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Birefringence Banding in a Micellar Solution or the Complexity of Heterogeneous Flows Sandra Lerouge,*,† Jean-Paul Decruppe,‡ and Peter Olmsted§ Laboratoire de Biorhe´ ologie et d’Hydrodynamique Physique, Universite´ Paris 7, 2 Place Jussieu, 75251 Paris, France, Laboratoire de Physique des Liquides et Interfaces, Groupe Rhe´ ophysique des Colloı¨des, Universite´ de Metz, 1 Bd. D.F. Arago 57078 Metz, France, and Polymer IRC and School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom Received July 22, 2004. In Final Form: September 24, 2004 We have investigated the shear flow behavior of a classical viscoelastic equimolar wormlike micellar system made of cetyltrimethylammonium bromide and potassium bromide in the semidilute regime using mechanical and optical measurements. The experimental flow curve of this surfactant solution exhibits, above a critical shear rate, a well-defined stress plateau, characteristic of a flow of the shear-banding type. We first focus on the rheological and rheo-optical transient response of the sample after the sudden startup of flow. The time-dependent stress profiles are strongly similar to those observed on various other systems with the occurrence of an overshoot at short times followed by a stretched exponential relaxation toward the steady state on a long time scale. This behavior is then correlated to the temporal evolution of the birefringence intensity and the extinction angle; the latter exhibits an undershoot just after the inception of the flow. Using direct visualizations of the sheared sample and spatially resolved flow birefringence accross the gap of the Couette cell, we have been able to highlight a peculiar banding structure made up of three distinct regions: two layers of homogeneous but strongly differing orientations located against the walls, separated by a mixed layer, the width of which can reach half of the gap as a function of the effective applied shear rate. The induced structures contained in the band adjacent to the inner moving cylinder are found to be almost fully aligned along the flow direction. The relative proportions of each region are derived from the orientation profiles and compared to the predictions of the lever rule. The results suggest that orientation bands and shear bands are not linked in an obvious way, and the observed band structure can finally be interpreted as the coexistence either of three distinct “phases” or of only two homogeneous phases separated by an interface which can be broad, or thin and fluctuating.
Introduction For a wide class of complex fluids of various microstructures, the application of a shear flow can induce dynamic instabilities and phase transitions. This is, for example, the case of wormlike micellar systems,1-5 lyotropic lamellar phases,6-9 telechelic polymer networks,10-12 and side-chain liquid-crystal polymers.13 The mechanical signature of this type of phenomenon is the existence of a non-monotonic constitutive relation between the shear stress (σ) and the shear rate (γ˘ ). In the case of a behavior of the shear-thinning type, the flow curve is made up of a lower branch of high viscosity supporting low shear rates * Corresponding author. Fax: 01 44 27 40 35. E-mail: slerouge@ ccr.jussieu.fr. † Universite ´ Paris 7. ‡ Universite ´ de Metz. § University of Leeds. (1) Berret, J. F.; Roux, D. C.; Porte, G.; Lindner, P. Europhys. Lett. 1994, 25, 521. (2) Schmitt, V.; Lequeux, F.; Pousse, A.; Roux, D. Langmuir 1994, 10, 955. (3) Berret, J. F.; Roux, D. C.; Porte, G. J. Phys. II 1994, 4, 1261. (4) Callaghan, P. T.; Cates, M. E.; Rofe, C. J.; Smeulders, J. B. A. F. J. Phys. II 1996, 6, 375. (5) Berret, J. F. http://xxx.lanl.gov/abs/cond-mat/0406681. (6) Roux, D. C.; Nallet, F.; Diat, O. Europhys. Lett. 1993, 24, 53. (7) Diat, O.; Roux, D.; Nallet, F. J. Phys. II 1993, 3, 1427. (8) Diat, O.; Roux, D.; Nallet, F. Phys. Rev. E 1995, 51, 3296. (9) Salmon, J. B.; Manneville, S.; Colin, A. Phys. Rev. E 2003, 68, 051503, 051504. (10) Tam, K. C.; Jenkins, R. D.; Winnik, M. A.; Bassett, D. R.; Macromolecules 1998, 31, 4149. (11) Berret, J. F.; Serero, Y.; Winkelman, B.; Calvet, D.; Collet, A.; Viguier, M. J. Rheol. 2001, 45, 477. (12) Berret, J. F.; Serero, Y. Phys. Rev. Lett. 2001, 87, 4. (13) Pujolle, C.; Noirez, L. Nature 2001, 409, 167.
and an upper branch of low viscosity supporting high shear rates, both being separated by a region of decreasing shear stress14-17 (see Figure 1a). Hence, if the applied mean shear rate is between γ˘ M and γ˘ m, the initially homogeneous flow becomes unstable and evolves toward a stationary state where regions subjected to different shear rates (corresponding to the low (γ˘ l) and high shear rate (γ˘ h) branches) coexist. A modification of the applied shear rate only affects the relative proportions of each layer (Rl and Rh) in order to satisfy the lever rule, which in the case of two bands takes the following form:16
γ˘ ) Rlγ˘ l + Rhγ˘ h
(1)
From an experimental point of view, the onset of a stress plateau beyond a critical shear rate in the steady-state flow curve has been related to the existence of such a non-homogeneous flow. Among the complex fluids evoked above and for which a stress plateau has been highlighted, the wormlike micellar systems have been the subject of the most intensive surveys. Giant micelles are locally cylindrical one-dimensional aggregates, the physical properties of which are linked to their flexibility and their reversible nature. Above the (14) Doı¨, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (15) McLeish, T. C. B.; Ball, R. C. J. Polym. Sci., Polym. Phys. Ed. 1986, 24, 1735. McLeish, T. C. B. J. Polym. Sci., Polym. Phys. Ed. 1987, 25, 2253. (16) Cates, M. E.; McLeish, T. C. B.; Marrucci, G. Europhys. Lett. 1993, 21, 451. (17) Spenley, N. A.; Cates, M. E.; McLeish, T. C. B. Phys. Rev. Lett. 1993, 71, 939.
10.1021/la0481593 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/16/2004
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Figure 1. (a) Schematic representation of a non-monotonic flow curve as predicted by Cates’s model. The dotted part corresponds to an unstable flow. (b) Schematic representation of the gap (plane (v b, 3 B v)) as viewed in flow birefringence experiments. The inhomogeneity of the flow field is deduced from the molecular alignment in each band. Only the relative proportions Rh and Rl of each band vary with the macroscopic shear rate in order to satisfy the lever rule.
overlap concentration, these elongated objects form a viscoelastic network showing thus strong analogies with polymer melts or semidilute solutions.18-20 However, contrary to polymeric chains, the size of the micelles fluctuates continuously in the course of time due to reversible scissions experienced by the chains. Hence, the dynamics of these structures is governed by the combination of two characteristic times, the reptation time and the breakage time. Using a microscopic approach, Cates21 has related this double mechanism of relaxation to the terminal time.16 This model extended to the nonlinear regime16 predicts in the semidilute concentration range a non-monotonic evolution of the shear stress as a function of the shear rate akin to that of Figure 1a. Parallel to these theoretical predictions, several experimental papers have reported the onset of a stress plateau above a threshold shear at once for concentrated1,2,5 and semidilute wormlike micellar solutions.3,22 In the first case, the combination of rheology and smallangle neutron scattering under shear (SANSUS) has led to identify an isotropic/nematic transition triggered by shear, whereas flow birefringence (FB) visualizations have supported the picture of coexisting bands containing respectively highly and weakly ordered entities.23-25 The state of shear banding consistent with the non-monotonic evolution of Figure 1a has been deduced from the molecular alignment in each band (see Figure 1b). For the semidilute solutions, the FB experiments carried out in a Couette cell on different samples have revealed that the discontinuity of slope in the flow curve is (18) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081. (19) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988, 4, 364. (20) Candau, S. J.; Hirsch, E.; Zana, R. Physics of complex supermolecular fluids; Wiley: New York, 1987; 569. (21) Cates, M. E. Macromolecules 1987, 20, 2289.
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associated with a macroscopic phase separation into two concentric layers of widely differing optical properties. However, the steady-state banding structure and consequently the organization state of micellar aggregates present important changes from one sample to another. In some cases, the bands seem homogeneous,27 while in other cases, the induced band is extremely striated and made up of subbands.26,28-30 Another technique of investigation, namely, nuclear magnetic resonance (NMR) velocimetry, appears to be useful for the study of constitutive properties. Like FB, it presents the great advantage of providing information on the flow properties at a local level. This technique has been developped by Callaghan and co-workers4 who mainly focused on the classical CPCl/NaSal solution originally studied by Rehage et al.22 In cone and plate flow, they found a three-band structure, with the width of the interfaces remaining practically constant on a wide range of imposed effective shear rates.31-33 In Couette geometry, they observed that the high shear rate band, which is not located near the inner rotating cylinder where the stress is maximum, is extremely narrow and slowly migrating on a time scale of several minutes.34,35 Very recently, a dynamic light scattering technique has been used to measure velocity profiles in wormlike micellar solutions.36 The authors demonstrated that the semidilute CPCl/NaSal sample, extensively studied by Berret et al.,3,26,38 follows the simplest scenario of shear banding with two bands and one sharp interface, with the thickness of the highly sheared band increasing linearly with increasing shear rate according to eq 1. They also emphasized the highly non-Newtonian character of the induced phase. Some authors have also investigated the kinetics of the formation of the band structure in response to an external applied steplike shear rate comprised between γ˘ l and γ˘ h. The time evolution of the shear stress (σ(t)) has been ascribed to a nucleation and one-dimensional growth process by Berret et al.,3,38 while Grand et al. put forward the formation and the slow migration of the interface between bands.39 In each case, the existence of a window of metastability ruled out the “top jumping” criterion for selection of the stationary shear stress. Let us note that recent works have demonstrated that top jumping de(22) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712; Mol. Phys. 1991, 74, 933. (23) Decruppe, J. P.; Cressely, R.; Makhloufi, R.; Cappelaere, E. Colloid Polym. Sci. 1995, 273, 346. (24) Cappelaere, E.; Berret, J. F.; Decruppe, J. P.; Cressely, R.; Lindner, P. Phys. Rev. E 1997, 56, 1869. (25) Berret, J. F.; Roux, D. C.; Lindner, P. Eur. Phys. J. B 1998, 5, 67. (26) Berret, J. F.; Porte, G.; Decruppe, J. P. Phys. Rev. E 1997, 55, 1. (27) Decruppe, J. P.; Cappelaere, E.; Cressely, R. J. Phys. II 1997, 7, 257. (28) Lerouge, S.; Decruppe, J. P.; Berret, J. F. Langmuir 2000, 16, 6464. (29) Decruppe, J. P.; Lerouge, S.; Berret, J. F. Phys. Rev. E 2001, 63, 2. (30) Lerouge, S.; Decruppe, J. P.; Humbert, C. Phys. Rev. Lett. 1998, 81, 5457. (31) Britton, M. M.; Callaghan, P. T. Phys. Rev. Lett. 1997, 78, 26. (32) Britton, M. M.; Callaghan, P. T. Eur. Phys. J. B 1999, 7, 237. (33) Britton, M. M.; Mair, R. W.; Lambert, R. K.; Callaghan P. T. J. Rheol. 1999, 43, 897. (34) Mair, R. W.; Callaghan, P. T. Europhys. Lett. 1996, 36, 719. (35) Mair, R. W.; Callaghan, P. T. J. Rheol. 1997, 41, 901. (36) Salmon, J. B.; Colin, A.; Manneville, S.; Molino, F. Phys. Rev. Lett. 2003, 90, 22. (37) Be´cu, L.; Manneville, S.; Colin, A. Phys. Rev. Lett. 2004, 93, 018301. (38) Berret, J. F. Langmuir 1997, 13, 2227. (39) Grand, C.; Arrault, J.; Cates, M. E. J. Phys. II 1997, 7, 1071.
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termines the onset of the instability in one CPCl/NaSal solution.41 In a previous work, we have tried to accurately correlate the particular kinetics of the mechanical behavior with the one of the microstructural properties.28,29 A strong decoupling between rheological and local optical properties has been highlighted. Such a decoupling has also been observed very recently for concentrated micellar solution by means of ultrasonic velocimetry.37 The authors have also observed some very interesting features such as oscillations of the interface position correlated with strong wall slip and transient nucleation of three-band flows. Despite a substantial quantity of experimental works, many aspects of flow instabilities and phase transitions in wormlike micelles are still subject to conjectures, including the exact steady-state banding structure or the precise mechanisms of formation of the heterogeneous flow. Numerous theoretical works have proposed explanations for most of these unclear points. Recent analyses have established that the incorporation of nonlocal diffusion terms (of stress or shear rate) in standard rheological models42 lifts the degeneracy of the steady-state flow and yields a single coexistence stress independent of flow history or imposed shear rate.43-48 Hence, the stress selection criterion is ensured by the stationarity condition of the interface. The behavior of such modified models has been studied in various flow geometries, providing information on the number of bands, the interface width, and even the stress transients ascribed to the slow migration of the interface.43-45 In cylindrical Couette geometry, the diffusive Johnson-Segalman model predicts a two-band profile whereas the interface width is found to scale with (DτR)1/2, where D governs the relaxation of stress gradients and has the form of a diffusion coefficient. The relaxation of the interface position is believed to follow an exponential evolution with a characteristic time notably depending on the diffusion coefficient.49 In the present paper, we report on rheological and optical properties in both stationary and transient flow of the well-known semidilute wormlike system cetyltrimethylammonium bromide/potassium bromide (CTAB/KBr) in deuterated water. Our first motivation came from the development of our optical device; the improvement of this device has been initiated at the time of a recent study of the shear-thickening transition which occurs in dilute surfactant solutions. We are now able to determine the spatial distribution of the optical quantities in the gap of the Couette cell with a non-negligible resolution (15 µm/pixel),50,51 with this new treatment of the data opening now interesting perspectives, since we are dealing with non-homogeneous flow. (40) Soltero, J. F. A.; Bautista, F.; Puig, J. E.; Manero, O. Langmuir 1999, 15, 1604. (41) Mendez-Sanchez, A. F.; Lopez-Gonzalez, J.; Rolon-Garrido, V. H.; Perez-Gonzalez, J.; de Vargas, L. Rheol. Acta 2003, 42, 56. MendezSanchez, A. F.; Perez-Gonzalez, J.; de Vargas, L.; Castrejon-Pita, J. R.; Castrejon-Pita, A. A.; Huelsz, G. J. Rheol. 2003, 47, 1405. (42) Larson, R. G. Constitutive Equations for Polymer Melts and Solutions; Butterworths: Boston, MA, 1988. (43) Radulescu, O.; Olmsted, P. D.; Lu, C. Y. D. Rheol. Acta 1999, 38, 606. (44) Olmsted, P. D.; Radulescu, O.; Lu, C. Y. D. J. Rheol. 2000, 44, 257. (45) Radulescu, O.; Olmsted, P. D. J. Non-Newtonian Fluid Mech. 2000, 91, 143. (46) Lu, C. Y. D.; Olmsted, P. D.; Ball, R. C. Phys. Rev. Lett. 2000, 84, 642. (47) Dhont, J. K. G. Phys. Rev. E 1999, 60, 4534. (48) Fielding, S.; Olmsted, P. D. Phys. Rev. Lett. 2003, 90, 2245011. (49) Radulescu, O.; Olmsted, P. D.; Decruppe, J. P.; Lerouge, S.; Berret, J. F.; Porte, G. Europhys. Lett. 2003, 62, 230-236.
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Our second motivation found its origin in the recent works of Fischer and Callaghan on the concentrated solution of CTAB in deuterated water originally studied in our own group. These authors, puzzled by the important contrast between the bandwidths observed in birefringence and NMR velocimetry imaging, have performed new experiments combining NMR spectroscopy and velocimetry. They showed that the optical birefringence effects are not correlated in a clear way with the shear bands and that finally the induced nematic phase is not a state of high shear rate and low viscosity.52,53 As already claimed in other papers,28,50,52,53 it became obvious that a complete understanding of these instabilities and phase transitions requires a local description of the phenomena. In this work, we describe the steady-state and transient behavior of the mean rheological and optical quantities. The time dependence of the stress in start-up experiments is similar to that obtained for other samples.3,28,38,39,40,51 The temporal evolution of the birefringence intensity and the extinction angle is consistent with the stress response and shows some features far from being clearly established up to now. We suggest by comparison with the available data in the literature that the qualitative transient behavior (associated with the plateau regime) of the mean rheological and optical quantities is generic. However, the direct visualization of the sheared sample reveals a peculiar band structure made up of three layers. The determination of the orientation profile accross the gap of the Couette cell confirms the existence of two homogeneous bands separated by a mixed layer, with the aggregates in the band located near the rotating inner cylinder being almost fully aligned in the flow direction. Experimental Protocols 1. Materials. In this study, we focus on the issue of the banding structure of a micellar wormlike system containing cetyltrimethylammonium bromide (CTAB) and potassium bromide (KBr) both at a concentration of 0.3 M in deuterated water. The surfactant (M ) 364.45 g/mol), sold by Janssen Chimica, is 99% pure and used without further purification, whereas the simple salt (M ) 119.01 g/mol) is an Alfa product purchased from Johnson Matthley Gmbh. After the preparation, the solution is stored at ∼40 °C in order to reach equilibrium. All the measurements presented here were realized at 34 °C. This classical system has been intensively studied in the past.54,55 It is well-known for forming long flexible wormlike micelles, the persistence length of which is typically ∼150 Å. At the concentration chosen here, the sample is semidilute, that is, made of entangled aggregates, the morphology of which is locally cylindrical with a radius of 21 Å. It is a Maxwellian fluid with a terminal time of τR ) 0.17 ( 0.2 s and a plateau modulus of G0 ) 235 ( 7 Pa. 2. Rheological Experiments. The steady-state data were obtained on a VT550 Haake rheometer and on a Carrimed CSL100 rheometer, using a cone-plane geometry (diameter 5 cm, angle 2°/4 cm, 0.5°) and working respectively in strain- and stress-controlled mode. Transient tests were performed on a rheometric fluid spectrometer (RFS II) using a cone and plane configuration (diameter 3 cm, angle 1.15°) under strain-controlled conditions. Transient tests consisted of recording the stress relaxation data toward the steady state, following the onset of simple shear flow, namely, a sudden step in shear rate. (50) Berret, J. F.; Lerouge, S.; Decruppe, J. P. Langmuir 2002, 18, 7279. (51) Lerouge, S. The`se de Doctorat, Universite´ de Metz, 2000. (52) Fischer, E.; Callaghan, P. T. Europhys. Lett. 2000, 50, 803. (53) Fischer, E.; Callaghan, P. T. Phys. Rev. E 2001, 64, 1. (54) Cates, M. E.; Candau, S. J. J. Phys.: Condens. Matter 1990, 2, 6869. (55) Kern, F.; Lemare´chal, P.; Candau, S. J.; Cates, M. E. Langmuir 1992, 8, 437.
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For each type of experiment, a solvent trap was used in order to minimize any evaporation effect. 3. Optical Experiments. The rheo-optical properties of the solution were studied with the help of the polarimeter built in our lab a few years ago and described in great detail in an earlier paper.56 The shear deformation was imposed using a Couette cell in which the inner cylinder rotates (such devices are usually called Searl systems). The diameters of the inner and outer coaxial cylinders are respectively 47 and 50 mm, giving a gap of 1.5 mm. Typically, a flow birefringence experiment consists of determining the following two main quantities as a function of shear rate, time, or space: the extinction angle (χ) and the birefringence intensity (4n). Indeed, the application of flow to a medium made of objects that are anisotropic from the electronic polarizability point of view generally induces an anisotropy of the refraction index tensor. In this case, the eigenaxes of this tensor are orientated at an angle (χ) compared with the laboratory axis, and the difference between the refraction index associated with each eigenaxis defines the birefringence intensity, 4n ) n1 - n2. In other words, the extinction angle characterizes the average dynamic orientation of the particles relative to the flow direction, whereas the birefringence intensity can simply be related to the retardation (φ) introduced by the biaxial solution by means of the following relation:
λφ 4n ) 2πe
(2)
where λ is the wavelength of the light and e is the optical path which has been fixed at 10 mm. To determine the properties mentioned above, different incident light sources and arrangements of the optical components were used. In each case, the direction of propagation is given by the vorticity axis, probing the birefringence in the (v b, 3 B v) plane. In all these optical experiments, the data are taken after the steady state is achieved. By operating with a He-Ne laser beam (wavelength 6328 Å), one can obtain steady-state and transient results. In this case, the measured quantities are averaged over the size of the laser beam which is ∼500 µm, allowing the measurements to be performed at three different positions in the gap of the Couette cell. In the steady-state regime, we used a null method: the extinction of the laser light by rotating the crossed polarizer/ analyzer pair provides the average orientation (χ) (the zone of extinction is called the isoclinic cross), whereas the birefringence intensity (4n) is deduced from the classical method of Senarmont.57 Transient results are collected according to a procedure similar to the one established by Osaki,58 which is based on the analysis of the transmitted light intensity. 4n is computed from the intensity transmitted through a circular polarizer, the sample under shear, and a circular analyzer (referred to in the following as “circular light assembling”) which can be expressed by
Ic(γ˘ , t) ) I′c sin2
φ(γ˘ , t) 2
(3)
where I′c is a coefficient that depends on the partial reflectivities of the various optical components and φ is related to 4n through eq 2. χ is calculated via the same configuration as the steadystate one, with the Couette cell being placed between crossed polarizers (referred to as “linear light assembling”). The transmitted intensity is given by
Iθ(γ˘ , t) ) I′ sin2
θ(γ˘ , t) sin2 2(χ(γ˘ , t) - θ) 2
(4)
where θ is defined as the angle between the polarization of the incident beam and the streamline. For two particular positions (56) Decruppe, J. P.; Hocquart, R.; Wydro, T.; Cressely, R. J. Phys. (Paris) 1989, 50, 3371. (57) De Se´narmont H. Ann. Chim. Phys. 1840, 73, 337. (58) Osaki, K.; Bessho, N.; Kojimoto, T.; Kurata, M. J. Rheol. 1979, 23, 457.
Figure 2. (9) Steady-state flow curve measured in straincontrolled mode. (4) The flow curve obtained by means of a stress sweep (3 min per data point) is given for comparison. of the crossed polarizer/analyzer (P/A) pair, namely, θ ) 0 and 45°, simple relations can be obtained:
I0(γ˘ , t) ) I′ sin2
φ(γ˘ , t) sin2 2χ(γ˘ , t) 2
(5)
I45(γ, t) ) I′ sin2
φ(γ˘ , t) cos2 2χ(γ˘ , t) 2
(6)
Because of the change of the optical components in this configuration, the prefactor I′ differs from I′c in eq 3. Hence, eq 3 leads to 4n according to
4n(γ˘ , t) )
λ arcsin πe
x
Ic(γ˘ , t) I′c
(7)
and the ratio I0/I45 yields χ:
χ(γ˘ , t) )
1 arctan 2
x
I0(γ˘ , t)
I45(γ˘ , t)
(8)
To increase the spatial resolution, we have also used an extended collimated white light source covering all the visible spectra for global visualizations of the flow field in the steady state. The sample is observed between crossed polarizers. The image of the entire gap is formed on the sensitive area of a chargecoupled device (CCD) camera with the help of a lens. Several snapshots were taken in order to follow the evolution of the band structure as a function of the shear rate. By adding a monochromatic filter (λ ) 5461 Å) we have been able to determine the spatially resolved orientation profiles according to a new procedure described in much more detail in a recent paper50 and elaborated for the study of dilute aqueous surfactant solutions exhibiting a shear-thickening transition. Indeed, as we are dealing with banding, all the optical quantities are found to depend on the spatial coordinate so that equations 3, 5, and 6 can be rewritten as a function of y which represents the spatial coordinate along the gradient direction. The treatment of the rheo-optical data under steady-state conditions is the following: For the shear rates studied here, photos are taken using the linear light assembly. Each picture is then digitized along a particular radius of the Couette cell, thus providing I0 and I45 as a function of y (and thus χ(y)), with a spatial resolution of 15 µm/pixel.
Results and Discussion A. Nonlinear Rheology: Flow Curve and Stress Relaxation. Figure 2 gives the variations, on a semilogarithmic plot, of the shear stress (σ) versus γ˘ obtained under controlled strain-rate conditions. Each point is taken after the steady state is achieved (we consider that the
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Langmuir, Vol. 20, No. 26, 2004 11359 Table 1. Characteristic Relaxation Time (τNG) and Exponent (r) Computed at Various Shear Rates from Rheological and Optical Transient Profiles
Figure 3. Transient response of the shear stress after the onset of flow for several applied shear rates (γ˘ ) 7-14 s-1). The inset represents the time-dependent behavior on a shorter time scale, and the open circles show a typical sigmoidal fit of the stress relaxation.
Figure 4. Steady-state variations of (9) the extinction angle (χ) and of (0) the absolute value of the birefringence intensity (4n) versus γ˘ . The full line illustrates the best fit of the extinction angle with the equation of Thurn.
steady state is reached when the relative variation (4σ/σ) is
