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Langmuir 1997, 13, 3902-3904
Tests of the Lodge-Meissner Relation in Anomalous Nonlinear Step Strain of an Entangled Wormlike Micelle Solution E. F. Brown and W. R. Burghardt* Department of Chemical Engineering, Northwestern University, Evanston, Illinois 60208 D. C. Venerus Department of Chemical & Environmental Engineering, Illinois Institute of Technology, Chicago, Illinois 60616 Received January 10, 1997
Introduction In the presence of salt, certain cationic surfactants can self-assemble to form wormlike micelles of extraordinary length. These solutions exhibit pronounced viscoelasticity, attributed to the formation of a transient entanglement network of micelles analogous to that found in linear polymers of sufficient molecular weight. Unlike synthetic polymers, however, surfactant micelles continually break and re-form, leading to an equilibrium distribution of chain length that is comparatively broad. In light of this omnipresent chain scission and recombination, such systems have been termed “living polymers”. These fluids have attracted considerable interest in recent years.1-9 One of the most intriguing aspects of the rheology of cationic surfactants is the simplicity of their linear relaxation spectrum, which often can be described by a single exponential relaxation process (i.e., as a Maxwell fluid). By combining the tube model for dynamics of entangled chains with a kinetic scheme accounting for chain scission and recombination, Cates10 demonstrated that such single exponential relaxation may be predicted when chain scission is fast compared to reptation. This approach has been further extended to extract specific estimates of the average breaking time, by probing deviations from single exponential behavior that occur at high frequencies.11 On the basis of the success of the modified reptation description of the linear viscoelasticity, Cates further extended the tube model for living polymers to the nonlinear regime.12 The resulting model gives nonlinear predictions that are nearly equivalent to the Doi-Edwards model,13 except for the prediction of single-exponential relaxation. Since polydispersity effects can obscure certain nonlinear rheological phenomena, these cationic surfactant solutions have been identified as model entangled systems. For instance, Spenley et al. have examined shear stress and first normal stress difference data of surfactants * Corresponding author:
[email protected]; 847-467-1401, phone; 847-491-3728, fax. (1) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712. (2) Rehage, H.; Hoffmann, H. Mol. Phys. 1991, 74, 933. (3) Candau, S. J.; Hirsch, E.; Zana, R.; Delsanti, M. Langmuir 1989, 5, 1225. (4) Kern, F.; Zana, R.; Candau, S. J. Langmuir 1991, 7, 1344. (5) Kern, F.; Lemarechal, P.; Candau, S. J.; Cates, M. E. Langmuir 1992, 8, 437. (6) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (7) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081. (8) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988, 4, 354. (9) Shikata, T.; Hirata, H.; Takatori, E.; Osaki, K. J. Non-Newtonian Fluid Mech. 1988, 28, 171. (10) Cates, M. E. Macromolecules 1987, 20, 2289. (11) Turner, M. S.; Cates, M. E. Langmuir 1991, 7, 1590. (12) Cates, M. E. J. Phys. Chem. 1990, 94, 371. (13) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon: Oxford, 1986.
S0743-7463(97)00037-1 CCC: $14.00
in steady shear flow to explore possible consequences of the non-monotonic dependence of shear stress on shear rate predicted by the tube model.14 However, despite superficial similarity in linear viscoelasticity and steady shear viscosity, some classes of cationic surfactant solutions have been shown to exhibit nonlinear rheology that is strikingly different from synthetic linear polymers. In particular, nonlinear step strain experiments of Shikata and co-workers9 have shown that aqueous solutions of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) can either show ordinary shear thinning or strain hardening behavior, depending on the relative salt and surfactant concentrations. Cates recognized that such anomalous rheological phenomena cannot be predicted in the context of the modified tube model12 and noted that they must arise from physical processes of as yet undetermined nature. Indeed, rheo-optical studies of the CTAB/NaSal system have shown failure of the stress-optical rule15 and pronounced structure formation16 in the nonlinear regime, accessed by steady shear at high rates. However, step strain deformations can often be the most direct way to probe nonlinear behavior, and in this note we extend the experiments of Shikata et al.9 to include measurements of N1, applying the test of the Lodge-Meissner relationship, which relates shear and normal stress relaxation following step strain deformations of a wide class of polymers. Experimental Section Cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) were purchased from Aldrich and used as received. In this note we report data on a solution with salt and surfactant concentrations both equal to 0.1 M, dissolved in ultrapure water. This solution mimics one described by Shikata and co-workers,7 although it is somewhat different in composition than the solution they studied in nonlinear step strain.9 Rheological measurements were performed on a Rheometrics RMS-800. The data presented here were obtained using a cone and plate fixture with radius of 25 mm and cone angle of 0.1 rad, allowing nonlinear step strain experiments to be performed with applied strains up to 5. Axial transducer compliance can interfere with collection of accurate transient normal force data due to squeeze flow.17 Using the known axial compliance of the rheometer and zero-shear viscosity, a characteristic normal force response time may be estimated. For the fixture used here, it is 890 times smaller than the fluid relaxation time, so results should be free from compliance artifacts. This was verified by repeating certain experiments with cones of different angles, which gave consistent results.
Results Figure 1 shows measurements of linear storage and loss moduli obtained in oscillatory flow. These data are typical of a wide variety of cationic surfactants that show nearly single exponential relaxation over a wide frequency range.1-8 The solid lines demonstrate that the dynamic rheology may be described by a single relaxation time to a high degree of approximation. Steady shear viscosity data on this solution are difficult to obtain due to expulsion of the sample from the gap at high shear rates; however, the solution shows a typical pattern of a Newtonian plateau followed by strong shear thinning once the shear rate exceeds the inverse of the fluid relaxation time. Figure 2a shows selected measurements of shear stress in step strain, reported as the nonlinear relaxation (14) Spenley, N. A.; Cates, M. E.; McLeish, T. C. B. Phys. Rev. Lett. 1993, 71, 939. (15) Shikata, T.; Dahman, S. J.; Pearson, D. S. Langmuir 1994, 10, 3470. (16) Wheeler, E. K.; Izu, P.; Fuller, G. G. Rheol. Acta 1996, 35, 139. (17) Venerus, D. C.; Kahvand, H. J. Rheol. 1994, 38, 1297.
© 1997 American Chemical Society
Notes
Langmuir, Vol. 13, No. 14, 1997 3903
determined from the data in Figure 1. Similar to the report of Shikata and co-workers,9 we observe strain hardening as the strain increases to values of about 3.5. That is, the nonlinear modulus increases, indicating that shear stress grows faster than linearly with applied strain. Within the strain hardening regime (γ e 3.5), the shear stress response is factorable, with independent time- and straindependencies:
G(t,γ) ) h(γ)G(t)
Figure 1. Storage and loss moduli vs frequency for CTAB/ NaSal solution; solid curve is prediction of Maxwell model, with Go ) 480 dyn/cm2, and λ ) 1.48 s.
(2)
While factorable behavior is typically observed in polymers, h(γ) is usually a strain softening function. Shikata et al. did not show extensive data in the strain hardening regime for their fluid; the limited data they presented do not show such factorability.9 For 3.5 < γ < 4.0, a sharp transition in behavior occurs. At a strain of 3.85, the relaxation modulus initially shows a further increase from that observed at γ ) 3.5. However, at around t ) 0.5 s, the relaxation modulus abruptly drops by about a factor of 5, after which the relaxation continues with roughly the same shape. At higher strains, the relaxation modulus drops even further, and it appears that a similar transition occurs at shorter times (eventually it occurs on times comparable to the application of the applied strain, t < 0.1 s, not shown in Figure 2a). Although the relaxation at long times for large strains appears to occur with a similar time dependence as that for smaller strains, the abrupt drop at intermediate strains means that the nonlinear modulus is not factorable according to eq 2 over the full range of strains. Figure 2b presents results resembling a damping function, plotting nonlinear modulus at fixed time vs strain. The strain hardening for γ e 3.5 is readily apparent, as is the abrupt onset of strain softening beyond this regime. These results are patently unlike those seen in linear polymers. Cates’ nonlinear model12 would predict
G(t,γ) ) Go e-t/λ hDE(γ)
(3)
where Go and λ are determined from linear viscoelasticity and hDE(γ) is the Doi-Edwards prediction of the shear damping function.13 As seen in Figure 2b, this description is inappropriate for this fluid. Most polymers obey the Lodge-Meissner relationship,18 which relates N1 and shear stress relaxation following a step shear strain
N1(t,γ) ) γ σ(t,γ)
Figure 2. Anomalous shear stress behavior in nonlinear step strain. (a) Nonlinear relaxation modulus for indicated strain values; solid curve is linear relaxation modulus. (b) Cross plot of nonlinear relaxation modulus measured at t ) 1 s vs applied strain. Line represents nonlinear step-strain predictions of the modified tube model, eq 3.
modulus
G(t,γ) ) σ(t,γ)/γ
(1)
where σ(t,γ) is the measured shear stress relaxation following a shear strain of magnitude γ. Also shown is the single exponential linear relaxation modulus, G(t),
(4)
Figure 3 presents a test of this relation for the step strain results of Figure 2a. The relationship is closely followed throughout the strain hardening range, γ e 3.5. At a strain of 3.85, the ratio of N1 to σ starts at the magnitude of the applied strain, consistent with eq 4, but at the point where the relaxation modulus drops, this ratio abruptly increases to a value roughly twice as large. At higher strains, where the fluid shows strain softening, the LodgeMeissner relation continues to fail, with normal stresses that are significantly too high in relation to the measured shear stresses. Shikata et al. report that their fluid slipped in the apparatus beyond strains of 4.0, preventing measurement.9 In our case, the free surface of the fluid at the edge of the cone and plate becomes visibly distorted at these strains, but it is not clear whether adhesive failure is occurring at the surfaces of the fixture. If it is, then the slippage is remarkably reproducible. Figure 4 presents (18) Lodge, A. S.; Meissner, J. Rheol. Acta 1972, 11, 351.
3904 Langmuir, Vol. 13, No. 14, 1997
Notes
successive experiments. At higher strains, the surface distortions become sufficiently severe that sample is expelled upon repeated deformation.
Figure 3. Experimental test of the Lodge-Meissner relation for CTAB solution, for indicated strain values.
Figure 4. Tests of reproducibility as sample is cycled between step strains of magnitude γ ) 1 and γ ) 4. Shear stress (____, O) and first normal stress difference (-----, 3) plotted as a function of relaxation time. Curves represent freshly loaded sample, while symbols represent results when sample has previously been subjected to a step strain with γ ) 4. Note that shear stress and normal stress data superimpose for γ ) 1, reflecting the Lodge-Meissner relation.
results of repeated experiments cycling between strains of 1 (basically linear) and 4 (beyond the critical strain). At both strains, measurements obtained after the fluid had been cycled through a strain of 4 are essentially identical to those obtained with a freshly loaded sample. This demonstrates that any structural breakdown resulting from large strains is healed within the 2 min between
Discussion Faithful adherence to the Lodge-Meissner relation and factorable time and strain dependence of G(t,γ) in the strain hardening regime is powerful evidence that, despite the unusual strain hardening, there is underlying order to the rheological response exhibited by the CTAB solution for γ e 3.5. In step strain, the reptation model allows initially affinely deformed Gaussian chains to relieve much of their stretching via the retraction mechanism, leading to strong shear softening of the damping function.13 The basic network theory of rubber elasticity, again assuming affine deformation of Gaussian chains, predicts neither strain hardening nor strain softening, since cross-linked chains are unable to retract. One possible rationalization of the strain hardening may be that, for the salt/surfactant ratios present here, there is insufficient “slack” between entanglements (or transient network junctions) to allow for a Gaussian response. Certainly, the abrupt transition between strains of 3.5 and 4 could result from catastrophic failure of a highly stretched, rather rigid network. It is not clear, however, how such a network rupture would lead to the observed Lodge-Meissner failure, where N1 is enhanced relative to shear stress following the transition. In synthetic polymers, Larson et al. observed failure of the LodgeMeissner rule in very highly entangled solutions.19 One possible explanation was shear banding into high and low strain layers, which could be anticipated from the nonmonotonic predictions of the tube model.20 Since σ and N1 have different strain dependencies, it is possible that a shear banding event could lead to a relative enhancement in N1 compared to the Lodge-Meissner prediction, following an analogous argument to that put forward by Spenley et al. with respect to shear and normal stress measurements in steady shear flow of a different surfactant solution.14 In the case of highly entangled polymer solutions, however, it appears that the ultimate cause of the Lodge-Meissner failure is from wall slip during the stress relaxation.21 It is worthwhile to note that by effectively reducing the applied strain throughout the fluid, such adhesive failure leads to negative deviations from the Lodge-Meissner relation.19 Together with the reproducibility seen above in Figure 4, the positive deviations observed in Figure 3 provide further evidence against slip as the direct cause of the abrupt strain softening observed at large strains in this fluid. Acknowledgment. We acknowledge financial support from the National Science Foundation (CTS-9457083) and a grant from the Exxon Educational Foundation. LA9700376 (19) Larson, R. G.; Khan, S. A.; Raju, V. R. J. Rheol. 1988, 32, 145. (20) Marrucci, G.; Grizzuti, N. J. Rheol. 1983, 27, 433. (21) Archer, L. A.; Chen, Y.-L.; Larson, R. G. J. Rheol. 1995, 39, 519.
