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Nonlinear Rheology of CTAB/NaSal Aqueous Solutions: Finite Extensibility of a Network of Wormlike Micelles Tadashi Inoue,* Yoshitaka Inoue, and Hiroshi Watanabe† Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan Received July 8, 2004. In Final Form: November 10, 2004 The nonlinear rheology of aqueous solutions of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) was investigated. The concentration of CTAB was fixed at 0.1 mol L-1, and the concentration of NaSal was varied from 0.07 to 0.4 mol L-1. For all test solutions, dynamic moduli were described with the Maxwell model having a single relaxation time, τ. Time evolutions of the shear stress, σ, and the first normal stress difference, N1, after inception of the steady shear flow were measured. For solutions having low NaSal concentrations, strain-hardening was observed and σ and N1 diverged at a certain strain when the shear rate, γ˘ , exceeded τ-1. For solutions with high NaSal concentrations, stress overshoot similar to that of ordinary entangled polymer solutions was observed at γ˘ between τ-1 and a certain critical rate, γ˘ C, while the strain-hardening was observed at γ˘ > γ˘ C. A simple relationship for elastic solids, N1/σ ) γ with γ being the strain imposed by shear flow, held for all the solutions in the strainhardening regime. The strain-hardening was attributable to the strain-dependent shear modulus and well described with the network theory considering the finite extensibility of network strands. The segment size of network strands was successfully determined. Thus, the stress-strain relationship obtained after the inception of fast flows is useful for characterizing the network properties.
Introduction Aqueous solutions of cationic surfactants show pronounced viscoelastic behavior at low concentrations when they contain certain aromatic acids or salts. One of the well-known examples is solutions of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal).1-3 In this system, a 1:1 complex of CTAB and NaSal formed wormlike or threadlike micelles having a uniform radius. The threadlike micelles can densely entangle with each other. According to Shikata et al., the features of the unique viscoelasticity of the CTAB/NaSal solutions can be classified into three types, I-III, depending on the concentration of NaSal, CS.4 For the type I solutions having a CS value much lower than the concentration of detergent (CTAB), CD, the viscoelastic properties of the solutions resembled those of dilute flexible polymer solutions without entanglement, and the relaxation spectrum can be well described with the bead-spring theories.5,6 For the viscoelastic behavior of the type II solutions having an intermediate CS value that was still less than CD, the viscoelastic behavior resembled that of well entangled polymer solutions. For the type III solutions with CS > CD, the viscoelastic properties were dramatically different and the above similarity to polymer solutions vanished. The properties of the type III solutions were essentially described by the Maxwell model, and therefore, the linear viscoelasticity can be characterized with one relaxation time, τ, and one plateau modulus, GN. This relaxation time was hardly dependent on the temperature or the detergent concen* To whom correspondence should be addressed. E-mail:
[email protected]. † E-mail:
[email protected]. (1) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 10811086. (2) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988, 4, 354-359. (3) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1989, 5, 398-405. (4) Shikata, T.; Hirata, H.; Takatori, E.; Osaki, K. J. Non-Newtonian Fluid Mech. 1988, 28, 171-182. (5) Rouse, P. E. J. Chem. Phys. 1953, 21, 1272-1280. (6) Zimm, B. H. J. Chem. Phys. 1956, 24, 269-278.
tration2 and was controlled only by the concentration of free salicylate ions in the bulk aqueous phase, CS* (not included in the CTAB/NaSal complex). From a consideration based on a quasi-network theory,7-9 Shikata et al. proposed an idea that the threadlike micelles exhibit thermal scission/reformation of entanglements and that the free salicylate ions in the aqueous phase behave as a catalyst for the scission reaction.1-3 The nonlinear rheological properties of the type III solutions were also studied by Shikata et al.4 According to their results, the nonlinear properties changed markedly with the free Sal concentration, CS*, which can be determined by the NMR technique. At low CS*, a kind of strain-hardening was observed: the shear stress, σ, increased enormously and exhibited a marked overshoot upon start-up of shear flow at high rates. At high CS*, the nonlinear behavior becomes somewhat similar to that of polymeric liquids; for example, the relaxation modulus, G(t,γ), decreased with increasing strain, γ, and the σ(t,γ)/γ˘ ratio under continuous flow decreased with γ˘ . In relation to the above rheological features of the CTAB/ NaSal systems, a comment may be needed for applicability of the concept of “reptation of living polymer” proposed by proposed by Cates.10 There is much literature reporting that the static properties and linear viscoelastic properties of surfactant solutions similar to the CTAB/NaSal system can be described with the model based on reptation of living polymer. On the other hand, some disagreements between the Cates model and experiments are noted for solutions having high salt concentrations; for example, to explain the anomalous decrease of relaxation time at a high salt concentration regime, the transition from entanglements (slip-link) to cross-links at high CS was suggested.11-13 More importantly, we note that the strainhardening behavior for type III solutions having low CS (7) Lodge, A. S. Elastic Liquids; Academic Press: London, 1964. (8) Lodge, A. S. Trans. Faraday Soc. 1956, 12, 25. (9) Yamamoto, M. J. Phys. Soc. Jpn. 1956, 11, 413. (10) Cates, M. E. Macromolecules 1987, 20, 2289-2296. (11) Khatory, A.; Kern, F.; Lequeux, F.; Appell, J.; Porte, G.; Morie, N.; Ott, A.; Urbach, W. 1993.
10.1021/la048292v CCC: $30.25 © 2005 American Chemical Society Published on Web 01/13/2005
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Table 1. Concentration and Viscoelastic Parameters for Test Solutions sample code
CD (mol L-1)
CS (mol l-1)
τ (s)
GN (Pa)
γ˘ C (s-1)
γF
γ˘ Cτ
n
Me/106
MS/104
ND
CTAB1 CTAB2 CTAB3 CTAB4 CTAB5 CTAB6
0.1 0.1 0.1 0.1 0.2 0.05
0.15 0.3 0.07 0.4 0.14 0.035
4.4 4.6 47 0.95 4.6 530
51 48 40 48 140 11
0.5 1 0.05 10 0.2 N/A
4.3 5.2 4.2 6.2 2.9 6.2
2.2 4.6 2.35 9.5 0.92 N/A
19 27 17 38 8.4 38
2.2 2.6 3.1 2.4 1.8 5.7
12 9.5 18 6.4 21 15
240 190 350 130 430 300
cannot be explained by the Cates model for nonlinear rheology of living polymers.14 In the original reptation theory, the entanglements are regarded as slip-links represented by a tube in which the chain can free slide.15 For fast flows, the chain can retract in the tube and keep its equilibrium length. Similarly, in the Cates model, the retraction of living chain is instantaneous compared to the terminal relaxation and therefore only shear thinning behavior was predicted. This discrepancy clearly indicates that the current molecular picture for surfactant systems is to be further refined. Brown and co-workers performed the stress relaxation measurements on the type III solution with low CS and found that the Lodge-Meissner relationship holds in the strain-hardening regime.16 This relationship indicates that the solution essentially behaves like an elastic material, although the modulus increases significantly with strain. They suggested that the molecular origin of the strainhardening is the finite extensibility of the network strands. However, unfortunately, they did not perform detailed analysis of the data. The stress relaxation measurement is a useful method for analyzing the nonlinear rheology, particularly for the case of systems obeying constitutive equation. However, some information such as the critical shear rate for strainhardening is not directly obtained from the relaxation experiments. In this paper, we report the first normal stress and the shear stress growth functions of the CTAB/ Sal solution after start-up of shear flow to investigate the molecular origin of the strain-hardening. We extend the range of shear rates compared to those in the previous study,4 and we demonstrate that the solution always shows a kind of strain-hardening irrespective of CS* at sufficiently high rates. We indicate that this strain-hardening can be attributed to the finite extensibility of the network strands. We also estimate the segment size of threadlike micelles which characterizes the network properties.
Figure 1. Frequency dependence of the complex shear modulus for CTAB1 ()sample C; CD ) 0.1 mol L-1 and CS ) 0.15 mol L-1) and CTAB4 ()sample F; CD ) 0.10 mol L-1 and CS ) 0.4 mol L-1) at 21 °C. solvent (water) evaporation. In the previous study by Shikata et al., it was reported that rheological properties did not vary noticeably with temperature in the range between 10 and 15 °C.4 In our experiments, the relaxation time slightly varied with temperature in the range between 20 and 25 °C, and we needed to control the temperature to obtain good reproducibility. Thus, all measurements were conducted at 22 ( 0.1 °C. The Kraft temperature was estimated to be below 20 °C for all the test solutions.
Results and Discussion Complex Modulus. Figure 1 shows the angular frequency (ω) dependence of the complex shear modulus for the samples CTAB1 (CD/CS ) 0.1:0.15) and CTAB4 (0.1:0.4) at 22 °C. These solutions are expected to be type III solutions for which the complex modulus is described with a single Maxwell model. Indeed, this expectation is confirmed in Figure 1 where the solid and dotted curves indicate the storage and loss moduli, G′ and G′′, of the Maxwell model:
iωτ G*(ω) ) G′(ω) + iG′′(ω) ) GN 1 + iωτ
(1)
Experimental Section Cetyltrimethylammonium bromide (CTAB, Wako Pure Chemical Industries, Ltd.) was purified by the recrystallization method with acetone/methanol. Sodium salicylate (NaSal, Wako Pure Chemical Industries, Ltd.) was used as received. Ion-exchanged water was used as the solvent. Concentrations of the solutions were determined by weighing. Table 1 specifies the CD and CS values of the examined sample solutions. All solutions were kept quiescently at room temperature prior to the tests for at least 2 days for equilibration. All measurements were performed with a Rheometric ARES system equipped with a force rebalance transducer (2K FRRN1). A cone-and-plate geometry with a 2.5 cm diameter and a 0.1 rad gap angle was used. A homemade solvent trap was used to prevent (12) Candau, S. J.; Khatory, A.; Lequeux, F.; Kern, F. J. Phys. IV 1993, 3, 197. (13) Kadoma, I. A.; Ylitalo, C.; van Egmond, J. W. Rheol. Acta 1997, 36, 1-12. (14) Cates, M. E. J. Phys. Chem. 1990, 94, 371-375. (15) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, U.K., 1986. (16) Brown, E. F.; Burghardt, W. R.; Venerus, D. C. Langmuir 1997, 13, 3902-3904.
Excellent agreement of the data and the model is noted at low frequencies. The plateau modulus, GN, and the relaxation time, τ, of all solutions were successfully determined with this Maxwell fitting and are summarized in Table 1. In Figure 1, we also note a deviation from the Maxwell behavior at high frequencies. Probably this deviation is due to the polymeric (Rouse) mode of the entangled threadlike micelles. Granek and Cates proposed a method to estimate the length of the micelle from this deviation from Maxwell behavior.17 However, as they noted in their paper, the current system is not adequate to use their model. We will not discuss this issue further. Shear Stress Growth after Inception of Steady Shear Flow. The shear stress growth function, σ+(t), of the sample solution CATB1 (0.1:0.15) after inception of steady shear flow is shown in Figure 2. The data are presented as a reduced quantity, the shear stress growth coefficient, η+(t) ) σ(t)/γ˘ . The filled circles indicate (17) Granek, R.; Cates, M. E. J. Chem. Phys. 1992, 96, 4758-4767.
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Figure 2. Shear stress response after inception of steady shear flow for CTAB1 at 21 °C. The numbers in the figure represent the shear rate (s-1).
η+
of the Maxwell-type solutions in the linear viscoelasticity limit
[
t τ
( )]
η+L(t) ) GNτ 1 - exp -
(2)
with the parameters τ and GN being given in Table 1. Inclusion of the faster modes observed in Figure 1 did not affect the calculated η+L values significantly. Obviously, at the lowest rate of shear, the shear stress agrees with this linear behavior. Thus, η+L was used to check the consistency of dynamic and transient measurements. The behavior of η+ seen in Figure 2 can be categorized into three types. The first type is the linear behavior, η+ ) η+L, observed at low shear rates. The second type is “the shear thinning behavior”, η+ < η+L at t > τ, observed at γ˘ ∼ 0.2 s-1 (∼τ-1). η+ shows an overshoot and then reaches the steady state. When the rate exceeds τ-1, the shear thinning behavior becomes remarkable. For example, η+ at γ˘ ) 0.3 is much lower than η+L at long t. The third type is “the strain-hardening behavior”, η+(t) > η+L(t), observed at high γ˘ g 0.5. For this type, η+(t) suddenly increases and deviates from η+L(t) at a certain time. After this deviation, η+ shows a sharp peak and then drastically decreases. From observation by the naked eye, it was found that the solution became unstable after the peak point and the test solution flew out from the cone-plate fixture. Thus, the solution did not reach the steadily flowing state when the strain-thickening was observed. Therefore, the data after the peak (dotted curve) are not reliable. We believe that the stress peak is related to the mechanical fracture of the elastic CTAB solution. As noted in Figure 2, the characteristic times for the onset of increases of η+ above η+L (onset of hardening) and for the fracture (η+ peak) are strongly dependent on the shear rate. Both times decrease with increasing shear rates. The fracture strain (peak strain) determined from the η+ peak time, γ˘ tpeak, is found to be constant (∼4) at high rates. Thus, the excess stress growth of η+(t) > η+L at high rates should be controlled by the strain. Similar behavior was observed for all solutions examined. From the above result, the limiting value of the fracture strain at high rates, γF, is considered to be a parameter describing the features of the fracture. For the solutions examined, the values of γF are summarized in Table 1. First Normal Stress Difference after Inception of Steady Shear Flow. The response of the first normal stress difference, N1, of CTAB1 (0.1:0.15) after the inception of steady shear flow is shown in Figure 3. The data are presented as the first normal stress growth
Figure 3. Normal stress difference response after inception of steady flow for CTAB1 at 21 °C. The numbers in the figure represent the shear rate (s-1).
coefficient, Ψ1+(t) ) N1/γ˘ 2. The data at t > tpeak (where the instability was observed) are not shown. In the low shear limit, the first normal stress growth coefficient should obey the following relation (deduced form the Lodge equation):18
( τt) - (τt) exp(- τt)]
[
Ψ1,L+(t) ) GNτ2 1 - exp -
(3)
Here, the parameters τ and GN are the same as those for η+ and given in Table 1. The data for γ˘ ) 0.1 s-1 are quite close to this Ψ1,L+ represented by filled circles. The qualitative feature seen in Figure 3 is very similar to that in Figure 2. At high rates, Ψ1+ deviates from the linear limit (eq 3). The peak of Ψ1+ was observed at the same time (same strain) as η+, and the peak strain was constant under fast shear. This feature is quite different from that of entangled polymer systems for which the peak time and peak strain for Ψ1+ are approximately twice those for η+.19 Elastic Strain and Modulus. In early studies of the normal stress effects, the following equations were employed to define an effective elastic strain, γeff, and effective elastic modulus, Geff, of an elastic liquid under flow.18
σ ) Geffγeff
(4)
N1 ) Geffγeff2
(5)
Thus, γeff can be evaluated from the η and Ψ1 data as
γeff )
N1 Ψ1 ) γ˘ σ η
(6)
The dependence of γeff on the macroscopically imposed strain, γ ()γ˘ t), is of our interest. For the purely elastic materials, γeff agrees with the macroscopic γ. To consider the relaxation effect, the ratio γeff/γ˘ ) Ψ1+/η+ for CTAB1 (0.1:0.15) is plotted against time in Figure 4. The linear limit calculated with eqs 2 and 3 is also shown (dotted curve). Note that the Ψ1+/η+ ratio in the linear limit, being (18) Larson, R. G. Constitutive Equations for Polymer Melts and Solutions; Butterworth: Sydney, 1988. (19) Osaki, K.; Inoue, T.; Isomura, T. J. Polym. Sci., Polym. Phys. Ed. 2000, 38, 1917-1925.
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Figure 4. Time dependence of effective elastic strain normalized by shear rate, γeff/γ˘ ) Ψ1+/η+, for CTAB1. The broken line indicates the dependence in the linear limits. See text for more details.
independent of the rate of shear, is proportional to t at short times and levels off at t > τ due to the relaxation effect. At γ˘ ) 0.1 s-1, the experimental data are close to this linear limit curve. The data taken at γ˘ ) 0.3 s-1 slightly deviate from the linear limit at long times. At γ˘ ) 0.5 s-1, the experimental data are well described with the straight line with a slope of 1. Note that respective data of η+ and Ψ1+ at this γ˘ value show the nonlinear strain-hardening (see Figures 2 and 3) but their ratio is not significantly different from that in the linear regime. Considering this result, we note that the Ψ1+/η+ ratio ()γeff/γ˘ ) at γ˘ ) 0.5 s-1 increases linearly with t even in a range of t > τ and the maximum γeff value observed on the fracture is a little larger than the γeff value ()γ˘ τ) in the steadily flowing state in the linear regime. This fact may be related to a shear-induced increase of τ, that is, a shearinduced retardation of the cycle of scission/reformation of the entanglement of the CTAB strands (that may reflect a change in the collision frequency of the strands under shear). At higher shear rates of γ˘ > 0.5 s-1, the following simple relationship holds up to the fracture time, tpeak, despite the highly nonlinear strain-hardening effect; see Figure 4.
γeff )t γ˘
(7)
Thus, the effective elastic strain, γeff, for the CTAB solutions agrees with the macroscopic strain imposed, γ ) γ˘ t. This fact means that the CTAB/NaSal solutions behave as an elastic solid under fast shear flow despite their remarkable strain-hardening. The present result is consistent with the study by Brown and co-workers who showed the validity of the Lodge-Meissner relationship for a very similar CATB solution.16 The effective elastic modulus, Geff, can be evaluated as
Geff )
σ+ η+ ) γ t
Figure 5. Strain dependence of the effective shear modulus for CTAB1 at various shear rates (s-1).
Thus, the effective modulus at high shear rates does not depend on the rates in an experimentally accessible shear range. This implies that no significant relaxation mechanism contributes to Geff(γ,∞). In other words, the structural change of micelles would not occur in a short time scale ( 0.01 s > τe and at γ˘ < 100 s-1 < τe-1, the present CTAB system can be regarded as being in a local-equilibrium state and the three-chain model can be safely applied. For biaxial deformation, the stress-strain relationship for the three-chain model can be written as
(8)
Figure 5 shows dependence of Geff(γ,γ˘ ) obtained at high rates, γ˘ , on the macroscopic strain, γ. With increasing shear rate, the Geff curves for various γ˘ values converge on one envelope that represents the limiting behavior of Geff(γ,γ˘ ). The limiting effective modulus, Geff(γ,∞), is approximately constant for small strain (γ < 4), while it rapidly increases with increasing strain (γ > 4). The fracture of the solution seems to be related to this increase (divergence) of the limiting modulus, Geff(γ,∞), at γ > 4.
t1 - t2 )
{ ( )
( )}
λ1 λ2 NkT 1/2 n λ1L-1 1/2 - λ2L-1 1/2 3 n n
(9)
Here, ti and λi, respectively, are the tensile stress and the stretch ratio in the direction of the principal axis i. L-1(x) is the inverse Lanvegin function of x, N is the number of chain strands per unit volume, and n is the number of (20) Wang, M. C.; Guth, E. J. Chem. Phys. 1952, 20, 1144. (21) Treloar, L. R. G. The Physics of Rubber Elasticity; Clarendon: Oxford, U.K., 1958.
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segments per strand. The model considers γ-independence N and n, which is consistent with the behavior of CTAB at short t < tpeak (where structural relaxation/adjustments cannot occur). For the simple shear deformation, λ1 and λ2 are given by21
λ1 )
γ + xγ2 + 4 2
(10)
1 λ1
(11)
λ2 )
Figure 6. Shear stress response after inception of steady shear flow for CTAB4 (CD ) 1.0 mol L-1 and CS ) 0.4 mol L-1) at 21 °C.
Correspondingly, the shear stress is given by
σ ) (t1 - t2) sin χ cos χ
(12)
Here, c is the mass concentration of the CTAB micelle. Assuming a 1:1 complex of CTAB and NaSal, we estimate
the Me value for CTAB1 to be 2.2 × 106. From this Me value and the n value ()19), the molecular weight of the segment is estimated as MS ) Me/n ∼ 1.2 × 105, which means that the number of CTAB molecules per segment, ND, is ∼240. Shikata et al. estimated the molecular weight per unit contour length of micelles, ML ) 8900 g mol-1 nm-1, and of the persistent length of the micelles, q ) 26 nm, for CTAB/NaSal solutions with rheo-optical and light scattering data.24 From these ML and q values, the molecular weight of the Kuhn segment is estimated as 2qML ) 4.6 × 105. This 2qML value is ∼4 times larger than our MS value. At the present time, we find no clear reason leading to this relatively large discrepancy. However, we should remember that Shikata et al. estimated ML from the flow birefringence measurement under an assumption that the threadlike micelles are perfectly orientated in the strainhardening region. This assumption may not be so accurately valid. Rothstein performed extensional measurements on CTAB/NaSal solutions with a filament stretching rheometer.25 The extensional data were well described with the FENE-PM model, while the shear experimental data could not be described with the model. For a similar solution with CD/(mol L-1) ) 0.1 and CS/(mol L-1) ) 0.1, he obtained n ∼ 30 from the FENE-PM model analysis. This value is ∼1.5 times larger than our value, n ) 19. The difficulty of the stretching experiments may cause the difference in n value between his and our experiments. We will come back to the issue of the segment size in the next section. Effect of NaSal Concentration. The viscoelastic properties of CTAB/NaSal solutions significantly depend on the salt concentration, CS. The relaxation time, τ, decreases with increasing CS > 0.1 mol L-1.4 For our solution, CTAB4 (0.1:0.4) having the largest CS value of 0.4 mol L-1, G* was described with the single Maxwell model with τ ) 0.1 s. Figure 6 shows the shear stress growth coefficient, η+, for CTAB4 at various shear rates. The characteristic behavior resembles that observed in Figure 2: At the lowest shear rate, η+ agrees with η+L (dotted curve). At very high shear rates, γ˘ > 20 s-1, the strain-hardening is observed. One important difference is that the critical rate, γ˘ C, at which the strain-hardening is observed for CTAB4 is much higher than 1/τ. For the samples CTAB1-CTAB4, the product γ˘ Cτ is summarized in Table 1. This product, giving a measure
(22) Porte, G.; Gomati, R.; El Haitami, O.; Appell, J.; Marignan, J. J. Phys. Chem. 1986, 90, 5746. (23) Drye, T. J.; Cates, M. E. J. Chem. Phys. 1992, 96, 1397-1375.
(24) Shikata, T.; Dahman, S. J.; Pearson, D. S. Langmuir 1994, 10, 3470. (25) Rothstein, J. P. J. Rheol. 2003, 47, 1227-1247.
with χ ) cot λ1. Then, the strain-dependent modulus of the network is obtained as -1
G(γ) )
σ (t1 - t2) sin χ cos χ ) γ γ
(13)
For the case of γ f 0, eq 13 reduces to G ) NkT. Thus, the strand number, N, in eq 9 can be unequivocally evaluated from the data for the plateau modulus, GN, measured in the linear regime. The remaining parameter, n, in eq 9 determines the strain dependence of G(γ), and therefore, n can be determined by fitting data with eqs 9-13. The thick solid curve in Figure 5 represents the strain-dependent modulus of the three-chain model with the parameters determined in this way, N/(mol m-3) ) 0.0224 and n )19. The strain dependence of Ge(γ,∞) under high shear flow is well described with this network model considering the finite extensibility of the network strands. The entangled linear polymers merely show strainhardening behavior in shear flow because the chain retraction process prevents the chain stretching. Similarly, the Cates theory for nonlinear rheology of entangled living polymers14 predicts shear thinning behavior. The strainhardening behavior due to the finite extensibility of the network strands strongly suggests that the entanglements for the CTAB systems are sticky and cannot be represented as the usual (freely slidable) tube, at least under high shear. Such sticky temporal cross-links may be formed due to complex formation1-3 between strands and/or branch formation22 in the strands. A statistical description of the occurrence of these cross-links in semidilute solutions of wormlike micelles has been proposed,23 but the relaxation dynamics of the cross-linked networks was not discussed. Segment Size of Threadlike Micelles. As discussed above, the strain-hardening behavior seen in Figures 2 and 3 can be excellently described with the three-chain model considering the finite extensibility of strands. The number of segments per strand giving this description, n ) 19 allows us to estimate the segment size in the following way. The molecular weight of the network strand between entanglements, Me, is estimated from the GN data.
Me )
cRT GN
(14)
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Figure 7. Time dependence of effective elastic strain normalized by shear rate, γeff/γ˘ ) Ψ1+/η+, for CTAB4. The broken line indicates the dependence in the linear limits. See text for more details.
of an effective strain for the onset of hardening, increases with increasing CS. For CTAB4, the shear thinning is also observed in Figure 6 in the range 1 < γ˘ /τ < 10, which is similar to the behavior reported by Shikata et al. 4 This feature is similar to that of the entangled polymeric solutions. However, we should also note that all samples having various CS values exhibit the strain-hardening under sufficiently fast shear. This feature was not found in the earlier study by Shikata et al.4 Similar results were obtained for the first normal stress growth coefficient, Ψ1+. The ratio γeff/γ˘ ) Ψ1+/η+ is plotted against time in Figure 7. At low shear rates, γ˘ < 1 s-1, this ratio (not shown in Figure 7) agreed with the linear viscoelasticity limit (thick dashed curve in Figure 7). Under fast shear at γ˘ ) 1-5 s-1 where the overshoot followed by thinning was observed for η+ and Ψ1+, the time dependence of Ψ1+/η+ is not significantly different from that in the linear limit, although a weak overshoot of Ψ1+/η+ is also noted (see Figure 7). The steady state value of Ψ1+/η+ is slightly lower than that in the linear limit but is insensitive to the shear rate examined. This insensitivity contrasts with the behavior of entangled polymer systems for which the steady value of Ψ1+/η+ decreases with increasing shear rate in the thinning regime.26 This result strongly suggests that the molecular origin of the stress overshoot is different for our CTAB systems and the entangled polymer systems. At higher shear rates (γ˘ ) 10 s-1), Ψ1+/η+ first deviates upward from the linear limits (filled circles) and follows the line representing the relationship Ψ1+/η+ ) t. This behavior of the solution CTAB4 possibly reflects the onset of strain-hardening and the suppression of the relaxation, as similar to the situation for CTAB1. At even higher shear rates (g20 s-1) where the strain-hardening was observed, eq 7 held for CTAB4 up to the fracture point. Equation 7 held for all the test solutions at high shear rates, and the limiting modulus at high rates, Geff(γ,∞), was successfully determined with the method explained in Figure 5. The strain dependence of Geff(γ,∞) for the solution of various CS values is shown in Figure 8. For constant CD (CTAB1-CTAB4), the strain at which the modulus diverges (fracture strain) slightly increases with increasing CS. This result suggests that the network structure may slightly change, although GN at low strains remains constant. According to the three-chain model, this change in the fracture strain can be related to the change of flexibility of the network strands. The present result may indicate that the segment size decreases a little with increasing CS (see the n values summarized in Table 1). On the other (26) Osaki, K.; Inoue, T.; Ahn, K. H. J. Non-Newtonian Fluid Mech. 1994, 54, 109-120.
Figure 8. Strain dependence of the shear modulus. The line shows the best fit result by the Edwards-Vilgis slip-link model with a number of cross-links of Ncross ) 0.004 22/(mol m-3), a number of slip-links of Nslip ) 0, and a finite extensibility parameter of R ) n-1 ) 0.154.
hand, the stress-optical coefficient of this system was reported to be essentially independent of CD and CS.24 This means that the segment size of micelle is essentially independence of CD and CS. Thus, there is some inconsistency in the absolute value/dependence of the segment size determined from rheo-optical measurement24 by Shikata et al. and our analysis of Geff data. Further studies are needed for this problem. As far as the authors know, the strong strain-hardening seen for the CTAB solutions is not observed for entangled networks around their characteristic relaxation time.16 For the case of well entangled polymer solutions, the chain retraction process is the major molecular origin of the nonlinear viscoelasticity and therefore shear thinning behavior is usually observed. This chain retraction process is suppressed only under very high shear (not so easily achieved in lab experiments). We believe that the significant hardening reported here is related to the temporal network structure of the CTAB micellar strands which has a special mechanism to suppress the retraction. In this context, we note that similar strain-hardening behavior is observed for some associating polymers.27,28 (In these systems, the linear modulus is not the single Maxwell type and the strain-dependent modulus can be expressed with G0(1 + Rγ2), where R is a parameter of the order of unity. Thus, the strain dependence is not as strong as the present case.) However, with increasing shear rate, the system first shows shear thickening behavior at the shear rate before showing the strain-hardening behavior, which is completely different from entangled systems. If the detergent micelles form some complex or associating structure at the entanglement point, we may observe shear thickening behavior. Thus, we believe that the molecular origin of the strain-hardening in the CTAB system is related to the suppression of the chain retraction. Detergent Concentration Dependence. The strain dependence of Geff(γ,∞) for different detergent concentrations, CD, is also displayed in Figure 8. The CD/CS ratio is fixed at 1:0.7 (for CTAB3 (0.1:0.07), CTAB5 (0.2:0.14), (27) Serero, Y.; Jacobsen, V.; Berret, J.-F.; May, R. Macromolecules 2000, 33, 1841-1847. (28) Berret, J.-F.; Serero, Y.; Wonkelman, B. J. Rheol. 2001, 45, 477492.
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Figure 10. Shear rate dependence of peak strain of stress overshoot for CTAB4. The horizontal axis is reduced by the relaxation time.
Figure 9. CTAB concentration dependence of the plateau modulus, GN, and fracture strains, γF, for the solutions with CD/CS ) 1:0.7.
and CTAB6 (0.05:0.035)). The shapes of the strainhardening curves are almost the same for these solutions except for the fracture strains. However, a careful inspection indicates a minor difference. For CTAB6, having the lowest CD, Geff slightly decreases with γ before showing the divergence. This strain thinning behavior cannot be described with the three-chain model. Motivated by the results by Urayama et al.,29,30 we tested the EdwardsVilgis slip-link model.31,32 This model can well explain the strain thinning behavior observed before the hardening, and the best fit results were obtained when all junctions were assumed to be the cross-linked junctions (no slip junction) (see Figure 8), The Edwards-Vilgis model without slip junctions (identical to slip-links utilized in the tube model) also provides good fitting results for Geff(γ,∞) for all other solutions. This result strongly suggests that under the fast flow all entangled junctions exhibit no slip and behave as cross-linked junctions. For the analysis of the data for CTAB at various CD and CS values and other detergent systems, the Edwards-Vilgis model is of interest. This analysis is considered to be future work. Figure 9 shows the CD dependence of the plateau modulus, GN, and the fracture strain, γF, at a fixed CD/CS ratio. GN is approximately proportional to CD2. This result is consistent with the previous result1-3 and also with the concentration dependence of GN of entangled polymer solutions. If the structure of the CTAB network strands does not vary with concentration, this dependence corresponds to a scaling relationship for the segment number per strand, n.
n ∼ CD-1
(15)
Figure 9 also demonstrates that the fracture strain decreases with increasing CD as
γF ∝ CD-1/2
(16)
According to the network models,20,21,31,32 the strain at the singular point, ∼γF, scales as n1/2. This prediction is consistent with eqs 15 and 16. Thus, the change of the strain-hardening behavior with CD is well described as a change due to dilution of the entangled network, and the (29) Urayama, K.; Kawamura, T.; Hojiya, S. Macromolecules 2001, 34, 8261-8269. (30) Urayama, K.; Kawamura, T.; Hojiya, S. J. Chem. Phys. 2003, 118, 5658-5664. (31) Edwards, S. F.; Vilgis, T. A. Polymer 1986, 27, 483. (32) Edwards, S. F.; Vilgis, T. A. Rep. Prog. Phys. 1988, 51, 243.
segment size of the network strand is insensitive to CD. The present study clearly shows that the type III CTAB/ NaSal solutions have a network structure composed of flexible strands having finite extensibility. Molecular Origin of Stress Overshoot Observed at High CS. For the solutions having low CS, the stress overshoot followed by thinning was observed in a very narrow range of shear rates, γ˘ (see Figure 2 for CTAB1 (0.1:0.15)). On the other hand, for CTAB4 (0.1:0.4) with the highest CS, marked stress overshoot was observed in a much wider range of shear rates (see Figure 6). Thus, the nonlinear response of the solutions becomes similar to those of ordinary entangled linear polymers with increasing CS. This feature is inconsistent with a previously proposed structural evolution of semidilute wormlike micelles from an entangled network to a multiconnected one upon the increase of CS.11-13 To consider the structural origin of the overshoot of the CTAB solutions in the thinning regime, we first summarize the stress overshoot mechanism for entangled polymer solutions. For the entangled solutions, the stress overshoot is related to the chain orientation and retraction.33 Under fast shear in a range of 1/τw < γ˘ < 1/τ*, with τw and τ* being the longest viscoelastic relaxation time and the characteristic time for chain length equilibration (a measure of the time required for the chain retraction), the chain is not significantly stretched but progressively oriented with increasing γ˘ . τ* can be related to the longest viscoelastic Rouse relaxation time, τR, as τ*)2τR. The overshoot seen in this range results from this large orientation. Indeed, in this range, the overshoot peaks of the shear stress, σ, and the first normal stress difference, N1, occur at constant strains, γ = 2 and 4, respectively. On the other hand, under faster shear at γ˘ > 1/(2τR), the chain is not only oriented but also stretched, and the overshoot has a different feature: The overshoot peak occurs at a constant time, t = 2τR for σ and t = 4τR for N1.19,34 Considering these features of the entangled polymers, Figure 10 shows plots of the strain at the overshoot peak, γm, against τγ˘ for CTAB4 having the highest CS. γm for σ is approximately twice γm for N1 in the whole range of shear rates. At low shear rates, γm for σ approaches 2, as similar to the behavior of entangled polymers at 1/τw < γ˘ < 1/(2τR). With increasing γ˘ , γm for both σ and N1 increases and γm for σ approaches γ˘ τ. This result means that the stress overshoot occurs at tm ) γm/γ˘ ∼ τ at high shear rates. This stress overshoot in the thinning regime seems to be controlled by the terminal relaxation time, τ. (33) Pearson, D.; Herbolzheimer, E.; Grizzuti, N.; Marrucci, G. J. Polym. Sci., Part B: Polym. Phys. 1991, 29, 1589-1597. (34) Osaki, K.; Inoue, T.; Uematsu, T. J. Polym. Sci., Part B: Polym. Phys. 2000, 38, 3271-3276.
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In other words, the stress overshoot for the CTAB systems is not controlled by the chain retraction at a rate of 1/(2τR) and is different from the overshoot of entangled polymer systems. According to the Cates theory, the terminal relaxation time of reptating living polymers is given by τ ∼ (τrepτbreak)1/2 and the chain equilibrium time scales by τ** ∼ 2τR for τR < τbreak and τ* ∼ (τRτbreak)1/2 for τR < τbreak.14 Here, τbreak is the characteristic lifetime of the chain. Thus, the theory always predicts τ* , τ. The present study demonstrates the relationship τ ∼ τ*, indicating that nonlinear rheology of CTAB/NaSal micelles cannot be explained by a simple reptation picture of living polymers. In addition, as shown in Figure 6, the strain-hardening behavior was observed even for the solution having high CS at sufficiently high shear rate. This result implies that the micelles cannot purely reptate, probably due to sticky temporal crosslinks. When a very small portion of entanglements is temporarily cross-linked, the relaxation times of reptation and chain retraction may be determined by the disassociation time of the temporal cross-links, τ*link. The reptation and retraction motions may be triggered by the dissociation of the cross-link. If τrep and τR of the chains are much smaller than τ*link, the retraction time should effectively coincide with τ*link, and the features seen in Figure 10 are regarded to be similar to those of entangled polymers. In addition, if the number of the temporal crosslinks increases with decreasing salt concentration, the chain retraction process may be restricted. For such a case, the shear thinning behavior will vanish and only strain-hardening will be observed. A further study is desirable for this potential similarity. In connection to this point, we should note that the product γ˘ Cτ characterizing an upper bound of the thinning regime is dependent on CS (see Table 1). This CS dependence suggests a decrease of temporal cross-links with increasing CS and may give us an important clue for discussing the above similarity between CTAB networks and the above entangled polymers. A comment may be needed on the shear-banding phenomena of the fast flows of surfactant solutions. For wormlike micelle solutions, the shear stress plateau was often observed in the flow curve (shear stress vs shear rate curve).35 With increasing shear rate, the solution first flows as a Newtonian fluid, and then, above a characteristic rate, the shear stress becomes independent of the shearing (35) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712.
Inoue et al.
field. The shear stress plateau is attributed to the shearbanding structure, in which two regions having different shear rates but the same shear stress coexist.36 An alternative explanation is the coexistence line of two thermodynamically stable phases present within the sheared solution.37 It should be noted that the above discussion on the shear-band structure is for the steady flows. According to Berret, shortly after the inception of shearing, t ∼ tm, the flow is homogeneous and becomes inhomogeneous as time evolves.37 Thus, our above analysis assuming the homogeneous flow should be valid even if the shear-banding structure is observed in the steady state. The relationship between shear-banding structures and finite extensibility is not clear at the present moment. Further studies including rheo-optical approaches are required. Concluding Remarks We examined the nonlinear rheology of CTAB/NaSal solutions showing the Maxwell-type relaxation. For all solutions having various CS and CD values, we observed strong strain-hardening under sufficiently fast shear, although some solutions showed significant shear thinning in a range of moderate shear rates. Under the sufficiently fast shear, the shear stress and the first normal stress difference satisfied the phenomenological relationship for the ideal elastic matter. The strain-hardening behavior could be attributed to the strong strain dependence of the effective modulus. The strain dependence of this modulus was well described with a permanent network theory considering the finite extensibility of the network strands. This analysis offers a reliable method for estimating the segment size of threadlike micelles. The obtained value is ∼4 times larger than the literature value estimated from rheo-optical measurement. From a molecular point of view, the simple entangled network similar to entangled polymers merely shows strain-hardening under the shear flow. The present result suggested the existence of sticky cross-links which effectively work as the long-lived entanglements and prevent the chain retraction particularly in CTAB/NaSal systems having low salt concentrations. Acknowledgment. This work was stimulated from a discussion with Emeritus Prof. K. Osaki. We thank Prof. T. Shikata for helpful discussion. LA048292V (36) Spenley, N. A.; Cates, M. E. Phys. Rev. Lett. 1993, 71, 939. (37) Berret, J. F. Langmuir 1997, 13, 2227-2234.
