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Articles Rheology of Viscoelastic Mixed Surfactant Solutions: Effect of Scission on Nonlinear Flow and Rheochaos Paloma Pimenta† and Eugene E. Pashkovski* Department of Chemical and Biochemical Engineering, Rutgers UniVersity, 98 Brett Road, Piscataway New Jersey 08854, and Colgate PalmoliVe Company, R&D, Technology Center, 909 RiVer Road, P.O. Box 1343, Piscataway, New Jersey 08855-1343 ReceiVed July 19, 2005. In Final Form: February 10, 2006 The linear and nonlinear rheology of viscoelastic mixed anionic-zwitterionic surfactant solutions has been systematically investigated. In the linear viscoelastic regime, these systems display nearly Maxwellian behavior with a unique relaxation time, τ0, and a characteristic elastic plateau modulus, G0. Linear rheological data were used to calculate the reptation and breaking times of the micelles, τrep and τb, respectively. Surprisingly, the elastic modulus G0 significantly increases with salt concentration cs, whereas τb decreases by 1 order of magnitude. The strong effect of cs on the material parameters and microstructure of rodlike micelles allowed for the systematic investigation of the effect of these parameters on nonlinear flow. For samples with relatively long τb, the quasi-static flow diagram (stress vs shear rate) shows a stress peak followed by a metastable branch (a region of decreasing shear stress), whereas for samples with relatively short τb, this phenomenon is not observed. Transient flow responses corroborate quasi-static flow findings and further reveal the significance of microscopic dynamic parameters on flow behavior. Shear stress time series were recorded at constant shear rates, and above a critical shear rate, γc2, stress fluctuations are observed. The amplitude of these stress fluctuations, ∆σ, was found to scale as ∆σ = G0(τb|γ˘ - γ˘ c2|)β with β ≈ 0.5. This scaling is observed for micellar systems with τb ranging from 0.12 to 0.01 s and G0 ranging from 1 × 103 to 7 × 103 dyn/cm2.
1. Introduction Rodlike micelles are semiflexible aggregates formed by surfactant molecules in solution. These systems are of great interest partly because of their numerous practical applications, for example, as suppressants of turbulence and as detergents and foam boosters in the personal care industry. These applications are based on the ability of micellar solutions to thin under shear flow but recover quickly after the cessation of flow, and this property is a result of the reaction of the micellar microstructure to the applied shear field. At small deformations, the dynamic response of some concentrated micellar solutions is similar to that of a Maxwellian fluid,1 with a unique relaxation time, τ0, and an elastic spring constant, G0, that define the Newtonian viscosity, η0 ≡ G0τ0. This remarkably simple linear viscoelastic behavior is observed for strongly polydisperse micellar systems2 if scission and recombination reactions are fast enough to make the micelles dynamically indistinguishable by thermally averaging their length. In a flow experiment, below shear rates on the order of 1/τ0, these systems display Newtonian behavior with the shear stress proportional to the shear rate, σ ) η0γ˘ ) G0η0γ˘ . At higher shear rates, a transition to a non-Newtonian regime, associated with a plateau on the stress diagram, σp, occurs. The existence of this stress plateau was first observed experimentally by Rehage and * Corresponding author. Present address: Unilever R&D, 40 Merritt Boulevard, Trumbull, Connecticut 06611. E-mail: eugene.pashkovski@ unilever.com. † Rutgers University and Colgate-Palmolive Company. (1) Cates, M. E. Macromolecules 1987, 20, 2289. (2) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press Inc.: San Diego CA, 1992.
Hoffmann3 and has since been observed for many systems. From a theoretical standpoint, Cates and co-workers were the first to recognize the physical importance of this stress plateau.4,5 A “top-jumping” mechanism, where σp is the highest possible stress value (σp ) σmax), was proposed as an explanation for the occurrence of the stress plateau,5 and on the basis of this mechanism, quantitative agreement between the predictions of the “reptation-reaction” model and Rehage and Hoffman’s data was observed.5 Later theoretical work, however, suggested that the top-jumping mechanism may not be appropriate to describe shear-banded flow.6,7 Furthermore, transient flow experiments by Berret and colleagues8,9 revealed behavior that clearly departs from the top-jumping scenario (i.e., a stress overshoot was observed at the onset of flow, with σmax . σp). A slow, sigmoidal relaxation to the plateau value was also observed and was interpreted as the nucleation and growth of a nematic band from the homogeneous, isotropic fluid.9-11 Similar behavior can be explained in terms of the existence of a metastable branch on the flow diagram. Under straincontrolled conditions, a peak on the quasi-static flow diagram (σmax > σp) was experimentally observed and immediately followed by a plateau in the stress.12 The segment of this stress (3) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712. (4) Cates, M. E. J. Phys. Chem. 1990, 94, 371. (5) Spenley, N. A.; Cates, M. E.; McLeish, T. C. B. Phys. ReV. Lett. 1993, 71, 939. (6) Spenley, N. A.; Yuan, X. F.; Cates, M. E. J. Phys. II 1996, 6, 551. (7) Olmstead, P. D.; Lu, C.-Y. D. Phys. ReV. E 1997, 56, R55 (8) Berret, J. F.; Roux, D. C.; Porte, G.; Lindner, P. Europhys. Lett. 1994, 25, 521. (9) Berret, J. F. Langmuir 1997, 13, 2227. (10) Berret, J. F.; Roux, D. C.; Lindner, P. Eur. Phys. J. B 1998, 5, 67. (11) Decruppe, J. P.; Lerouge, S.; Berret, J. F. Phys. ReV. E 2001, 63, 022501. (12) Grand, C.; Arrault, J.; Cates, M. E. J. Phys. II 1997, 7, 1071.
10.1021/la0519453 CCC: $33.50 © 2006 American Chemical Society Published on Web 03/24/2006
Viscoelastic Mixed Surfactant Solution Rheology
peak where stress increases with shear rate (before σmax is reached) is the metastable branch of the flow curve, and the segment where stress decreases with increasing shear rate from σmax to σp is a region of unstable flow. In the metastable region, the system changes so slowly that a steady flow is not achieved within the experimental time scale (with the typical measurement time being several minutes for each data point on the quasistatic flow diagram).12 The time constant associated with these changes diverges as τss ∝ (γ˘ - γ˘ c1)-p (with p ≈ 2), and in the shear rate window defining the metastable branch, it becomes several orders of magnitude longer than τ0. This explains why the system may exist, for a very long time, in the metastable state, and this very long process may be associated with the formation and structural evolution of the shear bands. In contrast to this, for some micellar solutions3 a smoother crossover to a plateau than for systems with a true metastable branch is observed. The fundamental difference between these two mechanisms is poorly understood; however, in both cases shear banding is considered to be the universal mechanism that explains the underlying flow diagram, and it has been observed by rheo-optical methods,13-15 NMR,16 light scattering,17 and, more recently, by ultrasonic velocimetry.18 Shear banding is similar, to some extent, to periodic pattern formation in nematic liquid crystals in a rotating magnetic field.19-21 Whereas in nematics periodic structures provide the fastest reduction of magnetic free energy, shear banding may result in the fastest reduction of accumulated elastic energy. The analogy between dynamic pattern-forming systems and complex fluids under shear is consistent with temporary fluctuations of shear stress reported for viscoelastic micellar solutions22 and lamellar phases.23 These fluctuating micellar systems were theoretically investigated by Fielding and Olmsted24 and by Cates et al.,25 and this type of flow, often called rheochaos,23 is defined as the occurrence of macroscopic chaos in viscoelastic materials at low Reynolds numbers. The transition from steady to chaotic flow occurs at a certain (global) shear rate γ˘ c2 that may be very close to the critical shear rate that corresponds to the onset of the stress plateau, γp. By analogy to liquid crystals,26,28 the critical value γ˘ c2 that controls the onset of fluctuations should depend on a certain combination of static and dynamic material parameters. The purpose of this article is to investigate the nonlinear flow and the onset of instabilities in micellar solutions and the parameters controlling the coupling between flow and micellar microstructure. Here, the effect of viscoelastic material parameters on the nonlinear flow of rodlike micellar mixtures of the anioniczwitterionic surfactants sodium dodecyl sulfate (SDS)/lauryl amidopropyl betaine (LAPB) is systematically investigated. For (13) Makhloufi, R.; Decruppe, J. P.; Ait-Ali, A.; Cressely, R. Europhys. Lett. 1995, 32, 253. (14) Berret, J. F.; Porte, G. Phys. ReV. E 1997, 55, 1668. (15) Cappelaere, E.; Berret, J. F.; Decruppe, J. P.; Cressely, R.; Lindner, P. Phys. ReV. E 1997, 56, 1869. (16) Callaghan, P. T.; Cates, M. E.; Rofe, C. G.; Smeulders, J. B. A. F. J. Phys. II 1996, 6, 375. (17) Hu, Y. T.; Boltenhagen, P.; Matthys E.; Pine, D. J. J. Rheol. 1998, 42, 1209. (18) Becu, L.; Manneville, S.; Colin, A. Phys. ReV. Lett. 2004, 93, 018301. (19) Migler, K. B.; Meyer, R. B. Physica D 1994, 71, 412. (20) Migler, K. B.; Meyer, R. B. J. Phys. ReV. Lett. 1991, 66, 1485. (21) Pashkovsky, E. E.; Stille, W.; Strobl, G.; Talebi. J. Phys. II France 1997, 7, 707. (22) Bandyopadhyay, R.; Basappa, G.; Sood, A. K. Phys. ReV. Lett. 2000, 84, 2022. (23) Salmon, J.-B.; Colin, A.; Roux, D. Phys. ReV. E 2002, 66, 031505. (24) Fielding, S. M.; Olmsted, P. D. Phys. ReV. Lett. 2004, 92, 084502. (25) Cates, M. E.; Head, D. A.; Ajdari, A. Phys. ReV. E 2002, 66, 025202(R). (26) Pashkovsky, E. E.; Stille, W. Strobl, G. J. Phys. II France 1995, 5, 397. (27) Pashkovsky, E.; Litvina, T. Macromolecules 1995, 28, 1818.
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these systems, we observe that the ionic strength has a significant effect on micellar microstructure. The elastic modulus monotonically increases with increasing ionic strength whereas the dynamic variables (i.e., the reptation and breaking times) change in a more complex way. Here, the effect of ionic strength on the microscopic parameters of this system is reported in great detail. Additionally, several representative solutions were selected in order to analyze the effect of material parameters on the nonlinear flow behavior. Quasi-static and transient stress responses were measured and interpreted in order to emphasize the effect of these parameters on nonlinear flow. The time dependence of the shear stress (i.e., stress fluctuations) was carefully analyzed, and the viscoelastic material parameters that affect the onset and the amplitude of stress fluctuations have been determined. 2. Experimental Section Materials. Sodium dodecyl sulfate (SDS) and laurylamidopropyl betaine (LAPB) were obtained from Aldrich and Hunstman Surface Sciences, respectively. The SDS used was a 99% pure ACS-grade reagent, and LAPB was obtained as a 30 wt % aqueous solution. Both surfactants were used as is, without any further purification. Sodium chloride was an ACS-grade reagent obtained from J. T. Baker. Several surfactant solutions containing a 2:1 ratio of SLS/ LAPB and different amounts of sodium chloride in the range of 0.5-5.0 wt % were prepared. The total surfactant concentration was varied in the range of 8-12 wt % in order to study surfactant concentration effects, and all solutions were prepared using DI water. Measurements. Samples were kept for at least 24 h before any rheological measurements were taken. To determine viscoelastic material parameters such as G0, τ0, and τb, dynamic oscillation experiments were carried out in a strain-controlled ARES LS rheometer in the frequency range of 0.1 e ω e 100 for solutions of different surfactant concentrations, φ, and NaCl concentrations, cs. The viscoelastic response of each system to a small oscillatory strain was measured in terms of G′ and G′′ and was used to calculate the material parameters. The nonlinear flow behavior of selected samples exhibiting different microstructural parameters has also been carefully studied. Steady-state flow experiments have been carried out for 0.1 e γ˘ e 100 s-1 with the time between measurements being ∆t . τ0. Transient (start-up) stress responses have also been recorded from step-rate experiments, where a constant shear rate is applied to the sample and the stress response is measured as a function of time for 1000 s. For all experiments conducted on the ARES rheometer, cone and plate geometry was used (50 mm cone diameter, 0.2 rad cone angle) with the gap set at 50 µm. Finally, shear stress time series were recorded using a stress-controlled AR2000 rheometer (TA Instruments), also using cone and plate geometry (40 mm cone diameter, 2° cone angle). In the strain-controlled regime, the shear stress time series were recorded at constant shear rate, which was found to be very stable (var(γ˘ )/〈γ˘ 〉t < 0.1%) with a typical feedback time of 60 ms. In the stress-controlled regime, the shear rate time series were recorded at constant stress. In this case, the feedback time is 25 ms. Typically, 5 data points per second were collected (200 ms per point) in both cases. In all experiments, the temperature was controlled by a Peltier at 23 ( 0.1 °C, and a solvent trap was used.
3. Results and Discussion Linear Viscoelasticity. In the semidilute concentration range, aqueous surfactant solutions exist as a network of entangled wormlike micelles. The viscoelastic properties of such systems are similar to those of reversibly breakable entangled polymer solutions, often called “living polymers”. As proposed by Cates,1 the dynamics of stress relaxation for these linear living polymer chains involves two main mechanisms: reptation and breaking. Whenever chain breakage occurs on a relevant time scale with respect to reptation, that is τb < τR, the relaxation time, τ0, depends on the reptation time, τR, and on the breaking time, τb, as τ0 = (τbτR)1/2. The reptation time scale is controlled by the diffusion
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Figure 1. (a) Normalized storage and loss moduli G′ and G′′ for SLS/LAPB systems (12 wt % total surfactant concentration) at various NaCl concentrations. Dashed symbols represent G′, and open symbols represent G′′. (b) Normalized Cole-Cole diagrams for the same samples. The legend for plot a also applies to plot b.
of micelles from a tube created by constraints imposed by other micelles,28,29 and the breaking time is governed by the kinetics of chain scission and recombination reactions.1 Whenever τb , τR, the chains break and recombine many times on the time scale of reptation, and in this case, the system behaves as a Maxwellian fluid, where the spectrum of relaxation times is reduced to a single Maxwellian relaxation time τ0.1 In this case, a model proposed by Turner and Cates30 can be used to calculate τb, and the value τR can be estimated from τ0 = (τbτR)1/2 provided that the behavior is Maxwellian (this was not applied to very low salt concentrations). In accordance with the Turner-Cates model, τb was determined here from the Cole-Cole plot using the relationship τb ) τ0(1 - 1/ζ), where ζ corresponds to the intercept between the line tangent to the semicircle and the x axis, G′/G0. A rapid decrease in the breaking time occurs in the regime where the micelles are linear (i.e., for cs < cmax, where cmax is the salt concentration corresponding to the maximum relaxation time). For higher salt concentrations, τb decreases slightly and ultimately plateaus at τb , τ0. The linear viscoelastic properties of an SDS/LAPB system were measured as a function of NaCl concentration (cs) and total surfactant concentration (φ). The behavior of the elastic and loss moduli, G′ and G′′, in the frequency range of 0.1 e ω e 100 rads/s is shown in Figure 1a. The Cole-Cole plots obtained from these data (Figure 1b) demonstrate that the system behaves as a Maxwellian fluid, except at very high frequencies where significant deviations from the model are observed for samples with cs e 1.0%. The elastic plateau modulus, G0, and the relaxation time, τ0, can be obtained from the crossover between G′ and G′′ (G0 ) 2Gcross and τ0 ) ωcross-1 31). For these systems, G′′ upturns associated with Rouse modes that usually occur at high frequencies are not detected, possibly because Rouse modes are outside the experimental frequency window. Our data shows that the value of the plateau modulus, G0, increases strongly with the addition of NaCl (Figure 2a). For well-studied cationic rodlike micellar systems such as cetyl trimethylammonium bromide (CTAB)/sodium salicylate (NaSal),32 the elastic modulus, which is inversely proportional to the network mesh size, G0 ) kBT/ξ3,29 is not usually affected by the increase in ionic strength. This is because the average contour length of the micelles is much longer then the mesh size, L˜ . ξ, thus the addition of salt, which promotes an increase in (28) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, U.K., 1986. (29) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca NY, 1979. (30) Turner, M. S.; Cates M. E. Langmuir 1991, 7, 1590. (31) Goodwin, J. W.; Hughes, R. W. Rheology for Chemists: An Introduction; Royal Society of Chemistry: Cambridge, U.K., 2000. (32) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081.
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Figure 2. Changes in material parameters (a) G0 and (b) τ0 as a function of salt concentration cs for three different surfactant concentrations, 12 (0), 10 (O), and 8% (4). cmax is defined as the salt concentration corresponding to the maximum τ0 value. The lines in plot b are simply to guide the eye.
Figure 3. Changes in the time scale for breaking (main graph) and reptation (inset) as a function of salt concentration for φ ) 12 (0), 10 (O), and 8% (4). The lines are simply to guide the eye.
L˜ , should not modify the local structure of the micellar network on the much shorter length scale characterized by ξ. However, there have been instances in which an increase in G0 with the addition of salt has been reported.33,34 In such instances, the authors suggested that this increase is due to a progressive screening of electrostatic repulsions due to the addition of salt.34 For the SDS/LAPB systems described here, three regions are observed on the graph of G0 versus cs for different surfactant concentrations, φ ) 8, 10, and 12% (Figure 2a). A rapid increase in G0 is observed for cs < cmax (Figure 2b), followed by a nearplateau region of very little increase in G0 (cs < cmax) and then another sharp increase at high salt concentrations, cs . cmax. The first region (0 < cs < cmax) corresponds to an increase in the population of long, linear rodlike aggregates and a depletion of small micellar structures due to the screening of electrostatic interactions between the surfactant headgroups. In this region, τ0 increases with increasing cs (Figure 2b). Existing rodlike structures grow, causing the reptation time, which scales with the micelle length approximately as τR ∝ L˜ 3.4, to increase (Figure 3, inset). The value of cmax decreases with φ most likely because of the contribution of surfactant counterions to the total value of the ionic strength. At cs ≈ cmax, all electrostatic repulsions have been screened out, and in this vicinity, the micellar network behaves as a classic linear micellar system. Therefore, the modulus changes only slightly in the region where cs < cmax. For cs . cmax, G0 increases again sharply, possibly because of micelle branching. For c > cmax, τ0 and τR both decrease, presumably because of micellar branching at high ionic strength.35,36 The screening of electrostatic interactions and the increase in micellar (33) Kern, F.; Lequeux, F.; Zana, R.; Candau, S. J. Langmuir 1994, 10, 1714. (34) Candau, S. J.; Hebraud, P.; Schmitt, V.; Lequeux, F.; Kern, F.; Zana, R. NuoVo Cimento 1994, 16, 1401. (35) Porte, G.; Gomati, R.; Haitamy, O; Appell, L.; Marignan, J. J. Phys. Chem. 1986, 90, 5746.
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Figure 4. Surfactant concentration dependence of the material parameters: (a) G0 and (b) τ0 for SDS/LAPB solutions at 1.0 wt % NaCl (0), 1.8 wt % NaCl (4), and 3.0 wt % NaCl (O).
length cause the breaking time, τb to decrease by nearly 1 order of magnitude (Figure 3). For the binary SDS/LAPB system in the region where cs < cmax, where the system is presumably very polydisperse, the dependence of the viscoelastic parameters on surfactant concentration is much stronger than that predicted for polymeric systems.29,36 For polymer systems, the elastic modulus scales as G0 ∝ φ2.25 for both linear and branched chains,37 and τ0 ∝ φ1.25 for linear chains and τ0 ∝ φ0.25 for branched ones.1,38 In our case, at cs ) 1%, G0 ∝ φ3.49, and τ0 ∝ φ4.63 (Figure 4). At cs ) cmax ) 1.8%, the dependencies of G0 and τ0 on φ are very close to those predicted for linear micelles (Figure 4). This indicates that at cs ) cmax the electrostatic interactions have been screened out, so the system behaves as a classic polymer network. At higher NaCl concentrations, the power law exponents are much smaller than the predicted values, and the exponent of the relaxation time is negative: for cs ) 3%, G0 ∝ φ1.75, and τ0 ∝ φ-1.25. The analysis presented here shows that the microstructural changes in the micellar network induced by salt (NaCl) affect both the elastic and dynamic material parameters of the system (G0 and τ0, τR, and τb) dramatically. In the following discussion, the intriguing nonlinear flow behavior of this mixed surfactant system will be analyzed in an effort to find a relationship between the flow behavior and the material parameters that define the microstructure of this system. Nonlinear Flow. The nonlinear flow of micellar systems represents an example of “anomalous rheology”.39 The common feature of anomalous flow is the nonmonotonic shape of the shear stress function σ12(t, γ˘ ), explained as a localization of strain or mechanical flow instability that is due to either wall slip40 or shear banding.41 For micellar systems, shear banding is considered to be the universal mechanism for explaining this anomalous flow behavior. Although shear bending has been extensively investigated, much remains to be explained. For instance, what determines the shear rates of the respective bands and the value of the global stress, σp, shared by the bands, and what are the mechanisms associated with the formation of a shear banded flow? Here, the effect of changes in the viscoelastic parameters (more specifically τb and G0) on the nonlinear flow behavior of the (36) Rubinstein, M. Theoretical Challenges in the Dynamics of Complex Fluids; NATO ASI Series; McLeish, T., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1997; p 21. (37) Lequeux, F.; Candau, S. J. Theoretical Challenges in the Dynamics of Complex Fluids; NATO ASI Series; McLeish, T., Ed.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1997; p 181. (38) Kern, F.; Zana, R.; Candau, S. J. Langmuir 1991, 7, 1344. (39) Larson, R. G. Structure and Rheology of Complex Fluids; Oxford University Press: Oxford, U.K., 1999; p 167. (40) Asher, L. A.; Chen Y.-L.; Larson, R. G. J. Rheol. 1995, 31, 519. (41) Makhloufi, R.; Decruppe, J. P.; Ait-Ali, A.; Cressely, R. Europhys. Lett. 1995, 32, 253.
Figure 5. Flow diagrams for 12 wt % SDS/LAPB solutions with 1.0% NaCl (a and c) and 2.7% NaCl (b and d). For these two systems, G0 ) 2146 and 3395 dyn/cm2 and τ0 ) 0.33 and 0.31 s, respectively. In plots a and b, σ/G0 (O) and N1/G0 (0) are plotted as a function of γ˘ τ0. The solid lines show the linear dependence of σ on γ˘ for low shear rates. Plots c and d show the flow behavior in the transition region as a function of ∆t. For plot c, ∆t ) 10 (0), 100 (O), 300 (4), and 500 s (]), and for plot d, ∆t ) 10 (0), 30 (+), and 100 s (O). Critical values are also shown.
surfactant system under study is discussed in detail. The quasistatic flow diagrams for two SDS/LAPB systems containing 1.0 and 2.7% NaCl are shown in Figure 5. The Maxwellian relaxation time for both systems is similar, τ0 = 0.3 s; however, τb and G0 are very different. For the low-salt system, τb ) 0.09 s and G0 ) 2146 dyn/cm2, and for the high-salt system, τb ) 0.02 s and G0 ) 3395 dyn/cm2. These samples were chosen so that the nonlinear flow behavior of two systems with essentially the same relaxation time but different microstructures could be systematically studied. The shear stress σ12(γ˘ ) and the first normal stress difference, N⊥ ≡ σ11 - σ12, have been measured for each shear rate γ˘ over a time interval of ∆t ) 100 s, which is much greater than τ0. At low shear rates, γ˘ < γ˘ c1, both systems exhibit Newtonian behavior with σ12 ∝ γ˘ , whereas at γ˘ c1 the flow becomes non-Newtonian and at γ˘ p a plateau in the shear stress, σp, is observed. Although this plateau is observed in both cases, the transition behavior seems remarkably different for the two systems under study (Figure 5). These differences will be discussed in detail in the paragraphs to follow. In the low-salt case (1.0 wt % NaCl), characterized by relatively long τb and small G0 values, very distinctive maxima are observed for both σ12 and N1, an indication of shear banding with the existence of a metastable branch.12 The reduced critical shear rate at which the transition from Newtonian to non-Newtonian behavior occurs is γ˘ c1τ0 = 0.15, and the occurrence of the stress plateau, σp = 0.25 G0, is first observed at γ˘ pτ0 = 0.39 (Figure 5a). Furthermore, σ12(γ˘ ) and N1(γ˘ ) cross over at γ˘ τ0 = 1.85, a shear rate that is far greater than γ˘ pτ0. The stress plateau value is well below σp = 0.67G0, which is the value predicted by the reptation-reaction model.5 This effect has been previously observed for charged polymers42,43 and for rodlike micelles near the isotropic-nematic transition34,44,45 and has been explained (42) Barrat, J. L.; Joanny, J. F. J. Phys. II 1994, 4, 1089. (43) Witten, T. A.; Pincus, P. A. J. Phys. II 1994, 4, 1103. (44) Berret, J. F.; Roux, D. C.; Porte, G. J. Phys. II France 1994, 4, 1261.
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Figure 6. Stress decay from start-up experiments for 12 wt % SDS/LAPB with 1.0% NaCl. (a) Metastable region; graphs correspond to γ˘ τ0 ) 0.17 (O), 0.20 (0), 0.22 (4), 0.23 (]), and 0.27 (+). Solid line indicates value of σp/G0. (b) Plateau region; curves shown correspond to γ˘ τ0 values of 0.50 (]), 0.67 (0), 1.00 (O), and 1.34 (4). Insets show a sigmoidal and an exponential fit to eq 1.
as the effect of electrostatic interactions of an orientational nature leading to the correlated motion of neighboring micelles and therefore to a decrease in the overall stress.33,34 The existence of a stress overshoot (σmax > σp) indicates that the response of the fluid to the applied shear rate near the transition to the nonlinear regime is a very slow one and that in the vicinity of γ˘ c1 the time required to achieve a steady state at each given γ˘ value can be very long and in fact should diverge near γ˘ ) γ˘ (σmax) if we are dealing with a true metastable branch.12 The selection of ∆t significantly affects the shape of the metastable region of the flow curve (Figure 5c). In the range of 10 s e ∆t e 500 s, the height of the stress peak decreases with increasing ∆t; however, the overshoot is still present for measuring times as large as 500 s, several orders of magnitude larger than τ0, indicating the strong possibility of a true metastable branch. Similar behavior has been observed for cationic surfactant systems including aqueous solutions of ethanediyl-R,ω-bis(dodecylmethylammonium bromide)/NaCl34 and of CPyCl/NaSal.12 Start-up flow experiments further confirm the existence of this metastable state. When the applied shear rate exceeds γ˘ c1, a stress overshoot is observed upon the inception of flow, and for a range of shear rates, γ˘ c1 < γ˘ < γ˘ max, corresponding to the increasing portion of the stress overshoot in the flow diagram (Figure 5a), an initial decay of the stress to a value larger than σP is observed. Within this range, the stress never decays to the plateau value (Figure 6a), at least not within the experimental time scale (t ) 1000 s). This proves the existence of a metastable branch in which the system reacts to the flow field very slowly and therefore remains in the metastable state for very long periods of time, perhaps indefinitely, as reported previously by Cates and co-workers.12 For γ˘ max < γ˘ < γ˘ p, the flow curve has a negative slope, and the flow is therefore unstable. For γ˘ g γ˘ p, σ(t) actually decays to the observed plateau value within the experimental time scale and τss, the time that it takes for σ(t) to reach σp, can be calculated by fitting each curve to the following equation14
[ ( )]
∆σ(t) ) ∆σ(0) exp -
t τss
ν
(1)
where ∆σ(0) is the initial excess stress, τss is the characteristic time required for the relaxation process to take place, and ν is the exponent characterizing the shape of σ(t) decay curves. The value of τss has been plotted as a function of γ˘ τ0 (Figure 7) for selected shear rates. At γ˘ τ0 ) 0.27, we find that τss ) 720 s, and as γ˘ increases, τss decreases rapidly. The data can be fitted (45) Schmitt, V.; Lequeux, F.; Pousse, A.; Roux, D. Langmuir 1994, 10, 955.
Figure 7. τss values for the low-salt sample (1.0% NaCl) from curve fits plotted against the reduced shear rate; τss diverges at γ˘ cτ0 = 0.22. The dashed line represents γ˘ c1τ0 = 0.15 obtained from the quasi-static flow diagram. The exponent ν is plotted in the inset for the same shear rate values.
according to the following relation: τss ≈ (γ˘ - γ˘ c)-p,12 where γ˘ c is the shear rate at which τss diverges and the exponent p ≈ 2.3. For the low-salt system studied here, τss diverges at γ˘ cτ0 ) 0.22, which is slightly higher than the critical shear rate obtained from the quasi-static flow diagram, γ˘ c1τ0 ) 0.15, leaving a window of shear rates, γ˘ c1 < γ˘ < γ˘ c, for which τss is unbounded (i.e., metastable branch). This explains the occurrence of the stress overshoot in the quasi-static flow diagram: within this γ˘ range, a true steady-state stress value is never achieved. Even for shear rates slightly larger than γc (located in the decreasing part of the overshoot), τss, although finite, is much larger than the measurement time; therefore, the true steady state cannot be observed within the experimental time scale. The exponent for this fit is p = 1.85, which is reasonable considering that even lower values have been reported for other systems.12 The exponent ν (Figure 7 inset) has also been obtained from the selected curve fits using eq 1. At moderate γ˘ values, the stress relaxation is sigmoidal and ν ≈ 2, and at high shear rates (i.e., γ˘ . γ˘ c1), a fast exponential relaxation is observed and ν ≈ 1 (examples of a sigmoidal and an exponential fit can be seen in the Figure 6b insets). This transition has been associated with a transition from a metastable to an unstable flow regime, by analogy to conventional phase transitions.14 As the shear rate is increased beyond the critical value γ˘ c1, the excess stress (σmax - σp) increases, and once it reaches a characteristic value, a transition from the metastable regime to a truly unstable regime is observed. We now shift our discussion to the SDS/LAPB system (12 wt % surfactants) with 2.7% NaCl, referred to here as the high-salt sample and characterized by a very short τb and high G0. In contrast to the low-salt system, a stress overshoot in the flow diagram is not detected for this system, but instead a slow, smooth transition into the plateau region is observed, with a slight inflection of the σ12(γ˘ ) curve in the region γ˘ c1 < γ˘ < γ˘ p (Figure 5b). For this system, γ˘ c1τ0 = 0.17 and γ˘ pτ0 = 0.79, and the shear and normal stresses cross over at γ˘ τ0 ≈ 0.85, with σ12/G0 ) N1/G0 = 0.48. In this case, one can argue that the metastable branch is not seen because the system’s response to the applied flow field is very fast, much faster than the measurement time, ∆t ) 100 s. Flow experiments were performed for ∆t values as low as 10 s (instrumental limitation), and the stress overshoot is still not observed (Figure 5d), a likely indication that the flow behavior here is truly different from that in the previous case. The values at which σ12(γ˘ ) and N1 cross over are very close to those predicted by the convective-constraint release (CCR) model developed by Marrucci et al. for the homogeneous flow of entangled polymer solutions.46,47 The model predicts a crossover at γ˘ τ0 = 0.70 and σ12/G0 ) N1/G0 = 0.51. Despite the
Viscoelastic Mixed Surfactant Solution Rheology
Figure 8. Transient stress response for the high-salt system (2.7%NaCl) at γ˘ τ0 ) (a) 0.31 and (b) 0.85.
fact that the CCR model was developed primarily for the entangled polymeric systems, it also seems to describe micellar systems with very short τb’s. It is possible, then, that the slight inflection of the stress curve can be due to the partial disentanglement and alignment of micelles within the homogeneous fluid, prior to the formation of shear bands. This would explain why the crossover values of the reduced stresses, σ12/G0 and N1/G0, are in such good agreement with the CCR model, which predicts the disentanglement of polymerlike structures due to the applied flow field.47 For polymers, the release of constrains may be due to the orientation and the effective stiffening of the chains whereas the micellar scission/recombination events seem to be an additional mechanism of relevance in surfactant solutions. In the transient regime, very interesting and complex stress responses are observed. For γ˘ < γ˘ c1, the behavior is Newtonian, and upon the inception of flow, the stress increases to its steadystate value and remains constant, as expected. In the range γ˘ c1 e γ˘ < γ˘ p (corresponding to the transition from Newtonian flow to the plateau region), σ12(t) increases to a steady-state value smaller than σp, and a slight overshoot is observed followed by relaxation to the steady-state value. This is illustrated in Figure 8a for γ˘ τ0 ) 0.31 and is also observed for all other shear rates within this range. The presence of a stress overshoot confirms that the flow is in fact non-Newtonian prior to the start of the plateau, as suggested by the inflection seen in the quasi-static flow diagram (Figure 5b). The time associated with this relaxation of the stress overshoot is on the order of τss ≈ 1 s, indicating that the system responds very quickly to the applied flow field, in contrast to the low-salt system described earlier. For γ˘ g γ˘ p, the behavior changes once again, and now two distinct decays are observed in the transient stress response. Figure 8b illustrates the decay of stress for γ˘ τ0 ) 0.85, near the beginning of the plateau region (γ˘ pτ0 = 0.79). A stress overshoot is observed, followed by a fast relaxation to some value greater than σp, which is then followed by a very slow sigmoidal relaxation to the steady-state stress value. The first decay remains and once again demonstrates the system’s ability to react very quickly to the applied flow field. The second decay, which characterizes the time required to achieve steady state, is well fitted by eq 1 with τss = 185 s and ν = 0.99. Notice that for the low-salt sample discussed previously ν ≈ 2 in the vicinity of the critical transition to the plateau regime, and in this case, ν ≈ 1. This provides additional evidence that for the high-salt system a metastable branch does not exist in the vicinity of this critical transition. Other curves, collected at higher shear rates, can also be fitted in the same manner (Figure 9 shows the fits for the final decay). The curves indicate that τss decreases with increasing γ˘ to a final value of (46) Marrucci, G. J. Non-Newtonian Fluid Mech. 1996, 62, 279. (47) Marrucci, G.; Ianniruberto, G. Macromol. Symp. 1997, 117, 233.
Langmuir, Vol. 22, No. 9, 2006 3985
Figure 9. Stress decay from start-up experiments for high-salt system (2.7% NaCl) at the following reduced shear rates: γ˘ τ0 = 3.1 (0), where τss ) 10.1 s and ν ) 1.1; γ˘ τ0 = 4.7 (O), where τss = 10.2 s and ν = 1.9; and γ˘ τ0 = 6.3 (+), where τss = 9.4 s and ν = 2.2. Solid lines represent fits to eq 1.
τss ≈ 9 s at γ˘ τ0 = 6.3. Additionally, τss values cannot be fitted as in Figure 7 and do not show divergence at any γ˘ . The most interesting phenomenon, however, is the sudden change observed in ν. The value of the exponent remains close to 1 for shear rates up to γ˘ τ0 = 3.1, and then suddenly at γ˘ τ0 = 4.7, it is closer to 2 (Figure 9). This may be an indication of the existence of a second critical point in the flow diagram for the high-salt system. In fact, as we will discuss in the next section, this sudden change in the exponent ν coincides with the appearance of flow instabilities in the form of observed stress fluctuations. Stress fluctuations are observed for both the high- and low-salt systems, and significant differences are observed both in the nature and amplitude of the fluctuations in each case. Rheochaos. In complex fluids, rheochaos has been observed at low Reynolds numbers in the form of fluctuations in stress (at constant shear rate) or shear rate (at constant stress). In such cases, a true steady state is never achieved, and the system displays temporally and spatially complex behavior.20-22 In the case of strain-controlled experiments, the fluctuations in stress can be recorded, and the time series may be analyzed using wellestablished statistical techniques. In a rheology experiment, however, because of the presence of slow transients, it is difficult to obtain a stable enough time series for such analysis to be reliable.18 Hence, to characterize such complex dynamics, we use the amplitude of stress fluctuations measured at constant shear rates. According to ref 25, the stress oscillates between two limits, which are the Hopf bifurcation points;48 therefore, for γ˘ > γ˘ c2, there is the onset of oscillations with the average amplitude varying as |γ˘ - γ˘ c2|1/2, where γ˘ c2 is the critical shear rate at the onset of fluctuations. It is our goal here to establish the relationship between the material parameters characterizing the micellar microstructure (i.e., G0, τ0, and τb) and the amplitude of oscillations near the onset, where significant visible changes to the sample do not take place. High-shear-rate chaotic behavior beyond the scope of this article. Instead, we concentrate here on the situation when no visible structural changes occur and the data are therefore reproducible. To achieve this reproducibility, samples are presheared at very low shear rates (γ˘ , γ˘ c2) for 500 s prior to the beginning of each experiment. Additionally, the first 100 s of data collected is not considered in the statistical analysis. At shear rates below the onset of oscillations, a very small amount of instrumental noise (
