Shear-Thickening in Salt-Free Aqueous Solutions of a Gemini Cationic

Gemini Cationic Surfactant: A Study by Small Angle Light. Scattering. V. Weber and F. ... gence,3,13,21,27 flow electrical conductivity,13 transmissio...
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Langmuir 2002, 18, 9705-9712

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Shear-Thickening in Salt-Free Aqueous Solutions of a Gemini Cationic Surfactant: A Study by Small Angle Light Scattering V. Weber and F. Schosseler* Laboratoire de Dynamique des Fluides Complexes, UMR 7506, (CNRS-ULP), 3 rue de l’Universite´ , 67084 Strasbourg Cedex, France Received July 17, 2002. In Final Form: August 28, 2002 The shear-thickening regime of dilute cationic surfactant solutions is studied by means of small-angle light scattering. The 2-dimensional intensity pattern measured in the flow velocity-vorticity plane exhibits an intense streak pattern perpendicular to the flow direction. The main features of this streak pattern are as follows: (i) a periodic modulation of the intensity in the vorticity direction, (ii) fluctuations of the intensity and orientation as a function of scattering volume, (iii) a strong correlation of the average intensity with the apparent shear viscosity, and (iv) an average shape independent of the applied shear rate. The corresponding anisotropic optical contrast profile with a characteristic modulation length about 33 µm in the vorticity direction cannot be interpreted in terms of the alignment of large anisotropic colloidal objects in the flow but, instead, is consistent with the existence of an aligned heterogeneous gellike layer in the gap.

I. Introduction The shear-thickening behavior of dilute ionic surfactant solutions has attracted much interest since its discovery in the early 1980s.1-28 It has been studied by different experimental techniques including strain and stress (1) Rehage, H.; Hoffmann, H. Rheol. Acta 1982, 21, 561. (2) Rehage, H.; Wunderlich, I.; Hoffmann, H. Prog. Colloid Polym. Sci. 1986, 72, 51. (3) Wunderlich, I.; Hoffmann, H.; Rehage, H. Rheol. Acta 1987, 26, 532. (4) Jindal, V. K.; Kalus, J.; Pilsl, H.; Hoffmann, H.; Lindner, P. J. Phys. Chem. 1990, 94, 3129. (5) Hu, Y. T.; Wang, S. Q.; Jamieson, A. M. J. Rheol. 1993, 37, 531. (6) Schmitt, V. the`se Universite´ Louis Pasteur, Strasbourg, 1994. (7) Schmitt, V.; Schosseler, F.; Lequeux, F. Europhys. Lett. 1995, 30, 31. (8) Hu, H. T.; Matthys, E. F. Rheol. Acta 1995, 34, 450. (9) Liu, C. H.; Pine, D. J. Phys. Rev. Lett. 1996, 77, 2121. (10) Pro¨tzl, B.; Springer, J. J. Colloid Interface Sci. 1997, 190, 327. (11) Boltenhagen, P.; Hu, Y. T.; Matthys, E. F.; Pine, D. J. Europhys. Lett. 1997, 38, 389. (12) Boltenhagen, P.; Hu, Y. T.; Matthys, E. F.; Pine, D. J. Phys. Rev. Lett. 1997, 79, 2359. (13) Oda, R.; Panizza, P.; Schmutz, M.; Lequeux, F. Langmuir 1997, 13, 6407. (14) Koch, S.; Schneider, T.; Ku¨ter, W. J. Non-Newtonian Fluid Mech. 1998, 78, 47. (15) Hu, Y. T.; Boltenhagen, P.; Pine, D. J. J. Rheol. 1998, 42, 1185. (16) Hu, Y. T.; Boltenhagen, P.; Matthys, E.; Pine, D. J. J. Rheol. 1998, 42, 1209. (17) Berret, J. F.; Gamez-Corrales, R.; Oberdisse, J.; Walker, L. M.; Lindner, P. Europhys. Lett. 1998, 41, 677. (18) Gamez-Corrales, R.; Berret, J. F.; Walker, L. M.; Oberdisse, J. Langmuir 1999, 15, 6755. (19) Oda, R.; Weber, V.; Lindner, P.; Mendes, E.; Schosseler, F. Langmuir 2000, 16, 4859. (20) Zheng, Y.; Lin, Z.; Zakin, J. L.; Talmon, Y.; Davis, H. T.; Scriven, L. E. J. Phys. Chem. B 2000, 104, 5263. (21) Berret, J. F.; Gamez-Corrales, R.; Lerouge, S.; Decruppe, J. P. Eur. Phys. J. E. 2000, 2, 343. (22) Weber, V., the`se Universite´ Louis Pasteur, Strasbourg, July 2001. (23) Berret, J. F.; Gamez-Corrales, R.; Se´re´ro, Y.; Molino, F.; Lindner, P. Europhys. Lett. 2001, 54, 605. (24) Truong, M. T.; Walker, L. M. Langmuir 2002, 18, 2024. (25) Fischer, P.; Wheeler, E. K.; Fuller, G. G. Rheol. Acta 2002, 41, 35. (26) Oelschlaeger, Cl.; Waton, G.; Buhler, E.; Candau, S. J.; Cates, M. Langmuir 2002, 18, 3076. (27) Berret, J. F.; Lerouge, S.; Decruppe, J. P. Langmuir 2002, 18, 7279. (28) For a review of recent work, see, for example: Walker, L. M. Curr. Opin. Colloid Interface Sci. 2001, 6, 451.

controlled rheology,1-3,5,6,8,9,11-13,15,16,21,24,26 flow birefringence,3,13,21,27 flow electrical conductivity,13 transmission electron microscopy (cryo-TEM),13,20 light-scattering microscopy,9,11,12,15,16 light scattering,9,10,22,25 small-angle neutron scattering (SANS),4,6,7,17-19,22,23,24 or particle image velocimetry (PIV).14 It is characterized by an increase of the steady shear viscosity by a factor up to 20-30 when the solutions are sheared above a critical shear rate γ˘ c. This increase occurs after an induction time that, for fresh solutions, decreases as the applied shear rate increases. It is however strongly dependent on the shear and temperature history of the samples.21 Despite this experimental effort, the debate about the origin and the mechanisms of this shear-thickening is still unsettled. Concerning its origin, one can conveniently distinguish two main trends in the current views. The shear-induced phase (SIP) transition picture states that the flow induces a phase separation into one gellike phase and one phase containing small aggregates.15,16 This transition bears features similar to a first-order phase transition, and the control parameter is the shear stress rather than the shear rate. The induction time then corresponds to a latency time for the nucleation of the highly viscous phase at one side of the shear cell. An inhomogeneous velocity gradient field within the gap is therefore expected with the position of the interface between the viscous and the dilute phases being determined by the dynamic equilibrium between the erosion and the growth of the gel phase at fixed imposed shear stress.29-31 Some experimental results indeed support the idea that the velocity gradient field can be inhomogeneous within the gap.9,11,12,14-16,25,27 The shear-induced structure (SIS) picture, on the other hand, considers that the shear-thickening results from shear-induced aggregation of the micelles present in the solutions at rest.1-4,10,13,17-19,21 This aggregation is controlled by hydrodynamics and can take place already at shear rates smaller than the critical value, but shear thickening occurs once the mean size of the aggregates has reached some critical value.10,13 In this picture, the (29) Ajdari, A. Phys. Rev. E 1998, 58, 6294. (30) Olmsted, P. D. Europhys. Lett. 1999, 48, 339. (31) Goveas, J. L.; Pine, D. J. Europhys. Lett. 1999, 48, 706.

10.1021/la026253i CCC: $22.00 © 2002 American Chemical Society Published on Web 11/09/2002

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induction time is related to the growth rate of the aggregates. It appears extremely difficult to model the shear-induced phase transition on microscopic grounds29-31 and most of the early as well as recent theoretical approaches32-37 have tried to explain the shear-thickening as a result of the shear-induced modification of the sizes or the architecture of the micellar aggregates existing at rest. Early models for the effect of shear flow on the growth of small rodlike micelles considered mechanisms where the growth can occur once the alignment of the rods by the flow can overcome the randomization of their orientations by rotational diffusion motion.32-35 As a consequence, the predicted critical shear rates are much larger than the experimental shear rate value corresponding to the onset of shear growth. Also these theories did not take into account the effect of electrostatic interactions between the micelles, although their presence seems to be a prerequisite for the shear-thickening behavior to be observed.24,28 Recently, it was proposed that counterion mediated attraction between the likecharged micelles could play a role to favor the formation of bundles of micelles19,36 and start the growth process at much smaller shear rate values, of the order of the inverse of the characteristic time needed to unbind two micelles.36 An alternate mechanism, involving the kinetics of linking and delinking ringlike micelles, which are predicted and sometimes observed in systems with high end-cap energy, could also yield much smaller critical shear rate values.37 Strikingly, the SIP picture accounts well for the experimental results obtained in stress-controlled experiments while the SIS description appears better suited to explain some results obtained while imposing the shear rate, and it is not clear whether there should exist a single unifying picture independent of the conditions of the flow. One can also mention that the concentration range of the experimental studies varies from the dilute regime, where the flow curve displays first a Newtonian behavior, with a shear viscosity very close to that of the solvent, followed by a shear-thickening behavior, to concentrations above the overlap concentration, where the flow curve displays first a shear-thinning behavior and then the shearthickening phenomenon. Here again, it is not clear that a unique scenario could be suited to any experimental conditions. In this paper, we use small-angle light scattering (SALS) under shear flow to investigate the large scale (∼1 µm) features of the structure of dilute cationic gemini surfactant solutions in the shear-thickening regime. This report completes previous papers on the same system6,7,13,19,38 for which recently (i) the structure of the solution at rest was studied by light and X-ray scattering and by dynamic light scattering38 and (ii) the structure of the solution under shear and after shear was studied by time-resolved small-angle neutron scattering.19 After the Experimental Section (section II), section III describes our results. Section IV outlines the theoretical background needed to discuss them in terms of the scattering intensity from elongated objects aligned in the flow. Section V is devoted to the Discussion and Conclusion. (32) Cates, M. E.; Turner, M. S. Europhys. Lett. 1990, 11, 681. (33) Wang, S. Q. Macromolecules 1991, 24, 3004. (34) Turner, M. S.; Cates, M. E. J. Phys.: Condens. Matter 1992, 4, 3719. (35) Bruinsma, R.; Gelbart, W. M.; Ben-Shaul, A. J. Chem. Phys. 1992, 96, 7710. (36) Barentin, C.; Liu, A. H. Europhys. Lett. 2001, 55, 432. (37) Cates, M.; Candau, S. J. Europhys. Lett. 2001, 55, 887. (38) Weber, V.; Narayanan, T.; Mendes, E.; Schosseler, F. Langmuir, submitted for publication.

Weber and Schosseler

Figure 1. Sketch of the flow and detection geometries.

II. Experimental Section Materials. The cationic gemini surfactant ethanediyl-1,2-bis(dodecyl dimethylammonium bromide),39 hereafter called 12-212, has been synthesized in our laboratory. Solutions are prepared by weighing the surfactant molecules in D2O. They are stirred at 50 °C for a few hours to ensure a complete dissolution and then filtered through 0.45 µm filters. Heavy water has been used as the solvent throughout this study to allow comparison with earlier studies.6,7,13,19,38 The critical micellar concentration of 122-12 in H2O at 25 °C is φcmc ) 0.84 ( 0.04 mM,39 and we have used this value as a good approximation for D2O. Light Scattering. Small angle light scattering (SALS) experiments under shear were performed with a novel apparatus whose complete description has been given elsewhere.40 The shear flow is generated by a concentric cylinders geometry (1 mm gap) at a controlled shear rate γ˘ , and the light source is an ionized Ar laser operating at λ ) 4880 Å, shining a vertically polarized beam along a diameter of the cylinders, i.e., along the velocity gradient direction. The intensity scattered from the sample is imaged directly through an optical train onto a cooled 12 bits digital CCD camera. The illuminated volume from which the scattering intensity is collected is approximately a cylinder with a diameter about 80 µm and a length equal to 1 mm. Figure 1 shows a sketch of the geometry together with the definition of the scattering angle θ and the azimuthal angle ψ that characterize the direction of the scattering wavevector q ) 4πn/λ sin(θ/2) (cos(ψ), 0, sin(ψ)), where n is the refractive index of the solution and the q component out of the velocity-vorticity plane has been safely approximated to 0 due to the small values of the scattering angle θ (e12°). The amplitude of the scattering wavevector lies in the range 5 × 10-5 < q (Å-1) < 3.5 × 10-4. Data treatment and calibration are described elsewhere.40

III. Results At rest, the dilute solutions of 12-2-12 are characterized by an isotropic weakly q-dependent scattering intensity about 10 times the intensity scattered by a toluene standard (Figure 2a). The most striking feature of the scattering intensity under shear flow is the appearance of a bright streak pattern in the direction perpendicular to the flow, when the shear rate exceeds a critical value γ˘ c. Figure 2b shows a three-dimensional view of this streak pattern that is obtained by averaging a large number (>100) of frames with short exposure time (≈1 ms). The streak appears after a latency time following the onset of flow above γ˘ c, this time being a decreasing function of shear rate and varying like γ˘ -1 for the highest shear rates (Figure 3). Snapshots with short exposure time (∼1 ms) reveal that the streak exhibits a fine pattern modulation consisting in a set of small intense lines perpendicular to the main direction of the streak (Figure 4). If we define an average intensity Iav as

Iav )

π/2+δψ q dψ∫q ∫π/2-δψ

I(q)q2 dq

max

min

(1)

we can appreciate qualitatively the onset of a steady state in the system, as shown in Figure 5. For a fresh sample, (39) Zana, R.; Benrraou, M.; Rueff, R. Langmuir 1991, 7, 1072. (40) Weber, V.; Schosseler, F. Rev. Sci. Instrum. 2002, 73, 2537.

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Figure 5. Growth of the streak average intensity defined in the text (eq 1) to the steady state. The time origin corresponds to the first apparition of the streak (φ ) 36.6 mM, T ) 20 °C, γ˘ ) 50.3 s-1). Figure 2. Perspective views of the 2-dimensional lightscattering intensities measured for a 12-2-12/D2O solution (φ ) 18.3 mM, T ) 20 °C): (a) at rest; (b) for γ˘ ) 10 s-1. Data are obtained by averaging a large number (>100) of frames with a short exposure time about 1 ms. The crossed lines come from the shadow of the threads suspending the beam stop.40 The circular intensity pattern corresponds to the full available q range and the q value at the periphery is 3.5 × 10-4 Å-1.

Figure 3. Shear rate dependence of the induction time for the first appearance of the streak (φ ) 18.3 mM, T ) 20 °C).

Figure 4. Typical modulation of the streak pattern visible in an individual frame with a short exposure time (1 ms). Top: zoom of the streak area. Bottom: intensity profile along the streak direction. The displayed area and the intensity profile correspond to -2.53 × 10-4 < qz (Å-1) < 2.53 × 10-4.

a steady value of Iav is obtained only after a rather long time (∼500 s) depending on the concentration and temperature conditions and on the shear rate value. The establishment of the steady state is much faster for samples that have already been submitted to shear rates above the critical value γ˘ c. The same trend is observed with the latency time corresponding to the first appearance of the streak. It can be noticed that, although Iav is an averaged quantity, it still displays large fluctuations around the mean steady-state value (Figure 5). The orientation of the streak pattern in the steady state is also fluctuating around its mean value ψ ) π/2 as shown in

Figure 6. Plots of the normalized distribution of streak orientation for different shear rate values. The continuous lines correspond to the fits by the empirical eq 2 in the text (φ ) 7.3 mM, T ) 20 °C).

Figure 6. The normalized distribution of the orientations p(ψ) can be well described by the empirical law:

p(ψ) )

1 exp(-|ψ - π/2|/∆ψ) 2∆ψ

(2)

with ∆ψ being an increasing function of the shear rate. It can be emphasized that the fine pattern modulation and the oscillations in the orientation of the streak are still observed once the steady-state value for Iav is obtained.

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Figure 7. Variation as a function of wavevector of the scattering intensity corresponding to the streak area (azimuthal average 85 < ψ (°) < 95). The numbers in front of each curve correspond to the shear rate values in s-1 (φ ) 18.3 mM, T ) 25 °C, Itol ) 4 × 10-5 cm-1).

Thus, they can be considered as equilibrium effects as far as a system under steady shear showing stress oscillations13 can be considered at equilibrium. The fluctuations in the intensity and in the orientation of the streak are averaged out when a mean picture is calculated from a large number (>100) of snapshots, as in Figure 2b. The resulting I⊥(q) in the direction perpendicular to the flow (ψ ) π/2) is then calculated as an azimuthal average (( 5°) around π/2 and is displayed as a function of the amplitude of the scattering wavevector q in Figure 7, for different shear rates. It can be seen that the main effect of a shear rate variation is to introduce a vertical shift in the corresponding scattering curves, without any marked change in the q dependence. Therefore, we can quantify the effect of the shear rate by computing an average value like in eq 1. Figure 8 compares the variations with shear rate of Iav and of the steadystate shear viscosities measured by Oda et al.13 on the same system. Upon a variation of surfactant concentration or temperature, the shear viscosity and the streak intensity shift in a similar way and remain correlated.22 Although these experiments were not performed simultaneously in the same cell, the absolute values of the critical shear rate appear to agree consistently for the two types of experiments.

Figure 8. Comparison of the evolution of the average streak intensity and of the apparent shear viscosity as a function of shear rate (φ ) 18.3 mM, T ) 25 °C). Viscosity values are taken from ref 13.

the solvent, respectively. The amplitude of the vertically polarized electric field scattered from a single rod is then

A(q) ∝

∫V

rod

(M‚zˆ ) exp(-iq‚r) d3r

(4)

Since M‚zˆ does not depend on the position r, the integration is straightforward and gives the standard result43

A(q,β) ∝ (b + δ cos2(ω))2π2LR2

sin(qL cos(β)/2) J1(qR sin(β)) qL cos(β)/2 qR sin(β) (5)

where J1(x) is the first-order Bessel function of first kind and β is the angle between q and n, i.e., cos(β) ) cos(ψ) sin(ω) cos(φ) + sin(ψ) cos(ω). For a set of N rods, the total scattering intensity will be the squared sum of their contributions to the scattered electric field N

I(q,{βi},{Ri}) )

N

∑ ∑A(q,βk)A(-q,βl) ×

k)1 l)1

exp(-iq‚(Rk - Rl)) (6) IV. Light Scattering from a Set of Rods in a Shear Flow 1. General Expression of the Total Intensity. We consider a set of N rods with length L and radius R in a shear flow. We use the set of axes defined in Figure 1. The director n of a rod is defined through the angles ω and φ as n ) (sin(ω) cos(φ), sin(ω) sin(φ), cos(ω)). The incident and the detected scattered light are both vertically polarized as in our experiments. The dipole moment induced in a rod by the incident wave with amplitude E0 is41,42

M ) E0(δ(zˆ ‚n)n + bzˆ )

(3)

where δ ) Rl - Rt and b ) Rl - Rs, with Rl, Rt, and Rs being the polarizabilities along the rod, across the rod, and in (41) Hashimoto, T.; Ebisu, S.; Inaba, N.; Kawai, H. Polym. J. 1981, 13, 701. (42) Patlazhan, S. A.; Riti, J. B.; Navard, P. Macromolecules 1996, 29, 2029.

where Ri is the position of the center of rod i. If the rods are strongly aligned in a shear flow, i.e., if the distributions of the angles βi are imposed by the flow and not by the respective positions of the rods, eq 6 can be simplified to44

I(q) ) 〈I(q,{βi},{Ri})〉{βi},{Ri} ) N〈A2(q,β)〉βS(q) S(q) ) 1 +

1

N

N

∑ ∑exp(-iq‚(Rk - Rl))〉{R } N k*1 l)1 〈

i

(7)

where 〈...〉β and 〈...〉{Ri} correspond to averages over the orientations and the positions of the rods, respectively. S(q) is the structure factor of the solution that characterizes the spatial organization of the centers of the rods. 2. Calculation of the Average Form Factor. The evaluation of 〈A2(q,β)〉β requires a probability function for (43) Guinier, A.; Fournet, G. Small Angle Scattering of X-rays; WileyInterscience: New York, 1955. (44) Hayter, J. B.; Penfold, J. J. Phys. Chem. 1984, 88, 4589.

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the angles β. The best known has been derived by Peterlin.45 The translation diffusion of the rods is neglected and the equilibrium distribution of orientations results from the competition between rotational diffusion motion, which tends to randomize the orientations, and the precession of the suspended rods, which depends on their relative orientations with respect to the streamlines. Thus, the equilibrium distribution P(ω,φ,Γ ) γ˘ /D) of slender rods with large aspect ratio will obey the equation

D

[

]

∂ ∂P 1 ∂2P 1 sin(ω) + ) 2 sin(ω) ∂ω ∂ω sin (ω) ∂φ2 ∂ ∂ 1 (Ω P sin(ω)) + (ΩφP) (8) ∂φ sin(ω) ∂ω ω

(

)

(

)

where D is the rotational diffusion coefficient and Ω(Ωω, Ωφ) is the angular velocity of the rods with components45,46

1 Ωω ) γ˘ sin(2ω) sin(2φ) 4 Ωφ ) -γ˘ sin2(φ) sin(ω)

(9)

Peterlin gave an analytic solution of eqs 8 and 9 as a series expansion in terms of spherical harmonics and calculated the first three terms, which provide a good approximation of P(ω,φ,Γ) up to Γ ) 6. Using Mathematica software and the recursion formulas established by Peterlin, we could calculate the next three terms, which extend the validity of the approximation to about Γ ) 20. Herbst et al.47 calculated numerically P(ω,φ,Γ) up to Γ ) 40 by the method of finite elements. However, as noted by Hayter and Penfold,44 Peterlin’s solution predicts a weaker alignment in the flow than experimentally observed for solutions of wormlike micelles. In fact, eqs 9 have been derived for rigid ellipsoids46 and are not necessarily satisfying for semiflexible objects that can be stretched by the flow. Therefore, they proposed the ad hoc function44

P(ω,φ,Γ) )

(1 - cos(2φ0))(1 + sin2(ω) cos(2φ0))3/2 4π(1 - sin2(ω) cos(2φ0) cos(2(φ - φ0)))2 2φ0 ) arctan(8/Γ)

(10)

Figure 9 shows the comparison between the distribution of Peterlin and that proposed by Hayter and Penfold. For the purpose of qualitative comparison with experimental data, we will use eq 10 later since it remains in a closed form well suited for numerical computation and, moreover, it agrees with the experimental observation of strong alignment of the micelles in the flow. 3. Evaluation of the Structure Factor. If no correlations exist between the positions of the rods, then the thermal average in S(q) (eq 7) involves random phases and vanishes. This implies S(q) ) 1. The validity of neglecting the correlations depends on the q window investigated, on the range of the interactions and on the mean distance between particles. It was found to be a good approximation for small-angle neutron scattering experiments on sheared neutral micellar solutions. For charged micelles in the absence of added salt, however, (45) Peterlin, A. Z. Phys. 1938, 111, 232. (46) Jeffery, G. B. Proc. R. Soc. London (A) 1922, 102, 161. (47) Herbst, L.; Hoffmann, H.; Kalus, J.; Thurn, H.; Ibel, K.; May, R. P. Chem. Phys. 1986, 103, 437.

Figure 9. Comparison of the distributions of rigid rods orientations given by Peterlin’s solution (a) and by the ad hoc Hayter and Penfold function (b) (Γ ) 10). (c) Variation with Γ of φmax, the φ position of the maximum of the distribution for both equations (ωmax ) π/2 for both).

a correlation peak is present for both unsheared and sheared solutions in the SANS q window. On the other hand, SALS experiments probe much larger length scales and setting S(q) ) 1 is anyway generally a poor approximation. Following references,42,48 we make the assumption that there is a characteristic distance ξ between the aligned rods, to explain the modulations of the streak pattern exhibited by the experimental snapshots (Figure 4). For the pattern in the vertical streak (ψ≈π/2), only correlations along the z axis contribute and

Sπ/2(q) ) 1 +

N

1 N

〈

N

∑ ∑exp(-iq(zk - zl))〉{z } i

k*1 l*1

(11)

To evaluate the average, we use a simple Gaussian form for the distribution of distances z between the centers of rods:48

pc(z) )

(

)

2

(z - ξ) 1 exp 2σξ2 σξx2π

(12)

(48) Hashimoto, T.; Ebisu, S.; Kawai, H. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 59.

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Figure 10. Distribution of the period of the oscillations in the modulation of the streak pattern. The distribution is averaged over several shear rate values (see text) (φ ) 18.3 mM, T ) 25 °C).

Straightforward integration gives then

(

Sπ/2(q) ) 1 + N cos(qξ) exp -

)

q2σξ2 2

(13)

V. Discussion and Conclusion The behavior of the streak pattern observed in our scattering experiments shares a number of common features with the behavior of the apparent macroscopic viscosity: (i) the existence of a critical shear rate, below which no streak pattern and no shear-thickening can be observed; (ii) above the critical shear rate, the existence of a latency time, after which the streak pattern begins to show up and the shear viscosity starts to increase; (iii) the behavior of this latency time, which decreases as the shear rate is increasing and can be as long as a few hundreds of seconds for shear rate values close to the critical value; (iv) the existence of hysteresis effects, where the latency time vanishes for solutions previously sheared above the critical value; (v) the variations of the critical shear rate with surfactant concentration and temperature are correlated in the SALS and in the rheological measurements; (vi) finally, the steady-state values of the average streak intensity and of the macroscopic viscosity exhibit the same behavior as a function of shear rate, showing a maximum for some optimal shear rate and a decrease at higher shear rates. Thus, there is no doubt that the streak pattern is linked to the shear-induced transformation of these dilute surfactant solutions that is responsible for their shearthickening behavior. In the following, we discuss two possible interpretations for this streak pattern: we consider first an homogeneous solution of large anisotropic objects (rigid rods) and then the possibility of a macroscopically inhomogeneous system. We first turn to the snapshots of the streak pattern. Given their short exposure time (≈1 ms), they are good approximations of instantaneous measurements of the light intensity scattered from the shear-induced structure. For the conditions φ ) 18.1 mM and T ) 20 °C, we have performed a statistical analysis of the oscillations seen in the streak intensity along the ψ ) π/2 direction (Figure 4) in the following way. The average distance between the maxima of the oscillations was computed for a given frame and this procedure was repeated for one to two hundred different frames to build an histogram of these average values. The histograms built for shear rates between 8 and 50 s-1 were identical within statistical errors and were therefore averaged for better readability (Figure 10). The characteristic oscillation period of about 15 pixels corresponds to a reciprocal space period ∆q ≈ 1.9 × 10-5 Å-1. Then if we use the simple approximation derived

above (eq 13) for rigid rods, this yields ξ ≈ 33 µm, i.e., a surprisingly high value, meaning that in our illuminated volume with diameter ≈ 80 µm, there are only a few elongated objects. In fact, Fourier transformation of 2-dimensional patterns shows that modulated streak patterns in the reciprocal space similar to the experimental ones are best generated when in real space only a few elongated objects oriented in about the same direction are present. It would have been worthwhile to explore this effect as a function of concentration and temperature, but this was out of the scope of the present work and has been left for the future. An important consequence of the streak modulation resulting from interference effects is that the variations of the streak intensity and orientation (Figures 5 and 6) result from collective effects and are not due to the variations of the size and the orientation of a single large object in the scattering volume. They reflect instead fluctuations of the average orientation and contrast in the solution, depending on the sampled scattering volume. The experiments show consistently that these fluctuations increase when the shear rate increases beyond the optimum value where the apparent shear viscosity is maximum. This suggests that the aligned structure induced by the flow at lower shear rates is perturbed at shear rates above the optimum value, resulting in domains with fluctuating average orientation. This could be linked to the decrease of the shear viscosity at the larger shear rates. We have however no model to justify the empirical law (eq 2) that is found to describe the fluctuations of the average orientation in the domains. The shear stress being the response of the whole sample to the applied shear rate, it is appropriate to correlate the measured shear viscosity with the ensemble average of the snapshot frames recorded for different scattering volumes as was done in Figure 7. As mentioned above, the applied shear does not introduce any marked change in the q dependence of the experimental curves but merely shifts them vertically by a numerical factor. From the preceding discussion on the modulation of the streak pattern, it is already clear that the ensemble averaged intensities contain a contribution from S(q), the structure factor of the solution, and that the curves in Figure 7 cannot be discussed in terms of form factors alone. For the purpose of qualitative discussion, however, it is interesting to examine the effects of alignment on the scattering intensity of noninteracting rigid rods (S(q) ) 1 in eq 7). Figure 11 displays the intensities scattered from monodisperse assemblies of rods, whose distributions of orientations are given by eq 10. The dimensions of the rods have been set to L ) 0.2 µm, R ) 100 Å (left) and to L ) 2 µm, R ) 100 Å (right), and three different values of Γ have been considered (Γ ) 1, 10, 20 from top to bottom). The last bottom row of graphics in Figure 11 corresponds to calculations performed for rods perfectly aligned in the flow. The parameters b and δ in eq 5 have been set arbitrarily to 1 and 0.1, respectively, and these relative magnitudes have no incidence for the qualitative discussion. In the q range of SALS, the small diameter of the rods plays a very minor role but this fact is hidden by the rotational average: only for perfectly aligned rods (ω ) π/2) is it clear that they have a small diameter through the stationery value of the intensity along qx ) 0 (ψ ) π/2) (bottom row of Figure 11). On the whole, as expected, the alignment in the flow of rods with a given length decreases (respectively increases) the q dependence in the direction perpendicular (respectively parallel) to the flow but does not change the scattering intensity at zero q value. This

Shear Thickening of a Cationic Surfactant

Figure 11. Calculated intensities scattered from a set of identical rods with distributions of orientations given by eq 10. The rod dimensions for the calculation are L ) 0.2 µm, R ) 100 Å (left) and L ) 2 µm, R ) 100 Å (right). Γ values are, from top to bottom: 1, 10, 20, ∞ (perfectly aligned rods). Wavevector values are given in 10-4 Å-1 and correspond to the experimental window. The scale of the intensity axis is different for the left and right columns.

is in contrast with our experimental curves (Figure 7), where the q dependence remains roughly the same but the intensity level increases by a factor about 100. Therefore, within the model of aligned rods, this increase must occur through the factor N(bLR2)2 in eqs 5 and 7. Any large change in the factor N can be ruled out from the analysis of the streak modulation, which has shown that the characteristic distance ξ does not change with the applied shear rate. Then, if we consider a change in L alone by a factor of 10, this would mean a huge decrease by a factor about 1000 in the rotational diffusion constant, D ∼ ln(L/2R)/L-3,49 and a corresponding increase in the value of Γ. We would then expect a complete alignment of the rods and obtain for high shear rates scattering intensities similar to the one in the bottom row (right) of Figure 11. However the marked decrease of the experi(49) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, England, 1986.

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mental scattering intensity along qx ) 0 would then indicate a rather large value of R and suggest bundles of rods, which is hard to reconciliate with a constant ξ, since if rods aggregate together as the shear rate increases, the mean distance between the objects tends to increase. Therefore, as a whole, it seems very difficult to make the aligned rods model consistent with the experiments. For sure, the hypothesis of rigid rods is rather crude and polydispersity has not been taken into account, but any analysis of the scattering intensity in terms of aligned anisotropic colloidal particles would fail on the same grounds as this rod model. Therefore, the alignment of individual objects cannot explain the modulation of the optical contrast observed in our experiments, and we have to search for an other origin of the streak pattern. Several studies have investigated the shear-induced state of wormlike micellar solutions in planes containing the velocity gradient direction by light scattering,9 lightscattering microscopy,9,11,12,15,16 or PIV14 and have concluded to the existence of a gel phase at the inner cylinder of the Couette gap. At the onset of the increase of the shear stress, this gel phase extends across the gap or displays a height smaller than the gap width, depending on the shear rate or the shear stress being imposed in the experiments.15 Temporal and spatial fluctuations of the associated light scattering or flow birefringence have been described in parallel with fluctuations in the measured shear stress or shear rate.16,21,27 In addition, visualization techniques have shown that the velocity profile is very different from that expected for an ideal Couette flow, showing in particular wall-slip phenomena14,16 and sample regions moving in opposite directions at the cessation of flow.9 For more concentrated solutions exhibiting first shear thinning and then shear thickening, the observation of a “stacked pancake” pattern involving alternately turbid and clear bands aligned along the flow direction has also been very recently reported.25 Also a modulation of the birefringence profile across the gap and the failure of the stress-optical law were reported, indicating that the level of stress in the solution is not correlated to the state of orientation of the wormlike chains.27 Our experiments measure in reciprocal space the spatial variations in the velocity-vorticity plane of an optical contrast averaged along the velocity gradient direction. The real-space pattern of these variations appears similar to a set of grooves aligned in the flow direction with a characteristic spacing about 33 µm in the vorticity direction. The exact relationship of this average optical contrast to the concentration profile can be rather intricate. However, a number of features of the streak pattern observed in these experiments suggest that it corresponds in fact to the scattering by a gellike phase structured/ aligned along the flow: (i) It appears at the same shear rate that defines the onset of stress growth. (ii) During the period of time needed to reach the steady state streak intensity (Figure 5), there is no change in the streak appearance in terms of its width and of the spacing of the modulation pattern. Simply, the average intensity increases. (iii) In the same way, the shear rate increase does not modify the shape of the streak but simply its intensity level, in good correlation with the evolution of the measured shear stress. (iv) The order of magnitude for the length scale of the modulation in the optical contrast is the same as the one reported27 for the spatially resolved flow birefringence experiments (∼50 µm).

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Therefore, possible scenarios consistent with our experimental observations could be (i) the formation and growth of a gellike layer for shear rates below γ˘ c and the subsequent structuration/alignment of this layer above γ˘ c or (ii) the formation of a structured gellike layer at γ˘ c consistent with the idea of homogeneous nucleation.15 As pointed out by several studies,14-16,25,27 the existence of a gel layer inside the gap and the converging evidence that the flow inside the gap is inhomogeneous when the shear stress increase is observed implies that only apparent shear viscosities are measured. The mechanism of structuration and alignment in the gellike layer remains uncertain. It could either follow from the succession of gel growth and fracture in the flow9,15,16 or involve the coupling between flow and fluctuations of concentration.25,50 The modeling of the flow inside the gap should then anyway imply elaborate models, although simpler phenomenological models for the description of the behavior of the shear stress have also been proposed.29-31 The questions of the microscopic structure of the gel phase and of the mechanisms involved in its formation remain still unsolved. It can be emphasized that most studies have focused on the characterization of the shearthickening regime and observations in the subcritical regime, for γ˘ < γ˘ c, are extremely scarce. We can only mention the appearance of a layer with enhanced contrast by light-scattering microscopy15 and the quantitative measurement of a 10% increase of the light-scattering intensity, with no indication for an orientation of the micelles in the subcritical regime, by Pro¨tzl and Springer.10 All other phenomena, like, e.g., SANS anisotropy, birefringence modulation, or streak patterns, are in fact observed only when the shear stress increases, i.e., after the flow field is already significantly different from an idealized Couette flow. The measurement by Pro¨tzl and Springer suggests the existence of a gellike phase below γ˘ c with a structure that is not dramatically different from that of the original solution before shear. An indirect confirmation of this result has been provided by timeresolved SANS experiments showing that the characteristic intensity peak of the original solution shifts to smaller q values under shear for γ˘ > γ˘ c19,23 and that, after the (50) Schmitt, V.; Marques, C. M.; Lequeux, F. Phys. Rev. E 1995, 52, 4009.

Weber and Schosseler

cessation of flow and the relaxation of the anisotropy in the intensity pattern, the peak remains at this same position for hours.19 These effects are observed only for solutions at concentrations below the overlap concentration and indicate that the gellike phase obtained at higher concentrations has the same structure as the original solution, at least in the q range probed in the SANS experiments. Our attempts to measure a difference in the SALS intensity for solutions before and after shear flow remained inconclusive because the intensity levels were close to the limits of our instrument. We can however infer that if such a difference exists, it must remain small, of the order of the instrumental resolution. For the same reasons, no significant effects could be measured for the steady-state light scattering in the regime γ˘ < γ˘ c. The observations that the typical time to reach steady stress value5 and the critical shear rate13 depend both on the size of the gap point to the fact that these quantities are only indirectly related to the microscopic structure of the gellike phase and to the mechanisms of its formation. Moreover, a dependence of the critical shear rate upon the size of the gap suggests also that the gel growth starts below γ˘ c. Therefore, the structure of this gel should be investigated in the subcritical regime to test the validity of any microscopic model. This indeed requires a lot of efforts to increase the sensitivity of the current experimental techniques. As a final remark, very recent results on quiescent solutions24,26,38,51 point to the presence of some very large micellar aggregates, even close to the critical micellar concentration, and suggest that our understanding of the thermodynamic equilibrium of these systems needs also some refinement.52 Indeed such large aggregates could trigger the gel growth at much smaller shear rates values than previously thought. Acknowledgment. We are indebted to O. Gavat for the kind synthesis of the samples. We thank also J. F. Berret for sending us a preprint of ref 27. We thank one reviewer for constructive remarks that helped to clarify some ambiguities in the original manuscript. LA026253I (51) Bernheim-Grosswasser, A.; Zana, R.; Talmon, Y. J. Phys. Chem. B 2000, 104, 4005. (52) May, S.; Ben-Shaul, A. J. Phys. Chem. B 2001, 105, 630 and references therein.