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Langmuir 2000, 16, 6464-6474
Correlations between Rheological and Optical Properties of a Micellar Solution under Shear Banding Flow Sandra Lerouge and Jean-Paul Decruppe* Laboratoire de Physique des Liquides et Interfaces, Groupe Rhe´ ophysique des Colloı¨des, Universite´ de Metz, 1 Bd. D. F. Arago, 57078 Metz, France
Jean-Franc¸ ois Berret Groupe de Dynamique des Phases Condense´ es, Unite´ Mixte de Recherche CNRS/Universite´ de Montpellier II n° 5581, Universite´ de Montpellier II, F-34095 Montpellier Ce´ dex 05, France Received February 25, 2000. In Final Form: May 11, 2000 We report on the nonlinear rheological and optical responses of a micellar viscoelastic solution of 0.3 M cetyltrimethylammonium bromide in water with a mineral salt, 1.79 M sodium nitrate , subjected to different steplike shear rates. This solution, which can be described in the regime of small deformations by the Maxwell model with a single relaxation time (τR), shows a stress plateau in the experimental flow curve (σ(γ˘ )) beyond a critical shear rate γc1. This behavior, characteristic of a system undergoing a phase transition of the shear banding type, is corroborated by steady-state flow birefringence experiments. The transient shear stress profiles (σ(t)) recorded after the startup of flow are very similar to those previously published on the cetylpyridinium chloride/sodium salycilate system. Just after the inception of the flow, σ quickly increases with time and then relaxes toward its stationary value. This relaxation mechanism occurs on two different time scales and has been interpreted as follows: for duration on the order of a few τR, the purely mechanical response of the fluid is observed (overshoot and eventually damped oscillations), while for t . τR, a long sigmoı¨dal decay toward the plateau value occurs, which is typical of nucleation and growth of shear-induced phase processes. Time-dependent measurements of the birefringence intensity and the extinction angle confirm the existence of such an evolution. However, direct visualization of the gap shows that the growth of the induced phase is not associated with the behavior of the stretched exponential, since it occurs after the steady state in stress and optical anisotropy is achieved. The spatial distribution of the transmitted light intensity through the gap indicates that the flow is locally nonhomogeneous: the shear-induced band is made up of small sub-bands closely aligned in the flow direction.
Introduction Under suitable concentration, temperature, and salinity conditions, surfactant molecules aggregate into long, flexible wormlike micelles, which can get entangled like polymers,1-3 conferring a highly viscoelastic character to the solutions. However, contrary to polymeric systems, the length of which is fixed by synthesis, micelles have their own dynamics, since they can reversibly break and reform. By combining these processes with the repetitive disengagement of the chains,4 Cates and co-workers5-7 have elaborated a microscopic model which leads in the regime of fast breaking times to a stress relaxation mechanism governed by a single characteristic relaxation time (Maxwell element) in the linear regime. In the nonlinear domain, this model predicted a nonmonotonic behavior of the shear stress σ as a function of the rate of strain γ˘ (see Figure 1) in the semidilute concentration range. The shear stress at first increases and then drops * Address for proofs: Jean-Paul Decruppe, Laboratoire de Physique des Liquides et Interfaces, Groupe Rhe´ophysique des Colloı¨des, Universite´ de Metz, 1 Bd. D. F. Arago, 57078 Metz, France. Fax: 03 87 31 58 01. E-mail:
[email protected]. (1) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1987, 3, 1081. (2) Shikata, T.; Hirata, H.; Kotaka, T. Langmuir 1988, 4, 364. (3) Candau, S. J.; Hirsch, E.; Zana, R. Physics of Complex Supermolecular Fluids; Wiley: New York, 1987; p 569. (4) Doı¨, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986. (5) Cates, M. E. Macromolecules 1987, 20, 2289. (6) Spenley, N. A.; Cates, M. E.; MacLeish, T. C. B. Phys. Rev. Lett. 1993, 71, 939. (7) Spenley, N. A.; Yuan, X. F.; Cates, M. E. J. Phys. II 1996, 6, 551.
Figure 1. Schematic representation of a nonmonotonic flow curve as predicted by Cates’ model. The dotted part corresponds to an unstable flow.
beyond a shear rate γM, corresponding to the maximum of the σ(γ˘ ) curve. If the applied shear rate lies in the decreasing part of the flow curve, the initially homogeneous flow becomes mechanically unstable. A system subjected to such a strain rate evolves toward a stationary state where bands of liquids supporting low and high shear rates (γ˘ c1 and γ˘ c2) coexist; the relative proportion of each layer changes with the macroscopic shear rate γ˘ . These theoretical predictions have promoted numerous experimental studies under steady-state flow conditions.
10.1021/la000269w CCC: $19.00 © 2000 American Chemical Society Published on Web 07/11/2000
Micellar Solution under Shear Banding Flow
The mechanical characterization of such a nonhomogeneous flow is the existence of a plateau (σ(γ˘ ) ) σp) in the measured flow curve. This type of behavior has been reported for concentrated solutions close to the equilibrium nematic transition and has clearly been identified as the signature of an isotropic-to-nematic firstorder phase transition through rheological measurements, small-angle neutron scattering (SANS) under shear,8-11 and birefringence microscopy.12 But similar mechanical properties have also been observed at lower concentrations (semidilute regime).13,14 Several methods of investigationsnamely, flow birefringence (F.B.)15-17 and, more recently, nuclear magnetic resonance (NMR) velocity imaging18-21slead to the same result: the emergence and the growth with γ˘ of a highly orientated liquid phase near the moving wall of the shearing device. Up to now, only a few authors14,22-27 have focused their attention on the kinetics of formation of the band structure. In particular, Berret et al.14,22 have carefully examined the transient evolution of the shear stress in the startup of flow experiments for the system of cetylpyridinium chloride (CPCl) and sodium salicylate (NaSal) in brine. They distinguished two relaxation processes in the σ(t) profiles: the purely mechanical response (overshoot and eventually damped oscillations), which occurs on a time scale on the order of a few Maxwell times, and a long-time sigmoı¨dal decay from the underlying constitutive curve toward the plateau value. This behavior in the stretched exponential has been interpreted in terms of nucleation and one-dimensional growth mechanisms of the (nematic) phase triggered by shearing so that the purely mechanical instability in the flow curve is preempted by a transition from a homogeneous to a nonhomogeneous flow. Grand et al.23 have found similar variations of the shear stress with time on the same micellar system but in a more dilute regime. They ruled out the “top-jumping” criterion for selection of steady-state shear stress and showed the existence of a metastable branch, on which the system can remain indefinitely or not, depending on the composition. Parallel to these experiments, some theoretical works7,28-32 tried with a modified Johnson-Segalman (8) Berret, J. F.; Roux, D. C.; Porte, G.; Lindner, P. Europhys. Lett. 1994, 25, 521. (9) Schmitt, V.; Lequeux, F.; Pousse, A.; Roux, D. Langmuir 1994, 10, 955. (10) Cappelaere, E.; Berret, J. F.; Decruppe, J. P.; Cressely, R.; Lindner, P. Phys. Rev. E 1997, 56, 1869. (11) Berret, J. F.; Roux, D. C.; Lindner, P. Eur. J. Phys., B 1998, 5, 67. (12) Decruppe, J. P.; Cressely, R.; Makhloufi, R.; Cappelaere, E. Colloid Polym. Sci. 1995, 273, 346. (13) Rehage, H.; Hoffmann, H. J. Phys. Chem. 1988, 92, 4712; Mol. Phys. 1991, 74, 933. (14) Berret, J. F.; Roux, D. C.; Porte, G. J. Phys. II 1994, 4, 1261. (15) Makhloufi, R.; Decruppe, J. P.; Aı¨t-Ali, A.; Cressely, R. Europhys. Lett. 1995, 32, 253. (16) Decruppe, J. P.; Cappelaere, E.; Cressely, R. J. Phys. II 1997, 7, 257. (17) Berret, J. F.; Porte, G.; Decruppe, J. P. Phys. Rev. E 1997, 55, 1. (18) Mair, R. W.; Callaghan, P. T. Europhys. Lett. 1996, 65, 241. (19) Callaghan, P. T.; Cates, M. E.; Rofe, C. J.; Smeulders, J. B. A. F. J. Phys. II 1996, 6, 375. (20) Mair, R. W.; Callaghan, P. T. Europhys. Lett. 1996, 36, 9. (21) Britton, M. M.; Callaghan, P. T. Phys. Rev. Lett. 1997, 78, 26. (22) Berret, J. F. Langmuir 1997, 13, 2227. (23) Grand, C.; Arrault, J.; Cates, M. E. J. Phys. II 1997, 7, 1071. (24) Lerouge, S.; Decruppe, J. P.; Humbert, C. Phys. Rev. Lett. 1998, 81, 5457. (25) Soltero, J. F. A.; Bautista, F.; Puig, J. E.; Manero, O. Langmuir 1999, 15, 1604. (26) Berret, J. F.; Porte, G. Phys. Rev. E 1999, 60, 4268. (27) Britton, M. M.; Callaghan, P. T. Eur. J. Phys., B 1999, 7, 237. (28) Olmsted, P. D.; Lu, C. Y. D. Phys. Rev. E 1997, 56, 55.
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model to explain the complex kinetics of shear banding transients. By incorporating diffusion terms, they elaborated reaction-diffusion models which describe banded flow in wormlike micelles and which show that slow transients could be controlled by the displacement of one or several interfaces in the gap of the shearing device. Another approach has been considered by Porte and coworkers.33 They proposed a phenomenological explanation of the formation of a nonhomogeneous flow, based on the existence of an effective nonequilibrium potential that accounts for the free energy stored by the material subjected to a constant shear rate. In this work, we report on the transient rheo-optical properties of a micellar solution of cetyltrimethylammonium bromide (CTAB) in water with a mineral salt, sodium nitrate (NaNO3), at a unique concentration (φ ) 11%, T ) 30 °C). This system has been chosen for several reasons. First, the flow curve of this almost perfect Maxwellian fluid exhibits a true plateau (no hysteresis) above a critical shear rate (γ˘ c ) 5 s-1). Second, the time dependence of the stress in startup experiments is very similar to that of the CPCl/NaSal solutions. This analogy led us to carefully investigate this transient behavior by the techniques of flow birefringence. The birefringence intensity and extinction angle profiles are consistent with the various types of responses identified in the stress evolution. The visualization of the gap allows us to follow as a function of time the phase separation into macroscopic layers in which the orientation of the particules relative to the flow direction differs. Our observations indicate that the long sigmoı¨dal decrease of the shear stress toward its plateau value is not associated with the growth of the shear-induced phase. The spatial distribution of the transmitted light intensity reveals that the induced band is divided in small sub-bands of various refractive index. Experimental Section Materials. The cationic surfactant used for this survey is cetyltrimethylammonium bromide (CTAB), which is sold by Janssen Chimica at >99% purity. The salt, sodium nitrate (NaNO3), is an Alfa product sold by Johnson Matthley Gmbh. The solution is prepared with water distilled twice in a quartz device and left standing for 2 days at ∼40 °C to reach equilibrium. In this work, we have studied a unique solution with surfactant and salt concentrations fixed respectively CD ) 0.3 M and CS ) 1.79 M. All the measurements were made at 30 °C. At this CTAB concentration, which corresponds to a weight fraction of about 11% and is higher than the overlap concentration C*, the aqueous solution contains strongly entangled wormlike micelles.1,2 The CTAB/NaNO3 system has been previously studied by Cappelaere et al.34 In particular, they report on the zero shear viscosity η0 versus the amount of added salt CS for CD ) 0.3 M. This curve shows a pronounced maximum for CS = 1.1 M, so the salt concentration chosen here (CS ) 1.79 M) is situated far beyond this maximum. Devices. The rheological measurements in steady-state and oscillatory flows have been performed with two cone-plane (diameter 4 cm, angle 0.5°; 5 cm, 2°) rheometers working in stresscontrolled (Carrimed CSL100) and strain-controlled (Haake VT550) modes, whereas the transient mechanical responses were obtained on a Rheometrics Fluid Spectrometer (RFS II) using a cone-plane configuration (diameter 30 mm, angle 0.02 rad) in controlled shear rate mode. The optical measurements were (29) Olmsted, P. D.; Radulescu, O.; Lu, C. Y. D. J. Rheol., submitted for publication, 1999. (30) Radulescu, O.; Olmsted, P. D. J. Non-Newtonian Fluid Mech., submitted for publication, 1999. (31) Radulescu, O.; Olmsted, P. D.; Lu, C. Y. D. Rheol. Acta, submitted for publication, 1999. (32) Yuan, X. F. Europhys. Lett. 1999, 46, 4. (33) Porte, G.; Berret, J. F.; Harden, J. L. J. Phys. II 1997, 7, 459. (34) Cappelaere, E.; Cressely, R. Colloid Polym. Sci. 1997, 275, 4.
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carried out on an apparatus built a few years ago in our lab and described in much detail in an earlier paper.35 The sample is placed in a conventional Couette cell with a height of 10 mm. The inner cylinder rotates and has a diameter of 47 mm, giving a gap of 1.5 mm. In all the experiments, the light (a laser beam at 6328 Å for the quantitative measurements and a source of white light for the qualitative observations of the gap) is propagated along the vorticity axis, probing the birefringence in the (υ b,∇ B υ) plane. The optical quantities of interest are the following: the extinction angle χ, which characterizes the average dynamic orientation of the particles, is defined as the smallest angle between the streamline and one of the two neutral axes of the solution. The birefringence intensity ∆n, defined as the difference between the two principal refractive indices of the biaxial medium, is related to the retardance φ introduced by the anisotropic solution by means of the simple relation
∆n )
λφ 2πe
(1)
where λ is the wavelength of the light and e is the thickness of the liquid. In steady-state regimes, χ can be determined by a null method, which consists of placing the solution between crossed polarizers, and the ∆n can be measured by the classical method of Senarmont. The transient optical results are also obtained with our homemade apparatus, which is monitored by a computer. The belt which sets the inner cylinder in rotation has been replaced by a rigid chain, avoiding any elasticity in the mechanical transmission. We have used different arrangements of the optical components in recording the transient optical properties. They all are based on the study of the transmitted light intensity.36 In this method, ∆n is calculated from the intensity transmitted by a circular polarizer, the medium, and a circular analyzer. The intensity of the emergent vibration is simply given by
Ic ) Ic′ sin2
φ 2
(2)
where Ic′ is a coefficient which depends on the partial reflections on the various optical components. As for χ, it is computed from the intensity transmitted by the material under flow, situated between crossed polarizer and the analyzer:
Iθ ) I′ sin2
φ sin2 2(χ-θ) 2
Figure 2. Storage modulus G′(O) and loss modulus G′′ (0) as functions of the angular frequency ω. The full lines result from the Maxwell fit using eqs 6 and 7.
(3)
where θ is defined as the angle formed by the polarization of the incident beam and the streamline. A judicious choice of the values of θ gives two other equations. Hence, for θ ) 45° and 0°, we obtain respectively
I45 ) I′ sin2
φ cos2 2χ 2
(4)
I0 ) I′ sin2
φ sin2 2χ 2
(5)
Equation 2 leads to φ (then to ∆n from eq 1), whereas the ratio of I0/I45 yields χ. The factor I′ plays the same role as the previous constant Ic′.
Results and Discussion Characterization of the Sample. Mechanical Rheology. The sample has been subjected to a sinusoı¨dal shear stress over a range of frequencies from 0.01 to 250 rad/s. This type of measurements allows for the determination (35) Decruppe, J. P.; Hocquart, R.; Wydro, T.; Cressely, R. J. Phys. II 1989, 50, 3371. (36) Osaki, K.; Bessho, N.; Kojimoto, T.; Kurata, M. J. Rheol. 1979, 23, 457.
Figure 3. Flow curves measured by an experimental procedure similar to that followed by Grand et al.23 Stress-controlled modes (6 s per data point): strain sweep (O, 25 s per data point); increasing strain sweep (2), decreasing strain sweep (0).
of the storage modulus G′ and the loss modulus G′′ against the angular frequency ω, the variations of which are given in Figure 2. The data can be fitted quite well over a wide range of frequencies by the following relations: 2
ω2 τR
G′(ω) ) G0 2 1 + ω 2 τR
(6)
ωτR G′′(ω) ) G0 2 1 + ω2 τR
(7)
indicating that the solution almost behaves as a perfect Maxwell element. Equations 6 and 7 provide the plateau modulus G0, which depends only on the degree of entanglement, and the terminal relaxation time τR, related to the dynamic processes of the colloı¨dal system. Hence, for our sample, τR ) 0.17 ( 0.01 s, and G0 ) 232 ( 7 Pa. Steady-state experiments yield the evolution of the shear stress σ as a function of the velocity gradient γ˘ , as shown on the semilogarithmic plot in Figure 3. At low shear rates, the behavior is Newtonian: σ increases linearly with γ˘ . Then, above a critical value of γ˘ c1, an important change in the slope of the σ(γ˘ ) curve happens. In fact, the
Micellar Solution under Shear Banding Flow
Figure 4. Variations of the extinction angle χ (9) and the absolute value of the birefringence intensity ∆n (0) versus γ˘ . The dotted lines represent the effective behavior of ∆n and χ in the region where the retardation is nearly 2π.
shear stress remains practically constant (σ ) σp) over about one decade of shear rates. Several experimental studies8-21 have demonstrated that this plateau behavior is typical of a sharply inhomogeneous flow. As will be confirmed with qualitative optical measurements, the system exhibits a phase transition of the shear banding type. Various types of runs have been performed according to a procedure similar to that described by Grand et al.23 in determining carefully the banding stress σ ) σp and the corresponding critical shear rate γ˘ c1, above which the macroscopic phase separation occurs. At first, the flow curve has been established under controlled-stress conditions (open circles) with about 6 s between each data point. The transition between the Newtonian and the plateau regimes is smooth, and the stress value ends up at σ = 190 Pa, which does not clearly correspond to the true steady state, as shown from carrying out the straincontrolled sweeps. In fact, when the sweep rate (about 25 s, namely 140 τR, per data point) is reduced, the plateau begins at γ˘ c1 = 5 ( 0.2 s-1 (see the decreasing sweep: open squares) to approach a quite different σ ) σp = 150 Pa. The existence of a slight slope in the plateau stress suggests, by analogy with more concentrated solutions (close to a nematic phase at rest), that the concentrations in the two phases may not be identical.8,28,37 In other respects, as already reported in ref 23, the increasing strain-rate sweep (full triangles) allows trapping of metastable states for which the stress is greater than σ ) σp, giving rise to a bump in the flow curve, which disappears in the decreasing scan. We shall see later with the help of startup measurements that all points for which σ > σp are transient data. Optical Considerations. The steady-state values of χ and ∆n are studied as functions of γ˘ for our sample at 30 °C (see Figure 4). The behavior of these two quantities is globally similar to that of the shear stress. We can distinguish two parts in the curves, which are characteristic of the existence of a band structure triggered by flow, as already pointed out by Decruppe et al.12,16 In fact, the absolute value of the birefringence, which appears to be negative for these solutions, increases with γ˘ , whereas in the same range of shear rates, the average orientation of the micellar aggregates decreases smoothly from 45°, indicating a better alignment of the particles along the (37) Schmitt, V.; Marques, C. M.; Lequeux, F. Phys. Rev. E 1995, 52, 4009.
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direction of the flow. Then, up to γc1 ) 5 ( 0.5 s-1, which is consistent with the previous value obtained by rheological measurements, χ and ∆n saturate to a plateau. However, we find a deviation from the usual behavior of the extinction angle curve observed in such a situation.16 In fact, a discontinuity in the experimental data is observed in the vicinity of γ˘ ) 3 s-1. For this value of the shear rate, the phase difference φ becomes equal to 2π. Just before this “fringe”, χ drops below the classical curve, whereas beyond the fringe, χ is somewhat too high. Hence, the determination of the average orientation of the particles is made difficult when the optical path difference is virtually zero. This type of evolution has already been observed for linear polymers.38 For all points in the zone where χ and ∆n are fairly constant, visual observations show that the gap of the Couette cell is split in two layers or shear bands, subjected to the same macroscopic shear stress σp but sheared at different velocity gradients. The band located near the moving wall, called the h-band, is bright and highly birefringent, which implies that the phase formed by the micellar entities is highly ordered. The other band (l-band) which supports a lower shear rate, is then more viscous than the h-band. The variations of χ with the shear rate γ˘ allow for the determination of the relaxation time of Maxwell. In fact, Thurn39 has shown that χ obeys the simple equation
χ)
π 1 - /2 arctan(γ˘ τR) 4
(8)
The curve χ(γ˘ ) can easily be fitted with the above expression (at least at low shear rates), thus providing τR with a value equal to 0.18 ( 0.01 s, which corroborates the rheological dynamic results. One of the most attractive sides of the flow birefringence experiments is their comparison with rheological data through the stress-optical law, which establishes a simple relation between the stress and the refractive index tensors. It is based on the hypothesis that these tensors are coaxial:
∆n sin 2χ ) 2Cσ
(9)
∆n cos 2χ ) CN1
(10)
where C is the so-called stress-optical coefficient and N1 is the first normal-stress difference. C is a constant which can easily be computed in steadystate or transient shear flows by plotting the product ∆n sin 2χ as a function of the shear stress σ or ∆n cos 2χ as a function of the first normal-stress difference. In our case, this method gives an optical coefficient equal to 2.78 × 10-7 Pa-1. This value is consistent with those obtained for other similar surfactant solutions.13,40-42 As shown in Figure 5, the stress-optical rule holds at low shear rates, indicating that the contribution to the global anisotropy of the intrinsic birefringence predominates over the form effect. However, at shear rates greater than γ˘ = 4.4 s-1, there is a strong departure between optical and mechanical data. The orientation of the strands of the micelles between entanglements becomes so high that they cannot bear the corresponding stress and no longer follow Gaussian chain statistics. The failure of this law can be (38) Gortemaker, F. H.; Hansen, M. G.; De Cindio, B.; JaneschitzKriegl, H. Rheol. Acta 1976, 15, 242. (39) Thurn, H.; Lo¨bl, M.; Hoffmann, H. J. Phys. Chem. 1985, 89, 517. (40) Shikata, T.; Dahman, S. J.; Pearson, D. S. Langmuir 1994, 10, 3470. (41) Humbert, C.; Decruppe, J. P. Colloid Polym. Sci. 1998, 276, 160. (42) Humbert, C.; Decruppe, J. P. Eur. J. Phys., B 1998, 6, 57.
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Figure 5. Validity of the stress-optical law: comparison between the steady-state shear stress obtained by mechanical rheology (2) and the stress computed from optical data (∆n sin 2χ/2C) (O).
Figure 6. Transient response of the shear stress after the onset of flow for several applied shear rates (γ˘ ) 5-10 s-1). The inset represents the same curves but on a shorter time scale.
explained by the apparition of a phase containing highly orientated entities.40,43 Analysis of the Rheo-Optical Transient Responses. In this section, we compare the mechanical and the optical responses of the medium recorded as a function of time during startup experiments. All the shear rates, which are applied to the solution, belong to the plateau region (namely, γ˘ g γ˘ c1) and are close to γc1 (the width of the plateau can be estimated at about 100 s-1). Quantitative Results. Figure 6 shows the transient shear stress for several shear rates chosen between 5 and 10 s-1 at the inception of the flow on short time (see the inset) and long time scales. This type of curve has already been observed by Berret et al.14,22 for a solution of CPCl/Sal with 0.5 M NaCl at a constant Sal/CPCl molar ratio of 0.5 and by Grand and co-workers23 for the same system (without brine) but in a more dilute concentration range (φ ) 4.5%). For all the investigated shear rates, an overshoot, the amplitude of which increases significantly with γ˘ , happens at t = 0.5 s. After this short initial period, during which the elastic response of the material occurs, the shear stress (43) Wheeler, E. K.; Pilar, I.; Fuller, G. G. Rheol. Acta 1996, 35, 139.
Lerouge et al
relaxes toward a stationary value, which is consistent with the plateau value established with the help of the steadystate rheological measurements. This relaxation process has a characteristic time much longer than the terminal relaxation time τR, and it exhibits various features as a function of the imposed shear rate. For γ˘ ) 5 and 6.3 s-1, the overshoot is followed by a short period, during which σ is constant; it lasts for 10 and 6 s, respectively. Then the shear stress decreases slowly and reaches its stationary limit σp after 120 and 40 s, respectively. A closer inspection of the curve at γ˘ ) 6.3 s-1 points out the apparition of a small undershoot around 30 s. Such a minimum in the transient stress profiles has already been detected in the CPCl/NaSal system for sufficiently high shear rates.14 At higher strain rates, i.e., for 7, 8, and 10 s-1, the long-time relaxation toward the stress plateau starts after a few damped oscillations. In each case, the first minimum after the overshoot arises at the same time t = 0.9 s. Some runs realized for γ˘ between 15 and 19 s-1 (not presented here) indicate that the period of this damping oscillations does not vary significantly with γ˘ , a feature already mentioned before.22,44 The undershoot in the stress curves persists for all the shear rates greater than 6.3 s-1. Figures 7 and 8 illustrate the time-dependent optical properties for two shear rates (6.3 and 10 s-1) which seem representative of the different features of the transient shear stress described previously. The variations of ∆n (see Figure 7a,b), χ (Figure 8a,b), and the shear stress with time look very much alike. At the inception of the flow, ∆n increases from zero to a maximum reached in ∼0.5 s. At γ˘ ) 6.3 and 10 s-1, this peak corresponds to retardations greater than 4π and 8π, respectively (for these retardations, ∆n = -14 × 10-5 and -28 × 10-5). Such overshoots have been observed in concentrated polymeric systems36,45,46 and, more recently, in micellar solutions but with a much smaller amplitude.40,47 As in the case of the shear stress, the relaxation processes, which follow this overshoot, occur on a long time scale in comparaison to that of the terminal relaxation time τR. In fact, at γ˘ ) 6.3 s-1, the birefringence intensity slowly decreases, and then shows a small undershoot before leveling off to its stationary value at about 30 s. In other respects, the run realized at γ˘ ) 10 s-1 reveals the presence of oscillations (see the inset for a more accurate inspection) of the optical anisotropy, which preempt the long-time relaxation. One can also note that the undershoot, which makes the transient response much longer, is still visible at this shear rate. The description of the evolution of the extinction angle with time appears more ambiguous, as can be seen in panels a and b of Figure 8 for γ˘ ) 6.3 and 10 s-1, respectively. The average orientation of the particules is computed from eqs 4 and 5; thus, when the retardation is equal to 2kπ (k ) 1, 2, ...), the calculation of χ is impossible. The discontinuity encountered in the steadystate data plotted in Figure 4 is an illustration of this problem for k ) 1. For time-dependent simple shear flows, the situation is even more complicated, since the retardation passes through multiple orders (k ) 2 at 6.3 s-1 and k ) 4 at 10 s-1), as emphasized above. This makes the (44) Shikata, T.; Hirata, H.; Takatori, E.; Osaki, K. J. Non-Newtonian Fluid Mech. 1988, 28, 171. (45) Zebrowski, B. E.; Fuller, G. G. J. Polym. Sci., Part B: Polym. Phys. 1985, 23, 575. (46) Lee, J. S.; Fuller, G. G. J. Non-Newtonian Fluid Mech. 1987, 26, 57. (47) Humbert, C. The`se de Doctorat, Universite´ de Metz, Metz, France, 1998.
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a
a
b
b
Figure 7. (a) Time-dependent evolution of the absolute value of the birefringence intensity ∆n after the startup of the flow at γ˘ ) 6.3 s-1. (b) Time-dependent evolution of the absolute value of the birefringence intensity ∆n at γ˘ ) 10 s-1. The inset shows the oscillations which preempted the sigmoı¨dal relaxation toward the steady state.
interpretation of the short-time response of the orientational angle difficult, in particular for relatively high applied shear rates. The instants at which the retardation reaches an order are denoted by dotted arrows in the insets of the graphs of Figure 8a,b. When the retardation goes over an order extremely quickly (typically a few hundredths of a second), the extinction angle presents an inflection point, while when it happens more slowly (i.e., several tenths of a second), χ exhibits strong oscillations which are not representative of the true behavior of the state of orientation. At γ˘ ) 6.3 s-1, χ starts at the expected value of 45°, typical of the isotropic state, decreases quickly, and then shows a minimum (∼17°) for t around 0.5 s (see the inset in Figure 8a). For t between 0.7 and 2.7 s, the relaxation of the extinction angle is highly perturbed, since in this time interval, the retardation introduced by the medium under flow is close to 4π. After this period during which the orientation of the particles is inaccessible, χ increases gradually and shows a small overshoot before reaching its stable value at t = 27 s. Shikata et al.40 have performed similar optical experiments on the same surfactant system (CTAB) but with a different salt (NaSal) and never found a minimum in the extinction curve. This feature is more commonly observed in concentrated polymer solutions.45
Figure 8. (a) Variations of the extinction angle χ as a function of time after the inception of the flow at γ˘ ) 6.3 s-1. The inset represents the short-time response of χ at this shear rate; the arrows indicate the moments at which the retardation reaches an order. (b) Variations of the extinction angle χ as a function of time after a step shear rate at γ˘ ) 10 s-1. The inset shows the short-time response of χ at this shear rate; the arrows indicate the multiple orders in the retardation.
The existence of an undershoot in the angle has been unambiguously confirmed by another study performed on the well-known CTAB/KBr system, which easily forms shear bands above a critical shear rate16 and for which the retardation does not exceed π/2.48 The variations of χ at 10 s-1 are much more chaotic due to the multiple orders in the retardation, which do not allow for the distinction between the true and the “artificial” oscillations. In particular, the undershoot is hidden, since it occurs between two orders (k ) 4). For 1 s e t e 2 s, the angle is out of reach because, as shown in the inset of Figure 7b, the retardation oscillates around 4π. Then for t > 2 s, χ increases smoothly, passes through a small maximum again, and stabilizes at its stationary value (30°). Hence, the more striking point of these measurements is the strong analogy between the evolution of the optical quantities and the profiles of the shear stress. As the flow is turned on, the growth of σ with time up to a maximum is associated with a sharp increase of the birefringence intensity and of the orientation of the particles with respect (48) Lerouge, S.; Decruppe, J. P., manuscript in preparation.
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Table 1. Characteristic Relaxation Time τNG and Exponent r Computed at Various Shear Rates from Rheological and Optical Transient Profiles γ˘ (s-1)
τrheo NG (s)
Rrheo
τopt NG (s)
Ropt
5 6.3 7 8 10
62 16.9 9.6 5.9 3.74
1.95 2.8 3.7 4.3 4.53
43.4 12 6.6 3.48
1.44 2.11 3.22 4.51
to the flow direction. This strong orientation is due to an important stretching of the strands of the entangled network, which then support enormous stresses, rather than an alignment of these strands in the shearing plane. Moreover, this hypothesis is consistent with the theoretical predictions of the model of Pearson et al. for entangled polymers subjected to high shear rates.49 After this brief period on the order of 2.5τR, the shear stress relaxes toward its stationary value. The optical experiments confirm the existence of the two regimes of stress relaxation evoked in ref 22, which occur on well-distinct time scales. The first one happens on the Maxwell time scale and corresponds to a quick decay of the shear stress and the optical anisotropy. The second one extends on a much longer time interval, depending on the applied shear rate, and during which the decay of σ comes with a gradual loss of orientation. These slow transients are consistent with the data obtained under increasing strain sweep. In fact, they point out the existence of a range of metastability, which characterizes a first-order phase transition. Such a remark led one of us to develop a nucleation and one-dimensional growth model.14 In this case, the time evolution of the shear stress can be fitted quite well by a sigmoı¨dal profile according to
((
σ(t) ) σp + (σM - σp) exp -
t
))
τNG(γ˘ )
R
(11)
where R ) 2 and σM is the value taken by the shear stress just before the onset of the long sigmoı¨dal decrease and τNG(γ˘ ) is the characteristic time for the nucleation and one-dimensional growth processes of the shear-induced orientated phase. The same type of formula has been used in fitting the optical responses. The results obtained from the best fits are summed up in Table 1. However, for our sample, a non-Gaussian behavior has been found under all circumstances except for γ˘ ) 5 s-1. A wider range of the exponent R has also been adopted in the study of the CPCl/NaSal system.23 Figure 9 compares the evolution of the relaxation time τNG computed from rheological and optical transient experiments. Both methods of investigation provide similar variations of the characteristic time τNG, even if it appears shorter in the optical measurements. This discrepancy is explained by the use of different cell geometriessnamely, a cone and a plane cell for the mechanical runs and a Couette cell for the optical tests. Grand et al.23 have emphasized the dependence of τNG with cell geometry, ascribing it to the greater inhomogeneity of the stress field in a Couette geometry. They show that the variations of τNG with γ˘ follow a power law divergence given by
τNG ≈ |γ˘ - γ˘ c|-p
(12)
(49) Pearson, D.; Herbolzheimer, E.; Grizzuti, N.; Marrucci, G. J. Polym. Sci., Part B: Polym. Phys. 1991, 29, 1589.
Figure 9. Comparison between the characteristic relaxation times τNG(γ˘ ) computed from rheological (0) and optical (b) measurements. The inset shows the comparison in a ln/ln plot with γ˘ cτR ) 0.76.
Our data can be fitted in this way (see the inset in Figure 9) with p = 1.5 and γ˘ c = 4.3 s-1 so that τ(γ˘ ) diverges below the strain γ˘ c1 at the onset of the two-phase region, which is similar to the results obtained on the 100 mM CPCl/75 mM NaSal solution.23 It should be noted that the value of γ˘ c approximately corresponds to the shear rate at which the stress-optical law is no longer valid. This agreement may be fortuitous, since the difference between the critical shear rates γ˘ c1 and γ˘ c is very small. Direct Observation of the Flow Birefringence. All the transient rheo-optical data indicate that the kinetics of formation of a new orientated phase is extremely complex. Thus, it seems interesting to follow in situ the apparition of the shear-induced phase by a direct observation of the gap of the Couette cell. The technique of flow birefringence is particularly appropriate in studying the microstructural changes which occur during the evolution of time in the solution. Figure 10 shows the visualization of the flow in the gap of the Couette cell placed between crossed polarizers at different moments during the process of inception of the flow at γ˘ ) 8 s-1. A source of white light was used, and the observation is realized in the (υ b,∇ B υ) plane. The letter A on the first photograph (t ) , i.e., just after the startup of the flow) indicates the inner rotating cylinder, while the letter B shows the fixed wall of the Couette cell. The crossed polarizers are placed so that one arm of the cross of isocline appears in the field of observation when the steady state is achieved. This improves the contrast between the shear-induced structure and the isotropic phase. After the inception of the flow and up to t ) 2 s, the gap is brightly illuminated, indicating a large increase of the birefringence intensity. Dark bands which are almost parallel to the walls and corresponding to multiple orders of retardation (φ ≈ 2kπ) spread very quickly through the gap from the inner rotating cylinder to the fixed wall at first (see photo 1 of Figure 10) and then from the outer cylinder to the moving wall (see photos 2, 3, and 4). The time resolution used for these recordings does not distinguish all the orders. However this behavior is consistent with the existence of the overshoots in the shear stress and the birefringence intensity. It points out that the retardation is very important and depends on both time and space. In photos 4 and 5, the gap becomes darker because of the presence of one branch of the cross of isocline, which
Micellar Solution under Shear Banding Flow
Figure 10. View of the flow in the annular gap of the Couette cell placed between crossed polarizers and illuminated with a source of white light at various moments during the process of inception of the flow (γ˘ ) 8 s-1). The observation is realized in the (υ b,∇υ b) plane. The inner rotating cylinder and the outer fixed wall of the Couette cell are indicated by the letters A and B, respectively (see photo 1).
enters in the field of observation. Then from t = 6 s, this dark band, which extends from the upper left “corner” to the lower right “corner” of the gap (if the visualized portion of the gap is roughly represented by a rectangle), has a quasistable position in the gap. This means that the extinction angle has almost reached its steady-state value. In fact, if the variation of the average orientation of the particles would be greater than 3°, the branch of the cross of isocline would leave the field of observation. This confirms the previous experiments, since the characteristic time of the sigmoı¨dal relaxation toward the steady state is close to 6 s. The tilting of the isocline could be a consequence of the heterogeneity of the flow field encountered in Couette geometry. In other respects, from t = 5-35 s, a very thin bright band confined against the moving wall can be distinguished. During this time interval, the width of this layer remains practically constant. At times longer than t ) 40 s, the band, constituted by the shear-induced phase, significantly grows and ends up holding approximately half of the gap (see the last picture). For this imposed shear rate (8 s-1), the steady state of the rheo-optical quantities is obtained at t = 15 s. In other words, the proportion of the shear-induced phase continues to grow,
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while the steady state in stress and optical anisotropy is achieved. This behavior has also been observed for all the shear rates investigated in this work, and it is quite reproducible. Thus, the snapshots show that the mechanism of formation of a nonhomogeneous flow is rather complex. This latter seems to occur in several stages. At first, a thin bright band, the size of which depends on the applied shear rate, forms in the vicinity of the rotating inner cylinder at a time on the order of τNG, which implies that the variations of σ with time are certainly related to a nucleation process. Then, after an “induction time” much longer than τNG and dependent on γ˘ , the width of the initial thin band considerably increases from the moving wall to reach a stationary value. This demonstrates that the long sigmoı¨dal relaxation of the shear stress toward its plateau value is not associated with the one-dimensional growth process predicted by Berret’s model. The last photo in Figure 10 (t ) 175 s) deserves additional comments. This picture is analogous to the ones reported on other surfactant systems12,17 and clearly shows that in the plateau regime of the stress curve, the flow is strongly nonhomogeneous. The band holds a wide part of the gap even though the investigated shear rate is very close to γ˘ c1. This is the reason the proportion of induced phase relative to the total sample volume in the stationary limit is important and cannot be ignored, contrary to the assumption of the one-dimensional growth model. A closer inspection of the bright band (h-band) shows that this latter appears extremely striated.17 Moreover, the interface between the highly orientated and the isotropic regions is unstable and difficult to localize with a great accuracy. Another type of experiments has been performed to examine more accurately the formation of the band structure. For these measurements, the gap is always illuminated with white light propagating along the vorticity axis, but the analyzer has been removed from the optical bench. A steplike velocity profile similar to the one used previously is applied to the sample and photos of the gap are taken as a function of time. Then these snapshots are digitized, thus providing transmitted light intensity profiles I(x,t) through the gap. The spatial resolution is about 15 µm per data point, while the temporal resolution can reach 0.15 s per photos. Figure 11 represents the intensity profiles I(x,t) recorded at γ˘ ) 8 s-1 as a function of the position in the gap of the Couette cell. The origin of the x-coordinates has been chosen on the fixed wall. Just after the inception of the flow (Figure 11a, t ) ) and apart from random small fluctuations due to dust or bubbles crossing the light beam, the intensity is constant over the entire gap. At t = 0.561.54 s, I(x,t) diminishes sharply. The decrease of the intensity is not homogeneous in the entire gap, and the minimum is reached near the rotating cylinder, while against the fixed wall, the intensity remains at its initial level in a small layer (x e 0.2 mm). A direct observation of the gap with a microscope shows that this diminution of I(x,t) corresponds to a sharp growth of the turbidity of the sample. Figure 11a points out that this phenomenon increases with t in the considered time interval and propagates from the moving wall to the fixed cylinder with a strong attenuation. From 1.54 to 10 s, the liquid becomes transparent again near the fixed wall of the cell first and then progressively over the entire gap. However, the propagation of the phenomenon occurs more slowly and in the opposite direction (i.e., from the fixed wall to the moving wall). The transmitted light intensity recovers
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Figure 11. (a-d) Transmitted light intensity I plotted against the spatial position x in the gap of the Couette cell and measured at different moments after the startup of flow at γ˘ ) 8 s-1.
its initial amplitude around 5.5 s, and its distribution appears again nearly homogeneous in the gap. A weak loss of intensity persists in a fine layer of approximately 250 µm confined near the rotating part. Then after 40 s (see Figure 11c,d), I strongly fluctuates as a function of the position in the gap. In fact, oscillations, the amplitude of which increases with time, seem to be born near the rotating wall and gradually move toward the outer cylinder, finally holding nearly half of the gap. At γ˘ ) 10 s-1 (Figure 12), the observed behavior is a bit more complicated. After the startup of the flow (Figure 12a, t e 0.75 s), the transmitted light intensity shows damped oscillations with time but remains practically homogeneous in the gap, which is totally turbid. At times longer than 0.75 s, the intensity profiles are much more chaotic and heterogeneities appear, particularly at t ) 1.5 s where a distinct drop in transmitted light intensity is observed in the vicinity of the inner rotating cylinder. Between 1.5 and 3 s, the minimum is reached, and the turbidity decreases in the regions of weaker curvature, namely, x < 1 mm. Then, between 3 and 6 s, the intensity progressively increases near the moving wall (x > 1 mm), while for x < 1 mm, it reaches a stationary value. On the whole, the intensity profiles between 6 and 30 s are similar. At times longer than 30 s, I(x,t) shows strong oscillations (see Figure 12d), the amplitude of which increases with time and holds about two-thirds of the gap.
These results are only preliminary, but they confirm the observations realized between crossed polarizers. At first, just after the inception of the flow, the sample becomes strongly turbid in a great part of the gap and even in all the gap if γ˘ is high enough. The evolution of the turbidity with γ˘ is more or less homogeneous and corresponds to the phenomenon of light scattering.50 The evolution of the extinction angle (Figure 8a,b) shows that at the onset of the flow, the entangled network is strongly stretched; this may lead to important concentration fluctuations responsible for the light diffusion process. The “milky phase” progressively disappears, near the fixed side of the cell first, and then in most of the gap in a time close to τNG( γ˘ ). Hence, the variations of the transmitted light intensity with time may be related to those of the shear stress (compare, for example, the oscillations in stress and intensity at γ˘ ) 10 s-1 for t e 3 s), but other experiments seem necessary to check this hypothesis. They seem to correspond to the formation of the “initial” thin band in the photos taken between crossed polarizers. The data at 10 s-1 are particularly appropriate in following the formation of this band, since its width is sufficiently large at this shear rate. This small band seems to form between 2 and 5 s by diffusion of sub-bands, which do not (50) Decruppe, J. P.; Lerouge, S.; Berret, J. F. Phys. Rev. Lett., submitted for publication, 2000.
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Figure 12. (a-d) Transmitted light intensity profiles I(x,t) at γ˘ ) 10 s-1.
transmit the same amount of light and which have their own dynamics. All these observations suggest that the nucleation process occurs in a great part of the gap so that the shear-induced phase coexists with the isotropic phase. Then a thin initial band forms by a phase separation mechanism, together with the diffusion of sub-bands from the moving wall. This last process could be responsible of the subsequent growth of this initial band, but this point is not very clear yet. Moreover, the behavior of I(x,t) at long times (t > 30 s, Figure 12d) shows that the flow is locally nonhomogeneous in the h-band. Indeed, the observed oscillations correspond to more or less intense sub-bands, indicating the existence of strong variations of the refraction index in the shearinduced phase. These sub-bands explain the striation between crossed polarizers (see Figure 10, photo 18). They are strongly aligned in the direction of the flow and produce a diffraction pattern50 formed by several distinct peaks and orientated parallel to the velocity gradient direction, when the laser beam propagates through the h-band. Such a pattern has already been detected on another similar system.24 The intensity spectra allow for the computation of the charateristic size of these sub-bands, which is typically estimated to be 100-150 µm. The portion of the gap held by the sub-bands widens with times longer than 40 s for γ˘ ) 8 s-1 and 30 s for γ˘ ) 10 s-1, which confirms that the growth of the initial thin band only occurs when the steady state in stress and optical anisotropy is achieved.
Conclusions We have investigated the nonlinear transient properties of a wormlike micellar solution undergoing a shearinduced phase transition above a critical shear rate using the techniques of rheology and flow birefringence. The σ(t) stress profiles have been found to be strongly similar to those for the intensively studied CPCl/NaSal system in various concentration ranges.14,22,23 One of the major findings of our paper is given by the quantitative flow birefringence measurements, which corroborate all the features observed in the stress relaxation toward a stationary value: an overshoot (and eventually damped oscillations at high strain rates) on short time scales and a sigmoı¨dal evolution on long time scales. Indeed, Figure 9, which compares the characteristic relaxation times τNG (much longer than the Maxwell times τR) of both optical and mechanical methods, points out the same phenomena. Berret et al.14 have suggested that such kinetics are associated with nucleation and one-dimensional growth processes. Another fundamental result is given by the direct vizualization of the gap of the Couette cell placed between crossed polarizers. The snapshots of the gap during startup experiments show that the mechanism of formation of the nonhomogeneous flow does not exactly correspond to the two above-mentioned processes. The sigmoı¨dal decay is certainly related to a nucleation process, since for t on the order of τNG, a bright band near the moving inner
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cylinder forms, but the growth of this initial small band happens well after the steady state is achieved in stress and in optical anisotropy. In other words, the long sigmoı¨dal relaxation toward a plateau value is not connected with the growth of the shear-induced phase. Moreover, the spatial distribution of the transmitted light intensity in the gap shows that the flow is locally nonhomogeneous, namely, that there are not only two bands, as in the classical picture of shear banding flow. The shear-induced band (h-band) appears extremely striated and is composed of many small sub-bands (100150 µm in width) strongly ordered in the direction of the flow, thus producing a diffraction pattern.
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Further experiments shall soon be performed in order to fully understand the kinetics of formation of the initial thin band and the mechanism which sets off the growth of this band, namely, what happens in this layer during the “induction time”. Acknowledgment. The authors thank C. Humbert, G. Porte, P. Olmsted, O. Radulescu, J. F. Tassin, and S. Cocard for their fruitful discussions and R. GamezCorrale`s for his suggestions on the data processing. This work was also supported by the GDR 1081 “Rhe´ophysique des Colloı¨des et Suspensions”, founded by the CNRS. LA000269W
