 0.3 M (O), 0.42 M (V), 0.56 M (©), 0.7 M (A), and 0.84 M (l-q).

The observation of a singly exponential stress relaxation function (see Fig. 1 and Materials and Methods) suggests that in the viscous flow the motion of the micelles is controlled by their breaking time rBr, which would be shorter than the viscous relaxation time TR (25). Recently reported results appear to agree with this conclusion. Thus, micelle fragmentation-coagulation processes (35) have been evidenced in the case of polydisperse micellar solutions and studied by means of chemical relaxation (35, 36) and time-resolved fluorescence probing (37). These processes were found to be fairly rapid, particularly for systems as concentrated as the ones investigated here. The reported data show that the mean time required for the miceUes to break in two parts (reciprocal of the fragmentation rate constant) can be much shorter than the TR values measured in the present investigation (0.1-1 s, see Fig. 4). Hoffmann et al., in several reports, have also been led to conclude that at high surfactant volume fractions, the viscous flow becomes controlled by the kinetic rate (22, 23).

The T dependences of TR and nr are also consistent with the inference that ZBr > TR. In Fig. 4, log TR is seen to increase linearly with 1/T. The straight lines obtained for different CTAB concentrations run nearly parallel, except that for the highest concentration. On the assumption that ~~r < TR, Eqs. [3] and [5] can be used to obtain from the slopes of these lines the value of K'/T. We thus found at T = 25°C, K'/T~- 24-35. These calculations took into account the change of the solvent viscosity with temperature. Figure 6 shows that log 7/r does not vary linearly with liT. This effect can be accounted for by the variation with T of the network structure evidenced by the variation of G. In the lower T range where G varies only slightly, log nr is seen to increase nearly linearly with 1/T. The analysis of the data in that range leads also to values of K'/T of ~_27-28. These values are large, compared to that obtained for the growth in the dilute

[

I I

I

I I

[

Ii

I

~r

10s

O

10~

103 ].109 0.153 &203 0.2540.305

o.3

i

i

0'.s 016

I

£ (Mlz} FIG, 7. Semilogarithmie variation of the relative zero shear viscosity~, with the CTAB concentration for solutions in H20-0.1 MKBr at 25°C (O), 30°C (V), and

35°C

(I-1). The straight lines going through the experimental points have the slope 5.42, predicted by the simple reptation model.

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CANDAU ET AL. i

1000(

5000 30013 2000 1000 500 30O 200 100

2b

36

T(*C',

FIG. 8. Semilogarithmic variation of the shear elastic modulus G with T for CTAB solution in H20-0. l M K B r at 0.42 M(O), 0.56 M(A), 0.7 M ([~), and 0.84 M(O).

range: K'/T - 15-19 in light scattering studies (1). They lead to physically unsound values of N, that is of the micelle aggregation number and length. On the other hand if the results are analyzed on the assumption that "rBr < T R the experimental stress relaxation time is then given by (25) TR.exp -----(rarTR) I/2,

[15]

where TR would be the viscoelastic relaxation time of the chain in the absence of breaking. As rB, is likely to strongly depend on temperature, Eq. [ 15] provides an explanation to the unrealistically large values of the slope of the lines of Figs. 4 and 6. Nevertheless, as noted above, the volume fraction dependences of ~r and TR obey power laws with exponents rather close to the theoretical predictions of the simple reptation model, that is for rBr > TR, and thus appear to be inconsistent with the above. Two explanations can be proposed to try to solve this contradiction.

by the size of the average micelle and not by the full micelle size distribution. This effect, however, would not account for the large T dependence of ~r and TR. (ii) The flow is controlled by the breaking mechanism and the high values of the exponents found from the • dependences ofnr and TR are due to an enhancement of the micellar growth associated with the simultaneous increases in volume fraction and ionic strength upon increasing surfactant concentration. The latter occurs because the micelles investigated are partly ionized and therefore the free counteflon concentration increases with the surfactant concentration. In our study the ionic strength was estimated to increase from about 0.11 M at C = 0.1 M to about 0.18 M at C = 0.8 M(38, 39). This change enhances the micellar growth. While the network properties are only slightly affected (see Fig. 2), the rheological behavior can be dramatically modified. This interpretation receives some support from the results of Hoffmann et al. (23) which clearly show power law behavior for l"/rand TR versus O with exponents of 8 and 5, respectively. Indeed these experiments refer to semidilute micellar solutions of cetylpyridinium salicylate in the presence of sodium chloride, at a mole ratio of 1. Thus, this study involved I

I 11

[

Ill

G

5000, 4000 3000i 2O0O I

50C I w I ' (i) The classical reptation model (with rB~ 03 014 01s '0~6 0.7 01a 0'.9 > TR) applies, but monomer additions at the C(MIx) ends of the micelles lead to motional narrowFIG. 9. Semilogarithmic variation of G with ~ for CTAB ing of the relaxation spectrum. Thus, the vis- solutions in H:O-0.1 M K B r at 25°C (O), 30°C (~7), and coelastic relaxation time would be determined 35°C (D). The straight lines have a slope 2.31.

Journal of Colloid and InterfaceScience, Vol. 122,No, 2, April 1988

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SEMIDILUTE SOLUTIONS OF CTAB a larger increase in ionic strength with the surfactant concentration than ours, and yielded larger exponents. This second explanation appears to be m o r e consistent with the results than the first, but m o r e experiments are needed to check its validity. Also, one c a n n o t discard the possibility that excluded v o l u m e effects modify the growth m e c h a n i s m in the semidilute range. CONCLUSION The results presented in this paper provide confirmatory evidence o f the analogy between semidilute miceUar solutions and entangled p o l y m e r solutions. The measurements o f the cooperative diffusion constant and of the shear m o d u l u s show that at volume fractions larger than the overlap v o l u m e fraction ~*, a transient network o f flexible cylindrical micelles form, whose mesh size decreases u p o n increasing v o l u m e fraction. This overlap o f the micelles does not prevent their growth, as evidenced by the viscosity and relaxation time measurements. The results at the present stage o f the work do not allow one to draw clearcut conclusions concerning the flow regime and the quantitative micellar growth as a function o f surfactant v o l u m e fraction at constant ionic strength. A temperature increase produces a breaking-up o f the micelles together with a loosening o f the network. This could be at the origin o f the departure from the power laws in the high temperature range. It must be added that for the systems investigated the power laws are obeyed only in a rather restricted range o f T (from 25 to 40°C, temperature where the micelles b e c o m e too small) and o f concentration (above 0.3 M). We are attempting to extend this work to other systems where the range o f applicability o f the power law would be wider. ACKNOWLEDGMENTS The authors thank Drs. P. Pincus and S. Safran for stimulating discussions and Dr. M. Cates for communicating his manuscript prior to publication and for many illuminating comments.

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