1466
Langmuir 1993,9, 1456-1464
Linear and Nonlinear Viscoelasticity of Semidilute Solutions of Wormlike Micelles at High Salt Content A. Khatory, F. Lequeux, F. Kern, and S. J. Candau* Laboratoire d’U1trason.s et de Dynamique des Fluides Complexes, Unit6 de Recherche Associ6e au CNRS No. 851, Universitb Louis Pasteur, 4 rue Blaise Pascal, 67070 Strasbourg Cedex, France Received December 10, 1992. In Final Form: March 3, 1993
The frequency-dependentshear modulus of semidilute aqueous solutions of hexadecyltrimethylammonium bromide (CTAB) in the presence of potassium bromide (KBr) has been measured as a function of surfactant and salt concentrations. The comparison between the experimental data and the recent results of a computer simulation provides an estimate of +b&, the time taken for a micelle of the mean length to break. The measured concentrationdependences of rheologicaland kinetic parameters suggest that intermicellar branching occurs at high salt content. An analysis of the data using the recent results of a Poisson renewal model leads to the determination of a characteristic length, identified as the ratio of the total length over the sum of the end-capsdensityand of twicethe connectionsdensity. The nonlinear behavior of the investigated systems is in good agreement with the recent theoretical predictions of Cates et al. as long as the stress relaxation of the system does not depart too much from a single exponential.
Introduction Recently,evidencehas accumulated about the formation in aqueous surfactant solutions of long flexiblecylindrical micelleswith a large spread in length and a behavior similar to that of polymer chains in so1ution.l In the semidilute range, Le., at surfactant concentrations large enough that the elongated micelles overlap, the systems exhibit a viscoelasticbehavior very reminiscent of that of transient polymeric networks. It is important to stress at this point the transient character of these wormlike micelles that can break and recombine on a time scale which is dependent on the system and on the physicochemical conditions. A recent model based on the tube model of polymer dynamics but including the effects of reversible scission kinetics has been derived by Cates to describe both the linear2 and nonlinear viscoelastic properties of these ~ystems.39~ In the linear regime, the model predicts several rheological regimes depending on the relative rates of diffusive polymer motion and reversible breakdown process. In particular, a nearly single exponentialstress decay function is predicted 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. Detailed results were obtained from a computer simulation by Tumer and Cates6 and from a Poisson renewal model by Granek and Catessfor various values of the ratio = q , d T ~where ~ T , b d is the time taken for a micelle of the mean length to break and T~~~ is its reptation time. The calculated Cole-Cole plots in which the imaginary part G”(o)of the frequency-dependentshear modulus is plotted against the real part G’(w), can be compared to the experimental ones providing a direct estimate of the parameter = ~ & T R , where TR is the terminal stress relaxation time. The Poisson renewal model was also applied to study the regimes that arise for small breaking time, andlor smalltime scaleswhen the dominant polymer
r
(1) See, for instance: Cabs, M. E.; Candau, 5.J. J. Phys.: Condens. Matter 1990,2, 6869, and references therein. (2) C a b , M. E. Macromolecules 1987,20,2289. Europhys. Lett. 1987, 4,497. J. Phys. (Paris) 1988,49, 1593. (3) Cam, M. E. J. Phys. Chem. 1990,94,371. (4) Spendley,N.; McLeieh, T.; C a b , M. E. to be published. (6) Turner, M. S.; Catee, M. E. Longmuir 1991, 7,1590. (6) Granek, R.; Catee, M. E. J. Chem. Phys. 1992,96,4758.
motion is not pure reptation, but either breathing (in which tube length fluctuations play a dominant role) or local Rouse m ~ t i o n .A~typical signature of the latter effect is a turnup of both G’(w)and G”(w) at high frequencywhereas the simple reptation picture predicts a constant asymptote for G’(o) and a decreasing G ’ ’ ( W ) . ~This ~ ~ results in a minimum in the C o l d o l e representation of the dynamic modulus,whose depth can be used to estimate the number of entanglements per chain in the system. The above model applies strictly to systems of entangled wormlike micelleswith entanglement length (i.e. the contour length between two successiveentanglements) much larger than the persistence length lpand in a regime where the breaking time is much larger than the Rouse time of a chain with the entanglementlength. This correspondsto a rather limited range of surfactant and salt concentration. Outside this range, deviations are expected for the shape of the C o l d o l e plots. The model of Cates predicts also scaling behaviors to dilution, of different rheological and kinetic parameters: zero-shear viscosity, plateau modulus, breaking time. However the theoretical predictions hold only in a limited range of salt concentration. In fact three different regimes can be considered. At zero or very low salt content, the Coulomb effects lead to a complicated behavior of the micellar length and therefore of the kinetic and rheological parameters upon increasingthe surfactant concentration.* This regime is characterized by a Debye-Hiickel length K - ~ 1 5; where f is the correlation length. At higher salt concentrations ( K - ~I 0 the electrostatic interactions are screened out and the Cates model applies. Finally, in the limit of high salt content, it has been recently suggested that intermicellar branching might occur (cf. Figure l), resulting in a strong modification of the rheological properties and more specifically in a reduction of the zero shear visco~ity.~J~ The boundary between the high and moderate salt content regimes cannot be defined in terms of K - ~but depends on the material. Light scattering (7) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (8) Mackintosh, F. C.; Safran, S. A.; P i n m , P. A. Europhys. Lett, 1990, 12, 697. (9) Candau, S. J.; Khatory, A.;Lequeux, F.; Kern, F. Proceedii of the workshop Complex Liquid System, Polistena 6 1 0 July 1992, to be published in J. Phys. IV.
0743-7463/93/2409-1456$04.00/0 0 1993 American Chemical Society
Wormlike Micelles at High Salt Content
Langmuir, Vol. 9, No.6, 1993 1457 4.0
30
(a)
Materials and Methods The samples of CTAB were the same as in previous investigationsll but different batches have been used as mentioned in the text and the tables. The rheological experiments were performed on a Carrimed CSL apparatus (controlled stress). We used stainless steel Couette and plane-plane geometries. For the Couette cell, the inner radius is 13.83 mm, the outer radius 15 mm, the immersed height 32 mm, and the sample volume approximately 9 cm3. The diameter of the plane is 40mm. Special devices were used to avoid evaporation. The frequency range investigated was 1q2Hz I f I 40 Hz and the shear-rate range was 103 s-l I y I 600 s-l. The ranges of CTAB and KBr concentrations investigated were 0.1 M I [CTAB] I0.6 M and 0.25 M 5 [KBr] I2 M. Experiments were performed at two temperatures: 31 and 35 "C.
I-
= 0.23
-_---- ( = 0.70 ---= 2.38
370
Table 111. Rheological and Kinetic Parameters: Effect of Salt Concentration; [CTAB] = 0.36 M KBr
~0
G',
(M) ( P a d (Pa) 0.4 1
1.5 2
0.4 1
1.5 2
52 200 410 400
370 290 300 295
30 140 220 200
560
430 320 360
G"& G'm
L(A)
T=31°C 0.177 294 0.077 337 0.033 331 0.047 334 Te35OC 0.175 303 0.066 272 0.048 321 0.062 301
L or Lc bm)
?
0.17 0.42 1.00 0.71
0.79 0.70
0.11 0.32 0.67 0.49
1.6 0.70
0.54
0.23
0.54
0.7
CTAB(M)
sb
(me)
emu/ C',
111
0.64
483 783 312
0.82 0.63
85 228 371 389
0.41 0.55 0.72 0.65
Cole plot is found between-two calculated ones, in which case we give intervals for { and nrd.
8
10;
1
7
O.*
CTAB(M)
1
Figure 10. Variation of the plateau modulus versus surfactant concentration: IKBrl = 1.5 M, T = 35 OC. The fiied circles refer to data obtained with a different batch. The full line is the best linear fit to the data (slope = 1.85).
(b) Effect of Surfactant Concentration. The effect of thesurfactant concentration on the different parameters determined by combining direct measurements(10, GI/-, G'4 and fits of the Cole-Cole plots (TbreadTrep) are illustrated in the Figures 9-11 relative to the experiments performed in the presence of 1.6 M KBr. Each of these parameters follows a behavior different from the predicted one: The zero-shear viscosity qo scales like instead of Cg45. The plateau modulus G ' m scales like 0.86, not too far from the theoretical prediction: c2-2.3. The ratio G"&G'm scales like C-OeB8 instead of the predicted C-7I4. These observations can fiid an explanation in the conclusion of a recent study on wormlike micelles on CPClOdC14Na that brought up the idea that an excess of salt could lead to a branching of the micelles and even to the formation of a saturated n e t w ~ r k . ~ JSuch ~ a conclusion was mainly suggested by the anomalously low viscosity of these systems that could not be described by the reptation model of entangled linear micelles. A statistical description of the occurrence of cross-links versus entanglementsin semidilute solutions of wormlike
Langmuir, Vol. 9, No,6, 1903 1461
Wormlike Micelles at High Salt Content 10000
;k
1
1
"
- 1 .4; 0.01 E
h
0.001
I CTAB(M) 10000
1
"'I
micelles has recently been proposed.17 It must be emphasized, that the connections in micellar systems are, by essence, very different from the cross-links encountered in classical polymers. In branched micelles, the surfactant molecules can flow through the branch points leading to a fluidization and to a decrease of the shear modulus of the system. Thus the effect of the intermicellar connections in a flow experiment is totally different from that of a c h i d interpolymer cross-linkand cannotbe described, at least for the systems considered here, by the rubber elasticity theory that has been used to explain the rheological behavior of some surfactant systems.18JQ The reptation process of branched micelles was also considered theoretically and it was shown by one of us that the branching of the micelles leads to a reduction of the zero-shear viscosity and of its surfactant concentration dependence" More precisely, all the resulta concerning the rheology of linear wormlike micelles can be ap lied to branched micelles, provided that one substitutes by L,, where L, represents the harmonic mean between the average distance from one point along the micelle to the first cross-link and the average distance from that point to the first end-cap, or equivalently, the ratio of the total length over the concentration of end-caps plus twice that of connections. In the present study, we deal with rather high salt contents and we cannot discard the presence of such an effect. Indeed if the micelles are branched, we do not expect a qualitative change in the shape of C o l d o l e plots. But then, the meaning of the dip would be different and
E
In a mean field approximation, E, is found to decrease with C according to if there are many connections (see Appendix).21 This is the opposite behavior of the one obtained for h e a r micelles for which L varies like C1/2. Therefore, in the presence of both connections and endcaps, the concentration dependence of G"mm/G/m is expected to be less than the C-7/4variation predicted from the model of entangled linear micelles and in the limit (17) Drye, T.J.; Catee, M.E. J. Chem. Phye. 1992, M,1367. (18) Janeschitz, H.; Papenhuijzen, J. M. P. Rheol. Acta 1971,10,461. (19) (a) Hoffmann, H.; Mbl, M.; Rehage, H.; Wunderlich, I. Tenside Deterg. 1986,22, 290. (b) Rehage, H.;Hof", H. J. Phys. Chem. 1988,92,4712. (20)Lequeux, F.Europhys. Lett. 1992,19 (8), 675. (21) Elleuch, K.; Lequeux, F.; Pfeuty, P. Manuscript in preparation.
B
1000 e
Figure 11. Variation of the ratio G"&G', versus surfactant concentration: [KBrl = 1.5 M,T = 35 "C. The fiied circles refer to data obtained with a different batch. The full line is the best linear fit to the data (slope = -0.88).
*
100,
3
kf
lo-
l /
0.1
KB;(M)
0
Figure 12. Variationoftheratioq,&~versus(a,top)surfactant concentration ([KBrl= 1.5 M, T = 36 OC,the fiied circles refer to data obtained with a different batch) and (b,bottom) salt concentration ([CTAB] = 0.35 M,T = 31 "C ( 0 ) )T = 35 O C
(fdt)).
-
where the connectionsare predominant,G"&G', PI4. In fact, the values of E, are found to be rather constant with the surfactant concentration (Table I). If it assumed that, like in the case of linear micelles, the breaking of branched micelles occurs along the micelle with the same probability per unit length, then ~dshould be inversely proportional not toL but to L,. Indeed by definition q - , d is the lifetime of the micelles of the mean length. As the system with connections is strictly equivalent to the one without connectionsprovided one substitutesL by L,,the time TI,& deduced from rheological measurements is simply TI,& = l/klL, instead of llklL, where k l is the probability of a scission per unit time and per unit arc length. Thus, q-,& is the lifetime of a virtual micelle of length L, and no longer has a physical meaning. However the constant kl can be deduced from measurements of L, and TI,&; kl must not depend on the concentration, but only on the salinity of the solution. The variation of q - , d represented in Table I is found to be consistent with that assumptionas illustrated by Figure 1%) where the product TI-&, is plotted versus the surfactant concentration. This product is found to be constant within the experimental accuracy. We have ale0 studied the rheological properties of a series of sampleswith CTAB concentration varying from 0.35 to 0.6 M in the presence of 0.25 M KBr. For these samples, that do not show a minimum in G"(w), we cannot estimateL (or E,). As for the general behavior concerning the concentration dependences of qo, G'm and TI,&, it reproduces qualitatively the results of a previous study." The values of l)o are smaller than the ones previously determined, but we know that such differences occur when changing of batch.llb The results obtained in the present
Khatory et al.
1462 Langmuir, Vol. 9, No.6, 1993
study, and more specifically those concerning G'- will be used for the study of nonlinear viscoelasticity. (c) Effect of Salt Concentration. It is generally admitted that at fixed surfactant concentration, an increase of salt content leads to a micellar growth, because of the enhanced screening of the interactions. It is then expected that G"JG', is a decreasing function of the salt concentration. The results reported in Table I11show that G"&G', goes through a minimum upon varying the KBr concentration. This can also be explained by the micellar branching. As mentioned above, L must be replaced by L, in eq 2 if the micelles are branched, which would tend to increase G"m,$G'm at high salt content. The behavior observed for G",&G', is closely correlated to that of the zero shear viscosity that exhibits a maximum at the salt concentration where G"&G', is minimum (cf. Table 111). In this respect, it must be noted that at that salt content the exponent of the scaling law to dilution for the zero-shear viscosity is lower than the exponent characteristic of the reptation of linear micelles, in favor of a branching mechanism. This behavior is quite similar to that reported for the system CPC103/C103Na.gJ0 One of the most convincing arguments in favor of the existence of connections is the following: The maximum of viscositycorrespondsto the maximum of L,. This means that at that salt content (1.5M)the number of cross-links should be close to that of end-caps. This is the situation where L, should be only slightly dependent on the surfactant concentration, because end-caps and connections vary smoothly, but in an opposite way with concentration. This is in fact the behavior observed experimentally (cf. Table I). Using the procedure described above, we have also determined the values of f and therefore of 7 b r d . The behavior of with salt concentration again follows qualitatively that of (LJ-1. One cannot however expect and (L,)-l since 7 b 4 is also proportionality between a function of the rate constant k that depends on the salt content. Indeed Figure 12b shows that the product nr& L, goes through a maximum at a salt content that depends on the temperature. 11. Nonlinear Viscoelaoticity
Theory. An extension of the linearized model presented in the previous section to describe nonlinear phenomena has been derived by C a t e ~ As . ~ a starting point, it is argued that, while strong flowshave important effects on the chain dynamics, they do not at a first approximation have any effect on the scissionfrecombinationreaction rates themselves. With this assumption, a nonlinear viscoelastic constitutive equation for the regime where q,+ is much shorter than 7mp,but long compared to 7 b W , has been derived by adapting the nonlinear theories of Doi and Edwards's2 and Marrucci,23relative to unbreakable polymers. The retraction process followingthe increase of the average tube length under deformation is treated as instantaneous, as for unbreakable polymers. However for breakable polymersthe retraction affectsall tube segments equally at time scales larger than TI,&. The equivalence of all tube segments, which leads to a single exponential behavior in the linear relaxation spectrum is maintained in the nonlinear regime. As a result, the nonlinear behavior is very close to that predictad for monodisperse unbreak(22) h i , hi.;Edwards, S. F.J. Chem. SOC.,Faraday Trans. 2 1979,74, 1789,1802,1818.
(23)Marrucci, G. J. Non-Newtonian Fluid Mech. 1986, 21, 329. Marrucci,G.; Grizzuti, N. J. Non-Newtonian Fluid Mech. 1986,21,319.
3
loa0
..
,o
.
0
. . . . . KKK BBB rrr === 01l .. S4MMM
0
a
~
* .
e
*
*
n
m
0
0
o
KBR=2 M
*
o m
1 I , ' 0.01
0
1 ' 1 ' 1 ' 1
I
"""'I
n
I
I
"""I
I
I
"""I
I
I
' I ' M
y )1:)1(
O.l
Figure 13. Variationof the stress versus shear rate for solutions at [CTAB] = 0.35 MIT = 35 "C, and different salt contents. IOoo
1
1001 3
(d
CL I b 10 z
3 001
'-i
0 1
> nl
In the general case where both end-caps and connections are present, one haam n2 L, = n, + 2n3
where B1 and Bz are two constants. Hence at equilibrium, in the absence of connections
All these features can be calculated using a well-known analogy between these systems and the magnetic Ising model.21
